Continuous Global Positioning System (GPS) networks record station position changes with millimeter-level accuracy and have revealed transient deformations on various spatial and temporal scales. However, the transient deformation may not be easily identified from the position time series because of the large number of sites in a network, low signal-to-noise ratios (SNR) and correlated noise in space and time. Here we apply state estimation and principal component analysis to the daily GPS position time series measured in Alaska sites of the Plate Boundary Observatory network. Our algorithm detects a transient signal, whose maximum displacement is ∼9 mm in horizontal and ∼11 mm in vertical, that occurred at Akutan volcano during the first half of 2008. A simple Mogi source inversion suggests inflation at shallow depth (∼3.9 km) beneath the volcano. Although the detection was not easy because the signal was aseismic, non-eruptive and weak (not apparent in raw daily time series), our detection method improves the SNR and therefore provides higher resolution for detecting the transient signal.
 Continuous Global Positioning System (GPS) networks, e.g., Plate Boundary Observatory (PBO; http://pbo.unavco.org/), provide data that allow a broad temporal and spatial spectrum of millimeter-level surface deformations to be studied. GPS position time series have revealed transient deformation due to earthquakes, volcanic activities, groundwater transport, and other processes. The large size of the networks, however, makes it time consuming to visually inspect each GPS time series to identify transient signals. Furthermore, GPS time series suffer from spatially [e.g., Wdowinski et al., 1997; Dong et al., 2006] and temporally correlated noise [e.g., Zhang et al., 1997; Mao et al., 1999; Williams et al., 2004; Langbein, 2008] that may mask transient signals depending on signal-to-noise ratios (SNR) of the transient. A transient signal detector needs to handle data from large networks while accounting for the correlated noise.
 We apply our detection algorithm to the PBO daily GPS position data collected in Alaska. The algorithm is based on state estimation in a Kalman filter formulation [e.g., Gelb, 1974; Anderson and Moore, 1979] and principal component analysis (PCA) [e.g., Jolliffe, 2002; Jackson, 2003]. The main purpose of the state estimation is to improve the SNR in the time domain by estimating secular velocity, seasonal sinusoids, time-correlated noise, and transient signals if any. PCA improves the SNR in the space domain by accounting for the spatial coherence of transient signals. PCA transforms the complex spatiotemporal structure of data into uncorrelated principal components (PCs) for the temporal variations and orthogonal sample eigenvectors for the spatial distributions. Due to the simple space-time separation and other advantages, PCA has been widely used in crustal deformation studies [e.g., Savage and Langbein, 2008; Savage and Svarc, 2009; Kositsky and Avouac, 2010; Lin et al., 2010].
 The algorithm has detected a transient signal that occurred at Akutan volcano, ∼1224 km southwest of Anchorage, during the first half of 2008. The transient signal is consistent with a volcanic inflation source. The signal, however, is not easy to identify in the original data (see Figure S1 of the auxiliary material) because of the low SNR and highly correlated noise.
2. Data and Method
 The PBO GPS analysis consists of two stages (details can be found at http://pboweb.unavco.org/?pageid=101). First, two analysis centers, Central Washington University using GIPSY/OASIS software [Webb and Zumberge, 1997] and New Mexico Institute of Mining and Technology using GAMIT/GLOBK software [Herring et al., 2009], process phase and pseudorange data to estimate station coordinates averaged over 24-hour durations. The Analysis Center Coordinator (Massachusetts Institute of Technology) combines the results from the first stage and aligns them to the PBO realization of the Stable North America Reference Frame (SNARF; see http://www.unavco.org/research_science/workinggroups_projects/snarf/snarf.htm). Final solutions are available with 14–20 day latency due to the availability of GPS orbits. The standard PBO GPS level-2 products are available from the PBO Archives. We used daily position time series of 151 Alaska sites between 1 January 2004 and 13 February 2010. The sample sizes are between 220 and 2235 with an average of 1065 ± 540.
 Our detection algorithm uses a smoother based on Kalman filtering and PCA. The Kalman filter state vector includes a secular rate, annual and semi-annual sinusoids, and a first-order Gauss-Markov (FOGM) process used to account for temporally correlated noise and also transient signals, if any. The output of the smoother, which are estimates of the FOGM stochastic process, is used as the input of the PCA. The FOGM process is not necessarily a physical process for both temporally correlated noise and transient signals. Other noise models have been reported such as random-walk, flicker, or power-law noise [e.g., Williams et al., 2004; Langbein, 2008]. The FOGM process provides a stochastic model that can accommodate correlated noise and signals, and it can be easily implemented in Kalman filters. When multiple FOGM processes are combined, they can approximate the spectral shape of many, but not all, noise processes [Langbein, 2008].
 PCA decomposes the FOGM state estimates into principal components (PCs) for the time history and sample eigenvectors for spatial distribution. We do not include the linear and sinusoidal components in the PCA. PCs have the unit of length and sample eigenvectors are dimensionless. Sample eigenvalues are the variances of PCs; larger sample eigenvalues imply more temporal variations in PCs. The PC uncertainties are obtained by error propagation. The sample eigenvector uncertainties are an asymptotic result derived with an effective sample size that accounts for temporal correlation (see the auxiliary material for details). These uncertainties allow us to infer the significance of the PCA results using a conventional χ2 test. When a PC contains a transient signal, chi-square per degree of freedom (χdof2) of the PC will be larger than that expected due to Gaussian noise in the estimates. Significant deviations from random behavior can also be tested with lag differences of a PC, i.e., differences between points of a PC at a given time lag (see the auxiliary material for details).
3. Spatially Correlated Noise
 GPS time series suffer from spatially correlated noise (see Figure S1); often referred to as “common mode error” (CME) that may result from orbital, reference frame and large-scale atmospheric errors [Wdowinski et al., 1997; Williams et al., 2004; Dong et al., 2006]. Discrimination between signal and noise in space can be made by the fact that tectonic sources produce localized and systematic pattern in space, while the CME has a relatively uniform pattern over large areas.
 As was found by Dong et al.  for the sites in southern California, we found that the Alaskan sites also contain spatially correlated noise (Figure 1a). The largest two horizontal PCs show two orthogonal CMEs whose temporal patterns are neither purely random nor purely systematic and whose spatial patterns look uniform over the network. Even though the CME is a function of the wavelengths of various systematic errors [Dong et al., 2006], the spatial uniformity of the CME is a valid assumption for the Alaska network, implying that the network size is smaller than the CME wavelength.
 To suppress the CME, one may simply ignore the PCs containing the CME and examine the next PC. Figure 1b shows that the third PC exhibits non-uniform spatial pattern but the temporal pattern is still noisy. Due to the orthogonality in PCA, the third PC may be contaminated by the first two PCs and vice versa.
 Another approach to suppressing the CME is a reference frame transformation by translation, rotation and scaling of the network (i.e., a seven-parameter Helmert transformation). We transformed data in SNARF into those in a regional frame that was realized by 5 ∼ 48 reference frame sites at each epoch, depending on the availability of data and the solution quality, whose spatial distribution is relatively uniform. The velocity uncertainties of these sites are less than 0.45 mm/yr in horizontal and 0.9 mm/yr in vertical when velocity uncertainties are estimated accounting for correlated noise. The resulting first PC shows a non-uniform spatial pattern and exposes a transient event starting in early 2008 and ending by mid-2008 (Figure 1b). For further investigation, we will use these data with the CME removed through a reference frame transformation.
4. Signal Identification and Significance
 We have detected a transient signal during early 2008, but the location of the signal is not clear. The signal can be isolated in space and time by including in the PCA only those sites and the time interval that experiences the target signal. Because the maximum amplitude of the spatial pattern in Figure 1b occurred at the station AV07, one of the Akutan volcano sites, and sites in this region also show large amplitudes, we performed PCA with 7 sites on the Akutan Island (see Figure 2 for site location and also see Figure S1 for the raw time series). We used a one-year time interval starting at 1 October 2007 that spans the detected signal. The station AV13 was not included in PCA because of the outliers at the beginning of 2008 in the east component (see Figure S1), that are believed due to snow on the antenna radome. Figure 2a shows that the spatial and temporal patterns of the resulting first PC were similar to those shown in Figure 1b but with improved SNR (i.e., increase in χdof2 from 3.5 to 4.3). The sample eigenvalue also increased from 20.8% to 64.5% of the total variance of the data set, resulting in a more focused signal. The maximum horizontal displacement of the signal is ∼9 mm, which is too small to detect by satellite radar interferometry (∼3.5 mm line-of-sight displacement).
 We were not able to identify from the vertical PC any signal related to the horizontal signal (Figure 2b). The vertical PC shows that all sites appear to move in the same direction with relatively uniform amplitudes, and the temporal pattern has a χ2 value consistent with simply random noise (χdof2 = 1.0). To examine if the identified transient signal also exists in vertical, we first fit an arctangent function to the horizontal PC (see Figure 2a and Table S1) and then obtained displacements by fitting the estimated arctangent function to the original time series with other parameters (see Figure S2 and Table S2). The horizontal displacements are similar to those from PCA but with slightly larger amplitudes and smaller uncertainties (Figure 2a). The vertical displacements are more concentrated in the near field (Figure 2b), which demonstrates that the vertical signal exists (also see Figure S2).
 The significance of the transient signal can be determined in several ways. Firstly, the first horizontal PC is the only PC having a systematic temporal pattern (Figure S3). Secondly, the PC explains ∼65% of the total data variance. Thirdly, χdof2 = 4.3 of the PC is larger than the 95% upper limit expected if the signal were simply random noise (Figure S3). Fourthly, lag differences of the PC show that early 2008 contains significant anomalies for lags of more than 100 days (Figure S4). Finally, the systematic radial pattern shows spatial coherence of the signal.
 Akutan volcano is one of the most active volcanoes in the Aleutian arc. The most recent eruption was a series of small steam and ash emissions from March to May 1992 [Lu et al., 2000]. The identified transient signal may be driven by volcanic sources (e.g., inflation or pressurization of magma bodies beneath the volcano). We used a Mogi point source [Mogi, 1958] to model the 3-dimensional surface displacements obtained from the arctangent parameterization (Figure 2 and Table S3). The estimated Mogi source locates near the volcano summit at a shallow depth (∼3.9 km below sea level) and explains most of the identified signal. The model prediction indicates that the uncertainties from PCA are more realistic than those of the arctangent fits. The uncertainties of the arctangent fits are too small because they are based on a white noise assumption whereas the PCA uncertainties account for temporally correlated noise. The station AV13 was not included in the source inversion. Its predicted displacements differ ∼2.5 mm in horizontal and ∼4 mm in vertical, which are consistent with their uncertainties. If the station AV13 is included in the source inversion, the horizontal and vertical differences decrease to ∼0.7 mm and ∼1.3 mm, respectively, and the source locates slightly deeper (∼4.3 km; see Table S3).
 Using satellite radar interferometry, Lu et al.  mapped ground cracks of more than 60 cm uplift on the northwest flank of the volcano associated with the March 1996 earthquake swarm and constrained the deformation as a composite source: shallow dike intrusion for the deformation associated with the ground cracks and a deep Mogi source for volcano-wide inflation. The Mogi source is deeper (∼13 km) than our estimates (∼3.9 km).
 The 2008 inflation signal was aseismic. The inflation was not associated with increase in sesimicity [Dixon et al., 2008; Dixon and Stihler, 2009]. The number of located earthquakes in 2008 was twice that in 2007, but at a similar level of seismicity prior to 2007. Furthermore, there was only one earthquake recorded while the inflation was ongoing [see Dixon and Stihler, 2009, Figure A21].
 The 2008 inflation event did not culminate in an eruption, and there were no inflation or deflation events in 2007 or in 2009. The 2008 inflation may be an episode that accumulates magma in a shallow reservoir. Many volcanoes inflate in an episodic fashion [e.g., Lu et al., 2002; Fournier et al., 2009]. When a threshold is reached through a series of episodes, magma will eventually erupt. The inflation associated with the March 1996 earthquake swarm can be regarded as such an episode. Continuous GPS monitoring of volcano deformation can help constrain the characteristics of the eruption cycle.
 We have detected a transient signal at Akutan volcano during the first half of 2008. A simple Mogi source at a shallow depth (∼3.9 km below sea level) can explain most of the displacement field. The event detected could in part of an inflationary sequence that will eventually lead to an eruption. There is no evidence for deflation of the volcano during the time span of the data analyzed here (2005–2010). The aseismic and non-eruptive inflation signal is difficult to detect in raw time series because of low SNR, the large size of the network and correlated noise in space and time. State estimation and PCA improved the SNR and successfully revealed the transient signal imbedded in the original time series. Our detection algorithm can provide higher resolution for accurately detecting transient signals and the resulting solutions can be further used for modeling and interpreting the transient deformation.
 This work was supported by NSF EAR-0734947, NASA NNX009AK68G, and the Southern California Earthquake Center and NSF cooperative agreement EAR-0529922. We also thank two anonymous reviewers for thoughtful and constructive comments, which improved this paper.
 The Editor thanks Jeffrey Freymueller and an anonymous reviewer for their assistance in evaluating this paper.