Generalized nonhyperbolic moveout approximation

NONHYPERBOLIC MOVEOUT APPROXIMATION

Let $t(x)$ represent the reflection traveltime as a function of the source-receiver offset $x$. We propose the following general form of the moveout approximation:

$$t^2(x) \approx (1-\xi)\,(t_0^2+a\,x^2) + \xi\,\sqrt{t_0^4 + 2\,b\,t_0^2\,x^2 + c\,x^4}\;.$$ (1)

The five parameters $a$, $b$, $c$, $\xi$, and $t_0$ describe the moveout behavior. By simple algebraic manipulations, one can also rewrite equation 1 as

$$t^2(x) \approx t_0^2+\frac{x^2}{v^2} + \frac{A\,x^4} {\displ... ... 2\,B\,t_0^2\,\frac{x^2}{v^2} + C\,\frac{x^4}{v^4}}\right)}\;,$$ (2)

where the new set of parameters $A$, $B$, $C$, $v$, and $t_0$ is related to the previous set by the equalities

$$a = \frac{A\,B+B^2-C}{v^2\,\left(A+B^2-C\right)}\;;$$ (3)

$$b = \frac{B}{v^2}\;;$$ (4)

$$c = \frac{C}{v^4}\;;$$ (5)

$$\xi = \frac{A}{C-B^2}\;.$$ (6)

The inverse transform is given by

$$v^2 = \frac{1}{a\,(1-\xi) + b\,\xi}\;;$$ (7)

$$A = \frac{\xi\,\left(c - b^2\right)} {\left[a\,(1-\xi) + b\,\xi\right]^2}\;;$$ (8)

$$B = \frac{b}{a\,(1-\xi) + b\,\xi}\;;$$ (9)

$$C = \frac{c}{\left[a\,(1-\xi) + b\,\xi\right]^2}\;.$$ (10)

The existence of the nonhyperbolic part in the traveltime approximation 1 and 2 is controlled by parameter $A$. When $A$ is zero (which implies that $\xi=0$ or $c=b^2$), approximation 1 is hyperbolic. When both $B$ and $C$ are very large, approximation 2 also reduces to the hyperbolic form.

Generalized nonhyperbolic moveout approximation