Comparison of Sampling Methods for Kriging Models

Abstract

This study aims to generate from a three-dimensional data set of carbon dioxide ux in the Southern Ocean, a sample set for use with Kriging in order to generate estimated carbon dioxide ux values across the complete three-dimensional data set. In order to determine which sampling strategies are to be used with the three-dimensional data set, a number of a-priori and a-posteriori sampling methods are tested on a two-dimensional subset. These various sampling methods are used to determine whether or not the estimated error variance generated by Kriging is a good substitute for the true error as a measure of error. Carbon dioxide is a well known "greenhouse gas" and is partially responsible for climate change. However, some anthropogenic carbon dioxide is absorbed by the oceans and as such, the oceans currently play a mitigating role in climate change by acting as a sink for carbon dioxide. It has been suggested that if the production of carbon dioxide continues unabated that the oceans may become a source rather than a sink for carbon dioxide. This would mean that the oceanic carbon dioxide ux (exchange of carbon dioxide between the atmosphere and the surface of the ocean) would invert. As such, modelling of the carbon dioxide ux is of clear importance. Additionally as the Southern Ocean is highly undersampled, a sampling strategy for this ocean which would allow for high levels of accuracy with small sample sizes would be ideal. Kriging is a geostatistical weighted interpolation technique. The weights are based on the covariance structure of the data and the distances between points. In addition to an estimate at a point, Kriging also produces an estimated error variance which can be used as an indication of uncertainty. This study made use of model data for carbon dioxide ux in the Southern i Ocean. This data covers a year by making use of averaged data for 5 day intervals. This results in a three-dimensional data set covering latitude, longitude and time. This study used this data to generate a covariance structure for the data after the removal of trend and using this covariance structure, tested various sampling strategies in two dimensions, sampling approximately 10% of the two-dimensional data subset. These sampling strategies made use of either the estimated error variance or the true error and included two simple heuristics, genetic algorithms, the Updated Kriging Variance Algorithm and Random Sampling. Two of the genetic algorithms tested were selected to maximise the error measure of interest, in order to determine the full range of errors that could be generated. The percentage absolute errors obtained across these methods ranged from 2:1% to 64:4%. Based on these strategies, the estimated error variance was determined to not be an accurate surrogate for true error and that in cases where absolute error is available, such as data minimisation, absolute error should be used as the measure of error. However, if no data is available then it does provide an easy to calculate measure of error. This study also concluded that Addition of a Point at Point of Maximum Absolute Error does provide a good validation sampling method to which other methods may be compared. Additionally, based on true errors and computational requirements, three methods were selected to be implemented on a three-dimensional subset of the data. These methods were Random Sampling, Addition of a Point at Point of Maximum Absolute Error and Addition of a Point at Point of Maximum Estimated Error Variance. Each of these methods for sampling were performed twice on the data, sampling up to approximately 5% of the data. Random Sampling produced percentage absolute errors of 21:02% and 20:98%, Addition of a Point at Point of Maximum Estimated Error Variance produced errors of 18:54% and 18:55% while Addition of a Point at Point of Maximum Absolute Error was able to produce percentage absolute errors of 14:33% and 14:32%.