Two high-precision gravity surveys were conducted to determine groundwater mass changes at a managed groundwater recharge site in northeastern Colorado. Gravity data were collected during pumping and two months after pumping ceased. During pumping, gravity was lower by as much as 46 μGal near the pumping wells and higher by as much as 90 μGal near the recharge ponds in comparison to data collected after pumping had ceased. These differences are attributed to changes in groundwater mass associated with drawdown and infiltration. Inverse modeling of the gravity data indicates a 5.1 × 105 m3 decrease in storage beneath the recharge ponds between the two surveys, which we attribute to dissipation of the groundwater mound created by recharge during pumping. This estimate of the change in groundwater storage is made independently of assumptions of physical properties of the aquifer. Dividing the change in water volume per unit area determined from the gravity modeling by the change in water level measured in wells provides an estimate of specific yield (Sy) of 0.21 ± 0.03, which is within the range of specific yield estimates derived from aquifer tests at the site. Water level changes predicted from the gravity data agree on average to within ±0.45 m of those measured, which we take to be an estimate of the uncertainty in water table depth predictions that can be obtained from gravimetric data in unconfined aquifers. The study covers a 3.2 km2 area, providing a prototype for similar studies at larger scales.
 A major uncertainty in water resource management is the response of unconfined aquifers to regional groundwater storage changes. Specific yield (Sy) and temporal changes in aquifer storage are conceptually well understood, but quantifying these components in the field from aquifer tests and applying these parameters to determine water availability poses challenges. One problem is the complex nature of drainage during aquifer tests in heterogeneous unconfined aquifers, which includes a three dimensional flow field [Moench, 1993; Chen et al., 2003] and delayed yield effects in the unsaturated zone [Nwankwor et al., 1984; Moench, 2004]. Nwankwor et al.  documented the discrepancies between type-curve and volume balance methods of analyzing aquifer test data and showed that Sy determined by type-curve methods, although commonly used, are much lower than the laboratory-derived specific yield.
 Another problem with quantifying three-dimensional regional aquifer storage change is the sparseness and reliability of available data. Often, only a small number of Sy and aquifer storage values are available and these are difficult to relate to the long-term yield of an aquifer, particularly over a large area. For example, Woodard et al.  measured Sy directly from core samples within the Denver Basin and found that these values were significantly lower than the Sy estimates derived from laboratory analysis of outcrop samples and drill core, and from aquifer tests along the basin margins that were previously used by water resource planners. Consequently, water levels in the basin have declined much faster than anticipated [Raynolds, 2004]. Conflicting results such as these illustrate the need for additional data to fill in gaps between monitoring wells and to improve estimates of the parameters used to predict recoverable groundwater at the regional scale.
 A complement to localized estimates of Sy derived from aquifer tests or laboratory measurements is to measure temporal changes in the Earth's gravity field. Changes in groundwater mass create changes in gravity that can be used to determine Sy and aquifer storage change over a regional area [Zohdy et al., 1974]. To test the accuracy to which Sy and storage changes can be estimated from gravity data we conducted two high-precision, spatially dense gravity surveys at the Tamarack Ranch State Wildlife Area (Tamarack) in northeastern Colorado in March and May 2005. Tamarack is a well-studied hydrogeological test site located along the South Platte River (Figure 1). Since 1997, a managed groundwater recharge project has been underway, extracting groundwater near the river and pumping it into upland recharge ponds located approximately one kilometer away. A groundwater mound forms beneath the recharge ponds, which causes an increase in groundwater discharge to the river at a later time as the mound dissipates. Water level drops of up to 8 m around the pumping wells and increases up to 10 m at the recharge ponds occur during the annual pumping period, usually January through March [Watt, 2003]. This paper describes efforts to correlate temporal variations in gravity with water table fluctuations at Tamarack and to assess the usefulness of the temporal gravity method as a means to estimate regional groundwater storage change and specific yield.
1.2. Previous Studies
 Several studies have shown that measurements of changes in gravity over time can be used to estimate variations in groundwater mass associated with a rise or fall in the water table [Montgomery, 1971; Pool and Eychaner, 1995; Pool and Schmidt, 1997; Howle et al., 2003]. Given a water density of 1.0 × 103 kg m−3, a 1 m change in the thickness of a horizontal layer of water of infinite lateral extent results in a 42 μGal (1 Gal = 10−2 m s−2) change in gravity. Accordingly, a 1 m change in the regional water table elevation in an aquifer with a specific yield of 0.20 produces an 8.4 μGal change in gravity. Modern relative gravity meters (for example the Scintrex CG-5 Autograv meter used in this study) are precise to within ±3 μGal. Thus, in principle, a temporal gravity survey can resolve submeter-scale changes in the water table elevation.
Pool and Eychaner  conducted four separate gravity surveys over two years to examine the correlation between time variations in gravity and water table elevations. They measured gravity at five wells in Pinal Creek Basin in central Arizona during February 1991, May 1991, June 1992, and March 1993. The well spacing varied from less than 0.5 km to over 5 km along an 11-km-long line. Changes in groundwater storage resulted from stream recharge and periodic groundwater withdrawals. Gravity measurements at each well location were compared to a reference station situated on crystalline bedrock where no groundwater mass changes were expected. By relating all data to this unchanging reference station, Pool and Eychaner  were able to compare gravity measurements between the five stations where water mass variations were unknown. Temporal gravity differences ranged from 77 to 159 μGal, and water table elevation changes ranged from 7.7 to 10.5 m. Assuming a flat water table, Pool and Eychaner  estimated Sy in the alluvial sediments by comparing the differenced gravity values to measured changes in the water table elevation. Their Sy values (0.16 to 0.21) overlap the range of values derived from aquifer tests in the area (0.20 to 0.22).
 The spatial scale and data coverage of Pool and Eychaner  were expanded by Pool and Schmidt  who recorded temporal gravity changes in a similar aquifer in southern Arizona. Pool and Schmidt  also used a gravity reference station situated on crystalline bedrock, but established a network of 43 gravity stations over a 15-km2 area, with station spacings of 300 to 400 m. Gravity data were recorded five separate times between December 1992 and January 1994. Water levels and gravity changed by as much as 9 m and 90 μGal, respectively, as a result of natural recharge to and discharge from the aquifer. Specific yield estimated at 12 well locations using the technique of Pool and Eychaner  range from 0.15 to 0.41 in the alluvial sediments. The spatial integral of the temporal change in the gravity field over the survey area provided an estimate of the change in groundwater mass during the survey period. Pool and Schmidt  divided the total mass change by water density to estimate a groundwater storage increase of 9.9 × 106 m3 between December 1992 and mid-May 1993, which was attributed to streamflow infiltration during that period.
 The studies of Pool and Eychaner  and Pool and Schmidt  involved naturally occurring storage changes. Howle et al.  measured temporal gravity changes at two injection wells at a site in Antelope Valley, California. The study of Howle et al.  differs from the previous investigations in that the geometry of the water table was irregular owing to water injection. In order to provide a detailed image of the groundwater mound, they used a cross-shaped pattern of 20 gravity stations in arrays extending 150 m to 700 m away from the injection wells. Over a five month period, gravity was measured prior to, during, and after injection. Using a planar water table model similar to that of Pool and Eychaner , Howle et al.  estimated Sy = 0.13 in the alluvial sediments. This was done by differencing gravity measurements at a station located 350 m from the center of the injection mound and comparing them to changes in the water table elevation. Howle et al.  also estimated total accumulated mass at the injection wells by integrating the change in the temporal gravity data over the survey area, and found that the gravity-derived accumulated mass accounted for approximately 40% of the total water mass injected into the aquifer. They concluded that most of the injected water had moved beyond the areal extent of the gravity network.
 All of the investigations discussed above assume a planar water table to estimate groundwater storage changes and Sy from the change in gravity between each measurement phase. Under this assumption, groundwater mass change can be parameterized as equivalent water thickness from the Bouguer slab equation [Telford et al., 1990]:
where Δz is the change in water table elevation, Δh is the thickness of an equivalent layer of water of infinite spatial extent that is required to produce the measured change in gravity (in other words, it is the change in groundwater volume per unit area), Δg is the change in gravity, ρ is the density of water, and δ is the universal gravitational constant (6.67 × 10−11 N m2 kg−2).
 The planar Bouguer slab model may not be adequate to predict irregular water table geometries formed by groundwater drawdown or mounding. The conical depression of the water table near a pumping well, for example, is not accurately characterized by a planar surface and requires a more realistic modeling approach. Several studies have attempted to model drawdown near pumping wells with simple geometric shapes, for which analytical solutions of the gravitational attraction have been found [e.g., Poeter, 1990; Damiata and Lee, 2006]. However, this is a time-consuming approach to modeling groundwater-induced gravitational changes over large areas and does not explicitly account for irregularities in the water table surface that may not be associated with isolated groundwater extraction or recharge sources. For example, Damiata and Lee  show that drawdown owing to pumping can produce changes in the gravity field in excess of 2 μGal several hundred meters distant from the pumping well location. Details of the gravitational response vary with pumping rate, static water table depth, and hydrological properties of the aquifer, but the analysis of Damiata and Lee  makes it clear that interactions of multiple drawdown cones (and recharge mounds) will create a complicated water table geometry that can produce temporal and spatial changes in gravity within the measurement precision of modern gravity surveys. Modeling the gravitational response to water table variations on a regional scale, even with simple assumptions regarding the shape of the drawdown cones and water table mounds, is time consuming if the water table surface is complicated by more than a few simple mounds and/or depressions.
 The purposes of this study are to assess the effect of nonplanar water table elevation changes on gravity measurements collected at different times, to investigate the use of a model not dependant on simple geometrical shapes to characterize the water table topography, and to identify a field protocol for conducting temporal gravity surveys of water mass in areas where no bedrock outcrop is available to establish a gravity reference station. Using temporal gravity data over an area of several square kilometers, our objectives are to (1) estimate groundwater volume change per unit area, (2) estimate specific yield by comparing water volume changes with water level changes measured in wells, (3) estimate changes in water table elevation between well locations, and (4) determine the precision to which specific yield, changes in storage, and changes in water table depth can be estimated from gravity data.
2. Site Description
 Tamarack is located in northeastern Colorado along the South Platte River (Figure 1). The survey encompasses a 3.2 km2 area within Tamarack, which is publicly owned land managed by the Colorado Division of Wildlife (CDOW).
 The Northern Colorado Water Conservancy District, in cooperation with CDOW, operates a managed groundwater recharge project at the Tamarack site where water is withdrawn from the alluvial aquifer of the South Platte River via groundwater pumping wells and discharged into upland depressions. The purpose is to augment river flow during low-flow periods in order to maintain wildlife habitat. The pumping wells normally operate during the winter months from January through March. In 2005, six pumping wells continuously withdrew groundwater at an average rate of 4 × 103 m3 day−1 from 14 February to 31 March. During this time three recharge ponds (Figure 1) were created: (1) a 3.9 ha main recharge pond, (2) a smaller recharge pond, and (3) a minor spillover pond to the east. Three smaller ponds (termed “F” ponds; see Figure 1) also received water, but these are clay-lined and have minimal infiltration.
 A network of 24 monitoring wells and piezometers provide a seven year record of water table elevation changes during the pumping and recharge periods. During natural conditions, when the pumping wells are off for an extended period of time, there is a gentle, east-northeast-directed hydraulic gradient of 0.002 (Figures 2 and 3a) [Hurr and Schneider, 1973]. During pumping operations the hydraulic gradient in the vicinity of the recharge ponds is generally directed northward toward the pumping wells located south of the river, and the water table at the recharge ponds is roughly 10 m higher than under natural conditions [Watt, 2003] (Figure 3b).
 The hydrologic system at Tamarack is composed of two primary components: surface water in the South Platte River and groundwater in the alluvial aquifer [Burns, 1985]. The alluvial aquifer is unconfined and is hydraulically connected to surface water in the river and backwater sloughs [Weston and Swain, 1979; Warner et al., 1994]. Under natural conditions the depth to water table ranges from 0 m at the river to 15 m in the southern portion of the study area [Watt, 2003]. The South Platte River flows year-round in distinct though extensively braided channels.
 The aquifer consists of Quaternary to Recent alluvium deposited by the South Platte River [Burns, 1985]. Alluvium covers approximately one third of the northern part of the site (Figure 1) and consists of predominantly sand and fine gravel, with minor interbedded silt and clay [Bjorklund and Brown, 1957]. The thickness of the alluvium ranges from less than a meter at the valley edge to almost 100 m at some places near the river [Warner et al., 1994]. Burns  reports a hydraulic conductivity of 60 m day−1 and Fox  reports values ranging from 60 to 200 m day−1 on the basis of aquifer tests at Tamarack. Underlying the alluvium is relatively impermeable shale bedrock at depths ranging from 20 to 60 m below ground surface [Hurr and Schneider, 1973].
 Vegetated, stable eolian sand of Pleistocene age forms extensive dunes which cover the southern two-thirds of the field site (Figure 1) [Bjorklund and Brown, 1957]. The dunes have variable topography with up to 25 m of relief relative to the alluvial plain. Total thickness of the sand ranges from 0 to 12 m with thickness increasing to the south [Poceta, 2005]. Infiltration rates are high because grain size is predominantly fine to medium sand [Bjorklund and Brown, 1957]. The recharge ponds are located within these dunes.
3.1. Principles of Temporal Gravity Surveys
 At least two gravity surveys collected at different times are necessary to estimate temporal water table fluctuations in an aquifer. Before temporal variations in gravity can be related to changes in groundwater mass, the data must be corrected for gravity changes caused by differences in the gravity meter height when a station is reoccupied between surveys, ground elevation changes between surveys, instrumental drift of the gravity meter, Earth tides (changes in gravity associated with the sun and moon), and barometric pressure changes. Other corrections that are commonly applied to gravity surveys, such as latitude, free air, Bouguer, and terrain corrections [Telford et al., 1990], are not necessary in temporal gravity surveys because measurements from different times are compared at the same locations, so these effects cancel out when the gravity data sets are differenced.
3.2. Survey Design
 A total of 145 gravity stations were established at Tamarack (Figure 1). This includes a grid of 45 gravity stations spaced 250 m apart, which we refer to as the background grid. The background grid is designed to have sufficient areal coverage to model groundwater mass changes over the entire field area. The station spacing was selected to minimize the time required to survey the study area while assuring sufficient station density to detect local changes in gravity associated with groundwater mounding or drawdown. Well data collected over the last decade indicates that pumping produces water table variations of a few 10 s of cm up to 10 m at depths of a few meters over distances of 50–100 m [Beckman, 2007]. Damiata and Lee  showed that local incremental water table changes of ∼30 cm at depths of 10–50 m produce measurable changes in gravity over lateral distances of a few hundred meters. Thus, a station spacing of 250 m represents a reasonable compromise between necessary spatial resolution and speed of surveying. The 250 m station spacing of the background grid is inadequate to resolve short-wavelength changes in gravity near the pumping wells, where there are changes in water table elevation of several meters over distances of a few tens of meters [Halstead and Flory, 2003]. Therefore, an additional series of stations spaced 25 m apart were located along a road between the pumping wells to provide a detailed two-dimensional image of the geometry of the drawdown cones. To better constrain the three-dimensional shape of the drawdown created by pumping, additional stations were deployed 25 m apart in a radial pattern extending 100–150 m outward from pumping wells P1 and P2 (Figure 1).
 Prior to the first deployment, each station in the network was surveyed with a Topcon laser total station for relative northing and easting to within 5 mm and elevation to within 25 mm. Gravity stations were marked with either a flag pushed into the soil or, when possible, an etching into concrete well platforms. The pond boundaries were surveyed to determine the total volume of surface water. At each station, gravity measurements were taken 6 times per second for twenty seconds using a Scintrex CG-5 Autograv meter. This was repeated three times, totaling 360 measurements per station which were averaged to obtain the final gravity value.
 Gravity data collection was timed to take advantage of maximum water level changes resulting from drawdown at the pumping wells and mounding at the recharge ponds. Data were collected in two phases: (1) from 11 to 16 March 2005, three weeks after the pumping wells were turned on and the recharge ponds were full, and (2) from 16 to 22 May 2005, six weeks after the pumping wells were turned off and the recharge ponds were dry. Summing the pumping rates measured with inline totalizing flowmeters indicate that the six wells pumped a total of 8.9 × 105 m3 of water into the recharge ponds prior to phase 1 data collection. An additional 6.3 × 105 m3 was pumped between phase 1 and phase 2.
 Water levels were measured in each of the monitoring wells using a water level indicator at approximately two week intervals, beginning two weeks prior to the onset of pumping and continuing for two weeks after the pumps were turned off [Beckman, 2007]. Water level data were also collected during each of the gravity surveys. It was assumed that the water table elevation corresponded to the water level measured in the wells. Water level data collected in previous years at more frequent intervals show that the major changes in water table depth in the area occur within 48 hours after the onset and cessation of pumping [Halstead and Flory, 2003]. Only small variations (<1 m) in a given well occurred during the intervening periods, indicating that the hydrogeologic system was near steady state during both data collection periods. Nested piezometers are present at wells T13, T17, and T18, located near the main recharge pond, and a single piezometer is present at well T9, located near pumping well P1 (Figure 1). These well sites are particularly important as these data are later used to constrain estimates of specific yield. The shallow wells at sites T17 and T18 are completed just above the base of the eolian sand and are screened over the lowest 1.5 m (Figure 2). The deep wells at these sites penetrate into the underlying alluvium and are also screened over the lowest 1.5 m. The shallow well at T13 is completed just above the base of the eolian sand and screened over the bottom 1.5 m. The T13 deep well was drilled to bedrock and screened within the alluvium over the bottom 6.5 m. The shallow wells at T13, T17, and T18 only have water in them when the recharge ponds are active. Well T9, located 100 m south of pumping well P1, is completed to 3.9 m depth in alluvium and is open at the bottom. Measurements of water table depth (relative to the top of the well casing) are repeatable to within 0.01 m during a 15 min period, which is taken to be the precision of the measurement [Beckman, 2007].
3.3. Gravity Data Processing
 The first gravity data processing step is to correct for differences in the height of the gravity meter relative to the ground surface between each occupation of a station. Minor variations in instrument height incurred when setting up the instrument can result in differences in gravity that are comparable to gravity changes associated with groundwater mass changes. For example, a 1-cm difference in instrument height between occupations of the same station produces a ∼3 μGal change in gravity [Telford et al., 1990]. In this survey, gravity meter heights were measured after the instrument was leveled, and the data were corrected to ground level by adding the product of the instrument height and free air gravity gradient to the measured gravity value [Telford et al., 1990]. The free air gravity gradient at Tamarack was determined by measuring the change in relative gravity at 0, 100, 112, and 140 cm above the ground surface using a LaCoste and Romberg Model G gravity meter. The average of three measurements at each height was used. The local free air gravity gradient was determined to be 3.054 μGal cm−1. For comparison, the global average free air gravity gradient is 3.086 μGal cm−1 [Telford et al., 1990]. The largest change in instrument height variations during the survey was 5.4 cm, resulting in a maximum instrument height correction of 17 μGal.
 Gravity changes associated with gravity meter drift and Earth tides were corrected by periodically repeating measurements at a common gravity reference station (the base station) and correcting for observed drift [Telford et al., 1990]. The base station was located on a stable concrete monitoring well platform located midway between the pumping wells and recharge ponds (Figure 1). Gravity measurements at the base station were repeated at one-hour intervals during each survey day, and a daily drift curve was obtained by linearly interpolating between hourly base station occupations. This approach has the advantage of simultaneously correcting for instrument drift, Earth tides, and daily barometric changes that occur during the survey period. It does not account for barometric changes that occurred between the two phases of the survey (see below). The maximum daily drift during the surveys was 42 μGal [Gehman et al., 2006]. A piecewise linear extrapolation between measurements was deemed acceptable given the short time elapsed between base station measurements (typically less than 1 hour) compared to the longer-term changes in instrument drift (nearly linear over a daily interval for the CG-5 instrument used in this study), Earth tides (∼12-hour cycle), and barometric pressure.
 Atmospheric pressure changes may produce gravity variations up to 0.4 μGal mbar−1 [van Dam and Francis, 1998]. In this survey, barometric pressure at the nearest weather station (in Akron, Colorado, located 90 km south of Tamarack) changed less than 3 mbar (equivalent to 1.2 μGal) during any survey day. The daily effect of atmospheric pressure changes on the gravity signal during each phase is removed by the drift correction since the base station experiences the same barometric loading as the field stations. Changes in barometric pressure between the two phases of the survey do not affect the survey results since all measurements are referenced to the base station, which is defined to have no change in gravity between each phase of the survey.
 Soil expansion or compaction between gravity surveys can potentially alter the ground surface elevation, resulting in a change in gravity between surveys that is not associated with groundwater mass changes [Romagnoli et al., 2003]. Soil expansion data were not available for this survey. However, as much as possible, this error source was minimized by locating gravity stations on stable measurement surfaces (concrete well platforms). Minimal ground elevation change between surveys is expected because (1) the soil and aquifer material have only minor clay content [Bjorklund and Brown, 1957]; and (2) the soils are coarse textured, have high infiltration and percolation rates, and have low shrink-swell potential [U.S. Department of Agriculture (USDA) Natural Resources Conservation Service, 1977]. Observations providing evidence that soil expansion was minor are (1) survey flags placed during the first survey were still vertical for the second survey, (2) well casings, fence posts, and telephone poles in place for decades remain near-vertical, and (3) concrete well platforms and building foundations in the area show only minimal cracking with no offset along fractures, indicating that the cracks can be attributed to aging of the concrete and not ground motion.
 The final data processing step was to correct for the gravitational effect of temporal changes in surface water in the recharge ponds. The boundaries of the ponds were surveyed using a total station, and the depths were estimated by comparison of the pond boundaries to a USGS 7.5 min topographic quadrangle. The surface water mass was then approximated as a three dimensional polygonal body and the associated vertical gravitational attraction at each station was calculated using the forward modeling algorithm of Singh and Guptasarma . The resulting gravitational acceleration was subtracted from each gravity measurement to remove the surface water effect. Because the recharge ponds are shallow (<2 m depth) and the topography is relatively flat, the gravitational attraction of the surface water is dominantly horizontal. In contrast, groundwater mass estimates are based on measurements of vertical gravitational acceleration. Consequently, changes in surface water volume do not greatly influence the data used in this study. The maximum correction to the vertical component of gravity arising from surface water changes was less than 1 μGal. Uncertainties in the surface water correction derive from uncertainties in surveying the pond geometry, and are less than 0.25 μGal.
3.4. Creating a Time-Lapse Image
 After applying the corrections described above, the gravity data were differenced by subtracting the data collected after pumping had ceased (May 2005) from the data collected when pumping was underway (March 2005). Positive gravity values in the differenced data indicate areas where groundwater mass was greater during the period in which pumping was underway in comparison to the period when the pumping wells were inactive. Negative gravity values represent areas where groundwater mass was less during the period in which pumping was underway. A uniform grid was then constructed from the differenced data for contouring and inverse modeling using a kriging algorithm with 25 m spacing. Other grid spacings (12.5 m to 50 m) and gridding algorithms (minimum curvature and spline fitting) were tested. Kriging on a 25 m grid was chosen because of its ability to best produce a smoothly varying gravity field (which is expected of potential field surfaces), theoretical arguments that kriging should best represent a surface with smooth variations superimposed on a background trend, and the ability of contours on the basis of the kriged data to accurately reproduce gravity values at key measurement points (the base station and well locations).
 Ideally, data differencing would be based on relating the two gravity surveys to a common base station located on bedrock, where no groundwater mass changes occurred between gravity surveys. No bedrock outcrop was present near the study area, so the measurements in the two gravity surveys were related by assuming no gravity change occurred at the base station. This assumption is rigorously valid only if no groundwater mass change occurred in the vicinity of the base station between surveys. Data from nearby wells indicate that water table elevations changed by no more than ±0.6 m between the two surveys at distances within 200 m of the base station, which corresponds to a gravity change at the base station of no more than 5 μGal (Figure 4). This is similar to the change in gravity that would be expected if the groundwater mass change resulted from a regional (planar) rise or fall of the mean water table, which is assessed using the Bouguer slab approximation (equation (1)). Assuming the mean specific yield of 0.20 estimated previously at Tamarack by Halstead and Flory , this model also predicts a change of no more than 5 μGal at the base station between surveys. A 5 μGal change in gravity at the base station would lead to a systematic over or underestimate of changes in water table elevations by as much as 0.6 m (again, assuming a specific yield of 0.20). For comparison, changes in gravity used to estimate specific yield and water table elevation in this study range from 17 to 90 μGal, corresponding to water table elevation changes of 2–11 m. We recognize that localized drawdown or mounding in areas adjacent to the base station may occur that are not captured in the well data, particularly east of the base station where no well data are available. However, we deem this unlikely because (1) the natural water table is a smooth, low gradient surface (Figure 3a) and (2) there are no pumping wells or recharge ponds within 500 m of the base station to produce artificially induced localized changes.
3.5. Inverse Modeling
 We used the three-dimensional inverse gravity modeling program 3DINVER.M [Gómez-Ortiz and Agarwal, 2005], which is based on the algorithm of Oldenburg , to assess aquifer properties and changes in groundwater storage between the two gravity surveys. Unlike previous studies that approximate the water table as a planar surface [Montgomery, 1971; Pool and Eychaner, 1995; Pool and Schmidt, 1997; Howle et al., 2003] or a simple geometrical shape [Poeter, 1990; Damiata and Lee, 2006], 3DINVER.M allows for a continuous, smoothly varying surface of arbitrary shape. We examine three applications: (1) if no a priori information is available other than the temporal gravity data, inversion of the gravity data results in an estimate of the storage change in the aquifer that occurs between the two gravity surveys; (2) if water table elevation changes are available (for example, from wells within the survey area), these can be combined with the estimates of storage change obtained from the inverse modeling to estimate specific yield; and (3) if estimates of specific yield are independently available or can be estimated at well sites as described above, the gravity data can be inverted to determine spatial variations in water table elevation between well locations.
 The 3DINVER.M program was originally written to determine the mean depth and topography on an interface separating two buried half-spaces of different density using nontemporal gravity data. In this study, we extend the range of applications by using as the input the difference between two temporal gravity data sets rather than nontemporal gravity data. Thus, instead of inverting for the topography on an interface that separates two half-spaces of different densities, we obtain the change in the topography that occurs between the two surveys. The interfaces of concern are an interface that represents in parametric form the change in groundwater storage and an interface representing the change in the water table elevation.
 Key input requirements in the model are the spatially varying gravity field represented on a regular grid, an estimate of the mean depth of the interface separating the half-spaces, and the density contrast across the interface. Some data preconditioning filters are also used (see discussion below), but experimentation shows these to have little effect on the results in this study. Following Parker , the gravity field is represented as a series approximation:
where G() is the Fourier transform of the gravity field, H() is the Fourier transform of the topography on the interface separating the two half-spaces, ρ is the density contrast across the interface, z0 is the mean depth to the interface, and H() is the wave number. Oldenburg  re-arranged equation (3) to formulate a method for inverting the gravity field:
The first term in equation (4) represents a planar interface between the two half-spaces that best fits the gravity data. In 3DINVER.M, higher-order terms are added in successive iterations, introducing successively shorter-wavelength topographic features to better describe the shape of the interface. Thus, the program first builds a rough planar solution, and then adds complexity as needed to improve the fit between the observed and modeled gravity fields, starting with the smoothest solution and progressing toward more complex undulating surfaces.
 Iterations are halted when the RMS change in the shape of the interface between successive iterations is below some user specified criterion. This stopping criterion, rather than one on the basis of the misfit between the modeled and observed gravity fields, is chosen because addition of higher-order terms often produce only marginal improvements to the model fit but may create unrealistically sharp gradients or localized high-amplitude topographic features on the interface. The stopping criterion used assures that we obtain the simplest (smoothest) model consistent with the observations. Further discussion of the stopping criterion is given by Gómez-Ortiz and Agarwal .
 To avoid edge effects associated with truncation of the model at the edges of the survey area, a cosine taper was applied to the last 10% of the gridded data at the edges of the model, as described by Gómez-Ortiz and Agarwal . The 3DINVER.M program also requires an estimate of the mean depth to the density interface and a filter to attenuate short-wavelength (potentially noisy) variations in the gravity field. Since this study deals with a shallow aquifer, a value of 0 m was used as the mean interface depth. The high cut filter was assigned to remove all wavelengths shorter than half the grid spacing (12.5 m) from the gravity signal before the data were inverted. We tested a range of choices for the taper (0–30%), interface depth (0–15 m), and high cut filter (10–30 m), and found that the model results, in this study, were not greatly sensitive to the parameter choices.
3.6. Estimating Changes in Groundwater Mass
 Assuming that all data processing steps are properly done, the difference in gravity between the two phases of the study is due solely to changes in groundwater mass. We equate this to a change in groundwater volume (storage). To capture spatial variability, we parameterize the change in storage as a change in water volume per unit area (Δh in equation (2)) which we refer to as the equivalent water layer thickness. The equivalent water layer thickness is the thickness of a layer of water that must be added to or removed from the aquifer to account for the observed changes in gravity.
 The interface H in equations (3) and (4) can be made to represent the equivalent water layer thickness if we specify a density contrast of 1 × 103 kg m−3 (the density of water) in the equations. Addition of higher-order terms used to describe H in this inversion was halted when the interface representing the thickness of the equivalent water layer on successive iterations agreed to within an RMS difference of 0.1 m. Once the equivalent water layer thickness is determined through inverse modeling, the net storage change in a given area (the change in groundwater volume between the two gravity surveys) can be estimated by spatially integrating the equivalent water layer thickness. The storage change estimate depends only on the gravity data and is independent of the physical properties of the aquifer (e.g., specific yield, porosity, permeability).
3.7. Estimating Specific Yield
 If water table changes (Δz) are constrained independently of the gravity data, we can use the change in water volume per unit area (Δh) determined from the inverse model to estimate Sy (equation (2)). In this study, water table changes measured at five wells are used to estimate Sy. Three of these wells, T13, T17, and T18 (Figure 1) are located near the main recharge pond. The other two wells, T6 and T9, are located near the pumping wells.
3.8. Estimating Changes in Water Table Elevation
 If an estimate of the specific yield is independently known, the gravity data can be used to estimate water table elevation changes throughout the study area. This is done by inverting the temporal gravity data using a density contrast equal to the change in density between saturated and unsaturated soil (i.e., the change in density across the water table surface). This density contrast is the product of Sy and the density of water. In this inversion, an RMS convergence criterion of 0.01 m was used in 3DINVER.M. This is an order of magnitude smaller than the convergence criterion used in inverting for water volume per unit area (section 3.6), but was required to force additional terms in the harmonic series that add shorter-wavelength components to the model. The higher-order terms are required because locally averaged gradients of the changes in water table elevation are greater than locally averaged gradients of the changes in water volume per unit area. All other inversion parameters used in this step are the same as those used to estimate the equivalent water mass as discussed previously.
4. Uncertainty Analysis
 The uncertainty in the gravity measurements at each station is the sum of the uncertainty associated with the correction for instrument height differences at each station between the two phases of the survey, uncertainties associated with possible elevation changes at each station between surveys owing to soil expansion or compaction, uncertainty associated with corrections for instrument drift and Earth tides, and uncertainty associated with the precision of the gravity meter. Instrument height is estimated to be accurate to ±0.5 mm (on the basis of repeated measurements involving re-seating and re-leveling of the instrument). On the basis of observations of offsets of cracks in well platforms and building foundations, we estimate soil expansion at each station between the two phases of the survey to be less than 2 mm. Thus, the total uncertainty in elevation changes at each station between phases of the survey is ±2.5 mm. Using the local free air gravity gradient of 3.054 μGal cm−1, this results in an uncertainty in the gravity measurements of ±0.76 μGal. Uncertainty in the correction for surface water and instrument drift, Earth tides, and daily barometric pressure changes are discussed above, and are estimated to be below ±1 μGal. Including the instrument precision (reported by Scintrex. Ltd. to be ±3 μGal), the total uncertainty in the gravity measurements is estimated to be ±4.76 μGal.
 An approximation of the uncertainty in the water volume per unit area calculation can be obtained by assuming a Bouguer slab model for the water table. This is rigorously correct only if the water table is flat, but as we show below, this gives a useful approximation to the uncertainty estimate. For a flat water table, a 1 m change in groundwater volume per unit area produces a 42 μGal change in gravity (equation (1)). Given the ±4.76 μGal uncertainty in the gravity measurements, the uncertainty in the water volume per unit area estimate is approximately 0.11 m.
 The precision of the estimates of water table changes (Δzerr) depends on the specific yield and the uncertainty in the water volume per unit area (Δherr):
Note that we are here referring to water table estimates derived from the gravity data, not water table elevations directly measured in the wells. Using Δherr = 0.11 m as determined above and a specific yield of 0.20 (the mean specific yield at Tamarack estimated by Halstead and Flory ), we find Δzerr = 0.55 m. For comparison, specific yields of 0.15 and 0.25 propagate to Δzerr = 0.73 m and 0.44 m, respectively. These uncertainties are similar to the results of Damiata and Lee , who developed an analytical model to consider the precision in gravity measurements required to detect conical drawdown associated with pumping in unconfined aquifers. The model of Damiata and Lee  depends on pumping rate, duration of pumping, and specific yield, but they estimate that approximately 0.30 m of drawdown within a few hundred meters of the pumping well would be detectable with gravity data with a precision of 2 μGal.
 Assuming that the variables Δz and Δh are uncorrelated, the uncertainty in the specific yield estimate is
where Δz and Δzerr are the water table measurements and respective uncertainties taken from well data [Bevington, 1969]. This uncertainty estimate depends on the specific yield and water table changes at specific well sites, so we defer this computation until after we present those results.
5.1. Gravity Differences
 Differences in gravity between the two surveys range from −46 μGal to 90 μGal, with the largest changes near the pumping wells and the main recharge pond (Figure 5). Gravity differences are negative where the water table was lower during phase 1 (during pumping) relative to phase 2 (when the pumps were inactive) and positive where groundwater levels were higher during phase 1 in comparison to phase 2.
 The negative gravity differences are centered on the pumping wells and are attributed to groundwater drawdown that occurred as a result of pumping, with the greatest difference (−46 μGal) at pumping well P7 (Figure 5). Gravity differences at the stations located on the line trending between the pumping wells are all negative, with the magnitude of difference decreasing with distance from each pumping well.
 Positive gravity differences near the recharge ponds are attributed to greater mass during the first survey (conducted during pumping) associated with buildup of the groundwater mound (Figure 5). In comparison to phase 2, gravity during phase 1 was 90 μGal higher at the station near well T13, adjacent to the northeast side the main recharge pond, 76 μGal higher at a station located on the western tip of this pond, and 12 μGal higher at the station adjacent to the smaller, middle recharge pond. A positive gravity difference of 44 μGal was also observed at a station immediately adjacent to pond F2.
5.2. Groundwater Mass Changes
 As discussed previously, the 3DINVER.M inverse modeling program [Gómez-Ortiz and Agarwal, 2005] was used to estimate the change in groundwater volume per unit area (Δh in equation (2)) between March and May 2005. This is represented as a map of the net thickness of a layer of stored groundwater (Figure 6). This should not be confused with changes in saturated thickness. Changes in saturated thickness are reflected in changes in the water table elevation, discussed in section 5.4. The RMS change in the equivalent water layer thickness map between successive iterations in 3DINVER.M was 1.7 × 10−3 m, achieved with 3 terms in the series approximation. The maximum difference between the observed and predicted gravity fields is less than 0.5 μGal over the study area (Figure 7). The poorest agreement occurs near areas where large water mass changes occur over a short distance. For example, near the pumping wells and the main recharge pond. Negative mass changes occur near the pumping wells, where drawdown occurred prior to and during phase 1 (Figure 6). The decrease in groundwater volume per unit area predicted at the pumping wells ranges from 0.12 m (well P1) to 0.89 m (well P7). Positive mass changes occur beneath the recharge ponds (Figure 6). This is attributed to a groundwater mound created during pumping prior to the first gravity survey. Once the pumps were turned off, the mound dissipated, resulting in a decrease in gravity in this area between the first and second gravity surveys.
 To estimate the storage change that occurred beneath the recharge ponds between surveys, the change in groundwater volume per unit area determined from the gravity measurements was integrated over the area of positive mass change surrounding the recharge ponds. The integrated volume is 5.1 × 105 m3. The total amount of water pumped into the recharge ponds between the two gravity measurement campaigns (taken to equal the volume of water recovered from the pumping wells) is 6.3 × 105 m3. The nearest precipitation data, from Crook, CO (located 3 km northwest of the study area) indicates 5.27″ (13.4 cm) of precipitation during the period elapsed between the gravity surveys (see the National Climate Data Center Web site at http://www.ncdc.noaa.gov/oa/ncdc.html). When multiplied by the area over which the modeled groundwater mass changes are integrated (1.6 × 105 m2), this results in a maximum of 2.1 × 104 m3 of additional water that may have infiltrated between the two gravity surveys. The nearest potential evapotranspiration data, from Haxtun, CO (located 35 km southeast of the study area) indicates potential evapotranspiration was approximately twice the precipitation (see the Colorado Agricultural Meteorological Network Web site at http://ccc.atmos.colostate.edu/~coagmet). Thus, groundwater added by precipitation totals only about 3% of the water pumped into the recharge pond at most, and if evapotranspiration is taken into consideration, is more likely to be near zero.
5.3. Specific Yield Estimates
 The change in water volume per unit area, Δh, determined in section 5.2 is used to estimate specific yield at wells where the change in water level is known (Table 1). We use wells T17, T13, and T18, located in the recharge area, and wells T6 and T9, located near the pumping wells. Δh is estimated to be 2.11 m, 1.20 m, and 0.61 m, at wells T17, T13, and T18 respectively, and is −0.42 m and −0.39 at wells T6 and T9. The change in water level measured at these wells is given in Table 1 and the specific yield estimates and associated uncertainties obtained using equations (2) and (6) are given in Table 2. In the recharge area (wells T17, T13, and T18), Sy ranges from 0.19 ± 0.03 to 0.29 ± 0.01, with a mean of 0.24 ± 0.02. For comparison, Sy estimated by the Bouguer slab method (on the basis of the measured changes in gravity at these wells and the assumption of a planar water table) ranges from 0.13 ± 0.03 to 0.29 ± 0.01 (equation (1)), with a mean of 0.20 ± .0.02 (Table 2). In the alluvium (wells T6 and T9) the mean specific yield on the basis of the inverted gravity data is 0.19 ± 0.04, compared to 0.22 ± 0.04 obtained from the Bouguer slab method.
Table 1. Measured and Predicted Water Level Changes
Change in Gravity (μGal)
Measured Water Level Change (m)
Predicted Water Level Change (m)
Water Level Difference (m)
Water Level Difference (%)
Table 2. Specific Yield Estimated Assuming Planar and Nonplanar Water Table
Sy Using Bouguer Slab Model (Planar Water Table)
Sy Using Modeled Nonplanar Water Table
0.29 ± 0.01
0.29 ± 0.01
0.17 ± 0.02
0.24 ± 0.02
0.13 ± 0.03
0.19 ± 0.03
0.26 ± 0.04
0.19 ± 0.03
0.18 ± 0.04
0.18 ± 0.04
5.4. Water Level Changes
 As described in section 3.7, changes in water table elevation can be predicted from the temporal gravity data if an estimate of specific yield is known a priori. To illustrate this, we will assume a uniform specific yield of 0.22, which is the mean of the specific yield calculated at wells T6, T9, T17, and T18 as discussed above. Data from well T13 are excluded from this average because there were fluctuations of the shoreline of the nearby recharge pond (see section 6 for further details). The mean specific yield of 0.22 is similar to the mean specific yield at the Tamarack site of 0.20 calculated by Halstead and Flory . If the specific yield estimate of Halstead and Flory  is more accurate than that obtained in this study, our approach will underestimate water table changes by 10% (the difference between the two mean specific yield estimates). A more problematic issue is that spatial changes in specific yield are likely, and are not accounted for in our analysis. Nonetheless, our analysis illustrates a procedure to use temporal gravity data to estimate spatial changes in water table elevation between well sites.
 Following the method described in section 3.7, the change in water table elevation that occurred between the two surveys is obtained by inverting the gravity data using a density contrast at the interface of 220 kg m−3 (the product of specific yield and the density of water) (Figure 8). The minimum RMS change in the water table depth map between successive iterations for this model is 6 × 10−3 m, achieved with 3 terms in the series approximation. The maximum difference between the observed and predicted gravity fields is generally less than 0.8 μGal. The poorest agreement is near the pumping wells and recharge ponds, where abrupt changes in the slope of the gravity field cannot be accurately described with the series approximation (Figure 9). The map of the water table elevation change obtained from the inverse modeling (Figure 8) is generally similar to that obtained from the well data (Figure 4a), although the map on the basis of the well data is necessarily smoother owing to the sparser data set and does not cover as large an area. Specifically, both maps show rises in the water table during the pumping phase in the area surrounding the recharge ponds, reaching a maximum of approximately 8 m at the main recharge pond. Water table elevation changes decrease northward away from the recharge ponds, with a slight drop in water Table (0.25 to 0.75 m) during pumping near the base station and a 2 m drop in the area just south of the pumping wells. The map on the basis of the gravity data covers a larger area, and shows localized drops in water table elevations at the pumping wells and a larger area of decreased water table elevation during the pumping phase in most of the eastern third of the map. The modeled decrease in water table elevation varies between pumping wells because pumping rates differ between wells, but is at least 2 m within a radius of 25 m from pumping wells P2, P3, P5, and P7 (Figure 8). The maximum decrease in water table elevation (4.0 m) is predicted at well P7, where the maximum negative gravity difference was measured (−46 μGal).
 The modeled water level change agrees to within 1.1 m or less with the water level changes measured at wells T13, T17, and T18, which are located in the recharge area, and to within 0.3 m at wells T6 and T9, located near the pumping wells (Table 1). Averaging the difference between the modeled and observed water table elevations for all these wells leads to an estimate of ±0.45 m for the uncertainty in the predictions of water table elevation change.
6.1. Interpretation of Changes in Gravity
 All stations located near the pumping wells have lower gravity relative to the base station during the pumping phase than during the period when the pumping wells were inactive. These negative gravity anomalies are focused within ∼50 m of the pumping wells, and in many cases are circular or semicircular in planform (Figure 5). These semicircular negative gravity anomalies overlap, creating a quasi-linear negative gravity anomaly extending along a line connecting the pumping well locations. The semicircular shape of the local anomalies around the wells is interpreted to be due to drawdown near each well [e.g., Damiata and Lee, 2006]. The quasi-linear negative gravity anomaly trending between wells is interpreted to be the result of the interaction of drawdown cones between the pumping wells.
 Stations that show higher gravity relative to the base station during the pumping phase than during the inactive phase are all located within 200 m of the recharge ponds. This is interpreted to indicate a rise in the water table owing to groundwater mounding beneath the recharge ponds. The large (44 μGal) increase in gravity adjacent to pond F2 indicates an increase in water mass beneath this pond during the pumping phase, suggesting that this clay-lined pond may be leaking.
 The estimate of the net change in storage (−5.1 × 105 m3) obtained in section 5.2 by integrating the equivalent water layer thickness in the vicinity of the recharge ponds allows us to construct an approximate water balance in the area surrounding the recharge ponds. The total volume of groundwater added to the system between the two gravity surveys is the sum of the water added by pumping (6.3 × 105 m3), water added by precipitation and evapotranspiration (approximately zero at this site), and water added by groundwater flow into or out of the area. Summing, we find that a net groundwater outflow of 1.14 × 106 m3 from the recharge area is required to account for the −5.1 × 105 m3 of groundwater storage change between the two surveys. This outflow includes water pumped into the recharge area between the two surveys and the volume of water lost from the recharge area by dissipation of the recharge mound.
 The total volume of water pumped into the recharge ponds during the entire period of operation (prior to and between gravity surveys) was 1.52 × 106 m3. Subtraction of the water that left the recharge area between the two phases of the survey (1.14 × 106 m3) leaves 3.8 × 105 m3 of water not accounted for by the gravity analysis. This is attributed to water needed to saturate pores where the initial water content was less than specific retention prior to the onset of pumping, to groundwater flow past the boundaries of the recharge area during the recharge period, and to groundwater pumped into the ponds during the early part of the operation that later returned to the pumping area and was potentially recycled into the system.
 The negative gravity differences in the east and southeast area of the study site (Figure 5) indicate that water levels beneath that area were higher after the pumps were turned off than they were during the pumping operation. We have identified two possible causes: (1) some of the water pumped into the recharge ponds may move eastward as the groundwater mound dissipated; and (2) seasonal water level fluctuations in that area range from 0.75 to 1 m, with the highest levels occurring during May [Halstead and Flory, 2003]. Both causes can result in higher water levels in this area following the cessation of pumping in comparison to water levels during pumping, resulting in the negative gravity difference east of the recharge area.
6.2. Estimates of Specific Yield
 In theory, the Bouguer slab model should systematically underestimate specific yield in areas where the water table is nonplanar. This is because the Bouguer slab model assumes the change in water mass is distributed over a wide area. Hence, small changes in water volume per unit area (equating to small specific yields) represent relatively large changes in total water mass and large changes in gravity. In contrast, models accounting for localized water mass changes require comparatively larger changes in equivalent water thickness to achieve the same change in total mass (and gravity) and therefore will predict a larger specific yield.
 Specific yield estimates may also be biased by the presence of water in the unsaturated zone. Changes in the amount of water in the unsaturated zone are detected as mass changes in the gravity data. In our parameterization, a change in water mass in the unsaturated zone is included in the estimate of changes in groundwater volume per unit area (Δh) derived from the gravity data. A change in water mass in the unsaturated zone is not accounted for in estimates of changes in the water table elevation (Δz) obtained from well data. This may potentially result in an overestimate of specific yield (equations (1) and (2)).
 The estimates of Sy on the basis of the Bouguer slab model is similar to or less than the Sy estimates on the basis of our model at wells T6, T13, T17, and T18 (Table 2). In contrast, at well T9 the Bouguer slab model predicts a higher specific yield than that obtained through inversion of the gravity data. The high specific yield obtained with the Bouguer slab method at well T9 may be due to the presence of water in the unsaturated zone. At wells T13, T17 and T18 the unsaturated zone is confined to the eolian sand, which has a low specific retention and therefore is expected to have low moisture content in the vadose zone. At well T9 changes in the water table elevation occur entirely in the alluvium, which has a greater specific retention than does the eolian sand and therefore may have relatively high moisture content in the unsaturated zone (although we note that well T6, also located in the alluvium, does not appear to be affected by this). Wells T13, T17, and T18 are all located near the main recharge pond. At these three sites the estimated Sy decreases with distance from the pond, from well T13 (closest to the pond) to T18 (furthest from the pond). During the 8 week period between gravity surveys, the surface area of the pond periodically expanded and contracted owing to variable pumping rates. This causes water to be stored in the unsaturated zone near the edges of the pond. Well T13 was within 5 m of the main recharge pond during phase 1 of the survey, and it is likely that water stored in the unsaturated zone at this time biases the specific yield estimate. Estimated groundwater mass changes at wells T17 and T18, conversely, are minimally affected by unsaturated zone groundwater storage because they are located over 35 m away from the recharge pond. Therefore, we do not include Sy estimated at site T13 in the mean value determination of Sy at Tamarack. The mean value of Sy in the eolian sand determined from wells T17 and T18 is 0.22 ± 0.03. The mean value of Sy in the alluvium, estimated at wells T6 and T9, is 0.19 ± 0.04. Averaging these values, we find a mean specific yield for the Tamarack site as a whole of 0.21 ± 0.04.
 The estimates of specific yield determined through inversion of the gravity data may be biased by changes in water table elevation at the base station since our analysis is predicated on the assumption that the water table elevation (and therefore gravity) at this location did not change between the two gravity surveys. As discussed in section 3.4, data from monitoring wells show water table elevation changes on the order of 0.6 m in the vicinity of the base station, corresponding to a change in gravity of up to 5 μGal (Figure 4). A 5 μGal change in gravity at the base station between surveys would produce a bias in the estimate of water mass per unit area (Δh in equation (1)) of up to 0.12 m. Considering this bias and following the method used previously to calculate specific yields, Sy at wells T6, T9, T13, T17, and T18, may be as high as 0.22, 0.24, 0.31, 0.26 and 0.23, respectively, or as low as 0.13, 0.13, 0.28, 0.22, and 0.15. Comparison to Table 2 shows that these Sy values lie close to or within the precision estimated earlier for specific yield, although the lower bound for the Sy estimates at wells T6, T9, and T18 may be slightly high (Table 2). The error range for specific yield at these wells may be underestimated because the change in water volume per unit area (Δh) at these wells is small in comparison to that at wells T13 and T17, so a 0.12 m error in Δh results in a relatively large bias in specific yield estimates.
7.1. General Statement
 The results presented in this paper provide a useful case study that can be used to help develop field and data processing protocols for similar temporal microgravity studies of other unconfined clastic aquifers. A crucial aspect of our methodology is that we re-occupy the same stations during each phase of the gravity survey and relate everything to the same reference station. Consequently, uncertainties in the terrain correction, Bouguer correction, and latitude correction that are usually required in analysis of gravity data cancel out when the gravity data are differenced.
 In regards to field protocol, this research helps to define: (1) the gravimeter to be used; (2) selection of a reference station; (3) station spacing; and (4) method of constraining instrument drift and Earth tide corrections. In regards to data analysis protocol, this research helps to define: (1) the method of gridding the data; and (2) how to invert the data to obtain the change in groundwater volume per unit area.
7.2. Field Protocols
 On the basis of the assessment of uncertainties discussed previously, a gravimeter with a nominal precision of 3–5 μGal is deemed to be sufficient to resolve water table elevation changes greater than approximately 0.5 m.
 Optimally, the reference station should be located on stable bedrock, several km distant from areas known to experience changes in water table elevation over time periods comparable to that of the survey. If this is not possible (as in the case of the study presented here), it is preferable that absolute gravity measurements be obtained at the reference station during each phase of the survey. In the absence of absolute gravity data, the reference station should be placed in an area where water table levels are static over periods comparable to that of the survey duration.
 The station spacing should be selected after the expected spatial scale of variations in the water table elevation of interest is considered. This is best done through preliminary forward modeling to determine the response of the gravity field to changes in the water table elevation that approximate those expected to occur during the survey. From the forward modeling, one can determine the maximum distance from the centroid of the water table elevation change that produces changes in gravity greater than the uncertainty in the measurements. In order to avoid spatial aliasing, this should be one-half the maximum spacing between gravity stations at most. For example, if the expected precision of the gravity measurements is 5 μGal and forward modeling indicates that changes in gravity resulting from the water table elevation change does not exceed 5 μGal at distances beyond 200 m from the well, then a station spacing of less than 100 m is required to assure detection of the gravity change A reasonable strategy (adopted in this study) is to consider the shortest-wavelength perturbation in the water table elevation that is expected and the smallest magnitude that it is required to resolve. In this study, we found the forward modeling method of Damiata and Lee  to be useful for this purpose since it provides the shortest-wavelength response in water table elevation changes that should be expected (point source groundwater extraction or addition), and allows control over the maximum magnitude of the water table elevation change. The method also has the advantage of using input parameters similar to those used by hydrogeologists to describe aquifer response to pumping.
 In regards to quantifying Earth tide variations during the study, a separate continuously recording gravimeter at the base station would help, but we do not believe this is necessary. That approach does not quantify instrument drift in the meter that is used to survey stations other than the one used at the base station. Instead, we recommend re-occupation of the base station at 1–2 hour intervals. In this study, use of a piecewise linear extrapolation of gravity measurements at the base station obtained by hourly re-occupation of the base station with the same instrument used to measure gravity at the other stations was adequate to capture both the Earth tide variations and instrument drift at the level of precision needed to resolve water table elevation changes.
7.3. Data Analysis Protocols
 The kriging method worked well in this study for data gridding, but gridding methods are sensitive to station spacing and the range of wavelengths over which the gridded surface changes. We suggest kriging as a first approach, but we also recommend that other gridding methods be tested. The optimal method can be identified by comparing the gridded data with measured data at key points and by assessing whether the gridded data represents a smoothly varying surface, which is expected from gravity data, as opposed to a surface that requires high curvatures (cusps) on the contours.
 The inversion method used here [Gómez-Ortiz and Agarwal, 2005] seems optimal for capturing the smoothly varying surface that is expected to represent changes in the water table elevation. The program requires an estimate of the magnitude of the mean change in water table elevation. Our results were not greatly sensitive to this parameter, but we recommend that a range of choices be examined to determine sensitivity of the outcome to this parameter choice. Our results indicated that other parameter choices required by the Gómez-Ortiz and Agarwal  algorithm can be the same as those we used, with the exception of the density contrast used to determine water table elevation changes. This parameter should be selected to be the product of the density of water (1 × 103 kg m−3) and the best estimate of specific yield of the aquifer that is available from independent studies.
 This research benefited by being located in an area where variations in water table elevation, barometric pressure, and ground surface elevations were minimal. Furthermore, spatially dense data constraining water table elevation changes were available at approximately two-week intervals, and the water budget can be reasonably balanced independently of the gravimetric data. Analysis of the uncertainties involved identify the following as important issues which should be considered in similar future studies: (1) changes in absolute gravity at the reference station; (2) relative elevation changes between the reference station and the field stations between gravimetric surveys; (3) the effect of barometric pressure changes, instrument drift, and Earth tides; (4) the influence of precipitation and evapotranspiration; and (5) the influence of water in the vadose zone.
 Variations in high-precision surface gravity measurements over time were compared to groundwater mass changes at the Tamarack managed groundwater recharge site. Positive gravity differences generally correlate with groundwater influx at recharge ponds and negative gravity differences correlate with the removal of water at pumping wells. Gravity differences range from −46 μGal near pumping well P7 to +90 μGal at well T17 (near the shore of the 3.9 ha recharge pond), where the water table rose 7.3 m. By integrating the gravimetrically constrained water mass change in the area of the recharge pond, we estimate that the decrease in groundwater storage that occurred between the first gravity survey (conducted when the pumps were running and the recharge mound was fully developed), and the second gravity survey (conducted six weeks after the pumps had been turned off and the recharge mound had fully dissipated) is about 5.1 × 105 m3. We believe this to be the volume of water sustained in the recharge mound in its steady state during pumping. Specific yield was estimated at four well sites by dividing the change in groundwater volume per unit area by the change in water table elevation. Mean specific yield in the eolian sand layer is estimated to be 0.22 ± 0.03 (wells T17 and T18). Sy in the alluvium is estimated to be 0.19 ± 0.04 (wells T6 and T9). These Sy estimates are within the range of those previously determined by aquifer tests. Specific yields determined by the planar Bouguer slab method, which neglects topography on the water table, are lower than those estimated using variable water table topography by as much as 0.09 (up to 41% difference).
 The water table elevation changes predicted from the gravimetric data agree to within 15% with water table changes measured in the wells. In the eolian sand, the difference between the predicted and measured water table change ranges from 0.3 m for a 3.20 m change in water table elevation to 1.1 m for a 7.28 m change in water table elevation. In the alluvium, the differences between the predicted and measured water table changes are 0.3 m for a 2.10 m change in water table elevation and 0.15 m for a 2.35 m change in water table elevation.
 Although gravity changes correlate well with groundwater mass changes, surface and vadose water mass adds complexity to this relationship. The gravitational effect of pond water at Tamarack was removed from the temporal gravity signal, but water in the vadose zone was not quantified. It is believed to be minimal in most of the field area owing to the lack of available unsaturated storage space.
 The results presented here demonstrate that temporal microgravity surveys can be used successfully to quantify groundwater storage changes in unconfined clastic aquifers independently of assumptions of the physical properties of the aquifer. By calibrating the storage change to changes in water levels in wells, specific yield can be estimated. Conversely, if specific yield is known, then the change in storage can be used to estimate changes in water levels.
 We thank three anonymous reviewers and the associate editor for comments that helped to improve this manuscript.