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

Forward modeling

Exact anisotropic wave equations are in the form of elastic wave equations. Acoustic anisotropic wave equations can be obtained by various approximations of the exact elastic equations. One way to do this is to set shear-wave velocity to zero in the exact elastic wave equations. Detailed derivations can be found in several papers (Duveneck et al., 2008; Zhang and Zhang, 2009; Crawley et al., 2010). The resulting acoustic anisotropic wave equations are a system of second-order equations:

$$\frac{\partial^2 p}{\partial t^2} = {v_p}^2\left(1+2\epsilon\right)\frac{\partial^2 p}{\partial x^2} \ + {v_p}^2\sqrt{1+2\delta}\frac{\partial^2 r}{\partial z^2}$$

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

where ${p}$ and ${r}$ are horizontal and vertical stress, respectively, ${v_p}$ is vertical p-wave velocity and ${\epsilon }$ and ${\delta }$ are anisotropic parameters (Thomsen, 1986).

To illustrate the traveltime sensitivity of data to parameter changes, I use part of the BP 2002 benchmark model. The original synthetic model has only p-wave velocity; the anisotropic parameters were created from the velocity model according to typical Gulf of Mexico anisotropic parameters. Velocity, ${\epsilon }$ and ${\delta }$ models are shown in Figure 1.

modelbw
Figure 1. Reference model for various modeling experiments. Top: velocity model; middle: ${\epsilon }$ model; bottom: ${\delta }$ model.

Four different modeling experiments with modeled shot records are shown in Figure 2. The first experiment is the VTI anisotropic modeling using all three fields shown in Figure 1. The second experiment is the same VTI modeling but with ${\delta =0}$ . The third experiment is isotropic modeling using the velocity field only, and the fourth is isotropic modeling using the horizontal p-wave velocity, which is defined as ${v_h=v_p\sqrt {1+2\epsilon }}$ . Figure 3 shows the refraction traveltime difference of the latter three shots compared to the first shot. It can be seen that traveltime is insensitive to ${\delta }$ changes, but is sensitive to ${\epsilon }$ changes. Also, isotropic modeling using the horizontal p-wave velocity results in non-trivial traveltime differences, which means that even using isotropic FWI, the retrieved model is not necessarily the horizontal p-wave velocity, as was previously thought (Ghilami et al., 2011).

record
Figure 2. Shots from different modeling experiments: a) VTI modeling using all three fields shown in Figure 1; b) same as a) except that ${\delta =0}$ ; c) isotropic modeling using only the velocity field; d) isotropic modeling using horizontal p-wave velocity ${v_h=v_p\sqrt {1+2\epsilon }}$ .

tdif
Figure 3. Refraction traveltime difference of shots compared to the VTI case.

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