Migration velocity analysis based on linearization of the two-way wave equation

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Theory

First, we start with the imaging condition as the following:

 (1)

where is the image, is the Green's function, is the surface data, and are the source and receiver coordinates, is the subsurface offset, is the Green's functions' coordinate and is frequency. Next, we define the Green's functions based on the two-way wave equation as follows:

 (2)

 (3)

where is slowness. Then, we can obtain the derivative of with respect to the slowness as follows;

 (4)

where is the slowness coordinate. Now, we can perturb the slowness:

 (5)

where is the background slowness. Then, we apply a first order approximation by squaring the slowness and ignoring the second order perturbation term as follows:

 (6)

We define a background Green's function that corresponds to the background slowness:

 (7)

By substituting this into the original wave equation, we arrive at the following:

 (8)

Now, we apply Born's approximation to simplify the previous equation to the following expression:

 (9)

where is the perturbed Green's function. Then, we solve for the perturbed Green's function as follows:

 (10)

which enables us to find the derivative of the Green's function with respect to slowness as shown in the following:

 (11)

We can follow the same steps for the receiver Green's function to get:

 (12)

Then, we substitute equations (11) and (12) in the image derivative to get the result:

 (13)

Finally, we can express the image perturbation as the following:

 (14)

Similarly, we can now compute the gradient, as given by:

 (15)

 Migration velocity analysis based on linearization of the two-way wave equation

Next: Interpretation Up: Almomin and Tang: WEMVA Previous: Introduction

2010-11-26