In 2001, a nearly five month long sequence of shallow, mostly small magnitude earthquakes occurred beneath the city of Spokane, a city with a population of about 200,000, in the state of Washington. During most of the sequence, the earthquakes were not well located because seismic instrumentation was sparse. Despite poor-quality locations, the earthquake hypocenters were likely very shallow, because residents near the city center both heard and felt many of the earthquakes. The combination of poor earthquake locations and a lack of known surface faults with recent movement make assessing the seismic hazards related to the earthquake swarm difficult. However, the potential for destruction from a shallow moderate-sized earthquake is high, for example Christchurch New Zealand in 2011, so assessing the hazard potential of a seismic structure involved in the Spokane earthquake sequence is important. Using interferometric synthetic aperture radar (InSAR) data from the European Space Agency ERS2 and ENVISAT satellites and the Canadian Space Agency RADARSAT-1, satellite we are able to show that slip on a shallow previously unknown thrust fault, which we name the Spokane Fault, is the source of the earthquake sequence. The part of the Spokane Fault that slipped during the 2001 earthquake sequence underlies the north part of the city, and slip on the fault was concentrated between ~0.3 and 2 km depth. Projecting the buried fault plane to the surface gives a possible surface trace for the Spokane Fault that strikes northeast from the city center into north Spokane.
 The Spokane area, an area of low background seismicity, is on the northeastern edge of the Columbia Basin (Figure 1), a physiographic province largely covered with Miocene flood basalts of the Columbia River Basalt Group [e.g., Tolan et al., 1989]. The earthquake sequence appears to have begun with an isolated magnitude 2 earthquake (all mentions of magnitude imply coda derived magnitude unless otherwise stated) on the 24th of May, 2001, but began in earnest with a magnitude 3.9 earthquake on the 25th of June, 2001 (Figure 2). A total of 105 earthquakes were recorded, the largest was a magnitude 4 (11 November, 2001), and the last recorded earthquake in the sequence was on the 23rd of November, 2001. Residents in small areas of Spokane reported feeling many of the earthquakes in the sequence and reports of hearing explosion-like noises associated with the earthquakes were also common.
Table 1. Parameters for Radar Scenes Used in Study
Interferogram No. & Figure
Master Scene Date
Slave Scene Date
Interferogram numbers are used in Figures 2 and 3. “ERS2” denotes European Space Agency (ESA)'s Earth Resource Satellite #2, “RSAT1” denotes the Canadian Space Agency RADARSAT-1 satellite, “ENV” denotes ESA's ENVISAT satellite. “Desc” denotes descending orbits whereas “Asc” denotes ascending orbits. B⊥ is the perpendicular baseline separation between the two satellite orbits during master and slave image acquisition.
 The Pacific Northwest Seismic Network (PNSN) in 2001 was not optimal for locating shallow, small magnitude earthquakes in Spokane. The network failed to record some of the earthquakes felt and heard by residents suggesting these earthquakes were small and very shallow. Personnel from the University of Washington and the USGS Seattle office, who jointly operate the PNSN, deployed temporary seismic stations, with varying coverage, which was best for covering the sequence (Figure 2a) in the month of July, 2001. The earthquakes that occurred during the July deployment are the best located in the sequence, and they are shown in red in Figure 2a—they are the only earthquakes shown in the subsequent figures.
 The Spokane area has no known faults that have been active in the last ~1.6 million years (Figure 1); however, evidence for a possible Miocene fault does exist. This possible fault, the Latah Fault [Derkey and Hamilton, 2001; Derkey et al., 2001], follows a ~50 km long, NNW trending lineament that skirts the west side of the city of Spokane (Figure 2a). Other surface evidence for active faults in and around Spokane is lacking, likely obscured by great Pleistocene floods from glacial Lake Missoula [e.g., Bretz, 1925]. Spokane is located in a throughway for the floods that resulted in the scouring and burying of possible fault scarps older than about 11 ka.
2 InSAR Data and Modeling
 The only geodetic data showing the surface deformation during the earthquake sequence are interferograms generated from satellite radar data acquired by the Canadian Space Agency RADARSAT-1 and the European Space Agency ERS2 satellites (Figures 2b and 3). The interferograms (Figure 3) for each pair of radar scenes (Table 1) were processed using a two-pass method [Massonnet and Feigl, 1998] using the SRTM one second digital elevation model [Farr et al., 2007], shown in shaded relief in Figure 3. Each interferogram was filtered [Goldstein and Werner, 1998] to increase the signal-to-noise ratio. The area of deformation is about 3 km by 2 km in size, it is located north of the center of Spokane, and the peak range change (change in distance from the ground to the satellite) is about 15 mm. An extra step was required to produce useful interferograms from the RADARSAT-1 data. Because orbits of the RADARSAT-1 are not accurate, the interferograms calculated from these data contained orbital fringes. The deforming area in the interferograms is small, so we removed the orbital fringes by masking out the deforming area, then fitting a quadratic surface to the masked interferogram [e.g., Hanssen, 2001]. We then removed this quadratic surface from the interferogram, and the results are shown in Figure 3. A comparison of interferograms that span the sequence with those before and after the sequence shows that the deformation is associated with the earthquake sequence (Figures 2b and 3). The geometry of the radar is different for the four interferograms that show deformation (Figures 3b–e), providing a good data set for modeling the deformation source.
 We have no a priori information about the geometry or detailed location of the deformation source, so it is important that we quantify our modeling efforts to find a best fit model and be able to say with some certainty which models can be rejected or accepted as viable. It is also important to find uncertainties on model parameters that are important for future geophysical studies of the Spokane Fault such as dip or the location of the surface trace where the fault plane intersects the surface. To do this, we choose to use an F-test to establish 95% uncertainties in model fit and model parameters [Lu and Wicks, 2010; Wicks et al., 2011a, 2011b]. To use the F-test requires that we reduce the redundancy of our deformation measurements to the minimum number of points needed to represent the deformation field. At this point, we can assume that the data points are linearly independent and the F-test is viable, at least in approximation. The following steps, some of which are presented in detail later, are used in our data reduction and modeling procedure:
 We modeled the interferograms in Figures 3b–e using a single rectangular dislocation [Okada, 1985] then a functional distribution of rectangular dislocations on a planar surface. In each case, we assumed the Earth to be a homogeneous isotropic half-space [Okada, 1985], and we included parameters to account for small orbital errors that approximate a tilted planar surface in each interferogram. We used a nonlinear inversion procedure in all the modeling we performed in this study.
 Since there is redundancy in the four interferograms in Figures 3b–e, we reduced the data to the two orthogonal components that are recoverable: the vertical and east-west components of the deformation field. To do this, we used the results of the preliminary modeling and followed the procedure of Wright et al. . Besides reducing the data redundancy, this also yielded a better signal-to-noise ratio. As we will explain later, the four interferograms are effectively “stacked” using the methods of Wright et al.  yielding an east-west and vertical interferogram.
 We reduced, or parsed, the number of data points in the east-west and vertical deformation fields using the quadtree windowing method of Jónsson et al.  and a model-based windowing scheme inspired by Lohman and Simons .
 The parsed data set was then used to find a best fit model which served as the base of comparison for F-tests to establish uncertainty levels and acceptability of other candidate models.
 Finally, because we assumed the north deformation component was negligible in calculating the east and vertical deformation components in step two above, we devised a synthetic test to show that this assumption was valid.
 We attempted to reduce the four interferograms into three orthogonal directions of deformation (east, north, and vertical) following the procedure of Wright et al. , but could only recover the east and vertical [see Wright et al., 2004]. In the preliminary modeling of the interferograms in Figure 3, we included a static shift and two tilt parameters (in the N-S and E-W directions) for each interferogram in the inversion. The static shift and tilt were subtracted from each interferogram at this point. Using the covariance matrix for the four interferograms and the four unit look vectors from the ground to each satellite (1), we calculated the standard errors in east, north, and up components to be 2.3 mm, 41.2 mm, and 4.9 mm respectively. The large error in the north component is mostly a result of the poor view of the north component afforded by the near-polar orbit of the satellites. The unit look vectors from the ground to the satellites are shown in the matrix in equation (1). Rows 1, 2, 3, and 4 of the matrix correspond, respectively, to the interferograms in Figures 3b–e. Columns 1, 2, and 3 in the matrix correspond to east, north, and up look directions, respectively. From the preliminary modeling of the four interferograms, we know that the peak deformation calculated from the best fit model in the east, north, and up components is about 7 mm, 8 mm, and 15 mm, respectively. The error in the north component is much larger than the signal, so we could not recover the north component of deformation. This is essentially the same as the results Wright et al.  found for the Nenana Mountain earthquake. We followed the methods and nomenclature of Wright et al.  by assuming the north component of the deformation field is negligible (we revisit this assumption later) and calculated proxies for the east and vertical components of deformation. The standard errors for these proxies are 2.1 mm and 2.7 mm in the respective east and vertical components, much less than the anticipated signal. The resulting maximum-likelihood estimates for east and vertical components of deformation are shown in Figure 4. The deforming area clearly corresponds to the epicentral region of the best located earthquakes (Figure 4) from which we infer that the process responsible for the surface deformation, which could be aseismic slip, is also responsible for the earthquake sequence.
 The next step is to parse the vertical and east-west deformation fields, shown in Figure 4, using the quadtree procedure of Jónsson et al. . This procedure reduces the data redundancy yielding a more manageable number of data points that approaches linear independence [Jónsson, 2002]. The variance cutoff value used in the quadtree procedure is somewhat qualitative; generally, such as we did in the preliminary modeling, we use the variance of the noise in a nondeforming part of the interferogram as a cutoff. Larger cutoff values yield less points, and smaller cutoff values yield more points. Hence, we devised a method to determine variance cutoff values based on the synthetic vertical and east deformation fields calculated from the model that best fit the four interferograms in Figure 3. We used an approach whereby we began by assuming a large variance value to quadtree parse the calculated deformation field then decreased the variance stepwise until we met a metric that showed the parsed data points accurately represented the deformation field. The metric we used is the difference between the volume under the deformation surface calculated using the best fit model and the volume under a surface fit to the quadtree-parsed data points using splines [Smith and Wessel, 1990]. Using a spline fit to the quadtree-parsed data points, we reconstituted the surface of the deformation field without knowledge of the model used to calculate the surface that was being parsed. When the volume beneath the spline-fit surface was 95% of the volume beneath the deformation surface calculated from the model, we considered the variance cutoff to be acceptable and the resulting parsed data points were accepted to be the minimum number required to represent the model surface. We applied this procedure first to the vertical deformation field then to the east deformation field finding a separate variance cutoff value for each field. The quadtree windows resulting from this process (shown in Figures S1a and b) were then used to parse the data in Figure 4 yielding the parsed vertical and east components of deformation shown in Figures S1a and b. This use of a preliminary model to determine the quadtree windowing was proposed by Lohman and Simons , with the use of a different metric, however. These data (Figures S1a and b) are used for subsequent modeling and evaluation of uncertainties.
 To model the data in Figures S1a and b, we adopted a simple procedure that has been used successfully before [Lu and Wicks, 2010; Wicks et al., 2011a, 2011b] where we assumed the slip followed a distribution of dislocation sources on a planar fault that can be described by a 2-D Weibull distribution as parameterized by Myrhaug and Rue  and written into code by Brodtkorb et al. . This parameterization includes a covariance parameter that allows communication between the two orthogonal Weibull distributions. This communication introduces the possibility of a wider range of slip distributions that can vary in the two orthogonal directions simultaneously [see Myrhaug and Rue, 1998 and references therein] allowing for a wider range of distribution shapes. We also introduced a model parameter that allows all or part of the slip in the distribution to be uniform in magnitude. This parameter truncates the slip value in the distribution where the level of truncation is determined in the inversion. In this procedure 16 model parameters were used. We used a nonlinear least squares procedure with multiple constrained Monte Carlo starting models in an Earth represented by a homogeneous, isotropic, elastic half-space [Okada, 1985]. The model that best fits the data is a shallow thrust fault that dips about 30° to the NW (Figure 5). We note here that the model that best fit the four interferograms in Figures 3b–e is essentially the same as is shown in Figure 5, complete with the tail to the north. The difference is that with this model (Figure 5), we have established a best fit model to which we can make comparisons using the F-test. The depth to the top of the slipping part of the fault is about 0.3 km, and the deepest slip is at a depth of about 2 km. The maximum slip in the best fit model is about 45 mm, and the slip has an along-strike component that yields a slip direction (the black arrow in Figure 5a) that is parallel to the Latah Fault [Derkey and Hamilton, 2001]. The fit to the data is good: the variance of the residual is ~0.53 mm2 (Figures 6, 7, and S1), and the model accounts for about 99% of the variance in the data.
 Finally, we evaluated our assumption that the north component of deformation was negligible in our calculation of proxy east and vertical components. Wright et al.  also made this assumption and justified it because the geometry of the fault they investigated should not have produced much displacement in the north direction. We have already shown that for our best fit model (Figure 5), the north component of deformation is nearly the same magnitude as the east component, so it might not be negligible. When we calculated the proxy deformation fields shown in Figure 4, the deformation was mapped into the east and vertical components. This means that some of the north component was mapped into the east and vertical deformation fields. We estimated the magnitude of this mapping error introduced into the proxy deformation fields by using the best fit model in Figure 5 to calculate synthetic interferograms in the geometry of the interferograms in Figures 3b–e and then used these synthetic interferograms to calculate synthetic proxy east and vertical deformation fields. Finally, we calculated synthetic east and vertical deformation fields for the best fit model and subtracted these from the synthetic proxy east and vertical deformation fields. The results, shown in Figure S2, indicate that the likely error introduced into the proxy deformation fields is small. The maximum error introduced into the vertical field is about 0.8 mm and the RMS error is about 0.24 mm. The maximum error introduced into the east field is about 0.18 mm, and the RMS error is about 0.05 mm. These small errors justify our assumption that the north component of deformation was negligible in our proxy calculations.
 To assess the slip model in Figure 5, we show an alternative model and estimated uncertainties (at the 95% level) of a few important model parameters. The model shown in Figure 8a is an allowable alternative model that does not have a northward tail. There is no tail because we have set the covariance parameter between the orthogonal Weibull distributions to zero preventing communication between the two distributions. Note in the two models shown (Figures 5a and 8a) that the projected surface trace of the fault plane passes through the Spokane Falls. When we calculate a range of allowable surface traces, however, we find that the traces do not all pass through the falls (Figure 8b). Although each model has a strike-slip component (Figures 5a and 8a), at the 95% level, only up-dip (thrust) slip is required. With the data set we have and the assumed Earth model we run multiple perturbations to the model and with the F-test determine a few general features of the fault: (1) slip of greater than 10 mm is constrained to depths greater than 0.1 km, (2) slip of up to 2 mm could occur at the surface, and (3) a small amount of slip (~10 mm) on the fault deeper than ~3 km is possible, but our interferometric synthetic aperture radar (InSAR) data is not sensitive enough to resolve it. Also, we find that the dip of the plane of the Spokane Fault is between 19 and 43° to the northwest (Figure 5b).
 The estimated geodetic moment of the modeled slip is between two and five times larger than the cumulative moment calculated from the recorded earthquake catalog. The cumulative moment magnitude of the earthquake sequence is ~4.1, whereas the geodetic moment magnitude for the model is between 4.3 and 4.53. Because of the often sparse coverage of seismic stations during the earthquake sequence, the cumulative seismic moment is almost certainly higher than recorded. Part of the moment discrepancy could be cause by deficiencies in our assumed Earth model. It is possible that some of the slip progressed aseismically, but we are not sure what fraction (if any) of the moment is from aseismic slip.
 The evidence for a previously unknown thrust fault beneath Spokane is clear, but given all available information currently at hand, it is not possible to assess the magnitude of the seismic hazard associated with the fault. We have a good estimate of the area of the fault that slipped (Figures 5 and 8a) but we do not know the down-dip or along-strike extent of the fault. To estimate the maximum size of an earthquake that could result from slip on the Spokane Fault, we need to know the possible area that could slip. There is no trace of the thrust fault at the surface except for possibly at Spokane Falls (Figures 4 and 5). If the falls are part of the surface trace, the fault last ruptured to the surface before ~11 ka and the part of the trace exposed at the falls was exposed by the Spokane River. The Spokane Fault might truncate at a possible fault scarp between the yellow arrows in Figures 5a and 8a. This possible fault scarp, with about 6 m of elevation change across it, is visible in the LiDAR derived DEM, and it is possibly a splay off the Latah Fault. There is no movement on this possible fault splay during the 2001 earthquake sequence that is visible in the InSAR data, but the SW edge of the slipping patches in the two different models in Figures 5a and 8 truncate near and parallel the possible scarp. This possible splay coincides with a feature in the flood deposit aquifer beneath Spokane known as the Trinity Trough [Hsieh et al., 2007]. This feature is a narrow channel in the aquifer across which the water level lowers in a small distance from east to west. However, the possible fault scarp could be a surface feature related to the geology (Figure 8c) and erosion/deposition during the Missoula floods.
 Shallow (less than ~250 m) well data in the area of the fault show a laterally heterogeneous lithology down to about 250 m depth [Derkey et al., 2001]. The shallow geology is defined by: (1) alternating layers of Grande Ronde Basalt of the Columbia River Basalt Group and poorly indurated Miocene lacustrine sediments of the Latah Formation (Figure 8c) [Derkey et al., 2001], and (2) glacial flood channel deposits. The geology at the depths of the modeled slip beneath Spokane is unknown, but a 2009 earthquake swarm in the Hanford Site, about 200 km to the southwest of Spokane, might offer some clues [Wicks et al., 2011a]. At Hanford, the subsurface geology at the depths of modeled slip is well known, and clay-rich sediments appear to have played a central role in promoting shallow aseismic slip. Sediments of the Latah Formation are similar to the interbed sediments at Hanford; however, a possible role of Latah sediments in the slip process during the Spokane sequence requires a better knowledge of the deeper geology beneath Spokane.
 There are two measures of regional strain in the Spokane area from geodetic measurements and geologic mapping that can be compared to the modeled geometry and slip on the Spokane Fault. Regional strain rates in the Spokane area, derived from ~18 years of GPS measurements, are very low [less than ~1 nanostrain/yr, McCaffrey et al., 2013, and personal communication] with contraction directed between the north and northeast, different from the slip direction we find in Figure 5a. However, long-term GPS measurements are sparse in the Spokane area. Hooper and Conrey  reviewed mapped structures across the Columbia River Plateau and determined a strain pattern in which the direction of contraction is directed NNE, consistent with the direction of slip we show in Figure 5a. They determined that the regional stress field has probably not changed substantially in the last 17 million years. Hooper and Conrey  also suggested that structures in the Columbia Basin that deviate from the pattern are most likely to be found at the edges of the basin where structures in the underlying, older crust will dominate. The NE strike of the Spokane Fault might be related to structures in the older underlying crust.
 Because of the occurrence of recent shallow destructive earthquakes in urban settings, it is particularly important to further evaluate the seismic potential of the Spokane Fault. The Mw 6.2 Christchurch, New Zealand earthquake occurred a few km from the city center of Christchurch [Holden, 2011], a city of about 350,000 inhabitants, on 22 February 2011. The earthquake occurred on a previously unknown fault in a region of low seismicity [Beavan et al., 2011]. The slip on the fault was a mixture of reverse-slip and strike-slip, slip was concentrated in a small 8 by 8 km area of the fault, and as much as 1 m of slip occurred at 1 km depth [Beavan et al., 2011; Holden, 2011]. More than 180 deaths were associated with the earthquake and more than 100,000 buildings were damaged or destroyed by the earthquake [Kalkan, 2011]. Further assessing the seismic hazard associated with the Spokane Fault will require additional geophysical work to map the extent of the Spokane Fault and the subsurface geology. Geodetic studies using a denser network of GPS stations to determine the convergence rate across the Spokane Fault could also help estimate the amount of elastic strain accumulation on the fault. Studies to evaluate the possible fault splay off the Latah Fault also need to be conducted. An accurate assessment of the seismic hazard posed by the Spokane Fault also requires an assessment of the earthquake vulnerability of the buildings in Spokane as well as an assessment of possible side effects.