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 Volume 40, Issue 2, 1992
Geophysical Prospecting  Volume 40, Issue 2, 1992
Volume 40, Issue 2, 1992

BOREHOLE RADAR APPLIED TO THE CHARACTERIZATION OF HYDRAULICALLY CONDUCTIVE FRACTURE ZONES IN CRYSTALLINE ROCK^{1}
Authors OLLE OLSSON, LARS FALK, OLOF FORSLUND, LARS LUNDMARK and ERIC SANDBERGAbstractThe borehole radar system, RAMAC, developed within the framework of the International Stripa Project, can be used in three different measuring modes; single‐hole reflection, cross‐hole reflection and cross‐hole tomography. The reflection modes basically provide geometrical data on features located at some distance from the borehole. In addition the strength of the reflections indicate the contrast in electrical properties. Single‐hole reflection data are cylindrically symmetrical with respect to the borehole, which means that a unique fracture orientation cannot be obtained. A method has been devised where absolute orientation of fracture zones is obtained by combining single‐hole reflection data from adjacent holes. Similar methods for the analysis of cross‐hole reflection data have also been developed and found to be efficient. The radar operates in the frequency range 20‐‐60 MHz which gives a resolution of 1–3 m in crystalline rock. The investigation range obtained in the Stripa granite is approximately 100 m in the single‐hole mode and 200‐‐300 m in the cross‐hole mode.
Variations in the arrival time and amplitude of the direct wave between transmitter and receiver have been used for cross‐hole tomographic imaging to yield maps of radar velocity and attenuation. The cross‐hole measurement configuration coupled with tomographic inversion has less resolution than the reflection methods but provides better quantitative estimates of the values of measured properties.
The analysis of the radar data has provided a consistent description of the fracture zones at the Stripa Cross‐hole site in agreement with both geological and geophysical observations. Comparison of the radar results with seismic cross‐hole data showed excellent agreement with respect to shape and location of the fracture zones in space. Comparison with hydraulic data shows that the features identified by radar are of hydrogeological significance.

SEPARATION OF REGIONAL AND RESIDUAL ANOMALIES BY LEAST‐SQUARES ORTHOGONAL POLYNOMIAL AND RELAXATION TECHNIQUES: A PERFORMANCE EVALUATION^{1}
Authors B. N. P. AGARWAL and CH. SIVAJIAbstractThe performances of least‐squares orthogonal polynomial and relaxation techniques in the separation of regional and residual anomalies have been evaluated with a view to minimizing personal biasing. The advantage of orthogonal over nonorthogonal polynomials is their ability to estimate an optimum order of polynomial to represent the predominant regional trend in the data using an approximate 2D difference table, the Z‐matrix. The correlation coefficients between residuals of two consecutive orders also give the same result. In the relaxation technique, a linear trend is assumed within each cell of the mesh of a square grid. A set of such linear segments can approximate any complicated regional trend. The performances of these two techniques have been evaluated using simulated gravity anomalies produced by 2D and 3D complex regional structures superimposed on residual fields due to cylinders and prismatic bodies, as well as three field examples taken from the published literature. The analyses have revealed that the relaxation technique produces excellent results when an optimum polynomial order rather than an arbitrary fixed one is used for computing the boundary conditions along the periphery of the map. Analyses have revealed that such boundary conditions provide minimum distortion near the two ends of the profile.

ACOUSTIC MODELLING ON A GRID OF VERTICALLY VARYING SPACING^{1}
Authors CORD JASTRAM and ALFRED BEHLEThe Fourier spectral method and high‐order differencing have both been shown to be very accurate in computing spatial derivatives of the acoustic wave equation, requiring only two and three gridpoints per shortest wavelength respectively. In some cases, however, there is a lack of flexibility as both methods use a uniform grid. If these methods are applied to structures with high vertical velocity contrasts, very often most of the model is oversampled. If a complicated interface has to be covered by a fine grid for exact representation, both methods become less attractive as the homogeneous regions are sampled more finely than necessary.
In order avoid this limitation we present a differencing scheme in which the grid spacings can be extended or reduced by any integer factor at a given depth. This scheme adds more flexibility and efficiency to the acoustic modelling as the grid spacings can be changed according to the material properties and the model geometry. The time integration is carried out by the rapid expansion method. The spatial derivatives are computed using either the Fourier method or a high‐order finite‐difference operator in the x‐direction and a modified high‐order finite‐difference operator in the z‐direction. This combination leads to a very accurate and efficient modelling scheme. The only additional computation required is the interpolation of the pressure in a strip of the computational mesh where the grid spacing changes.

INVERSION OF GRAVITY AND MAGNETIC DATA FOR THE LOWER SURFACE OF A 2.5 DIMENSIONAL SEDIMENTARY BASIN^{1}
Authors KEVIN L. MICKUS and WAYNE J. PEEPLESAbstractGravity and magnetic data have been inverted to obtain the continuous lower surface of a 2.5 dimensional sedimentary basin. The non‐linear problem is linearized and a solution is calculated through a recursive process until the predicted data matches the observed data. An average model is then calculated and a resolution analysis shows which features are uniquely determined. The results of individual inversion indicate that a final solution is initial model dependent but the average models are independent of the initial model except at the margins. The average model for the magnetic solutions have uniformly smaller spreads than the gravity solutions.
The algorithms were applied to data from the Sanford Basin in North Carolina. The results indicate that the basin is asymmetrical in shape with a maximum depth of 3.2 km. Comparing these results with those obtained from a generalized linear inverse (GLI) algorithm indicate that the higher‐frequency features determined from the GLI algorithm are not resolved.

SHEAR‐WAVE VELOCITY ESTIMATION IN POROUS ROCKS: THEORETICAL FORMULATION, PRELIMINARY VERIFICATION AND APPLICATIONS^{1}
Authors M. L. GREENBERG and J. P. CASTAGNAAbstractShear‐wave velocity logs are useful for various seismic interpretation applications, including bright spot analyses, amplitude‐versus‐offset analyses and multicomponent seismic interpretations. Measured shear‐wave velocity logs are, however, often unavailable.
We developed a general method to predict shear‐wave velocity in porous rocks. If reliable compressional‐wave velocity, lithology, porosity and water saturation data are available, the precision and accuracy of shear‐wave velocity prediction are 9% and 3%, respectively. The success of our method depends on: (1) robust relationships between compressional‐ and shear‐wave velocities for water‐saturated, pure, porous lithologies; (2) nearly linear mixing laws for solid rock constituents; (3) first‐order applicability of the Biot–Gassmann theory to real rocks.
We verified these concepts with laboratory measurements and full waveform sonic logs. Shear‐wave velocities estimated by our method can improve formation evaluation. Our method has been successfully tested with data from several locations.

SEISMIC PHASE UNWRAPPING: METHODS, RESULTS, PROBLEMS^{1}
More LessAbstractSix known methods of seismic phase unwrapping (or phase restoration) are compared. All the methods tested unwrap the phase satisfactorily if the initial function is a simple theoretical wavelet. None of the methods restore the phase of a synthetic trace exactly.
An initial validity test of the phase‐unwrapping method is that the sum of the restored wavelet phase spectrum and the restored pulse‐trace phase spectrum (assuming the convolutional model for the seismic trace) must be equal to the restored phase spectrum of the synthetic trace. Results show that none of the tested methods satisfy this test. Quantitative estimation of the phase‐unwrapping accuracy by correlation analysis of the phase deconvolution results separated these methods, according to their efficiency, into three groups. The first group consists of methods using a priori wavelet information. These methods make the wavelet phase estimation more effective than the minimum‐phase approach, if the wavelet is non‐minimum‐phase. The second group consists of methods using the phase increment Δø(Δω) between two adjacent frequencies. These methods help to decrease the time shift of the initial synthetic trace relative to the model of the medium. At the same time they degrade the trace correlation with the medium model. The third group consists of methods using an integration of the phase derivative. These methods do not lead to any improvement of the initial seismic trace.
The main problem in the phase unwrapping of a seismic trace is the random character of the pulse trace. For this reason methods based on an analysis of the value of Δø(Δω) only, or using an adaptive approach (i.e. as Δω decreases) are not effective. In addition, methods based on integration of the phase derivative are unreliable, due to errors in numerical integration and differentiation.

NATURAL SMOOTHNESS CONSTRAINTS IN CROSS‐HOLE SEISMIC TOMOGRAPHY^{1}
Authors M. PILKINGTON and J. P. TODOESCHUCKAbstractTomographic inversion problems are ill‐posed and therefore solutions must be either damped or regularized to produce results that are geologically reasonable. We introduce a priori information in terms of parameter covariances to constrain the solution. We use slowness logs to determine the appropriate parameter covariances for the inversion of traveltime data collected for a cross‐hole geometry. We find that the logs exhibit power spectra proportional to a power α of the frequency. The value of α controls the smoothness of the inversion solution. For α < 0, the solution smoothness is maximized. Thus, knowing the correct value of α, which can be found from well logs, we can specify the appropriate amount of smoothing, rather than using some arbitrary level.
Inversions were carried out on synthetic data for the case of α= 0 and α=−1. The use of α= 0 implies uncorrelated model parameters and is equivalent to standard damped least‐squares methods. We find that solutions for α= 0 show greater complexity than for α=−1 but this level of resolution can be illusory. An example from the Midale field, Saskatchewan, Canada, is inverted using both α= 0 and α=−2, given by the observed parameter covariances. The solution for α= 0 exhibits spurious detail which must be smoothed away. For α=−2, the solution smoothness is maximized and we recover only that structure which is required to fit the data.

NON‐LINEAR AVO INVERSION FOR A STACK OF ANELASTIC LAYERS^{1}
Authors TERJE DAHL and BJØRN URSINAbstractParameters in a stack of homogeneous anelastic layers are estimated from seismic data, using the amplitude versus offset (AVO) variations and the travel‐times. The unknown parameters in each layer are the layer thickness, the P‐wave velocity, the S‐wave velocity, the density and the quality factor.
Dynamic ray tracing is used to solve the forward problem. Multiple reflections are included, but wave‐mode conversions are not considered. The S‐wave velocities are estimated from the PP reflection and transmission coefficients. The inverse problem is solved using a stabilized least‐squares procedure. The Gauss‐Newton approximation to the Hessian matrix is used, and the derivatives of the dynamic ray‐tracing equation are calculated analytically for each iteration.
A conventional velocity analysis, the common mid‐point (CMP) stack and a set of CMP gathers are used to identify the number of layers and to establish initial estimates for the P‐wave velocities and the layer thicknesses. The inversion is carried out globally for all parameters simultaneously or by a stepwise approach where a smaller number of parameters is considered in each step.
We discuss several practical problems related to inversion of real data. The performance of the algorithm is tested on one synthetic and two real data sets. For the real data inversion, we explained up to 90% of the energy in the data. However, the reliability of the parameter estimates must at this stage be considered as uncertain.
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