Abstract

The validation of satellite surface soil moisture products requires comparisons between point-scale ground observations and footprint-scale (>100 km2) retrievals. In regions containing a limited number of measurement sites per footprint, some of the observed difference between the retrievals and ground observations is attributable to spatial sampling error and not the intrinsic error of the satellite retrievals themselves. Here, a triple collocation (TC) approach is applied to footprint-scale soil moisture products acquired from passive microwave remote sensing, land surface modeling, and a single ground-based station with the goal of the estimating (and correcting for) spatial sampling error in footprint-scale soil moisture estimates derived from the ground station. Using these three soil moisture products, the TC approach is shown to estimate point-to-footprint soil moisture sampling errors to within 0.0059 m3 m−3 and enhance the ability to validate satellite footprint-scale soil moisture products using existing low-density ground networks.

1. Introduction

The upcoming National Aeronautics and Space Administration (NASA) Soil Moisture Active Passive (SMAP) mission and the recently launched European Space Agency (ESA) Soil Moisture Ocean Salinity (SMOS) mission are designed to retrieve surface soil moisture at coarse spatial resolutions (100 km2 for SMAP and 1600 km2 for SMOS). Both missions include ground validation activities to verify that retrievals meet required root-mean-square error (RMSE) accuracy goals. However, these activities are hampered by the scale contrast between satellite-based sensor resolutions and the point-scale nature of ground-based instrumentation used for validation (Crow et al. 2005). Since the majority of the available ground-based soil moisture observations are from low-density networks in which one or two measurements are available per satellite footprint (T. J. Jackson 2010, personal communication), the direct comparison of ground networks to footprint-scale satellite soil moisture retrievals will yield mean-square differences (MSDs), which are a function of the intrinsic accuracy of the remote sensing product as well as the spatial representativeness of the ground observations (Cosh et al. 2008). Given the high levels of spatial variability typically observed in soil moisture fields (Famiglietti et al. 2008), poor representativeness may artificially inflate the measured MSD comparisons above mission accuracy goals.

Recently, Scipal et al. (2008) proposed the application of a triple collocation (TC) procedure (Stoffelen 1998; Caires and Sterl 2003; Janssen et al. 2007) to soil moisture. TC is based on the premise that uncertainty in three parallel estimates of a single variable can be deduced if the estimates possess mutually independent errors. Here, we describe the first application of a TC approach to ground-based soil moisture instrumentation and the estimation of sampling errors associated with the spatial upscaling of their measurements. As described above, the direct comparison of point-scale ground observations with satellite-based soil moisture retrievals yields an MSD that is inflated by the sampling error associated with acquiring footprint-scale means using sparse ground observations. Our goal here is to apply TC to estimate (and correct for) the portion of the total MSD between the ground observations and the retrievals attributable to the spatial sampling error and improve prospects for adequately validating soil moisture retrievals using existing ground-based instrumentation.

2. Triple collocation

Our TC approach is based on three separate time series assumed to approximate footprint-scale (>100 km2) surface soil moisture (θ): a microwave remote sensing product (θRS), a land surface model product (θLSM), and a ground-based product derived from a single point-scale observation within each footprint (θPOINT). All three products contain errors arising from mutually distinct sources. Remotely sensed estimates are impacted by instrument noise and uncertainty in microwave emission modeling. Model-based estimates suffer from a simplified parameterization of soil water loss and forcing data error. Coarse-scale soil moisture estimates obtained from a single point-scale observation are degraded by sensor calibration/measurement errors and representativeness errors due to the inherent spatial heterogeneity of surface soil moisture fields. Given the diversity of these sources, it appears reasonable to assume that the three products contain mutually independent errors.

Prior to the application of TC, each product is decomposed into its climatology mean and anomaly components:

 
formula

where is the climatological expectation for soil moisture at the day of year (D) associated with time step i and θi is the actual anomaly relative to this expectation. Values of are calculated through moving window averaging of multiyear data within a window size of N days centered on D. The implications of this decomposition are discussed further in section 5. In addition, θRS and θLSM are rescaled to match the temporal variance of θPOINT. Unless otherwise noted, N = 31 days and the subscript i is dropped in future references to time series variables.

Differences in the temporal anomalies estimated by the remote sensor and point-scale ground observations can be written as

 
formula

where θTRUE represents the true anomaly time series. Assuming mutual independence of error in the remote sensing observations (θRSθTRUE) and point observations (θTRUEθPOINT), the mean of the square of both sides is

 
formula

or equivalently

 
formula

where the measurable quantity MSD(θRS, θPOINT) differs from the true validation quantity of interest MSD(θRS, θTRUE) due to the spatial sampling error MSD(θPOINT, θTRUE). Our approach applies TC to estimate MSD(θPOINT, θTRUE) and uses (4) to correct estimates of MSD(θRS, θTRUE) based on measured values of MSD(θRS, θPOINT).

TC is based on expressing the relationship between temporal anomalies in all three available soil moisture estimates (θRS, θPOINT and θRS) and true soil moisture anomalies as

 
formula
 
formula
 
formula

where the ε terms denote times series errors relative to θTRUE.

Subtracting (6) and (7) from (5) yields

 
formula
 
formula

Multiplying (8) and (9) and averaging in time (denoted by “〈−〉”) gives

 
formula

Assuming mutually independent errors, (10) collapses to

 
formula

Our goal is validating estimates of MSD(θPOINT, θTRUE) from (11) with independent estimates of the same quantity acquired from ground-based soil moisture networks within data-rich watershed sites. TC-based estimates of MSD(θPOINT, θTRUE) can then be combined with (4) to improve estimates of MSD(θRS, θTRUE) without access to extensive ground-based measurements. Note that the success of both inferences hinges on the, as of yet untested, independent error assumptions underlying (4) and (11).

3. Data and processing

The verification analysis described above is conducted over four U.S. Department of Agriculture (USDA) Agricultural Research Service (ARS) experimental watersheds: the Little River (LR), Georgia (Bosch et al. 2007), the Little Washita (LW), Oklahoma (Allen and Naney 1991; Cosh et al. 2006), the Reynolds Creek (RC), Idaho (Slaughter et al. 2001), and the Walnut Gulch (WG), Arizona (Renard et al. 2008; Cosh et al. 2008). As a group, they provide a range of climate, land cover, and topographic conditions under which to evaluate TC. Each watershed also contains a network of about 20 Stevens Water Hydra Probe surface (0–5 cm) soil moisture sensors installed at USDA Micronet sites within each watershed. See Fig. 1 for watershed/network site locations and Table 1 for a summary of the watershed characteristics and soil moisture instrumentation. Further details are given in Jackson et al. (2010).

Fig. 1.

Location of USDA Micronet sites within the four USDA ARS experimental watersheds listed in Table 1.

Fig. 1.

Location of USDA Micronet sites within the four USDA ARS experimental watersheds listed in Table 1.

Table 1.

Summary of the physical characteristics and ground-based, surface (0–5 cm) soil moisture instrumentation for the four USDA ARS experimental watersheds.

Summary of the physical characteristics and ground-based, surface (0–5 cm) soil moisture instrumentation for the four USDA ARS experimental watersheds.
Summary of the physical characteristics and ground-based, surface (0–5 cm) soil moisture instrumentation for the four USDA ARS experimental watersheds.

To obtain a reference dataset to verify TC predictions, a Thiessen polygon approach is used to interpolate all available 1330 local solar time (LST) ground-based soil moisture measurements up to a single watershed-scale daily time series of θNETWORK (Jackson et al. 2010). While θNETWORK represents the best-available approximation of θTRUE for a watershed, small instrumental and spatial sampling errors in θNETWORK can still artificially inflate their MSD comparisons with other soil moisture products. To correct for this, the estimated error variance in θNETWORK(〈εNETWORK2〉) is correctively subtracted from MSD comparisons with θNETWORK to estimate MSD versus θTRUE:

 
formula

Based on comparisons with gravimetric soil moisture measurements obtained during field campaigns, 〈εNETWORK2〉 is assumed to be on the order of 0.0102 m6 m−6 (Cosh et al. 2006; Cosh et al. 2008). In addition, multiple sets of θPOINT time series are acquired by repeatedly selecting different individual sensor locations to represent each watershed. Only locations containing measurements for at least 50% of the days in the study period (2 February 2002 to end dates listed in Table 1) are used to represent θPOINT.

Our θRS estimates are based on 0.25° single-channel algorithm (Jackson et al. 2010) retrievals acquired from Advanced Microwave Scanning Radiometer (AMSR-E) 10.6-GHz brightness temperature observations. Only data from the 1330 LST AMSR-E overpass are considered. Time series of θRS for each watershed are extracted from the 0.25° pixel most closely matching each watershed. Validation work has demonstrated that the measurement depth of these retrievals is consistent with the ground measurements described above (Jackson et al. 2010).

Our θLSM estimates are based on 0–5-cm surface soil moisture predictions from a 0.125° Noah land surface model (LSM; Mitchell 2009) simulation run on a 30-min time step and driven by the North American Land Data Assimilation System (NLDAS) forcing dataset (Cosgrove et al. 2003), the Foreign Agricultural Office world soil classification with Reynolds et al. (2000) soil/clay fractions, and the 1-km global land cover classification of Hanson et al. (2000). Soil and vegetation parameter lookup tables are based on the Noah implementation for the NLDAS project (Robock et al. 2003). Soil moisture states are spun up for 18 months prior to the start of the analysis, and surface values for the multiple Noah 0.125° pixels corresponding to each watershed are spatially averaged to obtain a single θLSM for each watershed.

4. Results

As stated in section 1, our goal is the estimation of MSD(θPOINT, θTRUE) based solely on the availability of θRS, θLSM and θPOINT. By comparing TC-based estimates of MSD(θPOINT, θTRUE) from (11) with comparable statistics obtained from extensive ground-based measurements, we can verify the assumption of mutually independent errors that underlies the approach.

Figure 2 provides such a comparison by summarizing TC results for the four watersheds described above. Each point represents the use of a single sensor in a given watershed as θPOINT. Expressed in terms of the square root of MSD (RMSD), the plot compares RMSD(θPOINT, θTRUE) values calculated using θNETWORK and (12) to TC-based estimates acquired by taking the square root of (11). Relative to benchmark values of RMSD(θPOINT, θTRUE) obtained independently from (12), TC estimates of RMSD(θPOINT, θTRUE) utilizing only θRS, θLSM and θPOINT data are nearly unbiased and have an RMSE accuracy of 0.0059 m3 m−3. Despite their intrinsic variability, the TC approach appears to work equally well in all four watersheds. Problems with the underlying TC assumption of mutually independent errors would manifest themselves as nonzero covariance terms on the right-hand side of (10) and induce bias in (11) relative to (12). The lack of any apparent bias (and/or extensive scatter) in Fig. 2 implies that these assumptions have been adequately met within our three collocated θ′ estimates.

Fig. 2.

For N = 31 days in (1), the relationship between actual RMSD(θPOINT, θTRUE) based on (12) and TC-estimated RMSD(θPOINT, θTRUE) based on (11).

Fig. 2.

For N = 31 days in (1), the relationship between actual RMSD(θPOINT, θTRUE) based on (12) and TC-estimated RMSD(θPOINT, θTRUE) based on (11).

Values of RMSD(θPOINT, θTRUE) in Fig. 2 are based on 〈εNETWORK2〉 = 0.0102 m6 m−6 in (12). However, results are generally robust to assumptions concerning 〈εNETWORK2〉. Assuming 〈εNETWORK2〉 = 0, for instance, leads to only a slight increase in RMSE (from 0.0059 to 0.0062 m3 m−3). In addition, N = 31 days in (1) dictates that soil moisture anomalies are calculated relative to a seasonally varying climatology. In contrast, selecting N = 365 days means that anomalies are calculated relative to a fixed soil moisture mean across the entire seasonal cycle. Recreating Fig. 2 for N = 365 days (Fig. 3) increases the RMSE from 0.0059 to 0.0089 m3 m−3, suggesting that TC works better when variations in soil moisture climatology are taken into account.

Fig. 3.

For N = 365 days in (1), the relationship between actual RMSD(θPOINT, θTRUE) based on (12) and TC-estimated RMSD(θPOINT, θTRUE) based on (11).

Fig. 3.

For N = 365 days in (1), the relationship between actual RMSD(θPOINT, θTRUE) based on (12) and TC-estimated RMSD(θPOINT, θTRUE) based on (11).

Another relevant issue is the required accuracy of the LSM. Decreasing LSM accuracy (via, e.g., an excessive coarsening of the spatial resolution or the degradation of the model physics) increases εLSM and therefore the sampling uncertainty in the 〈εLSMεRS〉 and 〈εLSMεPOINT〉 covariance terms on the right-hand side of (10). Since (11) is based on neglecting these terms, such noise induces a greater amount of random error into TC estimates. To examine the magnitude of this impact, the Noah LSM is replaced with a simple antecedent precipitation index, where θLSM is generated via

 
formula

and Pi is a watershed-scale daily rainfall accumulation based on the Tropical Rainfall Measurement Mission’s (TRMM) 0.250° 3B42 rainfall product. Relative to Noah, (13) represents a degradation in resolution (from 0.125° to 0.250°), forcing data (from gauge-based NLDAS to satellite-based TRMM 3B42 rainfall), and the quality of the LSM physics. Nevertheless, duplicating Fig. 2 for θLSM from (13) (not shown) produces only a modest increase in the RMSE of TC-based RMSD(θPOINT, θTRUE) (from 0.0059 to 0.0082 m3 m−3). This implies that the approach is relatively tolerant to variations in LSM performance.

By combining TC-based estimates of MSD(θPOINT, θTRUE) from Fig. 2 with the observable quantity MSD(θRS, θPOINT), RMSD(θRS, θTRUE) can be estimated as the square root of (4). Figure 4b shows the relationship between such estimates and actual values of RMSD(θRS, θTRUE) obtained from (12). For comparison, Fig. 4a shows the same relationship but assumes MSD(θPOINT, θTRUE) = 0. Due to sub-watershed-scale soil moisture variability, the comparison of a single point-scale ground observation with a footprint-scale AMSR-E retrieval leads to an inflated estimate of RMSD(θRS, θTRUE) (Fig. 4a). Estimating MSD(θPOINT, θTRUE) via (11) and inserting it into (4) improves our ability to recover RMSD(θRS, θTRUE) and thus validate θRS (Fig. 4b).

Fig. 4.

The relationship between actual RMSD(θRS, θTRUE) from (12) and (a) estimated RMSD(θRS, θTRUE), assuming θPOINT measurements perfectly sample each watershed, and (b) estimated RMSD(θRS, θTRUE), utilizing TC and (4) to correct for sampling error in θPOINT.

Fig. 4.

The relationship between actual RMSD(θRS, θTRUE) from (12) and (a) estimated RMSD(θRS, θTRUE), assuming θPOINT measurements perfectly sample each watershed, and (b) estimated RMSD(θRS, θTRUE), utilizing TC and (4) to correct for sampling error in θPOINT.

5. Summary and discussion

The NASA SMAP and ESA SMOS missions are faced with RMSE-based error validation goals for their surface soil moisture products. For most areas, the sole basis for demonstrating such accuracies is comparisons with ground-based soil moisture networks that provide a very small number of observations per satellite footprint. Consequently, retrieval error estimates from such comparisons are spuriously inflated. Requiring only the added availability of a LSM simulation, the TC approach described here offers a viable approach for addressing this problem. In particular, results in Figs. 2 and 3 demonstrate the ability of a TC-based procedure to estimate spatial sampling errors associated with using low-density soil moisture observations to obtain footprint-scale soil moisture averages. When combined with (4), the accurate specification of these errors provides a robust basis for removing the positive bias in RMSD(θRS, θPOINT) associated with ground-based spatial sampling errors (Fig. 4). That is, application of the TC approach enhances our ability to validate remotely sensed soil moisture estimates using existing low-density soil moisture networks.

The presented approach is limited to recovering information about the accuracy of soil moisture temporal anomalies obtained from (1) and cannot therefore predict the long-term bias of a particular point-scale observation site relative to a footprint-scale average. However, such biases can potentially be estimated based on knowledge of local land surface conditions (Grayson and Western 1998) and/or the application of a spatially distributed LSM (Crow et al. 2005). In addition, most data assimilation systems require the rescaling of soil moisture retrieval products into a particular LSM’s unique climatology prior to analysis. In such cases, the information content of the remote sensing retrievals is based solely on their representation of anomalies (Koster et al. 2009). Consequently, the ability of a TC approach to validate the representation of soil moisture anomalies in remote sensing products is arguably addressing the most critical aspects of the problem.

Acknowledgments

This research was NASA-supported through Wade Crow’s membership on the NASA SMAP mission science definition team.

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Footnotes

* Current affiliation: Department of Hydrology and GeoEnvironmental Sciences, VU University Amsterdam, Amsterdam, Netherlands

Corresponding author address: Wade T. Crow, USDA Agricultural Research Service, Hydrology and Remote Sensing Laboratory, Rm. 104, Bldg. 007, BARC-W, Beltsville, MD 20705. Email: wade.crow@ars.usda.gov