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Finite-offset rays

Suppose that each independent $ i$ -th ray corresponds to ray parameters $ P_{xi}$ and $ P_{yi}$ and arrives at offset $ X_i$ and $ Y_i$ with reflection traveltime $ T_i$ . The $ i$ index ranges from 1 to 4 and denotes the associated ray direction of $ x$ -axis, $ y$ -axis, $ x=y$ , and $ x=-y$ respectively. Substituting moveout approximation 1 into equations $ t(X_1,0)=T_1$ and $ dt/dX_1=P_{x1}$ and solving for $ B_1 $ and $ C_1$ , we have, from the ray along $ x$ -axis ($ i=1$ ) (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, $ B_3$ and $ C_5$ can be found from solving equations $ t(0,Y_2)=T_2$ and $ dt/dY_2=P_{y2}$ , which is equivalent to replacing $ X_1$ , $ P_{x1}$ , and $ W_1$ with $ Y_2$ , $ P_{y2}$ , and $ W_3$ respectively in equations 15 and 16. The remaining coefficients: $ B_2$ , $ C_2$ , $ C_3$ , and $ C_4$ 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 $ x$ and $ y$ directions and traveltime $ T_3$ and $ T_4$ along rays in $ x=y$ and $ x=-y$ 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.


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

2017-04-20