Twenty Second Annual Meeting 20192020 Boundary Modeling with
Twenty Second Annual Meeting: 2019/2020 Boundary Modeling with Uncertainty using Indicator Interpolation and Thresholding Steve Mancell
Motivation • Boundary definition is an important step in geological modeling and resource estimation – Boundary: extent of domain where a transition occurs • Soft or hard • Demarcate geological features • Group geostatistical populations – Consequences of poor boundary control • Dilution • Grade smearing • Tonnage uncertainty From Martin & Boisvert 2017 • Current implicit modeling techniques widely utilize signed distance functions (SDF) – Problems with SDF algorithm 1. In presence of data asymmetry and/or sparseness of data • introduces conservative bias 2. Modification parameters for boundary uncertainty • simplistic • unrealistic From Carvalho 2018 2
Problem • In the presence of data asymmetry: – Inside data concentrated along a trend will read off the closest data of opposite indicator – Does not regard data on opposite, but equally pertinent boundary – Outside data opposite of the trend reads off the proper data – Conservative bias introduced – C-parameter for uncertainty is global additive that often gives simplistic and unrealistic uncertainty 3
Problem • In the presence of data asymmetry: – Inside data concentrated along a trend will read off the closest data of opposite indicator – Does not regard data on opposite, but equally pertinent boundary – Outside data opposite of the trend reads off the proper data – Conservative bias introduced – C-parameter for uncertainty is global additive that often gives simplistic and unrealistic uncertainty From Carvhalo, 2018 4
Indicator Thresholding Approach Use Nearest Neighbour (NN) model to calibrate indicator estimate to globally unbiased model volume that closely follows structure of the data • Nearest Neighbour models: – – Derived from Delaunay triangulations, and its dual, Voronoi cells Honour the spatial structure of the data Unbiased representation For any location, the closest data point is that which belongs to the Voronoi cell for which the location resides – Often used to check global volumes of boundary models • NN Steps: 1. Delaunay triangulations (red lines) connect data with line along Euclidean distance 2. Bisectors of the Euclidean lines are extrapolated out until they connect 3. Result is Voronoi cells (grey lines) that are an unbiased spatial representation of the input data. 5
Threshold Use Nearest Neighbour (NN) model to calibrate indicator estimate to globally unbiased model volume that closely follows structure of the data Global Krig of SDF- Threshold to iso-zero • Global/Dual Ordinary Indicator Kriging (GDOIK): – Global kriging uses all available data – Results show no artifacts – Same derivation as Radial Basis Functions (RBF). Equivalent estimation if parameterized the same • Threshold IK estimate to NN model volume proportions – – – Globally unbiased model of volume Threshold value, Th = 1 -(Vnn/Vtotal) FCDF(Th)=z-value for thresholding indicator estimate Any value above z is considered inside the boundary Follows structure of data Still need to find a way to assess uncertainty 6
Boundary Uncertainty Doubt is not a pleasant condition, but certainty is absurd Voltaire – • Uncertainty in boundary modeling is ubiquitous • Exists anywhere between 2 samples that do not have the same binary value • Further away from boundary locations, the higher the uncertainty • Originates from lack of knowledge and sparseness of data – Volumetric uncertainty (sparse data) • Current method: generate boundary realizations – Geometric uncertainty when a significant portion of data is comprised of contact orientations • Current method: multiple realizations of orientation data • Goal is to provide globally and locally unbiased and fair boundary uncertainty that fluctuates in width and honours structure of conditioning data 7
Boundary Uncertainty Probability Threshold Curve Experimental Workflow: 1. Simulate truths and sample 2. Estimate and threshold at varying percentiles 3. Compare threshold models to Truth – If Th > Truth, Th= 1. Else, Th=0 – For each drill spacing at particular threshold: • Sum values • Divide by T=100 • Returns probability of Th model being larger than truth for given spacing • Plot the 20 points from each drill hole spacing 4. Repeat for multiple scenarios over varying drill hole spacings Infer PTC for real geological data and determine acceptable threshold value for uncertainty 8
PTCs for Multiple Scenarios • • • Repeat PTC workflow for numerous different shapes and sizes Compare PTCs over hundreds of scenarios and drill spacings Standardize PTCs Assume linear PTC model Assume Delta (d)=h=l 9
PTCs for Multiple Scenarios • • • Assess varying delta values on scenarios Calculate volumetric uncertainty over all scenarios by standardizing to the true volumes Recommend uncertainty thresholding for +/-0. 15 of NN model threshold: • zeroded= z. NN + 0. 15 • zdilated = z. NN – 0. 15 10
Single Domain Porphyry Case Study • • • Comprised of 3, 276 DH data RBF for indicator interpolation NN model volume of 55, 898 blocks: • • • Indicator NN Threshold Base Case with DH Data Indicator Threshold with Uncertainty Th. NN = 1 -(VNN/Vtotal) Th. NN =0. 88; Fi*(Th. NN) = z. NN = 0. 52 Uncertainty thresholding for +/-0. 15 of NN model threshold: • • zeroded= 0. 52 + 0. 15 zdilated = 0. 52 – 0. 15 • Compared to SDF modeling: Indicator Threshold with Uncertainty SDF with Uncertainty 11
Multicategorical Domain Porphyry Case Study • • Three intrusions, an oxide and sulphide domain Each domain independently interpolated NN model volume thresholds for each domain Overlap of nodes: • • Drillhole Data Indicator Threshold with Uncertainty mitigated by selecting highest discrepancy between estimated probability and corresponding NN threshold Uncertainty thresholding for single intrusive domain +/-0. 15 of NN model threshold: • • • zeroded= 0. 45 + 0. 15 zdilated = 0. 45 – 0. 15 Overlap in nodes given precedence to domain with uncertainty (Intrusion 1) Plan View : Indicator Threshold with Uncertainty Plan View: SDF with Uncertainty • Compared to SDF modeling: 12
Vein Case Study • • Single vein with data from 14 drill holes Interpolated indicators with anisotropy modeled within Leapfrog software • NN model volume threshold (yellow) • Uncertainty thresholding for vein with +/0. 10 of NN model threshold: • zeroded= 0. 281 + 0. 10 (cyan) • zdilated = 0. 281 – 0. 10 (red) • Compared to SDF surface used in Leapfrog (no uncertainty) Drillhole Data Indicator Threshold with Uncertainty 13
Conclusions & Future Work • Signed Distance Function modeling is susceptible to bias • Conventional ‘C’ parameter uncertainty is unrealistic • Thresholding implicit indicator models addresses these concerns • Geologically realistic case studies and examples in 2 -D and 3 -D show efficacy of approach • Non-linear PTCs for boundary uncertainty • Establish reliable predictions of thresholding curve using Machine Learning 14
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