Reweighting strategies in seismic deconvolution

Reweighting strategies have been widely used to diminish the influence of outliers in inverse problems. In a similar fashion, they can be used to design the regularization term that must be incorporated to solve an inverse problem successfully. Zero-order quadratic regularization, or damped least squares (pre-whitening) is a common procedure used to regularize the deconvolution problem. This procedure entails the definition of a constant damping term which is used to control the roughness of the deconvolved trace. In this paper I examine two different regularization criteria that lead to an algorithm where the damping term is adapted to successfully retrieve a broad-band reflectivity.
Synthetic and field data examples are used to illustrate the ability of the algorithm to deconvolve seismic traces.     

What has seismic missed?

The relative efficiencies of alternative geometric patterns of both discrete borehole and continuous grid line search have been extensively discussed in the mathematical geology literature. However, an equally important problem has received virtually no attention: How to use a sample of properties of geologic anomalies detected by grid line search of a region to estimate systematically both the number and size distribution of geologic anomalies missed by the search. We show how estimation methods developed in the sample survey design literature can be adapted to this problem, and we apply these methods to data describing 94 anomalies identified by a seismic reconnaissance survey.     

Ellipticity corrections for seismic phases

The advent of broad-band seismology has meant that use is being made of a wide range of seismic phases, for many of which ellipticity corrections have not been readily available. In particular, when many seismic phases are used in location schemes, it is important that the systematic effects of ellipticity are included for each phase.
An efficient and effective procedure for constructing ellipticity corrections is to make use of the ray-based approach of Dziewonksi & Gilbert (1976), as reformulated by Doornbos (1988), in conjunction with the rapid evaluation of traveltimes and slownesses for a given range using the tauspline procedure of Buland & Chapman (1983).
Ellipticity coefficients have been tabulated for a wide range of seismic phases and are available in electronic form. The ellipticity correction procedures have been extended to include an allowance for diffraction phenomena, for example P _diff, S _diff diffracted along the core-mantle boundary. Corrections for additional phases can be generated by building the ellipticity coefficients from suitable combinations of the coefficients for different phase segments.     

One-way representations of seismic data

A general one-way representation of seismic data can be obtained by substituting a Green's one-way wavefield matrix into a reciprocity theorem of the convolution type for one-way wavefields. From this general one-way representation, several special cases can be derived.
By introducing a Green's one-way wavefield matrix for primaries , a generalized Bremmer series representation is obtained. Terminating this series after the first-order term yields a primary representation of seismic reflection data. According to this representation, primary seismic reflection data are proportional to a reflection operator, 'modified' by primary propagators for downgoing and upgoing waves. For seismic imaging, these propagators need to be inverted. Stable inverse primary propagators can easily be obtained from a one-way reciprocity theorem of the correlation type.
By introducing a Green's one-way wavefield matrix for generalized primaries , an alternative representation is obtained in which multiple scattering is organized quite differently (in comparison with the generalized Bremmer series representation). According to the generalized primary representation, full seismic reflection data are proportional to a reflection operator, 'modified' by generalized primary propagators for downgoing and upgoing waves. Internal multiple scattering is fully included in the generalized primary propagators {either via a series expansion or in a parametrized way). Stable inverse generalized primary propagators can be obtained from the one-way reciprocity theorem of the correlation type. These inverse propagators are the nucleus for seismic imaging techniques that take the angle-dependent dispersion effects due to fine-layering into account.     

Simultaneous inversion of seismic data

Summary. The resolving power of different data sets, consisting of surface-wave dispersion measurements and S travel times, are compared for a continental structure. The shear velocity in the low-velocity zone can be resolved in some detail with higher-mode phase-velocity data. Sufficient resolution for small density contrasts (0.03 g cm⁻³) until depths of ∼ 300 km can be reached if higher-mode group velocities are available as well, even at a precision as low as 0.10 km/s. At greater depths the density is not resolved, and here travel-time data are superior to higher modes in resolving the shear velocity.     

Synthesis of seismic surface waves

Summary. The reflection method is used to produce complete sets of Rayleigh wave dispersion curves in a given phase velocity—frequency window for horizontally stratified media, including modes of very high numerical order, and theoretical surface wave seismograms are then synthesized.
The difficulties encountered when attempting to complete large numbers of dispersion curves are discussed. Particular problems arise from models with a low-velocity zone, when curves in a certain portion of the dispersion diagram split into two families, the crustal and the channel modes. The reflection method provides a convenient framework in which to examine this phenomenon heuristically and so devise a method to overcome the difficulties.
Seismograms are produced by mode summation and it is found that body-wave behaviour, as well as surface-wave features can be synthesized. The effect of truncating the mode series at a number comparable with the number of modes used in previous studies is examined. It is found that although the S -coda at longer ranges is not adversely affected, the arrivals attributable to body-waves are severely distorted. This must call into question the validity of synthetic seismograms generated by summation of only a few modes.     

Diffraction phenomena in seismic models

Summary. Refraction seismic profiles in the Alpine area occasionally show seismic 'shadows' which cannot be explained by a horizontal low-velocity channel. An example is demonstrated where the low P -velocity zone (Flysch) dips below a wedge of high P -velocity (the Calcareous Alps) which is a typical Alpine structure. This feature has been studied by means of a seismic model experiment. The corner of the wedge works as a source of diffracted P -, S - and Rayleigh-waves, when met by seismic waves. The diffracted waves can be distinguished from conventional waves by their amplitude–distance relation when the extinction coefficient is well known by an independent experiment.     

The growing spherical seismic source

Summary. A solution is found for the seismic radiation from an arbitrarily growing spherical source in an inhomogeneously prestressed elastic medium. The general problem of the growing seismic source in a prestressed medium is formulated as a boundary value problem. For the special case of the growing spherical source, an expansion in vector spherical harmonics reduces the problem to a set of one-dimensional Volterra integral equations. The equations can be easily formed through the use of Bessel function recursion relations. The integral equations for a growing spherical cavity are solved numerically. Waveforms are then computed for homogeneous and inhomogeneous stress fields for several growth histories. The resulting waveforms are similar to the waveforms of the corresponding instantaneous problem, but stretched out in time and reduced in amplitude. The effects of diffraction and the overshoot of equilibrium are reduced with a reduction in growth rate. The effects caused by inhomogeneity of the stress field are quite strong for the growing as well as for the instantaneous seismic source.