C3 · Publication Volume 13

Spatial Geochemistry

trends, domains, support, interpolation and compositional effects

Learning goals

The learner should be able to distinguish sample location from source support; align coordinate reference, domain, medium, method and physical support before spatial comparison; describe trend, local variation and directional transport without assuming stationarity; choose interpolation only when its estimand and assumptions are defensible; display coverage and uncertainty; and design spatial validation that prevents nearby observations from leaking into evaluation.

Geochemical values are spatial observations with complex support. A point symbol often represents a soil volume, composite footprint, stream catchment, water interval or rock channel. The plotted coordinate is a reference location, not necessarily the geometry over which material was generated or mixed. Spatial analysis begins by restoring that support.

Coordinates, support and comparability

Verify coordinate reference, axis order, units, datum, collection method and precision. A small coordinate error can place a sample in the wrong catchment, lithology or landform. Retain original coordinates and store transformations as derived geometry. Do not infer high precision from decimal digits that exceed field accuracy.

Represent support explicitly where possible: channel line, composite polygon, catchment, depth interval or approximate footprint. Two samples at the same coordinate but different horizons or seasons are distinct observations. Two drainage samples far apart can have heavily overlapping catchments and therefore are not independent.

Spatial comparison requires compatible medium, fraction, preparation, method, unit, qualifier and process domain. If campaigns differ, map the boundary and test overlap before merging. A spatial step exactly matching a method change is an analytical hypothesis, not a geological contact.

Trend, continuity and directional process

Regional trend can arise from lithology, weathering, climate, grain size, drainage area, method or broad mineral-system processes. Removing a trend is an interpretive act. Document the trend model and show both original and residual values. A flexible surface can remove the target itself; an overly simple trend can leave broad background variation as an anomaly.

Spatial geochemistry separates observed support, process domains, transport direction, interpolation and validation blocks
Spatial geochemistry separates observed support, process domains, transport direction, interpolation and validation blocks

Spatial continuity may be anisotropic. Mechanical dispersion follows slope and drainage; alteration follows structure or stratigraphy; wind or water movement creates direction. Estimate direction from independent terrain or geological reasoning as well as data. A smooth variogram or map does not establish cause.

Stationarity means the statistical relationship is sufficiently stable within a modelled domain. It is not a property of all geochemistry. Bound domains at known process changes and test sensitivity. Sparse data cannot resolve a detailed continuity model; communicate that limitation rather than fitting an ornate curve.

Interpolation, mapping and uncertainty

Interpolation estimates a defined quantity between observations. For point-support soil data, a local concentration surface may be meaningful within a comparable horizon and process domain. For catchment-integrated sediment, treating outlet values as point samples and interpolating across divides can be physically misleading. A catchment or network representation is more appropriate.

Every map should show observations, sampling coverage, domains, censoring and quality status. Class breaks affect visual prominence. Use rules tied to data or decisions, state transformations, and include an unobserved class. Do not extrapolate beyond support without a visible mask and uncertainty.

Model uncertainty includes measurement, sampling, support mismatch, domain and spatial-model components. A narrow interpolation variance under an incorrect domain is false confidence. Scenario maps using alternative domains, trends or transport directions can be more informative than one nominal uncertainty surface.

Spatial validation and change of support

Randomly holding out individual points can give optimistic validation because nearby training samples reveal the same local process. Use spatial blocks, separated catchments, later sampling phases or prospective sites. Fit transformations, domains and interpolation settings using training data only, then evaluate the held-out observations.

Change of support matters when comparing sample types or mapping to targets. A composite averages small-scale variation; a catchment sample mixes sources; a narrow rock channel preserves local extremes. Values on different supports do not become comparable through interpolation. Use a support model, aggregate to a common support where defensible, or keep layers separate.

Validation metrics should match the action. Overall error can be low while high-value anomalies are missed. Report bias, interval coverage, rank or classification performance by domain and concentration range. Inspect spatial residuals; a coherent residual pattern means the model has not captured a process.

Worked synthetic example

Three adjacent synthetic soil cells have support areas 100, 200 and 300 m² and mean X concentrations 20, 50 and 80 mg/kg. The area-weighted mean over their combined 600 m² support is

$\bar{x}_A=\frac{100(20)+200(50)+300(80)}{600}=60\ \mathrm{mg/kg}.$

The unweighted mean is (20+50+80)/3=50 mg/kg. Treating each cell as an equal point underestimates the combined-support mean because the high cell represents more area. This calculation is valid only if the cell means and areas share a comparable material and the requested estimand is mass fraction over equal bulk density or another justified weighting basis.

Now suppose the 80 mg/kg observation is a stream outlet integrating 300 m²-equivalent catchment support, while the others are local soil cells. The same weighted calculation is no longer physically meaningful: media and source kernels differ. Keep the drainage observation as a separate directional constraint or construct an explicit mixing model.

If a spatial model is tuned and validated using alternating individual stations 20 m apart while continuity extends 100 m, the test set is not independent. Holding out 200 m-wide blocks provides a more demanding estimate of prediction beyond local neighbours.

Spatial audit workflow

  1. Verify coordinates, reference system, precision and transformation history.
  2. Represent collection location and source or integration support separately.
  3. Harmonise medium, fraction, method, unit, qualifier and quality status.
  4. Map sampling density, gaps, domains and transport direction before modelling.
  5. Separate broad trend from local variation with declared process assumptions.
  6. Choose point, polygon, catchment or network analysis to match support.
  7. Limit interpolation to defensible domains and show extrapolation masks.
  8. propagate measurement, sampling, support and domain uncertainty.
  9. Validate with spatially or temporally separated observations.
  10. Convert mapped patterns into process predictions and targeted follow-up.

Practice and review

  1. Recalculate the weighted example if the third cell area is 150 m² and state the weighting assumptions.
  2. Draw a map that distinguishes collection point, catchment support and inferred source corridor.
  3. Explain why interpolating stream-sediment outlet values across drainage divides can be misleading.
  4. Design a blocked validation scheme for samples collected on 50 m lines where continuity may extend 300 m.
  5. Compare two trend models and state which target footprints each might remove.

Review questions: What geometry does each value support? Are samples comparable? Which process creates direction? Where is the map interpolated or extrapolated? Does validation test genuinely new space?

Sources and further reading