InSAR time series analysis is increasingly used to image subcentimeter displacement rates of the ground surface. The precision of InSAR observations is often affected by several noise sources, including spatially correlated noise from the turbulent atmosphere. Under ideal scenarios, InSAR time series techniques can substantially mitigate these effects; however, in practice the temporal distribution of InSAR acquisitions over much of the world exhibit seasonal biases, long temporal gaps, and insufficient acquisitions to confidently obtain the precisions desired for tectonic research. Here, we introduce a technique for constraining the magnitude of errors expected from atmospheric phase delays on the ground displacement rates inferred from an InSAR time series using independent observations of precipitable water vapor from MODIS. We implement a Monte Carlo error estimation technique based on multiple (100+) MODIS-based time series that sample date ranges close to the acquisitions times of the available SAR imagery. This stochastic approach allows evaluation of the significance of signals present in the final time series product, in particular their correlation with topography and seasonality. We find that topographically correlated noise in individual interferograms is not spatially stationary, even over short-spatial scales (<10 km). Overall, MODIS-inferred displacements and velocities exhibit errors of similar magnitude to the variability within an InSAR time series. We examine the MODIS-based confidence bounds in regions with a range of inferred displacement rates, and find we are capable of resolving velocities as low as 1.5 mm/yr with uncertainties increasing to ∼6 mm/yr in regions with higher topographic relief.
 Interferometric Synthetic Aperture Radar (InSAR) time series analysis, with its broad spatial coverage and ability to image regions that are sometimes very difficult to access, is a powerful tool for imaging surface deformation and its temporal variations on land [e.g., Massonnet et al., 1993; Burgmann et al., 2000; Ferretti et al., 2001; Colesanti et al., 2003; Ferretti et al., 2007; Lanari et al., 2004]. These observations can inform modeling of subsurface tectonic and anthropogenic processes such as fluid withdrawal and injection [Lanari et al., 2004], earthquakes and transient aseismic fault slip [e.g., Massonnet et al., 1993; Fialko et al., 2001; Fielding et al., 2004; Lohman and McGuire, 2007; Barnhart et al., 2011; Barnhart and Lohman, 2013], volcanic inflation/deflation [e.g., Wicks et al., 2002; Fialko and Pearse, 2012; Henderson and Pritchard, 2013], and interseismic motion across faults [e.g., Wright et al., 2004; Elliott et al., 2008]. The precision of measurements in individual interferograms is impacted by several sources of noise, including spatially correlated signals caused by path delays as the radar signal propagates through the stratified and turbulent atmosphere and ionsophere [e.g., Williams et al., 1998]. The variations in atmospheric water vapor often introduce several centimeters of apparent deformation in the radar line-of-sight (LOS), correlated over short spatial scales (<10 km), referred to as the “wet delay” [Emardson et al., 2003; Lohman and Simons, 2005]. Wet delay signals are particularly problematic because, like the subsidence and uplift signals associated with tectonic deformation, they are often spatially correlated with topography.
 In regions with many SAR acquisitions and favorable imaging characteristics (i.e., low rates of signal decorrelation), InSAR time series techniques such as the Small BASeline (SBAS) approach [e.g., Berardino et al., 2002] can reduce the impact of these noise sources and allow characterization of signals with rates as low as 1–2 mm/yr in the radar line-of-sight over short spatial scales (<25 km) [e.g., Lanari et al., 2004; Finnegan et al., 2008; Barnhart and Lohman, 2012]. In theory, the contribution of wet delay to interferograms should be random in sign over time, with radar acquisitions equally sampling times where the atmosphere was drier or wetter than the annual average. InSAR time series techniques typically assume that, if the time series contains a large number (>50) of SAR acquisitions, the wet delays in individual interferograms will average out to near-zero displacement over the full observation time interval. In most locations, however, the number of available SAR acquisitions is far fewer than would be required to obtain the level of precision desired by most researchers (∼1 mm/yr line-of-sight rates), even under an idealized scenario where the noise was independently distributed with Gaussian characteristics. The spatial correlation of atmospheric effects within each interferogram results in a need for even more data to reach a particular precision threshold. In addition, the frequency of SAR acquisitions often varies greatly in time, leading to long temporal gaps and potential seasonal biases [e.g., Doin et al., 2009]. For instance, if the early part of a time series is dominated by winter acquisitions and the later portion by acquisitions during summer, the resulting time series may contain a bias that could easily overwhelm the background tectonic signals. More problematically, since signals associated with temporal biases may also be spatially coherent and correlated with topography, they could easily be misinterpreted as a real ground deformation due to motion along faults, anthropogenic activity, and so on.
 Here we explore how a priori knowledge of the dates of SAR acquisitions, combined with independent observations constraining the character of variations in atmospheric water vapor, can allow us to quantify the expected contribution of such biases to InSAR time series analysis of a particular set of interferograms. We introduce an approach for estimating error bounds on inferred deformation rates using constraints on atmospheric water vapor content from the Moderate Resolution Imaging Spectroradiometer (MODIS) instrument, housed on both of NASA's TERRA and AQUA satellites. Rather than attempting to correct and remove the atmospheric contribution to individual interferograms, an approach that has been shown to have limitations (described below), we explore a stochastic approach where we generate an ensemble of pseudo-InSAR time series based on many sets of MODIS data that sample the same temporal range as the available SAR imagery. We apply this technique to a set of SAR imagery spanning 2003–2010 over the Mojave Desert in Southern California—an arid region with significant topographic relief and good interferometric coherence (Figure 1). The mean 1σ error on the average inferred deformation rate over the entire study area is 1.5 mm/yr but may be as high as 6 mm/yr in regions where weather patterns and topographic relief result in larger amounts of variability in atmospheric water vapor content. Several regions within the study area are experiencing high rates of deformation associated with anthropogenic removal of subsurface fluids (Figure 1)—our error analysis allows us to state with confidence that these rates are significant given expected errors. Additional SAR acquisitions covering longer observation intervals will result in more precise estimates of ground velocity, in a manner that can be predicted using this style of analysis of historical observations of atmospheric characteristics in a given region.
1.1. Previous Work
 Previous efforts to mitigate the effects of atmospheric noise on interferogram time series products have largely focused on the process of estimating and removing wet delay signals in individual interferograms. Techniques explored have included: removing topographically correlated wet delay signal with an elevation-dependent model of phase delay across the full interferogram [e.g., Wicks et al., 2002; Elliott et al., 2008] or across shorter spatial scales using wavelet transforms or running windows [e.g., Lin et al., 2010; Shirzaei and Bürgmann, 2012; Béjar-Pizarro et al., 2013], estimating and correcting wet delays with independent observations from dense continuous GPS networks [e.g., Li et al., 2005; Onn and Zebker, 2006], and removing expected atmospheric delays using weather models [e.g., Wadge et al., 2002; Foster et al., 2006; Doin et al., 2009; Löfgren et al., 2010]. Other methods for removing the effects of wet delay rely on multispectral satellite images, including MODIS and the Medium Resolution Imaging Spectrometer (MERIS), to predict and remove wet delay signals [e.g., Li et al., 2005, 2009; Puysségur et al., 2007; Fournier et al., 2011a]. An alternative family of approaches have focused on characterizing the statistics of the noise to improve weighting of data in inversions for parameters of geophysical interest or for use in characterizing the error bounds on such parameters, including Monte Carlo tests using synthetic noise [e.g., Wright et al., 2003; Lohman and Simons, 2005].
 Each of these approaches has its strengths and weaknesses. Techniques for characterizing the noise run into difficulties when studies are expanded to length scales >100 s of km [e.g., Fournier et al., 2011a; Barnhart and Lohman, 2012; Wang and Wright, 2012], complicating studies of signals with a large spatial scale (e.g., interseismic motion or subduction zone earthquakes). Both the turbulent component and the topographically correlated wet delay signal in interferograms are generally highly nonstationary and anisotropic. Dense continuous GPS networks are not common globally and are therefore of limited utility for correction of interferograms, and significant artifacts can be found in between stations under even the most optimal scenarios [e.g., Onn and Zebker, 2006; Löfgren et al., 2010]. MODIS and MERIS multispectral imagery have shown promise as a tool for directly correcting interferograms because they both provide independent global measurements of precipitable water vapor and are acquired at a temporal frequency equal to or greater than SAR; however, they suffer from four key drawbacks. First, the spatial resolution of MODIS is significantly lower than that of most interferograms, requiring upsampling of MODIS/MERIS images (or downsampling of interferograms). Second, MODIS/MERIS precipitable water vapor products require daytime acquisitions with minimal cloud cover, whereas SAR imagery can be acquired during any weather conditions at any time of day. Third, even in cases where researchers attempt to mask out cloudy regions, wispy clouds missed by the algorithms and complicated atmospheric paths result in reflected “ghost” images of the clouds within the water vapor product that can be misinterpreted as deformation [e.g., Li et al., 2009]. Last, MODIS observations are not obtained simultaneously with SAR observations (Figure 2), although the data used in this study were acquired at approximately the same time of day. Because atmospheric water vapor characteristics can change significantly on short timescales (<1 h) [e.g., Onn and Zebker, 2006], correction of individual interferograms with MODIS images may lead to additional phase errors.
 In place of an interferogram correction methodology, we present an approach that evaluates expected uncertainties on inferred line-of-sight displacements using MODIS near-infrared observations of precipitable water vapor that sample approximately the same dates as those available for a particular InSAR time series. SAR images are obtained at the same time of day (to within a few seconds) on each pass. MODIS data (particularly from the Terra instrument) are acquired at approximately the same time of day as the SAR imagery (Figure 2), so it captures the stochastic behavior of the atmosphere during those acquisitions. SAR imagery acquired at a different time of day would not likely be comparable, and this technique cannot be used for nighttime SAR acquisitions. Below, we describe the process we use to generate an InSAR time series and then our approach for producing a population of MODIS time series with similar temporal sampling as the InSAR time series from which we generate noise statistics.
2. Time Series Generation and Error Analysis
2.1. Mojave Desert InSAR Time Series
 To construct our InSAR time series, we use 40 Envisat C-band SAR acquisitions from descending track 356, spanning the time period February 2003 to June 2010 and covering an area of 116 km × 208 km (Figure 1). We restrict interferogram pairs to those with perpendicular baselines <500 m and temporal baselines <4 years. Interferograms are generated at a resolution of 162 m × 162 m (“looked down” eight times) using the JPL/Caltech ROI_PAC processing package [Rosen et al., 2004] and topographic phase is removed with the 90 m Shuttle Radar Topography Mission (SRTM) DEM [Farr et al., 2007]. Filtered interferograms are unwrapped with the Statistical-cost, Network-flow Algorithm for PHase Unwrapping (SNAPHU) [Chen and Zebker, 2001], and unwrapping errors are manually corrected when apparent. All interferograms are coregistered to a single master interferogram (7 February 2003 to 27 June 2003) in the radar coordinate system.
 In the time series processing, we first remove a quadratic function from each interferogram. Signals at this scale could be due to errors in the satellite orbital positions, troposphere or actual deformation. Because of this inherent tradeoff, it is standard practice to remove a planar or quadratic function [e.g., Berardino et al., 2002], with the understanding that the output deformation field product has no sensitivity to features with scales greater than ∼50 km. Combinations of GPS and InSAR data can address this issue [e.g., Manzo et al., 2012], but the sensitivity to longer spatial scales then originates from the GPS data, not from the InSAR. As discussed below, we follow the same procedure during our processing of the MODIS data, since this process of removing a planar or quadratic function will also remove any effects of long-wavelength features in the distribution of atmospheric water vapor. We then generate the time-variable displacement history on a pixel-by-pixel basis using a methodology similar to Berardino et al. , inverting the set of interferograms (an interferogram “tree”) for the displacement history relative to the first date. When the tree of interferograms is completely connected (i.e., there are no subsets of dates that are not connectable through interferograms to the first date), the inverse problem is well-determined and we invert for the displacement history without regularization. For pixels where the inversion is not perfectly determined due to the existence of disconnected subsets in the available interferogram tree, a situation commonly encountered due to spatial and temporal variations in decorrelation, we assume that the velocities during time periods in between subsets are equivalent to the average linear rate during the rest of the time series. To impose this constraint, we augment our inversion with pseudo-data that penalizes inferred rates during the unconstrained periods of time that differ from the average rate during the time series as a whole [Berardino et al., 2002; Barnhart and Lohman, 2012]. We only determine deformation rates for pixels where >90% of the acquisitions dates (36 dates per pixel for this data set) are constrained by at least one coherent interferogram (e.g., a repeatedly incoherent pixel is not included due to poor conditioning in the inversion). We then infer average line-of-sight velocities through linear regression between displacement and time for each point (Figure 1).
2.2. MODIS Time Series and Error Estimation
 The MODIS instrument is a passive imaging spectroradiometer with five near-infrared bands that are sensitive to water vapor [Gao and Kaufman, 2003]. Precipitable water vapor (PWV) is estimated from the observed attenuation of reflected solar radiation [Gao and Kaufman, 2003]; hence, PWV measurements are only available during daytime, cloud-free acquisition. PWV products are provided at a resolution of 1 km ×1 km, along with cloud masks and georeferencing information.
 Here, our goal is to generate MODIS-based, pseudo-InSAR time series that characterize the variability in inferred rates that one would expect from variations in water vapor and biases in the seasonal coverage. To generate a MODIS-based time series, we select a MODIS PWV observation from a narrow date range (10 days before and after) spanning each SAR image used in our actual InSAR time series (Figure 3). We generate a suite of 100 MODIS-based time series using this approach, each using a different set of MODIS acquisition dates, from which we derive statistics on the expected noise characteristics, detection levels, and so on, on the InSAR time series product.
 We utilize MODIS near-infrared observations from both the TERRA and AQUA instruments (http://modis.gsfc.nasa.gov). Individual georeferenced MODIS scenes are registered to the interferograms in radar coordinates. We only use MODIS acquisitions with minimal cloud cover and mask individual portions of MODIS scenes that are flagged as clouds or that exhibit anomalous, short wavelength signal spikes. These features are attributed to small, unmasked clouds or highly reflective ground properties (e.g., dry lake beds). To cull these features, we delete pixels that fall outside three standard deviations of the mean signal magnitude of the MODIS scene. An additional imaging problem in MODIS scenes is streaking caused by the push-broom acquisition style of the sensor. We apply a two-dimensional median filter to reduce the impact of these artifacts.
 In several studies over southern California and other locations, MODIS has been shown to provide an accurate estimate of wet delay using continuous GPS observations [e.g., Gao and Kaufman, 2003; Li et al., 2003, 2005; Prasad and Singh, 2009; Ha et al., 2010]. To convert MODIS observations of precipitable water vapor into wet delay in the SAR line-of-sight, we use the relationship of Li et al. :
where IWD is the wet delay projected into the radar line-of-sight, ∏ is a dimensionless conversion factor, and θ is the radar incidence angle. ∏ may be determined empirically for a particular region using continuous GPS and records of ground temperature [Bevis et al., 1992]. We adopt a value of 6.2, which was previously used for studies in southern California [e.g., Li et al., 2003, 2005] but is shown to be more commonly 6.4 on a global scale and may vary by ∼2–3% [Bevis et al., 1992, 1994].
 After masking and scaling MODIS observations, we generate trees of “pseudo-interferograms” (Figures 3 and 4), generating a single MODIS interferogram by differencing two acquisitions. For each of the 100 MODIS time series, we generate the apparent displacement history due to the wet delay of each pixel following the same inversion procedure described above for the InSAR time series. So as not to introduce additional error from the inversion procedure, we retain the same subsets for each pixel in the MODIS time series that were present in the InSAR time series. To quantify the expected variability in InSAR time series results given the distribution of available SAR acquisitions, we extract the 1-σ uncertainties from the population of MODIS time series, including the time-averaged apparent deformation rate as well as the full history of apparent displacement at each pixel. The wet delay displacements exhibit non-Gaussian distributions, so we define the 1-σ error bounds as the 16th/84th percentiles of the displacement time series.
3. Error Characteristics
 In this section, we first comment on the characteristics of topographically correlated noise within single interferograms with short time spans, and the implications of these characteristics for interferogram corrections. We end with a description of the implications of our MODIS time series error analysis.
3.1. Nonstationary Noise in Interferograms
 A common strategy for mitigating the effects of topographically correlated atmospheric noise in individual interferograms is the subtraction of a topography-dependent model of phase delay, often using a linear relationship between topography and phase [e.g., Wicks et al., 2002; Cavalie et al., 2007; Elliott et al., 2008]. This method assumes that the degree of correlation between topography and noise is stationary across the entire interferogram or region of interest. Figure 5a shows a single interferogram from our time series that exhibits a strong positive correlation between signal and topography. Because of the short time interval (1 month) and short perpendicular baseline (44 m) of the interferogram, we infer that the apparent displacements are dominated by wet delay rather than by real ground deformation. The relationship between topography and phase can be highly heterogeneous on short spatial scales (<10 km)—in some cases, this relationship changes significantly across a single valley or ridge on the mountain (Figures 5c and 5d). Even approaches that allow for a spatially variable relationship between topography and phase across the interferogram [e.g., Shirzaei and Bürgmann, 2012; Béjar-Pizarro et al., 2013] may be biased by changes over such short spatial scales.
3.2. InSAR and MODIS Time Series
 The average displacement rates for the Mojave InSAR time series and a single pseudo-interferogram time series are shown in Figure 1. The InSAR constraints on average velocity are primarily within ±2 mm/yr, but there are notable regions of broad subsidence of 1–2 mm/yr in addition to significant subsidence of an individual farm that exceeds 8 mm/yr (Figure 1, point b). Profiles of velocities across the InSAR and several pseudo-interferogram time series (Figure 6) indicate that the surface velocities constrained with InSAR (Figure 6, blue profiles) are of similar magnitude as the apparent velocities from MODIS data (Figure 6, red profiles). Furthermore, the pseudo-interferogram velocities shown exhibit nonzero velocities that are collocated with several of the subsidence signals in the InSAR time series. One of these regions (c) corresponds to an area with larger topographic relief (vertical line in Figure 6), where we would expect the MODIS data to produce large signals with both positive and negative signs that would average out given sufficient data. In locations where the ensemble of MODIS time series consistently contain signals with similar magnitude and spatial wavelength as those seen in the InSAR time series, we infer that the these signals likely result from variations in water vapor content rather than actual ground displacements.
 We examine three locations that capture the main characteristics of the time series (Figure 7), marked a–c in Figure 1: (a) A location with an inferred LOS velocity of 0 mm/yr, (b) a rapidly subsiding small area related to agricultural activity (−8 mm/yr), and (c) apparent subsidence (−2 mm/yr) over a broad region. Because of the potential seasonal biases [e.g., Cavalie et al., 2007] of SAR acquisition dates, we do not expect the variation in the pseudo-interferogram time series to exhibit a Gaussian distribution centered on zero. As mentioned above, the distribution of wet delay displacements indeed do not exhibit a Gaussian distribution (Figure 7, scatter). Therefore, we determine 1-σ uncertaintiy envelopes for the time series using the 16th and 84th percentile of the spread of MODIS data for each point in space and time (Figure 7, gray). For each pixel, we compare the variation in all pseudo-interferogram time series in addition to the InSAR time series (Figures 7a–7c). In the near-zero velocity example (Figure 7a), the inferred error bounds indicate that the InSAR-derived velocity falls within the level of noise. In the case of the subsidence signal due to agricultural activity (Figure 7b), the InSAR-derived velocity is greater than our estimated 1-σ uncertainties, so we can confidently state that the subsidence for this pixel is statistically significant. In the final example, where the inferred InSAR velocity over a broad region is −2 mm/yr (Figure 7c), we observe that the InSAR-derived rate falls within the error bounds determined from the pseudo-interferogram time series. In this case, the nonzero inferred velocity falls within the uncertainty of the noise and cannot be significantly distinguished from zero.
 Figure 8 illustrates the correlation between topography and the magnitude of velocity errors (i.e., the error in slope of lines fit to time series such as those shown in Figure 7) inferred from the set of 100 MODIS time series. The greatest velocity uncertainties correlate to regions of relatively steep topographic relief, with the velocity uncertainties exceeding 1.5 mm/yr. There are several short (<1km) spatial scale artifacts evident in Figure 8, including subtle E-W residual striping that was not fully removed with spatial smoothing and highly irregular, meandering patterns. These meandering patterns result from variations in the number of dates that contribute to the time series inversion for each pixel. Adjacent pixels are not necessarily constrained by the same distribution of dates, owing to data loss from decorrelation or cloud cover.
 The Monte Carlo-style error estimation technique used here, while not providing an actual correction to the data, allows the user to place confidence bounds on the line-of-sight velocities and time-displacement histories inferred from a set of interferograms. By using MODIS PWV products to estimate these errors, the error bounds we generate reflect uncertainty due to the presence of correlated atmospheric noise that, while random overall, is not generally sampled adequately enough to average to zero. Two observations drawn from the Mojave data example suggest that several assumptions commonly made during InSAR time series analysis may not be valid: (1) Correlated atmospheric noise in single interferograms can be highly nonstationary, with relationships to topography that vary significantly within a single frame and (2) wet delay induced by correlated atmospheric noise does not necessarily average to <1 mm/yr displacement rates even with the fairly rich set of data acquisitions available in southern California. For our 7 year time series with 40 SAR acquisitions, we find that we are able to confidently resolve velocities greater than ∼1.5–2 mm/yr, as previously suggested by Finnegan et al. , in regions with low signal decorrelation, low relief, and few unwrapping errors. These uncertainties rapidly increase, though, with increasing topographic relief. InSAR time series methodologies will be able to attain higher precision if future SAR missions acquire data with higher frequency and regularity.
 This technique for uncertainty estimation does not account for other common sources of error in InSAR observations such as unwrapping errors, correlated noise due to ionospheric perturbation, digital elevation model errors, or errors in the orbital positions of the satellites. For example, in Figure 1a, high apparent velocities (>5 mm/yr) over one mountain peak likely result primarily from unwrapping errors over steep topography. Likewise, we do not account for errors introduced in the processing of MODIS-PWV products, which carry an accuracy of 5–10% [Gao and Kaufman, 2003], or errors introduced by our methodology of interpolating 1 × 1 km MODIS-PWV observations onto the InSAR grid. Last, the large daily fluctuations in wet delay mean that care must be taken when choosing a satellite-based PWV product. Planning for future SAR missions may wish to avoid orbits that would result in nighttime acquisitions or other times of day that are not nearly contemporaneous with near-infrared observations, since such observations would not benefit from independent assessments of wet delay. The ideal scenario would include high spatial resolution spectroradiometer sensors on future SAR satellite platforms for the purposes of both InSAR correction [e.g., Li et al., 2009; Fournier et al., 2011a; Agram et al., 2013] and error estimation. Despite these limitations, the stochastic approach described here can be used in the absence of detailed weather models or data acquired at the same time as the SAR imagery (such as MERIS data or models based on dense continuous GPS networks), and could be used with a wide variety of inputs to assess the impact of variations to future SAR satellite acquisition plans.
 S. Minson and two anonymous reviewers provided helpful commentary and guidance during the preparation of this manuscript. ENVISAT imagery was provided through a Category-1 proposal through the European Space Agency. This work was funded by the Southern California Earthquake Center, grant 12096. Several figures made using the Generic Mapping Tools [Wessel and Smith, 1998].