Inverse modeling of interbed storage parameters using land subsidence observations, Antelope Valley, California

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1 WATER RESOURCES RESEARCH, VOL. 39, NO. 2, 1031, doi: /2001wr001252, 2003 Inverse modeling of interbed storage parameters using land subsidence observations, Antelope Valley, California Jörn Hoffmann Department of Geophysics, Stanford University, Stanford, California, USA Devin L. Galloway U.S. Geological Survey, Sacramento, California, USA Howard A. Zebker Department of Geophysics, Stanford University, Stanford, California, USA Received 14 February 2002; revised 25 July 2002; accepted 25 July 2002; published 13 February [1] We use land-subsidence observations from repeatedly surveyed benchmarks and interferometric synthetic aperture radar (InSAR) in Antelope Valley, California, to estimate spatially varying compaction time constants, t, and inelastic specific skeletal storage coefficients, S kv *, in a previously calibrated regional groundwater flow and subsidence model. The observed subsidence patterns reflect both the spatial distribution of head declines and the spatially variable inelastic skeletal storage coefficient. Using the nonlinear parameter estimation program UCODE we estimate compaction time constants between 3.8 and 285 years. The S kv * values are estimated by linear estimation and range from 0 to almost We find that subsidence observations over long time periods are necessary to constrain estimates of the large compaction time constants in Antelope Valley. The InSAR data used in this study cover only a three-year period, limiting their usefulness in constraining these time constants. This problem will be alleviated as more SAR data become available in the future or where time constants are small. By incorporating the resulting parameter estimates in the previously calibrated regional model of groundwater flow and land subsidence we can significantly improve the agreement between simulated and observed land subsidence both in terms of magnitude and spatial extent. The sum of weighted squared subsidence residuals, a common measure of model fit, was reduced by 73% with respect to the original model. However, the ability of the model to adequately reproduce the subsidence observed over only a few years is impaired by the fact that the simulated hydraulic heads over small time periods are often not representative of the actual aquifer hydraulic heads. Errors in the simulated hydraulic aquifer heads constitute the primary limitation of the approach presented here. INDEX TERMS: 1829 Hydrology: Groundwater hydrology; 1894 Hydrology: Instruments and techniques; 8020 Structural Geology: Mechanics; KEYWORDS: subsidence, InSAR, storage, compaction, estimation Citation: Hoffmann, J., D. L. Galloway, and H. A. Zebker, Inverse modeling of interbed storage parameters using land subsidence observations, Antelope Valley, California, Water Resour. Res., 39(2), 1031, doi: /2001wr001252, Introduction [2] Land subsidence caused by the compaction of susceptible aquifer systems has been related to groundwater level declines accompanying the development of groundwater resources [e.g., Tolman and Poland, 1940; Riley, 1969; Poland et al., 1975; Bell and Price, 1991; Holzer, 1984, 1979; Ikehara and Phillips, 1994; Galloway et al., 1998a]. A large number of case studies have documented the global scale of the problem [e.g., Galloway et al., 1999; Johnson, 1991; Barends et al., 1995]. While no comprehensive assessment of the costs related to land subsidence Copyright 2003 by the American Geophysical Union /03/2001WR SBH 5-1 has been made, the annual costs from flooding and structural damage caused by land subsidence probably exceeds $125 million [National Research Council, 1991]. Galloway et al. [1999] argue convincingly that the actual cost would be much higher if all related effects were accounted for. The obstacles Galloway et al. [1999] cite in realistically assessing these costs include difficulties in mapping the affected areas and establishing cause-and-effect relations. The application of space-borne InSAR techniques to detect and monitor ongoing land subsidence [Galloway et al., 1998a; Amelung et al., 1999] has greatly enhanced our ability to observe and quantify land subsidence at a regional scale. Further, Hoffmann et al. [2001] have used time series InSAR observations and water levels measured in wells to estimate spatially varying aquifer system elastic storage

2 SBH 5-2 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE Figure 1. Antelope Valley, California. The area shown is the full extent of the MODFLOW grid. The blue frame encloses the area shown in Figures 3, 6, 7 and 8. The radar amplitude image covers the area within the model region that is covered by the interferograms. Major roads (red) and the playas are shown for reference. The thick black lines delineate groundwater subbasins conceptualized by Bloyd [1967]. The Lancaster subbasin is the largest and most important subbasin, both in terms of groundwater pumping and subsidence. The black circle indicates the location where the data shown in Figure 2 was obtained and the green symbols (A D) indicate locations for the curves in Figure 9. The yellow lines delineate the parameter zones used for inversion of the time constants. coefficients. Lu and Danskin [2001] and Bawden et al. [2001] have employed InSAR to help define the structure of groundwater basins. [3] A number of studies have used historical subsidence data obtained from borehole extensometers [e.g., Helm, 1975, 1976; Hanson and Benedict, 1994; Sneed and Galloway, 2000] and repeat surveys of benchmarks [e.g., Williamson et al., 1989; Hanson and Benedict, 1994; Leighton and Phillips, 2003; Nishikawa et al., 2001] in the calibration of groundwater flow and subsidence models. No work to date has attempted to use the available land subsidence data from InSAR to constrain models of regional groundwater flow and aquifer-system compaction. The main objective of this paper is to present an approach to use subsidence observations from benchmark leveling and InSAR in conjunction with aquifer heads simulated in a previously calibrated regional groundwater flow model to estimate spatially variable inelastic skeletal storage coefficients and compaction time constants of compressible interbeds. 2. Antelope Valley Aquifer System [4] Antelope Valley is a topographically closed, triangular basin located about 80 km northeast of Los Angeles, California (Figure 1). The sedimentary basin is bounded by the Tehachapi mountains in the northwest and the San Gabriel mountains in the southwest. The eastern boundary is less clearly defined by lower hills. The basin has been filled to depths of more than two kilometers with fluvial and lacustrine sediments forming the aquifer system that has provided much of the water supply for agriculture and the growing communities in Antelope Valley since the late 1800s. The valley floor has very little topographic relief and is dominated by two large playas, Rosamond Lake and Rogers Lake. [5] The first extensive investigation into the water resources of Antelope Valley was undertaken by Johnson [1911], who judged the water resources to be sufficient and accessible enough to merit agricultural development of much of the valley. Artesian conditions existed over large parts of the central Antelope Valley. Development of the groundwater resource during the early part of the twentieth century led to groundwater withdrawals vastly exceeding the natural recharge, causing large scale drawdowns of the hydraulic heads. Annual pumpage peaked at 400 hm 3 (300,000 acre-ft) in 1950 [Snyder, 1955]. During the second half of the twentieth century the declining groundwater levels made pumping increasingly uneconomical, resulting in a decline of both irrigated acreage and annual pumpage [Durbin, 1978]. The trend of decreasing pumping rates was discontinued in the 1990s, when groundwater use shifted from primarily agricultural to municipal-industrial to support the rapidly growing communities in the valley, primarily Lancaster and Palmdale [Galloway et al., 1998b]. [6] The groundwater basin has been conceptually subdivided into 12 subbasins [Bloyd, 1967]. The Lancaster subbasin (Figure 1) is the largest and most developed of these. All subsidence simulated in this study occurs within the Lancaster subbasin. The aquifer system in the Lancaster subbasin has been conceptualized using two or three aquifers. Earlier reports identified two aquifers, a principal and a deep aquifer [e.g., Bloyd, 1967; Durbin, 1978], which are

3 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE SBH 5-3 Figure 2. Simulated and observed hydraulic head and land subsidence at the location indicated by the black circle in Figure 1. The measured head values (crosses) are observations from a nearby well. The subsidence values shown on the right as circles are the kriged subsidence values, determined from repeat leveling surveys at benchmarks near that location. Subsidence values derived from the interferograms at that location are also shown. The water level observed in the well clearly shows a recovery of the water level between 1974 and During the same period the observed subsidence continues, albeit at decreasing rates. This can be explained by slow drainage of water from the compressible parts of the aquifer system. vertically separated by a laterally extensive lacustrine unit, where this is present. The lacustrine unit extends from Rogers Lake, where it is exposed at the land surface, down dip to the south west. Near the southern end of the valley the lacustrine unit is overlain by over 200 m of alluvium [Sneed and Galloway, 2000]. Referring to more recent data, Sneed and Galloway [2000], Nishikawa et al. [2001] and Leighton and Phillips [2003] use a conceptual model with three aquifers termed the upper, middle and lower aquifer. According to Durbin [1978], most of the water is produced from the principal (unconfined) aquifer, while Sneed and Galloway [2000], Nishikawa et al. [2001], and Leighton and Phillips [2003] consider the upper aquifer as generally unproductive and the middle (confined) aquifer to be the most productive. Low-permeability interbeds consisting of compressible, unconsolidated deposits are present throughout the aquifer system. 3. Delayed Compaction [7] In thick, compressible interbeds of low vertical hydraulic conductivity the pore fluid pressure does not instantaneously equilibrate with the head in the surrounding aquifer. Instead, a pore fluid pressure gradient develops across the interbed, driving the slow drainage of water from the interbeds into the aquifer. Because the compaction of an interbed is caused by the head change in the interbed itself, the compaction of the interbed and consequently the subsidence of the land surface lag the head declines measured in wells tapping the aquifer. This time delay leads to continuing subsidence despite static or recovering hydraulic heads. Most investigators to date have ignored these delays. However, this effect is important to simulate land subsidence in Antelope Valley. This is evident from commonly observed continuing subsidence in the presence of recovering water levels (Figure 2). [8] Delayed dissipation of overpressures from interbeds can be simulated with the subsidence package (SUB) [Hoffmann et al., 2003], developed by Leake [1990]. The package is based on the Terzaghi [1925] theory of onedimensional consolidation, which we refer to as compaction. The compaction of a single interbed can be written as (see also derivation in Appendix A) with 0 st ðþ¼s kv h p 2 k¼0 S kv ¼ S skv b 0 ; exp p2 4 t t k ð2k þ 1Þ 2 1 A; ð1þ where S kv is the inelastic skeletal storage coefficient, b 0 is the thickness of the interbed, S skv is the inelastic skeletal specific storage, h is a step change in hydraulic head, and t is time after the step change. t k is defined as t k ¼ ð2þ b 0 2Sskv 2 K v ð2k þ 1Þ 2 ; ð3þ where K v is the vertical hydraulic conductivity. The value t = t 0 is typically referred to as the time constant of the interbed. For a system with N individual interbeds with thicknesses b i and identical S skv and K v, Helm [1975] defined an equivalent thickness by an expression equivalent to vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 X N b equiv ¼ t b 2 i : ð4þ N The time constant of the system of N interbeds can then be computed by using b equiv in equation (3). To compute the subsidence history produced by all N interbeds combined, i¼1

4 SBH 5-4 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE the value computed using equation (1) can then be multiplied by the factor n ¼ 1 b equiv X N i¼1 b i ¼ S kv S skv b equiv ; where S kv * denotes the cumulative inelastic skeletal storage coefficient for all N interbeds. The second equality assumes that S skv is the same for all interbeds. Note that equation (5) does not require the knowledge of all b i, but only of their sum. The factor n can be used to parameterize the distribution of the total interbed storage into individual interbeds. By definition n is always greater than or equal to one. 4. Approach [9] We modified a previously calibrated regional groundwater flow and aquifer-system compaction model of Antelope Valley [Leighton and Phillips, 2003] to improve the agreement between the simulated aggregate compaction (land subsidence) and the observations at benchmarks and in InSAR-derived displacement maps. The Leighton and Phillips [2003] model uses MODFLOW [McDonald and Harbaugh, 1988] with the interbed storage package (IBS1) [Leake and Prudic, 1991] to simulate subsidence. It is based on a regular model grid with 1 by 1 mile grid cells, extending 60 miles (97 km) from west to east and 43 miles (69 km) from south to north. The model area is shown in Figure 1. Three model layers are used to simulate groundwater flow. All model boundaries are represented as no-flow boundaries. Small amounts of run-off entering the aquifer system along the foot of the mountains in the south, west and northwest represent the only natural recharge to the system. Evapotranspiration and groundwater pumping are the only simulated mechanisms for water to leave the system. The aquifer system is stressed by hydraulic head changes caused by pumping from the upper two model layers. Estimates of annual pumpage used in the model were based on measured values for municipal/industrial water use. For agricultural water use estimates were based on a water budget analysis for the types of crops that were cultivated and published in annual crop reports. Aquifersystem compaction caused by compaction of compressible sediments interbedded in the aquifer system is simulated in the upper two model layers. Our modifications to this original model were limited to the representation of interbed storage. We accounted for the time delays discussed in the previous section by using the SUB package to simulate compaction in the interbeds and resulting subsidence. The IBS1 package used by Leighton and Phillips [2003] assumes the instantaneous equilibration of the heads in the interbeds with the heads in the surrounding aquifers. This assumption is valid for interbeds with very short time constants, such as thin interbeds. The SUB package used in this study employs a numerical technique to compute the head profile across the interbeds at every MODFLOW iteration and the slow accrual of compaction as the residual pore fluid overpressures are dissipated Subsidence Data [10] We used two kinds of subsidence data in this study, leveling observations from repeated benchmark surveys and ð5þ InSAR-derived subsidence maps. Several benchmark leveling surveys have been conducted in Antelope Valley by different agencies and according to various standards. Ikehara and Phillips [1994] compiled a comprehensive data set of subsidence values from various surveys. Their table 9 contains subsidence magnitudes for over 250 benchmarks during the six time periods, , , , , , and Furthermore, they used the data acquired in these campaigns and an earlier subsidence map by Mankey [1963] to estimate land subsidence between In order to constrain the parameter estimates more easily spatially and to simplify the relative weighting of the benchmark and InSAR observations in the inversion, we interpolated these subsidence values at locations away from the benchmark locations using a kriging interpolator. We employed ordinary kriging [Deutsch and Journel, 1998] with an isotropic gaussian variogram model that was fit to the available subsidence data for each of the six time periods. [11] The second kind of subsidence data used to constrain the inversions in this study were InSAR-derived displacement maps. We formed 22 interferograms from 17 SAR scenes acquired by the European Remote-Sensing satellite 2 (ERS-2) between January 26, 1996 and May 1, 1999 (Figure 3). All interferograms were formed using precise orbit information [Scharroo and Visser, 1998] and corrected for topography using a digital elevation model (two-pass method [Massonnet et al., 1993]). Phase unwrapping was performed using a minimum-cost network-flow algorithm [Costantini, 1998]. Any residual linear phase slopes across the image were removed using tie-points outside the deforming areas. The resulting interferograms can be interpreted in terms of vertical surface displacements, if any horizontal displacements and atmospheric delay differences between the acquisition are assumed to be negligible. Horizontal displacements have been observed near pumping centers or near the boundaries of an aquifer system [Bawden et al., 2001; Watson et al., 2002] and neglecting these displacements may lead to an over- or underestimation of the vertical displacement where they are significant. However, the steep incidence angle of the ERS-2 satellite (23 ) ensures that the measurement sensitivity to horizontal displacements is at most about 42% of that to vertical displacements, depending on the direction of the horizontal displacement component with respect to the line of sight of the satellite. The resulting error from misinterpreting a horizontal deformation signal as vertical displacement is therefore both small and spatially localized and is unlikely to seriously affect our analysis. In order to assess the level of accuracy achieved in the InSAR-derived displacement maps, we compared them with compaction data from the two borehole extensometers operating in Antelope Valley (Figure 4). The agreement between the two measurements is very good. Only at a few times do the two measurements differ by more than 5 mm and often this can be traced back to an obvious artifact in the corresponding image. Seasonal displacements caused by seasonally fluctuating groundwater levels are clearly obvious in Figure 4. These seasonal observations can be used to estimate elastic storage coefficients [Sneed and Galloway, 2000; Hoffmann et al., 2001]. However, for this study we adopted the use of annual stress periods used in the Leighton and Phillips [2003] model, and

5 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE SBH 5-5 Figure 3. Interferometric data and derived composite subsidence maps. The bars indicate the times spanned by the 22 interferograms used in this study. The ERS-2 orbit numbers of the acquisitions used to form the interferograms are indicated. The interferograms underlain in color were used to create the three composite images shown on the right. These images were used in constraining the parameter inversion. The satellite line of sight (LOS) direction is indicated by the arrow in the subsidence maps. thus did not consider seasonal stresses. In general, seasonal head fluctuations only lead to recoverable, elastic displacements. Neglecting these fluctuations therefore does not significantly affect the estimation of parameters governing inelastic compaction. This assessment was verified by numerical simulations. We combined (stacked) 13 of the images to create 3 displacement maps approximately covering the years , and 1998 (Figure 3). This approach removes the seasonal signal from the observations (Figure 4) and decreases the level of (temporally noncoherent) atmospheric noise Parameterization [12] According to equations (1) and (2) the compaction in a system of interbeds due to a unit step decrease in hydraulic head depends on the time constant t, the inelastic skeletal specific storage, S skv, and the thickness of the compacting interbeds. The time constant only affects the Figure 4. Comparison of compaction measured by the Lancaster and Holly extensometers and the InSAR-derived subsidence. Each cross represents one SAR acquisition. The diamonds show the subsidence measured in the composite images used for the inversion. Water level observations from wells colocated with the extensometers are shown for reference (labeled with well name). The surface displacement measurements at the Lancaster site agree well within the expected accuracies, suggesting that little compaction is occurring below the anchoring depths of the extensometers (363 m). At the Holly site the seasonal displacement fluctuations are stronger in the InSAR-derived values, indicating elastic displacements below the anchoring depth of that extensometer (256 m).

6 SBH 5-6 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE timing of the compaction, but not the ultimate magnitude of the subsidence. The actual subsidence can be thought of as a convolution of the compaction in equation (1) and the drawdown history, assuming that the stress-dependence of K v and S skv is negligible. Although both K v and S skv have been shown to be stress-dependent for many geologic materials, the effect of neglecting this stress-dependence is small for stress-changes typical for water level drawdowns in deep aquifer systems [Leake and Prudic, 1991]. Because different sets of K v, S skv and b equiv result in the same time constant (equation (3)), and hence the same subsidence history, these parameters cannot be resolved independently using only land subsidence measurements. Similarly, the inelastic skeletal specific storage, S skv, of the interbeds cannot be separated from the interbed thickness using surface subsidence measurements alone. We therefore estimate the time constant, t, and the total inelastic skeletal storage coefficient, S kv *, which is the product of S skv and the cumulative interbed thickness. The estimated value for S kv * can be translated into an estimate of the cumulative interbed thickness if a value for S skv is either assumed or available from independent information (equation (5)). If the vertical hydraulic conductivity, K v, is also assumed or known, the equivalent thickness of the interbeds can be computed from the estimated time constants (using b equiv in equation (3)), yielding information on the distribution of the compressible sediments into individual interbeds (equation (4)) Zoning and Parameter Inversion [13] Because of the spatial heterogeneity of the skeletal storage coefficient in the aquifer system it is of interest to derive spatially varying estimates rather than average values for the entire aquifer system. This is particularly important in areas where relatively large subsidence gradients may point to heterogeneities. The InSAR-derived subsidence maps can afford the spatial detail necessary to make these spatially variable estimates. However, observations over periods on the order of the compaction time constants are necessary to reliably constrain these estimates. Unfortunately, SAR data suitable for interferometry have only been acquired for about 10 years, and the data used in this study only cover about a three year period. This proved to be too short to constrain the large time constants in Antelope Valley. Thus the inversion of the time constants has to rely primarily on benchmark data which cover a large part of the century, but do not afford the high spatial detail of the InSAR maps. To overcome this problem we defined six parameter zones (Figure 1). The definition of these zones was based on a zonation in the original Leighton and Phillips [2003] model, which we modified slightly based on spatial structure observed in the InSAR images and initial inversion results. These parameter zones were only used to estimate the time constants. The inelastic skeletal storage coefficient, S kv *, was allowed to vary at each of the 282 model cells within the 6 parameter zones. [14] For a given drawdown history and time constant, the subsidence is linearly related to the inelastic skeletal storage coefficient (equation (1)). While this is only approximately true in the presence of elastic compaction, it is a reasonable assumption for interbeds, where the ratio of S skv to S ske (the inelastic and elastic skeletal specific storage values) is generally large. This recognition enabled Figure 5. Flowchart for the estimation of S kv * and t. The S kv * values are estimated in a linear estimation (step 1) and the t values are estimated nonlinearly by UCODE. The aquifer heads, h it (x, y, t) are updated in every UCODE iteration (it) through a new MODFLOW simulation. the more efficient two-level inversion described in the following paragraph. A flow diagram of the estimation approach is shown in Figure 5. [15] After simulating the initial drawdown histories in a MODFLOW simulation using initial estimates for S kv * and t, we used the general-purpose inversion code UCODE [Poeter and Hill, 1998] to perform a nonlinear parameter estimation of the time constants in all 6 parameter zones (shaded box in Figure 5). UCODE employs a modified Gauss-Newton method to solve a general nonlinear regression problem iteratively. Thus it minimizes the weighted squared differences between the simulation and the constraining observational subsidence data (described above), repeatedly running the model in the process. Every model run consisted of the following steps (Figure 5). 1. Estimate the best S kv * value for all locations inside the parameter zones using linear least squares and an auxiliary program to compute one-dimensional subsidence for the local drawdown history. 2. Convert these S kv * estimates and the time constants to the input parameters required by the SUB package.

7 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE SBH 5-7 Figure 6. Final model-derived parameter values for compaction time constants and total inelastic skeletal storage coefficients of the interbeds. The time constants are not strictly constant within the parameter zones because the constraint that the total storage must amount to at least the equivalent thickness, b equiv (determined from the time constant using assumed values for S skv and K v ) was enforced. The estimated time constants can be expressed as equivalent thicknesses of the interbeds (equation (4)), which is indicated on the left of the first color bar (in m). Note that the color bar has two linear time intervals (right labels), above and below 100 years. Similarly, the S kv * values can be interpreted as the cumulative thickness of all interbeds (left labels on second color bar, in m). The black 20 m contours show the thickness and distribution of the lacustrine clay unit as mapped by Durbin [1978]. 3. Run the MODFLOW model to simulate groundwater flow and land subsidence for the entire model domain. [16] Several points need to be noted regarding these steps. Employing a linear inversion to estimate the inelastic skeletal storage coefficients (step 1) makes this step very efficient. However, it inherently decouples the interbed compaction from the flow system, which is affected by the water expelled from, or taken up by, the interbeds. The coupled system is solved in the MODFLOW simulation. To avoid biasing the solution we compared the drawdown history of the last iteration (used to compute the subsidence in step 1) to the final MODFLOW-computed drawdown history. As long as we observed significant differences, we repeated the estimation, using the new drawdown history. After 2 iterations the maximum difference in the drawdowns was less than 1 cm. [17] The parameters required for the SUB package are the vertical hydraulic conductivity of the interbeds, K v,the elastic and inelastic values of skeletal specific storage of the interbeds, S ske and S skv, the equivalent thickness of the system of interbeds, b equiv, and the factor n, defined in equation (5). We used K v = ft/d ( m/s), S ske = ft 1 ( m 1 ), and S skv = ft 1 ( m 1 ). These are the values for aquitards thicker than 18 ft, estimated by Sneed and Galloway [2000] from inverse modeling of extensometer compaction data and hydraulic head measurements in colocated wells at the Holly site, south of Rogers Lake. Using these values the estimated time constant and inelastic skeletal storage coefficient, S kv *, can be translated into an equivalent thickness of the interbeds and the factor n. At some locations, this led to a value of n that was smaller than 1, meaning that with the assumed values for S skv and K v,a single interbed of thickness b equiv would lead to an overestimation of the subsidence at that location. Instead of locally modifying S skv or K v we chose to decrease the time constant t at that location, which corresponds to a decrease in b equiv. We then repeated the linear estimation of S kv * until the factor n was at least 1. This prevents the time constants from being strictly constant within the parameter zones (Figure 6a). [18] It is important in both the UCODE inversion and the linear least squares inversion to assess the relative weights given to the different subsidence observations. Both the UCODE cost function and the linear inversion weight the data residuals according to the variance of the corresponding observation. It is difficult to assess measurement variances that adequately account for all error sources. Because of the high accuracy in precision leveling surveys the variance in the benchmark data is dominated by the interpolation and averaging over spatially varying values. We used the spatially varying kriging variance (from the kriging interpolation of the benchmark subsidence maps described above) at the center of each model cell as the measurement variance of the benchmark values at that location. This accounts for the increasing uncertainty distant from the actual benchmarks. We also enforced a minimum variance at the level of the theoretical variance within one model grid cell (using the variogram model) to avoid unreasonably small variances where a benchmark was very close to the cell center. [19] For the InSAR-derived subsidence values we assume a spatially constant variance of 50 mm 2. While this is a subjective choice it addresses an attempt to include two separate error sources. The first is the measurement variance mostly due to atmospheric disturbances. As discussed above the importance of random atmospheric errors in the images are reduced by the stacking of several individual interferograms to produce the composite images. We therefore assume a variance of 25 mm 2 for this error source. This corresponds to a standard deviation of 5 mm, which is approximately the level at which atmospheric artifacts in the InSAR-derived subsidence maps would become clearly visible. The agreement between the compaction trends measured by the extensometer instruments and the longterm subsidence values determined from the InSAR subsidence maps (Figure 4) supports this estimate of the achieved accuracy for the two extensometer locations. The

8 SBH 5-8 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE Table 1. Time Constants and UCODE-Determined Confidence Intervals for the Six Parameter Zones a Zone t, years 95% Confidence Interval [38.3, 43.5] [207.8, 550.3] [73.1, 81.6] [91.4, 97.7] [10.2, 30.7] [29.0, 49.3] a The estimates in zones 2 and 5 are very poorly constrained owing to the lack of sufficient historical subsidence information in these zones. second error source is related to the variance of the actual subsidence values within a model grid cell. Again, we assign a variance of 25 mm 2 to this error source. This value is approximately the mean experimental variance of the InSAR-derived subsidence within one grid cell. All three InSAR-derived displacement maps share a common reference year (1999) and common data acquisitions. This correlation is accounted for in the covariance matrix used in the linear inversions for S kv * at each location. 5. Results [20] The resulting parameter estimates for t and S kv * show significant spatial variability (Figure 6 and Table 1). The estimated compaction time constants range from 20.5 years in zone 5 to years in zone 2 (Table 1). Nonlinear 95% confidence intervals for the estimated time constants are also computed by UCODE (Table 1). The estimated compaction time constants are not strictly constant within each parameter zone and time constants as low as 3 years occur in the final results (Figure 6a), because the time constant was locally decreased to ensure physically reasonable results (see previous section). Using the S skv and K v estimates from Sneed and Galloway [2000], these values correspond (equation (3)) to an equivalent thickness between 3.8 m and 36.4 m (Figure 6a). [21] The resulting inelastic skeletal storage coefficients, S kv * range from zero at the boundaries of the estimation domain to almost 0.09 in zone 2 (Figure 6b). Again using S skv from Sneed and Galloway [2000] this translates (equation (5)) to cumulative interbed thicknesses up to 77 m (Figure 6b). Another result afforded by the linear inversions is a spatially varying estimation variance (not shown). This variance reflects the spatially variable variances of the observational data and the goodness of fit but does not include any uncertainties related to imprecise or unrepresentative drawdown histories used in the inversions. [22] The simulated subsidence agrees well with the kriged subsidence maps derived from the benchmark observations (Figure 7). Both the magnitude of the subsidence and the spatial pattern of the subsidence field are reproduced adequately. This agreement represents a significant improvement with respect to the original Leighton and Phillips [2003] model. The total UCODE cost function, computed as the sum of the weighted squared residuals between subsidence data and simulation decreased by about 73% with respect to that model. It is important to note that despite the spatial continuity suggested by the kriged subsidence maps (Figure 7), the data lack the spatial detail afforded by the InSAR data. Small-scale heterogeneities in the subsidence field are not observed at the sparsely distributed benchmarks or smoothed out in the interpolation, biasing the estimates derived from them. [23] The agreement of the simulated subsidence with the InSAR-derived subsidence maps (Figure 3) has also improved with respect to the original model. The cost function computed from the residuals for only these data decreased by 30%. The residuals (simulated subsidence minus InSAR maps) show that the simulation overestimates the subsidence during the time period covered by InSAR (Figure 8). This systematic disagreement can at least in part be explained by simulated drawdowns exceeding the actual drawdowns during this time period, as suggested by the comparison with kriged drawdowns derived from several wells in the area (Figure 8). As discussed in the following section, this should not be misinterpreted to mean that kriged heads are more representative of true aquifer heads in general though. The good agreement between simulated and observed subsidence is also shown for four point locations marked in Figure 1 (Figure 9). [24] Another important result of this study is the effect of our modifications to the interbed storage in the model on the simulated aquifer heads. We found that over the long term, these modifications did not significantly alter the flow field in the aquifers, although the simulated interbeds supplied large volumes of water to the wells. Over the simulated 84 years an average of 5% of the water pumped from wells originated from interbeds, with peak values of 10% in some years. Introducing delay in the interbed drainage caused larger initial drawdowns in the aquifer after the onset of pumping. These larger drawdowns increased the head gradient across the interbeds, causing them to drain at a higher rate. However, within a few years the head in our modified model returned to the level found in the original Leighton and Phillips [2003] model. 6. Discussion [25] The resulting parameter estimates reflect the spatial heterogeneity of interbed storage in the aquifer system, which causes the uneven distribution of subsidence when sufficiently large and widespread drawdowns occur. It is important to observe that the subsidence pattern is not solely due to the spatial distribution of drawdowns, but reflects the spatial variability of the skeletal storage coefficients of the interbeds. [26] The simulated land subsidence for the final parameter estimates agrees much more closely with the observations than the subsidence simulated by the original Leighton and Phillips [2003] model, as evidenced by the 73% decrease of the UCODE-computed cost function. Both the magnitude and spatial extent of the subsidence are reproduced much better in our modified model. However, while the simulated subsidence captures the timing and magnitude of the main subsidence features, some important differences still remain. These differences highlight the limitations of the approach presented here. Particularly over time periods of only a few years, the simulation does not adequately reproduce the observed subsidence (Figure 7, first row, and Figures 8 and 9). This is likely caused by short-term fluctuations in hydraulic head that are not reproduced in the flow simulation. In calibrating the regional flow model emphasis was placed on simulating regional-scale and long-

9 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE SBH 5-9 Figure 7. Comparison of simulated subsidence and drawdowns for the final parameter estimates with the kriged subsidence and drawdown values. All values are in meters. The location of the benchmarks used in the kriging are shown on the kriged maps. The location of the well observations used in kriging the drawdown maps are shown on the drawdown maps. The first six rows show time periods between major leveling surveys; the last row shows the estimate for the interval. The kriged map is based on estimated values for that time period by Ikehara and Phillips [1994]. The simulation reproduces the main observed subsidence features. However, the agreement is better over longer time periods, when the effect of short-term fluctuations in the aquifer hydraulic head, that are not reproduced in the simulation, are less important. term trends in groundwater flow at the expense of local or short-term changes. This can severely bias the short-term subsidence simulations and the parameter estimates resulting from our inversions. [27] Unreliable or unrepresentative simulated drawdowns over short and longer periods probably constitute the most important limitation of our inversion approach. While it was beyond the scope of this study to improve on the simulated heads in the flow-model, we recognize that the parameter estimates are strongly biased where the simulated heads in the aquifers are not representative of the head at the boundaries of the compacting interbeds. Replacing the simulated heads with measured heads in wells is not a reasonable alternative. The relatively few well observations would have to be interpolated in both space and time, introducing new, likely larger, inaccuracies. Notwithstanding these biases resulting at location where the true aquifer head history diverges from the simulated one meaningful parameter estimates can be determined where representative aquifer heads have been measured or can be simulated. [28] The fact that the simulated heads were not significantly altered by our modifications of the interbed storage

10 SBH 5-10 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE Figure 8. Simulated subsidence and data residuals (simulation minus InSAR-derived subsidence (Figure 3)) for final parameter estimates. All displacement values are in centimeters; aquifer drawdowns are in meters. The simulated and kriged drawdown maps are also shown. The agreement between the simulation and the observations is worse than for the longer time intervals in Figure 7. This is likely due to unreliable simulated drawdowns in the aquifer over short time periods. indicates that despite the hydrologic coupling of the compacting interbeds with the regional flow field, over the long term the flow field can be simulated without accounting for delayed interbeds. However, if the short-term response of hydraulic head to changes in pumping are to be simulated, the presence of delayed interbeds must be accounted for. The increased drawdown that occurs after the onset of pumping when delayed drainage of the interbeds is simulated has been observed previously by Leake [1990]. It occurs in response to the prescribed pumping rates and the slow release of water from storage in the interbeds. More water is initially required from storage in the aquifer to supply the pumping wells, resulting in larger drawdowns in the aquifer than would occur without delayed drainage of interbeds. These large drawdowns increase the hydraulic gradient between the centers of the interbeds and the aquifers, thereby accelerating the drainage. After some time the aquifer drawdowns approach those for the aquifer system with no-delay interbeds. [29] Another implication of the weak coupling between long-term drawdowns and interbed storage properties is that it may in many cases be possible to separate the flow simulations (in aquifers) from subsidence simulations (in compacting interbeds). This would simplify the simulations as the flow field need not be recomputed at every iteration of the inversion. [30] The distribution of final estimates for the inelastic skeletal storage coefficient are compared with the thickness of the lacustrine clay unit mapped by Durbin [1978] in Figure 6b (contour lines). The observed correlation between the two distributions suggests that compaction in the lacustrine clay, which confines the underlying part of the aquifer system, may be responsible for part of the observed subsidence. Compaction in this unit was not simulated by Leighton and Phillips [2003]. In their simulation of sediment compaction at the Holly site, Sneed and Galloway [2000] estimated that the confining unit was responsible for 31% of the historical land subsidence at that location. During this fraction increased to 42%. [31] Residual compaction and land subsidence in the presence of stabilizing or recovering hydraulic heads have been observed previously [e.g., Galloway et al., 1998a; Hoffmann et al., 2001; Sneed and Galloway, 2000]. The time constants determined in this study are bracketed by time constants less than 1 year estimated for thin (1.7 6 m) doubly-draining interbeds, 60 years for one thick (21 m) doubly-draining interbed, and 350 years for the 23 m-thick singly draining confining unit at the Holly site in Antelope Valley [Sneed and Galloway, 2000]. The large time constant determined for the confining unit indicates that compaction in this unit may have biased the time constants resulting from our inversion. Unrepresentative drawdown histories may also have biased the time constant estimates. [32] Deep compaction occurring below the anchored depth of an extensometer can cause significant differences in the subsidence observed by InSAR and the compaction measured by the extensometer [Hoffmann et al., 2001]. We also observe this phenomenon at the Holly site extensometer (Figure 4b), evidenced by the larger seasonal fluctuations of the surface displacement observed in the InSAR-derived displacements compared to the extensometer record. This disagrees with results by Sneed and Galloway [2000], where 95% of the total simulated compaction at the Holly site occurred within the depth interval spanned by the extensometer. The good agreement between the InSAR-derived subsidence and the extensometer measurements at the Lancaster extensometer (Figure 4a) indicates that most of the subsidence there is occurring at shallower depths. [33] InSAR-derived ground displacements data proved extremely valuable in mapping recent subsidence and defining parameter zones in the model. But owing to the limited

11 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE SBH 5-11 Figure 9. Subsidence and hydraulic head in the aquifer for the four locations indicated by green dots in Figure 1. The dashed lines on the right show the simulated subsidence and the solid lines correspond to observations in nearby wells. Note that observations from two separate wells around location B differ, demonstrating the difficulties in selecting a representative head for that location. In the subsidence plots (left), the dashed lines show the simulated results, the circles denote the benchmark-derived measurements, and the pluses denote InSAR-derived subsidence measurements. Surface displacement is measured as change in land surface elevation between two observations. As this only constitutes a relative measurement, a constant offset has been added to the observed subsidence values shown to better compare them with the simulations. The estimate is shown separately of the observations for shorter time intervals because of its much higher uncertainty. temporal coverage of the data used in this study (3 years), the use of InSAR observations to constrain the estimates of the time constants of compacting interbeds was limited. Because time constants of interbeds in Antelope Valley are on the order of decades the comparatively short time periods spanned by even the presently available SAR data (<10 years) are insufficient to reliably constrain the inversion. This is aggravated by the fact that the simulated head

12 SBH 5-12 HOFFMANN ET AL.: INVERSE MODELING OF INTERBED STORAGE histories used in the inversion far less adequately capture short-term head changes that are driving compaction over the time periods monitored by InSAR. As SAR data sets for longer time periods become available in the future this will become less limiting. Because InSAR data can be acquired much more easily and inexpensively than data from groundbased surveying, they can be provide both more frequent and spatially complete observations of surface displacements. For these reasons these data will likely replace benchmark data in many future studies. At present, InSAR is more helpful in parameter estimations where shorter time constants prevail. This has been successfully demonstrated for the case of predominantly elastic deformations in the Las Vegas Valley aquifer system [Hoffmann et al., 2001]. 7. Summary and Conclusions [34] Historical land subsidence observations at benchmarks and recent InSAR-derived subsidence observations were used to inverse model the compaction time constant and inelastic skeletal storage coefficient of compacting interbeds in a coupled regional groundwater flow and aquifer-system compaction model of Antelope Valley, California. An existing, calibrated model was modified to account for the effects of delayed drainage of thick interbeds. Spatially variable parameter estimates characterizing the heterogeneous interbed storage in the Antelope Valley aquifer system were determined. Time constants ranging from 10s to over 200 years were estimated for 6 parameter zones and inelastic skeletal storage coefficients up to 0.09 were estimated at 282 finite difference grid cells within the area of estimation. Using an independent estimate of the inelastic specific skeletal storage from a previous study an estimate of the thickness of compressible sediments in the aquifer system was derived. The large estimated time constants may be affected by compaction in a laterallyextensive, thick confining unit or biased by inaccurately simulated drawdown histories. Sufficient historical subsidence data were unavailable in some locations, causing the inversions for the interbed time constants there to be poorly constrained. [35] The resulting parameter estimates significantly improved the agreement between the model-simulated subsidence and the observations. The agreement was better over longer time periods. This can be explained by the fact that the simulated heads generally match the long-term head changes better than short-term head fluctuations which strongly influence subsidence over short periods. Interestingly, the long-term drawdown histories were relatively little affected by the modifications to account for the delayed drainage of water released from storage in the interbeds, suggesting that regional groundwater flow in Antelope Valley is relatively insensitive to groundwater contributed by the compaction of interbeds. [36] Due to the currently limited temporal coverage of SAR data the large time constants found in Antelope Valley cannot be determined from InSAR alone. As more SAR data become available in the future, the importance of InSAR in the study of aquifer-system properties, including compaction time constants, is likely to increase. However, the presently available data has proven useful in mapping and monitoring ongoing land subsidence, defining structural boundaries in aquifer systems, defining parameter zones within models, and estimating storage parameters where time constants are small. Appendix A: Derivation of Equation (1) [37] The dissipation of pore fluid pressure in an idealized interbed of infinite horizontal extent is described by the solution to a one-dimensional diffusion equation. This has been solved in the context of heat flow by Carslaw and Jaeger [1959]. Their solution can be expressed in terms of the analogous hydrologic parameters. Assuming an initially uniform head h 0 across the interbed and at the boundaries (i.e., in the surrounding aquifer), the head distribution across an interbed at time t after a step change in aquifer hydraulic head at the boundaries, h, can be written as ht; ð zþ ¼ h 0 þ h 4h X 1 p k¼0 with ð 1 ð2k þ 1 Þ k Þ e p2 4 t ð2k þ 1Þpz t k cos ; b 0 ða1þ b 0 2 t k ¼ 2 Dð2k þ 1Þ 2 : ða2þ Here z is the vertical distance from the center plane of the interbed, b 0 is the thickness of the interbed, and D = K v /S s is the vertical diffusivity. If the vertical hydraulic conductivity, K v and the specific storage S s are assumed to be constant (not stress-dependent), the compaction of the interbed can be computed by the integral Z b0=2 st ðþ¼ S s hðt; zþdz ða3þ b 0=2 or, using the symmetry about the center plane st ðþ¼2 Z b0=2 0 S s hðt; zþdz ða4þ Solving this integral gives equation (1), where we use the inelastic skeletal specific storage, S skv, instead of the specific storage S s, neglecting the storage components due to elastic deformations and the compressibility of the water. [38] Acknowledgments. This work was supported by NASA Headquarters under the Earth System Science Fellowship grant NGT References Amelung, F., D. L. Galloway, J. W. Bell, H. A. Zebker, and R. J. Laczniak, Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation, Geology, 27, , Barends, F. B. J., F. J. J. Brouwer, and F. H. Schröder (Eds.), Land Subsidence: By Fluid Withdrawal, by Solid Extraction: Theory and Modeling, Environmental Effects and Remedial Measures, Int.Assoc.of Hydrol. Sci., Gentbrugge, Belgium, Bawden, G. W., W. Thatcher, R. S. Stein, and K. Hudnut, Tectonic contraction across Los Angeles after removal of groundwater pumping effects, Nature, 412, , Bell, J. W., and J. G. Price, Subsidence in Las Vegas Valley, : Final project report, Open File Rep. 93-4, Nev. Bur. of Mines, Reno, 1991.