Interpolation

Overview

Geostatistical interpolation models can be used to predict variables over geospatial domains. These models are used within the Interpolate and InterpolateNeighbors transforms with advanced options for change of support, probabilistic prediction and neighborhood search.

Interpolate(domain; [options])

# nearest neighbor model
Interpolate(grid, model=NN())

# inverse distance weighting model
Interpolate(pset, model=IDW())

InterpolateNeighbors(domain; [options])

# polynomial model with 10 nearby samples
InterpolateNeighbors(grid, model=Polynomial(), maxneighbors=10)

# inverse distance weighting model with samples inside 100m radius
InterpolateNeighbors(pset, model=IDW(), neighborhood=MetricBall(100u"m"))

All models work with general Hilbert spaces, meaning that it is possible to interpolate any data type that implements scalar multiplication, vector addition and inner product:

The framework also provides a low-level interface for advanced users who might need to GeoStatsModels.fit and GeoStatsModels.predict (or GeoStatsModels.predictprob) models in non-standard ways:

fit(model, geotable)

predict(model, vars, gₒ)

predictprob(model, vars, gₒ)

For model validation, including cross-validation error estimates, please check the Validation section.

Models

We illustrate the models with the following geotable and grid:

gtb = georef((; z=[1.,0.,1.]), [(25.,25.), (50.,75.), (75.,50.)])

grid = CartesianGrid(100, 100)

100×100 CartesianGrid
├─ minimum: Point(x: 0.0 m, y: 0.0 m)
├─ maximum: Point(x: 100.0 m, y: 100.0 m)
└─ spacing: (1.0 m, 1.0 m)

NN

NN(distance=Euclidean())

itp = gtb |> Interpolate(grid, model=NN())

itp |> viewer

IDW

IDW(exponent=1, distance=Euclidean())

itp = gtb |> Interpolate(grid, model=IDW())

itp |> viewer

LWR

LWR(weightfun=h -> exp(-3 * h^2), distance=Euclidean())

itp = gtb |> Interpolate(grid, model=LWR())

itp |> viewer

Polynomial

Polynomial(degree=1)

itp = gtb |> Interpolate(grid, model=Polynomial())

itp |> viewer

Kriging

A Kriging model has the form:

\[\hat{Z}(\p_0) = \lambda_1 Z(\p_1) + \lambda_2 Z(\p_2) + \cdots + \lambda_n Z(\p_n),\quad \p_i \in \R^m, \lambda_i \in \R\]

with $Z\colon \R^m \times \Omega \to \R$ a random field.

This package implements the following Kriging variants:

• Simple Kriging
• Ordinary Kriging
• Universal Kriging

which can be materialized in code with the generic Kriging constructor.

Kriging(fun, mean)

Kriging(fun)

Kriging(fun, drifts)

Kriging(fun, deg, dim)

model = Kriging(GaussianVariogram(range=35.))

itp = gtb |> Interpolate(grid, model=model)

itp |> viewer

Simple Kriging

SimpleKriging(fun, mean)

# univariate model with mean 5.0
SimpleKriging(SphericalVariogram(), 5.0)

# multivariate model with mean [5.0, 10.0]
SimpleKriging(I(2) * SphericalVariogram(), [5.0, 10.0])

In Simple Kriging, the mean $\mu$ of the random field is assumed to be constant and known. The resulting linear system is:

\[\begin{bmatrix} cov(\p_1,\p_1) & cov(\p_1,\p_2) & \cdots & cov(\p_1,\p_n) \\ cov(\p_2,\p_1) & cov(\p_2,\p_2) & \cdots & cov(\p_2,\p_n) \\ \vdots & \vdots & \ddots & \vdots \\ cov(\p_n,\p_1) & cov(\p_n,\p_2) & \cdots & cov(\p_n,\p_n) \end{bmatrix} \begin{bmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n \end{bmatrix} = \begin{bmatrix} cov(\p_1,\p_0) \\ cov(\p_2,\p_0) \\ \vdots \\ cov(\p_n,\p_0) \end{bmatrix}\]

or in matricial form $\C\l = \c$. We subtract the given mean from the observations $\boldsymbol{y} = \z - \mu \1$ and compute the mean and variance at location $\p_0$:

\[\mu(\p_0) = \mu + \boldsymbol{y}^\top \l\]

\[\sigma^2(\p_0) = cov(0) - \c^\top \l\]

Ordinary Kriging

OrdinaryKriging(fun)

# univariate model with unknown mean
OrdinaryKriging(SphericalVariogram())

# multivariate model with unknown mean
OrdinaryKriging(I(2) * SphericalCovariance())

In Ordinary Kriging the mean of the random field is assumed to be constant and unknown. The resulting linear system is:

\[\begin{bmatrix} \G & \1 \\ \1^\top & 0 \end{bmatrix} \begin{bmatrix} \l \\ \nu \end{bmatrix} = \begin{bmatrix} \g \\ 1 \end{bmatrix}\]

with $\nu$ the Lagrange multiplier associated with the constraint $\1^\top \l = 1$. The mean and variance at location $\p_0$ are given by:

\[\mu(\p_0) = \z^\top \l\]

\[\sigma^2(\p_0) = \begin{bmatrix} \g \\ 1 \end{bmatrix}^\top \begin{bmatrix} \l \\ \nu \end{bmatrix}\]

Universal Kriging

UniversalKriging(fun, drifts)

UniversalKriging(fun, deg, dim)

# univariate model with mean 1 + β₁x + β₂y where x and y are Cartesian coordinates
UniversalKriging(SphericalVariogram(), [p -> 1, p -> coords(p).x, p -> coords(p).y])

# multivariate model with mean 1 + βx² where x is the first Cartesian coordinate
UniversalKriging(I(2) * SphericalVariogram(), [p -> 1, p -> coords(p).x^2])

In Universal Kriging, the mean of the random field is assumed to be a linear combination of known smooth functions. For example, it is common to assume

\[\mu(\p) = \sum_{k=1}^{N_d} \beta_k f_k(\p)\]

with $N_d$ monomials $f_k$ of degree up to $d$. For example, in 2D there are $6$ monomials of degree up to $2$:

\[\mu(x_1,x_2) = \beta_1 1 + \beta_2 x_1 + \beta_3 x_2 + \beta_4 x_1 x_2 + \beta_5 x_1^2 + \beta_6 x_2^2\]

The choice of the degree $d$ determines the size of the polynomial matrix

\[\F = \begin{bmatrix} f_1(\p_1) & f_2(\p_1) & \cdots & f_{N_d}(\p_1) \\ f_1(\p_2) & f_2(\p_2) & \cdots & f_{N_d}(\p_2) \\ \vdots & \vdots & \ddots & \vdots \\ f_1(\p_n) & f_2(\p_n) & \cdots & f_{N_d}(\p_n) \end{bmatrix}\]

and polynomial vector $\f = \begin{bmatrix} f_1(\p_0) & f_2(\p_0) & \cdots & f_{N_d}(\p_0) \end{bmatrix}^\top$.

The variogram determines the variogram matrix:

\[\G = \begin{bmatrix} \gamma(\p_1,\p_1) & \gamma(\p_1,\p_2) & \cdots & \gamma(\p_1,\p_n) \\ \gamma(\p_2,\p_1) & \gamma(\p_2,\p_2) & \cdots & \gamma(\p_2,\p_n) \\ \vdots & \vdots & \ddots & \vdots \\ \gamma(\p_n,\p_1) & \gamma(\p_n,\p_2) & \cdots & \gamma(\p_n,\p_n) \end{bmatrix}\]

and the variogram vector $\g = \begin{bmatrix} \gamma(\p_1,\p_0) & \gamma(\p_2,\p_0) & \cdots & \gamma(\p_n,\p_0) \end{bmatrix}^\top$.

The resulting linear system is:

\[\begin{bmatrix} \G & \F \\ \F^\top & \boldsymbol{0} \end{bmatrix} \begin{bmatrix} \l \\ \boldsymbol{\nu} \end{bmatrix} = \begin{bmatrix} \g \\ \f \end{bmatrix}\]

with $\boldsymbol{\nu}$ the Lagrange multipliers associated with the universal constraints. The mean and variance at location $\p_0$ are given by:

\[\mu(\p_0) = \z^\top \l\]

\[\sigma^2(\p_0) = \begin{bmatrix}\g \\ \f\end{bmatrix}^\top \begin{bmatrix}\l \\ \boldsymbol{\nu}\end{bmatrix}\]

CoKriging

All the Kriging variants are well-defined in the multivariate setting. We can use multivariate variograms (or covariances) in a linear model of coregionalization, or transiograms to perform multivariate interpolation:

# 3 variables
table = (; a=[1.0, 0.0, 0.0], b=[0.0, 1.0, 0.0], c=[0.0, 0.0, 1.0])
coord = [(25.0, 25.0), (50.0, 75.0), (75.0, 50.0)]

# 3x3 variogram
γ = [1.0 0.3 0.1; 0.3 1.0 0.2; 0.1 0.2 1.0] * SphericalVariogram(range=35.)

gtb = georef(table, coord)

itp = gtb |> Interpolate(grid, model=Kriging(γ))

itp |> Select("a") |> viewer

itp |> Select("b") |> viewer

itp |> Select("c") |> viewer

Heterotopic settings are automatically detected in the presence of missing values:

# 3 variables with missing values
table = (; a=[1.0, 0.0, 0.0], b=[0.0, 1.0, missing], c=[missing, 0.0, 1.0])

gtb = georef(table, coord)

itp = gtb |> Interpolate(grid, model=Kriging(γ))

itp |> Select("a") |> viewer

itp |> Select("b") |> viewer

itp |> Select("c") |> viewer