Migration velocity analysis/inversion

The macro-velocity field has a decisive influence on seismic-wave imaging. Unfortunately, it is not easy to accurately estimate the velocity field from the seismic data. Up to now, the residual depth/time difference in the common-image gathers has been used for migration-velocity analysis (MVA) or inverting the macro-velocity distribution. However, in the case of complex topography and gelogical structures, MVA is not a successful approach. Therefore, seismic-wave imaging in complex survey areas has a long way to go. We propose the following approach to inverting the macro-velocity field. The norm is defined as  

$$E=W_{1}\left(U_{S}^{k+1} -U_{S}^{k}\right) ^{2}+W_{2}\left(\... ...e S_{m}^{k} \right) ^{2}+W_{3}\left(R^{k+1}-R^{k} \right) ^{2},$$ (48)

where k stands for the iterative number; U_S is the calculated scattering wavefield. R is the position of the main reflectors, which can be identified from the migrated profile. $\triangle S_{m}$ is the slowness disturbance field. W₁,W₂ and W₃ are the different weights. According to Bleistein (2000,p.39), the calculated scattering wavefield can be given by  

$$U_{S}\left(\vec{x}_{g},\vec{x}_{s},\omega \right) =\omega^{2... ...mega \right)g\left(\vec{x},\vec{x}_{g},\omega \right)d\vec{x} ,$$ (49)

where $\alpha \left( \vec{x}\right)=\frac{c^{2} \left( \vec{x}\right) }{c^{2}\left(\vec{x} \right) } -1$.Alternatively, the calculated scattering wavefield () also can be given by  

$$\frac{\partial U_{S}\left(\omega, k_{x}, k_{y}, z \right) }{... ...left(x,y,z \right) U_{I}\left( \omega, x, y, z\right) \right] ,$$ (50)

where $\triangle S\left(x,y,z \right) =S\left( x,y,z\right) -S_{ref}\left( x,y,z\right)$ is the slowness disturbance, U_I is the incident wave field, and k_z is the vertical wavenumber. The incident wave field U_I can be calculated with the following equation:  

$$\frac{\partial U_{I}\left(\omega, k_{x}, k_{y}, z \right)}{\... ...ial z}=ik_{0} k_{z} U_{I}\left(\omega, k_{x}, k_{y}, z \right).$$ (51)

where $k_{0}=\frac{\omega}{v_{r}}$ , $k_{z}=\sqrt{1-\left( \frac{k_{T}}{k_{0}}\right)^{2}}$ and $k_{T}=\sqrt{k_{x}^{2}+k_{y}^{2}}$.