Hugh D. Geiger

The equivalent offset method of prestack time migration

A prestack time migration is presented that is simple, efficient, and provides detailed velocity information. It is based on Kirchhoff prestack time migration and can be applied to both 2-D and 3-D data. The method is divided into two steps: the first is a gathering process that forms common scatterpoint (CSP) gathers; the second is a focusing process that applies a simplified Kirchhoff migration on the CSP gathers, and consists of scaling, filtering, normal moveout (NMO) correction, and stacking. A key concept of the method is a reformulation of the double square‐root equation (of source‐scatterpoint‐receiver traveltimes) into a single square root. The single square root uses an equivalent offset that is the surface distance from the scatterpoint to a colocated source and receiver. Input samples are mapped into offset bins of a CSP gather, without time shifting, to an offset defined by the equivalent offset. The single square‐root reformulation gathers scattered energy to hyperbolic paths on the appropri...

Improving explicit seismic depth migration with a stabilizing Wiener filter and spatial resampling

We present a new approach to the design and implementation of explicit wavefield extrapolation for seismic depth migration in the space-frequency domain. Instability of the wavefield extrapolation operator is addressed by splitting the operator into two parts, one to control phase accuracy and a second to improve stability. The first partial operator is simply a windowed version of the exact operator for a half step. The second partial operator is designed, using the Wiener filter method, as a band-limited, least-squares inverse of the first. The final wavefield extrapolation operator for a full step is formed as a convolution of the first partial operator with the complex conjugate of the second. This resulting wavefield extrapolation operator can be designed to have any desired length and is generally more stable and more accurate than a simple windowed operator of similar length. Additional stability is gained by reducing the amount of evanescent filtering and by spatially downsampling the lower tempor...

Equivalent Offset CRP Gathers

Traditional seismic velocity analysis uses a gather of traces that have a common midpoint (CMP) between the source and receiver.A method is presented for creating a common reflection point (CRP) gather in which all input traces within the prestack migration aperture are used. The traces are sorted with an equivalent offset derived from the source to reflector and reflector to receiver distances. This equivalent offset CRP gather represents the traces that would prestack migrate to the output CRP location.

Fourier prestack migration by equivalent wavenumber

Fourier prestack migration is reformulated through a change of variables, from offset wavenumber to a new equivalent wavenumber, which makes the migration phase shift independent of horizontal wavenumber. After the change of variables, the inverse Fourier transform over horizontal wavenumber can be performed to create unmigrated, but fully horizontally positioned, gathers at each output location. A complete prestack migration then results by imaging each gather independently with a poststack migration algorithm. This equivalent wavenumber migration (EWM) is the Fourier analog of the space‐time domain method of equivalent offset migration (EOM). The latter is a Kirchhoff time‐migration technique which forms common scatterpoint (CSP) gathers for each migrated trace and then images those gathers with a Kirchhoff summation. These CSP gathers are formed by trace mappings at constant time, and migration velocity analysis is easily done after the gathers are formed. Both EWM and EOM are motivated by the algebrai...

Equivalent offset prestack migration for rugged topography

The process of prestack migration using equivalent offset and common scatter point gathers was initially based on input data converted to a horizontal datum (Bancroft et al. 1994). This short paper demonstrates how the process may be adapted for rugged topography, and allow the prestack migration to be from surface. Equivalent offset gathers were created for a model data set with a single scatter point in the subsurface. The results indicate that the algorithm works for both positive and negative datum shifts, although large shifts may move data of a single input trace over a wide range of offsets. An Alberta Foothills data set was prestack migrated using the rugged topography method. The initial results show excellent imaging of subsurface structure. INTRODUCTION Equivalent offset migration Equivalent offset prestack migration (Bancroft 1994 a,b) creates an intermediate step in the Kirchhoff process by creating a gather from all the input traces for each output trace. The input traces that are gathered for a given output location are sorted by an offset that is based on the distances of the scatter point from the source and receiver locations. The collection of input traces is referred to as the common scatter point (CSP) gather, and is similar in function to the common mid point (CMP) gather. Both are defined for an output location, and both contain input traces that are sorted with offset. There is no time shifting of the input samples when they are sorted into the CSP gathers. When the CSP gathers have been formed, each CSP gather may be scaled and filtered, or processed similarly to CMP gathers. Conventional algorithms such as noise and multiple removal, or velocity analysis, may also be used on the CSP gathers. Velocity analysis performed on the CSP gather will contain a more accurate v locity discrimination than those derived from CMP gathers. The improved discrimination results from using only one CSP gather, the high fold in the offset bins within the CSP gather, and offsets that are much larger than the source-receiver offset. Noise and multiples in the CSP gather now apply to the final prestack migration offsets, and not the offsets of CMP gathers in conventional processing. Processing datum and the surface elevation It has been recognized for a long time that migration and normal moveout (NMO) should be performed on time data with a processing datum as close to that actual surface as possible. Time migration and NMO assume that offset data are hyperbolic

Optimizing explicit depth migration with a stabilizing Wiener filter and spatial resampling

We present a new approach to the design of stable and accurate wavefield extrapolation operators needed for explicit depth migration. We split the theoretical operator into two component operators, one a forward operator that controls the phase accuracy and the other an inverse operator, designed as a Wiener filter that stabilizes the first operator. Both component operators are designed to have a specific fixed length and the final operator is formed as the convolution of the components. We utilize this operator design method to build an explicit, wavefield extrapolation method based on the migration of individual source records. Two other features of our method are the use of dual operator tables, with high and low levels of evanescent filtering, and frequency-dependent spatial down sampling. Both of these features improve the accuracy and efficiency of the overall method. Empirical testing shows that our method has a similar performance to the time-migration method called phase shift, meaning it scales as NlogN. We illustrate the method with tests on the Marmousi synthetic dataset. We call our method FOCI which is an acronym for forward operator conjugate inverse.

Amplitude-preserving weights for Kirchhoff prestack time migration

Summary In a ‘Kirchhoff’ (i.e. weighted diffraction stack) prestack migration, the summation over a complete diffraction surface can be thought of as an average of reflectivity estimates from migrated common-shot, common-receiver or common -offset gathers. The optimal weight for averaged reflectivity should be based on Bleistein et al’s (2001) β common-offset weight. In comparison, the β common-shot and common-receiver weights, although correct for individual gathers, produce average reflectivity estimates with a dip- and depth-dependent bias. Bleistein et al’s (2001) 1 β common-offset weight is more suitable as a basis for practical weights because it downweights by the cosine of the ray half-opening or obliquity angle at the reflector and hence accounts for the corresponding reduced spatial resolution as obliquity angle increases. In this paper, weights for optimal 2.5-D and 3-D diffraction stack migrations are i) tested using simple noise-free constant-wavespeed synthetics, ii) re-expressed for practical implementation as relative-amplitude-preserving prestack time migration weights, and iii) customized for the equivalent offset method (EOM) of prestack time migration.

A kinematic comparison of DMO‐PSI, and equivalent offset migration (EOM)

There is considerable interest in prestack migrations that include velocity analysis as part of their algorithms. These methods are based on the scatter point principle in which all the energy reflected from a scatter point is collected into a prestack migration gather. Two such methods are DMO-PSI (Gardner’s dip moveout followed by prestack imaging) and EOM or equivalent offset migration. Both methods create a superior prestack migration gather that is referred to as a common scatter point gather (CSP), and are formed from input data before NMO. All energy from a scatter point maps to a hyperbolic path on the CSP gather and is ideally suited for velocity analysis. Normal moveout correction and stacking complete the prestack migration process.

Automatic selection of reference velocities for recursive depth migration by peak search method

Wave equation depth migration methods such as phase-shift plus interpolation, extended split-step Fourier, or Fourier finite difference plus interpolation require a limited set of reference velocities for efficient wavefield extrapolation through laterally inhomogeneous velocity models. In basic implementations, reference velocities are selected as either a linear or a geometric progression spanning the range of model velocities. However, it is unlikely that the model velocities are distributed linearly or geometrically. If the reference velocities can be distributed statistically to more closely approximate the actual distribution, the accuracy of the extrapolation step can be improved. In this paper, we present a modification to a previously published algorithm for statistical selection of reference velocities (Bagaini et al. 1995). Key features of our automatic reference velocity selection algorithm are 1) division of the velocity distribution into clusters 2) entropy based statistical control to determine the minimal number of reference velocities required within a cluster, 3) a novel ‘greedy search’ that selects reference velocities within each cluster at or near peaks in the probability distribution, and 4) calculation of a single reference velocity if the velocity range within each cluster is less than a minimum threshold. We show that our method is superior to Bagaini et al.’s method, which includes only step (2) above, and - in place of step (3) - selection of reference velocities by piecewise constant interpolation of the probability distribution of the underlying velocities. Our automatic velocity selection algorithm can be run using a single parameter – the approximate desired velocity step expressed as a percentage – although the maximum step and minimum threshold can be specified if the defaults are not suitable. The automatic velocity selection algorithm is implemented as part of a prestack PSPI depth migration of the 2-D Marmousi model. The resulting images are clearly superior to images created using a linear or geometrical distribution of a similar number of reference velocities. The algorithm is suitable for 3-D data, where the tradeoff between accuracy and efficiency is more pronounced.

The FOCI method versus the WLSQ and Hale's wavefield extrapolation methods

Recursive wavefield extrapolation methods are more powerful than ray theory based methods because of their greater ability to handle strong lateral velocity variations. Wavefield extrapolation methods have two major problems: (1) the extrapolator instability and (2) the computational expense. The forward operator and conjugate inverse (FOCI) method is an appropriate method for designing accurate and efficient extrapolation operators that remain stable in a recursive algorithm. FOCI’s results are comparable with results obtained with other known methods such as Hale’s and the weighted least square (WLSQ) extrapolation methods. Further, the spatial resampling in the FOCI method offers computational advantages.