# Model-Based Probabilistic Inversion Using Magnetic Data: A Case Study on the Kevitsa Deposit

^{1}

^{2}

^{*}

^{†}

^{‡}

^{§}

^{‖}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Bayesian Framework

- 1.
- Parameterization of the system: Developing a set of model parameters $\theta $ that describe the system under study. An ideal parameterization should contain enough complexity to explain the data, without over-fitting.
- 2.
- Forward model: Using mathematical models $\mathcal{M}$ that, given the model parameters $\theta $, allow us to simulate the measurements of the observable parameters or data y.
- 3.
- Inference: Using measurements of observable parameters y to find the model parameters $\theta $ that are able to explain this data.

#### 2.2. Parameterization

#### 2.3. Forward Model

#### 2.3.1. Implicit 3D-Modeling

#### 2.3.2. Magnetic Forward Modeling

- 1.
- Magnetization $\overrightarrow{J}$ is only due to induction, so that $\overrightarrow{{J}_{r}}=0$ and $\overrightarrow{J}=\overrightarrow{{J}_{i}}=k\overrightarrow{B}$, where $\overrightarrow{B}$ is the magnetic induction and k is the magnetic susceptibility.
- 2.
- Susceptibility k of each modeled lithological unit in the geological model is known, and homogeneous and anisotropic throughout the lithological unit.
- 3.
- The anomalous magnetic field due to local subsurface conditions is small compared to the Earth’s natural magnetic field so that the direction of the total field is said to be in the same direction as the geomagnetic field, a general assumption in magnetic prospecting [24].

#### 2.4. Bayesian Inference

#### 2.4.1. Markov Chain Monte Carlo

#### 2.4.2. Gradient-Based MCMC Sampling

#### 2.4.3. Automatic Differentiation

## 3. Application to the Kevitsa Deposit

#### 3.1. Geological Setting

#### 3.2. Field Data

#### 3.3. Defining the Probabilistic Model

#### 3.4. Analysis of Model Uncertainties

#### 3.5. Numerical Implementation

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lelievre, P.; Carter-McAuslan, A.; Farquharson, C.; Hurich, C. Unified geophysical and geological 3D Earth models. Lead. Edge
**2012**, 31, 322–328. [Google Scholar] [CrossRef] [Green Version] - Ellis, R.G.; MacLeod, I.N. Constrained voxel inversion using the Cartesian cut cell method. ASEG Ext. Abstr.
**2013**, 1, 1–4. [Google Scholar] [CrossRef] [Green Version] - Özyıldırım, Ö.; Candansayar, M.E.; Demirci, İ.; Tezkan, B. Two-dimensional inversion of magnetotelluric/radiomagnetotelluric data by using unstructured mesh. Geophysics
**2017**, 82, E197–E210. [Google Scholar] [CrossRef] [Green Version] - Sheldon, H.; Micklethwaite, S. Damage and permeability around faults: Implications for mineralization. Geology
**2007**, 35. [Google Scholar] [CrossRef] [Green Version] - Okamoto, A.; Tsuchiya, N. Velocity of vertical fluid ascent within vein-forming fractures. Geology
**2009**, 37, 563–566. [Google Scholar] [CrossRef] - Gallardo, L.; Thébaud, N. New insights into Archean granite-greenstone architecture through joint gravity and magnetic inversion. Geology
**2012**, 40, 215–218. [Google Scholar] [CrossRef] - Wellmann, F.; Caumon, G. 3-D Structural geological models: Concepts, methods, and uncertainties. In Advances in Geophysics; Elsevier: Amsterdam, The Netherlands, 2018; Volume 59, pp. 1–121. [Google Scholar]
- Gelman, A.; Carlin, J.B.; Stern, H.S.; Rubin, D.B. Bayesian Data Analysis, 3rd ed.; Chapman and Hall/CRC: London, UK, 2013; p. 675. [Google Scholar]
- MacKay, D.J.C. Information Theory, Inference, and Learning Algorithms; Copyright Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Tarantola, A.; Valette, B. Inverse Problems = Quest for Information. J. Geophys.
**1982**, 50, 159–170. [Google Scholar] - Mosegaard, K.; Tarantola, A. Monte Carlo sampling of solutions to inverse problems. J. Geophys. Res.
**1995**, 1001, 12431–12448. [Google Scholar] [CrossRef] - Sambridge, M.; Mosegaard, K. Monte Carlo methods in geophysical inverse problems. Rev. Geophys.
**2002**, 40, 1009. [Google Scholar] [CrossRef] [Green Version] - Betancourt, M.J.; Byrne, S.; Livingstone, S.; Girolami, M. The Geometric Foundations of Hamiltonian Monte Carlo. arXiv
**2014**, arXiv:1410.5110. [Google Scholar] [CrossRef] - Jessell, M.; Ailleres, L.; Dekemp, E. Towards an integrated inversion of geoscientific data: What price of geology? Tectonophysics
**2010**, 490, 294–306. [Google Scholar] [CrossRef] - Wellmann, F.; Horowitz, F.; Schill, E.; Regenauer-Lieb, K. Towards incorporating uncertainty of structural data in 3D geological inversion. Tectonophysics
**2010**, 490. [Google Scholar] [CrossRef] - Lindsay, M.; Ailleres, L.; Jessell, M.; Dekemp, E.; Betts, P. Locating and quantifying geological uncertainty in three-dimensional models: Analysis of the Gippsland Basin, southeastern Australia. Tectonophysics
**2012**, 546-547, 10–27. [Google Scholar] [CrossRef] - De la Varga, M.; Wellmann, F.; Murdie, R. Adding geological knowledge to improve uncertain geological models: A Bayesian perspective. Geotecton. Res.
**2015**, 97, 18–20. [Google Scholar] [CrossRef] - Lindsay, M.; Jessell, M.; Ailleres, L.; Perrouty, S.; Dekemp, E.; Betts, P. Geodiversity: Exploration of 3D geological model space. Tectonophysics
**2013**, 594. [Google Scholar] [CrossRef] - Wellmann, J.F.; Regenauer-Lieb, K. Uncertainties have a meaning: Information entropy as a quality measure for 3-D geological models. Tectonophysics
**2012**, 526-529, 207–216. [Google Scholar] [CrossRef] - Wellmann, F.; Lindsay, M.; Poh, J.; Jessell, M. Validating 3-D Structural Models with Geological Knowledge for Improved Uncertainty Evaluations. Energy Procedia
**2014**, 59, 374–381. [Google Scholar] [CrossRef] [Green Version] - De la Varga, M.; Wellmann, J.F. Structural geologic modeling as an inference problem: A Bayesian perspective. Interpretation
**2016**, 4, SM1–SM16. [Google Scholar] [CrossRef] [Green Version] - Wellmann, J.F.; de la Varga, M.; Murdie, R.E.; Gessner, K.; Jessell, M. Uncertainty estimation for a geological model of the Sandstone greenstone belt, Western Australia—Insights from integrated geological and geophysical inversion in a Bayesian inference framework. Geol. Soc. Lond. Spec. Publ.
**2018**, 453, 41–56. [Google Scholar] [CrossRef] - Telford, W.M.; Telford, W.; Geldart, L.; Sheriff, R.E.; Sheriff, R.E. Applied Geophysics; Cambridge University Press: Cambridge, UK, 1990; Volume 1. [Google Scholar]
- Hinze, W.J.; von Frese, R.R.B.; Saad, A.H. Gravity and Magnetic Exploration: Principles, Practices, and Applications; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar] [CrossRef]
- Tarantola, A. Inverse Problem Theory and Methods for Model Parameter Estimation; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2005. [Google Scholar] [CrossRef] [Green Version]
- Lajaunie, C.; Courrioux, G.; Manuel, L. Foliation fields and 3D cartography in geology: Principles of a method based on potential interpolation. Math. Geol.
**1997**, 29, 571–584. [Google Scholar] [CrossRef] - Chiles, J.P.; Delfiner, P. Geostatistics: Modeling Spatial Uncertainty; John Wiley & Sons: Hoboken, NJ, USA, 2009; Volume 497. [Google Scholar]
- De la Varga, M.; Schaaf, A.; Wellmann, F. GemPy 1.0: Open-source stochastic geological modeling and inversion. Geosci. Model Dev.
**2019**, 12, 1–32. [Google Scholar] [CrossRef] [Green Version] - Talwani, M. Computation with the help of a digital computer of magnetic anomalies caused by bodies of arbitrary shape. Geophysics
**1965**, 30, 797–817. [Google Scholar] [CrossRef] - Plouff, D. Gravity and magnetic fields of polygonal prisms and application to magnetic terrain corrections. Geophysics
**1976**, 41, 727–741. [Google Scholar] [CrossRef] - Güdük, N. Model-Based Probabilistic Inversion Using Magnetic Data: A Case Study on the Kevitsa Deposit. Master’s Thesis, IdeaLeague, RWTH Aachen University, Aachen, Germany, 2020. Available online: http://resolver.tudelft.nl/uuid:f809a7ed-89e9-4e6f-abe4-e6683ba0fa78 (accessed on 2 March 2020).
- Blakely, R.J. Potential Theory in Gravity and Magnetic Applications; Cambridge University Press: Cambridge, UK; New York, NY, USA, 1995. [Google Scholar]
- Davidson-Pilon, C. Bayesian Methods for Hackers: Probabilistic Programming and Bayesian Inference, 1st ed.; Addison-Wesley Professional: Boston, MA, USA, 2015. [Google Scholar]
- Metropolis, N.; Ulam, S. The Monte Carlo method. J. Am. Stat. Assoc.
**1949**, 44, 335. [Google Scholar] [CrossRef] - Betancourt, M. A Conceptual Introduction to Hamiltonian Monte Carlo. arXiv
**2018**, arXiv:1701.02434. [Google Scholar] - Bellman, R. Dynamic Programming, 1st ed.; Princeton University Press: Princeton, NJ, USA, 1957. [Google Scholar]
- Neal, R.M. MCMC using Hamiltonian dynamics. arXiv
**2012**, arXiv:1206.1901. [Google Scholar] - Homan, M.D.; Gelman, A. The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res.
**2014**, 15, 1593–1623. [Google Scholar] - Baydin, A.G.; Pearlmutter, B.A.; Radul, A.A.; Siskind, J.M. Automatic differentiation in machine learning: A survey. arXiv
**2018**, arXiv:1502.05767. [Google Scholar] - Mutanen, T. Geology and Ore Petrology of the Akanvaara and Koitelainen Mafic Layered Intrusions and the Keivitsa-Satovaara Layered Complex, Northern Finland; Bulletin (Geologian tutkimuskeskus (Finland)); Geological Survey of Finland: Espoo, Finland, 1997. [Google Scholar]
- Le Vaillant, M.; Hill, J.; Barnes, S.J. Simplifying drill-hole domains for 3D geochemical modelling: An example from the Kevitsa Ni-Cu-(PGE) deposit | Elsevier Enhanced Reader. Ore Geol. Rev.
**2017**, 90, 388–398. [Google Scholar] [CrossRef] - Koivisto, E.; Malehmir, A.; Hellqvist, N.; Voipio, T.; Wijns, C. Building a 3D model of lithological contacts and near-mine structures in the Kevitsa mining and exploration site, Northern Finland: Constraints from 2D and 3D reflection seismic data: Kevitsa 3D geological model. Geophys. Prospect.
**2015**, 63, 754–773. [Google Scholar] [CrossRef] - Montonen, M. Induced and Remanent Magnetization in Two Boreholes of the Kevitsa Intrusion. Master’s Thesis, University of Helsinki, Helsinki, Finland, 2012. [Google Scholar]
- Gregory, J.; Journet, N.; White, G.; Lappalainen, M. Kevitsa Nickel Copper Mine, Lapland, Finland; Technical report; First Quantum Minerals Ltd.: London, UK, 2016. [Google Scholar]
- Hölttä, P.; Väisänen, M.; Väänänen, J.; Manninen, T. Paleoproterozoic metamorphism and deformation in Central Lapland, Finland. Spec. Paper Geol. Surv. Finl.
**2007**, 44, 7–56. [Google Scholar] - Fournier, D. Advanced Potential Field Data Inversion with l
_{p}-norm Regularization. Ph.D. Thesis, University of British Columbia, Vancouver, BC, Canda, 2019. [CrossRef] - Hinze, W.J. The Role of Gravity and Magnetic Methods in Engineering and Environmental Studies. In Geotechnical and Environmental Geophysics: Volume I, Review and Tutorial; Society of Exploration Geophysicists: Houston, TX, USA, 2012; pp. 75–126. [Google Scholar] [CrossRef]
- Li, Y.; Oldenburg, D.W. Separation of regional and residual magnetic field data. Geophysics
**1998**, 63, 431–439. [Google Scholar] [CrossRef] - National Oceanic and Atmospheric Administration (NOAA). Available online: https://www.ngdc.noaa.gov/geomag/calculators/magcalc.shtml#igrfwmm (accessed on 6 December 2019).
- Thiele, S.T.; Jessell, M.W.; Lindsay, M.; Ogarko, V.; Wellmann, J.F.; Pakyuz-Charrier, E. The topology of geology 1: Topological analysis. J. Struct. Geol.
**2016**, 91, 27–38. [Google Scholar] [CrossRef] - Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv
**2016**, arXiv:abs/1605.02688. [Google Scholar] - Salvatier, J.; Wiecki, T.; Fonnesbeck, C. Probabilistic Programming in Python using PyMC. arXiv
**2015**, arXiv:1507.08050. [Google Scholar] - Li, Y.; Oldenburg, D.W. Incorporating geological dip information into geophysical inversions. Geophsysics
**2000**, 65, 148–157. [Google Scholar] [CrossRef] [Green Version] - Wood, R.; Curtis, A. Geological prior information and its applications to geoscientific problems. Geol. Soc. Lond. Spec. Publ.
**2004**, 239. [Google Scholar] [CrossRef] [Green Version] - Calcagno, P.; Courrioux, G.; Joly, A.; Ledru, P.; Guillen, A. Geological modelling from field data and geological knowledge: Part II. Modelling validation using gravity and magnetic data inversion. Phys. Earth Planet. Inter. Phys. Earth Planet. Inter.
**2008**, 171, 1–4. [Google Scholar] - Joly, A.; Martelet, G.; Chen, Y.; Faure, M. A multidisciplinary study of a syntectonic pluton close to a major lithospheric-scale fault: Relationships between the Montmarault granitic massif and the Sillon Houiller Fault in the Variscan French Massif Central. Part II: Gravity, aeromagnetic investigations and 3D geologic modeling. J. Geophys. Res.
**2008**, 113. [Google Scholar] [CrossRef] [Green Version] - Lelievre, P.; Oldenburg, D.; Williams, N. Integrating geological and geophysical data through advanced constrained inversions. Explor. Geophys.
**2009**, 40. [Google Scholar] [CrossRef] - Jessell, M.; Ailleres, L.; Dekemp, E.; Lindsay, M.; Wellmann, F.; Hillier, M.; Laurent, G.; Carmichael, T.; Martin, R. Next Generation Three-Dimensional Geologic Modeling and Inversion. Soc. Econ. Geol. Spec. Publ.
**2014**, 18, 261–272. [Google Scholar] [CrossRef] - Bosch, M.; Guillen, A.; Ledru, P. Lithologic tomography: An application to geophysical data from the Cadomian belt of northern Brittany, France. Tectonophysics
**2001**, 331, 197–227. [Google Scholar] [CrossRef] - Giraud, J.; Pakyuz-Charrier, E.; Jessell, M.; Lindsay, M.; Martin, R.; Ogarko, V. Uncertainty reduction through geologically conditioned petrophysical constraints in joint inversionConditioned petrophysical constraints. Geophysics
**2017**, 82, ID19–ID34. [Google Scholar] [CrossRef] [Green Version] - Giraud, J.; Lindsay, M.; Ogarko, V.; Jessell, M.; Martin, R.; Pakyuz-Charrier, E. Integration of geoscientific uncertainty into geophysical inversion by means of local gradient regularization. Solid Earth
**2019**, 10, 193–210. [Google Scholar] [CrossRef] [Green Version] - Tikhonov, A.; Arsenin, V. Solution of Ill-Posed Problem; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1977; p. 258. [Google Scholar]
- Zhdanov, M. Geophysical Inverse Theory and Regularization Problem; Elsevier: Amsterdam, The Netherlands, 2002; Volume 36. [Google Scholar]
- Sun, J.; Li, Y. Adaptive Lp inversion for simultaneous recovery of both blocky and smooth features in a geophysical model. Geophys. J. Int.
**2014**, 197. [Google Scholar] [CrossRef] [Green Version] - Tensor Research. ModelVision Magnetic & Gravity Interpretation System, 17 ed. 1993. Available online: http://www.tensor-research.com.au/Geophysical-Products/ModelVision (accessed on 6 December 2019).
- Adams, C.; Dentith, M. Magnetic measurements on diamond drill core: Are we really measuring magnetic susceptibility? In Proceedings of the Exploration ’17 Sixth Decennial International Conference on Mineral Exploration, Toronto, ON, Canada, 22–25 October 2017; Volume 52, pp. 725–728. [Google Scholar]
- Clark, D. Methods for determining remanent and total magnetisations of magnetic sources—A review. Explor. Geophys.
**2014**, 45, 271. [Google Scholar] [CrossRef] - Schmidt, P.; Lackie, M. Practical considerations: Making measurements of susceptibility, remanence and Q in the field. Explor. Geophys.
**2014**, 45, 305. [Google Scholar] [CrossRef] - Lee, M.D.; Morris, W.A. Comparison of Magnetic-Susceptibility Meters Using Rock Samples from the Wopmay Orogen, Northwest Territories; Technical Note 5; Geological Survey of Canada: Vancouver, BC, Canada, 2013. [Google Scholar] [CrossRef]

**Figure 1.**The Bayesian system for our probabilistic model with prior model parameters (red); governing equations (black) that relate model parameters and observations; likelihood functions (blue); observations (blue), with ${y}_{1}$ being observations from magnetic measurements and ${y}_{2}$ from core logs; and the posterior model parameters (purple). The governing equations and the likelihood functions together form the mathematical model $\mathcal{M}$ (grey area). Parameters are stochastic (dashed) or deterministic (solid). The arrows indicate parent-child like hierarchies.

**Figure 2.**A 2D representation of the grid where the forward magnetics are built on. The observation point (purple) is at the center of the grid. From here, the grid spacing increases with a distance squared relationship. The magnetic field decreases cubically with distance as shown by the blue hemispheres.

**Figure 3.**(

**Left**) Geological surface map of the Kevitsa deposit with the described units (after Fournier [46]), and the location of Kevitsa in Finland and in the Central Lapland greenstone belt (CLGB) (in green). (

**Right**) Original airborne magnetic data with the measured total magnetic intensity.

**Figure 4.**Probability density function (pdf) of downhole measured susceptibilities representing the modeled lithologies ultramafic pyroxenite (UPX) and host rock. Note the logarithmic x-axis.

**Figure 5.**Processed magnetic data showing the anomalous magnetic field intensity, with the cross-section location (line) at which we evaluate our results; the locations where we evaluate the magnetic likelihood functions (filled dots); and the geological likelihood (star). We use the locations with an extra circle to invert for the susceptibility of the intrusion.

**Figure 6.**(

**Left**) Pdf with prior and posterior distribution of ${k}_{UPX}$, and the mean of the downhole measured susceptibility for ${k}_{UPX}$ (dashed line). (

**Right**) Pdf with prior and posterior predictive distribution as simulated at one of the observation locations, and the measured magnetic intensity at that observation location (dashed line).

**Figure 7.**(

**a**) Cross-section of the initial model; (

**b**) difference map between the forward modeled magnetic response of the initial model and the magnetic measurements; (

**c**) calculated entropy from all realized models based on assigned uncertainty in the initial model, where zero (white) corresponds to locations with no uncertainty (i.e., locations that have the same outcome in all realizations). (

**d**) Cross-section of the Maximum A Posteriori (MAP) geological model based on magnetic constraints; (

**e**) difference between the forward modeled magnetic response of the MAP model and the magnetic measurements; (

**f**) calculated entropy from all realized models in the probabilistic inversion using only magnetic likelihoods. (

**g**) Cross-section of the MAP model based on magnetic and additional geological constraints; (

**h**) difference between the forward modeled magnetic response of the MAP model and the magnetic measurements; (

**i**) calculated entropy from all realized models in the probabilistic inversion using magnetic and geological likelihoods.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Güdük, N.; de la Varga, M.; Kaukolinna, J.; Wellmann, F.
Model-Based Probabilistic Inversion Using Magnetic Data: A Case Study on the Kevitsa Deposit. *Geosciences* **2021**, *11*, 150.
https://doi.org/10.3390/geosciences11040150

**AMA Style**

Güdük N, de la Varga M, Kaukolinna J, Wellmann F.
Model-Based Probabilistic Inversion Using Magnetic Data: A Case Study on the Kevitsa Deposit. *Geosciences*. 2021; 11(4):150.
https://doi.org/10.3390/geosciences11040150

**Chicago/Turabian Style**

Güdük, Nilgün, Miguel de la Varga, Janne Kaukolinna, and Florian Wellmann.
2021. "Model-Based Probabilistic Inversion Using Magnetic Data: A Case Study on the Kevitsa Deposit" *Geosciences* 11, no. 4: 150.
https://doi.org/10.3390/geosciences11040150