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Kriging Neighborhood Analysis on Block Models (KNA)

Kriging Neighborhood Analysis (KNA) is used to avoid overly restrictive search neighborhoods and reduce conditional bias. It is performed for block locations and sizes, and in practice it is typically run for all blocks in the block model so results can be reviewed with maps and summary statistics.

The core question answered by KNA is:

How much smoothing is introduced to maintain unbiased estimates?

KNA results are stored in predefined block model variables that must be activated in the schema before running estimation.

For each estimated variable, activate the relevant predefined outputs in the Kriging category:

  • KNSF: Number of samples found for estimation.
  • KNSU: Effective number of samples used (weight-based).
  • KSE: Kriging standard error.
  • RSE: Relative kriging standard error (used in lognormal or indicator contexts).
  • KLM: Local mean used by simple kriging (when applicable).
  • KEF: Kriging efficiency.
  • KSR: Slope of regression.
  • KLA: Lagrange multiplier.
  • KFN: Fraction of negative weights.
  • SKWM: Weight of mean in simple kriging.
  • KSHN: Number of distinct drillholes contributing to the estimate.

Additional quality indicators commonly reviewed with KNA:

  • Sample_Avg_Dist: Average distance of found composites.
  • Sample_Weighted_Dist: Weighted average distance (accounts for sample weights).

Main KNA Review Criteria (Typical Priority)

  1. Slope of Regression (SR / KSR)
  2. Weight of Mean in Simple Kriging (WOMSK / SKWM)
  3. Fraction of Negative Weights (KFN)
  4. Kriging Variance (through KSE) and Kriging Efficiency (KEF)

Slope of Regression (SR)

SR represents the regression slope between true block grade and estimated block grade. Even though true block grades are unknown in practice, SR can be derived theoretically from the kriging system outputs.

  • Ideal target: close to 1
  • Typical range: 0 to 1
  • Interpretation: values closer to 1 indicate less smoothing bias in estimates.

WOMSK (Weight of Mean for Simple Kriging)

WOMSK indicates how much the estimate relies on the mean rather than nearby composites.

  • Ideal target: close to 0
  • Typical range: 0 to 1
  • Interpretation: high WOMSK means selected composites contribute little useful local information.

This metric is relevant to simple kriging workflows.

Fraction of Negative Weights

The fraction of negative (or very small) kriging weights helps assess whether the search neighborhood extends too far in some directions.

  • If negative weights appear, the neighborhood may already be sufficiently large in that direction.
  • If all weights remain significant, additional distant samples may still improve support.

In many practical models, negative weights remain limited, so this metric is often secondary, but it can still be useful for neighborhood tuning.

Kriging Efficiency (KE)

Kriging Efficiency is a scaled measure derived from block variance (BV) and kriging variance (KV):

[ KE = \frac{BV - KV}{BV} ]

Where:

  • BV = block variance (expected variance of real blocks for the selected variogram and block size)
  • KV = kriging variance of the block

Notes:

  • KE values are generally below 1.
  • KE can be negative when KV is greater than BV.

Over-Smoothing Ratio (OSR)

OSR compares expected block dispersion to estimated block dispersion. It is used to quantify smoothing introduced by kriging relative to unknown real block values.

  • SD Expected = expected standard deviation of real blocks ((\sqrt{BV}))
  • SD Estimated Blocks = standard deviation of estimated blocks

Typical interpretation examples:

  • Nearest neighbor: around -40% (too much dispersion / insufficient smoothing)
  • Ordinary kriging with dense data: around 15%
  • Ordinary kriging with sparse data: can increase to 70%–90%

General Quality Metrics

Effective Number of Samples Used (NbSampleUsed / KNSU)

This metric differs from NbSampleFound (KNSF). It accounts for weight distribution and expresses how many samples are effectively contributing.

For example, if 30 samples are found but only 5 carry most of the weight, the effective number of samples used is close to 5.

Formula:

[ K = \frac{1}{\sum_{i=1}^{n} w_i^2} ]

Sample Distance Metrics

  • Sample_Avg_Dist: simple mean distance to found composites.
  • Sample_Weighted_Dist: weighted mean distance using composite weights.

The weighted metric is usually more representative of estimate support quality because it reflects actual contribution strength.

Neighborhood Settings That Strongly Affect KNA

KNA responses are highly sensitive to how the neighborhood is configured. In practice, review results when changing:

  • Search mode (distance-based, octant-based, or irregular workflows).
  • Global minimum and maximum number of samples.
  • Per-octant minimum and maximum number of samples (when octant control is used).
  • Maximum samples per drillhole (to limit local clustering effects).
  • Search geometry options (elliptic or variable ellipsoid behavior).

These controls directly influence sample selection, smoothing, and all KNA indicators.

Practical Use

When reviewing KNA results:

  1. Activate required predefined variables before estimation.
  2. Run estimation on the full block model (or representative zones).
  3. Review maps and averages for KSR, SKWM, KFN, KEF, and distance-based metrics.
  4. Adjust neighborhood parameters and compare results to balance bias control and smoothing.