S U M M A R YFull waveform synthetic seismograms are now used as standard in the interpretation of marine refraction data, but only with a laborious trial-and-error fitting procedure. We show how the matching of observed waveforms with WKBJ synthetic seismograms can be efficiently and automatically performed for 1-D earth models. Automation of the inversion is difficult because of the multi-modal, irregular form of the misfit of data and synthetics. A sequence of three steps is required f0r.a complete inversion: location of the deepest valley of the misfit function, descent to the global minimum, and description of the neighbourhood of the minimum for error analysis. The first step is accomplished with a Monte Carlo search through a very large model space defined by poor prior knowledge of velocities and gradients of the solution. Weak traveltime constraints are used to eliminate poorly fitting models from waveform calculations. A comparison of the traveltime and waveform misfits of the Monte Carlo models clearly illustrates that waveforms are providing more information than traveltimes alone. Bayesian statistics are used to construct marginal probability distributions and the covariance matrix, which give a rough preliminary error analysis. In the second step, damped least-squares linearized inversion makes small adjustments to the best-fitting Monte Carlo model as it descends to the global minimum. Finally the immediate neighbourhood of the global minimum is explored with constrained least-squares inversion. Realistic error bounds on each parameter are defined from the resulting slices through the misfit function. These bounds are much narrower than traveltime inversion provides. Correlations between parameters are obtained from the covariance matrix constructed from models examined during this error analysis. The inversion methods are illustrated on the FF2 refraction data set of the Scripps Institution of Oceanography. The Monte Carlo search successfully locates the valley of the global minimum as well as a nearby secondary minimum. The error analysis puts realistic error bounds on the detailed inversion of this data set by Spudich & Orcutt.

The circular harmonic decomposition method for evaluating the inverse Radon transform is investigated. A discrete, finite set of projection data may be aliased and its interpretation is inevitably non-unique. When the inverse Radon transform is approximated by a summation, the filtered back projection, it is shown that as well as being non-unique, the reconstruction is inconsistent with the data. By contrast, the circular harmonic decomposition produces a consistent image. The stable form of the method is used to develop a simple and efficient numerical algorithm. This is illustrated with various simple examples and head phantoms.

Typical 3-D land and OBC seismic surveys are sampled finely in two spatial coordinates (e.g. receiver-x and source-y) and coarsely in the other two coordinates (receiver-y and source-x), so a simplistic discretization of the full 5-D Kirchhoff prestack migration integral leads to artifacts due to aliasing. Padhi and Holley (1997) suggested that imaging of well-sampled subsets of the data (minimal datasets) avoids the integration over the coarsely-sampled coordinates. For 3-D orthogonal acquisition geometries there are at least two types of minimal dataset that yield unaliased 1-fold images of the subsurface. A cross-spread is one type of minimal dataset (Vermeer, 1998a). A common-offset-vector (COV) gather is an alternative "basic building block" of 3-D wide-azimuth surveys. COV gathers (or volumes) are the simple extension of 2-D common-offset gathers (or sections) to orthogonal 3-D coordinates: the inline and crossline offsets are binned in such a way as to yield N 3-D volumes with a trace at each CDP, where N is the CDP fold. Traces within each COV gather share the same binned inline offset and binned crossline offset, and when they are sorted or stacked by CDP, the volume spans most of the survey area, as illustrated in Figure 1(a).Imaging COV volumes is like imaging common-offset sections in 2-D surveys: the exact offsets are not necessarily honored if they are not properly sampled (the binned offset values are often used instead), but each common-offset section or COV volume extends over most of the survey area, so the migrated image is complete except at the survey boundaries. Migrating 3-D crossspreads is like migrating shot gathers in 2-D surveys: the exact shot and receiver positions are honored but each shot gather or cross-spread has only small spatial extent, so much of the migrated image is dominated by edge effects.

The processing of converted‐wave (P-SV) seismic data requires certain special considerations, such as commonconversion‐point (CCP) binning techniques (Tessmer and Behle, 1988) and a modified normal moveout formula (Slotboom, 1990), that makes it different for processing conventional P-P data. However, from the processor’s perspective, the most problematic step is often the determination of residual S‐wave statics, which are commonly two to ten times greater than the P‐wave statics for the same location (Tatham and McCormack, 1991). Conventional residualstatics algorithms often produce numerous cycle skips when attempting to resolve very large statics. Unlike P‐waves, the velocity of S‐waves is virtually unaffected by near‐surface fluctuations in the water table (Figure 1). Hence, the P‐wave and S‐wave static solutions are largely unrelated to each other, so it is generally not feasible to approximate the S‐wave statics by simply scaling the known P‐wave static values (Anno, 1986).

When performing four-component surface-consistent deconvolution, it is assumed that the decomposition of amplitude spectra into source, receiver, offset, and common-depth-point components enables accurate deconvolution filters to be derived. However, relatively little effort has been put into the verification of this assumption. Some verification of the assumption is available by analyzing the results of the surfaceconsistent decomposition of real seismic data. The surface-consistent log-amplitude spectra of land seismic data are able to provide convincing evidence that the source component collects effects of the source signature and near-source structural effects, and that the receiver component collects receiver characteristics and near-receiver structural effects. In addition, the offset component collects effects due to ground roll and average reflectivity, and the CDP component collects mostly random noise unless it is constrained to be smooth. Based on the results of this analysis, deconvolution filters should be constructed from the source and receiver components, while the offset and CDP components are discarded.The four-component surface-consistent decomposition can be performed efficiently by making use of a simple rearrangement of the Gauss-Seidel matrix inversion equations. The algorithm requires just two passes through the prestack data volume, regardless of the sorted order of the data, so it is useful for both two-dimensional and three-dimensional (2-D and 3-D) data volumes.