Finite-difference migration

 

This chapter is a condensation of wave extrapolation and finite-difference basics from IEI which is now out of print. On the good side, this new organization is more compact and several errors have been corrected. On the bad side, to follow up on many many interesting details you will need to find a copy of IEI (http://sepwww.stanford.edu/sep/prof/).

In chapter we learned how to extrapolate wavefields down into the earth. The process proceeded simply, since it is just a multiplication in the frequency domain by $\exp [ ik_z ( \omega , k_x ) z ]$.In this chapter instead of multiplying a wavefield by a function of k_x to downward continue waves, we will convolve them along the x-axis with a small filter that represents a differential equation. On space axes, a concern is the seismic velocity v. With

lateral velocity variation, say v(x), then the operation of extrapolating wavefields upward and downward can no longer be expressed as a product in the k_x-domain. (Wave-extrapolation procedures in the spatial frequency domain are no longer multiplication, but convolution.) The alternative we choose here is to go to finite differences which are convolution in the physical x domain. This is what the wave equation itself does.