## Px xy Py ylxPx x Py y132

Equation (13.2) is a simple prescription for combining information from both the dynamics and the data. Given estimates of the errors in initial conditions, boundary forcing, or other model inhomogeneities, one can, in principle, find PX(x), the probability distribution of the oceanic state, in the absence of measurements. Knowledge of the measurement system determines PY(y | x), the probability distribution of the observations, conditioned on the oceanic state. With these quantities in hand, it is simply a matter of computation to find the posterior probability distribution of the oceanic state conditioned on the observations, PX(x |y). The denominator, PY(y)=]PY(y | x)PX(x)dx, can be computed; however, since this pdf is independent of x, it merely serves to normalize PX(x |y).

There is a choice regarding whether to use a maximum likelihood, mean, or median estimator, but these all coincide if the assumed statistics are multivari-ate Gaussian, and the mean is used almost universally. The following factors are generally of greater importance and vary widely among ocean data assimilation systems.

The Definition of Oceanic State Variables Implicit in the above discussion is the assumption that the oceanic state consists of the fields of momentum, buoyancy, and pressure within a region of the ocean, within some time interval. The number of state variables may be considerably reduced in practice, depending on context, by using diagnostic relations amongst the variables. Dimensionality is important. Consider, for example, the fields in a regional ocean model defined on a spatial grid of NX= 200 by NY = 200 horizontal grid points, and NZ = 30 vertical grid points, at NT = 1,000 time points. A sequential assimilation scheme might estimate initial conditions of sea-surface height at N = NX * NY grid points, resulting in a cardinality of N = 4T04 for the state variable X. Alternately, if X is taken as the initial conditions for the N = NX * NY * NZ * 4 values of the horizontal velocity, buoyancy, and pressure field (u, v, b, p), one has N = 4.8 • 106 unknowns in the state vector. In some versions of so-called "weak-constraint 4-D variational assimilation" (W4D-Var), one seeks an optimal state estimate of the above fields at all NT time steps, which yields a cardinality of N = 4.8 • 109 for the unknown state.

Complexity of the Error Models If the error in the initial conditions, boundary conditions, etc. can be adequately approximated by multivariate Gaussian distributions, then the implementation of the Bayesian analysis procedure is greatly simplified. But the specification of Gaussian distributions requires estimates of the means, variances, and cross-covariances of the relevant fields, in both space and time. Prescribing realistic error models can be a challenge.

Complexity of the Model Dynamics Even if the errors are correctly described by Gaussian distributions, the model dynamics may be sufficiently nonlinear to render the pdf of the model state PX(x) non-Gaussian. Differing treatments of the nonlin-

earity in the model dynamics yield both formal and practical differences between various data assimilation algorithms.

13.3.2 Example 1: Estimation of a Scalar

To make ideas concrete, consider first a trivial example, namely, the estimation of a scalar by combining information from a climatology and a single observation.

Assume that one wishes to estimate a scalar, say, temperature, denoted x. A climatology has been constructed, from which can be approximated the probability distribution function

where the background xb is the climatological mean. In other words, the climatology (which contains no dynamics, but is prescribed from prior data) is used for the background, with expected deviation ax.

A thermometer provides an observation of temperature with finite accuracy. Given temperature, x, the probability distribution of the observations is assumed to be a Gaussian also,

In other words, the measurements are assumed to be unbiased, and the standard deviation of the measurement error is a.

It is left as an exercise to the reader to use the definition to

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