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In probability theory, the cost function is defined as Tarantola (1984)
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(91) |

where *C*_{D} and *C*_{M} are the covariance matrices for the data sets and models respectively. It is relatively easy to analyze the role of the covariance matrix *C*_{M}. The second term of equation (91) can be expressed in the following matrix form:
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(92) |

The vector represents a stochastic process. The covariance matrix *C*_{M} stands for the linear relation of the stochastic process at two different times. If the value of the covariance is very small, the stochastic process at the two different times changes a little. Therefore, the uncertainty of the stochastic process is also low. Small values are chosen for the elements in the matrix *C*_{M} if we have enough a priori information of the model.
The similarity between two images produced by two successive iterative migration/inversion steps are used to generate the covariance matrix. The correlation coefficients of the two images can be used to fill the covariance matrix *C*_{M}. Of course, the reciprocal of the correlation coefficients of the two images can be used to fill the covariance matrix *C*_{M}^{-1}.
The local singular values (from SVD) of an image reflect the local continuity of an event, which can be used for constraints.

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Stanford Exploration Project

11/1/2005