Norman F. Uren

Analytical solution for the electric potential due to a point source in an arbitrarily anisotropic half-space

A very large class of important theory and applications in geophysics requires analytical solutions for the determination of the electric potential due to a point source in an arbitrarily anisotropic half-space. In this paper, a very clear and simple solution to the half-space problem has been developed from consideration of an arbitrarily anisotropic whole space. For the first time, the method of images is used to generate the solution for an arbitrarily anisotropic three dimensional half-space. Based on traditional extreme value theory the image source point has been determined and calculated for the half-space case.

Recovery of elastic parameters for a multi-layered transversely isotropic medium

A method for recovery of elastic anisotropic parameters of a multi-layered medium from three-component walkaway VSP data is presented. By analysing the multi-component signals recorded by any receiver in a walkaway VSP survey, the apparent transmission velocity field for the layered media above this receiver depth is inverted to yield the effective combined elastic constants of those layers. The interval elastic parameters and the velocity fields for the interval layer of interest between two receiver depths are also efficiently determined from these. The inversion technique adopted the Levenberg–Marquardt method to recover the elastic parameters from the qP-, qSV- and SH-wave transmission travel time data. For the layered media above a given receiver, the P- and S-wave velocities (α0, β0) along the symmetry axis, and the full set of Thomsens anisotropic parameters are recovered by our inversion approach. For the combined layers properties, the tilt angle (ψ) of the symmetry axis from the vertical is also inverted. The inversion software was first tested on synthetic data. It was then applied to actual field data and found to give plausible results. Anisotropic parameters recovered in this way may be used to improve data processing and imaging in the presence of transverse isotropy.

Poisson's ratio in transversely isotropic media and its effects on amplitude response: An investigation through physical modeling experiments

Amplitude interpretation is a seismic technique that has proven to be an effective tool in oil reservoir delineation, particularly in development areas. Amplitude variation with offset is a commonly used method in amplitude interpretation. It can distinguish a hydrocarbon bearing sand reservoir from a water bearing reservoir. The well-known simplification by Shuey (1985) of Zoeppritz’s equation is widely used in interpretation. It assumes that the media involved are isotropic. Blangy (1994) provided an expression for reflection coefficients in the presence of anisotropy. This is expressed in terms of the phase angle which does not correspond to the conventional angle of incidence obtained directly from ray path geometry. Poisson’s ratio also becomes directionally dependent in anisotropic media.

Analytical solution for direct current electric potential in an anisotropic half‐space with a vertical contact

The analytical solution for the electrical potential due to point source in an arbitrarily anisotropic . . half-space with a vertical contact is obtained, using a new image source method. The images of the point source consist of a reflection image and a transmission image arising from the vertical contact interface. The positions of the images depend on the resistivity tensors of the earth media and the point source position. Within this approach the solution provides a complete description of the potential in an arbitrarily anisotropic half-space with a vertical contact.

A method for determining in 3‐D, the true dip and azimuth of a reflecting horizon

A 3-D field method has been devised in which the true dip of a reflecting horizon and its azimuth can be measured within a survey area. The method requires a suitable horizon to which the velocity is known, and assumes that the reflecting horizon has constant dip throughout the area. Two orthogonal swath surveys are performed, and data are stacked using variable dip and strike values. A 3-D NM0 equation corrects data to a zero-offset reference point. An optimum value of dip and strike provides the best reflection line-up at the reference point, producing the value of true dip and its azimuth for that horizon. A physical model was used to demonstrate the method, which was able to predict dips and their azimuths to within half a degree.

3D Multiple Moveout Wavefield Transformation For Pre-conditioning Data For the Removal of Water Bottom Multiples

Summary Wavefield transformations have been used for many years to precondition seismic data to enhance useful features of the seismic record or to suppress unwanted signal or noise. With the recording of three dimensional seismic surveys it is appropriate to extend the transformations to utilise the 3D data. A 2D Multiple Moveout transformation has been applied previously to the removal of long period multiple reflections with significant success. This removal techniques relies on periodicity of the position of the multiples on the transformed seismic record and the stationarity of the time sequence. This work extends the MMO technique to three dimensions and applies it to simple synthetic data.

An experimental investigation of the small-offset P-wave NMO equation in transversely isotropic media.

Summary Numerical and physical modelling studies have been used to investigate the accuracy and limitations of the small-offset P-wave NMO velocity estimation in transversely isotropic media. An expression for the P-wave small-offset NMO velocity for a single horizontal reflector in a vertically transversely isotropic layer has been previously obtained: V nmo =+ αδ 0 12 where αο and δ respectively represent the vertical P-wave velocity and the P-wave critical anisotropy at oblique angles. The above equation is presented as being exact and valid for any degree of anisotropy and has been used in estimating the small-off set NMO velocity in the offshore Zaire seismic data acquired by Unocal in West Africa. The accuracy of this expression is subject t o practical testing in this paper. A one-layer horizontal model was used to carry out the numerical modelling studies. A computer program which calculates Pwave NMO velocities in transversely isotropic media was used. This program uses the exact equations for the phase and ray velocities in generating velocity functions to be used in ray tracing; it is known to give accurate results. Using the publishe d elastic parameters for anisotropic rocks, the small-offset NMO velocities are computed and results are compared with values obtained using the above expression. The degree of anisotropy varied from very weak to strong and contrasting velocity function s (positive and negative δ) were used in numerical modelling simulations. Phenolite materials with contrasting velocity functions were used to simulate transversely isotropic media with vertical axis of symmetry in physical modelling tests. Experimental reflection data were collected and velocity analysis was conducted for small-offset tr aces, and the NMO velocity determined by the maximum stack response. The elastic parameters αο and δ of the Phenolite materials were recovered by an anisotropic inversion technique. These parameters were used to estimate the small-offset P-wave NMO velocity in Phenolite with the above expression and also with our computer program. The NMO velocities obtained from these two methods were compared to that obtained from velocity analysis of the experimental seismic data. These comparisons enabled the validity of the above expressio n to yield accurate P-wave NMO velocity, in the limit of small offset, to be tested. Numerical and physical modelling results indicate that the accuracy of the small-offset P-wave NMO equation depends on the nature and degree of the anisotropy encountered in a given area, and is not valid for all degrees of P-wave anisotropy. The sma lloffset NMO (stacking) velocity from horizontal reflectors is known to influence the effects of vertical transverse isotropy on normal moveout (NMO), dip moveout (DMO) corrections and time migration. Determination of the correct small-offset P-wave NMO velocity is essential for obtaining accurate seismic images, and subsequently correct interpretations.

The dependence of lateral resolution on the orientation of symmetry axis and elastic parameter in transversely isotropic media

Similar models having positive and negative values and hence, contrasting wavefront curvatures were also used in conducting the experiments. Cylindrical holes of various sizes were drilled into the bottoms of the Phenolite models. The hole sizes ranged from 0.83 to 1.834 and from 0.86 to 2.054 times the expected Fresnel-zone diameters in the horizontally layered and vertically fractured Phenolites respectively.

Lateral Resolution In Anisotropic Seismic Reflection: Physical Modeling Approach

Cylindrical holes of progressively increasing sizes were drilled into the bottom centres of both models. These holes served as reflectors which ranged from 0.19 to 1.25 and from 0.35 to 2.10 times the expected Fresnel-zone diameter in phenolite and plexiglass respectively. Care was taken to ensure that all the drilled holes were flat at the top. The degree of resolution is determined by comparing the apparent extent of reflection information across the drilled hole with the true spatial dimension of the hole.