7.2 Inverse Distance & Ordinary Kriging Grade Estimation

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 $x_0$ the assay grade of the single drillhole composite closest to the block center:

$$Z^*_{NN}(x_0) = Z(x_{\text{nearest}})$$

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 $p$:

$$Z^*_{IDW}(x_0) = \frac{\sum_{i=1}^{n} \frac{Z(x_i)}{d_i^p}}{\sum_{i=1}^{n} \frac{1}{d_i^p}} = \sum_{i=1}^{n} w_i Z(x_i)$$

Where:

• $Z^*(x_0)$ = Estimated grade at target block location $x_0$.
• $Z(x_i)$ = Grade of composite sample $i$.
• $d_i$ = Euclidean distance between sample $i$ and target block centroid $x_0$.
• $p$ = Power exponent (typically $p = 2$ for Inverse Distance Squared $\text{IDW}^2$, or $p = 3$ for $\text{IDW}^3$).
• $w_i$ = Normalized weight assigned to sample $i$ ($\sum w_i = 1.0$).

As the power exponent $p$ increases ($p \to \infty$), 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 $n$ surrounding composite samples:

$$Z^*_{OK}(x_0) = \sum_{i=1}^{n} \lambda_i Z(x_i)$$

To satisfy the unbiasedness condition ($E[Z^*(x_0) - Z(x_0)] = 0$), the sum of kriging weights must equal one:

$$\sum_{i=1}^{n} \lambda_i = 1$$

To achieve the minimum estimation variance ($\sigma_k^2 = E[(Z(x_0) - Z^*(x_0))^2] \to \text{minimum}$), Ordinary Kriging incorporates a Lagrange multiplier $\mu$. The resulting system of $n+1$ linear equations (the Kriging System) is solved simultaneously:

$$\sum_{j=1}^{n} \lambda_j \gamma(x_i, x_j) + \mu = \gamma(x_i, x_0) \quad \text{for } i = 1, 2, \dots, n$$ $$\sum_{i=1}^{n} \lambda_i = 1$$

Where:

• $\gamma(x_i, x_j)$ = Semi-variance between sample $i$ and sample $j$ (from the fitted variogram).
• $\gamma(x_i, x_0)$ = Average semi-variance between sample $i$ and all points within block $x_0$.
• $\lambda_i$ = Kriging weight assigned to composite sample $i$.
• $\mu$ = Lagrange multiplier enforcing the unbiasedness constraint.

Once weights $\lambda_i$ and Lagrange multiplier $\mu$ are solved, the Kriging Variance $\sigma_{OK}^2$ is calculated as:

$$\sigma_{OK}^2 = \sum_{i=1}^{n} \lambda_i \gamma(x_i, x_0) + \mu$$

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.

                 Major Axis (Along Strike)
                    <----------------->
                   .-------------------.
                . '                     ' .
              .                             .
 Semi-Major  |              (x0)             |
   Axis      |          Block Center         |
 (Down Dip)   .                             .
                . '                     ' .
                   '-------------------'

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.