Introduction to Spatial Interpolation
Spatial interpolation is the process of estimating values at unsampled locations within an area covered by existing observations. Deterministic methods use mathematical functions to create surfaces from measured points, based on either the extent of similarity or the degree of smoothing.
Key Characteristics
- Based on exact mathematical formulas
- No statistical assumptions about data
- Reproduces sample points exactly (exact interpolators)
- Simple to implement and computationally efficient
Common Applications
- Elevation modeling (DEM creation)
- Temperature and precipitation mapping
- Soil property estimation
- Air pollution monitoring
Popular Deterministic Methods
Inverse Distance Weighting (IDW)
Assumes that each measured point has a local influence that diminishes with distance.
Formula:
Ẑ(s₀) = Σ[z(sᵢ)/dᵢᵖ] / Σ[1/dᵢᵖ]
Pros & Cons
Nearest Neighbor
Assigns the value of the nearest measured point to each prediction location.
Algorithm:
Ẑ(s₀) = z(sᵢ) where d(s₀,sᵢ) is minimized
Pros & Cons
Natural Neighbor
Uses area-based weights from Voronoi tessellation to interpolate values.
Formula:
Ẑ(s₀) = Σwᵢz(sᵢ), wᵢ = Aᵢ/Atotal
Pros & Cons
Radial Basis Functions
Uses basis functions centered at each point to create a smooth surface.
Formula:
Ẑ(s₀) = Σλᵢφ(||s₀-sᵢ||) + p(s₀)
Pros & Cons
Triangulation (TIN)
Creates a network of triangles connecting sample points with linear interpolation within each triangle.
Process:
1. Delaunay triangulation
2. Linear interpolation
Pros & Cons
Spline Interpolation
Uses piecewise polynomial functions that minimize surface curvature between points.
Minimizes:
∫∫[(∂²z/∂x²)² + 2(∂²z/∂x∂y)² + (∂²z/∂y²)²]dxdy
Pros & Cons
Method Comparison
| Method | Exact/Approximate | Smoothness | Computational Cost | Typical RMSE | Best For |
|---|---|---|---|---|---|
| IDW | Exact | Medium | Low | 0.8-1.2σ | Dense, uniform data |
| Nearest Neighbor | Exact | None | Very Low | 1.0-1.5σ | Discrete data |
| Natural Neighbor | Exact | High | Medium | 0.7-1.0σ | Irregular data |
| RBF | Exact/Approx | Very High | High | 0.6-0.9σ | Smooth surfaces |
| TIN | Exact | Low | Medium | 0.9-1.3σ | Terrain models |
| Spline | Exact/Approx | Very High | High | 0.5-0.8σ | Gentle gradients |
Statistical Performance Metrics
Common Validation Metrics
Root Mean Square Error (RMSE)
√(Σ(Ẑᵢ - zᵢ)²/n)
Measures average magnitude of errors, sensitive to outliers
Mean Absolute Error (MAE)
Σ|Ẑᵢ - zᵢ|/n
Average absolute difference, less sensitive to outliers
Coefficient of Determination (R²)
1 - Σ(Ẑᵢ - zᵢ)²/Σ(zᵢ - z̄)²
Proportion of variance explained by the model
Cross-Validation Approaches
Leave-One-Out Cross Validation (LOOCV)
Iteratively remove one point, predict it, and calculate error
Computationally intensive but uses all data
k-Fold Cross Validation
Divide data into k subsets, rotate validation set
Typical k=5 or 10, balances computation and reliability
Spatial Block Validation
Divide study area into regions, validate on entire blocks
Better assesses spatial autocorrelation effects