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Elastic Wave Propagation in Anisotropic Media: Source Theory, Traveltime Computations and Migration
Arcangelo G. Sena
Submitted
to the Department of Earth, Atmospheric, and Planetary Sciences on May 14,
1992 in partial fulfillment of the requirements for the degree of Doctor of
Philosophy
Abstract
In this thesis, we develop several theoretical methods for wave propagation
in anisotropic media. The main objectives are to established techniques to
interpret multicomponent seismic data in the presence of anisotropy. In this
context, we provide new nonconventional processing algorithms that improve
the quality of the seismic data.
First, we extend seismic source theory to general anisotropic media for the
numerical evaluation of spectral amplitudes for point sources in an anisotropic
crust. We obtain explicit representation of the elastodynamic Green’s
tensor in general homogenous anisotropic media as a sum of three integrals
over the corresponding three slowness surfaces. The multidimensional stationary
phase principle is then applied to derive an asymptotic approximation at the
far field.
The availability of the Green’s tensor in analytical form enables one
to obtain numerical solutions for sources in anisotropic media. First, we
show that the radiation field of an explosion has the following new features:
1) quasi-transverse waves are created with four- and eight-lobe patterns;
2) quasi-longitudinal waves are generated for the collatitudinal displacement
with four-lobe patterns; 3) the energy ratio SV/P may reach the value of 20
for more than 50% of the azimuths in crustal structures such as tuff and shales;
and 4) radiation patterns for vertical shear waves are created that are indistinguishable
from corresponding waves produced by earthquake faults. For the special case
of azimuthally isotropic media, we present an alternative representation of
the Green’s tensor and the displacement fields in the form of an exact
Hankel transform over the horizontal wave number variable. The total field
is specified in terms of two potentials: an SH potential and a mixed quasi-transverse/quasi-
longitudinal potential, both which assume the role of two scalar Green’s
functions. A Haskell-type matrix algorithm for a multilayered azimuthally
isotropic half-space can then be established, enabling us to calculate body
w aves and surface waves in real-earth crustal models.
Next, we derive analytical expressions for the traveltime-offset curves for
multilayered, weakly azimuthally isotropic and anisotropic media in terms
of the elastic properties of each layer. This method is based upon an approximate
skewed hyperbolic moveout formula involving three measured bulk velocities
for each reflector. The primary benefits of this technique are: 1) it allows
for fast traveltime computation; 2) it makes possible an extremely rapid estimation
of the interval elastic parameters; and 3) it provides physical insight into
wave propagation in anisotropic media.
Based on forward modeling discussed above, we develop a traveltime inversion
algorithm that estimates the five elastic constants together with the orientation
of the axis of symmetry for each layer. In the isotropic limit, this algorithm
reduces to the conventional one used for determining the interval velocities
from stacking velocity measurements. The inversion technique is applied to
surface seismic measurements and VSP field surveys. In both cases, the method
provides very good estimates of the six parameters. The results also show
that the orientation of the horizontal axis of symmetry, for a given azimuthally
anisotropic layer at depth, can be obtained using only quasi-P wave information.
This technique can be easily incorporated into conventional velocity analysis
algorithms.
Finally, we combine the theoretical ray amplitudes together with the traveltime
equations to generate a suitable Green’s tensor to perform Kirchhoff
migration in anisotropic media. This imaging scheme is applied to the case
of azimuthal isotropy. An anisotropic velocity analysis scheme is also established
in order to generate an appropriate velocity (elastic constant) model for
migration in azimuthal isotropic media for non converted and converted qP-qSV
waves. Synthetic examples showing migration of qP-qP and converted qP-qSV
sections are presented. In both cases, the method provides accurate images
of the subsurface. In quasi-compressional qP=qP case, we show that, even with
a weak to moderate percentage of anisotropy, an isotropic migration algorithm
cannot handle the anisotropy properly. In a real data example from South Texas
using converted qP-qSV waves, our anisotropic migration scheme improves the
delineation of a fault plane and the lateral continuity of the flat reflectors.
Furthermore, the anisotropic Kirchhoff migration algorithm enables us to manipulate
multicomponent data with an arbitrary geometry of sources and receivers. In
addition, the proper handling of anisotropy, together with the combination
of converted and nonconverted waves, provides more geological information
about the subsurface and better delineation of potential hydrocarbon reservoirs.
