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Double square root equations for teleseismic geometry

The theory developed here is similar to the traditional survey-sinking migration approach where the source and receiver wavefields are downward continued into the earth Claerbout (2001). However, in the traditional application of zero-offset survey sinking downward continued source wavefields do not have an intrinsic initial horizontal velocity. In the teleseismic case, planar teleseismic sources nearly always have non-zero horizontal velocities. This discrepancy is reconciled by deriving modified survey-sinking equations.

To help conceptualize the equations required for this context, consider a reference frame moving with a horizontal velocity equal to that of the source wavefield. In this reference frame a dipping plane wave propagates vertically. Hence, the downward continuation operators applied to recorded teleseismic data must transform to the double square root equations in a reference frame moving with a horizontal velocity equal to that of the source. To enable source horizontal propagation, wavefields downward continued to the next depth step are now dependent on horizontal position (x) in addition to depth (z). That is,
{\rm d}t={\rm d}t(x_g,z_g,x_s,z_s),\end{displaymath} (1)
where subscripts s and g correspond to source and receiver wavefields, respectively. To begin the derivation the differential is expressed explicitly,
{\rm d}t=\frac{\partial t}{\partial x_g} {\rm d}x_g + \frac{...
 ... x_s} {\rm d}x_s +
 \frac{\partial t}{\partial z_s} {\rm d}z_s,\end{displaymath} (2)
where partial derivatives with respect to the x and z coordinates may be readily associated with wavefield parameters. In a v(z) medium the constant horizontal slowness of the source wavefield, p, is defined by
\frac{\partial t}{\partial x_s} = p, \end{displaymath} (3)
the vertical slowness of source wavefield by
\frac{\partial t}{\partial z_s}= \sqrt{\frac{1}{v_s^2}-p^2},\end{displaymath} (4)
the variable horizontal slowness of the scattered receiver wavefield, pscat, by
\frac{\partial t}{\partial x_g} = p_{scat}, \end{displaymath} (5)
and the vertical slowness of receiver wavefield by
\frac{\partial t}{\partial z_g}= \sqrt{\frac{1}{v_g^2}-p_{scat}^2},\end{displaymath} (6)
where vs and vg represent the source and receiver wavefield velocities, respectively. Differential pairs ${\rm d}x_g$ and ${\rm d}z_g$, and ${\rm d}x_s$ and ${\rm d}z_s$ are linked geometrically by
{\rm d} x_g=\tan \theta_g {\rm d}z_g,\end{displaymath} (7)
{\rm d} x_s=\tan \theta_s {\rm d} z_s,\end{displaymath} (8)
where $\theta_s$ and $\theta_g$represent angles of the source and receiver propagation direction with respect to the z-axis, respectively. Replacing trigonometric expressions in equations (7) and (8) with ray parameter equivalents, and by downward continuing the source and receiver wavefields by equal depth steps (i.e. ${\rm d}z={\rm
 d}z_g={\rm d}z_s$) one may rewrite equation (2) as
\frac{\partial t}{\partial z}=
\frac{p^2 v_s} {\sqrt{1-p^2 v...
 ...t{1-p_{scat}^2 v_g^2}} +
\frac{\sqrt{1-p_{scat}^2 v_g^2}}{v_g}.\end{displaymath} (9)
Equation (9) demonstrates that the effects of the propagating source wavefield are transferred to the receiver wavefield. Accordingly, using the chain rule and algebraic manipulation, the partial derivative of the receiver wavefield, U, with respect to depth is  
\frac{\partial U}{\partial z}=\frac{\partial U}{\partial t}\...
 ...qrt{1-p_{scat}^2 v_g^2}} \right)
\frac{\partial U}{\partial t}.\end{displaymath} (10)
Relating Fourier domain variables to the angular components of the individual plane waves
p_{scat}(v_g(z))=\frac{sin \theta_g}{v_g}=\frac{k_x}{\omega},\end{displaymath} (11)
and applying a Fourier Transform over the temporal coordinate in equation (10) yields  
\frac{\partial U}{\partial z}= -{\rm i} \omega \left(
 ...2}} +
\frac{\omega} {v_g\sqrt{\omega^2-k_x^2 v_g^2}}
\right) U.\end{displaymath} (12)
The solution to differential equation (12) expressed in a discrete sense is,
U(z+\Delta z, \omega, p)=U(z,\omega,p) \hspace{0.1in}{\rm ex...
\frac{\omega} {v_g\sqrt{\omega^2-k_x^2 v_g^2}}\right) \right].\end{displaymath} (13)
With this expression, the receiver wavefield is downward continued and the image of the reflector at each model point (x,z) constructed through the usual imaging condition Claerbout (1971),
I(x,z)=\sum_{\omega} U(s=x,g=x,z,\omega).\end{displaymath} (14)
Although the method derived here is strictly for 2-D geometry, it may be extended to account for out-of-plane propagation by assuming an underlying 2-D medium and accounting for the effects of 2.5-D propagation through incorporation of the invariant cross-line horizontal slowness. This method is also potentially applicable in 3-D, but this is not explored here.

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