On the prediction error variance of three common spatial interpolation schemes

Abstract

Three forms of linear interpolation are routinely implemented in geographical information science, by interpolating between measurements made at the endpoints of a line, the vertices of a triangle, and the vertices of a rectangle (bilinear interpolation). Assuming the linear form of interpolation to be correct, we study the propagation of error when measurement error variances and covariances are known for the samples at the vertices of these geometric objects. We derive prediction error variances associated with interpolated values at generic points in the above objects, as well as expected (average) prediction error variances over random locations in these objects. We also place all the three variants of linear interpolation mentioned above within a geostatistical framework, and illustrate that they can be seen as particular cases of Universal Kriging (UK). We demonstrate that different definitions of measurement error in UK lead to different UK variants that, for particular expected profiles or surfaces (drift models), yield weights and predictions identical with the interpolation methods considered above, but produce fundamentally different (yet equally plausible from a pure data standpoint) prediction error variances.

Keywords:

• Spatial accuracy assessment
• Error propagation
• Linear interpolation
• Bilinear interpolation
• Geostatistics
• Trend surface models

Acknowledgements

We are grateful to the following researchers for stimulating our interest in this problem: C. Q. Zhu and G. X. Wang of the Zhengzhou Institute of Surveying and Mapping and the Chinese Academy of Sciences; W. Z. Shi and Tracy C. K. Cheung of the Hong Kong Polytechnic University; Q. Q. Li of Wuhan University; and Erfu Dai of the Chinese Academy of Sciences. We would like to thank Klaus Tempfli of the International Institute for Geo‐Information Science and Earth Observation (ITC) in the Netherlands for providing us with some early references on this problem. We extend our appreciation to three anonymous reviewers, whose constructive comments led to significant improvements in the original manuscript. We also gratefully acknowledge the funding that was provided by the National Geospatial‐Intelligence Agency (NGA) for the project Strategic Enhancement of NGA's Geographic Information Science Infrastructure.