By Carl Wunsch

Advent to inverse and kingdom estimation difficulties for graduate scholars and researchers in oceanography, weather technological know-how, and geophysical fluid dynamics.

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**Extra resources for Discrete Inverse and State Estimation Problems: With Geophysical Fluid Applications**

**Sample text**

Usually, when the dimension of I L is clear from the context, the subscript is omitted. IA = A, AI = I, for any A for which the products make sense. If there is a matrix B, such Basic machinery 24 that BE = I, then B is the “left inverse” of E. If B is the left inverse of E and E is square, a standard result is that it must also be a right inverse: EB = I, B is then called “the inverse of E” and is usually written E−1 . ” Analytical expressions exist for a few inverses; more generally, linear algebra books explain how to find them numerically when they exist.

1 Basic formulation Consider the elementary problem motivated by the “data” shown in Fig. 2. t is supposed to be an independent variable, which could be time, a spatial coordinate, or just an index. Some physical variable, call it θ(t), perhaps temperature at a point in a laboratory tank, has been measured at coordinates t = ti , i = 1, 2, . . , M, as depicted in the figure. 86) where n(t) is the inevitable measurement noise. The straight-line relationship might as well be referred to as a “model,” as it represents the present conception of the data structure.

Such a claim is almost always vacuous. , forming an average) that seems to be so unsophisticated that no model is present. Consider, however, a simple problem faced by someone trying to determine the average temperature in a room. A thermometer is successively placed at different three-dimensional locations, ri , at times ti . Let the measurements be yi and the value of interest be m˜ = 1 M M yi . 35) i=1 In deciding to compute, and use, m˜ the observer has probably made a long list of very sophisticated, but implicit, model assumptions.