Generalized migration impulse response in parametric form

Integral migration can be conceptually performed by spreading the data along spreading surfaces as well as by summing data along the summation surfaces discussed above. The spreading surfaces are duals of the summation surfaces and represent the impulse response of the migration operator. In homogeneous anisotropic medium the shape of the impulse responses of the generalized integral migration can be easily evaluated analytically as a function of the subsurface offset $h_\xi$, in addition to the usual image depth $z_\xi$ and midpoint $m_\xi$.Figure  illustrates the geometry used to evaluate this impulse response. Notice that the angles in this figure ($\alpha_x$ and $\gamma$) are missing a tilde because they are group angles, and not phase angles as in the previous section. In an isotropic medium these angles are the dip and aperture angles, but in an anisotropic medium these angles are not easily related to the geological dip and the reflection aperture angles. They can be thought of as convenient parameters to evaluate the impulse response.

 

imp-resp Figure 2 Geometry used for evaluating the impulse response of the generalized integral migration.

Simple trigonometry applied to Figure  allows us to express the impulse response in parametric form, as a function of $\alpha_x$ and $\gamma$.If we migrate an impulse recorded at time t_D, midpoint m_D and surface offset h_D, the migration impulse response can be expressed as follows:

$$\begin{eqnarray} z_\xi& = & L\left(\alpha_x,\gamma\right)\frac{\cos ^2 \alpha_x-... ...}- L\left(\alpha_x,\gamma\right)\frac{\sin \gamma}{\cos \alpha_x},\end{eqnarray}$$ (18) (19) (20)

with

$$L\left(\alpha_x,\gamma\right)= \frac{L_s+ L_r}{2}.$$ (21)

In a isotropic medium the half path-length L would be simply given by t_D/2S, but in an anisotropic medium it is function of the angles. Its two components L_s and L_r can be calculated by solving the following system of linear equations:

$$\begin{eqnarray} t_{D}&=& {S_sL_s+ S_rL_r}, \\ z_s-z_r&=& L_s\cos\left(\alpha_x- \gamma\right)- L_r\cos\left(\alpha_x+ \gamma\right)=0.\end{eqnarray}$$ (22) (23)

Equation 22 constraints the total traveltime to be equal to the impulse time, and equations 23 constraints the depth of the end point of the two rays (z_s and z_r) to be equal, since the subsurface offset is assumed to be horizontal. The solution of this system of equation yields the following for the half path-length:

 

$$L\left(\alpha_x,\gamma\right)= \frac{L_s+ L_r}{2} = \frac {t... ...S_r+S_s\right) + \left(S_r-S_s\right)\tan \alpha_x\tan \gamma}.$$ (24)

The combination of equation 24 and equations 18-20 enables the evaluation of the generalized migration impulse response in a arbitrary homogeneous anisotropic medium.

Figure  shows a 3-D rendering of the impulse response computed using the previous equations for an impulse with t_D=.9 seconds, m_D=0 kilometers, and h_D=.4 kilometers, and vertical slowness S_V=1 s/km. The anisotropic parameters correspond to the Taylor Sand as described by Tsvankin (2001) using the three Thomsen parameters: $\epsilon=0.110,\;\delta=-0.035,\;{\rm and}\; \eta=.155$.The gray line (green in color) superimposed onto the impulse response is the result of cutting the surface at zero subsurface offset, and thus corresponds to the conventional impulse response of prestack migration. The black line superimposed onto the impulse response is the result of cutting the surface at zero midpoint. In Figure  these two lines are superimposed onto the corresponding vertical sections cut from the images computed by an anisotropic wavefield source-receiver migration applied with the same parameters described above. Figure b shows the conventional migration impulse response, whereas Figure a shows the zero-midpoint section. The lines computed by applying the kinematic equations perfectly match the impulse responses computed using wavefield migration, confirming the accuracy of the kinematic equations.

 

surf_taylor_hxd_dot_4
Figure 3 Impulse response of generalized anisotropic prestack migration. The gray line (green in color) superimposed onto the impulse response corresponds to the conventional impulse response of prestack migration.

 

Surf-taylor_hxd_.4-overn Figure 4 Vertical sections cut from the impulse response computed by an anisotropic wavefield source-receiver migration. The lines superimposed onto the images correspond to the lines superimposed onto the surface shown in Figure  and are computed by applying the kinematic expressions presented in equations 18-24.