taken during the timespan of interest, as described in Section 13.2. To determine the state vector, x, we assume that y equals the observation model vector, g(x,t), based on the mathematical model of the observations plus additive random noise, v. Thus, for each element of y,

Loss Function. We will use Eq. (13-25) to estimate x°, given an a priori estimate x°, the observations y, the functional forms'of b(x0,t) and g(x,t), and the statistical properties of v. To accomplish this, we use the least-squares criterion as a measure of "goodness of fit"; the best value of x° minimizes the weighted sum of the squares of the residuals between the elements of the observation and observation model vectors. This is done quantitatively by minimizing the loss function, where the observation residual vector, p, is defined by p=y-g

W is an (nXn) symmetric, nonnegative definite matrix chosen to weight the relative contribution of each observation, according to its expected accuracy or importance. In the simplest case, W is the identity matrix indicating that equal weight is given to all observations. Throughout the rest of this section, we assume that W has the form

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