End-point contributions to synthetic seismograms

Abstract

Summary. Synthetic seismograms represented by integrals generally display signals associated with the limits of integration. Sometimes these ‘end-point’ contributions are spurious (e.g. in the WKBJ seismogram) and sometimes they are the main physical interest (e.g. the Kirchhoff integral for an edge). The end-point contributions may be asymptotically approximated using integration by parts or Laplace's method and it may then be possible to reduce them if desired. We describe examples in the WKBJ seism ogram for reflected or transmitted waves in homogeneous layers and for turning waves. We also study signals due to discontinuities in reflection coefficients, by partitioning the real slowness integral so that the discontinuities lie at end points. Examples are the head wave, which is a physically correct signal, and spurious diffractions caused by using plane-wave coefficients for grazing rays in the WKBJ seismogram.