Changes in groundwater levels cause water-bearing basins to deform. Here we provide a detailed history of horizontal surface displacements in the San Gabriel Valley, California, which we show are highly correlated with the water level changes measured at a nearby well: correlation coefficient of 0.96 ± 0.01. We use the surface response from a one year period during which water level change in the nearby well was over 16 m, to monitor the deformations over a 14 year period from 1998 to 2011. The water level changes lead surface deformation by 21 days, but the significance of the offset is only weakly supported (p-value = 0.129). The projection method can be used as a real-time monitoring or detection system for deformation caused by a variety of mechanisms such as fluid injection and removal (groundwater development, oil production, carbon sequestration), and by seismic and volcanic activity.
 The San Gabriel Valley is a water-bearing sedimentary basin located in eastern Los Angeles County, California (Figure 1). The basin consists of Quaternary alluvial sediments ranging from coarse gravel to fine-grained sands deposited by streams and rivers flowing out of the San Gabriel Mountains [California Department of Water Resources, 1966]. The area of the deposits is about 200 square km and the sediment thickness is more than 1 km at the center of the basin [King et al., 2007]. Major sources of groundwater include infiltration of rainfall over the basin and percolation of runoff from the adjacent mountains. Groundwater generally flows from the margins toward the center of the basin following topographic slope and mostly exits through the lowest point of the area and a narrow floodplain, Whittier Narrows. Faults and hills surrounding the basin form barriers to groundwater flow.
 Changes in recharge and discharge of groundwater in a basin affect groundwater elevation and cause the basin to deform [Galloway et al., 1999]. When groundwater levels increase during recharge, fluid pressure in the pores and cracks of aquifer systems, especially in unconsolidated rocks, increases (i.e., decrease in the effective stress). Thus the basin surface protuberates and expands. Conversely, when groundwater levels decrease during discharge, the pore fluid pressure decreases (i.e., increase in the effective stress) so that the basin surface subsides and contracts.
 Surface deformation attributed to hydrogeologic processes in groundwater basins has been successfully observed by Interferometric Synthetic Aperture Radar (InSAR) and Global Positioning System (GPS) [e.g., Bawden et al., 2001; Watson et al., 2002; Lanari et al., 2004; Argus et al., 2005; King et al., 2007; Motagh et al., 2007]. Using both techniques, King et al. reported an expansion (∼10 mm) and uplift (∼47 mm) of the San Gabriel Valley aquifers recharged after near-record rainfall (∼95 cm on downtown Los Angeles which is 2.5 times larger than the seasonal average of 38 cm [King et al., 2007]) in winter of 2004–2005 (hereinafter referred to as the 2005 rain event). The observed displacements are well correlated with the abrupt increase (∼16 m) in groundwater levels for about 5 months. Based on the history of abrupt water level increases at a nearby well, King et al.  predicted that other signals caused by large amount of groundwater recharge are likely to occur in the future.
 In this study, we test if surface deformation is an ongoing response of the San Gabriel Valley to smaller and/or slower changes in recharge and discharge in the aquifer system. We report a fine-resolution history of surface deformation over the period of 1998–2011 under the assumption that the basin is likely to produce the same spatial pattern of deformation as the basin responded to the 2005 rain event. We used only horizontal components of daily GPS time series which are more accurate than the vertical components and InSAR range changes, even though the latter two are also useful for monitoring land subsidence and uplift.
2. Data and Methods
 We used daily GPS time series from 11 sites (Figure 1) processed by the Scripps Orbital and Permanent Array Center (http://sopac.ucsd.edu/). The longest time series is from September 2, 1994 to July 31, 2011 at site CIT1, and the sample sizes are between 2416 and 6015 daily position estimates with an average number of 4342 ± 1055 days. To suppress common mode errors over the network in Southern California, the time series, originally in the ITRF2005 reference frame [Altamimi et al., 2007], were rotated to a North America fixed frame and aligned to a regional frame realized by 87 reference frame sites within 380 km of the San Gabriel Valley. These sites were chosen because their horizontal and vertical velocity uncertainties were less than 0.15 and 0.3 mm/yr, respectively, when a correlation noise is used.
 Since 1932, groundwater levels have been measured in the Baldwin Park Key Well located near the center of the basin (Figure 1). Although the water level data from the Key Well may not accurately represent the aquifer system over the entire area, the data have been used to monitor changes in the amount of water stored in the aquifer system. The water level data have been obtained monthly since 1970. Earlier data were more frequently sampled. The highest level above the sea level was 99.3 m on July 13, 1944 and the lowest was 57.8 m on December 9, 2009. Fluctuations in water level have been within 25 m over the last 20 years including about 16 m increase due to the 2005 rain event. The latest level is 71.4 m on June 28, 2011. The data are available from the U. S. Geological Survey groundwater information page (http://nwis.waterdata.usgs.gov/ca/nwis/gwlevels) with site name of 001S010W07R002S.
 Our primary assumption is that the groundwater-driven deformation in the San Gabriel basin is the same as that due to the 2005 rain event. Under the assumption, our approach to data analysis involves two steps: 1) obtaining a spatial pattern that represents the surface deformation caused by the 2005 rain event and 2) projecting the GPS data onto the 2005 spatial pattern over the whole period of time.
 We obtained the 2005 spatial displacement pattern by using a smoother and principal component analysis (PCA) with one-year interval of time series beginning October 1, 2004. The smoother estimates a secular rate, annual and semi-annual sinusoids, and a first-order Gauss-Markov (FOGM) process. We chose the FOGM process as a flexible stochastic representation in the Kalman filtering to account for the 2005 rain event as well as temporally correlated noise in the GPS time series [e.g.,Williams et al., 2004; Langbein, 2008]. Then PCA decomposes the FOGM state estimates into principal components (PCs) for temporal patterns and sample eigenvectors for spatial patterns. See Ji and Herring  and Ji  for more detailed description of the method.
 In order to obtain the entire time history (1994–2011) of surface deformation, we projected the data time series onto the principal axes (i.e., sample eigenvector) associated with the largest sample eigenvalue obtained in step (1). Before the projection, we removed a linear rate from the data time series but retained seasonal signals (i.e., annual and semiannual sinusoids) because hydrologic signals could be seasonal [e.g., Hoffmann et al., 2001; Watson et al., 2002]. Only horizontal components were used because vertical displacements are noisier and dominant at only one station, LONG (Figure 1). Because of their location at the margin of the valley, the sites except for LONG produce small vertical displacements. Therefore, the time series projected onto the principal axis are similar to the vertical displacements at LONG over the entire time period.
 A problem arises with the projection when data from one or more stations are not available. The projected values are underestimated in amplitude because of the contribution from the missing data. To compensate for the contribution from the missing data we rescaled the sample eigenvector using weighted least squares (i.e., by fitting the sample eigenvector to the data at each epoch). For weights, we used formal errors of the time series. In this way, we can mitigate the effect of missing data and make the projection more or less in a consistent framework from noisy data.
 The spatial PC pattern represents an expansion and uplift of the basin and the temporal pattern is highly correlated with the 2005 rain event (Figure 1). The stations, except for LONG, are sensitive to horizontal deformation because they are located at the margins of the basin. The maximum horizontal and vertical displacements are about 10 mm at LPHS and about 30 mm at LONG, respectively. A simple elastic model showed that vertical displacements dominate over the basin and horizontal displacements are largest near the margins up to one third of the vertical displacements consistent with the observations [Bawden et al., 2001; King et al., 2007]. The station CIT1, which has small displacements, is located in the Raymond basin and the small displacements support the premise that the Raymond Fault is an effective barrier between the Raymond and the San Gabriel basins along the southwest part of the fault [California Department of Water Resources, 1966]. The displacements of GVRS and RHCL are relatively small, which suggests different response between the stations in the hills and on the basin floor.
 The projection produces a time series (hereinafter referred to as the projected time series) that represents time history of horizontal surface deformation along the principal axes associated with the 2005 rain event (Figure 2). Since 1998.5 when more than 4 stations are available, the projected time series and water level are highly correlated with the correlation coefficient of 0.96 ± 0.01 (0.85 ± 0.04 from vertical components). The correlation coefficient was computed using averaged monthly values of the projected time series because the groundwater levels are available (total 145 samples). To determine the likely correlation between two random series, we simulated time series with the autocovariance structure of the groundwater level and the propagated time series, and cross-correlated these series. The 95% confidence interval of the correlation coefficients has a range of ±0.17.
 We also calculated correlation coefficients with time lags. The maximum correlation (0.9673) occurred at 21-day lag with the water level changes leading the surface deformation by 21 days. Analysis of the time series shows that noise could generate lags of this amount with the p-value of 0.13. At a 5% significance level, we cannot reject the null hypothesis that the lag is zero.
4. Discussion and Conclusions
 We obtained a horizontal surface deformation history in the San Gabriel Valley by projecting detrended data onto the principal axes derived from PCA with 1-year length data including the 2005 rain event. The projected time series provides an unprecedented temporal detail of surface deformation and results in high correlation (correlation coefficient of 0.96) with the changes in groundwater levels when there are four or more sites available.
 For the relationship between groundwater elevation and surface deformation in the San Gabriel Valley, previous studies have used the InSAR range changes, the vertical component at LONG, and the aerial dilatation from the horizontal components of 7 stations. Argus et al.  reported from the analysis of InSAR data that the basin center subsided at −8 mm/yr from May 1998 to May 2000, which is consistent with the decrease in the water level and confirmed by King et al.  from the vertical component at LONG. However, the vertical displacements are too noisy to provide detailed picture of temporal correlation. The horizontal aerial dilatation agreed with the decrease in water level since 2000 [King et al., 2007], but it represents only an isotropic response to the water level changes. This study used principal axes that represent a multi-dimensional space with different amplitudes of total 22 horizontal components (Figure 1), which accounts for heterogeneity and anisotropy of the aquifer system response.
 The projection method seeks a component of surface deformation along a principal axis. A surface pattern orthogonal to the principal axis will not project into the principal axis (e.g., a rotational signal with respect to a principal axis in a radial direction). Complex basins can behave differently in subsequent hydrologic events. In this case, the projection method is less useful. We expect that surface responses to changes in groundwater level are similar if the geometry and structure of basins have not severely altered over the time. Figure 2 demonstrates that weaker signals in the San Gabriel Valley have also occurred in a radial direction. Inelastic deformation (e.g., permanent compaction of aquitards) can change the way of surface response, but it dominantly occurs in vertical components and over a longer time period.
 The analysis of correlation coefficients with time lags indicates that changes in groundwater levels lead surface deformation by 21 days. In the Los Angeles basin, Watson et al.  also observed delayed surface response to changes in precipitation by about 3 months. These delayed surface responses would be expected when groundwater flows through the aquifer system with time. In the San Gabriel Valley, the groundwater may flow from the center of the basin at which the Baldwin Park Key Well is located to the margins of the basin at which the GPS stations are located. This interpretation, however, is not consistent with the observations that the groundwater flows are generally controlled by topographic slope, that is, flows from the margins toward the center [California Department of Water Resources, 2004]. According to the error analysis for the lagged correlation coefficients, the delayed response is weakly supported with a p-value of 0.129. More frequent sampling of water levels at the Baldwin Park Key Well and also at more wells distributed over the basin will be useful for detailed pattern of the groundwater flows.
 Lagged drainage of fine-grained aquitards into coarse-grained aquifers [Galloway et al., 1999], reducing the pore volume in aquitards and resulting in permanent (or inelastic) compaction when the stress exceeds the preconsolidation threshold, might be considered for the delayed response of surface deformation. However, the 21-day delay is not likely attributed to the lag in aquitard drainage because it may require decades or centuries to attain equilibrium in fluid pressures depending on the thickness, permeability, and compressibility of aquitards. The projected time series mostly show short-term elastic response to the recharge-pumping cycle of groundwater. Nevertheless, the unrecoverable compaction is an ongoing process supported by the negative long-term vertical rates of stations (e.g., −1.51 ± 0.17 mm/yr at LONG). Inelastic horizontal deformation cannot be easily verified because tectonic motions are dominant in horizontal rates.
 The projection method developed in this study can be used as a real-time monitoring system for transient deformation because it is simple and fast. The results in this study clearly show that the projection method works well for describing surface responses to changes in groundwater levels. The method can be applicable in a real-time fashion to monitoring hydrologic hazard (e.g., land subsidence, earth fissures, sinkholes, etc.) caused by fluid injection and removal (e.g., groundwater development, oil production, carbon sequestration) and also to other types of deformation mode (e.g., earthquake and volcano deformation). Normally, at least one previous deformation event (like the 2005 rain event used in this study) should be available for the derivation of principal axes along which GPS time series are projected, but principal axes can also be obtained from pre-defined models or from other sources other than PCA of GPS measurements.
 We thank Nancy King and Stephane Mazzotti for thoughtful and constructive comments. This work was supported by NSF EAR-0734947-03, NASA NNX009AK68G, and the Southern California Earthquake Center and NSF cooperative agreement EAR-0529922. Figures were created by using the Generic Mapping Tools [Wessel and Smith, 1998] and MATLAB.
 The Editor thanks Nancy King and Stephane Mazzotti for their assistance in evaluating this paper.