Stochastic Inversion of Geophysical Data by Sequential Bayesian Updating Under a Non-Stationary Gaussian Process Prior

Abstract

The acquisition of geophysical data is becoming increasingly important in the context of critical mineral exploration. Geophysical data and inversion products are essential to map many components of the critical mineral system by detecting geophysical anomalies that can be interpreted by expert geologists. However, the inversion of airborne geophysical data acquired along flightlines into subsurface petrophysical properties remains an outstanding challenge. Many inversion techniques rely either on 1D deterministic inversion or on stochastic inversion on a local scale. The outcome of our work is the stochastic inversion along flightlines of 2D panels (flightline direction vs. depth), while at the same time producing plausible spatial variation in the petrophysical properties. Our method relies on a sequential application of Bayesian inversion, where we invert a sequence of 2D panels such that the variation in petrophysical properties avoids generation of artifacts across the panel boundaries. We show that our method can be used in a practical setting in the context of mineral exploration in the Cape Smith Belt of Canada.

1. Introduction

In the drive to explore for critical minerals needed for the energy transition, many countries have been or are in pursuit of mapping the subsurface using large-spread airborne geophysics. Typically, several types of geophysical measurements are obtained, such as gravity, magnetic, and electromagnetic (EM). Data along flightlines are then interpolated to obtain a 2D (or in EM, a time-domain) map of measured magnetic, gravity, and EM. These maps are used to detect geophysical anomalies of interest; they may be used in mineral prospectivity mapping or may guide exploration companies in drilling certain targets.

In addition, the data contain information on the variation in petrophysical properties in the 3D subsurface, and when inverted, the data turn into 3D models using either deterministic or stochastic methods. In deterministic inversion, a single model is obtained by minimizing a regularized misfit between the predicted and observed data; this yields one smooth image but no assessment of the many alternative models that could explain the data equally well. Stochastic inversion instead poses the problem probabilistically and draws an ensemble of data-consistent models, so that the non-uniqueness of the solution is sampled and the uncertainty of the recovered properties is quantified, at the cost of greater computation [1,2,3,4,5,6]. This is usually done over areas of particular interest, which tend to be limited in extent. Stochastic inversion along extensive flightlines is not common. Some methods are used in the context of scientific studies of the Earth’s crust. Other methods rely on consecutive pseudo-1D or -2D inversions along flightlines [7,8,9]. These approaches are often deterministic [10]. “2.5D inversion”, or stacking the same 2D petrophysical models into 3D, is sometimes used by assuming petrophysical homogeneity perpendicular to the lines [11]. Some recent studies in large-scale potential field inversion were conducted directly on 3D voxel models with regularization [12]. Similarly, the authors in [13] attempted to invert the 3D conductivity of the contiguous US on a 70 km grid mesh from magneto-telluric (MT) data. All of the above results were obtained by minimizing the misfit between model-predicted geophysical maps and the observed data (often interpolated) with a model parameter regularization term, using gradient-based optimization. In these approaches, a well-defined large-scale 2D/3D reference (or initial model) is essential, often with weights used to control how the inverted results closely adhere to the reference model [14]. The resulting models are typically deterministic, and defining 3D reference models at large scales can be very difficult. What is lacking today are methods that can perform stochastic inversions along flightline data. While this work does not yet cover very large areas, such as entire countries, it proposes a method that can be extended to large-scale applications.

Several challenges need to be overcome to achieve this. A single-step inversion over a large area is not feasible in terms of memory and computational resources, as well as the challenge of formulating a single prior model. To address this, we will perform stochastic inversions per panel, along the direction of the flightline, making sure that panels are stitched together without creating artificial boundaries. Secondly, the spatial variability of the petrophysical properties may change dramatically over large distances due to changes in rock properties and the probable structure controlling the rock continuity. This means that the style of spatial variability of these physical properties is likely to change in their direction of continuity (e.g., azimuthal and dipping) and in their subsurface morphologies (layers, intrusions, and folding). This means that building comprehensive prior distribution models for multiple petrophysical properties required for Bayesian inversion is challenging. To address this issue, we use non-stationary Gaussian processes as prior, where the non-stationary mean and covariance function are also uncertain, i.e., have their own prior distributions. We intend to show that with these non-stationary Gaussian priors, a wide range of possible petrophysical variations can be generated. Additionally, correlation structures related to large-scale geological structures between the various petrophysical properties can be imposed. Our approach is applied to the Cape Smith Belt, which is currently under exploration for Ni-Cu-Co sulfide deposits.

2. Application Area

Our area of study is from the Cape Smith Belt (CSB) in the Nunavik region of Northern Québec, Canada. As an early Proterozoic thrust belt containing metamorphosed basaltic volcanic, mafic intrusive, and sedimentary rocks, the CSB has been widely explored for Ni-Cu-PGE magmatic sulfide deposits. We use publicly available airborne magnetic data acquired concurrently with the HeliTEM² survey and obtained through the Geomining Information System of Québec; these data were released into the public domain by KoBold Metals [15]. The survey was conducted by Xcalibur MPH (Canada) Ltd. using a HeliTEM² 35 m-diameter-loop electromagnetic system (Xcalibur MPH (Canada) Ltd., Toronto, ON, Canada) and a CS-3 cesium-vapor magnetometer (Scintrex Ltd., Concord, ON, Canada) [15]. The data were acquired between 18 June and 24 August 2022. The survey comprised 3626.0 km of traverse lines spaced 200 m apart and 381.7 km of tie lines spaced 1450–2000 m apart, for a total of 4007.7 km (Figure 1).

Prior to inversion, the HeliTEM² total-field magnetic data were processed following the standard airborne magnetic workflow: removal of diurnal variations using base-station records, subtraction of the International Geomagnetic Reference Field (IGRF), tie-line levelling and micro-levelling to remove residual line noise, interpolation of the levelled data onto a regular grid, and calculation of a vertical magnetic gradient to enhance near-surface magnetic units (see [15] for processing details). These steps produce the gridded magnetic-anomaly maps that are conventionally interpreted qualitatively or inverted deterministically into a single subsurface model. What such workflows do not deliver is a quantified uncertainty for the recovered petrophysical properties. The present work addresses precisely this shortcoming: rather than a single deterministic image, it produces an ensemble of data-consistent magnetic-susceptibility models from which uncertainty can be quantified.

3. Methodology

3.1. Sequential Bayesian Updating: A Review

Our approach aims to solve a larger inverse problem by the sequential inversion of smaller panels, areas, or blocks. Since the approach is Bayesian, we will review here some fundamental concepts of sequential Bayesian updating [16,17]. Consider a model parameter, $y$, that is informed by a sequence of data, $x_1,$ $x_2,\dots , x_n$. In a Bayesian context, we start with a prior distribution, $p\left(y\right)$, and at step $n$, we can decompose the likelihood as follows:

$$p\left(x_1,x_2,\dots , x_n|y\right)=p\left(x_1|y\right)p\left(x_2|x_1,y\right)\dots p(x_n|x_{n-1},y).$$

(1)

where

$$x_{n-1}=\left\{x_{n-1},x_{n-2},\dots ,x_1\right\}.$$

(2)

Because $x_{n-1}$ is treated as a set (the history of observations), the order of its elements is immaterial for the conditioning used here.

The update distribution at time $n$ then is the posterior:

$$p\left(y|x_n\right)=p_n\left(y\right) ~ p\left(y\right) p(x_n|y).$$

(3)

Then, after obtaining new information, $x_{n+1}$, we start anew and update

$$p_{n+1}\left(y\right)=p\left(y|x_{n+1}\right)~ p\left(y\right) p(x_{n+1}|y).$$

(4)

Essentially, we are using the $p_n\left(y\right)$ as the prior distribution in the next step of the Bayesian update to get the next posterior, $p_{n+1}\left(y\right)$. Indeed, we can derive that the posterior of $y$ at step $n$ becomes the prior at step $n+1$.

$$p_n\left(y\right)p\left(x_{n+1}|x_n,y\right)=p\left(y\right)p\left(x_n|y\right)p\left(x_{n+1}|x_n,y\right)=p\left(y\right)p\left(x_{n+1}|y\right) ~ p_{n+1}\left(y\right).$$

(5)

Note that we do not need a conditional independence assumption in this derivation. The revised posterior equals the current posterior times the likelihood of the new data. All historical knowledge about $y$ is captured by $p_n\left(y\right)$. The sequential application of Bayes’ rule is therefore recursive.

Consider now a problem with a dynamic $y$, changing in time (or in space, or both), and a Markovian (conditional independence) model is used:

$$p\left(y_{n+1}|y_n\right)= p\left(y_{n+1}|y_n\right),$$

(6)

where $y_n=\left\{y_n, y_{n-1}, \dots , y_1\right\}$.

The true state is not observed; instead, a sequence of data is acquired. Assuming conditional independence,

$$p\left(x_n|y_n,x_{n-1}\right)=p(x_n|y_n).$$

(7)

Then, the joint density becomes

$$p\left(x_n,y_n\right)=p\left(y_0\right){\displaystyle {{\displaystyle \prod }}_{i=1}^n}p\left(y_i|y_{i-1}\right)p(x_i|y_i).$$

(8)

The dynamic evolution of $y$ is modeled using two steps: predict $y$ at time $n$ from data at previous times (filter step) and predict a $y$ at the next step, $n+1$. The filter distribution is the posterior, $p(y_n|x_n)$. Using this filter distribution, the predictive distribution becomes

$$p\left(y_{n+1}|x_n\right)={\int }_{y_n}p\left(y_{n+1}|y_n\right) p(y_n|x_n) d{y}_n.$$

(9)

After a new observation, $x_{n+1}$, is obtained, application of recursive Bayes combines the predictive distribution with a new likelihood, $p(x_{n+1}|y_{n+1})$:

$$p\left(y_{n+1}|x_{n+1}\right)=p\left(y_{n+1}|x_{n+1}, x_n\right)=p\left(y_{n+1}|x_n\right) p(x_{n+1}|y_{n+1}).$$

(10)

where now $x_{n+1}$ is conditionally independent of $x_n,$ knowing $y_{n+1}.$ This formulation looks deceptively simple because calculating high-dimensional pdfs and dealing with non-linear dynamics require Monte Carlo simulations. Common approaches are the use of sequential Monte Carlo methods such as particle filtering [18]. However, we will show in the next section that such methods become difficult to apply in a large-scale spatial context, where the model and data are both high-dimensional, and the physical model is CPU demanding.

3.2. A Bayesian Formulation for Large-Scale Geophysical Inversion

To model large areas, we divide the area over which data need to be inverted into blocks (3D), or, here, into 2D panels (in x and z direction); see Figure 2. For each block, we assume the prior to be a Gaussian process but with unknown covariance and mean function [19,20]. Let $\phi$ be a stationary Gaussian process, and then for each panel, $P_i$, we have

$${\phi }_i=\phi \left(s\right), s ϵ P_i.$$

(11)

A stationary Gaussian process is fully defined by stationary mean and covariance:

$$\begin{array}{c}E\left[\phi \right(s\left)\right]=\mu \\ Cov\left(\phi \left(s_i\right),\phi \left(s_j\right)\right)=E\left[\left(\phi \left(s_i\right)-\mu \right)\left(\phi (s_j)-\mu \right)\right]=C\left(s_i-s_j\right),\end{array}$$

(12)

where $C$ is any positive definite function, and where the random vector of any set of spatial locations, s₁, …, s_n is

$$\mathit{\phi }=\left\{\phi \left(s_1\right),\phi \left(s_2\right),\dots ,\phi \left(s_n\right)\right\},$$

(13)

and is multivariate Gaussian:

$$\begin{array}{c}\phi ~{\displaystyle \frac{1}{{\left(2\pi \right)}^{n/2}{\left|\mathsf{\Sigma }\right|}^{1/2}}}\exp \left(-{\displaystyle \frac{1}{2}}{\left(\mathit{\phi }-\mathit{\mu }\right)}^T{\mathsf{\Sigma }}^{-1}\left(\mathit{\phi }-\mathit{\mu }\right)\right),\\ {\displaystyle {\mathsf{\Sigma }}_{ij}}=C\left(s_i-s_j\right).\end{array}$$

(14)

If we consider the mean and covariance function to be the (hyper)-parameters, $\theta$, of the stationary process, then we can introduce the notation ${\phi }_i\left({\theta }_i\right).$

Figure 2 shows the sequential Bayesian model that applies to this case. In comparison with the notation of the previous section, we have $y=\left\{\phi ,\theta \right\}$ and the $x = d$, the geophysical data. The difficulty in applying the sequential approach to our problem is that

• $\phi$ represents a process in space and is high-dimensional;
• $\theta$ that defines the process $\phi$ is unknown;
• The relationship between $\theta$ and $d$ is not known/intractable;
• The relationship between $\phi$ and $d$ is known but not linear;
• Spatial dependency exists between ${\phi }_n$ and ${\phi }_{n+1}$, as well as ${\theta }_n$ and ${\theta }_{n+1}$.

Still, we can apply the “predict plus filter” approach to this problem:

• Predict: Sample the posterior distribution of the next panel, $P_{i+1}$, given the previous panel, $P_i.$ This will require enforcing spatial correlation between ${\phi }_n$ and ${\phi }_{n+1}$, as well as ${\theta }_n$ and ${\theta }_{n+1}$.
• Filter: Sample the posterior distribution of the next panel, given the geophysical data of the next panel. This requires solving an inverse problem using the geophysical data of the next panel and the predictive distribution as the prior distribution (relying on the recursive Bayes model).

Consider the first panel, $P_0$. In the first step, using the (stated) prior distribution,

$$p\left({\theta }_0,{\phi }_0\right)=p\left({\theta }_0\right) p({\phi }_0|{\theta }_0),$$

(15)

we sample several realizations of ${\theta }_0$ and then of ${\phi }_0$, creating realizations ${\left({\theta }_0,{\phi }_0\right)}_i, i=1,\dots ,L$. In the next step, we need to filter the geophysical data, $d_0$; in other words, we need to solve an inverse problem. Solving such a problem requires sampling from $p({\theta }_0,{\phi }_0|d_0)$. This problem is treated in Section 3.3. After inversion, the next set of particles, ${\left({\theta }_1,{\phi }_1\right)}_i$, needs to be predicted from the updated set, ${\left({\theta }_0,{\phi }_0\right)}_i$. The relationship between ${\phi }_1$ and ${\phi }_0$ is linear, since they model the same property. In geostatistical jargon, any realization of ${\phi }_0$ serves as hard/exact data for constraining the Gaussian process, ${\phi }_1$. At the same time, we need to sample ${\theta }_1$, given ${\theta }_0$. This problem is treated in Section 3.4. Then, this procedure is recursively applied until all panels have been inverted.

Our approach will use Monte Carlo simulation, but it does not employ particle filtering, meaning that we sample directly from the posterior distribution without employing an importance sampler. Instead, we leverage high-performance computing to generate an ensemble of inverted models, computed in parallel. Next, we will detail both the filter and predict step.

3.3. Filter: Inversion of a Single Panel

We now focus on the filtering aspect of our sampler. $\theta$ and $\phi$need to be inverted (filtered) sequentially, honoring the hierarchical relationship between them. To do so, we propose the following decomposition of the joint inversion of $\theta$ and $\phi$ at step $n$:

$$p\left({\theta }_n,{\phi }_n|d_n\right)=p\left({\theta }_n|d_n\right) p\left({\phi }_n|d_n,{\theta }_n\right).$$

(16)

Consider first $p\left(\theta |d\right)$, whose dimension is low. For example, in 2D, the Gaussian process has a maximum of three parameters for the covariance function (one angle and two ranges), and a mean and variance, giving a total of five parameters. This suggests modeling $p\left(\theta |d\right)$ as a high-dimensional density function. To estimate this density function, we generate a training set by the Monte Carlo simulation, then estimate $p\left(\theta |d\right),$ and then sample from the estimated distribution with an exact method. More specifically, consider the joint distribution:

$$p\left({\theta }_n,d_n\right)=p\left({\theta }_n)p(d_n|{\theta }_n\right)=p\left({\theta }_n\right)p\left(d_n|{\phi }_n\right)p({\phi }_n|{\theta }_n),$$

(17)

using the Bayes’ network definition (Figure 2) in the last step, where $d_n$ is related to ${\theta }_n$, through the spatial model, ${\phi }_n$. To sample a joint distribution, $p\left(\theta ,d\right)$, for any panel ($n$) (in Equations (18)–(22), the panel subscript, n, is dropped for brevity), we therefore perform the following sequential sampling:

$${\theta }^{\left(j\right)}~p\left(\theta \right),$$

(18)

Then, we sample a Gaussian process realization:

$${\phi }^{\left(j\right)}~p(\phi |{\theta }^{\left(j\right)}),$$

(19)

And then, we forward-model the data:

$$d^{\left(j\right)}=g\left({\phi }^{\left(j\right)}\right).$$

(20)

Note that we use a superscript, $\left(j\right)$, to indicate a sample, while using a subscript to indicate a panel. We now have the training dataset:

$$\left\{{\theta }^{\left(j\right)},d^{\left(j\right)}\right\}, j=1,\dots ,J.$$

(21)

The training set is used to fit a high-dimensional probability density model, $\widehat{p}\left(\theta ,d\right)$, from which a conditional distribution can be calculated:

$$\widehat{p}\left(\theta |d\right)={\displaystyle \frac{\widehat{p}\left(\theta ,d\right)}{\widehat{p}\left(d\right)}}.$$

(22)

For the second term, $p\left(\phi |\theta ,d\right)$, we need to solve a non-linear inverse problem with a Gaussian prior whose hyperparameters are known. In Appendix A, we review two known solutions for this problem through the gradual deformation method (GDM) and the probability perturbation method (PPM) [21,22,23,24]. These methods were implemented using the fast Fourier transform for Gaussian process simulation with GeoStats.jl version 0.75.0 under Julia version 1.10.5 [19,20,25].

3.4. Predict: Generating a New Prior

At this stage, we need a predictive distribution that generates a new prior for the next inversion. Practically speaking, we have a set of realizations of the previous panel that match the geophysical data of that panel. Following Equation (15) of the predictive model, we need to specify $p\left({\phi }_{n+1},{\theta }_{n+1}|{\phi }_n,{\theta }_n\right)$, which can be further decomposed as

$$p\left({\phi }_{n+1},{\theta }_{n+1}|{\phi }_n,{\theta }_n\right)=p\left({\phi }_{n+1}|{\phi }_n,{\theta }_n,{\theta }_{n+1}\right) p\left({\theta }_{n+1}|{\phi }_n,{\theta }_n\right).$$

(23)

The first term is the distribution of a Gaussian process conditioned to a previously sampled Gaussian process in panel $P_n$, with known covariance parameters. This is a solved problem in geostatistics known as conditional simulation under a Gaussian model, for which several sampling algorithms are available, such as sequential Gaussian simulation [19,20,25].

In the second term, we need to impose spatial continuity between the hyperparameters. First, we assume conditional independence:

$$p({\theta }_{n+1}|{\theta }_n,{\phi }_n)\cong p({\theta }_{n+1}|{\theta }_n),$$

(24)

To determine $p({\theta }_{n+1}|{\theta }_i)$, we consider that ${\theta }_{n+1}$ depends on ${\theta }_n$, because the data, $d_{n+1}$, is correlated to $d_n$. In Section 3.3, we saw how we can estimate both $p\left({\theta }_n|d_n\right)$ and $p({\theta }_{n+1}|d_{n+1})$. We therefore approximate $p({\theta }_{i+1}|{\theta }_i)$ using a distribution $p^{*}({\theta }_{i+1}|{\theta }_i)$ estimated from samples ${\theta }_i^{*} \mathrm{and} {\theta }_{i+1}^{*}$ as follows:

$$p({\theta }_{i+1}|{\theta }_i)\cong p^{*}({\theta }_{i+1}|{\theta }_i)\mathrm{with}{\theta }_i^{*}~p\left({\theta }_i|d_i\right) and {\theta }_{i+1}^{*}~p\left({\theta }_{i+1}|d_{i+1}\right).$$

(25)

3.5. Complete Methodology

Our method is a Monte Carlo method, meaning that each spatial model that matches all the geophysical data is sampled independently. We do not employ a particle filter where an ensemble model is updated jointly. As outlined in the previous section, there are methodological challenges that make application of particle filtering challenging. Additionally, by leveraging parallel computing, we can run each sample on a different GPU/CPU. With 100 cores, the cost of 100 samples is the same as for one sample. A naïve Monte Carlo method is also much more straightforward to implement compared to a particle filter, which requires the additional modeling of an importance sampler, which is not trivial when dealing with spatial models.

Algorithm 1 summarizes, using the above notations, the two main steps of the process, which are as follows:

• Estimating a predictive model for the hyperparameters of the Gaussian process;
• Iterative application of the filter and predict steps.

Algorithm 1. For the proposed algorithm, $\widehat{p}$ $\mathrm{and}{p}^{*}$ are estimated within the algorithm steps.
Input	Prior $p\left({\theta }_0\right),$ $p\left({\phi }_0|{\theta }_0\right).$
	Geophysical data per panel:$d_{obs,0},d_{obs,1},\dots ,d_{obs,n}.$
Output	Posterior samples $\left({\phi }_{0,post},{\theta }_{0,post}\right),$ $\left({\phi }_{1,post},{\theta }_{1,post}\right),\dots ,\left({\phi }_{n,post},{\theta }_{n,post}\right).$
Estimate a predictive model for Gaussian process hyperparameters (training set):
	Sample:${\theta }^{\left(j\right)}~p\left({\theta }_0\right);{\phi }^{\left(j\right)}~p\left({\phi }_0|{\theta }_0\right),j=1,\dots ,J.$
	Calculate: $d^{\left(j\right)}=g\left({\theta }^{\left(j\right)}\right),j=1,\dots ,J.$
	Estimate:$\widehat{p}(\theta |d)$ $(\mathrm{Equation}(22))\mathrm{using}\mathrm{samples}({\theta }^{\left(j\right)},d^{\left(j\right)}),j=1,\dots ,J.$
	$\mathrm{Sample}{\theta }_0^{*,\left(j\right)}~p\left({\theta }_0|d_0\right)and{\theta }_1^{*,\left(j\right)}~p\left({\theta }_1|d_1\right),j=1,\dots ,J.$
	Estimate:$p^{*}({\theta }_1|{\theta }_0)$ $(\mathrm{Equation}(25))\mathrm{using}\mathrm{samples}({\theta }_0^{*,\left(j\right)},{\theta }_1^{*,\left(j\right)}),j=1,\dots ,J.$
Do Panel 0:
	Predict:${\theta }_{0,obs}~\widehat{p}\left(\theta |d_{obs,0}\right);{\phi }_{0,obs}~p\left({\phi }_0|{\theta }_0\right).$
	Filter:${\phi }_{0,post}~p\left({\phi }_0|{d_{obs,0},\theta }_{0,obs}\right)$ using GDM or PPM
$ForPaneli=1,\dots ,n$
	$\mathrm{Predict}{\theta }_{i,obs}^{*}~p^{*}\left({\theta }_i|{\theta }_{i-1}\right);$ ${\phi }_{i,obs}~p\left({\phi }_i|{{\phi }_{i-1},\theta }_i\right).$
	Filter:${\phi }_{i,post}~p\left({\phi }_i|{d_{obs,i},\theta }_{i,obs}\right)$ using GDM or PPM.
End	Repeat starting with Panel 0.

4. Application

4.1. Illustration of a Two-Panel Case

4.1.1. Falsification Test for the Prior, $p\left({\theta }_0\right)$

Before moving to the full real case, we present in detail the application of the above methodology to a two-panel case. Application to multiple panels is then the sequential application of the two-panel solution. Before starting the iterative filter, we need to generate a training set for predicting the conditional hyperparameter distribution, $\widehat{p}\left(\theta |d\right)$, and derive from this $p^{*}({\theta }_{i+1}|{\theta }_i)$. The training set should be able to predict the data along a line. In other words, a falsification test is needed on the prior distribution, $p\left({\theta }_0\right)$; see Figure 3. Such a falsification test requires sampling models from the prior, running the forward simulator and testing whether the data are an outlier with respect to the simulated data. Figure 3 shows the application of the falsification test on the two-panel case. The prior is taken to be a wide, uniform prior (

Table 1. A broad prior distribution on model variables, as an initial statement of uncertainty. This prior needs to be tested, in a falsification test, with actual observed data.

Prior Variables		Definition	Distributions
Gaussian process hyper-parameters 2D anisotropy	a	Length of the semi-major axis of ellipse, in meters	U [200, 700]
	bf	Length ratio of the semi-minor axis to the semi-major axis in an ellipse, the length of semi-minor axis b = abf	U [0.01, 1]
	γ	Azimuth angle of ellipse	U [1, 180]
Petrophysics	μms	Mean value magnetic susceptibility	U [0, 0.3]
	σms	Standard deviation magnetic susceptibility	U [0.05, 0.5]

). Samples of $\theta$ and $\phi$ are sampled following the hierarchical framework, and the simulated data are generated by forward modeling. We use a Robust Mahalanobis Distance test to define whether the data are an outlier with respect to the simulated data samples. Figure 3 shows that this is not the case.

4.1.2. Estimate Predictive Models for Gaussian-Process Hyperparameters

The purpose of this step is to learn, from the Monte Carlo training set, the two statistical relationships that drive the sequential scheme: (i) the conditional distribution, $\widehat{p}\left(\theta |d\right)$, which links the Gaussian-process hyperparameters, $\theta$, to the geophysical data, $d$, and allows the filter step to convert an observed datum into plausible hyperparameters (Equation (22)); and (ii) the predictive distribution, $p^{*}({\theta }_{i+1}|{\theta }_i)$, which links the hyperparameters of one panel to those of the next and allows the predict step to propagate the prior from panel to panel (Equation (25)). Figure 4 illustrates how both are obtained.

Figure 4 shows the various steps to estimate and then sample from $p^{*}({\theta }_{i+1}|{\theta }_i)$. First, the data are reduced in dimension using MDS (multidimensional scaling). In this case, the first five principal components contain more than 95% of variance. The dimension of $\theta$is five, so a total of 10 dimensions cover the hyperparameter and data variables. Figure 4 shows a projection into 2D data (first two principal directions for d) and 1D parameter (mean). The pdfs, $\widehat{p}\left(\theta |d\right)$, are estimated using kernel density estimation. Figure 4 shows the resulting correlation between the mean and the major range. Thus, a correlation exists between the hyperparameters of two consecutive panels.

4.1.3. Predict and Filter

Panel $P_0$ needs to be inverted from the geophysical data above that panel. This requires first sampling ${\theta }_0~ \widehat{p}\left({\theta }_0|d_{obs,0}\right)$ and then GDM (Appendix A) on the first panel, using a Gaussian process with hyperparameters, ${\theta }_0$. Figure 5 shows three representative initial realizations (out of the full ensemble of N) sampled from the prior. The second row shows the same three realizations with hyperparameters sampled from $\widehat{p}\left({\theta }_0|d_{obs,0}\right)$. Each realization is then perturbed using the GDM method to achieve models that match the observations; see the bottom two rows in Figure 5.

The next step is the prediction of the next panel (gray panel), $P_1$; see Figure 6. In this step, we first sample ${\theta }_1~p^{*}\left({\theta }_1|{\theta }_0\right)$ and then sample realization of panel 1, constrained (conditioned) to the magnetic susceptibility of the previous panel. These realizations are the new predictive distribution that goes into the gradual deformation method to generate models that match the data. This procedure is iterated if more panels need to be inverted.

4.2. Four-Panel Case: Uncertainty Analysis

Figure 7 shows three realizations of the full four-panel case (L2 flight line; see Figure 1), together with the ensemble mean and ensemble variance. Because of the averaging, the ensemble mean loses spatial correlation that might exist and represents a smooth model (much like a deterministic inversion), as shown in Figure 8. The ensemble variance can be seen as one metric that characterizes uncertainty (but not the only one). We observe here that the variance increases with depth, but not uniformly. For example, the area in the first 1 km has much higher uncertainty (and higher mean) than the rest of the panel.

4.3. Uncertainty Quantification with Multiple Parallel Panels

We ran the same procedure on four parallel lines (L1–L4 in Figure 1), each with four panels. Because each realization is an independent sample of the posterior, the ensemble provides, at every location, a Monte Carlo estimate of the probability that the magnetic susceptibility exceeds a given threshold. For example, in Figure 9, we plot the probability that the magnetic susceptibility is above 0.039 (which is the 75% quantile of the inverted magnetic susceptibility). High probability means high likelihood of exceeding this threshold. Regions of high probability correspond to zones likely to host elevated magnetic susceptibility within the domain. The results can be used by geologists/geophysicists for further interpretation of what these features mean and how they are to be connected in 3D.

From a geological standpoint, the elevated magnetic-susceptibility zones imaged along lines L1–L4 (Figure 1) are consistent with the mafic and ultramafic intrusive rocks that characterize the Cape Smith Belt. In this belt, the Ni-Cu-PGE magmatic sulfide mineralization is hosted by such mafic–ultramafic bodies, which commonly carry magnetite and therefore produce strong magnetic responses [27]. The high-probability susceptibility anomalies of Figure 9 can thus be interpreted as candidate mafic–ultramafic intrusions and their contacts, which are the primary exploration targets in the region. Because our approach quantifies the probability that the susceptibility exceeds a chosen threshold, it provides a probabilistic delineation of these prospective host rocks rather than a single deterministic image, helping to prioritize drilling targets while making the associated uncertainty explicit.

Figure 9. Probability for magnetic susceptibility to be above 0.039, calculated from multiple realizations and interpolated in 3D using Gaussian Process Regression [28].

Figure 9. Probability for magnetic susceptibility to be above 0.039, calculated from multiple realizations and interpolated in 3D using Gaussian Process Regression [28].

5. Discussion

Choosing a prior model for inversion is challenging, and then using such a prior model in the inversion is equally challenging. In this paper, we introduce a non-stationary Gaussian process prior model that provides compromises on both challenges. This Gaussian prior has the flexibility to match data well because the mean and covariance terms are considered uncertain, in addition to the spatial uncertainty of a petrophysical property. Gaussian models can easily be simulated in sequences of panels without artifacts, thereby making large-scale inversion possible. Additionally, many inversion algorithms exist for Gaussian priors. We used the probability perturbation method, but anyone’s favorite method can be used as well. The compromise lies in the fact that the Gaussian prior cannot capture more complex and detailed geological variability, such as faulting, stratigraphy, and other discrete features. Our procedure includes a falsification test to investigate if the Gaussian prior should be rejected, which may be the case with significant discrete changes in the data.

However, we believe it is a step forward in the context of mineral exploration, where the common practice is to first perform a deterministic smooth geophysical inversion, and then only do a manual interpretation of the results. Such a manual interpretation may however be biased because deterministic inversion may ignore important sources of variability, as we have shown through the many alternative data-matching inversions that look very different from a deterministic inversion. As a result, planning drilling based on deterministic inversion only may miss the very magnetic-susceptibility anomaly one wants to target. At a minimum, this method may help geologists interpret deterministic inversion, since now uncertainty is quantified. The latter is very useful to avoid overinterpreting detailed features of the deterministic inversion. A probability model such as shown in Figure 9 will help in avoiding such overinterpretation.

The proposed approach has a remarkably small computational footprint. The GDM iterations and panels along a flightline are computed sequentially and on a single thread. The method scales through parallelization at the level of independent realizations. Per-realization memory is in the order of 1.1 GB, so an ensemble of 100 realizations occupies approximately 110 GB of RAM and fits comfortably on a single multi-core workstation. On the AMD EPYC 7713 used for our experiments, with 64 physical cores and 128 hardware threads, the per-panel cost is dominated by the 150 GDM outer iterations needed to converge to a data-matching realization. A single realization spanning four panels completes in 45 min to 1 h, and because the 100 realizations run concurrently, the full ensemble completes in essentially the same wall-clock. The wall-clock scales linearly with the length of the flightline. Ensemble-based stochastic inversion at the line scale therefore runs on a single workstation in under an hour.

Because of its flexibility, the methodology can be extended. Importantly, the framework is agnostic to the geophysical forward operator: it only requires a forward model linking the petrophysical property to the measured response, so the same sequential Bayesian scheme applies to potential-field data (gravity and magnetic, as demonstrated here), as well as to electromagnetic data. For example, one may invert 3D blocks instead of panels and perform inversion of such blocks in a sequence such that a very large area is covered in 3D. The method can be extended to multi-physics inversion by adding a model of co-variation between two or more petrophysical properties. Several such models are available for Gaussian processes in the geostatistical literature [29,30,31].

6. Conclusions

An increase in the demand for critical minerals will necessitate the acquisition and inversion of massive amounts of geophysical data. In terms of inversion, the focus so far has mostly been on large-scale deterministic inversion, or local-scale stochastic inversion. Large-scale stochastic inversion, producing multiple models of petrophysical properties, has not yet been achieved. In this paper, we present a methodology that takes a step in that direction. Our methodology implements a sequential Bayesian inversion of blocks of data while avoiding the creation of artifacts at the block boundaries. The latter is achieved by formulating the prior distribution models as a non-stationary Gaussian process that is sequentially simulated from block to block. The filtering step is achieved by means of the probability perturbation method that inverts each block locally. Having knowledge of uncertainty may prevent overinterpreting variation in the inversion that is a risk with traditional deterministic inversion.