Acoustic wavefield evolution as function of source location perturbation

Introduction

The wave equation is the central ingredient in simulating and constraining wave propagation in a given medium. No other formulation, including the eikonal equation or ray tracing, can be as conclusive and elaborate (includes most traveltime and amplitude aspects) as does the full elastic wave equation [Aki and Richards (1980)]. Thus, the wave equation has the near complete far-field description of the wave behavior for a given velocity function. To reduce the computational cost, $P$-wave propagation in the Earth's subsurface is approximately simulated by numerically solving the acoustic wave equation. The wavefield in acoustic media is described by a scalar quantity. Kinematically, for $P$-waves in the far field of the source, the acoustic and elastic wave equations are similar; they both yield the same eikonal equation in isotropic media.

2The first-order dependence of wavefields, and specifically the acoustic wave equation, on media parameters is described by the Born approximation. It is a single scattering approximation, used in seismic applications to approximate the perturbed wavefield due to a small perturbation of the reference medium. It is also, in its inverse form, used to help infer medium parameters from observed wavefields (Cohen and Bleistein, 1977; Panning et al., 2009). In the spirit of the Born approximation, the partial differential equations, introduced here, describe the wavefield shape first and second order dependence on a source location perturbation. Data evolution as a function of changes in acquisition parameters goes back to the development of normal moveout and the transformation of common-offset seismic gathers from one constant offset to another (Bolondi et al., 1982). Bagaini and Spagnolini (1996) identified offset continuation (OC) with a whole family of prestack continuation operators, such as shot continuation (Bagaini and Spagnolini, 1993), dip moveout as a continuation to zero offset (Alkhalifah, 1996; Hale, 1991), and three-dimensional azimuth moveout (Biondi et al., 1998). Even residual 2operators between different medium parameters or acquisition configurations versions of these mapping equations are described by Alkhalifah and Biondi (2004) and Alkhalifah (2005). All these methods are based on a geometrical optics development using constant velocity approximations.

Alkhalifah and Fomel (2009) suggested a plane wave source perturbation expansion for the eikonal equation. Their approach rendered linearized forms of the eikonal equation capable of predicting the traveltime field for a shift in the source location represented by a shift in the velocity field in the opposite direction. The development here follows the same approach applied now to the wave equation. The major result in the eikonal application is the linearized forms of the new perturbation equations extracted from the conventionally nonlinear eikonal partial differential equation.

The source perturbation introduced here is based on a plane wave expansion around the source. For homogeneous or vertically inhomogeneous media, the wavefield calculated for a source in any location along the horizontal surface is valid in all other locations, and as a result no modifications are needed. For laterally inhomogeneous media, this statement is no longer true and the difference in the wavefield depends on the complexity of the lateral velocity variation. In this paper, I develop partial differential equations that approximately predict such changes, and thus, can be used to simulate wavefields for sources at other positions in the vicinity of the original source. Figure 1 illustrates the concept of wavefield evolution as a function of source perturbation in which the wavefield evolves as a function of source location perturbation $l$ granted that the velocity field changes laterally. In Figure 1, the wavefield wavefront schematic reaction depicts a general velocity decrease with $l$. 2In this paper, we consider source perturbation in the lateral direction, which adheres to our familiarity with surface seismic exploration applications. However, equivalent perturbations may be applied in the depth direction as well, as Alkhalifah and Fomel (2009) showed for the eikonal equation, which may have applications in datuming.

SourcePert2
Figure 1. Illustration of the evolution of the two-dimensional wavefield given by distance, $x$, and depth, $z$, as a function of the shift in the source location, $l$. The actual source shift is transformed to a new 3rd axis, where the $x$ axis now describes the offset from the source. The dot in the middle on the surface represents the source location.

Acoustic wavefield evolution as function of source location perturbation