The numerical solution of the Helmholtz equation is examined for a separated source and receiver over a model having a single planar interface. Expressions describing the construction and reconstruction of acoustical wave fields are derived in terms of Green’s functions. Their relation to the Fourier transform is briefly discussed. Three simple Green’s functions—free space, free surface, and rigid surface—are used to test the relative accuracy of the respective weighting factors by comparing the numerically calculated field for a simple model to a field obtained analytically by application of rigorous diffraction theory. The main purpose of this paper is to study the behavior of the total response (amplitude and phase) for models in which the aperture is not sufficiently sampled (e.g., close to half the wavelength). The degree of distortion in the response due to spatial undersampling is unacceptable for all three Green’s functions. A modified weighting factor relative to the free‐space Green’s function is introduced, which effectively reduces the degree of distortion in the total response under the same sampling condition. The importance of this finding to exploration geophysics in the construction of the synthetic seismograms by application of the Huygen’s principle and in seismic migration will be demonstrated.

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The first part of this paper examines a special case of acoustical imaging in which the source and the receiver coincide. The benefits of weighting and muting are studied in detail by means of computer modeling. The test model consists of a single planar interface z=z1, abruptly terminated at x= o. The amplitude and phase responses are computed in the plane z=z0= o for two separations of neighboring stations, Δx=λ/10 and Δx=λ/2. Six different weighting factors are used in the test. However, in this source‐receiver coincident case, three of the weighting factors produce identical responses, so that all six test factors may be represented by only four curves. It is found that when the spatial sampling at the aperture approaches the condition of critical sampling, i.e. Δx=λ/2, only the weighting factor which implicitly takes into account beam steering along the specular reflection path is acceptable. This factor alone keeps the amplitude and the phase curves undistorted until the difference 2 ·ΔR between two neighboring paths reaches approximately λ/2. If we set 2 ·ΔR=λ/2, we may construct a set of curves which we may call quite appropriately muting curves. These curves are physically interpretable only for station separation Δx > λ/4. The muting curves are symmetrical about the line x= 0 and their angular opening depends on spatial separation Δx, depth z, and wavelength λ (which may vary with depth). The second part of this paper suggests how the weighting factor with implicit beam steering can be applied to reconstruction of two and three‐dimensional wavefields. Seismic migration of common depth point (CDP) stacked line data is also discussed. This is a hybrid case which presents certain theoretical difficulties. We shall also mention the velocity problem which is inherent to migration of CDP stacked data. The third and final part concerns implementation of the migration of CDP stacked data. When the spatial sampling is between λ/4 and λ/2, the migration process will benefit from beam steering and from muting. The benefits are more subtle when the separation of the traces is less than λ/4. However, in that case the cost of data collection is considerable and often prohibitive. In either case the migration of seismic data can be expedited by use of precalculated tables of migration velocities, ray path distances, and weights (including muting).

Study of simple synthetic seismograms convinced both authors that the diffracted pulse shape shows virtually no variation with source‐receiver offset if the midpoint is held constant. First, I wish to corroborate their findings by means of a simple example and at the same time suggest that conclusions of both authors apply strictly only to planar (and planar dipping) reflectors. Secondly, I wish to show that because of the invariant behavior of the pure diffraction pulse and because of the equally invariant behavior of the pure reflection pulse, the shape of the reflected pulse for depth points close to the edge of the reflector can not be offset invariant.

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