Lecture Notes on Spatial Interpolation

Introduction to Spatial Interpolation

Spatial interpolation is the process of estimating attribute values (e.g., elevation) at unsampled locations.
The goal is to find a function that fits or approximates the sampled points closely.
Used in various engineering and scientific fields.

Exactness: The interpolant should honor or pass through the original sample points.
Continuity and Smoothness:
- C0 Interpolant: Continuous with a value at every point, but derivative may be undefined.
- C1 Interpolant: First derivative is known everywhere.
- C2 Interpolant: Second derivative is known everywhere.
Locality: Uses only nearby points for interpolation.
Adaptability: Should accommodate non-normal and anisotropic data distributions.
Computational Efficiency: Results should be obtained in reasonable time.
Automaticity: Ideally, no user-defined parameters should be needed.

Possible to find a polynomial of degree n-1 for n points.
High-degree polynomials may cause oscillations between points, especially above degree 10.
Splines: Piecewise polynomials that ensure continuity and smoothness at junctions, often preferred over high-degree polynomials.

More common than polynomials and splines in GIS.
Principles:
1. Determine which subset of points to use.
2. Assign weights to points based on importance (e.g., inverse distance).
Inverse Distance to a Power:
- Uses a circle to select nearby points.
- Importance is inversely proportional to distance.

Example: Interpolation of terrain in Tasmania using different methods.
Methods include linear interpolation with Delaunay triangles, natural neighbor, and others.
Provides a QGIS3 project for exploration and comparison of different methods.

Various methods exist for spatial interpolation, each with unique strengths and weaknesses.
Understanding these methods helps in selecting the appropriate one for specific applications.

Note: For detailed explanations and implementations, refer to the book and provided QGIS project.