Land data assimilation and estimation of soil moisture using measurements from the Southern Great Plains 1997 Field Experiment

Transcription

1 WATER RESOURCES RESEARCH, VOL. 38, NO. 12, 1299, doi: /2001wr001114, 2002 Land data assimilation and estimation of soil moisture using measurements from the Southern Great Plains 1997 Field Experiment Steven A. Margulis, 1 Dennis McLaughlin, Dara Entekhabi, and Susan Dunne Ralph Parsons Laboratory, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Received 10 December 2001; revised 10 May 2002; accepted 10 May 2002; published 19 December [1] Remotely sensed microwave measurements provide useful but indirect observations of surface soil moisture. Ground-based measurements are more direct but are very localized and limited in coverage. Model predictions provide a more regional perspective but rely on many simplifications and approximations and depend on inputs that are difficult to obtain over extensive areas. The only effective way to achieve soil moisture estimates with the accuracy and coverage required for hydrologic and meteorological applications is to merge information from satellites, ground-based stations, and models. In this paper we describe a convenient data merging (or data assimilation) procedure based on an ensemble Kalman filter. This procedure is illustrated with an application to the Southern Great Plains 1997 (SGP97) field experiment. It uses land surface and radiative transfer models to derive soil moisture estimates from airborne L band microwave observations and ground-based measurements of micrometeorological variables, soil texture, and vegetation type. The ensemble filter approach is appealing because (1) it can accommodate a wide range of models, (2) it can account for input and measurement uncertainties, (3) it provides information on the accuracy of its estimates, and (4) it is relatively efficient, making large-scale applications feasible. Results from our SGP97 application of the ensemble Kalman filter include large-scale maps (10,000 km 2 )ofsoil moisture estimates and estimation error standard deviations for the entire month long experiment and comparisons of filter soil moisture and latent heat estimates to ground truth measurements (gravimetric and flux tower observations). The ground truth comparisons show that the filter is able to track soil moisture fluctuations. The filter estimates are significantly better than those from an open loop simulation that includes the same ground-based data but does not incorporate radio brightness measurements. Overall, the results from this field test indicate that the ensemble Kalman filter is an accurate, efficient, and flexible data assimilation procedure that is able to extract useful information from remote sensing measurements. INDEX TERMS: 1866 Hydrology: Soil moisture; 3337 Meteorology and Atmospheric Dynamics: Numerical modeling and data assimilation; 1818 Hydrology: Evapotranspiration; 1836 Hydrology: Hydrologic budget (1655); KEYWORDS: remote sensing, soil moisture, data assimilation, L band measurements Citation: Margulis, S. A, D. McLaughlin, D. Entekhabi, and S. Dunne, Land data assimilation and estimation of soil moisture using measurements from the Southern Great Plains 1997 Field Experiment, Water Resour. Res., 38(12), 1299, doi: /2001wr001114, Introduction [2] Soil moisture plays a key role in a number of important land surface processes that affect weather, vegetation, and global chemical cycles. Near-surface soil moisture controls partitioning of incident energy into latent and sensible heat fluxes through its effect on evapotranspiration. It also controls partitioning of precipitation into infiltration 1 Now at Department of Civil and Environmental Engineering, University of California, Los Angeles, California, USA. Copyright 2002 by the American Geophysical Union /02/2001WR001114$ and runoff. Since soil moisture regulates the availability of water and nutrients to plants it has a significant influence on global element cycles. Large-scale variations in soil moisture can change the meteorological fluxes that drive soil moisture, creating a feedback mechanism that can have significant impacts on climate and land use change [e.g., Gallus and Segal, 2000; Fennessy et al., 1999; Beljaars et al., 1996; Atlas et al., 1993]. For all of these reasons, soil moisture characterization is an important research topic in hydrology and its related disciplines. [3] Although various methods are available for measuring soil moisture at a point there are currently no networks of in situ sensors that provide regional or global data sets. Since such networks are logistically infeasible and prohib-

2 35-2 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE EnKF itively expensive, attention has turned to remote sensing techniques, which may be able to provide large-scale information appropriate for regional and global applications. In particular, low frequency passive microwave radiation can be used to measure soil moisture because it is affected by the sharp contrast between the dielectric constants for water and soil minerals [e.g., Entekhabi et al., 1999; Njoku and Entekhabi, 1995]. Signals in the L band (1.4 GHz) show maximum sensitivity to surface moisture and are not limited by cloud cover or solar illumination. [4] Since the early 1970s field experiments comparing ground-based L band radio brightness measurements and in situ soil moisture observations have demonstrated that remote sensing of soil moisture is able to estimate volumetric water content in the top 5 cm of the soil column to within a few percent [Jackson et al., 1995, 1997, 1999]. Recently, there have been a number of airborne field campaigns to demonstrate soil moisture mapping capabilities over larger domains. An important example is the Southern Great Plains 1997 or SGP97 experiment, conducted during a 4 week period in the summer of 1997 [Jackson et al., 1999]. [5] Although remote sensing offers a promising source of information about large-scale soil moisture, this information is not perfect. Interpretation of remotely sensed passive and active microwave measurements is complicated by the effects of vegetation, small-scale fluctuations in surface elevation, and temperature variations. Also, remote sensing measurements are only one of many data sources that provide information about soil moisture. Precipitation, soil texture, topography, land use, and a variety of meteorological variables influence the spatial distribution and temporal evolution of soil moisture. Many recent observational and modeling studies at the SGP97 site have examined how these variables influence the spatiotemporal distribution of soil moisture and surface fluxes [Mohanty and Skaggs, 2001; Bindlish et al., 2001; Kustas et al., 2001; Wickel et al., 2001; Mohanty et al., 2000a, 2000b; Mohr et al., 2000; Famiglietti et al., 1999]. It seems reasonable to expect that uncertain soil moisture estimates derived from remote sensing measurements could be improved by incorporating additional sources of information into the estimation process. This can be done in a systematic way if a coupled model of the soil-vegetation-atmosphere system is used to relate all the measured variables to the soil moisture. The process of using models to merge and interpret measurements is commonly called data assimilation [Lorenc, 1986; Daley, 1991; Ghil and Melanotte-Rizzoli, 1991; Bennett, 1992; McLaughlin, 1995]. [6] In this paper we use ensemble Kalman filtering, a particularly flexible data assimilation approach, to characterize soil moisture and related land-atmosphere fluxes at the SGP97 site. Our filtering approach merges L band radio brightness measurements with micrometeorological, precipitation, land use, and soils data to take maximum advantage of all available sources of information. The merging process relies on generally available land surface and radiative transfer models. [7] Data assimilation offers several advantages over traditional methods for retrieving soil moisture from microwave radio brightness observations. First, the assimilation procedure estimates not only soil moisture but the various energy and mass fluxes that are related to soil moisture through the land surface model. Second, estimates of soil moisture and related fluxes are provided not only near the surface but throughout the soil column. Third, the soil moisture estimates are continuous in time and space since the dynamic models used in the assimilation procedure interpolate measurements taken at discrete sampling times (generally once per day for aircraft measurements and up to once every three days for spaceborne instruments). Fourth, the data assimilation procedure can generate estimates at finer resolution than the microwave observations. Such downscaling is possible if other data, such as vegetation type, soil classification, and precipitation are available at finer scales. Fifth, the data assimilation procedure provides information on the accuracy of its estimates. This information reflects the effect of uncertainties in both measurements and model predictions. All of these capabilities are discussed in more detail by Reichle et al. [2001a, 2001b, 2002]. [8] The selection of a data assimilation procedure appropriate for a given application is a balance between making the best use of available information (optimality), computational efficiency, flexibility, and robustness. These performance criteria often conflict and different researchers may reach different compromises. The options range from reinitialization of land surface models with the soil moisture values obtained from retrieval algorithms (direct insertion) through static updating techniques (statistical interpolation) to complex optimization algorithms that adjust uncertain parameters to obtain a good fit to observations (variational methods) [Houser et al., 1998; Reichle et al., 2001a, 2001b]. [9] The particular data assimilation method considered here, ensemble Kalman filtering, is popular in meteorology and has been successfully applied to the soil moisture problem in observing system simulation experiments (OSSEs) based on the SGP97 field study [Reichle et al., 2002]. The ensemble Kalman filter offers several tangible benefits for soil moisture data assimilation. It has a modular structure that makes it possible to use nearly any land surface and radiative transfer model to guide the measurement merging process. There is no need to derive a linear approximation to either of these models (as in variational approaches or extended Kalman filtering). Uncertainties in model inputs can be additive, multiplicative, or combinations of both. Error statistics can change over time or with the local value of soil moisture. Temporal and spatial correlation can be accommodated. Moreover, the filter provides approximate marginal probability distributions of all estimated variables. These distributions may be used to assess the likely variability of estimated quantities such as soil moisture or latent heat fluxes and to examine alternative point estimates (e.g., the means versus the mode of the distribution). Reichle et al. [2002] compare the ensemble Kalman filter with one of the most commonly used alternatives, a variational data assimilation algorithm. [10] The application of ensemble Kalman filtering described in this paper and in the work by Crow and Wood [2002] provides insight into the method s capabilities and limitations. Our results demonstrate that the ensemble filter can provide informative estimates of soil moisture over spatial scales comparable to those of interest in regional meteorological and hydrological studies. Although further development and testing will be required before the filter s

3 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE ENKF 35-3 operational capabilities are fully demonstrated, the SGP97 application described here represents an important first step. 2. Ensemble Kalman Filter [11] The ensemble Kalman filter is a natural application of Bayesian estimation concepts [Jazwinski, 1970]. Detailed discussions are provided by Evensen [1994], Houtekamer and Mitchell [1998, 2001], and Reichle et al. [2002]. The basic concepts may be illustrated if we consider a vector y(t) composed of N y uncertain hydrologic states. In our application, these states include soil moisture and temperature in several layers within each pixel of a regular computational grid. The basic goal of the data assimilation procedure is to estimate these uncertain states from available radio brightness and meteorological measurements. [12] The true states are described by a land surface model with uncertain inputs. This model is based on mass and energy conservation principles and can be written concisely as a vector-valued discrete time state equation: yt ðþ¼ayt ½ ð Þ; a; uðtþ; t; tš t > t 0 ; y 0 ðt 0 Þ ¼ y 0 ðaþ The vector a represents time-invariant parameters such as soil properties while the vector u(t) represents timedependent forcing variables such as precipitation. The initial condition at time t = t 0 is given by y 0 (a) and the function A describes how the state at a previous time t is related to the state at time t. We suppose the inputs and states are random variables in order to account for uncertainties in their values. [13] For this study we assume that the statistical properties of all random inputs are specified. In particular, the mean values of precipitation and other time-dependent inputs are interpolated from available ground-based micrometeorological measurements and the mean values of soil and vegetation properties are derived from field observations. The covariances of these inputs are assigned reasonable values based on observed variability. This topic is discussed in more detail in subsequent sections. [14] In order to estimate system states from radio brightness measurements we need a radiative transfer model that describes how observed radio brightness is related to soil moisture and temperature. This model can be concisely expressed as a vector-valued discrete-time measurement equation: ð1þ z i ¼ My; ½ w i ; t i Š ð2þ Here z i is the vector of radio brightness measurements obtained at time t i and w i is a random error that accounts for differences between the true and observed radio brightness values. This error may be additive, multiplicative, or of some other form, as required in a given application. We assume that its statistical properties are specified. [15] When we use data assimilation in a filtering/forecasting mode we seek the best estimate of y(t), given the set of all measurements Z m =[z 1, z 2,..., z m ] T taken through time t m t. Since y(t) is a random variable everything we know about it for a given Z m is contained in the conditional probability density function f [ y(t)jz m ]. The ensemble Kalman filter uses a Monte Carlo approach to describe how this density evolves over time (between measurements) and how it changes when new measurements are included. It does this by following individual realizations (or replicates) drawn from a small population (or ensemble). This process is illustrated in Figure 1. [16] At time t 0, the state vector y j (t) associated with replicate j is started with an initial condition y 0 (a j ) derived from a synthetically generated realization a j of the timeinvariant input vector a (here j =1,...N R, where N R is the number of replicates included in the ensemble). The state vector for each replicate is propagated forward in time to the first measurement time t 1, according to (1). Synthetically generated values (a j and u j (t)) of the random inputs a and u(t) are inserted into (1) as required. At t 1 each replicate is updated (or conditioned) to reflect the effect of the measurement z 1. The updated states, which are written y j (t 1 jz 1 )to indicate their dependence on all measurements collected through t 1, become the initial conditions for the next time period (t 1, t 2 ]. This process continues sequentially: first a propagation step over each interval between measurement times (e.g., t i and t i+1 ) and then an update step at each measurement time (e.g., t i +1 ). [17] Ideally, the update step of the sequential filtering procedure should be based on Bayes theorem, which specifies the probability density for the ensemble of updated replicates y j (t i +1 jz i +1 )att i +1. However, it is difficult to apply this theorem in practice, especially for large problems, unless the propagated state and the measurement vector are assumed to be jointly Gaussian. Then the propagated and updated densities are completely characterized by their means and covariances and Bayes theorem reduces to a pair of update equations for these two conditional moments. In the version of the ensemble Kalman filter used here each replicate is updated with the following expression[reichle et al., 2002]: y j ðt iþ1 jz iþ1 Þ¼ y j ðt iþ1 jz i ÞþK z iþ1 þ w j iþ1 M y j ðt iþ1 jz i Þ ð3þ where K is a weighting (or Kalman gain) matrix derived from the measurement operator M and the sample covariance C yy (t i +1 jt i ). It is the same for all replicates. The synthetic random measurement error w j i + 1 is generated by the ensemble Kalman filter routine and should have the same statistical properties as the error included in (2). The quantity M[ y j (t +1}jZ i ] appearing in the update equation is the filter s prediction of the measurement obtained at t i+1, given all information collected through t i. The update depends on the difference between the actual and predicted measurements. [18] The sample covariance C yy (t i+1 jt i ) used to derive the Kalman gain is computed as follows: C yy ðt iþ1 jt i Þ ¼ 1 Y N R 1 e ðt iþ1 jt i ÞeY ðt iþ1 jt i Þ T ð4þ where ey (t i +1 jt i )isan y by N R dimensional matrix whose columns are the deviations of the replicate state vectors from their respective sample means (see Burgers et al. [1998] for further details). This matrix can be viewed as a low rank square root of the sample covariance matrix. Propagation and updating of the ensemble is equivalent to propagation and updating of the sample mean and square root matrix.

4 35-4 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE EnKF Figure 1. Basic features of ensemble filtering in (a) the conditional probability density evolves between measurement times t i and t i+1 and is updated with new information at time t i+1 and (b) individual realizations drawn from an ensemble characterized by this density evolve and are updated. The ensemble mean can be used as an estimate and the standard deviation as a measure of estimation accuracy. [19] When the Gaussian assumption used in the update procedure is correct, the ensemble Kalman filtering algorithm outlined above gives an increasingly accurate characterization of the conditional probability density f [ y(t)jz m ]asthe number of replicates is increased. From this density one could compute means, covariances, and other distributional quantities of interest. However, in the soil moisture application the Gaussian assumption is generally imperfect because the nonlinearities in the state and measurement equations tend to produce complex y(t) distributions which are skewed at both the low and high ends of the range of possible moisture contents. Consequently, the ensemble Kalman filter only approximates f [ y(t)jz m ]. However, estimates based on this approximation may be quite useful since the propagated ensemble members used to derive the sample covariance are affected by nonlinearities in the state equation. [20] The practical value of the ensemble Kalman filtering approach must be established through application-specific tests such as those discussed below. However, it should be noted that this approach makes fewer assumptions and is more flexible than most alternatives, including the extended Kalman filter [Galantowicz et al., 1999], which uses a linearized model during the propagation step, and variational methods [Reichle et al., 2001a, 2001b], which generally offer less flexibility for dealing with model and measurement errors. For these reasons the ensemble filtering approach deserves careful consideration, especially in operational settings. 3. Using the Ensemble Kalman Filter to Assimilate SGP97 Data 3.1. SGP97 Experiment [21] The Southern Great Plains 1997 (SGP97) field experiment took place between 18 June and 17 July in central and eastern Oklahoma. The experimental objectives were to investigate near-surface soil moisture and temperature at different scales and to establish the feasibility of character-

5 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE ENKF 35-5 Figure 2. Location of the Southern Great Plains 1997 (SGP97) experiment in central Oklahoma. izing soil moisture with remotely sensed data. Figure 2 shows the experimental area, which covers 10,000 km 2 and includes long-term observing stations at the Central Facility (to the north), El Reno (in the central portion) and Little Washita (in the south). At these sites gravimetric soil moisture measurements were obtained for comparison with soil moisture estimates derived from airborne remote sensing instruments. Figure 3c shows the locations of the long-term observing stations as well as a number of Oklahoma Mesonet stations that also provide micrometeorological measurements. [22] The airborne remote sensing instrument used in our study is the Electronically Scanned Thinned Array Radiometer (ESTAR [LeVine et al., 1994]). ESTAR is a synthetic aperture, passive microwave radiometer operating at a center frequency of GHz (a wavelength of 21 cm) with a bandwidth of 20 MHz. The instrument was flown on a P3B aircraft at an altitude of 7.5 km [Jackson et al., 1999]. The ESTAR data are available over a uniform grid of 800 m by 800 m cells. Due to weather and aircraft constraints and instrument problems, brightness temperature images were obtained for only 16 of the 30 days in the SGP97 study period. Our grid was selected to insure that images are available at most pixels for most of these 16 days. This involved eliminating extensions to the north that occur on 20 June and 2, 12 and 16 July as well as some pixels at the edge of the study area. Finally, the ESTAR data were aggregated to 4km by 4km (twenty-five 800 m by 800 m cells) to reduce the number of computational pixels from to 770. This aggregation considerably reduces numerical computational effort while maintaining a pixel size sufficiently small to identify spatial patterns Land Surface and Radiative Transfer Models [23] In order to apply the ensemble Kalman filtering approach to the SGP97 data assimilation problem we need to specify (1) a land surface model for propagating ensemble replicates between measurement update times and (2) a radiative transfer model for computing the predicted radio brightness used in the update equation. In an ensemble Kalman filter the equations of the land surface model do not need to be explicitly incorporated into the filtering algorithm. Instead, the model can be treated as a stand-alone program which communicates with the filter through its input and output files. The filter provides a set of random initial conditions, parameters, and forcing variables to the land surface model. In turn, the model derives a timedependent state vector that is passed to the filtering algorithm. This modularity makes it possible to use nearly any land surface model in a data assimilation procedure based on an ensemble Kalman filter. There is no need to derive an adjoint or tangent linear approximation or to make substantive changes in the model code. Such flexibility offers many practical advantages and makes it feasible to develop a useful data assimilation algorithm in a relatively short time. [24] The land surface model used in this study is the NOAH model described by Chen et al. [1996]. The current NOAH model has a long heritage and is related to a number of hydrologic models that have been tested extensively over a wide range of climate regimes. It is based on a typical onedimensional soil-vegetation-atmosphere transfer (SVAT) approach that solves the coupled energy and water budgets at the land surface and within the unsaturated zone. A resistance approach is used to account for both aerodynamic and vegetation controls on energy fluxes. NOAH is being freely offered and promoted as a community model by its developers and is currently one of the land surface models being used in the NASA Land Surface Data Assimilation (LDAS) project. [25] The state variables of the NOAH model include soil moisture and temperature in several layers covering the

6 35-6 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE EnKF Figure 3. Maps showing distribution of (a) soil, (b) vegetation, and (c) meteorological inputs for the NOAH land surface model and (d) clay and (e) sand fraction inputs for the radiative transfer model. uppermost 2 3 meters of soil column, skin temperature (bare soil or vegetation), canopy water storage, and various storage variables related to snow processes. The required micrometeorological forcing variables are incoming solar and longwave radiation, air temperature and humidity, precipitation, wind speed, and surface pressure. Several time-invariant parameters are also specified for each of the soil and vegetation classes shown in Figures 3a and 3b, respectively. [26] In our SGP 97 application, where we are concerned with spatial variability over scales of hundreds of kilometers, the NOAH model is applied at each pixel of an extensive estimation grid. For simplicity we assume that there are no lateral (horizontal) moisture and heat fluxes within the unsaturated zone. This assumption is reasonable for a low-relief landscape like the Southern Great Plains, where vertical fluxes dominate over horizontal fluxes. As a result of this simplification, the domain is discretized horizontally into a large number of uncoupled vertical soil columns, one associated with each pixel. A unique set of model parameters is specified for each pixel. [27] The radiative transfer model required in the update step of the ensemble Kalman filter relates the states of the NOAH model to the radio brightness measurements used in the data assimilation procedure. Our radiative transfer model closely resembles the model used by Jackson et al. [1999] to retrieve soil moisture from SGP97 microwave observations. In our version, the effect of surface soil moisture on soil dielectric properties is described by the mixing model of Wang and Schmugge [1980]. These dielectric properties influence the bare soil microwave emissivity through the Fresnel equations. The emissivity and surface ground temperature give the brightness temperature of the bare soil surface. Surface roughness [Choudhury et al., 1978] and vegetation [Jackson and Schmugge, 1991] effects are included to model the brightness temperature observed by the airborne microwave sensor. [28] The required inputs for the radiative transfer model are surface soil moisture and temperature (obtained from the NOAH model), vegetation type and vegetation water content, soil mineralogy (percent sand and percent clay), and surface roughness. Vegetation classifications are shown in Figure 3b while mineralogical classifications are shown in Figures 3d and 3e. In general, the effects of vegetation tend to weaken the sensitivity of measured brightness temperature to soil moisture (i.e. the signal becomes weaker as the vegetation density increases). However, for the types of vegetation at the SGP site (mostly grasses and cultivated crops) measured brightness temperatures still provide significant information about the surface soil moisture Statistical Description of Model Input Uncertainties and Measurement Errors [29] The ensemble Kalman filtering approach generates individual replicates from random values of selected model inputs and measurement errors. In order to apply this approach we need to specify the probability distributions of all random variables (inputs and errors) included in our SGP97 application. The uncertain inputs for this application are initial soil moisture, porosity, wilting point, minimum stomatal resistance, saturated hydraulic conductivity and

7 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE ENKF 35-7 precipitation. The only random error included is the radio brightness measurement error. The statistical properties we have assumed for these random variables are briefly discussed in the following paragraphs. [30] The nominal initial soil moisture used in our study is based on a constant value of 50% relative saturation over the entire soil profile in every pixel of the estimation grid. This percentage is multiplied by the spatially variable porosity to give a spatially variable initial volumetric moisture content. The unknown true initial soil moisture will vary from the nominal, sometimes significantly. In order to account for this, a normally distributed random fluctuation with a mean of zero and a standard deviation of 20% is added to the initial moisture content in each layer and each pixel for each replicate generated in the ensemble Kalman filter. Random fluctuations at different pixels are uncorrelated. This produces a spatially variable initial condition with a mean in each pixel equal to the nominal value. [31] Uncertainties in porosity and wilting point control the upper and lower limits on volumetric soil moisture. The saturated hydraulic conductivity, which is highly variable over space, is the primary parameter controlling the infiltration rate. The minimum stomatal resistance is the primary parameter controlling the effect of vegetation on soil moisture. In our study we account for uncertainties in these parameters by multiplying nominal values specified at each pixel by lognormal spatially uncorrelated random variables with means of 1.0. This approach assures that the random time-invariant inputs generated by the ensemble Kalman filter are positive and unbiased. [32] The nominal values for all time-invariant inputs are derived from the soil and vegetation maps shown in Figure 3. The multiplicative random coefficients for porosity and wilting point are assigned standard deviations equal to 5% of the corresponding nominal values. The hydraulic conductivity and minimum stomatal resistance coefficient standard deviations are assigned values equal to 100% of the nominals, reflecting the high degree of uncertainty in these variables. The resulting time-invariant random inputs vary significantly around the nominal values. [33] Precipitation is the most important time-dependent forcing variable in the SGP application. Precipitation uncertainties, especially in ungaged areas, can be expected to have significant impacts on the evolution and distribution of soil moisture. The density and coverage of the meteorological network available at the SGP study site are much better than in most regions of the world. In order to provide a data assimilation test that is representative of conditions encountered in continental or global-scale applications we chose to use only the El Reno forcing data, which is located roughly in the center of the domain. Consequently, the nominal precipitation values at all pixels were set equal to the observed El Reno precipitation. The random precipitation at each time step was obtained by multiplying this nominal precipitation by a lognormally distributed random variable with a mean of 1.0 and a standard deviation equal to 50% of the nominal value. The random multipliers at different times and pixels are uncorrelated. Temporal correlation could be readily added if desired. [34] Radio brightness measurement errors are assumed to be additive and normally distributed, with a mean of 0.0 and a standard deviation of 3 Kelvin. Errors in different pixels and at different times are assumed to be uncorrelated. In reality, these errors are probably correlated but it is difficult to identify the correlation structure from available data. This is a worthwhile topic for future research. [35] In the ensemble Kalman filter random values of all the inputs described above are generated for each replicate and the corresponding hydrologic states are propagated over the time intervals between measurements. The resulting ensemble reflects the uncertainty introduced by input errors. In particular, the ensemble replicates span a wider range of values and the variances of the propagated states increase, as compared to the case where model input values are held fixed at their nominal values. This increased variability across the ensemble tends to makes the filter rely more on measurements and reduces the adverse impact of model bias. Our experience with synthetic test problems indicates that an ensemble of 100 replicates is sufficiently large to provide accurate estimates of soil moisture for the SGP conditions [Reichle et al., 2002]. For this reason we used an ensemble size of 100 in the application described here. However, it should be noted that the factors affecting the minimum number of replicates in any given application are not well understood. This is an appropriate topic for further research. [36] Our decision to generate spatially uncorrelated model inputs in the ensemble Kalman filter reduces computational time significantly since it allows the updates in each pixel to be made independently of other pixels (i.e. the sample covariance matrix becomes block diagonal and need not be stored in its entirety). It is possible, and perhaps advisable in some situations, to introduce spatial correlation. The price to be paid for this enhancement is storage of the large sample covariance matrices needed to update all pixels simultaneously. 4. Data Assimilation Results [37] The performance of our ensemble-based data assimilation procedure can be evaluated in a number of different ways. Here we focus on the conditional means of the filter s moisture content and brightness temperature ensembles. For brevity, we refer to the conditional mean simply as the filter estimate. We also consider soil moisture predictions made without the benefit of radio brightness measurements. These open loop estimates are based solely on nominal soil, vegetation, and precipitation inputs propagated forward in time with no updates. The filter and open loop estimates both rely on the same inputs, which are, in all likelihood, imperfect. However, the filter can also benefit from information contained in radio brightness measurements. Improvements in the performance of filter estimates over open loop results demonstrate the value added by the radio brightness information Precipitation Inputs [38] It is helpful to begin our discussion of results by considering the implications of using the El Reno rainfall record as the nominal precipitation input for the entire site. Figure 4 provides useful information on the heterogeneity of the precipitation field. The three plots in Figure 4 show the actual rainfall time series observed at the three ground truth stations at Central Facility (CF), El Reno (ER), and Little Washita (LW) (see Figure 3c). Several large storms occurred

8 35-8 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE EnKF Figure 4. Actual precipitation time series for Central Facility (CF), El Reno (ER), and Little Washita (LW) pixels. The ensemble filter and open loop simulation assume that the ER applies throughout the SGP domain. Brightness temperature times are marked by stars. over the SGP domain during the month of the field experiment. Overall, the cumulative rainfall in the northern region near Central Facility is of similar magnitude to that at El Reno, yet the timing and depth of the major events are different. Little Washita in the southern region is significantly drier than the other stations, with only small amounts of rainfall falling in the south during periods when El Reno is experiencing large storms. [39] The three time series in Figure 4 indicate that errors will be introduced if El Reno rainfall is applied everywhere in the SGP97 domain (especially in an open loop run which is completely dependent on ground-based meteorological observations). However, such errors are likely to arise in typical large-scale (i.e. continental or global scale) applications. Of course, radio brightness measurements can be expected to add more information in regions with sparse ground-based data than in regions where ground stations already provide a reasonably accurate description of meteorological forcing Comparison of Radio Brightness Estimates and Observations Over The SGP97 Region [40] Figure 5 shows selected maps of the measured (ESTAR), open loop, and ensemble filter brightness temperatures estimates in all of our 4 km SGP97 pixels. These maps are for day 169, which is the first measurement (update)

9 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE ENKF 35-9 Figure 5. Brightness temperature maps for the SGP97 site on selected days during the experiment. The first column is the measured brightness temperature, the second column is the open loop brightness temperature estimate, and the third column is the ensemble filter brightness temperature estimate just after a measurement update. The fourth column is a map of the normalized innovations after each update. time, day 178, which is just after the first major storm, day 184, which is during the drydown after major storms at El Reno and Central Facility; and day 194, which is two days after the only significant storm in Little Washita and after an extended period of missing ESTAR measurements. [41] The first row in Figure 5 (day 169) clearly shows the immediate improvement in the estimated brightness temperature provided by the ensemble filter update. While the open loop simulation is still dominated by the initial condition, the estimated brightness picks up the primary spatial features shown in the ESTAR map. The estimated field captures relatively cool (moist) regions in the north, middle, and extreme south of the domain, as well as some hotter (drier) patches in between. Note that the estimated brightness fields presume that there are measurement as well as model errors and thus will not exactly match the ESTAR measurements.

10 35-10 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE EnKF This is in contrast to a retrieval (or direct insertion) algorithm based on a one-to-one relationship between radio brightness and soil moisture. Retrieval algorithms implicitly assume that the measurements are perfect and ignore model predictions whenever measurements are available. [42] On day 178 (second row), the open loop map is still relatively homogeneous compared to the measured and filtered radio brightness maps. The modest heterogeneity that is present in the open loop map primarily reflects variability in soil properties (see Figure 3), which tend to control soil moisture immediately after a storm (via infiltration and soil moisture redistribution). [43] On day 184 (third row), the ESTAR measurement indicates that the southern two-thirds of the domain is very dry (except for the region localized around El Reno), with the northernmost region still moist. At this time, the open loop estimate is controlled mostly by the vegetation type (see Figure 3), which has the greatest influence on soil moisture during the inter-storm period. The ensemble filter brightness temperature estimates in this period are lower than the ESTAR measurements. This is due primarily to the excess precipitation specified in the region near Little Washita. Nevertheless, the filter estimate is far superior to the open loop simulation. [44] The spatial pattern in the ESTAR measurements on day 194 is much more patchy, primarily because storms at this time are highly localized. The open loop estimate begins to capture more spatial heterogeneity but continues to perform poorly compared to the measurements. While the ensemble filter estimate does relatively well in picking up the moist patches over the domain, it is still not dry enough in the driest patches. This seems to be a residual effect from earlier estimates (e.g., on day 184). [45] Maps of the normalized innovations are plotted on the far right in Figure 5. The innovations variable is the difference between the predicted and measured radio brightness at a given pixel divided by the standard deviation of the differences, taken over all pixels. When the filter s Gaussian assumption is correct the normalized innovations values should be normally distributed with a mean of 0.0 and a standard deviation of 1.0. In this case most of the innovations values should fall between 2.0 and Also, when the filter is working optimally the innovations values should be spatially uncorrelated (i.e., the innovations map should not exhibit any distinct spatial structure). While the innovations generally satisfy the normality requirement, they have enough structure to suggest that the filter is not truly optimal. This is most likely due to a combination of factors, including spatial correlation in measurement errors and parameter fluctuations (which was ignored) and incorrect error statistics. Overall, the filter does significantly better than the open loop simulation, confirming that radio brightness measurements provide a useful supplement to the limited ground-based data used in our SGP evaluation Surface Soil Moisture Estimates and Uncertainties Over the SGP97 Region [46] While the brightness temperature comparison provides a good visual test of filter performance, the true objective of the data assimilation algorithm is to provide estimates of soil moisture. The first two columns of Figure 6 show the surface volumetric moisture content estimates (conditional ensemble means) in all pixels before and after a measurement update for the same days discussed above. A major benefit of the ensemble filtering technique is its ability to provide information about variability in these estimates. In particular, the sample standard deviation across the ensemble is a convenient measure of the magnitude of the estimation error. The soil moisture standard deviations are shown in the third and fourth columns of Figure 6 immediately before and after the update, respectively. [47] On day 169, after the first measurement update, a significant change in the estimate (mean) occurs. The update merges the brightness temperature map with the propagated surface soil moisture field. The additional information causes the estimate uncertainty (the estimation error standard deviation) to decrease significantly. On subsequent days the changes are less dramatic but the measurements continue to provide a reduction in uncertainty Brightness Temperature and Surface Soil Moisture Time Series at Ground Truth Sites [48] Another perspective on the performance of the ensemble filter and the open loop simulation can be obtained by examining the temporal evolution of brightness temperature and soil moisture at the intensively monitored Central Facility (CF), El Reno (ER), and Little Washita (LW) sites, where ground truth measurements are available. Here we compare the brightness temperature estimates at each of these sites to ESTAR measurements for the 800 m by 800 m pixel containing the station. We also compare volumetric moisture content estimates to the spatial average of the 14 gravimetric measurements taken within the same pixel. The within-pixel standard deviation of the gravimetric measurements is a measure of subpixel soil moisture variability. [49] Figure 7 compares the filter and open loop brightness temperature estimates with ESTAR measurements at CF site 8 (CF08), ER site 5 (ER05), and LW site 13 (LW13), all shown in Figure 3c. The ensemble of brightness temperature values is also shown. The filter estimate at any given time is the mean of the ensemble values. The behavior of individual ensemble members during the propagation and update steps provides useful insight about the workings of the ensemble filter. Generally speaking, at each update the ensemble replicates (and the ensemble mean) move toward the measurement and the ensemble spread is reduced. The open loop estimate is not updated and tends to move away from the measurements during drydown periods, especially at the CF08 and LW13 sites. This divergence is the result of using El Reno precipitation throughout the SGP site. The filter can compensate for precipitation errors by taking advantage of the information provided by the radio brightness measurements but the open loop simulation must rely entirely on the specified precipitation. [50] Figure 8 compares the filter and open loop soil moisture estimates to gravimetric soil moisture at the ground truth sites. As in Figure 7, the filter s ensemble of soil moisture values is shown. The gravimetric ± one standard deviation values (over measurements within the pixel) are shown with vertical error bars centered about the mean. Recall that for all three sites the initial soil moisture content was based on a value of 50% saturation. This value is relatively close to the observations at CF08 and LW13 but is significantly drier than the actual ER05 initial soil

11 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE ENKF Figure 6. Ensemble filter volumetric surface soil moisture content estimate and the corresponding estimation error standard deviation at selected times during the SGP97 experiment. The first two columns show the estimated soil moisture immediately before and immediately after the update on the indicated day. The two columns on the right are the filter s predicted estimation error standard deviations before and after the update. moisture. Assimilation of the first radio brightness measurement generally removes any bias imposed by the specified initial condition. At both CF08 and ER05, the estimated surface soil moisture time series generally is within the observation error bars, with some exceptions at CF08 due to use of El Reno precipitation throughout the domain (e.g., note days 177 and 191). Also, the estimate is generally updated toward the observation at measurement times. At LW13, the estimate does reasonably well until the large El Reno storms on days 177 and 179. [51] As mentioned earlier, Little Washita experiences much less precipitation than either El Reno or Central Facility. This causes both the filter and open loop estimates to depart from the gravimetric observations. However, the filter estimate, which benefits from the ESTAR measurements, does significantly better than the open loop estimate

12 35-12 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE EnKF Figure 7. Comparison of the ensemble filter (black lines), open loop (red lines), and measured (ESTAR) brightness temperatures (solid blue circles) at the three validation sites. The brightness temperature realizations in the filter ensemble are plotted as dotted (cyan) lines. under these circumstances. This applies even during the period when no ESTAR measurements are available (days ), reflecting the residual benefits of information gained in previous updates. Overall, the match between the filter s soil moisture estimates and available gravimetric measurements lends credibility to the large-scale maps discussed in the preceding section Root Zone Soil Moisture and Latent Heat Flux Comparison [52] Up to this point all of our discussion has focused on surface soil moisture (moisture in the uppermost 5 cm). Equally, if not more, important is the profile of soil moisture throughout the root zone. This profile affects evapotranspiration and the partitioning of available energy at the surface. A significant benefit of the ensemble Kalman filter is its ability to instantaneously update moisture estimates and error standard deviations throughout the soil profile. This is possible because the filter Kalman gain (derived from samples propagated through the land surface model) relates radio brightness measurements at the surface directly to the soil moisture in each layer. By contrast, traditional retrieval algorithms or direct insertion data assimilation methods only update surface soil moisture. The subsurface effects of the surface update only become apparent slowly, as surface moisture changes redistribute throughout the soil column.

13 MARGULIS ET AL.: SOIL MOISTURE ESTIMATION USING THE ENKF Figure 8. Comparison of the ensemble filter (black lines), open loop (red lines), and measured gravimetric surface soil moisture values (solid blue circles) at the three validation sites. Measurement ± one standard deviation range is indicated with vertical bars. The soil moisture realizations in the filter ensemble are plotted as dotted (cyan) lines. ESTAR measurement times are denoted by stars. [53] It is difficult to obtain ground truth measurements for checking the accuracy of soil moisture estimates in subsurface layers. One alternative is to compare observed latent heat fluxes to the values estimated by the ensemble filter. These fluxes provide a useful aggregate measure of profile soil moisture. Latent heat observations from flux towers are available at selected ARM-CART flux stations in the SGP region. Figure 9 compares time series of ensemble filter (black line) and open loop (dashed red line) latent heat flux estimates to measurements (closed blue circles) from the ARM-CART Central Facility (site CF01) flux station. [54] In the early part of the experiment (first 10 days), the filter and open loop estimates both significantly underestimate the observed latent heat flux, with the filter estimate only marginally better than the open loop. From the observations it appears that the deeper soil layers in the rootzone were significantly wetter (giving a larger latent heat flux) than the specified initial moisture content. By the middle of the experiment (second 10 days) the filter estimate comes reasonably close to the peak latent heat flux measurement on most days but the open loop estimate is consistently too small. This change coincides with large