DMO operator

In order to represent the same subsurface picture, we have to map the nonzero offset isochrone to a set of zero offset isochrones. The circles corresponding to the zero offset data set have to be tangential to the nonzero offset ellipse. Superposition of these circles by migrating the zero offset data set will result in the picture of their envelope, which is the prestack migration result of the nonzero offset impulse. The zero offset data set is called the DMO impulse response.

The circle described by (A-4) is tangent to the ellipse of (A-3) when both equations have the same slope at P(x,z). (The slopes can be expressed by implicit partial differential terms. The difference of these slopes is zero if:)  

$$b = x(1 - {B^{2}\over{a^{2}}}).$$ (14)

Substituting for A and B using (A-1) (A-2) yields an expression for x. Eliminating x and z from the circle (A-4), ellipse (A-3) results in:  

$$t_{0}^{2}=(t_{n}^{2}-{4 h^{2}\over{V^{2}}})(1-{b^{2}\over{h^{2}}}).$$ (15)

Because the isochrone ellipse is the impulse response of the nonzero offset data set, replacing this ellipse by impulse responses of a zero offset data set gives us the impulse response of the DMO operator. DMO can be achieved by replacing each nonzero offset trace sample by its DMO impulse response, which is defined by equation (A-6).