Geostatistical interpolation by quantile kriging
Authors: Lebrenz Henning , Bárdossy András 
Published: March 2019

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 2-fold quantile–quantile transformation with the beta- and
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 show 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.