ARTICLE IN PRESS. Received 13 November 2006; received in revised form 27 April 2007; accepted 8 May 2007

ARTICLE IN PRESS Computers & Geosciences 34 (2008) 503 514 www.elsevier.com/locate/cageo Application of DInSAR-GPS optimization for derivation of three-dimensional surface motion of the southern California region along the San Andreas fault $ S.V. Samsonov a,, K.F. Tiampo a, J.B. Rundle b a Department of Earth Sciences, University of Western Ontario, London, Ont., Canada b Center for Computational Science and Engineering, University of California, Davis, CA, USA Received 13 November 2006; received in revised form 27 April 2007; accepted 8 May 2007 Abstract We present here a methodology that allows the combination of GPS and Differential InSAR data for the calculation of continuous three-dimensional (3D) high-resolution velocity maps with corresponding errors. It is based on analytic minimization of the Gibbs energy function, which is possible in the case when neighborhood pixels of the velocity maps are considered independent. By joining scalar DInSAR data and vector GPS data, the technique allows us to achieve significant improvement in accuracy in the components of the velocity vector in comparison with the GPS data alone. In the accompanying example, the method is used for the investigation of the creep motion of the southern San Andreas fault around the Salton Sea region. The velocity maps are calculated for two time periods (1992 1998 and 1997 2001) and also for 3D and 2D cases. The preliminary analysis of the optimized data suggests that creep on the San Andreas fault in this region is time-dependent. r 2007 Elsevier Ltd. All rights reserved. Keywords: GPS; InSAR; San Andreas fault creep 1. Introduction In this work we present a method of merging the data from continuous global positioning system (GPS) (Hofmann-Wellenhof et al., 2001) and differential interferometric synthetic aperture radar (DInSAR) (Massonnet and Feigl, 1998; Rosen et al., 2000) for derivation of high-resolution $ Code on server at: http://www.iamg.org/cgeditor/index. htm. Corresponding author. Current address: GNS Science, 1 Fairway Drive, Avalon, P.O. BOX 30368, Lower Hutt, New Zealand. Tel.: +64 4 570 4566; fax: +64 4 570 4600. E-mail address: s.samsonov@gns.cri.nz (S.V. Samsonov). three-dimensional (3D) surface velocity maps with the corresponding errors. The complementary nature of both geodetic techniques is evident and has been extensively exploited (Ge et al., 2000, 2001; Gudmundsson et al., 2002; Samsonov and Tiampo, 2006). The continuous GPS measures 3D coordinates of the GPS sites with high temporal but low spatial resolution. Currently, there are only a few GPS networks in the world, such as the Japanese GEONET (Tsuda et al., 1998) and the Southern California Integrated GPS Network (SCIGN) 1 1 Southern California Integrated GPS Network SCIGN, http:// www.scign.org/. 0098-3004/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2007.05.013

504 ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 (Bock et al., 1997), that have sufficiently dense spatial coverage and are capable of measuring the surface deformations with the resolution of approximately 10 30. DInSAR, on the other hand, measures the total deformation between two acquisition scenes projected in the line-of-site (LOS) with high spatial resolution over a wide area, but low temporal resolution. The best temporal resolution suitable for long term continuous measurements of 35 days was achieved by ERS-1/2 satellite missions between 1992 and 2005, not considering tandem pairs with temporal resolution of 1 day, which are not suitable for long term deformation studies but are very useful for coseismic measurements. In Samsonov and Tiampo (2006), we reconstructed a 3D synthetic velocity field from sparse GPS measurements and two differential interferograms (ascending and descending). It was shown that this method significantly improves the accuracy of the velocity measurements derived from the GPS data aloneandalsocanbeusedtoderivehigh-resolution continuous maps of the surface velocity field. Unfortunately, for many regions only one type of interferogram is available. In Samsonov et al. (2007), only descending ERS-2 interferograms along with the GPS data from the SCIGN were used to derive a 3D surface velocity field of the southern California region near Los Angeles. It was shown that by using this technique, it is possible to differentiate between various deformation signals, such as anthropogenic signal due to groundwater extraction or tectonic signal due to plate motions. The goal of this work is to present computer code that is used to perform the DInSAR-GPS optimization along with some additional routines that are used to prepare data for the optimization. The Generic Mapping Tools (GMT) software package (Wessel and Smith, 1998) was used for visualization of the intermediate and final results. Almost all computationally extensive routines are implemented in C/ C++ language (ANSI/ISO/IEC standard) and are called from bash script. 2 The computer code used in this work was compiled and tested under Linux (Red Hat, Fedora). Although this has not been tested specifically, it is anticipated that the C/C++ code can be easily recompiled to be used on Windows. Finally, the DInSAR-GPS optimization technique is used to investigate the surface motion across the southern part of the San Andreas fault near the 2 Introduction to bash script, http://www.gnu.org/software/ bash/. Salton Sea region. The work of Lyons and Sandwell (2003) suggest that the San Andreas fault in that region creeps with the approximately permanent velocity of 1.2 1.8 and its motion is approximately horizontal (strike-slip). In this work we calculate the 3D velocity field for two time periods: 1992 1998 and 1997 2001. From the results it is evident that both assumptions, permanent creep velocity and the absence of the vertical motion across the fault, are contradictory. We show that if one assumes that the creep rates are approximately constant over time then the motion across the fault cannot be considered strictly horizontal. On the other hand, if we assume that the motion across the fault is horizontal (strike-slip), then the creep rates vary greatly in the short time interval. The accuracy of the optimization technique proposed in this work greatly depends on the number and distribution of the GPS sites in the area (Samsonov and Tiampo, 2006). Since the linear regression of the time series is used to calculate an initial 3D model of the surface velocities, it is also necessary to have a sufficiently long time series in order to achieve good accuracy. In this work the data from seven GPS sites were used to calculate the initial velocity model for the 1992 1998 period and the data from 51 sites were used for the 1997 2001 period. These numbers seem to be insufficient for a good estimation of the initial velocity model. Also, some DInSAR signal is not completely understood; therefore, these results should be considered only preliminary. They require more detailed investigation and validation by other sources of data, such as locally installed strainmeters. 2. Implementation The theoretical aspects of DInSAR-GPS optimization along with the complete set of the optimization equations are explained in Samsonov et al. (2007) (Eqs. (8) (9) and (11) (12)) and are given here in a brief form. In order to perform the optimization, we construct an energy function of the following form and search for its minimum with regard to its arguments uðv x ; v y ; v z Þ¼ XN i¼1 fc i ins ðv i LOS Si x vi x Si y vi y Si z vi z Þ2 þ C i x ðv i x vi x Þ2 þ C i y ðv i y vi y Þ2 þ C i z ðv i z vi z Þ2 g ð1þ

ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 505 with coefficients C i ins ¼ 1 2ðs ; i Ci ins Þ2 x ¼ 1 2ðs ; i Ci x Þ2 y ¼ 1 2ðs, i y Þ2 C i z ¼ 1 2ðs, ð2þ i z Þ2 where ðs i x ; Si y ; Si zþ is the unit vector pointing from the ground to the satellite (LOS vector), V i LOS is the DInSAR interferogram, ðv i x ; V i y ; V i zþ is the interpolated GPS velocity vector, s s are the standard deviations of the measurements and ðv i x ; vi y ; vi zþ is the optimized velocity vector. Since each term of uðv x ; v y ; v z Þ is nonnegative, its global minimum is reached when each subgroup with the same index i is also minimal and, therefore, the first derivatives qu=qv i x, qu=qvi y, qu=qvi z are equal to zero. Thus, the solution of this problem is found analytically. 2.1. Differential InSAR There are a number of software packages currently available for differential InSAR processing, such as the commercial EarthView package developed by Atlantis Scientific or the freeware ROI_PAC from Caltech/Jet Propulsion Laboratory and DORIS from Delft University of Technology. All these packages perform similar steps in order to calculate the differential interferogram, such as image coregistration, interferogram computation, baseline estimation and interferogram, attending and removal of topographic phase. In this work all interferograms were processed by ROI_PAC package (Rosen et al., 2004). Delft precise orbits (Scharroo et al., 2000) were used to estimate the exact position of the satellite during the acquisition, and SRTM Digital Elevation Model (DEM) data were used to remove the topographic phase. From previous investigations, it is evident that differential interferograms carry significant amount of useful data, such as surface deformations due to seismic events (Jacobs et al., 2002; Massonnet et al., 1993, 1996) and volcanic activities (Lundgren et al., 2003) and anthropogenic subsidence due to mining (Perski, 1999) or groundwater extraction (Shmidt and Burgmann, 2003). Different types of errors are also present in the interferograms and these need to be estimated and, if possible, corrected. The most common error, which is easy to observe and correct, is the one due to inaccurate orbit estimation. This error manifests itself as a linear gradient across the interferogram (Massonnet and Feigl, 1998). Therefore, subtraction of the fitted plane from the interferogram usually successfully corrects it. Often ionospheric error is present, which also manifests itself as a linear trend across interferogram. In this case the ionospheric error also can be removed by subtraction of the fitted plane from the interferogram. However, in rare cases the linear trend is due to long-wavelength deformation signal. We found that in order to ensure the nature of the signal, it is sometimes useful to create a synthetic differential interferogram from interpolated GPS data and to analyze the residual between the synthetic and the differential interferograms in both (lat/long) directions. The atmospheric water vapor effects are perhaps the most challenging. It has been shown that even small variations in air temperature, density and humidity can significantly affect the signal and cause an LOS displacement of a few centimeters (Hanssen and Feijt, 1996). This error can manifest itself either as a localized disturbance, which is most probably due to a large cloud, or as a large area due to a large atmospheric front (Hanssen, 2001; Massonnet and Feigl, 1998). There are a few techniques available to compensate and reduce the atmospheric errors, such as GPS- MODIS correction (Li et al., 2005) as was done in Samsonov et al. (2007). An alternative method is to create an interferometric stack as it was done in Lyons and Sandwell (2003) and also in this work. The flow chart diagram of DInSAR-GPS optimization is presented in Fig. 1. The bash script STACK is used to calculate the interferometric stack from unwrapped interferograms. This procedure consist of a few steps. First the deformation values of each interferogram (in cm) are divided by the timespan (in years), thereby converting them to velocities. Then the stable region of the interferogram is identified and the average value for the region is calculated. Finally, all the interferograms are adjusted to the same average value by adding or subtracting the difference between the reference average value and the current average value, bringing all of the interferograms to a common vertical reference frame. Then velocity values of the interferograms are multiplied by timespan, thereby converting them back to displacements (in cm) and stack is calculated according to the technique proposed in Wright et al. (2004)

506 ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 Start Inspect each interferogram for presence of orbitalor ionosphericerrors.disregard noisy interferograms. Calculate DInSA Rstack (STACK) Calculate decorrelation and topographic errors and combine them together (STACK_ERROR) Select geographical region and time span Adjust DInSAR interferogram in order to match it with synthetic image by averaging over the whole area Calculate optimized DInSAR-GPS velocities with corresponding errors (OPTIMIZE) Plot results (PLOT) Stop Calculate sparse GPS surface velocities by linear regression of GPS time series Create continuous velocity maps by ordinary kriging of sparse velocities (KRIG) Combine regression and kriging errors Calculate synthetic interferogram Is DInSAR and GPS data for the same time period available? No Fig. 1. Schematic representation of processes used to perform DInSAR-GPS optimization. Some steps, such as selection of region and timespan and selection of interferograms are performed manually. In total, five bash scripts are used: STACK, STACK_ERROR, KRIG, OPTIMIZE, PLOT. Some computationally intensive subroutines are programmed in C/C++ and called from scripts. dividing the total deformation by the total timespan of all interferograms. In general, choosing a stable region is quite complicated (or impossible) due to water vapor effects and also due to the unknown nature of the deformation. However, we believe that, for this region, it is still can be done accurately because the analysis of over 100 differential interferograms for this region suggests that the deformation Yes process occurs in two areas: the south-west corner (the San Andreas fault) and the north-west corner (Hector Mine earthquake). Also, a small part at the south-east corner shows signs of deformation of an unidentified nature, which is likely due to atmospheric disturbances. The remaining area of the interferograms is stable and moving with the average plate velocity. Then, atmospheric noise can be significantly reduced if averaging is performed over a large spatial window. The bash script STACK_ERROR calculates the errors due to decorrelation as a Cramer Rao bound (Rodriguez and Martin, 1992) and due to residual topography (Massonnet and Feigl, 1998). It requires geocoded correlation files and perpendicular baseline information in order to estimate both sources of errors correctly. Finally, the total error is calculated as a sum of both errors: s total ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 decor: þ s2 top:. It is assumed that both sources of errors are uncorrelated. Both the stacked interferogram and error file are then converted to the GMT grid file format. 2.2. GPS data The calculation of the initial velocity maps from GPS data with corresponding errors is performed by KRIG script. In this work we used GPS time series from SCIGN. The data were preprocessed in order to remove outliers and offsets. Three components (north, east, up) of the velocity vector were calculated by applying linear regression (Press et al., 1992) over a period corresponding to the timespan of the interferometric stack. Then sparse velocity data were interpolated by ordinary kriging (Brooker, 1991; Deutsch and Journel, 1997) with the help of GSTAT (program for geostatistical modeling, prediction and simulation) (Pebesma and Wesseling, 1998) in order to form three continuous velocity maps with the same discretization and geocoding as the DInSAR image. In order to estimate the total GPS accuracy we combined the errors due to the regression analysis and also due to kriging. The errors due to the linear regression were estimated as a by-product of the regression analysis (Press et al., 1992) and then interpolated on the grid with the same discretization as the interferogram. These errors mainly depend on the linearity of the time series. The errors due to kriging were estimated as a by-product of the kriging (Deutsch and Journel, 1997) and are valid

ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 507 for each discretization point or pixel on the continuous GPS velocity map. It is equal to zero at the location of the GPS sites and increases with distance from the known position. Therefore, it reaches a maximum at the most remote areas. Finally, both errors were combined together in order to be used as weighting parameters for the optimization. 2.3. Optimization and error estimation After both DInSAR and GPS data were prepared and corresponding errors were estimated, the optimization was performed by running the OPTI- MIZE script. This script calls a compiled C++ binary program which performs the optimization. The output of the program is six binary files of the optimized velocity model (north, east, up) with corresponding errors (one standard deviation). The visualization is performed by the PLOT script, which uses GMT routines. The California faults and GPS sites that are used for the optimization are also plotted as black lines and black dots in order to enhance the appearance. from 1992 to 2001. The south-west corner of these interferograms also covers the portion of the San Andreas fault which was our primary interest. The creeping signal was observed on many interferograms, particularly on those with long temporal baselines. However, the value of the creep and even its direction differ greatly. Fig. 2 shows the results of the average jump in the LOS velocity (in ) across the San Andreas fault vs average time (in years) between two acquisitions. It was estimated that the value of the discontinuity for the 1992 1998 subset is 1:1 1:2cm=year, and for the 1997 2001 subset is 1:4 2:1cm=year. Even though the errors are quite large it is still evident that the motion at the San Andreas fault in this area is quite different for these two time periods and thus the velocity maps should be different. To calculate the velocity map for 1992 1998 time period, we stacked three interferograms (Table 1 Table 1 Differential interferograms used for optimization for 1992 1998 time period (left) and for 1997 2001 time period (right) 3. Reversal of motion across San Andreas fault Interferogram (1992 1998) B? Interferogram (1997 2001) B? In this section the results of the DInSAR-GPS optimization are presented. Originally over 100 differential interferograms were calculated for the Salton Sea region (track 356, frame 2925) spanning 930302 980424 9 970404 000428 140 950608 960802 365 980109 991210 154 951130 980109 63 980424 000602 568 990129 000915 20 8 6 The value of the average discontinuity of the LOS velocity across the San Andreas fault 1992-1998 1997-2001 LOS velocity, 4 2 0-2 -4 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Time, year Fig. 2. Discontinuities in LOS velocities across southern part of San Andreas fault vs average time of DInSAR interferogram. It is seen that values before 1998 are smaller ð 1:1 1:2Þ than values after 1997 ð1:4 2:1Þ and of opposite sign. This suggests a time-dependent reversal pattern of motion at San Andreas fault.

508 ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 33 45' -116 30' -115 45' -115 00' -116 30' -115 45' -115 00' -116 30' -115 45' -115 00' -116 30' -115 45' -115 00' -116 30' -115 45' -115 00' -116 30' -115 45' -115 00' 0.05 0.10 0.15 0.20-116 30' -115 45' -115 00' -116 30' -115 45' -115 00' Fig. 3. The 1992 1998 time period. Components of surface velocity field ((a) (c)) for southern California region near Salton Sea, based on local GPS measurements from SCIGN. Known faults are shown as thin black lines. Also shown are components of optimized velocity field ((d) (f)) and stack (g) of differential interferograms with corresponding errors (h). (a) Kriged V north component of velocity; (b) Kriged V east component of velocity; (c) Kriged V up component of velocity; (d) Optimized DInSAR+GPS component of v north ; (e) Optimized DInSAR+GPS component of v east ; (f) Optimized DInSAR+GPS component of v up ; (g) DInSAR interferogram; (h) Standard deviation of DInSAR interferogram.

ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 509 012.5 25-116 30' -115 45' -115 00' -116 30' -115 45' -115 00' -116 30' -115 45' -115 00' 012.5 25 012.5 25-116 30' -115 45' -115 00' -4-2 0 2 4 6-116 30' -115 45' -115 00' -116 30' -115 45' -115 00' -4-2 0 2 4 6 0.05 0.10 0.15 0.20-116 30' -115 45' -115 00' -116 30' -115 45' -115 00' Fig. 4. The 1997 2001 time period. Components of surface velocity field ((a) (c)) for southern California region near Salton Sea, based on local GPS measurements from SCIGN. Also shown are components of optimized velocity field ((d) (f)) and stack (g) of differential interferograms with corresponding errors (h). Signal in NW corner (Hector Mine earthquake) is colored in white in order to produce color scheme similar to previous figure. (a) Kriged V north component of velocity; (b) Kriged V east component of velocity; (c) Kriged V up component of velocity; (d) Optimized DInSAR+GPS component of v north ; (e) Optimized DInSAR+GPS component of v east ; (f) Optimized DInSAR+GPS component of v up ; (g) DInSAR interferogram; (h) Standard deviation of DInSAR interferogram.

510 ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 (left)). This interferometric stack with corresponding errors is presented in Figs. 3(g) (h). The creeping motion around the San Andreas fault in the southwest corner is well observed. The data from seven GPS stations from the SCIGN network were used for the calculation of the initial velocity model. The components of the initial velocity are shown in Figs. 3(a) (c). The results of the optimizations are presented in Figs. 3(d) (f) and the horizontal vector map of the velocity field is shown in Fig. 5a. The northern component of the velocity approximately ranges from 1 to 1 in the southwest to north-east direction and does not change significantly after optimization. This was expected, as the DInSAR dependence on the motion in the north south direction is insignificant ðs x ¼ 0:08Þ. The eastern component ranges from approximately 2:5 to 1:5cm=year also in the south-west to north-east direction and changes moderately after optimization ðs y ¼ 0:30Þ. The initial vertical signal is almost negligible. After the optimization, the vertical component changes greatly which is, of course, due to the configuration of the SAR sensor ðs z ¼ 0:95Þ. In this case almost all the signal observed on the differential interferogram around the San Andreas fault propagates to the vertical component. For the 1997 2001 time period we stacked four interferograms (Table 1 (right)). The interferometric stack with corresponding error is presented in Figs. 4(g) (h). The motion around the San Andreas fault is less observable in this case; however, it is evident that the direction of the motion is opposite in comparison with the first stack. The large signal in the north-west corner of the interferogram is due to the Hector Mine earthquake. The data from 51 GPS SCIGN sites were used to build the initial velocity model over a time a little larger than the 1997 2001 timespan in order to improve its accuracy. The components of the velocity are shown in Figs. 4(a) (c). The results of the optimizations are presented in the Figs. 4(d) (f) and the horizontal vector map of the velocity field is shown in Fig. 5b. For the first time period (1992 1998), the vertical component of velocity shows subsidence in the region below the San Andreas fault. However, for the second time period (1997 2001) the region below the San Andreas fault (as seen on the figure) uplifts in comparison with the rest of the interferogram. At the same time the differences in the horizontal velocity maps are insignificant for both time periods. In Lyons and Sandwell (2003), it was noted that strainmeters around the San Andreas fault do not observe any vertical motion in this area. Therefore, as a second option we chose to assume that all the signal around the fault that is observed in both interferograms is purely horizontal. In order to 1 2 3 4-116 30' -116 00' -115 30' -115 00' 1 2 3 4-116 30' -116 00' -115 30' -115 00' Fig. 5. Vector maps calculated from horizontal components of (3D) GPS-DInSAR optimized velocities for two time periods. The pattern of motion is certainly over-smoothed during interpolation of GPS data due to small number of sites available in this area. (a) 1992 1998; (b) 1997 2001.

ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 511 constrain this assumption, the vertical component of the velocity was excluded from the optimization process. These results are presented in Figs. 6 and 7. In this case, the entire signal from the interferometric stacks only propagates to the horizontal components, so that the changes after optimization are more significant. In the case of horizontal slip the motion around the San Andreas fault is very different for the two time periods. For 1992 1998, the motion is oriented mostly toward the west, but for 1997 2001 the motion is oriented mostly toward the east. 4. Conclusions We have presented a method which allows the combination of GPS and DInSAR data for the derivation of the 3D high-resolution velocity maps -10-5 0 5 10-10 -5 0 5 10-116 30' -115 45' -115 00' -116 30' -115 45' -115 00' -10-5 0 5 10-10 -5 0 5 10-116 30' -115 45' -115 00' -116 30' -115 45' -115 00' Fig. 6. Horizontal components of surface velocity for two different time periods (1992 1998 top row, 1997 2001 bottom row). In this case vertical component of velocity was excluded from optimization process (two-dimensional case). (a) Optimized DInSAR+GPS component of v north ; (b) Optimized DInSAR+GPS component of v east ; (c) Optimized DInSAR+GPS component of v north ; (d) Optimized DInSAR+GPS component of v east.

512 ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 0 2 4 6 8 10-116 30' -116 00' -115 30' -115 00' 0 2 4 6 8 10-116 30' -116 00' -115 30' -115 00' Fig. 7. Vector maps calculated from horizontal components of (2D) GPS-DInSAR optimized velocities for two time periods. It is seen that motion across San Andreas fault is different for both time periods, which suggests either a time-dependent reversal pattern of motion such as silent slip or presence of vertical motion across San Andreas fault. (a) 1992 1999; (b) 1997 2001. with corresponding errors. The methodology is implemented in C/C++ and is called from five bash files: STACK, STACK_ERROR, KRIG, OPTIMIZE and PLOT. The first two scripts calculate the interferometric stack from differential interferograms and corresponding errors due to decorrelation and residual topography. The KRIG script is used to interpolate GPS velocities data in order to form the initial velocity model with the same geocoding and spatial resolution as the differential interferograms. Finally, the OPTIMIZE script is used to combine both data sets using analytical optimization of the Gibbs energy function. The output of the program, plotted by PLOT script, is the refined velocity model of the region, which contains large-scale features from GPS data and small-scale perturbations from the DInSAR data. As an example of this technique, the DInSAR- GPS optimization is applied to data from the southern California region near the Salton Sea, which includes the southern part of the San Andreas fault. The creep signal is well observed on all interferograms from 1992 to 2001 (not shown here) but the value and direction of the creep are significantly different over that time period. Velocity maps were created for two time periods (1992 1998 and 1997 2001) and for 2D and 3D cases. For the 3D case, most of the signal from the differential stack propagates to the vertical component of the optimized velocity vector which is, of course, due to the configuration of the ERS-1/2 sensors. The horizontal components for both time periods in the 3D case do not differ significantly. For the 2D case, since the vertical component of the velocity is excluded from the optimization, all the signal from the DInSAR stacks propagates to horizontal components only and, therefore, they largely differ for both time periods. For example, for 1992 1998, the velocity field across the San Andreas fault does not change direction, but it does for 1997 2001. These results seem significant; however, they are presented here only as an example of the methodology. The DInSAR-GPS optimization simply merges two available data sets in such a way that each technique complements the other. Before the final decision can be made about the time-dependent motion of the San Andreas fault in this region, it is imperative to understand the nature of the signal presented in both data sets. First, one must clearly understand the nature of the DInSAR signal. For example, the signal in the south-east corner, which is uplift for 1992 1998 and subsidence for

ARTICLE IN PRESS S.V. Samsonov et al. / Computers & Geosciences 34 (2008) 503 514 513 1997 2001, is not easily explained. Also, it is not clear if the signal around the San Andreas fault directly corresponds to deformation signal. Part of this signal might be due to changes in the physical properties of the material around the fault such as density, temperature or water content. Second, it is evident from our previous work that seven (as in the first time period) and 51 (as in the second time period) GPS stations are not enough to create a good initial model. We believe that the initial velocity model which is used in this work is oversimplified and does not capture exactly the velocity pattern around and across the fault. Therefore, more GPS stations need to be installed and operated for a long time before a satisfactory initial velocity model of the region can be constructed. Despite the practical difficulties in the realization of this method, it can be used for the investigation of surface motion in many regions of the world and argues for the increased procurement and availability of these high-quality geodetic data sets. Acknowledgments This work was funded by an NSERC Discovery grant and also by CSEG, KEGS Pioneers and SEG scholarships received by S. Samsonov. The work of K. Tiampo was supported by an NSERC Discovery Grant. Research by J. Rundle has been supported by a grant from the US National Aeronautics and Space Administration under grant NAG5-13743 to the University of California, Davis. Technical support for this work has been provided by the POLARIS network. The ERS data were obtained from the WINSAR Consortium and processed by the Repeat Orbit Interferometry Package (ROI_PAC) developed at Caltech/Jet Propulsion Laboratory. The SRTM DEM data were provided by USGS and the ERS precise orbits were provided by Delft University of Technology. The GPS data were downloaded from the SCIGN web site (http:// www.scign.org) and interpolated using the GSTAT package. The images were plotted with the help of GMT software developed and supported by Paul Wessel and Walter H.F. Smith. References Bock, Y., Wdowinski, S., Fang, P., Zhang, J., Behr, J., Genrich, J., Williams, S., Agnew, D., Wyatt, F., Johnson, H., Stark, K., Oral, B., Hudnut, K., Dinardo, S., Young, W., Jackson, D., Gurtner, W., 1997. Southern California permanent GPS geodetic array: continuous measurements of crustal deformation between the 1992 Landers and 1994 Northridge earthquakes. Journal of Geophysical Research 108, 18013 18033. Brooker, P., 1991. A Geostatistical Primer. World Scientific, Singapore, 95pp. Deutsch, C., Journel, A., 1997. GSLIB Geostatistical Software Library and User s Guide, second ed. Oxford University Press, New York, 384pp. Ge, L., Han, S., Rizos, C., 2000. Interpolation of GPS results incorporating geophysical and InSAR information. 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