Derivation of forward and adjoint operators for least-squares shot-profile split-step migration

Abstract

The forward and adjoint operators for shot-profile least-squares migration are derived. The forward operator is demigration, and the adjoint operator is migration. The demigration operator is derived from the Born approximation. The process begins with a Green’s function that allows for a laterally varying migration velocity model using the split-step approximation. Next, the earth is divided into horizontal layers, and within each layer the migration velocity model is made to be constant with respect to depth. For a given layer, (1) the source-side wavefield is propagated down to its top using the background wavefield. This gives a background wavefield incident at the layer’s upper boundary. (2) The layer’s contribution to the scattered wavefield is computed using the Born approximation to the scattered wavefield and the background wavefield. (3) Next, its scattered wavefield is propagated back up to the measurement surface using, again, the background wavefield. The measured wavefield is approximated by the sum of scattered wavefields from each layer. In the derivation of the measured wavefield, the shot-profile migration geometry is used. For each shot, the resulting wavefield modeling operator takes the form of a Fredholm integral equation of the first kind, and this is used to write down its adjoint, the shot-profile migration operator. This forward/adjoint pair is used for shot-profile least-squares migration. Shot-profile least-squares migration is illustrated with two synthetic examples. The first uses data collected over a four-layer acoustic model, and the second uses data from the Sigsbee 2a model.