# Entropic Regularization to Assist a Geologist in Producing a Geologic Map

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Forward Problem

_{ij}of A, which are defined by:

**Figure 1.**Schematic representation of (a) geological map showing four geologic units (colored areas) and geologic faults (thick black lines) and (b) interpretation model consisting of a set of rectangular, 3D juxtaposed prisms. Each prism has known vertical and horizontal dimensions and horizontal top and bottom. The physical properties (density or magnetization intensity for a gravity or magnetic source, respectively) contrasts of all prisms are the parameters to be estimated from the potential-field data (gravity and magnetic data) measured over the Earth surface and produced by a set of unknown 3D geologic sources.

## 3. Formulation of the Inverse Problem Using the Concept of Entropic Regularization

^{−8}) used to guarantee the definition of the entropy measures, L is the number of adjacent pairs of parameters, and t

_{k}is the kth element of vector $t$ that represents the finite-difference approximation to the first derivative of $m$ along the horizontal directions.

_{0}and γ

_{1}are positive numbers controlling the trade-off between the “maximization” of ${Q}_{0}(m)$ and the minimization of ${Q}_{1}(m)$. The negative sign imposed on ${Q}_{0}(m)$ in Equation (8) guarantees the “maximization” of the zeroth-order entropy measure.

_{1}controls the number of discontinuities in the estimated physical-property distribution. An optimum value for γ

_{1}is the largest positive value producing no more oscillations or discontinuities than those expected for the physical-property distribution being interpreted. The variable γ

_{0}must be assigned the smallest value necessary to prevent an estimated physical-property distribution presenting unrealistic pattern with predominantly null values and a few unrealistically large nonnull values. Among the numerous estimated physical-property contrast distributions fitting the data with acceptable precision, the entropic regularization favors an estimated physical-property distribution with locally smooth regions separated by abrupt discontinuities.

## 4. Numerical Results

#### 4.1. Gravity Sources

^{3}[blue region in Figure 2(b)] and 0.1 g/cm

^{3}[green region in Figure 2(b)]. Hence, we simulated a geologic contact as a boundary surface between these two rocks where there is an abrupt lithological change around x = 0.4 km.

**Figure 2.**Test simulating a geologic contact. (a) Noise-corrupted gravity data in mGal (solid gray lines) produced by the true density-contrast distribution shown in b and fitted gravity data (dashed black lines) produced by the estimated density-contrast distribution shown in d using the entropic regularization. (b) True theoretical density-contrast distribution simulating a geologic contact. (c) Estimated density-contrast distribution produced by the first-order Tikhonov regularization. (d) Estimated density-contrast distribution produced by the entropic regularization with ${\gamma}_{1}$ = 1.8 and ${\gamma}_{0}\text{}$ = 1.2.

_{1}= 1.8 and γ

_{0}= 1.2. The values assigned to γ

_{1}and γ

_{0}were obtained by trial and error. Provisionally we assign to γ

_{0}a small positive value, including zero, and to γ

_{1}a large positive value. If the solution collapses into a source whose horizontal dimensions are substantially smaller than those expected for the true source, we must increase the value assigned to γ

_{0}. If the solution shows undefined discontinuities, we must increase γ

_{1}. On the other hand, if the solution shows several discontinuities, we must decrease γ

_{1}. Figure 2(c,d) shows the perspective views of the estimated density maps produced by the first-order Tikhonov regularization and by the entropic regularization, respectively.

^{3}. Larger differences (up to ± 0.05 g/cm

^{3}) coincide with the geologic contact around x = 0.4 km.

#### 4.2. Magnetic Sources

**Figure 4.**Test simulating two magnetic sources. (a) Noise-corrupted magnetic data in nT (solid gray lines) produced by the true magnetization-contrast distribution shown in b and fitted magnetic data (dashed black lines) produced by the estimated magnetization-contrast distribution shown in (d), using the entropic regularization. (b) True theoretical magnetization-contrast distribution simulating two magnetic sources. (c) Estimated magnetization-contrast distribution produced by the first-order Tikhonov regularization. (d) Estimated magnetization-contrast distribution produced by the entropic regularization with γ

_{1}= 20 and γ

_{0}= 3.

_{1}= 20 and γ

_{0}= 3. The values assigned to γ

_{1}and γ

_{0}were obtained by trial and error. Provisionally we assign to γ

_{0}a small positive value including zero and to γ

_{1}a large positive value. If the solution collapses into a source whose horizontal dimensions are substantially smaller than those expected for the true source, we must increase the value assigned to γ

_{0}. If the solution shows undefined discontinuities, we must increase γ

_{1}. On the other hand, if the solution shows several discontinuities, we must decrease γ

_{1}.

**Figure 5.**Test simulating two magnetic sources. Decay of functions ${Q}_{0}(m)$

**(a)**and ${Q}_{1}(m)$

**(b)**along the iterations for the solution shown in Figure 4(d).

## 5. Application to Real Data

**Figure 6.**Butte Valley Stock. (a) Observed total-field aeromagnetic anomaly (solid gray lines) and fitted (dashed black lines) total-field anomaly (in nT) using the entropic regularization solution shown in (c). (b) Estimated magnetization-contrast distribution produced by the first-order Tikhonov regularization [18]. (c) Estimated magnetization-contrast distribution produced by the entropic regularization with γ

_{1}= 3,385 and γ

_{0}= 100 [18].

_{1}= 3,385 e γ

_{0}= 100. The first-order Tikhonov regularization result [Figure 6(b)] presents, as expected, a smooth transition of the estimated magnetizations from the body center to the borders, preventing a clear delineation of the source outline. Moreover, a wide region of spurious negative magnetization-contrast values shows up around the source. Conversely, the estimated magnetization-contrast distribution by the entropic regularization [Figure 6(c)] displays steeper gradients close to the source’s borders allowing a better delineation of its horizontal projection. One also notes the conspicuous reduction of the region displaying spurious negative values around the source. The entropic regularization estimates a magnetization-contrast distribution with smaller horizontal length and higher (13 A/m) maximum values of the estimated magnetization contrast. Compared with the first-order Tikhonov regularization, it produces smaller (10 A/m) maximum values of the estimated magnetization contrast. Under the hypothesis of induced magnetization, a magnetization of 13 A/m is associated with an estimated susceptibility closer to the measured value of 0.63 SI than the estimate of 10 A/m produced by the first-order Tikhonov regularization. The fitted anomaly produced by the entropic regularization is shown in Figure 6(a) in dashed black lines.

**Figure 7.**Butte Valley Stock. Possible skarn emplacements explaining the absence of a ring-like shape of the magnetization-contrast map. (a) Exoskarn where the ion exchange between the intrusive and the host rock occurs along a limited portion of the intrusive border. (b) Exoskarn with top located below the erosion level. (c) Exoskarn whose intrusive rock has been emplaced with an irregular shape. (d) Endoskarn where the magnetic minerals are formed along faults and joints in the intrusive rock.

## 6. Conclusions

## Acknowledgements

## References

- Nabighian, M.N.; Asten, M.W. Metalliferous mining geophysics—State of the art in the last decade of the 20th century and the beginning of the new millennium. Geophysics
**2002**, 67, 964–978. [Google Scholar] [CrossRef] - Keating, P. Density mapping from gravity data using the Walsh transform. Geophysics
**1992**, 57, 637–642. [Google Scholar] [CrossRef] - Pilkington, M.; Crossley, D.J. A Kalman filter approach to susceptibility mapping. Geophysics
**1987**, 52, 655–664. [Google Scholar] [CrossRef] - Medeiros, W.E.; Silva, J.B.C. Geophysical inversion using approximate equality constraints. Geophysics
**1996**, 61, 1678–1688. [Google Scholar] [CrossRef] - Silva, J.B.C.; Hohmann, G.W. Airborne magnetic susceptibility mapping. Explor. Geophys.
**1984**, 15, 1–13. [Google Scholar] [CrossRef] - Cordell, L.; McCafferty, A.E. A terracing operator for physical property mapping with potential field data. Geophysics
**1989**, 54, 621–634. [Google Scholar] [CrossRef] - Tikhonov, A.N.; Arsenin, V.Y. Solutions of Ill-Posed Problems; V.H. Winston & Sons: Washington, DC, USA, 1977. [Google Scholar]
- Silva, J.B.C.; Medeiros, W.E.; Barbosa, V.C.F. Potential field inversion: Choosing the appropriate technique to solve a geologic problem. Geophysics
**2001**, 66, 511–520. [Google Scholar] [CrossRef] - Blakely, R.J. Potential Theory in Gravity and Magnetic Applications; Cambridge University Press: Cambridge, MA, USA, 1995. [Google Scholar]
- Muniz, M.B.; Ramos, F.M.; Campos Velho, H.F. Entropy- and Tikhonov-based regularization techniques applied to the backwards heat equation. Comput. Math. Appl.
**2000**, 40, 1071–1084. [Google Scholar] [CrossRef] - Ramos, F.M.; Campos Velho, H.F.; Carvalho, J.C.; Ferreira, N.J. Novel approaches on entropic regularization. Inverse Probl.
**1999**, 15, 1139–1148. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics II. Phys. Rev.
**1957**, 108, 171–190. [Google Scholar] [CrossRef] - Shannon, C.E. The mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Gill, P.E.; Murray, W.; Wright, M.H. Practical Optimization; Academic Press Inc.: London, UK, 1981. [Google Scholar]
- Oliver, D.S.; Reynolds, A.C.; Liu, N. Inverse Theory for Petroleum Reservoir Characterization and History Matching; Cambridge University Press: Cambridge, MA, USA, 2008. [Google Scholar]
- Silva, J.B.C.; Oliveira, F.S.; Barbosa, V.C.F.; Campos Velho, H.F. Apparent-density mapping using entropic regularization. Geophysics
**2007**, 72, I51–I60. [Google Scholar] [CrossRef] - Silva, J.B.C.; Vasconcelos, S.S.; Barbosa, V.C.F. Apparent-magnetization mapping using entropic regularization. Geophysics
**2010**, 75, L39–L50. [Google Scholar] [CrossRef] - Silva, J.B.C.; Oliveira, A.S.; Barbosa, V.C.F. Gravity inversion of 2D basement relief using entropic regularization. Geophysics
**2010**, 75, I29–I35. [Google Scholar] [CrossRef] - Barbosa, V.C.F.; Silva, J.B.C.; Medeiros, W.E. Gravity inversion of basement relief using approximate equality constraints on depths. Geophysics
**1997**, 62, 1745–1757. [Google Scholar] [CrossRef] - Barbosa, V.C.F.; Menezes, P.T.L.; Silva, J.B.C. Gravity data as a tool for detecting faults: In-depth enhancement of subtle Almada’s basement faults, Brazil. Geophysics
**2007**, 72, B59–B68. [Google Scholar] [CrossRef] - Barbosa, V.C.F.; Silva, J.B.C.; Medeiros, W.E. Practical applications of uniqueness theorems in gravimetry: Part II—Pragmatic incorporation of concrete geologic information. Geophysics
**2002**, 67, 795–800. [Google Scholar] [CrossRef] - Martins, C.M.; Barbosa, V.C.F.; Silva, J.B.C. Simultaneous 3D depth-to-basement and density-contrast estimates using gravity data and depth control at few points. Geophysics
**2010**, 75, I21–I28. [Google Scholar] [CrossRef] - Nunes, T.M.; Barbosa, V.C.F.; Silva, J.B.C. Magnetic basement depth inversion in the space domain. Pure Appl. Geophys.
**2008**, 165, 1891–1911. [Google Scholar] [CrossRef] - Silva, J.B.C.; Costa, D.C.L.; Barbosa, V.C.F. Gravity inversion of basement relief and estimation of density contrast variation with depth. Geophysics
**2006**, 71, J51–J58. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Barbosa, V.C.F.; Silva, J.B.C.; Vasconcelos, S.S.; Oliveira, F.S. Entropic Regularization to Assist a Geologist in Producing a Geologic Map. *Entropy* **2011**, *13*, 790-804.
https://doi.org/10.3390/e13040790

**AMA Style**

Barbosa VCF, Silva JBC, Vasconcelos SS, Oliveira FS. Entropic Regularization to Assist a Geologist in Producing a Geologic Map. *Entropy*. 2011; 13(4):790-804.
https://doi.org/10.3390/e13040790

**Chicago/Turabian Style**

Barbosa, Valeria C.F., João B.C. Silva, Suzan S. Vasconcelos, and Francisco S. Oliveira. 2011. "Entropic Regularization to Assist a Geologist in Producing a Geologic Map" *Entropy* 13, no. 4: 790-804.
https://doi.org/10.3390/e13040790