Fermat's Interferometric Principle of Convolutional Type Lab

Figure 1. Plot of actual and Fermat principle traveltimes for a surface shot gather in a 2-layer medium with a horizontal interface. The virtual SSP traveltimes were obtained by the VSP to SSP correlation transform.

Key Idea: Fermat's interferometric principle of convolutional type says VSP traveltimes can be transformed into SSP traveltimes by the convolutional form of Fermat's principle. For the VSP model in Figure 2, the sum of the direct VSP traveltime T_Bx (middle diagram) and the VSP primary traveltime T_{xx_o}+T_{x_oA} (left diagram) yields the diffraction SSP traveltime (rightmost diagram)

T_Bx + T_{xx_o}+T_{x_oA}

Minimizing the above over values of x along the well yields the SSP traveltime (see Chapter 6).

T_Bx* + T_{x*x_o}+T_{x_oA}

where x* is a stationary point in the well.

Objective: Execute program fermat1.m to reproduce above figure. The theoretical traveltimes are produced by straight ray tracing and the interferometric traveltimes are produced by Fermat's principle from VSP traveltimes.

Procedure:

Download the following programs: fermat1.m.
Type fermat1.m to generate the above picture for a two layer model with a horizontal interface. The animation displays the common shot gather traveltimes for different shot positions on the surface.