Kalman Filter Data Assimilation Research Articles

Assimilation of remotely sensed latent heat flux in a distributed hydrological model

This paper addresses the question of whether remotely sensed latent heat flux estimates over a catchment can be used to improve distributed hydrological model water balance computations by the process of data assimilation. The data used is a series of noaa-avhrr satellite images for the Drentse Aa catchment in the Netherlands for the year 1995. These 1×1 km resolution images are converted into latent heat flux estimates using sebal ( surface energy balance algorithm for land [J Hydrol 2000;229:87]). The physically-based distributed model simgro ( simulation of groundwater flow and surface water levels [J Hydrol 1997;192:158]) is used to compute the water balance of the Drentse Aa catchment for that same year. Comparison between model-derived and remotely sensed area-averaged evapotranspiration estimates show good agreement, but spatial analysis of the model latent heat flux estimates indicate systematic underestimation in areas with higher elevation. A constant gain Kalman filter data assimilation algorithm is used to correct the internal state variables of the distributed model whenever remotely sensed latent heat flux estimates are available. It was found that the spatial distribution of model latent heat flux estimates in areas with higher elevation were improved through data assimilation.

Three‐dimensional soil moisture profile retrieval by assimilation of near‐surface measurements: Simplified Kalman filter covariance forecasting and field application

The Kalman filter data assimilation technique is applied to a distributed three‐dimensional soil moisture model for retrieval of the soil moisture profile in a 6 ha catchment using near‐surface soil moisture measurements. A simplified Kalman filter covariance forecasting methodology is developed based on forecasting of the state correlations and imposed state variances. This covariance forecasting technique, termed the modified Kalman filter, was then used in a 1 month three‐dimensional field application. Two updating scenarios were tested: (1) updating every 2 to 3 days and (2) a single update. The data used were from the Nerrigundah field site, near Newcastle, Australia. This study demonstrates the feasibility of data assimilation in a quasi three‐dimensional distributed soil moisture model, provided simplified covariance forecasting techniques are used. It also identifies that (1) the soil moisture profile cannot be retrieved from near‐surface soil moisture measurements when the near‐surface and deep soil layers become decoupled, such as during extreme drying events; (2) if simulation of the soil moisture profile is already good, the assimilation can result in a slight degradation, but if the simulation is poor, assimilation can yield a significant improvement; (3) soil moisture profile retrieval results are independent of initial conditions; and (4) the required update frequency is a function of the errors in model physics and forcing data.

An Ensemble Adjustment Kalman Filter for Data Assimilation

A theory for estimating the probability distribution of the state of a model given a set of observations exists. This nonlinear filtering theory unifies the data assimilation and ensemble generation problem that have been key foci of prediction and predictability research for numerical weather and ocean prediction applications. A new algorithm, referred to as an ensemble adjustment Kalman filter, and the more traditional implementation of the ensemble Kalman filter in which “perturbed observations” are used, are derived as Monte Carlo approximations to the nonlinear filter. Both ensemble Kalman filter methods produce assimilations with small ensemble mean errors while providing reasonable measures of uncertainty in the assimilated variables. The ensemble methods can assimilate observations with a nonlinear relation to model state variables and can also use observations to estimate the value of imprecisely known model parameters. These ensemble filter methods are shown to have significant advantages over four-dimensional variational assimilation in low-order models and scale easily to much larger applications. Heuristic modifications to the filtering algorithms allow them to be applied efficiently to very large models by sequentially processing observations and computing the impact of each observation on each state variable in an independent calculation. The ensemble adjustment Kalman filter is applied to a nondivergent barotropic model on the sphere to demonstrate the capabilities of the filters in models with state spaces that are much larger than the ensemble size. When observations are assimilated in the traditional ensemble Kalman filter, the resulting updated ensemble has a mean that is consistent with the value given by filtering theory, but only the expected value of the covariance of the updated ensemble is consistent with the theory. The ensemble adjustment Kalman filter computes a linear operator that is applied to the prior ensemble estimate of the state, resulting in an updated ensemble whose mean and also covariance are consistent with the theory. In the cases compared here, the ensemble adjustment Kalman filter performs significantly better than the traditional ensemble Kalman filter, apparently because noise introduced into the assimilated ensemble through perturbed observations in the traditional filter limits its relative performance. This superior performance may not occur for all problems and is expected to be most notable for small ensembles. Still, the results suggest that careful study of the capabilities of different varieties of ensemble Kalman filters is appropriate when exploring new applications.

Distance-Dependent Filtering of Background Error Covariance Estimates in an Ensemble Kalman Filter

The usefulness of a distance-dependent reduction of background error covariance estimates in an ensemble Kalman filter is demonstrated. Covariances are reduced by performing an elementwise multiplication of the background error covariance matrix with a correlation function with local support. This reduces noisiness and results in an improved background error covariance estimate, which generates a reduced-error ensemble of model initial conditions. The benefits of applying the correlation function can be understood in part from examining the characteristics of simple 2 × 2 covariance matrices generated from random sample vectors with known variances and covariance. These show that noisiness in covariance estimates tends to overwhelm the signal when the ensemble size is small and/or the true covariance between the sample elements is small. Since the true covariance of forecast errors is generally related to the distance between grid points, covariance estimates generally have a higher ratio of noise to signal with increasing distance between grid points. This property is also demonstrated using a two-layer hemispheric primitive equation model and comparing covariance estimates generated by small and large ensembles. Covariances from the large ensemble are assumed to be accurate and are used a reference for measuring errors from covariances estimated from a small ensemble. The benefits of including distance-dependent reduction of covariance estimates are demonstrated with an ensemble Kalman filter data assimilation scheme. The optimal correlation length scale of the filter function depends on ensemble size; larger correlation lengths are preferable for larger ensembles. The effects of inflating background error covariance estimates are examined as a way of stabilizing the filter. It was found that more inflation was necessary for smaller ensembles than for larger ensembles.

A singular evolutive extended Kalman filter for data assimilation in oceanography

In this work, we propose a modified form of the extended Kalman filter (KF) for assimilating oceanic data into numerical models. Its development consists essentially of approximating the error covariance matrix by a singular low rank matrix, which amounts in practice to making no correction in those directions for which the error is the most attenuated by the system. This not only reduces the implementation cost but may also improve the filter stability as well. These `directions of correction' evolve with time according to the model evolution, which constitutes the most original feature of this filter and distinguishes it from other sequential assimilation methods based on the projection onto a fixed basis of functions. A method for initializing the filter based on the empirical orthogonal functions (EOF) is also described. An example of assimilation based on the quasi-geostrophic (QG) model for a square ocean domain with a certain wind stress forcing pattern is given. Although this is only a simple test case designed to assess the feasibility of the method, the results are very encouraging.