Probabilistic Interpolation

HES 505 Fall 2025: Session 20

Matt Williamson

Interpolation

Spatial interpolation is the process of predicting values of a target variable or attribute at an unobserved location, from a set of values at known (observed) locations

Deterministic vs. Probabalistic Interpolation

  • Deterministic- interpolated values calculated as a function of distance

  • Probabilistic- interpolated values are predicted based on statistical models

Kriging

  • Previous methods predict \(z\) as a (weighted) function of distance

  • Treat the observations as perfect (no error)

  • If we imagine that \(z\) is the outcome of some spatial process such that:

\[ \begin{equation} z(\mathbf{x}) = \mu(\mathbf{x}) + \epsilon(\mathbf{x}) \end{equation} \]

then any observed value of \(z\) is some function of the process (\(\mu(\mathbf{x})\)) and some error (\(\epsilon(\mathbf{x})\))

  • Kriging exploits autocorrelation in \(\epsilon(\mathbf{x})\) to identify the trend and interpolate accordingly

Autocorrelation

  • Correlation the tendency for two variables to be related

  • Autocorrelation the tendency for observations that are closer (in space or time) to be correlated

  • Positive autocorrelation neighboring observations have \(\epsilon\) with the same sign

  • Negative autocorrelation neighboring observations have \(\epsilon\) with a different sign (rare in geography)

Ordinary Kriging

  • Assumes that the deterministic part of the process (\(\mu(\mathbf{x})\)) is an unknown constant (\(\mu\))

\[ \begin{equation} z(\mathbf{x}) = \mu + \epsilon(\mathbf{x}) \end{equation} \]

Steps for Ordinary Kriging

  • Removing any spatial trend in the data (if present). ** Computing the experimental variogram, \(\gamma\), which is a measure of spatial autocorrelation.
  • Defining an experimental variogram model that best characterizes the spatial autocorrelation in the data.
  • Interpolating the surface using the experimental variogram.

(semi)Variograms

  • Relationship between observation variance and distance

  • nugget - the proportion of semivariance that occurs at small distances

  • sill - the maximum semivariance between pairs of observations

  • range - the distance at which the sill occurs

Variograms

  • Empirical vs. fitted.

Universal Kriging

  • Assumes that the deterministic part of the process (\(\mu(\mathbf{x})\)) is now a function of the location \(\mathbf{x}\)

  • Could be the location or some other attribute

  • Now y is a function of some aspect of x

Co-Kriging

  • relies on autocorrelation in \(\epsilon_1(\mathbf{x})\) for \(z_1\) AND cross correlation with other variables (\(z_{2...j}\))

  • Extending the ordinary kriging model gives:

\[ \begin{equation} z_1(\mathbf{x}) = \mu_1 + \epsilon_1(\mathbf{x})\\ z_2(\mathbf{x}) = \mu_2 + \epsilon_2(\mathbf{x}) \end{equation} \]

  • Note that there is autocorrelation within both \(z_1\) and \(z_2\) (because of the \(\epsilon\)) and cross-correlation (because of the location, \(\mathbf{x}\))

  • Not required that all variables are measured at exactly the same points

Why Krig?

  • Allows empirical estimate of autocorrelation

  • Incorporate actual spatial structure

  • Increased accuracy

  • Uncertainty estimates

Why Not Krig?

  • Computationally intensive

  • Smoothing not guarenteed

  • Variograms (somewhat) subjective