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Nonzero-offset ray

To determine uniquely the remaining coefficients $B$ and $C$, we propose to use just one additional ray reflected at a nonzero offset. Suppose that a reflection ray with the ray parameter $P$ arrives at the offset $X$ and traveltime $T$. Substituting approximation 2 into equations $t(X)=T$ and $dt/dX=P$ and solving for $B$ and $C$ produces the explicit analytical solution

$\displaystyle B$ $\textstyle =$ $\displaystyle \frac{t_0^2\,(X - P\,T\,v^2)}{X\,(t_0^2-T^2+P\,T\,X)} -
\frac{A\,X^2}{X^2 + v^2\,(t_0^2-T^2)}\;,$ (22)
$\displaystyle C$ $\textstyle =$ $\displaystyle \frac{t_0^4\,(X - P\,T\,v^2)^2}{X^2\,(t_0^2-T^2+P\,T\,X)^2} +
\frac{2\,A\,v^2\,t_0^2}{X^2 + v^2\,(t_0^2-T^2)}\;.$ (23)




2013-07-26