『Abstract
　The rate of nitrous oxide emissions was measured from 276 soil cores on a 7.5 -km transect, and then a subset of these data was used to compute geostatistical models in which land categories (land-use and soil type) were fixed effects. In one model the random effects were assumed to be second-order stationary. In the other models non-stationary random variation was modelled independently for the autocorrelation and variance of the spatially correlated component of emission rate, and for the nugget variance. This was done with the method of spectral tempering. Non-stationary variance parameters were modelled as functions of discrete or continuous auxiliary variables. Models in which spectral tempering was applied using quadratic functions of soil pH fitted the data significantly better than a stationary model and gave better estimates of the prediction error variances. A significantly better fit was also obtained using splines on location to model non-stationary, but mapped soil associations did not provide a basis for a significantly better variance model. Computational difficulties with spectral tempering are identified and strategies to overcome them are discussed.

Keywords: Empirical spectrum; Kriging variance; nitrous oxide; Non-stationary covariance; REML; Spectral tempering』

1. Introduction
2. Theory of spectral tempering for non-stationary variance models
3. Materials and methods
　3.1. The Bedfordshire transect
　3.2. Laboratory measurement of nitrous oxide flux
　3.3. Preliminary analysis to test for run effects
　3.4. Partitioning the data for modelling and validation
　3.5. Stationary variance models for subset M
　3.6. Non-stationary variance models for subset M
4. Results
　4.1. Preliminary analysis to test for run effects
　4.2. Stationary variance models for subset M
　4.3. Non-stationary variance models for subset M
5. Discussion and conclusions
Acknowledgements
Appendix A. Spline basis vectors
References