paper.tex

国外免费地震资料处理软件包

h\,t\,\left({4 \over {v^2}} + 
\left({{\partial t} \over {\partial h}}\right)^2\,-
\left({{\partial t} \over {\partial x}}\right)^2\right)\;.
\label{eq:OCeikonal}
\end{equation}

Differentiating (\ref{eq:length}) with respect to the velocity $v$ yields
\begin{equation}
- v^2\,{{\partial t} \over {\partial v}} = 
2\,h\,{\cos{\alpha} \over \sin{\theta}}\;.
\label{eq:dtdv} 
\end{equation}
Finally, dividing (\ref{eq:dtdv}) by (\ref{eq:snells3}) produces
\begin{equation}
v\,{{\partial z} \over {\partial v}} = 
{h \over {\cos{\theta}\,\sin{\theta}}}\;.
\label{eq:dzdv} 
\end{equation}
Equation (\ref{eq:dzdv}) can be written in a variety of ways with the help
of an explicit geometric relationship between the half-offset $h$ and
the depth $z$, 
\begin{equation}
h = z\,
{{\sin{\theta}\,\cos{\theta}} \over
{\cos^2{\alpha}-\sin^2{\theta}}}\;, 
\label{eq:z2h} 
\end{equation}
which follows directly from the trigonometry of the triangle in Figure
\ref{fig:vlcray} \cite[]{ofcon}. For example, equation (\ref{eq:dzdv}) can
be transformed to the form obtained by \cite{GEO60-01-01420153}:
\begin{equation}
v\,{{\partial z} \over {\partial v}} = 
{z \over{\cos^2{\alpha}-\sin^2{\theta}}} =
{z \over{\cos{\alpha_1}\,\cos{\alpha_2}}}\;.
\label{eq:liu} 
\end{equation}
In order to separate different factors contributing to the velocity
continuation process, one can transform this equation to the form
\begin{eqnarray}
\nonumber
v\,{{\partial z} \over {\partial v}} & = & 
{z \over {\cos^2{\alpha}}} +
{{h^2} \over z}\,\left(1-\tan^2{\alpha}\,\tan^2{\theta}\right) \\
& = & z\,\left(1 + \left({{\partial z} \over {\partial x}}\right)^2\right) +
{{h^2} \over z}\,\left(1-\left({{\partial z} \over {\partial x}}\right)^2\,
\left({{\partial z} \over {\partial h}}\right)^2\right)\;.
\label{eq:zeikonal} 
\end{eqnarray}
Rewritten in terms of the vertical traveltime $\tau = z/v$, it further
transforms to equation 
\begin{equation}
{{\partial \tau} \over {\partial v}} = 
v\,\tau\,\left({{\partial \tau} \over {\partial x}}\right)^2 +
{{h^2} \over {v^3\,\tau}}\,\left(1 - 
v^4\,
\left({{\partial \tau} \over {\partial x}}\right)^2\,
\left({{\partial \tau} \over {\partial h}}\right)^2\right)\;,
\label{eq:eikonal0} 
\end{equation}
equivalent to equation~(\ref{eq:eikonal}) in the main text. Yet
another form of the kinematic velocity continuation equation follows
from eliminating the reflection angle $\theta$ from equations
(\ref{eq:dzdv}) and (\ref{eq:z2h}). The resultant expression takes the
following form:
\begin{equation}
v\,{{\partial z} \over {\partial v}} = 
{{2\,(z^2 + h^2)} \over
{\sqrt{z^2 + h^2 \sin^2{2\,\alpha}} + z\,\cos{2\,\alpha}}} =
{z \over {\cos^2{\alpha}}} + {{2\,h^2} \over
{\sqrt{z^2 + h^2 \sin^2{2\,\alpha}} + z}}\;.
\label{eq:notheta} 
\end{equation}

\append{Derivation of the residual DMO kinematics}

This appendix derives the kinematical laws for the residual NMO+DMO
transformation in the prestack offset continuation process.

The direct solution of equation (\ref{eq:DMONMOeikonal}) is
nontrivial. A simpler way to obtain this solution is to decompose
residual NMO+DMO into three steps and to evaluate their contributions
separately. Let the initial data be the zero-offset reflection event
$\tau_0(x_0)$. The first step of the residual NMO+DMO is the inverse
DMO operator. One can evaluate the effect of this operator by means of
the offset continuation concept \cite[]{ofcon}. According to this
concept, each point of the input traveltime curve $\tau_0(x_0)$
travels with the change of the offset from zero to $h$ along a special
trajectory, which I call a {\em time ray}. Time rays are parabolic
curves of the form
\begin{equation}
x\left(\tau\right) =  x_0+{{\tau^2-\tau_0^2\left(x_0\right)} \over
{\tau_0\left(x_0\right)\,\tau_0'\left(x_0\right)}}\;,
\label{eq:xoftau}
\end{equation}
with the final points constrained by the equation
\begin{equation}
h^2 = \tau^2\,{{\tau^2-\tau_0^2\left(x_0\right)} \over
{\left(\tau_0\left(x_0\right)\,\tau_0'\left(x_0\right)\right)^2}}\;,
\label{eq:hoftau}
\end{equation}
where $\tau_0'\left(x_0\right)$ is the derivative of 
$\tau_0\left(x_0\right)$.
The second step of the cumulative residual NMO+DMO process is the
residual normal moveout. According to equation (\ref{eq:ResNMO}), residual
NMO is a one-trace operation transforming the traveltime $\tau$ to
$\tau_1$ as follows:
\begin{equation}
\tau_1^2 = \tau^2 + h^2\,d\;,
\label{eq:ResNMO2} 
\end{equation}
where
\begin{equation}
d = \left({1 \over v_0^2} - {1 \over v_1^2}\right)\;.
\label{eq:formers}
\end{equation}
The third step is dip moveout corresponding to the new velocity
$v_1$. DMO is the offset continuation from $h$ to zero
offset along the redefined time rays \cite[]{ofcon}
\begin{equation}
x_2\left(\tau_2\right) =  
x + {{h\,X} \over {\tau_1^2\,H}}\,\left(\tau_1^2-\tau_2^2\right)\;,
\label{eq:xoftau2}
\end{equation}
where $H = {{\partial \tau_1} \over {\partial h}}$, and $X =
{{\partial \tau_1} \over {\partial x}}$. 
The end points of the time rays (\ref{eq:xoftau2}) are defined by the
equation
\begin{equation}
\tau_2^2 = - \tau_1^2\,{{\tau_1\,H} \over {h\,X^2}}\;.
\label{eq:tauofh2}
\end{equation}
The partial derivatives of the common-offset traveltimes are
constrained by the offset continuation kinematic equation
\begin{equation}
h\,(H^2 - X^2) = \tau_1\,H\;,
\label{eq:OCequation}
\end{equation}
which is equivalent to equation (\ref{eq:OCeikonal}) in Appendix
A. Additionally, as follows from equations (\ref{eq:ResNMO2}) and the ray
invariant equations from \cite[]{ofcon},
\begin{equation}
\tau_1\,X = \tau\,{{\partial \tau} \over {\partial x}} = 
{{\tau^2\,\tau_0'\left(x_0\right)} \over {\tau_0\left(x_0\right)}}\;.
\label{eq:invariant}
\end{equation}
Substituting~(\ref{eq:xoftau}-\ref{eq:formers}) and
(\ref{eq:OCequation}-\ref{eq:invariant}) into equations
(\ref{eq:xoftau2}) and (\ref{eq:tauofh2}) and performing the algebraic
simplifications yields the parametric expressions for velocity
rays of the residual NMO+DMO process:
\begin{equation}
\left\{
\begin{array}{rcl}
x_2(d) & = & \displaystyle{x_0 + {{h^2\,\tau_0'(x_0)} \over T}\,
\left(1 - {T^2 \over T_2^2(d)}\right)}\;,
\\ 
\\
\tau(d) & = & \displaystyle{{{\tau_1^2(d)} \over {T_2(d)}}}\;,
\end{array}
\right.
\label{eq:ResDMO2}
\end{equation}
where the function
$T\left(h,\tau_0(x_0),\tau_0'\left(x_0\right)\right)$ is defined by
\begin{equation}
T\left(h,\tau,\tau_x\right) = 
{{\tau + \sqrt{\tau^2 + 4\,h^2\,\tau_x^2}} \over 2}\;,
\label{eq:CapT}
\end{equation}
\begin{equation}
T_2(d) = \sqrt{T\left(h,\tau_1^2(d),\tau_0'\left(x_0\right)\,
T\left(h,\tau_0(x_0),\tau_0'\left(x_0\right)\right)\right)}\;,
\label{eq:CapT2}
\end{equation}
and
\begin{equation}
\tau_1^2(d) = \tau_0\,T + d\,h^2\;.
\label{eq:tauofs}
\end{equation}

The last step of the cascade of inverse DMO, residual NMO, and DMO is
illustrated in Figure \ref{fig:vlcvoc}. The three plots in the figure show
the offset continuation to zero offset of the inverse DMO impulse
response shifted by the residual NMO operator. The middle plot
corresponds to zero NMO shift, for which the DMO step collapses the
wavefront back to a point.  Both positive (top plot) and negative
(bottom plot) NMO shifts result in the formation of the specific
triangular impulse response of the residual NMO+DMO operator. As
noticed by \cite{Etgen.sepphd.68}, the size of the triangular
operators dramatically decreases with the time increase.  For large
times (pseudo-depths) of the initial impulses, the operator collapses
to a point corresponding to the pure NMO shift.

\inputdir{Math}

\sideplot{vlcvoc}{width=0.9\textwidth}{Kinematic residual NMO+DMO
  operators constructed by the cascade of inverse DMO, residual NMO,
  and DMO. The impulse response of inverse DMO is shifted by the
  residual NMO procedure. Offset continuation back to zero offset
  forms the impulse response of the residual NMO+DMO operator. Solid
  lines denote traveltime curves; dashed lines denote the offset
  continuation trajectories (time rays). Top plot: $v_1/v_0 = 1.2$.
  Middle plot: $v_1/v_0 = 1$; the inverse DMO impulse response
  collapses back to the initial impulse. Bottom plot: $v_1/v_0 = 0.8$.
  The half-offset $h$ in all three plots is 1 km.}

\append{INTEGRAL VELOCITY CONTINUATION AND KIRCHHOFF MIGRATION}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The main goal of this appendix is to prove the equivalence between the
result of zero-offset velocity continuation from zero velocity and
conventional post-stack migration. After solving the velocity
continuation problem in the frequency domain, I transform the solution
back to the time-and-space domain and compare it with the conventional
Kirchhoff migration operator \cite[]{GEO43-01-00490076}. The frequency-domain
solution has its own value, because it forms the basis for an efficient
spectral algorithm for velocity continuation \cite[]{second}. 
 
Zero-offset migration based on velocity continuation is the solution
of the boundary problem for equation (\ref{eq:POMequation2}) with the
boundary condition
\begin{equation}
\left.P\right|_{v=0} = P_0\;,
\label{eq:POMbound} 
\end{equation}
where $P_0(t_0,x_0)$ is the zero-offset seismic section, and
$P(t,x,v)$ is the continued wavefield. In order to find the solution
of the boundary problem composed of (\ref{eq:POMequation2}) and
(\ref{eq:POMbound}), it is convenient to apply the function
transformation $R(t,x,v) = t\,P(t,x,v)$, the time coordinate
transformation $\sigma = t^2/2$, and, finally, the double Fourier
transform over the squared time coordinate $\sigma$ and the spatial
coordinate $x$:
\begin{equation}
\widehat{R}(v) = \int \int\,P(t,x,v)\,
\exp(i \Omega \sigma - i k x )\,t^2\,dt\,dx\;.
\label{eq:FTK} 
\end{equation}
With the change of domain, equation (\ref{eq:POMequation2}) transforms
to the ordinary differential equation
\begin{equation}
{{d\,\widehat{R}} \over {d\,v}} = 
i\,{k^2 \over \Omega}\,v\,\widehat{R}\;,
\label{eq:ODE} 
\end{equation}
and the boundary condition (\ref{eq:POMbound}) transforms to the initial
value condition
\begin{equation}
\widehat{R}(0) = \widehat{R}_0\;, 
\label{eq:ODEbound} 
\end{equation}
where 
\begin{equation}
\widehat{R}_0 = \int \int\,P_0(t_0,x_0)\,
\exp(i \Omega \sigma_0 - i k x_0 )\,t_0^2\,dt_0\,dx_0\;,
\label{eq:FTK0}
\end{equation}
and $\sigma_0 = t_0^2/2$.  The unique solution of the initial value
(Cauchy) problem (\ref{eq:ODE}) - (\ref{eq:ODEbound}) is easily found to be
\begin{equation}
\widehat{R}(v) = \widehat{R}_0\,
\exp\left( i\,{{k^2} \over {2\,\Omega}}\,v^2\right)\;.
\label{eq:ODEsolution} 
\end{equation}

In the transformed domain, velocity continuation appears to be a unitary
phase-shift operator. An immediate consequence of this remarkable fact is the
cascaded migration decomposition of post-stack migration
\cite[]{GEO52-05-06180643}:
\begin{equation}
\exp\left( i\,{{k^2} \over {2\,\Omega}}\,
(v_1^2 +  \cdots + v_n^2)\right) =
\exp\left( i\,{{k^2} \over {2\,\Omega}}\,v_1^2\right)\,\cdots\,
\exp\left( i\,{{k^2} \over {2\,\Omega}}\,v_n^2\right)\;.
\label{eq:cascaded} 
\end{equation}
Analogously, three-dimensional post-stack migration is decomposed
into the two-pass procedure \cite[]{GPR31-01-00340056}:
\begin{equation}
\exp\left( i\,{{k_1^2+k_2^2} \over {2\,\Omega}}\,v^2\right) =
\exp\left( i\,{{k_1^2} \over {2\,\Omega}}\,v^2\right)\,
\exp\left( i\,{{k_2^2} \over {2\,\Omega}}\,v^2\right)\;.
\label{eq:two-pass}
\end{equation}

The inverse double Fourier transform of both sides of equality
(\ref{eq:ODEsolution}) yields the integral (convolution) operator
\begin{equation}
P(t,x,v) = \int\int\,P_0(t_0,x_0)\,K(t_0,x_0;t,x,v)\,dt_0\,dx_0\;,
\label{eq:convolution}
\end{equation}
with the kernel $K$ defined by
\begin{equation}
K = {{t_0^2/t} \over {(2\,\pi)^{m+1}}}\,
\int\int\,\exp\left(
i\,{{k^2} \over {2\,\Omega}}\,v^2 + ik\,(x - x_0) - 
{{i\Omega} \over 2}\,(t^2 - t_0^2)
\right)\,dk\,d\Omega\;,
\label{eq:kernel}
\end{equation}
where $m$ is the number of dimensions in $x$ and $k$ ($m$ equals $1$
or $2$). The inner integral on the wavenumber axis $k$ in formula
(\ref{eq:kernel}) is a known table integral \cite[]{grad}. Evaluating this
integral simplifies equation (\ref{eq:kernel}) to the form
\begin{equation}
K = {{t_0^2/t} \over {(2\,\pi)^{m/2+1}\,v^m}}\,
\int\,(i\Omega)^{m/2}\,\exp\left[
{{i\Omega} \over 2}\,
\left(t_0^2 - t^2 - {{(x - x_0)^2} \over v^2}\right)\right]\,
d\Omega\;.
\label{eq:skernel}
\end{equation}
The term $(i\Omega)^{m/2}$ is the spectrum of the anti-causal
derivative operator ${d \over {d\sigma}}$ of the order $m/2$. Noting
the equivalence
\begin{equation}
\left({\partial \over {\partial \sigma}}\right)^{m/2} =
\left({1 \over t}\,{\partial \over {\partial t}}\right)^{m/2} =
\left({1 \over t}\right)^{m/2}\,
\left({\partial \over {\partial t}}\right)^{m/2}\;,
\label{eq:halfdif}
\end{equation}
which is exact in the 3-D case ($m=2$) and asymptotically correct in
the 2-D case ($m=1$), and applying the convolution theorem
transforms operator (\ref{eq:convolution}) to the form
\begin{equation}
P(t,x,v) = {1 \over {(2\,\pi)^{m/2}}}\,\int\,
{{\cos{\alpha}} \over {(v\,\rho)^{m/2}}}\,
\left(- {\partial \over {\partial t_0}}\right)^{m/2}
P_0\left({\rho \over v},x_0\right)\,dx_0\;,
\label{eq:Kirchhoff}
\end{equation}
where $\rho = \sqrt{v^2\,t^2 + (x - x_0)^2}$, and $\cos{\alpha} =
t_0/t$. Operator (\ref{eq:Kirchhoff}) coincides with the Kirchhoff operator
of conventional post-stack time migration \cite[]{GEO43-01-00490076}.

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