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Hydrology and Earth System Sciences An interactive open-access journal of the European Geosciences Union
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© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 30 May 2018

Research article | 30 May 2018

Review status
This discussion paper is a preprint. A revision of the manuscript is under review for the journal Hydrology and Earth System Sciences (HESS).

Geostatistical interpolation by Quantile Kriging

Henning Lebrenz1,2 and Andras Bárdossy2 Henning Lebrenz and Andras Bárdossy
  • 1University of Applied Sciences and Arts – Northwestern Switzerland, Institute of Civil Engineering, Switzerland
  • 2University of Stuttgart, Institute for Modelling Hydraulic and Environmental Systems, Germany

Abstract. The widely applied geostatistical interpolation methods of Ordinary Kriging (OK) or External Drift Kriging (EDK) interpolate the variable of interest to the unknown location, providing a linear estimator and an estimation variance as measure of uncertainty. The methods implicitly pose the assumption of Gaussianity on the observations, which is not given for many variables. The resulting best linear and unbiased estimator from the subsequent interpolation optimizes the mean error over many realizations for the entire spatial domain and, therefore, allows a systematic under- (over-) estimation of the variable in regions of relatively high (low) observations. In case of a variable with observed time-series, the spatial marginal distributions are estimated separately for one time step after the other, and the errors from the interpolations might accumulate over time in regions of relatively extreme observations.

Therefore, we propose the interpolation method of Quantile Kriging (QK) with a two step procedure prior to interpolation: we firstly estimate distributions of the variable over time at the observation locations and then estimate the marginal distributions over space for every given time step. For this purpose, a distribution function is selected and fitted to the observed time-series at every observation location, thus converting the variable into quantiles and defining parameters. At a given time step, the quantiles from all observation locations are then transformed into a Gaussian-distributed variable by a twofold quantile-quantile transformation with the Beta- and the Normal-distribution function. The spatio-temporal description of the proposed method accommodates skewed marginal distributions and resolves the spatial non-stationarity of the original variable. The Gaussian-distributed variable and the distribution parameters are now interpolated by OK and EDK. At the unknown location, the resulting outcomes are reconverted back into the estimator and the estimation variance of the original variable. As a summary, QK newly incorporates information from the temporal axis for its spatial marginal distribution and subsequent interpolation and, therefore, could be interpreted as a space-time version of Probability Kriging.

In this study, QK is applied for the variable of observed monthly precipitation from raingauges in South Africa. The estimators and estimation variances from the interpolation are compared to the respective outcomes from OK and EDK. The cross-validations shows that QK improves the estimator and the estimation variance for most of the selected objective functions. QK further enables the reduction of the temporal bias at locations of extreme observations. The performance of QK, however, declines when many zero-value observations are present in the input data. It is further revealed that QK relates the magnitude of its estimator with the magnitude of the respective estimation variance as opposed to the traditional methods of OK and EDK, whose estimation variances do only depend on the spatial configuration of the observation locations and the model settings.

Henning Lebrenz and Andras Bárdossy
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Henning Lebrenz and Andras Bárdossy
Henning Lebrenz and Andras Bárdossy
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Short summary
Many variables, e.g. in hydrology, geology, social sciences, etc., are only observed at a few distinct measurement locations and their actual distribution in the entire space remains unknown. We introduce the new geostatistical interpolation method of Quantile Kriging, providing an improved estimator and associated uncertainty. It can also host variables, which would not fulfill the implict presumptions of the traditional geostatistical interpolation methods.
Many variables, e.g. in hydrology, geology, social sciences, etc., are only observed at a few...