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DSR equation

Constant velocity prestack migration in offset-midpoint coordinates (Yilmaz, 1979) can be formulated as:  
 \begin{displaymath}
p(t=0,k_y,h=0,z)=
{\int d\omega \int d k_h \; e^{ik_z(\omega,k_y,k_h)z}
p(\omega,k_y,k_h,z=0)},\end{displaymath} (1)
where $p(\omega,k_y,k_h,z=0)$ is the 3-D Fourier transform of the field p(t,y,h,z=0) recorded at the surface, using Claerbout's (1985) sign convention:

\begin{displaymath}
p(\omega,k_y,k_h,z=0)= 
\int dt \; e^{i\omega t} \int dy e^{-ik_yy} \int dh e^{-ik_hh} 
p(t,y,h,z=0).\end{displaymath}

The phase $k_z(\omega,k_y,k_h)$ is defined in the dispersion relation as  
 \begin{displaymath}
{k_z(\omega,k_y,k_h)} \equiv
{ -{\rm sign}(\omega) \left\{ \...
 ...r v^2} - 
{1 \over 4}(k_y-k_h)^2\right]^{1 \over 2} \right\} },\end{displaymath} (2)
which is also referred to as the double square-root (DSR) equation. The two integrals in $\omega$ and kh in equation (1) represent the imaging condition for zero-offset and zero time (h=0,t=0).

In equation (1), the values of kz have to be real. Imaginary values of kz do not satisfy the downward continuation ordinary differential equation  
 \begin{displaymath}
{{\partial^2 P} \over {\partial z^2}}=
-k_z^2 P\end{displaymath} (3)
and have to be excluded. Real values of kz in equation (2) require both of the following conditions to be satisfied:

\begin{displaymath}
\left \{
\begin{array}
{lcl}
{2 \over v}\mid \omega \mid & \...
 ...mid \omega \mid & \geq & \mid k_y-k_h \mid .\end{array}\right .\end{displaymath}

By considering the four possible sign cases, given by assigning ky and kh positive and negative values, these two conditions can be reduced to the condition:  
 \begin{displaymath}
\begin{array}
{lcl}
{2 \over v}\mid \omega \mid & \geq & \mid k_y \mid + \mid k_h \mid.\end{array}\end{displaymath} (4)
Equation (4) will prove to be crucial in determining the source of artifacts that appear in constant-offset migrated sections (Figure [*]a).

The prestack DSR algorithm using constant sampling in kh can be summarized as:


		FFT along all axes $p(t,y,h) \rightarrow P(\omega,k_y,k_h)$ 
		do z 
		 		do ky 
		 		do $\omega$ 
		 		do kh 
		 		if$({2\over v}{\rm abs}(\omega)\gt{\rm abs}(k_y)+{\rm abs}(k_h))$ then
		 		 		$P(\omega,k_y,k_h)=P(\omega,k_y,k_h) e^{ik_z dz}$ 
		 		 		$M(z,k_y)=M(z,k_y)+P(\omega,k_y,k_h)$ 
		 		endif

Note that in order to perform an FFT along the offset axis, the variable kh is evenly sampled between the Nyquist negative and positive values. However, due to condition (4), for each set of $\omega,k_y$ the loop in kh will use either a subset or all of the possible kh sampled values.


previous up next print clean
Next: OFFSET SEPARATION Up: Introduction Previous: Introduction
Stanford Exploration Project
11/16/1997