Estimation and prediction

Landscape soils and surface environments - Week 3 Workshop 2a

Raphael Viscarra Rossel & Lewis Walden

2026-03-04

Week 3 Workshop – Estimation and Prediction

Recap from previous sessions:

  • Week 3A: HOW to measure (direct, remote, proximal sensing)
  • Week 3B: WHERE to sample (random, stratified, grid designs)

Today’s question: WHAT do we DO with the data?

Two general approaches to estimation and prediction

Two fundamental goals:

  1. Infer and compare unknown quantities → Classical statistics (mean, variance, totals)
  2. Quantify how unknown quantities vary across space → Spatial modelling, geostatistics

Tip

Choice depends on research, ecological, or management question…and data availability!

1. Infer and compare

When the objective is to compare, assess differences, estimate totals.

Methods used:

  • Unbiased sampling design (random, stratified) critical for valid inference
  • Inference is about summarising data, not mapping
  • Means, variances, confidence intervals, sums…

Classical statistics

  • Hypothesis tests (t-tests, ANOVA)…, etc.

Infer and compare, example A

Example 1: How much C stock do forest hold compared to woodlands?

  • Design: Stratified random sampling
  • Analysis: Calculate mean \(\pm\) SE for each stratum, ANOVA to test differences
  • Output: “Forest 45 \(\pm\) 5 Mg C/ha vs. woodland 28 \(\pm\) 4 Mg C/ha, p < 0.01”

Doesn’t tell you WHERE C is high/low.

Infer and compare, example B

Example 2: Estimate total aboveground biomass in catchment

  • Design: Random or stratified sampling
  • Analysis:
    • Simple random: Mean biomass × area
    • Stratified: \(\sum\)(mean\(_i\) × area\(_i\)) per stratum
  • Output: “Catchment stores 12,400 \(\pm\) 1,300 Mg aboveground biomass”

Doesn’t tell you WHERE biomass is high/low.

2. Quantify spatial variability

When the objective is to quantify spatial variability, predict values at unsampled locations, or create maps.

E.g. How does soil pH vary across the SCP? Where are the acidic soils?

Methods: Spatial modelling

  • Geostatistics (e.g. kriging):

    Uses spatial autocorrelation

  • Digital or Predictive Mapping:

    Uses environmental relationships

Quantifying spatial variability, geostatistics (e.g. kriging)

Data needed:

  • Point measurements (e.g. soil pH)
  • Spatial coordinates (Lat, Long, or X, Y)
  • Dense systematic sampling, e.g. grid

Method:

  1. Model spatial autocorrelation (semivariogram)
  2. Interpolate, predict at unsampled locations
  3. Provide estimates of uncertainty

Note

Uses a semivariogram (model of spatial variation), and gives estimates of uncertainty.

Tobler’s First Law of Geography

“Everything is related…but near things are more related than distant things.”

W. Tobler (1970)

Why?

  • Continuous processes (weathering, erosion) operate across space
  • Landscape features change gradually across space
  • Soil-forming factors, cl, o, r, p… all vary spatially

Dissimilarity increases with distance ➡️ spatial dependence (autocorrelation)

We can use spatial relationships to predict!

What is a semivariogram?

Tells you how much a measured quantity changes as you move farther away:

  • If nearby locations are very similar → the semivariogram value is small.
  • As distance increases → the semivariogram usually increases.
  • When it levels off → locations are no longer related (no spatial correlation).

Note

A semivariogram measures how spatial similarity decreases with distance.

Deriving a semivariogram

A semivariogram shows how the variance of a mesured quantity changes with distance.

To calculate it, we:

  1. Group pairs by separation distance (lag)

  2. Compare all pairs of sample points

  3. Calculate variance between each pair

  4. Average variance for each lag distance to get semivariance

  1. Plot semivariance vs lag distance to get the semivariogram

The equation to compute the semivariance is:

\[ \hat{\gamma}(h) = \frac{1}{2N(h)} \sum_{i=1}^{N(h)} \big[ Z(x_i) - Z(x_i + h) \big]^2 \]

where \(\hat{\gamma}(h)\) is the semivariance at lag distance \(h\), \(Z(x_i)\) and \(Z(x_i + h)\) are values at locations separated by \(h\), and \(N(h)\) is the number of such pairs.

You are not expected to memorise this formula, but to understand it conceptually

Important

Shows how pairs at distance \(h\) differ: small values → similar, large values → different.

The semivariogram: Show how variance changes with distance

  • At short distances ↓ variance → Tobler’s Law
  • At longer distances up to the range, ↑ variance
  • Beyond the range, variance levels off → no spatial correlation
  • Parameters (nugget, sill, range) → spatial structure

Note

The semivariogram model (dash line) provides the parameters for kriging

Variogram models

We fit a variogram model through the points.

  • Spherical: rises then clearly flattens at the sill

  • Exponential: rises then gradually reaches the sill

  • Gaussian: very smooth near the origin.

  • Different shapes describe different spatial continuity.

Note

Choice of model depends on the shape of the data and the underlying spatial processes.

Three key parameters: ⓵ nugget variance (\(C_0\) )

  • Variance at distance ≈ 0 (Y-intercept)
  • Represents:
    • measurement errors and
    • small-scale variation not captured by sampling
    • Large nugget → more error in kriging predictions

Note

Nugget depicts data quality. Large nugget → need better sampling or measurement.

Three key parameters: ② sill (C\(_0\) + C) variance

  • Partial sill (C\(_0\)) is the spatially structured variance
  • The Sill (C\(_0\) + C) is the maximum variance
  • Large sill - the property is highly variable across the study area
  • Small sill - the property is relatively homogeneous across the study area

Note

Sill shows property variation across the landscape. At sill spatial autocorrelation ends

Three key parameters: ③ range (\(a\))

  • Distance where sill is reached → beyond this, no spatial correlation
  • Reveals the scale at which processes operate:
  • Short range → spatial pattern is patchy
  • Long range → broad, smooth gradients.
  • Only points within roughly the range contribute to kriging

Important

Sample at appropriate distance (range) to capture variation

How the semivariogram is derived and fitted

Depending on the shape of the semivariogram, different models can be fitted (Exponential, Gaussian, linear, etc.).

Here is an example of the fitting process for spherical model:

From semivariograms to maps by Kriging

Kriging is a spatial interpolation method — it makes a map from point samples. It:

  1. Predicts values at unsampled locations as a weighted average of nearby samples

  2. Uses the variogram to decide the weights: closer samples get more weight, farther samples get less weight, and the weights also depend on the nugget, sill, range

  3. Tells you how confident the prediction is at every location (kriging variance)

How kriging works (1)

  1. Derive and fit semivariogram to quantify spatial variability and autocorrelation structure

  2. For each prediction location:

    • Identify nearby sample points within a certain distance (e.g. range)
    • Calculate weights based on (i) spatial autocorrelation and (ii) distance to samples
    • Make prediction as weighted averages of sample values and calculate uncertainty
  1. Create maps:
  • Prediction map (kriging estimates)
  • Uncertainty map (kriging variance)

How kriging works (2)

  • Kriging prediction is a weighted average of nearby samples
  • Weights depend on semivariogram parameters (nugget, sill, range)
  • Provides smooth predictions that respect spatial structure

Activity (10-15 min) Variogram fitting and kriging interpolation

Objective: Learn about semivariograms and kriging.

Link to activity Or copy and paste this URL into your browser:

https://ravr19.github.io/lsse_teaching/kriging_app.html

Follow instructions in the app to:

1. Use the app to simulate spatial data, fit variogram models, and see how they affect kriging predictions.

2. Changing one setting at a time to see its impact on the variogram and the kriging prediction and variance.

3. Answer questions to test your understanding.

Key takeaways

1. Two approaches for different questions:

  • Infer and compare → means, totals, tests (random/stratified sampling)
  • Quantify spatial variability → maps, patterns (one approach is geostatistics)

2. Tobler’s Law: Near things are more related → spatial autocorrelation

3. Semivariogram parameters:

  • Nugget = error + micro-variation | Sill = total variance | Range = scale of pattern

4. Kriging uses the semivariogram to map point data with uncertainty

5. Grid spacing ≤ half the range to capture spatial structure

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