REFERENCES

Castagna, J. P., Swan, H. W., and Foster, D. J., 1998, Framework for AVO gradient and intercept interpretation: Geophysics, 63, no. 3, 948-956.

Claerbout, J.

Geophysical Estimation by Example: Environmental soundings image enhancement:.
http://sepwww.stanford.edu/sep/prof/, 1998.

Clapp, R., 2000, Multiple realizations using standard inversion techniques: SEP-105, 67-78.

Clapp, R. G., 2001a, Geologically constrained migration velocity analysis: Ph.D. thesis, Stanford University.

Clapp, R. G., 2001b, Multiple realizations: Model variance and data uncertainty: SEP-108, 147-158.

Gratwick, D., 2001, Amplitude analysis in the angle domain: SEP-108, 45-62.

Guitton, A., 2000, Coherent noise attenuation using Inverse Problems and Prediction Error Filters: SEP-105, 27-48.

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A

We can follow a parallel definition for the data fitting goal in terms of the inverse noise covariance $\bf N$:

$$\sigma_{d} \bf \eta \approx \bf N( \bf d- \bf L\bf m) .$$ (5)

Noise covariance for velocity estimation Using the multiple realization methodology for velocity estimation problem posed in the manner results in several difficulties. First, what I would ideally like is a model of the noise. This poses the problem of how to get the noise inverse covariance. The first obstacle is that our data is generally a uniform function of angle $\theta$ and a non-uniform function of $\bf x$.What we would really like is a uniform function of just space. We can get this by first removing the angle portion of our data.

I obtain $\bf \Delta t$ by finding the moveout parameter $\gamma$that best describes the moveout in migrated angle gathers. I calculate $\bf \Delta t$ by mapping my selected $\gamma$ parameter back into residual moveout and the multiplying by the local velocity. Conversely I can write my fitting goals in terms of $\bf \gamma_i$ by introducing an operator $\bf S$ that maps $\bf \Delta t$ to $\bf \gamma$,

$$\begin{eqnarray} \bf \gamma_i &\approx&\bf S \bf T_{} \bf \Delta s\\ \nonumber \bf A\bf s_{0} &\approx&\epsilon \bf A\bf \Delta s.\end{eqnarray}$$ (6)

Making the data a uniform function of space is even easier. I can easily write an operator that maps my irregular $\bf \gamma_i$ to a regular function of $\bf \gamma_r$ by a simple inverse interpolation operator $\bf M$.I then obtain a new set of fitting goals,

$$\begin{eqnarray} \bf \gamma_r &\approx&\bf M \bf S \bf T_{} \bf \Delta s\\ \nonumber \bf A\bf s_{0} &\approx&\epsilon \bf A\bf \Delta s.\end{eqnarray}$$ (7)

On this regular field the noise inverse covariance $\bf N$ is easier to get a handle on. We can approximate the noise inverse covariance as a chain of two operators. The first, $\bf N_{1}$, f a fairly traditional diagonal operator that amounts for uncertainty in our measurements. For the tomography problem this translate into the width of our semblance blob. For the second operator we can estimate a Prediction Error Filter (PEF) on $\bf r_{d}$ Guitton (2000) after solving

$$\begin{eqnarray} \bf \bf 0&\approx&\bf r_{d}= \bf N_{1} ( \gamma_r \bf M \bf S \... ...er \bf A\bf s_{0} &\approx&= \bf r_{m} \epsilon \bf A\bf \Delta s.\end{eqnarray}$$ (8)

If we combine all these points and add in the data variance we get,

$$\begin{eqnarray} \sigma_{d} \bf \eta &\approx&\bf N_1 \bf N_2( \bf \gamma_r - \b... ...a_{m} \bf \eta \bf A\bf s_{0} &\approx&\epsilon \bf A\bf \Delta s.\end{eqnarray}$$ (9)