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.

Many environmental variables (e.g., precipitation, ore grades) are only
measured at some distinct observation locations, but possess a highly
variable and unknown spatial distribution

Therefore, the assessment of the distribution by geostatistical models arose,
whose theoretical fundamentals were firstly laid out by

The stated hypothesis entails three implications: the first condition of the
intrinsic hypothesis (Eq.

The theoretical extension of external drift kriging (EDK,

The non-parametric methods of indicator kriging and probability kriging

In summary, geostatistical methods have been derived in the past in order to
address the stated shortfalls of the intrinsic hypothesis. However, all
present methods only regard the observations from the one respective time
step for the estimation of their marginal spatial distribution, but do not
incorporate observations from other time steps. The inclusion of a temporal
behavior into the geostatistical models is mostly irrelevant for the original
geological variables. However, the temporal variability of a variable becomes
more prominent for other sciences, e.g., hydrology, where observations from
raingauges over several time steps are implemented into the geostatistical
models in order to generate spatial precipitation estimates. These estimates
subsequently serve as input to the hydrological modeling (e.g.,

In the following section, we introduce quantile kriging as a spatio-temporal
description of the variable

The theory of quantile kriging (QK) is outlined along with the major theoretical implications, followed by a general discussion of the underlying geostatistical model and a case study for the variable of monthly precipitation is presented.

Histogram from the times series
of observed monthly precipitation for two random stations:
“Laingsnek”,

A preliminary analysis of the selected variable exemplary reveals
(Fig.

QK presumes the existence of observations of the variable

At first, the distribution over time is estimated at every observation
location location

The marginal spatial distribution corresponding to a time step

The transformation via the quantiles into the variable

The outlined conversion of the variable

The inherent assumption of second-order stationarity implies the existence of
a constant spatial mean for the variable

The defining parameters

At last, the resulting Gaussian distribution of the random variable

Flowchart for the basic methodology of quantile kriging.

The basic methodology of the proposed QK is illustrated in Fig.

Since the proposed method of QK is applied for the variable of monthly
precipitation (see Sect.

The monthly (and even daily) precipitation amounts

The meteorological processes, which are generating precipitation, are usually of large spatial extent: if one location receives heavy precipitation, it is likely that other locations also receive heavy precipitation.

Correlations between time series of precipitation indicate a strong spatial dependence, while the spatial dependence of precipitation at one given time step (e.g., day, month) usually show a much weaker spatial dependence.

Let

For each individual time step

We use

However, this is not the case with observed data because wet and dry
conditions occur simultaneously over the entire domain. This is controlled by

The introduction of

We estimate and subsequently interpolate

Non-Gaussianity should be considered due to the usually skewed distribution
of precipitation amounts and it only applies to the marginal distribution at
a given time step

The distributions

The proposed method of QK is applied for the variable of monthly precipitation in South Africa and the outcomes are compared to those from OK and EDK.

The rectangular study area (

Study area, elevation and location of raingauges.

The observations of monthly precipitation were retrieved from raingauges of
four different sources: the Department of Water Affairs

The observed average monthly precipitation over the 12 calendar months

At first, we subdivided the observations of monthly precipitation into the
corresponding calendar month

Average monthly precipitation (in mm) and dry ratio (in

Thus, the original observations of monthly precipitation

We further selected EDK as non-stationary interpolation method for the
defining parameters

Scatterplot of sample mean

The principal component analysis allows for the transformation into the new
Cartesian coordinate system with the new coordinates

The proposed interpolation method of QK, using either a

The outcomes from the interpolation by OK, EDK and QK-

Spatial patterns of the estimator

The estimator

QK utilizes elevation for the interpolation of the two distribution
parameters

Spatial patterns of the estimator

The standard deviations

The estimation error from OK and EDK depends on the spatial configuration of
the observation locations

A relationship between the magnitude of the estimator

Evolution of
the Spearman rank correlation coefficient

The rank correlation varies over the calendar months for all implemented
interpolation methods and reach their seasonal maximums in June or July
(Fig.

An improvement in the relationship between the estimator

The inferior correlation coefficients of OK and EDK are nearly congruent due
to their inherent geostatistical definition: although the kriging weights are
altered by the drift, they influence the linear estimator

The estimator

The overall values of the six objective functions from all 32 226 original
observations, along with a separation into dry season (calender months: 4–9)
and wet season (calender months: 1–3 and 10–12) are given in
Table

Summary results from the cross-validation of the estimator

Errors of the estimator

The total values of the correlation coefficient

Complementary, OK and EDK have superior values for the biases B1 and B3 as a
result of the implicit definition as best linear and unbiased estimator. OK
(and to some extent EDK) optimize the spatial bias B3 for a given month by
adapting their global mean to the observed mean, according to
Eq. (

Raingauge “Wilgervier” (

Summary results from the cross-validation of the estimation error for the entire year, and split into dry (calender months: 4–9) and wet (calender months: 1–3 and 10–12) season.

The evolution of the temporal bias B2 at raingauge “Wilgervier” is
calculated from cross-validation according to Eq.

Evolution of two objective functions for the estimator over the 12
calendar months: Correlation coefficient

OK displays the highest systematic underestimation over time
(Fig.

Raingauge “Wilgervier” illustrates that OK and EDK might optimize the
spatial bias (Table

The effects of the increased occurrence of zero-value observations on the
Pearson correlation coefficient

QK-

The performance of QK is considerably influenced by the dry ratio. The presence of many zero values in the data leads to very steep or nearly vertical theoretical cdfs, hampering the allocation of the quantiles to the respective precipitation values.

Histograms for the

The estimated error distribution of the estimator

LEPS compares the values of the estimator

The test on uniformity verifies the estimated, conditional distribution

The values of the two objective functions from cross-validation of all
32 226 original observations of the entire year, and divided into dry
(calender months: 4–9) and wet season (calender months: 1–3 and 10–12)
are displayed in Table

The best overall LEPS values are received from the traditional EDK and
OK (Table

Evolution of the two objective functions for the error distribution
over 12 calendar months:
LEPS

However, the

QK-

The effect of many zero-value observations on the error distribution is
investigated by the differentiation into calendar months. The objective
functions are recalculated accordingly and illustrated in
Fig.

The temporal evolution of the LEPS values for the two versions of QK is
influenced by the presence of many zero-value observations.
QK-

The temporal evolution of the

The cross-validation for the uncertainty suggests an improvement by QK under
the prerequisite of a low dry ratio within the input data. This improvement
is attributed to the wider range of the error distribution and the increased
relation between the magnitude of the estimator and the spread of the
distribution (see Sect.

The geostatistical interpolation method of QK addresses the spatial
non-stationarity of a variable of interest by its conversion into quantiles
and defining distribution parameters. The spatial–temporal description of
the variable by QK is a novelty in applied geostatistics and can be regarded
as a temporal extension of probability kriging. Therefore, the proposed
method could be extended to spatially aggregated variables of streamflows,
requiring, however, further investigations. The proposed method accommodates
skewed marginal distributions and converts them into an ideal Gaussian
distribution prior to interpolation as a major theoretical advantage over the
traditional OK or EDK. QK describes an asymmetrical distribution of the
random variable

The variable of monthly precipitation, observed at

The cross-validation of the estimator revealed an improvement for most of the
selected objective functions. In particular, QK addresses the temporal bias,
which remains unattended by the traditional geostatistical methods, which
only optimize the mean spatial bias. In case of the estimator, QK-

Respective codes can be obtained from the corresponding author.

Precipitation and elevation data can be obtained from the
respective sources mentioned in Sect.

HL and AB set up, collaborated and designed the study. HL performed the computational work and wrote the paper. AB and HL interpreted the results and replied to the comments from the reviewers.

The authors declare that they have no conflict of interest.

This research was executed at the Institute for Modeling Hydraulic and Environmental Systems of the University of Stuttgart.

This paper was edited by Sally Thompson and reviewed by Marc F. Muller and two anonymous referees.