Introduction to Ocean Data Assimilation

Abstract

Data assimilation is the process of hindcasting, now-casting, and forecasting using information from both observations and ocean dynamics. Modern ocean forecasting systems rely on data assimilation to estimate initial and boundary data, to interpolate and smooth sparse or noisy observations, and to evaluate observing systems and dynamical models. Every data assimilation system implements an optimality criterion which defines how to best combine dynamics and observations, given an hypothesized error model for both. The realization of practical ocean data assimilation systems is challenging due to both the technical issues of implementation, and the scientific issues of determining the appropriate set of hypothesized priors. This chapter reviews methodologies and highlights themes common to all approaches.

“Conventional ocean modeling consists of solving the model equations as accurately as possible, and then comparing the results with observations. While encouraging levels of quantitative agreement have been obtained, as a rule there is significant quantitative disagreement owing to many sources of error: model formulation, model inputs, computation and the data themselves. Computational errors aside, the errors made both in formulating the model and in specifying its inputs usually exceed the errors in the data. Thus it is unsatisfactory to have a model solution which is uninfluenced by the data.”—Bennett (Inverse Methods in Physical Oceanography, 1st edn. Cambridge University Press, New York, p. 112, 1992)

Appendix

1.1 Software and WWW Resources

The Inverse Ocean Model (IOM) is a software toolkit designed to produce a custom variational data assimilation system from basic modeling components provided by the user (Bennett et al. 2008). The software features a graphical user interface (GUI) which is used to select assimilation or analysis algorithms and monitor program execution. The IOM has been used successfully with parallel and serial codes, including structured and unstructured grid finite-differece models, and finite-element models (Muccino et al. 2008). The website, http://iom.asu.edu, contains pedagogical material as well as software.

Another software system, the Data Assimilation Research Testbed (DART), provides a framework for developing, testing, and distributing ensemble data assimilation methodologies (Anderson et al. 2009). The DART system uses algorithms which do not require adjoint codes, so it has been used by a relatively large number of researchers and educators.

The Tangent linear and Adjoint Model Compiler (TAMC) was developed by Giering and Kaminski (1998) as a source-to-source Fortran translator for the generation of tangent-linear and adjoint codes needed in variational data assimilation and sensitivity studies. The company, FastOpt, maintains an active presence in this field, and is a good source of information on latest developments and conferences (http://fastopt.com/).

Another source of information and software is the ACTS-Adjoint Compiler project, which has created OpenAD, a tool for automatic differentiation of C and Fortran code (http://ww.mcs. anl.gov/OpenAD/). A recent highlight of their work is the development of adjoint codes from models which use domain decomposition via the parallel MPI library (Utke et al. 2009).

A brief search for “data assimilation” on the web will uncover many course materials and tutorials on data assimilation. One notable source is the European Center for Medium Range Weather Forecasting (ECMWF), which publishes online an excellent series of lecture notes on data assimilation and the use of satellite data. See http://www.ecmwf.int/newsevents/training/rcourse\_notes/ for their Meteorological Training Course Lecture Notes.

Glossary

Analysis

The analysis is the end-point or result of a data assimilation. It is the best estimate of the true state of the ocean at a given time, or within a given time interval, and, ideally, it is accompanied by an estimate of its errors. If the analysis is retrospective, i.e., it is the best estimate of the oceanic state at some past time conditioned upon measurements both before and after the analysis time, it is called a “re-analysis.” Typically the analysis is presented as a set of uniformly gridded oceanic state variables (sea-surface height, current vectors, temperature, salinity, etc.), on the same discrete grid as the ocean model. The analysis may be the end result of a forecast system, or it may provide input for the computation of other diagnostics, such as the computation of transport across transects. Sometimes the analysis is compared with new observations to either verify the analysis or assess the quality of the new observations.

Analysis increment

The analysis increment is the difference between the analysis field and the background. Equivalently, the analysis increment is the correction to the background field which results in the optimal analysis.

Background

The background state, sometimes called the “first guess,” is the prediction of the oceanic state prior to the assimilation of data. In the absence of other information, a climatology or other dynamics-free estimate of the ocean may serve as the background.

Control variables

The control variables, sometimes simply called “the controls,” are the independent quantities to be estimated in the data assimilation. The dynamical model consists of a set of diagnostic or prognostic relations which relate the control variables to the state variables. There is not a unique partition between control variables and state variables, but the controls are generally regarded as inputs while the state is regarded as an output. For example, in the 4D-Var algorithm, the model’s initial conditions are regarded as the control variable; although, these same initial conditions and the resulting forecast may be regarded as state variables. In the Kalman Filter, the system noise is regarded as the control variable.

Data assimilation

Data assimilation is the systematic methodology of incorporating information from an observing system into a dynamical model in such a manner that an optimality criterion is satisfied. Optimality criteria typically express a maximum likelihood or minimum mean square error criterion. In practice, many data assimilation systems find analysis states which only approximately satisfy the stated optimality criterion. This is usually considered acceptable because the optimality criteria are based on error models which are themselves approximate.

Dynamical model

It is assumed that the state of the ocean is predicted or modeled by a set of dynamics, e.g., Newton’s Laws expressed in the usual formulations of continuum mechanics such as the Navier-Stokes equations or the shallow water equations. The dynamical model is assumed to be formulated as a mathematically well-posed initial-boundary-value problem.

Error model

An error model is a description of the probability distribution of some possibly multivariate or field quantity. For example, an error model for a measurement of temperature might minimally declare that the errors have zero mean (are unbiased), known variance, σ, and are Gaussian distributed. An error model for an observing system would minimally consist of error models for the individual observations. An error model for a set of dynamics would minimally consist of error sub-models for the initial conditions, boundary conditions, and other model inhomogeneities. Each of these sub-models would be characterized by its own space-time covariance structure, as appropriate.

Generalized inversion

Because the oceanic state is, in principle, uniquely determined by the ocean dynamics, the addition of observational data in data assimilation makes the oceanic state an over-determined quantity. Alternately, if we consider the measurement error and the dynamics errors to be unknown quantities, which ought to be determined by the data assimilation, the problem of identifying both the oceanic state together with the error fields is an under-determined problem. From this perspective, data assimilation may be regarded as a generalized inversion of the ocean model dynamics. The generalized inverse of the dynamical model consists of the analysis fields, as well as estimates of the error fields, the statistics of which were specified a priori by the error models. Bennett (1992, 2002) uses this language to describe data assimilation, thus highlighting a unifying theme of mathematical inverse theory, statistical estimation, control theory, and non-parametric estimation methods.

Innovation vector or residual vector

The innovation vector is the difference between the observation vector and an observation of the ocean state, written as y − Hx in the notation of this article. Sometimes the background innovations, y − Hx ^b, may be distinguished from the analysis innovations, y − Hx ^a.

Objective analysis

The technique of objective analysis—called optimal interpolation, statistical interpolation, and Gauss-Markov smoothing—was originally applied to create a set of consistently gridded fields from sparse observations (Bretherton et al. 1976). When a background field is present, the corrections applied to the background may be called analysis increments. The technique has a close relationship with multivariate smoothing splines (Wahba 1990; Bennett 1992).

Observing system

An observing system produces observations or measurements of some subset of variables characterizing the ocean. Observing systems are defined by a set of observation operators, also called measurement kernels, one per measurement, which mathematically represent the mapping from the oceanic state to a finite set of real values.

State variables

State variables are those fields or quantities which characterize the oceanic state to be estimated. More abstractly, state variables are elements in the domain of the observation operators.

State vector

A state vector is a finite-dimensional state variable. Even when the dynamics are represented by a set of partial differential equations, the computational implementation usually requires projection onto a finite-dimensional vector.