PUBLICATIONS. Water Resources Research. Estimating temporal changes in hydraulic head using InSAR data in the San Luis Valley, Colorado

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1 PUBLICATIONS Water Resources Research RESEARCH ARTICLE 1.12/213WR14938 Key Points: InSAR data can be used to interpolate/extrapolate hydraulic head data in time Kriging allowed us to calibrate directly at head monitoring well locations Seasonal deformation must be much larger than the InSAR uncertainty Correspondence to: R. Knight, Citation: Reeves, J. A., R. Knight, H. A. Zebker, P. K. Kitanidis, and W. A. Schre uder (214), Estimating temporal changes in hydraulic head using InSAR data in the San Luis Valley, Colorado, Water Resour. Res., 5, , doi:1.12/213wr Received 22 OCT 213 Accepted 2 MAY 214 Accepted article online 8 MAY 214 Published online 29 MAY 214 Estimating temporal changes in hydraulic head using InSAR data in the San Luis Valley, Colorado Jessica A. Reeves 1, Rosemary Knight 1, Howard A. Zebker 1, Peter K. Kitanidis 2, and Willem A. Schre uder 3 1 Department of Geophysics, Stanford University, Stanford, California, USA, 2 Department of Civil and Environmental Engineering, Stanford University, Stanford, California, USA, 3 Principia Mathematica, Lakewood, Colorado, USA Abstract The sustainability of the confined aquifer system in the San Luis Valley, Colorado is of utmost importance to the valley s agricultural economy. There is a dearth of hydraulic head measurements in the confined aquifer to which the current groundwater flow model can be calibrated. Here we investigate the extent to which spatially and temporally dense measurements of deformation from Interferometric Synthetic Aperture Radar (InSAR) data can be used to fill in spatial and temporal gaps in the head data set by calibrating the InSAR data with head at the monitoring well locations. We conduct this calibration at 11 wells where we expect sufficient deformation for reliable InSAR measurement, given the accepted level of uncertainty (1 cm). In the San Luis Valley, crop growth degrades the quality of the InSAR signal, which means that the high-quality deformation data may not be collocated with the wells. We use kriging to estimate the deformation directly at the well locations. We find that the calibration is valid at three well locations where the seasonal magnitude of the deformation is much larger than the uncertainty of the InSAR measurement. At these well locations, we predict head prior to and within the temporal sampling window of the head measurements. We find that 59% of the InSAR-predicted hydraulic head values agree with the measured values, within the uncertainty of the data. Given our success in extending the hydraulic head data temporally, the next step in our research is to use InSAR data to interpolate spatially between head measurements. 1. Introduction The San Luis Valley (SLV), located in southern Colorado, is home to a vibrant agricultural economy. Crop production from the approximately 6, acres of irrigated land depends greatly on the effective management of limited water resources. In 24, the Confined Aquifer Rules Decision mandated that hydraulic head levels within the confined aquifer system of the SLV be maintained within the range experienced in the years between 1978 and 2 ( Thus, groundwater management of the SLV requires knowledge of both seasonal changes in hydraulic head as well long-term trends during this time period. Colorado State agencies have developed the Rio Grande Decision Support System (RGDSS) and a MODFLOW finite difference groundwater flow model to predict hydraulic head in the confined aquifer system. At present, the model is limited by a dearth of confined aquifer hydraulic head calibration points, that is, there are very few hydraulic head measurements from monitoring wells [RGDSS, 25]. Here we investigate the extent to which spatially and temporally dense measurements of deformation from Interferometric Synthetic Aperture Radar (InSAR) can be used to interpolate and extrapolate temporal gaps in the hydraulic head data set. InSAR is a remote sensing method that maps relative ground surface deformation. Synthetic Aperture Radar (SAR) is a microwave imaging system, which uses a radar antenna mounted on an airborne or satellitebased platform to transmit and receive electromagnetic (EM) waves. In this study, we use InSAR data, acquired every 35 days, from the ERS-1 and ERS-2 satellites. The difference in the phase of the EM wave as measured between two acquisitions is related to deformation of Earth s surface. We quantify the quality of the deformation measurement using the mutual coherence of the two radar signals. Here we assume that InSAR-derived deformation of the ground surface is directly related to changes in the thickness of the confined aquifer due to recharge and withdrawal of groundwater. The change in aquifer thickness Db is the product of the specific elastic skeletal storage of the aquifer system (S ske ), the thickness of the producing aquifer unit (b * ), and the change in hydraulic head (Dh)[Riley, 1969]: REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4459

2 1.12/213WR14938 Db5S ske b Dh: (1) This relationship is valid when the aquifer system is deforming elastically/recoverably, which we have previously shown to be the case in the SLV [Reeves et al., 211]. In previous research, we used the InSAR deformation time series to show that the sediments in the SLV were not experiencing inelastic/permanent deformation. We assume that Db is equal to the deformation of the ground surface (Dd) as measured by InSAR and that Dh is the change in head that would be measured in a monitoring well at the location of the InSAR data. Because the horizontal hydraulic conductivity is generally much larger than the vertical hydraulic conductivity, we can assume that the thickness of the producing aquifer unit is equal to the length of the screened interval for the well. Equation (1) can be rewritten as follows: Db5S ke Dh; (2) where the parameter S ke, the skeletal elastic storage coefficient, is equal to the product of S ske and b *. There are three main ways in which InSAR data have been previously used to address groundwater problems: (1) to map the spatial extent of aquifer system deformation, (2) to estimate aquifer compressibility parameters, and (3) to calibrate groundwater flow models. Preliminary research on this topic began with the use of InSAR deformation measurements to map aquifer system deformation [Galloway et al., 1998; Amelung et al., 1999; Watson et al., 22] as well as monitor deformation over time [Schmidt and Burgmann, 23]. Other work has used InSAR deformation and hydraulic head to estimate aquifer compressibility parameters, which are some of the necessary inputs to any transient groundwater flow model [Hoffmann et al., 21, 23; Bell et al., 28; Wisely and Schmidt, 21]. A number of authors have used InSAR deformation data as added constraints for transient groundwater flow models, with the goal of improving the predictability of the hydraulic head [Hoffmann et al., 23; Calderhead et al., 211; Gonzalez and Fernandez, 211]. It is important to note that these studies focused on areas where the authors observed inelastic/permanent deformation due to long-term high-intensity groundwater extraction. The InSAR deformation data were used to calibrate the model-predicted permanent deformation. The InSAR calibrated models were able to reproduce leveling line surveys measuring surface deformation. However, because of the overall complexity of the models, the ability to predict hydraulic head was not improved. In previous work, we investigated the relationship between the deformation measured with InSAR and hydraulic head measured in wells, at three locations in the SLV [Reeves et al., 211]. We found that the two data sets showed similar seasonal variation. We estimated hydraulic head, using InSAR-measured deformation, measurements of S ske from aquifer tests and b * from driller s logs, and found that 67% of the estimates agreed with measured hydraulic head values within the uncertainty of the data. However, the uncertainty in the InSARderived head values was on the order of meters, which is too large to be useful for groundwater applications. In this paper, we describe a new approach to using InSAR data to increase the temporal, rather than spatial, density of head measurements in the confined aquifer system of the SLV. We use the relationship between deformation and hydraulic head at well locations as a means of calibration to determine S ke in equation (2). We then use the InSAR data set and our calibrated values of S ke to predict head, filling in temporal gaps in the hydraulic head data set. We were not able to extend this technique to estimate the hydraulic head data spatially between the wells in the SLV because the network of hydraulic head monitoring wells and the magnitude of the seasonal deformation in some areas were not large enough. In arid or urban areas, it has generally been found that high-quality InSAR deformation measurements, collocated with hydraulic head measurements at monitoring wells, permit a relatively straightforward calibration approach [Hoffmann et al., 21; Schmidt and Burgmann, 23; Wisely and Schmidt, 21]. This is not the case in the SLV, where crop growth, irrigation, land erosion, and harvesting cycles can all seriously degrade the InSAR data by perturbing the positions of individual radar scatterers. As a result, high-quality InSAR data are often not available at all well locations. In Reeves et al. [211], we selected the highest quality deformation measurement within 1 km of the monitoring well for comparison with hydraulic head. However, we found no robust relationship between the InSAR deformation and the hydraulic head at the well for the reasons listed above. In the current work, in contrast, we calibrate for the relationship directly at the monitoring well locations. REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 446

3 1.12/213WR14938 N 3 km Track 98 Frame 2853 Colorado New Mexico Figure 1. The RGDSS model boundary (purple), available SAR data from track 98 frame 2853 (white), and the state line between Colorado and New Mexico (red) (source of background image: Google Earth map with European Space Agency track and frame overlays). We begin by selecting well locations within the SAR scene, with hydraulic head measurements during the SAR acquisition time period, with temporally dense time series and that sample the confined aquifer system. We determine the well locations where the predicted deformation is larger than the uncertainty in the InSAR deformation measurements (section 2). We use geostatistical techniques to analyze all of the high-quality InSAR deformation data and estimate the deformation at the confined aquifer well locations (section 3). The hydraulic head data and the deformation data are then used to estimate S ke at the well locations (equation (1)) and predict hydraulic head prior to and within the temporal sampling window of the hydraulic head measurements (section 4). 2. Analysis of Hydraulic Head Data The main goals for this section are: (1) to describe the preparation of hydraulic head data for comparison with InSAR deformation data and (2) to list the well locations where the seasonal deformation is larger than the uncertainty in the InSAR deformation measurement (1 cm) [Hoffmann et al., 23]. We begin by determining which well locations are within the SAR scene and have sufficient temporal sampling (greater than two measurements) of hydraulic head during the SAR acquisition time period, (section 2.1). Because we limit our analysis to hydraulic head change in the confined aquifer system, we can review driller s logs from the SLV to determine which hydrogeologic layer each well is sampling (section 2.2). The confined aquifer monitoring wells in the SLV are sampled at variable rates ranging from once per month to once per year. We proceed by temporally interpolating the hydraulic head for comparison with the irregularly sampled InSAR data set (section 2.3). Finally, in order to compare the deformation and hydraulic head measurements, we need to ensure that the amount of deformation at each well location can be accurately measured using InSAR. We calculate the average seasonal hydraulic head change at each well location, estimate the sampling interval of the well from driller s logs, and use estimates of S ske from the literature to predict the amount of deformation we expect to see at the surface using equation (1) (section 2.4) [Batu, 1998]. If the seasonal deformation predicted is below the accepted uncertainty of the InSAR deformation data, then the hydraulic head data from these well locations are not used for further analysis. At the end of this section, we note which well locations have hydraulic head measurements sufficient to overcome uncertainty in the InSAR measurements of deformation (1 cm) Select Wells Within SAR Scene and With Sufficient Temporal Sampling We acquired SAR data for the SLV from two sources: the Western North American Interferometric Synthetic Aperture Radar Consortium (WInSAR) and the European Space Agency (ESA). In this study, we used data from the ERS-1 and ERS-2 satellites (track 98 frame 2853 in Figure 1), with 3 scenes acquired from 1992 to 2. This time period is particularly relevant for the Confined Aquifer Rules decision of 24. Each pixel of data contains a complex number describing the reflected amplitude and phase of the EM wave from a resolution cell on the ground, which is averaged to 5 m by 5 m to improve the signal-to-noise ratio. The quality of the InSAR data can be quantified by the coherence of the EM wave between the two acquisition times [Zebker and Villasenor, 1992]. Figure 1 shows the RGDSS model boundary (purple) and the spatial REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4461

4 1.12/213WR14938 Latitude x layer 2 layer 3 layer 4 layer Longitude Figure 2. A plan view showing the locations of 5 monitoring wells. Each well is colored based on the hydrogeologic layer being monitored [RGDSS, 25]. The blue box shows the outline of the SAR data from track 98 frame extent of the SAR data (white). The sampling interval of ERS SAR data is 35 days, there can be gaps when using historic data as the satellite did not necessarily acquire data each time it revisited the area over the SLV. Our data set has fairly irregular sampling of approximately one acquisition every 3 months. We processed the InSAR deformation data using Small Baseline Subset (SBAS) analysis, to obtain a time series of deformation for all high-quality pixels, as quantified by the mean coherence through time [Berardino et al., 22; Reeves et al., 211]. We used a modified version of SBAS analysis to process these data; we did not filter the deformation time series, as is often done, to ameliorate uncertainty due to atmospheric phase effects (Reeves et al., submitted to IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing). Therefore, the deformation time series that we use in this paper may contain uncertainties due to atmospheric phase effects that we have not quantified. We will revisit this topic in more detail in section 3 when we investigate the spatial structure of the deformation data set. The RGDSS database contains information about the hydraulic head monitoring well network with tables that are linked to the location, the screened interval, and the ground level elevation of each well. Of the 328 wells, sampling the confined aquifer system only 69 wells had hydraulic head measurements from 1992 to 2 (time span of InSAR data) and were located within the SAR scene. An initial assessment of the data found that 19 wells had only one hydraulic head measurement from 1992 to 2. Because of the low temporal sampling at these 19 well locations, the hydraulic head data were not used for further analysis. In the section to follow, we discuss the sampling interval of the hydraulic head data at the remaining 5 wells. 5 x Select Wells Sampling the Confined Aquifer System In this study, we are only interested in the wells that are sampling hydraulic head in the confined aquifer system. Therefore, we must review the driller s logs to determine which hydrogeologic layer the wells are sampling. The Rio Grande Decision Support System (RGDSS) hydrogeologic model layers are labeled from 1 to 5, where 1 is the unconfined aquifer and 2 is the aquitard layer and layers 3 5 are for the confined aquifer system. In the SLV, the aquitard layer is in some locations considered part of the confined aquifer system so in the analysis to follow we will investigate the locations of wells screened in layers 2 5. In the SLV information on well construction, the depth and extent of the screened interval was not collected for all wells. Most of the wells in the SLV were initially drilled to extract water at a high flow rate. It is common practice that on encountering a high flowing interval, drilling is stopped. Consequently, we assumed that the lowest unit in the well is the interval of the aquifer being monitored. In Figure 2, we show the locations of the 5 wells with screened intervals in layers 2 5; the marker color depicts within which model layer the wells are screened. A majority of the wells are monitoring the aquitard layer (pink filled circles), which is a clay-rich confining unit throughout most of the SLV. There are 33 wells grouped together in the northeast corner of the SAR scene in an area known as the Closed Basin (the northeast corner of the scene) that are all monitoring layer 2. These wells were put in place by the United States Bureau of Reclamation (USBR) to monitor a reclamation project that extracts water from the unconfined aquifer. Because there are no wells extracting water in layers 3 5 in this area, we assumed the hydraulic head in these wells not to be connected to the confined aquifer system. There are two other wells monitoring layer 2: RIO3 (central portion of the scene) and NA361112BAB (south eastern portion of the scene). A driller s log from RIO3 indicated that the well is indeed screened in a clay-rich aquitard unit. The hydrogeology of the southeastern area is REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4462

5 1.12/213WR measurements interpolated interpolated with moving window Jun92 Jun93 Jun94 Jun95 Jun96 Jun97 Jun98 Jun99 Jun Figure 3. Hydraulic head measured at well EW71C shown with blue markers. The arrows signify hydraulic head measurements that may be affected by instrument errors or pumping at nearby wells. such that NA361112BAB is likely monitoring a thick volcanic unit that also acts as an aquitard. Because these two wells are also not monitoring hydraulic head in the confined aquifer system, we excluded them also from our analysis. The remaining 15 wells were used in the analysis to follow Interpolate Hydraulic Head Data The confined aquifer monitoring wells in the SLV are sampled at variable rates ranging from once per month to once per year. In order to compare these hydraulic head data with our irregularly sampled InSAR deformation data set, we interpolated the hydraulic head data in time. We reviewed the 15 hydraulic head time series and found large changes in hydraulic head over very short time periods at 1 out of 15 well locations. For example, the measurements made at well EW71C are shown in Figure 3. The hydraulic head measurements shown here are made relative to the land surface. In red is the interpolation of the measurements. At certain times, the hydraulic head changes drastically: June 1993, November 1993, April 1996, December 1996, November 1997, January 1997, December 1999, February 2, and June 2 (shown with arrows in Figure 3). These abrupt changes in hydraulic head may be due to a nearby well being used for groundwater extraction. In practice, for purposes of calibrating or adjusting a regional flow model, hydrologists attempt to make measurements of the hydraulic head that are indicative of stable conditions and not when nearby extraction wells are being pumped. However, in the SLV, this is particularly difficult during the irrigation season, i.e., the summer months, when most wells are being used for groundwater extraction. The locations of the 15 wells used in our analysis are shown in Figure 4. At most extraction wells in the SLV hydraulic head levels rebound to stable conditions within 2 3 h from the cessation of pumping (E. Harmon, personal communication, 213). It is this stable hydraulic head that the RGDSS model aims to model, and hence, we must Figure 4. Location of the 15 wells where the predicted deformation was calculated. The purple outline is the Rio Grande Decision Support System (RGDSS) model boundary. REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4463

6 1.12/213WR14938 Table 1. Estimated Range for the Seasonal Deformation (Dds) at 15 Well Locations Well Dh s (m) Producing Lithology S ske Range (m 21 ) b * (m) Dd s Range (cm) ALA6 7.6 Sand ALA7 6.1 Sand ALA8 4.3 Sand ALA Clay CON1 3 Basaltic rock ALA4 3 Hard rock/clay CON2 1.8 Basaltic rock RIO2 1.8 Sand ALA2.6 Unknown RIO1 N/A Sand RG88 <.5 Unknown ALA15 <.5 Unknown ALA14 <.5 Unknown ALA1 N/A Sand RIO4 N/A Sand evaluate to what extent InSAR deformation data are capturing these stable conditions. It is also possible that these rapid fluctuations could be due to errors with the measurement device. These types of errors are fairly common and hence interpolation and smoothing is generally required for hydraulic head data. It is also important to note that there may be some delay in the deformation from the time at which the actual hydraulic head change is occurring [Riley, 1969]. In clay-rich sediments, this time lag can be on the order of months or years. Most producing wells in the SLV will be screened in high hydraulic conductivity materials; in our analysis, we have therefore assumed that there is not time lag between head change and deformation. To remove the effects of rapid perturbations in the hydraulic head time series (blue markers are highlighted with arrows in Figure 3), we applied a moving window average to the interpolated data. The window length of the temporal filter was varied from 1 to 2 days. We found that a window length of 9 days allowed for the mitigation of these rapid perturbations without removing the entirety of the seasonal groundwater signal (see the green dashed line in Figure 3). We applied the same moving window average to the hydraulic head data from all 15 wells Select Wells With Large Seasonal Deformation In this section, we further limit our study to include only wells where the seasonal magnitude of the deformation is larger than the uncertainty in the InSAR deformation measurement. If the estimated seasonal magnitude of the deformation is larger than the uncertainty in the deformation measurement (1 cm), then we assumed that InSAR is accurately able to measure the deformation [Hoffmann et al., 23]. We also assumed that the deformation occurring in the SLV is elastic, which allowed us to use equation (1) to predict the seasonal deformation at each well from estimates of S ske, b *, and Dh. If the product of S ske, b *, and Dh was greater than 1 cm, then we assumed that InSAR can provide an accurate estimate of deformation at that well location. Our estimates of S ske, b *, and Dh at the 15 well locations are given in Table 1. Here we outline how we estimated these variables given the available data in the SLV. We found a range of values for S ske in the literature, which allowed us to use equation (1) to estimate a range for the magnitude of the seasonal deformation (Dd s ). The second column of Table 1 contains an aggregate variable seasonal hydraulic head change (Dh s ), which we define as the average seasonal peak-to-trough change in hydraulic head. We used this aggregate variable so that we could compare a single seasonal deformation estimate to the accepted uncertainty of the InSAR measurement. We assigned the seasonal hydraulic head change to negligible (N/A) at well locations where we did not observe any seasonal hydraulic head change (ALA1, RIO4, and RIO1). These wells were generally located around the edges of the SLV where not a lot of groundwater extraction is occurring. From Table 1, we can see that the seasonal hydraulic head changes vary from 7.6 to.5 m, with wells that have the largest seasonal hydraulic head change located in the center of the SLV. REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4464

7 1.12/213WR14938 Table 2. Range of S ske for Different Lithologies [Batu, 1998] Lithology S ske (m 21 ) Clay Sand Rock, fissured/jointed We obtained the lithology of the screened interval from driller s logs, and based on this lithology, we assigned upper and lower bounds for S ske from the literature (see Table 2). At well locations where driller s logs were not available, we show the lithology as unknown and used S ske values for sand; it is most likely that an extraction well would be screened in a sandy interval of the aquifer system. We used the S ske value for Rock fissured/jointed from Table 2 at wells CON1 and CON2 where basaltic rock is present. The screened interval of ALA4 had two lithologies present, so we used Rock fissured/jointed for the lower bound of the Dd s range and clay for the upper bound. Most of the well locations have some estimate of the screened interval length, which we used as a proxy for the thickness of the producing aquifer unit (b * ), from the RGDSS database and driller s logs. However, it is important to note that the values for b * are very difficult to estimate accurately. Driller s logs are difficult to interpret in terms of the depth of the sediments and that is when they are recorded at all. Therefore, it is possible that the screened interval length for a number of these wells is far different from the estimates we provide in Table 1 and that our estimates for the range of Dd s are not accurate as well. The sixth column of Table 1 shows the product of S ske, b *, and Dh s, as the range for the magnitude of the seasonal deformation (Dd s ). Based on the calculated range of Dd s in Table 1, we concluded that the deformation at eleven wells, ALA6, ALA7, ALA8, ALA13, CON1, ALA4, CON2, RIO2, RG88, ALA15, and ALA14, should be above our assumed accuracy limit for InSAR. We found that three well locations, ALA1, RIO4, and RIO1, had essentially zero seasonal hydraulic head change; therefore, no seasonal deformation is predicted at those locations. We concluded that the predicted deformation at well ALA2 was too small to be accurately measured by InSAR. We proceed with the remainder of our analysis using the 11 wells where the deformation was predicted to be large enough to be accurately measured by InSAR. 3. Spatial Analysis of InSAR Deformation Data Our approach requires a calibration using hydraulic head and InSAR deformation to determine S ke at 11 well locations. However, because we were working an agricultural area of the SLV, it was necessary to consider that the deformation measurements were not collocated with the hydraulic head data. In Reeves et al. [211], we found that using the deformation from the highest quality pixel within 1 km of the well location did not provide a robust relationship between deformation and hydraulic head. Therefore, the goal of this section is to determine an estimate of the deformation directly at the 11 confined aquifer well locations. We begin by quantifying the spatial variability of the InSAR deformation data set by using a geostatistical function called the semivariogram. We provide the theoretical background for variogram analysis from a geostatistical perspective (section 3.1). We then use the Stanford Geostatistical Modeling Software (SGeMS) to calculate the semivariograms for the deformation data at each time step (section 3.2). Finally, we estimate the deformation directly at the well locations using the calculated deformation semivariogram and a geostatistical technique called simple kriging (section 3.3) Geostatistical Definition of the Variogram The main geostatistical function that we use in this section is known as the semivariogram, c(v) (also referred to as a structure function). The semivariogram function shows the average dissimilarity of a random variable at specific lag distances (v). For irregularly spaced or sparse data, a lag tolerance is included. If the direction of the lag vector is not considered, then the semivariogram is referred to as omnidirectional. However, if anisotropy is expected in the spatial data, then directional semivariograms should be calculated. If the directional variogram is calculated for all angles from to 18, then a 2-D variogram map is obtained. The structure in a semivariogram is commonly fit using different analytical functions; this facilitates comparison and provides a model that can then be used for simple kriging. In section 3.2, we either used an exponential model or a linear model to characterize the different structures observed, as they tended to fit the data the best. The semivariance associated with an exponential plus nugget-effect model is given as follows: REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4465

8 1.12/213WR cðvþ5 c3ð12e ð2v=aþ Þ 1n; (3) semivariogram exponential model Figure 5. Experimental variogram for lag spacing 5 m, and tolerance 5 m. The red line shows an exponential variogram model fit of the semivariogram structure for lags distances up to 188 m. where the semivariogram is characterized by the sill (c), the range (a), and the nugget (n). The lag distance where the semivariogram model reaches 95% of the sill value is the range. Because of random observation error and variability below the sampling scale of the measurement (microvariability), it is often found that the semivariogram value closest to the origin is nonzero, which is modeled through the nugget term. If the semivariogram does not stabilize around a sill value, but continues to increase with lag distance, a linear model can be used to fit the spatial structure. The linear model only requires two parameters for characterization: a nugget and a slope. Because the semivariogram value continues to increase with lag distance, the random variable in question is considered to be nonstationary, i.e., the correlation length depends on the location of the random variable Analysis of Dd In this section, we calculate omnidirectional semivariograms of the deformation data at each InSAR acquisition time step using the Stanford Geostatistical Modeling Software (SGeMS) package. First we discuss at a single semivariogram and investigate the nested spatial structure with respect to uncertainty due to atmospheric phase effects in the final deformation time series. We then discuss whether it is valid to describe the spatial structure with an omnidirectional variogram by calculating a 2-D variogram map. Finally, we calculate the semivariogram for each InSAR acquisition time step. The InSAR deformation data used here were originally processed for Reeves et al. [211] using SBAS analysis. The end product of SBAS analysis is a deformation time series for each high-quality pixel. However, there are over 1 million high-quality pixels in the SLV and SGEMS cannot compute the semivariogram for more than 5, measurements. To overcome this issue, we used 5, randomly selected deformation measurements to compute the semivariogram for 1 different subsets of the data. We found that the overall shape of the semivariogram was the same for each subset. The omnidirectional semivariogram for the deformation from 19 June 2 to 24 July 2 is shown in Figure 5. We selected this semivariogram for our initial discussion because in the raw interferograms we observed a large deformation signal, which is indicative to a high signal-to-noise ratio for these acquisitions [Reeves et al., 211]. The lag separation selected was 5 m, as that is the approximate distance between pixel centers. We included a lag tolerance of 5 m to accommodate variability in the distance between pixel centers. The semivariogram in Figure 5 shows a nested structure: for shorter lag distances, we observe an exponential structure and for longer lag distances a linear structure. We are particularly interested in how the deformation varies close to the well locations, so we will focus our discussion on the exponential structure. The transition from exponential to linear structure occurs near 188 m. We used a least squares fit to the semivariogram data for lag distances up to 188 m to fit an exponential model. The number of observations per lag distance was used to weight the least squares fit, so that a lag distance with more observations affected the fit more than lag distances with less observations. We found that the best fit value for a m. These results reveal that the deformation is varying on a scales less that 234 m, and hence we should be estimating the deformation directly at the monitoring well locations and not comparing deformation measurements to hydraulic head measurements in monitoring wells 1 km away, as we did in Reeves et al. [211]. The next question we asked was, is the structure of the deformation semivariogram in Figure 5 entirely due to deformation or is some of the structure due to uncertainty due to atmospheric phase effects? The modified version of SBAS used to process the InSAR data does not filter the final deformation time series to REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4466

9 1.12/213WR14938 v north direction (km) v east direction (km) Figure 6. Two-dimensional variogram map of the deformation data from the time step 19 June 2 24 July γ(v) ameliorate this component of the uncertainty. Because we observed a nested structure in the semivariogram in Figure 5, it may be possible that one of the structures is caused by the deformation and the other is caused by the uncertainty due to atmospheric phase effects. However, we could not differentiate these two structures; therefore, we assumed that the structure due to atmospheric phase effects is negligible [Emardson et al., 23]. We also needed to determine whether it is accurate to describe the spatial structure of the deformation with an omnidirectional variogram. We calculated a 2-D variogram map for the time step: 19 June 2 24 July 2 (see Figure 6). We plotted the 2-D variogram map for each angle of the lag vector, where the color scale represents the semivariance in a particular lag direction. The variability in the color for a set lag distance across all angles can highlight important anisotropic spatial structures. Figure 6 shows that lag distances up to approximately 8 km show isotropic conditions. We performed this analysis for each deformation time step and found that the deformation was also isotropic on shorter length scales. This means that for the distances we are interested in, i.e., close to the well locations, the omnidirectional semivariogram will suffice for further analysis. As a final step, we calculated the semivariogram for the deformation data at each InSAR acquisition time. These semivariograms are needed for the kriging algorithm to estimate the deformation directly at the monitoring well locations. We used a least squares fit, as described earlier, with both exponential and linear models in this analysis. The semivariograms for the first eight acquisition times are shown in Figure 7. The structure of these semivariograms changes significantly from one acquisition time to the next. We fit four of the semivariograms with linear models, and 26 with exponential models. The exponential models show a lot of variability in the best fit parameters. The mean range for the variograms fit with an exponential model was 3.9 km, with a standard deviation of 2.5 km. This implies that the deformation estimates vary spatially on a number of different length scales, which is dependent on the surface deformation and the atmospheric conditions at that acquisition time. The mean nugget equals.253, with a standard deviation of.221. This indicates that a lot of variability exists in either: (a) the microvariability of the data, or (b) the uncertainty from one acquisition time to the next. Previously, we observed that the uncertainty in the deformation data changes from one acquisition time to the next due to changes in coherence; hence, the most likely source for the variability in the size of the nugget [Reeves et al., 214]. Now that we have semivariograms that describe the deformation at each InSAR acquisition time we can proceed with estimating the deformation directly at the monitoring well locations Kriged Estimates of Deformation at Well Locations We used SGeMS for kriging the deformation data, a form of generalized linear regression that provides an optimal spatial estimator by minimizing the mean-squared-error between measured data and the estimated data at a given location [Deutsch and Journel, 1992]. models, which characterize the spatial structure of the system, provide the optimal spatial estimator, which can then be used to estimate the variable in question at locations where it is unknown, i.e., at the well locations. In the Earth sciences, many variants of kriging have been implemented; however, the most basic form is known as simple kriging. We have chosen to use simple kriging over other more complicated forms because the technique implies that a valid mean for the data set can be calculated. The InSAR deformation data set is spatially dense enough that this is a valid assumption. Simple kriging, as implemented in SGeMS, does not incorporate the reliability of the data into the kriging algorithm. However, the kriged estimate of the deformation at the well location should be influenced by the reliability of the deformation measurements at surrounding pixels. We used the uncertainty of the deformation measurements, as calculated in Reeves et al. [214], as a measure of the reliability. We then REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4467

10 1.12/213WR range = , sill =.1982, nugget = range = , sill =.76733, nugget = July semivariogram linear model August September 1992 April 1993 September range = , sill =.34829, nugget = range = , sill =.42239, nugget = range = , sill =.15359, nugget = January 1993 July 1993 September range = , sill =.23698, nugget = range = , sill =.1332, nugget = range = , sill =.6233, nugget = Figure 7. s and models for deformation acquisition times July 1992 to September REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4468

11 1.12/213WR14938 A) Deformation (m) R =.875 S ke=.444 +/ Estimated Measured Interpolated Well ALA Jun98 Jan99 Jun99 Jan Jun Jan1 B) Deformation (m) R =.846 S ke=.795 +/ Well ALA7 8 Jun98 Jan99 Jun99 Jan Jun Jan1 C) Deformation (m) R =.814 S ke=.65 +/ Well ALA8 8 Jun98 Jan99 Jun99 Jan Jun Jan1 Figure 8. (left) Linear regression of Dh and Dd at (a) ALA6, (b) ALA7, and (c) ALA8. The data points show the uncertainty in the InSAR measurement (gray circles). The blue line is the best fit linear regression, and the dashed lines show the uncertainty in that best fit. (right) Estimated hydraulic head from InSAR deformation data (blue), hydraulic head measurements (green), and interpolated hydraulic head measurements (red dashed line) at (Figure 8a) ALA6, (Figure 8b) ALA7, and (Figure 8c) ALA8. implemented a stochastic technique in order to incorporate this uncertainty into the simple kriging algorithm in SGEMS. For a single acquisition of deformation data at a single pixel location, we randomly assigned a deformation from a Gaussian distribution. We created the Gaussian distribution with a mean equal to deformation measurement and a standard deviation equal to the uncertainty in the deformation REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4469

12 1.12/213WR14938 A) 1 Estimated head 8 Measured head Well ALA6 8 Jun92 Jun93 Jun94 Jun95 Jun96 Jun97 Jun98 Jun99 Jun Jun1 B) C) Well ALA7 Jun92 Jun93 Jun94 Jun95 Jun96 Jun97 Jun98 Jun99 Jun Jun1 Well ALA8 Jun92 Jun93 Jun94 Jun95 Jun96 Jun97 Jun98 Jun99 Jun Jun1 Figure 9. The estimated hydraulic head (blue markers) and the measured hydraulic head (red markers) at three well locations. measurement. We repeated this procedure 3 times, i.e., 3 realizations of the deformation datum. We then used simple kriging to estimate the deformation directly at the well location for each realization. We calculated the mean of the 3 realizations, which provided a deformation estimate at the well location, and the standard deviation, which provided the uncertainty in the deformation estimate at the well location. We found that the mean value of the 3 realizations was within 1% of the measured deformation. We repeated this process for each acquisition time and then finally for each of the 11 well locations. 4. Calibrate S ske and Predict Hydraulic Head In this section, we describe how we use a linear regression analysis of the kriged deformation values and the hydraulic head measurements at the 11 well locations from section 2.3 to estimate S ke. We then show how the estimates of S ke can be used to predict hydraulic head at specific well locations during the time period of InSAR acquisition (1992 2) Linear Regression of Kriged Deformation Data and Head Data We first describe the linear regression results at three wells, ALA6, ALA7, and ALA8. At these well locations, we observed a linear relationship between hydraulic head (Dh) and deformation (Dd). We then discuss the results of the linear regression analysis for the remaining eight wells. We use a form of linear regression analysis that takes into account the changing uncertainty in the deformation and hydraulic head measurements when estimating S ke [York et al., 24]. The uncertainty in the hydraulic head measurements is estimated to be 61 cm (E. Harmon, personal communication, 213). The slope of the linear best fit, between the Dh and Dd, provides an estimate of the elastic skeletal storage coefficient (S ke ). The goodness of fit is evaluated by calculating the coefficient of determination (R 2 ). In the paragraphs to follow, we use R 2 to evaluate the relationship between hydraulic head and deformation at the 11 well locations. The results of the linear regression of Dh and Dd at well ALA6 are shown in Figure 8a (left). The deformation and the hydraulic head appear to be linearly related at this location (R ). The gray circles signify the uncertainty in the InSAR measurement as calculated from the multiple realizations of the kriged deformation data. We then used the slope of the regression line (S ke ) to estimate the hydraulic head from the deformation measurements, shown in Figure 8a (right). REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 447

13 1.12/213WR14938 Table 3. The Average Seasonal Hydraulic Head Change and Regression Analysis Results for Each of the 11 Well Locations Well Dh s (m) R 2 S ke ALA ALA ALA ALA CON ALA CON RIO RG88 <.5 Negative N/A ALA15 <.5 Negative N/A ALA14 <.5 Negative N/A Figure 8a (right) shows the actual hydraulic head measurements with green markers and the interpolated hydraulic head values with a red dashed line. Because the InSAR deformation measurements were not acquired at the same time as the hydraulic head measurements we will compare the InSAR-derived estimates of hydraulic head to the interpolated hydraulic head values. We can see that 4% of the estimated hydraulic head data agree with the interpolated hydraulic head data (Figure 9a, right). An aquifer test 3 m away from well ALA6 resulted in an estimate of S If we assume that S is a proxy for S ke, i.e., the storage change due to the expansion and contraction of water (S w ) is small relative to S ke, then we can compare these two results. We find that S ke from the regression analysis is larger than S from the aquifer test. This could be due to a number of factors, e.g., uncertainty in the depth of the producing zone, spatial variability of S, and uncertainty due to atmospheric phase effects in the deformation data. However, it is encouraging that S and S ke are the same order of magnitude. Figure 8b (left) shows a linear relationship between the Dh and Dd data at ALA7 (R ). The slope of the regression line is S ke Wells ALA6 and ALA7 are at the same location but are sampling different depths of the aquifer system. We find that 6% of the estimated hydraulic head agree with the interpolated hydraulic head at (see Figure 8b, right). Well ALA8 is located approximately 2 km northwest of the ALA6/ALA7 wells. The regression of Dh and Dd in Figure 8c shows a linear relationship at well ALA8 (R ). The slope of the regression line is S ke , and we see that 4% of the estimated hydraulic head agree with the interpolated hydraulic head (Figure 8c, right). The results of the regression analysis for all 11 wells are shown in Table 3. In the paragraphs to follow we discuss how the quality of the linear relationship between the hydraulic head and the deformation varies based on magnitude of Dh, S ske, and b * (see equation (1)). If these variables are small at the well location in question, then we do not expect the deformation to be large enough to be accurately measured by InSAR. Table 3 shows the aggregate variable, seasonal hydraulic head change (Dh s ), which is the average seasonal peak-to-trough change in hydraulic head (used previously in section 2.3). The three wells at the top of Table 3 (ALA6, ALA7, and ALA8) show large seasonal hydraulic head change, and the regression analysis suggested a linear relationship between the deformation and the hydraulic head. These three wells are all located in the central part of the SLV where we observed the highest seasonal deformation signals in the raw interferograms [Reeves et al., 211]. However, the data from the eight other well locations produced low or negative values for R 2 from the linear regression analysis, which indicates that a linear relationship should not be used to relate the two data sets. In the remainder of this section we will investigate why a linear relationship was not found at these eight well locations. For the three wells at the bottom of Table 3, RG88, ALA15, and ALA14, the measured seasonal hydraulic head change was small (<.5 m). These were the three wells for which we did not have any information about the lithology of the producing aquifer system so assigned to a lithology of sand in section 2.3. Although our predicted deformation was larger than 1 cm at these well locations, it is possible that given more information about the lithology we would see that the S ske value would be much smaller than that of a typical sand. It is also possible that the aquifer thickness (b * ) for these wells was improperly estimated. If S ske or b * are smaller than we originally anticipated, then it is possible that the amount of deformation occurring at these well locations is too small to be accurately measured by InSAR. At wells CON1, ALA4, CON2, and RIO2, we found that the upper bound of Dd s was large enough to result in accurate measurements of deformation, but the lower bound values were below the sensitivity of the InSAR measurement. Because a linear relationship was not observed at these well locations, we believe that the REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4471

14 1.12/213WR14938 seasonal magnitude of the deformation was not large enough for the deformation to be accurately measured by InSAR. ALA13 is located in the Closed Basin, where water is being pumped only from the unconfined aquifer (see previous discussion in section 2.2). This well was drilled very deep, 544 m, for oil and gas exploration purposes. There are over 275 m of clay sediments recorded in the driller s log, as the well was not drilled for water production (as discussed in section 2.3). We suggest that the presence of these clays leads to a time lag between hydraulic head change and deformation [Riley, 1969]. The hydraulic head data were recorded only from 1999 to 212, which coincides with only seven SAR acquisition times over the span of the Therefore, a rigorous analysis of the time lag associated with the deformation of the clays is not likely to produce useful results, and we did not attempt it. It is important to note that we did not discuss atmospheric phase effects as a factor that may affect the relationship between Dh and Dd at some of the well locations. Because the modified version of SBAS analysis used on the data from the SLV does not include atmospheric temporal filtering, there will likely be some unaccounted for amount of atmospheric phase effects in each deformation time series [Reeves et al., 214]. However, given the results observed at wells ALA6, ALA7, and ALA8, it appears that given large changes in hydraulic head, and favorable hydrogeologic conditions (the magnitude of S ske and b * ), the InSAR data and hydraulic head data can exhibit a reliable linear relationship Predicting Hydraulic Head at Wells ALA6, ALA7, and ALA8 We used the estimated values of S ke from the previous section to estimate hydraulic head at three of the well locations, ALA6, ALA7, and ALA8, for the entire InSAR data time span (1992 2). Figure 9 shows the predicted hydraulic head at the three well locations. Because the InSAR deformation measurements are relative and not absolute, the hydraulic head shown in Figure 9 is relative to the first InSAR acquisition time. At each of the three well locations, we observe that a seasonal trend in the predicted hydraulic head values does exist back to We predicted that the hydraulic head was particularly high at all three wells in the winter of At other wells in the confined aquifer system: SAG7, CON1, EW71C, and ALA2 the measured hydraulic heads exhibited a larger high in 1996 than in and This implies that higher overall hydraulic head levels in the confined aquifer system may have caused the high InSAR-derived estimates of hydraulic head in The approach used here is not valid if (a) the aquifer system is undergoing inelastic deformation during the time periods without hydraulic head measurements, or (b) the uncertainty due to atmospheric phase effects is large. In the case that the aquifer system is undergoing inelastic deformation, the elastic skeletal storage coefficient would not allow for the prediction of hydraulic head back in time, and more hydraulic head data would be needed in order to understand the inelastic relationship. Also, if the InSAR deformation data from 1992 to 1998 contain significant uncertainty due to atmospheric phase effects, the estimates of the hydraulic head would also be incorrect. Although a valley wide comparison was not possible, the estimated hydraulic head at these three well locations provides information about the hydrologic system through time. The InSAR data have enabled us to look back through the 199s, an important time for the management of the confined aquifer system in the SLV and determine that the hydraulic head did not exhibit a negative linear trend at these locations from 1992 to 2. This has implications for improving the estimation of hydraulic head at wells in other areas, agricultural or otherwise, with incomplete or sparsely sampled hydraulic head time series. 5. Conclusions Our goal for this paper was to use InSAR measurements of deformation to interpolate and extrapolate the hydraulic head measurements temporally in the San Luis Valley, Colorado. We were able to extend hydraulic head measurements temporally with InSAR because the deformation was elastic/recoverable in nature and significantly larger than the uncertainty of the InSAR measurement. We note that the SLV was a relatively simple system, such that there was no considerable time lag between the hydraulic head change and the deformation. If a time lag were present, due to an abundance of clays in the producing zones of the wells, REEVES ET AL. VC 214. American Geophysical Union. All Rights Reserved. 4472