Evaluation of earthquake triggering during the earthquake sequence on Qeshm Island, Iran

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2010jb007710, 2010 Evaluation of earthquake triggering during the earthquake sequence on Qeshm Island, Iran R. B. Lohman 1 and W. D. Barnhart 1 Received 18 May 2010; revised 25 August 2010; accepted 9 September 2010; published 16 December [1] Coseismic static Coulomb stress changes may either advance or retard the timing of subsequent earthquakes on nearby faults in a manner that depends both on the relative orientations of faults within the vicinity of the primary rupture, as well as on the postseismic redistribution of stress. Here we examine interferometric synthetic aperture radar (InSAR) data spanning two earthquakes that occurred in close temporal and spatial proximity on Qeshm Island in southern Iran. Remote observations of earthquakes such as ground deformation or seismic waves allow us to infer earthquake source parameters, but in general, a nonunique family of solutions is consistent with the data, given our understanding of data noise and the limitations of our ability to model the real Earth. Here, we explore methods for assessing and propagating InSAR noise through our entire analysis of the stress triggering potential between these two earthquakes. We generate a range of source models for each earthquake based both on InSAR and seismic observations and assess whether the expected static Coulomb stress change during the first earthquake is likely to have brought the second event closer to or further from failure. Some combinations support the hypothesis that the first earthquake brought the second fault closer to failure, whereas some do not. Fault plane geometries that agree with InSAR constraints are more likely to suggest that the first earthquake increased the likelihood of the second earthquake, those based on the Harvard and global centroid moment tensor tend to not support triggering between the two events. Citation: Lohman, R. B., and W. D. Barnhart (2010), Evaluation of earthquake triggering during the earthquake sequence on Qeshm Island, Iran, J. Geophys. Res., 115,, doi: /2010jb Introduction [2] The broad anticlines of the Zagros Mountains (Figure 1) are ideal for studies of the interplay between seismicity and folding because of the high rate of seismic activity, arid environment, spectacular surface exposures and steadily increasing availability of satellite imagery. The Zagros fold and thrust belt is deforming as a result of approximately 3 cm/yr [e.g., DeMets et al., 1994] of ongoing convergence between Arabia and Eurasia. Very few earthquakes in the Zagros Mountains result in surface faulting [e.g., Berberian and King, 1981], although tension cracks are often seen in areas affected by folding [e.g., Nissen et al., 2007, 2010]. Most earthquakes in this region have shallow (<15 km) focal depths [e.g., Maggietal., 2000; Hatzfeld et al., 2003; Engdahl et al., 2006], making it likely that most seismicity occurs within the sedimentary cover overlying the basement rocks. In recent years, a variety of approaches employing both geodetic and seismic observations [e.g., Tatar et al., 2004; Talebian et al., 2004; 1 Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, New York, USA. Copyright 2010 by the American Geophysical Union /10/2010JB Lohman and Simons, 2005a] have improved constraints on the depth range of seismicity even further. [3] On 27 November 2005, a M w 5.9 earthquake struck the region around Qeshm Island, located near the mouth of the Persian Gulf (Figure 1). This earthquake caused damage extending across the Gulf to Qatar and resulted in several casualties and over one hundred injuries. This earthquake was followed by another M w 6.0 earthquake within 6 months (on 25 March 2006) that occurred several tens of kilometers to the north, and then by a third M w 6.0 earthquake less than 3 years later and in the immediate vicinity of the 2005 earthquake on 10 September This heightened seismic activity follows almost three decades of relative quiescence following a similar swarm of M 6 earthquakes that occurred in the late 1970s in the Qeshm Island region. Other significant earthquakes that have occurred in the past include M 6.5 and M s 6.9 events near Bandar Abbas in 1897 and 1977 [e.g., Talebian and Jackson, 2002]. [4] Here we examine interferometric synthetic aperture radar (InSAR) data spanning the 2005 and 2008 Qeshm Island earthquakes. The arid conditions in this region and resulting high interferometric coherence, as well as the existence of observations from multiple independent SAR tracks (locations in Figure 1) and from two satellite platforms utilizing different radar wavelengths, make the sur- 1of13

2 Figure 1. Shaded relief and background seismicity (black dots) [Engdahl et al., 2006] in the Zagros Mountains of southern Iran. Black boxes indicates locations and track names for interferograms used in this study (Table 1 and Figure 2), with Global CMT mechanisms and locations for the earthquakes in red (Table 2). NUVEL 1a plate motion vector is from DeMets et al. [1994]. The insert reference map indicates region in main figure (red box). face deformation field associated with this sequence of thrust earthquakes one of the best constrained to date. The close proximity of these earthquakes in space and times suggests that the second earthquake may have been triggered by the first. We explore whether the deformation observations support this hypothesis. 2. Assessing the Sequence [5] The effects of static Coulomb stress changes [e.g., Lin and Stein, 2004; Toda et al., 2005] on subsequent seismicity and postseismic deformation have been studied for a large number of earthquakes, with some examples including Northridge, California [e.g., Stein et al., 1994], Izmit, Turkey [e.g., Hearn et al., 2002b], and Hector Mine, California [e.g.,]. One of the first order questions that can be asked about the two Qeshm Island earthquakes that occurred in such close proximity is whether the region that slipped in 2008 was brought closer to or further from failure by the 2005 event. As mentioned previously, the last Qeshm Island earthquakes of this size occurred as part of a cluster as well, suggesting that sequences of M 6 earthquakes might be a common characteristic of strain release in this region. [6] The comparison of slip and static Coulomb stress changes that occurred during this earthquake sequence requires understanding of both the fault geometry and slip distributions, as well as information about the elastic structure of the crust surrounding the events. Many of the previous earthquake studies mentioned above found that vertical layering within the crust and upper mantle can result in significant differences in inferred slip and subsequent deformation compared with models generated using an elastic half space. Three dimensional variations in crustal rigidity variations can result in significant changes in the observed strain field and, therefore, our interpretation of the underlying geologic structures [e.g., Fialko, 2006; Hsu et al., 2003; Zhao et al., 2004]. In the Zagros Mountains, spatial heterogeneity in crustal properties likely exists due to a variety of features, such as the pronounced layering of evaporites and sediments above the Precambrian basement rocks, basin/fold contrasts, and the numerous large salt diapirs that are scattered throughout the fold belt. In this paper we will assess the variations in static Coulomb stress/ fault slip interactions that are consistent with the InSAR observations, accounting for the correlated atmospheric noise present in the data. Further variability introduced through unmodeled crustal elastic structure will be left for future work and is likely to affect the depth range of slip as well as the inferred orientation of each fault plane. 3. InSAR Observations [7] We generated 11 interferograms spanning the events using the JPL/Caltech developed ROI_PAC software [e.g., 2of13

3 Table 1. Interferograms Used in Study, With Envisat Swath Number (Bm), Perpendicular Baseline in Meters (Bp), and Number of Resampled Points (Np)a Sat Track ENV ENV ENV ENV T242 T328 T328 T435 ALOS ALOS ENV ENV P565 P566 T328 T435 Bm Bp Date 1 27 November 2005 Earthquake /07/ /01/ /05/ /11/24 10 September 2008 Earthquake /08/ /09/ /06/ /04/17 Date 2 Np 2005/12/ /12/ /06/ /12/ /10/ /10/ /10/ /10/ a Read 2005/12/15 as 15 December Rosen et al., 2000] and Envisat ASAR and ALOS PALSAR imagery. We focus on the eight independent interferograms that exhibit the highest coherence (Table 1). The availability of both ascending and descending tracks for each event, as well as different look angles, allows stronger constraints on characteristics of the fault geometry than could be obtained for similar amounts of data from only one viewing geometry. We remove topographic effects using the SRTM DEM [Farr and Kobrick, 2000]. Figure 2 illustrates a subset of the SAR interferograms that are available for the two earthquakes discussed here, both from Envisat track 435. Note that the interferograms spanning each earthquake show approximately the same magnitude of coseismic deformation when projected into the satellite line of sight (LOS) and that there are features in the InSAR observations throughout each image that are likely due to variations in atmospheric water vapor. One particularly strong and problematic feature is the region of negative LOS change that follows the southeastern coastline of Qeshm Island (Figure 2). We see many similar coast parallel signals in interferograms spanning short time intervals where no earthquake occurred, so we attribute this signal to steep gradients in atmospheric water vapor from the Persian Gulf. [8] We follow the methodology laid out by Lohman and Simons [2005b] to subsample the interferograms to a number of points that is manageable for seismic source inversions ( 1000/interferogram, Table 1) and to characterize the spatial correlation of the atmospheric noise present within each interferogram, with some variations that are described in Appendix A. Here, we include the azimuthal variation of atmospheric noise in our analysis, which is particularly important in the Persian Gulf region, given the observed variations in water vapor that tend to parallel the coast. Figure 3 illustrates the strength of the azimuthal dependence of spatial correlation that we observe in the Qeshm region. [9] Peak observed line of sight ground displacements are on the order of cm over areas spanning a few tens of square kilometers. The 2008 event has a more elongate deformation field that follows a more northeasterly trend than the 2005 ground displacements. The ground deformation for both events is bounded to the west in the vicinity of the Kuh e Namakdan fold (which may be cored by a salt diapir), with the peak curvature of ground deformation corresponding to a flexure in the fold. 4. Methods: Static Coulomb Stress Changes [10] In a simple system containing two parallel faults, slip along one fault will result in a negative static Coulomb stress change for that orientation of rupture on the immediately adjacent portion of the second fault. This means that the first earthquake will inhibit slip with the same rake on Figure 2. Interferograms from track 435 (Table 1) spanning (a) the 2005 earthquake (24 November 2005 to 29 December 2005) and (b) the 2008 earthquake (17 April 2008 to 9 October 2008), overlain on SRTM topography [Farr and Kobrick, 2000]. Color scale indicates line of sight (LOS) deformation in cm in the direction of the black arrow. 3 of 13

4 Figure 3. (a) Data residual used in covariance estimate for interferogram from track 435 spanning 24 November 2005 to 29 December Blank zone is due to our removal of data within 5 km of the fault trace (black line) before estimating spatial covariance. (b) Number of pixels that are utilized in the covariance estimate at each offset. (c) Azimuthally varying covariance estimated from data. (d) Covariance structure modeled using a best fit power law decay (characteristic length indicated by blue ellipse), with the same color scale as in Figure 3c. Note that we are not fitting any of the negative correlations that exist in the real covariance structure. the second fault and increase the likelihood of rupture with the opposite orientation. However, the two earthquakes discussed here, while proximal and similar in orientation, may differ enough in their fault geometry, position, or in the spatial variability of their slip distributions such that the slip that occurred in the 2005 event could have either encouraged or discouraged the faulting that occurred during the 2008 earthquake. [11] We examine the relationship between stress and slip during these two earthquakes by evaluating the static Coulomb stress changes due to inferred slip distributions on candidate fault planes for the 2005 event on the rupture planes associated with the 2008 event [e.g., Lin and Stein, 2004; Toda et al., 2005], and comparing the regions of positive or negative static Coulomb stress change to models of the location of slip during the 2008 event. Because the earthquakes occurred in such close proximity, the inferred fault planes that best fit the data can be subparallel or even intersect, depending on the choice of reasonable constraints on fault orientation and slip (described below). In some of these cases, the combination of faults is most likely unphysical, and we should not place any significance on characteristics of the inferred static Coulomb stress changes within the volume that is most dependent on small perturbations to the fault geometry or slip distribution. Here, we attempt to capture the range of variability in fault orientations and slip distributions that are consistent with InSAR observations for both events, based on estimates of the error in fault plane geometry and slip distributions in our analysis. We attach significance only to features of the relationship 4 of 13

5 Table 2. Earthquake Source Parameters for the Best Fit Single Fault Patch With Uniform Slip Derived From Seismic and Geodetic Observations, With the Dip Direction Appended to Distinguish Between the Two Potential Nodal Planes a Lon Lat Depth (km) Strike Dip Rake M w L (km) W (km) Slip (m) 27 November 2005 Earthquake LB SE LB NW HCMT SE HCMT NW Nissen September 2008 Earthquake LB SE LB NW GCMT SE GCMT NW Nissen a LB indicates the best fit InSAR derived solutions from this work. HCMT, GCMT, and Nissen indicate inversions seeking the best fit location, depth, and slip distribution, constrained to the focal mechanisms from the Harvard and global centroid moment tensor or Nissen et al. [2010] solutions. Lat/Lon and depth refer to the center of the fault patch. between static Coulomb stress change and slip during the 2008 event that are robust (defined in section 5) given these potential variations Earthquake Source Geometry [12] We sample a range of potential fault geometries in two ways. First, we include our best fit north and south dipping fault planes as well as our relocations of the focal mechanisms reported by the Harvard and global centroid moment tensor (CMT) catalogs and other reported fault geometries inferred for these events [Nissen et al., 2010]. We infer fault orientations within a uniform elastic halfspace [Okada, 1985] from the observed surface deformation using the Neighbourhood Algorithm [e.g., Sambridge, 1999; Lohman et al., 2002; Lohman and Simons, 2005a], a nonlinear inversion technique employing a global search that efficiently focuses in on regions with the lowest data misfit. Fault characteristics are allowed to vary in location, size and orientation (except for inversions where the orientation is fixed to a previously reported mechanism) to minimize the residual between the modeled and observed deformation signal. At every fault location explored, we perform a linear inversion for the best fit slip magnitude and for a linear ramp across each interferogram (three free parameters), which accounts for our lack of precise knowledge about the satellite orbits as well as for the unknown average offset associated with each interferogram. This results in 11 free parameters for the full inversions and 8 for the inversions where the focal mechanism is fixed. [13] As is common in inversions of geodetic data associated with dipping faults that lack an obvious surface expression, the constraints on fault orientation and rake are not as strong as the constraints on the centroid location and depth. Our best fit solutions for the events are consistent with those reported by Nissen et al. [2010], as well as with CMT mechanisms (Table 2). Figure 4 illustrates the range of misfits for our best fit inversions when the rake value was held fixed to a range of values, as well as for the relocated Nissen and CMT models. The SAR observations place essentially no constraint on the choice of nodal plane (e.g., dipping to the north versus the south) that ruptured for any particular mechanism. While one nodal plane may result in a lower RMS error when we use a single fault patch to fit the data, allowing for distributed slip results in essentially identical fits to the observations. Within fold and thrust faults underlain by salt (such as the Zagros Mountains), thrust faults dip both toward and away from the convergence zone [e.g., Davis and Engelder, 1985], so we do not feel that we can rule out potential rupture planes with either orientation. This may be particularly true in the area around Qeshm Island, which is proximal to the region where the Musandam Peninsula impinges on the southern Iranian coast and appears to complicate the geometry of plate convergence when compared with regions to the west. Therefore, we consider both potential fault orientations for both earthquakes in our analysis. Figure 4. Residual mean sum of squares (RMS) for candidate NW and SE dipping planes for the 2005 earthquake. Black and gray curves indicate the best fit fault plane for a range of inversions where the rake was held fixed. Open symbols indicate the best fit solutions where the position and size of the fault plane were allowed to vary but the mechanism was fixed to that of previous work (Table 2). 5of13

6 Table 3. The 1s Error Bounds on the Source Parameters in Table 2 a East (km) North (km) Depth (km) Strike Dip Rake Slip (m) Area (km 2 ) 27 November 2005 Earthquake LB SE LB NW HCMT SE NA NA NA HCMT NW NA NA NA Nissen NA NA NA September 2008 Earthquake LB SE LB NW GCMT SE NA NA NA GCMT NW NA NA NA Nissen NA NA NA a Derived using the Monte Carlo sensitivity test described in the text. Note that we report slip and area rather than moment, and errors on distance in km from the best fit location rather than in lat/lon. Trade offs are larger for the two LB models, as expected, since those inversions also allow strike, dip, and rake to vary. NA, not available. [14] Second, we assess our confidence on these best fit fault planes through a Monte Carlo approach and attempt to include these estimates in our final results by generating a family of 50 fault planes that are drawn from our 1s error bounds on each of the 5 best fit models (Table 3). We implement the Monte Carlo approach by producing a large number (>100) of synthetic sets of noise drawn from the noise covariance matrix estimated from each interferogram [Lohman and Simons, 2005b], that we then add to predicted deformation field produced by our best fit earthquake parameters [Lohman and Simons, 2005a]. This approach addresses specifically the estimates of errors due to the contribution of atmospheric noise (including the spatial structure of the noise covariance shown in Figure 3 within the interferograms, not those due to deviations of the realworld scenario from uniform slip on a simple rectangular fault plane in an elastic half space, or from any anthropogenic or aseismic deformation that may also contribute to the observations Earthquake Slip Distributions [15] To generate a family of slip distributions for each fault geometry, we subdivide the fault plane into rectangular patches, covering an area that is 5 km longer in width and length (truncated at Earth s surface, if necessary) than the inferred plane with constant slip. We impose roughness penalties through a finite difference approximation of the Laplacian on inversions with more than a single fault patch, using an approach for choice of smoothing weight which allows us to easily include positivity constraints on slip (Appendix B). [16] Given a particular source and target fault plane and slip distribution, we predict the associated static Coulomb stress change as in the work by Meade [2007], which uses the formulations given by Okada [1992]. This approach produces the same results as those of Lin and Stein [2004] and Toda et al. [2005] and has the ability to easily ingest triangular dislocations, which will allow for the use of more spatially complex fault models in the future when the data warrants it. [17] In order to assess the effect of reasonable variations in fault slip on the predicted static Coulomb stress changes, we not only consider the best fit fault slip distribution for each candidate fault plane but generate a family of 50 slip distributions that are consistent with the data and noise. Because the positivity constraints result in this being a nonlinear inversion, we follow a Monte Carlo approach similar to that in the previous section: we generate synthetic noise, add it to the input data and invert for the best fit slip. Therefore, we obtain not only the inferred static Coulomb stress change due to the best fit slip distribution, but also an estimate of the confidence that we can place on that stress change given the noise in the data. One example of the range of fault planes and slip distributions that we infer for each pair of events is shown in Figure 5, which illustrates each of the 50 2 candidate models generated by the Monte Figure 5. View toward the southwest of candidate fault planes and slip distributions used in analysis of the 2005 LB SE (right cluster) and 2008 LB NW (left cluster) static Coulomb stress interactions. 6of13

7 Figure 6. Stress interactions during one particular combination of source/target fault planes. (a) Color and arrows indicate inferred slip distribution for one inversion using the best fit south dipping rupture plane for the 2005 earthquake (LB SE, Table 2). (b) Color indicates estimated static Coulomb stress change [e.g., Lin and Stein, 2004; Toda et al., 2005] due to slip in Figure 6a, on our best fit north dipping candidate fault plane for the 2008 earthquake (LB NW, Table 2). Black arrows indicate inferred slip during the 2008 earthquake. White circles indicate fault patches where the 1s error bounds derived from our Monte Carlo tests are smaller than the positive static Coulomb stress change values shown here, i.e., where the positive values are robust given the parameters explored in our analyses, including both slip distributions and fault geometry variations. Carlo approach. This pair (2005 LB SE and 2008 LB NW solutions in Table 2) will also be referenced in following figures. 5. Results [18] Figure 6 illustrates a comparison of regions where our inferred slip distribution on the LB SE candidate fault plane for the 2005 earthquake (Figure 6a) and the associated positive or negative static Coulomb stress changes (in color, Figure 6b) on the LB NW candidate fault plane for the 2008 earthquake, with the inferred slip during the 2008 earthquake on that particular fault plane (black arrows, Figure 6b). For each of the 25 source target combinations possible between the two sets of five models listed in Table 2, we explored 50 perturbations to the geometry for each source and target, as well as 50 perturbations to each of those slip distributions, or pairings. [19] One way of distilling this information is to examine the percent moment that is released in regions of positive static Coulomb stress change for each set (red region in Figure 6b), and to then assess whether any of the sourcetarget combinations were more likely to reflect a situation where the second fault failed in a region that was brought closer to failure by the 2005 earthquake. We perform this calculation for each of the 25 best fit source target combinations and then examine the variability of this result given our perturbations to fault slip and geometry. White circles in Figure 6b indicate fault patches where the static Coulomb stress change for that patch (shown in color) is larger than the 1s error bounds on static Coulomb stress change that we inferred during our Monte Carlo test with multiple slip distributions and variations in fault geometry; that is, that the value is likely to be positive given all the permutations that we have explored here. These are the regions that where it is most likely that fault slip during the first work promoted failure with the observed rake during the second earthquake and are what we term robustly positive regions of positive static Coulomb stress change. [20] To come up with a more quantitative comparison of the 25 potential source/target fault combinations, we compute the moment release on a patch by patch basis and examine the statistics of whether moment was released in regions that show robustly positive Coulomb stress change. 7of13

8 Figure 7. Comparison of all pairs of source and receiver earthquakes. Color indicates the percent moment release, averaged over all combinations of candidate fault planes for each pair of fault orientations, that occurs in regions where the static Coulomb stress change is positive and above the 1s error bounds derived from our Monte Carlo slip distribution and geometry perturbation tests. The fault combination discussed in the text is outlined in red. For instance, for the pair of fault planes shown in Figure 6, if we average over all of our perturbations to slip and fault geometry, 35% of the moment release occurs in regions with positive static Coulomb stress change for the Monte Carlo slip test. The rest of the moment release is in regions with either negative or poorly constrained static Coulomb stress changes. Figure 7 illustrates this quantity for all possible combinations of the fault planes studied here. Note that the lowest percentages are found for the combinations that include the Global CMT solutions for the 2008 earthquake. These solutions involved strikes that differ by 30 from those inferred by either of the independent studies using InSAR observations. It is possible that the source may have involved a more complex, nonplanar rupture such as that suggested by Nissen et al. [2010], which could explain the observed difference between the source models inferred using geodetic versus seismic data. 6. Conclusions [21] Studies that evaluate static Coulomb stress changes associated with individual earthquakes often focus on the best fit fault plane and slip distribution. At distances greater than a fault width or two, small variations due to errors in fault geometry or due to the inherent smoothing of fault slip distributions during inversions of geodetic data may be small in comparison to the stress change values themselves. However, in regions that are proximal to the source fault, variations in fault location, orientation and slip can result in large changes in the magnitude or even the sign of the inferred static Coulomb stress change. In this work, we have explored a wide range of potential fault planes for two earthquakes and have evaluated the potential triggering of the second earthquake by the first. We choose our family of fault models to reflect variations that are consistent with the data given the level of noise within our observations. As mentioned above, model errors such as variations in elastic moduli, the existence of interseismic or postseismic deformation, and the fact that our simplified planar fault may not be a sufficient representation of the real world will likely increase the family of models that should be explored. Therefore, itis likely that the error bounds described here are overly optimistic and that a more conservative range is applicable. [22] A factor not included in our analysis is the redistribution of stresses through postseismic deformation processes [e.g., Freed and Lin, 2001; Hearn et al., 2002a] acting over the several years in between the two events examined here. The existence of the numerous salt domes in the region (which are observed to flow radially outward at various rates in interferograms) complicates the situation somewhat, but may be addressed through the use of finite element models in the future. [23] One conclusion that we draw from this study is that the fault planes and slip distributions consistent with the InSAR do not require that the 2008 earthquake was affected or caused by stress changes due to the 2005 event, since the models predict both positive and negative static Coulomb stress changes in the vicinity of where slip occurred on the target fault plane in However, if we make the assumption that the 2008 earthquake was less likely to initiate in a region of negative static Coulomb stress change, then our approach places some constraints on the likelihood of various nodal plane combinations, e.g., the 2005 SE and 2008 NW dipping planes from LB may be more likely than the two SE dipping CMT solutions, where most of the moment release during the 2nd event occurred in a region where we would predict negative static Coulomb stress changes. The InSAR data does not place constraints on the location where slip initiated, so inclusion of information about the actual hypocenter as constrained by seismic data would improve our ability to discuss whether the 2005 earthquake affected the occurrence of the 2008 event. Examination of future earthquake sequences in the area may help to determine whether one set of orientations consistently produces stress changes that are more favorable for the following earthquake sequence than the other set. Appendix A: Data Downsampling [24] Lohman and Simons [2005b] present a method for downsampling interferograms that contain several million observation points to a number of spatially averaged observations that is manageable for seismic source inversions ( 1000/interferogram). This approach relies on an a priori fault model and begins with a coarse grid of boxes where the data are averaged within each box. Given the forward relation Gm = D, where G is the design matrix of Green s functions associated with this fault parameterization: data pairing, m is the source model or slip distribution, and D are the locations of the spatially averaged observations within 8of13

9 Figure A1. (a c) Iterations 2, 4, and 8 of the data downsampling sampling algorithm applied to the InSAR data shown in Figure 2a, with color indicating the average LOS value in cm and the small black dot indicating the centroid location for all non NAN points contained within each box. White line indicates the fault plane used in the downsampling procedure. 9 of 13

10 Figure A2. (a) Width of best fit Gaussian in meters for the row of N associated with each data box for iteration 5 during downsampling procedure for data shown in Figure 2a. (b) Row of N associated with a poorly resolved data box (outlined in black) that will not be further downsampled at the next iteration. (c) Row of N associated with a data box that will be downsampled. 10 of 13

11 each box, we find a solution using a generalized inverse, G g, such that m * ¼ G g D; ða1þ where G g =(G T a G a ) 1 G T, where the augmented Green s function is G a = [G ls] T, S is the finite difference approximation of the Laplacian smoothing matrix, d a = [d 0] T, and m* is the output model inferred by the inversion [Menke, 1989]. [25] The data downsampling procedure then follows these steps: (1) Find all D i where the diagonal of the data resolution matrix N, where N = G g *, is above some preset threshold; (2) subdivide all of these boxes into4 smaller boxes, increasing the number of D i in areas that have more impact on the fault slip inversion; and (3) recalculate the new set of Green s functions and data resolution matrix. Figure A1 illustrates an example of this approach, using the data from Figure 2a. [26] Lohman and Simons [2005b] controlled the data downsampling by two tunable parameters: the strength of regularization that is included in the inversion, and by the threshold value for the diagonal of N that is used to determine whether a particular data box is subdivided at the next iteration. Here, we improve on this approach by replacing the somewhat arbitrary choice of threshold value with a more physically based measure of whether a box should be subdivided. At each iteration, we plot the row of N associated with each D i versus distance from that resample data point location. We then compare the width of the best fit Gaussian curve to this relationship to the actual size of the box represented by that D i. If the width of the best fit Gaussian is comparable to the size of the box, then that D i is subdivided at the next iteration. Figure A2 illustrates the relationship between N and the choice to downsample at one iteration. Appendix B: Regularization [27] We define a good choice of regularization strength, l, as one that would fit the underlying, noise free signal (d 0 ) as well as possible without introducing model characteristics that are merely fitting the noise. We use lower case d here to differentiate from the data downsampling procedure discussed in Appendix A. Increasing l decreases our ability to fit the underlying signal, as the inferred model is forced to be increasingly smooth. We call the error due to oversmoothing the regularization error, since it is a function of our regularization method and the input slip distribution, not of the noise [e.g., Hansen, 1998]. As we decrease l, the inferred model fits more and more of the noise, resulting in an increase in what we call the perturbation error. The total error for a given l is the sum of the perturbation and regularization errors. [28] To better understand the behavior of regularized slip inversions, we rely on the fact that we can treat our observed data, d i, as a sum of two parts: d i d 0 þ n i ; ðb1þ where d 0 = Gm 0 is the physical response of the earth to an input slip distribution (m 0 ) in the absence of noise, and n i are realizations of independent, identically distributed Gaussian noise with variance s 2 and zero mean. If the forward and inverse problems are linear, we can separate the inversion into the parts controlled by the exact data, d 0, and by the noise. Below, we will discuss the more general, nonlinear case which holds when positivity constraints are incorporated into the inversion. [29] The regularization error quantifies the degree to which the inferred slip models can fit the exact data in the absence of noise: m * 0 ¼ G g* d 0 ; d * 0 ¼ Gm* 0 ; ðb2þ ðb3þ where G g * is the generalized inverse for a regularized inversion with a given l [e.g., Menke, 1989], m 0 * is the smoothed version of m 0 that we would infer with a particular regularization, and d 0 * is the smoothed surface deformation predicted by m 0 *. In general we use the asterisk to signify inversion quantities where we have applied smoothing. We define the regularization error, 0 r 0, as the difference between the exact data (d 0 ) and the deformation predicted by the smoothed model (d 0 *): 0r 0 d 0 d * 0 ; ðb4þ The perturbation error is the degree to which a given realization of the noise in the data, n i, is mapped by the inversion into the inferred slip, m i *, and back into our predicted synthetic data, d i *. We separate the operation of G g * on the noisy data into a sum of its parts: m * i ¼ G g* d 0 þ G g* n i ; ðb5þ d * i ¼ d * 0 þ n* i ; ðb6þ where n i * is the noise filtered by our regularization. To help us separate out that part of the inversion that is only fitting the noise in the data, we define jr n i n j n * i ; ðb7þ where j r i n compares a smoothed set of noise (n i *) with a completely independent realization of the noise(n j ). This quantity, which we define as the perturbation error, increases as l decreases, because an inversion that fits one set of noise very well will not necessarily fit an independent set of noise. The total error when we compute a residual between one data set and a smoothed, independent data set is jr i d j d * i : Combining equations (B4) and (B8), we get jr i ¼ 0 r 0 þ j r n i ; ðb8þ ðb9þ indicating that the total error is equivalent to the sum of the perturbation and regularization errors. We define measures of the size of these residuals as jr i 1 X k j r2 i ðb10þ 0R 0 1 k X 0 r2 0 ; ðb11þ 11 of 13

12 an identity matrix. If we define a matrix M as [I N], then we can express the residual quantities as 0r 0 ¼ Md ½ 0 d 0 Š T ; ðb15þ T jr i ¼ Md j d i ; ðb16þ jr n i T ¼ Mn j n i ; ðb17þ We can expand equation (B10) using equation (B9): jr i ¼ 1=k X h ð0r 0 Þ 2 þ 2ð 0 r 0 Þ j r n i þ j r n i 2 i : ðb18þ Figure B1. Values of l (best values for individual data sets shown with dots) and j R i (1s error bounds based on family of data sets shown with solid lines) predicted for a simple scenario with a vertical strike slip fault and observations of fault parallel displacement along a line crossing the fault, using both the approach for exact data with no bound constraints, the j R a i approach described in the text (black line and dots, also with no bound constraints), and for inversions of many simulated synthetic data sets where we did impose positivity constraints on the solution (gray dots). where k is the number of observation points used in the inversion. The value of l that minimizes j R i is the optimum regularization that we use in our inversions. As l approaches 0, the perturbation error approaches 2s and the regularization error approaches 0. We use a script R to avoid confusion with R, the model resolution matrix. As l becomes large and smoothing increases, the perturbation error decreases and the regularization error increases. B1. Calculating j R i for Exact Data [30] One way to calculate the value of p or l that optimizes j R i for a synthetic system would be to create a large number of synthetic data sets with different realizations of the noise, and to calculate j R i numerically by computing all the permutations of d j d i *. However, we can be more efficient in these synthetic cases and capitalize on the fact that we know the input model, m 0, and the covariance structure, C d, of the noise. We can use these two quantities to find the value of j R i analytically for any (p, l). [31] We can also write equations (B2) (B4) as m * 0 ¼ Rm 0 d * 0 ¼ Nd 0 ðb12þ ðb13þ 0r 0 ¼ ½I NŠd 0 ; ðb14þ where R = G g *G and N = G g * are the model and data resolution matrices, respectively [e.g., Menke, 1989] and I is Since j r n i is a random variable with mean 0, the middle term disappears, and we are left with jr i ¼ 1=k X ð0r 0 Þ 2 þ 1=k X jr n 2: i ðb19þ The first term is equivalent to the definition of 0 R0. Since the mean of j r i n is 0, the expectation of the second term in equation (B19) is a sum over the variances of j r i n at each data point. If C d is the data covariance matrix, then by the law of covariance propagation: C r ¼ MC d M T ; ðb20þ where C r is the covariance matrix of j r i n. Therefore, we have jr i ¼ 0 R 0 þ 1 X diag ð Cr Þ: ðb21þ k This formulation of j R i depends only on the input model, m 0, and on the noise covariance, which are both quantities that we would know for a synthetic system. B2. Calculating j R a i for Real Data [32] In order to form an approximation of j R i in the case where we have only one data set and impose nonlinear constraints such as bounds on the inferred model, we attempt to infer the quantities described in section B1 from our inversion of the single data set and our knowledge of characteristics of the noise in the data. [33] Given the data covariance matrix C d, we can compute the second term in equation (B9) and are left with a need to approximate 0 R0. We can compute one realization of i r i = d i d i * and i R i 1 P k ir 2 i using our existing data set, and can then use the same process as described in equations (B1) (B19) to find where ir i ¼ 0 R 0 þ 1 X diag ð C2 Þ; ðb22þ k C 2 ¼ MI 2 M T ; ðb23þ where I 2 is a matrix constructed of 2 2 I matrices. At this point all values needed to solve for 0 R0 are at least approximated. We are left with our approximation of the j R i value: jr a i i R i 1 X diag ð C2 Þþ 1 X diag ð Cr Þ: ðb24þ k k 12 of 13

13 We find that for many fault slip inversions, the use of bound constraints during the initial calculation of i Ri, although it violates many of the assumptions made in constructing R, N, etc., results in a slightly lower choice of l, as would be expected. Synthetic tests using many realizations of noisy data sets and a known input model m 0 with bound constraints on m i * result in values of l that are consistent with the ones inferred using the j Ri a approach discussed here (Figure B1). [34] Acknowledgments. We thank R. Mellors, D. Sandwell, and E. Fielding for their development and maintenance of the software used to process the ALOS data used in this research. The manuscript was improved thanks to helpful suggestions made by J. Crider and one anonymous reviewer. ALOS imagery was acquired through the Alaska Satellite Facility (ASF), through a UPASS proposal supported through NASA. Envisat imagery was made available through a Category 1 proposal through the European Space Agency. 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