INVERSION OF EM DATA TO RECOVER 1-D CONDUCTIVITY ANDA GEOMETRIC SURVEY PARAMETERBySean Eugene WalkerB. Sc. (Honours), Geology & Physics, McMaster University, 1996a thesis submitted in partial fulfillment ofthe requirements for the degree ofMaster of Scienceinthe faculty of graduate studiesdepartment of earth and ocean sciences(geophysics)We accept this thesis as conformingto the required standard: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :the university of british columbiaJune, 1999c Sean Eugene Walker, 1999

In presenting this thesis in partial ful�lment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis for�nancial gain shall not be allowed without my written permission.Department of Earth and Ocean Sciences(Geophysics)The University of British Columbia129-2219 Main MallVancouver, CanadaV6T 1Z4Date:

AbstractThe presence of geometrical survey parameter errors can cause problems when attemptingto invert electromagnetic (EM) data. There are two types of data which are of particularinterest: airborne EM (AEM) and ground based horizontal loop EM (HLEM). Whendealing with AEM data there is a potential for errors in the measurement height. Thepresence of measurement height errors can result in distortions in the conductivity modelsrecovered via inversion. When dealing with HLEM there is a potential for errors in thecoil separation. This can cause the inphase component of the data to be distorted.Distortions such as these can make it impossible for an inversion algorithm to predict theinphase data. Examples of these types of errors can be found in the Mt. Milligan andSullivan data sets. The Mt. Milligan data are contaminated with measurement heighterrors and the Sullivan data are contaminated with coil separation errors.In order to ameliorate the problems associated with geophysical survey parametererrors a regularized inversion methodology is developed through which it is possible torecover both a function and a parameter. This methodology is applied to the 1-D EMinverse problem in order to recover both 1-D conductivity structure and a geometricalsurvey parameter. The algorithm is tested on synthetic data and is then applied to the�eld data sets.Another problem which is commonly encountered when inverting geophysical data isthe problem of noise estimation. When solving an inverse problem it is necessary to �tthe data to the level of noise present in the data. The common practice is to assign noiseestimates to the data a priori. However, it is di�cult to estimate noise by observationalone and therefore, the assigned errors may be incorrect. Generalized cross validation isii

a statistical method which can be used to estimate the noise level of a given data set. Anon-linear inversion methodology which utilizes GCV to estimate the noise level withinthe data is developed. The methodology is applied to 1-D EM inverse problem. Thealgorithm is tested on synthetic examples in order to recover 1-D conductivity and alsoto recover 1-D conductivity as well as a geometrical survey parameter. The strengthsand limitations of the algorithm are discussed.

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Table of ContentsAbstract iiList of Tables viiList of Figures viiiAcknowledgement xiii1 Introduction 11.1 Motivation for the Thesis : : : : : : : : : : : : : : : : : : : : : : : : : : : 21.2 Outline of the Thesis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 102 Background Information 122.1 De�ning the EM Data : : : : : : : : : : : : : : : : : : : : : : : : : : : : 122.1.1 Generic Horizontal Coplanar Systems : : : : : : : : : : : : : : : : 142.1.2 Speci�c Horizontal Coplanar Systems : : : : : : : : : : : : : : : : 182.2 Details of the Inverse Problem : : : : : : : : : : : : : : : : : : : : : : : : 242.2.1 The Forward Problem : : : : : : : : : : : : : : : : : : : : : : : : 242.2.2 The Inverse Problem : : : : : : : : : : : : : : : : : : : : : : : : : 293 Geometric Survey Parameter Errors 373.1 Measurement Height Errors in AEM : : : : : : : : : : : : : : : : : : : : 403.1.1 How do measurement height errors occur? : : : : : : : : : : : : : 403.1.2 How do measurement height errors a�ect the data? : : : : : : : : 42iv

3.1.3 How do measurement height errors a�ect the inversion results? : : 443.2 Coil Separation Errors in HLEM data : : : : : : : : : : : : : : : : : : : : 473.2.1 How do coil separation errors occur? : : : : : : : : : : : : : : : : 473.2.2 How do coil separation errors a�ect the data? : : : : : : : : : : : 483.2.3 How do coil separation errors a�ect the inversion results? : : : : : 533.3 Parameter Errors in the Field Data Sets : : : : : : : : : : : : : : : : : : 603.3.1 Mt. Milligan : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 603.3.2 Sullivan Data : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 614 Recovering 1-D Conductivity and a Geometric Survey Parameter 624.1 Changes due to the extra parameter : : : : : : : : : : : : : : : : : : : : : 634.1.1 The Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 634.1.2 The Data : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 644.1.3 The Sensitivities : : : : : : : : : : : : : : : : : : : : : : : : : : : 654.1.4 The Objective Function : : : : : : : : : : : : : : : : : : : : : : : 674.2 Inversion Algorithms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 684.2.1 Details of algorithm to recover 1-D conductivity with �xed p : : : 694.2.2 De�ning the Parameter Norm : : : : : : : : : : : : : : : : : : : : 804.2.3 Inversion algorithm to recover 1-D conductivity and p : : : : : : : 974.3 Synthetic Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1014.3.1 AEM Sounding : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1014.3.2 Trade o� between � and h : : : : : : : : : : : : : : : : : : : : : : 1034.3.3 HLEM sounding : : : : : : : : : : : : : : : : : : : : : : : : : : : 1074.4 Field Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1084.4.1 Mt. Milligan : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1104.4.2 Sullivan : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 114v

4.5 Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1145 Application of GCV to 1-D EM Inversion 1185.1 Estimating Noise using GCV : : : : : : : : : : : : : : : : : : : : : : : : : 1195.1.1 GCV in Linear Inverse Problems : : : : : : : : : : : : : : : : : : 1195.1.2 GCV in Non-linear Problems : : : : : : : : : : : : : : : : : : : : 1205.2 GCV and the 1-D EM inverse problem : : : : : : : : : : : : : : : : : : : 1225.2.1 GCV when p is �xed : : : : : : : : : : : : : : : : : : : : : : : : : 1225.2.2 GCV when p is included in the model : : : : : : : : : : : : : : : : 1305.2.3 Application of GCV to small data sets : : : : : : : : : : : : : : : 1335.3 Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1426 Summary 143References 145

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List of Tables3.1 Results from the inversion of AEM data with the incorrect hest values. : : 443.2 Results from the inversion of small separation HLEM data with the incor-rect s value. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 543.3 Results from the inversion of small separation HLEM data with the incor-rect s value and noise estimated to account for inphase modelling errors. 563.4 Results from the inversion of large separation HLEM data with the incor-rect s value and noise estimated to account for inphase modelling errors. 584.1 Comparison of results from the AEM inversions with �xed h and variable h.1054.2 Comparison of results from the overburden AEM inversions with �xed hand variable h. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1075.1 Comparison of parameter values from the AEM discrepancy principle andGCV inversions with a �xed value of h. : : : : : : : : : : : : : : : : : : : 1255.2 Comparison of results from AEM example discrepancy principle and GCVinversions with h included in the model. : : : : : : : : : : : : : : : : : : 1325.3 Comparison of results from the four frequency AEM example discrepancyprinciple and GCV inversions with a �xed value of h. : : : : : : : : : : : 137vii

List of Figures1.1 The inphase and quadrature components of the Sullivan data. : : : : : : 31.2 Inversion results from the Sullivan data. : : : : : : : : : : : : : : : : : : 41.3 Observed and predicted data from the inversion of the Sullivan data. : : 51.4 Comparison of the inphase and quadrature data from two soundings alongLine 1 of the Sullivan data. : : : : : : : : : : : : : : : : : : : : : : : : : : 61.5 The inphase and quadrature components of the Mt. Milligan data. : : : : 71.6 Inversion results from the Mt. Milligan data. : : : : : : : : : : : : : : : : 82.1 The four common types of transmitter-receiver geometries used in loop-loop EM surveys. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 132.2 Geometry of the generic horizontal coplanar EM system. : : : : : : : : : 142.3 Cartoon showing the behavior of a horizontal coplanar EM system in freespace and in the presence of a conductive body. : : : : : : : : : : : : : : 152.4 Cartoon of a typical airborne EM system. : : : : : : : : : : : : : : : : : 192.5 The bucking coil system used in AEM. : : : : : : : : : : : : : : : : : : : 202.6 Cartoon of a typical HLEM system. : : : : : : : : : : : : : : : : : : : : : 222.7 Discretization of the problem domain for the forward problem. : : : : : : 262.8 Cartoon of a typical Tikhonov curve. : : : : : : : : : : : : : : : : : : : : 343.1 Conductivity models and data from the synthetic examples. : : : : : : : 393.2 Values used to determine measurement height. : : : : : : : : : : : : : : : 413.3 The two situations that will result in measurement height errors. : : : : : 423.4 Synthetic data from AEM soundings with htrue = 24 m, 30 m, and 36 m. 43viii

3.5 Results from the inversion of AEM data with the incorrect h values. : : : 453.6 The two situations that will result in coil separation errors. : : : : : : : : 483.7 Synthetic data from small separation HLEM soundings with sest = 10 mand strue = 9 m, 10 m, and 11 m. : : : : : : : : : : : : : : : : : : : : : : 503.8 Synthetic data from large separation HLEM soundings with sest = 50 mand strue = 49 m, 50 m, and 51 m. : : : : : : : : : : : : : : : : : : : : : 523.9 Results from the inversion of small separation HLEM data with the incor-rect s value. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 553.10 Results from the inversion of small separation HLEM data with the incor-rect s value and the noise was estimated to account for inphase modellingerrors. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 573.11 Results from the inversion of large separation HLEM data with the incor-rect s value and the noise was estimated to account for inphase modellingerrors. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 594.1 Flowchart for the �xed � inversion algorithm with a �xed value of p. : : : 724.2 Recovered models and predicted data from AEM example �xed � inversionwith a �xed value of h. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 744.3 Convergence curves from AEM example �xed � inversion with a �xed valueof h. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 754.4 Flowchart for the discrepancy principle inversion algorithm with a �xedvalue of p. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 784.5 Recovered models and predicted data from AEM example discrepancyprinciple inversion with a �xed value of h. : : : : : : : : : : : : : : : : : 804.6 Convergence curves from AEM example discrepancy principle inversionwith a �xed value of h. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 81ix

4.7 Results from eleven inversions of the AEM sounding data with �xed valuesof h ranging from 45 m to 15 m. : : : : : : : : : : : : : : : : : : : : : : : 834.8 Results from eleven inversions of the small separation HLEM soundingdata, strue = 11 m, with �xed values of s ranging from 12.5 m to 7.5 m. : 844.9 Results from eleven inversions of the large separation HLEM soundingdata, strue = 51 m, with �xed values of s ranging from 52.5 m to 47.5 m. 854.10 Values calculated during the line search for � during the �rst iteration ofthe inversion of the AEM data with �p =1 and 0. : : : : : : : : : : : : 884.11 Values calculated during the line search for � during the �rst iteration ofthe inversion of the AEM data with �p =1, 10�6, and 10�3. : : : : : : : 904.12 Results from the search for a best �t pref value for the AEM example. : : 924.13 Results from the search for a best �t pref value for the HLEM examples. 934.14 Contour plots of �d for a range of �ref and p values for the AEM exampleand both HLEM examples. : : : : : : : : : : : : : : : : : : : : : : : : : : 954.15 Flowchart for the �xed � inversion algorithm with p included in the model. 984.16 Recovered models and predicted data from AEM example �xed � inversionwith h included in the model. : : : : : : : : : : : : : : : : : : : : : : : : 994.17 Convergence curves from AEM example �xed � inversion with h includedin the model. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1004.18 Flowchart for the discrepancy principle inversion algorithm with p includedin the model. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1024.19 Recovered models and predicted data from AEM example discrepancyprinciple inversion with h included in the model. : : : : : : : : : : : : : : 1034.20 Convergence curves from AEM example discrepancy principle inversionwith h included in the model. : : : : : : : : : : : : : : : : : : : : : : : : 104x

4.21 Comparison of models recovered from the inversion of AEM data with�xed h and variable h. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1054.22 Recovered models and predicted data from small separation HLEM ex-ample discrepancy principle inversion with s included in the model. : : : 1084.23 Convergence curves from small separation HLEM example discrepancyprinciple inversion with s included in the model. : : : : : : : : : : : : : : 1094.24 Recovered models and predicted data from large separation HLEM ex-ample discrepancy principle inversion with s included in the model. : : : 1104.25 Convergence curves from large separation HLEM example discrepancyprinciple inversion with s included in the model. : : : : : : : : : : : : : : 1114.26 Inversion results from Mt. Milligan Data with h included in the model. : 1124.27 Recovered �h superimposed on the �xed h inversion results from Mt. Mil-ligan. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1134.28 Observed and predicted data from the inversion of the Sullivan data withs included in the model. : : : : : : : : : : : : : : : : : : : : : : : : : : : 1154.29 Inversion results from Sullivan Data with s included in the model. : : : : 1165.1 Flowchart for the GCV inversion algorithm with a �xed value of p. : : : : 1235.2 Comparison of recovered models and predicted data from AEM examplediscrepancy principle and GCV inversions with a �xed value of h. : : : : 1255.3 Convergence curves from AEM example discrepancy principle inversionwith a �xed value of h. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1265.4 Convergence curves from AEM example GCV inversion with a �xed valueof h. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1275.5 GCV Curves and recovered models from iterations 1, 3, and 6 of the AEMexample GCV inversion with a �xed value of h. : : : : : : : : : : : : : : 129xi

5.6 Flowchart for the GCV inversion algorithm with p included in the model. 1315.7 Comparison of recovered models and predicted data from AEM examplediscrepancy principle and GCV inversions with h included in the model. : 1325.8 Convergence curves from AEM example discrepancy principle inversionwith h included in the model. : : : : : : : : : : : : : : : : : : : : : : : : 1345.9 Convergence curves from AEM example GCV inversion with h includedin the model. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1355.10 GCV Curves and recovered models from iterations 1, 3, and 6 of the AEMexample GCV inversion with h included in the model. : : : : : : : : : : : 1365.11 Comparison of recovered models and predicted data from the four fre-quency AEM example discrepancy principle and GCV inversions with a�xed value of h. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1385.12 Convergence curves from four frequency AEM example discrepancy prin-ciple inversion with a �xed value of h. : : : : : : : : : : : : : : : : : : : : 1395.13 GCV Curves and recovered models from iterations 1, 2, and 3 of the AEMexample GCV inversion with a �xed value of h. : : : : : : : : : : : : : : 141xii

AcknowledgementFirst of all I would like to thank Doug Oldenburg for opening my eyes to the wonderfulworld of inversion. The amount that I have learned by being a part of his research groupover the past three years is startling. I would also like to thank him for his supportthroughout my degree. I would be hard pressed to �nd a more enthusiastic, positive, andunderstanding person anywhere.I would like to thank Tanya for her love and support.Special thanks go to Len for making up the second half of the 400 lb. o�ce. Theresearch might have gone faster without him, but it wouldn't have been nearly as enjoy-able.Thanks also go to all of my friends in the Geophysics building for providing theopportunity for many not-so-scienti�c experiences.I must also thank Colin Paton who was omitted from thank yous in a previous workof mine. Thanks for giving me an excuse to go to Edmonton twice a year.xiii

Chapter 1IntroductionBoth airborne and ground based frequency domain electromagnetic (EM) geophysicalmethods were initially developed as mineral exploration tools. The early EM systemswere rather crude and their applications were limited to locating highly conductive orebodies within resistive host rocks. Further research and development, coupled with theuse of digital technology, resulted in modern EM instruments with greater measurementprecision and a wider range of measurement frequencies. These advancements have madeit possible to apply EM methods to a wide variety of problems which include the esti-mation of sea ice thickness (Kovacs et al., 1995), the detection of contaminant plumes(Sauck et al., 1998), and the mapping of hydrogeologic structures (Wynn & Gettings,1998).The widespread use of geophysical EM methods has generated a need for new datainterpretation and processing tools. While traditional curve matching techniques are stillused, the importance of EM inversion is now being realized. The development of fastinversion techniques and the a�ordability of high powered personal computers has helpedto make EM inversion a feasible processing option for most practicing geophysicists.The ideal inversion scheme would have the ability to invert multiple lines of EMdata in order to recover a 3-D distribution of conductivity. However, fast and reliable3-D forward modelling codes are still in the development stages. Therefore, most of theavailable codes are restricted to inverting data from individual soundings to recover1-Dconductivity structures. The recovered 1-D models are then plotted one next to the other1

Chapter 1. Introduction 2to produce a pseudo 2- or 3-D conductivity structure.The theoretical details of solving the 1-D EM inverse problem are well established(Fullagar & Oldenburg, 1984 and Zhang & Oldenburg, 1999) and the algorithms havebeen proven to be successful when applied to �eld data sets. However, it is alwaysimportant to be aware of the problems that can arise when attempting to invert �elddata.1.1 Motivation for the ThesisThe motivation for this thesis comes from problems encountered when attempting toinvert two �eld data sets. The �rst data set was from a Max-Min ground based EMsurvey performed over a tailings pond at Cominco's Sullivan Mine in Kimberley, BritishColumbia. The goal of the survey was to investigate the shallow subsurface conductivitystructure around the pond in order to determine the e�ects of groundwater ow on thetailings (Jones, 1996). The data were collected using the horizontal coplanar con�gura-tion of the Max-Min system. Measurements of the inphase and quadrature componentof the secondary �elds were taken at 4 frequencies ranging from 7040 Hz to 56320 Hz.The coil separation was preset to be 5 m for each measurement. I will concentrate onLine 1 which was 350 m long with stations every 5 m. The inphase and quadrature dataare shown in Figure 1.1.Upon inspection the Sullivan data appear to defy interpretation. The measurementsoscillate from one station to the next and the inphase component is almost entirelynegative. The inphase component of the data generated from a horizontal coplanarEM survey over top of a 1-D conductivity structure should asymptote to zero as themeasurement frequency decreases however, the Sullivan data do not exhibit this behavior.Therefore, in order to invert the Sullivan data with existing algorithms it is necessary to

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%)Figure 1.1: The inphase and quadrature components of the Sullivan data. 7040 Hz(circles), 14080 Hz (squares), 28160 Hz (diamonds), and 56320 Hz (stars).assign error estimates to the inphase measurements that are large enough to counteractthe problems caused by the negative inphase values. This is equivalent to discarding thenegative inphase values.When inverting the Sullivan data I assigned standard deviations of 20% and 1% ofthe primary �eld to the inphase and quadrature components of the data respectively.The target data mis�t for each sounding was equal to 8. The results from invertingeach sounding in the Sullivan data are shown in Figure 1.2. The top panel of Figure 1.2shows that it was possible to achieve the target mis�t at most stations. The recoveredconductivity models plotted in the bottom panel of Figure 1.2 show a trend within thesubsurface from a resistive zone to a more conductive zone as the stations move from eastto west along the line. The survey information provided with the Sullivan data indicatedthat the eastern end of Line 1 (0E to 40E) lies on top of a bedrock outcrop while theremainder of the line extends over the tailings pond itself. Therefore, the recovered modelseems to be in agreement with the available a priori information. These results suggestthat the Sullivan data have been successfully inverted. However, by plotting the observeddata and the data predicted by the recovered model (Figure 1.3) it is clear that I have

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−1Figure 1.2: Inversion results from the Sullivan data. Top panel: the �nal data mis�tvalues from each inversion plotted by station location. Bottom panel: recovered 1-Dconductivity models from each inversion plotted by station location.only predicted the quadrature component of the data.While in some sense the inversion has been successful I am unsure as to whether Ishould believe the results because such large errors have been assigned to the data. I amalso left to wonder whether the information contained in the inphase component could beimportant. These problems prompted me to determine the cause of the negative inphasevalues. A potential cause could have been the presence of magnetizable materials withinthe tailings however, the negative inphase data is seen at all station locations along Line1. The data from soundings at two stations are plotted for comparison in Figure 1.4.The sounding in Figure 1.4(a) is from station 5E at the east end of Line 1 on a bedrockoutcrop on the edge of the tailings pond. It is fairly safe to assume that this rock will notcontain large amounts of magnetite. Figure 1.4(b) shows a sounding from station 150Ewhich is located in the center of the tailings pond. Both soundings show comparablenegative shifts of the inphase data. This suggests that these shifts are due to somethingother than magnetic susceptibility.

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Figure 1.4: Comparison of the inphase and quadrature data from two soundings alongLine 1 of the Sullivan data. (a) Data from Station 5E and (b) Data from Station 150E.Inphase (circles) and quadrature (crosses).The Sullivan data was collected using a 5 m coil separation which is the smallest coilseparation at which it is possible to perform a Max-Min survey. When performing sucha survey it is not uncommon for the inphase component of the data to be a�ected bycoil positioning errors (Alumbaugh & Newman, 1997). This suggested the possibility ofcoil separation errors as the cause of the shifts in the inphase data. Examination of thereference cable used to collect the Sullivan data revealed that it was chained correctlyto be 5 m in length however, due to the design of the Max-Min system, when the cablewas pulled tight, the coils were in fact 5.5 m apart. If this were to happen during asurvey it would result in a coil separation error of 0.5 m. Such an error is negligiblewhen considering a survey using a 100 m coil separation however, at a separation of 5 mit represents a 10% error. The negative inphase measurements in the Sullivan data aretherefore, likely the result of coil separation errors.The second set of �eld data that prompted this research was from a Dighem airborneEM survey performed over the Mt. Milligan deposit in north central British Columbia.Mt. Milligan is a Cu-Au porphyry deposit and the goal of the survey was to delineate

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900 Hz IP × .1 900 Hz Q × .1

7200 Hz IP × .1+257200 Hz Q × .1+25

56 kHz IP × .1+75 56 kHz Q × .1+75

Figure 1.5: The inphase and quadrature components of the Mt. Milligan data. 900Hz (circles), 7200 Hz (squares), and 56000 Hz (diamonds). Inphase (solid line) andquadrature (dashed line).the intrusive stock and the surrounding zone of mineralization (Oldenburg, et al., 1997).The data were collected using three horizontal coplanar transmitter and receiver coil pairswith measurement frequencies of 900 Hz, 7200 Hz, and 56000 Hz and coil separations of8 m, 8 m, and 6.4 m respectively. I will concentrate on Line 12625E which is 1 km longwith measurements approximately every 10 m. The inphase and quadrature componentsof the data are shown in Figure 1.5. In contrast to the Sullivan data the Mt. Milligandata appear to be interpretable and free of any serious errors. The inphase componentof the Mt. Milligan data asymptotes to zero at low frequency and both components ofthe data vary smoothly from station to station.When inverting the Mt. Milligan data I assigned a standard deviation of 25 Dighemunits (5 PPM) to the 900 Hz measurements and a standard deviation of 50 Dighem units

Chapter 1. Introduction 80

10

20

Dat

a M

isfit

9000 9200 9400 9600 9800

0

100

200

300

Northing (m)

Dep

th (

m)

log10

σ−3.5

−3

−2.5

−2

−1.5

Figure 1.6: Inversion results from the Mt. Milligan data. Top panel: the �nal data mis�tvalues from each inversion plotted by station location. Bottom panel: the recovered 1-Dconductivity models from each inversion plotted by station location.(10 PPM) to the 7200 Hz and 56000 Hz measurements. The target data mis�t for eachsounding was equal to 6. I can see from the inversion results in the top panel of Figure 1.6that it has been possible to achieve the desired mis�t at most stations. However, therecovered conductivity model shown in the bottom panel of Figure 1.6 shows some regionsof anomalously low conductivity near the surface. This has been described by Ellis &Shehktman (1994) as an \air layer" e�ect and it is commonly seen in inversion resultswhen the incorrect measurement height has been used. The measurement heights usedin the inversions are the values that were measured by the Dighem system and it iswell known that these values can be incorrect in certain situations. In Fraser (1986) itwas suggested that both extreme terrain and tree cover can lead to measurement heighterrors. I know that the region of the Mt. Milligan deposit has topographical variationand it is covered by large trees in some areas. Therefore, it is likely that the presence ofmeasurement height errors in the Mt. Milligan data is the cause of the distortions in therecovered conductivity models.

Chapter 1. Introduction 9From a mathematical viewpoint the Sullivan and Mt. Milligan data sets are simi-lar in that each is di�cult to interpret because a survey parameter (coil separation ormeasurement height) was not accurately provided. Therefore, these two �eld data setsinvite the solution of the general problem of inverting EM data when geometrical surveyparameters are inaccurate. Thus an inversion methodology must be generated to �ndboth the electrical conductivity and the erroneous survey parameter. The major part ofthe work contained in this thesis is concerned with solving this problem.A second problem which will be investigated concerns the estimation of noise wheninverting geophysical data. Correctly estimating the amount of random noise associatedwith a given data set can increase the amount of information that can be extracted viainversion. Noise estimates for inverse problems are usually made by inspecting the dataand assigning the level to which the data should be �t. By examining the erratic natureof the Sullivan data shown in Figure 1.1 it is clear there is a fair bit of noise presenthowever, the task of estimating the amount of noise is an extremely di�cult one. Whileinformation about the accuracy of Max-Min measurements is available, it can not provideme with details about the amount of random error associated with a given measurement.A more desirable method of estimating noise would make use of the information containedin the data itself. Generalized cross validation (GCV) is such a method.GCV is a statistical method which can be used to estimate the amount of randomnoise associated with a given data set. It has been applied to widely used to estimatenoise in linear problems (Wahba, 1990) and has recently been applied to geophysicalinverse problems (Haber, 1997). The application of GCV to the 1-D EM inverse problemwould help to ameliorate some of the problems caused by estimating errors by inspection.

Chapter 1. Introduction 101.2 Outline of the ThesisThe goal of this thesis is to develop an inversion methodology that will be able to recovera 1-D conductivity distribution as well as a geometrical survey parameter from frequencydomain EM data.In order to address this problem e�ectively some background information is needed.The necessary information is provided in Chapter 2. This section includes a descriptionof frequency domain EM data and how it is generated using airborne and ground basedEM systems. It also includes a discussion of how the 1-D EM inverse problem is currentlysolved. This review should provide the tools needed to e�ectively attack the problem.Chapter 3 presents a discussion of the problems associated with parameter errors inEM data. It describes the way in which parameter errors occur and how these errorsa�ect the EM data. A description of how these errors a�ect inversion results is alsoprovided. Each of these topics is covered in terms of both AEM and HLEM surveys.The �nal section of this chapter attempts to con�rm the presence of geometrical surveyparameter errors in the two �eld data sets.Chapter 4 deals with the development of an inversion methodology which has theability to recover a function and a parameter. The changes that must be made to theinversion methodology in order to include an extra parameter are addressed �rst. Themost important of these changes is made to the model objective function. The way inwhich the model objective function is de�ned greatly a�ects the stability of the algorithmtherefore, it must be de�ned carefully. Once all of the changes are completed the newalgorithm is applied to both synthetic and �eld data sets.Chapter 5 is concerned with the application of GCV to the 1-D EM inverse problem. Abrief description of how GCV is used to estimate random noise is presented. A discussionof how it has been applied in both linear and non-linear inverse problems is also included.

Chapter 1. Introduction 11An inversion algorithm is developed that uses GCV to estimate noise. The algorithm isdeveloped for the 1-D EM inverse problem to recover a conductivity structure, as well asto recover both conductivity and a geometric survey parameter. The algorithm is thentested on synthetic data. A discussion of the applicability of GCV to small data sets isalso presented.The �nal chapter will summarize the results of the entire thesis.

Chapter 2Background InformationAs was stated in Chapter 1 this thesis will address a number of problems that can arisewhen attempting to invert problematic �eld data sets. In order to do this it is necessaryto have a good understanding of all aspects of the inversion process. The purpose of thischapter is to provide the required background information. I will begin by de�ning theEM data used here. This will be followed by a discussion of how the 1-D EM inverseproblem is currently solved. This will provide the tools needed to deal with the proposedinverse problems.2.1 De�ning the EM DataThis discussion of frequency domain geophysical EM methods will be focused towardsmall loop EM surveys. These surveys employ a small loop of wire as a transmitter. Inorder for a transmitter loop to be considered small it must have a radius that is muchsmaller than the distance that separates the transmitter and the receiver. The receiver istypically a small coil that measures the time rate of change of the magnetic ux density.Methods which employ this type of transmitter receiver pair are commonly called loop-loop EM surveys. While it is possible to measure the magnetic ux density itself using a ux-gate magnetometer as a receiver, both the airborne and ground based surveys I amconcerned with use coils as receivers and therefore, I will not discuss the details of usingmagnetometers, but the alteration of my methodology to work with ux is simple.12

Chapter 2. Background Information 13HC VC

CA PP

Tx Rx

Tx

Tx

Tx

Rx

Rx RxFigure 2.1: The four common types of transmitter-receiver geometries used in loop-loopEM surveys. The �gure is labeled as follows : Tx - the transmitter, Rx - the receiver,HC - horizontal coplanar, VC - vertical coplanar, CA - coaxial, and PP - perpendicular.There are many possible con�gurations of transmitter and receiver coils in loop-loop surveys. The four most common geometries are horizontal coplanar (HC), verticalcoplanar (VC), coaxial (CA), and perpendicular (PP). Examples of these transmitter-receiver pairs are shown in Figure 2.1. In the HC geometry the axes of both coils areperpendicular to the surface of the earth and therefore, the coils lie in a common planeparallel to the surface of the earth. The VC geometry can be thought of as the HCgeometry shown in Figure 2.1 rotated 90o out of the page. The VC coils lie in a commonplane perpendicular to the surface of the earth and the axes of the coils are parallelto the surface. The coils in the CA geometry lie on a common axis which is parallelto the surface of the earth. Finally, in the PP geometry, the axes of the two coils areperpendicular to one another and both of these axes lie in a plane perpendicular to thesurface of the earth. Each of these coil geometries enables the EM system to sample theearth in a di�erent way and therefore they will each generate di�erent data.Since the �eld data sets I am investigating contain data collected using only the HCgeometry I will not discuss any of the other geometries within the thesis. However, itis important to remember that the principles of loop-loop EM measurements remain the

Chapter 2. Background Information 14S

Tx Rx

h

x

zFigure 2.2: Geometry of the generic horizontal coplanar EM system. Tx and Rx denotethe transmitter and receiver coils respectively, s is the coil separation, and h is themeasurement height above the surface of the earth.same for any geometry, and all of the methods that are developed within the thesis couldeasily be applied to other transmitter-receiver geometries (Zhang, et al., submitted forpublication).2.1.1 Generic Horizontal Coplanar SystemsA diagram of the generic horizontal coplanar EM system is shown in Figure 2.2. In thiscon�guration the transmitter coil is located a vertical distance h above the surface of theearth and is oriented with its axis perpendicular to the surface of the earth. The receiverand transmitter coil lie in the same horizontal plane and are separated by a horizontaldistance s.While I want to understand what goes on in a measurement taken over a givenconductivity structure it is always good practice to start simple. Therefore, I will beginby considering the case when the transmitter and receiver pair are in free space, shownin Figure 2.3(a). A time varying current I is set up in the transmitter loop. This currenthas the formI(!) = I0ei!t; (2.1)where I0 is the amplitude of the current, ! is the angular frequency, and t is time. Thiscurrent produces the primary magnetic �eld HP . It is known from Faraday's Law that

Chapter 2. Background Information 15Tx Rx

Tx Rx

a)

b)

HP

HS

IIS

σFigure 2.3: Cartoon showing the behavior of a horizontal coplanar EM system (a) in freespace and (b) in the presence of a conductive body. HP is the primary �eld, HS is thesecondary �eld, I is the source current, and IS are the eddy currents.

Chapter 2. Background Information 16a loop in the presence of a time varying magnetic �eld will have an electromotive force(emf) associated with it. The emf measured in the receiver coil is referred to as theprimary voltage V0, and it has the formV0 = �@�@t ; (2.2)where � is the magnetic ux through the coil which is de�ned as� = NR ZSB � dS; (2.3)where NR is the number of turns in the receiver coil, S is the surface de�ned by thereceiver coil, and B is the magnetic ux density. By substituting Equation 2.3 intoEquation 2.2, and by making use of both the constitutive relation B = �0H and thetime dependence of the magnetic �eld, I am left withV0 = �i!�0NR ZSHP � dS; (2.4)where �0 is the magnetic permeability of free space. Since the radius of the receiver coilis assumed to be much smaller than the separation of the transmitter and receiver coil Ican assume that HP is constant across the surface S. This simpli�es the expression forthe primary voltage toV0 = �i!�0NRARHPz (s); (2.5)where AR is the area enclosed by the receiver coil and HPz is the vertical component ofthe primary �eld at the center of the receiver coil. The primary voltage is commonlynormalized by the amplitude of the current in the transmitter. This gives me the mutualimpedance of the two coils in free spaceZ0 = V0I0 : (2.6)

Chapter 2. Background Information 17Z0 is often referred to simply as the free space impedance.The next step is to introduce the e�ect of a conductive body near by the transmitter-receiver pair, shown in Figure 2.3(b). When HP interacts with a conductive body � theprimary �eld will induce eddy currents IS in the conductor. These currents will in turnproduce a secondary magnetic �eld HS. The �eld sensed at the receiver coil HR will bea combination of both HP and HS. Following the same steps I used to calculate V0 Isee that the voltage V measured in the receiver coil will have the formV = �i!�0NRAR �HPz (s) +HSz (!; �; h; s)� : (2.7)In Equation 2.7 the primary �eld at the receiver coil only depends upon the coil separationwhile the secondary �eld is a function of the frequency, the conductivity structure, themeasurement height, and the coil separation. The voltage V is normalized by I0 to givean expression for the mutual impedance Z of the two coils in the presence of a conductivebody Z = VI0 : (2.8)The quantity which the EM system measures is V . However, a plot of V as a functionof frequency is not very informative. In order to glean some information about subsurfaceconductivity structure from EM measurements the data are presented as the mutualimpedance ratio Z=Z0 or more commonly the relative change in the impedance�ZZ0 = ZZ0 � 1; (2.9)where �Z = Z � Z0. By substituting Equations 2.5 and 2.7 into the de�nitions of Z0and Z, I getZZ0 � 1 = HSz (!; �; h; s)HPz (s) : (2.10)

Chapter 2. Background Information 18This is the value that is recorded by the EM system. However, it is important to note thatHS lags HP , due to the inductive interaction of HP and the conductor, and thereforeeach datum is a complex quantity. A single datum from Equation 2.10 can be splitinto a real and imaginary part. The real part is called the inphase component and theimaginary part is called the quadrature component. The �nal form of the data from thegeneric horizontal coplanar EM system isInphase = Re �HSz (!; �; h; s)�HPz (s) � �; (2.11)and Quadrature = Im �HSz (!; �; h; s)�HPz (s) � �: (2.12)where � is a multiplicative factor. This factor is usually set equal to 100 or 106 such thatthe data units are either percent or parts per million of the primary �eld. The units andmultiplicative factors have not been de�ned in the above equations so as to make themas generic as possible.2.1.2 Speci�c Horizontal Coplanar SystemsAs mentioned previously, the voltage V is actually measured by the system. Thereforein order to generate the data in the form shown in Equations 2.11 and 2.12 the primaryvoltage V0 must be calculated. When using the horizontal coplanar geometry the primary�eld value at the receiver is easy to calculate. It is equal to the vertical component ofthe magnetic �eld from a dipole source.HPz (s) = �mT4�s3 ; (2.13)where mT is the dipole moment of the transmitter coil which is de�ned asmT = IATNT ; (2.14)

Chapter 2. Background Information 19h

sFigure 2.4: Cartoon of a typical airborne EM system. h is the measurement height ands is the coil separation.where AT is the area enclosed by the transmitter coil, and NT is the number of turns inthe transmitter coil. By substituting Equation 2.13 into Equation 2.5 I getV0 = i!�0NRAR mT4�s3 : (2.15)While the horizontal coplanar geometry is used widely in both airborne and groundbased surveys there are slight di�erences in the way the data are generated. The di�erencelies in how the value for the primary voltage is attained. Therefore I will present a briefdiscussion of the details of airborne and ground based EM surveys.Airborne EMThere are many types of airborne EM (AEM) systems in use. I will concentrate on aDighem type of AEM system in which the coil pairs are housed in a \bird" which is towedbehind a helicopter. A cartoon of a typical AEM system is shown in Figure 2.4. The birdcontains a number of coils at �xed separations, each of which can make measurements ata particular frequency. Since it is not possible for the measurement height h to be �xed ata known value while the survey is own, it is determined using a laser altimeter mounted

Chapter 2. Background Information 20S

Tx RxBx

SB

- +αVB V

αVB

V - αVB

gaincontrolFigure 2.5: The bucking coil system used in AEM. The coils are labeled as follows: Txis the transmitter, Rx is the receiver, and Bx is the bucking coil. The separations arelabeled as follows: s the separation between Tx and Rx and sB is the separation betweenTx and Bx. V is the voltage measured at the receiver coil and VB is the voltage measuredin the bucking coil. The gain control � is adjusted such that �VB is equal to the primaryvoltage at the receiver V0. The output of the system V � �VB is equal to the voltageinduced in the receiver coil by the secondary �elds.on the helicopter. The measurement height is calculated by subtracting the estimateddistance between the bird and the helicopter from the reading on the altimeter. Thecalculated value of h is included in the data output from the system.The fact that the coils are rigidly mounted within the bird ensures that the coilseparation is constant throughout the survey. This enables the use of a bucking coil toremove the e�ect of the primary �eld at the receiver. A discussion of the types of buckingcoils commonly used in EM systems is presented in Norton, et al. (1999). My discussionwill concentrate on the type used in Dighem AEM systems as described in Fitterman(1998).The bucking coil, shown in Figure 2.5, is located between the transmitter and thereceiver. Since the goal is to remove the primary �eld from the �eld sensed at thereceiver coil, I want the voltage VB in the bucking coil to be entirely due to the primary

Chapter 2. Background Information 21�eld. Therefore, the distance sB between the transmitter and the bucking coil is chosensuch that the amplitude of the primary �eld at the bucking coil is much larger thanthe amplitude of the secondary �eld. In order to satisfy this condition the bucking coilis placed in close proximity to the transmitter coil. The bucking coil voltage is thenadjusted using a gain control � (shown in Figure 2.5) such that V0 = �VB. This gaincalibration is done prior to the survey on the ground. It is assumed that the groundover which the AEM system is calibrated is highly resistive and therefore, the e�ect ofthe secondary �elds on the system is negligible. Fitterman (1998) discusses problemsthat can be encountered when the subsurface is in fact conductive. However, for thisinvestigation I will assume that the calibration procedure is successful. Therefore, thevalue �VB will provide an accurate estimate of V0.Figure 2.5 shows that the bucking coil and receiver coil are wired together. Thisallows the estimated primary voltage to be removed from the measured voltage to leavethe secondary voltage. The AEM system also normalizes the secondary voltage by �VB.This is equivalent to substituting the gain adjusted bucking coil voltage into Equation 2.10such thatZZ0 � 1 = V � �VB�VB ; (2.16)and since I am assuming that �VB = V0 I getZZ0 � 1 = HSz (!; �; h; s)HPz (s) : (2.17)Due to the fact that the secondary �eld values are very small in AEM surveys, the dataare expressed in parts per million (PPM) of the primary �eld. Therefore, a single AEMmeasurement will result in data of the following formInphase(PPM) = Re �HSz (!; �; h; s)�HPz (s) � 106; (2.18)

Chapter 2. Background Information 22h

sFigure 2.6: Cartoon of a typical HLEM system. h is the measurement height and s isthe coil separation.and Quadrature(PPM) = Im �HSz (!; �; h; s)�HPz (s) � 106: (2.19)Ground based EMThere are many types of ground based horizontal loop EM (HLEM) systems in use. I willconcentrate on a Max-Min type of system in which the transmitter and receiver coils areindependent and are carried by two people. This makes it possible to take measurementsat various preset coil separations as well as at a range of frequencies. A cartoon of atypical HLEM system is shown in Figure 2.6.While the transmitter and receiver are not rigidly connected they are linked via thereference cable.The reference cable is used to pass information about the phase and amp-litude of the primary �eld to the receiver, and to provide the operators with an estimateof the coil separation. The HLEM system can be used for both parametric soundings(when the frequency is varied at a �xed coil separation) and geometric soundings (whenthe coil separation is varied at a �xed frequency). While analysis will focus on paramet-ric sounding results all of the methods developed in the thesis are equally applicable togeometric sounding data.Since the coils are being carried by people the height of the transmitter coil maynot be equal to the height of receiver coil. It is also possible for the coils to be tilted

Chapter 2. Background Information 23incorrectly with respect to one another such that they are not coplanar. However, formy investigation, I assume that the e�ects of such errors are negligible and hence it isassumed that the measurement height is constant and the coils are oriented properly.When taking a measurement, the transmitter and receiver coils must be positioned apreset distance apart from one another. Their separation is commonly estimated using thelength of the reference cable or by using station location markers. The preset separationvalue is used to calculate the voltage VC that would be induced in the receiver by theprimary �eld alone. This gives meZZ0 � 1 = V � VCVC (2.20)and since VC is calculated using Equation 2.5 I can assume V0 = VC such thatZZ0 � 1 = HSz (!; �; h; s)HPz (s) : (2.21)Due to the fact that both the transmitter and receiver coils are positioned close tothe surface of the earth, the amplitude of HLEM data is much larger than AEM data.Therefore, HLEM data are usually expressed in units of percent of the primary �eld. Asingle HLEM measurement will result in data of the following formInphase(%) = Re �HSz (!; �; h; s)�HPz (s) � 100; (2.22)and Quadrature(%) = Im �HSz (!; �; h; s)�HPz (s) � 100: (2.23)Comparison of AEM and HLEMWhile both AEM and HLEM surveys can be carried out with horizontal coplanar coilsthere is an important di�erence between them which should be reinforced. As was men-tioned AEM surveys use coils which are rigidly mounted such that the coil separation s

Chapter 2. Background Information 24is �xed. The measurement height h in these surveys is determined by the AEM system.It is important to note that this value of h may not be correct. HLEM surveys di�erin the fact that the transmitter and receiver coils are suspended at a constant heighth above the earth. Another di�erence is that the coils are independent and therefore,they must be positioned at the expected separation s for each measurement. In prac-tice, accurately positioning the coils at a �xed distance s can be di�cult. Therefore,both AEM and HLEM each have an important geometrical survey parameter that isnot accurately known at the time of data collection. While the presence of these errorsis important, most inversion algorithms do not recognize their existence and therefore,AEM and HLEM data are treated identically.2.2 Details of the Inverse ProblemIn order to understand the problems that can occur when inverting EM data it is impor-tant to understand the details of the inversion process. I will treat this in two sections:the forward problem and the inverse problem.2.2.1 The Forward ProblemHaving discussed the details of how the data from the two EM systems are generated Iwould like to be able to simulate the response of an EM experiment numerically. The dataI collect has the generic form shown in Equations 2.11 and 2.12. The set of measurementsfrom a given sounding which I wish to invert will be stored in a data vector d. The entriesof this vector will be the real and imaginary parts of the data values. The total numberof data N will be equal to twice the number of measurements and the two parts of thedata from the ith measurement di will have the formRe(di) = Re �HSz (!i; �; hi; si)�HPz (si) � �; (2.24)

Chapter 2. Background Information 25and Im(di) = Im �HSz (!i; �; hi; si)�HPz (si) � �: (2.25)where each of the subscripted parameters represent the parameter value associated withthe ith measurement and � is some multiplicative factor. It has been shown in Equa-tion 2.13 that the primary �eld value at the receiver can easily be calculated using theseparation of the transmitter and receiver. However, the task of calculating the resultantsecondary magnetic �elds for a given coil separation, measurement height, frequency, andconductivity distribution is a little more involved.While I know that the subsurface conductivity structure is usually 3-D I choose in-stead to model the response of a 1-D structure. The reason for this is that in many casesthe volume of the earth that is being sampled by the EM system can be adequately rep-resented by a 1-D conductivity structure. The validity of this assumption depends uponthe size of the \footprint" of the system relative to the scale of the lateral inhomogene-ity I am dealing with. AEM systems have quite a large \footprint"; however, they arecommonly used for mapping large scale features. The \footprint" of an HLEM systemdepends upon the coil separation being used and the separation is chosen according tothe size of the feature that is being mapped. Surveys with small coil separations areused to map small scale structures and larger separations are used to map larger struc-tures. In many of these situations the 1-D assumption is acceptable and therefore, I willuse a forward modelling code that generates the secondary �elds from an arbitrary 1-Dconductivity structure. The choice of a 1-D model instead of a 3-D model also providesme with a faster forward modelling routine which is important since I am planning toincorporate it into an inversion scheme in which multiple forward modellings will be ne-cessary. The general solution of the 1-D frequency domain EM problem can be found inWard & Hohmann (1988) and I have based my solution on their work. I will present an

Chapter 2. Background Information 26s

Tx Rx

h

σ1

σi

σM

σM+1

x

z

surface

i = 2, ... M-1

h1

hi

hMFigure 2.7: Discretization of the problem domain for the forward problem. Tx and Rxdenote the transmitter and receiver coils respectively, s is the coil separation, h is themeasurement height, hi is the thickness of the ith layer, and �i is the conductivity of theith layer.abbreviated description of the speci�c method used to calculate the secondary �elds.The details of the experiment are the same as in Section 2.1. The horizontal coplanarcoil geometry is shown in Figure 2.7 above a layered subsurface. The transmitter andreceiver coils are separated by a distance s and lie in a common horizontal plane at aheight h above the surface of the earth. The current I in the transmitter is de�ned inEquation 2.1. In this coordinate system the z direction is downwards. The subsurfaceis discretized into M layers. Each of these layers is assigned a thickness, hi, and aconductivity, �i. These M layers overlie a half-space of conductivity �M+1. I haveassumed that the magnetic permeability of each layer is equal to that of free space.The region between the surface and the coils will be referred to as the 0th layer andthe thickness of this layer is equal to the measurement height such that h0 = h. Theconductivity of the air is �0 = 0.As with any EM problem I start with Maxwell's equations. I will make use of the

Chapter 2. Background Information 27frequency domain quasi-static Maxwell's equations in which the e�ects of displacementcurrents are neglected. The assumption that the e�ects of displacement currents arenegligible is described in Weaver (1994) to be equivalent to the assumption that thetime taken by an EM wave to travel over the region of interest (i.e. the region of thesounding) is much less than the scale time scale over which the EM �elds are changing(i.e. a time scale of t = 2�=! ). For the length scales and measurement frequenciesconsidered in my EM experiments the quasi-static assumption holds for any region withinthe problem domain. Therefore, the equations that describe the �elds in a region ofconstant conductivity � arer�E + i�0!H = 0 (2.26)r�H � �E = JS ; (2.27)where E is the electric �eld, H is the magnetic �eld, and JS is an electric currentsource. The symmetry of the loop source allows me to pose the problem in cylindricalcoordinates. Since the current source is in the � direction, and all of the conductivityboundaries are parallel to the r � z plane, all of the induced current is forced to owin the � direction. The vertical and radial components of the electric �eld will thus beequal to zero and therefore I see from Equation 2.26, that the tangential component ofthe magnetic �eld is also equal to zero. This leaves me with the following set of equations!�0Hr = @E�@z ; (2.28)!�0Hz = �1r @@r (rE�); (2.29)@Hr@z � @Hz@z = �E� + Js: (2.30)Each of the E and H �eld components is a function of r, z, and ! even though ithas not been explicitly stated. By substituting Hr from Equation 2.28, and Hz from

Chapter 2. Background Information 28Equation 2.29 into Equation 2.30, I get the following partial di�erential equation for thetangential electric �eld@2@z2 (E�)� @@r��1r @@r (rE�)�+ k2E� = i!�0Js; (2.31)where k2 = �i!�0�. Following Ryu et al. (1970) I can transform Equation 2.31 into theHankel domain to get an ordinary di�erential equation for the tangential component ofthe electric �eld� @2@z2 � u2� ~E� = i!�0 ~Js; (2.32)where u2 = �2 � k2, and ~E� and ~Js are the Hankel transforms of E� and Js.As mentioned previously, these equations only hold for regions of constant conductiv-ity. Therefore, I must solve Equation 2.32 in each of the M +2 regions that were de�nedin Figure 2.7. Then by matching the interface conditions I will be able to propagate thesolution from the lower half-space up to the surface.The expression for the secondary electric �eld at the receiver loop, in the Hankeldomain is~E�(�; !; h) = �i!�04� e�2u0h0u0 RTEJ1(�r)�2d�; (2.33)where J1 is a 1st order Bessel function and RTE is the transverse electric (TE) �eld modere ection coe�cient. The re ection coe�cient is de�ned asRTE = Z1 � Z0Z1 + Z0 ; (2.34)where the intrinsic impedance of the ith interface Zi is de�ned asZi = �i!�0ui

Chapter 2. Background Information 29and the input impedance of the ith layer Z i is de�ned by the recursion relationZ i = Zi 2666664Z i+1 + Zi (1� e�ui2hi)(1 + e�ui2hi)Zi + Z i+1 (1� e�ui2hi)(1 + e�ui2hi)3777775 i = M; : : : ; 1; (2.35)whereZM+1 = ZM+1: (2.36)However, the value I want to calculate is Hz(r; !; z). Using Equation 2.29, and applyingthe inverse Hankel transform, I getHz(r; !; h0) = mT4� Z 10 e�2u0h0u0 RTEJ0(�r)�3d�: (2.37)The expression for Hz(r; !; h0) in Equation 2.37 is equal to the secondary magnetic �eldHSz . In order to convert these values into either percent or PPM of the primary �eldformat it is just a matter of normalizing the values to the primary �eld and multiplyingby the appropriate factor. Therefore, I now have a method of calculating the data foruse in the inversion algorithm.2.2.2 The Inverse ProblemI have assumed that within the region of the survey the conductivity can be represented asa one-dimensional function of depth �true(z). The observed data dobs is a vector containingN data which are the result of an EM experiment overtop of �true(z). The data includesome unknown amount of measurement noise. The goal of the inversion is to recover thefunction �true(z) however, this problem is seriously underdetermined since I have onlyN data constraints and a function has in�nitely many degrees of freedom. Therefore,the solution to the inverse problem has a non-unique solution. This means that if one

Chapter 2. Background Information 30�(z) can be found which reproduces the data there exist an in�nite number of other �(z)functions which can also �t the data. In order to deal with the problem non-uniquenessit is necessary to provide information about the speci�c type of conductivity model Iwant to recover. This is where it is possible to incorporate any a priori information thatis available about the model.The �rst pieces of information that I can include in the model are that conductivityis always positive and that values for earth materials can vary over several orders ofmagnitude. Therefore, I will de�ne the continuous model M to be equal toM = log(�(z)): (2.38)This form of the conductivity is also a good choice because it treats relative conductivitycontrasts well.Now that I have chosen a model, more information can be added by introducing amodel objective function �m of the form�m(M) = �s Z ws(z)(M�Mref )2dz + �z Z wz(z)�@(M�Mref )@z �2dz; (2.39)whereMref is some reference model. The �rst term of �m is a measure of how closeMis to the reference modelMref . This term is referred to as the smallest model norm. Thesecond term provides a measure of the derivative of M in the vertical direction. Thisterm is referred to as the attest model norm. The parameters �s and �z determine therelative importance of the smallest and attest components of the model norm and thefunctions ws(z) and wz(z) are spatial weighting functions which can be used to includefurther information about the model. This formulation provides the ability to recoverthe smallest model when �s = 1 and �z = 0, the attest model �s = 0 and �z = 1, or anycombination of the two. Usually I want to �nd a model that is as featureless as possibleand hence �s and �z are selected such that the second term in �m is dominant over the

Chapter 2. Background Information 31�rst. When no extra information is available it is common to set ws(z) = wz(z) = 1.This is the general de�nition of �m that I will adopt throughout the thesis.While I would like to be able to recover the continuous functionM, it is necessary todiscretize my model in order to both solve the inverse problem and calculate the forwardmodelling. Thus the model is divided into M layers of constant conductivity (as shownin Figure 2.7 ) such that m, the discrete representation of M, will have the formm = [log(�1); : : : ; log(�M)]T : (2.40)It is important to note that M >> N and therefore, the problem remains underdeter-mined. The discrete form of �m in Equation 2.39 will be�m(m) = (m�mref)T ��sW Ts Ws + �zW Tz Wz� (m�mref ); (2.41)where Ws is the smallest model weighting matrix and Wz is the attest model weightingmatrix. These terms can be combined such that�m(m) = (m�mref)TW TmWm(m�mref ); (2.42)where Wm is the model weighting matrix which encompasses all of the details incor-porated into my model objective function. The model objective function can also beexpressed as the following�m(m) = jjWm(m�mref)jj2; (2.43)where jj � jj is the Euclidean 2-norm.The design of the model objective function is such that if I simply minimize �m, andWm is invertible, the model I recover will be equal to the reference model.Now that I have de�ned the character of the model I want to recover I can usethe observed data to re�ne the model further. The observed data can be expressed

Chapter 2. Background Information 32mathematically asdobs = F [mtrue] + �; (2.44)where mtrue is the discrete representation of the true model, F is the forward operatorwhich generates the data, and � is a vector of length N which contains the noise associatedwith the data. I will assume that � is Gaussian random noise.The fact that the observed data are contaminated with noise suggests that �tting thedata exactly is a bad idea. Therefore, I introduce a data mis�t function �d. As its nameindicates �d is a measure of how well the data predicted from a given model m �ts theobserved data. Since I have assumed that the errors are Gaussian, a sensible mis�t termwould be�d = NXi=1�Fi[m]� dobsi�i �2; (2.45)where �i is an estimate of the standard deviation of the noise on the ith datum. I canrewrite �d in matrix notation as follows�d = jjWd(F [m]� dobs)jj2; (2.46)where Wd is the diagonal data weighting matrix which is equal toWd = diag� 1�1 ; : : : ; 1�N�: (2.47)Di�erent mis�t criteria may be required in some problems and they can be easily de�nedby choosing a di�erent Wd, a di�erent norm, or by completely rede�ning �d. Using thede�nition of �d from Equation 2.46 I see that it is a random variable with a �2 distributionand therefore, it will have an expected value approximately equal to N . Thus the targetmis�t �?d should also be approximately equal to N .

Chapter 2. Background Information 33Now that I have dealt with the noise in the data and the problem of non-uniquenessI can express the goal of the inversion as:Find m which minimizes �m such that �d = �?d: (2.48)This can also be expressed in the form of an optimization problem where I want tominimize a global objective function � which is equal to�(m) = jjWd(F [m]� dobs)jj2 + �jjWm(m�mref )jj2; (2.49)where � is the regularization or trade-o� parameter. The problem will then become:Minimize � = �d + ��m such that �d = �?d; (2.50)The methods used to solve this problem depend upon the details of the forward modelling.When the forward modelling F [m] is linear the data can be expressed in general asF [m] = Gm; (2.51)where G is an N � M matrix. Substituting Equation 2.51 into the global objectivefunction from Equation 2.49 results in the objective function for the linear inverse problem�(m) = jjWd(Gm� dobs)jj2 + �jjWm(m�mref )jj2: (2.52)The minimization of the linear functional in Equation 2.52 for a particular � is solved bycalculating the gradient of � and setting it equal to zero.The gradient g is obtained by di�erentiating � with respect to the model mg(m) = 2GTW Td Wd(Gm� dobs) + 2�W TmWm(m�mref ): (2.53)Setting g(m) = 0 leads to the matrix system of equations[GTW Td WdG + �W TmWm]m = GTW Td Wddobs + �W TmWmmref ; (2.54)

Chapter 2. Background Information 34★

Figure 2.8: Cartoon of a typical Tikhonov curve. �d is the data mis�t, �?d is the targetmis�t, �m is the model objective function, and � is the trade o� parameter.which can be solved to provide m which minimizes Equation 2.52. By carrying outthe minimization at a number of � values it is possible to generate a plot of �d(�)versus �m(�). This plot is called the Tikhonov curve. A cartoon of a typical Tikhonovcurve is shown in Figure 2.8. This curve illustrates how the choice of � dictates thevalues of �d and �m and therefore, the character of the recovered model. When � getslarge, �(m) � ��m and the minimization is strictly searching for the minimum structuremodel. This can result in models that do not �t the data very well. Whereas, when� approaches zero, �(m) � �d. In this case the minimization will attempt to �nd themodel that provides the best �t to the data however, this can result in models with largeamounts of structure. In between these two extremes the relative importance of �m and�d is determined by the value of �. The Tikhonov curve quanti�es the way in which thechoice of � regularizes the solution.In linear inverse problems the process of �nding � which provides the desired �d valueis straight forward. By plotting the Tikhonov curve it is possible to determine whethera � exists such that �d = �?d. If such a � does exist, as shown in Figure 2.8, it is a simple

Chapter 2. Background Information 35matter to �nd it. In the case that the forward modelling F [m] is non-linear however, thetask of �nding m which minimizes the objective function from Equation 2.49 for a given� is more involved. It is necessary to linearize the problem and solve it iteratively. Thisis done using a standard Gauss-Newton methodology.As in the linear case the minimization of the functional in Equation 2.49 for a partic-ular � is solved by calculating the gradient of � and setting it equal to zero. The gradientg of � is equal tog(m) = 2JTW Td Wd(F [m]� dobs) + 2�W TmWm(m�mref ); (2.55)where J is the Jacobian matrix de�ned asJ = @F [m]@m : (2.56)The gradient from Equation 2.55 is a non-linear equation in m and as a result, it isnecessary to linearize g about some current model mk. The linearized version of g isexpressed asg(mk + �m) = g(mk) +H(mk)�m; (2.57)where H(mk) is the Hessian matrix, evaluated at mk, which is de�ned asH(mk) = @g(mk)@mk = 2@J(mk)T@mk W Td Wd(F [mk]� dobs) + : : :2J(mk)TW Td WdJ(mk) + 2�W TmWm: (2.58)In order to avoid evaluating the derivative of the Jacobian, the Hessian is approximatedas H(mk) � 2J(mk)TW Td WdJ(mk) + 2�W TmWm: (2.59)

Chapter 2. Background Information 36Setting g(mk + �m) = 0 yields the matrix system of equations[J(mk)TW Td WdJ(mk) + �W TmWm]�m = : : :J(mk)TW Td Wd(dobs � F [mk])� �W TmWm(m�mref ); (2.60)which can be solved to give �m such that the model for the next iteration will be equalto mk+1 = mk + �m. The process of linearization and iteration is continued until thesolution converges to the minimum of Equation 2.49. This process can be repeated for anumber of � values, as in the linear case, to �nd the � which satis�es �d = �?d.

Chapter 3Geometric Survey Parameter ErrorsIn order to interpret EM data it is necessary to know the parameter values that de�nethe survey from which the data were collected. Without knowledge of these parametersthe data are merely a set of numbers whose relationship to one another is incomprehens-ible. For this reason the pertinent survey parameter values are recorded with the dataoutput from the EM system. However, both AEM and HLEM surveys are susceptible togeometric survey parameter errors that can cause problems in the interpretation and/orinversion of the data.In Section 1.1 I conjectured that some of the problems associated with the interpreta-tion and inversion of the �eld data sets may be the result of geometric survey parametererrors. The goal of this chapter is to show that these problems are indeed due to geomet-ric survey parameter errors. I will discuss the details of how geometric survey parametererrors occur and how they a�ect both EM data and the inversion results.This will provide me with the ability to identify the presence of geometric surveyparameter errors in the �eld data sets and inversion results. However, before I beginit is necessary to introduce the notation I will use throughout the thesis as well as thesynthetic models with which I will be concerned.NotationIn order to introduce notation I will consider a generic survey parameter p. The valueof p that the EM system records in the data output is referred to as the expected or37

Chapter 3. Geometric Survey Parameter Errors 38estimated value of p and it is represented with the variable pest. The true value of p isrepresented with the variable ptrue. The parameter error is represented with the variable�p and is de�ned as�p = ptrue � pest: (3.1)This notation will be applied to all of the parameters discussed in the thesis.Synthetic ExamplesThroughout this chapter I will use synthetic data from three EM surveys. The �rst surveysimulates a typical AEM survey, the second simulates an HLEM survey performed at asmall coil separation, and the third simulates an HLEM survey performed at a larger coilseparation. A description of the details of each survey and the conductivity models areprovided along with an example of synthetic data from each survey. The data shown arefree of parameter errors and no noise has been added so that it is possible to becomefamiliar with the uncontaminated form of the data before I begin the investigation ofparameter errors.The AEM survey consists of measurements of the inphase and quadrature compo-nents of the secondary �elds at 10 frequencies ranging from 110 Hz to 56320 Hz. Eachmeasurement is taken at a precise coil separation of 10 m and at some height above theearth. I will refer to this type of measurement as the AEM sounding. The conductivitymodel over which the AEM soundings will be simulated is shown in Figure 3.1(a). Itconsists of a 20 m thick 0.1 S/m layer buried 30 m deep in a 0.01 S/m halfspace. Thedata shown in Figure 3.1(b) are acquired at a height of 30 m. It is important to notethat to simulate an AEM sounding it is necessary to de�ne the true measurement heightvalue.Both of the HLEM surveys consist of measurements of the inphase and quadrature

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Chapter 3. Geometric Survey Parameter Errors 40components of the secondary �elds at 10 frequencies ranging from 110 Hz to 56320 Hz,each with a measurement height of 1 m. The di�erence between the two synthetic HLEMexamples is the coil separation; the �rst survey is performed with a coil separation of10 m and the second has a coil separation of 50 m. The conductivity model used forthe small separation HLEM soundings is shown in Figure 3.1(c). It consists of a 20 mthick 0.1 S/m layer buried 10 m deep in a 0.01 S/m halfspace. Figure 3.1(d) shows thedata from a measurement with a coil separation of 10 m. Since the depth of penetrationincreases with coil separation it is possible to look at deeper structures therefore, I chosea di�erent model for the large coil separation example. The conductivity model I usefor the large separation HLEM soundings is shown in Figure 3.1(e). It consists of a 30m thick 0.1 S/m layer buried 20 m deep in a 0.01 S/m halfspace. The synthetic datagenerated with a coil separation of 50 m is shown in Figure 3.1(f).Now that I have introduced the notation and synthetic examples it is possible to beginthe discussion of parameter errors.3.1 Measurement Height Errors in AEMAs mentioned in Section 2.1.2 the transmitter-receiver pairs in AEM surveys are rigidlymounted within the bird and therefore, the coil separation is assumed to be �xed. Becausethe bird is towed below the helicopter variations in height are unavoidable and therefore,the geometric survey parameter of interest in AEM surveys is the measurement height.3.1.1 How do measurement height errors occur?The true measurement height htrue is de�ned as the distance from the bird to the surfaceof the earth. However, the measurement height which is recorded in the data �le is a valuecalculated by the AEM system. Since this value may not be equal to htrue I will refer

Chapter 3. Geometric Survey Parameter Errors 41h

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bFigure 3.2: Cartoon of an AEM system showing the values used to calculate measurementheight h. a is the altitude of the helicopter and b is the distance from the helicopter tothe bird.to it as the estimated measurement height hest. Using Equation 3.1 the measurementheight error �h can be expressed as�h = htrue � hest: (3.2)A cartoon of the AEM system and the values used to calculate hest are shown inFigure 3.2. The altitude of the helicopter a is determined using a laser altimeter and thedistance from the helicopter to the bird b is assumed to be constant. The assumptionabout the �xed value of b is based on the fact that the bird, and the cable that connectsit to the helicopter, are designed so that b is equal to a known value when the helicopteris traveling at a constant speed. Using the values of a and b, h is calculated to beh = a� b: (3.3)By substituting the subscript est and subscript true versions of Equation 3.3 into Equa-tion 3.2 I get the expression�h = �a� �b; (3.4)

Chapter 3. Geometric Survey Parameter Errors 42Case 1. Case 2.Figure 3.3: The two situations which that result in measurement height errors. Case1. �h > 0, measurement height underestimated because of tree cover. Case 2. �h < 0,measurement height overestimated due to extreme terrain.which de�nes the measurement height error in terms of errors in a and b.Errors possibly exist in a and b however, for my purposes the bottom line is the valueof �h. When �h > 0 the bird is farther away from the surface than expected and when�h < 0 the bird is closer to the surface than expected. Figure 3.3 illustrates two possiblecases in which measurement height errors can exist.3.1.2 How do measurement height errors a�ect the data?In order to assess the e�ect of measurement height errors I must �rst investigate how thedata are a�ected by measurement height. Letting htrue denote the true height, the AEMdata from Equations 2.18 and 2.19 areInphase(PPM) = Re�HSz (!; �; htrue; s)HPz (s) �� 106; (3.5)and Quadrature(PPM) = Im�HSz (!; �; htrue; s)HPz (s) �� 106: (3.6)By generating synthetic data from AEM soundings with di�erent htrue values it willbe possible to investigate the interpretation problems caused by measurement heighterrors. Figure 3.4 shows the inphase and quadrature components of AEM sounding data

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Figure 3.4: Synthetic data from AEM soundings with htrue = 24 m, 30 m, and 36 m. (a)The inphase component of the data (htrue = 24 m - diamonds, htrue = 30 m - circles,and htrue = 36 m - squares) (b) The quadrature component of the data (htrue = 24 m -diamonds, htrue = 30 m - crosses, and htrue = 36 m - squares).with htrue values of 24 m, 30 m, and 36 m. The inphase and quadrature componentof the htrue = 30 m data are represented by circles and crosses respectively in order toshow that they are identical to the data plotted in Figure 3.1(b). Both the inphase andquadrature component of the htrue = 24 m data are represented by diamonds and thehtrue = 36 m data are represented by squares.The curves in Figure 3.4 clearly illustrate the e�ect of measurement height on AEMdata. In general, the closer the bird is to the surface, the greater the signal, and viceversa. However, this e�ect does exhibit some frequency dependence. At lower frequenciesthe di�erence in the data is not large while at the highest frequency a �6 m change inmeasurement height can lead to a change of greater than �30% of the data amplitude.It follows that if htrue = 30 m, and if the AEM data were interpreted with an incorrectmeasurement height (either 24 m or 36 m), the high frequency data would need to beassigned an error of at least 30% of the data amplitude. Otherwise, any interpretationwould be the result of �tting \noise" caused by the presence of measurement heighterrors.

Chapter 3. Geometric Survey Parameter Errors 44hest �m �d �?d24 m 0.136 20.0 20.030 m 0.127 20.0 20.036 m 0.353 25.1 20.0Table 3.1: Results from the inversion of AEM data with hest values of 24 m, 30 m, and36 m when htrue is equal to 30 m. �m is the recovered model norm, �d is the data mis�tachieved by the inversion and �?d is the target mis�t. These values correspond to resultsplotted in Figure 3.53.1.3 How do measurement height errors a�ect the inversion results?In order to obtain some insight about the e�ects of assuming an incorrect measurementheight I will invert a single data set, acquired at a true height of 30 m, three di�erenttimes. The results of an actual sounding were simulated by adding Gaussian randomnoise with a standard deviation equal to 5% of the data amplitude + 10 PPM was addedto each synthetic datum. The noisy data were then inverted three times using hest valuesequal to 24 m, 30 m and 36 m respectively. The reference model for each inversion wasa 0.01 S/m halfspace and the target mis�t was equal to 20.The conductivity models recovered from the three inversions are plotted in Fig-ures 3.5(a), 3.5(c), and 3.5(e) and the observed and predicted data are plotted in Fig-ures 3.5(b), 3.5(d), and 3.5(f). The �nal data mis�t and model norm values from eachinversion are shown in Table 3.1. From the data mis�t values, and the data plotted inFigure 3.5, I can see that the inversion algorithm has been able to recover models that�t the data to the desired �2 mis�t level of 20 for both hest = 24 and 30 m and has donefairly well (�d � 25) for hest = 36 m. However, I can also see from both the model normvalues and the recovered models plotted in Figure 3.5 that the conductivity structurerecovered for each hest is di�erent. Since the inversion results from hest = 30 m (�h = 0)are free of measurement height errors I will adopt them as the results against which Icompare the other inversions.

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Figure 3.5: Results from the inversion of AEM data with h values of 24 m, 30 m, and 36m when htrue is equal to 30 m. Panels (a), (c), and (e) show the true (dashed line) andrecovered (solid line) conductivity models and panels (b), (d), and (f) show the observeddata (inphase - circles, quadrature - crosses) and predicted data (inphase - solid line,quadrature - dashed line) from the three inversions.

Chapter 3. Geometric Survey Parameter Errors 46The model recovered with an estimated measurement height of 24 m (�h > 0) plottedin Figure 3.5(b) has a region of low conductivity at the surface which is not present inFigure 3.5(d). This has been described as the \air layer" e�ect by Ellis & Shekhtman(1994) and is a result of the estimated height being smaller than the true value. I alsonotice that, as a result of the \air layer", the conductivity structure recovered using hest= 24 m is slightly pushed down in comparison to the structure recovered with hest = 30m. This distortion is an attempt to decrease the amplitude of the predicted data. The\air layer" is trying to provide the extra distance between the surface and the bird thatis needed recreate the survey that generated the observed data. The distance by whichthe recovered conductivity structure is pushed downwards is approximately equal to �h.I can see from the recovered model plotted in Figure 3.5(f) that the opposite e�ectis observed when the data is inverted with an estimated measurement height of 36 m(�h < 0). In this case there is a region of high conductivity at the surface which is notpresent in Figure 3.5(d). It also appears that in the model recovered using h = 36 mthe conductivity structure below the high conductivity region is distorted and pulled uptowards the surface in comparison to the structure recovered using h = 30 m. In thiscase the algorithm has a harder time correcting the problems caused by �h. Since it isnot possible to decrease the distance between the bird and the surface it is necessary to�nd another way to increase the amplitude of the data. This is achieved by increasingthe conductivity of the model near the surface. The extra structure at the surface causesthe rest of the recovered conductivity structure to be pulled up towards the surface. Thepresence of these features are due to the estimated measurement height being larger thanthe true value.It has been shown that the presence of measurement height errors in AEM datacan have a de�nite e�ect upon inversion results. From the above examples I see thatwhile it is still possible to achieve the target mis�t the recovered models are distorted by

Chapter 3. Geometric Survey Parameter Errors 47the incorrect measurement height estimates. While this makes it di�cult to assess thepresence of measurement height errors when the true conductivity model is not knownat least I have an idea of the ways in which the inversion results can be a�ected.3.2 Coil Separation Errors in HLEM dataAs mentioned in Section 2.1.2 the transmitter and receiver coils in HLEM surveys areindependent of each other. The coils are carried separately by two individuals and varia-tions in measurement height are considered to be negligible and as such the measurementheight is assumed to be �xed. However, the separation of the coils is variable. Therefore,the geometric survey parameter of interest in HLEM surveys is the coil separation.3.2.1 How do coil separation errors occur?Prior to collecting data the HLEM system must be set to operate at a particular coilseparation. This preset value is the expected coil separation sest and this is the coilseparation value recorded in the data �le. The distance which actually separates thetransmitter and receiver is the true coil separation strue. Using Equation 3.1, I canexpress the coil separation error �s as�s = strue � sest: (3.7)When a survey is performed on level ground separation errors are likely due to in-correct chaining of the station markers or reference cable. When a survey is performedin extreme terrain separation errors can occur even when the reference cable is chainedcorrectly. These types of errors can occur when passing over topographic features suchas gullies, cli�s, and valleys. Figure 3.6 shows the two situations that will result in coilseparation errors.

Chapter 3. Geometric Survey Parameter Errors 48Case 1. Case 2.Figure 3.6: The two situations that will result in coil separation errors. Case 1. �s > 0,the coils are too far apart, and Case 2. �s < 0 the coils are too close together.3.2.2 How do coil separation errors a�ect the data?In order to assess the e�ect of coil separation errors I must determine the roles that bothstrue and sest play in the HLEM data. The magnetic �elds sensed at the receiver coildepend upon strue. Therefore, the voltage V that is induced in the receiver is equal toV = �i!�0ms �HPz (strue) +HSz (strue)� : (3.8)However, the estimate of the primary voltage calculated by the HLEM system dependsupon sest. Therefore, the voltage VC is de�ned asVC = �i!�0msHPz (sest): (3.9)By substituting V , from Equation 3.8, and VC , from Equation 3.9, into the expressionfor the HLEM measurement, from Equation 2.20, I am left withZZ0 � 1 = HSz (!; �; h; strue) +HPz (strue)�HPz (sest)HPz (sest) : (3.10)From this expression it is clear that when sest 6= strue the HLEM measurements can notbe represented by the data from Equations 2.22 and 2.23. Therefore, the HLEM datamust be rewritten in terms of strue and sest asInphase(%) = Re�HSz (!; �; h; strue)HPz (sest) �� 100 + �HPz (strue; sest); (3.11)

Chapter 3. Geometric Survey Parameter Errors 49and Quadrature(%) = Im�HSz (!; �; h; strue)HPz (sest) �� 100 (3.12)where �HPz is the primary �eld error and is de�ned as�HPz (strue; sest) = HPz (strue)�HPz (sest)HPz (sest) � 100: (3.13)These equations show that both strue and sest have a de�nite e�ect upon HLEM data.This e�ect can be subdivided into the e�ects on the normalized secondary �elds and thee�ects on the primary �eld error. I can see from Equations 3.11 and 3.12 that the realand imaginary parts of the normalized secondary �elds behave in a fashion similar to theinphase and quadrature components of AEM data in Equations 3.5 and 3.6. However,this problem is compounded by the presence of �HPz in the inphase component.In order to get more insight into the behavior of �HPz I substitute the de�nition ofthe primary �eld from Equation 2.13 into Equation 3.13 and end up with�HPz (strue; sest) = �� seststrue�3 � 1� � 100: (3.14)This shows that the presence of the coil separation errors has a direct e�ect upon theinphase component of HLEM data by way of the �HPz term. From Equation 3.14 I seethat when �s > 0 (strue > sest) the value of �HPz will be negative and therefore the inphasedata will be shifted in the negative direction. Similarly when �s < 0 (strue < sest) thevalue of �HPz will be positive and the inphase data will be shifted in the positive direction.All of the e�ects on the data can be illustrated using the two HLEM examples. I willbegin by considering a survey in which three synthetic small separation HLEM soundingsare taken over the conductivity model shown in Figure 3.1(c). The sest value for all ofthe small separation soundings is set to 10 m and the strue value for each sounding is11 m, 10 m, and 9 m respectively. These values correspond to coil separation errors of

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Frequency (Hz)Figure 3.7: Synthetic data from small separation HLEM soundings with sest = 10 m andstrue = 9 m, 10 m, and 11 m. (a) Inphase component of the data (strue = 9 m - diamonds,strue = 10 m - circles, strue = 11 m - squares), (b) quadrature component of the data(strue = 9 m - diamonds, strue = 10 m - crosses, strue = 11 m - squares), (c) real part ofnormalized secondary �eld (as in panel (a)), and (d) primary �eld error (as in panel (a)).+1 m, 0 m, and -1 m. The synthetic data from each of the soundings, with no noiseadded, are plotted in Figure 3.7. The inphase and quadrature components of the strue =10 m data are represented by circles and crosses respectively in order to show that theyare identical to the data plotted in Figure 3.1(d). This data will be referred to as theexpected data. Both the inphase and quadrature component of the strue = 11 m dataare represented by squares and the strue = 9 m data are represented by diamonds.The plots in Figures 3.7(a) and (b) show the inphase and quadrature components of

Chapter 3. Geometric Survey Parameter Errors 51the data respectively. The shift due to �HPz seen in the inphase component, is muchlarger than the variation of the response with frequency. The two portions of the inphaseresponse, as per Equation 3.11, are plotted separately in Figures 3.7(c) and (d).Figures 3.7(c) and (b) illustrate that the closer the coils are to one another the largerthe amplitude normalized secondary �elds. These errors are similar to those seen in theAEM case. However, the e�ect of the primary �eld errors (Figure 3.7(d)) has a directe�ect upon the data. These shifts are independent of frequency and therefore, a�ect theentire inphase component of each sounding. This type of error can render the inphasecomponent of HLEM data (Figure 3.7(a)) uninterpretable.From Equation 3.14 it can be seen that the magnitude of the shift depends upon therelative size of the coil separation error. Therefore, the shift caused by �s = 1 m in asurvey with sest = 10 m is equivalent to the shift caused by �s = 5 m in a survey withsest = 50 m. However, it is unlikely to encounter coil separation errors larger than afew metres, even when considering surveys with large coil separation values. Therefore,surveys performed with small coil separations are more likely to be detrimentally a�ectedby coil separation errors. This di�erence can be illustrated with the synthetic data fromthe large separation HLEM example.Consider a survey in which three large separation HLEM soundings are taken overthe conductivity model shown in Figure 3.1(e). For each of the soundings sest is equal to50 m and the value of strue at each sounding is 51 m, 50 m, and 49 m respectively. Theseseparation values correspond to relative errors of +2%, 0%, and -2%. The syntheticdata from each of the soundings, with no noise added, are plotted in Figure 3.8. Theinphase and quadrature components of the strue = 50 m data are represented by circlesand crosses respectively in order to show that they are identical to the data plotted inFigure 3.1(f). This data will be referred to as the expected data. Both the inphase andquadrature components of the strue = 51 m data are represented by squares and the strue

Chapter 3. Geometric Survey Parameter Errors 52(a)

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= 51mFigure 3.8: Synthetic data from large separation HLEM soundings with sest = 50 mand strue = 49 m, 50 m, and 51 m. (a) Inphase component of the data (strue = 49 m -diamonds, strue = 50 m - circles, strue = 51 m - squares), (b) quadrature component ofthe data (strue = 49 m - diamonds, strue = 50 m - crosses, strue = 51 m - squares), (c)real part of the normalized secondary �eld (as in panel (a)), and (d) primary �eld error(as in panel (a)).

Chapter 3. Geometric Survey Parameter Errors 53= 49 m data are represented by diamonds.The inphase and quadrature components of the data are plotted in Figures 3.8(a) and(b). The plots in Figures 3.8(c) and (d) are the real part of the normalized secondary �eldand the primary �eld errors respectively. By comparing the primary �eld error plottedin Figure 3.8(d) to the error plotted in Figure 3.7(d) I see that the shift due to a 2%coil separation error is a factor of �ve smaller than the shift due to a 10% coil separationerror. Although the shift errors in the large separation HLEM example are comparativelysmall they still dominate the inphase component of the data at low frequency.Using these two HLEM examples I have shown that the presence of coil separationerrors will have a direct e�ect upon HLEM data. These errors manifests themselves ina shift of the inphase data. While it is more likely that the inphase component of thedata from surveys performed at small coil separations will be rendered uninterpretable bycoil separation errors, these problems can also occur in large separation HLEM. I haveshown that coil separation errors can also cause discrepancies between the measuredand expected secondary �eld values. This result is similar to the AEM case in that itcreates problems with the interpretation, and, while it is present in both the inphaseand quadrature components of the data, the inphase portion is usually overshadowed byprimary �eld errors. This means that even if someone were to attempt to deal with theshift in the inphase by only interpreting the quadrature part of an HLEM data set theresults may still be incorrect.3.2.3 How do coil separation errors a�ect the inversion results?The previous results show that substantial di�erences exist between the expected dataand those obtained when the coil separation is incorrect. Since many inversion algo-rithms can not account for these di�erences the inversion of data contaminated with coilseparation errors may be di�cult. In order to explore these di�culties I will attempt to

Chapter 3. Geometric Survey Parameter Errors 54strue �m �d �?d11 m 3.46 �10�3 2.20 �104 20.010 m 0.116 20.0 20.09 m 0.377 3.58 �104 20.0Table 3.2: Results from the inversion of small separation HLEM data with an s valueof 10 m when strue is equal to 11 m, 10 m, and 9 m. �m is the recovered model norm,�d is the data mis�t achieved by the inversion and �?d is the target mis�t. These valuescorrespond to results plotted in Figure 3.9invert the data from the two HLEM examples.I will start by considering the data from the three small separation HLEM soundings.In order to simulate the data from an actual sounding Gaussian noise with a constantstandard deviation of 0.5% of the estimated primary �eld at the receiver was added toeach synthetic datum. Noise with a standard deviation which is a percentage of the dataamplitude was not added in this case due to the fact that the soundings contaminatedwith separation errors would have been assigned unrealistically large amounts of noise.The noisy data are the result of soundings with strue values of 11 m, 10 m, and 9 m, andsest = 10 m. Therefore, they were each inverted using a coil separation of s = 10 m.The reference model for each inversion was 0.01 S/m. Each datum was assigned an errorestimate of 0.5% of the primary �eld and the target mis�t was set equal to 20.The �nal data mis�t and model norm values from the three inversion are shown inTable 3.2 and the recovered conductivity models are plotted in Figures 3.9(a), 3.9(c),and 3.9(e) and the observed and predicted data are plotted in Figures 3.9(b), 3.9(d),and 3.9(f).From the data mis�t values in Table 3.2 it can be seen that it was not possible toachieve the target mis�t when strue is equal to either 9 m or 11 m. In both cases therecovered data mis�t was much larger than the target of 20. Figures 3.9(b) and 3.9(f)indicate that the algorithm was not able to �t the inphase data. This suggests that the

Chapter 3. Geometric Survey Parameter Errors 55strue = 11 m. (a)

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IP Obs. IP Pred.Q Obs. Q Pred. Figure 3.9: Results from the inversion of small separation HLEM data with an s valueof 10 m when strue is equal to 11 m, 10 m, and 9 m. Panels (a), (c), and (e) show thetrue (dashed line) and recovered (solid line) conductivity models and panels (b), (d), and(f) show the observed data (inphase - circles, quadrature - crosses) and predicted data(inphase - solid line, quadrature - dashed line) from the three inversions.

Chapter 3. Geometric Survey Parameter Errors 56strue �m �d �?d11 m 0.086 20.0 20.010 m 0.116 20.0 20.09 m 0.061 20.0 20.0Table 3.3: Results from the inversion of small separation HLEM data with an s valuesof 10 m when strue is equal to 11 m, 10 m, and 9 m and noise estimated to account forinphase modelling errors. �m is the recovered model norm, �d is the data mis�t achievedby the inversion and �?d is the target mis�t. These values correspond to results plottedin Figure 3.10di�erences between the expected data and the erroneous data, which are in fact modellingerrors, must be dealt with. One way to handle them in the inversion is to assign to eachdatum a standard deviation that is about the same size as the modelling error. For thesmall HLEM sounding this means assigning errors of 25%, 0.5%, and 38% of the primary�eld to the inphase components of the strue = 11 m, 10 m, and 9 m data respectively.An error of 0.5% of the primary �eld was assigned to all of the quadrature data. Each ofthe inversions was carried out again with a reference model of 0.01 S/m and the targetmis�t set equal to 20.The �nal data mis�t and model norm values from the three inversion are shown inTable 3.3 and the recovered conductivity models are plotted in Figures 3.10(a), 3.10(c),and 3.10(e) and the observed and predicted data are plotted in Figures 3.10(b), 3.10(d),and 3.10(f).I see from the data mis�t values in Table 3.3 that the algorithm has been able toachieve the target mis�t of 20 for each sounding. However,the data plotted in Fig-ures 3.10(b) and 3.10(f) show that it is not possible to adequately reproduce the inphaseobservations for the strue = 11 m and 9 m soundings. Only when �s = 0 is it possible topredict both the inphase and quadrature data(shown in Figure 3.10(d)). The recoveredmodels are quite similar. This is good since it shows that the inversion result is notextremely sensitive to bias errors in the inphase data. Nevertheless, the fact that I have

Chapter 3. Geometric Survey Parameter Errors 57strue = 11 m. (a)0 50 100 150

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IP Obs. IP Pred.Q Obs. Q Pred. Figure 3.10: Results from the inversion of small separation HLEM data with an s value of10 m when strue is equal to 11 m, 10 m, and 9 m and the noise was estimated to accountfor inphase modelling errors. Panels (a), (c), and (e) show the true (dashed line) andrecovered (solid line) conductivity models and panels (b), (d), and (f) show the observeddata (inphase - circles, quadrature - crosses) and predicted data (inphase - solid line,quadrature - dashed line) from the three inversions.

Chapter 3. Geometric Survey Parameter Errors 58strue �m �d �?d51 m 0.203 20.0 20.050 m 0.171 20.0 20.049 m 0.152 39.7 20.0Table 3.4: Results from the inversion of large separation HLEM data with an s valuesof 50 m when strue is equal to 51 m, 50 m, and 49 m and noise estimated to account forinphase modelling errors. �m is the recovered model norm, �d is the data mis�t achievedby the inversion and �?d is the target mis�t. These values correspond to results plottedin Figure 3.11not predicted the observations more closely is unsettling. It leaves me to wonder if thereare artifacts in the recovered models and if there is possibly more information about theearth conductivity that could be extracted if the modelling errors were accounted for.Similar problems can be found in the large separation HLEM example. In this case theshifts caused by the primary �eld errors are much smaller however, in order to achieve thetarget mis�t it is still necessary to account for modelling errors in the inphase component.In order to simulate actual HLEMmeasurements Gaussian noise with a constant standarddeviation of 0.5% of the primary �eld was added to the data from each of the largeseparation HLEM soundings. Then the noisy data from the three soundings were eachinverted using a coil separation of s = 50 m even though the true coil separations foreach of the soundings were actually 51 m, 50 m, and 49 m. The reference model for eachinversion was 0.01 S/m. The error estimates assigned to the inphase data were 8% of theprimary �eld for both strue = 51 m and 49 m and the target mis�t was set to 20.The �nal data mis�t and model norm values from the three inversion are shown inTable 3.4 and the recovered conductivity models are plotted in Figures 3.11(a),(c), and(e)and the observed and predicted data are plotted in Figures 3.11(b), (d), and (f).In all three inversion results the quadrature component of the data appears to havebeen predicted quite well and while the shape of the inphase response has been predictedwell, only the strue data looks to have been reproduced completely. I notice from Table 3.4

Chapter 3. Geometric Survey Parameter Errors 59strue = 51 m. (a)0 50 100 150

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Q Pred. Figure 3.11: Results from the inversion of large separation HLEM data with an s value of50 m when strue is equal to 51 m, 50 m, and 49 m and the noise was estimated to accountfor inphase modelling errors. Panels (a), (c), and (e) show the true (dashed line) andrecovered (solid line) conductivity models and panels (b), (d), and (f) show the observeddata (inphase - circles, quadrature - crosses) and predicted data (inphase - solid line,quadrature - dashed line) from the three inversions.

Chapter 3. Geometric Survey Parameter Errors 60that the target mis�t was achieved for strue = 51 m and 50 m, and the value recovered forstrue = 49 m was close (�d � 40). All of the recovered models shown in Figures 3.11(a),(b), and (c) are good representations of the true model.I have shown that it is possible to achieve the target mis�t by assigning large errorsto the inphase data. However, when this is done the modelling errors in the inphase dataare not being adequately dealt with. This suggests that there may be a better way todeal with coil separation errors.3.3 Parameter Errors in the Field Data SetsNow that I have discussed the details of measurement height and coil separation errors Ishould be able to con�rm the role, if any, that geometric survey parameter errors play inthe �eld data sets. For both the Mt. Milligan and Sullivan �eld data sets I will addressthe possible sources of geometrical survey parameter errors, how the errors a�ect thedata, and how they a�ect the inversion results.3.3.1 Mt. MilliganMt. Milligan is an area with topographical variation and with variable tree cover, bothof which can introduce measurement height errors. The conductivity model recovered byinverting the data (Figure 1.6) shows regions of very low conductivity near the surface.These were identi�ed as \air layers" which can arise when hest is smaller than htrue. Itwould appear, that at least in some areas, measurement height errors caused by treecover could be a problem.

Chapter 3. Geometric Survey Parameter Errors 613.3.2 Sullivan DataThe possibility of coil separation errors due to topography at Sullivan was discardeddue to the fact that the tailings pond is relatively at. However, it was discovered thatthe design of the Max-Min system used to collect the data presented the possibility ofa maximum coil separation error of +0.5 m. Since the expected coil separation in theSullivan survey was equal to 5 m, this represented a possible +10% coil separation error.In Section 3.2 the small separation HLEM example was used to model synthetic datawith a +10% relative coil separation error. This error resulted in a shift in the inphasecomponent of the data by approximately -25% of the primary �eld while there was noevidence of distortion in the quadrature component. When I re-examine the Sullivandata shown in Figure 1.1 I see that the 7040Hz measurements of the inphase componenthave values between -20% and -30% of the primary �eld. The model that was recoveredby including large error estimates on the inphase component is shown in Figure 1.2. Fromthis plot it is seen that the target mis�t of 8 has been achieved in most cases, and thesubsurface conductivity agrees with the available a priori information. However, the factthat it is not possible to predict the inphase component of the data (Figure 1.3) leavessomething to be desired. It appears that coil separation errors are causing problems withthe inversion of the Sullivan data.

Chapter 4Recovering 1-D Conductivity and a Geometric Survey ParameterThe previous chapter contained a discussion of the problems that erroneous survey para-meters can cause when inverting both AEM and HLEM data. In this chapter a solutionto these problems is proposed in the form of an inversion methodology which can recoverboth a 1-D conductivity distribution and a geometric survey parameter p. The idea ofinverting geophysical data to recover a property distribution as well as information aboutthe experiment has been done previously by Pavlis & Booker (1980) and deGroot-Hedlin(1991). The approach I choose is to include the survey parameter in the model vectorand solve the inverse problem using a methodology similar to that used in the absenceof the extra parameter. There is no reason that p could not be a vector of parametervalues. The formulation and numerical solution that I will present is designed in sucha way that this could be accommodated. However, in my case a single parameter is allthat is needed to correct the �eld data sets and hence I will treat p as a single parameterfrom now on.I will begin with a discussion of the changes that are necessary when dealing withan extra parameter. These include changes to the model, the data, and the sensitivitycalculations. This will be followed by the development of a modi�ed objective function.and a discussion of some of the problems and possible solutions encountered when at-tempting to solve the inverse problem. I will show synthetic results from the AEM andHLEM examples and apply the new inversion methodology to the two �eld data sets.62

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 634.1 Changes due to the extra parameterIn this section I will discuss how the model, the forward modelling and the sensitivitycalculations are a�ected by the inclusion of an extra parameter. When referring specif-ically to details that relate to either AEM or HLEM data I will use a superscript A or asuperscript H respectively (e.g. dA will represent the forward modeled AEM data). How-ever, outside of the discussion in this section, the methodology will not specify betweenthe inversion of AEM and HLEM data. Each inversion will be carried out in terms ofgeneric models, data, and sensitivities.4.1.1 The ModelTo deal with survey parameter errors I have included an erroneous parameter p in themodel vector I am attempting to recover. The generic model m now has the formm = [log(�1); : : : ; log(�M); p]T ; (4.1)where M is equal to the number of layers in the conductivity model. The vector mhas a length of M + 1. The parameter is either measurement height or coil separationdepending upon whether the data are AEM or HLEM. As a result the two speci�c modelswhich must be de�ned in order to deal with parameter errors aremA = [log(�1); : : : ; log(�M ); h]T ; (4.2)which is used when inverting AEM data, andmH = [log(�1); : : : ; log(�M); s]T : (4.3)which is used when inverting HLEM data.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 644.1.2 The DataAs in Section 2.2.2 the data values are stored in the vector d which has the formd = [Re(d1); Im(d1); : : : ; Re(dN ); Im(dN)]T ; (4.4)where N is equal to the number of data. Both AEM and HLEM data, as usually de�ned,are normalized by an estimate of the primary �eld value at the receiver and multiplied bya constant. For my purpose it is simpler to work with the denormalized data. Therefore,I de�ne the generic form of the ith measurement in a given data set to beRe(di) = Re(Fi[m]); (4.5)and Im(di) = Im(Fi[m]); (4.6)where the details of the forward modelling Fi depend upon the survey being considered.In the AEM case I denormalize the data from Equations 3.5 and 3.6 and substituteh for htrue to get the following expression for the data from the ith AEM measurementRe(dAi ) = Re(HSz (!i; si;�; h)); (4.7)and Im(dAi ) = Im(HSz (!i; si;�; h)): (4.8)Since the estimated measurement height hest has no direct e�ect upon AEM data, theinclusion of htrue does not change the forward modelling calculations.Similarly, by denormalizing Equations 3.11 and 3.12 substituting s for strue the datafrom the ith HLEM measurement will have the formRe(dHi ) = Re(HSz (!i; hi;�; s)) +HPz (s)�HPz (sest); (4.9)

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 65and Im(dHi ) = Im(HSz (!i; hi;�; s)): (4.10)From Equation 4.9 it is clear that the presence of a shift in the inphase data is due to acoil separation error.4.1.3 The SensitivitiesIn order to solve the inverse problem it is necessary to be able to calculate the sensitivitiesof the data to changes in the model parameters. These quantities are expressed inthe form of the Jacobian, or sensitivity, matrix J . The ijth entry in J represents thesensitivity of the ith datum to a change in the jth model parameter as de�ned byJij = @di@mj : (4.11)As a result the generic sensitivity matrix will have the formJ = [J�; Jp]; (4.12)where J� is the N �M conductivity sensitivity matrix and Jp is the N � 1 parametersensitivity matrix.From the expressions for both dA and dH it is clear that the subsurface conductivitystructure only a�ects the secondary �eld terms in AEM and HLEM data. Therefore,regardless of type of data being inverted it is necessary to calculate the sensitivities of thesecondary �elds to changes in conductivity. The sensitivities for the type of frequencydomain EM surveys I am dealing with have been determined in Zhang & Oldenburg(1994) using the adjoint Green's function method. The sensitivity of the ith complexdatum to a change in the conductivity of the jth layer can be expressed as@di@�j = �2�i!�0mT Z 10 �Z zj+1zj ~E2�dz�J0(�s)�3d�; (4.13)

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 66where zj is the depth to the top of the jth layer, ~E� is the tangential component of theelectric �eld in the Hankel transform domain, and J0 is a zeroth order Bessel function.However, the model parameter I am concerned with is log(�j) therefore, the sensitivityvalue I want is de�ned byJij = 1�j @di@�j : (4.14)These values will �ll J� which makes up the �rst M columns of the sensitivity matrix.The �nal column of the matrix will contain the sensitivities of the data to the surveyparameter p. The details of these sensitivities depend upon the survey being considered.In the AEM case I must determine the sensitivity of the data to a change in measure-ment height h. Since the AEM data in Equations 4.7 and 4.8 are simply the secondarymagnetic �eld we can use Equation 2.37 to express dA in complex form asdAi = mT4� Z 10 e�2u0hu0 RTE�3J0(�si)d�: (4.15)By di�erentiating Equation 4.15 with respect to h I can calculate the sensitivity of theith datum to a change in measurement height. The complex sensitivity is equal to@dAi@h = �mT2� Z 10 e�2u0h RTE�3J0(�si)d�: (4.16)These values will �ll Jp when inverting AEM data.When dealing with HLEM data I want to calculate the sensitivity to changes in thecoil separation s. I can expand the HLEM data in Equations 4.9 and 4.10 in complexform asdHi = mT4� Z 10 e�2u0hiu0 RTE�3J0(�s)d� + ��mT4�s3 � � � �mT4�s3est� : (4.17)The sensitivity of the ith datum due to a change in the true coil separation can becalculated by di�erentiating Equation 4.17 with respect to s as follows@dHi@s = mT4� Z 10 e�2u0hiu0 RTE�3 @@s (J0(�s)) d� � mT4� @@s � 1s3� : (4.18)

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 67Expanding the partial derivative in the integrand I get@J0(�s)@s = @@�s (J0(�s)) @@s (�s) ;= J 00(�s)�;= �2 [J�1(�s) � J1(�s)] : (4.19)Using the identityJ�n(�s) = (�1)n Jn(�s); (4.20)I am left with@J0(�s)@s = ��J1(�s); (4.21)which can be substituted back into the integrand to give the complex sensitivity value@dHi@s = �mT4� Z 10 e�2u0hiu0 RTE�4J1(�s)d� + 3mT4�s4 : (4.22)These values will �ll Jp when inverting HLEM data.4.1.4 The Objective FunctionSince I have modi�ed the model it is necessary to modify the objective function accord-ingly. I want the solution of the modi�ed problem to remain as close to the parametererror free version as possible. Therefore, I choose a global objective function � similar tothe one de�ned in Section 2.2.2. The objective function has the form� = �d + ��m; (4.23)where �d is the data mis�t, �m is the model objective function, and � is the regularizationparameter. I will choose �d to be the same as the mis�t term de�ned in Equation 2.46.However, since the model has changed I have to rede�ne �m as�m = �� + �p�p; (4.24)

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 68where �� is the conductivity model norm, �p is the parameter norm, and �p is weightingparameter which de�nes the relative importance of �p within the model norm. The con-ductivity norm determines the character of the conductivity structure I want to recover,and is de�ned as�� = jjW�(log(�)� log(�ref))jj2; (4.25)where W� is the conductivity weighting matrix and �ref is the reference conductivitymodel. W� is chosen to be identical to the model weighting matrix de�ned in Equa-tion 2.43.The parameter norm determines the closeness of the recovered parameter to somereference value pref , and is de�ned as�p = (p � pref )2: (4.26)Using the model m from Equation 4.1 it is possible to express �m as�m = jjWm(m�mref )jj2; (4.27)where Wm is equal toWm = 24W� 00 p�p35 : (4.28)Having de�ned the new objective function it is now possible to solve the inverseproblem.4.2 Inversion AlgorithmsNow that I have made all of the changes needed to deal with the extra parameter it ispossible to design my inversion algorithm. This thesis, so far, has avoided discussing the

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 69details of my inversion algorithm. Therefore, I begin this section with a description of thealgorithm used to invert EM data when p is �xed. The modi�cations to the algorithmthat are necessary in order to accommodate the extra parameter will depend upon myde�nition of �p. Hence, I will examine the best possible choice of �p and how this choicea�ects the performance of the current inversion algorithm. The insight gained throughthese investigations is used to design the algorithm which can invert EM data to recoverboth conductivity and a geometric survey parameter.4.2.1 Details of algorithm to recover 1-D conductivity with �xed pIn order to provide as much detail as possible I will present a description of the twoinversion algorithms that can be used to recover a 1-D conductivity structure with a�xed value of p. The �rst algorithm is a detailed description of the method providedin Section 2.2.2 to solve the non-linear inverse problem for a particular � value. It wasstated that this process could be repeated for a number of � values in order to �ndthe one which satis�es �d = �?d. In practice however, this is computationally expensive.Therefore, I will describe a second algorithm in which a � is varied at each iteration, suchthat the eventual solution will satisfy �d = �?d. These two algorithms will be referred torespectively as the �xed � and discrepancy principle algorithms.Details of �xed � AlgorithmThis section will present the details of the algorithm I use to invert EM data with a �xedvalue of �. As mentioned in Section 2.2.2 the inverse problem can be solved by �ndingthe model m that minimizes �. This is done using a damped Gauss-Newton (DGN)approach. The theory of this approach can be found in Dennis and Schnabel (1996) andsome background on its application to non-linear inverse problems can be found in Haber(1997). In my algorithm the DGN approach proceeds as follows: at each iteration the

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 70Gauss-Newton step �m is checked to see whether or not it has actually decreased �. Thisdone by calculating the objective function corresponding to the updated model�(mk+1) = �k+1d + ��k+1m ; (4.29)and the objective function corresponding to the model from the previous iteration�(mk) = �kd + ��km: (4.30)If �(mk+1) < �(mk) then �m is accepted since it decreases the global objective function.However, when �(mk+1) > �(mk) the step �m should not be taken. Therefore, the stepis corrected by damping it as follows�m = !�m; (4.31)where ! is a constant between 0 and 1. In my algorithm ! = 0:5. After �m is damped,the model mk+1 is updated and �(mk+1) is recalculated and compared to �(mk). Thisprocess is repeated until a �m, that decreases the objective function, is found. When anacceptable mk+1 is found the algorithm must decide whether to continue on to the nextiteration or to accept the model mk+1 as the solution to the minimization problem. Thestopping criteria used in the algorithm are (1) the relative gradient of �, i.e. is mk+1 at aminimum, and (2) the relative change betweenmk+1 and mk, i.e. is the model stationary.The relative gradient of � is considered small whenmax jg(i)(mk+1) m(i)k+1j�(mk+1) ; i = 1; : : : ;M ! < �1; (4.32)where g(mk+1) is the gradient of � and �1 is a tolerance level. The model is consideredto be stationary whenjjmk+1 �mkjjmax(jjmk+1jj; jjmkjj) < �2;

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 71where �2 is a tolerance level. The algorithm is said to have converged when either ofthese conditions are satis�ed. In my algorithm both of the tolerance levels are set equalto 10�3. With all of these pieces explained in detail it is now possible to map out theentire algorithm.Each of the steps taken during the inversion is summarized in the ow chart shownin Figure 4.1 and each entry in the following list corresponds to a step in the ow chart.While some of the steps may seem trivial for the �xed � algorithm, they are includedsuch that the numbering scheme will be consistent for subsequent algorithms.1. Input pertinent information for the inverse problem. This includes: (1) parametersthat concern the data such as forward modelling details (measurement frequencies,measurement heights, coil separations, etc.) and the observed data, (2) parameterswhich concern the model such as discretization details (number of layers, depth tolayer interfaces),mstart, andmref , and (3) parameters needed to de�ne the objectivefunctions such as �s and �z, and error estimates for the data. The value of � mustalso be provided.2. The initial values of d, �d, �m, and J are calculated using mstart.3. Iteration counter k is set equal to zero and the variables calculated in Step 2. arelabeled to belong to the kth iteration as mk, dk, �kd, �km, and J(mk).4. Begin k + 1th iteration.5. The value of � supplied in Step 1. is selected as the regularization parameter.6. The Gauss-Newton step �m is calculated using Equation 2.60. The model for thisiteration is de�ned as mk+1 = mk + �m.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 72Set k = 0Initialize Variables

Begin iter. k+1

Use fixed value of β

Convergence ?

Exit

Set k=k+1Update variables

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Fixed βInversion Algorithmfor fixed value of p

User Inputincluding β

1

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Figure 4.1: Flowchart for the �xed � inversion algorithm with a �xed value of p.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 737. The values that were calculated in Step 2. are calculated using mk+1, to be dk+1,�k+1d , �k+1m , and J(mk+1).8. The values of �(mk+1) and �(mk) are calculated using Equations 4.29 and 4.30. If�m decreases �, then go to Step 9. However, if �m results in an increase in � go toStep 8A and damp �m, then go back to Step 7.9. Check for convergence. If solution has converged go to Step 10. Otherwise go toStep 9A and set k = k + 1 and switch the index of all values then go to Step 4.10. Exit algorithm. Return the recovered model, predicted data, and the �nal datamis�t and model norm values.This algorithm was used to invert the AEM sounding data with a �xed value of �.The value of � was selected, through trial and error, such that the goal �d = �?d wasachieved. The measurement height was set equal to 30 m and the rest of the details ofthe inversion were the same as those used in previous AEM inversions. Figure 4.2 showsthe recovered model and the predicted data from the inversion and Figure 4.3 shows thevalues of �d, �m, � and �, as a function of iteration. It is shown that at each iterationthe objective function was decreased until it reached a minimum value. Since � had beenproperly selected the solution achieves the target mis�t of 20. However, when the proper� is not known it can be time consuming to solve a number of problems in order to �ndit. Therefore, next step is to construct an algorithm that will �nd the desired � value.Details of discrepancy principle AlgorithmThe desired algorithm will �nd the � value that satis�es �d = �?d. The best plan isto decrease � gradually such that the addition of unwanted structure to the model isavoided. Therefore, the algorithm will start with a large � value and decrease it slightly

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 74(a)0 50 100 150

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Q Pred. Figure 4.2: Recovered models and predicted data from AEM example �xed � inversionwith a �xed value of h. The measurement height was set equal to 30 m. (a) True conduct-ivity (dashed line) and recovered conductivity (solid line) and (b) observed data (inphase- circles, quadrature - crosses) and predicted data (inphase - solid line, quadrature -dashed line).at each iteration until the desired � value is reached. The next step is to decide howto decrease �. The method that is adopted is referred to as the discrepancy principle.The following is the inversion algorithm I use to invert EM data using the discrepancyprinciple for a �xed value of p.The algorithm proceeds such that at iteration k + 1 the value �k+1 is chosen suchthat it will result in decreasing �d by a �xed amount. This is continued until the targetfor the current iteration �?(k+1)d is equal to the target for the inversion �?d. Therefore, ateach iteration the target �?(k+1)d is de�ned as�?(k+1)d = max( �kd; �?d); (4.33)where is a constant which should have a value between 0.5 and 1.0. In my algorithm is set equal to 0.5. It is important to note that when searching for �?(k+1)d , the curve of�d versus � usually has a parabolic shape. Therefore, there may well exist two values of� that give �d = �?(k+1)d . In this case the smaller of the two � values should be chosen

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 75(a)

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Figure 4.3: Convergence curves from AEM example �xed � inversion with a �xed valueof h. The measurement height was set equal to 30 m. (a) �d (circles) and target �d(crosses) vs. iteration, (b) �m vs. iteration, (c) �xed � vs. iteration, and (d) �(mk+1)(circles) and �(mk) (crosses) vs. iteration.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 76as �k+1 such that the least amount of structure is added to the model. In the case thatit is not possible to �nd a � value which satis�es �d = �?(k+1)d the value of � which givesthe minimum possible �d value is chosen as �k+1.After having found �k+1 the Gauss-Newton step �m is calculated using Equation 2.60and the model is updated to be mk+1 = mk+ �m. At this point it is necessary to check ifthe step �m is bene�cial. Since � is changing at each iteration, the � values from iterationk, evaluated with �k, can not be compared with those from iteration k + 1, evaluatedwith �k+1. Therefore, in order to evaluate the changes taking place in � at successiveiterations both �(mk) and �(mk+1) are calculated with � = �k+1 such that the valuesare de�ned as�(mk) = �d(mk) + �k+1�m(mk); (4.34)and �(mk+1) = �d(mk+1) + �k+1�m(mk+1): (4.35)In practice, it has been shown that the choice of �k+1, corresponding to a smalldecrease in �d, can result in a solution that both minimizes the global objective functionand satis�es �d = �?d (Parker, 1994). However, this is not always the case. When �?dis set too small then the target mis�t is not attainable and the algorithm can reach astage at which a decrease in �d can only be achieved by choosing small � values that addlarge amounts of structure to the model. This can cause the recovered model to be quitepoor. The problem comes from the fact that there is no guarantee that the step �m willprovide a new model mk+1 that is better than the model mk from the previous iteration.Requiring �(mk+1) < �(mk) is a consistency check on the algorithm. It at leastensures that the previous model mk is not a better model than the updated modelmk+1 = mk + �m. If �(mk+1) > �(mk), then a number of actions are possible. First the

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 77size of the perturbation step may be damped to see if �(mk + !�m) < �(mk) for somefactor !. This did not seem to be necessary here since the step sizes have already beenlimited by carrying out successive iterations with the goal of slowly reducing the mis�t(�?(k+1)d = �(k)d ). Empirically, the case that �(mk+1) > �(mk) occured when the mis�twas beginning to plateau out and could no longer be signi�cantly reduced, even with alarge perturbation �m. The achieved mis�t �mind was inferred to be a representation ofthe minimum mis�t. Accordingly �?d should lie above this value. Therefore, when thealgorithm detects that �(mk+1) > �(mk), the value of �?d is increased and the inversionis restarted. In my algorithm the restart value of �?d is set equal to 1:1� �mind .The convergence criteria for the discrepancy principle algorithm are the same as forthe �xed � case. With all of these pieces explained in detail it is now possible to mapout the entire algorithm. Each of the steps taken during the inversion is summarized inthe ow chart shown in Figure 4.4 and each entry in the following list corresponds to astep in the ow chart.1. Input pertinent information for the inverse problem. This includes: (1) parametersthat concern the data such as forward modelling details (measurement frequencies,measurement heights, coil separations, etc.) and the observed data, (2) parameterswhich concern the model such as discretization details (number of layers, depth tolayer interfaces),mstart, andmref , and (3) parameters needed to de�ne the objectivefunctions such as �s and �z, and error estimates for the data. The target mis�t �?dmust be provided.2. The initial values of d, �d, �m, and J are calculated using mstart.3. Iteration counter k is set equal to zero and the variables calculated in Step 2. arelabeled to belong to the kth iteration as mk, dk, �kd, �km, and J(mk).

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 78Begin iter. k+1Set target φd for this iter. φ = max(γ φ ,φ )

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Figure 4.4: Flowchart for the discrepancy principle inversion algorithm with a �xed valueof p.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 794. Begin k + 1th iteration. Calculate �?(k+1)d using Equation 4.33.5. The � value which satis�es �d = �?(k+1)d is selected as �k+1.6. The Gauss-Newton step �m, corresponding to �k+1, is calculated using Equa-tion 2.60. The model for this iteration is de�ned as mk+1 = mk + �m.7. The values that were calculated in Step 2. are calculated using mk+1, to be dk+1,�k+1d , �k+1m , and J(mk+1).8. The values of �(mk+1) and �(mk) are calculated using Equations 4.34 and 4.35. Ifthe step �m decreases � go to Step 9. If, however, �m results in an increase in � goto Step 8A and increase the global target mis�t �?d. Then the inversion is restartedby going to step 3.9. Check for convergence. If solution has converged go to Step 10.Otherwise go to Step 9A and set k = k + 1 and switch the index of all values thengo to Step 4.10. Exit algorithm. Return the recovered model, predicted data, and �nal data mis�tand model norm values.This algorithm was used to invert the AEM sounding data using the discrepancyprinciple. The target mis�t was set equal to 20. The measurement height was set equalto 30 m and the rest of the details of the inversion were the same as those used in previousAEM inversions. Figure 4.5 shows the recovered model and the predicted data from theinversion and Figure 4.6 shows the values of �d, �m, � and �, as a function of iteration.The inversion results are identical to those presented in Section 3.1 when the AEM datawere inverted with the correct h value. Figure 4.6(a) shows that after each iteration thedata mis�t was decreased by a constant amount until reaching the target value of 20. It

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Figure 4.5: Recovered models and predicted data from AEM example discrepancy prin-ciple inversion with a �xed value of h. The measurement height was set equal to 30m. (a) True conductivity (dashed line) and recovered conductivity (solid line) and (b)observed data (inphase - circles, quadrature - crosses) and predicted data (inphase - solidline, quadrature - dashed line).can also be seen in Figure 4.6(b) that when the target level of the mis�t was attainedat iteration 8, the model norm was then decreased slightly until the solution converged.This results in a model which both �ts the data and has minimum structure. The plotof � in Figure 4.6(d) shows that at each iteration the objective function was decreased.In order to de�ne the details of the modi�ed inversion algorithms it is necessary tode�ne the parameter norm.4.2.2 De�ning the Parameter NormThe new objective function, which includes a geometric survey parameter, requires thespeci�cation of �p and pref . A discussion of the ways in which each of these values a�ectthe problem is presented in this section. The goal of this discussion is to determine aneducated way in which to assign values to both �p and pref . The discussions in thissection will deal strictly with the discrepancy principle inversion algorithm.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 81(a)

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Figure 4.6: Convergence curves from AEM example discrepancy principle inversion witha �xed value of h. The measurement height was set equal to 30 m. (a) �d (circles) andtarget �d (crosses) vs. iteration, (b) �m vs. iteration, (c) � vs. iteration, and (d) �(mk+1)(circles) and �(mk) (crosses) vs. iteration.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 82Choosing �pIn order to choose a good �p value it is necessary to understand how this choice a�ectsthe problem. The value of �p determines the relative importance of the parameter normterm. Therefore, it controls how strictly the requirement that the recovered p is close topref will be enforced. To get a feeling for the e�ects I set �p equal to its two extremevalues: 0 and 1. As �p approaches in�nity, p is �xed at the value of pref . However,when �p is equal to zero, p is not required to be close to the pref value at all. Looking athow the problem behaves at these two extremes provides insight into choosing the valueof �p.I begin by looking at the case where �p approaches in�nity. Since it is not mathem-atically possible to set �p = 1 I simply �x p = pref for each inversion. For each of thesynthetic examples I carried out a series of inversions, each with a di�erent pref value.For each inversion I attempted to �t the data to a speci�ed target mis�t. The abilityof the algorithm to achieve the desired mis�t level, as well as the resultant �m value,provided insight into the importance of p in both AEM and HLEM inversions.The �rst example I looked at was the noisy AEM data with htrue = 30 m. The datawere inverted 11 times with h �xed equal to di�erent values of href ranging from 45 mto 15 m. For each of the inversions the data were assigned an error of 5% of the dataamplitude + 10 PPM, the target mis�t was equal to 20, and a reference model of 0.01S/m was used.The results from all of the inversions are plotted together in Figure 4.7. The valuesplotted in the top panel are the �nal �m values from each inversion. These values areequal to �� since �p is equal to zero as a result of p being set equal to pref . The middlepanel shows the �nal �d values from each inversion and the bottom panel shows therecovered conductivity models plotted next to one another.

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−0.5Figure 4.7: Results from eleven inversions of the AEM sounding data with �xed valuesof h ranging from 45 m to 15 m. The top panel is �nal �m value, middle panel is �nal�d value, and the bottom panel is the recovered conductivity models plotted next to oneanother.These results are identical to those described in Section 3.1. When the h value used bythe inversion is smaller than htrue, the recovered model has a low conductivity structureat the surface and it is possible to achieve the target mis�t level. When the h valueused by the inversion is larger than htrue, the recovered model has a high conductivitystructure at the surface, and, as h increases, it becomes impossible to �t the data.The success of any inversion is gauged, in part, on the ability of the recovered modelto predict the observed data. Figure 4.7 shows that there are a number of h values thatsatisfy this condition. By considering only the �m values that correspond to this subset,h = 33 m to 15 m, it can be seen that there exists a minimum �m value that correspondsto a particular value of h. For this example, this value of h is approximately equal to 27m which is close to htrue.I also inverted the noisy data from the HLEM examples. From the small separation

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−0.5Figure 4.8: Results from eleven inversions of the small separation HLEM sounding data,strue = 11 m, with �xed values of s ranging from 12.5 m to 7.5 m. The top panel is�nal �m value, middle panel is �nal �d value, and the bottom panel is the recoveredconductivity models plotted next to one another.case I chose to invert the data that had an strue value of 11 m. Eleven inversions werecarried out with sref values ranging from 12.5 m to 7.5 m. For each of the inversionsthe data were assigned an error of 0.5% of the primary �eld, the target mis�t was equalto 20, and a reference model of 0.01 S/m was used. The results are plotted next to oneanother in Figure 4.8. The middle panel shows that the data could only be �t to thedesired mis�t level when s was equal to 11 m. This value is near a distinct minimumin the �d versus s curve. Both the model norm values and the recovered conductivitystructures from the other s values provide little information since it was not possible to�t the data.The inversion results from the large separation HLEM example show the same e�ect.The noisy data with strue = 51 m were inverted eleven times with sref values rangingfrom 52.5 m to 47.5 m. Again the data were assigned an error of 0.5% of the primary

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−1Figure 4.9: Results from eleven inversions of the large separation HLEM sounding data,strue = 51 m, with �xed values of s ranging from 52.5 m to 47.5 m. The top panel is�nal �m value, middle panel is �nal �d value, and the bottom panel is the recoveredconductivity models plotted next to one another.�eld, the target mis�t was equal to 20, and a reference model of 0.01 S/m was used.Figure 4.9 shows that the results are similar to the small separation case. The targetdata mis�t was only achieved when s was set equal to 51 m and the inversion results fromevery other s are inconclusive. These HELM examples show that there exists a narrowrange of p values for which a conductivity model can be found that will satisfy the data.In both of these examples the values of s were close to strue.Each of the three examples suggest that if p were allowed to vary freely it wouldchoose a satisfactory value. In the AEM example, when there is a wide range of h valuesthat can achieve the target mis�t, the selection of that p that provides a minimum modelnorm value is a good estimate of the true value. In both of the HLEM examples a narrowrange of s values that could achieve the target mis�t were found. Therefore, it appearsthat setting �p = 0 is a good choice.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 86Instabilities when �p = 0While it seems that setting �p = 0 is a good idea it can cause problems during the �rstfew iterations. As described in Section 4.2.1 the goal of the inversion algorithm is to addstructure to the model slowly. Therefore, the starting model is usually set equal to thereference model. This means that in the �rst iteration, as � !1, the recovered modelreverts to the reference model. Hence, both �d and �m return to their starting values. Bydecreasing � slowly the problem is regularized and any poor steps that may have beentaken are avoided. However, if �p is set equal to zero, �p will not be regularized by �and there is no guarantee that the step �p will be bene�cial. By examining the systemof equations that are solved to give the Gauss-Newton step it is possible to see whathappens when �p = 0. In order to do this I will rede�ne the terms in Equation 2.60 inaccordance with changes made in Section 4.1.The model perturbation �m becomes,�m = [�log(�); �p]: (4.36)For ease of presentation I will assume that Wd is equal to the identity matrix. As a resultJTW Td WdJ becomes JTJ which can de�ned explicitly, using Equation 4.12, asJTJ = 24JT�JTp 35hJ� Jpi = 24JT� J� JT� JpJTp J� JTp Jp35 ; (4.37)where the matrix has a block format, and the upper left hand block of the matrix isM �M , the upper right hand block is 1 �M , lower left hand block is M � 1, and thelower right hand block is 1� 1. From Equation 4.1.4 the matrix W TmWm, is de�ned asW TmWm = 24W T� W� 00 �p35 ; (4.38)

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 87Substituting all of these details into Equation 2.60 we can write out the Gauss-Newtonstep in matrix form as240@JT� J� JT� JpJTp J� JTp Jp1A+ �0@W T� W� 00 �p1A3524�log(�)�p 35= 24JT�JTp 35 �d� � 24W T� W� 00 �p3524log(�)� log(�ref )p � pref 35 ; (4.39)where �d is equal to dobs � F [m].For �p = 0 and � !1 Equation 4.39 becomes0@W T� W� 00 JTp Jp1A24�log(�)�p 35 = 24�W T� W� (log(�)� log(�ref ))JTp �d 35 ; (4.40)such that the conductivity and parameter perturbations are decoupled and the solutionhas the form�log(�) = � (log(�)� log(�ref )) (4.41)�p = (JTp Jp)�1JTp �d: (4.42)From Equation 4.41 it is clear that �� is equal to zero when log(�) = log(�ref ). Theparameter perturbation however, is equal to the Gauss-Newton minimization of �d withrespect to p which is independent of �. This clearly shows that �p is not being regularizedat all. Using the AEM sounding data the possible problems associated with settling�p = 0 can be shown.For this example I chose an href value of 42 m and de�ned the rest of the detailsgoverning the inversion such that they were identical to those used when the data wasinverted with a �xed h value in Section 3.1. The �rst iteration of the inversion was thenperformed for both the �p = 1 and the �p = 0 cases. Figure 4.10 shows plots of thevalues of �d, ��, �, and h evaluated at a series of � values during the search for �?(1)d .

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Figure 4.10: Values calculated during the line search for � during the �rst iteration ofthe inversion of the AEM data. The starting height was set equal to 42 m. The two linesin the plot correspond to values of �p =1 (circles) and 0 (plusses). (a) �d vs. �, (b) ��vs. �, (c) h vs. �, and (d) � vs. �.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 89The solid lines with circles refer to the case where �p = 1 and the dashed lines withplusses represent �p = 0.When �p = 1 the problem behaves as expected. At large � values the �d curveasymptotes to a constant value. By taking note of the �� versus � curve, which goesto zero as � ! 1, and the fact that h is �xed, it is evident that as � gets large m isreverting to mref and therefore, �d is approaching its starting value. However, setting�p = 0 causes a change in the behavior. The �� versus � curve still approaches zero as� increases however, h plateaus to a value approximately equal to 20 m. This causes the�d values, and hence the � values, at large �, to be larger than their initial values (whichare known from the �p = 0 curves). Therefore, the linearized Gauss Newton step �p istoo large and it is necessary to introduce some additional regularization to ensure thatthe step is acceptable. A simple solution is to set �p equal to some non zero value.When �p is non zero and � !1 the Gauss-Newton step will have the form0@W T� W� 00 11A24�log(�)�p 35 = 24�W T� W� (log(�)� log(�ref ))� (p� pref ) 35 ; (4.43)and the solution will be�log(�) = � (log(�)� log(�ref )) (4.44)�p = � (p� pref ) ; (4.45)such that when log(�) = log(�ref ) and p = pref , �p will be equal to zero. It is importantto note however, that while �log(�) goes to zero as a function of �, �p goes to zero as afunction of the product ��p. Therefore, any non-zero value of �p will result in �p ! 0as � gets su�ciently large, but the point at which this occurs depends upon �p. Thisis shown by the plots of �d, ��, �, and h versus � from the �rst iteration of the AEMsounding inversion shown in Figure 4.11. The solid lines with circles refer to the case

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Figure 4.11: Values calculated during the line search for � during the �rst iteration of theinversion of the AEM data. The starting height was set equal to 42 m. The three linesin the plot correspond to values of �p = 1 (circles), 10�6 (plusses), and 10�3 (crosses).(a) �d vs. �, (b) �� vs. �, (c) h vs. �, and (d) � vs. �.where �p = 1, the dashed lines with plusses represent �p = 10�6, and the dotted lineswith crosses represent �p = 10�3. From the curves in Figure 4.11 it is clear that as �pincreases the value of � at which �p goes to zero decreases.For the algorithm to be stable it is necessary for �p to be non zero at early iterations.However, it has been shown that �p should be equal to zero in the late stages of theinversion. Therefore, we propose a cooling schedule for �p. This entails decreasing �pgradually at each iteration such that in the �nal iterations it is equal to zero. This is, in

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 91fact, a simplistic application of the Levenberg-Marquardt method (Dennis & Schnabel,1996). Implementing a cooling schedule for �p provides the algorithm with the stabilityit needs at early iterations as well as permitting the parameter to roam freely at the endof the inversion.The �nal step needed to �nalize the algorithm is to determine a method to choosepref .Choosing prefDuring the �rst few iterations the inversion algorithm will demand that p be close topref therefore, it is a good idea to have a realistic estimate for its value. The problemof selecting a value for pref has some inherent di�culties associated with it. While,both the AEM and HELM systems provide an estimate of the parameter of interest it isunclear as to whether or not this value should be used as pref . The fact that an inversionalgorithm is being used to recover the true parameter value indicates that it has alreadybeen assumed that pest may be incorrect. Therefore, it may be useful to �nd a betterestimate of pref .A simple choice of pref is to �nd the value that minimizes �d for the reference con-ductivity model �ref provided by the user. The �rst step in �nding this value of p is to�x the conductivity model equal to �ref and forward model data for a range of p values.Using the data generated from each p value, the data mis�t is plotted as a function ofp and the value that corresponds to the minimum in the �d versus p curve is chosen aspref . This estimate of pref will depend upon the de�nition of �ref . In the case that thereference conductivity model is a good �rst order approximation to the true conductivitymodel, the choice of pref should be accurate. However, when the guess of �ref is not asgood the results may vary. I investigated the in uence of �ref on the choice of pref foreach of the three synthetic examples.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 92AEM (a)0 20 40 60

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σTrue Figure 4.12: Results from the search for a best �t pref value for the AEM sounding datawhere htrue = 30 m. (a) �d versus h curves for the three halfspaces: 0.1 S/m (solid line),0.01 S/m (dashed line), and 0.001 S/m (dotted line). and (b) the halfspace models (asin (a)) and the true conductivity structure (thick gray line) .The data mis�t values, calculated for the AEM sounding data at a series of h valuesranging from 0 m to 60 m, are plotted in Figure 4.12(a). The true conductivity modelas well as the corresponding halfspace �ref models (0.1 S/m, 0.01 S/m, and 0.001 S/m)are plotted in Figure 4.12(b). It can be seen that when �ref = 0.01 S/m (dashed line)the �d versus h curve has a broad minimum. However, the minimum does appear tobe centered close to the true measurement height of 30 m. On the other hand, when�ref is varied by one order of magnitude in either direction the results are not as good.In fact, the �ref = 0.1 S/m (solid line) and the �ref = 0.001 S/m (dotted line) haveminima at approximately 50 m and 11 m respectively. This leads to the assumption thatthe in uence of the measurement height on the data can easily be overpowered by theconductivity structure. Therefore, attempting to estimate a good reference value for hin this manner is unreliable. Before addressing a solution to this problem I will presentthe results from the HLEM examples.Since the in uence of s on HLEM data is very strong I would expect the �d versuss curves to be much di�erent than those shown in Figure 4.12. For the small HLEM

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 93Small HLEM (a)8 9 10 11 12

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Figure 4.13: Results from the search for a best �t pref value for the HLEM examples.Small HLEM sounding data where strue = 11.0 m : (a) �d versus h curves for the threehalfspaces: 0.1 S/m (solid line), 0.01 S/m (dashed line), and 0.001 S/m (dotted line),and (b) the halfspace models (as in (a)) and the true conductivity structure (thick grayline). Large HLEM sounding data where strue = 51.0 m : (c) �d versus h curves forthe three halfspaces: (as in (a)), and (b) the halfspace models (as in (a)) and the trueconductivity structure (thick gray line).sounding I attempted to estimate s using the noisy data generated with strue = 11 m.Values of the data mis�t were calculated at s values ranging from 7.5 m to 12.5 forconductivity values of 0.1 S/m, 0.01 S/m, and 0.001 S/m. The resulting �d versus scurves are shown in Figure 4.13(a). Each of the �d curves have sharp minima and theyare all located at about 11 m. Choosing s which corresponds to the minimum �d valuehas provided a good estimate of strue in this case due to the fact that the e�ect ofcoil separation errors on the data greatly overpowers that of the conductivity structure.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 94However, as the coil separation increases, and the importance of primary �eld errorsdecreases, this method of estimating s becomes less e�ective.When trying to estimate the value of s for the large separation HLEM data the resultsare similar. The minima for each curve shown in Figure 4.13(c) are quite broad and thebest estimate is made when �ref = 0.01 S/m. However, each of the minimum values liewithin 2 m of the correct separation.By looking at the curves on the �d versus p plots shown in Figures 4.12 and 4.13 indi-vidually, this method of estimation appears to work best when the parameter dominatesthe response. Otherwise it seems that a poor choice of �ref could result in an incorrectestimate. However, by examining the three curves from each plot together it is possibleto see that the best estimate of ptrue not only corresponds to the best �ref choice, butalso to the minimum with the smallest �d value. This suggests that it may be possibleto get an accurate estimate of ptrue by searching for the pair of pref and �ref values thatcorrespond to the minimum �d value. In order to investigate this possibility the datamis�ts for a range of �ref halfspace values and p can be calculated and contoured overthe �ref and p values.From the preliminary investigation, the AEM data appears to be more sensitive tovariation in ground conductivity than the HLEM data. Since the goal is to estimate h the�d value used for the contour plot was calculated using the data from the two frequenciesalone. It is hoped that this will limit the in uence of the conductivity structure at depthon the data. The contour plot for the AEM example in Figure 4.14 shows �d values fora range of �ref halfspaces from 0.001 S/m to 0.1 S/m, and a range of h values from 15m to 45 m.It can be seen that a minimum �d value exists at approximately 0.01 S/m and 30m. This minimum lies within a relatively shallow bowl that trends diagonally from lowconductivity and small height values to high conductivity and large height values. This

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Coil Separation (m)

log 10

Con

duct

ivity

(S

/m)

log10

φd

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Figure 4.14: Contour plots of �d for a range of �ref and p values for the AEM exampleand both HLEM examples. (a) AEM example in which only two highest frequencieswere used to calculate �d, (b) small HLEM example and (c) large HLEM in which everyfrequency was used to calculate �d

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 96is in general agreement with the inversion results when p is incorrect from Section 3.1.When the bird is too close to the earth the data can best be predicted by a less conductivestructure, and vice versa. This behavior explains the shape and depth of the bowl in �dcontour plot.For each of the HLEM examples, all of the data were included in the parametricinversion. This was done because it appears to be impossible to overpower the e�ectsof coil separation errors with incorrect conductivity estimates. The contour plot for thesmall HLEM example in Figure 4.14(b) shows �d values for a range of �ref halfspaces from0.001 S/m to 0.1 S/m, and a range of s values from 7.5 m to 12.5 m. The minimum �dvalue is located near 0.01 S/m and 11 m. This minimum lies in a deep trough which runsparallel along the s = 11 m value for each conductivity value. This is a clear indicationthat the true value of s can not be masked by an incorrect �ref value.The results from the large HLEM example are similar. The contour plot for the smallHLEM example in Figure 4.14(c) shows �d values for a range of �ref halfspaces from0.001 S/m to 0.1 S/m, and a range of s values from 47.5 m to 52.5 m. The minimum�d value is located near 0.01 S/m and 51 m. The pattern of the contours about theminimum appear more spread out than those from the small HLEM example. This isdue to the fact that the distortions caused by coil separation errors are a function of therelative coil separation error. Since the small separation example represents a 10% error,and the large example only represents a 2% error, the di�erence in the patterns makesense.These examples have shown that accurate estimates of both �ref and pref , can beobtained from either AEM or HLEM data. Therefore, the inclusion of a pre-processingstep in which a best-�t 2-parameter model is determined would be bene�cial to theinversion algorithm.The inversion could then be carried out using the calculated value of pref . The decision

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 97to discard the calculated �ref value is based on the fact that while the estimate of �refmay be a good one, since the �ref value provided by the user was most likely chosen fora particular reason. One reason could be that the sounding is part of line of inversionsbeing performed, and for consistency it is best to use the same reference model for eachinversion along the line. While the option to choose between using the estimated andsupplied pref value could easily be included in an inversion algorithm, for my purposes Ihave decided to always use the user supplied value of �ref .This section has provided a reliable way in which to choose pref and a stable wayin which to de�ne �p, and hence it is now possible to de�ne the inversion algorithm torecover both conductivity and a parameter.4.2.3 Inversion algorithm to recover 1-D conductivity and pIn order to provide as much detail as possible I will present a description of the twoinversion algorithms that can be used to recover a 1-D conductivity structure and asurvey parameter p. The �rst algorithm solves the problem for a �xed � value and thesecond algorithm uses the discrepancy principle.Details of �xed � AlgorithmThis algorithm is very similar to the one presented Section 4.2.1 for the �xed p solution.The only modi�cations are the inclusion of a cooling schedule for �p that a�ects Steps1 and 9A, and the inclusion of the step in which pref is determined as described inSection 4.2.2. A ow chart of the algorithm is shown in Figure 5.6. The changes in thealgorithm are as follows:1. The user must provide a starting value of �p.2. The starting pref value is calculated as discussed in Section 4.2.2.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 98Begin iter. k+1

Use fixed value of β

Convergence ?

Exit

Calc GN step δmSet m = m + δm

Y

Y

N

N

Calc k+1 values

Fixed βInversion Algorithmwith p included inthe model

Has φ decreased?φ +β φ < φ +β φ

3

4

5

6

7

8

9

10

(k)(k+1)

d(k+1)

m m(k+1) (k) (k)

d

Use DGN δm = ω δm andm = m + δm

8A

(k)(k+1)

Calc initial valuesincluding pref.

User Inputincluding αp

1

2

Set k=k+1Update variablesαp = θ αp

9A

Set k = 0Initialize Variables

Figure 4.15: Flowchart for the �xed � inversion algorithm with p included in the model.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 99(a)0 50 100 150

−3

−2.5

−2

−1.5

−1

−0.5

0

Depth (m)

log

10 C

on

du

ctiv

ity (

S/m

)True Model

Recovered Model (b)10

210

310

410

50

500

1000

1500

2000

2500

3000

Frequency (Hz)

Inph

ase

and

Qua

d.(P

PM

) IP Obs.

IP Pred.

Q Obs.

Q Pred. Figure 4.16: Recovered models and predicted data from AEM example �xed � inversionwith h included in the model. The recovered measurement height was equal to 27.2m. (a) True conductivity (dashed line) and recovered conductivity (solid line) and (b)observed data (inphase - circles, quadrature - crosses) and predicted data (inphase - solidline, quadrature - dashed line).9A. The value of �p is decreased by a factor of 10 after each iteration.This algorithm was used to invert the AEM sounding data with a �xed value of �.The value of � was selected such that the goal �d = �?d was achieved. The value of �p wasset equal to 1.0. The rest of the details of the inversion were the same as those used inprevious AEM inversions. Figure 4.16 shows the recovered model and the predicted datafrom the inversion and Figure 4.17 shows the values of �d, ��,p, �p, � and �, as a functionof iteration. The convergence curves �d, ��, and � curves are very similar to those fromthe �xed p example. However, the important curve is the plot of p versus iteration inFigure 4.17(c). The value of p decreases gradually from it starting value of 30.5 m toeventually plateau at a �nal value of 27.2 m. While this value is not equal to htrue, itdoes recover a model that can �t the data, has minimum structure, and reproduces thetrue model quite well. Therefore, the inversion using �xed � has been successful. Thenext step is to test the methodology using the discrepancy principle.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 100(a)0 2 4 6 8

101

102

103

104

Iteration

Dat

a M

isfit

Recovered Misfit

Target Misfit (b)0 2 4 6 8

10−2

10−1

Con

duct

ivity

Nor

m

(c)0 2 4 6 8

26.5

27

27.5

28

28.5

29

29.5

30

30.5

31

Mea

sure

men

t Hei

ght (

m)

Recd hTrue h (d)

0 2 4 6 810

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10−10

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10−6

10−4

10−2

Par

amet

er N

orm

(e)0 2 4 6 8

102

103

104

105

Iteration

Reg

ular

izat

ion

Par

am. (f)

0 2 4 6 810

1

102

103

104

Iteration

Glo

bal O

bjec

tive

Fun

c.

φ new

φ old

Figure 4.17: Convergence curves from AEM example �xed � inversion with h includedin the model. (a) �d (circles) vs. iteration and target �d (crosses), (b) �� vs. iteration,(c) recovered h (circles) vs. iteration and htrue (crosses), (d) �p vs. iteration, (e) �xed �,and (f) �(mk+1) (circles) and �(mk) (crosses) vs. iteration.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 101Details of discrepancy principle AlgorithmThis algorithm is very similar to the one presented in Section 4.2.1 for the discrepancyprinciple solution. The only modi�cations that had to be made were the inclusion ofa cooling schedule for �p that a�ects Steps 1 and 9A, and the inclusion of the step inwhich pref is determined as described in Section 4.2.2, which a�ects Step 2. Since thesechanges were outlined with respect to the �xed � algorithm they will not be listed againhere. However, a ow chart of the algorithm is shown in Figure 4.18. The discrepancyprinciple algorithm will be used for the rest of the chapter to invert EM data, henceforthit will simply be referred to as the inversion algorithm. The next step is to test it on thesynthetic examples.4.3 Synthetic ExamplesIn this section the algorithm is tested on each of the synthetic AEM and HLEM examplesthat have been used throughout the thesis.4.3.1 AEM SoundingThe algorithm was used to invert the AEM sounding data. The value of �p was setequal to 1.0. The rest of the details of the inversion were the same as those used inprevious AEM inversions. Figure 4.19 shows the recovered model and the predicteddata from the inversion and Figure 4.20 shows the values of �d, ��,p, �p, � and �, as afunction of iteration. Figure 4.19 shows that the algorithm has also been able to recover aconductivity structure that is a good representation of the true model. The measurementheight recovered using the discrepancy principle has a value of 27.2 m, which is close tothe true height of 30 m. The plot of p in Figure 4.20 illustrates the way in which thecooling schedule for �p has ensured that p is not able to wander too far from pref during

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 102Begin iter. k+1Set target φd for this iter. φ = max(γ φ ,φ )

Line Search for βensure δp not too large

Convergence ?

Exit

Set k=k+1Update variablesαp = θ αp

Choose largerφ and restart

Calc GN step δmSet m = m + δm

Y

Y

N

N

Calc initial valuesincluding pref.

Calc k+1 values

Discrepancy PrincipleInversion Algorithmwith p included inthe model

User Inputincluding αp

1

Has φ decreased?φ +β φ < φ +β φ

2

3

4

5

6

7

8

9

10

8A

9A

(k+1)

★ ★

★

(k+1)

d

d d(k)

d

(k)(k+1)

d(k+1) (k+1)

m m(k+1) (k+1)(k) (k)

d

Set k = 0Initialize Variables

Figure 4.18: Flowchart for the discrepancy principle inversion algorithm with p includedin the model.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 103(a)0 50 100 150

−3

−2.5

−2

−1.5

−1

−0.5

0

Depth (m)

log

10 C

on

du

ctiv

ity (

S/m

)True Model

Recovered Model (b)10

210

310

410

50

500

1000

1500

2000

2500

3000

Frequency (Hz)

Inph

ase

and

Qua

d.(P

PM

)

IP Obs. IP Pred.Q Obs. Q Pred.

Figure 4.19: Recovered models and predicted data from AEM example discrepancy prin-ciple inversion with h included in the model. The recovered measurement height wasequal to 27.2 m and the true measurement height was equal to 30 m. (a) True conductiv-ity (dashed line) and recovered conductivity (solid line) and (b) observed data (inphase -circles, quadrature - crosses) and predicted data (inphase - solid line, quadrature - dashedline).the early iterations when � is large. The remaining curves show a character similar tothose seen in the discrepancy principle inversions when p was �xed (Figure 4.6).It is satisfying to see the inversion recover both measurement height and conductivityfrom AEM data has been successful. However, the fact that the recovered height isdi�erent from the true height by an amount of 3 m opens up the results to furtherinvestigation. In particular, the question of the nature of the trade o� which takes placebetween measurement height and conductivity structure when p is allowed to vary shouldbe explored.4.3.2 Trade o� between � and hIn order to determine the nature of the trade o� between measurement height and con-ductivity model the inversion results from the �xed p and variable p, inversions of theAEM data will be compared. Figure 4.21(a) shows the recovered models from the �xed

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 104(a)0 5 10 15

101

102

103

104

Iteration

Dat

a M

isfit

Recovered Misfit

Target Misfit (b)0 5 10 15

10−3

10−2

10−1

Con

duct

ivity

Nor

m

(c)0 5 10 15

27

28

29

30

31

32

33

34

Iteration

Mea

sure

men

t Hei

ght (

m)

Recd hTrue h (d)

0 5 10 1510

−12

10−10

10−8

10−6

10−4

10−2

Par

amet

er N

orm

(e)0 5 10 15

103

104

105

Iteration

Reg

ular

izat

ion

Par

am. (f)

0 5 10 1510

1

102

103

104

Iteration

Glo

bal O

bjec

tive

Fun

c.

φ new

φ old

Figure 4.20: Convergence curves from AEM example discrepancy principle inversion withh included in the model. (a) �d (circles) vs. iteration and target �d (crosses), (b) �� vs.iteration, (c) recovered h (circles) vs. iteration and htrue (crosses), (d) �p vs. iteration,(e) � vs. iteration, and (f) �(mk+1) (circles) and �(mk) (crosses) vs. iteration.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 105(a)0 50 100 150

−3

−2.5

−2

−1.5

−1

−0.5

0

Depth (m)

log

10 C

on

du

ctiv

ity (

S/m

)m

rec:p

true = 30m

mrec

:prec

= 27.17mm

true:p

true = 30m (b)

0 50 100 150−3

−2.5

−2

−1.5

−1

−0.5

0

Depth (m)

log

10 C

on

du

ctiv

ity (

S/m

)

mrec

:ptrue

= 30m m

rec:p

rec = 29.88m

mtrue

:ptrue

= 30m Figure 4.21: Comparison of models recovered from the inversion of AEM data with �xedh (dashed line) and variable h (solid line). The true model is also plotted (dotted line).(a) Models from the synthetic AEM data used throughout the chapter. (b) Models froma di�erent set of AEM in which the model was more complex.h �m �d �?dFix h 30.0 2:50 � 10�2 20.0 20.0Var. h 27.2 2:17 � 10�2 20.0 20.0Table 4.1: Comparison of results from the AEM inversions with �xed h and variable h.h is the recovered measurement height, �m is the recovered model norm, �d is the datamis�t achieved by the inversion, and �?d is the target mis�t. Values are tabulated forboth the �xed h (Fix h) and variable h (Var. h) inversions.p (dashed line) and variable p (solid line) inversions. The true model is also plotted(dotted line). The �nal measurement height, model norm, data mis�t, and target mis�tvalues for both of the inversions are shown in Table 4.1. As was mentioned previously,the recovered h value is close but not equal to htrue. By comparing the recovered modelsplotted in Figure 4.21 it appears that the di�erence in measurement height is accountedfor by di�erences in the recovered conductivity structure. While the amplitude of therecovered conductive layer at 40 m is similar for both algorithms, the other regions ofthe structure are di�erent. The model recovered from the variable p inversion appearsto have been smoothed out both above and below the peak conductivity. This can be

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 106con�rmed by comparing the �m values from Table 4.1.These types of distortions should be expected in the AEM case. In Figure 4.7 it wasshown that there are a wide range of h values that can �t the data. The algorithm selectedthe one with the minimummodel norm value, which, in this case, corresponded to a heightof 27.2 m. The size of the di�erence between the recovered height and the true heightwill depend a great deal on the model being considered. In the AEM example shownabove, the structure above and below the conductive layer had a secondary in uenceon the data. Therefore, perturbations to the data due to changes in ight height werecompensated by changes to the recovered conductivity structure. In this example theconductivity model was smoothed out and a smaller measurement height was recoveredhowever, a di�erent synthetic model will produce a di�erent result.I will now attempt to invert data collected above a slightly more complex structureto examine the trade o� between conductivity and h. In this case a 5 m overburden witha conductivity of 0.1 S/m was placed at the surface and the conductivity of the buriedlayer was increased to 0.5 S/m. The data were simulated at ten frequencies ranging from110 Hz to 56320 Hz with a coil separation of 10 m and measurement height of 30 m. Thisexample will be referred to as the overburden AEM example. In order to simulate actualmeasurements, random noise with a standard deviation equal to 5% of the data amplitude+ 10 PPM was added to each datum. Figure 4.21(b) shows the models recovered fromthe �xed p (dashed line) and variable p (solid line) inversions of the overburden AEMdata. The �nal measurement height, model norm, data mis�t, and target mis�t valuesfor both of the inversions are shown in Table 4.2. It can be clearly seen that there isvery little trade o� possible in this example. This is a direct result of the presence of theoverburden at surface. In order to �t the data, it is necessary to recover the conductivelayer at the surface accurately. Therefore, it is not possible to vary the ight height inorder to recover a model with a smaller �m value.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 107h �m �d �?dFix h 30.0 2:50 � 10�1 20.0 20.0Var. h 29.9 2:49 � 10�2 20.0 20.0Table 4.2: Comparison of results from the overburden AEM inversions with �xed h andvariable h. h is the recovered measurement height, �m is the recovered model norm,�d is the data mis�t achieved by the inversion, and �?d is the target mis�t. Values aretabulated for both the �xed h (Fix h) and variable h (Var. h) inversions.The insight provided by these two examples is encouraging. It has been shown thatthe trade o� that can occur between the conductivity and the measurement height willonly a�ect parts of the conductivity structure that are of secondary importance to thedata. Therefore, while these trade o�s can result in incorrect recovery of the measurementheight, they will not distort the important features of the conductivity model.4.3.3 HLEM soundingThis section will present the inversion results from both of the HLEM examples. Firstof all, the algorithm was used to invert the small HLEM data. The value of �p wasset equal to 1.0. The rest of the details of the inversion were the same as those usedin previous small HLEM inversions. Figure 4.22 shows the recovered model and thepredicted data from the inversion and Figure 4.23 shows the values of �d, ��,p, �p, � and�, as a function of iteration. Figure 4.22 shows that the algorithm has also been ableto recover a conductivity structure that is a good representation of the true model. Thecoil separation recovered using the discrepancy principle has a value of 10.98 m, whichis very close to the true separation of 11 m. The plot of p in Figure 4.23 shows that pnever wandered very far from the value determined by the best �t half space. This typeof behavior is expected since the HLEM is dominated by the true coil separation value.Similar results are seen when the large HLEM data is inverted. Figure 4.24 shows therecovered model and the predicted data from the inversion and Figure 4.25 shows the

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 108(a)0 50 100 150

−2.5

−2

−1.5

−1

−0.5

Depth (m)

log

10 C

on

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)True Model

Recovered Model (b)10

210

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5−2.5

−2

−1.5

−1

−0.5

0

0.5x 105

Frequency (Hz)

Inph

ase

and

Qua

d.(P

PM

)

IP Obs. IP Pred.Q Obs. Q Pred. Figure 4.22: Recovered models and predicted data from small separation HLEM ex-ample discrepancy principle inversion with s included in the model. The recovered coilseparation was equal to 10.98 m and the true coil separation was equal to 11 m. (a)True conductivity (dashed line) and recovered conductivity (solid line) and (b) observeddata (inphase - circles, quadrature - crosses) and predicted data (inphase - solid line,quadrature - dashed line).values of �d, ��,p, �p, � and �, as a function of iteration.The algorithm has been successful in recovering both coil separation and conductivityfor HLEM data. Due to the fact that the data is dominated by the e�ects of coil separationthere is very little trade o� between the recovered conductivity structure and the coilseparation value. This is as expected since it was shown previously that there exists anarrow range of separation values that will be able to �t the data (Figures 4.8 and 4.9).Now that the algorithm has been tested on synthetic examples it is time to apply itto the �eld data sets.4.4 Field ExamplesI have already determined that the �eld data sets may be contaminated with geometricalsurvey parameter errors. Therefore, I will now apply the modi�ed inversion methodology

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 109(a)0 5 10 15

101

102

Iteration

Dat

a M

isfit

Recovered Misfit

Target Misfit (b)0 5 10 15

10−5

10−4

10−3

10−2

10−1

Con

duct

ivity

Nor

m

(c)0 5 10 15

10.94

10.95

10.96

10.97

10.98

10.99

11

11.01

Co

il S

ep

. (m

)

Recd sTrue s

(d)0 5 10 15

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Par

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(e)0 5 10 15

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Iteration

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Par

am. (f)

0 5 10 1510

1

102

Iteration

Glo

bal O

bjec

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Fun

c.

φ new

φ old

Figure 4.23: Convergence curves from small separation HLEM example discrepancy prin-ciple inversion with s included in the model. (a) �d (circles) vs. iteration and target �d(crosses), (b) �� vs. iteration, (c) recovered s (circles) vs. iteration and strue (crosses),(d) �p vs. iteration, (e) � vs. iteration, and (f) �(mk+1) (circles) and �(mk) (crosses) vs.iteration.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 110(a)0 50 100 150

−2.5

−2

−1.5

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Depth (m)

log

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)True Model Recovered Model (b)

102

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Frequency (Hz)

Inph

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and

Qua

d.(P

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)

IP Obs. IP Pred.Q Obs. Q Pred. Figure 4.24: Recovered models and predicted data from large separation HLEM examplediscrepancy principle inversion with s included in the model. The recovered coil sepa-ration was equal to 50.93 m and the true coil separation was equal to 51 m. (a) Trueconductivity (dashed line) and recovered conductivity (solid line) and (b) observed data(inphase - circles, quadrature - crosses) and predicted data (inphase - solid line, quadra-ture - dashed line).to each of the data sets. When inverting the Mt. Milligan data I will attempt to recoverthe true measurement height and when inverting the Sullivan data I will attempt torecover the true coil separation.4.4.1 Mt. MilliganWhen inverting the Milligan data I assigned a standard deviation of 25 Dighem units(5 PPM) to the 900 Hz measurements and a standard deviation of 50 Dighem units (10PPM) to the 7200 Hz and 56000 Hz measurements. The target data mis�t for eachsounding was equal to 6. The results from the inversion are plotted in Figure 4.26.The top panel shows the �nal �d values achieved when h was �xed (solid line), and the�nal �d values achieved when h was included in the model (dashed line). The modi�edinversion algorithm has �t the data to approximately the same degree at each station.This is expected in the case where the bird is too high, which is what was assumed in

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 111(a)0 2 4 6 8 10

101

102

103

104

Iteration

Dat

a M

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Target Misfit (b)0 2 4 6 8 10

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duct

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(c)0 2 4 6 8 10

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. (m

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(d)0 2 4 6 8 10

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(e)0 2 4 6 8 10

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0 2 4 6 8 1010

2

103

104

Iteration

Glo

bal O

bjec

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c.

φ new

φ old

Figure 4.25: Convergence curves from large separation HLEM example discrepancy prin-ciple inversion with s included in the model. (a) �d (circles) vs. iteration and target �d(crosses), (b) �� vs. iteration, (c) recovered s (circles) vs. iteration and strue (crosses),(d) �p vs. iteration, (e) � vs. iteration, and (f) �(mk+1) (circles) and �(mk) (crosses) vs.iteration.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 1120

10

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Dat

a M

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Fixed hRecd h

50

100

Hei

ght(

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9000 9200 9400 9600 9800

0

100

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300

Northing (m)

Dep

th (

m)

log10

σ−3.5

−3

−2.5

−2

−1.5

Figure 4.26: Inversion results from Mt. Milligan Data with h included in the model. Thetop panel: Data mis�t values from the �xed h inversion (solid line) and the variable hinversion (dashed). Middle panel: The estimated (solid line) and recovered (dashed line)measurement height. Bottom panel: The recovered conductivity models plotted next toone another.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 1139000 9200 9400 9600 9800

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th (

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log10

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Figure 4.27: Plot of the recovered �h superimposed on the �xed h inversion results fromMt. Milligan. The solid line is �h = h� hest.the Mt. Milligan area. The plots of the recovered (dashed line) and estimated (solidline) measurement heights in the middle panel of Figure 4.26 indicate the presence ofmeasurement height errors as large as +20 m. However, errors of this size should beexpected in an area of dense tree cover such as Mt. Milligan.The conductivity models plotted in the bottom panel of Figure 4.26 also show abig di�erence from those recovered using the estimated measurement. The \air layers"that were present in Figure 1.6 have now been removed. There is a direct correspondencebetween the removal of the air layers and the recovery of a measurement height error. Thedi�erence between the recovered h values and the measured hest values are, in fact, a goodapproximation of the thickness of the air layers. In order to illustrate this point I haveplotted �h along the survey line and have superimposed it upon the conductivity sectionrecovered using the estimated measurement height. This plot is shown in Figure 4.27.Overall, the inversion of the Mt. Milligan data with the modi�ed inversion method-ology has been successful. While the algorithm has not been able to recover any newstructures at depth, it has been able to remove the distortions in the model that werecaused by the presence of measurement height errors.

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 1144.4.2 SullivanWhen inverting the Sullivan data I assigned standard deviations of 1 percent of theprimary �eld to both the inphase and quadrature components of the data. The targetdata mis�t for each sounding was equal to 8. The observed data are plotted with thedata predicted by the modi�ed inversion algorithm in Figure 4.28. The improvement,when compared to the predicted data from the �xed s inversion (Figure 1.3), is evidentimmediately. By allowing s to vary we have been able to predict the inphase componentof the data and we have also been able to �t the quadrature data better. The rest of theresults from the inversion are plotted in Figure 4.29. The data mis�t values plotted in thetop panel indicate that the algorithm has been able to �t the data to the desired level atmost stations. The middle panel shows the estimated (solid line) and recovered (dashedline) coil separation value. It appears that the coil separation is approximately equal to+0.5 m at most stations along the line. This is in accordance with the error that wasestimated by examining the Max-Min system used to collect the data. It is reassuringto see that the results match well with the expectations. The recovered conductivitystructure varies slightly at depth from the models recovered from the inversion with�xed s (Figure 1.2). However, the fact that the model can now predict the inphase dataimproves my con�dence in the recovered model.The use of the modi�ed inversion algorithm has been able to improve the results fromboth the Mt. Milligan and Sullivan data.4.5 SummaryIn this Chapter a general methodology was developed to recover both a function and aparameter. It was then applied to the problem of recovering a 1-D conductivity distrib-ution and a survey parameter from EM data. The details of the problem were developed

Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 115(a)(b)(c)(d)0 50 100 150 200 250 300 350

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Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 11610

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Chapter 4. Recovering 1-D Conductivity and a Geometric Survey Parameter 117for both AEM and HLEM data. An investigation into e�ects of adding the parameterto the inversion resulted in a modi�ed inversion algorithm. The algorithm was testedon synthetic AEM and HLEM data sets. The results have shown that the algorithm isstable and provides good results.In the AEM case a good estimate of the measurement height was recovered. Theaccuracy of the estimate depends in part upon the model being recovered. However, theheight that is recovered corresponds to the minimum structure model that �ts the data.For the examples shown the conductivity structure that is recovered is a good repres-entation of the true model. When inverting HLEM data the algorithm has recoveredan accurate estimate of the coil separation and a good representation of the subsurfaceconductivity structure.The use of the modi�ed inversion algorithm has been able to improve the results fromboth the Mt. Milligan and Sullivan data. The inversion results from the Mt. Milligandata show that the distortions due to the \air layers" have been removed by recoveringa better estimate of the ight height. This has generated a conductivity model which ismuch easier to interpret. The inversion of the Sullivan data made it possible to predictthe inphase component of the data by recovering an estimate of the true coil separation.While the inclusion of the coil separation in the model does not drastically change therecovered conductivity structure, the plausibility of the inversion result has been increasedby the fact that the algorithm has been able predict the inphase component of the data.

Chapter 5Application of GCV to 1-D EM InversionWhen inverting geophysical data the presence of noise is always an issue. The amountof information that can be extracted from a noisy set of data is dependent upon ourknowledge of the noise. The standard way to deal with noise is to provide an estimateof the noise and attempt to �t the data accordingly. Since it is di�cult to estimate theamount of noise associated with the data simply by inspection, this method can causeproblems.Generalized cross validation (GCV) is a method of estimating the regularization pa-rameter that will account for the noise associated with a given data set. This techniquehas been widely used in statistics (Wahba, 1990) and has recently been applied to geo-physical inverse problems (Haber, 1997). This chapter will present a brief review of GCVin both the linear and non-linear inverse problems. Using this methodology, a non-linearGCV inversion algorithm will be developed for the 1-D frequency domain EM inverseproblem. Both the problem of when p is �xed and when it is included in the model willbe dealt with. Each of the algorithms will be tested on synthetic data sets.Since �eld data sets, such as the Sullivan and Mt. Milligan data sets, consist of limitednumbers of data, the performance of a statistical method such as GCV is uncertain.Therefore, the applicability of GCV to data sets with small numbers of data will also beinvestigated. 118

Chapter 5. Application of GCV to 1-D EM Inversion 1195.1 Estimating Noise using GCVThis section will begin with a discussion of GCV in a linear inverse problem. After thebasics have been covered, a methodology by which GCV can be applied to nonlinearinverse problems will be presented.5.1.1 GCV in Linear Inverse ProblemsThe idea behind GCV is that a good model should be able to predict measurements thathave yet to be taken. However, it is not feasible to take new measurements after eachinversion in order to test the recovered model. A simpler approach would be to leaveone datum out of the data vector and search for the model that is best able to predictthe missing datum. This is the method used by GCV to determine the model which isa�ected least by any given datum.The theoretical background of GCV can be found in Wahba (1990) and will not becovered here. However, an overview of how GCV can be used in inverse problems willbe discussed. I will begin by considering a generic linear inverse problem in which I amattempting to invert the observed data dobs, that are contaminated by random noise witha uniform, but unknown, standard deviation �. I want to recover the model m with theminimum amount of structure that �ts dobs to the correct amount. I will begin as usualby de�ning a global objective function,�(m) = �d + ��m = jjGm� dobsjj2 + �jjWmmjj2 (5.1)where �d is the data mis�t, � is the regularization parameter, �m is the model norm, Gis a linear forward operator, and Wm is a weighting matrix which penalizes a certain typeof structure in the model. As mentioned in Section 2.2.2 the solution of this problem canbe found by minimizing � at the correct value of �, such that our expectations for the

Chapter 5. Application of GCV to 1-D EM Inversion 120data mis�t are satis�ed. GCV is a method which can be used to get an estimate of thecorrect value of �.The GCV function is de�ned as,GCV (�) = jj(I �G(GTG + �W TmWm)�1GT )dobsjj2[trace(I �G(GTG + �W TmWm)�1GT )]2 : (5.2)It has been shown that the � value corresponding to the minimum of the GCV functionwill provide a good estimate for the optimal regularization parameter in problems with theform shown in Equation 5.1. The model recovered using the GCV estimate of � sometimesover-�ts the data, but generally provides a reliable estimate of the noise associated withdobs.When each datum within dobs is contaminated by Gaussian noise with the samestandard deviation, the GCV estimate of � requires no information about the noise fromthe user. However, in the case where the standard deviation of the noise varies with eachdatum it is necessary to scale the data such that the relative noise levels are equal foreach datum. This is due to the fact that the mathematics of GCV are done in terms ofnoise with a single standard deviation. However, it is important to note that the userneed only provide the relative noise estimates, the GCV estimate of � will determine theabsolute noise level associated with the data. It is also important to note that the GCVestimate of � can not account for correlated noise. This fact is what suggests that theapplication of GCV to nonlinear problems could be useful.5.1.2 GCV in Non-linear ProblemsThe non-linear problem is approached in a manner similar to the linear case. The non-linear objective function is de�ned as�(m) = �d + ��m = jjF [m]� dobsjj2 + �jjWmmjj2; (5.3)

Chapter 5. Application of GCV to 1-D EM Inversion 121where F [m] is a non-linear forward operator. The problem of not knowing �? still exists,but now the problem is compounded by the fact that the function being minimized isnonlinear. A common way to proceed in a case like this is to linearize the system andsolve it iteratively. Therefore, I assume that the current model is equal to mk. My goalnow, is to �nd the perturbation �m to the model such that mk+1 = mk+�m will minimizethe objective function �. By expanding the operator F in a Taylor series I am left withF (mk+1) = F (mk + �m) = F (mk) + J(mk)�m+N (O(�m2)); (5.4)where J(mk) is the Jacobian de�ned at mk and N represents the higher order terms ofthe expansion. Neglecting higher order terms, the data mis�t �d can be approximatedby the linearized mis�t �lind which has the form�dlin = jjF (mk) + J(mk)mk+1 � J(mk)mk � dobsjj2: (5.5)This results in a linearized objective function of the form�lin(mk+1) = �dlin + ��m = jjJ(mk)mk+1 � rkjj2 + �jjWmk+1jj2; (5.6)where rk = dobs+J(mk)mk�F (mk). The fact that the linear objective function �lin hasa form similar to the objective function shown in Equation 5.1 suggests that using GCVto estimate � might be a good idea. The usual problem of choosing a regularizationparameter is made more di�cult in the nonlinear case by the fact that the equationsbeing minimized are linear approximations to the real ones. It is not unreasonable tothink of the inaccuracies that are introduced by linearization as correlated noise, and ifso the GCV estimate of � might not be a�ected by these terms. Therefore, GCV shouldprovide an estimate of � which will be able to �t the noise associated with dobs properly.After selecting � using GCV, the perturbation to the model is calculated using Equa-tion 2.60. However, the errors due to the linearization may cause problems with the

Chapter 5. Application of GCV to 1-D EM Inversion 122Gauss-Newton step. Since these errors are of O(�m2) they may be su�ciently large suchthat the step �m will not result in a decrease in the non-linear objective function. Thisproblem is dealt with by taking a damped Gauss-Newton (DGN) approach, as was donein Section 4.2.1. By incorporating these pieces into the �xed � inversion algorithm it willbe straightforward to design the GCV algorithm.5.2 GCV and the 1-D EM inverse problemHaber (1997) successfully applied a GCV based inversion methodology to a non-linearproblem in EM (viz magnetotellurics) using a DGN approach. This methodology will beapplied to the 1-D frequency domain EM problem. It will �rst be applied to the casein which the parameter p is �xed during the inversion, and then the algorithm will bemodi�ed to accommodate the parameter p as part of the model. Each of these algorithmswill be tested on synthetic AEM data. The �nal part of the section will deal with theapplication of GCV to small data sets.5.2.1 GCV when p is �xedWhen investigating a new inversion methodology the best starting point is to solve thesimplest problem possible. Therefore, I will begin by applying GCV to the inversion ofEM data when p is �xed.Inversion AlgorithmThe inversion algorithm for GCV with a �xed value of p is very similar to the algorithmusing a constant � for �xed p that was presented in Section 4.2.1. Each of the stepstaken during the inversion are summarized in the ow chart shown in Figure 5.1. Thedi�erences between this algorithm and the constant � algorithm are in Steps 5 and 8,

Chapter 5. Application of GCV to 1-D EM Inversion 123Begin iter. k+1

Estimate β using GCV

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Figure 5.1: Flowchart for the GCV inversion algorithm with a �xed value of p.

Chapter 5. Application of GCV to 1-D EM Inversion 124and the modi�cations are summarized in the following list:5. The GCV estimate of � that corresponds to the minimum of the GCV function isselected as �k+1.8. The values of �(mk+1) and �(mk) are evaluated using �k+1, as in the discrepancyprinciple algorithm in Section 4.2.1.Now, having designed the algorithm, it can be tested on a synthetic example.Synthetic AEM ExampleA set of synthetic AEM data were generated over the conductivity model shown inFigure 3.1(a). The data were simulated at ten frequencies ranging from 110 Hz to 56320Hz with a coil separation of 10 m and measurement height of 30 m. In order to simulateactual measurements, random noise with a standard deviation equal to 5% of the dataamplitude + 10 PPM was added to each datum.The noisy data were then inverted using the GCV algorithm. The noise estimates usedfor the scaling matrix were 5% of the data amplitude + 10PPM and the reference modelwas 0.01 S/m. Figure 5.2 shows a comparison of the recovered models and predicted datafrom the discrepancy principle inversion and the GCV inversion. The �nal �, model norm,data mis�t, and target mis�t values for both of the inversions are shown in Table 5.1.All of these results indicate that the two inversion results are both qualitatively andquantitatively similar. In comparison with the discrepancy principle however, the GCValgorithm has over-�t the data slightly. The convergence curves from both inversions areplotted in Figures 5.3 and 5.4. It is clear from the curves in these Figures that the pathstaken by the two algorithms to reach the solution were quite di�erent. While the datamis�t is decreased at each iteration in both inversions, the decreases in the GCV case arenot at a constant rate. The model norm does not begin small and increase gradually, as

Chapter 5. Application of GCV to 1-D EM Inversion 125�2 (a)0 50 100 150

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Q Pred. Figure 5.2: Comparison of recovered models and predicted data from AEM examplediscrepancy principle and GCV inversions with a �xed value of h. The measurementheight was set equal to 30 m. Discrepancy principle (�2) results: (a) True conductivity(dashed line) and recovered conductivity (solid line) and (b) observed data (inphase -circles, quadrature - crosses) and predicted data (inphase - solid line, quadrature - dashedline). GCV results: (c) conductivity models (labeled as in (a)) and (d) data (labeled asin (b)) � �m �d �?d�2 1020.4 1:87� 10�2 20.0 20.0GCV 257.5 2:35� 10�2 17.4 N/ATable 5.1: Comparison of parameter values from the AEM discrepancy principle andGCV inversions with a �xed value of h. � is the �nal regularization parameter value, �mis the recovered model norm, �d is the data mis�t achieved by the inversion, and �?d isthe target mis�t. Values are tabulated for both the discrepancy principle (�2) and GCVinversions.

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Figure 5.4: Convergence curves from AEM example GCV inversion with a �xed value ofh. The measurement height was set equal to 30 m. (a) �d (circles) and target �d (crosses)vs. iteration, (b) �m vs. iteration, (c) �xed � vs. iteration, and (d) �(mk+1) (circles)and �(mk) (crosses) vs. iteration.

Chapter 5. Application of GCV to 1-D EM Inversion 128it does in the discrepancy principle example, instead it remains close to the same valuethroughout the inversion. The way in which � varies through the inversion is opposite inthe two algorithm. When the discrepancy principle algorithm is used, � generally startsout large and decreases at each iteration. Conversely the GCV algorithm usually startsout by estimating small values of � which gradually increase as the inversion proceeds.The di�erences in the �m and � curves in the GCV results are caused by the fact that theGCV tends to over-�t the noise associated with the linearized problem at each iteration.This can be seen by plotting the GCV function values as a function of � for a particulariteration along with the model that was calculated using the � chosen by the GCV. Theplots were generated for iterations 1, 3, and 6 and are shown in Figure 5.5. In the �rstiteration the minimum in the GCV function lies at a small � value. The perturbation �mcalculated using this � was quite large and had to be damped once. While the dampingensured that the step decreased �, the model recovered at this iteration still has a lotof structure. By the third iteration however, the GCV curve has selected a larger �value and is calculating good perturbations. In fact, by the sixth and �nal iteration theminimum of the GCV curve has settled to a relatively constant value.As it stands, the GCV algorithm must work to remove the structure added to themodel due through the selection of small � values during early iterations. While this isnot the path the discrepancy principle algorithm takes, it leads to an acceptable result.The recovered conductivity structure is a good representation of the true conductivityand the data is �t to an appropriate level. Since the GCV algorithm has be successfulfor a �xed value of p there is no reason why it should not work when p is included in themodel.

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Chapter 5. Application of GCV to 1-D EM Inversion 1305.2.2 GCV when p is included in the modelThe next step is to modify the algorithm to include p in the model, such that it is possiblerecover both conductivity and a parameter value.Inversion AlgorithmThis algorithm is almost identical to the one presented Section 5.2.1 for the �xed psolution. The only modi�cation is the inclusion of a cooling schedule for �p. Thesechanges lie in Steps 1,2, and 9A. A ow chart of the algorithm is shown in Figure 5.6.The changes in the algorithm are as follows:1. The user must provide a starting value of �p.2. The starting pref value is calculated as discussed in Section 4.2.2.9A. The value of �p is decreased by a factor of 10 after each iteration.Now having designed the algorithm it can be tested on a synthetic example.Synthetic AEM ExampleThe same synthetic AEM data used in the �xed p example was inverted with p as part ofthe model. For this inversion the starting value of �p was set equal to 1. Figure 5.7 showsa comparison of the recovered models and predicted data from the discrepancy principleinversion and the GCV inversion. The �nal measurement height, �, model norm, datamis�t, and target mis�t values for both of the inversions are shown in Table 5.2. Theseresults show that the GCV algorithm behaves similarly when p is included in the model.The discrepancy principle inversion recovered a measurement height of 29.1 m and theGCV recovered a value of 28.9 m, both of which are very good estimations of the trueheight of 30 m. In both cases the recovered conductivity models are good representations

Chapter 5. Application of GCV to 1-D EM Inversion 131Calc initial valuesincluding pref.

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Q Pred. Figure 5.7: Comparison of recovered models and predicted data from AEM example dis-crepancy principle and GCV inversions with h included in the model. The measurementheight recovered with discrepancy principle was 29.1 m, with GCV was 28.9 m, and thetrue value was 30 m. Discrepancy principle (�2) results: (a) True conductivity (dashedline) and recovered conductivity (solid line) and (b) observed data (inphase - circles,quadrature - crosses) and predicted data (inphase - solid line, quadrature - dashed line).GCV results: (c) conductivity models (labeled as in (a)) and (d) data (labeled as in (b))h � �m �d �?d�2 29.1 1235.3 1:84 � 10�2 20.0 20.0GCV 28.9 259.9 2:26 � 10�2 17.4 N/ATable 5.2: Comparison of results from AEM example discrepancy principle and GCVinversions with h included in the model. h is the recovered measurement height, � isthe �nal regularization parameter value, �m is the recovered model norm, �d is the datamis�t achieved by the inversion, and �?d is the target mis�t. Values are tabulated forboth the discrepancy principle (�2) and GCV inversions.

Chapter 5. Application of GCV to 1-D EM Inversion 133of the true model. However, once again the GCV algorithm has over-�t the data slightly.The convergence curves from both of the inversions are plotted in Figures 5.8 and 5.9.Upon comparison, the convergence curves from these inversion are quite similar to thosefrom the �xed p inversions. The plots of �d, ��, and � versus iteration from the GCVinversion exhibit a character similar to that seen when the height was held �xed at 30m. In the GCV inversion the p versus iteration plot follows a much more direct routeto the �nal parameter value than that of the discrepancy principle solution. This is afunction of the small values of � values being selected by the algorithm. As discussed inSection 4.2.2, when � is large the equations that govern �log(�) and �p are decoupledand the steps taken by p. This is the reason why in the early iterations of the discrepancyprinciple inversions, p tends to wander slightly. However, the small � values selected bythe GCV algorithm ensure that both parts of the step �m are in uenced by one another,and as a result p approaches its �nal value in a direct manner. Plots of the GCV functionversus � and the resultant models were generated for iterations 1, 3, and 6 and are shownin Figure 5.10. The GCV curves and the recovered models from this inversion are quitesimilar to those from the �xed p inversion. The algorithm introduces unwanted structurein the �rst iteration by selecting a small value of �, and the remaining iterations are usedto remove the structure until a model which �ts the data is found. For this example theGCV algorithm has worked well to recover good representations of both conductivity aswell as a parameter value.5.2.3 Application of GCV to small data setsThe theory of GCV requires a large number of data. The previous examples have shownthat GCV can work well with ten frequencies, that is twenty data. However, both of the�eld examples which have motivated this thesis consist of soundings with a small numberof measurement frequencies. A sounding from the Sullivan data has four frequencies and

Chapter 5. Application of GCV to 1-D EM Inversion 134(a)0 5 10 15

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Chapter 5. Application of GCV to 1-D EM Inversion 135(a)0 2 4 6

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Figure 5.9: Convergence curves from AEM example GCV inversion with h included inthe model. (a) �d (circles) and target �d (crosses) vs. iteration, (b) �� vs. iteration, (c)h (circles) vs. iteration and htrue (crosses), (d) �p vs. iteration, (e) � vs. iteration, and(f) �(mk+1) (circles) and �(mk) (crosses) vs. iteration.

Chapter 5. Application of GCV to 1-D EM Inversion 136it=110

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Figure 5.10: GCV Curves and recovered models from iterations 1, 3, and 6 of the AEMexample GCV inversion with h included in the model. The measurement height was setequal to 30 m. Panels (a), (c), and (e) GCV function vs. � (solid line), � selected byGCV algorithm (circle). Panels (b), (d), and (f) model calculated using GCV estimateof � (solid line), mode after it has been damped according to DGN approach (dashedline), true model (dotted line), and ndamp is the number of dampings performed.

Chapter 5. Application of GCV to 1-D EM Inversion 137� �m �d �?d�2 2231.7 3:10� 10�2 8.0 8.0GCV 1:44 � 10�3 2.61 12.5 N/ATable 5.3: Comparison of results from the four frequency AEM example discrepancy prin-ciple and GCV inversions with a �xed value of h. � is the �nal regularization parametervalue, �m is the recovered model norm, �d is the data mis�t achieved by the inversion,and �?d is the target mis�t. Values are tabulated for both the discrepancy principle (�2)and GCV inversions.the Mt. Milligan data has three. In this Section I will apply the GCV algorithm for �xedp to a set of data collected using four frequencies. This will shed some light onto theapplicability of my GCV algorithm to the �eld data sets.Four Frequency AEM ExampleIn order to investigate how GCV behaves for a small number of data it is applied to asynthetic AEM data set. The data were generated over the conductivity model shown inFigure 3.1(a). The data were simulated at four frequencies ranging from 110 Hz to 56320Hz with a coil separation of 10 m and measurement height of 30 m. In order to simulateactual measurements random noise with a standard deviation equal to 5% of the dataamplitude + 10 PPM was added to each datum.In order to invert the data using GCV the data were scaled by the standard deviationsof the errors which were equal to 5% of the data amplitude + 10PPM. The data wereinverted with a reference model equal to 0.01 S/m. Figure 5.11 shows a comparison ofthe recovered models and predicted data from the discrepancy principle inversion andthe GCV inversion of the four frequency data. The �nal �, model norm, data mis�t,and target mis�t values for both of the inversions are shown in Table 5.3. These resultsshow that the GCV algorithm has not been successful. The plot of the recovered modelin Figure 5.11(c) does not resemble the true model, and the values from Table 5.3 show

Chapter 5. Application of GCV to 1-D EM Inversion 138�2 (a)

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Chapter 5. Application of GCV to 1-D EM Inversion 139(a)0 10 20 30

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Figure 5.12: Convergence curves from four frequency AEM example discrepancy principleinversion with a �xed value of h. The measurement height was set equal to 30 m. (a)�d (circles) and target �d (crosses) vs. iteration, (b) �m vs. iteration, (c) �xed � vs.iteration, and (d) �(mk+1) (circles) and �(mk) (crosses) vs. iteration.that the GCV solution in no way resembles the solution reached using the discrepancyprinciple algorithm.It is clear from this that my GCV algorithm is unstable when attempting to invertthe four frequency data. However, determining the cause of the instability may help indeveloping a better GCV algorithm. In order to gain some insight into why the algorithmfailed, it is useful to examine the convergence curves from the inversion. The plots of �d,�m, �, and � versus iteration shown in Figure 5.12 are jagged. The most important curve,which is also the most erratic, is the regularization parameter. The GCV function wasnever able to reach a constant minimum value. A possible reason for this can be seen by

Chapter 5. Application of GCV to 1-D EM Inversion 140plotting the GCV curves from various stages during the inversion. GCV function versus� and the resultant models, for the �rst three iterations, are plotted in Figure 5.10. TheGCV curve from the �rst iteration has a well de�ned minimum and appears to be normal;however, the perturbation �m which it generates is extremely large. The size of the stepwas damped once, such that the objective function decreased. The damped model shownin Figure 5.13(b) was physically unacceptable; however, the algorithm accepted the stepand carried on to the next iteration. It is at this point that the problems continued. TheGCV curve from the second iteration, which is plotted in Figure 5.13(c), does not havea unique minimum. As a result the algorithm chooses an even smaller � value than itchose in the �rst iteration. As would be expected, the step taken by the model in thisiteration is another unphysical one. This type of behavior continued until the algorithmstopped after the thirtieth iteration. The behavior exhibited in this example could beconstrued as the failure of GCV however, this is not the case.When working with a linearized problem the solution that is attained can only beas good as the linearization itself. Therefore, when evaluating the GCV function duringa given iteration, the chance of success depends in part on the model about which theproblem is linearized. Therefore, it should not come as a surprise that the GCV func-tion calculated during the second iteration was problematic. If the model about whichthe problem was linearized was physically unreasonable the GCV had little chance ofaccurately estimating the noise.These results suggests that the instabilities of my algorithm could be ameliorated byadjusting the choice of � during the �rst few iterations. While choosing the GCV estimateof � in the �rst iteration may well predict the noise level correctly, it may also result inan unacceptable amount of structure being added to the model. A better choice of � forthe �rst iteration would be a large value which decreased the data mis�t slightly withoutadding too much structure, such as the step taken by the discrepancy principle. This can

Chapter 5. Application of GCV to 1-D EM Inversion 141it=110

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Figure 5.13: GCV Curves and recovered models from iterations 1, 2, and 3 of the AEMexample GCV inversion with a �xed value of h. The measurement height was set equalto 30 m. Panels (a), (c), and (e) GCV function vs. � (solid line), � selected by GCValgorithm (circle). Panels (b), (d), and (f) model calculated using GCV estimate of �(solid line), mode after it has been damped according to DGN approach (dashed line),true model (dotted line), and ndamp is the number of dampings performed.

Chapter 5. Application of GCV to 1-D EM Inversion 142help to ensure that the recovered model will remain sensible and that the GCV estimateof � will be reliable. An algorithm in which the value of � was gradually cooled withthe eventual goal of using the GCV estimate of � could help deal with the instabilitiesof the algorithm as it is currently de�ned. The implementation of such an algorithm isnecessary in order to e�ectively assess the applicability of GCV to small data sets.5.3 SummaryIn this section a GCV based inversion algorithm which makes use of the damped Gauss-Newton approach was developed. The algorithm was shown to work well when invertingfrequency domain EM soundings which consist of twenty data. The GCV inversion algo-rithm was applied to both the problem of recovering conductivity as well as the problemof recovering conductivity and a survey parameter. The results from the synthetic AEMcompared well with those attained using the the discrepancy principle algorithm devel-oped in Chapter 4. When applied to data sets with smaller numbers of data the GCValgorithm, as it has been designed, was unstable. This suggests the need for a newalgorithm that combines the stability of the discrepancy principle and the noise estimat-ing properties of GCV. Overall, the use of GCV to estimate noise in non-linear inverseproblems looks promising.

Chapter 6SummaryThis thesis seeks a method to improve the inversion results of the Mt. Milligan andSullivan �eld data sets. It was conjectured that the problems associated with the datasets were due to geometric survey parameter errors. In the Mt. Milligan data, meas-urement height errors were causing distortions in the conductivity structure and in theSullivan data, coil separation errors were making it impossible to predict the inphasedata. A solution was proposed in the form of an algorithm that would recover both 1-Dconductivity and a geometric survey parameter.Making use of background information on electromagnetic (EM) methods and the1-D EM inverse problem, presented in Chapter 2, as well as the insight into the nature ofgeometric parameter errors, gained in Chapter 3, the stage was set to solve the problem.In Chapter 4 a modi�ed inversion methodology was proposed to incorporate the extraparameter into the model recovered during the inversion. The necessary changes weremade to the current algorithm and the new algorithm was designed. The algorithm wastested on synthetic AEM and HLEM data sets. The results from these tests showed thatit was possible to recover both good estimates of parameter values as well as realisticrepresentations of the subsurface conductivity structure. When applied to the �eld datasets the problems that were encountered before were corrected. The recovery of correctmeasurement heights in the Mt. Milligan case removed the distortions from the conduc-tivity models and in the Sullivan case the recovery of correct coil separations made itpossible to predict the inphase data. 143

Chapter 6. Summary 144The problem of noise estimation was also addressed in the thesis. Chapter 5 presenteda discussion of the use of generalized cross validation (GCV) to estimate noise in non-linear inverse problems. A GCV based inversion algorithm was developed for the 1-DEM inverse problem. The algorithm worked well when applied to synthetic AEM dataset with 20 data. However, some of the algorithms instabilities were revealed when itwas applied to smaller data sets. While it appears that GCV worked well to estimate thenoise associated with the data, the algorithm needs modi�cation in order to fully exploitthe potential of GCV. The result of these investigation was a better understanding ofhow GCV should be used within the context of non-linear inverse problems.

References[1] Alumbaugh, D.L., and Newman, G.A., 1997, 3-D electromagnetic inversion for envi-ronmental site characterization: Proceedings of the Symposium on the Applicationof Geophysics to Environmental and Engineering Problems, 407-416.[2] deGroot-Hedlin, C., 1991, Removal of static shift in two dimensions by regularizedinversion: Geophysics, 56, 2102-2106.[3] Dennis, J.E., and Schnabel, R.B., 1996, Numerical Methods for Unconstrained Op-timization and Nonlinear Equations: SIAM Philadelphia.[4] Ellis, R.G., and Shehktman, R., 1994, ABFOR1D & ABINV1D: Programs for for-ward and inverse modelling of airborne EM data, in JACI 1994 Annual Report:Dept. of Geophysics & Astronomy, University of British Columbia.[5] Fitterman, D.V., 1998, Sources of Calibration Errors in Helicopter EM Data: Proc.of the International Conference on Airborne Electromagnetics, 1.8.1-1.8.10.[6] Fraser, D.C., 1986, Dighem resistivity techniques in airborne electromagnetic map-ping: in Palacky, G.J., Ed., Airborne Resistivity Mapping: Geol. Survey of Canada,Paper 86-22, 49-54.[7] Fullagar, P.K., and Oldenburg, D.W., 1984, Inversion of horizontal loop electromag-netic frequency soundings: Geophysics, 49, 150-164.[8] Haber, E., 1997, Numerical Strategies For the Solution of Inverse Problems: Ph. D.Thesis, Univ. of British Columbia. 145

References 146[9] Jones, F.H.M., 1996, Preliminary Report: Geophysical Surveys at Sullivan MineTailings: Dept. of Earth & Ocean Sciences, Univ. of British Columbia.[10] Kovacs, A., Holladay, J.S., and Bergeron, C.J.Jr., 1995, The footprint/altitude ratiofor helicopter electromagnetic soundings of sea-ice thickness: Comparison of theo-retical and �eld estimates: Geophysics, 60, 374-380.[11] Norton, S.J., San Filipo, W.A., and Won, I.J., 1999, Reciprocity formulas for elec-tromagnetic induction systems: Proc. of the Symposium on the Application of Geo-physics to Environmental and Engineering Problems, 961-970.[12] Oldenburg, D.W., Li, Y., and Ellis, R.G., 1997, Inversion of geophysical data overa copper gold porphyry deposit: A case history for Mt. Milligan: Geophysics, 62,1419-1431.[13] Parker, R.L., 1994, Geophysical Inverse Theory: Princeton University Press.[14] Pavlis, G.L., and Booker, J.R., 1980, The mixed discrete-continuous inverse prob-lem: application to the simultaneous determination of earthquake hypocenters andvelocity structures: Journal of Geophysical Research, 85, 4801-4810.[15] Ryu, J., Morrison, H.F., and Ward, S.H., 1970, Electromagnetic �elds about a loopsource of current: Geophysics, 35, 862-896.[16] Sauck, W.A., Atekwana, E.A., and Bermejo, J.L., 1998, Characterization of a newlydiscovered LNAPL plume at Wurtsmith AFB, Oscoda, Michigan: Proc. of the Sym-posium on the Application of Geophysics to Environmental and Engineering Prob-lems, 399-407.[17] Wahba, G., 1990, Spline models for observational data: SIAM Philadelphia.

References 147[18] Weaver, J.T., 1994, Mathematical methods for geo-electromagnetic induction: JohnWiley & Sons Inc.[19] Ward, S.H., and Hohmann, G.W., 1988, Electromagnetic theory for geophysicalapplications: in Nabighian, M.N., Ed., Electromagnetic Methods in Applied Geo-physics: Soc. Expl. Geophys., Vol 1, 131-311.[20] Wynn, J., and Gettings, M., 1998, An airborne electromagnetic (EM) survey used tomap the upper San Pedro River aquifer, near Fort Huachuca in southeastern Arizona:Proc. of the Symposium on the Application of Geophysics to Environmental andEngineering Problems, 993-998.[21] Zhang, Z., and Oldenburg, D.W., 1994, Inversion of airborne data over a 1-D earth,in JACI 1994 Annual Report: Dept. of Geophysics & Astronomy, University ofBritish Columbia.[22] Zhang, Z., and Oldenburg, D.W., 1999, Simultaneous reconstruction of 1-D suscep-tibility and conductivity from electromagnetic data: Geophysics, 64, 33-47.[23] Zhang, Z., Routh, P.S., Oldenburg, D.W., Alumbaugh, D.L., and Newman, G.A.,submitted for publication, Reconstruction of 1D conductivity from dual-loop EMdata: Geophysics.