Iterative electromagnetic migration for 3D inversion of ... · Geophysical Prospecting doi: 10.1111/j.1365-2478.2011.00991.x Iterative electromagnetic migration for 3D inversion of

Geophysical Prospecting doi: 10.1111/j.1365-2478.2011.00991.x

Iterative electromagnetic migration for 3D inversion of marinecontrolled-source electromagnetic data

Michael S. Zhdanov1,2, Martin Cuma1,2, Glenn A. Wilson1∗, Evgeny P. Velikhov3,Noel Black1 and Alexander V. Gribenko1,2

1TechnoImaging, 4001 South, 700 East, Suite 500, Salt Lake City, UT 84107, USA, 2University of Utah, 115 South, 1460 East, Room 383,Salt Lake City, UT 84112, USA, and 3Russian Research Center Kurchatov Institute, 1 Akademika Kurchatova pl., Moscow 123182, Russia

Received December 2010, revision accepted June 2011

ABSTRACTThe key to deriving a reliable quantitative interpretation from marine controlled-source electromagnetic data is through the integration of shared earth modeling androbust 3D electromagnetic inversion. Subsurface uncertainty is minimized throughefficient workflows that use all available subsurface data as a priori information andwhich permit multiple resistivity models to explain the same observed data. To thisend, we present our implementation of an iterative migration method for controlled-source electromagnetic data that is equivalent to rigorous 3D inversion. Our iterativemigration method is based on the 3D integral equation method with inhomogeneousbackground conductivity and focusing regularization with a priori terms. We willshow that focusing stabilizers recover more geologically realistic models with sharperresistivity contrasts and boundaries than traditional smooth stabilizers. Addition-ally, focusing stabilizers have better convergence properties than smooth stabilizers.Finally, inhomogeneous background information described as a priori resistivity mod-els can improve the fidelity of the final models. Our method is implemented in a fullyparallelized code. This makes it practical to run large-scale 3D iterative migrationon multicomponent, multifrequency and multiline controlled-source electromagneticsurveys for 3D models with millions of cells. We present a suite of interpretationsobtained from different iterative migration scenarios for a 3D controlled-source elec-tromagnetic feasibility study computed from a detailed model of stacked anticlinestructures and reservoir units of the Shtokman gasfield in the Russian sector of theBarents Sea.

Key words: CSEM, Electromagnetics, Inversion, Migration, Regularization,Stabilizer.

INTRODUCTION

The premise of the various marine controlled-source electro-magnetic (CSEM) methods is sensitivity to the lateral extentsand thicknesses of resistive bodies embedded in conductivehosts. For this reason, CSEM methods were initially appliedto de-risking exploration and appraisal with direct hydrocar-

∗E-mail: [email protected]

bon indication. However, CSEM methods represent just onepart of an integrated exploration strategy for hydrocarbons.The value of CSEM data is only realized when those mod-els derived from CSEM inversion are integrated, with soundgeological understanding, in a shared earth model. The mostsuccessful applications of CSEM to date have been in com-plement to those seismic interpretations where lithological orfluid variations cannot be adequately discriminated by seis-mic methods alone (Hesthammer et al. 2010). Hydrocarbonreserves and resources are estimated with varying confidence

C© 2011 European Association of Geoscientists & Engineers 1

2 M.S. Zhdanov et al.

(or risk) from volumetrics calculated from different subsurfacemodels. Confidence is only established through the testing ofmultiple subsurface models that satisfy all available data. Re-sistivity models derived from 1D or 2.5D CSEM inversionprovide the least confidence because of their inherently re-duced dimensionality; those models derived from 3D inver-sion provide the most confidence.

Here, we distinguish between inversion and interpretation,two terms often used interchangeably. Inversion is a quanti-tative process whereby a model is recovered from observeddata, regardless of whether any a priori knowledge was usedor not. Interpretation is a more qualitative process wherebygeological insight is inferred from at least one model. CSEMinterpretation is inherently reliant upon models derived frominversion methods since CSEM data cannot simply be sep-arated or transformed with linear operators as per seismicmethods. Methods for inverting CSEM data are complicatedby the very small, non-unique and non-linear responses ofhydrocarbon-bearing reservoir units when compared to thetotal fields.

In 3D CSEM inversion, geological prejudice is introducedvia regularization (Tikhonov and Arsenin 1977), whether thisis an a priori model, data or model weights, model boundsand/or by the choice of a stabilizing functional (Zhdanov2002). Most often, resistivity models are obtained from regu-larization with a smooth stabilizing functional (e.g., Mackie,Watts and Rodi 2007; Commer and Newman 2008; Støren,Zach and Maaø 2008; Ramananjaona and MacGregor 2010).This means the first or second derivatives of the resistivity dis-tribution are minimized, resulting in smooth distributions ofthe resistivity. These types of solution allegedly satisfy Oc-cam’s razor since they are claimed to produce the most ‘sim-plistic’ models for the data. Unfortunately, this approach hasled to a tendency to deliver a single resistivity model as ‘the’ so-lution. Moreover, these classes of models are the least relevantto economic geology, which is anything but smooth. When theresistivity distribution is discontinuous, smooth stabilizers canalso produce spurious oscillations and artefacts. Therefore, amodel that is smooth in the first or second derivative of theanomalous resistivity distribution is not any ‘simpler’ than,say, a model that has the minimum volume of the anoma-lous resistivity distribution. As we will show below, the useof focusing stabilizers allows us to recover stable and geologi-cally realistic models with sharper geoelectric boundaries andcontrasts (Fig. 1 ).

In this paper, we demonstrate that the best practice is torun multiple 3D inversion scenarios with differing regulariza-tion in order to enable interpreters to explore alternative 3D

Figure 1 A smooth stabilizer will recover a model with a smoothdistribution of model parameters. A focusing stabilizer will recover amodel closer to the true model, with sharper boundaries and highercontrasts.

resistivity models and to select the most geologically plausibleones for further analysis. Such practice identifies any artefactsthat may arise from interpreting a single resistivity model.Alternative models may be used to reveal which additionaldata, if any, are needed to further constrain the interpreta-tion. To this end, it is important to develop rigorous but fast3D inversion methods. The requirement for ‘fast’ 3D inver-sion makes the various stochastic methods computationallyprohibitive. Rigorous inversion methods are also not practi-cal, as the sensitivity matrix needs to be constructed at eachiteration for the many transmitter-receiver combinations in aCSEM survey. To minimize computation costs, various incar-nations of the Born approximation that linearize the adjointoperators are often used (e.g., Gribenko and Zhdanov 2007;Støren et al. 2008). However, for nonlinear inverse problemssuch approximations can result in inaccurate calculations ofthe gradient directions, model updates and thus final models.Our more pragmatic approach to 3D CSEM inversion is basedon iterative electromagnetic migration. Iterative migration isimplemented in a reweighted regularized conjugate gradientmethod to rigorously compute the gradient directions with-out needing to explicitly construct the sensitivity matrix orits products (Zhdanov 2002). Modelling is based on the 3Dintegral equation method with inhomogeneous backgroundconductivity that can capture arbitrary geological complex-ity. We have implemented our iterative migration method infully parallelized software that allows us to invert entire 3DCSEM surveys for models with millions of cells. Our approachmakes it practical to run multiple inversion scenarios as de-scribed above and as we shall demonstrate in our Shtokmancase study.

C© 2011 European Association of Geoscientists & Engineers, Geophysical Prospecting, 1–13

Iterative electromagnetic migration for 3D inversion 3

ELECTROMAGN E T I C M I GR A T I ON

The physical principles of electromagnetic migration parallelthose underlying optical holography and seismic migration;i.e., the recorded fields scattered by an object form a holo-gram, which one can subsequently use to reconstruct an imageof the object by ‘illuminating’ the hologram with a referencefield (Zhdanov 1988). In traditional formulation of migrationimaging (e.g., Zhdanov and Cuma 2009), in order to producethe migration field, we replace a set of the receivers with aset of auxiliary transmitters located in the receiver’s positions.These transmitters generate an electromagnetic (EM) field,which is called a backscattering or migration field, Em. Thevector cross-power spectrum of the background field, Eb andthe migration field, Em, produces a numerical reconstructionof a volume image of the conductivity distribution (Zhdanov2002, 2009). In the case of the marine CSEM method, it isconvenient to use a reciprocity principle both for forwardmodelling and imaging/inversion of the marine CSEM data.In the framework of this approach, the multiple computationsof the EM field in the sea-bottom receiver, generated by mul-tiple transmitter positions, is substituted by one calculationof the EM fields in the locations of the true transmitters dueto one reciprocal source located in the true receiver position.Taking into account this ‘reciprocal’ application of the reci-procity method, the principles of marine CSEM migration canbe summarized as follows (Zhdanov 2009):

1 We ‘illuminate’ the background media by a reciprocal elec-tric dipole (in the case of electric observations) located inthe positions of the actual receivers to generate the ‘electricmode’ background EM fields, EbE and HbE. Alternatively,we ‘illuminate’ the background media by a reciprocal mag-netic dipole (in the case of the magnetic field observations)located in the positions of the actual receivers to generatethe ‘magnetic modes’ background EM fields, EbH and HbH.

2 We ‘illuminate’ the background media by artificial trans-mitters located in the positions of the true transmittersand represented by equivalent (fictitious) electric currentdipoles. In the case of electric observations, the current mo-ments are determined by the complex conjugate anomalouselectric field observed in the true receiver for a given trans-mitter position. The electromagnetic field produced by thissystem of artificial electric dipoles generates the ‘electricmode’ migration anomalous fields, EmE and HmE. In thecase of magnetic observations, the current moments are de-termined by the complex conjugate anomalous electric fieldmultiplied by the factor −iωμ. The electromagnetic fieldproduced by this system of artificial transmitters generates

the ‘magnetic mode’ migration anomalous fields, EmH andHmH.

3 In the case of electric field observations, the conductivitymodel of the subsurface lE

0 is formed by summation of thecross-power spectrum of the ‘electric mode’ backgroundand migration fields:

lE0 = Re

∑ω

EbE · EmE, (1)

where summation is done over all frequencies of therecorded fields.

4 In the case of magnetic field observations, the conductivitymodel of the subsurface lH

0 is formed by calculating thecross-power spectrum of the ‘magnetic mode’ backgroundand migration fields:

lH0 = Re

∑ω

EbH · EmH. (2)

5 In the case of joint migration of electric and magnetic ob-served data, the conductivity model of the subsurface lEH

0 isformed by summation of the ‘electric mode’ and ‘magneticmode’ images

lEH0 = Re

∑ω

[EbE · EmE + EbH · EmH

]. (3)

Note that, in the case of multireceiver observations, thefinal image is produced by summation of all the migrationfields generated for each receiver. The conductivity modelsgenerated by equations (1)–(3) are usually slightly distorteddue to the different spatial sensitivities of the observed datain the model. To account for different sensitivities of the datato the conductivity distribution, we use an additional modelweighting function, Wm:

σ ≈ −k(W∗mW)−1l0, (4)

where l0 stands for any of the migration images introducedabove and k is some scaling coefficient. The model param-eter weighting matrix Wm is computed using the integratedsensitivity S as follows:

Wm =√

S, (5)

where the integrated sensitivity is determined from:

S = diag√

F∗F. (6)

In this last equation, F is the Frechet matrix of the for-ward modeling operator. By weighting the migration imagelEH0 with the integrated sensitivity, we make sure that the ob-

served data are equally sensitive to the conductivity variationswithin every part of the domain of investigation.



Electromagnetic migration for controlled-sourceelectromagnetic methods

Let us consider a typical CSEM survey consisting of a set ofelectric and magnetic field receivers located on the seafloorand an electric bipole transmitter towed at some elevationabove the seafloor. We assume that the electrical conductiv-ity in the model can be represented as the sum of the back-ground conductivity σb and an anomalous conductivity �σ

distributed within some local inhomogeneous domain D as-sociated with the reservoir structures.

The background conductivity is formed by a horizontallylayered model that consists of nonconductive air and a layeredsubsurface including the conductive sea column. The receiversare located at the points r j , where j = 1, 2, 3, . . . , J, in a Carte-sian coordinate system. Every receiver Rj records electric andmagnetic field components generated by an electric bipoletransmitter moving above the receivers. We denote these ob-served fields as Ei (r j ) and Hi (r j ) where i is the index of thecorresponding transmitter Ti located at the point ri where i =1, 2, 3, . . . , I.

Let us consider the data observed by one receiver, Rj. Wealso introduce two auxilary electric current dipoles, q and p.According to reciprocity, the electric field component excitedat r j in the direction of p by an electric current element q at ri

is identical to the electric field component excited at ri in thedirection of q by an electric current element p at r j (Zhdanov2002, p. 226):

Ei (r j ) · p = EEj (ri ) · q. (7)

Similarly, the magnetic field component excited at r j in thedirection of p by an electric current element q at ri is equalto the electric field component (multiplied by the minus sign)excited at ri in the direction of q by a magnetic current elementp at r j :

Hi (r j ) · p = −EHj (ri ) · q. (8)

Therefore, one can substitute a reciprocal survey configu-ration for the original survey, assuming that we have electricTE

j and magnetic THj dipole transmitters located in the posi-

tions of the receivers, Rj and a set of receivers measuring thereciprocal electric fields EE

j (ri ) and EHj (ri ) in the positions of

the original transmitters, Ti.We can calculate the migration field for the data collected

by one fixed seafloor receiver, Rj. Consider, for example, thereciprocal electric fields EE

j (ri ). These fields can be representedas the sum of the background (b) and anomalous (a) parts:

EEj (ri ) = Eb

j (ri ) + Eaj (ri ), (9)

where the background electric field Ebj (ri ) is generated by the

electric dipole transmitter TEj in a model with a background

conductivity σ b(z). The residual fields REj (ri ) are equal to thedifference between the background and ‘observed’ reciprocalfield:

REj (ri ) = Ebj (ri ) − EE

j (ri ) = −EEaj (ri ). (10)

According to the definition by Zhdanov (2002, 2009), themigrated residual field is a field generated in the backgroundmedium by a combination of all electric dipole transmitterslocated at points ri with current moments determined by thecomplex conjugate residual field R∗

Ej(ri ) according to the fol-

lowing:

Emj (r) = Em

j (r; R∗Ej

) =I∑

i=1

GE(r, ri ) · R∗Ej

(ri ), (11)

where the lower subscript j shows that we migrate the fieldobserved by the receiver Rj and GE(r, ri ) is the electric Green’stensor for the background conductivity model, σ b. Therefore,the migration field can be computed as a superposition of1D responses weighted by the corresponding receiver residualand generated by electric dipoles with unit moments locatedat every transmitter position in the model with backgroundconductivity σ b. This 1D electric dipole modelling is a veryfast process, which results in a fast migration algorithm.

Equation (11) allows us to reconstruct the migration fieldeverywhere in the medium under investigation. It can beshown that this transformation is stable with respect to thenoise in the observed data (Zhdanov 2009). At the same time,the spatial distribution of the migration field is closely relatedto the conductivity distribution in the medium. However, oneneeds to apply imaging conditions to enhance the conductivitymodel produced by migration. We will discuss this problemin a later section.

In the general case of multiple receivers, the migration fieldis generated in the background medium by all electric dipoletransmitters located above all receivers, Rj, having currentmoments determined by the complex conjugate residual field,R∗

Ej(ri ):

Em(r) =J∑

j=1

I∑i=1

GE(r, ri ) · R∗Ej

(ri ). (12)

According to equation (11), we have:

Em(r) =J∑

j=1

Emj (r). (13)



Therefore, the total migration field for all receivers can beobtained by summation of the corresponding migration fieldcomputed for every individual receiver. A remarkable fact isthat the migration of both electric and magnetic field data isactually reduced to the same forward problem for the electricfield generated by the electric dipole transmitters. The onlydifference is that, in the case of the electric field receivers,we use the electric observed data to determine the electriccurrent moment in the reciprocal transmitters. In the case ofthe magnetic receivers, the observed magnetic data are usedto determine the electric current moments of the receivers.

Regularized iterative migration

It can be demonstrated that migration can be treated as thefirst iteration in the solution of an electromagnetic inverseproblem (Zhdanov 2009). Obviously, we can obtain betterimages if we repeat migration iteratively. Following Zhdanov,Cuma and Ueda (2010), we can describe the method of iter-ative migration as follows: on every iteration, we calculatethe predicted electromagnetic response En for the given con-ductivity model σ n, obtained from the previous iteration. Wethen calculate the residual field between this response and theobserved data, Rn

E, and then we migrate the residual field. Theupdated conductivity model is obtained, according to equa-tion (1), as a sum over the frequencies of the dot productof the migrated residual field and the predicted electromag-netic response En. This conductivity model is then correctedby the integrated sensitivity S to produce the new conductivitymodel σ n on the basis of equation (4). The iterative migrationis terminated when the residual field becomes smaller than therequired accuracy level of the data fitting. In fact, the iterativemigration results in rigorous inversion.

It has been demonstrated that migration provides an al-ternative method for evaluation of adjoint operators. Whenapplied iteratively, migration is analogous to inversion by pro-viding a rigorous solution to the corresponding inverse prob-lem (Zhdanov 2002, 2009).

Note that every iteration of the migration algorithm re-quires two forward modeling computations: one to computethe migration field and another to compute the predicted datain the receivers. In this work, we use a recently developedmigration code that is parallelized over the z dimension ofthe migration domain. For calculation of the migration andpredicted fields, we use the integral equation method withinhomogeneous background conductivity. This enables us toconsiderably reduce computation time and also model largerproblems by increasing the migration domain size and the

number of the cells used for the migration domain discretiza-tion.

CHOOSING A STABIL IZ ING FUNCTIONAL

Another advantage of iterative migration is based on the factthat it allows us to include an a priori model of the target inthe iterative process in a way similar to the case of conven-tional inversion. The details of this technique can be found inZhdanov (2002).

Regardless of the iterative scheme used, all our regularizedinversions seek to minimize the Tikhonov parametric func-tional, Pα(m):

Pα(m) = φ(m) + αs(m) → min, (14)

where φ(m) is a misfit functional of the observed and pre-dicted data, s(m) is a stabilizing functional and α is the reg-ularization parameter that balances (or biases) the misfit andstabilizing functional (Zhdanov 2002). The stabilizing func-tional incorporates information about the class of models usedin the inversion. The choice of stabilizing functional shouldbe based on the user’s geological knowledge and prejudice.Using an inappropriate type of stabilizer is akin to lookingfor an inappropriate solution. In this section, we will brieflydescribe different smooth and focusing stabilizers in order todemonstrate the results from the iterative migration of thesame CSEM data produced by each.

A minimum norm (MN) stabilizer will seek to minimize thenorm of the difference between the current model and an apriori model:

sMN(m) =∫

V(m − mapr )2dv, (15)

and usually produces a relatively smooth model for a differ-ence (m − mapr). The Occam (OC) stabilizer implicitly in-troduces smoothness with the first derivatives of the modelparameters:

sOC(m) =∫

V(∇m − ∇mapr )2dv, (16)

and produces smooth resistivity models that bear little re-semblance to economic geology. Moreover, it can result inspurious oscillations and artefacts when the resistivity is dis-continuous.

Alternatively, the use of focusing stabilizers makes it possi-ble to recover models with sharper geoelectric boundaries andcontrasts. We briefly describe this recently introduced fam-ily of stabilizers here and refer the reader to Zhdanov (2002,



2009) and Zhdanov, Gribenko and Cuma (2008) for furtherdetails.

First, we present the minimum support (MS) stabilizer:

sMS(m) =∫

V

(m − mapr )2

(m − mapr )2 + e2dv, (17)

where e is a focusing parameter introduced to avoid singularitywhen m = mapr. The minimum support stabilizer minimizesthe volume with nonzero departures from the a priori model.Thus, a smooth distribution of all model parameters witha small deviation from the a priori model is penalized. Afocused distribution of the model parameters is penalized less.Similarly, we present the minimum vertical support (MVS)stabilizer:

sMVS(m) =∫

V

(m − mapr )2∫S(m − mapr )2ds + e2

dv, (18)

where S is a horizontal section from the inversion domain (Zh-danov et al. 2008). This minimizes the thickness of the volumewith non-zero departures from the a priori model. The MVSstabilizer is specifically designed to invert thin subhorizontalstructures, such as hydrocarbon-bearing reservoirs.

Finally, we present the minimum gradient support (MGS)stabilizer:

sMGS(m) =∫

V

∇(m − mapr ) · ∇(m − mapr )∇(m − mapr ) · ∇(m − mapr ) + e2

dv, (19)

which minimizes the volume of model parameters with non-zero gradient.

Note that, the focusing parameter e controls the degree offocusing the inverse images. The smaller the focusing param-eter e, the sharper the contrasts of conductivities in the in-verse images. However, parameter e should not be too small,because it could result in singularities of the expressions forfocusing stabilizers when m = mapr. At the same time it shouldnot be too large, because in this case the image will not be fo-cused. Thus, the problem of selecting the focusing parametere is very similar to the problem of choosing the regularizationparameter alpha. Zhdanov and Tolstaya (2004) introduceda method to estimate the optimum value of e, similar to theL-curve method for selection of the regularization parameterα (Hansen 1998).

Figure 2 Observed (red) and background (blue) fields for the in-line electric and transverse magnetic field components at a receiver close to thecentre of the Shtokman model.



C A S E S T U D Y – S H T O K M A N

The Shtokman gasfield lies in the centre of the Russian sec-tor of the Barents Sea, approximately 500 km north of theKola Peninsula. It is currently operated by a joint venture be-tween Gazprom, Total and StatoilHydro. Shtokman is one

of the world’s largest known natural gasfields, with reservesof 3.8 tcm of gas and 37 mln t of gas condensate (Gazprom2009). The water depth gently varies between 320–340 mover the field. The overburden sequence contains Jurassic andCretaceous siliciclastics of shallow marine origin. From ap-proximately 1800 m depth, the Shtokman reservoir sequence

Figure 3 Vertical cross-section of the Shtokman resistivity model obtained from the iterative migration of in-line electric field data with ahomogeneous half-space for a) the true resistivity, b) Occam inversion, c) minimum norm, d) minimum support, e) minimum vertical supportand f) minimum gradient support. All models fit the observed data within 5.5%.



consists of four Middle and Upper Jurassic sandstone hori-zons. The gas is trapped in an anticlinal four-way dip struc-ture that is faulted in the crest. The reservoir horizons varyfrom 10–80 m in thickness. The porosity is between 15–20%,and permeability ranges from hundreds of millidarcies to overa darcy (Zakharov and Yunov 1994).

We constructed a 3D geoelectric model of the Shtokmanfield from available geological and geophysical information.This model was used to simulate a multifrequency 3D CSEMsurvey at 0.25 Hz, 0.5 Hz and 0.75 Hz using the 3D integralequation method. The survey consisted of 345 receiver posi-tions distributed over a 2 km × 2 km grid draped over theseafloor. The transmitter was towed 50 m above the seaflooralong 50 km long lines that were spaced 2 km apart. Forthe entire survey, we prepared different combinations of themultifrequency data for migration: in-line electric field only,in-line electric and transverse magnetic fields and in-line andvertical electric and transverse magnetic fields. No noise wasadded to any of the data, so we could effectively compare theperformance of each stabilizer.

As noted in the introduction, the anomalous fields are gen-erally very weak compared to the background fields in theCSEM measurements. Figure 2 shows the observed and back-ground fields for a receiver near the centre of the Shtokmanfield. The observed field, which includes the anomaly contri-bution, shows only a small departure from the backgroundsignature at offsets under 10 km for the Ex and Hy compo-nents. For the Ez component the difference is larger but themagnitude of the amplitude is much smaller.

A number of iterative migration scenarios were considered.In each, the model domain was 44 km × 40 km × 3 km ineasting and northing and at depth. The migration was doneon a grid with a cell size of 200 m by 200 m in the hor-izontal directions and with a relatively fine vertical size of20 m. Thus, the vertical cell size of the migration grid wassmaller than the 50–100 m thicknesses of the reservoir layers.The data sets corresponding to each data combination werethen migrated with different stabilizers: Occam (OC), mini-mum norm (MN), minimum support (MS), minimum verticalsupport (MVS), and minimum gradient support (MGS). Thefocusing parameter was equal to 1e-12 for all the stabilizers.For the purpose of benchmarking performance, all scenarioswere run to a common misfit of 5.5%. In the first part ofour study, all scenarios commenced with no a priori modelsso as to not bias the effectiveness of any stabilizer. With noa priori model, we do not expect to be able to resolve thestacked reservoir units of the Shtokman gasfield. What we doexpect, however, is to recover a feature with a general shapeand a conductivity-thickness product that is comparable tothe stacked reservoir units. This is a well-known limitation ofthe CSEM method’s resolution.

Figures 3, 5 and 6 present the results for the different it-erative migration scenarios at their final iterations. Thoughthe actual resistivity models are 3D, we show only commonvertical cross-sections through each model for ease of visualinspection of model quality. Panel (b) of Fig. 3 shows that iter-ative migration of the Ex component only with the Occam sta-bilizer converged to produce a very smooth resistivity model,

Figure 4 Convergence of misfit for inversion of in-line electric field data with minimum vertical support (MVS), minimum gradient support(MGS), minimum support (MS), Occam (OC) and minimum norm (MN) stabilizers.



one that bears the least resemblance to the actual resistivitymodel as shown in panel (a) of the same figure. As shownin panel (c) of the same figure, iterative migration with theminimum norm stabilizer also produced a smooth resistivitymodel. Models with sharper geoelectric boundaries and con-trasts were obtained using the family of focusing stabilizers, as

shown in panels (d)–(f). Migration with the minimum support,minimum vertical support and minimum gradient support sta-bilizers produced compact resistivity models. These resistivitymodels bear the most geological relevance to the actual geol-ogy, as they recovered the anticlinal trends of the Shtokmanreservoir units. As expected, the minimum vertical support

Figure 5 Vertical cross-section of the Shtokman resistivity model obtained from the iterative migration of in-line electric field and transversemagnetic field data with a homogeneous half-space for a) the true resistivity, b) Occam inversion, c) minimum norm, d) minimum support, e)minimum vertical support and f) minimum gradient support. All models fit the observed data within 5.5%.



recovered the thinnest resistivity model. The minimum gradi-ent support spreads the anomaly out somewhat in an effort toaccount for the multiple stacked true anomalies. As shown inFigs 5 and 6, addition of the Hy and Ez components improvesthe Occam and minimum norm results but they still produceoverly smooth models.

We have compared the convergence of the inversion mis-fit, which we define as the norm of difference between thenormalized observed and predicted data (Fig. 4). This resultis representative of the other scenarios whereby all stabilizersexhibited near-quadratic convergence. As shown in previousfigures, our results show that there is noticeable improvement

Figure 6 Vertical cross-section of the Shtokman resistivity model obtained from the iterative migration of in-line and vertical electric field andtransverse magnetic field data with a homogeneous half-space for a) the true resistivity, b) Occam inversion, c) minimum norm, d) minimumsupport, e) minimum vertical support and f) minimum gradient support. All models fit the observed data within 5.5%.



in the quality of the recovered resistivity models as the trans-verse magnetic and then vertical electric fields are added to theCSEM data prepared for migration. It follows that, as the in-dustry moves towards acquiring 3D surveys with the intent ofdefining 3D structure, the ability to invert all components of

data along multiple lines for 3D resistivity models will proveto be essential.

In the second part of our study, we evaluate the use ofan inhomogeneous background resistivity model rather thana homogeneous half-space (Zhdanov and Cuma 2009). We

Figure 7 Vertical cross-section of the Shtokman resistivity model obtained from the iterative migration of in-line and vertical electric field andtransverse magnetic field data with an inhomogenous a priori model for a) the true resistivity, b) inhomogeneous a priori resistivity, c) Occaminversion, d) minimum support, e) minimum vertical support and f) minimum gradient support. All models fit the observed data within 5.5%.



constructed an inhomogeneous background resistivity modelfrom the original resistivity model but removed the reservoirsso as to present the anticlinal structures. Again, we ran it-erative migration for all types of stabilizers and data combi-nations. Figure 7 shows results for iterative migration of theEx, Hy, and Ez data. Interestingly, the stacked reservoir unitsemerge from the smooth stabilizers but are far better definedwith the focusing stabilizers. In Fig. 8, we compare modelsobtained using the minimum vertical support stabilizer fordifferent data combinations. It is remarkable that it is possi-ble to recover stacked resisters in these images that correlateto the stacked reservoir horizons.

CONCLUSION

3D inversion of CSEM data is inherently nonunique; multiplemodels will satisfy the same observed data. Multiple inversion

scenarios must be investigated in order to explore different apriori models, data combinations and stabilizers, all with theintention of minimizing subsurface uncertainty. For such prac-ticality, it is important to use rigorous but fast 3D inversionmethods. Our approach to this is based on iterative migra-tion, theoretically equivalent to but more efficient than iter-ative inversion. As we have demonstrated with our syntheticexample for the Shtokman field, we are able to effectivelyinvert multicomponent, multifrequency and multiline CSEMsurveys for models with millions of cells. This makes it prac-tical to run multiple scenarios in order to build confidence inthe robustness of features in the resistivity models, as well asto discriminate any artefacts that may arise from the interpre-tation of a single resistivity model. We have shown that re-liance on regularization with smooth stabilizers may produceresistivity models that bear little resemblance to petroleum ge-ology. We have shown that focusing stabilizers recover more

Figure 8 Vertical cross-section of the Shtokman resistivity model obtained from the iterative migration using the minimum vertical supportstabilizer with an inhomogeneous a priori model for a) the true resistivity, b) inversion of in-line electric field data, c) inversion of in-line electricfield and transverse magnetic field data and d) inversion of in-line and vertical electric field and transverse magnetic field data. All models fitthe observed data within 5.5%.



realistic resistivity models with sharper geoelectric contrastsand converge to lower misfits in fewer iterations. Finally, usinga known non-horizontally layered structure as an inhomoge-neous background resistivity a priori model to the iterative mi-gration can improve model fidelity and recover stacked reser-voir units. In order to extract the most reliable informationfrom geophysical data, multiple inversion scenarios should beapplied and the inversions should be run both with differ-ent a priori models and with different regularization param-eters. The final selection of the most geologically meaningfulmodel should be based on integrated analysis/interpretationof all available geological and/or geophysical information inthe area of investigation.

ACKNOWLEDGE ME N T S

The authors acknowledge TechnoImaging for support of thisresearch and permission to publish. The authors acknowl-edge the Russian Research Centre Kurchatov Institute for as-sistance in preparation of the Shtokman geoelectric model.M. Zhdanov, M. Cuma and A. Gribenko also acknowledgethe support of the University of Utah’s Consortium for Elec-tromagnetic Modeling and Inversion (CEMI) and Center forHigh Performance Computing (CHPC).

REFERENCES

Commer M. and Newman G.A. 2008. New advances in controlledsource electromagnetic inversion. Geophysical Journal Interna-tional 172, 513–535. doi: 10.1111/j.1365-246X.2007.03663.x

Gazprom. 2009. http://old.gazprom.ru/eng/articles/article21712.shtml (accessed 19 December 2009).

Gribenko A. and Zhdanov M.S. 2007. Rigorous 3D inversion of ma-rine CSEM data based on the integral equation method. Geophysics72, WA73–WA84. doi: 10.1190/1.2435712

Hansen C. 1998. Rank-deficient and Discrete Ill-posed Problems: Nu-

merical Aspects of Linear Inversion. SIAM Monographs on Math-ematical Modeling and Computation, SIAM.

Hesthammer J., Stefatos A., Boulaenko M., Fanavoll S. and DanielsenJ. 2010. CSEM performance in light of well results. The LeadingEdge 29, 34–41. doi: 10.1190/1.3284051

Mackie R., Watts M.D. and Rodi W. 2007. Joint 3D inversion ofCSEM and MT data. 77th SEG meeting, San Antonio, Texas, USA,Expanded Abstracts, 574. doi:10.1190/1.2792486

Ramananjaona C. and MacGregor L.M. 2010. 2.5D anisotropic in-version of marine CSEM survey data – an example from WestAfrica. 72nd EAGE meeting, Rome, Italy, Expanded Abstracts.

Støren T., Zach J.J. and Maaø F.A. 2008. Gradient calculations for 3Dinversion of CSEM data using a fast finite-difference time-domainmodeling code. 72nd EAGE meeting, Rome, Italy, Expanded Ab-stracts.

Tikhonov A.N. and Arsenin Y.V. 1977. Solution of Ill-posed Prob-lems. Winston.

Zakharov Y.V. and Yunov A.Y. 1994. Direction of exploration forhydrocarbons in Jurassic sediments on the Russian shelf of theBarents Sea. Geologiya Nefti i Gaza 2, 13–15.

Zhdanov M.S. 1988. Integral Transforms in Geophysics. SpringerVerlag.

Zhdanov M.S. 2002. Geophysical Inverse Theory and RegularizationProblems. Elsevier.

Zhdanov M.S. 2009. Geophysical Electromagnetic Theory and Meth-ods. Elsevier.

Zhdanov M.S. and Cuma M. 2009. Electromagnetic migrationof marine CSEM data in areas with rough bathymetry. 79th

SEG meeting, Houston, Texas, USA, Expanded Abstracts, 684.doi:10.1190/1.3255847

Zhdanov M.S., Gribenko A. and Cuma M. 2008. Regularized focus-ing inversion of marine CSEM data using minimum vertical supportstabilizer. 78th SEG meeting, Las Vegas, Nevada, USA, ExpandedAbstracts, 597. doi:10.1190/1.2792487

Zhdanov M.S., Cuma M. and Ueda T. 2010. 3D electromagneticholographic imaging in active monitoring of sea-bottom geoelectricstructures. In: Active Geophysical Monitoring (eds J. Kasahara, V.Korneev and M.S. Zhdanov), pp. 317–342. Elsevier.

Zhdanov M.S. and Tolstaya E. 2004. Minimum support nonlinear pa-rameterization in the solution of 3D magnetotelluric inverse prob-lem. Inverse Problems 20, 937–952.


Documents

Iterative electromagnetic migration for 3D inversion of ... · Geophysical Prospecting doi: 10.1111/j.1365-2478.2011.00991.x Iterative electromagnetic migration for 3D inversion of