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In consideration of the authors’ reply, I must clarify the following: This discussion is solely concerned with pointing out that the eigenstate expansion method in the first 4 pages of their paper [the material encompassing equations (1) to (17)] is essentially identical to my GHM/LEP method; this discussion is not at all concerned with the Chebychev expansion method in the latter part of their paper [the material following equation (18)], nor is this discussion concerned with advocating one method over another.

A fundamental mathematical algorithm is presented for solving the wave equation in inhomogeneous media. This method completely generalizes the Haskell matrix method, which is the standard method for solving the wave equation in laterally homogeneous media. The Haskell matrix method has been the mathematical basis for many seismic techniques in exploration geophysics. In the method presented the medium is divided into layers and vertically averaged within each layer. The wave equation, within a layer, is then decoupled into an eigenvalue equation of the horizontal coordinate and a wave equation of the vertical coordinate. The eigen‐value equation is solved numerically. The vertical equation is solved analytically, once the eigenvalues are found. The solution throughout the medium is constructed by matching layer solutions at layer interfaces. The solution process of this method is “modular,” in the sense that each layer corresponds to an independent module and all the modules together form the final, total solution. Such a modular solution process has the following advantages. First, in a 2-D problem, for example, each module is a 1-D problem, which is a much simpler problem numerically than the original full 2-D problem. Second, the module solutions can be used repeatedly to form the solution corresponding to different problems. For example, in modeling. only those layers which differ between two models require recalculation. The solution to plane‐wave diffraction by a cylinder is obtained using this method, and it agrees well with the analytical solution.

Migration requires one‐way wave continuation. In the spatial domain, one‐way wave equations are derived based on various approximations to an assumed dispersion relation. In the frequency‐wavenumber domain, the well known f-k method and the phase‐shift method are strictly valid only within homogeneous models and layered models, respectively. In this paper, a frequency‐wavenumber domain method is presented for one‐way wave continuation in arbitrarily varying media. In the method, the downward continuation is accomplished, not with plane waves individually as in the f-k or the phase‐shift method, but with the whole spectrum of plane waves simultaneously in order to account for the coupling among the plane waves due to lateral inhomogeneity. The method is based on a matrix integral equation. The method has the following merits: (1) The method is a generalization of the f-k and the phase‐shift methods, valid in arbitrarily varying models. (2) The method has physical interpretations in terms of upgoing and downgoing plane waves, and as such the method has well defined steps leading from full‐wave continuation (two‐way wave) to one‐way wave continuation for migration. (3) The method is rigorous; the only approximations in the method—other than the one‐way wave approximation necessary for migration—are the discretization of a continuous system (which is necessary in computer methods) and imperfections associated with the limited spatial aperture of the data; as such, the method can achieve high solution accuracy. (4) The method can be fast, since computations are mainly matrix‐vector multiplications, which are easily vectorizable. In particular, compared to spatial domain methods, I contend that the method is (1) more rigorous in one‐way wave theory, (2) more accurate in migration of high‐dip events, and (3) faster for smooth models. I applied the method to a few examples of zero‐offset data migration, including imaging a point diffractor below a dipping interface, migration with sharp lateral variations in velocity, and migration with smooth lateral variations in velocity.

In induction logging, both vertical and radial conductivity variations exist in the logged medium; elementary separation of variables cannot be applied directly to the partial differential equation describing the electromagnetic field in such a medium. A modal expansion method of solution can be designed using modes defined with respect to the vertical coordinate. This method breaks the original two‐dimensional (2-D) problem into a set of independent one‐dimensional (1-D) problems. The receiver signal is decomposed into a direct signal traveling within the borehole and a reflection signal bearing information about the conductivity distribution outside the borehole. The solution is obtained as a coupled‐mode, multiple‐reflection series accounting for repeated mode reflections at the vertical boundaries in the medium. Each term in the series represents the next higher order reflection in a sequence of multiple reflections; results show two terms are sufficient to reach accuracy acceptable for log modeling.