Predicting rugged water-bottom multiples through wavefield extrapolation with rejection and injection

Recursive Kirchhoff wavefield extrapolation

Theoretically, wavefield rejection and injection can be carried out for any of the wavefield extrapolation algorithms. However, the computational efficiency can differ greatly. Recursive Kirchhoff wavefield extrapolation in both space-frequency and space-time domain is the most suitable algorithm.

Because the basic theory and extrapolation operator based on the Kirchhoff integral are widely known (Schneider, 1978; Margrave and Daley, 2001; Berkhout, 1981), I derive only the formulae corresponding to upward and downward wavefield extrapolation of both the up-going and down-going waves.

In general, the wavefield extrapolation from depth ${\bf z}$ to ${\bf z+\Delta{z}}$ can be written as follows:

$$P^+\left(k_x,k_y,z\pm\Delta{z},\omega\right) = P^+\left(k_x,k_y,z,\omega\right)e^{-ik_z(\pm\Delta{z})},$$ (1)

$$P^-\left(k_x,k_y,z\pm\Delta{z},\omega\right) = P^-\left(k_x,k_y,z,\omega\right)e^{+ik_z(\pm\Delta{z})},$$ (2)

where $P^+\left(k_x,k_y,z,\omega\right)$ and $P^-\left(k_x,k_y,z,\omega\right)$ are the Fourier transform over $x$ , $y$ and $t$ of the down-gong and up-going waves at position ($x$ ,$y$ ,$z$ ), respectively. The terms $k_x$ , $k_y$ and $k_z$ are the three components of the wavenumber vector, and $\omega$ is the angular frequency. The sign $\pm$ before the depth interval $\Delta{z}$ relates to the upward and downward wavefield extrapolations. Thus, there are four types of wavefield extrapolations in total.

On the other hand, the Kirchhoff integral in the space-frequency domain is

$$\tilde{P}\left(r_A,\omega\right) = \oint_{s}\left[\tilde{G}\left(... ...ht) \ \frac{\partial\tilde{G}\left(r,r_A,\omega\right)}{\partial{n}}\right]ds,$$ (3)

where $r$ and $r_A$ are the shorthand notations of ($x$ ,$y$ ,$z$ ) and ($x_A$ ,$y_A$ ,$z_A$ ) . The term $n$ is the outward normal of the surface $S$ . The wavefield $\tilde{P}\left(r,\omega\right)$ and Green function $\tilde{G}\left(r,r_A,\omega\right)$ satisfy the following Helmholtz equations:

$$\nabla^2{\tilde{P}\left(r,\omega\right)} + \frac{\omega^2}{c^2}\tilde{P}\left(r,\omega\right) =0,$$ (4)

$$\nabla^2{\tilde{G}\left(r,r_A,\omega\right)} + \frac{\omega^2}{c^2}\tilde{G}\left(r,r_A,\omega\right)= \delta\left(r-r_A)\right),$$ (5)

where $c$ is wave propagation velocity. Obviously, it is not easy to relate equation 3 to the upward and downward wavefield extrapolation of both the up-going and down-going waves. Suppose $S$ consists of a horizontal $S_0$ surface at $z=z_n$ and a hemisphere $S_1$ which contains point A and satisfies the Sommerfeild radiation condition. Transforming the Kirchhoff integral from the ($x$ ,$y$ ,$z$ ,$\omega$ ) domain into ($k_x$ ,$k_y$ ,$z$ ,$\omega$ ) domain, equation 3 becomes (Berkhout, 1989)

$$P\left(r_A,\omega\right) = \frac{1}{\left(2\pi\right)^2}\int\int\... ...artial{P}}{{\partial{z}}} \ - P\frac{\partial{G}}{\partial{z}}\right]dk_xdk_y,$$ (6)

where $G$ and $P$ represent $G\left(-k_x,-k_y,z,r_a,\omega\right)$ and $P\left(k_x,k_y,z,\omega\right)$ . Furthermore, equation 6 can be mathematically expressed by

$$P\left(r_A,\omega\right) = \frac{1}{\left(2\pi\right)^2}\int\int\... ...(P^++P^-\right)\frac{\partial\left(G^++G^-\right)}{\partial{z}}\right]dk_xdk_y,$$ (7)

where $P^+$ and $G^+$ are the down-going wave and Green’s function respectively, and $P^-$ and $G^-$ are the up-going wave and Green’s function respectively.

Using the definitions of the down-going and up-going waves,

$$\frac{\partial{P^\pm}}{{\partial{z}}}=\mp{k_z}P^\pm,$$ (8)

$$\frac{\partial{G^\pm}}{{\partial{z}}}=\mp{k_z}G^\pm,$$ (9)

Equation 7 can be simplified to

$$P\left(r_A,\omega\right) = \frac{1}{\left(2\pi\right)^2}\int\int\... ...tial{z}}} \ -P^-\frac{\partial{G^+}}{{\partial{z}}}\right)\right]dk_xdk_y, \$$ (10)

and furthermore to

$$P\left(r_A,\omega\right) = \frac{1}{2\pi^2}\int\int\left[P^+\frac... ...{{\partial{z}}} \ +P^-\frac{\partial{G^+}}{{\partial{z}}} \right]dk_xdk_y, \$$ (11)

Thus the four types of wavefield extrapolations in terms of the Kirchhoff integral can be easily defined based on equation 11, as shown in figure 3.

In practice, generally we transform the equation 11 from the ($k_x$ ,$k_y$ ,$z$ ,$\omega$ ) domain back into the($x$ ,$y$ ,$z$ ,$\omega$ ) or ($x$ ,$y$ ,$z$ ,$t$ ) domain In this case, the down-going and up-going Green’s functions will change accordingly, as shown in figure 3. The terms $G^{out}$ and $G^{in}$ are the causal and anticausal Green’s functions , or the “out-going ” and “in-going ” waves respectively.

After the above derivation, we have all types of wavefield extrapolations in terms of the Kirchhoff integral. In water-bottom multiple prediction, we use only types (a) and (c), that is, the downward wavefield extrapolation of the down-going wave and the upward wavefield extrapolation of the up-going wave.

vel
Figure 4. The structure and interval velocity of the geological model. [NR]

shotmult1
Figure 5. The shot gather at the leftmost location(left) and predicted multiples (right). [NR]

shotmult2
Figure 6. The shot gather at the central location (left) and predicted multiples (right). [NR]

Predicting rugged water-bottom multiples through wavefield extrapolation with rejection and injection