BLEISTEIN'S DERIVATION

Bleistein (1987) proposed an inversion operator to process the seismic data:

 

$$\beta(\vector_seg{x}) \approx \frac{1}{8 \pi^3} \int \!\!\!\int_{S_\xi} d^2\xi \frac{\left\vert... ...x}_s\right)+ \nabla\tau\left(\vector_seg{x},\vector_seg{x}_r\right)\right\vert}$$

$$\times \int i \omega d\omega F(\omega) \: exp \left\{-i\omega\lef... ...\vector_seg{x},\vector_seg{x}_r\right)\right]\right\} D\left(\xi,\omega\right).$$ (6.1)

where $\xi=(\xi_1,\xi_2)$ is a pair of parameters that determines the source and receiver coordinates:

$$\vector_seg{x}_s=\vector_seg{x}_s(\xi),~~~~~\vector_seg{x}_r=\vector_seg{x}_r(\xi).$$

The domain of integration $S_\xi$ is the set of $\xi$ values which are required to cover the source-receiver array. $F(\omega)$ is regarded as a smoothly tapered version of the source wavelet. The functions $\tau(\vector_seg{x},\vector_seg{x}_s)$ and $A(\vector_seg{x},\vector_seg{x}_s) \quad [ \tau(\vector_seg{x},\vector_seg{x}_r)$ and $A(\vector_seg{x},\vector_seg{x}_r)]$ are the solutions to the ray-theoretic eikonal equation and transport equation with the source at $\vector_seg{x}_s~~[\vector_seg{x}_r]$ and observation point at $\vector_seg{x}$, and $D(\xi,\omega)$ is the seismic data. The term $h(\vector_seg{x},\xi)$ is the determinant, i.e.,

$$h(\vector_seg{x},\xi) = \det \left\vert \begin{array}{l} \nab... ...ctor_seg{x},\vector_seg{x}_r)\right] \end{array}\right \vert .$$ (6.2)

The weight function is chosen to give the true ampltude image:

$$w(\vector_seg{x},\xi) = \frac{\left\vert h\left(\vector_seg{x... ...a\tau\left(\vector_seg{x},\vector_seg{x}_r\right)\right\vert}.$$ (6.3)

The integration in equation 6.1 is carried out over the entire data space, that is, the set of $\xi$ that is required to cover the entire source-receiver array. This inversion operator is a Kirchhoff-type integration. Bleistein applied this operator to the upward reflected field which is given by the Kirchhoff approximation :

 

$$D(\xi,\omega) \approx i \omega \int_S R(\vector_seg{x}^\prime,\vector_seg{x}_s)A(\vecto... ...or_seg{x}_s) + \nabla^\prime\tau(\vector_seg{x}^\prime,\vector_seg{x}_r)\right]$$

$$\times \: exp\left \{ i \omega \left [ \tau(\vector_seg{x}^\prime... ...+\tau(\vector_seg{x}^\prime,\vector_seg{x}_r) \right ] \right \} \: dS^\prime ,$$ (6.4)

where $R(\vector_seg{x}^\prime,\vector_seg{x}_s)$ is the angular-dependent reflection coefficient and $\tilde{\boldmath {n}}$ is the upward normal unit vector. Equation 6.4 was plugged into equation 6.1 to yield

 

$$\beta(\vector_seg{x}) \approx \frac{1}{8 \pi^3}\int \!\!\!\int_{S_\xi} d^2\xi \frac{\left\vert ... ...tor_seg{x},\vector_seg{x}_r\right)\right\vert} \int \omega ^2 d\omega F(\omega)$$

$$\times \int_S R(\vector_seg{x}^\prime,\vector_seg{x}_s)A(\vector_... ...vector_seg{x},\vector_seg{x}^\prime,\vector_seg{x}_s,\vector_seg{x}_r)\right \}$$

$$\times \tilde{\boldmath {n}} {\scriptscriptstyle \stackrel{\bulle... ...+ \nabla^\prime\tau(\vector_seg{x}^\prime,\vector_seg{x}_r)\right] \: dS^\prime$$ (6.5)

where

$$\Phi(\vector_seg{x},\vector_seg{x}^\prime,\vector_seg{x}_s,\v... ...ctor_seg{x}_s)+ \tau(\vector_seg{x},\vector_seg{x}_r) \right].$$ (6.6)

Then Bleistein evaluated equation 6.5 at the stationary phase point which yields

 

$$\beta(\vector_seg{x}) \approx - R(\vector_seg{x}^\prime,\vector_seg{x}_s) \frac{A(\vector_seg{x... ...g{x}_r)} {A(\vector_seg{x},\vector_seg{x}_s)A(\vector_seg{x},\vector_seg{x}_r)}$$

$$\times \frac{\left\vert h(\vector_seg{x},\xi)\right\vert} {\left\... ...eg{x},\vector_seg{x}_s)+\nabla\tau(\vector_seg{x},\vector_seg{x}_r)\right\vert}$$

$$\times \tilde{\boldmath {n}} {\scriptscriptstyle \stackrel{\bulle... ...e\tau(\vector_seg{x}^\prime,\vector_seg{x}_r)\right] \sqrt{g}I(\vector_seg{x}).$$ (6.7)

The condition for the stationary phase point is

 

$$\frac{d\Phi}{d\xi_m} \nabla_s\left[\tau(\vector_seg{x}^\prime,\vector_seg{x}_s) -\tau(... ...ht] {\scriptscriptstyle \stackrel{\bullet}{{}}}\frac{d\vector_seg{x}_s}{d\xi_m}$$ =

$$+\nabla_r\left[\tau(\vector_seg{x}^\prime,\vector_seg{x}_r) -\tau... ...t] {\scriptscriptstyle \stackrel{\bullet}{{}}}\frac{d\vector_seg{x}_r}{d\xi_m},$$

= 0 , (6.8)

and

 

$$\frac{d\Phi}{d\sigma_m} \nabla^\prime\left[\tau(\vector_seg{x}^\prime,\vector_seg{x}_s) +... ...tyle \stackrel{\bullet}{{}}} \frac{d\vector_seg{x}^\prime}{d\sigma_m},~~~~m=1,2$$ =

= 0, (6.9)

where $\sigma=(\sigma_1,\sigma_2)$ is a pair of parameters which determines the reflector coordinates

$$\vector_seg{x}^\prime = \vector_seg{x}^\prime(\sigma).$$ (6.10)

The integration in equation 6.5 reduces to equation 6.7 which is evaluated at the stationary point. The evaluation at the stationary point gives the following result:

$$\beta(\vector_seg{x}) \approx R(\vector_seg{x}^\prime,\vector... ...ma _B(\vector_seg{x}), \mbox{ $\vector_seg{x}$\space on $S$ }.$$ (6.11)

where $\gamma _B(\vector_seg{x})$ is a band-limited delta function with its support on the reflector. This equation shows that the output image of inversion operator 6.1 is a singular function scaled by the geometric-optics angular-dependent reflection coefficients. The term $\beta(\vector_seg{x})$ reaches its peak amplitude at the reflector, and the peak amplitudes are proportional to the reflection coefficients.

Note that in fact it is the stationary phase approximation (i.e. equation 6.7) to the diffraction stack formula given in equation 6.1 that leads to the true amplitude image given in equation 6.11. In other words, equation 6.7 is more close to equation 6.11 than equation 6.1. Once we know the stationary point, equation 6.7 directly yields a result described by equation 6.11.