ILL-POSEDNESS OF ABSORBING BOUNDARY-CONDITIONS FOR MIGRATION

Absorbing boundary conditions for wave-equation migration were introduced by Clayton and Engquist. We show that one of these boundary conditions, the B2 (second-order) condition applied with the 45 degree (third-order) migration equation, is ill-posed. In fact, this boundary condition is subject to two distinct mechanisms of ill-posedness: a Kreiss mode with finite speed at one boundary and another mode of a new kind involving wave propagation at unbounded speed back and forth between two boundaries. Unlike B2, the third-order Clayton-Engquist boundary conditions B3 is well-posed. However, it is shown that it is impossible for any boundary condition of Clayton-Engquist type of order higher than one to be well-posed with a migration equation whose order is higher than three.