Seismic Reflection Moveout for Azimuthally Anisotropic Media

AbdulFattah Al-Dajani

 

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy (2002)

Abstract

Anisotropy can exist in the subsurface as an intrinsic property, or as an induced property, or as a combination of both. Induced anisotropy can result due to preferred orientation of grains, thin layering, and/or the presence of fractures. In this thesis, we study the reflection moveout for Compressional and Shear waves in horizontally stratified azimuthally anisotropic media.

Reflection moveout in common-midpoint (CMP) gathers is generally treated by an azimuthally-isotropic hyperbolic equation. Recently, special treatment for the reflection moveout in azimuthally anisotropic media has been introduced due to the fact that the moveout in such cases is azimuthally dependent. Here, the normal moveout (NMO) equation parameterized by the exact normal-moveout (NMO) velocity is studied. We verify numerically (through synthetic data) the accuracy of the exact (i.e., analytic) azimuthal description of the NMO velocity. We show that the azimuthal variation of the NMO velocity has, in general, a relatively simple elliptical form. We also show that the NMO equation is sufficiently accurate for $P$- and Shear-wave propagation on conventional spreadlengths (e.g., lenghts close to the reflector depth).

The influence of anisotropy causes the deviation of the moveout curve from a hyperbola, even in a homogeneous anisotropic layer with a horizontal interface. Hence, reflection moveout in azimuthally anisotropic media is not only azimuthally dependent but it is also nonhyperbolic. To account for the nonhyperbolic moveout, we have derived an exact expression for the azimuthally dependent quartic coefficient of the Taylor series expansion for the two-way traveltimes [$t^2(x^2)$] that is valid for any pure mode of wave propagation. As a result, we introduce an analytic representation for the quartic coefficient for pure mode reflection in anisotropic medium with an arbitrary strength of anisotropy. In addition, we present an analytic expression for large-offset, nonhyperbolic reflection moveout (NHMO).

Special attention is given in this study toward $P$-wave propagation in orthorhombic (ORT) and horizontal tranverse isotropic (HTI) medium with horizontal interfaces. The presence of azimuthal anisotropy causes shear waves to split into fast and slow shear waves. As a result, the quartic coefficient and the reflection moveout for shear wave propagation are briefly addressed in this thesis.

The quartic coefficient has a relatively simple form, especially for shear waves. The quartic moveout coefficient and the normal-moveout velocity are substituted into the nonhyperbolic moveout equation, which is originally designed for vertical transverse isotropic (VTI) media. Synthetic examples show that this equation accurately describes the azimuthally dependent $P$-wave reflection traveltimes, even for spreadlengths (offsets) twice as large as the reflector depth. The reflection moveout for shear-wave propagation (i.e., fast and slow), on the other hand, is purely hyperbolic in the direction normal to the polarization. In addition, the nonhyperbolic portion of the moveout for shear-wave propagation reaches its maximum along the polarization direction, and it decreases rapidly away from the direction of polarization. Hence, the anisotropy-induced nonhyperbolic reflection moveout for shear-wave propagation is only significant in the vicinity of the polarization directions (e.g., $\pm 30^\circ$) and for large offset-to-depth ratios.

In multilayered azimuthally anisotropic media, the NMO velocity and the quartic moveout coefficient can be calculated with good accuracy using Dix-type averaging (e.g., the known averaging equations for VTI media). The interval NMO velocities and the interval quartic coefficients, however, are azimuthally dependent. This allows us to extend the nonhyperbolic moveout (NHMO) equation, originally designed for VTI media, to more general horizontally stratified azimuthally anisotropic media. Numerical examples, from reflection moveout in orthorhombic and HTI media, show that this NHMO equation accurately describes the azimuthally dependent $P$-wave reflection traveltimes, even on spreadlengths twice as large as the reflector depth.

In 3-D (or, 2-D) land seismic data acquisitions (also in water-bottom cable surveys), we have relatively full control on offset and azimuthal coverage. In conventional marine surveys, however, the azimuthal coverage is quite limited. Except in limited cases, current seismic data acquisition surveys do not utilize the azimuthal variation of the reflection moveout and the NMO velocity to obtain better medium parameter-estimation or imaging. Here, we address the inverse problem of using the azimuthal dependence of the NMO velocity in azimuthally anisotropic media. The error and sensitivity analysis provide insight into the inverse problem and how it relates to seismic data acquisition. We explore the limitations involved in using the azimuthal variation in the NMO velocity to invert for the medium parameters. Moreover, we address issues such as the effect of sectorization in 3-D seismic data surveys on such inversion. Finally, we study how the resolution of the inverse problem is influenced by the number of NMO velocities that enter the inversion process.

Parameter estimation from the elliptical variations in the normal-moveout (NMO) velocity in azimuthally anisotropic media is sensitive to the angular separation between the survey lines in 2D, or equivalently source-to-receiver azimuth in 3D, and to the set of azimuths used in the inversion procedure. The accuracy in estimating the orientation of NMO ellipse is also sensitive to the strength of anisotropy. On the other hand, the accuracy in estimating the semi-axes of the NMO-velocity ellipse is about the same for any strength of anisotropy.

To invert for the NMO ellipse parameters at least three NMO-velocity measurements along distinct azimuth directions are needed. In order to maximize the accuracy and stability in parameter estimation, it is best to have the azimuths for the three source-to-receiver directions $60^\circ$ apart. Having more than three distinct source-to-receiver azimuths (e.g., full azimuthal coverage) enhances the quality of the estimates.

The azimuthal variation of the NMO velocity in azimuthally anisotropic media can be utilized to invert for some of the medium parameters which can be useful in characterizing the zone of interest. In HTI media, for example, and using the $P$-wave reflection moveout, we can estimate three key parameters: the vertical velocity $V_{\rm Pvert}$, anisotropy parameter $\delta^{\rm (V)}$, and the azimuth $\alpha$ of the symmetry-axis. Similarly, in orthorhombic media, inverting for the semi-axes of the NMO-ellipse by using $P$--wave normal-moveout (NMO) velocity, allows the computation of the difference in the anisotropic parameters $\delta^{\rm (1)}$ and $\delta^{\rm (2)}$. With the nonhyperbolic portion of the reflection moveout, we can obtain the $\epsilon$ parameters (and $\eta$) which can be used to estimate or at least constrain the fracture density.

The NMO velocity exhibits similar azimuthal variation for different azimuthally anisotropic models (e.g., HTI, orthorhombic, and monoclinic). Hence, it is not possible to distinguish between the different azimuthal anisotropic models solely on the behavior of the NMO velocity. Additional information such as well logs and cores can be helpful to classify the medium. Unlike the NMO velocity, the NHMO coefficient manifests different azimuthal behavior in different anisotropic models. This distinct azimuthal variations can lead to distinguishing different types of anisotropic media by studying the nonhyperbolic portion of the reflection moveout.

To maximize quality in the inversion process, it is recommended that at the design stage of seismic data acquisition to ensure having small sector sizes ($\leq 10^\circ$) with adequate fold and offset distribution.

Using three NMO-velocity measurements, $60^\circ$ apart, an azimuthally anisotropic layer overlain by an azimuthally isotropic overburden (as might happen for fractured reservoirs) should have a relative thickness (in time) to the total thickness of at least equal to the ratio of the error in the NMO (stacking) velocity to the interval anisotropy strength of the fractured layer. Coverage along more than three azimuths, however, improves this limitation, which is imposed by Dix differentiation, by at most 50$\%$ depending on the number of observations (NMO Velocities) that enter the inversion procedure.

An application of this work to field data is performed using ground penetrating radar (GPR) surveys in azimuthally anisotropic media. Ground penetrating radar (GPR) signatures, such as reflection moveout, are sensitive to the presence of azimuthal anisotropy. Ignoring the effect of anisotropy on the normal-moveout (NMO) velocity, that characterizes the reflection moveout, causes erroneous images and time-to-depth conversions. Similar to surface seismology in azimuthally anisotropic media, the GPR normal moveout (NMO) velocity, along different orientations of common-midpoint (CMP) gathers, varies elliptically with azimuth.

Here, we introduce an exact representation for the GPR NMO-velocity in azimuthally anisotropic media (e.g., HTI, and orthorhombic symmetries). The NMO-velocity representation is valid for arbitrary strength of anisotropy. A field data example is presented in which three GPR CMP gathers are acquired along three different azimuths, $60^\circ$ apart, over a fractured medium. The data are obtained for the transverse mode of electromagnetic wave propagation in which the polarization is normal to the incidence plane of the CMP gathers. Our data analysis demonstrates the azimuthal variation of the GPR NMO velocity, which is utilized to invert for the local orientation of the fracture system in the near surface. As a result, this study has important 3D applications in imaging near surface geologic structures and in the determination of tectonic-induced fractures in the near surface.

Finally, this thesis provides analytic insight into the behavior of the reflection moveout, specially the nonhyperbolic moveout and the NMO velocity, in azimuthally anisotropic media. In addition, this work has important 3D applications in imaging, modeling, and inversion of reflection moveout in azimuthally anisotropic media.

Note: This abstract is written in LaTex.
\pm means plus or minus
\circ means degrees
\delta is the Greek letter delta
\epsilon is the Greek letter epsilon
\eta is the Greek letter eta
^ means power
$ needs to be erased after you make the changes.