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Linear equations used to approximate reflection coefficient-versus-angle curves are usually valid only for small seismic parameter changes across reflectors, and they are rather inaccurate close to the critical angle. These inaccuracies affect the quality of AVO analysis and cause systematic errors when estimating relative seismic-parameter variations at reflectors, especially for density. We present an optimal model-based approach to build more accurate linear AVO approximations. Their basis functions are calculated by applying singular value decomposition to realistic modeled AVO curves. By extending the validity range of linear approximations to larger angles, this approach helps when using information contained at near-critical offsets. It alsooffers several advantages in other situations. The basis functions of the new approximations are orthogonal. Their coefficients represent new AVO attributes that can be used either to classify AVO responses directly, or to obtain more accurate estimates of usual AVO attributes (intercept, gradient, and possibly a third coefficient). This leads to a better estimation of seismic-parameter contrasts at reflecting interfaces. These coefficients are naturally sorted in decreasing order of importance. Therefore, the proper number of terms in the proposed equations can be chosen easily to offer an optimal compromise between noise and the information carried by each coefficient. Synthetic tests confirm the robustness of the method. This flexible and robust approach will be particularly well adapted for three-parameter AVO analysis.

Available high-order travel time equations allow us to flatten seismic gathers in a larger range of offsets than the normal moveout hyperbola, but they require the estimation of at least one extra parameter and they are still insufficiently accurate in certain situations, e.g. when strong ray bending occurs. To approximate reflection travel times very accurately with as few parameters as possible, I have derived new time-offset series, following a simple model-based approach: using information on velocities and on the recording configuration available for a given survey, I build a set of reference velocity models and calculate the associated travel time-offset functions. These functions define the vectorial space of all realistic time-offset curves. A set of basis functions for this vectorial space can be obtained by singular value decomposition. The new travel time series basically consist of weighted sums of these basis functions. A simple test in a plane-layered medium with strong heterogeneity and large aperture shows that the new series may be more efficient than other types of high-order travel time equations: with less coefficients, they provide more accurate approximations of the exact travel time curve (also at large offsets), and more reliable estimates of parameters of the velocity model, such as the RMS velocity and the quartic coefficient a4 of the Taner and Koehler series. On a practical level, the estimation of coefficients is simplified because the bases of functions are othogonal. The proposed procedure can be applied to many types of problems, e.g. to handle non-hyperbolic travel times resulting from anisotropy, wave conversions, high heterogeneity, wide apertures, etc. It should be useful for optimizing velocity analysis and different processing techniques that use travel time approximations, e.g. moveout correction, multiple attenuation and time migration.

The deterioration of prestack migration images caused by an erroneous velocity or dissipation model can be represented by a space‐variant filter. We have derived an approximate expression of this spatial filter for the case of a plane horizontal reflector between two homogeneous half‐spaces, using analytical expressions for the Green’s functions and a stationary phase approximation of the Kirchhoff integral. We have used complex, frequency‐dependent velocities to represent a general homogeneous viscoacoustic medium. Hence, the filter can describe the effects of errors in phase velocity and absorption on the image of the reflector for different situations. A pure phase velocity error deforms the reflector, affects the value of the recovered reflectivity, and creates a spurious amplitude versus offset (AVO) effect. Neglecting dissipation destroys the phase of the image, decreases the resolution, and causes an underestimation of the reflectivity. To have a true‐amplitude, good‐resolution migration with Claerbout’s crosscorrelation imaging principle, the forward extrapolation of the downgoing wavefield should be done with the complex conjugate of the velocity in the viscoacoustic medium, and the backward extrapolation of the upgoing wavefield should be done with the complex velocity. In both cases, this amplifies the wavefields to compensate for the effects of attenuation on wave propagation.

Intrinsic absorption in the earth affects the amplitude and phase spectra of the seismic wavefields and records, and may degrade significantly the results of acoustic full‐waveform inversion. Amplitude distortion affects the strength of the scatterers and decreases the resolution. Phase distortion may result in mislocated interfaces. We show that viscoacoustic gradient‐based inversion algorithms (e.g., steepest descent or conjugate gradients) compensate for the effects of phase distortion, but not for the effects of amplitude distortion. To solve this problem at a reasonable numerical cost, we have designed two new forms of preconditioning derived from an analysis of the inverse Hessian operator. The first type of preconditioning is a frequency‐dependent compensation for dispersion and attenuation, which involves two extra modeling steps with inverse absorption (amplification) at each iteration. The second type only corrects the strength of the recovered scatterers, and requires two extra modeling steps at the first iteration only. The new preconditioning methods have been incorporated into a finite‐difference inversion scheme for viscoacoustic media. Numerical tests on noise‐free synthetic data illustrate and support the theory.

Summary We propose a method, namely OptAVO to build enhanced linear AVO approximations. The basis functions of the approximation are orthogonal and their coefficients represent a new set of AVO attributes. These attributes can directly be used for AVO classification or to obtain better estimates of the usual coefficients (e.g., intercept, gradient). The method will be illustrated for class I reflectors using large reflection angles. A real data example shows the applicability of the proposed approach.

Summary Non-hyperbolic approximations of offset-traveltime curves usually have at least three terms. This complicates the practical calculation and interpretation of velocity spectra. We show here how conventional velocity spectra (i.e. computed using hyperbolae) can be improved using a nonhyperbolic traveltime approximation with only two terms.