Peter M. Manning

Analyzing the effectiveness of receiver arrays for multicomponent seismic exploration

This paper uses an experimental seismic line recorded with three‐component (3C) receivers to develop a case history demonstrating very little benefit from receiver arrays as compared to point receivers. Two common array designs are tested; they are detrimental to the P‐S wavefield and provide little additional benefit for P‐P data. The seismic data are a 3C 2‐D line recorded at closely spaced (2 m) point receivers over the Blackfoot oil field, Alberta. The 3C receiver arrays are constructed by summing five (one group interval) and ten (two group intervals) point receivers. The shorter array emphasizes signal preservation while the longer array places priority on noise rejection. The effectiveness of the arrays versus the single geophones is compared in both the t−x and f−k domains of common source gathers. The quality of poststack data is also compared by analyzing the f−x spectra for signal bandwidth on both the vertical receiver component (P‐P) and radial receiver component (P‐S) structure stacks produc...

Finite Difference Modelling Analysis, Dispersion And Stability

Fourier analysis is used to present conclusions about stability and dispersion in finite-difference modelling. The most elementary finite-difference model is presented, one dimension in space with second-order differencing in space and time. For this one spatial dimension case, formulae are derived to correct for the dispersion and instability caused by finite grid sampling. The conclusions drawn are compatible with other discussions of stability in one dimension, but an increased understanding of the mechanisms for instability is obtained.

Analysis of the Effectiveness of 3-C Receiver Arrays For Converted Wave Imaging

During November 1997 the CREWES Project at the University of Calgary recorded a 3C-2D seismic survey at the Blackfoot field east of Calgary. This survey consisted of recording dynamite shots into a combination of conventional 20 m and high-resolution 2 m receiver intervals. We used this high-resolution data to examine two alternative approaches to array design by simulating 3-C receiver arrays via convolution in the t-x domain. The effectiveness of each approach was then evaluated by analyzing the response in both the t-x and f-k domains. The post-stack effect was also compared by analyzing the f-x response of both the final P-P and P-S structure stacks produced using these two array design philosophies.

An integrated VSP case history of coal‐seam anisotropy in Alberta

A four-source multioffset three-component (3-C) VSP experiment was carried out in a 3-D prospect area. The data were processed with a view to studying coal anisotropy and comparing the resolution of multioffset-VSP and 3-D surface data. Converted-wave processing is carried out in order to enhance any anisotropic effects due to coal fracturing. Imaging ambiguities on conventional seismic data caused by coal measures are also discussed for the case of a reef lying underneath coal. Shear-wave birefringence or splitting occurs only in anisotropic solids and is diagnostic of anisotropy. This can he related to rock properties (such as aligned fractures or cracks. thin layering, aligned grains. etc.) and to the symmetry properties of these fabric elements. Significant energy has shown up on the minimized (trensverse) component for both the A and B sources after data rotation. This is not believed to he due to rotation error because there is a significant difference (about IO ms on the first trace) between the arrival times of the A single three-component geophone was moved up converted-wave coal reflection on the two rotated horizontal from 1460 m to 700 m at 10-m intervals. At each location. components. This does not happen for sources C and D. This four different sweeps were applied simultaneously. The arrival-time difference between radial and transverse survey geometry is shown in Figure 1. components on the CCP maps is about 6 ms for both sources A and B. The birefringence analysis method presented here is embodied in an algorithm developed hy Harrison (1992).

Finite-difference Modelling With Correction Filters In Variable Velocity

The theory of correction filters for one and two dimensional finite-difference modelling has been presented by the authors in earlier papers (Manning and Margrave 2000, 2001). In those papers we showed how a correction filter could compensate for finite-difference sampling effects (with inherently stable conditions), and thereby obtained results that matched analytic modelling. These papers considered media with a single uniform velocity.

Optimum Boundaries For Finite-difference Modelling of Waves

A technique is described to generate an optimum set of points beyond the boundary of a finite-difference model; points which can be used for finitedifference operations, and which eliminate reflections from that boundary. The rationale is explained, the mathematics are developed, and the final formulae are given. Examples are shown for the case of a two-dimensional elastic pressure wave. The limitations of the technique are shown to be consistent with the general limitations of simulating continuous waves by finite-difference techniques. INTRODUCTION There have been many techniques developed to reduce the reflections from the boundaries of finite-difference models, and simulate the infinite real earth. These techniques are useful for economic reasons, so that a model size can remain small and yet simulate the effects of specified internal boundaries without interference from the model edges. The basic technique is to provide extra rows and columns of points around the edges of a model. The amplitudes at these points are needed to allow the finitedifference operations to be executed within the model, but cannot themselves be generated by the same techniques because of their edge position. Unique algorithms, or in some cases unique conditions, must be used to calculate these amplitudes. The earlier techniques used to reduce boundary effects were called absorbing boundaries, and simulated the effects of having a highly attenuating material around the model. This technique is very practical where the modelling already accounts for viscous effects on the particle motion (Kelly and Marfurt, 1990). The viscosity is simply made very high for several rows and columns around the model’s area of interest. Another absorbing technique that can be used is to taper, at each time step, the amplitudes toward the model edge by a minimal amount. Cerjan et al. (1985) got very successful results by tapering to a maximum of 0.92 across a boundary zone of 20 points in width. With absorbing boundaries, the edge-point amplitudes are calculated by an approximate algorithm, but any errors that this introduces is shielded by the attenuating zone. The increased overhead caused by providing the attenuating zone is usually not a major barrier with modern computers. An alternative to absorbing boundary conditions can be called transmitting boundary conditions. Reynolds (1978) called his boundaries transmitting, and although Clayton and Engquist (1977) called their boundaries absorbing, they used algorithms similar to Reynolds. These algorithms project amplitudes into the boundary zones from the values already calculated for the zone of interest. Clayton and Engquist adapted a migration algorithm to project boundary values. Reynolds factored the wave equation and then used approximations for finitedifferencing. A requirement of these techniques is to select only those solutions that advance into the boundary, and suppress solutions that advance out of the boundary (the reflections). They are found to work very well with waves moving directly toward the boundary, but not so well with waves approaching the boundary at an acute angle. This paper describes a transmitting boundary solution for the second-order finitedifference elastic wave equation. In this space of digital values, two unknowns must be found. The first unknown is the extra boundary value amplitude, and the second unknown is the advanced time-step amplitude that is calculated using the extra boundary value. The first of the two equations that is required for a solution is, of course, the time stepping equation. We have found that the second required equation is the one that relates all the first derivatives of an unimpeded advancing wave (the eikonal equation). Any solution that does not satisfy this equation must involve some reflected energy. The above simultaneous solution takes the form of a quadratic. The root of the quadratic must be chosen so that the slope of the wave toward the boundary is consistent with the slope in time of an advancing wave. In particular, a slope down toward the boundary must accompany more positive amplitudes with time, and vice-versa. THEORY The development of the theory starts with the definition of a scalar plane-wave, which may be chosen to advance with time ( ) ( ) t k x z F P ω θ θ − + = sin cos . (1) Then an equation relating the derivatives of the function may be shown to be

Elastic finite difference modeling in two dimensions: Stability and dispersion corrections

This paper presents a general method to calculate and correct for the dispersion and instability inherent within finite-difference elastic modelling in two dimensions. The method is based on an extension of the Von Neumann stability analysis. For a fixed frequency an analytic relationship is derived between the continuous derivatives in the elastic wave equation and their second-order finite-difference approximations. Typically, the continuous derivative is equal to a finite-difference result divided by a correction factor that is a squared sinc function dependant on frequency and grid size. When the continuous derivatives are replaced by these expressions, an equivalent to the elastic wave-equation results that involves finite differences and correction factors. These correction factors are all dependent on either wavenumber or frequency. The frequency dependence can be converted to wavenumber dependence using P and S wave velocities. The correction factors can then be seen and applied as spatial filters. Numerical tests show that these correction factors compensate for a wide range of dispersion and instability.

Rayleigh Wave Modelling By Finite Difference

Finite difference modelling of elastic wave fields is a practical method for elucidating features of records obtained for exploration seismic purposes, including surface waves. This has become possible because of the major increase in computing power available at reasonable cost. To take advantage of this power a 2D finite difference modelling program for elastic displacements has been written in Matlab. Two techniques for representing realistic boundaries are presented in some detail. Examples are provided of the program’s use to propagate waves at the surface of the earth through shallow lateral and vertical velocity changes. It can be seen that surface wave reflections and transmissions have similarities to body wave reflections and transmissions.