Geostatistical interpolation by quantile kriging
<p>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 a...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Copernicus Publications
2019-03-01
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| Series: | Hydrology and Earth System Sciences |
| Online Access: | https://www.hydrol-earth-syst-sci.net/23/1633/2019/hess-23-1633-2019.pdf |
| _version_ | 1818021438256316416 |
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| author | H. Lebrenz H. Lebrenz A. Bárdossy |
| author_facet | H. Lebrenz H. Lebrenz A. Bárdossy |
| author_sort | H. Lebrenz |
| collection | DOAJ |
| description | <p>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.</p>
<p>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.</p>
<p>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.</p> |
| first_indexed | 2024-04-14T08:18:49Z |
| format | Article |
| id | doaj.art-f910b0b117704a19a0bd5acc334104c8 |
| institution | Directory Open Access Journal |
| issn | 1027-5606 1607-7938 |
| language | English |
| last_indexed | 2024-04-14T08:18:49Z |
| publishDate | 2019-03-01 |
| publisher | Copernicus Publications |
| record_format | Article |
| series | Hydrology and Earth System Sciences |
| spelling | doaj.art-f910b0b117704a19a0bd5acc334104c82022-12-22T02:04:19ZengCopernicus PublicationsHydrology and Earth System Sciences1027-56061607-79382019-03-01231633164810.5194/hess-23-1633-2019Geostatistical interpolation by quantile krigingH. Lebrenz0H. Lebrenz1A. Bárdossy2University of Applied Sciences and Arts – Northwestern Switzerland, Institute of Civil Engineering, Muttenz, SwitzerlandUniversity of Stuttgart, Institute for Modelling Hydraulic and Environmental Systems, Stuttgart, GermanyUniversity of Stuttgart, Institute for Modelling Hydraulic and Environmental Systems, Stuttgart, Germany<p>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.</p> <p>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.</p> <p>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.</p>https://www.hydrol-earth-syst-sci.net/23/1633/2019/hess-23-1633-2019.pdf |
| spellingShingle | H. Lebrenz H. Lebrenz A. Bárdossy Geostatistical interpolation by quantile kriging Hydrology and Earth System Sciences |
| title | Geostatistical interpolation by quantile kriging |
| title_full | Geostatistical interpolation by quantile kriging |
| title_fullStr | Geostatistical interpolation by quantile kriging |
| title_full_unstemmed | Geostatistical interpolation by quantile kriging |
| title_short | Geostatistical interpolation by quantile kriging |
| title_sort | geostatistical interpolation by quantile kriging |
| url | https://www.hydrol-earth-syst-sci.net/23/1633/2019/hess-23-1633-2019.pdf |
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