High-order kernels for Riemannian Wavefield Extrapolation

Mixed-domain extrapolation

Mixed-domain solutions to the one-way wave equation consist of decompositions of the extrapolation wavenumber defined in equation (13) in terms computed in the Fourier domain for a reference of the extrapolation medium, followed by a finite-differences correction applied in the space-domain. For equation (13), a generic mixed-domain solution has the form:

$$k_\tau \approx {k_\tau }_0+ \omega \left (a-a_0\right )+ \... ...{\mu-\rho \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;,$$ (21)

where $a_0$ and $b_0$ are reference values for the medium characterized by the parameters $a$ and $b$, and the coefficients $\mu$, $\nu$ and $\rho$ take different forms according to the type of approximation. As for usual Cartesian coordinates, ${k_\tau }_0$ is applied in the Fourier domain, and the other two terms are applied in the space domain. If we limit the space-domain correction to the thin lens term, $\omega \left (a-a_0\right )$, we obtain the equivalent of the split-step Fourier (SSF) method (Stoffa et al., 1990) in Riemannian coordinates.

Appendix A details the derivations for two types of expansions known by the names of pseudo-screen (Huang et al., 1999), and Fourier finite-differences (Biondi, 2002; Ristow and Ruhl, 1994). Other extrapolation approximations are possible, but are not described here, for simplicity.

• Pseudo-screen method:

  The coefficients for the pseudo-screen approximation to equation (21) are
  

  $$\nu = a_0 \left [c_1 \left (\frac{a}{a_0}-1 \right )- \left (\frac{b}{b_0}-1 \right )\right ] \left (\frac{b_0}{a_0}\right )^2\;,$$ (22)

  $$\mu = 1 \;,$$ (23)

  $$\rho = 3c_2 \left (\frac{b_0}{a_0}\right )^2\;,$$ (24)

  
  
  where $a_0$ and $b_0$ are reference values for the medium characterized by parameters $a$ and $b$. In the special case of Cartesian coordinates, $a=s$ and $b=1$, equation (21) with coefficients equation (22) takes the familiar form
  

  $$k_\tau \approx {k_\tau }_0+ \omega \left [1+ \frac{ \frac{... ...ma }{ \omega } \right )^2} \right ] \left (s-s_0 \right )\;,$$ (25)

  
  
  where the coefficients $c_1$ and $c_2$ take different values for different orders of the finite-differences term: $(c_1,c_2)=(0.50,0.00)$, $(c_1,c_2)=(0.50,0.25)$, etc. When $(c_1,c_2)=(0.00,0.00)$ we obtain the usual split-step Fourier equation (Stoffa et al., 1990).

• Fourier finite-differences method:

  The coefficients for the Fourier finite-differences solution to equation (21) are
  

  $$\nu = \frac{1}{2}\delta_1^2 \;,$$ (26)

  $$\mu = \delta_1 \;,$$ (27)

  $$\rho = \frac{1}{4}\delta_2 \;,$$ (28)

  
  
  where, by definition,
  

  $$\delta_1 = a \left (\frac{b }{a }\right )^2- a_0 \left (\frac{b_0}{a_0}\right )^2\;,$$ (29)

  $$\delta_2 = a \left (\frac{b }{a }\right )^4- a_0 \left (\frac{b_0}{a_0}\right )^4\;.$$ (30)

  
  
  $a_0$ and $b_0$ are reference values for the medium characterized by the parameters $a$ and $b$. In the special case of Cartesian coordinates, $a=s$ and $b=1$, equation (21) with coefficients equation (26) takes the familiar form:
  

  $$k_\tau \approx {k_\tau }_0+ \omega \left [1+ \frac{ \frac{... ...ma }{ \omega } \right )^2} \right ] \left (s-s_0 \right )\;,$$ (31)

  
  
  where the coefficients $c_1$ and $c_2$ take different values for different orders of the finite-differences term: $(c_1,c_2)=(0.50,0.00)$ for $15^\circ$, $(c_1,c_2)=(0.50,0.25)$ for $45^\circ$, etc. When $c_1=c_2=0.0$ we obtain the usual split-step Fourier equation (Stoffa et al., 1990).

High-order kernels for Riemannian Wavefield Extrapolation