THEORY OF 2.5-D KIRCHHOFF INVERSION IN V(x,z) MEDIA

In the spatial and frequency domain, the 3-D acoustic wave equation can be formulated as

$$\left [\nabla^2+\frac{\omega^2}{c^2({\bf x})} \right ] G({\bf x},{\bf x}_s,\omega) = -\delta({\bf x}-{\bf x}_s)$$ (1)

where $G({\bf x},{\bf x}_s,\omega)$ can be approximated by the WKBJ Green's function

$$G({\bf x},{\bf x}_s,\omega) \sim A({\bf x},{\bf x}_s) e^{i\omega \tau({\bf x}, {\bf x}_s)}$$ (2)

where $\tau({\bf x}, {\bf x}_s)$ is the traveltime from source ${\bf x}_s$ to an arbitrary point ${\bf x}$. Using the WKBJ Green's function, Beylkin 1985 gave an inversion formula in 3-D media

$$\alpha({\bf x}) \sim \frac{c^2({\bf x})}{8\pi^3}\int \int_{S... ...},{\bf x}_s)+\tau({\bf x},{\bf x}_r) \right ]} D(\omega,\xi).$$ (3)

In the above formula, $\alpha({\bf x})$ is the perturbation to the background velocity $c({\bf x})$. The updated velocity model is given by

$$v^{-2}({\bf x})=c^{-2}({\bf x})\left [1+\alpha({\bf x}) \right ].$$ (4)

S₀ is the 2-D integral surface. $h({\bf x},\xi)$ is introduced by Beylkin 1985, which is associated with the ray curvature. $A({\bf x},{\bf x}_s)$ and $A({\bf x},{\bf x}_r)$ are the WKBJ Green's function. $F(\omega)$ is a high-pass filter determined the source. $D(\omega,\xi)$ represents the observed data at ${\bf x}_r$ due to the source ${\bf x}_s$.

Bleistein et al.1987 specialize the 3-D formula to the 2.5-D geometry using the method of stationary phase. The corresponding 2.5-D inversion formula is

$$\begin{eqnarray} \alpha({\bf x}) & \sim & 2\sqrt{\frac{2}{\pi}} \int d\xi \lef... ...{i\omega}} F(\omega) e^{-i\omega (\tau_s+\tau_r)} D(\omega, \xi)\end{eqnarray}$$ (5)

Here, ${\bf p}_s$ and ${\bf p}_r$ are the slowness vectors at the imaging location pointing to the source and receiver respectively. $\sigma_{s_0}$ and $\sigma_{r_0}$ are the parameters defined by the following equations

$$\begin{eqnarray} \sigma_{s_0} = \int_0^{\tau_s} c({\bf x}) d\tau, & & \sigma_{r_0} = \int_0^{\tau_r} c({\bf x}) d\tau.\end{eqnarray}$$ (6)

$\hat{\bf n}_s$ and $\hat{\bf n}_r$ are unit downward normals at the source and receiver points respectively. ${\bf p}_{s_0}$ and ${\bf p}_{r_0}$ are the slowness vectors at the source and receiver points respectively.

This inversion formula is only valid in the high-frequency limit. Under such circumstances, it is better to process data for the upward normal derivative ${{\partial{\alpha}}/{\partial{n}}}$ at each discontinuity surface of $\alpha({\bf x})$. ${{\partial{\alpha}}/{\partial{n}}}$ is a sum of weighted singular functions with peaks on the reflectors. Therefore, ${{\partial{\alpha}}/{\partial{n}}}$ actually provides an image of the subsurface. Using the Fourier transform, we can obtain the following 2.5-D formula for ${{\partial{\alpha}}/{\partial{n}}}$.

$$\begin{eqnarray} \frac{\partial{\alpha}}{\partial{n}}({\bf x}) & \sim & \frac{... ...qrt{i\omega} F(\omega) e^{-i\omega(\tau_s+\tau_r)} D(\omega, \xi)\end{eqnarray}$$ (7)

Bleistein et al.1987 also shows that ${{\partial{\alpha}}/{\partial{n}}}$ can be related to the reflection coefficient on the interface by

$$\frac{\partial{\alpha}}{\partial{n}} \sim 4\cos^2{\theta} R({\bf x}, \theta) \gamma({\bf x})$$ (8)

$\gamma({\bf x})$ in the singular function of the model space. $R({\bf x}, \theta)$ is determined by the changes of velocity and density above and below the interface and the incident angle on the interface

$$R({\bf x}, \theta) = \frac{c_{down}({\bf x})\rho_{down}\cos{... ...bf x})-c^2_{down}({\bf x}) \sin^2{\theta}}}. \EQNLABEL{r_coeff}$$ (9)

In order to determine $R({\bf x}, \theta)$ from ${{\partial{\alpha}}/{\partial{n}}}$, we have to determine $\cos{\theta}$. In their paper, Bleistein et al.1987 proposed another inversion operator $\beta(x,z)$ with a kernel slightly modified from that in ${{\partial{\alpha}}/{\partial{n}}}$.

$$\begin{eqnarray} \beta({\bf x})& \sim & \frac{2}{\sqrt{\pi}c({\bf x})}\int d\xi ... ...qrt{i\omega} F(\omega) e^{-i\omega(\tau_s+\tau_r)} D(\omega, \xi)\end{eqnarray}$$ (10)

There is a simple relation between ${{\partial{\alpha}}/{\partial{n}}}$, $\beta(x,z)$, and $\cos{\theta}$, that is

$$\cos^2{\theta} = \frac{\frac{\partial{\alpha}}{\partial{n}}(peak)} {4\beta(peak)}.$$ (11)

With $\cos{\theta}$ known, we can use ${{\partial{\alpha}}/{\partial{n}}}$and $\cos{\theta}$ to calculate the reflection coefficient $R({\bf x}, \theta)$. From $R({\bf x}, \theta)$ and $\cos{\theta}$, we can further estimate the AVO coefficients: intercept and slope.

Instead of using ${{\partial{\alpha}}/{\partial{n}}}$ and $\beta(x,z)$, we propose another pair of inversion operators that can determine $\cos{\theta}$ in a similar, but more straightforward and physically meaningful manner. The first operator gives the reflection coefficient at the specular incident angle

$$\begin{eqnarray} R({\bf x},\theta)& \sim &\frac{1}{\sqrt{\pi}c({\bf x})}\int d\x... ...i\omega(\tau_s+\tau_r)}D(\omega,\xi) \EQNLABEL{kirchhoff-integral}\end{eqnarray}$$ (12)

The second gives the reflection coefficient multiplied by $\cos{\theta}$

$$\begin{eqnarray} R^{\prime}({\bf x},\theta) & \sim & \frac{1}{\sqrt{2\pi}c({\bf ... ...sqrt{i\omega} F(\omega) e^{-i\omega(\tau_s+\tau_r)} D(\omega, \xi)\end{eqnarray}$$ (13)

From $R({\bf x}, \theta)$ and $R^{\prime}({\bf x},\theta)$, we can easily calculate $\cos{\theta}$

$$\cos{\theta} = \frac{R^{\prime}({\bf x},\theta)}{R({\bf x},\theta)}.$$ (14)

In order to reduce the sensitivity of $\cos{\theta}$ to noise in the data, we use a least-squares procedures to estimate $\cos{\theta}$. First, we define a small window (n_x $\times$ n_z). Within the window, we can get a series of equations

$$\begin{array} {lll} R(x_1,z_1,\theta) \cos{\theta} & = & R^... ...eta) \cos{\theta} &=& R^{\prime}(x_n,z_n,\theta) \\ \end{array}$$ (15)

the least-squares sense estimate of $\cos{\theta}$ is then

$$\cos{\theta} = \frac{\sum_{x_i,z_i}^{n} R(x_1,z_1,\theta) R... ...rime}(x_1,z_1,\theta)}{\sum_{x_i,z_i}^{n} R^2(x_1,z_1,\theta)}.$$ (16)