GPU-accelerated Bayesian inference for block-cave mine monitoring via muon tomography

Abstract

We describe a Bayesian framework for an inverse problem arising from monitoring block caving operations via muon tomography. We work with a low dimensional surface-based representation of the geometry of the block cave, which dramatically reduces the computational requirements of the model while allowing realistic geometries. Adopting a Bayesian approach, we define a prior distribution on the space of geometries that favors realistic cave shapes. Pairing this prior with a likelihood based on the muon tomography forward model, we obtain a posterior distribution over cave geometries using Bayes rule. We obtain approximate samples from this posterior distribution using Markov chain Monte Carlo algorithms running on GPUs, resulting in fast and accurate sampling. We test the fidelity of our methodology by applying it to a simulated block caving scenario for which the ground truth is known. Results show that our method produces a diverse array of sensible geometries that are simultaneously compatible with the data.

Figures (10)

• Figure 1: Three-dimensional illustration of block caving. Left: depiction of the three main units: solid rock, muck pile, and air gap. Funnels that capture broken ore can be seen below the muck pile. Right: muons penetrating through the mine, with some intersecting the field of view of sensors placed at the bottom of the cave.
• Figure 2: A two-dimensional visualization of a layer model with $3$ distinct units representing rock, muck pile, and possible air gap.
• Figure 3: Four perspectives of the true layers of the simulated example dataset. Axis labels have been omitted for clarity. Base $x$-$y$ grid is of size $19\times19$ with an extent of $760$ meters in each direction, while the $z$ axis has a range of $(-25, 625)$ meters. The two layers have been plotted with transparency in order to show the non-trivial air gap at the top of the formation, and the location of the muon sensors (at constant $z=-25$).
• Figure 4: Two-dimensional illustration of the field of view of two sensors (semi-circles at the bottom) imaging a rectangular object from two different positions. Rays emanating from a sensor indicate the partition of the field of view into distinct pixels.
• Figure 5: An example of reshaping a three-dimensional array of size $(2,3,2)$ into a vector of length $12$, using the convention in which the right-most index varies the fastest.