Finite-offset rays

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Finite-offset rays

Suppose that each independent -th ray corresponds to ray parameters $P_{xi}$ and $P_{yi}$ and arrives at offset and with reflection traveltime . The index ranges from 1 to 4 and denotes the associated ray direction of -axis, -axis, , and respectively. Substituting moveout approximation 1 into equations and $dt/dX_1=P_{x1}$ and solving for and , we have, from the ray along -axis ( ) (Fomel and Stovas, 2010):

$\displaystyle B_1$	$\displaystyle =$	$\displaystyle \frac{t_0^2(W_1X_1-P_{x1} T_1)}{X_1 (t_0^2-T_1^2+P_{x1} T_1X_1)} + \frac{W_1A_1 X_1^2}{T_1^2-t^2_0-W_1X_1^2}~,$	(15)
$\displaystyle C_1$	$\displaystyle =$	$\displaystyle \frac{t_0^4(W_1X_1-P_{x1} T_1)^2}{X_1^2 (t_0^2-T_1^2+P_{x1} T_1X_1)^2} + \frac{2A_1 t^2_0}{t^2_0-T_1^2+W_1X_1^2}~.$	(16)

Analogously,

and

can be found from solving equations

and $dt/dY_2=P_{y2}$ , which is equivalent to replacing

, $P_{x1}$ , and

with

, $P_{y2}$ , and

respectively in equations 15 and 16. The remaining coefficients:

, and

can be solved numerically from the four conditions given below:

$\displaystyle \frac{\partial t}{\partial y}\vert _{y=0,~x=X_1} = P_{y1} ~,$			(17)
$\displaystyle \frac{\partial t}{\partial x}\vert _{x=0,~y=Y_2} = P_{x2} ~,$			(18)
$\displaystyle t(X_3,X_3) = t(Y_3,Y_3) = T_3 ~,$			(19)
$\displaystyle t(X_4,-X_4) = t(-Y_4,Y_4) = T_4 ~.$			(20)

They represents matchings of $P_{y1}$ and $P_{x2}$ along rays in

and

directions and traveltime

and

along rays in

and

directions. Provided the above information from the zero-offset ray and four finite-offset rays, we can define the remaining parameters appearing in the proposed moveout approximation (equation 1) in a systematic manner.

Next: Accuracy tests Up: General method for parameter Previous: Zero-offset ray

2017-04-20