crustal deformation mapped by combined gps and insar

CRUSTAL DEFORMATIONMAPPED BYCOMBINED GPS AND INSARSverrir GuðmundssonLYNGBY 2000SUBMITTED FOR THE DEGREE OF M.Sc. E.IMM

PrefaceThis thesis was made in collaboration between the Institute for MathematicalModelling at the Technical University in Denmark (IMM at DTU) and the NordicVolcanological Institute in Iceland (NORDVULK), and is a partial fulfilment of therequirements for my degree of M.Sc. in engineering.Various image analysis methods are used to combine two complimentary geodeticobservation techniques to map the earth surface movements.The work was funded and data was supplied by NORDVULK. Expertise in InSARand GPS observation techniques was provided by NORDVULK, and expertise inimage analyses by IMM at DTU. Supervisors were Jens Michael Carstensen fromIMM at DTU and Freysteinn Sigmundsson from NORDVULK.Lyngby, March 1 2000______________________Sverrir GuðmundssonI

AbstractIn this thesis, an attempt is made to extract the maximum amount of information fromtwo complementary geodetic technique and to infer high resolution maps of threedimensionalground movements.The first technique uses Interferometric analysis of Synthetic Aperture Radar(InSAR), acquired from a satellite Synthetic Aperture Radar (SAR). Aninterferogram is formed by combining two synthesised SAR images acquired atdifferent times (from Master and Slave tracks), and is in its simplest form a record ofa phase difference between two signals. The interferograms contains a modulatedmeasure of the one-dimensional change in range from the ground to the satellite, witha typical resolution of 20x20 m 2 .The other technique uses NAVigation Satellite Timing And Ranging GlobalPositioning System (NAVSTAR GPS) for geodetic observations. A GPS network ofground control points, typically spaced few km or tens of km apart, is created for thepurpose of measuring ground movements. Surface movements are detected as achange in range of the ground control points, between two or more sets ofobservations. The GPS geodetic observations can be used to provide threedimensionalmeasurements of ground displacements at sparse locations, with accuracywithin 1 cm.Test data from the Reykjanes Peninsula, Iceland is used for experimentation.Available are both GPS and interferometric observations, recorded from a descendingsatellite pass, with various elapsed time intervals within the period 1992 to 1998. Theground movements at the Reykjanes Peninsula consist of both crustal deformation andplate movements. A test image that describes a surface iceflow at an axisymmetricalglacier cauldron, is created for experimental purposes. The image includes no errorsand all deformation components are known, which gives an opportunity to obtainestimation errors.Both InSAR and GPS observations may include several error factors. Some errorfactors need to be reduced before combining the two complementary geodetictechniques. The activity of the atmosphere generates noise errors in themeasurements. A vectorized filtering is efficiently used for a noise reduction ofmodulated (wrapped) interferograms. The Master and Slave tracks of aninterferogram include two different viewpoints and distances to the same object. Thisresults in a systematic error that can be described with a phase plane. The GPSobservations are not expected to include significant systematic errors, and aretherefore 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 usesMarkov Random Field (MRF) based regularisation and simulating annealingoptimisation is used to unwrap InSAR images. The unwrapping process utilises therelationship of the interferometric and GPS observations, both in the MRF modellingand for initialisation. The MRF regularisation also uses an assumption about surfacesmoothness. For the purpose of initialising the process, virtual InSAR images arecreated by ordinary kriging of GPS observations.II

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

Table of contents1. INTRODUCTION.........................................................................................................................32. GROUND MOVEMENT OBSERVATIONS...............................................................................42.1. INTERFEROMETRIC OBSERVATIONS............................................................................................42.2. GPS OBSERVATIONS.................................................................................................................72.3. INSAR IMAGES AND GPS MEASUREMENTS FROM THE REYKJANES PENINSULA, ICELAND............72.3.1. InSAR images...............................................................................................................82.3.2. GPS measurements......................................................................................................82.3.3. Accounting for differences in elapsed time intervals.................................................103. TEST IMAGE.............................................................................................................................115. NOISE REDUCTION.................................................................................................................146. EXTRACTING REGION OF INTEREST .................................................................................157. GPS TILTING OF WRAPPED INSAR IMAGES.....................................................................167.1. ELIMINATING A PHASE PLANE FROM WRAPPED INSAR IMAGES.................................................167.2. LEAST SQUARE ESTIMATION OF COEFFICIENTS .........................................................................177.3. UNWRAPPING LINE PROFILES...................................................................................................187.3.1 Some error effects........................................................................................................187.4. THE TILTING ALGORITHM........................................................................................................208. MOTION FIELD IMAGES CREATED FROM A SPARSELY LOCATED GPSOBSERVATIONS..........................................................................................................................248.1. ORDINARY KRIGING................................................................................................................248.2. SEMIVARIOGRAM ...................................................................................................................258.3. SEMIVARIOGRAM MODEL........................................................................................................258.4. KRIGING RESULTS...................................................................................................................268.5. ESTIMATION OF UNCERTAINTY................................................................................................288.6. COMPARISON OF INTERPOLATION METHODS.............................................................................289. USING KRIGING OF GPS MEASUREMENTS AND MARKOV RANDOM FIELDREGULARISATION TO UNWRAP INSAR IMAGES.................................................................309.1. PROBLEM DESCRIPTION...........................................................................................................309.2. ESTIMATION OF THE INITIAL WAVE NUMBERS ..........................................................................319.3. SIMULATING ANNEALING AND MRF REGULARISATION.............................................................339.4. THE SIMULATING ANNEALING UNWRAPPING ALGORITHM AND MRF MODELS............................349.4.1. Simulating annealing iteration algorithm....................................................................349.4.2. Penalizing the second derivative................................................................................359.4.3. Detecting area of interest............................................................................................369.4.4. Guiding with GPS measurements. .............................................................................389.5. UNWRAPPING PROCESS ...........................................................................................................3910. COMBINATION OF GPS AND INTERFEROMETRIC OBSERVATIONS TO INFERTHREE-DIMENSIONAL GROUND MOVEMENTS ....................................................................4410.1. PROBLEM DESCRIPTION.........................................................................................................4410.1.1. Simplifying the problem.............................................................................................4410.2. OPTIMIZATION OF TWO-DIMENSIONAL DEFORMATION ............................................................4510.2.1. Two-dimensional simulating annealing algorithm ...................................................4510.2.2. Energy functions........................................................................................................4610.2.3. Further utilisation of GPS observations ...................................................................4810.3. INFERRING THE THREE DIMENSIONAL MOTION FIELD AT THE REYKJANES PENINSULA ..............5110.3.1. Corrections of the data observations .......................................................................5210.3.2. Inferred three-dimensional motion maps .................................................................52

11. RESULTS AND DISCUSSIONS............................................................................................6011.1. PRE-PROCESSING METHODS...................................................................................................6011.1.1. Noise reduction in interferometric observations......................................................6011.1.2. Extraction of area of interest.....................................................................................6011.1.3. Utilisation of GPS observations to correct wrapped InSAR images ......................6011.1.4. Creation of virtual InSAR images with interpolation of GPS data..........................6111.1.5. Unwrapping process..................................................................................................6111.1.6. Utilisation of GPS observations to correct unwrapped InSAR images ..................6211.2. CONSTRUCTION OF THREE-DIMENSIONAL HIGH RESOLUTION MOTION MAPS.............................6311.3. PRE-PROCESSING AND AVERAGED MOTION MAPS AT THE REYKJANES PENINSULA ...................6412. CONCLUSION ........................................................................................................................66REFERENCES ..............................................................................................................................67APPENDIXES................................................................................................................................68A. GPS TILTING OF WRAPPED INSAR IMAGES; PROJECTION INTO THE COMPLEXUNIT CIRCLE ................................................................................................................................68A.1. THE PROCEDURE....................................................................................................................68A.1.1. The first objective function..........................................................................................69A.1.2. The second objective function....................................................................................70A.2. CORRECTION OF THE INSAR DATA.........................................................................................71A.3. REMOVING UNWANTED AREAS FROM THE CORRECTED IMAGES ................................................72B. PROCESSED IMAGES............................................................................................................75C. MATLAB FUNCTIONS ............................................................................................................80C.1. CONTENT LIST .......................................................................................................................80C.2. OPENING.............................................................................................................................81C.3. MA.......................................................................................................................................81C.4. LOLA2PIX...........................................................................................................................82C.5. MASK..................................................................................................................................83C.6. MK.......................................................................................................................................83C.7. PROFILE_TILT....................................................................................................................84C.8. KRIG1D...............................................................................................................................85C.9. PATTERN............................................................................................................................86C.10. UNWRAP_GPS..................................................................................................................87C.11. UNWRAP_SMOOTHN......................................................................................................88C.12. UNWRAPPED....................................................................................................................90C.13. TILT_UNWRAP.................................................................................................................90C.14. SIMUL_2D.........................................................................................................................91C.15. WEIGHTSIMUL_2D..........................................................................................................93C.16. LINE_UNWRAP................................................................................................................94C.17. WRAP................................................................................................................................94C.18. FYLKI................................................................................................................................95C.19. TILT...................................................................................................................................962

1. IntroductionThe earth surface is in continuos shaping due to various forces. Surface movementsconsist for example of divergent plate displacements, crustal deformations or glaciericeflow. 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 theearth 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 twocomplementary geodetic techniques. The first technique uses Interferometric analysisof a satellite born Synthetic Aperture Radar (InSAR). An InSAR image contains amodulated measure of the one-dimensional change in range from the ground to thesatellite The other technique uses Global Positioning System (GPS) for geodeticobservations, which provides accurate three-dimensional measurements of grounddisplacements at sparse locations.Various image analysis methods are used to combine GPS and InSAR. Methods andresults are presented in the following chapters.3

2. Ground movement observationsMany methods are available to measure ground movements. In this thesis, resultsfrom two observation techniques are combined. The first method is an interferometricobservation that is used to generate high-resolution map of one-dimensional groundmovement. The second method is GPS observations used to measure threedimensionalground movements at sparse locations. The methods are explainedbriefly in this chapter. Further details can be found about the interferometrictechnique in [1] and GPS observation in [2].2.1. Interferometric observationsSynthetic Aperture Radar (SAR) images are recorded with radar satellites. In thisstudy 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-daydifference, and an orbital repetition cycle of 35-day each. Each satellite scans thesame area twice per 35-day interval; once from an acceding and once from adescending orbit pass, which gives two different view angles of the same area. ERSradar transmits electromagnetic waves with wavelength of λ = 56,7 mm, andmeasures the reflected signal from the Earth.A SAR image is a synthesised image, generated from the earth back-scattered radarsignal. It consists of complex pixel values (amplitude and phase), with a typicalresolution about 20x20 meters. SAR images can be used to form an interferogram. Aninterferogram (InSAR image) is generated from the phase difference of two pairs ofSAR 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 anInSAR image, generated from ERS-1 and ERS-2, can vary from one day up to severalyears. Figure 2.1 explains the InSAR geometry where S 1 is the Master track and S 2 isthe 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 * isthe 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 isgiven by the formula [1]I =*∑ f ( M) f ( S ) exp( 2πjG).22∑ f ( M) ⋅ ∑ f ( S)(2.1)G is designed to eliminate topographical and orbital phase errors and the filter freduces the difference in radar impulse response perceived by each satellite track. Themagnitude of (2.1) ranges from 0 to 1 and is called a choerence, and is used tomeasure the reliability of the measurement. A value of 1 mean that every pixel agreedwith the phase within its cell and 0 indicates a meaningless phase.4

Figure 2.1. Geometry of an interferometric measurement. From [3].One phase shift ( 2π ) in ERS interferogram corresponds to a displacement of λ 2 =2 .835 cm in the direction from the satellite to the ground, since the wave travelsfrom the satellite to the ground and back again. Displacement of more than onewavelength ∆ θ = φ + n2π, where 0 < φ < 2πand n ≠ 0 is an integer, is registered asa 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 fringecorresponds to a scalar change of ∆ρ ≈ 2.8 cm in the direction from the satellite tothe ground. A periodic or modulo-2π interferograms are called wrapped InSARimages and images that have been corrected for the modulo-2π effect are calledunwrapped InSAR images. Figure 2.2 explains the difference of a wrapped and anunwrapped signal.A phase change ∆ θ in an interferogram can be due to number of effects, includingcontribution due to difference in orbital trajectory, topography and several noisefactors.∆ θ= ∆θdisplacement+ ∆θorbital+ ∆θtopography+ ∆θnoise(2.2)Difference between viewpoints and distances of the Master and Slave orbit to thesame ground object can lead to gradual phase change or regularly distributed orbitalfringes in the InSAR image. Orbital fringes can be eliminated by subtracting a phaseplane from the image. This can be done by using a knowledge of the satellitetrajectories S 1 and S 2 (Figure 2.1). However, the knowledge is usually not accurate tothe scale of wavelength, which can leave a few regular fringes uncorrected. Figure 2.3explains the effect of suppressing the orbital fringes.Topographical fringes can be extracted from interferograms by means of syntheticinterferograms, calculated from digital elevation model (DEM) and orbit parameters.5

Figure 2.2. Wrapped and unwrapped signals. The wrapped signal is modulo-2π, where 2πcorrespond to displacement of ∆ρ ≈ 2.8 cm.Height sensitivity measurements for the interferograms are used to estimate theimpact of possible errors due to the topography. This is done by estimating thesocalled “altitude of ambiguity” ha[1] (a measure of stereoscopic effects). Thealtitude of ambiguity represent surface altitude needed to produce one topographicalfringe in the interferogram and is a characteristic number for an interferogram. Highnumbers of harepresent an insensitivity of the interferometric measurement tovariations in the surface elevation.Noise can for example be induced from difference in atmosphere conditions and backscattering characteristics of the surface, between the two times of observations. Anattempt is made to keep the internal phase contribution constant between the Masterand Slave images by acquiring them from similar surface conditions. Extreme casesinclude water-covered surfaces, which include no stability in the back scatteringcharacteristics. Noise errors induce random speckles in the image, which aremeasured as a decorrelation or an inchoherence (see (2.1)). Also, large movementscan result in an ambiguity between neighbouring pixels that will blur fringes and givea low choerence.If all error effects can be corrected for, then the scalar displacementsatellite towards the ground can be described mathematically by [1]∆ ρ from theFigure 2.3. Effect of removing orbital fringes. The image to the left is before correction and theone to the right is after correction. From [1].6

λ 2∆ρ= ∆2πθ displaceme nt= −u⋅s,(2.3)where u is the three dimensional displacement vector and s is the unit vector pointingfrom the ground toward the satellite.2.2. GPS observationsHighly precise navigation measurements are done by using the NAVigation SatelliteTiming And Ranging Global Positioning System (NAVSTAR GPS), established andoperated by the U.S. military [2]. The system consists of 24 GPS satellites, orbitingaround the earth at an altitude about 20200 km. The satellites are located on sixalmost circular orbital planes, each inclined about 55° with respect to equator andwith an orbital period around 12 hours. At all time, four to eight satellites areavailable for navigating, which is enough to give a position in three-dimensionalspace. GPS navigation systems use accurate measurements of travel times of wavesignals to estimate the distance between the satellites and the GPS receivers. Thephase differences between transmitted signals and waves generated in the receiversare 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 ofmeasuring surface movements. Surface movements at each site in the network aredetermined as a difference in position of a fixed ground control point between two ormore times of observations. During an observation, one GPS receiver collects satellitedata at a reference point, while the points in the GPS network are continuouslymeasured over some period of time. Therefore, one point observation consists of atime average of regularly sampled measurements. A post-processing the informationfrom the measured signals along with a precise information about the satellite orbitscan be used to calculate an accurate location of point in space, at millimetre levelrelative 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 solidrock with only a small piece of the bolt standing out. The position of the benchmark isthen measured accurately, by putting a receiver antenna on a tripod located directlyabove the stick. The tripod is used to put the antenna accurately at the same positionabove 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 thebenchmark.2.3. InSAR images and GPS measurements from the ReykjanesPeninsula, IcelandReykjanes Peninsula is located at the SW-part of Iceland. Iceland is located on themid-Atlantic Ridge, which makes it an ideal place for studying mechanics ofdivergent plate movement and crustal deformation. The plate boundary between theNorth-American and the Eurasian plate runs ashore at the SW tip of the ReykjanesPeninsula [2,4]. Figure 2.4 (a) shows the location of fault and eruptive fissures andthe outline of central volcanoes cite area. The image shows also an approximatelocation of the central axis of the plate boundary as inferred from seismicity.7

2.3.1. InSAR imagesSeven wrapped InSAR images from the Reykjanes Peninsula were available for thisstudy. The data have been coded to 8-bits. The images are all acquired from adescending satellite passes. Topographical effects have been eliminated by means of aDEM data and known orbital effects removed. The images includes an informationabout the crustal deformation component in the ground to satellite direction (∆ρ or theSlant-Range-Shift), where each pixel have a resolution of 1/600° in longitude and1/1200° in latitude (approximately 93x82 m 2 area).Fore these interferograms, the descending unit vector s pointing from ground towardsthe satellite is given as [4]s = [ 0.34E,−0.095N,0.935V],(2.4)where E, N and V means East, North and vertical respectively. Figure 2.4 (b) shows aplot of the unit vector s. The highest contribution to the interferometric signal is fromthe vertical ground deformation, whereas the north-south movements gives a very lowcontribution.Table 2.1 shows the tracks of the Master and Slave orbits for all the seven wrappedInSAR images and the time of observation. The time intervals represented by theinterferograms vary from one month up to four years. Figure 2.5 shows the imageswith the highest altitude of ambiguity. The images may include a noise caused by theatmosphere and by change in surface back scatter characteristics between the time ofobservation of the Master and Slave images, and also an error induced by incompletecorrection of the orbital and topographical effects [4].2.3.2. GPS measurementsGPS measurements of the three-dimensional displacement vector u in (2.3) areavailable at sparse points at the Reykjanes Peninsula. The same GPS network wasfirst measured 1993 and again 1998 (5 years interval). The locations of the GPSpoints are shown in Figure 2.6, and also the one-year average of GPS measureddisplacement field over the elapsed five years interval from 1993 to 1998. A GPSmeasure 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 measuredcomponent of the displacement field are also available [2].Table 2.1. Characteristics of the InSAR imagesMaster orbit Date ofobservationSlave orbitDate ofobservationElapsed timeAltitude ofambiguityh a5064 04.07. ’92 5565 08.08. ‘92 35 days 35.6 m5064 04.07. ’92 10575 24.07. ‘93 0.93 years 25.1 m5064 04.07. ’92 21941 25.09. ‘95 3.22 years 19.6 m5565 08.08. ‘92 10575 24.07. ‘93 0.83 years 59.0 m5565 08.08. ‘92 21941 25.09. ‘95 3.12 years 43.6 m5565 08.08. ‘92 7278 09.10. ‘96 4.17 years 22000 m10575 24.07. ‘93 21941 25.09. ‘95 2.29 years 166.0 m8

(a)Figure 2.4. Reykjanes Peninsula. (a): The image shows the locations of faults and eruptivefissures and outline of central volcanoes (circled areas). The thick line indicates theapproximate 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 ReykjanesPeninsula(b)Figure 2.5. Wrapped InSAR images from the Reykjanes Peninsula. The same colorbarapplies to all the images.9

applies to all the images.Figure 2.6. Displacement rate in the period from 1993 to 1998. The GPS locations are shownas ∆. The figure is from [2].2.3.3. Accounting for differences in elapsed time intervalsThe InSAR images and the GPS measurements are observed at different times. InSARimages represent a displacement at various elapsed time intervals within 1992 to1996, while the GPS measurements represent the deformation during the period from1993 to 1998. The crustal deformation at the Reykjanes Peninsula can be assumed tobe smoothly continuos (no abrupt changes in the surface), since no large earthquakeshave 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 variationof the deformation field as a function of time for some parts of the image areas. Thishas to be considered before the interferometric and GPS data can be combined, seeChapter 9 and 10.10

3. Test imageIn addition to the image pairs from the Reykjanes Peninsula (Chapter 2), a 164x164pixel test image was created to experiment on (Figure 3.1). The image was created byusing a mass balance equation to describe a surface iceflow at an axisymmetricalglacier cauldron [5]. The image includes no error factors and all the deformationcomponents are known. This image was created for process testing and to obtain anestimation 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 ( − [. 34, −.095,.935] ⋅ [ V , V , V ] Timages in (a), (b), (c), (d).ENV), (e): plots of line 82 (central line) out of the11

4. Angular difference between the InSAR and GPSmeasured Slant-Range-ShiftThis chapter present some comparison between the unprocessed GPS and InSAR data.A process of projecting both the GPS and interferometric measurements into the unitcomplex circle can be used to get a rough comparison of the consistency between theinterferometric and GPS measured Slant-Range-Shift ( ∆ ρ ).Before reading any further, it should be noted that a bold letter is used to represent amatrix (image) or a vector, while a non-bold indexed letter refer to a single value ofthe matrix. A single index is used when referring to a pixel number, but two indexeswill also be when referring to row and column numbers.For the GPS measurements, a sparse Slant-Range-Shift image can be calculated asIGPS iT= yr ⋅ ( u ⋅ s ),i∀i(4.1)where i is sparse pixel values corresponding to the GPS locations, u is the threedimensional GPS measured crustal deformation (in cm/yr), s is the unit vectorpointing from the ground towards the satellite and yr is the elapsed time representedby the interferogram. The process requires the assumption of having lineardeformation with time. IGPSis projected into the complex unit circle withCGPS i⎛= exp⎜⎝j ⋅ 2π⋅ Iλ 2GPS i⎞⎟,⎠∀i,(4.2)and the wrapped InSAR image valuescircle byIInSARare projected into the complex unitiCInSARi⎛= exp⎜⎝j ⋅ 2π⋅ Iλ 2InSARi⎞⎟,⎠∀i.(4.3)If both the GPS and interferometric measurements are describing exactly the samethen∠ CInSARi− ∠CGPS i= 0,∀i,(4.4)It should be noted that this method compares the interferometric and the GPS signalson periodical forms. Therefore, this does not give an idea about the absolutedifference. The histogram in Figure 4.1 shows an example of an angular differencebetween the GPS and interferometric measurements; (a) shows the difference betweenGPS (1993-1998) and 2.29 years interferogram (1993-1995), and (b) between GPSand 3.12 years interferogram (1992-1995).12

Distributions of the angular difference ∠ CInSAR− ∠Ci GPS(Figure 4.1) shows that theierror can be large for some of the points or up to 180°. This can be a consequence ofseveral 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 theorbital and topographical effects,4. big jumps in the deformation at some areas within the image as a result of smallearthquakes or5. the assumption of having linear deformation does not hold for some periods or atsome 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 dueto insufficient topographical correction (Table 2.1).3. Insufficient correction for orbital effects should lead to a systematic error that canbe described by a phase plane. One attempt to make the InSAR and GPS datamore consistent could be to find some optimal planar correction of the InSARimages by using time scaled GPS measurements (Chapter 7).Effects due to non-linear crustal deformation could be reduced by selectinginterferometric and GPS measurements that represent approximately the same timeperiods. Consistency between the InSAR images and GPS measurements will beconsidered further in Chapter 9 and 10.(a)(b)Figure 4.1. Distribution of∠C− ∠Cat the sparse pixels i defined by the GPSInSARiGPS ilocations. The interferometric and GPS measurements are made time consistent by assuminga liner deformation with time. If GPS and InSAR observations are fully consistence theseobservations should group around 0 and 2π.13

5. Noise reductionThe wrapped InSAR images can include high noise factors. Filtering of the wrappedinterferograms can have many practical applications and will be widely used infurther processing of the data. The modulate characteristics of the image values needto be considered before filtering. This is handled by projecting the modulated signalinto two vectors, perpendicular to each other (cosine- and sinusoidal). The twovectors are then filtered separately (vectorized filtering). The filtered interferogram isgiven asλ 2If= ∠(f ∗C),2π(5.1)where f is some linear filter coefficients (e.g. moving average window), ∠ means theangle and C is a complex intensity image given by⎛C = exp⎜⎝j ⋅ 2π ⋅ Iλ 2⎞⎟⎠(5.2)with I as the wrapped InSAR image with the modulated (wrapped) intensity interval[ 0,λ 2].Figure 5.1 shows the 4.17 years InSAR images with the Master and Slave tracks 5565and 7278, respectively, before and after filtering. The image is filtered with a 0.5x0.5km 2 moving average window. The figure shows how efficiently the vectorizedfiltering can reduce noise speckles.Figure 5.1. The 4.17 years wrapped InSAR image from Reykjanes Peninsula before (a) andafter (b) filtering. (c): Profiles from (a) and (b) (the profile location is shown as thick black lineon the images in (a) and (b)).14

6. Extracting region of interestThe Reykjanes Peninsula is surrounded by an ocean, which is displayed as a noinformationin the InSAR images (zero valued pixels). Binomial images that separatesinformation areas (foreground) from no information areas (background) will be usedin further processing of the InSAR images. A process that separates the images into abackground and a foreground is given as:1. A binomial image is created by thresholding the original InSAR image, such thatall pixels distinct from zero are assigned the value one (foreground), and the zerovalued pixels are kept as zero (background).2. The foreground is cleaned with a morphological cleaning process; a closeoperation that consists of dilation followed by erosion.Further details about morphological cleaning are given in [6,7,8]. Figure 6.1 showsbinomial images generated from thresholded InSAR image both before and after themorphological 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.15

7. GPS tilting of wrapped InSAR imagesMaster and Slave orbital trajectories of an InSAR image can have two differentviewpoints and distances to the same object. Correction for orbital errors is notaccurate to a scale of a wavelength. Hence, the InSAR images can include a residualorbital error that can be described by a tilted plane and an offset (a phase plane). TheGPS measurements are on the other hand not expected to have a significantsystematic error. An approach to correct residual orbital errors in InSAR images is touse 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 isassumed that the only systematic error in the interferometric measurements can bedescribed 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 thenfound by optimisation in the complex domain. This has the drawback of changing aunique solution into periodical solutions. Furthermore, this can also lead to wrongoptimal solutions since the minimum difference between each individual GPSmeasurements and the interferogram becomes also periodic. Experiments have shownthat this algorithm can easily result in wrong tilting.A tilting method that use unwrapped interferometric profiles located between pointsdefined by the sparse GPS locations is presented in this chapter. The algorithm usesthe unwrapped profiles to estimate a sparse unwrapped values of the interferogramthat correspond to the sparse GPS locations, and Least-Square (LS) method toestimate the optimal tilting.7.1. Eliminating a phase plane from wrapped InSAR imagesFor the GPS measured three-dimensional crustal deformation, a sparse Slant-Range-Shift image is calculated asIGPS( i,j)=yr ⋅ ( u(i,j)⋅ sT),(7.1)where u is the three dimensional GPS measured rate of displacement (in cm/yr), s isthe unit vector pointing from the ground towards the satellite, yr is the elapsed timeinterval of the interferogram and i,j are the sparse row and column numbers definedby the GPS locations. The relationship between the sparsely located GPSmeasurements and the corresponding interferometric measurements is then expectedto be on the formIGPS[ i,j,1] ⋅ , ∀i,j,( i,j)= I ( i,j)+ xInSARUw(7.2)whereIInSARis the unwrapped InSAR image andUwx =[ , x x ] Tx1 2,3(7.3)16

is vector including the coefficient of the two dimensional phase plane. For a known x,tilting of a wrapped interferogram I can be calculated asInSARWλ 2IInSAR= c2 InSARW Plan2 π( l,c) { ∠( C ( l,c) C ( l,c))},∀l,,(7.4)where l and c are the line and columns numbers of I , respectively, ∠ stands forInSARWthe phase,CInSARW⎛= exp⎜⎝j ⋅ 2π⋅ Iλ 2InSARW⎞⎟,⎠(7.5)andCPlan( l,c)⎛= exp⎜⎝j ⋅ 2π xλ( [ ] ) T⋅ l,c,1⎞⎟,∀l,d .2⎟⎠(7.6)'Note that if x3is a solution of the optimal offset in (7.3), then x3= x3+ nλ2,for allinteger numbers n, are also solutions of the optimal offset of the wrappedinterferogram. The offset can also be set as x 0 in (7.3) and estimated after tiltingwith x1and x2as3 =xλ 2 1*( C ( i , j ) C ( i , j ))k'3= ∑ ∠GPS n n InSAR n n,22πk n=1(7.7)where k is the number of GPS measurements andCGPS⎛= exp⎜⎝j ⋅ 2π⋅ Iλ 2GPS⎟ ⎞.⎠(7.8)7.2. Least square estimation of coefficientsIf the absolute or unwrapped values are known for both the GPS and interferometricmeasurements at sparse locations i,j, then the coefficient in (7.3) can be estimatedwith the standard usual LS estimatorx =T −1T( A A) A y,(7.9)with⎡i1j1A =⎢⎢M M⎢⎣i kj k1⎤M⎥⎥1⎥⎦(7.10)and17

( i , j )⎡ IGPS( i1, j1)− IInSARUw 1 1⎢y =⎢M⎢⎣IGPS( ik, jk) − IInSARUw k k( i , j )⎤⎥⎥,⎥⎦(7.11)where k is the number of GPS measurements. In order to use (7.9) to (7.11), the sparsevalues of I are estimated from unwrapped line profiles.InSARUw7.3. Unwrapping line profilesThe relationship between unwrapped and wrapped profilesis given aspUwand p , respectively,p = p + k λUw2,(7.12)where k is the wave number vector and λ is the wavelength of the SAR radar. Asimple line-unwrapping process is used in the tilting algorithm. For a wrapped lineprofile p of size m, the procedure can be described as:Algorithm 7.11. n=2.2. Calculate d p − p ), d p + λ 2 − p ) and d p − λ 2 − p ).1= (n n−11min( d1,d2, d33d1,d2, d2= (nn−13= (nn−1d2= min( d1,d2, d3= 1,3. If d = ), go to step 5. Else if ), k go to step4. Else if d = min(3), k = −1,go to step 4.4. For h = n to h = m,ph = ph+ k λ 2.5. n = n +1.If n > m , stop. Else if n ≤ m,go to step 2.Figure 7.1 shows an example of a line profile before and after unwrapping withAlgorithm 7.1. It is evident that the algorithm uses the first profile value p1as thereference for the absolute value of the whole profile when unwrapping.7.3.1 Some error effectsErrors from high- and low frequency noise factors need to be considered whenunwrapping the interferogram profiles from the Reykjanes Peninsula. One effect of alow frequency noise is explained in Figure 7.2. On the images in (a) are shownlocation of profiles (between the points ABC), located over both light and dark areasof the wrapped interferogram. The sudden changes in colours are due to lack of onewave number (see (7.12)) in the darker area. This lack of wave number is expected toresults in abrupt leaps of ~ λ / 2 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 profileBC. The reason could be a low frequency atmospheric noise that can smooth leaps of~ λ / 2 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 wouldalso effect the unwrapping process.18

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.19

7.4. The tilting algorithmThe tilting process uses unwrapped profiles between sparse locations in the InSARimage defined by GPS sites (Figure 7.3). The tilting coefficients x1and x2are thenestimated by the LS algorithm in Section 7.2, and the offset x3is estimated with (7.7).Error effects are reduced by pre-oversmoothing the interferogram with a vectorizedfiltering (see Chapter 4). The pre-oversmoothing does reduce the high frequencynoise. Furthermore, experiments have shown that this does also reduce errors of anunwanted smoothing of ~ λ 2 edges due to low-frequency errors (see definition of anunwanted smoothing of ~ λ 2 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 tomask the tilted output image.The tilting algorithm is given in Algorithm 7.2. In the tilting algorithm IGPSis avector including k samples of a GPS measured Slant-Range-Shift, IInSARis theWwrapped InSAR image, i, j are line and column numbers and n, h reefers to thespatial location of the n th and h th sample of IGPS.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 locationi , j ) to all the other sparse GPS locations ( i , j ), ∀h≠ n in theii)(n ninterferogram. The unwrapped values of the interferogram ( I ( i , j ))hhInSARUware estimated from the unwrapped profiles with I ( i , j ) as thereference absolute value, whereInSAROWIInSARis smoothed IOWInSARWnnhh. The tiltingcoefficients x1and x2are then estimated from the unwrapped values, the GPSmeasurements and the LS algorithm.This process is then repeated for all the k GPS locations and the final tiltingcoefficients 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 errordue to low frequency atmospheric errors. The process in (i) to (ii) does not give aninformation about the offset between the GPS and interferometric measurements sinceI ( i , j ) is used as the reference absolute value in each step. The offset isInSARUwnntherefore calculated with (7.7) in Algorithm 7.2, after tilting with x1and x2.Algorithm 7.2.1. Oversmooth IInSAR(10x10 pixel window) which gives I .WInSAROW2. Calculate mask image M.3. n = 1.4. h = 1,t = 0,I = empty vector,G = empty vector,l = empty vector andc = empty vector.5. If h ≠ n , then create an unwrapped line profile p of size m from the locationsi , j ) to i , j ) of I , go to step 6. Else if h = n,t = t +1,(n n(h hInSAROWI(t ) = I ( i , j ), G(t)= I ( h),and l( t ),c( t))= ( i h, j ), go to step 7.InSAROWhhGPS(h20

6. If the line profile does not intersect with the background, t = t +1,I(t ) = p( m),G(t)= ∆ρ(h)and ( l( t ),c( t))= ( i h, jh).7. h = h +1.If h ≤ k,go to step 5. Else if h > k,go to step 8.8. Calculate the slopes x 1(n ) and x ( n)2(see (7.3)) by using (7.9) to (7.11) and l, c, Iand G. n = n +1.If n ≤ k,go to step 4. Else if n > k,go to step 9.''9. Calculate average slopes as x = x (1) + K x ( k))k and x = (1)+ K1(1+12k= 3x1 2,3to tilt the unsmoothed imageInSARW'' 'x3by (7.7) and by setting x1= x2= 1.' ' ''x = x1 , x2, x3( x ', x from step 9 and '1 23+ x ( k)), and set x '0.' ' '10. Use = [ x , x x ]11. Calculate12. Use [ ]imageI by using (7.4) to (7.6).InSARW2( x 2I by using (7.4) to (7.6).x from step 11) to tilt the unsmoothed13. Create an output image by pixelvies multiplication of the tilted InSAR image andthe mask image (M).Experiments indicate that tilting of wrapped interferograms is more risky than tiltingof unwrapped interferograms. Algorithm 7.2 will therefore only be used to tiltwrapped interferograms before unwrapping (see unwrapping in Chapter 9). Toachieve safer tilting before inferring the three dimensional crustal deformation, theunwrapped 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 tiltingparameters for an error free wrapped interferogram. Another experimental example isgiven in Table 7.1 and Figure 7.3. In this example the 3.12 years 400x750 pixelsInSAR 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 themethods described in Chapter 9 and 10 (see also Appendix B). Input image (Figure7.3 (a)) with known tilting parameters x1,x2and x3was then created by adding alinear plane to the image in (f). The tilting procedure is explained in Figure 7.3 andthe 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 inlatitude (rows) and (0.0056-0.0038)⋅750/2.835≈0.48 fringes in longitude (columns).The offset error is estimated as − 5 .081+2* 2.835 + 0.035 = 0. 624 or 0.624/2.835≈0.22 fringes (note that the solution of the offset is periodical with the periodλ 2 = 2.835 cm).A dataflow diagram of Algorithm 7.2 is given in Figure 7.4.Table 7.1. Comparison of correct and estimated tilting parameters.x 1 (rows) x 2 (columns) x 3 (offset)Correct 0.0115 cm/pixel 0.0056 cm/pixel -5.081 cmEstimated with Algorithm 7.2 0.0091 cm/pixel 0.0038 cm/pixel -0.035 cm21

Figure 7.3. Tilting by using Algorithm 7.2; (a): the untilted input image, (b): the input imageoversmoothed with a vectorized filtering, (c): the image in (a) after tilting, (d): mask calculatedfrom the image in (a), (e): the image in (c) masked with the mask in (d), (f): the correct tiltedimage.22

Figure 7.4. Dataflow diagram of the tilting process.23

8. Motion field images created from a sparsely locatedGPS observationsInterpolation of a sparse GPS data can be used to create motion field images. Amethod that uses a Markov Random Field regularisation to unwrap InSAR images ispresented in Chapter 9. This method offer a possibility of initialisation of theunwrapped interferogram, that can for example be created from an interpolation ofsparse GPS measured Slant-Range-Shift. A method that optimises three dimensionalmotion field is then presented in Chapter 10. The method also uses Markov RandomField regularisation, where the three-dimensional motion fields images are initialisedfrom 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 thetest image has been achieved with the kriging.Kriging algorithms use geo-statistical measurement the dispersion matrix to find anoptimal set of weights, used for the interpolation [9,10,11]. An ordinary onedimensional (1D) kriging algorithm is used to interpolate the sparse GPS data, wherethe dispersion matrix is approximated with use of an estimated semivariogram. 2Dkriging[11] have also been considered for use in estimation of the three-dimensionalmotion field. The main drawback of 2D-kriging is that it requires estimation of onesemivariograms and two cross-semivariograms compared to one semivariogram for1D-kriging, which makes it much more complicated to use. 2D-kriging is thereforenot used in this thesis.The 1D-kriging method used for the implementation is only described briefly in thischapter, but further details about it are given in [9].8.1. Ordinary krigingGiven a set of sparsely located M-measurementsz =T[ z z ] ,1 , 2 ,...,z M(8.1)an unbiased estimator of an arbitrary point z0iswhere [ ] TTzˆ0= ω z,with ∑ωi= 1,ω = ω1,ω2,...,ω Mis the set of optimal weights. The variable zican beinterpreted as an outcome of a random variable Zi.By requiring Ε{ Z ˆ0− Z0}= 0 andminimising the error variance Var{Z ˆ0− Z0}the optimal solution of the weights in(8.2) is found byNi=1(8.2)⎡ C⎢⎢M⎢C⎢⎣ 111M 1LOLLCC1MMMM11⎤⎡ωM⎥⎢⎥⎢M1⎥⎢ω⎥⎢0⎦⎣λ1M⎤ ⎡ C⎥ ⎢⎥ = ⎢M⎥ ⎢C⎥ ⎢⎦ ⎣ 1010M⎤⎥⎥,⎥⎥⎦(8.3)24

whereCijis the covariance between the points ziand zj.8.2. SemivariogramAn estimation of the covariance C h)= C , where h is the distance vector between(ijpoints ziand zj, is done by using a semivariogram. The semivariogram is defined aswhere r is some point in the space and Z is the set of the random variables Zi. If thespatially distributed measurements are assumed or forced to be first and second orderstationary then γ ( r,h)= γ ( h)and2C ( 0) = σ is the variance of the stochastic variables. An estimator for thesemivariogram is given as2{[ Z(r)− Z(r h)] },1γ ( r , h)= Ε +(8.4)2γ ( h)= C(0)− C(h).(8.5)N( h)1γ ˆ( h)= ∑[ z −+]2 kz k h,(8.6)2N( h)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 h ± ∆hto increase N(h ) (thenumber of points). For this purpose ∆ h is implemented as an option in the krigingalgorithm.8.3. Semivariogram modelA Gaussian model is used to fit the data calculated by (8.6). The model is written ask=1γ*⎧ 0⎪( h)= ⎨C⎪⎩0h = 02⎡ ⎛ 3h⎞⎤+ C1⎢1− exp⎜ −⎟⎥h < 0.2⎣ ⎝ R ⎠⎦(8.7)The coefficients = [ C C , R] Tfunctionθ are calculated iteratively by using the objective0 , 1and the Pattern Search iteration algorithm. The Pattern Search algorithm was chosenfor its properties of being independent of derivatives. Further details aboutimplementation of the Pattern Search is given in [12]. C0+ C1in (8.7) is called the2*Sill and is equal to C(0)= σ = lim γ ( h).The Gaussian semivariogram model doesh→∞approach the Sill asymptotic without ever reaching it.*θ = min γˆ(h)− γ (θ , h)(8.8)θThe estimation given in (8.6) to (8.8) is used to estimate the variances C(h)in (8.5),where C (0)is approximated by calculating γ * ( h ) for some very large number of h.25

The estimation of C (h),∀ h,is used to approximate the coefficients in (8.3), which isthen used to calculate the weights for (8.2).8.4. Kriging resultsThe result of kriging sparsely located Slant-Range-Shift values from the test image isshown in Figure 8.1 and kriging result of GPS measured Slant-Range-Shift from theReykjanes Peninsula is shown in Figure 8.2. It can be an advance, when calculatingγ * ( h), 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 anoption in the kriging algorithm.The assumption of having spatially first or second order stationary ground movementsis not expected to hold in general. Furthermore, the correlation is likely to be also afunction of direction between points. But Figure 8.1 (d) shows that the ordinarykriging algorithm can be used to interpolate between sparse deformation values with avery good result, by limiting h in (8.6) to some maximum number before estimatingthe semivariogram model in (8.7). Three-dimensional crustal deformation inferredfrom kriging of GPS measurements is shown in Figure 8.3. The motion field isdisplayed 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 markedas + 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 ofinterpolating between the sparse values, (d): the error difference between the images in (b)and (c).26

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 pointsmarked 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 GPSmesurements. (a), (c) and (e): vector plot of the sparse GPS measured Vertical, East andNorth crustal deformation respectively, (b), (d) and (f): Kriging of the Vertical, East and Northcrustal deformation respectively,27

8.5. Estimation of uncertaintyA spatial uncertainty estimation is implemented into the kriging algorithm. This isdone 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 iscalculated as the inverse proportional to the Gaussian semivariogram model.3. The uncertainty image is scaled to the interval [0,1] where 1 means nouncertainty.8.6. Comparison of interpolation methodsThe result of kriging was compared to two other interpolation methods. The firstmethod use a delaunay triangulation cubic splining of the data (an available functionin MatLab) and the second one use distance weighted average. A distance weightedaverage estimation of an arbitrary point z0in given asN0= ∑ iz i,i=1zˆ ω(8.9)whereωi= N1 d∑i=1i1 di,(8.10)ziis the i th sample of the N numbers of GPS observations, ωiis the weight of the i thGPS sample and diis the distance from the i th sample to the observation point d0.The comparison is given in Figure 8.4. It is evident that the best result is achievedwith the ordinary kriging process.28

Figure 8.4. Comparison of interpolation methods, estimated by using the test image; (a): thecorrect 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 thesubimages.29

9. Using kriging of GPS measurements and MarkovRandom Field regularisation to unwrap InSAR imagesThe InSAR images are unwrapped before inferring the tree-dimensional crustaldeformation by combined GPS and interferometric observations. InSAR images fromthe Reykjanes Peninsula can include large noise factors like previously described.Also, the interferometric signal is expected to be relatively weak compared to theerror factors, since it represent slow crustal deformation. Furthermore, high- and lowfrequency atmospheric noise can drastically affect the image information, since thePeninsula is surrounded by an ocean. Large atmospheric noise factors can give ariseto number of problems when using unwrapping processes.A process that uses Markov Random field (MRF) regularisation and simulatingannealing optimisation was designed to unwrap InSAR images. The process can beinitialised and guided by sparsly located “correct” values like GPS measurements. Forthe initialisation, the ordinary kriging method, described in Chapter 8, is used tointerpolate between the sparse GPS measured Slant-Range-Shift and create a virtualunwrapped InSAR image. One advance of using simulating annealing optimisation ofthe MRF regularisation is its capability of unwrapping images despite of largeatmospheric noise factors. Results of applying the method to both the test image, andthe interferometric and GPS measurements from the Reykjanes Peninsula arepresented in this chapter.9.1. Problem descriptionThe relationship between an unwrapped ( I Uwbe written as) and a wrapped ( I W) InSAR image canI I +Uw= WλN2,(9.1)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 thewave numbers in N . Figure 9.1 (a) and (b) shows the wrapped interferogram I and Wthe 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 wholeInSAR image, and will be considered in Section 9.5. Figure 9.1 (c) and (d) shows awrapped and unwrapped interferogram of the same area, highly influenced byatmospheric noise. The wrapped interferogram is periodical with no informationavailable about the wave numbers. This can be seen as abrupt changes from light todark coloured areas or reverse (see Figure 9.1. (c))30

Figure 9.1. Wrapped and unwrapped format; (a): the test image on a wrapped format, (b): thewave numbers for the image in (a), (c) and (d): a wrapped and unwrapped interferogram fromthe Reykjanes Peninsula, respectively.9.2. Estimation of the initial wave numbersThe ordinary kriging algorithm described in Chapter 8 is used to calculate a virtualinterferogram from a sparsely located GPS measured Slant-Range-Shift. The virtualinterferogram ( I V) along with the wrapped interferogram ( I W) can be used toestimate the wave number matrix N in (9.1) as( ⎛ IV− IW) .2⎟ ⎞N = round⎜⎝ λ ⎠(9.2)This estimation will be used as an initial step in further calculations.Estimation of N by (9.2) with IVas the kriged test image in Figure 8.1 (c) and I as Wthe wrapped test image in Figure 9.1 (a), is shown in Figure 9.2.31

Figure 9.2. Estimation of wave numbers by help of the ordinary kriging algorithm; (a): wavenumbers estimated by (9.2), (b): the wrapped test image plus the estimated wave numbers( I Uw= I W+ N λ 2 , IWis shown in figure 9.1 (a)).As previously described, the assumption of having linear deformation with time at theReykjanes Peninsula does not always hold (Section 2.3.3 and Chapter 4). This canresult in a bad consistency between the time scaled GPS measurements and theinterferometric measurements. Comparisons have though shown that a tolerable fitcan be achieved between most of the 1993 to 1998 GPS observations and the tiltedinterferometric observations that represent various elapsed time intervals between1992 to 1996. The algorithm offers the opportunity to be initialised and guided with aGPS measured Slant-Range-Shift. Here, the following is done to ensure consistencybetween the GPS data and the InSAR images before unwrapping: Excluding from theGPS data set before kriging and unwrapping1. GPS measurements in disagreement with other neighbouring GPS measurementsand2. GPS measurements in poor agreement with the tilted interferogram.The disagreement between the GPS points and the interferogram can be due to severalreasons like a non-linear deformation and errors and noise in GPS and interferometricmeasurements.Figure 9.3 shows a result of estimating the wave number matrix N for the 2.29 yearsinterferogram 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 goodestimation of the initial wave numbers (Figure 9.2 and 9.3). The result is though verydependent on the consistency between the sparse GPS measurements and the tiltedinterferogram. This is evident by comparing the kriging result in Figure 8.2 (b) and9.3 (b). Kriging all the GPS points available (Figure 8.3 (b)) would lead to large errorwhen estimating the wave numbers in (9.2) for the 2.29 years tilted interferogram inFigure 9.3 (a).32

Figure 9.3. Estimation of wave numbers by help of the Kriging algorithm; (a): the wrapped andtilted 2.29 years interferogram (1993-1995), (b): a result of Kriging the GPS pints with thelocation shown as ⊕ in (a), (b) and (d), (c): the wave number estimated by (9.2); (d): thewrapped and tilted interferogram in (a) plus the estimated wave numbers ( I I + N λ 2Uw= W).9.3. Simulating annealing and MRF regularisationThe problem task is modelled by using a MRF based regularisation. The wrappedinterferograms are then unwrapped by a simulated annealing optimisation. Theoptimisation process uses a Maximum a posteriori (MAP) estimate to represent anoptimal realisation image x of a random field X , for a given image y [13]. The MAPestimation is given asxˆ= arg max P ( X = x | Y = y).x(9.3)For convenient P ( X = x)will be written as P (x)when expressing the likelihood.The Bayesian theorem givesP(x)P(y | x)P( x | y)=∝ P(x)P(y | x),P(y)(9.5)33

where P (x)represent a prior expectations about the random field X (oftensmoothness assumptions) and P ( y | x)is the likelihood of the image y given theimage x (the relation to the observations). The simulating annealing optimisation canbe described as a sampling of the densityP ( x | y)∝T1[ P(x)P(y | x)] T ,(9.6)where the temperature T starts at some “high” value T > 00 and falls towards 0during the iteration steps. One of the great advances of using simulating annealingoptimisation process is its relatively low risk of running into a local minima comparedto other optimisation algorithms.A Markov random field X with a realisation image x is defined with respect to itsneighbourhood system (see definition of MRF and Gibbs random field (GRF) in[13,14]). By using the Hammersley-Clifford theorem (MRF-GRF equivalencetheorem) [13], the density function in (9.6) can be written asP T⎛ 1 ⎞( x | y ) ∝ exp⎜− U ( x | y ) ⎟,⎝ T ⎠(9.7)where U ( x | y)is an energy function defined with respect to the neighbourhoodsystem in the image x. Due to the MRF-GRF equivalent theorem, the MRF modellingcan be regarded as defining a suitable energy function that leads to global minima forunwrapped interferograms.9.4. The simulating annealing unwrapping algorithm and MRF modelsThe 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 GPSand interferometric observations, e.g. by (9.1) with N estimated from (9.2). The goalis then to minimise an energy function U ( x | y)= U ( N | I).By defining a suitableMRF regularisation models for the InSAR images, the illdefined unwrapping problemcan turned into well defined. A suitable energy function is designed with respect tothe 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 algorithmSimulating annealing optimisation algorithm is used to minimise of the energyfunction U ( N | I),and hence, find the unwrapped interferogram (the realisationimage). For a wrapped interferogram IW, with M-numbers of pixels, the algorithmcan 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 ( T = T0).2. k=1, where k is a pixel number.3. Increased or decreased the wave number Nkby 1 with equal probability, whichgives a new wave number matrix'N .34

( T ).''4. Calculate r = ( p ( N | I)p ( N | I)) = exp − ( U ( N | I)−U( N | I))5. If r > µ [ 0,1 ]kTT'k, then Nt= Nk k, else Nt= Nk.k6. k=k+1, if k ≤ M go to step 3, else go to the next step.7. N = , I = I W+ N λ 2 except maybe at sparse GPS points, T = T ⋅ cool , whereN tcool < 1 is constant.8. Go to step 2.[ 0,1]µ is random number within the interval [0,1], selected from a uniformdistribution. Algorithm 9.1 can be implemented both as a non-recursive and a codedrecursive.Update of coded-recursive algorithm can be explained with help of thepixel-grid in Figure 9.4, i.e. step 3 to 7 in Algorithm 9.1 could first be done on pixelmarked as X and then repeated for pixels marked as O, before lowering thetemperature. A coded recursive algorithm is often more efficient and faster than nonrecursive,but on the cost of being more complicated.implemented as non-recursive.Here, the algorithm is9.4.2. Penalizing the second derivativeSeveral energy functions have been tested that requires the image surface to besmooth. The best results have been achieved by requiring smoothness of the firstderivative, implemented as a penalization on the second derivative with theapproximationU21(N ) = γ1∑∑( fi+ 1, j+ fi−1,j− 4 fi,j+ fi,j+1+ fi,j−1) ,i∈uj∈v(9.8)where γ1is a constant, u and v is the row and column space respectively andf = I + N 2. Algorithm 9.1 can be used to minimise (9.8) by settingi , j W i , j i , jλU ( N | I ) = U 1(N ).If the temperature is lowered slowly enough, then (9.7) will assign maximumprobability state to the annealed image [13]. Experiments indicate that hightemperature ( T > 070 in Algorithm 9.1) is needed for the unwrapping. Experimentshave also shown that too high temperature can easily result in damage of correctlyunwrapped image areas, which is not easy to overcome. To avoid this, withoutreducing the probability of finding the global solution, the following is done:1. The temperature T0is set reasonable high and one annealing is done by using step1 to 8 in Algorithm 9.1.2. If T ≤ T 1, 0 < T1

Figure 9.4. Pixel-grid explaining the coded-recursive update.Figure 9.5 explains this procedure when using the test image and the parametersγ = 2, T = 90,T = 2 and = 0.99.1 0 1cool No sparsely located GPS points were used.Figure 9.5 (a) shows the initial wave numbers, (b) the initial image and (c) the resultafter one annealing. The correct solution is reached after ~25000 iterations on eachpixel or 65 reannealing (Figure 9.5 (e) and (f)). The initial wave numbers used in thisexample 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 estimatedsemivariogram calculated by (8.6), instead of using the Gaussian model in (8.7). Thiswas 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 interestIt is not necessary to do reannealing on all the image pixels. A method that finds thearea of interest for each reannealing can be described as follow:1. A thresholded edge detection is used to find large edges in the resulting InSARimage after each reannealing.2. The areas of interest in the resulting binomial image are then expanded by a 5x5constructive element dilation, to create a mask for next reannealing.Several edge detection methods have been tested for this purpose, both methods thatsearch for the highest gradient by approximating the first derivative, and methods thatlooks for zero-crossing by approximating the second derivative. The best result hasbeen achieved with the socalled Prewitt operator [7]. The Prewitt operator finds thesteepest gradient by approximate the first derivative. Here, the gradient is estimatedfor two direction (x and y) with the two maskshy⎡ 1⎢⎢0⎢⎣−111 ⎤⎥⎥−1⎥⎦⎡−1⎢⎢⎢⎣−11⎤⎥⎥1⎥⎦[] 0 0 and h = −1[] 0 1 ,=x−100(9.9)where the small parenthesis indicates the kernel origin. The maximum gradient is thengiven by the mask that gives the maximal respond. The binomial edge image can becreated for example by thresholding the edge image to half the maximum imagevalue.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 Figure9.5 (c) (the result after one annealing). Only pixels within the black areas of the maskare reannealed.By using only the penalization on the second derivative and the no initialisation of thewave numbers, the wrapped test image (Figure 9.1 (a)) can be unwrapped in ~50000iterations or 128 reannealings, which is much slower than if the kriged virtual InSAR36

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 tounwrap a smoothed version of the 3.12 years InSAR image from the ReykjanesPeninsula is shown in Figure 9.6.Figure 9.5. Result of unwraping by only penalizing the second derivative; (a): the initial wavenumbers N; (b): the initial image ( I = I W+ N λ 2 ), (c): the result after one annealing, (d): amask used when reannealing the image in (c), (e): the resulting wave numbers after ~25000updates of each pixel or 65 reannealings, (f): the resulting image after 65 reannealings.37

Figure 9.6. Unwrapping an InSAR image by penalizing the second derivative; (a): thewrapped 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 thereforeno information about the absolute pixel values. Hence, it tends to keep the wavenumbers of the most dominant joined area unchanged. It is explained in Section 9.2how the kriging of sparse GPS data can be used to initialise the unwrappingalgorithm. The GPS points can also be utilised further to guide the algorithm, since itincludes information about the absolute pixel values at sparse locations. This can bedone in several ways.The method presented here uses energy function that penalizes only pixels in thenearest neighbourhood of the GPS pixel. The GPS pixel domain will be called adomain of frozen (fixed) pixels, since they are not updated in the simulating annealingoptimisation algorithm. The domain of frozen pixels is then expanded before eachreannealing. This method utilises the knowledge from GPS observations.The energy function used to give extra penalization for pixels in the neighbourhood tothe correct domain is written asU2( I | N)= γ2 ∑k ↔n2(( f − f ) W ) ,knn(9.10)where f I + N λ 2,k=W k k2γ is constant,k ↔ n means all pixels k in theneighbourhood of the pixel n and Wnis 1 if n is within the domain of correct pixelsand zero otherwise. The relationship to the observed image I is gained by freezing theGPS pixels (The GPS observations). The energy function in (9.10) is used along withthe energy function in (9.8), with γ = 12 and γ = 70,i.e.2U ( N | I)= U1(N)+ U2(I | N).(9.11)γ1was optimised by using only the process described in Section 9.4.2, for T = 90 0,T = 2 and cool = 0.99.The high value of 1γ2result from experimenting with U2on38

pixel domains close to the domain of frozen pixels, done by keeping all otherparameters unchanged.The following masking is done before each reannealing process:1. the domain of frozen pixels is expanded with a dilation, by using a structuredelement with four nearest neighbourhood pixels and the pixel.2. in step 6 in Algorithm 9.1 I is updated by I = I W+ N λ 2 except within thedomain of frozen pixels and3. the reannealing is done only at areas masked by the thresholded edge detectionfollowed by dilation.Figure 9.7 explains how the domain of correct pixels expands when T = 90 0, T = 2 1and cool = 0. 99 in Algorithm 9.1. The initial wave numbers used in this example areshown in Figure 9.5 (a), and the resulting image and the corresponding mask after 10reannealing in Figure 9.7. The final solution was reached after ~14000 iterations or 38reannealing.The main advances of utilising the sparsely located GPS points are:1. information about the absolute pixel values have been included into theunwrapping algorithm,2. the algorithm runs into the expected solution with more safety and faster (aroundtwo times faster for the test image).9.5. Unwrapping processThe methods described so far were used to build up an unwrapping process for InSARimages that may include high noise factors. The algorithm uses the penalization on thesecond derivative and expansion of the domain of frozen pixels. A virtual InSARimage is created by ordinary kriging of the sparse GPS pixels, and the initial wavenumbers are estimated by (9.2).Figure 9.7. Expansion of the domain of frozen pixels; (a): the result after 10 reannealing andthe expansion of the frozen domain (the frozen domain are shown as light rhombus which areexpanding from its centre, marked as +), (b): the mask used for reannealing of the image in(a).39

Error factors need to be considered before unwrapping. To be able to use the GPSmeasurements the following is done:1. time scale GPS measurements are time scaled to fit the elapsed time intervalrepresented by the InSAR image,2. use the tilting process described in Chapter 7 to eliminate a phase plane in theInSAR observations and3. remove measurements from the GPS data set that are in bad consistency with thetilted InSAR image.High frequency noise in the InSAR images can influence the penalization on thesecond derivative by both generating errors and slow down the process. Furthermore,low frequency error can smooth edges of the size ~ λ 2 (that correspond to 2π leapsdue to periodical properties of the signal, see explanation in Section 7.3.1), which inturn can lead to failure when using the expansion of the frozen pixel domain. Thefollowing unwrapping process is designed to reduce the effects from those noisefactors:1. The InSAR image IWis oversmoothed by the vectorized filtering (Chapter 5),which gives the image IO.This smoothes the surface and makes the penalizing ofthe second derivative easier. Furthermore, this reduces also the errors of unwantedsmoothing of ~ λ 2 edges.2. A mask is created (Chapter 6).3. Simulated reannealing is done by using (9.11), and the domain of frozen pixels isexpanded before each reannealing.4. After full expansion of the domain of frozen pixels, errors generated by unwantedsmoothing of ~ λ 2 edges are reduced by a simulating annealing process thatuses only penalization on the second derivative (like explained in Section 9.4.2).5. The resulting unwrapped oversmoothed InSAR image IOis then used toUwestimate the wave numbers for the unfiltered InSAR image IWby using⎛ IN'= ( round)⎜⎝(9.12)'6. An unwrapped unsmoothed interferogram is calculated as I Uw= I W+ N λ 2.7. The resulting unwrapped interferogram is masked by pixelvise multiplication ofthe interferogram and the mask.A dataflow diagram of the unwrapping process is shown in Figure 9.8. It is possible touse the mask image as an input into the iteration process, and do only updates on theforeground area. Experiments have shown that this can influence or damage the outerboarder of the area of interest. Also, this is not needed when the thresholded edgedetection process, described in Section 9.4.3, is used. Here, a masking is done afterunwrapping as explained in Figure 9.8.Figure 9.9 shows the result of applying the process to the 2.29 years interferogramfrom the Reykjanes Peninsula, where each step in the unwrapping algorithm isexplored. The effect of oversmoothing the wrapped interferogram before unwrappingOUwλ− I2W⎟ ⎞.⎠40

is evident by comparing Figure 9.3 (d) and Figure 9.9 (e). It is also evident fromFigure 9.9 (i), that an accurate estimation of the wave numbers for a high frequencywrapped InSAR image can be done by using the corresponding oversmoothedunwrapped InSAR image and (9.12). Indeed, experiments indicate that the wavenumber 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 anassumption about the surface, by penalizing the second derivative. The GPS measuredSlant-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 isdone 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, whichmakes 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 tobe efficient in unwrapping the InSAR images from the Reykjanes peninsula.41

Figure 9.8. Dataflow diagram of the unwrapping process.42

Figure 9.9. Exploring the unwrapping procedure by using 2.29 years interferogram from theReykjanes Peninsula. (a): Tilted and unsmoothed wrapped interferogram, (b): the image in (a)oversmoothed, (c): ordinary kriging of sparsely located GPS observations, (d): estimatedinitial wave numbers by using (b) and (c), (e): the wave numbers in (d) added to (b), (f): theresult 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 secondderivative, (h): wave numbers estimated from (a) and (g), (i): the wave numbers in (h) addedto (a). Sparse location of GPS points are shown as + on some of the subimages.43

10. Combination of GPS and interferometricobservations to infer three-dimensional groundmovementsThe InSAR images include high-resolution maps of one-dimensional Slant-Rangeground movements, while GPS measurements contain information about threedimensionalmotions at sparse locations. In Chapter 9 it was shown how an ordinarykriging of sparse GPS measurements, MRF based regularisation and simulatingannealing optimisation can be used to unwrap interferograms. The ordinary krigingalgorithm, MRF based regularisation and simulating annealing optimisation is utilisedfurther in this chapter to estimate high-resolution maps of the three-dimensionalground movements from combined GPS and interferometric observations. Result ofapplying the method to both the 164x164 pixels test image and interferometric andGPS observations from the Reykjanes peninsula are presented.10.1. Problem descriptionThe satellite measure the one dimensional Slant-Range-Shift (SRS) given asVSRS kT[ V ,V , V ] ⋅[ u , u , u ] ∀k= −,E kN kV kENV(10.1)for each pixel k in the InSAR imageV . ,SRSVE NV andVertical of deformation images, respectively, and [ ]VVare the East, North ands = u , u u is a given unitE N,vector pointing from the ground towards the satellite. For the descending satellite passInSAR images from the Reykjanes PeninsulaVs ≈ [ 0.34E,−0.095N,0.935V].(10.2)The task is then to find the three motion field images VE, VNand VV, for knownVSRSimage (InSAR observations) and sparse values of VE, VNand VV(GPSobservations).10.1.1. Simplifying the problemThe three-dimensional problem can be changed into two two-dimensional problems,for example asVSRS k= −T[ V , V ] ⋅[ u , u ] , ∀k,L kV kLV(10.3)whereu = u + uL2E2N(10.4)andVL k=[ V , V ] ⋅[ u , u ]E kN kuLENT,∀k.(10.5)44

Figure 10.1. The geometry of (10.3). V V and V L are the Verticaland horizontal look-direction motion fields, respectively.VLis the deformation in the Horizontal look-direction of the satellite. The geometryof (10.3) is explained in Figure 10.1. For the InSAR images from the ReykjanesPeninsula [ u u ] [ 0.935,0.353].V,L=The East and North motion field imagesan optimization of VV, and VLwith (10.5) or by usingV and V , respectively, can be found afterENVSRS kT[ V ,V ] ⋅ [ u , u ] ∀k+ u V = −,VV kE kN kEN(10.6)10.2. Optimization of two-dimensional deformationA simulated annealing process is used to optimise a MRF based regularisation of thetwo-dimensional ground deformation given in (10.3) and (10.6). The process isinitialised with two motion field images generated with an ordinary kriging of thesparsely located GPS observations (see explanation of simulating annealing and MRFmodelling in Section 9.3, and ordinary kriging in Chapter 8).10.2.1. Two-dimensional simulating annealing algorithmA general form of (10.3) and (10.6) isVk= −T[ V ,V ] ⋅ [ u , u ] , ∀ ,1 2 1 2kk k(10.7)where Vkis known for all pixels k, V1and Vk2are only known at sparse locationskand u1and u2are constants. The MRF models can therefore be designed forregularisation of two-dimensional motion field images.The simulating annealing process used for the optimisation of two realisation imagesis given in Algorithm 10.1. The algorithm is used to optimise a combined energystage ( U1(V1,V2|V)and U2( V1,V2| V)) of two images in the same iterationprocess. A 1/0-switch s2is added to the algorithm. If the switch value is 1 then boththe motion field images are updated, but if the value is 0 then only the motion image45

V1is updated by keeping V2unchanged. This can be an advance if one of the motionfield images is known with a high certainty.Algorithm 10.1.1. Choose initial images V1and V2(e.g. by kriging). Extract region of interest(Chapter 6) and set the initial temperature T = T 0.2. Choose a switch, s = 21 or s = 0.23. k=1, where k is a pixel number.4. Increase or decrease V1with equal probability by a value of ∆ V, which gives aknew image V' .1'5. Calculate r1= ( p1( V1, V2| V)p1( V1, V2| V)) =k TT'( − ( U ( V , V | V)−U( V , V | ))T )exp1 21 1 21V6. If r1> µ [ 0,1],then V = 'V , else V = V .k1t k 1k1t k 1k7. k=k+1, if k ≤ M go to step 4, else go to the next step, (M is the total number ofpixels).8. V1= V 1t, except maybe at the sparse GPS points.9. If s = 0 2, go to step 17. Else if s = 1 2, go to the next step.10. k=1, where k is a pixel number.11. Increase or decrease V2with equal probability by a value of ∆ V, which gives ak'new image V .2'12. Calculate r2= ( p2( V1, V2| V)p2( V1, V2| V)) =k TT'( − ( U ( V , V | V)−U( V , V | V)) ).exp1 22 1 22T13. If r2> µ [ 0,1],then V = 'V , else V = V .k2t k 1k1t k 2 k14. k=k+1, if k ≤ M go to step 11, else go to the next step.15. V2= V 2t, except maybe at the sparse GPS points.16. T = T ⋅ cool,where cool < 1 is a constant.17. Go to step 3.[ 0,1]µ is a random number within the interval [0,1], selected from an uniform randomgenerator. Like Algorithm 9.1 (Section 9.4), Algorithm 10.1 can be implemented bothas a non-recursive and a coded-recursive. Here it is implemented as a non-recursive.10.2.2. Energy functionsThe main energy functions used in Algorithm 10.1 uses the relationship between thetwo-dimensional motion field images and the known image V (e.g. InSAR image)given in (10.7), and is writtenT 2( V + [ V , V ] [ u , u ] ) ,U12 ( V | V1, V2) = γ12∑ n 1 2⋅nnn12(10.8)where n is a pixel number. An additional constraint is the smoothness of the firstderivative of V1and V2, implemented as a penalization on the second derivative as46

and( V + V − 4V + V )∑∑ 1+i− 1, j 1i+1, j 1i,j 1i,j−11i,j+U11(V1)= γ1Vij( V + V − 4V + V)∑∑ 2+i− 1, j 2 i+1, j 2i,j 2i,j−12i,j+U22( V2) = γ2Vij1122(10.9)(10.10)where i, j are the row and column numbers respectively. γ12, γ1and γ2areconstants in (10.8), (10.9) and (10.10). The purpose of the smoothness requirements isto keep neighbouring pixels connected or preserve the correlated relationship of theimage pixels. If those energy terms are excluded the pixel values can turn out to bespatially chaotic.By using (10.8) to (10.10), the energy functions U1(V1,V2| V)and U2( V1,V2| V)inAlgorithm 10.1 are written asU1(V1,V2| V)= U12( V | V1, V2) + U11(V1)(10.11)andU2( V1,V2| V)= U12( V | V1, V2) + U22( V2)(10.12)It should be noted that infinite set of optimal solutions exist for (10.8) to (10.10) whenthe correct values of the images V1and V2are only known at few sparse locations.Therefore, solution from any optimisation process must be very dependent on thequality of the initial values. Hence, the quality of the solution is very dependent on thedensity and quality of the GPS measurements and the performance of the interpolationprocess.For Algorithm 10.1, the final maximum probability state assigned to the annealedimages is rather independent of the initial temperature value, as long as it is set highenough at the initial state. This is due to the strong relationship of the motion fieldimages to the interferometric observations. This was not the case the MRFregularisation, 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 andHorizontal look-direction motion field images (with V1= VLand V2= VV). In thisexample, ∆ V = 0.1,γ = 10 12, γ = 1,γ = 2,T = 5,= 0.99,1 2 0cool u 0.1= uL= 353and u2= uV= 0.935.Sparse GPS values were not updated during the iteration run(frozen GPS values). The program was terminated for T < 0.1.Table 10.1 shows theestimated errors for the example given in Figure 10.2. The process errors areestimated as the mean (µ ) and standard deviation (σ ) of the difference between thecorrect and predicted motion field images. Table 10.1 shows that the kriging errorsare decreased by Algorithm 10.1, both for the Vertical and the Horizontal lookdirectiondeformation images. The predicted uncertainty is though much less for theVertical deformation component. This can be explained by the high contribution of47

the Vertical component to the Slant-Range-Shift observation ( u = 0. V935 versusu = 0.353L), which makes it more dominated in the iteration process. The balancebetween the two components are though partly taken care of by setting γ2> γ 1. Thereason for the higher kriging error of VLthan VVin this example is the location ofthe GPS points, which are more favourable for the vertical component.Table 10.1. Estimated kriging and simulation errors. ∆ V = 0.1,γ = 10,γ = 1,γ = 2,12 1 2T = 5, cool = 0.99,u = u = 0.3530 1 Land u 2= u V= 0.935.Vertical Horizontal look-direction Slant-RangeKriging errorµ 0.07 0.09 -0.10σ 0.44 0.79 0.93Simulation errorµ -0.02 0.03 0.002σ 0.28 0.73 0.0410.2.3. Further utilisation of GPS observationsBoth the interferometric and GPS measurements can include error factors. The lowestkriging uncertainty of the multi-dimensional motion fields is though expected to be atthe sparse GPS location and decrease as a function of the distance from them.Uncertainty images can be created along with the kriging of motion fieldmeasurements (Section 8.5). The uncertainty images can be utilised when penalizingthe motion field images for deviating from the GPS observations. This is done byusing the energy functionsUK1( VK1| V ) = γ1K12∑ ( W1( − ) )nVK1nV1nn(10.13)andUK 2( VK2| V2) = γK22∑ ( W2( − ) )nVK2nV2n,n(10.14)where n is pixel number, W1and W2are the uncertainty images corresponding to theresulting kriged motion field images VK1and VK2, respectively, and γK1and γK2areconstants. The uncertainty images includes the intensity interval [0,1], where 1 meansno uncertainty (at GPS locations) and 0 means no certainty. An example ofuncertainty image is given in Figure 10.3, where the sparse GPS pixel locations arealso marked on the image.The purpose of (10.13) and (10.14) is to give an extra force to penalize the images fordeviating 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 thecorrect multi-dimensional motion field at sparse locations and if the interferometricsignal is weak or includes high error factors.48

Figure 10.2. A result of applying Algorithm 10.1 to the 164x164 test image. (a): Differencebetween the correct and initial values (kriged) of the vertical motion field, (b): differencebetween the correct and simulated values of the Vertical motion field, (c): difference betweenthe correct and initial values of the Slant-Range-Shift, (d): difference between the correct andinitial values (kriged) of the Horizontal look-direction motion field, (e): difference between thecorrect and simulated values of the Horizontal look-direction motion field, (f): differencebetween the correct and simulated result of the Slant-Range-Shift. (g): Plot of line 82 out ofthe initial, simulated and correct vertical images, (h): plot of line 82 out of the initial, simulatedand correct Horizontal look-direction images, (i): plot of line 82 out of initial, simulated andcorrect Slant-Range images. The locations of the spares GPS sites are shown as + on theimage in (c).49

Figure 10.3. Uncertainty image created with the kriging algorithmin Chapter 8. Sparse GPS loacations are marked as +.The energy functions in (10.13) and (10.14) are added to Algorithm 10.1 by usingandU1(V1,V2| V,VK1) = U12( V | V1, V2) + U11(V1) + UK1(VK1| V1)U2( V1, V2| V,VK2) = U12( V | V1, V2) + U22( V2) + UK2( VK2| V2)(10.15)(10.16)instead of ( V , V | ) and ( V , V | ), respectively, into the algorithm.U1 1 2VU2 1 2VA 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 to10.4. In these examples, the coefficients are all the same as fore Table 10.1, with thealgorithm terminated when T < 0.1.In addition γK1= γK2= 10 in (10.15) and(10.16). For all the tables, the Vertical and Horizontal look-direction components areoptimised by using (10.3) and Algorithm 10.1, and the East and North components arethen estimated afterwards by using the optimised Vertical image, (10.6) andAlgorithm 10.1. No frozen pixels were used when the uncertainty images wereutilised in the MRF regularisation. Table 10.2 shows the estimated errors (the meanand standard deviation) between the correct and predicted motion fields images, whenno errors are included in the interferometric and GPS observations. Table 10.3 showsthe same, except a Gaussian noise with µ = 0 and σ = 0. 5 has been added to theinterferogram. Table 10.4 shows then the result of adding also a Gaussian noise withµ = 0 and σ = 0. 2 to the sparse GPS measured motion fields.The results in Table 10.2 to 10.3 indicate that the method offers high certaintyestimation of the Vertical motion field. This is a consequence of the high contributionfrom the Vertical ground deformation component to the observed Slant-Range-Shift(see (10.2)). Optimisation results for the other motion field image may depend onnoise errors in the measurements. Experiments have also shown that the optimisationresults for each individual motion image can be very dependent on the spatialdistribution of the GPS locations. Despite of that, the tables indicate that the krigingerrors can be reduced by using the relationship of the motion field images to theinterferometric observations. Also a better result can be achieved for the East andNorth components (motion fields with the lowest contribution to the interferometric50

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 theenergy functions (10.15) and (10.16). No errors are included in the interferogram or the GPSmeasurements.Vertical Horizontal lookEastNorthdirectionKriging errorµ 0.07 0.09 0.18 0.32σ 0.44 0.79 1.03 0.66Simulation error by using (10.11) and (10.12) and Algorithm 10.1µ -0.02 0.03 0.13 0.23σ 0.29 0.76 0.85 0.57Simulation error by using (10.15) and (10.16) and Algorithm 10.1µ -0.02 0.05 0.11 0.24σ 0.28 0.72 0.83 0.56Table 10.3. An error estimation of the energy functions in (10.11) and (10.12) versus theenergy functions (10.15) and (10.16). A Gaussian random noise have been added to theinterferogram.Vertical Horizontal lookEastNorthdirectionKriging errorµ 0.07 0.09 0.18 0.32σ 0.44 0.79 1.03 0.66Simulation error by using (10.11) and (10.12) and Algorithm 10.1µ -0.03 0.06 0.13 0.06σ 0.31 0.75 0.84 0.75Simulation error by using (10.15) and (10.16) and Algorithm 10.1µ -0.03 0.06 0.11 0.24σ 0.31 0.73 0.86 0.58Table 10.4. An error estimation of the energy functions in (10.11) and (10.12) versus theenergy functions (10.15) and (10.16). A Gaussian random noise have been added to theinterferogram and the GPS measurements.Vertical Horizontal lookEastNorthdirectionKriging errorµ 0.06 0.04 0.18 0.05σ 0.54 0.83 1.04 0.93Simulation error by using (10.11) and (10.12) and Algorithm 10.1µ -0.02 0.03 0.11 0.22σ 0.36 0.89 0.95 0.87Simulation error by using (10.15) and (10.16) and Algorithm 10.1µ -0.01 0.03 0.07 0.23σ 0.36 0.86 0.82 0.8610.3. Inferring the three dimensional motion field at the ReykjanesPeninsulaAlgorithm 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 thebackground are kept frozen during the iteration run. This makes the algorithm faster.51

The first program uses the energy functions given in (10.11) to (10.12) and frozenGPS pixels. The second one uses the energy functions in (10.15) to (10.16) and nofrozen foreground pixels. The programs were then tested to infer the threedimensionalmotion fields at the Reykjanes Peninsula.10.3.1. Corrections of the data observationsChapter 7 presented a method to tilt wrapped interferograms by using a GPSmeasured Slant-Range-Shift. This tilting process can be used before unwrapping. AGPS tilting of an unwrapped InSAR image is though expected to be much safer thantilting of a wrapped InSAR image, since modulated effects have been removed. Here,the InSAR images are tilted again after unwrapping, with the GPS observations andthe Least Square (LS) algorithm. This is done to ensure save tilting (more accurateelimination of the phase plane) and consequently better consistency between the GPSand interferometric measurements, before inferring the three-dimensional groundmovements.The LS tilting is done by using (7.9) to (7.11) (see Section 7.2). Here, IInSARinUw(7.11) is the resulting interferogram from the unwrapping process described in Section9.5. The LS tilting process uses a mask detection (Chapter 6), to extract region ofinterest after tilting of the unwrapped InSAR image.It is possible to use all the GPS measurements available for the kriging and the MRFoptimisation. Also, it is not necessary to have all the motion components measured atsame 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 theInSAR image,2. remove GPS observations that are in bad consistency with the unwrapped andtilted InSAR image.Dataflow diagram describing the usage of the ordinary kriging algorithm and theoptimisation 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 mapsFigure 10.6 to 10.9 shows the result of inferring the three-dimensional crustaldeformation for the 4.17 years interferogram (Table 2.1). Figure 10.6 present the GPSobservations and an ordinary kriging of the GPS observations. Figure 10.7 shows theresult of combining the GPS observations and the 4.17 years interferogram. Theoriginal wrapped interferogram was pre- filtered with 3x3 moving average window.Results of inferring the crustal deformation from other images are given in AppendixB. Figure 10.7 (a), (c), (e) and (g) shows the result of inferring three-dimensionalcrustal deformation by using the energy functions in (10.11) and (10.12), and Figure10.7 (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 asinitial images in both optimisations. The residual errors between the interferogram( VSRS) and the combined predicted motion field images ( − { uVVV+ uEVE+ uNVN})are also included in Figure 10.7.52

The predicted Vertical, East and North motion field images describe similardeformation patterns in shape when using energy functions (10.11) to (10.12) versusthe energy functions in (10.15) to (1016) (Figure 10.7). The main difference is that themotion field images observe more detailed characteristics from the interferogram inFigure 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 theprocess presented in the latter case, that tends to keep updated images close to theinitial kriged images (see (10.13) and (10.14)). This does also explain why the motionfield 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 ( VSRS) and the combined motion field images ( − { uVVV+uEVE),+ u V }, N Nand also between the combined kriged motion fields images ( − { u V +u V + u V ) and the combined motion field images. Stronger relationship toEKENKN }VSRSis 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 GPSobservations).Figure 10.8 shows a comparison of line 300 out of the 450x750 pixels Vertical, Eastand 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( VSRS) and the combined predicted motion field images ( − { uVVV+ uEVE+ uNVN})are shown in Figure 10.9. It is also evident by comparing the plots in Figure 10.8, thatthe energy functions in (10.15) to (10.16) tends to keep the signal shape closer to theinitial kriged result than the energy functions in (10.11) to (10.12). The residual errorsin Figure 10.9 (b) are close to be white random noise, which indicates a strongrelationship between the interferogram and the predicted motion fields.The process that utilises uncertainty images, is recommended if the GPS observationsare 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 excludinguncertainty images is recommended (Figure 10.4).Table 10.5. The mean and standard deviation of the difference between the interferogram( V ) and the simulated motions field images ( V , V and V ), and also the kriged motionSRSV ENfield images ( V , V and V ) and the simulated motion field images.KV KEKNVSRS+ u V + u V + u V u V + u V + u V ) − ( u V + u V + u V )VVEENN(V KV E KE N KN V V E E N NUsing the energy functions in (10.11) and (10.12)µ 0.005 0.405σ 0.30 1.26Using the energy functions in (10.15) and (10.16)µ -0.055 0.466σ 0.40 1.18VKV53

Figure 10.4. Dataflow diagram of the process of using combined GPS and interferometricobservations to infer three-dimensional motion maps. The two-dimensional inferring of motionfield images used in the diagram, are described by Algorithm 10.1 and the energy functions in(10.11) and (10.12).54

Figure 10.5. Dataflow diagram of the process of using combined GPS and interferometricobservations to infer three-dimensional motion maps. The two-dimensional inferring of motionfield images used in the diagram, are described by Algorithm 10.1 and the energy functions in(10.15) and (10.16).55

Figure 10.6. Infer of the three-dimensional crustal deformation at the Reykjanes Peninsula, byusing an ordinary kriging of sparsely located GPS observations. (a), (c), (e): The sparselylocated GPS measured Vertical, East and North motion vectors, respectively. (b), (d), (f): Theresult of inferring the Vertical, East and North motion maps, respectively.56

Figure 10.7. Infer of the three-dimensional crustal deformation at the Reykjanes Peninsula byusing GPS measurements and 4.17 years interferogram. The figure shows comparison ofmotion field images optimised by using the energy functions in (10.11) to (10.12) versus theenergy functions in (10.15) to (10.16). (a), (c), (e): The Vertical, East and North motion fieldimages, 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).57

Figure 10.8. Line 300 out of the estimated 450x750 pixels motion field images in Figure 10.6and 10.7; (a): the result from kriging, (b): the result of using optimisation with the energyfunctions (10.11) to (10.12) into Algorithm 10.1, (c): the result of using optimisation with theenergy functions (10.15) to (10.16) into Algorithm 10.1.58

Figure 10.9. Residual errors between the 4.17 years interferogram and the motion fieldprofiles in Figure 10.6 and 10.7; (a): the residuals from kriging, (b): the residuals by usingoptimisation with the energy functions (10.11) to (10.12) into Algorithm 10.1, (c): the residualsby using optimisation with the energy functions (10.15) to (10.16) into Algorithm 10.1.59

11. Results and discussionsIn this thesis methodologies have been developed to combine interferometric and GPSmeasurements of ground movements. Two of the methods utilise a kriging algorithm,Markov Random Field regularisation and simulating annealing optimisation tounwrap the interferometric measurements and to infer high resolution maps of threedimensionalground movements. Several other procedures have also been constructedfor various purposes. For example algorithms that uses the GPS measured Slant-Range-Shift to eliminate a phase plane from interferometric observations, and amethod that uses a vectorized filtering to reduce noise errors in wrapped InSARimages. InSAR images may include areas with no information. It has been usefulwhen processing the images, to separate them into information and no informationareas. For this purpose, a method that uses a threshold and a morphological cleaningalgorithm was designed to extract areas of interest. All the methods discussed in thischapter, have been tested both on a test data and a real data set with very promisingresults.For the purpose of using the GPS and wrapped interferometric observations to inferhigh-resolution ground motion maps, the methods can be separated into a preprocessingand infer of three-dimensional high-resolution motion maps.11.1. Pre-processing methodsPre-processing of interferometric and GPS observations can have several purposes,e.g. reducing error effects and to obtain a good consistency between the interferometricand GPS observations.11.1.1. Noise reduction in interferometric observationsWrapped interferometric signal can be projected into two vectors perpendicular toeach other (cosine- and sinusoidal), which can also be interpreted as a projection intothe complex unit circle. Here, the cosine- and sinusoidal are filtered separately with amoving average window before doing the inverse projection. The vectorized filteringhas turn out to be very efficient in reducing error effects in the interferometricmeasurements. Furthermore, it has been a very powerful tool to use in all the GPS andinterferometric combinations.11.1.2. Extraction of area of interestAnother tool that has been useful in all the GPS and interferometric combinations, isan 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 imageis created by thresholding a wrapped or an unwrapped interferogram, followed by amorphological cleaning process. The foreground areas can include error pixels afterthresholding of the InSAR images Those areas can be fully removed with amorphological cleaning process. The mask images have been useful, both for maskingof the processed images, and to restrict the data processing to only the areas of interestand hence, make the algorithms faster.11.1.3. Utilisation of GPS observations to correct wrapped InSAR imagesA methodology that uses kriging of sparsely located GPS data, MRF basedregularisation and simulating annealing optimisation have been developed to unwrapinterferograms. The Master and Slave track of an interferogram can include two60

different viewpoints and distances to the same object, which results in a systematicerror that can be described by a phase plane [1]. The phase plane needs to beeliminated from the wrapped interferograms before utilising the GPS data in theunwrapping process. The GPS data is not expected to include significant systematicerrors, and can therefore be used to eliminate the phase plane in interferometricmeasurements.Two methods have been designed for this purpose. The first method uses a projectionof both the interferometricly and GPS measured Slant-Range-Shift into the complexunit circle, and uses an optimisation in the complex domain to find the optimalrotation and offset of the interferogram. The drawback is that the method changes aunique solution into periodical solutions, which can lead to wrong estimation of thephase plan. The second method uses an unwrapping of line profiles between sparseGPS locations in the InSAR image, to estimate sparse unwrapped values thatcorrespond to the GPS locations. The sparse unwrapped values and the sparse GPSmeasured Slant-Range-Shift are then used to find the phase plane by a Least Squareestimation. The InSAR images from the Reykjanes Peninsula can include highatmospheric noise that can disturb the profile unwrapping and hence, lead to someerrors when tilting the unwrapped interferograms. Despite of that, this process can beused before utilising the GPS measurements in the unwrapping process.11.1.4. Creation of virtual InSAR images with interpolation of GPS dataThe unwrapping process can be initialised with a virtual InSAR image, for examplecreated by an interpolation of a sparse GPS measured Slant-Range-Shift. Severalmethods have been tested for this purpose. The best result has been achieved with anordinary kriging method, when a Gaussian semivariogram model is used to estimatethe dispersion matrix for the sparse data.The ordinary kriging method assumes the sparse spatially distributed data to be firstand second order stationary. This is not expected to hold in general for groundmovements. However, very good results have been achieved by using the krigingmethod to interpolate between a sparse data of a test motion field image.11.1.5. Unwrapping processThe unwrapping process estimates the missing wave numbers of a wrappedinterferogram, by using a MRF modelling and simulating annealing optimisation. Theunwrapped interferogram (the optimal realisation image) is found by using anassumption about the surface smoothness and by using its relationship to the GPSobservations. Several energy functions have been tested that requires the imagesurface to be smooth. The best results have been attained by requiring smoothness ofthe first derivative, implemented as a penalization on the second derivative. Thesparse GPS measurements are the only data that is directly related to the wavenumbers. The unwrapping process is therefore mainly controlled by the smoothnessrequirements. Indeed, experiments have shown that an interferogram can beunwrapped by only using a penalization on the second derivative, as long as the highfrequency (both noise and information) is within certain limit.The weak relationship of the wave number estimation to the observations is expectedto result in need for very slow temperature drop (cooling) in the simulated annealingoptimisation. It has been shown for this case, that a reannealing procedure with a61

faster cooling can be successively used instead. The reannealing procedure usesrepeated simulated annealing on the images until the optimal realisation image isfound. It is not necessary to update all the image pixels during each reannealing. Athresholded edge detection followed by dilation have been used with good result todetect areas of interest before each reannealing. Only the detected pixels are thenupdated in each iteration, which makes the algorithms much faster.The GPS observations are utilised by freezing the pixel values corresponding to thesparse GPS locations (frozen pixel domains) during the optimisation, and to applyextra smoothness requirement for its neighbouring pixels. The frozen pixel domainsare then expanded before each reannealing. The main advances of using the GPSobservations are that an information about absolute pixel values have been includedinto the procedure, and the expected solution is reached with higher reliability.The InSAR images from the Reykjanes Peninsula are surrounded by an ocean, andcan 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 errorswhen expanding the frozen pixel domains. Experiments have shown that vectorizedfiltering can successfully be used to reduce high frequency noise. Furthermore, thiscan also reduce certain type of the unwanted low frequency error effects that mayappear in interferograms. Experiments do also indicate that a wave number matrix fora non-smoothed wrapped interferogram can be estimated both accurately and in asimple way from a corresponding oversmoothed unwrapped interferogram.A process has been designed that take advantage of this. Before unwrapping, thewrapped InSAR images are oversmoothed by vectorized filtering. This makes thepenalization on the second derivative more effective and reduces errors that can begenerated during the expansion of the frozen pixel domains. Errors that may appearafter full expansion of the frozen pixel domains are reduced by using a reannealingprocess that only uses penalization on the second derivative. This procedure has beenused to unwrap the InSAR images from the Reykjanes peninsula with good results.11.1.6. Utilisation of GPS observations to correct unwrapped InSAR imagesMethods have been constructed that uses MRF based regularisation and simulatedannealing 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 aspossible.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 muchmore accurate to estimate the phase plane with GPS observations along with theunwrapped version of the interferogram. Therefore, a method that uses GPSobservations and a Least Square optimisation have also been developed to eliminate aphase plane. This procedure can be used after unwrapping of the interferograms, togain more accurate consistency between the two observation methods, beforeinferring the motion maps. Results indicate that a safer phase plane elimination can beachieved by using tilting of unwrapped interferograms than wrapped.62

11.2. Construction of three-dimensional high resolution motion mapsThe interferometric observations include a high-resolution maps of a one dimensionalSlant-Range-Shift, while the GPS observations include information about threedimensionalmotion fields at sparse locations. A methodology have been developed toinfer high-resolution three-dimensional motion maps from combined GPS andinterferometric observations.The problem of optimising the three-dimensional motion field can be separated intoan optimisation of two two-dimensional motion fields images, and thereby making thealgorithms simpler. The simulating annealing algorithm is then designed to find tworealisation images in the same process. Here, the optimisation have been separated asfollows. First the Vertical and Horizontal look-direction motion maps (Horizontallook-direction of the SAR satellite observations) are estimated from combined GPSand InSAR. Then the East and North motion maps are estimated by using combinedGPS and InSAR along with the optimised Vertical motion map. A MRF basedregularisation and a simulating annealing optimisation have been used for the twodimensionalcombination with good results.The two-dimensional motion field optimisation procedure has to be initialised, e.g. bya kriging of two-dimensional GPS observations of the same motion fields. Theordinary kriging algorithm have shown to be very efficient in estimating motion fieldimages from sparse observations. Hence, it has been used with good results toestimate the initial images for the combination procedure. One advantage of theordinary kriging algorithm is it simplicity compared to other kriging algorithms. But,for further improvements, cokriging algorithms should also be considered and testedfor the estimation of initial two-dimensional motion field images.The multidimensional motion fields have a strong relationship to the interferometricobservations, which can be used as the main constraint in the MRF regularisation.More constraint are also needed, e.g. to preserve the neighbouring pixel relationshipin the image structure. A smoothness requirement, implemented as a penalization onthe second derivative, has been efficiently used for this purpose. It has also beenshown that the relationship to the sparse GPS observations can also be utilised, e.g. byusing an uncertainty estimation of the kriged images along with the MRFregularisation. The relationship to the GPS observations has successfully been used toreduce errors in the multidimensional motion field images at areas close to the sparseGPS locations.The optimal solutions of the multidimensional motion field images are illdefinedwhen the correct values are known only at sparse locations. Therefore, the result fromthe combination optimisation can be very depending on the quality of the initialvalues. Hence, the quality of the solution is very dependent of the locations anddensity of the GPS observations. Despite of that, experimental results have shown thatthe kriging errors in estimated motion field images can be reduced by using theinterferometric observations in the MRF regularisation, especially in the Verticalmotion field due to its high contribution to the interferometric signal.The optimisation procedure uses a one-dimensional interferometric measurementalong with the GPS observations, which can be either from ascending or descendingorbit passes. In some cases, it is possible to use two interferograms from the same63

area, recorded from both ascending and descending satellite orbit passes. Theascending and descending orbit passes have two different viewpoints to the sameobject, and can in some cases be interpreted as a two dimensional interferometricobservations. In that case, and additional constraint can easily be added to the MRFregularisation that makes advantage of the two-dimensional interferometricobservations. The results of inferring multidimensional motion maps are expected tobe more accurate if both the satellite passes can be utilised in the process.11.3. Pre-processing and averaged motion maps at the ReykjanesPeninsulaA 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 ashoreat the SW tip of the Peninsula (Figure 2.4 and 2.6). The Peninsula consists also oferuptive fissures and volcanoes. The ground motion consists therefore of crustaldeformations and plate movements. Available GPS measurements of groundmovements describe a deformation for the elapsed time interval from 1993 to 1998.The interferograms are all recorded from a descending satellite pass, with variouselapsed time intervals within 1992 to 1996.Measurements from the Peninsula indicate some non-linearity in the groundmovements 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 andinterferometric observations. Experiments have though shown that a tolerable fit canbe achieved between most of the 1993 to 1998 GPS observations and theinterferograms with elapsed time intervals within 1992 to 1996. Here this is handledby using all the GPS observations when eliminating a phase plane from theinterferograms, and reject GPS measurements that are inconsistent with theinterferogram before unwrapping and inferring the multidimensional groundmovements. It is though not necessary to reject any GPS data before inferring themultidimensional 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 motionfields, 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 estimatedindependently for each of the four interferograms. The averages were then created byweighting the images with its corresponding elapsed time intervals.The surface movements consist of complex mixture of subsidence, uplifting andsurface rifting [2,4]. The plate movements are clearly seen in the East motion fieldimage, where the South part of the Peninsula is moving towards the East, and theNorth part of it towards the West (a rift of a ~2 cm/yr). Also, some subsidences areclearly evident, especially at the cauldroun around Svartsengi. The subsidence atSvartsengi is estimated as ~2 cm/yr. The subsidence can also been seen from theSlant-Range-Shift motion field image. The East and North motion fields at theSvartsengi area does show that the cauldron surface is also moving towards the centreof the cauldron. Those effects of the subsidence can not be visualised by only usingthe interferometric observations. Furthermore, some subsidence is evident around theEast part of the Peninsula towards the Bláfjallahryggur (Bláfjallhryggur is a long64

idge). At this area, the East and North motion field images does also indicate sometendency 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 intervals1992-1993, 1992-1995, 1992-1996 and 1993-1995. (b), (c) and (d): Average Vertical, Eastand North motion maps, respectively, inferred by the 1992-1993, 1992-1995, 1992-1996 and1993-1995 interferometric measurements, combined with the 1993-1998 GPS observations.Motion maps were created independently for each of the four interferograms, and then theaverages were created afterwards for each of the motion field components. For all the motionimages, the average was created by weighting the motion field images with the correspondingelapsed time intervals. On the image in (a) is also shown its approximate location of theSvartsengi area and the Bláfjallahryggur ridge.65

12. ConclusionVarious methods have been created and tested to process InSAR images and tocombine GPS and InSAR observations, for the purpose of visualising the earthsurface deformations. InSAR images contains a modulated measure of onedimensionalchange in range from the ground to the satellite (Slant-Range-Shift),while GPS include accurate measurements of the three-dimensional deformation atsparse locations. It has been possible by combining those two complementarygeodetic techniques, to design methods to eliminate errors in observations, correct thedata and construct high resolution maps of three-dimensional ground movements.Those methods have been error tested with good results and used to combine GPS andInSAR observations from the Reykjanes Peninsula, Iceland, to infer high resolutionmotion maps of crustal deformations and plate movements.A method that uses vectorized filtering have been efficiently used to reduce noiseerrors in wrapped InSAR images. It has also been shown how systematic errors inInSAR observations can be eliminated by using GPS observations.A Markov Random Field (MRF) regularisation and simulating annealing optimisationhave 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 ininitialisation of the unwrapping process. The utilisation of GPS observations in theunwrapping process gives an opportunity of including sparse measurements of theunwrapped image values, which results in a faster and safer optimisation process. Anordinary kriging of GPS measured Slant-Range-Shift have been very successfullyused to create virtual InSAR images used to initialise the process.A method that uses a kriging of sparse GPS measurements, a MRF regularisation andsimulating annealing optimisation have been developed and used to construct highresolution maps of three-dimensional motion fields. The procedure is initialised withthe kriging of the GPS measurements, and the MRF regularisation and the simulatingannealing is then used for further optimisation. The method has been error tested withgood result. Very reasonable results have also been achieved when inferring the threedimensionalground movements at the Reykjanes Peninsula.A one-dimensional ordinary kriging method has been used to create motion fieldimages, both for the initialisation of the unwrapping algorithm and to constructmotion field images. The ordinary kriging algorithm has the advance of being simplecompared to other kriging methods, and has shown to be a very efficient tool toestimate the motion field images. Cokriging methods could also be tested for furtherimprovements.The MRF regularisation, used for the construction of multidimensional motion fieldimages, utilises a one-dimensional interferometric observation recorded from eitherascending or descending satellite passes. It is often possible to combine InSARimages from both ascending and descending satellite passes, which can be interpretedas a two-dimensional interferometric observation. An additional constraint can easilybe added to the MRF regularisation that would take advantage of this.66

References[1] D.Massonnet, K.L.Figel. Radar Interferometry and its Applications toChanges in the Earth’s Surface. Reviews of Geophisic, Vol 36, No. 4, 1998.[2] S.Hreinsdóttir. GPS Geodetic Measurements on the Reykjanes Peninsula, SWIceland: Crustal Deformation from 1993 to 1998. A thesis submitted to theUniversity 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 ofGlaciology, Vol 44, No 148, 1998.[4] H.Vadon, F.Sigmundsson. Crusatal Deformation from 1992 to 1995 at theMid-Atlantic Ridge, SouthWest Iceland, Mapped by Satellite RadarInterferometry. Science, Vol. 257, 1997.[5] S.Jónsson, N.Adam, H.Björnsson. Effects of Subglacial Geothermal ActivityObserved 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-WesleyPublishing 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 forMathematical Modelling, Technical University of Denmark, 1998. Internethttp://www.imm.dtu.dk/documents/users/aa/publications.html.[10] A.G.Journel, 1989. Fundamentals of Geostatistic in Five Lessons. ShortCourse Presented at the 28 th International Geological Congress Washington,D.C. American Geophysical Union, Washington, D.C., 1989.[11] M.Gumpertz 1999. Applied Spatial Statistics. Internet http://www.eos.ncsu.edu/eos/info/st/st733_info/www/oldindex.html.[12] V.Torczon 1999. Pattern Search Methods for Nonlinear Optimization.[13] J.M. Carstensen. Description and Simulation of Visual Texture. Ph.D. work.Institute for Mathematical Modelling, Technical University of Denmark, 1992.[14] S.Z.Li. Markov Random Field Modelling in Computer Vision. ComputerScience Workbench, 1995.67

AppendixesA. GPS tilting of wrapped InSAR images; projectioninto the complex unit circleThis appendix explains a titling method that uses optimisation of the differencebetween 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 tiltingof the InSAR image. Both the interferometricly and GPS measured Slant-Range-Shiftare projected into the unit complex circle. The optimisation is then done in thecomplex domain. Experiments indicate that this method can easily result in wrongtilting for images including a large tilting error. Therefore, another method has alsobeen designed for this problem task, that estimates unwrapped values of theinterferogram at sparse locations corresponding to the GPS sites (see Chapter 7).A.1. The procedureThe procedure is as follows. A sparse unwrapped image I GPS is created by calculating∆ρ = u⋅s for all the sparse pixels, where u is the measured three dimensional GPSdisplacement and s is the unit vector pointing from the ground towards the satellite. Anew unwrapped image is then calculated asIGPS2T( i,j) = I ( i,j) + x ⋅[ i,j,1] , ∀i,j,GPS(A.1)where x =[x 1 ,x 2 ,x 3 ] is a vector including the coefficient of a two dimensional planeand i, j are the sparse row and columns numbers of I GPS , respectively. The vector xfulfils the conditionsx = min f x( x),(A.2)where the objective function f is chosen to have global minimum when the residualI i j and a wrapped version of the imageserror of the wrapped InSAR image ( , InSAR)( i j)GPS,2I is at minimum for all i and j. A pre-filtered InSAR image, created with theprocedure given in (A.1) and (A.2), can be used to generate IInSAR. An optimal planarcorrection of the InSAR image is then calculated asIInSAR2λ 22π( l,c) = ( ∠C( l,c) − ∠C( l,c)),∀l,c,InSARPlan(A.3)where l and c are the line and columns numbers of I InSAR respectively,CInSAR⎛= exp⎜⎝j2Iλ 2π InSAR⎞⎟⎠(A.4)and68

CPlan( l,c)⎛= exp⎜⎝j2π xλ( [ ] ) T⋅ l,c,1⎞⎟,∀l,d ,2⎟⎠(A.5)where λ is the wavelength of the SAR satellite and x is given in (A.2).A.1.1. The first objective functionThe optimal value of x is found by using some arbitrary chosen initial values for x andan 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) isf= −∑C1 GPS 2∀i,j( i,j)+ CInSAR( i,j) ,(A.6)whereC⎛= exp⎜⎜⎝j2πIGPS 2GPS2 λ2⎞⎟⎟⎠(A.7)and C InSAR is given by (A.4). The entire complex values of both CGPS 2and CInSARin(A.6) lies on the complex unit circle. The function f 1 has a global minimum when thei j C i j are as close as possible for all i and jcomplex unit vectors C ( ) and ( )(lign up).GPS,2InSAR ,The objective function in (A.6) leads to a periodical solution of (A.2), i.e.x = x + λ 2, x + n λ 2, x + n 2 , for all integer numbers n 1 , n 2 and n 3 . All the[ ]1n12 2 3 3λ3+ n 3λsolutions of the offset x 2 are equivalent. The difference between themaximum and the minimum values of the optimal plane x⋅[l,c,1] T , ∀l, c, is thoughexpected to be less than of order of three to four fringes, where one fringe correspondto displacement of λ/2. This means that the optimal solutions of the planar slopes''''given by x1and x2should fulfil x1

Figure A.1. The periodical behaviour of the function f 1 in equation (A.6). In this particularx = 2,−1,3and the wavelength λ = 10.example the optimal parameters are known to be [ ]The main advance of f 1 is that it can be used to optimise all the three coefficientneeded for the planar correction in (A.3) in the same operation. A noise InSAR testimages were created to evaluate the result of applying the function f 1 in (A.2). Theresults indicate that the initial values of x 1 and x 2 need to be selected close to theoptimal solution for the algorithm to not run into some local extreme solutions. Thefunction is on the other hand insensitive for initial values of x 3 .A.1.2. The second objective functionAnother objective function that can be used to find the optimal slopes x 1 and x 2 wasalso tested. The function minimises the standard deviation of the angular difference∠ CInSAR − ∠C GPS 2where CInSARand CGPS 2are complex unit vectors given in (A.4)and (A.7), respectively. The function is written asf2= Var( ∠C)(A.8)where C is given asC∗( i, j) = C ( i,j)C ( i,j),∀i,j.2InSARGPS(A.9)70

Figure A.2. The periodical behaviour of the function f 2 in equation (16). In this particularx = 2,−1,0and the wavelength λ = 10.example the optimal parameters are known to be [ ]The function f 2 can only be used to find the optimal slopes x 1 and x 2 , but not optimaloffset x 3 . The vector x is therefore written as [x 1 ,x 2 ,0] in (A.2) when using f 2 . Theoptimal offset x 3 can be found as the mean value of ∠ C in (A.9) after optimisation ofx 1 and x 2 with f 2 . Results of applying the Pattern Search algorithm and f 2 on noiseexperimental InSAR images indicated that the optimal solution of the slopes is muchless dependent of the initial values of x 1 and x 2 than it is for the objective function f 1 .The function f 2 leads also to periodical solution of the slopes. The image in Figure A.2explains the behaviour of f 2 as a function of x 1 and x 2 . The optimal solution in thisparticular example is known to be x 1 = 2, x 2 = -1 and λ = 10, which is the same as forFigure A.1.A.2. Correction of the InSAR dataThe process described in (A.1) to (A.3), with the objective function in (A.8) wasimplemented into a program and used to correct the InSAR data from Reykjanespeninsula. The program allows a pre-filtering of the InSAR images beforeoptimisation. Initial values of the slopes x 1 and x 2 need to be selected in forehand forthe iteration process. Since the difference between the maximum and the minimumvalues of the optimal plane are expected to be less than of order of two to threefringes (conditional iteration), the initial values can be found as follow:1. Two vectors1= [ x11, x12, K,x1n] = [ minx, min , ,max ,max ]1 x+ ∆1 xK1x− ∆1 x1x1= [ x x , K,x ] = [ min , min + ∆ , K,max − ∆ , max ]x andx are created by2 21, 22 2mx2x2x2x2x2x2xmaxx 1, min xand22the program, where min1,maximum allowed values of the slopes of x 1 and x 2 respectively andmax xare the minimum and∆x 1andare the step size of the two vectors x 1 and x 2 respectively.x , x = min f x x for all2. The value selected as initial values are [ ] [ ] ( )possible combination of i and j.1 i 2 jx , 2 1i,1i x2 j2 j∆x 271

Figure A.3. Iteration run of the Pattern Search algorithm for the objective function f2and0.83 years InSAR image.The iteration algorithm continues to search for optimal values of x 1 and x 2 by using theinitial values x 1i and x 2j . An example of the iteration run of the Pattern Searchalgorithm is shown in Figure A.3, when using the 0.83 years InSAR image. FigureA.4 shows two InSAR images before and after correction with the program and thecorresponding angular residuals ∠ CInSAR− ∠CGPSand ∠CInSAR− ∠C2 GPS, whereC InSAR is given asC⎛= exp⎜⎝j2π⋅ IInSAR2 λInSAR22⎞⎟⎠(A.10)and IInSAR 2is given in (A.3). The images present 3.12 years and 0.83 years ofdeformation and have a master and slave orbits 5565 and 21941, and 5565 and 10575respectively. The residuals include more random noise after correction and someinformation that was not evident in the InSAR images becomes clearer.A.3. Removing unwanted areas from the corrected imagesThe Reykjanes peninsula is surrounded by sea, which are displayed as no informationin the InSAR images (zero valued pixels). Those areas with no information do notnecessarily consist of zero pixel values after the planar correction of the InSARimages. The method described in Chapter 6 is used to create a mask image. Theresulting tilted image is then masked by pixelvice multiplication of the mask andtilted InSAR images.The process described in this appendix is shown as dataflow diagram in Figure A.5.72

Figure A.4. Result of correcting InSAR images with GPS measurements; (a) and (b) showsthe original 3.12 years InSAR image and the angular difference between the original 3.12years InSAR and GPS measurements respectively, and (c) and (d) shows the same aftercorrection of the 3.12 years InSAR image after correction. (e) to (f) shows the same for 0.83years InSAR image.73

Figure A.5. Dataflow diagram of the process.74

B. Processed imagesThis appendix shows the result of unwrapping, tilting of unwrapped images andconstructing three-dimensional motion field maps, for the four interferograms thathave the highest altitude of ambiguity (Table B.1 and Table 2.1). Results aredisplayed in Figure B.1 to B.4. The original wrapped interferograms are shown in allthe figures. Before processing, the interferograms were vectorized filtered with 3x3moving average window (see vectorized filtering in Chapter 5). The images in (c) inFigure B.1 to B.4 shows unwrapped tilted interferograms (see the tilting described inSection 10.3.1). The images in (d) in Figure B.1 to B.4 shows then wrapped version ofthe unwrapped and tilted interferograms in (c). Wrapped versions of the tiltedinterferograms are included for comparison to the original wrapped untiltedinterferograms. Here, the inferring of the three-dimensional motion fields was doneby 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 todepend on the signal strength and the noise level of the data, which varies a lotbetween the interferograms. Also, some errors may appear due to discrepancybetween the GPS and interferometric observations, since they are not describingcrustal deformation at the same time intervals. The inferred Vertical, East and Northmotion maps do though describe on general similar deformation pattern in shape,especially for the area around Svartsengi where the signals are strong due to largeground movements (see the location of Svartsengi in Figure 2.6). At this area, theimage shows subsiding cauldron, approximately axisymmetric in shape [4]. Thesubsidence is clearly seen in the Vertical motion maps, but the East and North motionmaps indicates a horizontal motions towards the centre of the cauldron. Thesehorizontal 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 ofobservationSlave orbit Date ofobservationElapsed timeAltitude ofambiguityh a5565 08.08. ‘92 10575 24.07. ‘93 0.83 years 59.0 m5565 08.08. ‘92 21941 25.09. ‘95 3.12 years 43.6 m5565 08.08. ‘92 7278 09.10. ‘96 4.17 years 22000 m10575 24.07. ‘93 21941 25.09. ‘95 2.29 years 166.0 m75

Figure B.1. Uncorrected and corrected InSAR images, and inferred crustal deformation bycombined GPS and InSAR measurements. The GPS measurement represent deformationfrom 1993 to 1998 and the InSAR observations from 1992 to 1993 (0.83 yr). (a): Originalwrapped interferogram, (b): original wrapped interferogram vectorized filtered with 3x3 movingaverage window, (c): unwrapped and tilted interferogram, (d): wrapped version of theinterferogram in (c). (e), (f) and (g): Inferred Vertical, East and North crustal deformationmaps, respectively. (h): Residual error between the unwrapped InSAR image and thecombined Vertical, East and North deformation images.76

Figure B.2. Uncorrected and corrected InSAR images, and inferred crustal deformation bycombined GPS and InSAR measurements. The GPS measurement represent deformationfrom 1993 to 1998 and the InSAR observations from 1992 to 1995 (3.12 yr). (a): Originalwrapped interferogram, (b): original wrapped interferogram vectorized filtered with 3x3 movingaverage window, (c): unwrapped and tilted interferogram, (d): wrapped version of theinterferogram in (c). (e), (f) and (g): Inferred Vertical, East and North crustal deformationmaps, respectively. (h): Residual error between the unwrapped InSAR image and thecombined Vertical, East and North deformation images.77

Figure B.3. Uncorrected and corrected InSAR images, and inferred crustal deformation bycombined GPS and InSAR measurements. The GPS measurement represent deformationfrom 1993 to 1998 and the InSAR observations from 1992 to 1996 (4.17 yr). (a): Originalwrapped interferogram, (b): original wrapped interferogram vectorized filtered with 3x3 movingaverage window, (c): unwrapped and tilted interferogram, (d): wrapped version of theinterferogram in (c). (e), (f) and (g): Inferred Vertical, East and North crustal deformationmaps, respectively. (h): Residual error between the unwrapped InSAR image and thecombined Vertical, East and North deformation images.78

Figure B.4. Uncorrected and corrected InSAR images, and inferred crustal deformation bycombined GPS and InSAR measurements. The GPS measurement represent deformationfrom 1993 to 1998 and the InSAR observations from 1993 to 1995 (2.29 yr). (a): Originalwrapped interferogram, (b): original wrapped interferogram vectorized filtered with 3x3 movingaverage window, (c): unwrapped and tilted interferogram, (d): wrapped version of theinterferogram in (c). (e), (f) and (g): Inferred Vertical, East and North crustal deformationmaps, respectively. (h): Residual error between the unwrapped InSAR image and thecombined Vertical, East and North deformation images.79

C. MatLab functionsThis appendix explains the applications some of the MatLab functions created andused in this study, and includes help manuals available for each function. Some usageexamples are given. The MatLab functions are located in a new toolbox, which iscalled “deform”. The toolbox can be added to MatLab by defining the path to it inthe MatLab path browser.C.1. Content listThe functional content list available for the toolbox “deform” can be listed in theMatLab window by typing “help deform”. Usage information for each of thefunctions can then be listed by typing ”help name”, where “name” refers to thename of a MatLab functions. The toolbox content list displayed in the MatLabwindow is as follow:InSAR and GPS combination.BACKGROUND: Finds a background in an image, correct area of interestand applies the same background to another image.MASK: Finds a mask for an image.VINSAR: Create virtual unwrapped InSAR images calculated by spline ofsparse three-dimensional GPS measured displacement.VINSAR2: Create virtual unwrapped InSAR images calculated by splineof sparse three-dimensional GPS measured displacement.TILT: Function that that finds an optimal planar rotation of wrappedinput InSAR image. Uses wrapped format of both GPS and InSARmeasurementsPROFILE_TILT: Function that that finds an optimal planar rotation ofan input InSAR image. Use profiles to estimate unwrapped InSARvalues.LINE_UNWRAP: Function that unwrap a single line or profileKRIG1D: 1D kriging of sparsely located dataKRIG2D: 2D kriging of sparsely located dataLOLA2PIX: Converts longitude and latitude values to pixel values.MA: Filtering of wrapped InSAR images (projected onto the complexunit 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 wrappedInSAR image.PLAN_CORRECT2: Function that subtracts a linear plane from a wrappedInSAR image.UNWRAP_GPS: Unwrap InSAR images with help of GPS measurements.UNWRAP_SMOOTHN: Unwrap InSAR images by only using smoothnessrequirements.UNWRAPPED: Wave numbers estimated from oversmoothed unwrappedinterferogramSIMUL_2D: Function that estimates 2D crustal deformation by combinedInSAR and GPSWEIGHTSIMUL_2D: Function that estimates 2D crustal deformation bycombined InSAR and GPSWRAP: Function that calculates the wrapped version of an unwrappedInSAR image.TILT_UNWRAP: Use GPS measurements to and LS estimation to tiltunwrapped InSAR image.FYLKI: Puts GPS measurements into three-dimensional sparse matrix.80

DIFFERENCE: Angular difference between GPS measured dhro and InSARmeasured dhro.HORN: Function that finds the magnitude and the angle of thelongitude and latitude displacement fields.C.2. OPENINGThe function “opening” reads into MatLab an image file with an unsigned integerformat. The output is a matrix (image) including 8 bit values within the interval[0,255]. The function call isx=opening('name',N,M,headerbite);where “'name'” is a string including the file name. Further information about inputand output variables and parameters can be listed by using the help manual in theMatLab 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 nameN,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 track5565 and 21941, respectively (3.12 years of deformation), has the name“a5565_21941.pbm”. The file includes 16 bite of header information. The file iswritten into the 900x1200 matrix “insar” by using:insar=opening('a5565_21941.pbm',900,1200,16);C.3. MAThe function “ma” is used for vectorized moving average filtering of a wrappedInSAR image (see Chapter 5). The function call isinsar_out=ma(insar_in,n,m);Further information about input and output variables and parameters can be listed byusing the help manual in the MatLab window by typing “help ma”. The helpmanual is as follow:MA: Filtering of wrapped InSAR images (projected onto the complexunit circle).insar_out=ma(insar_in,n,m);Function that convert an input modulo InSAR image with pixel valueslying in the interval [0,max_value] into unity complex vector,81

y converting 0 to 0*pi and max_value to 2*pi. The functionthen calculates moving average of the Re and Im part of thecomplex vector by using a window of the size nxm. Finally thefunction return a filtered version of InSAR image with pixelvalues in the interval [0,max_value].Input:insar_in:n,m:An InSAR image to be filtered.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 usinginsar =ma(insar,3,3);C.4. LOLA2PIXThe function “lola2pix” is used to convert longitude and latitude coordinate valuesof GPS points into corresponding row and line numbers of an InSAR image. Thefunction call isgps_out=lola2pix(gps,n,m,lo1,la1,dlo,dla);Further information about input and output variables and parameters can be listed byusing the help manual in the MatLab window by typing “help lola2pix”. Thehelp 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:n,m:lo1:lo1:dlo:dla:Matrix with the columns information:[(1. lo. position), (2. la. position),(3. movement lo.), (4. movement la.),(5. movement up)].Number of lines and columns of the reference InSAR image(nxm matrix).Longitude co-ordinate corresponding to the lower left cornerof the InSAR image.Latitude co-ordinate corresponding to the lower left cornerof the InSAR image.The pixel resolution in lo. direction.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)].82

Example C.3A GPS file from the Reykjanes Peninsula includes three point measurements. Thecolumn 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 inExample C.1lo1=-23;la1=63.75;dlo=1/600;dla=1/1200;The longitude and latitude coordinates are converted into pixel location correspondingto the InSAR image by usinggps=lola2pix(gps,900,1200,lo1,la1,dlo,dla)which change the MatLab variable “gps” togps=[711 981 0.2800615 921 -0.8280528 909 -1.1660].C.5. MASKThe function “mask” is used to find a binomial mask image for an InSAR image (seeChapter 6). The function call ismask_out=mask(insar_in);Further information about input and output variables and parameters can be listed byusing the help manual in the MatLab window by typing “help mask”. The helpmanual 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 valuesfor bacground and 1 values for forground.C.6. MKThe function “mk” is used to find Least-Square (LS) parameter estimation for a linearplane (see Section 7.2). The function call isa=mk(x,y,z);83

Further information about input and output variables and parameters can be listed byusing the help manual in the MatLab window by typing “help mk”. The helpmanual is as follow:MK:Least square solution of a1, a2 and a3 for the linear planegiven 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 coefficientsin z=a1x+a2y+a3.C.7. PROFILE_TILTThe function “profile_tilt” is used to find a planar correction of a wrappedInSAR image by using the GPS measured Slant-Range-Shift. The function usesunwrapping of image profiles to estimate unwrapped values of the InSAR image.Least-Square (LS) method is then used to estimate the planar correction (see Chapter7). The function call isinsar_out=profile_tilt(insar,gps,tol);Further information about input and output variables and parameters can be listed byusing 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 aninput InSAR image.insar_out=profile_tilt(insar,gps,tol);Uses unwrapping of profiles to estimate unwrapped InSAR values. Alldata should be described in the same unit (e.g. cm). anddescribe deformation over the same time interval.Input:insar:gps:tol:Output:insar_out:The input InSAR image with pixel values lying in theinterval [0,tol].The input GPS file with values defined atsparse points. The column order should be[(1. line position), (2. column position),(3. Slant-Range-Shift)].The tolerance (f.ex. 2.835 cm for ERS-1 and ERS-2).The corrected InSAR image with pixel values lying in theinterval [0,tol].Example C.4.84

The 3.12 years InSAR image (matrix) in Example C.1 can be scaled to the interval[0,2.835] by usinginsar=insar*2.835/255;If the resulting variable “gps” from Example C.3 describe the Slant-Rangedeformation in cm/yr, then it can be used to tilt “insar” with the function callinsar =profile_tilt(insar,[gps(:,1:2) gps(:,3)*3.12],2.835);C.8. KRIG1DThe function “krig1d” is used to interpolate between sparsely located data. Thefunction uses ordinary kriging and semivariogram model to calculate optimal weights(see Chapter 8). The semivariogram model is calculated with help of Pattern Searchiteration algorithm (see Section C.9). An uncertainty image can also be created withthe function (see Section 8.5). The function call is[krig,uncer]=krig1d(gps_input,N_row,N_column,delta_h,h_max,test);orkrig1d(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 tothe 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 andparameters 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)orkrig1d(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 toestimate the semivariogram. (gamma(h)=1/(2N(h))sum_k(z(k)-z(k+h)))^2.h_max:Maximum length (in pixels) used to estimate thesemivariogram model.test:If test=1, then a plot is made of the estimatedsemivariogram and the semivariogram model. Thisoption can be used to optimise or test the bestsuitable parameters used for the kriging model. Iftest=0, then a kriging is done with the parametersdefined in function.Output:krig:The resulting kriged image. Note: only available85

uncer:if test=0, if test=1 then the krig matrix include noinformation.An uncertainty matrix. the value 1 is the lowestuncertainty and 0 the highest.Figure C.1. Two example of semivariogram models displayed by the function “krig1d” . Notethat ∆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 thefunction callkrig1d(gps,900,1200,delta_h,h_max,1)where “gps” is the GPS measure of the Slant-Range-Shift at the ReykjanesPeninsula, with locations in pixels that correspond to the 900x1200 InSAR image inExample C.1 (see Section C.1 and C.4). The semivariogram models in (a) and (b) areresult of usingdelta_h=20; h_max=600;anddelta_h=20; h_max=200;respectively. The semivariogram model in (b) is in more agreement with the data. Inthis example, the kriged Slant-Range-Shift could be calculated by the function callkrig=krig1d(gps,900,1200,20,200,0);C.9. PATTERNThe Pattern Search iteration method for non-linear optimisation is implemented intothe 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.86

Example for the semivariogram model in (8.7), [ C , C , R] TX0 1= and*F = γ ( C0, C1,R).Further information about input and output variables andparameters 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' shouldreturn 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)) forall x).y: The final value of the optimal point (y=min_over_x(f(x))for all x).f_iter:x_iter:Is f as a function of iterations (convergens of f).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_GPSThe function “unwrap_gps” use simulating annealing and Markov Random Field(MRF) modelling to unwrap a tilted InSAR image (see Chapter 9). The function usesparsely located GPS measured Slant-Range-Shift (or reference points) to guide theprocess. The process can also be initialised for example by using kriging of GPSmeasured Slant-Range-Shift, created by the MatLab function “krig1d” (see SectionC.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 isdetected by thresholded edge detection after each annealing (see Section 9.4.2 and9.4.3). The function call is[vinsar,n]=unwrap_gps(insar,gps,krig,la);Further information about input and output variables and parameters can be listed byusing “help unwrap_gps” in the MatLab window. During the iteration, theprocess temperature (90 at the start and 2 at the end of each annealing), number ofiterations and number of reannealing are displayed in the MatLab window.Furthermore, after each annealing, the image status is displayed on the computerscreen. 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)87

The function use Simulating Annealing and MRF modelling tounwrap InSAR image, and also guiding with GPS measured Slant-Range-Shift. During the iteration, the function does display thetemperature constant (90 at the beginning of one annealing and 2 atthe end), the iteration number, the number of annealing. The imagestatus is displayed after each annealing.Input:gps:insar:krig:la: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 bein consistency with the InSAR pixel numbers and theGPS measured Slant-Range-Shift should be scaledTo the same time interval as the InSAR image.The wrapped InSAR image.Kriged virtual unwrapped InSAR images created by thefunction 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 timeinterval as the InSAR image.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 correctedfurther by the function UNWRAP_SMOOTHN. Seehelp unwrap_smoothn in the MatLab windown: the wave number added to the input image insar to createvinsar.Note: after each reannealing, the function does also save theresulting image in a MatLab file called vinsar.mat. This file isavailable when the function is interrupted.Example C.6An example of function call isvinsar=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 (ExampleC.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_SMOOTHNThe function “unwrap_smoothn” use simulating annealing and Markov RandomField (MRF) modelling to unwrap a tilted InSAR image (see Chapter 9). The processcan 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 isdone by using Algorithm 9.1 and the energy function in (9.7) (only penalization onthe second derivative). The function uses rennealing process, where the area ofinterest is detected by thresholded edge detection after each annealing (see Section9.4.2 and 9.4.3). The function call isvinsar=unwrap_smoothn(insar,krig,la,max_rean);88

Further information about input and output variables and parameters can be listed byusing “help unwrap_smoothn” in the MatLab window. During the iteration, theprocess temperature (90 at the start and 2 at the end of each annealing), number ofiterations and number of reannealing are displayed in the MatLab window.Furthermore, after each annealing, the image status is displayed on the computerscreen. 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 smoothnessrequierments.vinsar=unwrap_smoothn(insar,krig,la,max_rean)The function use Simulating Annealing to unwrap InSAR image.During the iteration, the function does display thetemperature constant (90 at the beginning of one annealingand 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 thefunction 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 timeinterval as the InSAR image. If the process is notinitilised with kriged image, then krig=insar in thefunction 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 theresulting image in a MatLab file called vinsar.mat. This file isavailable when the function is interrupted.Example C.7.The function can be used for further correction of the resulting unwrapped imagefrom 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 forexample over-smoothed 3.12 years InSAR image “insar” (Example C.2), that couldbe initialised by “krig” created by the MatLab function “krig1d” (Example C.5).The function call in that case isvinsar=unwrap_smoothn(insar,krig,2.835,max_rean);89

C.12. UNWRAPPEDThe function “unwrapped” estimates missing wave numbers for non-smoothedwrapped InSAR image, by using oversmoothed unwrapped version of the sameInSAR image (see Section 9.5 and (9.12)). The unwrapped version of the nonsmoothedInSAR image is then calculated by adding the missing wave numbers to thenon-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 byusing the help manual in the MatLab window by typing “help unwrapped”. Thehelp manual is as follow:UNWRAPPED: Wave numbers estimated from oversmoothed unwrappedinterferogram[insar_out,n]=unwrapped(insar,vinsar,la);Function that calculates a wave number matrix fora wrapped interferogram by using oversmoothed unwrappedversion of the same interferogram.Input:insar:vinsar:la:Wrapped non-smoothed interferogram.Unwrapped oversmoothed interferogram.Note that vinsar should be oversmoothedand unwrapped version of insar.Half the wave length in cm (2.835 for ERS).Output:insar_out: The unwrapped version of the non-smoothedinterferogram 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 MatLabfunction “ma” (see Example C.2). The oversmoothed InSAR image can then beunwrapped by using the MatLab functions “unwrap_gps” and“unwrap_smoothn” (see Example C.6 and C.7). If “vinsar” is the oversmoothedunwrapped 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_UNWRAPThe function “tilt_unwrap” is used to find a planar correction of an unwrappedInSAR 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). Thefunction call isinsar_out=tilt_unwrap(insar_in,gps);90

Further information about input and output variables and parameters can be listed byusing 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 tiltunwrapped InSAR image.insar_out=tilt_unwrap(insar_in,gps);Input:insar_in:gps:Unwrapped InSAR image.GPS measured deformation. gps should include thecolumn [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 andin the same unit.Example C.9.The unwrapped 3.12 years InSAR image “insar_uw” in Example C.8 can be tiltedwith the resulting GPS variable “gps” (cm/yr) from Example C.3 withinsar_out=tilt_unwrap(insar_uw,[gps(:,1:2) 3.12*gps(:,3)]);C.14. SIMUL_2DThe function “simul_2d” infer two-dimensional deformation by combining GPSobservations 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 twodimensionaldeformation, while the InSAR image includes high-resolution map(image) of the Slant-Range-Shift (“Uw”). Simulating annealing algorithm is used forthe iteration (see Algorithm 10.1 in Section 10.2.1). The function is initialised withtwo motion field images (“V1_init” and “V2_init”), created for example withthe MatLab function “krig1d” (see Section C.8), and returns high-resolution maps“V1” and “V2”, optimised by combined GPS and InSAR. The function minimise thedifference “(Uw+c1*V1+c2*V2)^2”, where “[c1,c2]” is a unit vector (seeSection 10.1), and additional constrains are smoothness of the second derivative ofthe image surfaces (see the energy functions in (10.11) and (10.12) in Section10.2.2.). Values corresponding to sparse GPS locations in the image are not updatedduring the iteration. The function offers the possibility of optimising only one of themotion fields by keeping the other unchanged. This can be useful if one of the twomotion 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 statusis displayed on the screen, updated after each 10 iterations.91

Further information about input and output variables and parameters can be listed byusing the help manual in the MatLab window by typing “help simul_2d”. Thehelp manual is as follow:SIMUL_2D: Function that estimates 2D crustal deformation by combinedInSAR 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. Additionalconstrains are smoothness of the first derivative. Simulatingannealing iteration process is used to optimise V1 and V2.Input:Uw:gps:Unwrapped interferogram or image that fulfilsUw=-[V1,V2][c1,c2]^T. The image must be error corrected(offset and tilting) and include pixel information in cm.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 equallyLong time intervals.V1_init: The initial V1 image (in cm) (f.ex vertical deformationimage).V2_init: The initial V2 image (in cm) (f.ex. horizontal lookdirectiondeformation 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 iterationprocess.Output:V1,V2:Note:Optimised motion images.The initial input images should be ordered in the way thatc1>=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 InSARimage “insar_out” in Example C.9 is given as s = [ 0.34E,−0.095N,0.935V].If theVertical motion field image “Vv” is known (for example from two-dimensionaloptimisation 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 theMatLab function “krig1d” (see Section C.8).92

C.15. WEIGHTSIMUL_2DThe function “weightsimul_2d” infer two-dimensional deformation bycombining 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 twodimensionaldeformation, while the InSAR image includes high-resolution map(image) of the Slant-Range-Shift (“Uw”). Simulating annealing algorithm is used forthe iteration (see Algorithm 10.1 in Section 10.2.1). The function is initialised withtwo motion field images (“V1_init” and “V2_init”), created for example withthe MatLab function “krig1d” (see Section C.8), and returns high-resolution maps“V1” and “V2”, optimised by combined GPS and InSAR. The function minimise thedifference “(Uw+c1*V1+c2*V2)^2”, where “[c1,c2]” is a unit vector (seeSection 10.1). Additional constrains are smoothness of the second derivative of theimage surfaces and penalisation of deviations from the initial kriged images, weightedas a function of distance from the sparse GPS locations in the images (see Section10.2.3 and the energy functions in (10.15) and (10.16)). The uncertainty images“uncer1” and “uncer2”, created with the MatLab function “krig1d”, are usedfor this purpose. The function offers the possibility of optimising only one of themotion fields by keeping the other unchanged. This can be useful if one of the twomotion 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 statusis displayed on the screen, updated after each 10 iterations.Further information about input and output variables and parameters can be listed byusing the help manual in the MatLab window by typing “helpweightsimul_2d”. The help manual is as follow:WEIGHTSIMUL_2D: Function that estimates 2D crustal deformation bycombination 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. Additionalconstrains are smoothness of the first derivative and weighting asa function from gps locations. Simulating annealingiteration process is used to optimise V1 and V2.Input:Uw:Unwrapped interferogram or image that fulfilUw=-[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 deformationimage).V2_init: The initial V2 image (in cm) (f.ex. horizontal lookdirectiondeformation image). V1_init and V2_init can befound f.ex. by kriging.c1,c2:[c1,c2] is a two dimensional unit vector.uncer1:Uncertenty matrix for V1_init, scaled to the interval93

[0 1]. 1 is high certenty (close to GPS locations) and 0low. The uncertenty matrix can be f.ex. created along withthe krigigng of GPS measurements (see the MatLab functionkrig1d).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:Note:Optimised motion images.The initial input images should be ordered in the way thatc1>=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” (seeSection C.8).C.16. LINE_UNWRAPThe function “line_unwrap” unwraps a wrapped line profile. The function call isprof_uw=line_unwrap(prof_w,tol);where “prof_w” is a wrapped line profile. Further information about input andoutput variables and parameters can be listed by using the help manual in the MatLabwindow by typing “help line_unwrap”. The help manual is as follow:LINE_UNWRAP: Function that unwrap a single line or profileprof_uw=line_unwrap(prof_w,tol)The function uses the first point (prof_W(1)) as areference point and unwraps the image by requiringsmoothness of the surface.Input:prof_w:tol:The wrapped profile.The reference tolerance (f.ex. lambda/2=2.835 for ERS 1 and 2).Output:prof_uw: The unwrapped profile.C.17. WRAPThe function “wrap” finds a wrapped version of unwrapped InSAR image. Theoutput matrix (image) includes 8-bit values scaled to the interval [0,255]. Thefunction call is94

winsar=wrap(insar,tol);where “insar” is an unwrapped InSAR image. Further information about input andoutput variables and parameters can be listed by using the help manual in the MatLabwindow by typing “help wrap”. The help manual is as follow:WRAP: Function that calculates wrapped version of an unwrapped InSARimage.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. FYLKIThe function “fylki” is used to put GPS observations longitude and latitudecoordinate into a three-dimensional image. This function is used along with thefunction “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 byusing “help fylki” in the MatLab window. The help information listed in theMatLab 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 gpsinto three dimentional matrix, equal in sizeto the InSAR image.Input:gps:insar:lo1:lo1:dlo:dla:Matrix with the columns informations:[(1. lo. position), (2. la. position),(3. movement lo.), (4. movement la.),(5. movement up)].the INSAR image.First pixel longitude co-ordinateFirst pixel latitude co-ordinateThe pixel resolution in lo. direction.The pixel resolution in la. direction.Output:gps_out: three dimentional matrix including valuesof the dispalcement in (lo,la,up)directions at sparse pixels and zeros othervise.lo: vector including the longitude value that correspondsto each column.la: vector including the latitude value that corresponds95

to each line.C.19. TILTThe function “tilt” is used to find a planar correction of wrapped InSAR image byusing the GPS measured Slant-Range-Shift. The function use wrapped form of boththe InSAR and GPS data to estimate the planar correction (see Appendix A). Thefunction 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 byusing “help tilt” in the MatLab window. The help information listed in theMatLab window is as follow:TILT: Function that that finds an optimal planar rotation of theinput 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 theinterval 0-255.gps_in: The input three dimensional GPS image with valuesdefined 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 thesatellite.Output:insar_out:res1:res2:The corrected InSAR image with no background correction.The residuals between the InSAR and GPS beforecorrection.The residuals between the InSAR and GPS aftercorrection.96