1
This paper was presented at the Applied Reservoir Characterization Using Geostatistics: The Value of Spatial
Modeling conference, December 2000, Houston, Texas, and published in the conference proceedings. It was
accepted in 2001for publication in the book Stochastic Modeling II being published by the American
Association of Petroleum Geologists, but still not published.
Available at http://www.esri.com/software/arcgis/arcgisxtensions/geostatistical/research_papers.html


A New Method for Handling the Nugget Effect in Kriging

Konstantin Krivoruchko13, Alexander Gribov1, and Jay M. Ver Hoef2

1 Environmental Systems Research Institute, 380 New York Street, Redlands, CA 92373-8100
2 Alaska Department of Fish and Game, 1300 College Road, Fairbanks, AK 99701
3 All correspondence to first author

Abstract

This chapter discusses the semivariogram parameter called the nugget effect. Commonly used exact and filtered
kriging methods are compared with a new method, which predicts a new value at the sampled location. Using
this new method, prediction at a location where data have been collected involves predicting the smooth
underlying value plus a new observation from the measurement error process. This is exactly what is necessary
for validation and cross-validation diagnostic. Example of the decision-making using new value kriging is
presented using radiocesium soil contamination data, collected in Belarus after the Chernobyl accident.

2
Introduction

Kriging is a spatial interpolation method used first in meteorology, then in geology, environmental sciences, and
agriculture, among others. It operationally and theoretically depends on models of spatial autocorrelation, which
can be formulated in terms of covariance or semivariogram functions. In either formulation, the models of spatial
autocorrelation often exhibit similar characteristics, which are called nugget, sill, and range. Sill represents the
total variation in the data, and range is the distance where autocorrelation vanishes or nearly vanishes. The
nugget effect refers to the situation when sampling locations are close to each other, but the difference between
measurements is not zero. Generally speaking, the nugget effect is unlikely to be a physical reality, because in
many physical processes, when observations become infinitely close they should converge in value, creating a
smooth process. The primary reason that discontinuities occur near the origin for semivariogram/covariance
models is the presence of measurement error or variation at scales too fine to detect (microscale variation).
In geological applications, kriging is usually associated with exact interpolation. Exact interpolation means that
when semivariogram/covariance models have a nugget effect, there will potentially be a discontinuity at
prediction locations where data have been collected. That is, the kriging predictions will change gradually and
relatively smoothly in space until they get to a location where data have been collected, and then there is a jump
in the prediction to the exact value that is measured. Because there is a jump in predictions to the exact value,
there is also a discontinuity in the prediction standard error, which jumps to zero at measured locations. When
multiple values are measured at a single location and these measurements are different, which is a common
situation in earth science applications, exact kriging cannot be used.
There are variations of kriging that can produce noiseless (or “filtered”) predictions (Gandin, 1959).
Interpolation based on filtered kriging produces smoother maps without the jumps. A consequence of filtering is
that the prediction standard error is smaller since measurement error is not included in the nugget effect.
This chapter presents an alternative kriging method, which predicts a new value at locations where data have
been observed called “new value kriging”. This method causes no discontinuities in predictions nor in their
standard errors, and the standard error is equivalent to that of exact kriging. Cross-validation is an application for