Variational data assimilation for very large environmental problems

Variational data assimilation is commonly used in environmental forecasting to estimate the current state of the system from a model forecast and observational data. The assimilation problem can be written simply in the form of a nonlinear least squares optimization problem. However the practical solution of the problem in large systems requires many careful choices to be made in the implementation. In this article we present the theory of variational data assimilation and then discuss in detail how it is implemented in practice. Current solutions and open questions are discussed.

Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

Weak constraints in four-dimensional variational data assimilation

The formulation of four-dimensional variational data assimilation allows the incorporation of constraints into the cost function which need only be weakly satisfied. In this paper we investigate the value of imposing conservation properties as weak constraints. Using the example of the two-body problem of celestial mechanics we compare weak constraints based on conservation laws with a constraint on the background state.We show how the imposition of conservation-based weak constraints changes the nature of the gradient equation. Assimilation experiments demonstrate how this can add extra information to the assimilation process, even when the underlying numerical model is conserving.

Four-dimensional variational assimilation of ozone profiles from the Microwave Limb Sounder on the Aura satellite

Ozone profiles from the Microwave Limb Sounder (MLS) onboard the Aura satellite of the NASA's Earth Observing System (EOS) were experimentally added to the European Centre for Medium-range Weather Forecasts (ECMWF) four-dimensional variational (4D-var) data assimilation system of version CY30R1, in which total ozone columns from Scanning Imaging Absorption Spectrometer for Atmospheric CHartographY (SCIAMACHY) onboard the Envisat satellite and partial profiles from the Solar Backscatter Ultraviolet (SBUV/2) instrument onboard the NOAA-16 satellite have been operationally assimilated. As shown by results for the autumn of 2005, additional constraints from MLS data significantly improved the agreement of the analyzed ozone fields with independent observations throughout most of the stratosphere, owing to the daily near-global coverage and good vertical resolution of MLS observations. The largest impacts were seen in the middle and lower stratosphere, where model deficiencies could not be effectively corrected by the operational observations without the additional information on the ozone vertical distribution provided by MLS. Even in the upper stratosphere, where ozone concentrations are mainly determined by rapid chemical processes, dense and vertically resolved MLS data helped reduce the biases related to model deficiencies. These improvements resulted in a more realistic and consistent description of spatial and temporal variations in stratospheric ozone, as demonstrated by cases in the dynamically and chemically active regions. However, combined assimilation of the often discrepant ozone observations might lead to underestimation of tropospheric ozone. In addition, model deficiencies induced large biases in the upper stratosphere in the medium-range (5-day) ozone forecasts.

A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances

Variational data assimilation for parameter estimation: application to a simple morphodynamic model

Data assimilation is a sophisticated mathematical technique for combining observational data with model predictions to produce state and parameter estimates that most accurately approximate the current and future states of the true system. The technique is commonly used in atmospheric and oceanic modelling, combining empirical observations with model predictions to produce more accurate and well-calibrated forecasts. Here, we consider a novel application within a coastal environment and describe how the method can also be used to deliver improved estimates of uncertain morphodynamic model parameters. This is achieved using a technique known as state augmentation. Earlier applications of state augmentation have typically employed the 4D-Var, Kalman filter or ensemble Kalman filter assimilation schemes. Our new method is based on a computationally inexpensive 3D-Var scheme, where the specification of the error covariance matrices is crucial for success. A simple 1D model of bed-form propagation is used to demonstrate the method. The scheme is capable of recovering near-perfect parameter values and, therefore, improves the capability of our model to predict future bathymetry. Such positive results suggest the potential for application to more complex morphodynamic models.