Generalized nonhyperbolic moveout approximation

Connection with other approximations

Equations 1-2 reduce to some well-known approximations with special choices of parameters.

• If $A=0$, the proposed approximation reduces to the classic hyperbolic form
  

  $$t^2(x) \approx t_0^2 + \frac{x^2}{v^2}\;,$$ (11)

  
  
  which is a two-parameter approximation.

• The choice of parameters $A=(1-s)/2$; $B=s/2$; $C=0$ reduces the proposed approximation to the shifted hyperbola (Malovichko, 1978; de Bazelaire, 1988; Castle, 1994), which is the following three-parameter approximation:
  

  $$t(x) \approx t_0\,\left(1-\frac{1}{s}\right) + \frac{1}{s}\,\sqrt{t_0^2+s\,\frac{x^2}{v^2}}\;.$$ (12)

  
  

• The choice of parameters $A = - 4\,\eta$; $B = 1 + 2\,\eta$; $C=(1 + 2\,\eta)^2$ reduces approximation 2 to the form proposed by Alkhalifah and Tsvankin (1995) for VTI media, which is the following three-parameter approximation:
  

  $$t^2(x) \approx t_0^2 + \frac{x^2}{v^2} - \frac{2\,\eta\... ...yle v^4\,\left[t_0^2 + (1+2\,\eta)\,\frac{x^2}{v^2}\right]}\;.$$ (13)

  
  

• The choice of parameters $A = - 2\,\gamma\,t_0^2\,v^2$; $B = -A/2$; $C=A^2/4$ reduces approximation 2 to the following three-parameter approximation suggested by Blias (2007) and reminiscent of the ``velocity acceleration'' equation proposed by Taner et al. (2005,2007):
  

  $$t^2(x) \approx t_0^2 + \frac{x^2}{v^2\,(1+\gamma\,x^2)}\;.$$ (14)

  
  

• The choice of parameters $A=(1-s)/2$; $B=1$; $C=2-s$ reduces the proposed approximation to the following three-parameter approximation suggested by Blias (2009):
  

  $$t(x) \approx \frac{1}{2}\,\sqrt{t_0^2+\left(1-\sqrt{s-1... ...\,\sqrt{t_0^2+\left(1+\sqrt{s-1}\right)\,\frac{x^2}{v^2}} \;.$$ (15)

  
  

• The choice of parameters $B=0$; $C=2\,A$ reduces the proposed approximation to the following three-parameter approximation also suggested by Blias (2009):
  

  $$t^2(x) \approx \frac{t_0^2}{2} + \frac{x^2}{v^2} + \frac{1}{2}\,\sqrt{t_0^4 + \frac{2\,A\,x^4}{v^4}}\;.$$ (16)

  
  

• The choice of parameters $A = 2\,\tan^2{\theta}$, $B=1-\tan^2{\theta}$, $C=1/\cos^4{\theta}$ reduces the proposed approximation to the double-square-root expression
  

  $$t(x) \approx \frac{1}{2}\, \sqrt{t_0^2 + \frac{x\,(x+t_0\,v\,\sin{2\theta})}{... ...}{2}\, \sqrt{t_0^2 + \frac{x\,(x-t_0\,v\,\sin{2\theta})}{v^2\,\cos^2{\theta}}}$$

  $$= \frac{\sqrt{z^2 + (y+x/2)^2}}{V} + \frac{\sqrt{z^2 + (y-x/2)^2}}{V}\;,$$ (17)

  
  
  where $V = v\,\cos{\theta}$, $z=(t_0\,V/2)\,\cos{\theta}$, and $y=(t_0\,V/2)\,\sin{\theta}$. Equation 17 describes moveout precisely for the case of a diffraction point in a constant velocity medium.

Thus, the proposed approximation encompasses some other known forms but introduces more degrees of freedom for optimal fitting.

Generalized nonhyperbolic moveout approximation