Wave-equation tomography for anisotropic parameters

Parameterization

In the VTI medium, Thomsen parameters $\epsilon$ and $\delta$ are commonly used to characterize the anisotropic seismic velocity. These two parameters define the relationships between the vertical velocity ($V_V$ ), the horizontal velocity ($V_H$ ), and the NMO velocity ($V_N$ ) as follows:

$$V_{H}^{2} = V_V^2(1+2\epsilon),$$ (1)

$$V_{N}^{2} = V_V^2(1+2\delta).$$ (2)

In the practice of surface seismic exploration, it is impossible to estimate the vertical velocity because depth of the reflectors is unknown, and there is no vertical offset information in the data. However, if we have long enough in-line and cross-line offsets, it may be possible to resolve the horizontal velocity and the NMO velocity. Therefore, the anellipticity parameter $\eta$ is used to provide a direct link between $V_N$ and $V_H$ :

$$V_{H}^{2} = V_N^2(1+2\eta),$$ (3)

where $\eta$ is defined by the Thomsen parameters as follows:

$$\eta = \frac{\epsilon-\delta}{(1+2\delta)}.$$ (4)

To reduce the number of parameters, and thereby the null space of the resulting inversion procedure, we make an arbitrary assumption that $\delta = 0$ . Hence, there are only two independent parameters:

$$\eta = \epsilon$$ (5)

and

$$V_V = V_N.$$ (6)

Therefore, we choose to use $V_V$ and $\eta$ as the model parameters that we will estimate during the inversion.

Wave-equation tomography for anisotropic parameters