**1. Introduction**

The subsurface image, more specifically, a detailed image of the oil reservoir, is essential in oil and gas exploration and production and requires appropriate data acquisition, processing to remove unwanted information, building a velocity model to use in an appropriate migration algorithm. The quality of the image obtained is, generally, controlled by the subsurface velocity model. When the geology of the area of interest is composed of salt bodies with complex geometrical shapes, the construction of a precise velocity model is more complicated. Full waveform inversion (FWI) is a tool that can provide us with a velocity model with greater precision and resolution. Wang and Rao [1] is, for the first time, applying FWI for the industrial standard reflection seismic data. Through the use of amplitude and travel time content of the acquired seismic data, this technique, theoretically, has the potential to be the most accurate method for the construction of subsurface velocity models [2,3]. Wang and Houseman [4] proposed the joint inversion that uses both the

**Citation:** Cruz, D.S.; de Araújo, J.M.; da Costa, C.A.N.; da Silva, C.C.N. Adding Prior Information in FWI through Relative Entropy. *Entropy* **2021**, *23*, 599. https://doi.org/ 10.3390/e23050599

Academic Editor: Amelia Carolina Sparavigna

Received: 21 April 2021 Accepted: 8 May 2021 Published: 13 May 2021

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amplitude and travel time data simultaneously, so as to mitigate the ambiguity of reflector geometry and the interval velocities between reflectors.

FWI is a nonlinear and ill-posed data fitting method that usually uses local optimisation methods and, thereforem its solution depends heavily on the initial model. In order to avoid the cycle skipping, the initial model should predict errors in the arrival times less than half the wavelength [3]. One can minimise the issue of non-linearity and the cycle skipping problem with the use of a multi-scale strategy, where we begin from the lowest to the highest frequencies, helping the convergence to the global minimum [5].

The effects of non-uniqueness of the ill-posed inverse problem are usually decreased by the use of regularisation techniques. These regularisation techniques help to stabilise the inversion scheme by incorporating a specific structure or characteristic of the model (e.g., smoothness, sparsity). The most used regularisation scheme is the one that was proposed by Tikhonov and Arsenin [6]. This method incorporated in the inversion scheme aims to find a smooth model that can justify the data. In FWI, in some cases, an *l*1 model penalty is used as a regularisation strategy to preserve edges and contrasts [7]. However, in some cases, prior information, such as sonic records, stratigraphic data, or geological restrictions, about the model is available. To mitigate the problems of non-uniqueness and stability of the solution, we can use a regularisation scheme that is composed of the model norm and first-order Tikhonov regularisation to act as a smoothing operator, as proposed in [8]. They also sugges<sup>t</sup> that the weighting of the term that is responsible for adding the prior information be done dynamically. This strategy proved to be useful in avoiding local minimums in the FWI. Peters and Hermmann [9] showed a way of including constraints on spatial variations and values ranges of the inverted velocities in FWI.

One way to add prior information to the inversion scheme is through relative entropy. In Thermodynamics, we can introduce the notion of entropy to characterise the degree of disorder of a system. The notion of entropy has been the subject of many controversies and different formulations [10]. Here, we will use entropy just as nomenclature used to restrict the solutions of the inverse problem. In a minimisation problem, when we compare entropy with the model norm, the logarithmic operation will suppress the lowintensity ripples. Thus, the entropy method can deliver images with better resolution in some cases [11]. In other words, adding entropy to the FWI formalism, we introduce a smoothness characteristic in the objective function, which will lead to smoothed solutions that are consistent with the available data [12].

The principle of minimum relative entropy (PMRE) was introduced in [13], and it was first applied in the context of geophysicist by Shore [14] in spectral analysis. Other works applied the PMRE in the seismic deconvolution to make limited band extrapolation [15], diffraction seismic tomography [16], and different geophysical problems, such as inversion of interval velocities, removal of the alias effect, and seismic deconvolution [17]. In the context of the FWI, the entropy concept was applied by Chen and Peter [18], who proposed a misfit function based on entropy regularised optimal transport. da Silva et al. [19] proposed a misfit function for FWI based on Shannon entropy for deeper velocity model updates. None of them made use of prior information and, all of them, in some way, use statistical formalism.

Recently, many works have been developed formulating the FWI in terms of Bayesian formalism. In this sense, Zhu et al. [20] show a Bayesian approach to estimate uncertainty for full-waveform inversion using a priori information from depth migration. Singh et al. [21] propose a robust way to constrain the inversion workflow using per-facies rock-physics relationships derived from well logs. Carvalho et al. [22] show Full-waveform inversion with subsurface fractal information and variable model uncertainties. Zhang and Curtis [23] present the first application of variational full-waveform inversion (VFWI) to seismic reflection data where they imposed realistically weak prior information on seismic velocity.

Usually, when working with entropy, we use probability distribution information or Bayesian formalism, which requires some additional step in the formulation of the problem, such as representing initial and prior models as posterior and prior probability distributions [17]. Our proposal is to add prior information to the FWI using relative entropy without explicitly using the concept of a probability distribution or Bayesian formalism. We will do this in a deterministic and direct way. We present three distinct ways to add prior information to the FWI formalism through relative entropy. We will discuss some aspects of the relative entropy for obtaining velocity models and show how these ways of adding prior information will contribute to recovering the velocity model in areas that are affected by the presence of bodies of salt through a synthetic application on the BP model.
