Explicit wavefield extrapolators are based on direct analytic mathematical formulae that express the output as an extrapolation operator acting on the input. Implicit techniques require an equation to be solved, or a matrix decomposed or inverted, to accomplish the extrapolation. Typically, explicit methods are faster and often give more insight into the physics of the wave propagation; but can suffer from instability. Implicit operators are often unconditionally stable.

Four different explicit extrapolators based on Fourier theory are presented and analyzed. They are: PS (ordinary phase-shift), PSPI (phase shift plus interpolation), NSPS (nonstationary phase shift), and, new in this paper, SNPS (symmetric nonstationary phase shift). The PS extrapolator is well known to be exact for constant velocity, unconditionally stable, but inapplicable for variable velocity. PSPI was originally formulated as an interpolation between sets of PS wavefields but it can be formulated as a nonstationary combination filter (or equivalently a pseudodifferential operator). NSPS is similar to PSPI but is a nonstationary convolution filter that gives it different properties. PSPI and NSPS both adapt very rapidly to local lateral velocity changes. When the lateral velocity variation is piecewise constant, NSPS and PSPI reduce to simple operations involving spatial windowing, extrapolation with the PS operator, and superposition. For NSPS, windowing precedes extrapolation while for PSPI it is the reverse.

NSPS and PSPI both lead to analytic expressions for wavefields in heterogeneous media that approximately solve the variable velocity wave equation. Their error terms are fundamentally different in character but vanish for constant velocity.

A formal proof is given that NSPS in a direction orthogonal to the velocity gradient is the mathematical adjoint process to PSPI in the opposite direction. This motivates the construction of SNPS that combines NSPS and PSPI in a symmetric fashion. This symmetry (under interchange of input and output lateral coordinates) is required by reciprocity arguments. PS and SNPS are symmetric while NSPS and PSPI are not.

An extensive numerical stability study using SVD (singular value decomposition) shows that all of these extrapolators can become unstable for strong lateral velocity gradients. Unstable operators allow amplitudes to grow unphysically in a recursion. Stability is enhanced by introducing a small (~3%) imaginary component to the velocities. This causes a numerical attenuation that tends to stabilize the operators but does not address the cause of the instability. For the velocity model studied (a very challenging case) PSPI and NSPS have exactly the same instability while SNPS is always more stable. Instability manifests in a complicated way as a function of extrapolation step size, frequency, velocity gradient, and strength of numerical attenuation. The SNPS operator can be stabilized over a wide range of conditions with considerably less attenuation than is required for NSPS or PSPI.

View full article as PDF (0.43 Mb)