S U M M A R YModern wide-angle surveys are often multi-fold and multi-channel, with densely sampled source and receiver spacings. Such closely spaced data are potentially amenable to multi-channel techniques involving wavefield propagation methods, such as those commonly used in reflection data processing. However, the wide-angle configuration requires techniques capable of handling very general wave types, including those not commonly used in reflection seismology. This is a situation analogous to that faced in cross-borehole seismics, where similar wave types are also recorded. In a real crossborehole example, we compare pre-stack migration, traveltime tomography and wavefield inversion. We find that wavefield inversion produces images that are quantitative in velocity (as are the tomograms) but are of significantly higher resolution; the wavefield inversion results have a resolution comparable to that of the (qualitative) pre-stack migration images. We seek to extend this novel development to the largerscale problem of crustal imaging.An essential element of the approach we adopt is its formulation entirely within the temporal frequency domain. This has three principal advantages: (1) we can choose to 'decimate' the data, by selecting only a limited number of frequency components to invert, thus making inversion of data from large numbers of source positions feasible;(2) we can mitigate the notorious non-linearity of the seismic inverse problem by progressing from low-frequency components in the data to high-frequency components; and (3) we can include in the model any arbitrary frequency dependence of inelastic attenugtion factors, Q(w), and indeed solve for the spatial distribution of Q.An initial synthetic test with an anomaly located within the middle crust yields a velocity image with the correct structural features of the anomaly and the correct magnitude of velocity anomaly. This is related to the fact that the reconstruction is obtained from forward-scattered waves. Under these conditions, the method thus behaves much like tomography. A second test with a deeper, more extensive anomaly yields an image with the correct velocity polarity and the correct location, but with a deficiency in low and high wavenumbers. In this case, this is because the reconstruction is obtained from backscattered waves; under these conditions the method behaves not like tomography, but like migration.A more extensive test, based on a large wide-angle survey in south-eastern California and western Arizona, demonstrates a real potential for high-resolution imaging of crustal structures. Although our results are limited by the acoustic approximation and by the relatively low frequencies that we can model today, the images are sufficiently encouraging to warrant future research. The problem of local minima in the objective function is the most significant practical problem with our method, but we propose that appropriate 'layer' stripping methods can handle this problem.

In full-wave inversion of seismic data in complex media it is desirable to use finite differences or finite elements for the forward modeling, but such methods are still prohibitively expensive when implemented in 3-D. Full-wave 2-D inversion schemes are of limited utility even in 2-D media because they do not model 3-D dynamics correctly. Many seismic experiments effectively assume that the geology varies in two dimensions only but generate 3-D (point source) wavefields; that is, they are Utwo-and-one-half-dimensional" (2.5-D), and this configuration can be exploited to model 3-D propagation efficiently in such media. We propose a frequency domain full-wave inversion algorithm which uses a 2.5-D finite difference forward modeling method. The calculated seismogram can be compared directly with real data, which allows the inversion to be iterated. We use a descents-related method to minimize a least-squares measure of the Manuscript received by the Editor

Crosshole experiments usually have sources and receivers confined to a plane, and it is assumed that there is negligible variation in the properties of the medium normal to this plane. Therefore, the problem appears two‐dimensional, except for the sources which are 3-D points rather than lines. This configuration is denoted as two‐and‐one‐half‐dimensional. We present a frequency‐domain approach to modeling acoustic wave propagation in such situations which allows correct treatment of point sources but takes advantage of the assumed 2-D nature of the medium to avoid full 3-D simulations. The approach uses a Fourier transform with respect to the out‐of‐plane coordinate to reduce the problem of modeling in 3-D to repeatedly solving a 2-D equation, which we accomplish using finite differences. The discrete inverse Fourier transform from the out‐of‐plane wavenumber implies the existence of an infinite number of spurious “ghost sources” spaced periodically in the out‐of‐plane direction. These sources generate significant artifacts on the time‐domain traces, even when their spatial period is much greater than the (in‐plane) dimensions of the survey area, because of time‐wrapping in the transform from the frequency domain. We describe two methods for reducing these artefacts, the more effective of which entails exponential damping by adding a positive imaginary part to the frequency, compensated by ramping of the wrapped time domain records. We check the modeling scheme by analysis of direct and scattered arrivals from simple models. The observed seismograms agree well with those calculated using Born theory, and so confirm the potential of this modeling method for use in inversion.

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