*3.5. Discussion*

We have shown that the addition of a priori information to the FWI scheme can be an effective strategy for driving the inversion process to converge toward the global minimum. Here, we have incorporated the priori information in different ways and, to quantify the accuracy of the inversion results for each one, we have computed the normalized model misfit using the following equation:

$$\varepsilon = \frac{\left[\sum\_{i=1}^{n} \left(m\_i^{true} - m\_i^{inv}\right)^2\right]^{1/2}}{\left[\sum\_{i=1}^{n} \left(m\_i^{true}\right)^2\right]^{1/2}},\tag{15}$$

where *mtrue* is the true model and model and *minv* is the inverted model using an FWI method [35,36]. An close to 0 means low error. The values are shown in the Table 1.

In the analysis of the results that are presented in Table 1, we observe that axiomatic form (third case) We note that the third case provides the model with the smallest error in both cases. Although it does not provide the smallest error in the model, our proposal to use the quadratic form of the symmetric form of the relative entropy (second case) is also robust. It is possible to see that the model error with this strategy is less than using the model norm. Finally, it is possible to observe that our proposal to use the quadratic form of relative entropy (first case), although its result depends on the inversion path, can also provide a good result. For the first part of the model, we were not successful in our tests, but we calculated the error for the results that are shown in Figure 9a,b, respectively. For the second part of the model, we can observe that the model error is less than the conventional case.


**Table 1.** Misfit model of the FWI results. (∗) This is the model misfit for the result that is illustrated in Figure 9a, while (∗∗) is the model misfit for the result illustrated in Figure 9b.

## **4. Conclusions**

In this work, we propose adding the relative entropy in the FWI formalism. We use relative entropy to include priori information in the FWI to reduce the difficulty of the uniqueness of the solution in this kind of inverse problem. The addition of the prior information was done in a deterministic way, which is, it was done in a direct way, avoiding the formulation in terms of a probability distribution. We have applied this scheme in two regions of the BP model, which presents a complex lateral velocity variation due to the halokinesis of salt layers. The numerical tests show the quality improvement in the result that was obtained when compared with the conventional FWI. In addition to the visual analysis, we calculated the misfit model to show the improvement that was brought by our proposal.

We present three different ways of introducing prior information through entropy to the FWI formalism: the literature as Kullback–Leibler's distance and its symmetrical form and an axiomatic form. In the first case, we show that in the deterministic approach, the Kullback–Leibler's distance is not positively defined in its entire domain. To avoid this misfortune, we sugges<sup>t</sup> using its quadratic form. We have seen that this quadratic form will not always help to solve the local minimum problem. We graphically illustrate

that the function described by the quadratic form of the Kullback–Leibler's distance has two regions of minimum. Therefore, the result will depend on the path taken by the FWI throughout the iterations: while it was not possible to obtain a satisfactory solution for the first part of the model after several tests, for the second part of the model, the result was satisfactory.

The second case was the symmetrical form of Kullback–Leibler's distance. We graphically illustrate that this function is also not positively defined, which makes it difficult to define the problem as a maximization or minimization. In addition, the symmetrical form has no minimum region. To avoid these inconveniences, we propose the use of its quadratic form. We have shown graphically that this quadratic shape has characteristics that can help the FWI to avoid local minimums. In addition to being positively defined, it presents a minimum region. The addition of previous information in the FWI through this quadratic form enables the FWI to deliver a satisfactory result for both cases.

The third case that we studied was the addition of prior information through an axiom of relative entropy. Graphically, we show that this shape is positively defined and it has a minimum region. These features allow this form to be used to add information prior to the FWI formalism in a straightforward manner. The results of the FWI with this regularization scheme were also satisfactory for both models.

The FWI results that were obtained using the relative entropy were compared to the result with the model norm. We observed that for the first part of the BP model, the FWI result with real entropy is slightly better (mainly on the left-hand side of the model). This fact is corroborated by the analysis of the velocity profiles in the position of the wells. We saw that the FWI result with the relative entropy provides an adjustment closer to the desired result when compared to the FWI result with the model norm. In addition, we saw that the misfit data are less when we add prior information through entropy. For the second part of the BP model, the results are visibly similar. We have seen that the adjustment of the data is close, although the first and third cases of relative entropy require a little more iteration. However, when comparing the well profiles, we observed that the adjustment of the FWI with the addition of prior information through entropy provides a better fit early in the deepest region of the model.

**Author Contributions:** Conceptualization, D.S.C., C.C.N.d.S. and J.M.d.A.; methodology, D.S.C., C.A.N.d.C., C.C.N.d.S. and J.M.d.A.; software, D.S.C. and J.M.d.A.; validation, D.S.C. and J.M.d.A.; formal analysis, D.S.C., C.A.N.d.C., C.C.N.d.S. and J.M.d.A.; investigation, D.S.C., C.A.N.d.C., C.C.N.d.S. and J.M.d.A.; writing–original draft preparation, D.S.C., C.A.N.d.C., C.C.N.d.S. and J.M.d.A.; visualization, D.S.C., C.A.N.d.C., C.C.N.d.S. and J.M.d.A.; supervision, C.C.N.d.S. and J.M.d.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work did not have external funding.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** The authors acknowledge the support of Petrobras. We thank the High Performance Computing Center at UFRN for providing the computational facilities to run the simulations. J.M.A. thanks CNPq for his productivity fellowship (grant No. 313431/2018-3). The authors also thank BP and Frederic Billette who provided the BP velocity model.

**Conflicts of Interest:** The authors declare no conflict of interest.
