Backprojection versus backpropagation in multidimensional linearized inversion: SHORT NOTE

Seismic migration can be viewed as either backprojection (diffraction-stack) or backpropagation (wave-field extrapolation) (e.g., Gazdag and Sguazzero, 1984). Migration by backprojection was the view supporting the first digital methods-the diffraction and common tangent stacks of what is now called classical or statistical migration (Lindsey and Hermann. 1970; Rockwell, 1971; Schneider, 1971; Johnson and French, 1982). In this approach, each data point is associated with an isochron surface passing through the scattering object. Data values are then interpreted as projections of reflectivity over the associated isochrons. Dually, each image point is associated with a reflection-time surface passing through the data traces. The migrated image at that point is obtained as a weighted stack of data lying on the reflection-time surface (Rockwell, 1971; Schneider, 1971). This amounts to a weighted backprojection in which each data point contributes to image points lying on its associated isochron. With the introduction of wave-equation methods by Claerbout (1971), this backprojection view was largely replaced by a backpropagation approach in which the recorded waves are extrapolated either downward in space or backward in time and the image extracted from the extrapolated wave field by an imaging condition (e.g., Berkhout, 1984; Stolt and Weglein, 1985). In the case of a single-source, multiplereceiver experiment. the simplest imaging condition consists of reading the value of the extrapolated field at each image point at the time of illumination from the source. This reverse propagation constitutes the solution of a boundaryvalue problem and can be accomplished either by means of a finite-difference simulation constrained by data values along the receiver array (Whitmore, 1983; Baysal et al., 1983; Chang and McMechan, 1986; Whitmore and Lines, 1986) or by means of a Kirchhoff integral (Schneider, 1978; Wiggins, 1984). The Kirchhoff formulation provides a key to the reconciliation of backprojection migration with the wave equation. It is formulated in terms of the backpropagation approach, but implemented as a backprojection. In this way one obtains the weights and prefiltering operations that are required to make the two algorithms equivalent. Recently, these classical migration methods have been reformulated in terms of a theory of multidimensional linearized inversion (Born inversion). Within this theory, the seismic imaging problem is recast from that of extrapolating a scattered wave field to one of recovering the perturbations of material parameters (the scattering potential) that gave rise to the scattered field. As with the earlier methods, multidimensional Born inversion can be formulated either in terms of backprojection (Miller et al., 1984: Beylkin, 1985; Miller et al.. 1987) or (for certain experimental geometries) in terms of backpropagation (Cheng and Coen, 1984: Esmersoy, 1986: Esmersoy and Oristaglio, 1988). For the case of zero-offset geometry, the two inversion formulas agree in the far field and differ from classical wave-equation migration only by a one-dimensional filter which is applied to the data traces before backprojection or backpropagation (Jakubowicz and Miller, 1989). For a single-source, multiple-receiver geometry, backprojection inversion differs from standard backprojection migration by adding to the integrand an extra “obliquity factor” that depends on the angle between the source and receiver rays at each image point. The backpropagation inversion method (Esmersoy, 1986; Esmersoy and Oristaglio, 1988) differs from standard backpropagation migration by adding an extra spatial diff‘erential operator which is applied to the extrapolated wave field before imaging. In this note, we discuss the relationship between these two formulations of single-source migration. We show that the two linearized methods satisfy the same formal equivalence as the earlier methods. In particular, the extra stacking weights applied in backprojection inversion are identical to