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By William Menke

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Key Features
* up-to-date and carefully revised edition
* extra fabric on geophysical/acoustic tomography
* precise dialogue of software of inverse idea to tectonic, gravitational and geomagnetic reviews

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This matrix describes how well the predictions match the data. If Ν — I, then d = d and the prediction error is zero. On the other hand, if the data resolution matrix is not an identity matrix, the prediction error is nonzero. If the elements of the data vector d possess a natural ordering, then the data resolution matrix has a simple interpretation. Consider, for example, the problem of fitting a straight line to (z, d) points, where the data have been ordered according to the value of the auxiliary variable z- If Ν is not an identity matrix but is close to an identity matrix (in the sense that its largest elements are near its main diago­ nal), then the configuration of the matrix signifies that averages of neighboring data can be predicted, whereas individual data cannot.

This problem is often solved by the so called method of least squares. In this method one tries to pick the model parameters (intercept and slope) so that the predicted data are as close as possible to the observed data. For each observation one defines a prediction error, or misfit, e = df* — d? The best fit-line is then the one with 51 pre 51 K t 35 3 Linear, Gaussian Inverse Problem, Viewpoint 1 36 (a) (b) d ζ ζ. Fig. 1. 1) The total error Ε (the sum of the squares of the individual errors) is exactly the squared Euclidean length of the vector e, or Ε = e e.

10a). Conversely, if E(m) has a broad minimum, we expect that m has a large variance (Fig. 10b). 12 Variance and Prediction Error of the Least Squares Solution (a) (b) model parameter m model parameter m Fig. 10. (a) The best estimate m of model parameter m occurs at the minimum of E(m). If the minimum is relatively narrow, then random fluctuations in E(m) lead to only small errors Am in w . (b) If the minimum is wide, then large errors in m can occur. est est its minimum. The curvature of the prediction error can be measured by its second derivative, as we can see by computing how small changes in the model parameters change the prediction error.

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