Integral formulation

The theory of the principal of diffraction summation can be derived using the inhomogeneous acoustic wave equation (Schneider, 1978; Claerbout, 1986). The simplified solution to the wave equation in an arbitrary volume is given by the Kirchhoff integral. A wavefield D can be expressed at any time and at any point as a surface integral over the measured data D(x₀,y₀,z₀,t). Here x₀,y₀,z₀ denote the coordinates of a coincident source-receiver pair, and S is the surface on which the data were recorded.

Evaluating the extrapolated recorded wavefield for all depths at time zero results in D(x,y,z,0), a three-dimensional image of the subsurface:

$$D(x,y,z,0) = -{1\over {2 \pi}} {\partial \over {\partial z}} \int_S \int {{D(x_0,y_0,z_0,t={R\over V})} \over {R}} dx_0 dy_0 ,$$ (1)

where $R = 2 {\lbrace (x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 \rbrace }^{1\over 2}$.This is the final formula used to migrate 3D zero-offset seismic data. Therefore, in order to image the subsurface, the first step is to carry out the integration over the diffraction surface. Let x,y,z be Cartesian coordinates with the origin at the surface of the earth and the z-axis pointing vertically downwards. A single scattering point (x₁,y₁,z₁) in the subsurface has an arrival time T for a coincident shot and receiver,

$$T~=~{\lbrace T_0^2 + {4\over {V^2}}~({(x-x_1)^2 + (y-y_1)^2})\rbrace} ^{1\over 2},$$ (2)

where V is the velocity of propagation and T₀=${2 z_1}\over V$. Equation (1) represents the mapping process of a point in the spatial domain (x,y,z) into a surface in the space-time domain (x,y,t). If V is a function of T₀ only, the diffraction surface is a surface of revolution. The time slice T=const cuts the diffraction surface in a circle with center at (x₁,y₁) and with radius $R~=~{V\over 2}{\lbrace T^2 - T_0^2 \rbrace}^{1\over 2} .$ In mapping from one domain to the other, scattering points are replaced by circles. The convolution which replaces points by circles can be expressed in the frequency domain by a multiplication with a cosine filter.