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Journal ArticleDOI

A complex point-source solution of the acoustic eikonal equation for Gaussian beams in transversely isotropic media

01 May 2020-Geophysics (Society of Exploration Geophysicists)-Vol. 85, Iss: 3
TL;DR: The complex traveltime solutions of the complex eikonal equation are the basis of inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography as discussed by the authors.
Abstract: The complex traveltime solutions of the complex eikonal equation are the basis of inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography. We have d...

Summary (1 min read)

Introduction

  • Inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography.
  • The analytical formulas, as outlined, are efficient methods for the computation of complex traveltimes from the complex eikonal equation.
  • Effectively, solutions for Hamiltonian formulations require solving a system of differential equations numerically.

COMPLEX EIKONAL EQUATIONS FOR INHOMOGENEOUS

  • The authors derive TTI complex eikonal equations in TTI media.
  • As pointed out earlier the medium is considered acoustic and the authors only consider P-wave propagation.
  • The complex traveltimes 𝜏 can be expressed in terms of the real part 𝜏R and the imaginary part 𝜏I 𝜏 = 𝜏R + 𝑖𝜏I , (2) where represents the imaginary unit.
  • The rotation matrix for the complex eikonal equation from VTI media to TTI media is given by , (5) where is the angle between the symmetry axis and the vertical-axis.
  • The components of that vector describe the projection of the symmetry axis on each plane.

COMPLEX ASYMPTOTIC SOLUTIONS USING COMPLEX

  • According to the research reported in Deschamps (1971) and in Felsen (1976, 1984), when the source of the wave is located at a complex point location, the angular spectrum of the wave is that of a plane wave whose wave vectors are close to the central ray.
  • Figures 2a and 2b show the real and imaginary parts of the complex traveltimes with the proposed method.
  • Because the region on the side of the source (the starting point of the central ray) is relatively far from the central ray, these values are relatively large.
  • Those formulas are also effective methods for benchmarking numerical solutions in such media and investigating seismic anisotropy.
  • This exact solution, and the complex analysis associated to it thus become a reference standard for the analysis and comparison of other practical and very important solutions obtained under certain approximations.

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A complex point-source solution of the acoustic eikonal equation
for Gaussian beams in transversely isotropic media
Running head: Complex point-source solutions
Xingguo Huang
1
, Hui Sun
2,3
, Zhangqing Sun
4
and Nuno Vieira da Silva
5
1
University of Bergen, Department of Earth Science, Allegaten 41, 5020 Bergen, Norway. Emails:
Xingguo.Huang@uib.no,xingguo.huang19@gmail.com.
2
Southwest Jiaotong University, Faculty of Geosciences and Environmental Engineering, Chengdu
610031, China; (Corresponding author) Email: sunhui@swjtu.edu.cn.
3
University of California, Earth and Planetary Sciences, Modeling and Imaging Laboratory, 1156 High
Street, Santa Cruz, CA 95064, USA.
4
Jilin University, College for Geo-exploration Science and Technology, 938 Ximinzhu Street,
Changchun 130026, China. Email: sun_zhangq@jlu.edu.cn.
5
Formerly Imperial College London, Department of Earth Science and Engineering, South Kensington
Campus, London SW7 2AZ, UK. Email: nuno.vdasilva@gmail.com.
ABSTRACT: The complex traveltime solutions of the complex eikonal equation are the basis of
inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography.
We present the analytic approximations for complex traveltime in transversely isotropic media with a
titled symmetry axis (TTI), which is defined by a Taylor series expansion over the anisotropy parameters.
The formulation for complex traveltime is developed using perturbation theory and the complex point
source method. The real part of the complex traveltime describes the wavefront and the imaginary part
of the complex traveltime describes the decay of amplitude of waves away from the central ray. We
derive the linearized ordinary differential equations for the coefficients of the Taylor-series expansion

2
using perturbation theory. The analytical solutions for the complex traveltimes are determined by
applying the complex point source method to the background traveltime formula and subsequently
obtaining the coefficients from the linearized ordinary differential equations. We investigate the influence
of the anisotropy parameters and of the initial width of the ray tube on the accuracy of the computed
traveltimes. The analytical formulas, as outlined, are efficient methods for the computation of complex
traveltimes from the complex eikonal equation. In addition, those formulas are also effective methods
for benchmarking approximated solutions.

3
INTRODUCTION
The complex eikonal equation plays an important role in wave propagation problems of
inhomogeneous plane waves (Červený, 2001). The inhomogeneous plane wave is a solution of the
elastodynamics equation if the traveltime is complex-valued (Krebes and Le, 1994). Studies on the
complex eikonal equation fall into two groups. In the first group, the complex eikonal equation is used
to describe the traveltimes in attenuating media. The complex eikonal equation is used for ray-based
imaging and inversion (Huang et al, 2019) for an attenuating medium (i.e. the stiffness coefficients in
the frequency-domain are complex-valued). Early studies on the complex traveltimes are based on the
complex ray tracing method (Hearn and Krebes, 1990a, 1990b; Zhu and Chun, 1994; Thomson, 1997;
Chapman et al., 1999; Kravtsov et al., 1990; Egorchenkov and Kravtsov, 2001). The computation of the
complex ray tracing is very expensive because it is implemented in a high dimensional space. The
introduction of the real ray tracing method based on perturbation theory (Vavryčuk, 2008a, 2008b, 2010,
2012; Klimeš and Klimeš, 2011) into the complex traveltimes computation enables computing complex
traveltimes numerically.
The research work carried out in the 1980s is principally focused on developing a linear system of
equations for asymptotic solutions concentrated in the vicinity of a ray utilizing a complex eikonal
(Babich and Nomofilov, 1981; Nomofilov, 1982). That early reported work shows Hamiltonian
formulations for the complex solution in some special systems of local coordinates. Although the theory
is well established, numerical implementation for Hamiltonian formulations is not known. Effectively,
solutions for Hamiltonian formulations require solving a system of differential equations numerically.
In the second group, the complex eikonal equation is used to describe the complex traveltimes of
Gaussian beams. Since Deschamps’s (1971), Wang and Deschamps(1974) and Felsen’s (1976) original
formulations of the complex phase and the complex eikonal equation method, the complex traveltimes
method has long attracted interest. Magnanini and Talenti (1999, 2003, 2006) use the Bäcklund transform
to conduct some extensive research on the complex eikonal equation. The Bäcklund transform constitutes
the basis of soliton theory (Terng and Karen, 2000). More recently, Li et al. (2011) develop a fast
marching method to solve the complex eikonal equation without the limitation of the paraxial ray
approximation. Accordingly, Huang et al. (2016b) design a local algorithm to compute the local complex
traveltimes by combining the fast marching method with the nonuniform grid-based finite difference
method. The complex traveltimes of the evanescent wave, in which their real part describes the wavefront
while their imaginary part describes the decay away from the central ray, play an important role in local
plane wave. Because of its advantages for handling caustics and multipath resulting from ray tracing in
complex media, the complex traveltimes have been widely used for Gaussian beam modelling (Červený,

4
1982; Popov, 1982; Huang et al., 2016a; Huang, 2018) and seismic imaging (Hill, 1990, 2001; Gray,
2005, 2009).
The work outlined herein sits in the second group. In this paper, we focus on solving the Gaussian
beam problem in a non-attenuating medium (i.e., the medium behaves elastically, and the stiffness
coefficients are real-valued). We use a “real-valued” eikonal equation, but we force the traveltimes to be
complex-valued, while Hearn and Krebs (1990a, 1990b), Zhu and Chun (1994), Thomson (1997),
Chapman et al. (1999), Kravtsov et al. (1999), Egorchenkov and Kravtsov (2001) and Hao and
Alkhalifah (2017a, b) use complex-valued eikonal equation without any enforcement of the traveltimes.
Previously, the condition that the gradient of the real part of the traveltimes is orthogonal to that of the
imaginary part is taken to solve the complex eikonal equation. That condition does not hold true for
traveltimes determined from the complex eikonal equation for an attenuating isotropic medium. A key
advantage of basing our discussion on Gaussian beam theory is that our expressions are readily applicable
to complex heterogeneous media which is paramount for accurate seismic imaging (Hill, 2001).
When media are anisotropic, the isotropic approximation is no longer adequate (Huang and
Greenhalgh, 2019). Areas with thin layering and fractures are particularly common in the geophysical
exploration context, therefore the effect of anisotropy needs to be considered in realistic applications of
the complex eikonal equation. This paper outlines the computation of the traveltimes for TTI media.
Červený (2001) introduces the complex ray tracing method for general anisotropic elastic media. It is
important to note that the application of such formulations is not usually feasible as the vast majority of
the recorded datasets do not contain complete information on the elements of the full anisotropic stiffness
tensor. In addition, it is unfeasible to consider the full anisotropic characterization of elastic media . It is
often verified that disregarding shear waves from the constitutive law for an anisotropic medium yields
an accurate representation of the kinematics of waves at the expense of sacrificing the accuracy of the
computed amplitudes. However, it is important to note that modeling seismic waves with the true
amplitude is generally not feasible as the exact source wavelet is unknown. In addition, the dynamic
response of an acoustic medium is different from that of an elastic medium. The acoustic approximation
has been widely used in seismic inversion and imaging (Alkhalifah, 2000, Grechka, 2004, Zhang et al.,
2011, Duveneck and Bakker, 2011, da Silva et al, 2016). The work follows the same rationale regarding
the constitutive law. The real eikonal approximation method is originally proposed as a means of
computing the real-valued traveltimes for anisotropic media (Alkhalifah, 2011a, 2011b, 2013; Stovas
and Alkhalifah, 2012; Waheed et al., 2013; Hao and Alkhalifah, 2017a, 2017b; Hao et al., 2018). In these
methods, a Taylor expansion of the traveltimes with respect to the anisotropy parameters and to the
background traveltimes for tilted elliptically isotropic media is carried out. The major advantage of these
methods is that if we know the background traveltimes and anisotropy parameters, we can obtain
traveltimes for general anisotropic media directly at a relatively low computational cost. More recently,

5
Huang et al. (2018) extend the mentioned above perturbation theory to the real and imaginary parts of
the complex traveltimes, and they solve successfully the complex eikonal equation in VTI and
orthorhombic media (Huang and Greenhalgh, 2018). Unlike the real eikonal approximation method, we
take Taylor expansions of both the real and imaginary parts of the complex traveltimes. Then, we
construct the linearized complex eikonal equations by substituting the Taylor’s expansions into the
complex eikonal equations.
According to the recent results of Huang et al. (2016b), solving the complex eikonal equation leads
to an inverse process for computing the imaginary slowness. This process is especially complicated for
anisotropic media. This is because the complex eikonal equation for anisotropic media includes several
parameters: the normal moveout (NMO) velocity, velocity along the symmetry axis, and the anellipticity
parameter, .
This paper focus on two principal aspects. The first aspect is deriving the linearized complex eikonal
equations based on perturbation theory (Alkhalifah, 2011a, 2011b, 2013; Stovas and Alkhalifah, 2012;
Waheed et al., 2013; Hao et al., 2016; Hao and Alkhalifah, 2017a, 2017b; Hao et al., 2018). A key
advantage of the linearized complex eikonal equations is providing possible method of solving the TTI
complex eikonal equation using a finite-difference method, which can be used to address the computation
of traveltimes in general heterogeneous media. Once the TTI complex eikonal equation can be solved
directly, the paraxial ray approximation can be avoided while solving the complex eikonal equation.
Following the recent methods (Huang and Greenhalgh, 2018; Huang et al., 2018), we develop a linear
partial differential equation for the TTI complex eikonal equation. However, we use a tilted elliptically
isotropic background medium for the perturbation expansion associated with the tilt angle, which allows
us to perform the perturbation expansion in η only. This reduces uncertainty in the computed complex
traveltimes.
The second aspect is obtaining the analytical formulas for homogeneous TTI media. The analytical
solutions can be used to benchmark the accuracy of alternative methods, as well as, providing insight on
the kinematics of wave propagation in TTI media. As pointed out earlier the latter has found many
relevant applications in applied geophysics, especially in seismic imaging and inversion. Even though
the theory developed in this paper applies to the general complex traveltimes for TTI media in the
complex space, our closed form (analytical) solution is for a specific Gaussian beam. This is based on
the fact that, for a complex point source (Choudhary and Felsen, 1974; Wang and Deschamps, 1974;
Felsen, 1976, 1984; Wu, 1985), the beam has an initial constant phase front and a Gaussian field profile,
and the wavefield has a Gaussian decay along the direction perpendicular to the central ray. In this case,
the contours of the real part of the traveltimes define the equiphase contours, while the contours of the
imaginary part define the phase paths. In addition, the theoretical development of the analytical formulas
is based on another assumption that the distance is limited to the region of the paraxial ray approximation.
η

Citations
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Journal ArticleDOI
TL;DR: In this paper, the eikonal-equation-based method outperforms traditional ray methods by producing more accurate traveltimes and raypaths for anisotropic tomography inversions.
Abstract: Computation of traveltimes and raypaths is important for anisotropic tomography inversions. The eikonal-equation-based method outperforms traditional ray methods by producing more accurate ...

6 citations

References
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Journal ArticleDOI
TL;DR: In this paper, it was shown that the field of a Gaussian beam can be represented by a function G(P) =eikr/r, where r is the distance from the observation point P to a fixed point having a complex location.
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557 citations

Frequently Asked Questions (1)
Q1. What are the contributions in "A complex point-source solution of the acoustic eikonal equation for gaussian beams in transversely isotropic media running head: complex point-source solutions" ?

The authors present the analytic approximations for complex traveltime in transversely isotropic media with a titled symmetry axis ( TTI ), which is defined by a Taylor series expansion over the anisotropy parameters. The authors investigate the influence of the anisotropy parameters and of the initial width of the ray tube on the accuracy of the computed traveltimes.