Traveltime and conversion‐point computations and parameter estimation in layered, anisotropic media by τ‐ptransform

Abstract

Anisotropy influences many aspects of seismic wave propagation and, therefore, has implications for conventional processing schemes. It also holds information about the nature of the medium. To estimate anisotropy, we need both forward modeling and inversion tools. Forward modeling in anisotropic media is generally done by ray tracing. We present a new and fast method using the τ‐p transform to calculate exact reflection‐moveout curves in stratified, laterally homogeneous, anisotropic media for all pure‐mode and converted phases which requires no conventional ray tracing. Moreover, we obtain the common conversion points for both P‐SV and P‐SH converted waves. Results are exact for arbitrary strength of anisotropy in both HTI and VTI media (transverse isotropy with a horizontal or vertical symmetry axis, respectively). Since inversion for anisotropic parameters is a highly nonunique problem, we also develop expressions describing the phase velocities that require only a reduced number of parameters for both types of anisotropy. Nevertheless, resulting predictions for traveltimes and conversion points are generally more accurate than those obtained using the conventional Taylor‐series expansions. In addition, the reduced‐parameter expressions are also able to handle kinks or cusps in the SV traveltime curves for either VTI or HTI symmetry.

Azimuthal τ-panalysis in anisotropic media

“…In most of these processes, a truncated rms type equation in traveltime offset ( x – t ) domain is used to perform normal moveout (NMO) correction and parameter estimation. This method has several drawbacks and has met with little success (Van der Baan & Kendall, 2002, 2003, Sen & Mukherjee, 2003, Taner et al , 2005). Problem starts with the fact of non‐availability of exact NMO correction equation for the TI medium in the offset‐time domain.…”

“…One approach to avoid these errors introduced by truncated rms type equation is conversion of x–t domain data to delay‐time and slowness (τ– p ) domain. Approximation‐free moveout correction equations can be derived in τ– p domain for both VTI and HTI media (Van der Baan & Kendall, 2002, 2003). Though the equations are exact, they are difficult to use to estimate all the anisotropic parameters for both the media because of the presence of inherent non‐uniqueness in them.…”

“…Though the equations are exact, they are difficult to use to estimate all the anisotropic parameters for both the media because of the presence of inherent non‐uniqueness in them. To overcome this problem, a reduced parameter approximate expression is used to estimate anisotropic parameters and for NMO correction (Van der Baan & Kendall, 2003). Van der Baan & Kendall (2003) develop two different reduce parameter equations for VTI and HTI medium considering weak anisotropy.…”

“…To overcome this problem, a reduced parameter approximate expression is used to estimate anisotropic parameters and for NMO correction (Van der Baan & Kendall, 2003). Van der Baan & Kendall (2003) develop two different reduce parameter equations for VTI and HTI medium considering weak anisotropy. In their equations, the number of reduced parameters for the VTI medium is two, whereas the number of reduced parameters is four for the HTI medium.…”

“…This equation is a generalization of the VTI τ– p NMO equation of Sen & Mukherjee (2003) to HTI media. Since seismic response over a HTI medium depends on another parameter ϕ, it is generally difficult to construct a reduced parameter expression for such a medium (Van der Baan & Kendall, 2003); however, we successfully reduce the number of parameters to two, and thus, reduce the non‐uniqueness. We check the performance of our expression for two single layer cases, one for weak anisotropy and another for standard anisotropy with three different azimuths, and report its satisfactory performance over the previous estimation.…”