Statistical Surface Modeling
Definition
Statistical surface modeling builds continuous surfaces from discrete observations using statistical frameworks rather than purely geometric methods. Examples include kriging, thin-plate splines, Gaussian processes, and Bayesian spatial models that produce predictions and uncertainty maps. These approaches allow covariates, anisotropy, and nonstationary variance, and are widely used where surfaces drive decisions.
Application
Meteorology models temperature and precipitation fields; environmental agencies map pollution; geoscience estimates ore grades; and public health interpolates health indicators. Uncertainty maps guide where to add sensors or collect additional samples for the biggest information gain.
FAQ
How do Gaussian process models relate to kriging?
Ordinary kriging is a special case of Gaussian processes with a chosen covariance kernel; GP implementations generalize this with flexible kernels and Bayesian inference.
What role do covariates play in improving surfaces?
Including terrain, land cover, or distance to sources explains systematic variation, reducing residual variance and sharpening predictions (universal kriging/regression-kriging).
How do you avoid over-smoothing peaks and sharp transitions?
Use kernels that allow shorter correlation lengths near fronts, or partition the domain and fit local models; validate with out-of-sample points near boundaries.
When should you invest in more data versus a fancier model?
Learning curves reveal diminishing returns from model complexity when sampling is too sparse; often, better coverage beats algorithmic sophistication.