Multiple realization methodology

Clapp (2004) suggested breaking up the tomography problem into two portions: creating several realizations of ${\bf \gamma}$ maps and using them as input to the tomography problem. Estimating the ${\bf \gamma}$ field is in itself difficult. The standard approach is to calculate semblance over a range of move-out values. The move-out at given point is then the maximum semblance at the location. To reduce noise, the semblance field is often smoothed. This is still far from in ideal solution. We are constantly fighting a battle between selecting local minima (not enough smoothing) and missing important move-out features (too much smoothing).

In Clapp (2004) the various ${\bf \gamma}$ maps were created by selecting a smooth set of random number and converting them into $\gamma$ values based on a normal score transform Isaaks and Srivastava (1989a). This approach was somewhat successful, but suffered from the fact that we don't, and effectively can't scan over an infinite set of move-outs. Therefore our distribution function is misleading. Methods to correct for the limited range proved adhoc.

Instead I am going to start from the approach outlined in Clapp (2003b). My goal is to estimate a smooth set of semblance values $\bf g_{{\rm smooth}}$. I begin by selecting the maximum semblance at each point $\bf g_{{\rm max}}$.I solve the simple minimization problem

$$\begin{eqnarray} \bf 0&\approx&\bf r_{data} = \bf W_g( \bf g_{{\rm max}}- \bf g_... ...pprox&\bf r_{model} = \epsilon \bf A\bf g_{{\rm smooth}}\nonumber,\end{eqnarray}$$ (6)

where $\bf A$ is again a steering filter, and $\bf W_g$is a function of the semblance value at each location. After estimating $\bf g_{{\rm smooth}}$ I select the maximum within a range around $\bf g_{{\rm smooth}}$ to form a new $\bf g_{{\rm max}}$,and repeat the estimation. At each iteration, the window I search around and the amount of smoothing ($\epsilon$)decreases. To create a series of models I introduce random noise into $\bf r_{data}$ scaled by the variance in the semblance at each location. With different sets of random noise I get different realistic models.