### Abstract

Simple and ordinary kriging assume a constant mean and variance of the soil variable of interest. This assumption is often implausible because the mean and/or variance are linked to terrain attributes, parent material or other soil forming factors. In kriging with external drift (KED) non-stationarity in the mean is accounted for by modelling it as a linear combination of covariates. In this study, we applied an extension of KED that also accounts for non-stationary variance. Similar to the mean, the variance is modelled as a linear combination of covariates. The set of covariates for the mean may differ from the set for the variance. The best combinations of covariates for the mean and variance are selected using Akaike's information criterion. Model parameters of the selected model are then estimated by differential evolution using the Restricted Maximum Likelihood (REML) in the objective function. The methodology was tested in a small area of the Hunter Valley, NSW Australia, where samples from a fine grid with gamma K measurements were treated as measurements of the variable of interest. Terrain attributes were used as covariates. Both a non-stationary variance and a stationary variance model were calibrated. The mean squared prediction errors of the two models were somewhat comparable. However, the uncertainty about the predictions was much better quantified by the non-stationary variance model, as indicated by the mean and median of the standardized squared prediction error and by accuracy plots. We conclude that the non-stationary variance model is more flexible and better suited for uncertainty quantification of a mapped soil property. However, parameter estimation of the non-stationary variance model requires more attention due to possible singularity of the covariance matrix.

Language | English |
---|---|

Pages | 138-147 |

Number of pages | 10 |

Journal | Geoderma |

Volume | 324 |

DOIs | |

Publication status | Published - 15 Aug 2018 |

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### Keywords

- Geostatistics
- Kriging
- Non-stationarity
- Pedometrics
- REML
- Uncertainty assessment

### Cite this

*Geoderma*,

*324*, 138-147. https://doi.org/10.1016/j.geoderma.2018.03.010

}

*Geoderma*, vol. 324, pp. 138-147. https://doi.org/10.1016/j.geoderma.2018.03.010

**Accounting for non-stationary variance in geostatistical mapping of soil properties.** / Wadoux, Alexandre M.J.C.; Brus, Dick J.; Heuvelink, Gerard B.M.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Accounting for non-stationary variance in geostatistical mapping of soil properties

AU - Wadoux, Alexandre M.J.C.

AU - Brus, Dick J.

AU - Heuvelink, Gerard B.M.

PY - 2018/8/15

Y1 - 2018/8/15

N2 - Simple and ordinary kriging assume a constant mean and variance of the soil variable of interest. This assumption is often implausible because the mean and/or variance are linked to terrain attributes, parent material or other soil forming factors. In kriging with external drift (KED) non-stationarity in the mean is accounted for by modelling it as a linear combination of covariates. In this study, we applied an extension of KED that also accounts for non-stationary variance. Similar to the mean, the variance is modelled as a linear combination of covariates. The set of covariates for the mean may differ from the set for the variance. The best combinations of covariates for the mean and variance are selected using Akaike's information criterion. Model parameters of the selected model are then estimated by differential evolution using the Restricted Maximum Likelihood (REML) in the objective function. The methodology was tested in a small area of the Hunter Valley, NSW Australia, where samples from a fine grid with gamma K measurements were treated as measurements of the variable of interest. Terrain attributes were used as covariates. Both a non-stationary variance and a stationary variance model were calibrated. The mean squared prediction errors of the two models were somewhat comparable. However, the uncertainty about the predictions was much better quantified by the non-stationary variance model, as indicated by the mean and median of the standardized squared prediction error and by accuracy plots. We conclude that the non-stationary variance model is more flexible and better suited for uncertainty quantification of a mapped soil property. However, parameter estimation of the non-stationary variance model requires more attention due to possible singularity of the covariance matrix.

AB - Simple and ordinary kriging assume a constant mean and variance of the soil variable of interest. This assumption is often implausible because the mean and/or variance are linked to terrain attributes, parent material or other soil forming factors. In kriging with external drift (KED) non-stationarity in the mean is accounted for by modelling it as a linear combination of covariates. In this study, we applied an extension of KED that also accounts for non-stationary variance. Similar to the mean, the variance is modelled as a linear combination of covariates. The set of covariates for the mean may differ from the set for the variance. The best combinations of covariates for the mean and variance are selected using Akaike's information criterion. Model parameters of the selected model are then estimated by differential evolution using the Restricted Maximum Likelihood (REML) in the objective function. The methodology was tested in a small area of the Hunter Valley, NSW Australia, where samples from a fine grid with gamma K measurements were treated as measurements of the variable of interest. Terrain attributes were used as covariates. Both a non-stationary variance and a stationary variance model were calibrated. The mean squared prediction errors of the two models were somewhat comparable. However, the uncertainty about the predictions was much better quantified by the non-stationary variance model, as indicated by the mean and median of the standardized squared prediction error and by accuracy plots. We conclude that the non-stationary variance model is more flexible and better suited for uncertainty quantification of a mapped soil property. However, parameter estimation of the non-stationary variance model requires more attention due to possible singularity of the covariance matrix.

KW - Geostatistics

KW - Kriging

KW - Non-stationarity

KW - Pedometrics

KW - REML

KW - Uncertainty assessment

U2 - 10.1016/j.geoderma.2018.03.010

DO - 10.1016/j.geoderma.2018.03.010

M3 - Article

VL - 324

SP - 138

EP - 147

JO - Geoderma

T2 - Geoderma

JF - Geoderma

SN - 0016-7061

ER -