# Optimum interpolation

Developed optimum interpolation for meteorology at the same time that Kriging was developed in the geosciences.

It is based on a spatial correlation function, assumes second order stationarity, and requires a first guess field, similar to the output of numerical weather prediction models. In that configuration, it is appropriate for interpolations at low time resolutions.

A. Ordinary Kriging

Ordinary Kriging is the most fundamental type of Kriging. Ordinary Kriging predicts a linear combination of the measured values.

The weights are determined by the spatial correlation between the data, as described by the variogram.

The method is based on the assumption of intrinsic stationarity. Sadly, meteorological variables are not always stationary.

In some cases, this problem can be avoided by varying the size and shape of the search neighbourhood. Ordinary Kriging is frequently used in meteorology, usually as part of residual or indicator Kriging.

B. Simple Kriging

Ordinary Kriging with a known mean is referred to as Simple Kriging. As a result, it is slightly more powerful than standard Kriging; however, the mean is frequently difficult to calculate.

C. Cokriging

Cokriging is an extension of standard Kriging that makes use of a multivariate variogram or covariance model as well as multivariate (ancillary) data. Estimates on a location are based on a linear weighted sum of all variables examined in cokriging.

When more than one covariable is taken into account, the method can become extremely complex.
Cokriging is frequently used in meteorology.

In general, cokriging produces better results when the number of covariables is (significantly) greater than the number of variables of interest and the spatial correlation between variables and covariables is high.

The assumptions are the same as for ordinary Kriging, with the addition of cross-variogram model estimation assumptions.

D. Universal Kriging

Kriging with a trend/external drift is another name for Universal Kriging. As part of the Kriging process, it employs a regression model to model the mean value expressed as a linear or quadratic trend.
In meteorology, kriging with an external drift is a very common method.

It is based on the assumption of intrinsic stationarity, and the number of “drift” variables should be much greater than the variable of interest.

E. Residual Kriging
Residual kriging is also known as detrended kriging. With residual Kriging, the residuals from a previously fitted regression are interpolated using ordinary Kriging.

The assumptions are the same as for universal kriging.

Both linear regression and Kriging methods can be used to assess success.

Residual Kriging is widely used in meteorology

F. Indicator kriging

Indicator Kriging is the interpolation of a categorical variable, such as the occurrence of precipitation. It generates binary data using thresholds and then interpolates using standard Kriging.

The results indicate the likelihood that a certain threshold will be exceeded. Indicator kriging can be thought of as a subset of disjunctive kriging.

G. Probability kriging

Probability kriging is a non-linear method that employs indicator variables. The method can be thought of as a variation on cokriging, with the first variable being the indicator and the second variable being the original untransformed data.
Probability kriging is more powerful than indicator kriging, but it necessitates more calculations due to the need to fit cross variances.
The method, like indicator Kriging, is not appropriate for data with a trend.