Norm Uren

Normal moveout in anisotropic media

Elliptical anisotropy may be considered a more general case of isotropy, in which the medium has been stretched in one direction (Helbig, 1983; Dellinger and Muir, 1988; Verwest, 1989). As a consequence, certain principles and relationships hold for both isotropy, where wavefronts are circular, and elliptical anisotropy, where the wavefronts are circles which have been stretched in one direction (i.e., ellipses); while they may not hold for the general case of anisotropy, where wavefronts have more complex shapes. Examples are the dip moveout (DMO) relation (Uren et al., 1990) and the method of images (Dellinger and Muir, 1988).

The migrator's equation for anisotropic media

The migrator’s equation, which gives the relationship between real and apparent dips on a reflector in zero‐offset reflection seismic sections, may be readily implemented in one step with a frequency‐domain migration algorithm for homogeneous media. Huygens’ principle is used to derive a similar relationship for anisotropic media where velocities are directionally dependent. The anisotropic form of the migrator’s equation is applicable to both elliptically and nonelliptically anisotropic media. Transversely isotropic media are used to demonstrate the performance of an f-k implementation of the migrator’s equation for anisotropic media. In such a medium SH-waves are elliptically anisotropic, while P-waves are nonelliptically anisotropic. Numerical model data and physical model data demonstrate the performance of the algorithm, in each case recovering the original structure. Isotropic and anisotropic migration of anisotropic physical model data are compared experimentally, where the anisotropic velocity fun...

Inversion technique for recovering the elastic constants of transversely isotropic materials

A least‐squares iterative inversion technique has been developed for the determination of the elastic parameter δ* of any transversely isotropic modeling material in the laboratory. For most applications in petroleum geophysics, the elastic parameter δ* is very important and is the crucial anisotropic parameter for near‐vertical P‐wave propagation. Despite the potential importance of δ* in seismic exploration and for resolution in an anisotropic medium, the conventional procedures adopted in estimating its value unfortunately are faced with many ambiguities and the reliability of its measurement is doubtful prior to the development of this technique. The anisotropic inverse modeling technique finds the best fitting solution. To optimize the accuracy of the results presented in this paper, analytical rather than numerical differentiations were implemented and the modeling procedures allow for controlled iterative adjustments in resolving the parameter δ*. Inversion of the first‐arrival traveltimes obtained...

Dip moveout in anisotropic media

When a common‐midpoint gather is collected above a dipping reflector, the point at which reflection occurs moves updip as the source‐receiver offset increases. Stacking velocity is constant, but it is a function of dip. Hence a stacked trace is not equivalent to a zero‐offset recorded trace. Dip moveout (DMO) is a processing step which converts traces to the equivalent of true zero‐offset records, making migration after stack (MAS) equivalent to migration before stack (MBS). The theory of velocity‐independent Gardner DMO is extended to triaxially elliptically anisotropic media in this paper. It is shown that the transformation is exact for homogeneous elliptically anisotropic media and that, after DMO, the stacking velocity is the horizontal component of the elliptically anisotropic velocity function. By taking three distinct seismic lines, three of the six constants of triaxial elliptical anisotropy may be determined. The remaining three cannot be obtained from surface seismic measurements. A simple nume...

Kirchhoff diffraction mapping in media with large velocity contrasts

Finite‐difference methods for calculating traveltimes are superior to ray‐tracing methods in inhomogeneous media. However, when these techniques are applied to Kirchhoff migration, a severe problem occurs in the presence of large velocity contrasts. If finite‐difference traveltime methods are used to calculate first arrivals, an incomplete image is created because substantial subsurface information is often carried by direct body waves. We propose a solution to this problem by developing a new method of calculating later arrival times and applying both first and later arrival times to a Kirchhoff diffraction mapping algorithm. A comparison shows that the implementation of both first arrivals and later arrivals in Kirchhoff migration can substantially improve the images in media with large velocity contrasts.

Fresnel zones and spatial resolution for P- and SH-waves in transversely isotropic media

In an elastically anisotropic medium where the seismic wave velocity is a function of direction, the wavefront shape is nonspherical and, in most cases, assumes a nonelliptical shape. Numerical modelling techniques have been used to calculate the Fresnel‐zone diameter for compressional (P) and shear (SH) waves in transversely isotropic and isotropic media, respectively. The size of the Fresnel zone is found to be predominantly dependent on the curvatures and wavelength of the wavefront as well as the dip angle of the reflector. In addition, the anisotropic elastic parameters δ* (critical near‐vertical anisotropy), e (the P-wave anisotropy), and γ (the SH-wave anisotropy) are found to significantly affect the size of the Fresnel zone. Numerical modeling results show considerable differences between the Fresnel zones for anisotropic and isotropic velocity functions at various reflector dips. In addition, the Fresnel‐zone dimensions for anisotropic media exhibit asymmetry and considerable change with dip. By...

Anisotropic NMO corrections for long offset P‐wave data from multi‐layered isotropic and transversely isotropic media

In this paper, we tested these expressions with numerical data and real data. It can be seen that the reflected traveltime formulas for a VTI model with a horizontal reflector and for a tilted TI model with its symmetry axis perpendicular to the reflector have the same form. The results show that the travel time formulas are also suitable for multi-layered models, and that both a multi-layered isotropic and an anisotropic model may behave anisotropically. The results also show that the events can be completely flattened even for very large offset numerical data and real data.

The modelling of direct current electric potential in an arbitrarily anisotropic half-space containing a conductive 3-D body

Abstract A new approach to the computation of the D.C. potentials in an arbitrarily anisotropic half-space in which there is embedded a 3-D equipotential (perfect) conductor is presented. With the aid of a Fredholms integral equation of the first kind developed for the potential, the problem is formulated in which the unknown function is the normal component of the current density on the surfaces of the conductor and it is solved numerically by means of a method of subareas. A Greens function for the point source potential in the arbitrarily anisotropic half-space is the kernel of the integral equation, which is developed by a new method. The integrated Greens function defined for the integral calculation of the subareas can be calculated using a one dimension analytical method followed by another multi-dimensional numerical approach when the body is represented by the cube cell accumulation method, in which the cell surfaces are parallel to the Cartesian coordinate system. The effects of the arbitrary anisotropy and the conductive body on the electric potential are illustrated with the aid of several numerical examples, where the point current source is located at different positions in the half-space. Forward modelling of the nature shown in this paper is valuable as an interpretation aid. An extension of this work to the cases of typical field survey geometries is relatively straight forward.

A study of the effects of transducer size on physical modeling experiments for recovering anisotropic elastic parameters

The effect of finite size transducers on elastic parameters recovered from laboratory measurements has been studied in this paper. Ultrasonic transmission experiments in the physical modeling laboratory were used to acquire the first arrival traveltime data at different ray angles, to simulate walkaway VSP surveys in the field. The elastic parameters of the materials were then recovered by our developed inversion techniques, using velocities at different directions inverted from the first arrival traveltimes. In our laboratory experiments, “point” transducers and “large” transducers were used to carry out the transmission experiments. The finite size of the transducers introduced systematic errors in the first break picks along the survey line, and hence the determined velocity field was inaccurate. These systematic errors cannot be corrected by the inversion technique. Numerical simulations were also conducted to study the differences between experiments using large and point transducers. From our experimental measurements, accurate elastic parameter values are obtained when the ratio of the sample-thickness to transducer-diameter is greater than 36. Decreasing the size of the transducers is highly recommended when conducting laboratory experiments to recover elastic parameters. A method of offset correction is also implemented in an attempt to decrease the measurement errors.

P-wave Ray Velocity Functions, Semblance Analysis and NMO corrections in Transversely Isotropic Media

Summary Sedimentary rocks encountered in exploration can be regarded as transversely isotropic (TI) media having only 5 independent elastic constants and a single symmetrical axis. Transversely isotropic phase velocity expressions exist for P-, SH- and SV-waves. Approximate explicit analytical Pwave ray velocity functions in transversely isotropic (TI) media have now been developed (Zhang and Uren, 2001 a, b). These ray velocity functions include several dimensionless constants. Unfortunately, no explicit exact analytical ray velocity functions have been formulated for P- or SV-waves propagating in TI media. In exploration seismology, explicit ray velocity functions are needed for efficient migration, velocity analysis and other seismic processing procedures. The formulae for calculating the travel times between any two points in homogeneous transversely isotropic (TI) media and for the reflection travel times from a single horizontal reflector overlain by transversely isotropic media with a vertical symmetrical axis (VTI) have been derived (Zhang and Uren, 2001 a, b). New traveltime expressions for HTI media are presented here. A new technique for velocity semblance analysis is presented to estimate the dimensionless parameters in the new NMO equation, using surface seismic survey data. The results of NMO corrections with these parameters flatten events over a large range of offsets.