Study guide
7.2 Inverse Distance & Ordinary Kriging Grade Estimation
PRC Mining Engineer Licensure Examination Philippines
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1 Chapter 1: Surface Mining Methods & Pit Optimization
1.1 Surface Mining Unit Operations & Method Selection 1.2 Bench Geometry, Slope Stability & Geotechnical Pit Design 1.3 Cut-Off Grade Optimization & Lerchs-Grossmann Pit Limits
2 Chapter 2: Underground Mining Methods & Strata Control
2.1 Underground Mining Method Classification & Selection 2.2 Rock Mechanics, Stress Analysis & Strata Control 2.3 Underground Ground Support Systems & Excavation Design
3 Chapter 3: Materials Handling, Equipment & Mine Economics
3.1 Drilling, Blasting & Explosives Engineering 3.2 Mine Loading, Haulage & Materials Handling Systems 3.3 Mine Capital (CAPEX), Operating Costs (OPEX) & Financial Evaluation
4 Chapter 4: Mine Safety, Ventilation & Philippine Mining Laws
4.1 Mine Ventilation Network Analysis & Environmental Quality 4.2 Philippine Mine Safety Rules, Health & Disaster Prevention 4.3 Philippine Mining Act of 1995 (RA 7942), DAO 2010-21 & Mining Ethics
5 Chapter 5: Physical Geology, Mineralogy & Philippine Deposit Types
5.1 Structural Geology, Tectonics & Ore Controls 5.2 Igneous, Metamorphic & Sedimentary Ore Mineralogy 5.3 Major Philippine Ore Deposit Models
6 Chapter 6: Mineral Exploration, Geophysics & Core Logging
6.1 Geochemical & Remote Sensing Exploration Techniques 6.2 Geophysical Exploration Methods in Mining Geology 6.3 Diamond Drilling, Core Logging & QA/QC Protocols
7 Chapter 7: Geostatistics, Resource Estimation & PMRC Code
7.1 Ore Body Block Modeling & Spatial Variography 7.2 Inverse Distance & Ordinary Kriging Grade Estimation 7.3 Philippine Mineral Reporting Code (PMRC 2020) & Resource Classification
8 Chapter 8: Comminution, Sizing & Physical Mineral Separation
8.1 Crushing, Grinding & Bond Work Index Calculations 8.2 Particle Screening, Hydrocyclone Classification & Sizing 8.3 Gravity Concentration & Magnetic/Electrostatic Separation
9 Chapter 9: Froth Flotation & Extractive Hydrometallurgy
9.1 Froth Flotation Chemistry, Reagents & Cell Kinetics 9.2 Leaching Technologies (Heap, Agitation & Tank) 9.3 Precious Metal Recovery: CIP, CIL, Merrill-Crowe & Electrowinning
10 Chapter 10: Process Plant Flowsheets & Environmental Tailings Management
10.1 Metallurgical Plant Flowsheets & Pulp Mass Balances 10.2 Tailings Storage Facility (TSF) Design & Dam Safety 10.3 Acid Mine Drainage (AMD) Mitigation & Environmental Protection (EPEP/FMR/DP)
11 Chapter 11: 2025 PRC Blueprint, Prospecting Tools & Exploration Decisions
11.1 Official 2025 TOS, Three-Paper Structure & Exam Strategy 11.2 Prospecting Instruments, Base Maps & Surface Geological Mapping 11.3 Integrating Direct and Indirect Evidence for Exploration Decisions
12 Chapter 12: Mine Infrastructure, Sustainable Development & Climate
12.1 Power, Water, Drainage & Support-Facility Selection 12.2 Sustainable Development, Progressive Rehabilitation & Community Impact 12.3 Mining, Climate, Reforestation, Coal & Energy-Transition Minerals
13 Chapter 13: Mine Valuation, Financial Analysis & Due Diligence
13.1 Time Value of Money, Future Value & Present Value 13.2 Financial Statements, Mine Economics & External Price Drivers 13.3 Exploration Provenance, Technical Review & Legal Due Diligence
14 Chapter 14: Mine Organization, Communication & Accountable Leadership
14.1 Line and Staff Functions in a Mining Organization 14.2 Communication With Superiors, Peers & Subordinates 14.3 Authority, Responsibility, Delegation & Safety Accountability
15 Chapter 15: Philippine Mining Law, Indigenous Rights & Professional Ethics
15.1 Legal Hierarchy, Mineral Tenure & Compliance Analysis 15.2 Constitutional Participation, IPRA, FPIC & Ancestral Domains 15.3 RA 4274, Professional Scope, Ethics & Document Integrity
16 Chapter 16: Applied Mathematics & Mine Surveying
16.1 Engineering Mathematics for Safe, Efficient Mine Decisions 16.2 Survey Control, Instruments, Traverses & Error Adjustment 16.3 Topographic Maps, Underground Set-Out & Survey Timeliness
17 Chapter 17: Feasibility Studies, Data Quality & Project Decisions
17.1 Scoping, Pre-Feasibility & Final Feasibility Study Maturity 17.2 Data Accuracy, Assumption Control & Go/No-Go Decisions 17.3 Multidisciplinary Integration, Interfaces & Independent Review
18 Chapter 18: Mining Computer Applications, Digital Control & Global Developments
18.1 Computer Functions, Office Applications & Data Governance 18.2 Geological Modeling, Mine Planning, Scheduling & Valuation Software 18.3 Real-Time Control, Critical Minerals & Regional Mining Developments
19 Chapter 19: Earth Systems, Geologic Time & Philippine Hazards
19.1 Solar-System Origin, Earth Interior & Atmospheric Layers 19.2 Rock Cycle, Landforms & the Geologic Time Scale 19.3 Philippine Tectonics, Geologic Hazards & Mitigation
20 Chapter 20: Metallurgy, Foundry Practice & Assaying
20.1 Extractive, Physical & Adaptive Metallurgy Principles 20.2 Foundry Materials, Casting, Solidification & Defects 20.3 Sampling, Fire Assay, Cupellation, Parting & Assay Calculations
21 Chapter 21: Solid–Liquid Separation, Environmental Law & Mine Ecology
21.1 Thickening, Filtration & Solid–Liquid Separation 21.2 Philippine Air, Water, Waste & Hazardous-Substance Controls 21.3 Mine Ecology, Closure, Tailings, Waste & Community Livelihood
7.2 Inverse Distance & Ordinary Kriging Grade Estimation
Key Takeaways Explain This Topic AI
Classical interpolation includes Nearest Neighbor (step-like boundary assignment) and Inverse Distance Weighting (IDW), where sample weight is inversely proportional to distance raised to power exponent p.
Ordinary Kriging (OK) is the Best Linear Unbiased Estimator (BLUE), minimizing estimation variance while guaranteeing unbiasedness through the constraint sum(lambda_i) = 1.
The Ordinary Kriging system of equations utilizes a Lagrange multiplier (mu) to solve sample weights based on sample-to-sample and sample-to-block variogram semi-variances.
Search ellipsoids are aligned with variogram anisotropy axes, utilizing octant/quadrant constraints to prevent over-weighting clustered drillhole data.
Kriging smooths local extremes; its effect on recoverable tonnes and grade at a cutoff depends on support, selectivity, information effect, estimation plan, and validation, so a universal direction of tonnage bias should not be asserted.
Last updated: August 2026
Grade estimation transforms sparse drillhole sample assay data into a continuous 3D block model of estimated grades. Selecting the appropriate spatial interpolation technique is vital to ensuring accurate tonnage-grade predictions, preventing systematic global or local bias, and properly representing the spatial uncertainty of the mineral deposit.
Classical Estimation Methods: Nearest Neighbor & Inverse Distance Weighting
Before advanced geostatistical methods were established, classical geometric interpolation techniques were standard in mining engineering.
1. Polygonal and Nearest Neighbor (NN) Estimation
The Nearest Neighbor method assigns to an un-sampled block x 0 x_0 x 0 the assay grade of the single drillhole composite closest to the block center: Z N N ∗ ( x 0 ) = Z ( x nearest ) Z^*_{NN}(x_0) = Z(x_{\text{nearest}}) Z NN ∗ ( x 0 ) = Z ( x nearest )
While simple and useful for constructing declustered global grade distributions, NN estimation creates abrupt step-like grade boundaries and ignores spatial trends across multiple surrounding samples.
2. Inverse Distance Weighting (IDW)
Inverse Distance Weighting estimates block grades as a weighted average of surrounding composite samples, where sample weights are inversely proportional to the Euclidean distance between sample locations and the block centroid raised to a power exponent p p p : Z I D W ∗ ( x 0 ) = ∑ i = 1 n Z ( x i ) d i p ∑ i = 1 n 1 d i p = ∑ i = 1 n w i Z ( x i ) Z^*_{IDW}(x_0) = \frac{\sum_{i=1}^{n} \frac{Z(x_i)}{d_i^p}}{\sum_{i=1}^{n} \frac{1}{d_i^p}} = \sum_{i=1}^{n} w_i Z(x_i) Z I D W ∗ ( x 0 ) = ∑ i = 1 n d i p 1 ∑ i = 1 n d i p Z ( x i ) = i = 1 ∑ n w i Z ( x i )
Where:
Z ∗ ( x 0 ) Z^*(x_0) Z ∗ ( x 0 ) = Estimated grade at target block location x 0 x_0 x 0 .
Z ( x i ) Z(x_i) Z ( x i ) = Grade of composite sample i i i .
d i d_i d i = Euclidean distance between sample i i i and target block centroid x 0 x_0 x 0 .
p p p = Power exponent (typically p = 2 p = 2 p = 2 for Inverse Distance Squared IDW 2 \text{IDW}^2 IDW 2 , or p = 3 p = 3 p = 3 for IDW 3 \text{IDW}^3 IDW 3 ).
w i w_i w i = Normalized weight assigned to sample i i i ( ∑ w i = 1.0 \sum w_i = 1.0 ∑ w i = 1.0 ).
As the power exponent p p p increases ( p → ∞ p \to \infty p → ∞ ), IDW converges toward the Nearest Neighbor result. Basic IDW weights are driven by distance rather than a covariance model. Anisotropic distance, sectoring, and sample limits can be imposed, but IDW does not intrinsically model spatial covariance or deliver the kriging optimality and variance associated with a valid geostatistical model.
Ordinary Kriging (OK) Principles & System Equations
Under its stationarity assumptions and a valid covariance or variogram model, Ordinary Kriging (OK) provides a best linear unbiased estimate for the defined support and neighborhood. Unlike IDW, Ordinary Kriging accounts for distance, sample clustering, spatial orientation, and structural continuity defined by the variogram model.
Mathematical Formulation of Ordinary Kriging
The Ordinary Kriging estimator computes block grade as a linear combination of n n n surrounding composite samples: Z O K ∗ ( x 0 ) = ∑ i = 1 n λ i Z ( x i ) Z^*_{OK}(x_0) = \sum_{i=1}^{n} \lambda_i Z(x_i) Z O K ∗ ( x 0 ) = i = 1 ∑ n λ i Z ( x i )
To satisfy the unbiasedness condition ( E [ Z ∗ ( x 0 ) − Z ( x 0 ) ] = 0 E[Z^*(x_0) - Z(x_0)] = 0 E [ Z ∗ ( x 0 ) − Z ( x 0 )] = 0 ), the sum of kriging weights must equal one: ∑ i = 1 n λ i = 1 \sum_{i=1}^{n} \lambda_i = 1 i = 1 ∑ n λ i = 1
To achieve the minimum estimation variance ( σ k 2 = E [ ( Z ( x 0 ) − Z ∗ ( x 0 ) ) 2 ] → minimum \sigma_k^2 = E[(Z(x_0) - Z^*(x_0))^2] \to \text{minimum} σ k 2 = E [( Z ( x 0 ) − Z ∗ ( x 0 ) ) 2 ] → minimum ), Ordinary Kriging incorporates a Lagrange multiplier μ \mu μ . The resulting system of n + 1 n+1 n + 1 linear equations (the Kriging System) is solved simultaneously: ∑ j = 1 n λ j γ ( x i , x j ) + μ = γ ( x i , x 0 ) for i = 1 , 2 , … , n \sum_{j=1}^{n} \lambda_j \gamma(x_i, x_j) + \mu = \gamma(x_i, x_0) \quad \text{for } i = 1, 2, \dots, n j = 1 ∑ n λ j γ ( x i , x j ) + μ = γ ( x i , x 0 ) for i = 1 , 2 , … , n ∑ i = 1 n λ i = 1 \sum_{i=1}^{n} \lambda_i = 1 i = 1 ∑ n λ i = 1
Where:
γ ( x i , x j ) \gamma(x_i, x_j) γ ( x i , x j ) = Semi-variance between sample i i i and sample j j j (from the fitted variogram).
γ ( x i , x 0 ) \gamma(x_i, x_0) γ ( x i , x 0 ) = Average semi-variance between sample i i i and all points within block x 0 x_0 x 0 .
λ i \lambda_i λ i = Kriging weight assigned to composite sample i i i .
μ \mu μ = Lagrange multiplier enforcing the unbiasedness constraint.
Once weights λ i \lambda_i λ i and Lagrange multiplier μ \mu μ are solved, the Kriging Variance σ O K 2 \sigma_{OK}^2 σ O K 2 is calculated as: σ O K 2 = ∑ i = 1 n λ i γ ( x i , x 0 ) + μ \sigma_{OK}^2 = \sum_{i=1}^{n} \lambda_i \gamma(x_i, x_0) + \mu σ O K 2 = i = 1 ∑ n λ i γ ( x i , x 0 ) + μ
Kriging variance reflects the modeled continuity, block support, and data configuration but not the observed grade values, geological-domain uncertainty, bias, or all estimation risk. It can be one classification diagnostic, never a stand-alone classifier; classification must integrate geology, data quality, continuity, estimation performance, and reasonable prospects.
Search Ellipsoid Selection & Neighborhood Parameters
To execute estimation efficiently, a search ellipsoid is defined around each block centroid to select surrounding samples.
- Search Parameter Operational Definition Selection and validation
- Ellipsoid Axes Major, Semi-Major, Minor radii (meters) Oriented parallel to variogram anisotropy axes
Search Ranges Maximum search distances along 3 axes Start from modeled continuity and test with neighborhood analysis, coverage, weights, slope of regression, and validation; they need not equal the variogram ranges
Min / Max Samples Minimum and maximum samples for a block estimate Treat numeric counts as trial parameters; validate by domain, drill-hole representation, redundancy, coverage, weights, and estimation quality
Octant / Quadrant Max samples allowed per octant/quadrant Prevents over-weighting clustered holes on one side Major Axis (Along Strike)
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- Grade-Tonnage Curves & The Smoothing Effect of Kriging
Because Ordinary Kriging acts as a minimum-variance estimator, it exerts a low-pass smoothing effect on estimated block grades. Kriged estimates commonly have less dispersion than the unknown true local block distribution and much less than point samples, but variance comparisons must use consistent support and validation rather than assume one universal inequality.
Impact on Grade-Tonnage Relationships
The smoothing effect has significant consequences for mine planning and cutoff grade selection:
Smoothing reduces the variance of estimated grades relative to unsmoothed local reality.
The resulting tonnes and grade above a cutoff can be biased in either direction depending on block support, selectivity, information effect, sampling, estimator, and cutoff. Validate grade-tonnage behavior against declustered data, swath plots, reconciliation, change-of-support analysis, or conditional simulation rather than memorizing one universal sign.
To evaluate kriging smoothing and mining selectivity, practitioners may use appropriate change-of-support methods or conditional simulation, provided distributional assumptions, support, conditioning, and validation are defensible. Such methods characterize uncertainty or recoverable resources; they do not automatically correct a weak model.