Non-hyperbolic common reflection surface

Hyperbolic and nonhyperbolic CRS

If $P(t,m,h)$ represents the prestack seismic data as a function of time $t$ , midpoint $m$ and half-offset $h$ , then conventional stacking can be described as

$$S(t_0,m_0) = \int P\left(\theta(h;t_0),m_0,h\right)\,d h\;,$$ (1)

where $S(t,m)$ is the stack section, and $\theta(h;t_0)$ is the moveout approximation, which may take a form of a hyperbola

$$\theta(h;t_0) = \sqrt{t_0^2 + \frac{4\,h^2}{v^2}}$$ (2)

with $v$ as an effective velocity parameter or, alternatively, a more complicated non-hyperbolic functional form, which involves other parameters (Fomel and Stovas, 2010).

The MF or CRS stacking takes a different form,

$$\widehat{S}(t_0,m_0) = \iint P\left(\widehat{\theta}(m-m_0,h;t_0),m,h\right)\,d m\,d h\;,$$ (3)

where the integral over midpoint $m$ is typically carried out only over a limited neighborhood of $m_0$ . The multifocusing approximation of Gelchinsky et al. (1999a) takes the form

$$\widehat{\theta}_{MF}(d,h;t_0) = t_0 + T_{(+)}(d,h) + T_{(-)}(d,h)\;,$$ (4)

where, in the notation of Tygel et al. (1999),

$$T_{(\pm)} = \frac{\sqrt{1+2\,K_{(\pm)}\,(d \pm h)\,\sin{\beta} + K_{(\pm)}^2\,(d \pm h)^2}-1}{V_0\,K_{(\pm)}}\;,$$ (5)

$$K_{(\pm)} = \frac{K_N \pm \sigma\,K_{NIP}}{1 \pm \sigma}\;,$$ (6)

and

$$\sigma(d,h) = \frac{h}{d + K_{NIP}\,\sin{\beta}(d^2-h^2)}\;.$$ (7)

The four parameters $\{K_N,K_{NIP},\beta,V_0\}$ have clear physical interpretations in terms of the wavefront and ray geometries (Gelchinsky et al., 1999a). $V_0$ represents the velocity at the surface and is typically assumed known and constant around the central ray. One important property of the MF approximation is that, in a constant velocity medium with velocity $V_0$ , it can accurately describe both reflections from a plane dipping interfaces and diffractions from point diffractors.

The CRS approximation (Jäger et al., 2001) is

$$\widehat{\theta}_{CRS}(d,h;t_0) = \sqrt{F(d) + b_2\,h^2}\;,$$ (8)

where $F(d) = (t_0 + a_1\,d)^2 + a_2\,d^2$ , and the three parameters $\{a_1,a_2,b_2\}$ are related to the multifocusing parameters as follows:

$$a_1 = \frac{2\,\sin{\beta}}{V_0}\;,$$ (9)

$$a_2 = \frac{2\,\cos^2{\beta}\,K_N\,t_0}{V_0}\;,$$ (10)

$$b_2 = \frac{2\,\cos^2{\beta}\,K_{NIP}\,t_0}{V_0}\;.$$ (11)

Equation (8) is equivalent to a truncated Taylor expansion of the squared traveltime in equation (4) around $d=0$ and $h=0$ . In comparison with MF, CRS possesses a fundamental simplicity, which makes it easy to extend the method to 3-D. However, it looses the property of accurately describing diffractions in a constant-velocity medium.

We propose the following modification of approximation (8):

$$\widehat{\theta}(d,h;t_0) = \sqrt{\frac{F(d) + c\,h^2 + \sqrt{F(d-h)\,F(d+h)}}{2}}\;,$$ (12)

where $c=2\,b_2+a_1^2-a_2$ . Equation (12), which we call non-hyperbolic common reflection surface, is derived in Appendix A. A truncated Taylor expansion of the squared traveltime from equation (12) around $d=0$ and $h=0$ is equivalent to equation (8).

There are two important special cases:

1. If $a_2=0$ or $K_N=0$ , equation (12) becomes equivalent to equation (8), with $F(d) = (t_0 + a_1\,d)^2$ . In a constant-velocity medium, this case corresponds to reflection from a planar reflector.
2. If $a_2=b_2$ or $K_{NIP}=0$ , equation (12) becomes equivalent to

   $$\widehat{\theta}(d,h;t_0) = \frac{\sqrt{F(d-h)}+\sqrt{F(d+h)}}{2}\;.$$ (13)

   
   
   In a constant-velocity medium, this case corresponds to a point diffractor.

Non-hyperbolic common reflection surface