### Abstract

This paper launches a hybrid sampling approach, entailing a design-based approach in space followed by a model-based approach in time, for estimating temporal trends of spatial means or totals. The underlying space–time process that generated the soil data is only partly described, viz. by a linear mixed model for the temporal variation of the spatial means. The model contains error terms for model inadequacy (model or process error) and for the sampling error in the estimated spatial means. The linear trend is estimated by Generalized Least Squares. The covariance matrix is obtained by adding the matrix with design-based estimates of the sampling variances and covariances and the covariance matrix of the model errors. The model parameters needed for the latter matrix are estimated by REML. The error variance of the estimated regression coefficients can be decomposed into the model variance of the errorless regression coefficients and the model expectation of the conditional sampling variance. In a case study on forest soil eutrophication, inclusion of the model error led to a considerable increase of the error variance for most variables. In the topsoil the contribution of the process error to the standard error of the estimated trend was much larger than that of the sampling error. For pH there was no contribution of the model error. Important advantages of the presented approach over the fully model-based approach are its simplicity and robustness to model assumptions.

Original language | English |
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Pages (from-to) | 241-248 |

Journal | Geoderma |

Volume | 173-174 |

DOIs | |

Publication status | Published - 2012 |

### Keywords

- soil properties
- optimization
- variograms
- patterns

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## Cite this

Brus, D. J., & de Gruijter, J. J. (2012). A hybrid design-based and model-based sampling approach to estimate the temporal trend of spatial means.

*Geoderma*,*173-174*, 241-248. https://doi.org/10.1016/j.geoderma.2011.12.010