Kriging Interpolator

Description

Interpolation using the Kriging method. Find the best fit curve for a given set of data. You can either compute the semivariance or the variogram (expected value).

The various variogram models can be interpreted as kernel functions for 2-dimensional coordinates a, b and parameters nugget, range, sill and A. Reparameterized as a linear function, with w = [nugget, (sill-nugget)/range], this becomes:

• Gaussian: k(a,b) = w[0] + w[1] * ( 1 - exp{ -( ||a-b|| / range )2 / A } )
• Exponential: k(a,b) = w[0] + w[1] * ( 1 - exp{ -( ||a-b|| / range ) / A } )
• Spherical: k(a,b) = w[0] + w[1] * ( 1.5 * ( ||a-b|| / range ) - 0.5 * ( ||a-b|| / range )3 )

Notice the σ2 (sigma2) and α (alpha) variables, these correspond to the variance parameters of the gaussian process and the prior of the variogram model, respectively. A diffuse α prior is typically used; a formal mathematical definition of the model is provided below.

The variance parameter α of the prior distribution for w should be manually set, according to:

• w ~ N(w|0, αI)

Using the fitted kernel function hyperparameters and setting K as the Gram matrix, the prior and likelihood for the gaussian process become:

• y ~ N(y|0, K)
• `t|y ~ N(t|y, σ2I)

Usage

import { KrigingInterpolator } from 'gis-tools-ts';
import type { VectorPoint } from 'gis-tools-ts';

// We have m-value data that we want to interpolate
interface TempData { temp: number; }

// given a point we are interested in
const point: VectorPoint = { x: 20, y: -40 };
//  get a collection of points relative to the point
const data: VectorPoint<TempData>[] = [...];

// interpolate
const interpolator = new KrigingInterpolator<TempData>(data, 'gaussian', (p) => p.m.temp);
const interpolatedValue = interpolator.predict(point);
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Links

• https://pro.arcgis.com/en/pro-app/latest/tool-reference/3d-analyst/how-kriging-works.htm