Dennis McLaughlin

Hydrologic Data Assimilation with the Ensemble Kalman Filter

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.

A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow

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 log10 (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.

Stochastic analysis of nonstationary subsurface solute transport: 1. Unconditional moments

This paper applies stochastic methods to the analysis and prediction of solute transport in heterogeneous saturated porous media. Partial differential equations for three unconditional ensemble moments (the concentration mean, concentration covariance, and velocity concentration cross covariance) are derived by applying perturbation techniques to the governing transport equation for a conservative solute. Concentration uncertainty is assumed to be the result of unmodeled small-scale fluctuations in a steady state velocity field. The moment expressions, which describe how each moment evolves over time and space, resemble the classic deterministic advection-dispersion equation and can be solved using similar methods. A solution procedure based on a Galerkin finite element algorithm is illustrated with a hypothetical two-dimensional example. For this example the required steady state velocity statistics are obtained from an infinite domain spectral solution of the stochastic groundwater flow equation. The perturbation solution is shown to reproduce the statistics obtained from a Monte Carlo simulation quite well for a natural log conductivity standard deviation of 0.5 and moderately well for a natural log conductivity standard deviation of 1.0. The computational effort required for a perturbation solution is significantly less than that required for a Monte Carlo solution of acceptable accuracy. Sensitivity analyses conducted with the perturbation approach provide qualitative confirmation of a number of results obtained by other investigators for more restrictive special cases.

Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media

This paper presents a numerical method for simulating flow fields in a stochastic porous medium that satisfies locally the Darcy equation, and has each of its hydraulic parameters represented as one realization of a three-dimensional random field. These are generated by using the Turning Bands method. Our ultimate objective is to obtain statistically meaningful solutions in order to check and extend a series of approximate analytical results previously obtained by a spectral perturbation method (L. W. Gelhar and co-workers). We investigate the computational aspects of the problem in relation with stochastic concepts. The difficulty of the numerical problem arises from the random nature of the hydraulic conductivities, which implies that a very large discretized algebraic system must be solved. Indeed, a preliminary evaluation with the aid of scale analysis suggests that, in order to solve meaningful flow problems, the total number of nodes must be of the order of 106. This is due to the requirement that Δxi ≪ gli ≪ Li, where Δxi is the mesh size, λi is a typical correlation scale of the inputs, and Li is the size of the flow domain (i = 1, 2, 3). The optimum strategy for the solution of such a problem is discussed in relation with supercomputer capabilities. Briefly, the proposed discretization method is the seven-point finite differences scheme, and the proposed solution method is iterative, based on prior approximate factorization of the large coefficient matrix. Preliminary results obtained with grids on the order of one hundred thousand nodes are discussed for the case of steady saturated flow with highly variable, random conductivities.

An integrated approach to hydrologic data assimilation: interpolation, smoothing, and filtering.

The hydrologic data assimilation problem can be posed in a probabilistic framework that emphasizes the need to account for uncertainty when combining different sources of information. This framework indicates where approximations need to be introduced and provides a way to compare alternative data assimilation methods. When discussing data assimilation it is useful to distinguish interpolation, smoothing, and filtering problems. Interpolation is illustrated here with an example based on multi-scale estimation of rainfall during the TOAGA-COARE field experiment. Smoothing is illustrated with a variational soil moisture estimation algorithm applied to the SGP97 field experiment. Filtering is illustrated with an ensemble Kalman filter, also applied to the SGP97 experiment. All of these data assimilation algorithms implicitly rely on linear Gaussian assumptions that can only be expected to apply in special cases. Although more general nonlinear data assimilation methods are available they are not practical for the very large problems frequently encountered in hydrology. Future research in hydrologic data assimilation will be need to focus on the issue of high dimensionality and on the need for more realistic descriptions of model and measurement error. This effort will be most successful if the modeling and data assimilation problems are approached in a coordinated way.

Downscaling of radio brightness measurements for soil moisture estimation: A four‐dimensional variational data assimilation approach

This paper investigates the feasibility of estimating large-scale soil moisture profiles and related land surface variables from 1.4 GHz (L-band) passive microwave measurements, using variational data assimilation. Our four-dimensional assimilation algorithm takes into account both model and measurement uncertainties and provides dynamically consistent interpolation and extrapolation of remote sensing data over space and time. The land surface hydrologic model which forms the heart of the variational algorithm was expressly designed for data assimilation purposes. This model captures key physical processes while remaining computationally efficient. We test our algorithm with a series of synthetic experiments based on the Southern Great Plains 1997 Hydrology Experiment. These experiments provide insights about three issues that are crucial to the design of an operational soil moisture assimilation system. Our first synthetic experiment shows that soil moisture can be satisfactorily estimated at scales finer than the resolution of the brightness images. This downscaling experiment indicates that brightness images with a resolution of tens of kilometers can yield soil moisture profile estimates on a scale of a few kilometers, provided that micrometeorological, soil texture, and land cover inputs are available at the finer scale. In our second synthetic experiment we show that adequate soil moisture estimates can be obtained even if quantitative precipitation data are not available. Model error terms estimated from radio brightness measurements are able to account in an aggregate way for the effects of precipitation events. In our third experiment we show that reductions in estimation performance resulting from a decrease in the length of the assimilation time interval are offset by a substantial improvement in computational efficiency.

Variational data assimilation of microwave radiobrightness observations for land surface hydrology applications

Our ability to accurately describe large-scale variations in soil moisture is severely restricted by process uncertainty and the limited availability of appropriate soil moisture data. Remotely sensed microwave radiobrightness observations can cover large scales but have limited resolution and are only indirectly related to the hydrologic variables of interest. The authors describe a four-dimensional (4D) variational assimilation algorithm that makes best use of available information while accounting for both measurement and model uncertainty. The representer method used is more efficient than a Kalman filter because it avoids explicit propagation of state error covariances. In a synthetic example, which is based on a field experiment, the authors demonstrate estimation performance by examining data residuals. Such tests provide a convenient way to check the statistical assumptions of the approach and to assess its operational feasibility. Internally computed covariances show that the estimation error decreases with increasing soil moisture. An adjoint analysis reveals that trends in model errors in the soil moisture equation can be estimated from daily L-band brightness measurements, whereas model errors in the soil and canopy temperature equations cannot be adequately retrieved from daily data alone. Nonetheless, state estimates obtained from the assimilation algorithm improve significantly on prior model predictions derived without assimilation of radiobrightness data.

Stochastic modeling of large‐scale flow in heterogeneous unsaturated soils

A simulation experiment is used to test the stochastic theory of unsaturated flow proposed by Mantoglou and Gelhar (1987a, b, c). Tension means and variances derived from this theory are compared to tension distributions obtained from a detailed three-dimensional model developed by Ababou (1988). The synthetically generated unsaturated hydraulic conductivity functions used in the detailed model vary significantly over relatively small distances. Results from a simulated infiltration event indicate that the stochastic theorys predictions reproduce the overall trend of the heterogeneous tension distribution. Deviations from the predicted mean generally lie within confidence intervals derived from the predicted variance. The stochastic theory predicts less vertical moisture movement and somewhat more horizontal spreading than a comparable deterministic analysis. This reflects the fact that the effective hydraulic conductivity function used in the stochastic approach is anisotropic. The magnitude of this anisotropy increases with increasing tension.

Assessing the Performance of the Ensemble Kalman Filter for Land Surface Data Assimilation

Abstract The ensemble Kalman filter provides an easy-to-use, flexible, and efficient option for data assimilation problems. One of its attractive features in land surface applications is its ability to provide distributional information about variables, such as soil moisture, that can be highly skewed or even bimodal. The ensemble Kalman filter relies on normality approximations that improve its efficiency but can also compromise the accuracy of its distributional estimates. The effects of these approximations can be evaluated by comparing the conditional marginal distributions and moments estimated by the ensemble Kalman filter with those obtained from a sequential importance resampling (SIR) particle filter, which gives exact solutions for large ensemble sizes. Comparisons for two land surface examples indicate that the ensemble Kalman filter is generally able to reproduce nonnormal soil moisture behavior, including the skewness that occurs when the soil is either very wet or very dry. Its conditional m...

A NONSTATIONARY SPECTRAL METHOD FOR SOLVING STOCHASTIC GROUNDWATER PROBLEMS : UNCONDITIONAL ANALYSIS

Stochastic analyses of groundwater flow and transport are frequently based on partial differential equations which have random coefficients or forcing terms. Analytical methods for solving these equations rely on restrictive assumptions which may not hold in some practical applications. Numerically oriented alternatives are computationally demanding and generally not able to deal with large three-dimensional problems. In this paper we describe a hybrid solution approach which combines classical Fourier transform concepts with numerical solution techniques. Our approach is based on a nonstationary generalization of the spectral representation theorem commonly used in time series analysis. The generalized spectral representation is expressed in terms of an unknown transfer function which depends on space, time, and wave number. The transfer function is found by solving a linearized deterministic partial differential equation which has the same form as the original stochastic flow or transport equation. This approach can accomodate boundary conditions, spatially variable mean gradients, measurement conditioning, and other sources of nonstationarity which cannot be included in classical spectral methods. Here we introduce the nonstationary spectral method and show how it can be used to derive unconditional statistics of interest in groundwater flow and transport applications.