FWI/JMI with TV and Directional TV

In this paper, we consider anisotropic TV as the basic regularization method, since TV can smooth the model and at the same time preserve edges by enhancing the sparsity of the spatial gradient of the velocity difference. In addition, the anisotropic version is easier to minimize compared to the isotropic one. Furthermore, we restrict ourselves to the 2D case, although extension to the full 3D situation is relatively straightforward.

The extended misfit function with a TV constraint can be expressed as

$$\begin{split} J_{tot} = \mu J + \lambda C_{TV} \left(\mathbf{... ...\lambda \vert\vert\nabla_z \mathbf{p}\vert\vert _1. \end{split}$$ (3)

Here, $J$ is $J_{FWI}$ or $J_{JMI}$. $\mathbf{p}$ is the parameter constrained by TV( $\mathbf{p}=\mathbf{m}$ for FWI, and $\mathbf{p} = \mathbf{v}$ for JMI). $\nabla_x$ and $\nabla_z$ represent horizontal- and vertical-gradient operator, respectively. For one gridpoint $\left (i,j\right )$ in a cartesian coordinate $\left(x,z\right)$, $\nabla_x \mathbf{p}\left(i,j\right) = p_{i+1,j} - p_{i,j}$ and $\nabla_z \mathbf{p}\left(i,j\right) = p_{i,j+1} - p_{i,j}$ (illustrated in figure 2a with the black dashed arrows). $\mu$ is the weight parameter of the fidelity term. $\lambda$ is the coefficient of the constraint term. The latter two together control the balance between the regularization and the misfit function.

However, this conventional TV regularization only tends to reduce the horizontal- and vertical-gradients of each gridpoint in the model, regardless of the geological direction of the model. Therefore, TV is not suitable where the local structure has a dominant direction. Unlike general digital images, the spatial changes in the subsurface always follow some specific geological structures, e.g., tilted layers, faults, and edges of a salt body. In this case, we propose FWI/JMI with directional TV and we design the directional TV based on the local dip estimated from a rough reflection image using the plane-wave destruction (PWD) algorithm (Fomel, 2002).

The misfit function with directional TV can be formulated as

$$\begin{split} J_{tot} = \mu J + \lambda C_{DTV} \left(\mathbf... ...\lambda \vert\vert\nabla_2 \mathbf{p}\vert\vert _1, \end{split}$$ (4)

where $\nabla_1$ and $\nabla_2$ are the gradient operators of the dominant direction and the direction perpendicular to the dominant direction, respectively. From the viewpoint of physical meaning, $\nabla_1$ and $\nabla_2$ are the rotated and scaled version of $\nabla_x$ and $\nabla_z$, according to the estimated local dip and a weighting parameter. Mathematically, for one point $\left (i,j\right )$, $\nabla _1 \mathbf {p}\left (i,j\right )$ and $\nabla _2 \mathbf {p}\left (i,j\right )$ can be represented as

$$\begin{split} \begin{pmatrix} \nabla_1 \mathbf{p}\left(i,j\ri... ...\theta \\ \sin \theta & \cos \theta \end{pmatrix}, \end{split}$$ (5)

where $\Lambda$ and $\mathbf{R}$ represent scaling matrix and rotation matrix, respectively. $\alpha_1$ and $\alpha_2$ represent the weight on the gradient of the dominant direction and its perpendicular direction, respectively, and $\theta$ is the dip of the local structure. An illustration of such a directional TV is shown in figure 2a with the red solid arrows.

Please note that if we assume $\alpha_1 = \alpha_2 = 1$ and $\theta = 0^o$, then $\Lambda$ turns into an identity matrix, which means the same weights are put on both directions, and $\mathbf{R}$ also becomes an identity matrix, indicating that the target directions are horizontal and vertical. Therefore, we can see that the conventional TV is actually a special case of the directional TV, and in turn, the directional TV is a more general version of the conventional TV and more suitable to a model with complex geologic structures. In this paper, we solve both FWI/JMI with the conventional TV and FWI/JMI with the directional TV effectively using the split-Bregman iterative algorithm (Goldstein and Osher, 2009). We only show the framework of solving FWI with the directional TV in Algorithm 1, because, as mentioned before, we treat the conventional TV as a special case of directional the TV, and JMI with the conventional TV/directional TV will follow a similar algorithm.