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As recognized recently by Tygel et al. 1994, true-amplitude migration Goldin (1992); Schleicher et al. (1993) is the asymptotic inversion of seismic modeling represented by the Kirchhoff high-frequency approximation. The Kirchhoff approximation for a reflected wave Bleistein (1984); Haddon and Buchen (1981) belongs to the class of stacking-type operators (1) with the summation path  
\theta(x;t,y) = t - T\left(s(y),x\right) - T\left(x,r(y)\right)\;,\end{displaymath} (45)
the weighting function  
w(x;t,y) = {1\over{\left(2\,\pi\right)^{m/2}}} \,
\over {R\left(s(y),x\right)\,R\left(x,r(y)\right)}}\;,\end{displaymath} (46)
and the additional time filter $\left({\partial \over {\partial
z}}\right)^{m/2}$. Here x denotes a point at the reflector surface, s is the source location, and r is the receiver location at the observation surface. The parameter y corresponds to the configuration of observation. That is, $s(y) = s\,,\;r(y) = y$ for the common-shot configuration, s(y) = r(y) = y for the zero-offset configuration, and $s(y) = y - h\,,\;r(y) = y + h$ for the common-offset configuration (where h is the half-offset). The functions T and R have the same meaning as in the datuming example, representing the one-way traveltime and the one-way geometric spreading, respectively. The function C(s,x,r) is known as the obliquity factor. Its definition is  
C(s,x,r) = {1 \over 2}\,
\left({{\cos{\alpha_s(x)}} \over {v_s(x)}} +
 {{\cos{\alpha_r(x)}} \over {v_r(x)}}\right)\;,\end{displaymath} (47)
where the angles $\alpha_s(x)$ and $\alpha_r(x)$ are formed by the incident and reflected waves with the normal to the reflector at the point x, and vs(x) and vr(x) are the corresponding velocities in the vicinity of this point. In this paper, I leave the case of converted (e.g., P-SV) waves outside the scope of consideration and assume that vs(x) equals vr(x) (e.g., in P-P reflection). In this case, it is important to notice that at the stationary point of the Kirchhoff integral, $\alpha_s(x) = \alpha_r(x) = \alpha(x)$ (the law of reflection), and therefore  
C(s,x,r) = {{\cos{\alpha(x)}} \over {v(x)}}\;.\end{displaymath} (48)
The stationary point of the Kirchhoff integral is the point where the stacking curve (45) is tangent to the actual reflection traveltime curve. When our goal is asymptotic inversion, it is appropriate to use equation (48) in place of (47) to construct the inverse operator. The weighted function (46) can include other factors affecting the leading-order (WKBJ) ray amplitude, such as the source signature, caustics counter (the KMAH-index), and transmission coefficient for the interfaces Cerveny et al. (1977); Chapman and Drummond (1982). In the following analysis, I neglect these factors for simplicity.

The model M implied by the Kirchhoff modeling integral is the wavefield with the wavelet shape of the incident wave and the amplitude proportional to the reflector coefficient along the reflector surface. The goal of true-amplitude migration is to recover M from the observed seismic data. In order to obtain the image of the reflectors, the reconstructed model is evaluated at the time z equal to zero. The Kirchhoff modeling integral requires explicit definition of the reflector surface. However, its inverse doesn't require explicit specification of the reflector location. For each point of the subsurface, one can find the normal to the hypothetical reflector by bisecting the angle between the s-x and x-r rays. Born scattering approximation provides a different physical model for the reflected waves. According to this approximation, the recorded waves are viewed as scattered on smooth local inhomogeneities rather than reflected from sharp reflector surfaces. The inversion of Born modeling Bleistein (1987); Miller et al. (1987) closely corresponds with the result of Kirchhoff integral inversion. For an unknown reflector and the correct macro-velocity model, the asymptotic inversion reconstructs the signal located at the reflector surface with the amplitude proportional to the reflector coefficient.

As follows from the form of the summation path (45), the integral migration operator must have the summation path
\widehat{\theta}(y;z,x) = z + T\left(s(y),x\right) + T\left(x,r(y)\right)\end{displaymath} (49)
to reconstruct the geometry of the reflector at the migrated section. According to (7), the asymptotic reconstruction of the wavelet requires, in addition, the derivative filter $\left(- {\partial
\over {\partial t}}\right)^{m/2}$. The asymptotic reconstruction of the amplitude defines the true-amplitude weighting function in accordance with (9), as follows:  
\widehat{w}(y;z,x) = 
\over {\partial x\,\partial y}}\right\vert\;.\end{displaymath} (50)

In the case of common-shot migration, we can simplify equation (50) with the help of Gritsenko's formula (39) to the form  
\widehat{w}_{CS}(r;z,x) = {1\over{\left(2\,\pi\right)^{m/2}}...
 ...{\cos{\alpha(r)}} \over {v(r)}}\,
{{R(s,x)} \over {R(r,x)}} \;,\end{displaymath} (51)
where the angle $\alpha(r)$ is measured between the reflected ray and the normal to the observation surface at the reflector point r. Formula (51) coincides with the analogous result of Keho and Beydoun 1988, derived directly from Claerbout's imaging principle Claerbout (1970). An alternative derivation is given by Goldin 1987. Docherty 1991 points out a remarkable correspondence between this formula and the classic results of Born scattering inversion Bleistein (1987).

In the case of zero-offset migration, Gritsenko's formula simplifies the true-amplitude migration weighting function (50) to the form  
\widehat{w}_{ZO}(y;z,x) = {{2^m}\over{\left(2\,\pi\right)^{m/2}}} \,
{{\cos{\alpha(y)}} \over {v(y)}}\;.\end{displaymath} (52)
In a constant-velocity medium, one can accomplish the true-amplitude zero-offset migration by premultiplying the recorded zero-offset seismic section by the factor $\left(v \over 2 \right)^{m-1}\,\left(t
\over 2\right)^{m/2}$ [which corresponds at the stationary point to the geometric spreading R(x,y)] and downward continuation according to formula (44) with the effective velocity v/2 Goldin (1987); Hubral et al. (1991). This conclusion is in agreement with the analogous result of Born inversion Bleistein et al. (1985), though derived from a different viewpoint.

In the case of common-offset migration in a general variable-velocity medium, the weighting function (50) cannot be simplified to a different form, and all its components need to be calculated explicitly by dynamic ray tracing Cerveny and de Castro (1993). In the constant-velocity case, we can differentiate the explicit expression for the summation path
\widehat{\theta}(y;z,x) = z + 
{{\rho_s(x,y) + \rho_r(x,y)} \over v}\;,\end{displaymath} (53)
where $\rho_s$ and $\rho_r$ are the lengths of the incident and reflected rays:
\rho_s(y,x) & = & 
\sqrt{x_3^2 + (x_1 - y_1 + h_1)^2 + (x_2 - y...
 ...& = & 
\sqrt{x_3^2 + (x_1 - y_1 - h_1)^ 2+ (x_2 - y_2 - h_2)^2}\;.\end{eqnarray} (54)
For simplicity, the vertical component of the midpoint y3 is set here to zero. Evaluating the second derivative term in formula (50) for the common-offset geometry leads, after some heavy algebra, to the expression  
\left\vert{{\partial^2 T\left(s(y),x\right)} 
\over {\partia...
 ...r} \over {v\,\rho_s\,\rho_r}}\right)^{m-1}
\,\cos{\alpha(x)}\;.\end{displaymath} (56)
Substituting (56) into the general formula (50) yields the weighting function for the common-offset true-amplitude constant-velocity migration:  
\widehat{w}_{CO}(y;z,x) = {1\over{\left(2\,\pi\right)^{m/2}}...
 ...(\rho_s^2 + \rho_r^2)} \over 
{v\,(\rho_s\,\rho_r)^{m/2+1}}}\;.\end{displaymath} (57)
Formula (57) is similar to the result obtained by Sullivan and Cohen 1987. In the case of zero offset h=0, (57) reduces to formula (52). Note that the value of m=1 in (57) corresponds to the two-dimensional (cylindric) waves recorded on the seismic line. A special case, valuable in practice, is the 2.5-D inversion, when the waves are assumed to be spherical, while the recording is on a line, and the medium has cylindric symmetry. In this case, the modeling weighting function (46) transforms to Bleistein (1986); Deregowski and Brown (1983)
w(x;t,y) = {1\over{\left(2\,\pi\right)^{1/2}}} \,
\over {\sqrt{\rho_s\,\rho_r\,(\rho_s + \rho_r)}}}\;,\end{displaymath} (58)
and the time filter is $\left({\partial \over {\partial
z}}\right)^{1/2}$. Combining this result with formula (56) for m=1, we obtain the weighting function for the 2.5-D common-offset migration in a constant velocity medium Sullivan and Cohen (1987):
\widehat{w}_{CO;2.5D}(y;z,x) = {1\over{\left(2\,\pi\right)^{...
 ..._s^2 + \rho_r^2)} \over 
{\sqrt{v}\,(\rho_s\,\rho_r)^{3/2}}}\;.\end{displaymath} (59)
The corresponding time filter for 2.5-D migration is $\left(-
{\partial \over {\partial t}}\right)^{1/2}$.

The weighting function of the asymptotic pseudo-unitary migration is found analogously to (38) as  
w^{(+)} = w^{(-)} = {1\over{\left(2\,\pi\right)^{m/2}}} \,
\over {\partial x\,\partial y}}\right\vert^{1/2}\;.\end{displaymath} (60)
Unlike true-amplitude migration, this type of migration operator doesn't change the dimensionality of the input. For common-shot migration, pseudo-unitary weighting coincides with the weighting of datuming and corresponds to the downward continuation of the receivers. In the zero-offset case, it reduces to downward pseudo-unitary continuation with a velocity of v/2. In the common-offset case, the pseudo-unitary weighting is defined from (60) and (56) as follows:
w^{(-)}_{CO}(y;z,x) = {1\over{\left(2\,\pi\,v\right)^{m/2}}}...
 ..._s^2 + \rho_r^2}} \over 
{(\rho_s\,\rho_r)^{{m+1} \over 2}}}\;,\end{displaymath} (61)
\cos{\alpha} = \left({
{(x - y)^2 + \rho_s\,\rho_r - h^2} \over {2\,\rho_s\,\rho_r}}
\right)^{1/2}\;.\end{displaymath} (62)

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Next: Post-Stack Time Migration Up: EXAMPLES Previous: Datuming
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