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APPENDIX B - THE FORWARD LINEARIZED DOWNWARD CONTINUATION OPERATOR

Downward continuation can be done in the Fourier domain as a phase shift applied to Uz, the wavefield at depth z Claerbout (1985):  
 \begin{displaymath}
U_{z + \Delta z} = e^{i k_z \Delta z} U_z\end{displaymath} (6)
The vertical wavenumber kz depends on the laterally varying velocity. In order for us to be able to implement the formula, we have to decompose kz into a part not influenced by lateral velocity variations (kzo) and a part influenced by them (kzx):  
 \begin{displaymath}
k_z = {k_z}_o + {k_z}_x \Rightarrow U_{z + \Delta z} = e^{i ...
 ...+ \Delta z} = e^{i {k_z}_o
\Delta z} e^{i {k_z}_x \Delta z} U_z\end{displaymath} (7)
The Born approximation is equivalent to a linearization of the exponential $e^x \approx 1 + x$, and therefore  
 \begin{displaymath}
 U_{z + \Delta z} \approx e^{i {k_z}_o \Delta z} U_z ( 1 + i
 {k_z}_x \Delta z ) \end{displaymath} (8)
In the case of the complexified local Born-Fourier (complexified pseudo-screen) method, with the notations in equations (12) and (13) of Sava (2000), we can rewrite it as:  
 \begin{displaymath}
 U_{z + \Delta z} \approx {\mathcal T} U_z \left[ 1 + {\mathcal S} (s
 - s_o) \right]\end{displaymath} (9)
where ${\mathcal T}$ is the background wavefield downward continuation operator applied in the $\omega - {\bf k_m}$domain:  
 \begin{displaymath}
{\mathcal T} = e^{i \Delta z \sqrt{{\omega}^2{s_o}^2 - {(1 - i
\eta)}^2 {\bf \left\vert k_m \right\vert}^2}} ,\end{displaymath} (10)
${\mathcal S}$ is the scattering operator, applied in the $\omega - x$domain:  
 \begin{displaymath}
{\mathcal S} = \frac{i \Delta z {\omega}^2
s_o}{\sqrt{{\omeg...
 ...{s_o}^2 - {(1 - i \eta)}^2 {\bf \left\vert k_m \right\vert}^2}}\end{displaymath} (11)
and where s is the slowness at the depth $z + \Delta z$, ${\bf k_m}$is the wavenumber across the midpoint direction (scalar for 2D, vector for 3D), so is the constant background slowness, $\omega$ is the frequency, and $\eta$ is a small dimensionless quantity introduced for numerical stability; ${\bf k_m}$ and $\omega$ must contain a $2 \pi$constant. The output of this operator can be seen in Figure 10.

Although with equation (9) we went a step towards linearity with respect to the slowness perturbation term, it is not fully linear because the slowness perturbations compose with themselves. This is visible if we examine the first two steps of the downward continuation. At z=0, Uz=0 = Data (Ricker wavelet at zero-time in the middle of the x axis). At $z = \Delta z$, 
 \begin{displaymath}
U_{z = \Delta z} = {\mathcal T} U_{z=0} + {\mathcal S}{\Delta s}_{z =
\Delta z} {\mathcal T} U_{z=0}.\end{displaymath} (12)
At $z = 2 \Delta z$, 
 \begin{displaymath}
U_{z = 2 \Delta z} = {\mathcal T} U_{z = \Delta z} + {\mathcal
S}{\Delta s}_{z = \Delta z} {\mathcal T} U_{z = \Delta z},\end{displaymath} (13)
and by plugging in the expression for $U_{z = \Delta z}$ and because ${\mathcal T}$ and ${\mathcal S}$ do not commute,  
 \begin{displaymath}
U_{z = 2 \Delta z} = {\mathcal T}{\mathcal T}U_{z=0} + {\mat...
 ...al T}{\mathcal S}{\Delta s}_{z = 2 \Delta
z}{\mathcal T}U_{z=0}\end{displaymath} (14)
In order to obtain a downward continuation that is linear in the slowness perturbations $\Delta s$, we have to drop the last term at each step. Thus, after the nth depth step, the wavefield will be:  
 \begin{displaymath}
U_{z = n\Delta z} = \left( {\prod\limits_1^n {\mathcal T} }
...
 ...t( {\prod\limits_1^j {\mathcal T} } \right)U_{z = 0} } \right]}\end{displaymath} (15)
The above formula is equivalent with stating that at each level, we compute the scattered wavefield only from the background wavefield from the previous level, then we propagate it down until the last level with the background operator. The results of this approach are visible in Figure 5.


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Next: : APPENDIX C - Up: Vlad: Velocity estimation for Previous: APPENDIX A - Derivation
Stanford Exploration Project
11/11/2002