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

Here, is or . is the parameter constrained by TV( for FWI, and for JMI). and represent horizontal- and vertical-gradient operator, respectively. For one gridpoint in a cartesian coordinate , and (illustrated in figure 2a with the black dashed arrows). is the weight parameter of the fidelity term. 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

where and are the gradient operators of the dominant direction and the direction perpendicular to the dominant direction, respectively. From the viewpoint of physical meaning, and are the rotated and scaled version of and , according to the estimated local dip and a weighting parameter. Mathematically, for one point , and can be represented as

where and represent scaling matrix and rotation matrix, respectively. and represent the weight on the gradient of the dominant direction and its perpendicular direction, respectively, and 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 and , then turns into an identity matrix, which means the same weights are put on both directions, and 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.

2020-12-07