CRUSTAL DEFORMATION MAPPED BY

COMBINED GPS AND INSAR

Sverrir Guðmundsson

LYNGBY 2000 SUBMITTED FOR THE DEGREE OF M.Sc. E.

IMM

I

Preface

This thesis was made in collaboration between the Institute for Mathematical Modelling at the Technical University in Denmark (IMM at DTU) and the Nordic Volcanological Institute in Iceland (NORDVULK), and is a partial fulfilment of the requirements for my degree of M.Sc. in engineering. Various image analysis methods are used to combine two complimentary geodetic observation techniques to map the earth surface movements. The work was funded and data was supplied by NORDVULK. Expertise in InSAR and GPS observation techniques was provided by NORDVULK, and expertise in image analyses by IMM at DTU. Supervisors were Jens Michael Carstensen from IMM at DTU and Freysteinn Sigmundsson from NORDVULK. Lyngby, March 1 2000 ______________________ Sverrir Guðmundsson

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Abstract In this thesis, an attempt is made to extract the maximum amount of information from two complementary geodetic technique and to infer high resolution maps of three-dimensional ground movements. The first technique uses Interferometric analysis of Synthetic Aperture Radar (InSAR), acquired from a satellite Synthetic Aperture Radar (SAR). An interferogram is formed by combining two synthesised SAR images acquired at different times (from Master and Slave tracks), and is in its simplest form a record of a phase difference between two signals. The interferograms contains a modulated measure of the one-dimensional change in range from the ground to the satellite, with a typical resolution of 20x20 m2. The other technique uses NAVigation Satellite Timing And Ranging Global Positioning System (NAVSTAR GPS) for geodetic observations. A GPS network of ground control points, typically spaced few km or tens of km apart, is created for the purpose of measuring ground movements. Surface movements are detected as a change in range of the ground control points, between two or more sets of observations. The GPS geodetic observations can be used to provide three-dimensional measurements of ground displacements at sparse locations, with accuracy within 1 cm. Test data from the Reykjanes Peninsula, Iceland is used for experimentation. Available are both GPS and interferometric observations, recorded from a descending satellite pass, with various elapsed time intervals within the period 1992 to 1998. The ground movements at the Reykjanes Peninsula consist of both crustal deformation and plate movements. A test image that describes a surface iceflow at an axisymmetrical glacier cauldron, is created for experimental purposes. The image includes no errors and all deformation components are known, which gives an opportunity to obtain estimation errors. Both InSAR and GPS observations may include several error factors. Some error factors need to be reduced before combining the two complementary geodetic techniques. The activity of the atmosphere generates noise errors in the measurements. A vectorized filtering is efficiently used for a noise reduction of modulated (wrapped) interferograms. The Master and Slave tracks of an interferogram include two different viewpoints and distances to the same object. This results in a systematic error that can be described with a phase plane. The GPS observations are not expected to include significant systematic errors, and are therefore used to eliminate a phase plane from InSAR images. The modulated effects of the interferometric observations are removed (unwrapped) before the three-dimensional motion maps are constructed. A methodology that uses Markov Random Field (MRF) based regularisation and simulating annealing optimisation is used to unwrap InSAR images. The unwrapping process utilises the relationship of the interferometric and GPS observations, both in the MRF modelling and for initialisation. The MRF regularisation also uses an assumption about surface smoothness. For the purpose of initialising the process, virtual InSAR images are created by ordinary kriging of GPS observations.

III

The error corrected and unwrapped interferograms are used along with sparsely located GPS measurements to infer high resolution motions maps of the three-dimensional ground motion. The problem of inferring the three-dimensional motion field is separated into two two-dimensional problems. MRF based regularisation and simulating annealing optimisation is used for the construction of high resolution two-dimensional motion maps from combined GPS and interferometric observations. The MRF model utilises the relationship of the two-dimensional motion fields to both the interferometric and GPS observations. An additional constraint is an assumption about surface smoothness of the motion field images. The process is initialised by motion field images created by ordinary kriging of GPS observations. Very promising results are achieved when the methods are applied to both the data from the Reykjanes Peninsula and the test image.

Table of contents 1. INTRODUCTION ......................................................................................................................... 3

2. GROUND MOVEMENT OBSERVATIONS............................................................................... 4

2.1. INTERFEROMETRIC OBSERVATIONS............................................................................................ 4 2.2. GPS OBSERVATIONS ................................................................................................................. 7 2.3. INSAR IMAGES AND GPS MEASUREMENTS FROM THE REYKJANES PENINSULA, ICELAND ............ 7

2.3.1. InSAR images ............................................................................................................... 8 2.3.2. GPS measurements ...................................................................................................... 8 2.3.3. Accounting for differences in elapsed time intervals................................................. 10

3. TEST IMAGE............................................................................................................................. 11

5. NOISE REDUCTION................................................................................................................. 14

6. EXTRACTING REGION OF INTEREST ................................................................................. 15

7. GPS TILTING OF WRAPPED INSAR IMAGES..................................................................... 16

7.1. ELIMINATING A PHASE PLANE FROM WRAPPED INSAR IMAGES ................................................. 16 7.2. LEAST SQUARE ESTIMATION OF COEFFICIENTS ......................................................................... 17 7.3. UNWRAPPING LINE PROFILES................................................................................................... 18

7.3.1 Some error effects ........................................................................................................ 18 7.4. THE TILTING ALGORITHM ........................................................................................................ 20

8. MOTION FIELD IMAGES CREATED FROM A SPARSELY LOCATED GPS OBSERVATIONS .......................................................................................................................... 24

8.1. ORDINARY KRIGING ................................................................................................................ 24 8.2. SEMIVARIOGRAM ................................................................................................................... 25 8.3. SEMIVARIOGRAM MODEL ........................................................................................................ 25 8.4. KRIGING RESULTS................................................................................................................... 26 8.5. ESTIMATION OF UNCERTAINTY ................................................................................................ 28 8.6. COMPARISON OF INTERPOLATION METHODS............................................................................. 28

9. USING KRIGING OF GPS MEASUREMENTS AND MARKOV RANDOM FIELD REGULARISATION TO UNWRAP INSAR IMAGES ................................................................. 30

9.1. PROBLEM DESCRIPTION........................................................................................................... 30 9.2. ESTIMATION OF THE INITIAL WAVE NUMBERS .......................................................................... 31 9.3. SIMULATING ANNEALING AND MRF REGULARISATION............................................................. 33 9.4. THE SIMULATING ANNEALING UNWRAPPING ALGORITHM AND MRF MODELS ............................ 34

9.4.1. Simulating annealing iteration algorithm .................................................................... 34 9.4.2. Penalizing the second derivative ................................................................................ 35 9.4.3. Detecting area of interest ............................................................................................ 36 9.4.4. Guiding with GPS measurements. ............................................................................. 38

9.5. UNWRAPPING PROCESS ........................................................................................................... 39

10. COMBINATION OF GPS AND INTERFEROMETRIC OBSERVATIONS TO INFER THREE-DIMENSIONAL GROUND MOVEMENTS .................................................................... 44

10.1. PROBLEM DESCRIPTION ......................................................................................................... 44 10.1.1. Simplifying the problem............................................................................................. 44

10.2. OPTIMIZATION OF TWO-DIMENSIONAL DEFORMATION ............................................................ 45 10.2.1. Two-dimensional simulating annealing algorithm ................................................... 45 10.2.2. Energy functions........................................................................................................ 46 10.2.3. Further utilisation of GPS observations ................................................................... 48

10.3. INFERRING THE THREE DIMENSIONAL MOTION FIELD AT THE REYKJANES PENINSULA .............. 51 10.3.1. Corrections of the data observations ....................................................................... 52 10.3.2. Inferred three-dimensional motion maps ................................................................. 52

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11. RESULTS AND DISCUSSIONS............................................................................................ 60

11.1. PRE-PROCESSING METHODS................................................................................................... 60 11.1.1. Noise reduction in interferometric observations ...................................................... 60 11.1.2. Extraction of area of interest..................................................................................... 60 11.1.3. Utilisation of GPS observations to correct wrapped InSAR images ...................... 60 11.1.4. Creation of virtual InSAR images with interpolation of GPS data .......................... 61 11.1.5. Unwrapping process.................................................................................................. 61 11.1.6. Utilisation of GPS observations to correct unwrapped InSAR images .................. 62

11.2. CONSTRUCTION OF THREE-DIMENSIONAL HIGH RESOLUTION MOTION MAPS............................. 63 11.3. PRE-PROCESSING AND AVERAGED MOTION MAPS AT THE REYKJANES PENINSULA ................... 64

12. CONCLUSION ........................................................................................................................ 66

REFERENCES .............................................................................................................................. 67

APPENDIXES................................................................................................................................ 68

A. GPS TILTING OF WRAPPED INSAR IMAGES; PROJECTION INTO THE COMPLEX UNIT CIRCLE ................................................................................................................................ 68

A.1. THE PROCEDURE .................................................................................................................... 68 A.1.1. The first objective function.......................................................................................... 69 A.1.2. The second objective function.................................................................................... 70

A.2. CORRECTION OF THE INSAR DATA ......................................................................................... 71 A.3. REMOVING UNWANTED AREAS FROM THE CORRECTED IMAGES ................................................ 72

B. PROCESSED IMAGES............................................................................................................ 75

C. MATLAB FUNCTIONS ............................................................................................................ 80

C.1. CONTENT LIST ....................................................................................................................... 80 C.2. OPENING............................................................................................................................. 81 C.3. MA....................................................................................................................................... 81 C.4. LOLA2PIX ........................................................................................................................... 82 C.5. MASK .................................................................................................................................. 83 C.6. MK....................................................................................................................................... 83 C.7. PROFILE_TILT.................................................................................................................... 84 C.8. KRIG1D ............................................................................................................................... 85 C.9. PATTERN ............................................................................................................................ 86 C.10. UNWRAP_GPS .................................................................................................................. 87 C.11. UNWRAP_SMOOTHN ...................................................................................................... 88 C.12. UNWRAPPED.................................................................................................................... 90 C.13. TILT_UNWRAP................................................................................................................. 90 C.14. SIMUL_2D......................................................................................................................... 91 C.15. WEIGHTSIMUL_2D .......................................................................................................... 93 C.16. LINE_UNWRAP ................................................................................................................ 94 C.17. WRAP ................................................................................................................................ 94 C.18. FYLKI ................................................................................................................................ 95 C.19. TILT ................................................................................................................................... 96

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1. Introduction The earth surface is in continuos shaping due to various forces. Surface movements consist for example of divergent plate displacements, crustal deformations or glacier iceflow. Knowledge about the earth activities can be of interest for many reasons, both theoretical and economical. Measurements of ground displacements are widely used to get an insight into the earth deformation and to increase the understanding of the nature forces. In this thesis, an attempt is made to extract the maximum amount if information from two complementary geodetic techniques. The first technique uses Interferometric analysis of a satellite born Synthetic Aperture Radar (InSAR). An InSAR image contains a modulated measure of the one-dimensional change in range from the ground to the satellite The other technique uses Global Positioning System (GPS) for geodetic observations, which provides accurate three-dimensional measurements of ground displacements at sparse locations. Various image analysis methods are used to combine GPS and InSAR. Methods and results are presented in the following chapters.

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2. Ground movement observations Many methods are available to measure ground movements. In this thesis, results from two observation techniques are combined. The first method is an interferometric observation that is used to generate high-resolution map of one-dimensional ground movement. The second method is GPS observations used to measure three-dimensional ground movements at sparse locations. The methods are explained briefly in this chapter. Further details can be found about the interferometric technique in [1] and GPS observation in [2].

2.1. Interferometric observations Synthetic Aperture Radar (SAR) images are recorded with radar satellites. In this study a data from the two European Earth Remote Sensing Satellites ERS-1 and ERS-2 is used. The two satellites orbit around the earth along the same track, with one-day difference, and an orbital repetition cycle of 35-day each. Each satellite scans the same area twice per 35-day interval; once from an acceding and once from a descending orbit pass, which gives two different view angles of the same area. ERS radar transmits electromagnetic waves with wavelength of mm,7,56=λ and measures the reflected signal from the Earth. A SAR image is a synthesised image, generated from the earth back-scattered radar signal. It consists of complex pixel values (amplitude and phase), with a typical resolution about 20x20 meters. SAR images can be used to form an interferogram. An interferogram (InSAR image) is generated from the phase difference of two pairs of SAR images acquired from about the same position in space, but at different times (called the Master track and the Slave track). The time interval represented by an InSAR image, generated from ERS-1 and ERS-2, can vary from one day up to several years. Figure 2.1 explains the InSAR geometry where S1 is the Master track and S2 is the Slave track. In its simplest form, an interferogram is the phase difference between the complex Master image (M) and Slave image (S) or the angle of ,*MS where * is the complex conjoint. Here, bold letters will be used to represent matrixes (images) unless other is stated. The interferogram is given as ∠I, where ∠ means the phase or the angle and I is given by the formula [1]

( ) ( ) ( )( ) ( )

.2exp

22

*

∑∑∑

⋅=

SM

GSMI

ff

jff π

G is designed to eliminate topographical and orbital phase errors and the filter f reduces the difference in radar impulse response perceived by each satellite track. The magnitude of (2.1) ranges from 0 to 1 and is called a choerence, and is used to measure the reliability of the measurement. A value of 1 mean that every pixel agreed with the phase within its cell and 0 indicates a meaningless phase.

(2.1)

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One phase shift )2( π in ERS interferogram corresponds to a displacement of =2λ

cm835.2 in the direction from the satellite to the ground, since the wave travels from the satellite to the ground and back again. Displacement of more than one wavelength ,2πφθ n+=∆ where πφ 20 << and 0≠n is an integer, is registered as a phase shift of φ in the interferogram, i.e. the displacement measurement is periodic, or modulated. An ERS interferogram consists therefore of fringes, where each fringe corresponds to a scalar change of cm8.2≈∆ρ in the direction from the satellite to the ground. A periodic or modulo-2π interferograms are called wrapped InSAR images and images that have been corrected for the modulo-2π effect are called unwrapped InSAR images. Figure 2.2 explains the difference of a wrapped and an unwrapped signal. A phase change θ∆ in an interferogram can be due to number of effects, including contribution due to difference in orbital trajectory, topography and several noise factors.

noisetopographyorbitalntdisplaceme θθθθθ ∆+∆+∆+∆=∆ Difference between viewpoints and distances of the Master and Slave orbit to the same ground object can lead to gradual phase change or regularly distributed orbital fringes in the InSAR image. Orbital fringes can be eliminated by subtracting a phase plane from the image. This can be done by using a knowledge of the satellite trajectories S1 and S2 (Figure 2.1). However, the knowledge is usually not accurate to the scale of wavelength, which can leave a few regular fringes uncorrected. Figure 2.3 explains the effect of suppressing the orbital fringes. Topographical fringes can be extracted from interferograms by means of synthetic interferograms, calculated from digital elevation model (DEM) and orbit parameters.

(2.2)

Figure 2.1. Geometry of an interferometric measurement. From [3].

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Figure 2.2. Wrapped and unwrapped signals. The wrapped signal is modulo-2π, where 2π correspond to displacement of ∆ρ ≈ 2.8 cm. Height sensitivity measurements for the interferograms are used to estimate the impact of possible errors due to the topography. This is done by estimating the socalled “altitude of ambiguity” ah [1] (a measure of stereoscopic effects). The altitude of ambiguity represent surface altitude needed to produce one topographical fringe in the interferogram and is a characteristic number for an interferogram. High numbers of ah represent an insensitivity of the interferometric measurement to variations in the surface elevation. Noise can for example be induced from difference in atmosphere conditions and back scattering characteristics of the surface, between the two times of observations. An attempt is made to keep the internal phase contribution constant between the Master and Slave images by acquiring them from similar surface conditions. Extreme cases include water-covered surfaces, which include no stability in the back scattering characteristics. Noise errors induce random speckles in the image, which are measured as a decorrelation or an inchoherence (see (2.1)). Also, large movements can result in an ambiguity between neighbouring pixels that will blur fringes and give a low choerence. If all error effects can be corrected for, then the scalar displacement ρ∆ from the satellite towards the ground can be described mathematically by [1] Figure 2.3. Effect of removing orbital fringes. The image to the left is before correction and the one to the right is after correction. From [1].

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,2

2 su ⋅−=∆=∆ ntdisplacemeθπ

λρ

where u is the three dimensional displacement vector and s is the unit vector pointing from the ground toward the satellite.

2.2. GPS observations Highly precise navigation measurements are done by using the NAVigation Satellite Timing And Ranging Global Positioning System (NAVSTAR GPS), established and operated by the U.S. military [2]. The system consists of 24 GPS satellites, orbiting around the earth at an altitude about 20200 km. The satellites are located on six almost circular orbital planes, each inclined about 55° with respect to equator and with an orbital period around 12 hours. At all time, four to eight satellites are available for navigating, which is enough to give a position in three-dimensional space. GPS navigation systems use accurate measurements of travel times of wave signals to estimate the distance between the satellites and the GPS receivers. The phase differences between transmitted signals and waves generated in the receivers are also used to determine changes in distance from the receivers to the satellites. A GPS network of ground control points has to be created for the purpose of measuring surface movements. Surface movements at each site in the network are determined as a difference in position of a fixed ground control point between two or more times of observations. During an observation, one GPS receiver collects satellite data at a reference point, while the points in the GPS network are continuously measured over some period of time. Therefore, one point observation consists of a time average of regularly sampled measurements. A post-processing the information from the measured signals along with a precise information about the satellite orbits can be used to calculate an accurate location of point in space, at millimetre level relative to the reference station. Points in the GPS network can be created by putting “benchmarks” into the ground. For example, when measuring crustal deformation, a copper bolt is put into a solid rock with only a small piece of the bolt standing out. The position of the benchmark is then measured accurately, by putting a receiver antenna on a tripod located directly above the stick. The tripod is used to put the antenna accurately at the same position above the benchmark, at each time of observation. Uncertainty in GPS measurements consists of both errors affecting the GPS signals, as well as errors resulting from an inaccuracy in antenna positioning over the benchmark.

2.3. InSAR images and GPS measurements from the Reykjanes Peninsula, Iceland Reykjanes Peninsula is located at the SW-part of Iceland. Iceland is located on the mid-Atlantic Ridge, which makes it an ideal place for studying mechanics of divergent plate movement and crustal deformation. The plate boundary between the North-American and the Eurasian plate runs ashore at the SW tip of the Reykjanes Peninsula [2,4]. Figure 2.4 (a) shows the location of fault and eruptive fissures and the outline of central volcanoes cite area. The image shows also an approximate location of the central axis of the plate boundary as inferred from seismicity.

(2.3)

8

2.3.1. InSAR images Seven wrapped InSAR images from the Reykjanes Peninsula were available for this study. The data have been coded to 8-bits. The images are all acquired from a descending satellite passes. Topographical effects have been eliminated by means of a DEM data and known orbital effects removed. The images includes an information about the crustal deformation component in the ground to satellite direction (∆ρ or the Slant-Range-Shift), where each pixel have a resolution of 1/600° in longitude and 1/1200° in latitude (approximately 93x82 m2 area). Fore these interferograms, the descending unit vector s pointing from ground towards the satellite is given as [4]

[ ],V935.0,N095.0,E34.0 −=s where E, N and V means East, North and vertical respectively. Figure 2.4 (b) shows a plot of the unit vector s. The highest contribution to the interferometric signal is from the vertical ground deformation, whereas the north-south movements gives a very low contribution. Table 2.1 shows the tracks of the Master and Slave orbits for all the seven wrapped InSAR images and the time of observation. The time intervals represented by the interferograms vary from one month up to four years. Figure 2.5 shows the images with the highest altitude of ambiguity. The images may include a noise caused by the atmosphere and by change in surface back scatter characteristics between the time of observation of the Master and Slave images, and also an error induced by incomplete correction of the orbital and topographical effects [4]. 2.3.2. GPS measurements GPS measurements of the three-dimensional displacement vector u in (2.3) are available at sparse points at the Reykjanes Peninsula. The same GPS network was first measured 1993 and again 1998 (5 years interval). The locations of the GPS points are shown in Figure 2.6, and also the one-year average of GPS measured displacement field over the elapsed five years interval from 1993 to 1998. A GPS measure of a one local position consists of one-day average of 15 seconds samples. This is done to eliminate random noise errors. Estimated errors for each measured component of the displacement field are also available [2].

(2.4)

Table 2.1. Characteristics of the InSAR images Master orbit Date of

observation Slave orbit Date of

observation Elapsed time

Altitude of ambiguity

ha 5064 04.07. ’92 5565 08.08. ‘92 35 days 35.6 m 5064 04.07. ’92 10575 24.07. ‘93 0.93 years 25.1 m 5064 04.07. ’92 21941 25.09. ‘95 3.22 years 19.6 m 5565 08.08. ‘92 10575 24.07. ‘93 0.83 years 59.0 m 5565 08.08. ‘92 21941 25.09. ‘95 3.12 years 43.6 m 5565 08.08. ‘92 7278 09.10. ‘96 4.17 years 22000 m 10575 24.07. ‘93 21941 25.09. ‘95 2.29 years 166.0 m

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Figure 2.5. Wrapped InSAR images from the Reykjanes Peninsula. The same colorbar applies to all the images.

Figure 2.4. Reykjanes Peninsula. (a): The image shows the locations of faults and eruptive fissures and outline of central volcanoes (circled areas). The thick line indicates the approximate location of central axis of the plate boundary as inferred from seismicity. From [4] (b): Plot of the unit vector s = [0.34, -0.094, 0.935] for the interferogram from the Reykjanes Peninsula

(a) (b)

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applies to all the images.

Figure 2.6. Displacement rate in the period from 1993 to 1998. The GPS locations are shown as ∆. The figure is from [2]. 2.3.3. Accounting for differences in elapsed time intervals The InSAR images and the GPS measurements are observed at different times. InSAR images represent a displacement at various elapsed time intervals within 1992 to 1996, while the GPS measurements represent the deformation during the period from 1993 to 1998. The crustal deformation at the Reykjanes Peninsula can be assumed to be smoothly continuos (no abrupt changes in the surface), since no large earthquakes have been recorded at the Reykjanes Peninsula during the period from 1992 to 1998, [2,4]. The image pairs in Figure 2.5 do though strongly indicate a non-linear variation of the deformation field as a function of time for some parts of the image areas. This has to be considered before the interferometric and GPS data can be combined, see Chapter 9 and 10.

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3. Test image In addition to the image pairs from the Reykjanes Peninsula (Chapter 2), a 164x164 pixel test image was created to experiment on (Figure 3.1). The image was created by using a mass balance equation to describe a surface iceflow at an axisymmetrical glacier cauldron [5]. The image includes no error factors and all the deformation components are known. This image was created for process testing and to obtain an estimation of errors.

Figure 3.1. Axisymmetrical test image, consisting of 164x164 pixels; (a), (b) and (c): the East, North and Vertical deformation components respectively, (d): the unwrapped Slant-Range-Shift =∆ρ ( [ ] [ ]T

VNE V,V,V935,.095.,34. ⋅−− ), (e): plots of line 82 (central line) out of the images in (a), (b), (c), (d).

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4. Angular difference between the InSAR and GPS measured Slant-Range-Shift This chapter present some comparison between the unprocessed GPS and InSAR data. A process of projecting both the GPS and interferometric measurements into the unit complex circle can be used to get a rough comparison of the consistency between the interferometric and GPS measured Slant-Range-Shift ( ρ∆ ). Before reading any further, it should be noted that a bold letter is used to represent a matrix (image) or a vector, while a non-bold indexed letter refer to a single value of the matrix. A single index is used when referring to a pixel number, but two indexes will also be when referring to row and column numbers. For the GPS measurements, a sparse Slant-Range-Shift image can be calculated as

iyr TiiGPS ∀⋅⋅= ),(I su

where i is sparse pixel values corresponding to the GPS locations, u is the three dimensional GPS measured crustal deformation (in cm/yr), s is the unit vector pointing from the ground towards the satellite and yr is the elapsed time represented by the interferogram. The process requires the assumption of having linear deformation with time. GPSI is projected into the complex unit circle with

,,2I2

expC ij iGPS

iGPS ∀

⋅⋅=

λπ

and the wrapped InSAR image values iInSARI are projected into the complex unit circle by

.,2I2

expC ij iInSAR

iInSAR ∀

⋅⋅=

λπ

If both the GPS and interferometric measurements are describing exactly the same then

,,0CC iiGPSiInSAR ∀=∠−∠ It should be noted that this method compares the interferometric and the GPS signals on periodical forms. Therefore, this does not give an idea about the absolute difference. The histogram in Figure 4.1 shows an example of an angular difference between the GPS and interferometric measurements; (a) shows the difference between GPS (1993-1998) and 2.29 years interferogram (1993-1995), and (b) between GPS and 3.12 years interferogram (1992-1995).

(4.2)

(4.3)

(4.4)

(4.1)

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Distributions of the angular difference iGPSiInSAR CC ∠−∠ (Figure 4.1) shows that the error can be large for some of the points or up to 180°. This can be a consequence of several reasons like: 1. noise in the measurements (mainly the InSAR images), 2. GPS measurements may be wrong at some sites, 3. a systematic error in the InSAR images due to insufficient correction for the

orbital and topographical effects, 4. big jumps in the deformation at some areas within the image as a result of small

earthquakes or 5. the assumption of having linear deformation does not hold for some periods or at

some areas within the Reykjanes Peninsula (see Chapter 2). The following can be done to reduce error effects: 1. The wrapped InSAR images can be pre-filtered before processing. (Chapter 5) 2. Interferograms with high altitude of ambiguity can be used to minimise errors due

to insufficient topographical correction (Table 2.1). 3. Insufficient correction for orbital effects should lead to a systematic error that can

be described by a phase plane. One attempt to make the InSAR and GPS data more consistent could be to find some optimal planar correction of the InSAR images by using time scaled GPS measurements (Chapter 7).

Effects due to non-linear crustal deformation could be reduced by selecting interferometric and GPS measurements that represent approximately the same time periods. Consistency between the InSAR images and GPS measurements will be considered further in Chapter 9 and 10. Figure 4.1. Distribution of iGPSiInSAR CC ∠∠ − at the sparse pixels i defined by the GPS locations. The interferometric and GPS measurements are made time consistent by assuming a liner deformation with time. If GPS and InSAR observations are fully consistence these observations should group around 0 and 2π.

(a) (b)

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5. Noise reduction The wrapped InSAR images can include high noise factors. Filtering of the wrapped interferograms can have many practical applications and will be widely used in further processing of the data. The modulate characteristics of the image values need to be considered before filtering. This is handled by projecting the modulated signal into two vectors, perpendicular to each other (cosine- and sinusoidal). The two vectors are then filtered separately (vectorized filtering). The filtered interferogram is given as

),(2

2 CI ∗∠= ff πλ

where f is some linear filter coefficients (e.g. moving average window), ∠ means the angle and C is a complex intensity image given by

⋅⋅=

22expλ

π IC j

with I as the wrapped InSAR image with the modulated (wrapped) intensity interval [ ].2,0 λ Figure 5.1 shows the 4.17 years InSAR images with the Master and Slave tracks 5565 and 7278, respectively, before and after filtering. The image is filtered with a 0.5x0.5 km2 moving average window. The figure shows how efficiently the vectorized filtering can reduce noise speckles. Figure 5.1. The 4.17 years wrapped InSAR image from Reykjanes Peninsula before (a) and after (b) filtering. (c): Profiles from (a) and (b) (the profile location is shown as thick black line on the images in (a) and (b)).

(5.1)

(5.2)

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6. Extracting region of interest The Reykjanes Peninsula is surrounded by an ocean, which is displayed as a no- information in the InSAR images (zero valued pixels). Binomial images that separates information areas (foreground) from no information areas (background) will be used in further processing of the InSAR images. A process that separates the images into a background and a foreground is given as: 1. A binomial image is created by thresholding the original InSAR image, such that

all pixels distinct from zero are assigned the value one (foreground), and the zero valued pixels are kept as zero (background).

2. The foreground is cleaned with a morphological cleaning process; a close operation that consists of dilation followed by erosion.

Further details about morphological cleaning are given in [6,7,8]. Figure 6.1 shows binomial images generated from thresholded InSAR image both before and after the morphological cleaning. Figure 6.1. Binomial images including a foreground and a background; (a): Interferogram, (b): thresholding of the image in (a), (c): The image in (b) after the morphological cleaning.

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7. GPS tilting of wrapped InSAR images Master and Slave orbital trajectories of an InSAR image can have two different viewpoints and distances to the same object. Correction for orbital errors is not accurate to a scale of a wavelength. Hence, the InSAR images can include a residual orbital error that can be described by a tilted plane and an offset (a phase plane). The GPS measurements are on the other hand not expected to have a significant systematic error. An approach to correct residual orbital errors in InSAR images is to use the sparse GPS measurements of change of range from the ground to satellite (Slant-Range-Shift). Two methods have been developed and tested for this purpose. The methods assume the crustal deformation to be linear with time. Furthermore, it is assumed that the only systematic error in the interferometric measurements can be described by a phase plane. The first method a projection of the GPS and interferometric measured Slant-Range-Shift into the complex unit circle, and is given Appendix A. The phase plane is then found by optimisation in the complex domain. This has the drawback of changing a unique solution into periodical solutions. Furthermore, this can also lead to wrong optimal solutions since the minimum difference between each individual GPS measurements and the interferogram becomes also periodic. Experiments have shown that this algorithm can easily result in wrong tilting. A tilting method that use unwrapped interferometric profiles located between points defined by the sparse GPS locations is presented in this chapter. The algorithm uses the unwrapped profiles to estimate a sparse unwrapped values of the interferogram that correspond to the sparse GPS locations, and Least-Square (LS) method to estimate the optimal tilting.

7.1. Eliminating a phase plane from wrapped InSAR images For the GPS measured three-dimensional crustal deformation, a sparse Slant-Range-Shift image is calculated as

),),((),(I TGPS jiyrji su ⋅⋅=

where u is the three dimensional GPS measured rate of displacement (in cm/yr), s is the unit vector pointing from the ground towards the satellite, yr is the elapsed time interval of the interferogram and i,j are the sparse row and column numbers defined by the GPS locations. The relationship between the sparsely located GPS measurements and the corresponding interferometric measurements is then expected to be on the form

[ ] ,,,1,,),(I),(I jijijiji UwInSARGPS ∀⋅+= x where UwInSARI is the unwrapped InSAR image and

[ ]Txxx 321 ,,=x

(7.1)

(7.2)

(7.3)

17

is vector including the coefficient of the two dimensional phase plane. For a known x, tilting of a wrapped interferogram WInSARI can be calculated as

( ) ( ) ( )( ){ } ,,,,C,C2

2,I 2 clclclcl PlanWInSARInSAR ∀∠=π

λ

where l and c are the line and columns numbers of ,WInSARI respectively, ∠ stands for the phase,

,2

2exp

⋅⋅=

λπ WInSAR

WInSARj I

C

and

( ) [ ]( ) .,,2

1,,2exp,CT

dlcljclPlan ∀

⋅⋅=

λπ x

Note that if 3x is a solution of the optimal offset in (7.3), then ,23

'3 λnxx += for all

integer numbers n, are also solutions of the optimal offset of the wrapped interferogram. The offset can also be set as 03 =x in (7.3) and estimated after tilting with 1x and 2x as

( )∑=

∠=k

nnnInSARnnGPS jiCji

kx

12

*'3 ,),(),(C1

22

πλ

where k is the number of GPS measurements and

.2

2exp

⋅⋅=

λπ GPS

GPSj I

C

7.2. Least square estimation of coefficients If the absolute or unwrapped values are known for both the GPS and interferometric measurements at sparse locations i,j, then the coefficient in (7.3) can be estimated with the standard usual LS estimator

( ) ,T1T yAAAx −=

with

=

1

111

kk ji

jiMMMA

and

(7.4)

(7.5)

(7.6)

(7.9)

(7.10)

(7.7)

(7.8)

18

( )

( ),

,I),(I

,I),(I 1111

−

−=

kkUwInSARkkGPS

UwInSARGPS

jiji

jijiMy

where k is the number of GPS measurements. In order to use (7.9) to (7.11), the sparse values of UwInSARI are estimated from unwrapped line profiles.

7.3. Unwrapping line profiles The relationship between unwrapped and wrapped profiles Uwp and ,p respectively, is given as

,2λkpp +=Uw where k is the wave number vector and λ is the wavelength of the SAR radar. A simple line-unwrapping process is used in the tilting algorithm. For a wrapped line profile p of size m, the procedure can be described as: Algorithm 7.1 1. n=2. 2. Calculate ),pp( 11 −−= nnd )p2p( 12 −−+= nnd λ and ).p2p( 13 −−−= nnd λ 3. If ),,,min( 3211 dddd = go to step 5. Else if ),,,min( 3212 dddd = ,1=k go to step

4. Else if ),,,min( 3213 dddd = ,1−=k go to step 4. 4. For nh = to ,mh = .2λkpp hh += 5. .1+= nn If mn > , stop. Else if ,mn ≤ go to step 2. Figure 7.1 shows an example of a line profile before and after unwrapping with Algorithm 7.1. It is evident that the algorithm uses the first profile value 1p as the reference for the absolute value of the whole profile when unwrapping. 7.3.1 Some error effects Errors from high- and low frequency noise factors need to be considered when unwrapping the interferogram profiles from the Reykjanes Peninsula. One effect of a low frequency noise is explained in Figure 7.2. On the images in (a) are shown location of profiles (between the points ABC), located over both light and dark areas of the wrapped interferogram. The sudden changes in colours are due to lack of one wave number (see (7.12)) in the darker area. This lack of wave number is expected to results in abrupt leaps of ~ 2/λ in the line profiles AB and BC (Figure 7.2 (b) and (c)). This is evident from the wrapped profile AB, but not from the wrapped profile BC. The reason could be a low frequency atmospheric noise that can smooth leaps of ~ 2/λ in the image. Error like this could lead to a large failure when unwrapping. Another possible error in an interferogram is a low coherence in the signal that would also effect the unwrapping process.

(7.11)

(7.12)

19

Figure 7.1. Line profile before and after unwrapping. Figure 7.2. Wrapped InSAR image from the Reykjanes Peninsula; (a): wrapped interferogram, (b), (c) and (d): plot of the profiles AB, BC and CA respectively.

20

7.4. The tilting algorithm The tilting process uses unwrapped profiles between sparse locations in the InSAR image defined by GPS sites (Figure 7.3). The tilting coefficients 1x and 2x are then estimated by the LS algorithm in Section 7.2, and the offset 3x is estimated with (7.7). Error effects are reduced by pre-oversmoothing the interferogram with a vectorized filtering (see Chapter 4). The pre-oversmoothing does reduce the high frequency noise. Furthermore, experiments have shown that this does also reduce errors of an unwanted smoothing of ~ 2λ edges due to low-frequency errors (see definition of an unwanted smoothing of ~ 2λ edges in Section 7.3.1.). A mask image (see Chapter 5) is used to automatically reject profiles that are located over background areas, and to mask the tilted output image. The tilting algorithm is given in Algorithm 7.2. In the tilting algorithm GPSI is a vector including k samples of a GPS measured Slant-Range-Shift, WInSARI is the wrapped InSAR image, ji, are line and column numbers and hn, reefers to the spatial location of the nth and hth sample of .GPSI Note that the tilting slopes are calculated in the following way in Algorithm 7.2: i) For each GPS sample n, the algorithm unwraps line profile from the location

),( nn ji to all the other sparse GPS locations nhji hh ≠∀),,( in the interferogram. The unwrapped values of the interferogram ( ),(I hhUwInSAR ji ) are estimated from the unwrapped profiles with ),(I nnOWInSAR ji as the reference absolute value, where OWInSARI is smoothed WInSARI . The tilting coefficients 1x and 2x are then estimated from the unwrapped values, the GPS measurements and the LS algorithm.

ii) This process is then repeated for all the k GPS locations and the final tilting coefficients are estimated as an average over all individual estimations.

The reason for doing the repetition step in (ii) is to minimise the risk of a tilting error due to low frequency atmospheric errors. The process in (i) to (ii) does not give an information about the offset between the GPS and interferometric measurements since

),(I nnUwInSAR ji is used as the reference absolute value in each step. The offset is therefore calculated with (7.7) in Algorithm 7.2, after tilting with 1x and .2x Algorithm 7.2. 1. Oversmooth WInSARI (10x10 pixel window) which gives .OWInSARI 2. Calculate mask image M. 3. .1=n 4. ,1=h ,0=t ,vectorempty=I ,vectorempty=G vectorempty=l and

.vectorempty=c 5. If nh ≠ , then create an unwrapped line profile p of size m from the locations

),( nn ji to ),( hh ji of ,OWInSARI go to step 6. Else if ,nh = ,1+= tt ),,(I)(I hhOWInSAR jit = ),(I)(G ht GPS= and ),,())(c),(l( hh jitt = go to step 7.

21

6. If the line profile does not intersect with the background, ,1+= tt ),(p)(I mt = )()(G ht ρ∆= and ).,())(c),(l( hh jitt =

7. .1+= hh If ,kh ≤ go to step 5. Else if ,kh > go to step 8. 8. Calculate the slopes )(1 nx and )(2 nx (see (7.3)) by using (7.9) to (7.11) and l, c, I

and G. .1+= nn If ,kn ≤ go to step 4. Else if ,kn > go to step 9. 9. Calculate average slopes as kkxxx ))()1(( 11

'1 ++= K and K+= )1(( 2

'2 xx

,))(2 kkx+ and set .0'3 =x

10. Use [ ]'3

'2

'1 ,, xxx=x to tilt the unsmoothed image WInSARI by using (7.4) to (7.6).

11. Calculate '3x by (7.7) and by setting .1'

2'1 == xx

12. Use [ ]'3

'2

'1 ,, xxx=x ( '

2'1 , xx from step 9 and '

3x from step 11) to tilt the unsmoothed image WInSARI by using (7.4) to (7.6).

13. Create an output image by pixelvies multiplication of the tilted InSAR image and the mask image (M).

Experiments indicate that tilting of wrapped interferograms is more risky than tilting of unwrapped interferograms. Algorithm 7.2 will therefore only be used to tilt wrapped interferograms before unwrapping (see unwrapping in Chapter 9). To achieve safer tilting before inferring the three dimensional crustal deformation, the unwrapped interferograms will be tilted further with the GPS measured Slant-Range-Shift and the LS estimation (see Section 10.3.1). Experiments on the test image have shown that Algorithm 7.2 can find correct tilting parameters for an error free wrapped interferogram. Another experimental example is given in Table 7.1 and Figure 7.3. In this example the 3.12 years 400x750 pixels InSAR image is used, with the Master and Slave tracks 5565 and 21941 respectively. The interferogram in Figure 7.3 (f) has been unwrapped and tilted further with the methods described in Chapter 9 and 10 (see also Appendix B). Input image (Figure 7.3 (a)) with known tilting parameters ,1x 2x and 3x was then created by adding a linear plane to the image in (f). The tilting procedure is explained in Figure 7.3 and the correct and estimated tilting parameters are given in Table 7.1. In this example, the maximum tilting error is estimated as (0.0115-0.0091)⋅400/2.835≈0.34 fringes in latitude (rows) and (0.0056-0.0038)⋅750/2.835≈0.48 fringes in longitude (columns). The offset error is estimated as 624.0035.0835.2*2081.5 =++− or 0.624/2.835≈ 0.22 fringes (note that the solution of the offset is periodical with the period

835.22 =λ cm). A dataflow diagram of Algorithm 7.2 is given in Figure 7.4. Table 7.1. Comparison of correct and estimated tilting parameters. x1 (rows) x2 (columns) x3 (offset) Correct 0.0115 cm/pixel 0.0056 cm/pixel -5.081 cm Estimated with Algorithm 7.2 0.0091 cm/pixel 0.0038 cm/pixel -0.035 cm

22

Figure 7.3. Tilting by using Algorithm 7.2; (a): the untilted input image, (b): the input image oversmoothed with a vectorized filtering, (c): the image in (a) after tilting, (d): mask calculated from the image in (a), (e): the image in (c) masked with the mask in (d), (f): the correct tilted image.

23

Figure 7.4. Dataflow diagram of the tilting process.

24

8. Motion field images created from a sparsely located GPS observations Interpolation of a sparse GPS data can be used to create motion field images. A method that uses a Markov Random Field regularisation to unwrap InSAR images is presented in Chapter 9. This method offer a possibility of initialisation of the unwrapped interferogram, that can for example be created from an interpolation of sparse GPS measured Slant-Range-Shift. A method that optimises three dimensional motion field is then presented in Chapter 10. The method also uses Markov Random Field regularisation, where the three-dimensional motion fields images are initialised from interpolated GPS data. Several methods have been tested for the interpolation, like a linear and cubic spline, weighted average and kriging. The best result for the test image has been achieved with the kriging. Kriging algorithms use geo-statistical measurement the dispersion matrix to find an optimal set of weights, used for the interpolation [9,10,11]. An ordinary one dimensional (1D) kriging algorithm is used to interpolate the sparse GPS data, where the dispersion matrix is approximated with use of an estimated semivariogram. 2D-kriging [11] have also been considered for use in estimation of the three-dimensional motion field. The main drawback of 2D-kriging is that it requires estimation of one semivariograms and two cross-semivariograms compared to one semivariogram for 1D-kriging, which makes it much more complicated to use. 2D-kriging is therefore not used in this thesis. The 1D-kriging method used for the implementation is only described briefly in this chapter, but further details about it are given in [9].

8.1. Ordinary kriging Given a set of sparsely located M-measurements an unbiased estimator of an arbitrary point 0z is where [ ]T

Mωωω ,...,,ω 21= is the set of optimal weights. The variable iz can be

interpreted as an outcome of a random variable .iZ By requiring 0}ˆ{ 00 =−Ε ZZ and

minimising the error variance }ˆ{Var 00 ZZ − the optimal solution of the weights in (8.2) is found by

[ ] ,,...,, 21T

Mzzz=z

,1with,ωˆ1

0 == ∑=

N

ii

Tz ωz

(8.1)

(8.2)

,

10111

1

0

011

1

111

=

MMMMM

M

C

C

CC

CCMM

L

L

MMOM

L

λω

ω

(8.3)

25

where ijC is the covariance between the points iz and .jz

8.2. Semivariogram An estimation of the covariance ,)( ijChC = where h is the distance vector between points iz and ,jz is done by using a semivariogram. The semivariogram is defined as where r is some point in the space and Z is the set of the random variables iZ . If the spatially distributed measurements are assumed or forced to be first and second order stationary then )(),( hhr γγ = and

2)0( σ=C is the variance of the stochastic variables. An estimator for the semivariogram is given as where )(N h is the number of points separated by the distance of h. The estimation in (8.6) can also be calculated from a cross section of hh ∆± to increase )(N h (the number of points). For this purpose h∆ is implemented as an option in the kriging algorithm.

8.3. Semivariogram model A Gaussian model is used to fit the data calculated by (8.6). The model is written as The coefficients [ ]TRCC ,, 10=θ are calculated iteratively by using the objective function and the Pattern Search iteration algorithm. The Pattern Search algorithm was chosen for its properties of being independent of derivatives. Further details about implementation of the Pattern Search is given in [12]. 10 CC + in (8.7) is called the Sill and is equal to ).(lim)0( *2 hC

hγσ

∞→== The Gaussian semivariogram model does

approach the Sill asymptotic without ever reaching it. The estimation given in (8.6) to (8.8) is used to estimate the variances )(hC in (8.5), where )0(C is approximated by calculating )(* hγ for some very large number of h.

[ ]{ },)()(21),( 2hrZrZhr +−Ε=γ

(8.4)

).()0()( hCCh −=γ (8.5)

[ ]∑=

+−=)(N

1

2 ,)(N2

1)(ˆh

khkk zz

hhγ (8.6)

<

−−+

== .03exp1

00)(

2

2

10

*

hRhCC

hhγ (8.7)

),θ()(ˆminθ *

θhh γγ −= (8.8)

26

The estimation of ),(hC ,h∀ is used to approximate the coefficients in (8.3), which is then used to calculate the weights for (8.2).

8.4. Kriging results The result of kriging sparsely located Slant-Range-Shift values from the test image is shown in Figure 8.1 and kriging result of GPS measured Slant-Range-Shift from the Reykjanes Peninsula is shown in Figure 8.2. It can be an advance, when calculating

)(* hγ , to have the resulting data set from (8.6) limited to some maximum value of h, like explained in Figure 8.1 (a) and Figure 8.2 (a). This is also implemented as an option in the kriging algorithm. The assumption of having spatially first or second order stationary ground movements is not expected to hold in general. Furthermore, the correlation is likely to be also a function of direction between points. But Figure 8.1 (d) shows that the ordinary kriging algorithm can be used to interpolate between sparse deformation values with a very good result, by limiting h in (8.6) to some maximum number before estimating the semivariogram model in (8.7). Three-dimensional crustal deformation inferred from kriging of GPS measurements is shown in Figure 8.3. The motion field is displayed as one-year average of the 1993-1998 observation. Figure 8.1. Result of kriging the sparse Slant-Range-Shift values from the test image marked as + on (b), (c) and (d); (a): semivariogram estimated by (8.6) and the fit of the model in (8.7) (by only using the points marked as ⊕ , i.e. h ≤ 75 pixels), (b): the test image, (c): result of interpolating between the sparse values, (d): the error difference between the images in (b) and (c).

27

Figure 8.2. Kriging of all the GPS measured Slant-Range-Shift from the Reykjanes Peninsula; (a): semivariogram estimated by (8.6) and the fit of the model in (8.7) (by only using the points marked as ⊕ , i.e. h ≤ 200 pixels), (b): kriging result and location of the GPS sites (+). Figure 8.3. Result of inferring the three dimensional motion field by kriging of GPS mesurements. (a), (c) and (e): vector plot of the sparse GPS measured Vertical, East and North crustal deformation respectively, (b), (d) and (f): Kriging of the Vertical, East and North crustal deformation respectively,

28

8.5. Estimation of uncertainty A spatial uncertainty estimation is implemented into the kriging algorithm. This is done by using the following: 1. No uncertainty is assigned to the sparse GPS locations. 2. The uncertainty increases as a function of distance from GPS locations, and is

calculated as the inverse proportional to the Gaussian semivariogram model. 3. The uncertainty image is scaled to the interval [0,1] where 1 means no

uncertainty.

8.6. Comparison of interpolation methods The result of kriging was compared to two other interpolation methods. The first method use a delaunay triangulation cubic splining of the data (an available function in MatLab) and the second one use distance weighted average. A distance weighted average estimation of an arbitrary point 0z in given as

∑=

=N

iii zz

10 ,ˆ ω

where

,1

1

1∑

=

= N

ii

ii

d

dω

iz is the ith sample of the N numbers of GPS observations, iω is the weight of the ith

GPS sample and id is the distance from the ith sample to the observation point .0d The comparison is given in Figure 8.4. It is evident that the best result is achieved with the ordinary kriging process.

(8.9)

(8.10)

29

Figure 8.4. Comparison of interpolation methods, estimated by using the test image; (a): the correct Slant-Range-Shift; (b): the result of kriging, (c): the result from the cubic splining, (d): the result from the weighted averaging. The sparse GPS locations are shown as + on all the subimages.

30

9. Using kriging of GPS measurements and Markov Random Field regularisation to unwrap InSAR images The InSAR images are unwrapped before inferring the tree-dimensional crustal deformation by combined GPS and interferometric observations. InSAR images from the Reykjanes Peninsula can include large noise factors like previously described. Also, the interferometric signal is expected to be relatively weak compared to the error factors, since it represent slow crustal deformation. Furthermore, high- and low frequency atmospheric noise can drastically affect the image information, since the Peninsula is surrounded by an ocean. Large atmospheric noise factors can give arise to number of problems when using unwrapping processes. A process that uses Markov Random field (MRF) regularisation and simulating annealing optimisation was designed to unwrap InSAR images. The process can be initialised and guided by sparsly located “correct” values like GPS measurements. For the initialisation, the ordinary kriging method, described in Chapter 8, is used to interpolate between the sparse GPS measured Slant-Range-Shift and create a virtual unwrapped InSAR image. One advance of using simulating annealing optimisation of the MRF regularisation is its capability of unwrapping images despite of large atmospheric noise factors. Results of applying the method to both the test image, and the interferometric and GPS measurements from the Reykjanes Peninsula are presented in this chapter.

9.1. Problem description The relationship between an unwrapped ( UwI ) and a wrapped ( WI ) InSAR image can be written as

,2λNII += WUw

where N is wave number matrix and λ is the wavelength of the SAR radar. Unwrapping an InSAR image can therefore be regarded as a problem of finding the wave numbers in .N Figure 9.1 (a) and (b) shows the wrapped interferogram WI and the corresponding wave number matrix N for the 164x164 test image. Some errors that may effect unwrapping processes were described in Section 7.3.1. Those error effects need also to be taken into account when unwrapping the whole InSAR image, and will be considered in Section 9.5. Figure 9.1 (c) and (d) shows a wrapped and unwrapped interferogram of the same area, highly influenced by atmospheric noise. The wrapped interferogram is periodical with no information available about the wave numbers. This can be seen as abrupt changes from light to dark coloured areas or reverse (see Figure 9.1. (c))

(9.1)

31

Figure 9.1. Wrapped and unwrapped format; (a): the test image on a wrapped format, (b): the wave numbers for the image in (a), (c) and (d): a wrapped and unwrapped interferogram from the Reykjanes Peninsula, respectively.

9.2. Estimation of the initial wave numbers The ordinary kriging algorithm described in Chapter 8 is used to calculate a virtual interferogram from a sparsely located GPS measured Slant-Range-Shift. The virtual interferogram ( VI ) along with the wrapped interferogram ( WI ) can be used to estimate the wave number matrix N in (9.1) as

.2

)(

−=

λWVround

IIN

This estimation will be used as an initial step in further calculations. Estimation of N by (9.2) with VI as the kriged test image in Figure 8.1 (c) and WI as the wrapped test image in Figure 9.1 (a), is shown in Figure 9.2.

(9.2)

32

Figure 9.2. Estimation of wave numbers by help of the ordinary kriging algorithm; (a): wave numbers estimated by (9.2), (b): the wrapped test image plus the estimated wave numbers ( 2λNII += WUw , WI is shown in figure 9.1 (a)).

As previously described, the assumption of having linear deformation with time at the Reykjanes Peninsula does not always hold (Section 2.3.3 and Chapter 4). This can result in a bad consistency between the time scaled GPS measurements and the interferometric measurements. Comparisons have though shown that a tolerable fit can be achieved between most of the 1993 to 1998 GPS observations and the tilted interferometric observations that represent various elapsed time intervals between 1992 to 1996. The algorithm offers the opportunity to be initialised and guided with a GPS measured Slant-Range-Shift. Here, the following is done to ensure consistency between the GPS data and the InSAR images before unwrapping: Excluding from the GPS data set before kriging and unwrapping 1. GPS measurements in disagreement with other neighbouring GPS measurements

and 2. GPS measurements in poor agreement with the tilted interferogram. The disagreement between the GPS points and the interferogram can be due to several reasons like a non-linear deformation and errors and noise in GPS and interferometric measurements. Figure 9.3 shows a result of estimating the wave number matrix N for the 2.29 years interferogram by using (9.2) and kriging of time scaled GPS measurements. The result of using the kriging algorithm along with (9.2) gives a reasonable good estimation of the initial wave numbers (Figure 9.2 and 9.3). The result is though very dependent on the consistency between the sparse GPS measurements and the tilted interferogram. This is evident by comparing the kriging result in Figure 8.2 (b) and 9.3 (b). Kriging all the GPS points available (Figure 8.3 (b)) would lead to large error when estimating the wave numbers in (9.2) for the 2.29 years tilted interferogram in Figure 9.3 (a).

33

Figure 9.3. Estimation of wave numbers by help of the Kriging algorithm; (a): the wrapped and tilted 2.29 years interferogram (1993-1995), (b): a result of Kriging the GPS pints with the location shown as ⊕ in (a), (b) and (d), (c): the wave number estimated by (9.2); (d): the wrapped and tilted interferogram in (a) plus the estimated wave numbers ( 2λNII += WUw ).

9.3. Simulating annealing and MRF regularisation The problem task is modelled by using a MRF based regularisation. The wrapped interferograms are then unwrapped by a simulated annealing optimisation. The optimisation process uses a Maximum a posteriori (MAP) estimate to represent an optimal realisation image x of a random field X , for a given image y [13]. The MAP estimation is given as

).|(maxargˆ yxxx

=== YXP

For convenient )( x=XP will be written as )(xP when expressing the likelihood. The Bayesian theorem gives

),|()()(

)|()()|( xyxy

xyxyx PPPPPP ∝=

(9.3)

(9.5)

34

where )(xP represent a prior expectations about the random field X (often smoothness assumptions) and )|( xyP is the likelihood of the image y given the image x (the relation to the observations). The simulating annealing optimisation can be described as a sampling of the density

[ ] ,)|()()|(1TT PPP xyxyx ∝

where the temperature T starts at some “high” value 00 >T and falls towards 0 during the iteration steps. One of the great advances of using simulating annealing optimisation process is its relatively low risk of running into a local minima compared to other optimisation algorithms. A Markov random field X with a realisation image x is defined with respect to its neighbourhood system (see definition of MRF and Gibbs random field (GRF) in [13,14]). By using the Hammersley-Clifford theorem (MRF-GRF equivalence theorem) [13], the density function in (9.6) can be written as

,)|(1exp)|(

−∝ yxyx U

TPT

where )|( yxU is an energy function defined with respect to the neighbourhood system in the image x. Due to the MRF-GRF equivalent theorem, the MRF modelling can be regarded as defining a suitable energy function that leads to global minima for unwrapped interferograms.

9.4. The simulating annealing unwrapping algorithm and MRF models The object of unwrapping can be viewed as finding the wave number matrix N in (9.1) for a given image I . The image I can for example be created by combined GPS and interferometric observations, e.g. by (9.1) with N estimated from (9.2). The goal is then to minimise an energy function ).|()|( INyx UU = By defining a suitable MRF regularisation models for the InSAR images, the illdefined unwrapping problem can turned into well defined. A suitable energy function is designed with respect to the neighbourhood structure in the images, which is equivalent to MRF modelling, due to the relationship given in (9.7). 9.4.1. Simulating annealing iteration algorithm Simulating annealing optimisation algorithm is used to minimise of the energy function ),|( INU and hence, find the unwrapped interferogram (the realisation image). For a wrapped interferogram ,WI with M-numbers of pixels, the algorithm can be written as: Algorithm 9.1. 1. Choose initial wave number matrix N (e.g. by (9.2)), interferogram I (e.g. by

(9.12)) and initial temperature ( 0TT = ). 2. k=1, where k is a pixel number. 3. Increased or decreased the wave number kN by 1 with equal probability, which

gives a new wave number matrix 'N .

(9.6)

(9.7)

35

4. Calculate ( ) ( )( ).)|()|(exp)|()|( '' TUUppr TTk ININININ −−== 5. If [ ]1,0µ>kr , then '

kkt NN = , else .kkt NN = 6. k=k+1, if Mk ≤ go to step 3, else go to the next step. 7. ,tNN = 2λNII += W except maybe at sparse GPS points, coolTT ⋅= , where

1<cool is constant. 8. Go to step 2.

[ ]1,0µ is random number within the interval [0,1], selected from a uniform distribution. Algorithm 9.1 can be implemented both as a non-recursive and a coded-recursive. Update of coded-recursive algorithm can be explained with help of the pixel-grid in Figure 9.4, i.e. step 3 to 7 in Algorithm 9.1 could first be done on pixel marked as X and then repeated for pixels marked as O, before lowering the temperature. A coded recursive algorithm is often more efficient and faster than non-recursive, but on the cost of being more complicated. Here, the algorithm is implemented as non-recursive. 9.4.2. Penalizing the second derivative Several energy functions have been tested that requires the image surface to be smooth. The best results have been achieved by requiring smoothness of the first derivative, implemented as a penalization on the second derivative with the approximation

( ) ,4)( 21,1,,,1,111 ∑∑

∈ ∈−+−+ ++−+=

ui vjjijijijiji fffffU γN

where 1γ is a constant, u and v is the row and column space respectively and

.2,,, λjijiWji NIf += Algorithm 9.1 can be used to minimise (9.8) by setting

).()|( 1 NIN UU = If the temperature is lowered slowly enough, then (9.7) will assign maximum probability state to the annealed image [13]. Experiments indicate that high temperature ( 700 >T in Algorithm 9.1) is needed for the unwrapping. Experiments have also shown that too high temperature can easily result in damage of correctly unwrapped image areas, which is not easy to overcome. To avoid this, without reducing the probability of finding the global solution, the following is done: 1. The temperature 0T is set reasonable high and one annealing is done by using step

1 to 8 in Algorithm 9.1. 2. If ,1TT ≤ ,0 01 TT <<< then 0TT = and another annealing is done, where 1T is a low temperature, used to terminate Algorithm 9.1. This reannealing process is repeated until the global solution is found.

(9.8)

36

Figure 9.5 explains this procedure when using the test image and the parameters

,21 =γ ,900 =T 21 =T and .99.0=cool No sparsely located GPS points were used. Figure 9.5 (a) shows the initial wave numbers, (b) the initial image and (c) the result after one annealing. The correct solution is reached after ~25000 iterations on each pixel or 65 reannealing (Figure 9.5 (e) and (f)). The initial wave numbers used in this example were calculated by the ordinary kriging algorithm and (9.2). In this example, the kriging was done by using linear interpolation between points in the estimated semivariogram calculated by (8.6), instead of using the Gaussian model in (8.7). This was done to increase initial errors, which explains the difference between Figure 9.2 (b) and Figure 9.5 (b). 9.4.3. Detecting area of interest It is not necessary to do reannealing on all the image pixels. A method that finds the area of interest for each reannealing can be described as follow: 1. A thresholded edge detection is used to find large edges in the resulting InSAR

image after each reannealing. 2. The areas of interest in the resulting binomial image are then expanded by a 5x5

constructive element dilation, to create a mask for next reannealing. Several edge detection methods have been tested for this purpose, both methods that search for the highest gradient by approximating the first derivative, and methods that looks for zero-crossing by approximating the second derivative. The best result has been achieved with the socalled Prewitt operator [7]. The Prewitt operator finds the steepest gradient by approximate the first derivative. Here, the gradient is estimated for two direction (x and y) with the two masks

[ ] [ ] ,101101101

and111

000111

−−−

=

−−−= xy hh

where the small parenthesis indicates the kernel origin. The maximum gradient is then given by the mask that gives the maximal respond. The binomial edge image can be created for example by thresholding the edge image to half the maximum image value. The reannealing is done only on the masked areas, which makes the algorithm faster. An example of a mask is given in Figure 9.5 (d), generated from the image in Figure 9.5 (c) (the result after one annealing). Only pixels within the black areas of the mask are reannealed. By using only the penalization on the second derivative and the no initialisation of the wave numbers, the wrapped test image (Figure 9.1 (a)) can be unwrapped in ~50000 iterations or 128 reannealings, which is much slower than if the kriged virtual InSAR

Figure 9.4. Pixel-grid explaining the coded-recursive update.

(9.9)

37

image is used to initialise the algorithm. The result of using the energy function in (9.8) with no initialisation of wave numbers, and using the reannealing process to unwrap a smoothed version of the 3.12 years InSAR image from the Reykjanes Peninsula is shown in Figure 9.6. Figure 9.5. Result of unwraping by only penalizing the second derivative; (a): the initial wave numbers N; (b): the initial image ( 2λNII += W ), (c): the result after one annealing, (d): a mask used when reannealing the image in (c), (e): the resulting wave numbers after ~25000 updates of each pixel or 65 reannealings, (f): the resulting image after 65 reannealings.

38

Figure 9.6. Unwrapping an InSAR image by penalizing the second derivative; (a): the wrapped interferogram, (b): the unwrapped interferogram. 9.4.4. Guiding with GPS measurements. The energy function in (9.8) includes no relationship to the observations and therefore no information about the absolute pixel values. Hence, it tends to keep the wave numbers of the most dominant joined area unchanged. It is explained in Section 9.2 how the kriging of sparse GPS data can be used to initialise the unwrapping algorithm. The GPS points can also be utilised further to guide the algorithm, since it includes information about the absolute pixel values at sparse locations. This can be done in several ways. The method presented here uses energy function that penalizes only pixels in the nearest neighbourhood of the GPS pixel. The GPS pixel domain will be called a domain of frozen (fixed) pixels, since they are not updated in the simulating annealing optimisation algorithm. The domain of frozen pixels is then expanded before each reannealing. This method utilises the knowledge from GPS observations. The energy function used to give extra penalization for pixels in the neighbourhood to the correct domain is written as

( )( ) ,)|( 222 ∑

↔

−=nk

nnk WffU γNI

where ,2λkkWk NIf += 2γ is constant, nk ↔ means all pixels k in the neighbourhood of the pixel n and nW is 1 if n is within the domain of correct pixels and zero otherwise. The relationship to the observed image I is gained by freezing the GPS pixels (The GPS observations). The energy function in (9.10) is used along with the energy function in (9.8), with 21 =γ and ,702 =γ i.e.

).|()()|( 21 NININ UUU +=

1γ was optimised by using only the process described in Section 9.4.2, for ,900 =T 21 =T and .99.0=cool The high value of 2γ result from experimenting with 2U on

(9.10)

(9.11)

39

pixel domains close to the domain of frozen pixels, done by keeping all other parameters unchanged. The following masking is done before each reannealing process: 1. the domain of frozen pixels is expanded with a dilation, by using a structured

element with four nearest neighbourhood pixels and the pixel. 2. in step 6 in Algorithm 9.1 I is updated by 2λNII += W except within the

domain of frozen pixels and 3. the reannealing is done only at areas masked by the thresholded edge detection

followed by dilation. Figure 9.7 explains how the domain of correct pixels expands when ,900 =T 21 =T and 99.0=cool in Algorithm 9.1. The initial wave numbers used in this example are shown in Figure 9.5 (a), and the resulting image and the corresponding mask after 10 reannealing in Figure 9.7. The final solution was reached after ~14000 iterations or 38 reannealing. The main advances of utilising the sparsely located GPS points are: 1. information about the absolute pixel values have been included into the

unwrapping algorithm, 2. the algorithm runs into the expected solution with more safety and faster (around

two times faster for the test image).

9.5. Unwrapping process The methods described so far were used to build up an unwrapping process for InSAR images that may include high noise factors. The algorithm uses the penalization on the second derivative and expansion of the domain of frozen pixels. A virtual InSAR image is created by ordinary kriging of the sparse GPS pixels, and the initial wave numbers are estimated by (9.2). Figure 9.7. Expansion of the domain of frozen pixels; (a): the result after 10 reannealing and the expansion of the frozen domain (the frozen domain are shown as light rhombus which are expanding from its centre, marked as +), (b): the mask used for reannealing of the image in (a).

40

Error factors need to be considered before unwrapping. To be able to use the GPS measurements the following is done: 1. time scale GPS measurements are time scaled to fit the elapsed time interval

represented by the InSAR image, 2. use the tilting process described in Chapter 7 to eliminate a phase plane in the

InSAR observations and 3. remove measurements from the GPS data set that are in bad consistency with the

tilted InSAR image. High frequency noise in the InSAR images can influence the penalization on the second derivative by both generating errors and slow down the process. Furthermore, low frequency error can smooth edges of the size ~ 2λ (that correspond to 2π leaps due to periodical properties of the signal, see explanation in Section 7.3.1), which in turn can lead to failure when using the expansion of the frozen pixel domain. The following unwrapping process is designed to reduce the effects from those noise factors: 1. The InSAR image WI is oversmoothed by the vectorized filtering (Chapter 5),

which gives the image .OI This smoothes the surface and makes the penalizing of the second derivative easier. Furthermore, this reduces also the errors of unwanted smoothing of ~ 2λ edges.

2. A mask is created (Chapter 6). 3. Simulated reannealing is done by using (9.11), and the domain of frozen pixels is

expanded before each reannealing. 4. After full expansion of the domain of frozen pixels, errors generated by unwanted

smoothing of ~ 2λ edges are reduced by a simulating annealing process that uses only penalization on the second derivative (like explained in Section 9.4.2).

5. The resulting unwrapped oversmoothed InSAR image UwOI is then used to estimate the wave numbers for the unfiltered InSAR image WI by using

6. An unwrapped unsmoothed interferogram is calculated as .2λ'NII += WUw 7. The resulting unwrapped interferogram is masked by pixelvise multiplication of

the interferogram and the mask. A dataflow diagram of the unwrapping process is shown in Figure 9.8. It is possible to use the mask image as an input into the iteration process, and do only updates on the foreground area. Experiments have shown that this can influence or damage the outer boarder of the area of interest. Also, this is not needed when the thresholded edge detection process, described in Section 9.4.3, is used. Here, a masking is done after unwrapping as explained in Figure 9.8. Figure 9.9 shows the result of applying the process to the 2.29 years interferogram from the Reykjanes Peninsula, where each step in the unwrapping algorithm is explored. The effect of oversmoothing the wrapped interferogram before unwrapping

(9.12) .2

)('

−=

λWUwOround

IIN

41

is evident by comparing Figure 9.3 (d) and Figure 9.9 (e). It is also evident from Figure 9.9 (i), that an accurate estimation of the wave numbers for a high frequency wrapped InSAR image can be done by using the corresponding oversmoothed unwrapped InSAR image and (9.12). Indeed, experiments indicate that the wave number matrix can always be estimated in that way. The errors in Figure 9.9 (f) explains why further simulation is done by only penalizing the second derivative (Figure 9.9 (g)), after full expansion of the domain of frozen pixels. The MRF regularisation used in the unwrapping process utilises mainly an assumption about the surface, by penalizing the second derivative. The GPS measured Slant-Range-Shift includes information about the wave numbers at sparse locations, which is the only relationship of the wave numbers to the observations. An attempt is done to use this observed relationship, by implement the energy function in (9.10). The main drawback is that those values are only known at sparse locations, which makes the relationship of the wave number matrix N to the observations weak. Despite of that, the unwrapping process described in this chapter have turned out to be efficient in unwrapping the InSAR images from the Reykjanes peninsula.

42

Figure 9.8. Dataflow diagram of the unwrapping process.

43

Figure 9.9. Exploring the unwrapping procedure by using 2.29 years interferogram from the Reykjanes Peninsula. (a): Tilted and unsmoothed wrapped interferogram, (b): the image in (a) oversmoothed, (c): ordinary kriging of sparsely located GPS observations, (d): estimated initial wave numbers by using (b) and (c), (e): the wave numbers in (d) added to (b), (f): the result of simulating annealing optimisation after full expansion of the domain of correct pixels, (g): simulating annealing optimisation with (f) as an input, by only penalizing the second derivative, (h): wave numbers estimated from (a) and (g), (i): the wave numbers in (h) added to (a). Sparse location of GPS points are shown as + on some of the subimages.

44

10. Combination of GPS and interferometric observations to infer three-dimensional ground movements The InSAR images include high-resolution maps of one-dimensional Slant-Range ground movements, while GPS measurements contain information about three-dimensional motions at sparse locations. In Chapter 9 it was shown how an ordinary kriging of sparse GPS measurements, MRF based regularisation and simulating annealing optimisation can be used to unwrap interferograms. The ordinary kriging algorithm, MRF based regularisation and simulating annealing optimisation is utilised further in this chapter to estimate high-resolution maps of the three-dimensional ground movements from combined GPS and interferometric observations. Result of applying the method to both the 164x164 pixels test image and interferometric and GPS observations from the Reykjanes peninsula are presented.

10.1. Problem description The satellite measure the one dimensional Slant-Range-Shift (SRS) given as

[ ] [ ] kuuukkkk ∀⋅−= ,,,V,V,VV TVNEVNESRS

for each pixel k in the InSAR image SRSV . ,EV NV and VV are the East, North and Vertical of deformation images, respectively, and [ ]VNE ,, uuu=s is a given unit vector pointing from the ground towards the satellite. For the descending satellite pass InSAR images from the Reykjanes Peninsula

[ ].V935.0,N095.0,E34.0 −≈s

The task is then to find the three motion field images ,EV NV and ,VV for known

SRSV image (InSAR observations) and sparse values of ,EV NV and VV (GPS observations). 10.1.1. Simplifying the problem The three-dimensional problem can be changed into two two-dimensional problems, for example as

[ ] [ ] ,,,V,VV TVLVLSRS kuukkk ∀⋅−=

where

2N

2EL uuu +=

and

[ ] [ ].,

,V,VV

L

TNENE

L ku

uukkk ∀

⋅=

(10.1)

(10.2)

(10.3)

(10.4)

(10.5)

45

LV is the deformation in the Horizontal look-direction of the satellite. The geometry of (10.3) is explained in Figure 10.1. For the InSAR images from the Reykjanes Peninsula [ ] [ ].353.0,935.0, LV =uu The East and North motion field images EV and ,NV respectively, can be found after an optimization of ,VV and LV with (10.5) or by using

[ ] [ ] kuuu kkkk ∀⋅−=+ ,,V,VVV TNENEVVSRS

10.2. Optimization of two-dimensional deformation A simulated annealing process is used to optimise a MRF based regularisation of the two-dimensional ground deformation given in (10.3) and (10.6). The process is initialised with two motion field images generated with an ordinary kriging of the sparsely located GPS observations (see explanation of simulating annealing and MRF modelling in Section 9.3, and ordinary kriging in Chapter 8). 10.2.1. Two-dimensional simulating annealing algorithm A general form of (10.3) and (10.6) is

[ ] [ ] ,,,V,VV T2121 kuukkk ∀⋅−=

where kV is known for all pixels k, k1V and k2V are only known at sparse locations and 1u and 2u are constants. The MRF models can therefore be designed for regularisation of two-dimensional motion field images. The simulating annealing process used for the optimisation of two realisation images is given in Algorithm 10.1. The algorithm is used to optimise a combined energy stage ( )|,(U 211 VVV and )|,(U 212 VVV ) of two images in the same iteration process. A 1/0-switch 2s is added to the algorithm. If the switch value is 1 then both the motion field images are updated, but if the value is 0 then only the motion image

(10.6)

(10.7)

Figure 10.1. The geometry of (10.3). VV and VL are the Vertical and horizontal look-direction motion fields, respectively.

46

1V is updated by keeping 2V unchanged. This can be an advance if one of the motion field images is known with a high certainty. Algorithm 10.1. 1. Choose initial images 1V and 2V (e.g. by kriging). Extract region of interest

(Chapter 6) and set the initial temperature .0TT = 2. Choose a switch, 12 =s or .02 =s 3. k=1, where k is a pixel number. 4. Increase or decrease k1V with equal probability by a value of V,∆ which gives a

new image .'1V 5. Calculate ( ) == )|,()|,( 2112

'111 VVVVVV TTk ppr

( )( )TUU )|,()|,(exp 2112'

11 VVVVVV −−

6. If [ ],1,01 µ>kr then ,VV '11t kk = else .VV 11t kk =

7. k=k+1, if Mk ≤ go to step 4, else go to the next step, (M is the total number of pixels).

8. ,1t1 VV = except maybe at the sparse GPS points. 9. If ,02 =s go to step 17. Else if ,12 =s go to the next step. 10. k=1, where k is a pixel number. 11. Increase or decrease k2V with equal probability by a value of V,∆ which gives a

new image .'2V

12. Calculate ( ) == )|,()|,( 212'2122 VVVVVV TTk ppr

( )( ).)|,()|,(exp 212'212 TUU VVVVVV −−

13. If [ ],1,02 µ>kr then ,VV '12t kk = else .VV 21t kk =

14. k=k+1, if Mk ≤ go to step 11, else go to the next step. 15. ,2t2 VV = except maybe at the sparse GPS points. 16. ,coolTT ⋅= where 1<cool is a constant. 17. Go to step 3.

[ ]1,0µ is a random number within the interval [0,1], selected from an uniform random generator. Like Algorithm 9.1 (Section 9.4), Algorithm 10.1 can be implemented both as a non-recursive and a coded-recursive. Here it is implemented as a non-recursive. 10.2.2. Energy functions The main energy functions used in Algorithm 10.1 uses the relationship between the two-dimensional motion field images and the known image V (e.g. InSAR image) given in (10.7), and is written

[ ] [ ]( ) ,,V,VV),|(U2T

2121122112 ∑ ⋅+=n

nnn uuγVVV

where n is a pixel number. An additional constraint is the smoothness of the first derivative of 1V and ,2V implemented as a penalization on the second derivative as

(10.8)

47

( )∑∑ +−+− ++−+=i j

jijijijiji2

1,11,1,1,11,111111 VVV4VV)(U γV

and

( )∑∑ +−+− ++−+=i j

jijijijiji2

1,21,2,2,12,122222 VVV4VV)(U γV

where ji, are the row and column numbers respectively. ,12γ 1γ and 2γ are constants in (10.8), (10.9) and (10.10). The purpose of the smoothness requirements is to keep neighbouring pixels connected or preserve the correlated relationship of the image pixels. If those energy terms are excluded the pixel values can turn out to be spatially chaotic. By using (10.8) to (10.10), the energy functions )|,(U 211 VVV and )|,(U 212 VVV in Algorithm 10.1 are written as

)(U),|(U)|,(U 1112112211 VVVVVVV +=

and

)(U),|(U)|,(U 2222112212 VVVVVVV += It should be noted that infinite set of optimal solutions exist for (10.8) to (10.10) when the correct values of the images 1V and 2V are only known at few sparse locations. Therefore, solution from any optimisation process must be very dependent on the quality of the initial values. Hence, the quality of the solution is very dependent on the density and quality of the GPS measurements and the performance of the interpolation process. For Algorithm 10.1, the final maximum probability state assigned to the annealed images is rather independent of the initial temperature value, as long as it is set high enough at the initial state. This is due to the strong relationship of the motion field images to the interferometric observations. This was not the case the MRF regularisation, given in Chapter 9, was used for unwrapping. Figure 10.2 shows the result of applying Algorithm 10.2 to the 164x164 test image, with the energy functions in (10.8) to (10.10) used to predict the Vertical and Horizontal look-direction motion field images (with L1 VV = and V2 VV = ). In this example, ,1.0V =∆ ,1012 =γ ,11 =γ ,22 =γ ,50 =T ,99.0=cool 353.0L1 == uu and .935.0V2 == uu Sparse GPS values were not updated during the iteration run (frozen GPS values). The program was terminated for .1.0<T Table 10.1 shows the estimated errors for the example given in Figure 10.2. The process errors are estimated as the mean ( µ ) and standard deviation (σ ) of the difference between the correct and predicted motion field images. Table 10.1 shows that the kriging errors are decreased by Algorithm 10.1, both for the Vertical and the Horizontal look-direction deformation images. The predicted uncertainty is though much less for the Vertical deformation component. This can be explained by the high contribution of

(10.9)

(10.10)

(10.11)

(10.12)

48

the Vertical component to the Slant-Range-Shift observation ( 935.0V =u versus 353.0L =u ), which makes it more dominated in the iteration process. The balance

between the two components are though partly taken care of by setting .12 γγ > The reason for the higher kriging error of LV than VV in this example is the location of the GPS points, which are more favourable for the vertical component. Table 10.1. Estimated kriging and simulation errors. ,V 0.1=∆ ,12 10=γ ,1 1=γ ,2 2=γ

,0 5=T ,0.99=cool 0.353== Luu1 and 0.935.== Vuu 2 Vertical Horizontal look-direction Slant-Range

Kriging error µ 0.07 0.09 -0.10 σ 0.44 0.79 0.93

Simulation error µ -0.02 0.03 0.002 σ 0.28 0.73 0.04 10.2.3. Further utilisation of GPS observations Both the interferometric and GPS measurements can include error factors. The lowest kriging uncertainty of the multi-dimensional motion fields is though expected to be at the sparse GPS location and decrease as a function of the distance from them. Uncertainty images can be created along with the kriging of motion field measurements (Section 8.5). The uncertainty images can be utilised when penalizing the motion field images for deviating from the GPS observations. This is done by using the energy functions

( )( )∑ −=n

nnn2

1K111K1K11K VVW)|(U γVV

and

( )( )∑ −=n

nnn ,VVW)|(U 22K222K2K22K γVV

where n is pixel number, 1W and 2W are the uncertainty images corresponding to the resulting kriged motion field images 1KV and ,2KV respectively, and 1Kγ and 2Kγ are constants. The uncertainty images includes the intensity interval [0,1], where 1 means no uncertainty (at GPS locations) and 0 means no certainty. An example of uncertainty image is given in Figure 10.3, where the sparse GPS pixel locations are also marked on the image. The purpose of (10.13) and (10.14) is to give an extra force to penalize the images for deviating from the kriging images in points at, and close to, the sparse GPS pixels. This force does then vanish when diverging from the GPS positions, see Figure 10.3. This is of advance if the GPS observations are the best estimation available for the correct multi-dimensional motion field at sparse locations and if the interferometric signal is weak or includes high error factors.

(10.13)

(10.14)

49

Figure 10.2. A result of applying Algorithm 10.1 to the 164x164 test image. (a): Difference between the correct and initial values (kriged) of the vertical motion field, (b): difference between the correct and simulated values of the Vertical motion field, (c): difference between the correct and initial values of the Slant-Range-Shift, (d): difference between the correct and initial values (kriged) of the Horizontal look-direction motion field, (e): difference between the correct and simulated values of the Horizontal look-direction motion field, (f): difference between the correct and simulated result of the Slant-Range-Shift. (g): Plot of line 82 out of the initial, simulated and correct vertical images, (h): plot of line 82 out of the initial, simulated and correct Horizontal look-direction images, (i): plot of line 82 out of initial, simulated and correct Slant-Range images. The locations of the spares GPS sites are shown as + on the image in (c).

50

The energy functions in (10.13) and (10.14) are added to Algorithm 10.1 by using

)|(U)(U),|(U),|,(U 1K1K11112112K1211 VVVVVVVVVV ++=

and )|(U)(U),|(U),|,(U 2K2K22222112K212 VVVVVVVVVV 2 ++=

instead of )|,(U 211 VVV and ),|,(U 212 VVV respectively, into the algorithm. A result of experimenting with the test image, and the energy functions in (10.11) and (10.12) versus the energy functions in (10.15) and (10.16), are given in Table 10.2 to 10.4. In these examples, the coefficients are all the same as fore Table 10.1, with the algorithm terminated when .1.0<T In addition 102KK1 == γγ in (10.15) and (10.16). For all the tables, the Vertical and Horizontal look-direction components are optimised by using (10.3) and Algorithm 10.1, and the East and North components are then estimated afterwards by using the optimised Vertical image, (10.6) and Algorithm 10.1. No frozen pixels were used when the uncertainty images were utilised in the MRF regularisation. Table 10.2 shows the estimated errors (the mean and standard deviation) between the correct and predicted motion fields images, when no errors are included in the interferometric and GPS observations. Table 10.3 shows the same, except a Gaussian noise with 0=µ and 5.0=σ has been added to the interferogram. Table 10.4 shows then the result of adding also a Gaussian noise with

0=µ and 2.0=σ to the sparse GPS measured motion fields. The results in Table 10.2 to 10.3 indicate that the method offers high certainty estimation of the Vertical motion field. This is a consequence of the high contribution from the Vertical ground deformation component to the observed Slant-Range-Shift (see (10.2)). Optimisation results for the other motion field image may depend on noise errors in the measurements. Experiments have also shown that the optimisation results for each individual motion image can be very dependent on the spatial distribution of the GPS locations. Despite of that, the tables indicate that the kriging errors can be reduced by using the relationship of the motion field images to the interferometric observations. Also a better result can be achieved for the East and North components (motion fields with the lowest contribution to the interferometric

Figure 10.3. Uncertainty image created with the kriging algorithm in Chapter 8. Sparse GPS loacations are marked as +.

(10.15)

(10.16)

51

observations) by using the energy functions (10.15) and (10.16) rather than (10.11) and (10.12). Table 10.2. An error estimation of the energy functions in (10.11) and (10.12) versus the energy functions (10.15) and (10.16). No errors are included in the interferogram or the GPS measurements. Vertical Horizontal look

direction East North

Kriging error µ 0.07 0.09 0.18 0.32 σ 0.44 0.79 1.03 0.66

Simulation error by using (10.11) and (10.12) and Algorithm 10.1 µ -0.02 0.03 0.13 0.23 σ 0.29 0.76 0.85 0.57

Simulation error by using (10.15) and (10.16) and Algorithm 10.1 µ -0.02 0.05 0.11 0.24 σ 0.28 0.72 0.83 0.56 Table 10.3. An error estimation of the energy functions in (10.11) and (10.12) versus the energy functions (10.15) and (10.16). A Gaussian random noise have been added to the interferogram. Vertical Horizontal look

direction East North

Kriging error µ 0.07 0.09 0.18 0.32 σ 0.44 0.79 1.03 0.66

Simulation error by using (10.11) and (10.12) and Algorithm 10.1 µ -0.03 0.06 0.13 0.06 σ 0.31 0.75 0.84 0.75

Simulation error by using (10.15) and (10.16) and Algorithm 10.1 µ -0.03 0.06 0.11 0.24 σ 0.31 0.73 0.86 0.58 Table 10.4. An error estimation of the energy functions in (10.11) and (10.12) versus the energy functions (10.15) and (10.16). A Gaussian random noise have been added to the interferogram and the GPS measurements. Vertical Horizontal look

direction East North

Kriging error µ 0.06 0.04 0.18 0.05 σ 0.54 0.83 1.04 0.93

Simulation error by using (10.11) and (10.12) and Algorithm 10.1 µ -0.02 0.03 0.11 0.22 σ 0.36 0.89 0.95 0.87

Simulation error by using (10.15) and (10.16) and Algorithm 10.1 µ -0.01 0.03 0.07 0.23 σ 0.36 0.86 0.82 0.86

10.3. Inferring the three dimensional motion field at the Reykjanes Peninsula Algorithm 10.1 was implemented into two programs. In addition to Algorithm 10.1, the programs calculate a mask image (Chapter 6), and pixels belonging to the background are kept frozen during the iteration run. This makes the algorithm faster.

52

The first program uses the energy functions given in (10.11) to (10.12) and frozen GPS pixels. The second one uses the energy functions in (10.15) to (10.16) and no frozen foreground pixels. The programs were then tested to infer the three-dimensional motion fields at the Reykjanes Peninsula. 10.3.1. Corrections of the data observations Chapter 7 presented a method to tilt wrapped interferograms by using a GPS measured Slant-Range-Shift. This tilting process can be used before unwrapping. A GPS tilting of an unwrapped InSAR image is though expected to be much safer than tilting of a wrapped InSAR image, since modulated effects have been removed. Here, the InSAR images are tilted again after unwrapping, with the GPS observations and the Least Square (LS) algorithm. This is done to ensure save tilting (more accurate elimination of the phase plane) and consequently better consistency between the GPS and interferometric measurements, before inferring the three-dimensional ground movements. The LS tilting is done by using (7.9) to (7.11) (see Section 7.2). Here, UwInSARI in (7.11) is the resulting interferogram from the unwrapping process described in Section 9.5. The LS tilting process uses a mask detection (Chapter 6), to extract region of interest after tilting of the unwrapped InSAR image. It is possible to use all the GPS measurements available for the kriging and the MRF optimisation. Also, it is not necessary to have all the motion components measured at same location, when using Algorithm 10.1 along with the energy terms in (10.15) and (10.16). But here, the following is done before utilising the GPS measurements: 1. time scale the GPS observations to fit the elapsed time interval represented by the

InSAR image, 2. remove GPS observations that are in bad consistency with the unwrapped and

tilted InSAR image. Dataflow diagram describing the usage of the ordinary kriging algorithm and the optimisation of the energy functions in (10.11) to (10.12) is given in Figure 10.4. Figure 10.5 shows the same when using the energy functions in (10.15) to (10.16), instead of the energy functions in (10.11) to (10.12). 10.3.2. Inferred three-dimensional motion maps Figure 10.6 to 10.9 shows the result of inferring the three-dimensional crustal deformation for the 4.17 years interferogram (Table 2.1). Figure 10.6 present the GPS observations and an ordinary kriging of the GPS observations. Figure 10.7 shows the result of combining the GPS observations and the 4.17 years interferogram. The original wrapped interferogram was pre- filtered with 3x3 moving average window. Results of inferring the crustal deformation from other images are given in Appendix B. Figure 10.7 (a), (c), (e) and (g) shows the result of inferring three-dimensional crustal deformation by using the energy functions in (10.11) and (10.12), and Figure 10.7 (b), (d), (f) and (h) shows the same when using the energy functions in (10.15) and (10.16). The ordinary kriged motion field images in Figure 10.6 were used as initial images in both optimisations. The residual errors between the interferogram ( SRSV ) and the combined predicted motion field images ( }{ NNEEVV VVV uuu ++− ) are also included in Figure 10.7.

53

The predicted Vertical, East and North motion field images describe similar deformation patterns in shape when using energy functions (10.11) to (10.12) versus the energy functions in (10.15) to (1016) (Figure 10.7). The main difference is that the motion field images observe more detailed characteristics from the interferogram in Figure 10.7 (a), (c) and (e), and from the initial kriged images in Figure 10.7 (b), (d) and (f). This is as expected since an additional spatial varying force is added to the process presented in the latter case, that tends to keep updated images close to the initial kriged images (see (10.13) and (10.14)). This does also explain why the motion field images look more smoothed in Figure 10.7 (b), (d) and (f) than in Figure 10.7 (a), (c) and (e). Table 10.5 shows the mean ( µ ) and standard deviation (σ ) between the Slant-Range-Shift ( SRSV ) and the combined motion field images ( EEVV{ VV uu +− ),

},NN Vu+ and also between the combined kriged motion fields images ( +− KVV{ Vu }KNNKEE VV uu + ) and the combined motion field images. Stronger relationship to

SRSV is achieved for the energy functions (10.11) to (10.12) than in (10.15) to (10.16). This is reversed when looking at the relationship to the kriging result (The GPS observations). Figure 10.8 shows a comparison of line 300 out of the 450x750 pixels Vertical, East and North motion maps, where (a), (b) and (c) shows the estimation from kriging, energy functions (10.11) to (10.12) and energy functions (10.15) to (10.16), respectively. The corresponding residual profile errors between the interferogram ( SRSV ) and the combined predicted motion field images ( }{ NNEEVV VVV uuu ++− ) are shown in Figure 10.9. It is also evident by comparing the plots in Figure 10.8, that the energy functions in (10.15) to (10.16) tends to keep the signal shape closer to the initial kriged result than the energy functions in (10.11) to (10.12). The residual errors in Figure 10.9 (b) are close to be white random noise, which indicates a strong relationship between the interferogram and the predicted motion fields. The process that utilises uncertainty images, is recommended if the GPS observations are assumed to include less error than the interferometric observations (Figure 10.5). If the interferometric signal is strong and error free then the process of excluding uncertainty images is recommended (Figure 10.4). Table 10.5. The mean and standard deviation of the difference between the interferogram ( SRSV ) and the simulated motions field images ( ,VV EV and NV ), and also the kriged motion

field images ( ,KVV KEV and KNV ) and the simulated motion field images.

NNEEVVSRS VVVV uuu +++ )()( NNEEVVKNNKEEKVV VVVVVV uuuuuu ++−++

Using the energy functions in (10.11) and (10.12) µ 0.005 0.405 σ 0.30 1.26

Using the energy functions in (10.15) and (10.16) µ -0.055 0.466 σ 0.40 1.18

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Figure 10.4. Dataflow diagram of the process of using combined GPS and interferometric observations to infer three-dimensional motion maps. The two-dimensional inferring of motion field images used in the diagram, are described by Algorithm 10.1 and the energy functions in (10.11) and (10.12).

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Figure 10.5. Dataflow diagram of the process of using combined GPS and interferometric observations to infer three-dimensional motion maps. The two-dimensional inferring of motion field images used in the diagram, are described by Algorithm 10.1 and the energy functions in (10.15) and (10.16).

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Figure 10.6. Infer of the three-dimensional crustal deformation at the Reykjanes Peninsula, by using an ordinary kriging of sparsely located GPS observations. (a), (c), (e): The sparsely located GPS measured Vertical, East and North motion vectors, respectively. (b), (d), (f): The result of inferring the Vertical, East and North motion maps, respectively.

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Figure 10.7. Infer of the three-dimensional crustal deformation at the Reykjanes Peninsula by using GPS measurements and 4.17 years interferogram. The figure shows comparison of motion field images optimised by using the energy functions in (10.11) to (10.12) versus the energy functions in (10.15) to (10.16). (a), (c), (e): The Vertical, East and North motion field images, respectively, inferred by using (10.11) to (10.12). (b), (d), (f): The same when using (10.15) to (10.16). (g): Residual error between the 4.17 years interferogram and the images in (a), (c) and (e). (h): The same for the images in (b), (d) and (f).

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Figure 10.8. Line 300 out of the estimated 450x750 pixels motion field images in Figure 10.6 and 10.7; (a): the result from kriging, (b): the result of using optimisation with the energy functions (10.11) to (10.12) into Algorithm 10.1, (c): the result of using optimisation with the energy functions (10.15) to (10.16) into Algorithm 10.1.

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Figure 10.9. Residual errors between the 4.17 years interferogram and the motion field profiles in Figure 10.6 and 10.7; (a): the residuals from kriging, (b): the residuals by using optimisation with the energy functions (10.11) to (10.12) into Algorithm 10.1, (c): the residuals by using optimisation with the energy functions (10.15) to (10.16) into Algorithm 10.1.

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11. Results and discussions In this thesis methodologies have been developed to combine interferometric and GPS measurements of ground movements. Two of the methods utilise a kriging algorithm, Markov Random Field regularisation and simulating annealing optimisation to unwrap the interferometric measurements and to infer high resolution maps of three-dimensional ground movements. Several other procedures have also been constructed for various purposes. For example algorithms that uses the GPS measured Slant-Range-Shift to eliminate a phase plane from interferometric observations, and a method that uses a vectorized filtering to reduce noise errors in wrapped InSAR images. InSAR images may include areas with no information. It has been useful when processing the images, to separate them into information and no information areas. For this purpose, a method that uses a threshold and a morphological cleaning algorithm was designed to extract areas of interest. All the methods discussed in this chapter, have been tested both on a test data and a real data set with very promising results. For the purpose of using the GPS and wrapped interferometric observations to infer high-resolution ground motion maps, the methods can be separated into a pre-processing and infer of three-dimensional high-resolution motion maps.

11.1. Pre-processing methods Pre-processing of interferometric and GPS observations can have several purposes, e.g. reducing error effects and to obtain a good consistency between the interfero-metric and GPS observations. 11.1.1. Noise reduction in interferometric observations Wrapped interferometric signal can be projected into two vectors perpendicular to each other (cosine- and sinusoidal), which can also be interpreted as a projection into the complex unit circle. Here, the cosine- and sinusoidal are filtered separately with a moving average window before doing the inverse projection. The vectorized filtering has turn out to be very efficient in reducing error effects in the interferometric measurements. Furthermore, it has been a very powerful tool to use in all the GPS and interferometric combinations. 11.1.2. Extraction of area of interest Another tool that has been useful in all the GPS and interferometric combinations, is an extraction of areas of interest. This is handled by creating a binary image (a mask) that separates the InSAR images into a foreground and a background. A binary image is created by thresholding a wrapped or an unwrapped interferogram, followed by a morphological cleaning process. The foreground areas can include error pixels after thresholding of the InSAR images Those areas can be fully removed with a morphological cleaning process. The mask images have been useful, both for masking of the processed images, and to restrict the data processing to only the areas of interest and hence, make the algorithms faster. 11.1.3. Utilisation of GPS observations to correct wrapped InSAR images A methodology that uses kriging of sparsely located GPS data, MRF based regularisation and simulating annealing optimisation have been developed to unwrap interferograms. The Master and Slave track of an interferogram can include two

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different viewpoints and distances to the same object, which results in a systematic error that can be described by a phase plane [1]. The phase plane needs to be eliminated from the wrapped interferograms before utilising the GPS data in the unwrapping process. The GPS data is not expected to include significant systematic errors, and can therefore be used to eliminate the phase plane in interferometric measurements. Two methods have been designed for this purpose. The first method uses a projection of both the interferometricly and GPS measured Slant-Range-Shift into the complex unit circle, and uses an optimisation in the complex domain to find the optimal rotation and offset of the interferogram. The drawback is that the method changes a unique solution into periodical solutions, which can lead to wrong estimation of the phase plan. The second method uses an unwrapping of line profiles between sparse GPS locations in the InSAR image, to estimate sparse unwrapped values that correspond to the GPS locations. The sparse unwrapped values and the sparse GPS measured Slant-Range-Shift are then used to find the phase plane by a Least Square estimation. The InSAR images from the Reykjanes Peninsula can include high atmospheric noise that can disturb the profile unwrapping and hence, lead to some errors when tilting the unwrapped interferograms. Despite of that, this process can be used before utilising the GPS measurements in the unwrapping process. 11.1.4. Creation of virtual InSAR images with interpolation of GPS data The unwrapping process can be initialised with a virtual InSAR image, for example created by an interpolation of a sparse GPS measured Slant-Range-Shift. Several methods have been tested for this purpose. The best result has been achieved with an ordinary kriging method, when a Gaussian semivariogram model is used to estimate the dispersion matrix for the sparse data. The ordinary kriging method assumes the sparse spatially distributed data to be first and second order stationary. This is not expected to hold in general for ground movements. However, very good results have been achieved by using the kriging method to interpolate between a sparse data of a test motion field image. 11.1.5. Unwrapping process The unwrapping process estimates the missing wave numbers of a wrapped interferogram, by using a MRF modelling and simulating annealing optimisation. The unwrapped interferogram (the optimal realisation image) is found by using an assumption about the surface smoothness and by using its relationship to the GPS observations. Several energy functions have been tested that requires the image surface to be smooth. The best results have been attained by requiring smoothness of the first derivative, implemented as a penalization on the second derivative. The sparse GPS measurements are the only data that is directly related to the wave numbers. The unwrapping process is therefore mainly controlled by the smoothness requirements. Indeed, experiments have shown that an interferogram can be unwrapped by only using a penalization on the second derivative, as long as the high frequency (both noise and information) is within certain limit. The weak relationship of the wave number estimation to the observations is expected to result in need for very slow temperature drop (cooling) in the simulated annealing optimisation. It has been shown for this case, that a reannealing procedure with a

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faster cooling can be successively used instead. The reannealing procedure uses repeated simulated annealing on the images until the optimal realisation image is found. It is not necessary to update all the image pixels during each reannealing. A thresholded edge detection followed by dilation have been used with good result to detect areas of interest before each reannealing. Only the detected pixels are then updated in each iteration, which makes the algorithms much faster. The GPS observations are utilised by freezing the pixel values corresponding to the sparse GPS locations (frozen pixel domains) during the optimisation, and to apply extra smoothness requirement for its neighbouring pixels. The frozen pixel domains are then expanded before each reannealing. The main advances of using the GPS observations are that an information about absolute pixel values have been included into the procedure, and the expected solution is reached with higher reliability. The InSAR images from the Reykjanes Peninsula are surrounded by an ocean, and can therefore be highly influenced by a high- and low frequency atmospheric noise. This can influence the penalization on the second derivative, and also leads to errors when expanding the frozen pixel domains. Experiments have shown that vectorized filtering can successfully be used to reduce high frequency noise. Furthermore, this can also reduce certain type of the unwanted low frequency error effects that may appear in interferograms. Experiments do also indicate that a wave number matrix for a non-smoothed wrapped interferogram can be estimated both accurately and in a simple way from a corresponding oversmoothed unwrapped interferogram. A process has been designed that take advantage of this. Before unwrapping, the wrapped InSAR images are oversmoothed by vectorized filtering. This makes the penalization on the second derivative more effective and reduces errors that can be generated during the expansion of the frozen pixel domains. Errors that may appear after full expansion of the frozen pixel domains are reduced by using a reannealing process that only uses penalization on the second derivative. This procedure has been used to unwrap the InSAR images from the Reykjanes peninsula with good results. 11.1.6. Utilisation of GPS observations to correct unwrapped InSAR images Methods have been constructed that uses MRF based regularisation and simulated annealing optimisation to infer high resolution two-dimensional motion field images, by combing GPS observations and an unwrapped interferogram. For the combination, a good consistency is needed between the GPS and interferometric observations. Hence, systematic errors like a phase plane need to be eliminated as accurately as possible. Attempts have been made to eliminate a phase plane from unwrapped interferograms. Experiments have shown that the phase plane elimination of a wrapped interferogram, designed and used in this thesis, may result in some errors. It is expected to be much more accurate to estimate the phase plane with GPS observations along with the unwrapped version of the interferogram. Therefore, a method that uses GPS observations and a Least Square optimisation have also been developed to eliminate a phase plane. This procedure can be used after unwrapping of the interferograms, to gain more accurate consistency between the two observation methods, before inferring the motion maps. Results indicate that a safer phase plane elimination can be achieved by using tilting of unwrapped interferograms than wrapped.

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11.2. Construction of three-dimensional high resolution motion maps The interferometric observations include a high-resolution maps of a one dimensional Slant-Range-Shift, while the GPS observations include information about three-dimensional motion fields at sparse locations. A methodology have been developed to infer high-resolution three-dimensional motion maps from combined GPS and interferometric observations. The problem of optimising the three-dimensional motion field can be separated into an optimisation of two two-dimensional motion fields images, and thereby making the algorithms simpler. The simulating annealing algorithm is then designed to find two realisation images in the same process. Here, the optimisation have been separated as follows. First the Vertical and Horizontal look-direction motion maps (Horizontal look-direction of the SAR satellite observations) are estimated from combined GPS and InSAR. Then the East and North motion maps are estimated by using combined GPS and InSAR along with the optimised Vertical motion map. A MRF based regularisation and a simulating annealing optimisation have been used for the two-dimensional combination with good results. The two-dimensional motion field optimisation procedure has to be initialised, e.g. by a kriging of two-dimensional GPS observations of the same motion fields. The ordinary kriging algorithm have shown to be very efficient in estimating motion field images from sparse observations. Hence, it has been used with good results to estimate the initial images for the combination procedure. One advantage of the ordinary kriging algorithm is it simplicity compared to other kriging algorithms. But, for further improvements, cokriging algorithms should also be considered and tested for the estimation of initial two-dimensional motion field images. The multidimensional motion fields have a strong relationship to the interferometric observations, which can be used as the main constraint in the MRF regularisation. More constraint are also needed, e.g. to preserve the neighbouring pixel relationship in the image structure. A smoothness requirement, implemented as a penalization on the second derivative, has been efficiently used for this purpose. It has also been shown that the relationship to the sparse GPS observations can also be utilised, e.g. by using an uncertainty estimation of the kriged images along with the MRF regularisation. The relationship to the GPS observations has successfully been used to reduce errors in the multidimensional motion field images at areas close to the sparse GPS locations. The optimal solutions of the multidimensional motion field images are illdefined when the correct values are known only at sparse locations. Therefore, the result from the combination optimisation can be very depending on the quality of the initial values. Hence, the quality of the solution is very dependent of the locations and density of the GPS observations. Despite of that, experimental results have shown that the kriging errors in estimated motion field images can be reduced by using the interferometric observations in the MRF regularisation, especially in the Vertical motion field due to its high contribution to the interferometric signal. The optimisation procedure uses a one-dimensional interferometric measurement along with the GPS observations, which can be either from ascending or descending orbit passes. In some cases, it is possible to use two interferograms from the same

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area, recorded from both ascending and descending satellite orbit passes. The ascending and descending orbit passes have two different viewpoints to the same object, and can in some cases be interpreted as a two dimensional interferometric observations. In that case, and additional constraint can easily be added to the MRF regularisation that makes advantage of the two-dimensional interferometric observations. The results of inferring multidimensional motion maps are expected to be more accurate if both the satellite passes can be utilised in the process.

11.3. Pre-processing and averaged motion maps at the Reykjanes Peninsula A test data set from the Reykjanes Peninsula, SW Iceland, has been used in this thesis. The plate boundary between the North-American and the Eurasian plates runs ashore at the SW tip of the Peninsula (Figure 2.4 and 2.6). The Peninsula consists also of eruptive fissures and volcanoes. The ground motion consists therefore of crustal deformations and plate movements. Available GPS measurements of ground movements describe a deformation for the elapsed time interval from 1993 to 1998. The interferograms are all recorded from a descending satellite pass, with various elapsed time intervals within 1992 to 1996. Measurements from the Peninsula indicate some non-linearity in the ground movements with time. Due to the differences in elapsed time intervals of the data set, the non-linearity needs to be considered before combining the GPS and interferometric observations. Experiments have though shown that a tolerable fit can be achieved between most of the 1993 to 1998 GPS observations and the interferograms with elapsed time intervals within 1992 to 1996. Here this is handled by using all the GPS observations when eliminating a phase plane from the interferograms, and reject GPS measurements that are inconsistent with the interferogram before unwrapping and inferring the multidimensional ground movements. It is though not necessary to reject any GPS data before inferring the multidimensional motion field maps. Figure 11.1 shows a result of inferring motion field maps at the Reykjanes Peninsula, by using data at various time intervals. The figure shows a one-year average motion fields, estimated by using the 1993 to 1998 GPS observations along with the 1992-1993, 1992-1995, 1992-1996 and 1993-1995 interferograms, or an elapse time of 5, 0.83, 3.12, 4.17 and 2.29 years respectively. Motion maps were estimated independently for each of the four interferograms. The averages were then created by weighting the images with its corresponding elapsed time intervals. The surface movements consist of complex mixture of subsidence, uplifting and surface rifting [2,4]. The plate movements are clearly seen in the East motion field image, where the South part of the Peninsula is moving towards the East, and the North part of it towards the West (a rift of a ~2 cm/yr). Also, some subsidences are clearly evident, especially at the cauldroun around Svartsengi. The subsidence at Svartsengi is estimated as ~2 cm/yr. The subsidence can also been seen from the Slant-Range-Shift motion field image. The East and North motion fields at the Svartsengi area does show that the cauldron surface is also moving towards the centre of the cauldron. Those effects of the subsidence can not be visualised by only using the interferometric observations. Furthermore, some subsidence is evident around the East part of the Peninsula towards the Bláfjallahryggur (Bláfjallhryggur is a long

65

ridge). At this area, the East and North motion field images does also indicate some tendency in the horizontal surface movement towards the ridge. Figure 11.1. Estimated one-year average motion field maps at the Reykjanes Peninsula. (a): Average Slant-Range-Shift, estimated by using interferograms with the elapsed time intervals 1992-1993, 1992-1995, 1992-1996 and 1993-1995. (b), (c) and (d): Average Vertical, East and North motion maps, respectively, inferred by the 1992-1993, 1992-1995, 1992-1996 and 1993-1995 interferometric measurements, combined with the 1993-1998 GPS observations. Motion maps were created independently for each of the four interferograms, and then the averages were created afterwards for each of the motion field components. For all the motion images, the average was created by weighting the motion field images with the corresponding elapsed time intervals. On the image in (a) is also shown its approximate location of the Svartsengi area and the Bláfjallahryggur ridge.

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12. Conclusion Various methods have been created and tested to process InSAR images and to combine GPS and InSAR observations, for the purpose of visualising the earth surface deformations. InSAR images contains a modulated measure of one-dimensional change in range from the ground to the satellite (Slant-Range-Shift), while GPS include accurate measurements of the three-dimensional deformation at sparse locations. It has been possible by combining those two complementary geodetic techniques, to design methods to eliminate errors in observations, correct the data and construct high resolution maps of three-dimensional ground movements. Those methods have been error tested with good results and used to combine GPS and InSAR observations from the Reykjanes Peninsula, Iceland, to infer high resolution motion maps of crustal deformations and plate movements. A method that uses vectorized filtering have been efficiently used to reduce noise errors in wrapped InSAR images. It has also been shown how systematic errors in InSAR observations can be eliminated by using GPS observations. A Markov Random Field (MRF) regularisation and simulating annealing optimisation have been tested, for the purpose of unwrapping InSAR images, with good results. The GPS measurements include information about the absolute values of the Slant-Range-Shift at sparse locations, which can be used in the MRF model and in initialisation of the unwrapping process. The utilisation of GPS observations in the unwrapping process gives an opportunity of including sparse measurements of the unwrapped image values, which results in a faster and safer optimisation process. An ordinary kriging of GPS measured Slant-Range-Shift have been very successfully used to create virtual InSAR images used to initialise the process. A method that uses a kriging of sparse GPS measurements, a MRF regularisation and simulating annealing optimisation have been developed and used to construct high resolution maps of three-dimensional motion fields. The procedure is initialised with the kriging of the GPS measurements, and the MRF regularisation and the simulating annealing is then used for further optimisation. The method has been error tested with good result. Very reasonable results have also been achieved when inferring the three-dimensional ground movements at the Reykjanes Peninsula. A one-dimensional ordinary kriging method has been used to create motion field images, both for the initialisation of the unwrapping algorithm and to construct motion field images. The ordinary kriging algorithm has the advance of being simple compared to other kriging methods, and has shown to be a very efficient tool to estimate the motion field images. Cokriging methods could also be tested for further improvements. The MRF regularisation, used for the construction of multidimensional motion field images, utilises a one-dimensional interferometric observation recorded from either ascending or descending satellite passes. It is often possible to combine InSAR images from both ascending and descending satellite passes, which can be interpreted as a two-dimensional interferometric observation. An additional constraint can easily be added to the MRF regularisation that would take advantage of this.

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References [1] D.Massonnet, K.L.Figel. Radar Interferometry and its Applications to

Changes in the Earth’s Surface. Reviews of Geophisic, Vol 36, No. 4, 1998. [2] S.Hreinsdóttir. GPS Geodetic Measurements on the Reykjanes Peninsula, SW

Iceland: Crustal Deformation from 1993 to 1998. A thesis submitted to the University of Iceland for M.Sc. in Geophysics, 1999.

[3] D.E. Fatland, C.S. Lingle 1998. Analysis of the 1993-95 Bearing Glacier

(Alaska) Surging using Differential SAR Interferometry. Journal of Glaciology, Vol 44, No 148, 1998.

[4] H.Vadon, F.Sigmundsson. Crusatal Deformation from 1992 to 1995 at the

Mid-Atlantic Ridge, SouthWest Iceland, Mapped by Satellite Radar Interferometry. Science, Vol. 257, 1997.

[5] S.Jónsson, N.Adam, H.Björnsson. Effects of Subglacial Geothermal Activity

Observed by Satellite Radar Interferometry. Geophysical Research Letters, Vol. 25, No. 7, p 1059, 1998.

[6] J.M.Carstensen . Digital Image Processing, Technical University of Denmark,

IMM 1998. [7] R.C.Conzales and R.E.Woods, Digital Image Processing, Addison-Wesley

Publishing Company, Inc., 1993. [8] J.C.Russ., The Image Processing Handbook, CRC Press Inc., 1992. [9] A.A. Nielsen. Geostatistik og Analyse af Spitielle Data. Institute for

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Science Workbench, 1995.

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Appendixes

A. GPS tilting of wrapped InSAR images; projection into the complex unit circle This appendix explains a titling method that uses optimisation of the difference between wrapped version of interferometricly and GPS measured Slant-Range-Shift. The method assumes the Slant-Range crustal deformation to be linear with time. Furthermore, it is assumed the only systematic error in the measurements to be tilting of the InSAR image. Both the interferometricly and GPS measured Slant-Range-Shift are projected into the unit complex circle. The optimisation is then done in the complex domain. Experiments indicate that this method can easily result in wrong tilting for images including a large tilting error. Therefore, another method has also been designed for this problem task, that estimates unwrapped values of the interferogram at sparse locations corresponding to the GPS sites (see Chapter 7).

A.1. The procedure The procedure is as follows. A sparse unwrapped image IGPS is created by calculating ∆ρ = u⋅s for all the sparse pixels, where u is the measured three dimensional GPS displacement and s is the unit vector pointing from the ground towards the satellite. A new unwrapped image is then calculated as

( ) ( ) [ ] ,,,1,,,, T2

jijijiji GPSGPS ∀⋅+= xII where x =[x1,x2,x3] is a vector including the coefficient of a two dimensional plane and i, j are the sparse row and columns numbers of IGPS, respectively. The vector x fulfils the conditions

),(min xx x f= where the objective function f is chosen to have global minimum when the residual error of the wrapped InSAR image ( )jiInSAR ,I and a wrapped version of the images

( )jiGPS ,2

I is at minimum for all i and j. A pre-filtered InSAR image, created with the procedure given in (A.1) and (A.2), can be used to generate InSARI . An optimal planar correction of the InSAR image is then calculated as

( ) ( ) ( )( ) ,,,,,2

2,2

clclclcl PlanInSARInSAR ∀∠−∠= CCIπ

λ

where l and c are the line and columns numbers of IInSAR respectively,

=

22

expλπ InSAR

InSARj I

C

and

(A.1)

(A.2)

(A.3)

(A.4)

69

( ) [ ]( ) ,,,2

1,,2exp,T

dlcljclPlan ∀

⋅=

λπ xC

where λ is the wavelength of the SAR satellite and x is given in (A.2). A.1.1. The first objective function The optimal value of x is found by using some arbitrary chosen initial values for x and an iteration algorithm to iterate to the minimum extreme of the objective function f(x). An example of objective function that can be used in (A.2) is

,),(),(,

1 2∑∀

+−=ji

InSARGPS jijif CC

where

=

2

2exp 2

2 λπ GPS

GPS

j IC

and CInSAR is given by (A.4). The entire complex values of both

2GPSC and InSARC in (A.6) lies on the complex unit circle. The function f1 has a global minimum when the complex unit vectors ( )jiGPS ,

2C and ( )jiInSAR ,C are as close as possible for all i and j

(lign up). The objective function in (A.6) leads to a periodical solution of (A.2), i.e.

[ ]2n,2n,2n 332211 λλλ +++= xxxx , for all integer numbers n1, n2 and n3. All the solutions of the offset 2n 33 λ+x are equivalent. The difference between the maximum and the minimum values of the optimal plane x⋅[l,c,1]T, ∀l, c, is though expected to be less than of order of three to four fringes, where one fringe correspond to displacement of λ/2. This means that the optimal solutions of the planar slopes given by '

1x and '2x should fulfil 2'

1 λ<<x and 2'2 λ<<x for high resolution

InSAR images. This leads to only one possible solution of the slopes given by 2nmin 112n

'1 11

λλ += + xx x and 2nmin 222n'2 22

λλ += + xx x for all n1 and n2. Images that explain the periodical behaviour of the function in (A.6) are shown in Figure A.1. In this particular example the optimal parameters are known to be [ ]3,1,2 −=x , the wavelength λ = 10 and the constant d = 1 in (A.1). The three images in Figure A.1 shows f1 as a function of x1 and x2 with x3 = 3, f1 as a function of x1 and x3 with x2 = -1 and f1 as a function of x2 and x3 with x1 = 2. The figure shows the periodical properties of f1, where the period λ/2 = 5 for all the coefficients. Pattern Search iteration algorithm [12] was used to find the optimal solution of (A.2) by using the function given in (A.6). The main reason for choosing the Pattern Search algorithm is its properties of being independent of derivatives. It is though on the cost of being relatively slower than other iteration algorithms that make use of derivatives or estimated derivatives.

(A.5)

(A.6)

(A.7)

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Figure A.1. The periodical behaviour of the function f1 in equation (A.6). In this particular example the optimal parameters are known to be [ ]3,1,2 −=x and the wavelength λ = 10. The main advance of f1 is that it can be used to optimise all the three coefficient needed for the planar correction in (A.3) in the same operation. A noise InSAR test images were created to evaluate the result of applying the function f1 in (A.2). The results indicate that the initial values of x1 and x2 need to be selected close to the optimal solution for the algorithm to not run into some local extreme solutions. The function is on the other hand insensitive for initial values of x3. A.1.2. The second objective function Another objective function that can be used to find the optimal slopes x1 and x2 was also tested. The function minimises the standard deviation of the angular difference

2GPSInSAR CC ∠−∠ where InSARC and 2GPSC are complex unit vectors given in (A.4)

and (A.7), respectively. The function is written as

( )C∠= Var2f where C is given as

( ) .,),,(),(,2

jijijiji GPSInSAR ∀= ∗CCC

(A.8)

(A.9)

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The function f2 can only be used to find the optimal slopes x1 and x2, but not optimal offset x3. The vector x is therefore written as [x1,x2,0] in (A.2) when using f2. The optimal offset x3 can be found as the mean value of C∠ in (A.9) after optimisation of x1 and x2 with f2. Results of applying the Pattern Search algorithm and f2 on noise experimental InSAR images indicated that the optimal solution of the slopes is much less dependent of the initial values of x1 and x2 than it is for the objective function f1. The function f2 leads also to periodical solution of the slopes. The image in Figure A.2 explains the behaviour of f2 as a function of x1 and x2. The optimal solution in this particular example is known to be x1 = 2, x2 = -1 and λ = 10, which is the same as for Figure A.1.

A.2. Correction of the InSAR data The process described in (A.1) to (A.3), with the objective function in (A.8) was implemented into a program and used to correct the InSAR data from Reykjanes peninsula. The program allows a pre-filtering of the InSAR images before optimisation. Initial values of the slopes x1 and x2 need to be selected in forehand for the iteration process. Since the difference between the maximum and the minimum values of the optimal plane are expected to be less than of order of two to three fringes (conditional iteration), the initial values can be found as follow: 1. Two vectors [ ] [ ]

111111max,max,,min,min,,, n112111 xxxxxxxxx ∆−∆+== KKx and

[ ] [ ]222222

max,max,,min,min,,, m222212 xxxxxxxxx ∆−∆+== KKx are created by the program, where ,min

1x ,max1x

2min x and

2max x are the minimum and

maximum allowed values of the slopes of x1 and x2 respectively and 1x∆ and

2x∆ are the step size of the two vectors x1 and x2 respectively.

2. The value selected as initial values are [ ] [ ] ( )j2i12,j2i1 ,min,j2i1

xxfxx xx= for all possible combination of i and j.

Figure A.2. The periodical behaviour of the function f2 in equation (16). In this particular example the optimal parameters are known to be [ ]0,1,2 −=x and the wavelength λ = 10.

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Figure A.3. Iteration run of the Pattern Search algorithm for the objective function 2f and 0.83 years InSAR image. The iteration algorithm continues to search for optimal values of x1 and x2 by using the initial values x1i and x2j. An example of the iteration run of the Pattern Search algorithm is shown in Figure A.3, when using the 0.83 years InSAR image. Figure A.4 shows two InSAR images before and after correction with the program and the corresponding angular residuals GPSInSAR CC ∠−∠ and GPSInSAR CC ∠−∠

2, where

CInSAR is given as

⋅=

22

exp 2

2 λ

π InSARInSAR

j IC

and

2InSARI is given in (A.3). The images present 3.12 years and 0.83 years of deformation and have a master and slave orbits 5565 and 21941, and 5565 and 10575 respectively. The residuals include more random noise after correction and some information that was not evident in the InSAR images becomes clearer.

A.3. Removing unwanted areas from the corrected images The Reykjanes peninsula is surrounded by sea, which are displayed as no information in the InSAR images (zero valued pixels). Those areas with no information do not necessarily consist of zero pixel values after the planar correction of the InSAR images. The method described in Chapter 6 is used to create a mask image. The resulting tilted image is then masked by pixelvice multiplication of the mask and tilted InSAR images. The process described in this appendix is shown as dataflow diagram in Figure A.5.

(A.10)

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Figure A.4. Result of correcting InSAR images with GPS measurements; (a) and (b) shows the original 3.12 years InSAR image and the angular difference between the original 3.12 years InSAR and GPS measurements respectively, and (c) and (d) shows the same after correction of the 3.12 years InSAR image after correction. (e) to (f) shows the same for 0.83 years InSAR image.

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Figure A.5. Dataflow diagram of the process.

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B. Processed images This appendix shows the result of unwrapping, tilting of unwrapped images and constructing three-dimensional motion field maps, for the four interferograms that have the highest altitude of ambiguity (Table B.1 and Table 2.1). Results are displayed in Figure B.1 to B.4. The original wrapped interferograms are shown in all the figures. Before processing, the interferograms were vectorized filtered with 3x3 moving average window (see vectorized filtering in Chapter 5). The images in (c) in Figure B.1 to B.4 shows unwrapped tilted interferograms (see the tilting described in Section 10.3.1). The images in (d) in Figure B.1 to B.4 shows then wrapped version of the unwrapped and tilted interferograms in (c). Wrapped versions of the tilted interferograms are included for comparison to the original wrapped untilted interferograms. Here, the inferring of the three-dimensional motion fields was done by using Algorithm 10.1 and the energy functions given in (10.11) and (10.12) (described in Section 10.2). The reliabilities of the results given in Figure B.1 to Figure B.4 are expected to depend on the signal strength and the noise level of the data, which varies a lot between the interferograms. Also, some errors may appear due to discrepancy between the GPS and interferometric observations, since they are not describing crustal deformation at the same time intervals. The inferred Vertical, East and North motion maps do though describe on general similar deformation pattern in shape, especially for the area around Svartsengi where the signals are strong due to large ground movements (see the location of Svartsengi in Figure 2.6). At this area, the image shows subsiding cauldron, approximately axisymmetric in shape [4]. The subsidence is clearly seen in the Vertical motion maps, but the East and North motion maps indicates a horizontal motions towards the centre of the cauldron. These horizontal motions information are not clearly seen in the original interferograms. Table B.1. Characteristics of the InSAR images used in this appendix. Master orbit Date of

observation Slave orbit Date of

observation Elapsed time

Altitude of ambiguity

ha 5565 08.08. ‘92 10575 24.07. ‘93 0.83 years 59.0 m 5565 08.08. ‘92 21941 25.09. ‘95 3.12 years 43.6 m 5565 08.08. ‘92 7278 09.10. ‘96 4.17 years 22000 m 10575 24.07. ‘93 21941 25.09. ‘95 2.29 years 166.0 m

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Figure B.1. Uncorrected and corrected InSAR images, and inferred crustal deformation by combined GPS and InSAR measurements. The GPS measurement represent deformation from 1993 to 1998 and the InSAR observations from 1992 to 1993 (0.83 yr). (a): Original wrapped interferogram, (b): original wrapped interferogram vectorized filtered with 3x3 moving average window, (c): unwrapped and tilted interferogram, (d): wrapped version of the interferogram in (c). (e), (f) and (g): Inferred Vertical, East and North crustal deformation maps, respectively. (h): Residual error between the unwrapped InSAR image and the combined Vertical, East and North deformation images.

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Figure B.2. Uncorrected and corrected InSAR images, and inferred crustal deformation by combined GPS and InSAR measurements. The GPS measurement represent deformation from 1993 to 1998 and the InSAR observations from 1992 to 1995 (3.12 yr). (a): Original wrapped interferogram, (b): original wrapped interferogram vectorized filtered with 3x3 moving average window, (c): unwrapped and tilted interferogram, (d): wrapped version of the interferogram in (c). (e), (f) and (g): Inferred Vertical, East and North crustal deformation maps, respectively. (h): Residual error between the unwrapped InSAR image and the combined Vertical, East and North deformation images.

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Figure B.3. Uncorrected and corrected InSAR images, and inferred crustal deformation by combined GPS and InSAR measurements. The GPS measurement represent deformation from 1993 to 1998 and the InSAR observations from 1992 to 1996 (4.17 yr). (a): Original wrapped interferogram, (b): original wrapped interferogram vectorized filtered with 3x3 moving average window, (c): unwrapped and tilted interferogram, (d): wrapped version of the interferogram in (c). (e), (f) and (g): Inferred Vertical, East and North crustal deformation maps, respectively. (h): Residual error between the unwrapped InSAR image and the combined Vertical, East and North deformation images.

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Figure B.4. Uncorrected and corrected InSAR images, and inferred crustal deformation by combined GPS and InSAR measurements. The GPS measurement represent deformation from 1993 to 1998 and the InSAR observations from 1993 to 1995 (2.29 yr). (a): Original wrapped interferogram, (b): original wrapped interferogram vectorized filtered with 3x3 moving average window, (c): unwrapped and tilted interferogram, (d): wrapped version of the interferogram in (c). (e), (f) and (g): Inferred Vertical, East and North crustal deformation maps, respectively. (h): Residual error between the unwrapped InSAR image and the combined Vertical, East and North deformation images.

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C. MatLab functions This appendix explains the applications some of the MatLab functions created and used in this study, and includes help manuals available for each function. Some usage examples are given. The MatLab functions are located in a new toolbox, which is called “deform”. The toolbox can be added to MatLab by defining the path to it in the MatLab path browser.

C.1. Content list The functional content list available for the toolbox “deform” can be listed in the MatLab window by typing “help deform”. Usage information for each of the functions can then be listed by typing ”help name”, where “name” refers to the name of a MatLab functions. The toolbox content list displayed in the MatLab window is as follow: InSAR and GPS combination. BACKGROUND: Finds a background in an image, correct area of interest and applies the same background to another image. MASK: Finds a mask for an image. VINSAR: Create virtual unwrapped InSAR images calculated by spline of sparse three-dimensional GPS measured displacement. VINSAR2: Create virtual unwrapped InSAR images calculated by spline of sparse three-dimensional GPS measured displacement. TILT: Function that that finds an optimal planar rotation of wrapped input InSAR image. Uses wrapped format of both GPS and InSAR measurements PROFILE_TILT: Function that that finds an optimal planar rotation of an input InSAR image. Use profiles to estimate unwrapped InSAR values. LINE_UNWRAP: Function that unwrap a single line or profile KRIG1D: 1D kriging of sparsely located data KRIG2D: 2D kriging of sparsely located data LOLA2PIX: Converts longitude and latitude values to pixel values. MA: Filtering of wrapped InSAR images (projected onto the complex unit circle). MK: Least square solution of z=a1x+a2y+a3. OPENING: Opens an uint8 InSAR image. PATTERN: Implementation of Pattern Search method; PLAN_CORRECT: Function that subtracts a linear plane from a wrapped InSAR image. PLAN_CORRECT2: Function that subtracts a linear plane from a wrapped InSAR image. UNWRAP_GPS: Unwrap InSAR images with help of GPS measurements. UNWRAP_SMOOTHN: Unwrap InSAR images by only using smoothness requirements. UNWRAPPED: Wave numbers estimated from oversmoothed unwrapped interferogram SIMUL_2D: Function that estimates 2D crustal deformation by combined InSAR and GPS WEIGHTSIMUL_2D: Function that estimates 2D crustal deformation by combined InSAR and GPS WRAP: Function that calculates the wrapped version of an unwrapped InSAR image. TILT_UNWRAP: Use GPS measurements to and LS estimation to tilt unwrapped InSAR image. FYLKI: Puts GPS measurements into three-dimensional sparse matrix.

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DIFFERENCE: Angular difference between GPS measured dhro and InSAR measured dhro. HORN: Function that finds the magnitude and the angle of the longitude and latitude displacement fields.

C.2. OPENING The function “opening” reads into MatLab an image file with an unsigned integer format. The output is a matrix (image) including 8 bit values within the interval [0,255]. The function call is x=opening('name',N,M,headerbite); where “'name'” is a string including the file name. Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help opening”. The help manual is as follow: OPENING: Opens an uint8 InSAR image. x=opening('name',N,M,headerbite) Input: name: A string including the file name N,M: The row and column numbers of the image, respectively. headerbite: The bite used in the header. Output: x: Matrix including the NxM image. Example C.1. The data file including 900x1200 pixel InSAR image with Master and Slave track 5565 and 21941, respectively (3.12 years of deformation), has the name “a5565_21941.pbm”. The file includes 16 bite of header information. The file is written into the 900x1200 matrix “insar” by using: insar=opening('a5565_21941.pbm',900,1200,16);

C.3. MA The function “ma” is used for vectorized moving average filtering of a wrapped InSAR image (see Chapter 5). The function call is insar_out=ma(insar_in,n,m); Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help ma”. The help manual is as follow: MA: Filtering of wrapped InSAR images (projected onto the complex unit circle). insar_out=ma(insar_in,n,m); Function that convert an input modulo InSAR image with pixel values lying in the interval [0,max_value] into unity complex vector,

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by converting 0 to 0*pi and max_value to 2*pi. The function then calculates moving average of the Re and Im part of the complex vector by using a window of the size nxm. Finally the function return a filtered version of InSAR image with pixel values in the interval [0,max_value]. Input: insar_in: An InSAR image to be filtered. n,m: The size of the moving average window (line and column). Output: insar_out: Filtered InSAR image. Example C.2. The InSAR image in Example C.1 can be filtered with 3x3 window by using insar =ma(insar,3,3);

C.4. LOLA2PIX The function “lola2pix” is used to convert longitude and latitude coordinate values of GPS points into corresponding row and line numbers of an InSAR image. The function call is gps_out=lola2pix(gps,n,m,lo1,la1,dlo,dla); Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help lola2pix”. The help manual is as follow: LOLA2PIX: Converts longitude and latitude values to pixel values. gps_out=lola2pix(gps,n,m,lo1,la1,dlo,dla); Input: gps: Matrix with the columns information: [(1. lo. position), (2. la. position), (3. movement lo.), (4. movement la.), (5. movement up)]. n,m: Number of lines and columns of the reference InSAR image (nxm matrix). lo1: Longitude co-ordinate corresponding to the lower left corner of the InSAR image. lo1: Latitude co-ordinate corresponding to the lower left corner of the InSAR image. dlo: The pixel resolution in lo. direction. dla: The pixel resolution in la. direction. Output: gps_out: Matrix with the columns information: [(1. line position), (2. column position), (3. movement lo.), (4. movement la.), (5. movement up)].

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Example C.3 A GPS file from the Reykjanes Peninsula includes three point measurements. The column order is [1. Longitude coordinates, 2. Latitude coordinates, 3. Slant-Range-Shift (cm/yr)]. The GPS measurements are given as: gps=[-21.3654 63.9086 0.2800 -21.4645 63.9884 -0.8280 -21.4847 64.0611 -1.1660]. The following parameters are known for the 900x1200 pixel InSAR image in Example C.1 lo1=-23; la1=63.75; dlo=1/600; dla=1/1200; The longitude and latitude coordinates are converted into pixel location corresponding to the InSAR image by using gps=lola2pix(gps,900,1200,lo1,la1,dlo,dla) which change the MatLab variable “gps” to gps=[711 981 0.2800 615 921 -0.8280 528 909 -1.1660].

C.5. MASK The function “mask” is used to find a binomial mask image for an InSAR image (see Chapter 6). The function call is mask_out=mask(insar_in); Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help mask”. The help manual is as follow: MASK: Finds a mask for an image. mask_out=mask(insar_in); Input: insar_in: Input image. Output: mask_out: The binomial mask image including 0 values for bacground and 1 values for forground.

C.6. MK The function “mk” is used to find Least-Square (LS) parameter estimation for a linear plane (see Section 7.2). The function call is a=mk(x,y,z);

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Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help mk”. The help manual is as follow: MK: Least square solution of a1, a2 and a3 for the linear plane given by z=a1x+a2y+a3. a=mk(x,y,z); Input: x is input vector. y is input vector. z is output vector. Note: x, y and z need to be a column vectors of the same size. Output: a = [a1;a2;a3] is the LS solution of the coefficients in z=a1x+a2y+a3.

C.7. PROFILE_TILT The function “profile_tilt” is used to find a planar correction of a wrapped InSAR image by using the GPS measured Slant-Range-Shift. The function uses unwrapping of image profiles to estimate unwrapped values of the InSAR image. Least-Square (LS) method is then used to estimate the planar correction (see Chapter 7). The function call is insar_out=profile_tilt(insar,gps,tol); Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help profile_tilt”. The help manual is as follow: PROFILE_TILT: Function that finds an optimal planar correction of an input InSAR image. insar_out=profile_tilt(insar,gps,tol); Uses unwrapping of profiles to estimate unwrapped InSAR values. All data should be described in the same unit (e.g. cm). and describe deformation over the same time interval. Input: insar: The input InSAR image with pixel values lying in the interval [0,tol]. gps: The input GPS file with values defined at sparse points. The column order should be [(1. line position), (2. column position), (3. Slant-Range-Shift)]. tol: The tolerance (f.ex. 2.835 cm for ERS-1 and ERS-2). Output: insar_out: The corrected InSAR image with pixel values lying in the interval [0,tol]. Example C.4.

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The 3.12 years InSAR image (matrix) in Example C.1 can be scaled to the interval [0,2.835] by using insar=insar*2.835/255; If the resulting variable “gps” from Example C.3 describe the Slant-Range deformation in cm/yr, then it can be used to tilt “insar” with the function call insar =profile_tilt(insar,[gps(:,1:2) gps(:,3)*3.12],2.835);

C.8. KRIG1D The function “krig1d” is used to interpolate between sparsely located data. The function uses ordinary kriging and semivariogram model to calculate optimal weights (see Chapter 8). The semivariogram model is calculated with help of Pattern Search iteration algorithm (see Section C.9). An uncertainty image can also be created with the function (see Section 8.5). The function call is [krig,uncer]=krig1d(gps_input,N_row,N_column,delta_h,h_max,test); or krig1d(gps_input,N_row,N_column,delta_h,h_max,1); If “test=1” then a plot of the estimated semivariogram model and its agreement to the data is displayed on the screen. This can be used to optimise the two parameters “delta_h” and “h_max”. Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help krig1d”. The help manual is as follow: KRIG1D: 1D kriging of sparsely located data [krig,uncer]=krig1d(gps_input,N_row,N_column,delta_h,h_max,test) or krig1d(gps_input,N_row,N_column,delta_h,h_max,1) Input: gps_input: A data matrix including the three columns [1. row-number, 2. column-number, 3. sparse-pixel values]. N_row,N_column: Number of rows and columns of the output image krig (i.e. the output image size is N_row x N_column). delta_h: Average combination of points (N(h)) used to estimate the semivariogram. (gamma(h)= 1/(2N(h))sum_k(z(k)-z(k+h)))^2. h_max: Maximum length (in pixels) used to estimate the semivariogram model. test: If test=1, then a plot is made of the estimated semivariogram and the semivariogram model. This option can be used to optimise or test the best suitable parameters used for the kriging model. If test=0, then a kriging is done with the parameters defined in function. Output: krig: The resulting kriged image. Note: only available

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if test=0, if test=1 then the krig matrix include no information. uncer: An uncertainty matrix. the value 1 is the lowest uncertainty and 0 the highest. Figure C.1. Two example of semivariogram models displayed by the function “krig1d” . Note that ∆h=∆h stands for the delta_h used in the function call. Example C.5. Figure C.1 (a) and (b) shows an example of semivariogram models created with the function call krig1d(gps,900,1200,delta_h,h_max,1) where “gps” is the GPS measure of the Slant-Range-Shift at the Reykjanes Peninsula, with locations in pixels that correspond to the 900x1200 InSAR image in Example C.1 (see Section C.1 and C.4). The semivariogram models in (a) and (b) are result of using delta_h=20; h_max=600; and delta_h=20; h_max=200; respectively. The semivariogram model in (b) is in more agreement with the data. In this example, the kriged Slant-Range-Shift could be calculated by the function call krig=krig1d(gps,900,1200,20,200,0);

C.9. PATTERN The Pattern Search iteration method for non-linear optimisation is implemented into the function “pattern” (see [12], Chapter 7 and Appendix A). The function call is [f,y,iter,f_iter,x_iter,delta_iter]=pattern('FUN',x,delta,Iter); where “'FUN'” is a string including a name of a MatLab function. The function “FUN” should return a scalar function value (F=FUN(X)), where X is an input vector.

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Example for the semivariogram model in (8.7), [ ]TRCC ,,X 10= and ( ).,,F 10

* RCCγ= Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help pattern”. The help manual is as follow: PATTERN: Implementation of Pattern Search method; [f,y,iter,f_iter,x_iter,delta_iter]=pattern('FUN',x,delta,Iter) Input: 'FUN': String including the name the function to be minimised (an M-file: FUN.M). The function 'FUN' should return a scalar function value: F=FUN(X). x: The initial value (vector including initial parameters). Delta: The initial steplenght. Iter: Maximum number of iterations. Output: f: The final function value of f(y) (y=min_over_x(f(x)) for all x). y: The final value of the optimal point (y=min_over_x(f(x)) for all x). f_iter: Is f as a function of iterations (convergens of f). x_iter: Is the value of x as a function of iteration (convergence of x to the optimal value y). delta_iter: Is the value of delta as a function of iterations. Iter: The number of iterations used.

C.10. UNWRAP_GPS The function “unwrap_gps” use simulating annealing and Markov Random Field (MRF) modelling to unwrap a tilted InSAR image (see Chapter 9). The function use sparsely located GPS measured Slant-Range-Shift (or reference points) to guide the process. The process can also be initialised for example by using kriging of GPS measured Slant-Range-Shift, created by the MatLab function “krig1d” (see Section C.8). The implementation is done by using Algorithm 9.1 and the energy functions in (9.7) and (9.8). The function uses rennealing process, where the area of interest is detected by thresholded edge detection after each annealing (see Section 9.4.2 and 9.4.3). The function call is [vinsar,n]=unwrap_gps(insar,gps,krig,la); Further information about input and output variables and parameters can be listed by using “help unwrap_gps” in the MatLab window. During the iteration, the process temperature (90 at the start and 2 at the end of each annealing), number of iterations and number of reannealing are displayed in the MatLab window. Furthermore, after each annealing, the image status is displayed on the computer screen. Also, the resulting image after each annealing is saved into the MatLab file “vinsar.mat”, that is available if the process has to be terminated before finishing. The help information listed with “help unwrap_gps” is as follow: UNWRAP_GPS: Unwraps InSAR images with help of GPS measurements. [vinsar,n]=unwrap_gps(insar,gps,krig,la)

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The function use Simulating Annealing and MRF modelling to unwrap InSAR image, and also guiding with GPS measured Slant-Range-Shift. During the iteration, the function does display the temperature constant (90 at the beginning of one annealing and 2 at the end), the iteration number, the number of annealing. The image status is displayed after each annealing. Input: gps: Matrix with the columns information: [1. row numbers, 2. column numbers, 3. Slant-Range- Shift(cm)]. Note that the row- and column numbers need to be in consistency with the InSAR pixel numbers and the GPS measured Slant-Range-Shift should be scaled To the same time interval as the InSAR image. insar: The wrapped InSAR image. krig: Kriged virtual unwrapped InSAR images created by the function KRIG1D. See help krig1d in the MatLab window. Note that krig and insar need to be of the same size, with same unit and include the same deformation time interval as the InSAR image. la: Half the wave length in cm (2.835 for ERS). Output: vinsar: The unwrapped InSAR image. Note that the image can include errors that can be corrected further by the function UNWRAP_SMOOTHN. See help unwrap_smoothn in the MatLab window n: the wave number added to the input image insar to create vinsar. Note: after each reannealing, the function does also save the resulting image in a MatLab file called vinsar.mat. This file is available when the function is interrupted. Example C.6 An example of function call is vinsar=unwrap_gps(insar,[gps(:,1:2) gps(:,3)*3.12],krig*3.12,2.835); where “insar” is for example over-smoothed 3.12 years InSAR image (Example C.2), “gps” (cm/yr) is the GPS measured Slant-Range-Shift (Example C.4) and “krig” is the kriging of “gps” (Example C.5).

C.11. UNWRAP_SMOOTHN The function “unwrap_smoothn” use simulating annealing and Markov Random Field (MRF) modelling to unwrap a tilted InSAR image (see Chapter 9). The process can be initialised for example by using kriging of GPS measured Slant-Range-Shift, created by the MatLab function “krig1d” (see Section C.8). The implementation is done by using Algorithm 9.1 and the energy function in (9.7) (only penalization on the second derivative). The function uses rennealing process, where the area of interest is detected by thresholded edge detection after each annealing (see Section 9.4.2 and 9.4.3). The function call is vinsar=unwrap_smoothn(insar,krig,la,max_rean);

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Further information about input and output variables and parameters can be listed by using “help unwrap_smoothn” in the MatLab window. During the iteration, the process temperature (90 at the start and 2 at the end of each annealing), number of iterations and number of reannealing are displayed in the MatLab window. Furthermore, after each annealing, the image status is displayed on the computer screen. Also, the resulting image after each annealing is saved into the MatLab file “vinsar.mat”, that is available if the process has to be terminated before finishing. The help information listed with “help unwrap_smoothn” is as follow: UNWRAP_SMOOTHN: Unwrap InSAR images by using only smoothness requierments. vinsar=unwrap_smoothn(insar,krig,la,max_rean) The function use Simulating Annealing to unwrap InSAR image. During the iteration, the function does display the temperature constant (90 at the beginning of one annealing and 2 at the end), the iteration number, the number of reannealing. The image status is displayed after each annealing. Input: insar: Wrapped or unwrapped InSAR image. krig: Kriged virtual unwrapped InSAR images created by the function KRIG1D. See help krig1d in the MatLab window. Note that krig and insar need to be of the same size, with same unit and include the same deformation time interval as the InSAR image. If the process is not initilised with kriged image, then krig=insar in the function call. la: Half the wave length in cm (2.835 for ERS). max_rean: Maximum number of reannealing. Output: vinsar: The unwrapped InSAR image. Note: after each reannealing, the function does also save the resulting image in a MatLab file called vinsar.mat. This file is available when the function is interrupted. Example C.7. The function can be used for further correction of the resulting unwrapped image from the unwrapping function “unwrap_gps” (Example C.6), e.g. vinsar=unwrap_smoothn(vinsar,vinsar,2.835,max_rean); where “max_rean” has to be chosen. The function can also bee used to unwrap for example over-smoothed 3.12 years InSAR image “insar” (Example C.2), that could be initialised by “krig” created by the MatLab function “krig1d” (Example C.5). The function call in that case is vinsar=unwrap_smoothn(insar,krig,2.835,max_rean);

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C.12. UNWRAPPED The function “unwrapped” estimates missing wave numbers for non-smoothed wrapped InSAR image, by using oversmoothed unwrapped version of the same InSAR image (see Section 9.5 and (9.12)). The unwrapped version of the non-smoothed InSAR image is then calculated by adding the missing wave numbers to the non-smoothed wrapped InSAR image. The function call is [insar_out,n]=unwrapped(insar,vinsar,la); Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help unwrapped”. The help manual is as follow: UNWRAPPED: Wave numbers estimated from oversmoothed unwrapped interferogram [insar_out,n]=unwrapped(insar,vinsar,la); Function that calculates a wave number matrix for a wrapped interferogram by using oversmoothed unwrapped version of the same interferogram. Input: insar: Wrapped non-smoothed interferogram. vinsar: Unwrapped oversmoothed interferogram. Note that vinsar should be oversmoothed and unwrapped version of insar. la: Half the wave length in cm (2.835 for ERS). Output: insar_out: The unwrapped version of the non-smoothed interferogram insar. n: the estimated wave number (n=round((vinsar-insar)/la)) Note that insar_out=n*la+insar. Example C.8. An oversmoothed wrapped InSAR image can be created by using the MatLab function “ma” (see Example C.2). The oversmoothed InSAR image can then be unwrapped by using the MatLab functions “unwrap_gps” and “unwrap_smoothn” (see Example C.6 and C.7). If “vinsar” is the oversmoothed unwrapped InSAR image and “insar” is the wrapped non-smoothed InSAR image, then the unwrapped version of “insar” can be calculated as: insar_uw=unwrapped(insar,vinsar);

C.13. TILT_UNWRAP The function “tilt_unwrap” is used to find a planar correction of an unwrapped InSAR image by using the GPS measured Slant-Range-Shift. Least-Square (LS) method is used to estimate the planar correction (see Section 10.3.1 and 7.2). The function call is insar_out=tilt_unwrap(insar_in,gps);

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Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help tilt_unwrap”. The help manual is as follow: TILT_UNWRAP: Use GPS measurements to and LS estimation to tilt unwrapped InSAR image. insar_out=tilt_unwrap(insar_in,gps); Input: insar_in: Unwrapped InSAR image. gps: GPS measured deformation. gps should include the column [1. (row position) 2. (column position) 3. (Slant-Range-Deformation)]. Output: insar_out: The tilted unwrapped InSAR image. Note: both InSAR and GPS should represent the same time interval and in the same unit. Example C.9. The unwrapped 3.12 years InSAR image “insar_uw” in Example C.8 can be tilted with the resulting GPS variable “gps” (cm/yr) from Example C.3 with insar_out=tilt_unwrap(insar_uw,[gps(:,1:2) 3.12*gps(:,3)]);

C.14. SIMUL_2D The function “simul_2d” infer two-dimensional deformation by combining GPS observations and unwrapped InSAR image. The function call is [V1,V2]=simul_2d(Uw,gps,V1_init,V2_init,c1,c2,s2); The GPS measurements include sparsely located measurements of the two-dimensional deformation, while the InSAR image includes high-resolution map (image) of the Slant-Range-Shift (“Uw”). Simulating annealing algorithm is used for the iteration (see Algorithm 10.1 in Section 10.2.1). The function is initialised with two motion field images (“V1_init” and “V2_init”), created for example with the MatLab function “krig1d” (see Section C.8), and returns high-resolution maps “V1” and “V2”, optimised by combined GPS and InSAR. The function minimise the difference “(Uw+c1*V1+c2*V2)^2”, where “[c1,c2]” is a unit vector (see Section 10.1), and additional constrains are smoothness of the second derivative of the image surfaces (see the energy functions in (10.11) and (10.12) in Section 10.2.2.). Values corresponding to sparse GPS locations in the image are not updated during the iteration. The function offers the possibility of optimising only one of the motion fields by keeping the other unchanged. This can be useful if one of the two motion images is known with high certainty. During the iteration, the process temperature (5 at the beginning and 0.1 at the end) and the iteration number is displayed on the MatLab window. Also, the image status is displayed on the screen, updated after each 10 iterations.

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Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help simul_2d”. The help manual is as follow: SIMUL_2D: Function that estimates 2D crustal deformation by combined InSAR and GPS [V1,V2]=simul_2d(Uw,gps,V1_init,V2_init,c1,c2,s2) The function optimise the difference (Uw+c1*V1+c2*V2)^2. Additional constrains are smoothness of the first derivative. Simulating annealing iteration process is used to optimise V1 and V2. Input: Uw: Unwrapped interferogram or image that fulfils Uw=-[V1,V2][c1,c2]^T. The image must be error corrected (offset and tilting) and include pixel information in cm. gps: A data matrix including the four columns [1. row-number, 2. column-number, 3. sparse pixel-values of V1 (in cm), 4. sparse pixel-values of V2 (in cm)]. Note: the InSAR and GPS measurement must describe equally Long time intervals. V1_init: The initial V1 image (in cm) (f.ex vertical deformation image). V2_init: The initial V2 image (in cm) (f.ex. horizontal look- direction deformation image). V1_init and V2_init can be found f.ex. by kriging. c1,c2: [c1,c2] is a two dimensional unit vector. Uw=-[V1,V2][c1,c2]^T; s2: If s2=0, then only V1 is optimised by keeping V2=V2_init. If s2=1, then both V1 and V2 are updated in the iteration process. Output: V1,V2: Optimised motion images. Note: The initial input images should be ordered in the way that c1>=c2, when both the images are optimised (s2=1); Example C.10. The three-dimensional unit vector for the unwrapped and corrected 3.12 years InSAR image “insar_out” in Example C.9 is given as [ ].V935.0,N095.0,E34.0 −=s If the Vertical motion field image “Vv” is known (for example from two-dimensional optimisation of Vertical and Horizontal look-direction motion field images, see (10.3) in Section 10.1.1), then the East and North motion field images “Ve” and “Vn”, respectively, can be optimised by [Ve,Vn]=simul_2d(insar_out+0.935*Vv,gps,Ve_init,Vn_init,0.34, … -0.095,1); where “gps” includes GPS measured East and North deformation, respectively, and “Ve_init” and “Vn_init” are the initial motion field images created by the MatLab function “krig1d” (see Section C.8).

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C.15. WEIGHTSIMUL_2D The function “weightsimul_2d” infer two-dimensional deformation by combining GPS observations and unwrapped InSAR image. The function call is [V1,V2]=weightsimul_2d(Uw,V1_init,V2_init,c1,c2,uncer1,uncer2,s2); The GPS measurements include sparsely located measurements of the two-dimensional deformation, while the InSAR image includes high-resolution map (image) of the Slant-Range-Shift (“Uw”). Simulating annealing algorithm is used for the iteration (see Algorithm 10.1 in Section 10.2.1). The function is initialised with two motion field images (“V1_init” and “V2_init”), created for example with the MatLab function “krig1d” (see Section C.8), and returns high-resolution maps “V1” and “V2”, optimised by combined GPS and InSAR. The function minimise the difference “(Uw+c1*V1+c2*V2)^2”, where “[c1,c2]” is a unit vector (see Section 10.1). Additional constrains are smoothness of the second derivative of the image surfaces and penalisation of deviations from the initial kriged images, weighted as a function of distance from the sparse GPS locations in the images (see Section 10.2.3 and the energy functions in (10.15) and (10.16)). The uncertainty images “uncer1” and “uncer2”, created with the MatLab function “krig1d”, are used for this purpose. The function offers the possibility of optimising only one of the motion fields by keeping the other unchanged. This can be useful if one of the two motion images is known with high certainty. During the iteration, the process temperature (5 at the beginning and 0.1 at the end) and the iteration number is displayed on the MatLab window. Also, the image status is displayed on the screen, updated after each 10 iterations. Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help weightsimul_2d”. The help manual is as follow: WEIGHTSIMUL_2D: Function that estimates 2D crustal deformation by combination of InSAR and GPS [V1,V2]=weightsimul_2d(Uw,V1_init,V2_init,c1,c2,uncer1,uncer2,s2) The function optimise the difference (Uw+c1*V1+c2*V2)^2. Additional constrains are smoothness of the first derivative and weighting as a function from gps locations. Simulating annealing iteration process is used to optimise V1 and V2. Input: Uw: Unwrapped interferogram or image that fulfil Uw=-[V1,V2][c1,c2]^T. The image must be error corrected (offset and tilting) and include pixel information in cm. V1_init: The initial V1 image (in cm) (f.ex vertical deformation image). V2_init: The initial V2 image (in cm) (f.ex. horizontal look- direction deformation image). V1_init and V2_init can be found f.ex. by kriging. c1,c2: [c1,c2] is a two dimensional unit vector. uncer1: Uncertenty matrix for V1_init, scaled to the interval

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[0 1]. 1 is high certenty (close to GPS locations) and 0 low. The uncertenty matrix can be f.ex. created along with the krigigng of GPS measurements (see the MatLab function krig1d). uncer2: Uncertenty matrix for V2_init. s2: If s2=0, then only V1 is optimised and V2=V2_init. If s2=1, then both V1 and V2 are updated in the iteration process. Output: V1,V2: Optimised motion images. Note: The initial input images should be ordered in the way that c1>=c2, when both the images are optimised (s2=1); Example C.11. The problem described in Example 10.11 can also be solved with [Ve,Vn]=weightsimul_2d(insar_out+0.935*Vv,Ve_init,Vn_init,0.34, … -0.095,uncere,uncern,1); “Ve_init” and “Vn_init” are the initial motion field images, and “uncere” and “uncern” are the corresponding uncertainty images. “Ve_init”, “uncere”, “Vn_init” and uncern” can be created by the MatLab function “krig1d” (see Section C.8).

C.16. LINE_UNWRAP The function “line_unwrap” unwraps a wrapped line profile. The function call is prof_uw=line_unwrap(prof_w,tol); where “prof_w” is a wrapped line profile. Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help line_unwrap”. The help manual is as follow: LINE_UNWRAP: Function that unwrap a single line or profile prof_uw=line_unwrap(prof_w,tol) The function uses the first point (prof_W(1)) as a reference point and unwraps the image by requiring smoothness of the surface. Input: prof_w: The wrapped profile. tol: The reference tolerance (f.ex. lambda/2 =2.835 for ERS 1 and 2). Output: prof_uw: The unwrapped profile.

C.17. WRAP The function “wrap” finds a wrapped version of unwrapped InSAR image. The output matrix (image) includes 8-bit values scaled to the interval [0,255]. The function call is

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winsar=wrap(insar,tol); where “insar” is an unwrapped InSAR image. Further information about input and output variables and parameters can be listed by using the help manual in the MatLab window by typing “help wrap”. The help manual is as follow: WRAP: Function that calculates wrapped version of an unwrapped InSAR image. winsar=wrap(insar,tol); Input: insar: Unwrapped InSAR image. tol: the tolerance (f.ex. 2.835 cm for ERS-1 and ERS-2). output: winsar: The wrapped InSAR image with pixel values within the interval [0 255].

C.18. FYLKI The function “fylki” is used to put GPS observations longitude and latitude coordinate into a three-dimensional image. This function is used along with the function “tilt” in Section C.19. The function call is [gps_out,lo,la]=fylki(gps,insar,dlo,dla); Further information about input and output variables and parameters can be listed by using “help fylki” in the MatLab window. The help information listed in the MatLab window is as follow: FYLKI: Puts GPS measurements into three-dimensional sparse matrix. [gps_out,lo,la]=fylki(gps,insar,dlo,dla); Function that puts the GPS vectors in gps into three dimentional matrix, equal in size to the InSAR image. Input: gps: Matrix with the columns informations: [(1. lo. position), (2. la. position), (3. movement lo.), (4. movement la.), (5. movement up)]. insar: the INSAR image. lo1: First pixel longitude co-ordinate lo1: First pixel latitude co-ordinate dlo: The pixel resolution in lo. direction. dla: The pixel resolution in la. direction. Output: gps_out: three dimentional matrix including values of the dispalcement in (lo,la,up) directions at sparse pixels and zeros othervise. lo: vector including the longitude value that corresponds to each column. la: vector including the latitude value that corresponds

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C.19. TILT The function “tilt” is used to find a planar correction of wrapped InSAR image by using the GPS measured Slant-Range-Shift. The function use wrapped form of both the InSAR and GPS data to estimate the planar correction (see Appendix A). The function call is [insar_out,diff,res]=correct(insar_in,gps_in,s,y,tol); Further information about input and output variables and parameters can be listed by using “help tilt” in the MatLab window. The help information listed in the MatLab window is as follow: TILT: Function that that finds an optimal planar rotation of the input InSAR image by using sparse GPS measurements. [insar_out,res1,res2]=tilt(insar_in,gps_in,tol,y,s); Input: insar_in: The input InSAR image with pixel values lying in the interval 0-255. gps_in: The input three dimensional GPS image with values defined at sparse points. tol: The tolerance (f.ex. 2.835 cm for ERS-1 and ERS-2). y: The number of years used to find the deformation (defined by the InSAR image). s: The unit vector pointing from ground toward the satellite. Output: insar_out: The corrected InSAR image with no background correction. res1: The residuals between the InSAR and GPS before correction. res2: The residuals between the InSAR and GPS after correction.