Let the layered earth response from shot s to geophone g be u(s,g,t)=u(0,g-s,t)=u(g-s,t) or in Fourier space, or simply u(g-s). When an upcoming wave hits the earth surface at g1 it encounters a scattering object which reflects the primary wave with a random scaling .The signal at g1 then takes off for a second flight like a multiple reflection, but departing in all directions. We are going to build the theoretical 2-D spectrum of this surface scattered wave w from the theoretical 2-D spectrum of u, the layered media primary reflection.
First we express the cascade of the two bounces. The arrival w at g2 at time t is the sum of the time of each bounce, and .Since this is a convolution in the time domain, we express it as a product in the frequency domain. Then we form the complex conjugate of this expression in preparation for autocorrelation on the x-axis.
We Fourier transform over x. The first factor above is not a function of space. It is merely a function of , say a filter .Thus our main result:
We see that in principle, for each shot point s, we measure the spectrum of the impulse response of the layered medium.