Submitted to JGR on February 12, 1998

Revised: June 16, 1998

Abstract Maps of surface displacement following the 1992, Landers, California earthquake, generated by interferometric processing of ERS-1 Synthetic Aperture Radar (SAR) images, reveal effects of various post-seismic deformation processes along the 1992 surface rupture. The large scale pattern of the post-seismic displacement field includes large lobes, mostly visible on the west side of the fault, comparable in shape with the lobes observed in the co-seismic displacement field. This pattern and the steep displacement gradient observed near the Emerson-Camprock fault cannot be simply explained by after-slip on deep sections of the 1992 rupture. Models show that horizontal slip occurring on a buried dislocation in a Poisson's material produces a characteristic quadripole pattern in the surface displacement field with several centimeters of vertical motion at distances of 10-20 km from the fault, yet this pattern is not observed in the post-seismic interferograms.

As previously proposed to explain local strain in the fault step-overs [Peltzer et al., 1996], we argue that poro-elastic rebound caused by pore fluid flow may also occur over greater distances from the fault, compensating the vertical ground shift produced by fault after-slip. Such a rebound is explained by the gradual change of the crustal rocks' Poisson's ratio value from undrained (co-seismic) to drained (post-seismic) conditions as pore pressure gradients produced by the earthquake dissipate. Using the Poisson's ratio values of 0.27 and 0.31 for the drained and undrained crustal rocks, respectively, elastic dislocation models show that the combined contributions of after-slip on deep sections of the fault and poro-elastic rebound can account for the range change observed in the SAR data and the horizontal displacement measured at GPS sites along a 60 km-long transect across the Emerson fault [Savage and Svarc, 1997].

Using a detailed surface slip distribution on the Homestead Valley, Kickapoo, and Johnson Valley faults, we modeled the poro-elastic rebound in the Homestead Valley pull-apart. A Poisson's ratio value of 0.35 for the undrained gouge rocks in the fault zone is required to account for the observed surface uplift in the 3.5 years following the earthquake. This large value implies a seismic velocity ratio Vp/Vs of 2.1, consistent with the observed low Vs values of fault-zone guided waves at shallow depth [Li et al., 1997]. The SAR data also reveal post-seismic creep along shallow patches of the Eureka Peak and Burnt Mountain faults with a characteristic decay time of 0.8 years. Co-seismic, dilatant hardening (locking process) followed by post-seismic, pore pressure controlled fault creep provide a plausible mechanism to account for the decay time of the observed slip rate along this section of the fault.

This study shows that spatially dense geodetic data, combined with a good accuracy in the vertical component, is essential to understand the complexity of faulting processes, especially in a strike-slip environment.

Introduction

The Mw=7.3, Landers earthquake of 28 June 1992 produced a 75 km long surface rupture with ~ 3 m average and up to 6.2 meters of right-lateral displacement (Figure 1) (Hart et al., 1992; Sieh et al., 1992). In the months and years following the earthquake, post-seismic surface displacement has been monitored by repeated surveys of GPS networks, trilateration arrays and creep-meters. Fault slip models based on the GPS data in the first year after the earthquake indicate that less than 10 cm of post-seismic displacement occurred along the northern and central sections of the fault, and up to 18 cm along the southern Johnson Valley and Eureka Peak faults [Shen et al., 1994]. This amount of after-slip accounts for a post-seismic strain release equivalent to 15% of the co-seismic moment, and is associated with a decay time of 34 days [Shen et al., 1994]. Data recorded by remote (65-100 km) stations of the Southern California Permanent GPS Geodetic array (PGGA) also show post-seismic displacement of up to 15% of the co-seismic signal [Bock et al., 1997; Wdonwinki et al., 1997]. Post-seismic displacements of the nearest sites indicate a decay time of 22±10 days, consistent with the previous estimate made by Shen et al. [1994], superimposed on a longer-term, interseismic trend [Wdonwinski et al., 1997]. Savage and Svarc [1997] interpreted surface displacement data from a linear GPS array across the Emerson fault in the 3.4 years after the earthquake as resulting from up to 1 m of right-lateral slip on the section of the fault below a depth of 10 km. The temporal behavior of their data is described by a short-term (84±23 days) exponential relaxation superimposed on an apparently linear trend. Surveys of small aperture trilateration arrays revealed minor horizontal displacement across the 1992 surface rupture in the 5 months following the earthquake. Virtually no displacement was recorded along the Emerson Camp-Rock fault, ~9 mm along the Johnson Valley fault and up to 40 mm along the Eureka Peak fault [Sylvester, 1993]. Creepmeters installed on the Eureka Peak fault after the earthquake have recorded up to 23 cm of surface slip in one year [Behr et al., 1994].

However, except the continuous GPS measurements at remote stations of the PGGA, all instruments listed above provide no or poor estimates of the vertical displacement of the ground. Furthermore, point positioning geodetic techniques are limited by the spatial range they are able to sample given the number and spacing of geodetic monuments they use. In this paper we take a new look at post-seismic processes using surface displacement maps of the Landers 1992 earthquake area generated by SAR interferometry (InSAR). The advantages and complementary character of SAR interferometry are to provide a quasi-continuous map view of the displacement field over broad areas, allowing us to detect and analyze surface displacement patterns of various spatial scales, and to have great sensitivity to vertical ground displacement. The largest post-seismic ground displacements and displacement gradients observed in the radar maps lie within 10 km of the 1992 surface break, at a scale that is both too short to be observed using the existing GPS arrays and too long to be observed in the near-field trilateration and creep-meter data. In a first section we briefly describe the approach and the data analysis strategy for the detection of slow deformation processes. In the following sections we discuss successively (1) after-slip and poro-elastic rebound along the Emerson-Camprock fault, (2) poro-elastic rebound in the Homestead Valley pull-apart structure, and (3) surface creep along the southern sections of the 1992 break.

Approach

The phase of each pixel of a SAR complex image provides a measure (modulo half the radar wavelength) of the path length between the antenna and the ground. The technique of SAR interferometry consists of combining two SAR complex images of a given area to form an interferogram [Gabriel et al., 1989]. The phase of each pixel in the interferogram is the difference between the phase of the corresponding pixels in each of the two original images. Its variation throughout the scene depicts the variation of the antenna-ground path length difference between the two images. If the images are acquired from slightly different locations, the interferometric phase is sensitive to the topography within the scene because of the small parallax between the two lines of sight [Zebker et al., 1994a; Rosen et al., 1995]. If the images are acquired at different times, the interferometric phase is also sensitive to any displacement of the ground along the radar line of sight that occurred during the time interval spanned by the image pair. The sensitivity of the phase to the topography increases with the distance separating the two antenna locations (baseline) at times of data acquisition. To generate a line of sight surface displacement map it is necessary to remove the topographic signal from the interferogram. This can be achieved by either (1) simulating the topographic phase using a digital elevation model and with knowledge of the geometry of the interferometric system (2-pass method, e.g., Massonnet et al. [1993]; Murakami et al. [1996]), or (2) by generating two interferograms out of three or four SAR images of the same area, and computing the phase difference of the two to eliminate the topographic phase signal common to both interferograms (3- or 4-pass method, e.g., Gabriel et al. [1989]; Zebker et al. [1994b]; Peltzer and Rosen [1995]). This method requires one phase field to be unwrapped and scaled to the same sensitivity to topography as the other phase field by the ratio of their baselines. In the absence of additional signal such as that produced by variations of the phase propagation delay in the wet troposphere [Massonnet and Feigl, 1995; Goldstein, 1996; Zebker et al., 1997], the remainder is a map of the component of the surface displacement field parallel to the radar line of sight.

To measure surface deformation related to the slow, post-seismic deformation processes after the 1992 Landers earthquake with ERS SAR data, we selected pairs of images spanning long time intervals in order to integrate surface deformation over sufficient time to be detectable by the radar. We also chose pairs having a small spatial baseline to minimize the sensitivity to topography and removed the topographic phase using either one of the methods mentioned above depending on orbit configuration. For image pairs with spatial baselines smaller than 20 m, we found the USGS 90 m-digital elevation model accuracy of 30 m acceptable to remove the topographic phase using the 2-pass approach. In fact, for the ERS C-band radar (wavelength = 56 mm), an elevation error of 30 m in the topographic map would produce a phase shift of 0.3 radians in a 20 m-baseline interferogram, corresponding to an error of 2 mm in the range displacement map.

Analysis of Near Field, Post-seismic Deformation

We analyzed SAR data acquired from the ERS tracks 399 and 127, which both cover the Landers earthquake area. The time intervals spanned by the data are shown in Figure 2. The images cover the Eastern Mojave Shear Zone, formed of several NW-SE, strike-slip faults, including the Emerson-Camprock and the Johnson Valley faults that ruptured during the 1992 earthquake. Geologic and geodetic data indicate that the shear zone accommodates ~15% of the Pacific-North America plate motion [Dokka and Travis, 1990; Savage et al., 1990; Sauber et al., 1994]. If approximately 10 mm/yr of right-lateral shear are distributed across the 100 km wide, NW-SE shear zone, such a rate would project into a line of sight displacement rate of 3.2 mm/yr. In three years, this change rate would imply 9.6 mm (1/3 of a phase cycle) of line of sight displacement, distributed across the 100 km wide SAR swath. Although it should be taken into account by proper modeling in the analysis of the far field post-seismic displacements, the small amplitude of the long-term signal and the width over which it is distributed suggest that such a correction is not required for the near and intermediate field displacement analysis presented here.

Figure 3 shows the interferogram covering the time interval between 27 September, 1992 and 23 January, 1996. Surface deformation patterns of various spatial scales are visible in the interferogram. (1) A large lobe, west of the Emerson-Camprock fault indicates that the ground moved away from the satellite in this area. Preliminary interpretations of this pattern suggested that it might be due to after-slip on deep sections of the fault [Massonnet et al., 1996; Peltzer et al., 1996a]. (2) Zones of large strain concentrated in the step-overs of the 1992 rupture have been explained by post-seismic rebound caused by the dissipation of pore fluid pressure gradients produced by the earthquake in the few years following the event [Peltzer et al., 1996b]. (3) Sharp cuts observed in the displacement maps along the Eureka Peak and Burnt Mountain faults indicate that these faults underwent post-seismic surface creep [Behr et al., 1994; Peltzer et al., 1996a]. In the following sub-sections we discuss in more detail these features of the post-seismic displacement field, and compare them with GPS data and predictions of elastic and poro-elastic models.

Right-lateral After-slip and/or Poro-elastic Rebound Along the Emerson-Camprock Fault

Right-lateral after-slip has been advocated to explain the surface displacement observations after the Landers, 1992 earthquake made with both campaign GPS measurements [Shen et al., 1994; Savage and Svarc, 1997] and continuous GPS measurements at remote PGGA stations [Bock et al., 1997; Wdonwinski et al., 1997]. Savage and Svarc [1997] proposed a model of after-slip based on repeated GPS measurements of a 40 km-long transect across the Emerson fault in the 3.4 years after the earthquake (Figure 3). The GPS array runs through the large, lobed pattern of deformation observed in the SAR data, and crosses the 1992 rupture a few kilometers south of the compressive jog of the Emerson-Camprock fault where post-seismic subsidence was observed during the same time period [Peltzer et al., 1996b]. Figure 4 shows range change profiles for 7 different time intervals (Profile 1 in Figure 3) running approximately parallel to the GPS array used by Savage and Svarc [1997]. The discrepancies between these profiles result from the sensitivity of SAR measurements to error sources such as variations in tropospheric conditions and surface conditions. For example, the large bump observed in profiles 2 and 3 between 10 and 20 km east of the fault is a topography residual that is probably caused by an anomalous phase propagation delay in the image of September 27, 1992, common to both profiles. Because water vapor density in the lower atmosphere decreases exponentially with increasing elevation [Gill, 1982], changes in atmospheric conditions between the epochs of data acquisitions produce a signal that, in some instances, correlates with topography [e.g., Delacourt et al., 1998]. The phase appears to be generally noisier in the 15 km-long section west of the fault than along other sections of the profiles. This region corresponds to the area where Zebker et al. [1994b] observed distributed surface cracks in the co-seismic interferogram. Nevertheless, several features appear to be stable between these profiles and can be related with confidence to actual ground displacements in the 4 years following the Landers earthquake. All profiles clearly show that the ground moved away from the satellite west of the fault and toward the satellite east of it, a pattern that is apparently consistent with right-lateral shear parallel to the fault [Massonnet et al., 1996, Peltzer et al., 1996a]. Overall, the displacement profiles are generally symmetric with respect to a point slightly east of the 1992 surface break, with a maximum amplitude in the four years following the earthquake of approximately 5% of the co-seismic signal amplitude. The profiles also show a steep displacement gradient in the 1-5 km section east of the 1992 surface break, with an amplitude that depends on both the starting date and the duration of the corresponding time interval. A decay time of 1.6±0.4 years can be derived by fitting the amplitude data to an exponentially decaying function. If these characteristics of the observed range change were due to pure, right-lateral shear parallel to the direction of the fault, the amplitude and steep gradient near the fault would require a source with its upper edge as shallow as 1.5 km and with approximately 0.3 m of right-lateral after-slip [Peltzer et al., 1996a].

However, we argue below that the steep gradient and part of the observed range change near the fault are more likely due to vertical (and, to a lesser extent, fault-perpendicular, horizontal) motion induced by poro-elastic rebound rather than due to right-lateral shear as previously inferred. Our argument is based on the general result of elastic modeling that any reasonable distribution of horizontal slip on a vertical dislocation buried in an elastic medium will produce four lobes of vertical displacement at the surface in the four quadrants determined by the fault and its perpendicular direction. In the particular case of strike-slip on a shallow fault patch, these lobes are located near the ends of the dislocation patch. Because the SAR is more sensitive to vertical ground shifts than it is to horizontal displacements, if after-slip were the only post-seismic process accounting for the observed displacement field, such lobes should be prominent in the interferograms. However, they are not observed in the SAR data.

Comparison between SAR data, GPS data, and after-slip model

To compare the displacements measured by GPS along the transect across the Emerson-Camprock fault (diamonds in Figure 3) with the SAR data (profile 1, Figure 3), we projected onto the radar line of sight both the horizontal and vertical components of the GPS-estimated vectors (Table 1 and Figure 6b in Savage and Svarc [1997]) after appropriate scaling for matching the observation time period of the SAR data (Figure 5a). Unfortunately this comparison appeared to be meaningless because of the quite large error bars of the vertical displacement rate estimates in the GPS data. Because the ERS SAR line of sight incidence angle at mid-swath is 23° off the vertical, the ±12 mm/yr error bars on the vertical rates obtained with the GPS lead to ±5 cm of error in range change over the 4 years of observation with the SAR, an error that is larger than the largest signal observed along the profile in the SAR data (Figure 5a). The line of sight projection of the horizontal components of GPS vectors however, seems to be in agreement with the SAR range change in the far field but a large discrepancy between the two data sets occurs within 10 km from the fault, where vertical motion is likely to have taken place.

Savage and Svarc [1997] interpreted the signal seen with seven surveys of the GPS array in the first 3.4 years after the earthquake as being caused by right-lateral after-slip on the downward projection of the 1992 rupture plane between depths of 10 and 30 km. The preferred after-slip solution (Model A) of Savage and Svarc [1997] inversion implies up to 1 m of slip on the Emerson section of the fault. A comparison between the line of sight surface displacement observed in the SAR data and that predicted by this model shows evident inconsistency (Figures 5b, 6a). In the area west of the Emerson-Camprock fault the SAR data show a range increase (movement away from the satellite) (Figures 3, 4) and the after-slip model predicts a range decrease (movement toward the satellite) (Figures 6a). Along the GPS transect line, the observed and modeled range displacement profiles have opposed polarities: the SAR data indicate a range increase west of the fault and range decrease east of it; the model predicts the opposite. Figure 5b also shows the independent contributions to the range change profile of the vertical and horizontal components of the displacement predicted by the model. It is clear that the line of sight displacement is dominated by the vertical surface motion, a contribution that is ignored in Savage and Svarc [1997] inversion because of the poor accuracy of the GPS data in the vertical.

Poro-elastic rebound

If after-slip has to be advocated to explain the right-lateral shear parallel to the fault observed in the GPS horizontal displacement vectors [Savage and Svarc, 1997], one has to explain why the contribution of the vertical motion predicted by the model is not depicted in the SAR data. A possible way of reconciling after-slip models and SAR observations is to advocate poro-elastic rebound due to pore fluid flow in the shallow crust. As in the fault step-overs [Peltzer et al., 1996b], the volume of rocks adjacent to the fault undergoes compression or stretching during the earthquake. This strain results in small volume changes and generates pore fluid pressure gradients in the shallow crust. As time passes after the earthquake, fluid flow allows pore pressure gradients to dissipate, and the volume of rock eventually reaches a drained condition. Theoretical models of elastic media containing holes [e.g., MacKenzie, 1950; Sato, 1952] and laboratory experiments on a wide variety of crustal rocks [e.g., Rice and Cleary, 1976] indicate that the Poisson's ratio of a porous media under undrained conditions (co-seismic) is larger than its value under drained conditions (post-seismic). Hence, the post-seismic relaxation of pore fluid pressure gradients induced by the co-seismic volume change of the country rock produces a rebound phenomenon. In the case of a strike-slip dislocation, vertical displacements are essentially proportional to the Poisson's ratio of the elastic crust and poro-elastic effects are expected to be large in the vertical component of the post-seismic displacement field.

The constitutive relation describing a fluid infiltrated, poro-elastic material is the same under undrained (no fluid flow) and drained (constant pore pressure) conditions, provided that the correct value of the Poisson's ratio (drained or undrained) is used. Conversely, the same shear modulus applies to both cases [Biot, 1956; Roeloffs, 1996]. We simply modeled the poro-elastic rebound subsequent to the Landers earthquake by computing the difference between two co-seismic dislocation models [Okada, 1985], using undrained and drained values of the Poisson's ratio. We used the fault geometry and co-seismic slip distribution of the joint inversion model of Wald and Heaton [1994] to predict the poro-elastic rebound (Figure 6b). Using values of n_d=0.27 and n_u=0.31 for the drained and undrained Poisson's ratio, respectively [Rice and Cleary, 1976], the combination of this rebound and the range change predicted by Savage and Svarc [1997] Model A results in a range change profile bearing the same polarity and comparable amplitude as the range change profile observed in the SAR data (Figures 6c, 7a). One has to note that the Poisson's ratio values used in the elastic model represent average values over the depth of approximately 15 km (the depth of the modeled dislocation) of parameters that vary with depth.

However, this combined model produces a steep gradient in the profile closer to the fault than it is observed in the SAR data. In the observed profiles, the gradient occurs between 1 and 5 km east of the 1992 surface break (Figures 4, 7a). A possible reason to explain the observed shifted location of the steep gradient is that the 1992 surface break has two branches, just north of the studied profile, suggesting that the actual shear may have occurred at depth slightly east of the surface break where the profile intersects with the fault. Because the poro-elastic rebound is directly derived from a model representing the Emerson-Camprock fault as a vertical, straight fault segment [Wald and Heaton, 1994], such complexities of the fault geometry and near-field displacements are not taken into account in the present poro-elastic model.

Another important feature of the poro-elastic rebound model is the significant horizontal displacement occurring near the fault along the Emerson-Camprock segment (Figure 6b). This horizontal rebound is similar to the vertical rebound discussed above and has important geodetic implications. Any strike-slip dislocation in an elastic half-space produces a surface displacement field with four characteristic lobes. In the case of the Landers earthquake, the volume of rock in the north-west quadrant, west of the Emerson-Camprock fault, experienced horizontal compression during the earthquake and expanded both vertically and horizontally, perpendicular to the fault direction. Conversely, the volume of rock in the north-east quadrant, east of the Emerson-Camprock fault, experienced horizontal extension during the earthquake and shrank both vertically (subsidence) and horizontally, perpendicular to the fault. The combined effects of the compression west of the fault and the extension east of it resulted in particular in an east north-east shift of the ground in the vicinity of the fault during the earthquake. Because this horizontal shift is proportional to the Poisson's ratio of the elastic crust, the poro-elastic process discussed above produces a post-seismic, horizontal shift of the ground near the Emerson fault in the opposite direction (Figure 6b). A symmetric pattern is observed along the southern branch of the 1992 dislocation, where poro-elastic rebound implies a fault-perpendicular, eastward movement of the surface near the Johnson Valley fault (Figure 6b).

It is interesting to note that the fault-perpendicular component of the displacement field, a feature that is not explained by the after-slip model alone, may account for the discrepancy between the after-slip model proposed by Savage and Svarc [1997] and the GPS data in the N50°E component. Notably, the poro-elastic rebound produces a negative notch in the N50°E displacement component profile that is qualitatively similar to the fault-perpendicular motion observed in the GPS data near the fault (Figure 7b). A quantitative agreement between the observed and modeled N50°E profiles would require a different co-seismic slip distribution on shallow fault patches than in the Wald and Heaton [1994] solution used here to compute the poro-elastic rebound. In addition, the poro-elastic model also predicts a small component of right-lateral shear parallel to the Emerson Camprock fault (Figure 7b). This suggests that, by having not taken into account poro-elastic effects, the Savage and Svarc [1997] solution may be an over-estimate of the amount of deep after-slip by approximately 30%.

Finally, it is important to note the failure of the combined model to predict the observed range change along the southern half of the 1992 break. As in the northern half of the modeled displacement field, the after-slip model produces two large lobes on both sides of the Johnson Valley fault (Figure 6a). The range displacement predicted by the poro-elastic rebound model (Figure 6b) compensates partially the lobed pattern west of the fault and only marginally that east of the fault (Figure 6c). The resulting displacement pattern differs markedly from the observed range change in this area (Figure 3). The SAR data show a region of range decrease west of the fault and a relatively flat signal east of it. In the absence of ancillary geodetic data in this area, we may tentatively argue that the amount of after-slip along the southern branch of the 1992 rupture may be greatly over-estimated in the Savage and Svarc [1997] solution.

Visco-elastic relaxation

An alternative explanation that would also reconcile the right-lateral shear observed in the GPS horizontal vectors with the SAR observations is that the shear could result from visco-elastic relaxation in the lower crust/upper mantle layers rather than from deep after-slip on the fault. Recent developments of 2-layer models have shown that vertical movements of the surface during post-seismic relaxation can be negligible compared to horizontal movements when gravity is taken into account [Yu et al., 1996; Pollitz, 1997]. Such models might then explain the fault-parallel shear observed in the GPS data without producing the vertical displacement quadripole pattern that is characteristic of strike-slip dislocations, yet not observed in the SAR data. However, visco-elastic relaxation is generally associated with time constants longer than the duration of our observations, and hence may have little influence during the first 4 years after a large earthquake. Furthermore, the steep gradient in range change observed near the 1992 rupture requires a relatively shallow source; this feature cannot be accounted for by visco-elastic relaxation in the lower crust, as it would lead to much broader and smoother surface displacement patterns [Yu et al., 1996; Pollitz, 1997]. A critical test of the visco-elastic hypothesis would be to characterize the time dependence of the long-term post-seismic displacement processes using geodetic observations covering a decade after the 1992 event.

Poro-elastic Rebound in the Homestead Valley Fault Step-over

We have proposed that the intense surface strains observed in the interferograms in the step-overs of the 1992 surface rupture were attributable to the poro-elastic response of the shallow crust to co-seismic strain [Peltzer et al., 1996]. In this section we develop a model based on a dislocation in an elastic half space to quantify such rebound in the Homestead Valley pull-apart structure and discuss implication of this model on the seismic velocity structure of the fault zone.

In the pull-apart basins between the Homestead Valley and Johnson Valley faults and between the Emerson and Homestead Valley faults, the observed ground displacement produced range decreases, consistent with surface uplift (Figure 3, 8). In the compressive jog between the Camprock and Emerson faults, the observed displacement produced range increase, consistent with ground subsidence (Figure 3, 8). Analysis of several interferograms covering various time intervals within the 4 years after the earthquake indicates that the decay time associated with this process is 0.75 ± 0.12 years (Figure 9), similar to the characteristic time describing earthquake-associated phenomena that are often explained by pore fluid flow in the upper crust [Nur and Booker, 1972; Booker, 1974; Anderson and Whitcomb, 1975; Li et al., 1987; Muir-Wood and King, 1993]. These observations led us to propose that the post-seismic rebound signal observed in fault step-overs was due to changes in mechanical properties of the shallow crustal rocks in the fault zone, as pore pressure gradients caused by the earthquake dissipated [Peltzer et al., 1996b].

Using the same approach as for the Emerson-Camprock fault, we modeled the poro-elastic rebound in the Homestead Valley pull-apart structure by computing the difference between two elastic dislocation models based on a co-seismic slip distribution and using different values of the Poisson's ratio, corresponding to the undrained (co-seismic) and drained (post-seismic) conditions. A more realistic fault geometry and slip distribution than that modeled in the Wald and Heaton [1994] global solution was necessary to take into account the local complexity of the displacement field in the pull-apart structure. The fault model includes 1-2 km-long, 4 km-deep, vertical fault patches aligned with the fault traces along the Homestead Valley, the Kickapoo and the Johnson Valley sections of the 1992 surface rupture [Sieh et al., 1993]. The horizontal, co-seismic slip for each patch was then interpolated between the data points mapped by Sowers et al. [1994] from latitude 34.28°N to 34.37°N, and by Sieh et al. [1993] and Hart et al. [1993] for the southern section of the Johnson Valley fault and the northern section of the Homestead Valley fault (Figure 10a). The "slip gap" section in the southern Homestead Valley fault where no surface offset was observed in the field [Sieh et al., 1993; Spotilla and Sieh, 1996] has been assigned a slip of 1 m based on the Hudnut et al. [1994] co-seismic slip model.

The areal distribution of predicted surface uplift agrees reasonably well with the SAR observations (Figure 3, 10a). As expected, the model predicts surface uplift in the valley between the overlapping sections of the 1992 rupture (Figure 10a). The model also predicts surface uplift near the bend in the Johnson Valley fault (Lat. 34.26°N, Figure 10a). Post-seismic motion there may result from co-seismic strain of the adjacent volume of rock due to curvature of the fault and the large along-strike variation of co-seismic slip in this section of the fault (Figure 10a). However, the model also predicts range decrease east of the Homestead Valley fault near the northern tip of the Kickapoo fault (Figure 10a), an area where no such displacement is observed in the SAR data (Figure 3). Improper modeling of the fault geometry and slip at depth along this complicated section of the 1992 rupture [e.g. Spotilla and Sieh, 1996] is the most likely explanation for the model's failure in this area.

Figure 10b shows the observed and predicted slant range components of displacement along the profile across the Homestead Valley pull-apart (Figure 3) using two sets of Poisson's ratios values. First, we have assumed values 0.27 and 0.3 for the drained and undrained Poisson's ratios of the shallow rocks, respectively [Rice and Cleary, 1976; Li et al., 1992; Peltzer et al., 1996b]. These values imply a post-seismic rebound accounting for less than 2 cm of range change in the pull-apart. A greater contrast between undrained and drained Poisson's ratio values would be needed to account for the observed displacement in the SAR data.

Recent seismological studies have revealed abnormally low S-wave velocities and correspondingly high Vp/Vs ratios within fault zones at shallow depth [e.g., Michelini and McEvilly, 1991; Li et al., 1997]. Reports on fault-zone guided waves from near-surface explosions in the San Andreas fault at Parkfield indicate values of up to 2.5 for the Vp/Vs ratio in the upper 3 km of the fault zone [Li et al., 1997; Y.G. Li, personal communication]. Such low values contrast with crustal velocity values estimated for the overall Mojave block [e.g., Li et al., 1992] and are commonly attributed to intense fracturing, brecciation, and fluid saturation of gouge rocks within the fault zones. Because the undrained Poisson's ratio is directly related to the ratio of seismic waves velocities Vp/Vs according to

a large Vp/Vs ratio indicates a large value of the undrained Poisson's ratio. In particular, Vp/Vs = 2.5 implies n_u = 0.4. Using the value of 0.35 for the undrained Poisson's ratio, within the range discussed by Li et al. [1997] for a brecciated fault zone, and a value of 0.27 for the drained Poisson's ratio [Rice and Cleary, 1976], the model leads to a poro-elastic induced range decrease of ~5 cm in the Homestead Valley pull-apart, in good agreement with the displacement observed in the SAR data (Figure 10b). This result supports our earlier interpretation that pore fluid flow provides a plausible mechanism to account for the observed surface movements in the fault step-overs after the Landers, 1992 earthquake [Peltzer et al., 1996b]. It is interesting to note that, with the large value of 0.35 for the undrained Poisson's ratio, the dislocation model predicts a co-seismic subsidence of ~20 cm in the Homestead Valley pull-apart structure, consistent with subsidence measured by leveling before and after the earthquake (A. Sylvester, personal communication).

Surface Creep Along the Burnt Mountain and Eureka Peak Faults

Sharp discontinuities in the displacement field are readily visible in the interferograms along the Burnt Mountain and the Eureka Peak faults' 1992 surface breaks (Figure 3, 11a). These features result from fault creep occurring in the years following the earthquake. Creepmeters installed along the Eureka Peak fault indicated that up to 23 cm of after-slip occurred on the Eureka Peak fault in the first year following the earthquake [Behr et al., 1994]. No instrument was installed on the Burnt Mountain fault and post-seismic creep had not previously been recognized along this fault to our knowledge. The phase profile across the creeping sections of the two faults indicates that after-slip is limited to shallow patches on the faults (Figure 11a). In fact, the distance from the faults over which the displacement vanishes does not exceed 5 km for the Eureka Peak fault and 2 km for the Burnt Mountain faults, corresponding to down-dip widths of the creeping patches of approximately 3 km and 1 km, respectively. The box-profile parallel to the Burnt Mountain fault shows that creep is nearly uniformly distributed along the 9-km long creeping section of that fault and it produced ~1-1.5 cm of line of sight displacement. If the observed offset is due to purely horizontal slip on the north-south striking fault, the observed change corresponds to 12-17 cm of right-lateral slip for the observation period spanned by the data. The box-profile parallel to the Eureka Peak fault indicates that slip gradually increases from north to south and abruptly stops near the southern end of the 1992 surface rupture (Figures 11a). Figure 11b shows the slip distribution along the creeping section of the Eureka Peak fault for the 27 September 1992 - 23 January 1996 and 10 January 1993 - 23 May 1995 time periods, assuming the observed range change corresponds to purely horizontal strike-slip on the N160°E striking fault [Behr et al., 1994; A. Sylvester, personal communication]. If the observed slip is distributed uniformly over a depth of ~ 3 km, the along-strike slip-distributions shown in Figure 11b correspond to geodetic moments of 9.3 x 10¹⁶ Nm and 5.4 x 10¹⁶ Nm for the two time periods, respectively. It is interesting to note that the cumulative seismic moments released by aftershocks within 5 km from the Eureka Peak fault during the same time periods over the entire seismogenic depth are only 9.0 x 10¹⁴ and 1.6 x 10¹⁴, respectively, more than two orders of magnitude lower than the geodetic moments above.

The proportionality of slip distributions for the two intervals shown in Figure 11b indicates a consistent temporal behavior of the slip rate distribution along strike. Using data covering four time intervals between September 1992 and March, 1997, we have adjusted an exponential function of the type

to the observed after-slip and derived a characteristic time of 0.8 years (0.55 years < t < 1.3 years) (Figure 12). As with the decay time associated with the post-seismic rebound in the fault step-overs, a decay time of 0.8 years also suggests a possible dependence on fluid flow in the shallow crust. Recent studies have emphasized the role of fluids in explaining the behavior of seismic and creeping faults [e.g., Rudnicki and Chen, 1988; Blanpied et al., 1992; Sleep and Blanpied, 1992; Lockner and Byerlee, 1994; Sleep and Blanpied, 1994]. Following these studies, we propose a scenario involving dilatant hardening followed by pore pressure controlled creep to explain the shallow after-slip observed along the Eureka Peak fault. Frictional slip is often accompanied by dilatancy, causing a local pore pressure decrease and an increase in the effective normal stress on the fault plane, thus inhibiting further slip [Rudnicki and Chen, 1988]. If such a mechanism (dilatant hardening) is responsible for the locking of the shallow part of the Eureka Peak fault during the 1992 earthquake, the following conditions should exist immediately after the earthquake: (1) a residual shear stress equivalent to a quantity of slip D is stored as elastic strain in the country rock over a distance L on each side of the fault (Figure 13), (2) the local pore pressure is below hydrostatic equilibrium. As pore fluid flow gradually restores hydrostatic pressure in the fault zone, the effective normal stress is reduced on the fault and creeping begins. As creep goes on, the Coulomb criterion

expresses the relation between the shear stress t and normal stress s_n on the fault plane, and the local pore pressure pf. Slip on the fault causes the shear traction to decrease as the post-seismic displacement d increases according to

where m is the elastic shear modulus of the adjacent rocks (Figure 13). If we assume that the normal stress remains constant during the time period of observation, equations (3) and (4) show that d and p_f are linearly related and should therefore have the same time dependence. If the temporal behavior of the pore pressure after the earthquake is described by an exponential increasing function, the post-seismic displacement d must increase similarly with time. By gradually reducing the effective normal stress on the fault, the pore pressure increase controls the creep rate on the fault.

Conclusion

We have analyzed intermediate and near-field, post-seismic surface displacements following the Landers, 1992 earthquake using ERS-1 SAR data covering various time intervals between September, 1992 and March, 1997. The interferometric maps revealed transient displacement patterns of various spatial scales that were either not observed or only partially captured by other geodetic techniques. In particular, the SAR maps depict clearly vertical displacements of the ground surface, a component of the displacement field that has been ignored in previous studies using other geodetic measurements. Analysis of the range change maps covering 4 years after the 1992 earthquake suggests that poro-elastic rebound, resulting from the change of the Poisson's ratio value of the strained rocks from undrained to drained conditions as pore fluid flow allows pore pressure gradients caused by the earthquake to dissipate, not only occurred in the fault step-overs of the 1992 rupture [Peltzer et al., 1996b] but also at distances of up to 15 km from the fault, an area where the country rock experienced large pore volume changes during the earthquake.

Our combined analysis of GPS and SAR data along the Emerson-Camprock fault array indicates that after-slip models alone cannot account for the observations because they would produce vertical displacement patterns that are not observed in the SAR data. We show that the added effects on the displacement field of poro-elastic rebound caused by pore fluid flow and after-slip on deep sections of the fault can account for the observed displacements near the Emerson-Camprock fault in both the radar line of sight and the horizontal directions. In particular, the analysis shows that, when combined with poro-elastic rebound, the after-slip model of Savage and Svarc [1997] over-estimates the amount of slip by approximately 30% on the Emerson-Camprock fault. The combined model, however, fails to explain the range change pattern observed along the southern half of the 1992 surface break.

Forward modeling of the poro-elastic rebound previously recognized in the Homestead Valley - Johnson Valley faults pull-apart structure [Peltzer et al., 1996b] requires a large undrained Poisson's ratio value (n_u = 0.35) to successfully account for the observed post-seismic uplift in the pull-apart. High values of n_u are independently suggested by the observed, abnormally low S-wave velocities in the upper 3-5 km of fault zones, where values as high as 2.5 have been estimated for the Vp/Vs ratio of fault-zone guided waves [Li et al., 1997].

Finally, SAR interferometric maps revealed that two sections of the 1992 rupture, the Eureka Peak fault and the Burnt Mountain fault, have undergone surface creep in the years following the earthquake. Clear cuts in the displacement field aligned with the fault traces allowed us to map the along-strike surface-slip distribution along these two faults. A simple model, involving dilatant hardening (fault locking process) followed by pore pressure controlled normal stress release, is proposed to explain the similarity between the observed decay time of fault creep and the relaxation times that describe percolation of fluids in the shallow crust. Along the Eureka Peak fault, the geodetic moment released by shallow after-slip exceeds the cumulative seismic moment released by aftershocks in the vicinity of the fault over the entire seismogenic depth during the same time periods by more than two orders of magnitude.

The spatial scale of the most intense post-seismic deformation features observed in the radar interferograms ranged from a hundred meters to a few kilometers. Such a scale range is typically too small to be observed by GPS [e.g., Shen et al., 1994] and too large to be detected by small aperture trilateration arrays across the fault [Sylvester, 1993]. Except along the Eureka Peak fault where creep has been monitored in 3 places during the year after the earthquake [Behr et al., 1994], most of the near-field, post-seismic deformation observed in the SAR data has been missed by the ground-based geodetic techniques. Furthermore, it has been possible to detect the various effects of post-seismic fluid flow because of the great sensitivity of SAR measurements to vertical displacements of the ground, a component that was poorly estimated, or not estimated at all, by other geodetic techniques used after the Landers earthquake.

These remarks emphasize the importance of measuring the three components of the surface displacement field continuously in space along active faults susceptible of generating earthquakes. If measurable precursory transient processes ever occur before some earthquakes, as it has been proposed in several instances [e.g., Allen et al., 1991; Kanamori and Anderson, 1975; Linde et al., 1988; Shifflett et al., 1995; Thurber, 1996], the associated ground movements may have remained undetected because of the inherent limitations of available geodetic techniques. In that sense, SAR interferometry has a great potential to efficiently complement point positioning geodetic techniques in the study of earthquakes and related processes.

Acknowledgments: We would like to thank E. Ivins for discussions on post-seismic processes, W. Thatcher for his comments on an early version of the manuscript, and C.Y. Wang, H. Zebker and D. Massonnet for their reviews. The ERS radar data were provided by the European Space Agency. The work presented in this paper has been performed at the Jet Propulsion Laboratory, California Institute of Technology under contract with NASA.

Figures:

Figure 1. Map of the 28 June 1992, Landers earthquake area. Solid lines is the 28 June, 1992 surface rupture [Sieh et al., 1993]. Shade depicts topography from USGS digital elevation model. White dots are aftershocks between 8/7/92 and 1/23/96 [Hauksson, 1993]. Box indicates area covered by SAR data shown in Figure 3.

Figure 2. Time intervals covered by SAR data used in this study. Numbers indicate perpendicular component of baselines (distance between orbits, perpendicular to radar line of sight) in meters at latitude 34°20'.

Figure 3. Interferometric map of the Landers area generated with SAR image covering the 27 September 1992 - 23 January 1996 time period. Colors overlaying backscatter radar image represent ground displacement in the direction of the satellite. Coordinates of vector pointing to ERS satellite at mid-swath in local east-north-up reference frame are 0.381, -0.088, 0.920 [European Space Agency, 1992]. Gray areas are zones where phase could not be unwrapped due to signal decorrelation between two SAR images. Black lines depict 1992 surface rupture. White lines indicate profiles shown in Figures 4, 5, 8, 10b, and 11a. Yellow diamonds show locations of GPS stations used by Savage and Svarc [1997]. Concentric fringes on left side of image result from M5.4, shallow aftershock of 4 December, 1992 [Feigl et al., 1995].

Figure 4. Line of sight surface displacement along 40 km-long, 800 m-wide profile across Emerson-Camprock fault (profile 1 in Figure 3) observed in 7 interferometric change maps. Numbers refer to corresponding time intervals in Figure 2.

Figure 5a. Comparison of range change along profile 1 between August 7, 1992 and September 24, 1995 observed in SAR data (solid line) with surface displacement estimated with GPS measurements along Emerson fault transect [Savage and Svarc, 1997]. Light gray dots and bars are projections of 3 components of GPS vectors and 1-sigma error bars along radar line of sight. Dark gray dots and bars are projection of GPS vectors horizontal components only. GPS data are scaled to match time interval covered by radar data the following way: horizontal components (Table 1, Savage and Svarc [1997]) are scaled by [f(s2)-f(s1)]/[f(t2)-f(t1)], where f is temporal function describing post-seismic GPS displacement [Savage and Svarc, 1997], t1 and t2 are epochs of first and last GPS survey of transect and s1 and s2 are epochs of SAR passes for data shown; GPS uplift rates (Figure 6 in Savage and Svarc [1997]) are scaled by (s2-s1).

Figure 5b. Line of sight displacement observed in SAR data (solid line) compared with range change predicted by Savage and Svarc [1997] model A (long dashes) along profile 1 (Figure 3). Dotted (resp. short dashed) line shows independent contribution of vertical (resp. horizontal) component of surface displacement predicted by model to range change displacement. Modeled curves are scaled as GPS data in Figure 5a.

Figure 6. Surface displacement predicted by (a) Savage and Svarc [1997] after-slip model, (b) poro-elastic model, and (c) the combination of (a) and (b). One color cycle represents 5.6 cm of surface displacement toward radar. Black arrows depict horizontal displacement vectors. White lines show fault geometry of Savage and Svarc [1997] after-slip model (a) and of Wald and Heaton's [1994] co-seismic model (b and c). White dots are locations of GPS stations used in Savage and Svarc [1997]. Models are scaled as in Figure 5a to represent surface displacement during time interval spanned by SAR data shown in Figure 5a.

Figure 7a. Observed (gray) and modeled (black) range displacement along profile 1 (Figure 3). Dashed line is Savage and Svarc [1997] after-slip model, dotted line is poro-elastic rebound model, and black solid line is the sum of the two.

 Figure 7b. Comparison of observed N50°E and N40°W displacement at GPS stations in the Landers array with that predicted by the Savage and Svarc [1997] after-slip model (dashed lines), the poro-elastic rebound (dotted lines), and the combination of the two (solid lines) plotted as a function of the N50°E distance from the rupture trace. Error bars on GPS data are 1-s. Modified from Savage and Svarc [1997].

Figure 8. Line of sight surface displacement along profiles 2, 3 and 4 shown in Figure 3 observed in interferograms spanning August 7, 1992 - September 24, 1995 (black), September 27, 1992 - January 23, 1996 (dark gray), and January 10, 1993 - May 25, 1995 (light gray) time intervals. Dots are displacement of individual image pixels within 400 m from profile line and solid curves indicate averaged values in 160 m-long bins along profiles strike. After Peltzer et al. [1996].

Figure 9. Post-seismic uplift in Homestead Valley pull-apart plotted as a function of time. Curve is function, adjusted to data (w₀=6.2 cm, t=0.75 years). Solid lines are uplift estimates using SAR data of Figure 5. Vertical bars indicate ±0.5 cm error on uplift estimates. Dashed lines are same as solid lines, shifted to match adjusted curve w(t).

Figure 10a. Modeled line of sight surface displacement resulting from poro-elastic rebound in the Homestead Valley pull-apart structure. Elastic model is based on co-seismic fault slip distribution shown on the left. Slip data are after Sowers et al. [1994], Sieh et al. [1993], Hart et al. [1993], and Hudnut et al. [1994].

Figure 10b. Observed and predicted line of sight displacement along profile 4 (Figure 3) across Homestead Valley pull-apart structure for the 7 August 1992 - 24 September 1995 time period. Light shade and dark shade modeled profiles correspond to sets of drained/undrained Poisson's ratios of 0.27/0.30 and 0.27/0.35, respectively.

Figure 11 a. Line of sight surface displacement along profiles across (5) and parallel to (6 and 7) Eureka Peak and Burnt Mountain faults (see location in Figure 3). Displacement values of individual image pixels in boxes along profiles parallel to faults fall into two groups depending on pixel location with respect to faults. Offset between two populations of dots indicates slip distribution along faults.

Figure 11 b. Post-seismic slip distribution observed along Eureka Peak fault for time intervals 27 September 92 - 23 January 1996 (upper curve) and 10 January 1993 - 23 May 1995 (lower curve). Right-lateral component of slip is derived from line of sight displacement assuming purely horizontal, strike-slip movement on fault.

Figure 12. Same as Figure 9 for after-slip data along Eureka Peak fault. Exponential curve fit to data indicates a relaxation time of 0.8 years.

Figure 13. 3-D sketch of creeping fault defining parameters D, d, t, s_n, and L discussed in text.

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