denote the model dynamics. To emphasize the dependence of the solution on the parameters , we explicitly denote the state system state as . Given the current operating point , the solution denotes the model forecast. Let be a mapping such that (2)

denotes the model predicted observations. It is tacitly assumed that the model and the map are exact, i. e., there are no errors in the model or in the relation that links observations to the model state. Assume observations are available at a specific time where these observations are denoted by . The difference between observations and model counterparts to the observations is given by . (3)

This constitutes a measure of the forecast error. Since and are assumed to be exact, a little reflection reveals that the forecast error is only due to errors or biases in the parameter vector . Given the forecast error in (3), our goal is to find a correction to the parameter such that closely resembles. We start by quantifying the first variation in resulting from the perturbation of by . To this end, first define the Jacobian, , of with respect to , given by (4)

The matrix is also known as the first-order sensitivity matrix. The variation in induces a variation in that is given by (6)

where the Jacobian is an matrix given by (7)

and . Combining (5) and (6), we obtain . (8)

Given the operating point and in (3), we search for such that . (9)

Combining (8) and (9), given , we seek such that . (10)

Then (10) becomes . (12) Since in general, this equation is solved as a linear least squares problem. From Lewis et al. (2006), we get the solution as if (13)

A number of observations are in order.

1. The matrix defined in (11) is a product of two Jacobian matrices and . Given , the matrix can be explicitly computed. However, to compute the second factor , we need to know the dependence of on . That is, we need to know the solution of (1). Since (1) is in general non-linear, except for special cases, we cannot obtain the solution . Hence clever numerical schemes must be designed to compute the evolution of with .

2. There is a vast literature dealing with the computation of first-order sensitivity matrices. The anthology edited by J. B. Cruz (1979) contains many of the classic results developed in the context of control theory. The review paper by Rabitz et al. (1983) contains a very readable account of the key results developed in the context of chemical kinetics. The recent monographs by Cacuci (2003) and Cacuci et al. (2005) contain an up to date coverage of topics related to sensitivity analysis.

3. Our goal is to demonstrate the utility of this framework. Accordingly, we have chosen the 3-variable nonlinear mixed layer model mentioned above. In a companion paper, we plan to illustrate the numerical process of computing using well-known methods discussed in the literature.

4. Since we have the luxury of an exact solution in the mixed-layer model case, we also illustrate the computation of second-order sensitivity.

We have generated systematic error in the mixed layer model that mimics the results found in operational practice. The methodology outlined above has succeeded in identifying the sources of error in a variety of cases.

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