From Folded Rocks to Refined Algorithms: A Journey Through Geostatistical and Machine Learning Unfolding

Folds and fractures may be present in ore bodies that have been subjected to structural geological processes (such as faults or various types of stress), which interfere with the continuity of the mineralisation. For complicated geology with non-linear features caused by faulting, folding, and other deformations, the use of a Cartesian coordinate system and Euclidean distances is inappropriate; continuity in these kinds of deposits is frequently thought to follow the geology.?? In certain kinds of ore bodies, variogram interpretation becomes challenging, which has an impact on resource evaluation, variographic analysis, estimation, and subsequent categorisation. It is customary to perform that interpretation with the presumption that the measured distance between any two points lies on the Euclidean norm.

This statement is untrue for ore bodies with folds and faults, where the distance between points must take into account the deposit geometry and incorporate all relevant geological information. To enable geostatistical methods that rely on linear two-point pairings, such as kriging or sequential Gaussian simulation (SGS), which use variograms to describe spatial continuity, one approach under these circumstances is to change the domain to an alternative coordinate system. Utilising locally variable anisotropy and associated techniques in domains with non-linear characteristics is another strategy. I've previously used two commercial software to unfold blocks for estimating mineral resources.

However, two and a half weeks ago, I started to wonder how these algorithms function because they produced a variety of outcomes; for example, some kept the volume after unfolding but deformed the geometry. However, some overlooked the geometry while producing the finest stratigraphic restoration. I next conducted a literature search and discovered the following papers:

Boisvert, J., Manchuk, J., & Deutsch, C. (2009). Kriging in the presence of locally varying anisotropy using non-euclidean distances. Mathematical Geosciences, 41(6), 585–601. https://doi.org/10.1007/s11004-009-9229-1

Dunbar, J.A., & Cook, R.W. (2003). Palinspastic reconstruction of structure maps: an automated finite element approach with heterogeneous strain. Journal of Structural Geology, 25(7), 1021–1036.

Garrido et al. (2016). Resource assessment with unfolding methodology: a case study. MiNiN2016.

Griffiths, P., Jones, S., Salter, N., Schaefer, F., Osfield, R., & Reiser, H. (2002). A new technique for 3-D flexural-slip restoration. Journal of Structural Geology, 24(5), 773–782.

Latifi, M., & Boisvert, J. (2022). Stratigraphic Coordinate Transformation. Retrieved from https://geostatisticslessons.com/lessons/stratcoords

Lovely, P., Flodin, E., Guzofski, C., Maerten, F., & Pollard, D.D. (2012). Pitfalls among the promises of mechanics-based restoration: Addressing implications of unphysical boundary conditions. Journal of Structural Geology, 41, 47–63. https://doi.org/10.1016/j.jsg.2012.02.020

Maerten, L., & Maerten, F. (2006). Chronologic modeling of faulted and fractured reservoirs using geomechanically based restoration: Technique and industry applications. AAPG Bulletin, 90(8), 1201–1226. https://doi.org/10.1306/02240605116

Williams, G.D., Kane, S.J., Buddin, T.S., & Richards, A.J. (1997). Restoration and balance of complex folded and faulted rock volumes: flexural flattening, jigsaw fitting and decompaction in three dimensions. Tectonophysics, 273(3-4), 203–218. https://doi.org/10.1016/S0040-1951(96)00282-X

I also read a LinkedIn article (https://www.dhirubhai.net/pulse/maths-unfulding-adrian-martinez-vargas/) by Adrian Martinez Vargas on unfolding, of which there was a comment by Ryan Shackleton who provided useful problem statement and stated that “In general, the problem is not straight forward, especially when geological realism is desired, or when the unfolding scenario must be used to model rock deformation….. There are a variety of other accepted methods to solve the restoration problem for volumes, many of which involve using geomechanical finite element techniques. These are quite a bit more involved, and also have their issues with geologic realism despite the fact that they use elastic and non-elastic rock properties to model deformation”

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As a result, I realised that there is no general solution to the unfolding problem, but there may be alternative, flexible approaches. Many papers present methods that unfold and restore geologic bodies under the assumption that the deposition occurred prior to the folding process without rheological influences (grade distribution does not depend on stress distribution). Garrido et al., 2016 noted that unfolding and modelling under the above assumptions improved the grade estimation and classification of mineral resources.

Among the more flexible solutions, Rafael Moniz Caixeta and José Felipe Coimbra Leite Costa presented “A robust unfolding approach for 3-D domains” in their paper published in 2021. Computers & Geosciences. Volume 155,104844,https://doi.org/10.1016/j.cageo.2021.104844. Using Julia, a general-purpose, high-level dynamic programming language, they implemented their approach. ?

I was primarily motivated by their 3-step approach when implementing a Python solution.

As a first step, I analysed their mathematical pipeline and translated their code into Python. In step 2, the translation did not work, and I reached out to friends in engineering and software development (i.e., academia and industry) for help. They struggled to provide the solution that I imagined. I quickly understood that this was a complex issue. Different experts were either not finding a fitting solution or tailoring their methods based on their specific field of study. For instance, material scientists examined how developed/fabricated materials such as steel and equipment deform, while geologists studied the behaviour of geological structures under various natural conditions of realism. Geostatisticians, on the other hand, focused on estimating resources by ensuring a continuous field. In essence, I had to navigate this challenge independently and determine a suitable approach.

During this journey, I had to revisit various mathematical ideas and understand how to apply them, especially Multidimensional Scaling. I noticed that different programming libraries in Python and Julia handle this differently. To get a clearer understanding, I found these two articles particularly helpful: https://jessicastringham.net/2018/05/20/Multidimensional-Scaling/ and https://towardsdatascience.com/mds-multidimensional-scaling-smart-way-to-reduce-dimensionality-in-python-7c126984e60b coupled with IsoMap: https://www.hindawi.com/journals/mpe/2017/3930957/.

Learning Julia was another task. While its syntax was somewhat familiar, there were aspects that threw me off, especially the way reference indices are used.

With the help of a friend (Jing Lee), we took a closer look at the mathematics and its relevance to resource estimation. After much effort, we've developed a functioning solution. Now, the focus shifts to refining and optimising the Python Unfolding algorithm. A highlight of this work is the ability to maintain volume during the unfolding process, ensuring no data loss and minimising geometric fragmentation. Below, you'll find images that showcase our results:

Three lessons that I learned from this project:

(1)?? Generative AI won't save you from solving complex geological problems, but it can assist in debugging the code quicker,

(2)?? Discipline-specific geology and geostats knowledge is key to linking mathematical concepts with realism, and

(3)?? Coding will teach you to be patient – I have been working on this daily and failed several times to find a practical solution but eventually managed to find a flexible solution pending optimisation. Anyone who spends hours coding will know how you lose yourself in the process, and the more things come together, the more you want to keep going – learn to take long breaks!

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