Kristopher A. Innanen

Inversion of the seismic AVF/AVA signatures of highly attenuative targets

Frequency-dependent seismic field data anomalies, appearing in association with low- Q targets, have, on occasion, been attributed to the presence of a strong absorptive reflection coefficient. This “absorptive reflectivity” represents a potent, and largely untapped, source of information for determining subsurface target properties. It would most likely be encountered where a predominantly elastic/nonattenuating overburden suddenly is interrupted by a highly attenuative target. Series expansions of absorptive reflection coefficients about small parameter contrasts and incidence angles can expose these anomalies to analysis, either frequency-by-frequency (amplitude variation with frequency [AVF]) or angle-by-angle (amplitude variation with angle of incidence [AVA]). Within this framework, variations in P-wave velocity and Q can be estimated separately through a range of direct formulas, both linear and with nonlinear corrections. The latter come to the fore when a contrast from an incidence medium Q≈∞ (i....

Seismic data interpolation using a fast generalized Fourier transform

We have found a fast and efficient method for the interpolation of nonstationary seismic data. The method uses the fast generalized Fourier transform (FGFT) to identify the space-wavenumber evolution of nonstationary spatial signals at each temporal frequency. The nonredundant nature of FGFT renders a big computational advantage to this interpolation method. A least-squares fitting scheme is used next to retrieve the optimal FGFT coefficients representative of the ideal interpolated data. For randomly sampled data on a regular grid, we seek a sparse representation of FGFT coefficients to retrieve the missing samples. In addition, to interpolate the regularly sampled seismic data at a given frequency, we use a mask function derived from the FGFT coefficients of the low frequencies. Synthetic and real data examples can be used to examine the performance of the method.

On the construction of an absorptive–dispersive medium model via direct linear inversion of reflected seismic primaries

Arbitrary multi-dimensional distributions of two absorptive–dispersive acoustic medium parameters (P-wave velocity and Q) may be determined to first order from reflected seismic primary data with an inverse scattering formulation. The problem can be considered a form of Q-estimation, but one that distinguishes itself by using the dispersive reflection coefficient as its primary source of information. If the overburden above a target to be characterized is known, synthetic Q estimates are successfully derived, albeit using subtle data variations. If the overburden is unknown, the linear estimate is instead best interpreted as the starting point of a nonlinear inverse scattering procedure. Simple analytic and numerical examples may be used to characterize associated issues of conditioning, leakage, detectability and transmission error.

Adaptive separation of free-surface multiples through independent component analysis

We present a three-stage algorithm for adaptive separation of free-surface multiples. The free-surface multiple elimination (FSME) method requires, as deterministic prerequisites, knowledge of the source wavelet and deghosted data. In their absence, FSME provides an estimate of free-surface multiples that must be subtracted adaptively from the data. First we construct several orders from the free-surface multiple prediction formula. Next we use the full recording duration of any given data trace to construct filters that attempt to match the data and the multiple predictions. This kind of filter produces adequate phase results, but the order-by-order nature of the free-surface algorithm brings results that remain insufficient for straightforward subtraction. Then we construct, trace by trace, a mixing model in which the mixtures are the data trace and its orders of multiple predictions. We separate the mixtures through a blind source separation technique, in particular by employing independent component analysis. One of the recovered signals is a data trace without free-surface multiples. This technique sidesteps the subtraction inherent in most adaptive subtraction methods by separating the desired signal from the free-surface multiples. The method was applied to synthetic and field data. We compared the field data to a published method and found comparable results.

Direct nonlinear Q-compensation of seismic primaries reflecting from a stratified, two-parameter absorptive medium

Q -compensation of seismic primaries that have reflected from a stratified, absorptive-dispersive medium may be posed as a direct, nonlinear inverse scattering problem. If the reference medium is chosen to be nonattenuating and homogeneous, an inverse-scattering Q -compensation procedure may be derived that is highly nonlinear in the data, but which operates in the absence of prior knowledge of the properties of the subsurface, including its Q structure. It is arrived at by (1) performing an order-by-order inversion of a subset of the Born series, (2) isolating and extracting a component of the resulting nonlinear inversion equations argued to enact Q - compensation, and (3) mapping the result back to data space. Once derived, the procedure can be understood in terms of nonlinear interaction of the input primary reflection data: the attenuation of deeper primaries is corrected by an operator built (automatically) using the angle- and frequency variations of all shallower primaries. A simple synthetic exam...

Extension of the non-linear depth imaging capability of the inverse scattering series to multidimensional media: strategies and numerical results

The inverse scattering series (ISS) has proven, and continues to prove, to be a highly effective formalism for the separate and isolated accomplishment of several key tasks of reflection seismic processing and inversion. In particular, Weglein et al. (2000), Shaw et al. (2003), and Shaw (2005) describe the development of an algorithm distilled from the ISS that concerns itself with the location of subsurface reflectors with no prior knowledge, or related intervening estimation, of the medium wavespeed. The specific non-linear data activity that accomplishes this goal has been investigated by Shaw as such for an idealized 1D pre-stack acoustic experiment; we here describe the extension of those ideas to accommodate media with lateral variation. This is a non-trivial step. Nevertheless, beneath the added algebraic complexity, recognizable patterns and mechanisms are visible. Analysis of these terms and patterns suggests that certain portions of the 2D reflector location mechanisms of the ISS are a good starting point for the creation of algorithms for the accurate depth location of reflectors with a moderate level of lateral variability. The partial 2D imaging capability within the ISS is examined in this paper for the special case of a constant density acoustic medium and taking kh=0. We demonstrate numerical implementations of these forms and discuss ongoing work towards capturing further imaging capability residing within the ISS, especially with regards to the accommodation of larger levels of contrast and rapidity of spatial variation in medium properties.

A direct nonlinear inversion of primary wave data reflecting from extended, heterogeneous media

Wave events that have reflected once from an extended, heterogeneous medium may be approximated, to varying degrees of accuracy, with subsets of terms of the forward scattering series and a homogeneous reference medium. Two such primary approximations, when inverted order-by-order, produce nonlinear, series-form algorithms that directly determine the medium from primary data. These forward and inverse expressions are remarkably symmetric; in fact, under the assumption of a depth-varying perturbation, they form pairs of nonlinear integral transforms back and forth between the data and the perturbation.

Multi-dimensional seismic imaging using the inverse scattering series

The inverse scattering series (ISS) is a comprehensive multidimensional theory for processing and inverting seismic reflection data, that may be task-separated such that meaningful sub-problems of the seismic inverse problem may be accomplished individually, each without an accurate velocity model. We describe a task-separated subseries of the ISS geared towards accurate location in depth of reflectors, in particular the mechanisms of the series that act in multiple dimensions. We show that some 2D ISS imaging terms have analogs in previously developed 1D ISS imaging theory (e.g., Weglein et al., 2002; Shaw, 2005) and others do not; the former are used to create a 2D depth-only imaging prototype algorithm which is tested on synthetic salt-model data, and the latter are used to discuss ongoing research into reflector location activity within the series that acts only in the case of lateral variation and the presence of, e.g., diffraction energy in the data. Numerical tests are encouraging and show clear added value.

Multiresolution modeling and seismic wavefield reconstruction in attenuating media

The propagation of the seismic wavefield through a viscoelastic medium is a multiresolution process, in which depth of propagation and scale are closely associated with one another. We propose the multiresolution wavefield reconstruction (MRWR) as a means to directly integrate such concepts of scale into the backpropagation component of an imaging method. MRWR produces a reconstruction of the wavefield (at some fixed depth d), in which each scale term reconstitutes the resolution that was lost as the wavefield propagated some step‐length towards the measurement surface, away from d. Concurrently, MRWR provides a stable platform for this removal of the effects of absorption in propagation. In a multiresolution model of propagation, the viscoelastic propagation kernel is seen, mathematically, to fill the role of the scale function in multiresolution theory, as it operates on a wavefield to propagate it through some distance. The suppression of high‐resolution components of the wavefield via this scaling/pro...

Characterizing the degree of amplitude-variation-with-offset nonlinearity in seismic physical modelling reflection data

The nonlinearity of the seismic amplitude-variation-with-offset response is investigated with physical modelling data. Nonlinearity in amplitude-variation-with-offset becomes important in the presence of large relative changes in acoustic and elastic medium properties. A procedure for pre-processing physical modelling reflection data is enacted on the reflection from a water-plexiglas boundary. The resulting picked and processed amplitudes are compared with the exact solutions of the plane-wave Zoeppritz equations, as well as approximations that are first, second, and third order in , , and . In the low angle range of 0°–20°, the third-order plane-wave approximation is sufficient to capture the nonlinearity of the amplitude-variation-with-offset response of a liquid-solid boundary with , , and ρ contrasts of 1485–2745 m/s, 0–1380 m/s, and 1.00–1.19 gm/cc respectively, to an accuracy value of roughly 1%. This is in contrast to the linear Aki–Richards approximation, which is in error by as much as 25% in the same angle range. Even-order nonlinear corrective terms are observed to be primarily involved in correcting the angle dependence of , whereas the odd-order nonlinear terms are involved in determining the absolute amplitude-variation-with-offset magnitudes.