Corresponding author: Kang Hyeun Ji, Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. (firstname.lastname@example.org)
 Long Valley Caldera in eastern California is an active volcanic area and has shown continued unrest in the last three decades. We have monitored surface deformation from Global Positioning System (GPS) data by using a projection method that we call Targeted Projection Operator (TPO). TPO projects residual time series with secular rates and periodic terms removed onto a predefined spatial pattern. We used the 2009–2010 slow deflation as a target spatial pattern. The resulting TPO time series shows a detailed deformation history including the 2007–2009 inflation, the 2009–2010 deflation, and a recent inflation that started in late-2011 and is continuing at the present time (November 2012). The recent inflation event is about four times faster than the previous 2007–2009 event. A Mogi source of the recent event is located beneath the resurgent dome at about 6.6 km depth at a rate of 0.009 km3/yr volume change. TPO is simple and fast and can provide a near real-time continuous monitoring tool without directly looking at all the data from many GPS sites in this potentially eruptive volcanic system.
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 Long Valley Caldera (LVC) is an oval-shaped topographically depressed crater in eastern California (Figure 1a). From the late 1970s to 2000, LVC had experienced multiple episodes of uplift (1979–1980, 1983, 1989–1990, 1997–1998), and the resurgent dome in the central section of the caldera had developed nearly 80 cm of uplift with respect to its pre-1980 level [Langbein, 2003; Hill, 2006]. Since 2000, space geodetic techniques (e.g., Global Positioning System (GPS) and Interferometric Synthetic Aperture Radar) have revealed a series of inflation and deflation events over the caldera (2002–2003 uplift, 2004–2007 slow subsidence, and 2007–2009 slow uplift) [Feng and Newman, 2009; Liu et al., 2011; Hetland et al., 2012]. These recent events are much weaker than the ones before 2000 and suggest that the LVC magmatic system may be waning [Liu et al., 2011]. Testing this suggestion, however, essentially requires continuous monitoring of this active volcanic region.
 We have developed a technique for monitoring surface deformation with continuous GPS data [Ji and Herring, 2012]. The technique uses a predefined target pattern of surface deformation along which daily GPS position time series are projected (hereafter referred to as Targeted Projection Operator (TPO)). Ji and Herring  applied TPO to the data collected in the San Gabriel basin, California, and showed high correlation between the projected time series and changes in groundwater level in the basin. TPO is simple and fast so that it can be efficiently used as a near real-time monitoring system with 1 day latency for rapid processing and 14–20 day latency when final IGS GPS orbits are used. We have monitored surface deformation using TPO with LVC data from the Plate Boundary Observatory (PBO) network and found an inflation signal that started in late-2011 and is continuing at the time of writing (November 2012). The rate of inflation is faster than the 2007–2009 event, suggesting the source of deformation is not decreasing in strength at least over of the short time scale of a few years.
2 Data and Methods
 This study consists of three steps of processing: (1) reference frame stabilization, (2) selection of a target surface pattern, and (3) TPO. Since the reference frame used in PBO processing was designed for the entire western US and Alaska, the position estimates in small regions have spatially correlated noise caused by orbital, reference frame, and large-scale atmospheric errors [Wdowinski et al., 1997; Dong et al., 2006]. In the first step, the spatially correlated noise is reduced by translating, rotating, and scaling the reference frame to realize a more stable local frame [Ji and Herring, 2011]. For better performance, reference frame sites are selected in a broad area that covers and extends beyond the LVC region, distributed as uniformly as possible over the area, and have linear motions that can be modeled with small velocity uncertainties. Here we select stable sites based on the uncertainty of velocity estimates determined with a first-order Gauss-Markov noise model. We chose 83 stable sites between 35°N–40°N and 123°W–114°W whose velocity uncertainties are less than 0.1 mm/yr in horizontal and 0.3 mm/yr in vertical. The sites in the LVC region were not used as reference frame sites.
 The second step is to define a target pattern of surface deformation. A fundamental assumption in this step is that the target pattern is maintained with time and only its amplitude varies. This assumption is likely to be valid for the recent relatively quiet phases in LVC. Liu et al.  showed that the 2002–2003 uplift, the 2004–2007 subsidence, and the 2007–2009 uplift have similar surface pattern, and they may have the same source located at depths of ~6–8 km beneath the resurgent dome.
 There are 25 PBO sites available in the LVC area (Figure 1a). The site JNPR was not used in this study because it had operated until June 2007. The other sites began operation between 2006 and 2008; they were not fully available until September 2008. We also removed the site P628 because it shows a typical effect of snow covering the antenna during winter, and the sites P630, P631, and P642 because they have large time-varying annual signals. We used data up to January 2011 to simulate a monitoring system that would be used after that date. The average sample size is 1307 ± 255 days, and the average missing data is 0.58%.
 We applied a Kalman filter [Ji and Herring, 2013] to obtain residual time series from each component of the remaining 20 GPS sites (Figure S1). We removed from the raw data a secular rate, annual and semi-annual sinusoids, and outliers defined as five times larger than the standard deviation of the residual time series. We used a smoothing Kalman filter to smooth the time-series residuals.
 We applied principal component analysis (PCA) to the smooth time-series residuals to obtain a target spatial pattern. PCA uses the smooth time series from Kalman filtering and produces eigenvectors that represent spatial patterns. We selected the time interval between 2009.0 and 2011.0 because it provides better spatial coverage of GPS sites than the 2007–2009 inflationary event (i.e., more PBO sites had been installed by 2009). The eigenvector associated with the largest eigenvalue represents the slow deflation with a radial pattern from the resurgent dome (Figure 1a). This component was determined to be the only dominant and significant component. Only horizontal components were used in PCA because the vertical components are normally ~3 times noisier than the horizontal components. However, we will look at the vertical components for calculating rates of uplift.
 The final step is to apply TPO with the 2009–2010 deflation pattern through the whole time interval up to the present. The residual time series were projected onto the eigenvector pattern (i.e., principal axes) by using weighted least squares estimation. Suppose that xij is the residual time series of the jth component at the ith epoch where j = 1, 2, ⋯, p, εij is the error of xij, and aj is the jth element of the eigenvector. Then, the TPO solution and its standard error at the ith epoch are obtained, respectively, by
where is chi-square per degrees-of-freedom of the residuals rij to the TPO, rij = xij − yiaj, and pi is the number of components used in TPO at the ith epoch. When there are q(< p − 1) number of missing components in xij, the corresponding eigenvector elements are replaced with zero and pi = p − q. The uncertainty σi increases as misfits grow and/or q increases. Without significant number of missing components, large uncertainty (and also large ) may indicate different surface response from the target deformation pattern.
 TPO provides a high-resolution temporal history of surface deformation at LVC (Figure 1b). The outliers with large errors at the beginning are due to the small number of components (< 10 out of 40). The TPO time series shows a slow inflation event in 2007–2009, even though the inflation seems to slow down around 2008. In 2009–2011, the period used for our target pattern, LVC deflated between mid-2009 and mid-2010 and went back to the level before the 2007–2009 event. An inflation event began in late 2011 and slowed down slightly in mid-2012 and then has increased its rate again. This event is about four times faster than the 2007–2009 event, suggesting that the LVC deformation source increases in strength.
 Figure 1c shows the values of in equation (2). The increased values of between 2007 and mid-2008 may indicate a surface deformation pattern different from the target and/or small number of available components. Because all of the 20 sites have been fully operated since mid-2008, the increased after 2011 is more likely caused by differences in surface pattern between the recent event and the target surface pattern.
 We calculated the velocities associated with the recent inflation in both horizontal and vertical directions by fitting a straight line to the residual time series only after 2011.75 (Table S1). Since LVC is actively deforming, it is difficult to account for the effect of temporally correlated noise in the time series on the estimation of velocity uncertainties. Instead, we simply scaled the velocity uncertainties, which are based on white noise assumption, by a factor of 5 which is given by Williams  for one-year duration of data and under the assumption of flicker noise. The observed horizontal velocity field (Figure 1a) shows subtle differences in both directions and relative amplitudes among sites that caused the increase in , but the differences are mostly not significant with respect to 95% error ellipses. The observed vertical velocities clearly show a significant uplift in LVC (Figure 2b). The maximum horizontal and vertical velocities are, respectively, 10.1 ± 1.9 mm/yr and 19.9 ± 3.9 mm/yr at P639 which is located on but near the south edge of the resurgent dome.
 A simple Mogi source [Mogi, 1958; McTigue, 1987] was used to model the observed velocity field. Model parameters were estimated by using the Levenberg-Marquardt nonlinear weighted least squares algorithm [Seber and Wild, 2003]. Because of the trade-off between depth and volume change, we fixed the horizontal location of the source estimated in the previous step and then applied grid search for the depth and volume change to minimize the chi-square statistics. The 95% confidence intervals for the two parameters were obtained from 10,000 Monte Carlo simulations with velocity fields perturbed by the velocity uncertainties. The deformation source in our estimates is located under the resurgent dome (longitude: −118.916, latitude: 37.694; see Figure 2) at a depth of about 6.6 km with a rate of volume change of 0.9 × 107 m3/yr. Figure 2 shows that the Mogi model explains most of the observed velocities within the velocity uncertainties (), even though the far-field vertical velocities are not well matched.
 The observed velocities at P632 and P635 seem to be faster in the west direction than those inferred from the model (Figure 2a). These sites are located in the west side of the Mono-Inyo craters. Feng and Newman  detected an opening of the craters possibly triggered by the 2002–2003 inflation, suggesting interaction between the craters and the LVC magma. Even though the velocity uncertainties are large at the two sites, the faster westward motions may suggest another opening due to the recent event.
4 Discussion and Conclusions
 Our TPO approach used PCA to obtain the 2009–2010 slow deflation along which residual time series were projected. However, TPO can be applicable to any surface deformation pattern as a target. For example, we performed TPO with the calculated velocities from the Mogi model with and without the sites P630, P631, and P642 to see how those sites affect the TPO results. The three sites were not included in the PCA-based target because of their large time-varying annual signals. The TPO time series without the sites is similar to the PCA-based result because of similar spatial patterns. In contrast, the TPO time series with the sites is rotated about the center of the time interval (Figure S2) because the secular rates of the sites were incorrectly estimated due to the annual signals. Therefore, the TPO performance is poor at the beginning and end of time period as indicated by increased (Figures S2b and S2d). We can also see the annual signals mapped into the TPO time series during summer periods between 2008 and 2011 (Figure S2c). Finally, both cases have the minimum around 2010.0, which suggests that the horizontal surface pattern of the Mogi model, obtained from the recent inflation, better represents the 2009–2010 slow deflation.
 More data from the U.S. Geological Survey (USGS) are available particularly inside of the caldera (see http://earthquake.usgs.gov/monitoring/gps/LongValley/), but we do not currently include them in our monitoring system for the following reasons. First, data processing is not consistent between PBO and USGS products. Such inconsistency may raise an issue for small variations in the data. Second, we are routinely operating the monitoring system with the PBO data. It may not be guaranteed that new data are simultaneously available at the system operation time. Third, the Mogi models inferred from both data types do not significantly differ within the 95% confidence interval (Figure S3). Although the USGS sites are denser in the south part of the caldera, these southern sites are less useful because they suffer from large time-varying annual variations; most of model misfits are from the southern sites (Figure S3). When we characterize the annual variations, the USGS data will improve the network geometry and place better constraints on deformation models at LVC. We are currently investigating the nature of the time-dependent annuals.
 We also applied a vertical ellipsoidal model [e.g., Yang et al., 1988], a preferred model in previous studies [e.g., Fialko et al., 2001; Langbein, 2003; Tizzani et al., 2009], to both PBO and USGS data. With fixed horizontal location at the Mogi source, we used grid search for depth, volume change, and an axis ratio (i.e., semi-minor/semi-major). In both cases, the ellipsoidal model is identical to the Mogi model with the axis ratio of 0.99 within the uncertainty; the vertical ellipsoidal model may not be resolvable for this weak event.
 We have not seen a significant increase in seismicity rate associated with the recent inflation event in the LVC region (Figure S4). The Epidemic-Type Aftershock Sequence (ETAS) model [Ogata, 1988], a stochastic seismicity rate model, is able to fit the seismicity since 2007 with constant background rate and triggering parameters. There has been a low level of seismic activity in the caldera since the 1997–1998 inflation event [Liu et al., 2011], occasionally punctuated by brief earthquake swarms. Earthquake swarms have been observed primarily within the south moat and beneath Mammoth Mountain, the largest of them occurring in the 1980s and 1990s. As suggested by Hill , because of their smaller spatial scale and delayed onset by several weeks relative to deformation, earthquake swarms could be a secondary effect associated with the caldera inflation, triggered by magmatic fluid intrusion. Prejean et al.  suggest that inflation of the resurgent dome produced only relatively small stress perturbations to the regional stress field, and so as in the post-2000 events, the recent inflation may be too small to trigger earthquakes.
 TPO is fast and simple because it does not involve any large-scale matrix inversion. With about 6 year long time series of 20 sites, TPO takes less than a minute including reference frame stabilization on a midspeed computer. Therefore, TPO is potentially applicable in a near real-time analysis environment. Furthermore, TPO provides one time series that shows a high-resolution deformation history so that we do not need to examine three components of a number of GPS sites in hand. The TPO time series in Figure 1b directly shows that the recent inflation event is larger than previous events since 2007 and is continuing at the present time (November 2012). To determine whether the increase in source strength is a short-term variation, until when the event will be continuing, and more importantly, whether future caldera unrest could culminate in magmatic eruption will require continuous monitoring of LVC surface deformation; TPO can play an important role in this regard.
 We thank John Langbein, David Hill, Andy Michael, and two anonymous reviewers for their constructive comments which improved this paper. 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].