paper.tex

国外免费地震资料处理软件包

\title{On anelliptic approximations for $qP$ velocities \\ in
\new{transversally isotropic} media}

\email{[e-mail]}
\author{Sergey Fomel}

\maketitle

\begin{abstract}
I develop a unified approach for approximating phase and group
  velocities of $qP$ seismic waves in a transversally isotropic medium
  with the vertical axis of symmetry (VTI). While the exact phase
  velocity expressions involve four independent parameters to
  characterize the elastic medium, the proposed approximate
  expressions use only three parameters. This makes them more
  convenient for use in surface seismic experiments, where estimation
  of all the four parameters is problematic. The three-parameter
  phase-velocity approximation coincides with the previously published
  ``acoustic'' approximation of Alkhalifah. The group velocity
  approximation is `new and noticeably more accurate than some of the
  previously published approximations. I demonstrate an application of
  the group velocity approximation for finite-difference computation
  of traveltimes.
\end{abstract}

\section{Introduction}

Anellipticity (deviation from ellipse) is an important characteristic of
elastic wave propagation. One of the simplest and yet practically important
cases \new{of anellipticity occurs} in transversally isotropic media with
the vertical axis of symmetry (VTI). In this type of media, the phase
velocities of $qSH$ waves and the corresponding wavefronts are elliptic, while
the phase and group velocities of $qP$ and $qSV$ waves may exhibit strong
anellipticity \cite[]{tsvankin}.

The exact expressions for the phase velocities of $qP$ and $qSV$ waves in VTI
media involve four independent parameters. However, it has been observed that
only three parameters influence wave propagation 
% at small angles from the vertical axis 
and are of interest to surface seismic methods
\cite[]{GEO60-05-15501566}. Moreover, the exact expressions for the group
velocities in terms of the group angle are difficult to obtain and too
cumbersome for practical use. This explains the need for developing practical
three-parameter approximations for both group and phase velocities in VTI
media.

Numerous different successful approximations have been previously developed
\cite[]{GEO54-12-15641574,jse,GEO60-05-15501566,GEO63-02-06230631,GEO65-04-13161325,GEO65-03-09190933,ANI00-00-03490361,SEG-2001-01060109}.
In this paper, I attempt to construct a unified approach for deriving
anelliptic approximations.

The starting point is the anelliptic approximation of Muir
\cite[]{Muir.sep.44.55,jse}. Although not the most accurate for immediate
practical use, this approximation possesses remarkable theoretical properties.
The Muir approximation correctly captures the linear part of anelliptic
behavior. It can be applied to find more accurate approximations with
nonlinear dependence on the anelliptic parameter. A particular way of
``unlinearizing'' the linear approximation is the shifted hyperbola approach,
familiar from the isotropic approximations in vertically inhomogeneous media
\cite[]{malov,Sword.sep.51.313,GEO53-02-01430157,GEO59-06-09830999} and from the
theory of Stolt stretch \cite[]{GEO43-01-00230048,mystolt}. I show that applying
this idea to approximate the phase velocity of $qP$ waves leads to the known
``acoustic'' approximation of \cite{GEO63-02-06230631,GEO65-04-12391250},
derived in a different way.  Applying the same approach to approximate the
group velocity of $qP$ waves leads to a new remarkably accurate
three-parameter approximation.

One practical use for the group velocity approximation is traveltime
computations, required for Kirchhoff imaging and tomography. In the last part
of the paper, I show examples of finite-difference traveltime computations
utilizing the new approximation.

\section{Exact expressions}

Wavefront propagation in the general anisotropic media can be
described with the anisotropic eikonal equation
\begin{equation}
  \label{eq:eikonal}
  v^2\left(\frac{\nabla T}{|\nabla T|},\mathbf{x}\right)\,|\nabla T|^2 = 
  1\;,
\end{equation}
where $\mathbf{x}$ is a point in space, \new{$T(\mathbf{x})$} is the
traveltime \new{at that point for} a given source, and
$v(\mathbf{n},\mathbf{x})$ is the \emph{phase velocity} in the phase direction
$\mathbf{n} = \frac{\nabla T}{|\nabla T|}$.

In the case of VTI media, the three modes of elastic wave propagation
($qSH$, $qSV$, and $qP$) have the following well-known explicit
expressions for the phase velocities \cite[]{gassmann}:
\begin{eqnarray}
  \label{eq:qsh}
  v_{SH}^2(\mathbf{n},\mathbf{x}) & = & 
  m\,\sin^2{\theta} + l\,\cos^2{\theta}\;; \\
  \nonumber
  v^2_{SV}(\mathbf{n},\mathbf{x}) & = &
  \frac{1}{2}\,\left[(a+l)\,\sin^2{\theta} + (c+l)\,\cos^2{\theta}\right] -  \\
  & & \frac{1}{2}\,\sqrt{\left[(a-l)\,\sin^2{\theta} - 
      (c-l)\,\cos^2{\theta}\right]^2 +
      4\,(f+l)^2\,\sin^2{\theta}\,\cos^2{\theta}}\;; 
  \label{eq:qsv} \\
  \nonumber
  v^2_{P}(\mathbf{n},\mathbf{x}) & = &
  \frac{1}{2}\,\left[(a+l)\,\sin^2{\theta} + (c+l)\,\cos^2{\theta}\right] +  \\
  & & \frac{1}{2}\,\sqrt{\left[(a-l)\,\sin^2{\theta} - (c-l)\,\cos^2{\theta}\right]^2 +
      4\,(f+l)^2\,\sin^2{\theta}\,\cos^2{\theta}}\;,
    \label{eq:qp}
\end{eqnarray}
where, in the notation of \cite{backus} and \cite{GEO44-05-08960917},
$a=c_{11}$, $c=c_{33}$, $f=c_{13}$, $l=c_{55}$, $m=c_{66}$, $c_{ij}(\mathbf{x})$
are the density-normalized components of the elastic tensor, and $\theta$ is
the phase angle between the phase direction $\mathbf{n}$ and the axis of
symmetry.
  
The group velocity describes the propagation of individual ray trajectories
$\mathbf{x}(\tau)$.  It can be determined from the phase velocity using the
general expression
\begin{equation}
  \mathbf{V} = \frac{d \mathbf{x}}{d \tau} =
  v \mathbf{n} + \left(\mathbf{I} - \mathbf{n}\, \mathbf{n}^T\right) 
  \nabla_{\mathbf{n}} v\;,
  \label{eq:group}
\end{equation}
where $\mathbf{I}$ denotes the identity matrix, $\mathbf{n}^T$ stands for the
transpose of $\mathbf{n}$, \new{and $\nabla_{\mathbf{n}} v$ is the gradient
  of $v$ with respect to $\mathbf{n}$}. The two terms in
equation~(\ref{eq:group}) are clearly orthogonal to each other. Therefore, the
group velocity magnitude is
\cite[]{GEO20-04-07800806,GEO44-05-08960917,GEO49-11-19081914}
\begin{equation}
  \label{eq:f}
  V = |\mathbf{V}| = \sqrt{v^2 + v_{\theta}^2}\;,
\end{equation}
where 
\begin{equation}
  \label{eq:falpha}
  v_{\theta}^2 =  \left|\left(\mathbf{I} - \mathbf{n}\,
      \mathbf{n}^T\right) \nabla_{\mathbf{n}} v\right|^2 =
  \left|\nabla_{\mathbf{n}} v\right|^2 - 
  \left|\mathbf{n} \cdot \nabla_{\mathbf{n}} v\right|^2\;.
\end{equation}

The group velocity has a particularly simple form in the case of elliptic
anisotropy. \new{Specifically}, the phase velocity squared \new{has} the
quadratic form
\begin{equation}
  \label{eq:ellips}
  v_{\mbox{ell}}^2(\mathbf{n},\mathbf{x}) = 
  \mathbf{n}^T\,\mathbf{A}(\mathbf{x})\,\mathbf{n}
\end{equation}
with a symmetric positive-definite matrix $\mathbf{A}$, and the group
velocity is 
\begin{equation}
  \label{eq:ellf}
  \mathbf{V}_{\mbox{ell}} = \mathbf{A}\,\mathbf{p}\;, 
\end{equation}
where $\mathbf{p} = \nabla T = \mathbf{n}/v(\mathbf{n},\mathbf{x})$.
The corresponding group slowness squared has the explicit expression
\begin{equation}
  \label{eq:ellv}
  \frac{1}{V_{\mbox{ell}}^2(\mathbf{N},\mathbf{x})} = 
    \mathbf{N}^T\,\mathbf{A}^{-1}(\mathbf{x})\,\mathbf{N}\;, 
\end{equation}
where $\mathbf{N}$ is the group direction, and $\mathbf{A}^{-1}$ is the matrix
inverse of $\mathbf{A}$. For example, the elliptic expression~(\ref{eq:qsh}) for
the phase velocity of $qSH$ waves in VTI media transforms into a completely
analogous expression for the group slowness
\begin{equation}
  \label{eq:ellsh}
  \frac{1}{V_{SH}^2(\mathbf{N},\mathbf{x})}  = 
  M\,\sin^2{\Theta} + L\,\cos^2{\Theta}
\end{equation}
where $M=1/m$, $L=1/l$, and $\Theta$ is the angle between the group direction
$\mathbf{N}$ and the axis of symmetry.

\inputdir{Math}

The situation is more complicated in the anelliptic case.
Figure~\ref{fig:exph} shows the $qP$ and $qSV$ phase velocity profiles in a
transversely isotropic material -- Greenhorn shale \cite[]{GEO46-03-02880297},
which has the parameters $a=14.47\,\mbox{km}^2/\mbox{s}^2$,
$l=2.28\,\mbox{km}^2/\mbox{s}^2$, $c=9.57\,\mbox{km}^2/\mbox{s}^2$, and
$f=4.51\,\mbox{km}^2/\mbox{s}^2$. Figure~\ref{fig:exgr} shows the
corresponding group velocity profiles. The non-convexity of the $qSV$ phase
velocity causes a multi-valued (triplicated) group velocity profile. The
shapes of all the surfaces are clearly anelliptic.


\plot{exph}{height=3in}{Phase velocity profiles for $qP$ (outer curve) and
  $qSV$ (inner curve) waves in a transversely isotropic material (Greenhorn
  shale).}

\plot{exgr}{height=3in}{Group velocity profiles for $qP$ (outer curve) and
  $qSV$ (inner curve) waves in a transversely isotropic material (Greenhorn
  shale).}

A simple model of anellipticity is suggested by the Muir approximation
\cite[]{Muir.sep.44.55,jse}, reviewed in the next section.

\section{Muir approximation}

\cite{Muir.sep.44.55} suggested representing anelliptic $qP$ phase
velocities with the following approximation:
\begin{equation}
\label{eq:muph}
  v_P^2(\theta) \approx e(\theta) + 
\frac{(q-1)\,a\,c\,
\sin^2{\theta}\,\cos^2{\theta}}{e(\theta)}
\;,
\end{equation}
where $e(\theta)$ is the elliptical part of the velocity, defined by
\begin{equation}
\label{eq:muel}
e(\theta) = a\,\sin^2{\theta} + c\,\cos^2{\theta}\;,
\end{equation}
and $q$ is the anellipticity coefficient ($q=1$ in case of elliptic
velocities). Approximation~(\ref{eq:muph}) uses only three parameters to
characterize the medium ($a$, $c$, and $q$) as opposed to the four parameters
($a$, $c$, $l$, and $f$) in the exact expression.

There is some freedom in choosing an appropriate value for the coefficient
$q$. Assuming near-vertical wave propagation and the vertical axis of symmetry
(a VTI medium) and fitting the curvature ($d^2 v_P/d \theta^2$) of the exact
phase velocity~\eqref{eq:qp} near the vertical phase angle ($\theta = 0$),
leads to the definition \cite[]{jse}
\begin{equation}
  \label{eq:qpdef}
  q = \frac{l\,(c-l) + (l+f)^2}
  {a\,(c-l)}\;.
\end{equation}
In terms of Thomsen's elastic parameters $\epsilon$ and $\delta$
\cite[]{GEO51-10-19541966} and the elastic parameter $\eta$ of
\cite{GEO60-05-15501566},
\begin{equation}
  \label{eq:eta2q}
  q = \frac{1 + 2\,\delta}{1 + 2\epsilon}
= \frac{1}{1 + 2\,\eta}\;.
\end{equation}
This confirms the direct relationship between $\eta$ and anellipticity.  If we
were to fit the phase velocity curvature near the horizontal axis
$\theta=\pi/2$ (perpendicular to the axis of symmetry), the appropriate value
for $q$ would be