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

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## 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

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**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.

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**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