Deterministic Interpolation

HES 505 Fall 2025: Session 19

Matt Williamson

Objectives

  • Describe reasons for interpolating data

  • Distinguish between deterministic and probabilistic methods for interpolation

  • Characterize the difference between distance- and spline-based interpolation

  • Implement two common interpolation methods in R

How do you view the world?

…As a Continuous Field

  • The earth is a single entity with properties that vary continuously through space

  • Spatial continuity: Every cell has a value (including “no data” or “not here”)

  • Self-definition: the values define the field{.semi-transparent}

  • Space is tessellated: cells are mutually exclusive

…but what about missing data?

Spatial data as a stochastic process

\[ {Z(\mathbf{s}): \mathbf{s} \in D \subset \mathbb{R}^d} \]

Interpolation

  • Goal: estimate the value of \(Z\) at new points in \(\mathbf{D}\)

  • Deterministic: Surfaces created directly from measured points

  • Probabilistic: Relies on statistical properties of measured points

Inverse-Distance Weighting

  • Unknown point estimated as weighted average

  • Weight based on distance (Tobler)

  • Can include all points or a subset

Inverse-Distance Weighting

\[ \begin{equation} \hat{z}(\mathbf{x}) = \frac{\sum_{i=1}w_iz_i}{\sum_{i=1}w_i} \end{equation} \] where

\[ \begin{equation} w_i = | \mathbf{x} - \mathbf{x}_i |^{-\alpha} \end{equation} \] \(\alpha\) is known as the power and controls the weight.

Thinking about Power

  • Decay = negative exponent (1=linear, 2=quadratic, etc)

  • Larger values = more localized results

  • Lower values = more smoothed (averaged) results

IDW and you

  • Assumes linear relationship between distance and the unknown value

  • Bounded estimation - can’t predict values it’s never seen

  • Interpolated values are local-ish averages

  • Particularly useful for dense, equally distributed data

Splines

  • A mathematical function for a curve that passes near/through points

  • Piecewise polynomials that are smoothed between endpoints

  • Allows for nonlinearity in relationships between distance and unknown values

Splines, so many

  • Tension Spline: flatter, forces estimated values to stay closer to control points

  • Regularized Spline: more elastic, more likely to produce values above and below the control points

  • Thin-plate Spline: a bit of middle-ground between the two.

Splines, so many

Splines and You

  • Great when you know you’ve undersampled the extreme data points

  • Smooth, continuous surface is the goal

  • Bad when close points have very different values