Parametric geological models such as implicit or kinematic models provide low-dimensional, interpretable representations of 3-D geological structures. Combining these models with geophysical data in a probabilistic joint inversion framework provides an opportunity to directly quantify uncertainty in geological interpretations. For best results, care must be taken with the intermediate step of rendering parametric geology in a finite-resolution discrete basis for the geophysical calculation. Calculating geophysics from naively voxelized geology, as exported from commonly used geological modeling tools, can produce a poor approximation to the true likelihood, degrading posterior inference for structural parameters. We develop a simple integrated Bayesian inversion code, called Blockworlds, showcasing a numerical scheme to calculate anti-aliased rock properties over regular meshes for use with gravity and magnetic sensors. We use Blockworlds to demonstrate anti-aliasing in the context of an implicit model with kinematic action for simple tectonic histories, showing its impact on the structure of the likelihood for gravity anomaly.

Geological modeling of subsurface structures is critical to decision-making
across numerous application areas, including mining, groundwater, resource exploration, natural hazard assessment, and engineering, yet is also subject to considerable uncertainty

Bayesian methods provide a probabilistically coherent framework for reasoning about uncertainty

Bayesian methods are used in both geology and geophysics, but with different framing for inference problems. In the geophysics community, where quantitative inversion frameworks have been in use for decades

In contrast, the core problem in geology is to interpret observations in terms of geological histories and processes

Conditioning geological forward models on geophysical observations makes it tractable to perform full posterior inference in an interpretable parameter space of reduced dimension. This can be done using sampling methods such as Markov chain Monte Carlo methods

Probabilistic inversion workflows that incorporate forward geophysics based on forward-modeled 3-D structural geology are, to date, still uncommon, owing in part to computational challenges faced by sampling algorithms. The pyNoddy code

This paper describes a simple Bayesian inversion framework, called
Blockworlds, which to our knowledge is the first to perform full posterior
inference based on forward geophysics from an implicit geological model
with kinematic elements. Although not yet intended to serve as a production-ready geological modeling package, Blockworlds' design addresses an obstacle for inversion workflows that resample the parametrized 3-D geometry of geological units onto a discrete volumetric mesh for geophysical calculations. Discontinuous changes in rock properties across unit interfaces may become undersampled unless the mesh is chosen to align with those interfaces

To address the issue of undersampled interfaces, Blockworlds incorporates
numerical anti-aliasing into its discretization step. Anti-aliasing to
address undersampling has a long history in computer graphics

This paper is organized as follows. Section

The forward modeling of observations falls under the calculation of the likelihood

While some measurements, such as structural observations, can be computed
directly from a continuous functional form for the geological forward model, the calculation of likelihoods based on simulation of geophysical sensors may require discretization of the geology. The likelihood terms in this case each involve the composition of two forward models:

a mapping

a mapping

Although discretization methods such as adaptive meshes

To show a straightforward example, we calculate the posterior density for the parameters of a uniform-density spherical inclusion (sphere radius and mass density) given the gravitational signal. An analytic solution exists, so we can compare directly to the true posterior, and with only two parameters we can simply evaluate over a grid of parameter values rather than using MCMC. We model a 1 km

Figure

Calculation of the posterior distribution for the radius and density of
a uniform spherical inclusion from gravity inversion. Top row: cross section of 3-D rock density field for the true model from which the data are generated. Second row: simulated gravity field at the surface. Third row: residuals of simulated gravity field from the analytic solution. Bottom row: cross section through the posterior for an independent Gaussian likelihood. Discretization schemes shown (columns, left to right): coarse aliased mesh (15

When inverting on the coarse mesh with no anti-aliasing, the density in a given cell changes only when the sphere radius crosses over the center of one or more mesh cells, and then it changes discontinuously (see last row, first column of Fig.

The inversions with anti-aliased geology result in continuous, even smooth,
posteriors that trace the analytic solution and run at nearly the same speed.
A closer look reveals that the coarse-mesh anti-aliased posterior is slightly
offset toward the bottom left relative to the analytic curve, while the
fine-mesh posterior is not. This is a product of low mesh resolution, which the anti-aliasing approximation only partially mitigates (see Sect.

We will later directly test the influence of aliasing on the performance of MCMC, since the capacity for full posterior inference is one of the major advantages of parametric geological inversions. However, aliasing will cause problems for a broad variety of estimation algorithms. Even for fine mesh resolution, the likelihood is piecewise constant along spatially aliased geological parameters, and so components of the likelihood gradient in those directions are zero almost everywhere in a measure-theoretic sense. Methods such as Nelder–Mead (for optimization) and Metropolis random walk (for sampling) that do not rely on posterior gradient information will seem to work with a fine enough mesh, although at greater computational cost and with no warning of the extent of aliasing effects unless run multiple times from different starting points. Gradient-based methods, such as stochastic gradient descent or Hamiltonian Monte Carlo, will fail catastrophically, since the zero gradient of an aliased likelihood will not reflect its geometry. The covariance matrix used to estimate local uncertainty in the fit from an optimization algorithm will be singular, since the components of the Hessian along aliased directions will be zero. Finally, while the gravity inverse problems we consider in this paper are linear, an attempt to invert for parametric geology based on a nonlinear sensor would face discontinuous changes in the sensitivity kernel with parameters, with unpredictable effects on parameter estimation.

One could argue that the likelihood might be less badly aliased if the data constraints were weaker. Inflating the data errors can be viewed as a form of tempering, which we would expect to merge the multiple aliased modes.
Figure

Influence of data error on the posterior distribution for the sphere inversion problem. The panels from left to right show the effects of multiplying the standard deviation of Gaussian noise on gravity observations by factors of 1 (baseline), 2, 5, and 10.

This kind of catastrophic breakdown occurs because of the restrictive
conditioning of rock properties on parametric geology, so we do not expect the same kind of effect to arise in flexible geophysical inversions that derive their priors from discretized geological models

While our example may seem artificial, widely used geological modeling tools such as Noddy and GeoModeller usually export on rectangular meshes, with lithology or rock properties evaluated at the center of each mesh cell precisely as described above. Any ongoing development or support of geological modeling tools intended for use in probabilistic workflows should keep examples like this in mind. Complex structure imparted by faults, folding, dykes, and sills is highly sensitive to discretization. The situation may arise where the causative body for a strong geophysical anomaly, such as a thin and relatively magnetic or remanently magnetized dyke, is removed from the geological prior due to overly coarse discretization parameters, while a resulting strong magnetic response remains. Inversion schemes are not intended to address situations where the target is unintentionally removed from the data.

The main idea behind anti-aliasing is to produce rock property values in each mesh cell such that the response of a sensor to the anti-aliased model best approximates the action of the same sensor on the same geology exported to a higher-resolution mesh. In this section, we will develop an anti-aliasing prescription for gravity anomaly, which responds linearly to mass density. We frame the problem as a regression that uses the position of a geological interface within a mesh cell to predict what the mean mass density would be for that cell in a high-resolution model.

The top panel of Fig.

To train the regression, we generate a synthetic dataset of 1000
(

Figure

Since the anti-aliasing approximation amounts to a strong prior on sub-mesh
structure, understanding its limitations is critical in practical modeling. The approximation giving rise to Eqs. (

Another drawback in anti-aliasing is that it ignores the relative orientations of interfaces in voxels spanning multiple interfaces. If only one interface passes through a voxel, Eq. (

Once an inversion has been performed, the best practice to assess bias will still be to repeat it with a different cell size. In the next section we will discuss metrics to evaluate the global agreement of posteriors based on different mesh sizes, allowing quantitative evaluation, online or offline, of the value to the user of running at a given mesh resolution.

To illustrate how anti-aliasing interacts with more realistic geological structures, Blockworlds implements a simplified kinematic model inspired by Noddy

The action of kinematic events in Blockworlds is recursive, with each new event modifying the geology resulting from the events before it. Since the notation describing this action is complex, we include a summary of symbols defined in this subsection in Table

Table of symbols for kinematic event notation.

Each event is parametrized by a collection of parameters

The calculation begins with a basement layer of uniform density,

Calculation of forward geology for Models 1 and 8 through a sequence of tectonic events: the addition of two stratigraphic layers and two faults. Top: geology; bottom: anti-aliased, voxelized rendering.

The Blockworlds code covers four elementary event types, each parametrized by its own scalar field that takes a zero value at the relevant interface:

Figure

We construct a series of 15 kinematic models with true configurations as shown in Fig.

Voxelized final states of all models in the suite, displaying 3-D geology in cross section.

The first set of models focus on planar dip-slip faults, with final configurations as follows:

We also include some additional folded configurations, selected to test the anticipated limitations of anti-aliasing described in Sect.

Parameter true values and prior distributions for fault-focused kinematic models labeled M1 to M7. The prior mean is set to the true value for each parameter.

Parameter true values and prior distributions for fold-focused kinematic models labeled M8 to M15.

We choose prior distributions for the parameters to simulate the realistic incorporation of structural geological knowledge, as shown in Tables

For each model, we examine four regimes of resolution and aliasing. We consider a “coarse”

We generate synthetic geophysics data based on the true model parameters, with measurements spaced evenly in a

As with the earlier synthetic examples for a spherical intrusion, we used SimPEG (version 0.14.0;

To illustrate some of the effects caused by aliasing and some of the limits of anti-aliasing, we visualize a series of two-dimensional slices through the posterior distribution. Pairs of free parameters are scanned on a regular

We also perform MCMC over the parameters of the kinematic models to demonstrate the impact of anti-aliasing on chain mixing. The aliased posterior has no useful derivative information for structural parameters, so we are limited to random walk proposals. For our tests, we choose the adaptive Metropolis random walk algorithm of

We use the coarse

We use the integrated autocorrelation time to measure MCMC efficiency within each chain, and the potential scale reduction factor

Finally, to quantify the overall accuracy of the posterior density calculation, we use the Kullback–Liebler divergence

Two-dimensional slices through the posterior distribution for selected variable pairs are shown for Model 1 (in Fig.

The log posterior distribution of pairs of parameters, denoted by

The log posterior distribution of pairs of parameters for Model 6.
All plot properties are the same as in Fig.

Model 1 is relatively well behaved; the aliased posterior has a single
dominant mode near the true value. This is a situation in which the aliasing
effects function mainly to make the likelihood appear blocky and terraced.
Although proposals that require derivatives of the likelihood will fail on
this posterior, it can still be navigated by appropriately tuned random walks, or by the discontinuous Hamiltonian Monte Carlo sampler of

Nevertheless, the benefits of anti-aliasing are still clear. Merely increasing the mesh resolution of the coarse aliased model by a factor of 5 (and the computational cost by over

Model 6 presents a more challenging case where a coarse mesh does not fully resolve structures close to the surface, resulting in bias for the low-resolution models. The layer thicknesses of the coarse aliased model are explicitly multi-modal as the interfaces hover between depths, and the modes for several variables are significantly offset from their true values. These values become sharp modes in a broader parameter space, easily missed by any inappropriately scaled MCMC proposal. The fine aliased model reproduces the overall posterior shapes, with some distortions relative to the fine anti-aliased reference model. The coarse anti-aliased model produces a smooth posterior and recovers the correct overall correlations between parameters, but are somewhat offset from the true values.

The differences between low- and high-resolution models are, predictably, more dramatic for folded models where the priors do not exclude fold wavelengths close to the Nyquist limit for the coarse mesh scale, such as Models 10 and 11. Other models, such as Model 9, show that anti-aliasing gives much better results for fold wavelengths greater than the mesh scale. In Model 11, where the undersampled structure is positioned farther beneath the surface, the apparent effects are less apparent than in Model 10, but still appear in directly linked variables such as the fold wavelength.

The biases caused by aliasing seem to be strongest in angular variables such
as fault directions. For Model 6, the best-fit dip-slip angles are offset from the true values by more than 10

These results confirm that our algorithm can reproduce the continuous behavior in an underlying posterior distribution, and can enable significant computational savings even for complex models as long as the interfaces are smooth on the mesh scale. It cannot – and is not intended to – recover more complex structures beneath the mesh scale that violate the core assumptions under which it was derived.

Table

Performance metrics for each MCMC run: mean (best-case, worst-case) integrated autocorrelation time

Models 5, 6, 10, 14, and 15 are challenging models for which the anti-aliased chain also has trouble mixing fully. An example trace plot from Model 5 is shown in Fig.

Trace plots for Fault 1 basement density during MCMC sampling of Model 5 (coarse mesh). The four colors in each plot represent four separate chains started from different prior draws. Traces begin at the end of burn-in and are thinned by a factor of 100 (every 100th trace point is shown).

Complementing the trace plots and MCMC performance metrics as a global measure of posterior inference

To fully characterize uncertainty in truly multi-modal problems, parallel tempering should be used to sample from all modes in proportion to their posterior probability. However, without anti-aliasing, the low-temperature chain in the tempering scheme will mix poorly and will rely more heavily on swap proposals to explore the posterior. The modes that do appear in anti-aliased posteriors are also more likely to represent distinct interpretations, rather than poorly resolved strong correlations between variables.

We have experimentally demonstrated the use and benefit of anti-aliasing only for kinematic models acting on a regular rectangular mesh. However, we see several natural extensions to this work that extend the range and impact of anti-aliasing for parametric geological models. We describe these future directions in the following section.

Although Blockworlds forward models geology through the action of parametrized tectonic events, it is still an implicit model in the sense that
the unit interfaces are defined to be level sets of a scalar field, which we
parametrize simply in terms of the distance to the interface. The mathematics of our anti-aliasing prescription generalizes to any differentiable scalar field, including those used in co-kriging models that interpolate structural measurements

For implicit geological models, each interface is defined as the level set

The development of the anti-aliasing mapping in Sect.

Another useful avenue of future work would be to extend Eq. (

In this paper we have focused on gravity, since it is among the most widely used geophysical sensors in an exploration context and its linear response makes an anti-aliasing treatment straightforward. All of our results extend immediately to magnetic sensors given the strong mathematical equivalence, and may also be appropriate for linear forward models of other sensors such as thermal or electric conductivity, or for the slowness in travel-time tomography.

Useful schemes to anti-alias forward-modeled geology may exist for other sensors, as long as the sensor action at scales smaller than the mesh spacing can be usefully approximated by a computationally simple function of the rock properties and the interface geometry. Frequency-dependent sensors that probe a range of physical scales may require a frequency-dependent anti-aliasing function, and similarly for sensors with anisotropic interactions with interfaces. While there is nothing fundamental that prevents anti-aliasing for sensors that respond nonlinearly to rock properties, mesh refinement may be more important in these cases to ensure the numerical accuracy, as well as the continuity, of the posterior. The framework in this paper treats only quantities defined at mesh cell centers; while it may be possible to derive similar relations for quantities defined on mesh cell faces and edges, as in finite-volume treatments of electromagnetic sensors, this represents future work.

Our experiments show the potential pitfalls of using oversimplified projections of 3-D structural geology onto a volumetric basis for calculation
of synthetic geophysics, and demonstrate an intuitive, efficient solution.
Anti-aliasing reproduces the smooth behavior of the underlying posterior with respect to the geological parameters to enhance the convergence of optimization or the mixing of sampling methods and to enable the use of derivative information in these methods. Anti-aliasing enables a fixed volumetric basis to more faithfully represent sensor response to latent discrete geology with interfaces that are flat at the scale of a mesh cell, reducing the computational burden of forward geophysics in an inversion loop. Our algorithm is also coordinate invariant and can be combined with curvilinear meshes, and with mesh refinement techniques such as octrees

We have focused on calculations over 3-D Cartesian volumetric meshes because implicit and kinematic models are naturally volumetric, and because existing geological modeling codes already export rock properties onto meshes as block models. Introducing anti-aliasing into these geological models is a minimally invasive modification to enhance their use in MCMC-based Bayesian inversions. Other discretization schemes built to align with geological interfaces, such as pillar grids

The accurate and efficient projection of these simple geological models onto meshes for geophysics calculations is a prerequisite to inversion for structural and kinematic parameters in more realistic situations. We can now take clear next steps towards MCMC sampling of kinematic histories for richer, higher-dimensional models. In addition, we can now move towards the sampling of hierarchical geophysical inversions that use a parametric structural model as a mean function. This will enable voxel-space inversions for which the geological prior is expressed in terms of uncertain interpretable parameters, or inversions for geology that include uncertainty due to residual rock property variations constrained by geophysics. These would represent more complete probabilistic treatments of geological uncertainty in light of available constraints from geophysics.

In the definitions to follow, we follow notation introduced in

The autocorrelation function, measuring the correlation between parameter
draws separated by a lag

The potential scale reduction factor (PSRF) was introduced as a metric for chain convergence by

The version of the Blockworlds model code used in this paper is archived with Zenodo (

The study was conceptualized by SC, who, with MJ, provided funding and resources. RS was responsible for project administration, designed the methodology, wrote the Blockworlds code resulting in v0.1, and carried out the main investigation and formal analysis under supervision from SC, EC, and MJ. RS and ML validated and visualized the results. RS wrote the original draft text, with contributions from ML, for which all coauthors provided a review and critical evaluation.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Parts of this research were conducted by the Australian Research Council Industrial Transformation Training Centre in Data Analytics for Resources and the Environment (DARE), through project number IC190100031. The work has been supported by the Mineral Exploration Cooperative Research Centre, whose activities are funded by the Australian Government's Cooperative Research Centre Programme. This is MinEx CRC Document 2021/34. Mark Lindsay acknowledges funding from ARC DECRA DE190100431 and MinEx CRC. We thank Alistair Reid and Malcolm Sambridge for useful discussions.

This research has been supported by the Australian Research Council (grant nos. IC190100031 and DE190100431).

This paper was edited by Lutz Gross and reviewed by Florian Wellmann and two anonymous referees.