## W m 2 x x2 y y2

Figure 13.4 shows the general scale of flow features which can be identified from this rather limited, and idealized, observational array. The spatial density of the measurements has been chosen to be compatible with the 12.5 km correlation scale of the sea-surface height. Figure 13.5 illustrates what happens when the observing array either over- or under-samples the variability of the unknown field. In the case where the unknown field has a correlation scale of L = 3 km x

(Fig. 13.5, left panel), there is not enough information in the measurements to estimate the sea-surface height field. Although data assimilation can potentially improve out knowledge of unknown or uncertain oceanic fields, it cannot add significant information if the observations fail to constrain the dominant scales of variability in the fields. A more ideal situation is shown in the right panel of Fig. 13.5, where the correlation scale of the field is generally larger than the spacing of the observations.

The determination of whether a given observing system constrains the variability of the field to be estimated is an important topic in the analysis of data assimilation systems. Results in this area are found in the antenna analysis of (Bennett 1992). A

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Fig. 13.5 Impact of Correlation Scale, L . The left panel illustrates the outcome of attempting to reconstruct a field which is severely undersampled. The correlation scale of the unknown field is Lx=3 km, which is less than the spacing between the observations. The right panel illustrates the opposite situation, where the observations generally well-sample the field, which, in this case, has a correlation scale of 25 km

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Fig. 13.5 Impact of Correlation Scale, L . The left panel illustrates the outcome of attempting to reconstruct a field which is severely undersampled. The correlation scale of the unknown field is Lx=3 km, which is less than the spacing between the observations. The right panel illustrates the opposite situation, where the observations generally well-sample the field, which, in this case, has a correlation scale of 25 km complementary description in terms of information content and degrees of freedom is found in Stewart et al. (2008).

An important generalization of the above is considered next, with particular attention paid to the forecast cycle, which leads to an evolution equation for P (the Kalman Filter), as well as a consideration of nonlinearity in both the ocean dynamics and the measurement operators.

### 13.3.3 Sequential Filtering Algorithms

Consider now the problem of sequential estimation, where one assumes that initial conditions x(t.) at t. propagate forward to time t.+1 according to x(t.+j) = M(t.+p t) [x(t.)]+],, where ]i is model noise with zero mean and known covariance. One also has a vector of observations y i collected in the interval [t., t.+J. Assume one intends to cycle the assimilation beginning with a previous analysis at t,, leading to a background forecast at t.+p and ending with an analysis t.+p as depicted in Fig. 13.1. The key idea with sequential filters is that the analysis covariance from step becomes the forecast covariance at step i + 1; hence, the covariance is evolved together with the state itself.

The notation xf denotes the background forecast and x" the analysis, both at t .. For consistency with notation in the literature, pf is used for the forecast covariance, and Pa is used for the analysis covariance at time t ..

Because ] is unknown, the forecast is computed from the previous analysis with

Assuming (ninf) = Qi is the model noise covariance, the forecast error covariance evolves according to

With these two pieces of information, one can find the analysis at t.+1 using the previously-derived results from Bayes' Theorem, xf+i = ti )xai.

where the Kalman gain matrix K. is

The analysis error covariance,

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