Into the Noddyverse: A massive data store of 3D geological models for 1 Machine Learning & inversion applications

15 16 Unlike some other well-known challenges such as facial recognition, where Machine Learning and Inversion algorithms are

are drawn randomly and independently from the event list comprised of folds, faults, unconformities, dykes, plugs, shear 147 zones and tilts. The likelihood of folds, faults and shear-zones are double the other events as we found that they had a bigger 148 impact of changing the overall 3D geology, and hence we wished to sample more of these events. This means we can have 149 7 3 =343 distinct deformation histories, although the specific parameters for each event can also vary, so the actual 150 dimensionality of the system is much higher. For clarification, the one million models are not the result of a combinatorial 151 approach, but of one million independent draws using a Monte Carlo sampling of the model space.

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The initial stratigraphy as well as new, above-unconformity stratigraphies, are defined to randomly have between two and five 153 units to keep the systems relatively simple, but this could of course be increased if desired. The lithology of each unit in a 154 stratigraphy is chosen to be coherent with the specific event and other units in the same sequence, so that we do not, for 155 example, mix high-grade metamorphic lithologies and un-metamorphosed mudstones in the same stratigraphic series (Table   156 2) nor do we assign the petrophysical properties of a sandstone to an intrusive plug. Once a lithology is chosen, the density 157 and magnetic susceptibility is randomly sampled from a table defining the Gaussian distribution of properties (linear for 158 density, log-linear for magnetic susceptibility) for that rock type. In the case of densities this may result in occasional negative 159 values, however since the gravity field is only sensitive to density contrasts this does not invalidate the calculation. Some rock 160 types have bimodal petrophysical properties to reflect real-world empirical observations, so we draw from a Gaussian mixture 161 in these cases. The petrophysical data is drawn from aggregated statistics (mean and standard deviation of one or two peaks) 162 of the approximately 13,500 sample British Columbia petrophysical database (Geoscience BC, 2008).

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The parameters which can be varied for each type of event, together with the range of these parameters, is shown in Table 1.

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Any subset of the geology can be calculated for any sub-volume of an infinite Cartesian space using Noddy, but we limit 169 ourselves to a 4x4x4 km volume of interest in this study. Similarly, although the geology within this volume can be calculated 170 at an arbitrary resolution, we have chosen to sample it using equant 20 m voxels as this is well below the typical resolved 171 measurement scale for these types of data when collected in the field.

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Geophysical forward models were calculated using a Fourier Domain formulation using reflective padding to minimise (but

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A study of geophysical image variability using a simple 2D correlation or maximal information coefficient between 206 pairs of images of different histories would be illuminating. Do we have images which are the same (or at least very 207 similar and within the noise tolerance of the geophysical fields) to each other, but belong to very different histories? 208 If these exist, the ambiguity of the histories can be examined, and we then know where we would expect poor 209 performance from ML techniques which rely on easily discriminated images. The systems of equations characterising 210 geophysical inverse problems often? have a non-unique solution. In ML research, if we only use magnetic data or 211 gravity data for inversion, we will be troubled by the non-uniqueness of the solution. However, because we have both 212 gravity data and magnetic data, we can extract features from multi-source heterogeneous data at the same time, and 213 then classify or regress after feature fusion. This could greatly reduce the influence of the non-unique solution.

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Having a large set of models will allow clustering of models accordingly to their geophysical response and identifying 215 subsets of geological models that are geophysically equivalent and cannot be distinguished using geophysical data.

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The analysis of diversity of such subsets of models will give an estimate of the severity of non-uniqueness and allow 217 the derivation of posterior statistical indicators conditioned by geological plausibility.

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The analysis of geophysically equivalent models will also enable us to estimate how significantly joint inversion or 233 interpretation can reduce the non-uniqueness of the solution, with the potential to identify families of geological 234 scenarios more suited to joint inversion than others. It is obvious that some 3D geological models will be 235 geologically more complex than others, and that some could be used for the benchmark of deterministic 236 geophysical inversion of gravity and magnetic data, but also of other geophysical techniques relying on wave 237 phenomena.

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We have limited ourselves to five deformation events in this study, and no more than five units in any one stratigraphy. These 256 decisions were based on an idea to "keep it simple" whilst simultaneously allowing a great variety of models to be built. We 257 recognise that these are somewhat arbitrary choices. We could have true randomly complex 3D histories, leading to models 258 with, for example, nine phases of folding, however the utility of over-complicating the system is not clear, and would rarely 259 or ever be discernible in natural systems. Similarly, we limited the parameter ranges of each deformation event, again on the 260 basis that the ranges chosen made models that are more interesting. For example, there did not seem much interest in having 261 folds with very large wavelengths or very low amplitudes, as they are equivalent to small translations of the geology and 262 would translate in the geophysical measurements into a regional trend that is often approximated and removed from the

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The primary goal of this study was to build a large dataset to provide a wide range of possible models for use in training ML 283 systems and to test more traditional geophysical inversion systems. The models here, whilst simpler than the large test models 284 mentioned earlier, represent to our knowledge the largest suite of 3D geological models with resulting potential field data and 285 tectonic history, which has its own utility. This usage applies equally well to classical geophysical inversion codes, which 286 have traditionally been tested on only a handful of synthetic models prior to being applied to real-world data, for which there 287 is no ground truth available.

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• Allow more events to increase the range of outcomes. We arbitrarily restricted ourselves to two started events (STRAT 306 and TILT) followed by three randomly chosen events, and an extension to the model suite could consider any number 307 of events in the sequence.

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• Include magnetic remanence and anisotropy effects. At present we only model scalar magnetic susceptibility but the 309 Noddy modelling engine can calculate variable remanence and anisotropic magnetic susceptibility as well.

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• Model larger volumes as large, or deep features cannot currently be modelled due to the 4 km model dimensions.

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• Build more models. We in no way believe we have explored the range of possible models in the present model suite, 316 and if we start in include more events, or more complex event definitions, we will certainly have to generate many more 317 models, perhaps orders of magnitude more, in order to provide robust training suites and inversion scenarios.

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• Include topographic effects. In this study, we have ignored the effect of topography on the models, although again this 323 could be included in the future, as it is supported by Noddy.

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We also need to be clear that a model built in Noddy is not capable of predicting all geological settings, as all Noddy models 325 are plausible geology, but not all plausible geology can be modelled by Noddy. To improve this situation, we would need to 326 improve the modelling engine itself. Similarly, the logic of trying to predict geology from geophysical datasets in this study 327 is only partially fulfilled: the geometry comes from geological events sequence, but identical geometries can be produced by 328 different event sequences.