[7] The regularization of the migration/inversion algorithms

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

$$2S\left( \textbf{m}\right)=\left(g\left(\textbf{m} \right)-\... ... \right)C_{M} ^{-1} \left(\textbf{m}-\textbf{m}_{prior} \right)$$ (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:  

$$\left( \begin{array} {ccccc} m_{1}&m_{2}&m_{3}&\cdots &m_{N}... ... m_{1} \\ m_{2} \\ m_{3} \\ \vdots \\ m_{N}\end{array} \right).$$ (92)

The vector $\textbf{m}-\textbf{m}_{prior}$ 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.