THE BULLETPROOFING OF MUIR AND GODFREY

Stable extrapolation can be assured by preserving certain symmetries. It will be shown that stability is assured in both the differential equation  

$${d q\over dz } \eq - \ R\ q$$ (80)

and its Crank-Nicolson approximation  

$${q_{n+1} \ -\ q_n \over \Delta z } \eq -\ {R\over 2 }\ ( q_{n+1} \ +\ q_n )$$ (81)

provided that $R\ +\ R^ { {\rm *} \, }$ is a positive definite (actually, semidefinite) matrix. When stability was studied in the previous section the operator R was a scalar Z-transform. Because Z-transforms were used, the mathematics of that section was particularly suitable for time domain migrations. Because R was a scalar, the mathematics of that section was particularly suitable when data has been Fourier transformed over x. Here we will focus on the matrix character of R. Thus we are concerning ourselves with migration in the x-domain. Our purpose in doing this theoretical work is to gain the ability to write a ``bulletproof'' program for migrating seismic data in the presence of lateral velocity variation. As an example, the familiar 45$^\circ$ extrapolation equation will be put in the bulletproof form. This section, combined with the previous one, gives a general theory for stable migration in (t,x)-space.