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Kirchhoff datuming

Based on Green's second identity, the integral expression for a pressure wavefield $P({\bf r},\omega)$within a source free volume bounded by surface S is

P({\bf r},\omega)\;=\;\frac{-1}{4\pi}\oint_S
\left(P \nabla G - G\nabla P\right) \cdot {\bf n}\;

G is a Green's function satisfying the scalar wave equation, and ${\bf n}$ is the inward pointing normal to the bounding surface (Figure [*]). For seismic data, the integral can be split into two parts so that S = S1 + S2 and the wavefield can be considered as arising from causal sources below S1, which represents the data collection surface. Since seismic data are recorded for a finite time, the surface S2 can be chosen in such a way that the contribution from this part of the integral is negligible. The choice of Green's function that vanishes on surface S1 reduces the integral to

P({\bf r},\omega)\;=\;\frac{-1}{4\pi}\int_{S_1}
P \frac{\partial G}{\partial n} dS_1.\end{displaymath}

When the undulations of the surface are small over a wavelength, the Green's function can be approximated by


where $k = \omega/v$ and $r = ({(x-x_s)^2 + (y-y_s)^2 + (z-z_s)^2})^\frac{1}{2}$. The subscript s indicates that the coordinate is on surface S1. $\omega$ is angular frequency and v is propagation velocity.

Figure 1
Geometry for derivation of the Kirchhoff integral (after Berkhout, 1982).

The surface integral then becomes

P({\bf r},\omega)&=&\frac{-1}{4\pi}\int_...
 ...\partial n}\,
 \frac{1 - ikr}{r^2}\,e^{ikr} \; dS_1.\end{array}\end{displaymath}

The integral can be split into near-field and far-field contributions by writing  
P({\bf r},\omega)\;=\;\frac{1}{2\pi}\int_{S_1}
 ...rac{1}{r^2}\,P({\bf r_s},\omega)e^{i\omega \tau} 
\right] dS_1,\end{displaymath} (1)
where $\tau = r / v$ represents the traveltime from the surface S1 to the observation point ${\bf r}$.This equation can be recast in the time domain by using the relationships

-i\omega\,P(\omega) & \Longleftrightarro...
 ...^{i\omega \tau} & \Longleftrightarrow & P(t - \tau),\end{array}\end{displaymath}

to obtain  
P({\bf r},t)\;=\;\frac{1}{2\pi}\int_{S_1}
\frac{\partial r}{...
 ...},t - \tau)
\frac{1}{r^2} P({\bf r_s},t - \tau)
\right] dS_1.\end{displaymath} (2)
Equations ([*]) and ([*]) are the 3-D frequency and time domain equations for the upward continuation of a scalar wavefield. For computational efficiency it is common to make the far-field approximation $\omega/rv \gt \!\! \gt 1/r^2$, and to drop the second term in the equations.

If the wavefield is invariant in one space coordinate, the frequency domain equation can be rewritten as

P({\bf r},\omega)\;=\;
P({\bf r_s},\om...
 ...rtial r}{\partial n}\, 
\frac{1 - ikr}{r^2}\,e^{ikr} \; dy\;dx.\end{displaymath}

Integrating out the y dependence by the method of stationary phase yields  
P({\bf r},\omega)\;=\;
P({\bf r_s},\omega)\,e^{i\omega t} 
\;dx.\end{displaymath} (3)
Making the far-field approximation yields  
P({\bf r},\omega)\;=\;\frac{1}{\sqrt{2\pi v}}
\int_x \frac{\...
 ...({\bf r_s},\omega)\,
\frac{ e^{i\omega \tau} } {\sqrt{r}}\; dx.\end{displaymath} (4)
The equivalent time domain expression is  
P({\bf r},t)\;=\;\frac{1}{\sqrt{2\pi v}}
\int_x \frac{\parti...
P({\bf r_s},t - \tau)\; dx.\end{displaymath} (5)
For the 2-D versions of the integrals, $r = ((x-x_s)^2 + (z-z_s)^2)^\frac{1}{2}$.The term $D_{\frac{1}{2}}$ is a convolutional operator corresponding to frequency domain multiplication by $\sqrt{-i\omega}$. Rather than making the far-field approximation, Berryhill (1979) obtains a 2-D version in the time domain by integrating out the y dependence of equation ([*]). His expression is similar to equation ([*]), except that his convolutional operator depends on r. Berryhill's time domain formulation is accurate since it includes the near-field term, but in practice, the added computation of the delay-time variant convolution is often unnecessary.

Equations ([*]) through ([*]) are the Kirchhoff integral equations for upward continuation. Wave-equation datuming is performed by implementing these equations. Downward continuation is performed by means of the adjoint processes. The concept of adjoint operators will be more fully described in a subsequent section, but for now, it is only important to note that the adjoint forms of the integral equations will have the opposite sign in $i\omega$ or in the time delay factor $\tau = r / v$. The adjoint forms of the time domain integrals, along with the imaging condition, result in Schneider's (1978) Kirchhoff migration operators. Wiggins (1984) uses the far-field approximation in his implementation of Kirchhoff datuming and migration.

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