Velocity-independent time-domain seismic imaging using local event slopes

Non-hyperbolic moveout

The hyperbolic model 1 is not accurate at large offsets in the case of non-hyperbolic moveouts, caused by vertical or lateral heterogeneity, reflector curvature, or anisotropy (Fomel and Grechka, 2001). One popular model for describing non-hyperbolic moveouts was developed by Malovichko (1978) and has the form of a shifted hyperbola (de Bazelaire, 1988; Castle, 1994; Siliqi and Bousquié, 2000)

$$t(l) = t_0\,\left(1-\frac{1}{S(t_0)}\right) + \frac{1}{S(t_0)}\sqrt{t_0^2 + S(t_0)\,\frac{l^2}{v_n^2(t_0)}}\;,$$ (9)

Equation 9 contains an additional parameter $S$, which is related to heterogeneity and anisotropy of seismic velocities. We can eliminate this parameter by differentiating the equation twice and defining the second derivative $q = \partial p/\partial l = \partial^2 t/ \partial l^2$. Eliminating both $v_n$ and $S$ from equation 9 and equations

$$p = \frac{l}{\left[t_0 + S(t_0)\,\left(t-t_0\right)\right]\,v_n^2}\;,$$ (10)

$$q = \frac{t_0^2}{\left[t_0 + S(t_0)\,\left(t-t_0\right)\right]^3\,v_n^2}\;$$ (11)

leads to the velocity-independent non-hyperbolic moveout equation

$$t_0 = t - \frac{p\,l}{1 + \sqrt{\frac{q\,l}{p}}}\;.$$ (12)

If moveout parameters $v_n$ and $S$ are required for subsequent interpretation, one can easily extract them as special data attributes

$$\frac{1}{v_n^2} = t_0\,\sqrt{\frac{p^3}{q\,l^3}}\;,$$ (13)

$$S = 1 + \frac{p\,(t - p\,l) - q\,l\,t}{\sqrt{q\,p^3\,l^3}}\;.$$ (14)

One could estimate the function $q(t,l)$ in practice by numerically differentiating the local slope field $p(t,l)$.

Velocity-independent time-domain seismic imaging using local event slopes