Early-arrival waveform inversion for near-surface velocity and anisotropic parameters: modeling and sensitivity kernel analysis

Velocity parametrization

Another parametrization is to use velocities for both variables. Defining ${m_1={v_p}^{2}}$ and ${m_2={v_p}^2\left(1+2\epsilon\right)}$ , equation 1 can be rewritten as follows:

$$\frac{\partial^2 p}{\partial t^2} = m_2\frac{\partial^2 p}{\partial x^2} \ + m_1\sqrt{1+2\delta}\frac{\partial^2 r}{\partial z^2}$$

$$\frac{\partial^2 r}{\partial t^2} = m_1\sqrt{1+2\delta}\frac{\partial^2 p}{\partial x^2} \ + m_1\frac{\partial^2 r}{\partial z^2},$$ (7)

Using similar procedure to the one described in the previous section, we can obtain a matrix form expression:

$$\begin{vmatrix}{ \frac{\partial^2 }{\partial t^2} - m_{2,0}\frac{... ...left\vert\begin{array}{c} {\Delta{m_1}} \\ {\Delta{m_2}}\end{array}\right\vert,$$ (8)

This is the linear relationship between model perturbation and data perturbation, and can be used to calculate the sensitivity kernel. Figure 6 shows the relative sensitivity kernel of the two model parameters. Both figures are clipped to the same value. Since both variables are parametrized as velocities, their updates are of similar strength, and this is an effective parametrization.

relimpv
Figure 6. Relative sensitivity kernel of parameter: a) ${{v_p}^{2}}$ ; b) ${{v_p}^2\left (1+2\epsilon \right )}$ . Both figures are clipped at the same value.

Early-arrival waveform inversion for near-surface velocity and anisotropic parameters: modeling and sensitivity kernel analysis