Soil moisture controls the partitioning of moisture and energy fluxes at the land surface and is a key variable in weather and climate prediction. The performance of the ensemble Kalman filter (EnKF) for soil moisture estimation is assessed by assimilating L-band (1.4 GHz) microwave radiobrightness observations into a land surface model. An optimal smoother (a dynamic variational method) is used as a benchmark for evaluating the filter's performance. In a series of synthetic experiments the effect of ensemble size and non-Gaussian forecast errors on the estimation accuracy of the EnKF is investigated. With a state vector dimension of 4608 and a relatively small ensemble size of 30 (or 100; or 500), the actual errors in surface soil moisture at the final update time are reduced by 55% (or 70%; or 80%) from the value obtained without assimilation (as compared to 84% for the optimal smoother). For robust error variance estimates, an ensemble of at least 500 members is needed. The dynamic evolution of the estimation error variances is dominated by wetting and drying events with high variances during drydown and low variances when the soil is either very wet or very dry. Furthermore, the ensemble distribution of soil moisture is typically symmetric except under very dry or wet conditions when the effects of the nonlinearities in the model become significant. As a result, the actual errors are consistently larger than ensemble-derived forecast and analysis error variances. This suggests that the update is suboptimal. However, the degree of suboptimality is relatively small and results presented here indicate that the EnKF is a flexible and robust data assimilation option that gives satisfactory estimates even for moderate ensemble sizes.

This paper presents a functional formulation of the groundwater flow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms. Unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a (possibly infinite) basis function expansion with random coefficients. The unknown parameter function is related to the measurements used for estimation by a “forward operator” which describes the measurement process. In the particular case considered here, the parameter of interest is the large‐scale log hydraulic conductivity, the measurements are point values of log conductivity and piezometric head, and the forward operator is derived from an upscaled groundwater flow equation. The inverse algorithm seeks the “most probable” or maximum a posteriori estimate of the unknown parameter function. When the measurement errors and parameter function are Gaussian and independent, the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness‐of‐fit and prior terms. When the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate. This constraint tends to make the problem well‐posed by limiting the range of admissible solutions. The Gaussian maximum a posteriori problem may be solved with variational methods, using functional generalizations of Gauss‐Newton or gradient‐based search techniques. Several popular groundwater inverse algorithms are either special cases of, or variants on, the functional maximum a posteriori algorithm. These algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use (linear versus nonlinear). The accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a Bayesian extension of the Cramer‐Rao lower bound on the estimation error covariance. This bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements.

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 km2) of soil 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.

Abstract. This paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity. The ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant. Seven geostatistical methods (fast Fourier transform (FF), fractal simulation (FS), linearized cokriging (LC), linearized semianalytical (LS), maximum likelihood (ML), pilot point (PP), and sequential self-calibration (SS)) were compared on four synthetic data sets. Each data set had specific features meeting (or not) classical assumptions about stationarity, amenability to a geostatistical description, etc. The comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km. Four of the methods, LS, ML, PP, and SS, were identified as being approximately equivalent for the specific problems considered. The magnitude of the variance of the transmissivity fields, which went as high as 10 times the generally accepted range for linearized approaches, was not a problem for the linearized methods when applied to stationary fields; that is, their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods. Nonstationarity of the "true" transmissivity field, or the presence of "anomalies" such as high-permeability fracture zones was, however, more of a problem for the linearized methods. The importance of the proper selection of the semivariogram of the 1og•0 (T) field (or the ability of the method to optimize this variogram iteratively) was found to have a significant impact on the accuracy and precision of the travel time predictions. Use of additional transient information from pumping tests did not result in major changes in the outcome. While the methods differ in their underlying theory, and the codes developed to implement the theories were limited to varying degrees, the most important factor for achieving a successful solution was the time and experience devoted by the user of the method. •2Stanford University, Stanford, California.•3Duke Engineering and Services, Inc., Austin, Texas.•4University of Arizona, Tucson.•Slnstitut Franqais du Pftrole, Rueil-Malmaison, France.•6University of California, Berkeley.Copyright 1998 by the American Geophysical Union. Paper number 98WR00003.0043-1397/98/98WR-00003509.00 tion, or performance assessment of planned waste disposal projects, it is no longer enough to determine the "best estimate" of the distribution in space of the aquifer parameters. A measure of the uncertainty associated with this estimation is also needed. Geostatistical techniques are ideally suited to filling this role. Basically, geostatistics fits a "structural model" to the data, reflecting their spatial variability. Then, both "best estim...