Multiple suppression using a variant of the Radon transform is discussed. This transform differs from the classical Radon transform in that the integration surfaces are hyperbolic rather than planar. This specific hyperbolic surface is equivalent to parabolae in terms of computational expense but more accurately distinguishes multiples from primary reflections. The forward transform separates seismic arrivals by their differences in traveltime moveout. Multiples can be suppressed by an inverse transform of only part of the data. Examples show that multiples are effectively attenuated in prestack and stacked seismograms.

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General characterization of physical systems uses two aspects of data analysis methods: decomposition of empirical data to determine system parameters and reconstruction of the system attributes using these characteristic parameters. Spectral methods, involving a frequency-based representation of data, usually assume stationarity. These methods, therefore, extract only average information and, hence, are not suitable for analyzing data with isolated or deterministic discontinuities, such as faults or fractures in reservoir rocks or image edges in computer vision.Wavelet transforms provide a multiresolution framework for data representation. They are a family of basis functions that separate a function or a signal into distinct frequency packets that are localized in the time domain. Thus, wavelets are well suited for analyzing nonstationary data. In other words, a function or a discrete data set when transformed into a time-scale space using wavelets shows how it behaves at different scales of measurement. Because wavelets have compact support, it is easy to apply this transform to large data sets with minimal computations.We apply wavelet transforms to one-dimensional and twodimensional permeability data to determine the locations of layer boundaries and other discontinuities. By binning in the timefrequency plane with wavelet packets, permeability structures of arbitrary size are analyzed. Wavelets are also applied to scaling up spatially correlated heterogeneous permeability fields. F͑x ͒ϭ ͵Ϫϱ ϱ ͑u͒ f ͑ uϪx ͒du, ͑1͒

The double square root equation for laterally varying media in midpoint‐offset coordinates provides a convenient framework for developing efficient 3‐D prestack wave‐equation depth migrations with screen propagators. Offset‐domain pseudoscreen prestack depth migration downward continues the source and receiver wavefields simultaneously in midpoint‐offset coordinates. Wavefield extrapolation is performed with a wavenumber‐domain phase shift in a constant background medium followed by a phase correction in the space domain that accommodates smooth lateral velocity variations. An extra wide‐angle compensation term is also applied to enhance steep dips in the presence of strong velocity contrasts. The algorithm is implemented using fast Fourier transforms and tri‐diagonal matrix solvers, resulting in a computationally efficient implementation. Combined with the common‐azimuth approximation, 3‐D pseudoscreen migration provides a fast wavefield extrapolation for 3‐D marine streamer data. Migration of the 2‐D Marmousi model shows that offset domain pseudoscreen migration provides a significant improvement over first‐arrival Kirchhoff migration for steeply dipping events in strong contrast heterogeneous media. For the 3‐D SEG‐EAGE C3 Narrow Angle synthetic dataset, image quality from offset‐domain pseudoscreen migration is comparable to shot‐record finite‐difference migration results, but with computation times more than 100 times faster for full aperture imaging of the same data volume.