Using multiple realization method to get multiple equal-probable velocity models

Regularized geophysical inversion problems include two fitting goals: data fitting and model styling. They can be written as:

$$\begin{eqnarray} 0 & \approx {\bf r_d} = & {\bf d}-{\bf Lm} \\ 0 & \approx {\bf r_m} = & \epsilon {\bf Am}\end{eqnarray}$$ (1) (2)

An ideal regularization operator $\bf A$ should be the inverse model covariance. In practice, according to the difficulty to get the explicit model covariance, $\bf A$ is usually approximated as Lapacian, PEF or steering filter.

Generally, the regularization operator only describes the two point statistics. The first order statistics, spatial variance, is not included in it. Like in geostatistics, we can add normal noise vector into model styling goal so that we can get the comparable variance in poorly determined regions as in well determined regions Claerbout (1999); Clapp (2000).

The fitting goals including both first and second model statistics can be written as:

$$\begin{eqnarray} 0 \approx {\bf r_d}= {\bf d}-{\bf Lm}\\ {\sigma}_m \mbox{\boldmath$\eta$} \approx {\bf r_m}= \epsilon {\bf Am}\end{eqnarray}$$ (3) (4)

Scalar ${\sigma}_m$ can be approximated as the variance of the model residual acquired by applying regularization operator $\bf A$ to first estimated model Claerbout (1999). By changing normal noise , we can get equal-probable models from which we can evaluate the variability of the model.

When we perform velocity analysis, the data we used are the value picked from semblance. So there also exist data uncertainty. Similar to the modification of model styling goal, we can add normal noise into the data fitting goal in terms of noise covariance to include this effect on our model evaluation. The noise inverse covariance can be approximated as the chain of a diagonal operator and a PEF on ${\bf r}_d$. A detailed discussion on how to include data and model uncertainty to evaluate velocity was given by Clapp 2002.