A numerical tour of wave propagation

Imaging condition

The cross-correlation imaging condition can be expressed as

$$I(\textbf{x})=\sum_{s=1}^{ns}\int_{0}^{t_{\max}}\mathrm{d}t \sum_{g=1}^{ng} p_s(\textbf{x},t;\textbf{x}_s)p_g(\textbf{x},t;\textbf{x}_g)$$ (55)

where $I(\textbf{x})$ is the migration image value at point ${\bf x}$ ; and $p_s(\textbf{x},t)$ and $p_g(\textbf{x},t)$ are the forward and reverse-time wavefields at point ${\bf x}$ . With illumination compensation, the cross-correlation imaging condition is given by

$$I(\textbf{x})=\sum_{s=1}^{ns}\frac{\int_{0}^{t_{\max}}\mathrm{d}t... ...thrm{d}t p_s(\textbf{x},t;\textbf{x}_s)p_s(\textbf{x},t;\textbf{x}_s)+\sigma^2}$$ (56)

in which $\sigma^2$ is chosen small to avoid being divided by zeros.

There exists a better way to carry out the illumination compensation, as suggested by Guitton et al. (2007)

$$I(\textbf{x})=\sum_{s=1}^{ns}\frac{\int_{0}^{t_{\max}}\mathrm{d}t... ...}t p_s(\textbf{x},t;\textbf{x}_s)p_s(\textbf{x},t;\textbf{x}_s)\rangle_{x,y,z}}$$ (57)

where $\langle\rangle_{x,y,z}$ stands for smoothing in the image space in the x, y, and z directions.

Yoon et al. (2003) define the seismic Poynting vector as

$$\textbf{S}=\textbf{v}p=\nabla p \frac{\mathrm{d}p}{\mathrm{d}t}p=(v_xp, v_zp).$$ (58)

Here, we denote $S_s$ and $S_r$ as the source wavefield and receiver wavefield Poynting vector. As mentioned before, boundary saving with split PML is a good scheme for the computation of Poynting vector, because $p$ and $(v_x,v_z)$ are available when backward reconstructing the source wavefield. The angle between the incident wave and the reflected wave can then be obtained:

$$\gamma=\arccos\frac{\textbf{S}_s\cdot \textbf{S}_r}{\vert\textbf{S}_s\vert\vert\textbf{S}_r\vert}$$ (59)

The incident angle (or reflective angle) is half of $\gamma$ , namely,

$$\theta=\frac{\gamma}{2}=\frac{1}{2}\arccos\frac{\textbf{S}_s\cdot \textbf{S}_r}{\vert\textbf{S}_s\vert\vert\textbf{S}_r\vert}$$ (60)

Using Poynting vector to confine the spurious artefacts, Yoon and Marfurt (2006) propose a hard thresholding scheme to weight the imaging condition:

$$I(\textbf{x})=\sum_{s=1}^{ns}\frac{\int_{0}^{t_{\max}}\mathrm{d}t... ...thrm{d}t p_s(\textbf{x},t;\textbf{x}_s)p_s(\textbf{x},t;\textbf{x}_s)+\sigma^2}$$ (61)

where

$$W(\theta)=\begin{cases}1 & \theta<\theta_{\max}\\ 0 & otherwise\\ \end{cases}$$ (62)

Costa et al. (2009) modified the weight as

$$W(\theta)=\cos^3(\frac{\theta}{2}).$$ (63)

These approaches are better for eliminating the backward scattering waves in image.

A numerical tour of wave propagation