MODELING IN A V(Z) MEDIUM

In a general medium, the scattered wavefield received at position g, on the surface from a shot at position s, is given by,

$$P(g,s,\omega) = \int \int G(x,z;s,\omega) G(x,z;g,\omega) S(x,z)\ dx\ dz$$

where S(x,z) is the scattering function in the subsurface. This formula suggests that we need to calculate a Green's function for every surface position. However if the velocity field is only a function of depth and not of x, we only need one Green's function. The integral can then be expressed as a convolution of the Green's function over x and an integral over depth.

$$P(g,s,\omega) = \int \int G(x-s,z;\omega) G(x-g,z;\omega) S(x,z)\ dx\ dz$$

For a constant offset panel in CMP coordinates (y,h) this can be further simplified to a convolution over a constant offset Green's function, and an integral over depth.

$$P(y,\omega;h) = \int \int G( x-y,z; \omega, h ) S(x,z) \ dx \ dz$$

Where $G(x-y,z; \omega, h ) = G( x-y+h/2, z; \omega) G(x-y-h/2, z; \omega )$.The constant offset Green's function is merely the one-way Green's function multiplied by a shifted copy of itself.

Figure  is the result of constant offset modeling for three different offsets. Again only three frequencies were calculated to generate the Green's function. The first panel is the scattering function, two spikes in the center of the panel. The second panel is the zero offset modeling result, the third panel is for an offset of 300m and the fourth panel is at an offset of 750m. In the zero offset panel the upward curvature of the limbs of the hyperbola and the phase change due to overturned waves can clearly be seen. In the far offset panel an interesting effect is visible, the top of the table-top is concave. The traveltime to a point under the midpoint is greater than the traveltime to a point to one side.

 

const-mod
Figure 5 Constant offset modeling in a V(z) medium