DERIVATION OF THE FOUR-PASS EQUATION

Starting with the 3-D dispersion relation, we derive the equation for the four-pass technique in the wavenumber domain. We then make a contour plot to study the isotropy of the operator. The exact dispersion relation in three dimensions in terms of the dimensionless quantity $vk/\omega$ is expressed by the following equation:  

$${v k_z \over \omega} = {\sqrt{1 - \frac{v^2k_x^2}{\omega^2} - \frac{v^2k_y^2}{\omega^2}}}$$ (1)

The splitting approximation of the 3-D dispersion relation Brown (1983) is given by  

$${v k_z \over \omega} = {\left [\sqrt{1 - \frac{v^2k_x^2}{\omega^2}} + \sqrt{1 - \frac{v^2k_y^2}{\omega^2}} - 1 \right ] }$$ (2)

The 45-degree approximation of the 3-D dispersion relation with splitting is  

$${v k_z \over \omega} = 1 - {\frac{ {v^2 \over \omega^2}k_x^2... ...{v^2 \over \omega^2}k_y^2} {2 - \frac{v^2k_y^2 }{2\omega^2} } }$$ (3)

In Figure 1, the solid line shows the contours of $vk_z/\omega$ for the exact dispersion relation of equation (1); the dashed line, the splitting approximation for the 3-D dispersion relation of equation (2); and the dotted line, the 45-degree splitting approximation for the 3-D dispersion relation of equation (3). The splitting approximations show the familiar anisotropy in relation to the azimuth. The greatest error appears at 45 degree azimuth.

 

exac-split-45 Figure 1 Contours of constant amplitude and phase for the exact 3-D dispersion relation described in equation (1) (solid line), the splitting approximation in equation (2) (dashed line), and the 45-degree approximation to splitting in equation (3) (dotted line).

After an azimuth rotation of 45 degrees, as shown in Figure , the 45-degree approximation for the split 3-D dispersion relation can be obtained as follows:

 

coord
Figure 2 The rotation from (x,y) coordinate system to (x',y') system.

$$k_x' = k_x\cos\theta + k_y\sin\theta$$ (4)

$$k_y' = -k_x\cos\theta + k_y\sin\theta$$ (5)

where k_x' and k_y' are the new axes after a rotation by angle $\theta$. For $\theta=45$ degrees, substituting the following relations

$$k_x' = {k_x \over \sqrt{2}} + {k_y\over \sqrt{2}}$$ (6)

$$k_y' = -{k_x \over \sqrt{2}} + {k_y\over \sqrt{2}}$$ (7)

into Equation (3) gives

 

$${v k_z \over \omega} = 1 - {\frac{{1\over2}{v^2 \over \omega... ...} (-k_x + k_y)^2} {2 - \frac{v^2(-k_x + k_y)^2 }{4\omega^2} } }$$ (8)

Figure 3 shows the contours of $vk_z/\omega$ for the 45-degree approximation, with an azimuth rotation of 45 degrees (the dashed line) and without (the solid line). The curves of the two approximations have their greatest errors at different places. For the one without rotation, the worst error is at the 45 degree azimuth; for the one with 45 degree rotation, the worst error is along x and y axes. Therefore, the average of the two should give us a more isotropic operator.

 

rot-45 Figure 3 Contours of constant amplitude and phase for the 45-degree splitting approximation described in equation (3) (solid line) and the rotated 45-degree splitting approximation in equation (8)(dashed line).

The operator obtained by averaging equation (3) and (8) gives

 

$${v k_z \over \omega} = {1\over2} {\left [{2 - {\frac{{v^2 \o... ...y)^2} {2 - \frac{v^2(-k_x + k_y)^2 }{4\omega^2}}} }\right ] }$$ (9)

In Figure , the solid line indicates the contour for the exact 3-D dispersion relation equation (1); the dot-dashed line, the four-pass operator of equation (9). The dispersion relation for the four-pass method looks isotropic but has an error that increases with dip, which is caused by the 45-degree approximation.

 

exac-rotavg Figure 4 Contours of constant amplitude and phase for the exact 3-D dispersion relation described in equation (1) (solid line) and the averaged 45-degree splitting approximation obtained from the four-pass operator, as in equation (9)(dotted line).