InSAR constraints on soil moisture evolution after the March 2015 extreme precipitation event in Chile

Constraints on soil moisture can guide agricultural practices, act as input into weather, flooding and climate models and inform water resource policies. Space-based interferometric synthetic aperture radar (InSAR) observations provide near-global coverage, even in the presence of clouds, of proxies for soil moisture derived from the amplitude and phase content of radar imagery. We describe results from a 1.5 year-long InSAR time series spanning the March, 2015 extreme precipitation event in the hyperarid Atacama desert of Chile, constraining the immediate increase in soil moisture and drying out over the following months, as well as the response to a later, smaller precipitation event. The inferred temporal evolution of soil moisture is remarkably consistent between independent, overlapping SAR tracks covering a region ~100 km in extent. The unusually large rain event, combined with the extensive spatial and temporal coverage of the SAR dataset, present an unprecedented opportunity to image the time-evolution of soil characteristics over different surface types. Constraints on the timescale of shallow water storage after precipitation events are increasingly valuable as global water resources continue to be stretched to their limits and communities continue to develop in flood-prone areas.

Because the region has little to no vegetation, we are able to examine how the interferometric phase characteristics between SAR images vary over time and attribute much of the variation to temporal variations in soil properties.

Results
We form interferograms between all pairs of available dates in three Sentinel-1A tracks (see Methods and  Supplementary Tables 1-3), covering a 100 km × 150 km area located where the March 2015 precipitation was  particularly heavy (Fig. 1b), and examine the phase coherence as a function of time. We show that the effects on coherence from other sources (timespan, orbital geometry) can be separated from those that are due to a proxy for soil moisture, for a result that is consistent between independent, overlapping tracks. Typically, interferograms that span longer time intervals typically have lower coherence than interferograms with shorter time spans. However, for this dataset, the least coherent interferograms are those that include the dates in the months immediately after the rain event, regardless of timescale. Even interferograms with very long time spans (>1 year) between two "dry" dates have much higher coherence than any of those that include dates immediately after the rain event (Fig. 2).
Each of the three independent tracks we examine here contains 11-14 dates acquired at least six months after the event, allowing us to capture the full temporal evolution from dates that are clearly affected by the rain event to those where coherence behavior appears to have returned to "normal". The interferograms (Supplementary  Tables 1-3) have a wide range of spatial and temporal baselines, both of which adversely affect coherence as they become larger 16 . We examine the decrease of coherence as a function of time and baseline in order to verify that the changes in the immediate aftermath of the rain event are not associated with those variables. At each pixel, we solve for the decrease in coherence as a function of temporal and spatial baseline using only interferograms between the subset of dates that are at least six months after the event (Fig. 3b, see Methods). We find that the dependence on spatial baseline is negligible (i.e., not significantly different from zero) except for over some higher-relief areas, which we hereafter omit from consideration in our conclusions. The dependence on temporal baseline results in decreases of less than 0.1 (coherence ranges from 0 to 1) except for along the coastal escarpment and other high relief areas that are not a focus of this study.
We attribute the remaining variation in coherence to changes in soil moisture following the rain events as well as permanent loss of coherence due to surface disruption and sediment transport. The former effect results in lower coherence only in interferograms that include dates immediately after the rain event (i.e., interferograms with time intervals of six months or more spanning the event have high coherence), while the latter results in lower coherence for any interferogram that spans the event. Over most of the study region interferograms between dates before the March rain event and dates long afterwards (six months or more) are also very coherent, indicating that the loss of coherence was "temporary" and was associated with variations in soil moisture. However, at other locations, interferograms between the pre-event and the later, post-event dates are consistently lower. We interpret this effect as a permanent loss of coherence, C p .
At each pixel, we invert for the "relative coherence", C r , at every date such that the predicted coherence of the interferogram between each pair of dates (after correction for temporal decay) depends on the absolute value of the difference in C r at those dates (  relationship between coherence and differences in soil moisture should depend on the absolute amount of soil moisture as well as the difference in moisture between the two SAR acquisition dates 19,31,33,36 . However, our approach captures most of the variability in the coherence signal (Fig. 3c), including the months-long temporal decay following the rain event. (Fig. 4). The poorest fits to the data are for the first dates acquired immediately after the rain event for each track (Fig. 3d). The change in relative coherence can be fit with an exponential decay with time constants of ~50 days (Fig. 4). The second rain event in August of 2015 is associated at some locations ( Fig. 4b) with a second pulse of relative coherence variation with a similar amplitude but shorter decay time than that observed for the March event. The interferograms do not robustly support inversion for a second permanent loss of coherence during the second event, but there is likely some permanent impact that does occur below the noise level in our data.
In regions that are covered by multiple, independent tracks of data (including imaging from different flight directions), the time series of relative coherence is remarkably consistent between tracks (Fig. 4). Coherence between dates that were both acquired before the March rain event is high (Track 54, Fig. 2a, black dots in Fig. 4). Permanent losses of coherence, C p , associated with the rain event are apparent in all tracks and agree in regions of overlap. The largest losses of coherence appear to be due to sediment transport in river channels (Fig. 4b) and at sites that had been disturbed by mining (Fig. 5). Other sites (Fig. 4a,c) do not show any appreciable permanent loss of coherence. Prominent NNW-trending streaks in C p (Fig. 5) are consistent with inferred prevailing wind patterns during the storm 41 and suggest a high degree of heterogeneity in land surface disturbance.

Discussion
The rich set of satellite-based SAR imagery spanning this event illustrates the potential of dense InSAR time series for use in soil moisture research. We image both the coherence loss associated with land disturbance during the event and its immediate aftermath, as well as a months-long perturbation to soil moisture over the region. Such observations, particularly when combined with field measurements or other remote sensing data (such as variations in SAR amplitude), are valuable input into efforts to understand the budget of infiltration and evaporation that impact soil moisture and groundwater recharge in arid environments.
The region examined here is hyperarid, with little to no vegetation. In areas with vegetation, particularly where crops are grown, we would expect the coherence behavior to have a large contribution from properties of the vegetation and their temporal changes 26,34,42,43 . However, even if we restrict ourselves to arid regions, there are many interesting problems that can be addressed with the rich, openly available catalogs of SAR data that are now becoming available through the Sentinel-1 and other satellite remote sensing acquisition missions. For instance, there has been an outstanding question of whether variations in coherence and phase such as those we see here are due to swelling of clays and soils at the surface, or due to changes in the strength of the scatterers distributed beneath the surface under changing moisture conditions 30,31,33,44 . Our data set contains at least one observation that may shed light on this question. At rain gauge locations where the second, August, event is detected (pink line in Fig. 4b), the observed precipitation was much smaller in magnitude than it was in March. Interestingly, the peak magnitude of our inferred C r variation is similar, while the temporal decay is shorter for the second event. If the variation in coherence is due to swelling of clays, our observations require that the soils somehow "unswell" to the exact same state they were in before, so that longer time interval interferograms that span the event are associated with high coherence. Additionally, we may expect the clays at the immediate Earth surface to dry out at a fairly uniform and fast rate, particularly in such a hyperarid environment, rather than the longer and nonuniform rates that we observed. The change in rates may be explained to some degree by seasonality, since the two events occurred in late summer (March) and winter (August), respectively.
On the other hand, if the coherence is related to the integration of radar backscatter within a layer with finite depth (on order of one to a few wavelengths), then alternatives exist which may better fit the observations. For example, alluvial fans where the moisture content of the soil was increased significantly by the rain, capillary action may move moisture upward toward the surface layer, replenishing evaporated near-surface moisture; this may continue over many weeks. Another example may exist in the regions whose surface is dominated by calcium sulfate, where there is likely a cycle in which rain water that infiltrated the highly porous soil will initially partially dissolve the mineral salts which form pore walls, and then during evaporation precipitate new calcium sulfate crystals. This dissolution-precipitation cycle changes the physical state of materials in the upper tens of centimeters below the surface. By either of those mechanisms, the smaller event would change the soil moisture at very shallow depths by a similar amount as did the larger event, but the lack of deeper penetration of moisture into the subsurface would result in a shorter time scale for return to the original conditions. Apart from the potential for characterization of hydrology and surface characteristics, InSAR observations, particularly those from modern satellite platforms with regular repeat times and open access to data, are also increasingly valuable for rapid characterization of damage associated with events such as earthquakes and large storms 21,22,45,46 . While the computational requirements are high, our results suggest that analysis of background variations in coherence associated with storms or seasonal effects may improve such forecasts. This may be particularly true in arid environments where variations in moisture, rather than infrastructure damage, may dominate the change in correlation over time. This work suggests that ongoing, multi-year SAR observations of arid and hyperarid regions (including coherence, phase delays and amplitude) and the development of a catalog of precipitation events can provide constraints on the temporal evolution of soil moisture and support future field or laboratory efforts.

Methods
InSAR data and processing. We use data acquired by the European Space Agency's Sentinel-1A satellite in the Terrain Observation by Progressive Scans (TOPS) mode (run in the interferometric wide swath mode) for three independent, overlapping tracks ( Supplementary Tables 1-3), for the subswaths that cover our study region. Images from a single track do not image exactly the same area on each pass, resulting in some pixels that are in one subswath on some dates and the adjacent subswath on other dates. We only perform our analysis on pixels that are located within the same subswath on all dates, resulting in a small gap between the subswaths that can be seen in Fig. 1b. We process all potential interferometric pairs with data acquired between January 2015 and September 2016 using the InSAR Scientific Computing Software (ISCE) package 47 . We produce between 153 and 190 interferograms for each of the three tracks, including up to two of the available swaths for each track (Fig. 1b). Most of the images were acquired in VV polarization, but we see no variation between those and images acquired in VV + HV.
Interferometric Coherence. Interferometric coherence 16  where s i is the complex-valued SAR backscattered field (magnitude and phase) recorded for each acquisition. The ensemble averages 〈⋅〉 are typically evaluated by averaging in space over some number of pixels, N. We estimate γ over a box with dimensions of 7 pixels in the range direction and 3 pixels in the azimuth direction. Full resolution pixels from the Sentinel mission in the interferometric wide swath mode have a ground dimension of approximately 5 meters in range and 20 meters in azimuth. In our results shown here we do not use the amplitude weighting in Equation 1, since we find that the natural, unvegetated terrain in this research area contains amplitude variations that are completely uncorrelated with the variance of the phase (i.e., dark, smooth surfaces adjacent to bright, rough surfaces with identical phase stability). We are primarily interested in the temporal behavior of phase stability in a region where the temporal decorrelation is nearly zero -i.e., where surface roughness is not changing over time.
Surface roughness strongly affects coherence and backscattered amplitude 48 , and would need to be examined in order to derive proxies for soil moisture in an absolute sense. Below, we refer to coherence as C = abs(γ), which will have low values (near 0.3) for regions that have experienced significant ground disturbance and high value (near 1) for regions that have not experienced much change in the position or characteristics of scatterers. The small box size used in our estimate of phase variance allows us to capture sharp spatial gradients and steps between different lithologies. However, small box sizes have been shown to bias the estimate of coherence slightly upwards, particularly for low values 19,49 . Our focus is on the temporal evolution of coherence, and focuses primarily on high coherence regions where the expected bias is small (<0.04 in correlation units for our least coherent pairs). We repeated the analysis described below at several locations using the correction for effective averaging over N = 9 and N = 25 and with and without amplitude-weighting (Supplementary Figure 6), but found no significant variation in the temporal evolution of relative coherence.
Products derived from these values (described below) are generated in the slant-range coordinate system for each track (i.e., not terrain corrected) and then the final products are georeferenced before comparison between different tracks.
Coherence modeling. Processes occurring before, during, and after the rain event are all expected to reduce interferometric coherence, C at a particular pixel, i, and between times t 1 and t 2 : r r 1 2 0 1 2 C 0 captures the loss of coherence over time-periods shorter than the available Sentinel-1 repeat interval of 12 days (6 days if Sentinel-1a and -1b data are available), including the effects of surface roughness on coherence. Δt is the timespan of individual interferograms, and τ describes the slope of the time-dependent loss of coherence. Here we model this decay as linear, although clearly that relationship does not hold at very long time intervals (coherence would become negative). Coherence in all interferograms between dates 6 months or longer after the rain event (presumably unbiased by coherence loss associated with soil moisture) decays approximately linearly with time, with a magnitude that is small compared with the variations in C r . We impose a negativity constraint on τ, such that coherence must decrease as the time-span between SAR acquisitions increases. This constraint does not significantly change the final result, but does keep the predicted coherence values from having non-physical values greater than one.
The C r (i, t j ) term involves the changes in soil moisture that the affect strength of interaction between the radar signal and the scatterers within the shallow subsurface. Coherence between two dates decreases when the set of scatterers change following either an increase or decrease in soil moisture on one of the two dates. Since the effect is the same regardless of the sign of that increase or decrease (i.e. a wet first date vs. a dry second date gives the same change as a dry vs. wet pair), we use the absolute value of the difference between C r on two dates, as in Equation 2. The inversion for C r is nonlinear because of the absolute value term, and has an ambiguity in sign, and in terms of an absolute shift (i.e., you could add a constant to all values and get the same fit). We normalize the inversion by constraining the mean of values at dates 6 months or more after the rain event to be zero and requiring that the first post-event date have positive C r . This normalization does not affect the differences between C r on individual dates. We use a trust-region reflective algorithm to perform the inversion, with multiple starting models, and find that it is robust to within the 0.1 coherence unit errors that we find for other aspects of our analysis. The inferred C r is shown in Fig. 4 as a function of time at several pixels and all available tracks, with the fit to the full coherence dataset at one pixel and one track shown in Fig. 3 and Supplementary Figures 3-5.
We also infer a coherence loss (C p ) that reflects permanent changes in the radar scatterers, likely associated with processes such as overland flow and debris transport during and shortly after the rain event. We define C p as the difference between C r averaged over pre-event dates and the C r averaged over dates 6 months or later after the rain event. Because two pre-rain event SAR acquisitions are available for Track 54, this permanent change can be easily seen in Fig. 4b (black dots) where the values of C r for the two pre-event dates are both negative, but very similar to each other -interferograms between those two dates have high coherence. The loss in coherence for the other two tracks on the single pre-event date relative to the rest of the time series could be assigned either to a permanent coherence loss or to a variation in soil moisture. However, the similarity to Track 54 (Fig. 5) and uniformly high coherence of dates 6 months after the rain event supports the idea that this loss of coherence was permanent and was likely related to permanent changes in the surface.
Temporal decay rates. C r decays to the background, near-zero values over a time period of >1 month at most pixels, and includes a signal from the secondary, smaller rain event in many locations (Fig. 4). To characterize the temporal decay of C r , we fit a piecewise exponential function to the inferred z t values of the following form, Rainfall and atmospheric data. Rainfall (Figs 1 and 4) is reported at stations distributed throughout the research area 37 and models of horizontal wind orientations 37,38 are consistent with the NNW-SSE orientation of streaks of lower correlation apparent in the C p product (Fig. 5). Most of the precipitation data is available to the public at http://dgasatel.mop.cl/, via the Dirección General de Aguas (DGA), a Chilean governmental agency.
Other sources are described in Jordan et al. 37 , Table S2.