HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-1693-2017A combined statistical bias correction and stochastic downscaling method for precipitationVolosciukClaudiacvolosciuk@geomar.deMaraunDouglasVracMathieuWidmannMartinhttps://orcid.org/0000-0001-5447-5763GEOMAR Helmholtz Centre for Ocean Research Kiel, Kiel, GermanyWegener Center for Climate and Global Change, University of Graz, Graz, AustriaLaboratoire des Sciences du Climat et de l'Environnement (LSCE), CNRS/IPSL, Gif-sur-Yvette, FranceSchool of Geography, Earth and Environmental Sciences, University of Birmingham, Birmingham, UKClaudia Volosciuk (cvolosciuk@geomar.de)22March2017213169317195August201615September20163March20176March2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/21/1693/2017/hess-21-1693-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/1693/2017/hess-21-1693-2017.pdf
Much of our knowledge about future changes in precipitation relies on
global (GCMs) and/or regional climate models (RCMs) that have resolutions
which are much coarser than typical spatial scales of precipitation,
particularly extremes. The major problems with these projections are both
climate model biases and the gap between gridbox and point scale.
developed a model to jointly bias
correct and downscale precipitation at daily scales. This approach, however,
relied on pairwise correspondence between predictor and predictand for
calibration, and, thus, on nudged simulations which are rarely available.
Here we present an extension of this approach that separates the downscaling
from the bias correction and in principle is applicable to free-running
GCMs/RCMs. In a first step, we bias correct RCM-simulated precipitation
against gridded observations at the same scale using a parametric quantile
mapping (QMgrid) approach. In a second step, we bridge the scale
gap: we predict local variance employing a regression-based model with
coarse-scale precipitation as a predictor. The regression model is calibrated
between gridded and point-scale (station) observations. For this concept we
present one specific implementation, although the optimal model may differ
for each studied location. To correct the whole distribution including
extreme tails we apply a mixture distribution of a gamma distribution for the
precipitation mass and a generalized Pareto distribution for the extreme tail
in the first step. For the second step a vector generalized linear gamma
model is employed. For evaluation we adopt the perfect predictor experimental
setup of VALUE. We also compare our method to the classical QM as it is
usually applied, i.e., between RCM and point scale (QMpoint).
Precipitation is in most cases improved by (parts of) our method across
different European climates. The method generally performs better in summer
than in winter and in winter best in the Mediterranean region, with a mild
winter climate, and worst for continental winter climate in Mid- and
eastern Europe or Scandinavia. While QMpoint performs similarly
(better for continental winter) to our combined method in reducing the bias
and representing heavy precipitation, it is not capable of correctly modeling
point-scale spatial dependence of summer precipitation. A strength of this
two-step method is that the best combination of bias correction and
downscaling methods can be selected. This implies that the concept can be
extended to a wide range of method combinations.
Introduction
To assess the impacts of hydrometeorological extremes in a changing climate,
high-quality precipitation projections on the point scale are often demanded.
Much of our knowledge about future changes in precipitation is based on
global (GCMs) and/or regional climate models (RCMs). These have resolutions
which are much coarser than typical spatial scales of processes relevant for
precipitation. This concerns particularly extreme precipitation, which is far
more sensitive to resolution than mean precipitation .
Although horizontal resolution of GCMs has successively increased since the
first assessment report of the Intergovernmental Panel on Climate Change
, resolving all important spatial and temporal scales remains
beyond current computational capabilities for transient global climate change
simulations . The simulation of precipitation depends
heavily on processes that are parameterized in current GCMs, and also in most
RCMs . Biases related to parameterization schemes and
unresolved processes thus remain in addition to systematic biases related to
the large-scale circulation e.g.,.
Different approaches have been employed to downscale and/or reduce biases of
simulated precipitation, particularly extremes: (a) high-resolution GCMs,
(b) dynamical downscaling using RCMs that are nested in the GCMs
, and (c) statistical downscaling including
post-processing with bias-correction methods . But even
though high-resolution GCMs and RCMs improve the representation of extreme
precipitation by better resolving mesoscale atmospheric processes, biases
remain and there is still a scale gap between the simulated gridbox values of
precipitation and point-scale data (i.e., rain gauges). Hence, statistical
bias-correction methods are also applied to such high-resolution simulations.
These so-called model output statistics (MOS) approaches employ a correction
function derived in present-day simulations to future simulations of the same
model .
Quantile mapping , one example MOS approach, is widely
applied to statistically post-process simulated precipitation. While this
might be a reasonable approach for correcting biases on the same spatial
scale, variability on local scales is not fully determined by grid-scale
variability, e.g., the exact location, size, or intensity of a thunderstorm.
This is part of the representativeness problem between gridbox and point
values . Quantile mapping is a deterministic approach that
cannot add random variability. It simply inflates the variance leading to an
overestimation of spatial extremes, and too smooth a variance in space and
also in time . Gridbox precipitation, e.g., is
the area average of sub-grid precipitation. The aggregation averages local
variations in time such that gridbox time series are smoother in time than
local time series. Quantile mapping can not overcome this mismatch in
temporal structure (apart from correcting the drizzle effect). This temporal
effect is more difficult to trace than the spatial effect .
Standard downscaling approaches in turn have a limited ability to correct
systematic biases. developed a model that jointly bias
corrects and downscales precipitation at daily scales. However, this approach
relies on pairwise correspondence between predictor and predictand for
calibration that is only provided by nudged GCM/RCM simulations, and is not
able to post-process standard, free-running GCM simulations .
Here we present a modification of the approach that is
designed to also work in principle for free-running GCM/RCMs, such as those
available from ENSEMBLES or CORDEX
e.g.,. With the aim of combining their respective
advantages we combine a statistical bias correction and a stochastic
downscaling method. Thereby we separate bias correction from downscaling by
inserting a gridded observational dataset as a reference between these two
steps. In particular, as a first step we apply a parametric quantile mapping
approach between an RCM and a gridded observational dataset. In a second step
we bridge the scale gap between gridded and point scale by employing a
stochastic regression-based model that is calibrated between gridded and
station observations and then applied to the bias-corrected precipitation
from the first step.
In Sect. the general concept is introduced; the data used
are described in Sect. . In Sect. we present
the bias correction and the stochastic downscaling model. Results of the
evaluation of our model for example stations across Europe are provided in
Sect. and, finally, Sect. contains the
conclusion.
Schematic of (black) our combined statistical bias correction and
stochastic downscaling model, and (grey) the
model.
General concept
We separate bias correction from downscaling into two steps to overcome the
shortcomings of each method and to combine their respective strengths. Our
concept is illustrated schematically in Fig. . In the first step, we
use the advantage of distribution-wise bias correction (i.e., the correction
function is calibrated on long-term distributions) to eliminate systematic
biases in the RCM. While this distribution-wise setting may correct
systematic RCM biases, it cannot bridge the gap between gridbox and point
scale for two reasons. First, a considerable portion of subgrid variability
is random for precipitation and has to be modeled as stochastic noise.
However, distribution-wise MOS methods are deterministic and do thus not add
unexplained random variability. Second, distribution-wise methods cannot
separate local variability into systematic explained variability and
small-scale unexplained variability. Moreover, when simulated short-term
variability is inflated to match local variability, long-term trends are also
inflated . Therefore, we only apply this distribution-wise
method to correct biases on the same spatial scale; i.e., as a reference we
use gridded observations on the same grid as the RCM. In the second step, we
employ a stochastic regression-based model to overcome the representativeness
problem. This regression model corrects systematic local effects (e.g.,
whether a rain gauge is positioned on the lee or windward side of a
mountain). It also adds random (unexplained) small-scale variability, in
contrast to approaches of combined methods that employ spatial interpolation
for downscaling or rescale the grid-scale
precipitation with a factor to match the observations . We
calibrate the probabilistic regression model between gridded and point-scale
observations and then apply it to the corrected grid-scale time series in the
validation period. This corresponds to a perfect prog (PP) setting for the
regression, while the bias is corrected in the first step. This combined
approach is an extension of the model by that jointly bias
corrects and downscales precipitation (see Fig. ). They employ a
probabilistic regression model that is calibrated between RCM and point-scale
observations (MOS approach). It requires nudged RCM simulations for
calibration since temporal correspondence is essential.
With this concept in place, basically in the first step any reasonable
distribution-wise MOS approach, and in the second step any adequate
stochastic model, can be employed. A strength of this concept is its
flexibility; i.e., the best suitable combination of statistical models for a
given location and season can be determined. In this study, we employ a
quantile mapping (QMgrid) approach based on the mixture
distribution of a gamma and generalized Pareto distribution
in the first step. The model used in the second step consists of a logistic
regression for wet day probabilities and a vector generalized linear model
predicting the parameters of a gamma probability distribution (VGLM gamma)
for precipitation intensities. Note that this combination of methods may not
be optimal in all studied locations. However, the aim of this study is to
introduce and evaluate the concept of this combined approach rather than to
find the optimal specific implementation for all studied locations.
To evaluate and illustrate our method, we adopt the perfect predictor
experimental setup of the VALUE framework . Employing the
same evaluation framework as VALUE allows for comparison of our method to all
models participating in the VALUE experiment. In this context, a
reanalysis-driven RCM is used which allows us to evaluate the
ability of the method to correct RCM biases, before evaluating GCM-driven
simulations where biases of both GCMs and RCMs need to be corrected. We note
that although this is a pairwise setup where simulated and observed weather
states are in principle synchronized (with the exception of the internal
variability generated within the RCM), we only use the simulated and observed
distributions for the bias correction. Thus, as explained above, the approach
can be transferred to any simulation setup, e.g., GCM-driven RCM simulations
or GCM simulations. For comparison we also applied the classical QM approach,
i.e., directly between RCM and point scale (QMpoint).
The method is evaluated by 5-fold cross-validation for the time
period 1979–2008; i.e., five 6-year long periods are predicted by the model
that was fitted to the remaining 24 years. Artificial predictive skill is
thus not present as the predicted period is not part of the training period.
The model is fitted and evaluated for each season separately; 86 stations
across Europe are studied (as selected for the VALUE experiment; see
Fig. ) representing different climates. In the evaluation of our
model we compare eight European subdomains (dashed lines in Fig. ):
the British Isles (BI), the Iberian Peninsula (IP), France (FR),
Mid-Europe (ME), Scandinavia (SC), the Alps (AL), the Mediterranean (MD), and
eastern Europe (EA). These domains have been defined within the PRUDENCE
project and are often used for RCM evaluation
e.g.,. Although climatic differences within these
subdomains remain, they summarize European climate zones and intercomparison
amongst them allows for study of large-scale gradients (e.g., from maritime
(west) to continental (east) or from cold (north) to mild (south) winters).
We slightly extended the PRUDENCE regions SC, AL, and MD such that all
studied rain gauges are included in the analysis.
Location and IDs of used rain gauges from ECA&D. IDs of red
marked stations from left to right: 244, 243, 4002, 58, and 13. Stations for
detailed analysis are marked in blue. Dashed lines represent European
subdomains for analysis as defined by the PRUDENCE project
: the British Isles (BI), the Iberian Peninsula (IP),
France (FR), Mid-Europe (ME), Scandinavia (SC), the Alps (AL, dashed red
line), the Mediterranean (MD), and eastern Europe (EA).
Data and gridbox selection
As prescribed by the perfect predictor experiment within the VALUE framework,
we use the RACMO2 RCM from the KNMI to test our
method for the time period from 1979 to 2008. The RCM has been driven with
ERA-Interim reanalysis within the EURO-CORDEX framework
. The simulation has been carried out at a horizontal
resolution of 0.44∘ (∼ 50 km) over a rotated grid. Note that
the resolution we employ (0.44∘) differs from the resolution used in
the VALUE experiment (0.11∘).
As the gridded observational dataset, E-OBS version 10 is
used, also at 0.44∘ resolution. The reason for choosing the
0.44∘ horizontal resolution for both RCM and E-OBS is that the actual
resolution of E-OBS might in some regions be lower than the nominal
0.22∘ due to sparse rain gauge density included in the
dataset
For station density of actual E-OBS versions, refer to the
ECA&D website: http://www.ecad.eu/dailydata/datadictionary.php.
.
Gridding very few rain gauges to a high resolution might in particular result
in overly smooth extremes
. Hence, too high a
resolution of a gridded dataset may be an unreliable reference for bias
correction, at least for summer extreme events. Moreover, this could cause
artificial smoothing of extremes by bias correction. In some regions where
station density is very sparse, this might even hold true for the chosen
resolution. Although E-OBS is probably not an appropriate reference in some
regions, it is the best available gridded dataset covering the whole
EURO-CORDEX domain.
The E-OBS reference gridbox for both steps (bias correction and downscaling)
is generally the closest gridbox to the respective station. If the closest
gridbox is an ocean gridbox (i.e., for coastal and island stations) and only
contains missing values, we select the gridbox with the highest correlation
in winter between daily precipitation at the given station and the five
closest E-OBS gridboxes. In winter the spatial decorrelation length of
precipitation is generally large, implying that several gridboxes are often
affected by the same weather system, and, thus, the gridbox with the most
similar climate can be reliably identified.
The RCM gridbox that is bias-corrected and downscaled is generally chosen as
the closest gridbox to the E-OBS reference gridbox – also for coastal and
island stations where the chosen RCM gridbox might thus differ from the
closest RCM gridbox to the final reference (i.e., rain gauge). For locations
in the rain shadows we choose the RCM gridbox which best represents the
climate at the given location to correct overly low precipitation values
caused by not enough windward air masses crossing the mountain range
“location bias”,. To this end, the highest
correlation between the winter seasonal mean of RCM and gridded observations
within 250 km around the closest gridbox to the observations is determined.
Note that when transferring this approach to free-running RCM simulations
this gridbox selection step needs to be carried out employing a
reanalysis-driven simulation of the same RCM to ensure temporal
correspondence.
For local-scale observations we used 86 stations across Europe from
ECA&D selected by the VALUE experimental framework
. The locations and ids of these stations are illustrated
in Fig. . A detailed analysis is carried out for some example
stations representing different climates (highlighted in blue in
Fig. ).
Statistical modelStep 1: bias correction
In our model we correct several biases. In a first step, the “location
bias” is corrected by gridbox selection (see Sect. for
details). In the second step, the “drizzle” effect is corrected by
increasing the wet day threshold for the RCM such that the number of wet days
(closely) matches the gridded observations, with a threshold of
0.1 mm day-1. Finally, we correct precipitation intensities of wet
days (i.e., exceeding the corrected wet day threshold) using a quantile
mapping (QM) approach which is described in the following. The correction
function y=f(x) between the simulated (x) and corrected (y)
values of daily precipitation intensities such that the corrected values
match the observations is based on the cumulative distribution
functions (cdfs) as cdfobs(f(x))= cdfRCM(x). To allow for extrapolation in a future climate to
unobserved precipitation intensities and to avoid deterioration of future
extremes that might occur with an approach that relies on empirical cdfs, we
chose a parametric QM approach.
To model precipitation intensities the gamma distribution is commonly used
. While the bulk of precipitation is generally well
represented, the tail of the gamma distribution is usually too light to
capture high and extreme rainfall intensities
e.g.,. Thus, an extreme value distribution,
such as the generalized Pareto (GP) distribution , might be
required to model the extremes of the precipitation distribution. To correct
the whole precipitation distribution, including extreme tails, we apply the
mixture distribution of which consists of a gamma
distribution for the precipitation mass and a GP distribution for the extreme
tail. This model is a variant of . The
distribution lϕ(x) of observed precipitation x on wet days is
modeled as
lϕ(x)=c(ϕ)1-wm,τ(x)fλ,γ(x)+wm,τ(x)gξ,σ(x),ϕ=(λ,γ,ξ,σ,m,τ),
where fλ,γ is the probability density function (pdf) of the
gamma distribution with the rate parameter λ and the shape parameter γ,
fλ,γ(x)=λγΓ(γ)xγ-1e-λx,λ,γ>0,
and gξ,σ is the pdf of the GP distribution:
gξ,σ(x)=1σ1+ξ(x-u)σ-1ξ-1forx≥u,
with the scale parameter σ> 0 and the shape parameter ξ which
determines the tail behavior of the GP distribution as follows:
ξ< 0: bounded tail; ξ→ 0: exponential
distribution (light tailed); and ξ> 0: infinite heavy tail. Here, we
constrain ξ≥ 0 to ensure that our model can be applied to a
future climate that may experience higher values than those observed during
the present-day training period for the model. The function wm,τ is a
weight function that determines the transition between the gamma and GP pdfs
as
wm,τ(x)=12+1πarctanx-mτ,m,τ>0,
with the location parameter m denoting the location of the center of this
transition and the transition rate τ influencing the rapidity of the
transition between the two distributions. To finally obtain the mixture pdf,
the mixture function (Eq. ) must be normalized, which is
carried out here by multiplying the mixture function by a
constant c(ϕ). In the mixture pdf (Eq. )
the threshold u in the GP distribution (Eq. ) is set to zero,
as the location parameter m of the weight function (Eq. )
fulfills the purpose of a threshold in Eq. (). Moreover,
setting the threshold to zero and applying a weight function instead also
solves the problem of threshold selection with unsupervised estimation and
avoids discontinuity in the mixture pdf lϕ(x) (Eq. )
. The parameters for lϕ(x) are estimated using
maximum likelihood estimation (MLE). For technical details on the
implementation of this model, please refer to
Appendix .
Since the mixture model is a complex model with six free parameters, a
thorough statistical model selection is necessary. We select between the
mixture model and the simpler gamma-only model separately for the
observed (Fobs) and RCM-simulated (FRCM)
distributions. For the selection, we apply the Akaike information criterion
AIC,, which asymptotically selects the model that
minimizes the mean squared error between prediction and observation
. The AIC is defined as -2log(L)+ 2k with the
likelihood L corresponding to the maximum likelihood estimate of the
k model parameters. The AIC is dominated by the most densely populated
region of the distribution. Hence, a good fit for the bulk of the
distribution (and thus a low AIC) might nevertheless come with large biases
in the extremes (see Appendix for an example). To
avoid a model choice with unreasonably high extremes, we therefore introduce
a criterion based on a comparison between the 100 season return levels
estimated by the mixture model (Eq. ) and by the GP
distribution (Eq. ) before the AIC-based model selection is
applied. For technical details on these model selection procedures, please
refer to Appendix .
To strictly avoid bias correction deteriorating the predictor and introducing
biases, both the complete cross-validated corrected time series and the raw
RCM output are compared to gridded observations as a reference using the
Cramér–von Mises (CvM) criterion. The CvM is a measure of the distance
between two empirical cdfs cdf bias hereafter; and has
been used to evaluate cdf-based correction models before
e.g.,. If cdfref(x) is the
empirical cdf of observations as a reference (i.e., the perfect bias
correction would match this reference) and cdfcorr(x) is the
empirical cdf of the bias-corrected time series, the CvM statistics is
defined as the integrated squared difference between cdfref and
cdfcorr as follows:
CvM=∫-∞∞|cdfcorr(x)-cdfref(x)|2dx.
Here, the CvM is computed for both the corrected daily precipitation time
series and the uncorrected RCM-simulated precipitation time series with E-OBS
as a reference. The predictor for downscaling is selected based on the lower
CvM. In other words, the bias-corrected time series is only used as a
predictor for the downscaling step if it improves the predictor compared to
the raw uncorrected RCM.
Step 2: stochastic downscaling
To bridge the scale gap we apply the regression model developed by
as follows. We determine the statistical relationship
between gridded and station observations. This statistical relationship is
then applied to coarse-scale precipitation as a predictor which is selected
in the first step, i.e., QMgrid-bias corrected or uncorrected
RCM-simulated precipitation. To be able to estimate the distribution of
precipitation as a function of a given predictor, a stationary distribution
is not sufficient. The family of generalized linear models (GLMs) extends
linear regression to such purposes e.g.,. In this
framework the time-dependent expectation of a random variable is linked via a
monotonic link function to a linear combination of predictors. The logistic
regression model belongs to the class of GLMs and is often used to model the
changing probability of rainfall occurrence . We model
the probability pi of a day i being wet (i.e., greater than the
threshold selected earlier at 0.1 mm day-1) as a function of
coarse-scale precipitation xi as
hpi=logpi1-pi=αxi+β,
where h(⋅) is the logit link function and the parameters α and β
are estimated by MLE. The logit link function gives the logarithm of the odds.
Subsequently, precipitation intensity on wet days is modeled using a vector
generalized linear model (VGLM) as a regression model . VGLMs are an extension of GLMs. While GLMs describe the conditional
mean of a wide range of distributions, VGLMs allow for prediction of a vector
of parameters from the same set of predictors, which is useful if one is also
interested in the variance or the extremes of a distribution.
implemented a mixture model version (see
Eq. ) and a gamma model version (see Eq. )
employing a VGLM. Here we apply the VGLM gamma version since the calibration
and model selection procedure for the VGLM mixture model is computationally
rather expensive. The simpler gamma model might be sufficient here as in the
downscaling step a predictor is employed that already explains a large
portion of the variance. The quality of downscaled precipitation does not
only depend on the chosen model, but also on the quality of the predictor.
Employing the mixture model for the bias-correction step is thus meaningful
to ensure a good representation of higher quantiles and extremes in the
predictor, although downscaling is performed with a simpler gamma model. The
scale θ (the inverse of λ in Eq. ) and the shape
γ parameters depend linearly on the predictor (coarse-scale
precipitation) xi. The model has the form
θi=θ0+ψθxi,γi=γ0+ψγxi,
where the regression parameters ψθ, and ψγ are
estimated by MLE.
Combining the probability of wet day occurrence and the gamma model
distribution defining the precipitation intensities, we get the probability
that observed precipitation on a given day (Ri) is less than or equal to
a particular precipitation intensity (r):
Prθ,γRi≤r=Γθ,γRi≤r|Wpi+1-pi,
where Γθ,γ(Ri≤r|W) is the gamma cdf and
pi is the probability of that given day being wet.
Mean bias. (a, b) Uncorrected RCM,
(c, d) QMpoint-corrected RCM to the point scale,
and (e, f) combined model selected predictor (RCM or QMgrid-corrected RCM)
and VGLM. Reference is station data.
Evaluation metrics
We evaluate our combined model based on the following metrics.
Mean bias: absolute difference between seasonal means as
(model - reference).
cdf bias: CvM criterion which represents the
mean squared error of a cdf compared to a reference cdf (for details see
Sect. ).
%sim >perc95obs: percentage of simulated wet days
exceeding the observed 95th percentile.
QQ plots: the quantiles (i.e., sorted time series) of modeled
precipitation are plotted against the quantiles of the reference. For the
evaluation of the second step (downscaling) standardized QQ plots are used
which are explained in Sect. .
Spatial autocorrelation: correlation of a variable with itself in
geographical space. The correlogram is estimated by centered Mantel
statistics using R package ncf . The correlation for a
set of distances at discrete distance classes is calculated. Significance is
assessed by 1000 random permutations. The correlogram is estimated for daily
values and then averaged. For the VGLM the correlogram is computed for
100 realizations of the stochastic model and then averaged. The correlogram
is centered on zero; i.e., zero represents similarity across the region.
Crossing the zero line implies thus that the pair of distances is not more
similar than what would be expected by chance alone across the region.
Results
We first evaluate the mean bias of our combined model (selected predictor and
VGLM) against station observations and compare it to the raw uncorrected RCM
and to classical QMpoint (between RCM and point scale). Then the
performance of the two steps (bias correction and downscaling) is assessed
individually and in combination. Finally, all analyzed models are compared. The evaluation is carried out for the
time period 1979–2008 by analyzing the cross-validated (5-fold) time series.
The first step (bias correction) is evaluated against the gridded E-OBS
dataset, although E-OBS might underrepresent the extremes in some regions
where station density is sparse. The second step (downscaling) and the
combined model (steps 1 and 2) are evaluated against station observations.
Step 1: bias correction to grid scale. (a, b) CvM score for
the selected cross-validated predictor against E-OBS. Threshold for values
under which the model cdf is not statistically significantly different at the
95 % level from the reference cdf: 0.461. (c, d) Percentage of wet
days in the CvM-selected cross-validated predictor exceeding the
95th percentile of wet days in E-OBS (%sim > perc95obs).
Selected model: circles: QMgrid-corrected RCM; triangles:
uncorrected RCM.
Evaluation of mean precipitation bias
Figure shows the mean bias of precipitation (against station
observations) as modeled by (a, b) the RCM and (c, d) the classical
QMpoint approach applied directly between RCM and station
observations and (e, f) our combined model. The RCM has a stronger bias in
DJF than in JJA. In DJF it is rather too wet, whereas in JJA many locations
have a dry bias. In both seasons the bias is improved by QMpoint,
with a slight remaining wet bias. Our combined model also improves the mean
bias of the RCM in JJA. However, in DJF wet biases remain and got even worse
in some locations. This raises the question why the results become worse when
statistical post-processing is applied. However, the bias of the seasonal
mean does not give information on how the precipitation distribution is
represented or the predictive power of the model. These issues are evaluated
in the following.
Evaluation of the combined model
First, both steps of the combined model are evaluated individually. Second,
the combination of both steps is evaluated. In this combined model the
predictor selected in the first step is used for the regression model in the
second step.
Evaluation of Step 1: bias correction vs. E-OBS
Figure shows the cross-validated selected predictor (uncorrected
RCM: triangles; QMgrid-corrected RCM: circles) that is used in
the second step for downscaling. For predictor selection we apply the
Cramér–von Mises score (CvM, Eq. , Sect. )
which represents the mean squared error of a cdf compared to a reference cdf
(cdf bias hereafter). The predictor is selected based on the lowest CvM score
of the cross-validated QMgrid-corrected time series and the raw
uncorrected RCM with gridded observations as a reference. Generally our bias
correction often improves precipitation. It is selected 73 times in
December–February (DJF) and 49 times in June–August (JJA) out of 86 rain
gauges.
Step 1: bias correction to grid scale. Boxplots of
(a, b) CvM score and (c, d) percentage of simulated wet
days exceeding the observed 95th percentile
(%sim > perc95obs) for the CvM-selected cross-validated
predictor in European subdomains: British Isles (BI), Iberian Peninsula (IP),
France (FR), Mid-Europe (ME), Scandinavia (SC), Alps (AL),
Mediterranean (MD), and eastern Europe (EA). Outlier out of range
in (b) AL and all: 12.95.
The CvM values of the selected predictor (Fig. a and b) indicate that
the cdf bias is generally lower in JJA than in DJF. In DJF the cdf bias is
lowest in the Mediterranean region, with a mild winter climate. However, the
CvM criterion is quite sensitive to small deviations between the cdfs. The
highest selected CvM values are found for Graz (Austria) in JJA, and Leba
(northern Poland), Siedlce (eastern Poland), and Dresden (eastern Germany) in
DJF. QQ plots for these high CvM values (see Appendix ) suggest
that the corrected time series are still usable and show improvements
compared to the raw RCM, although they are of course not a perfect match of
the observations. These remaining inaccuracies of the QMgrid
approach can be related to both a time-varying correction function and the
parametric correction function. Figure summarizes Fig. over
the European subdomains by boxplots. Spatial variability throughout the
subdomain is quantified by CvM variability represented by the box. In DJF the
boxplots confirm the lowest cdf bias in the Mediterranean region (MD and IP)
that is already visible on the map (Fig. a). The highest median is in
ME. However, although the median is slightly lower than in ME, spatial
variability is largest in EA, extending to the highest CvM values. This
indicates that there are problems with continental winter climate which
persist after bias correction as in ME and EA mostly the bias-corrected model
is selected (Fig. a). QQ plots of the two worst examples in EA (Leba
and Siedlce; Appendix ) show that the complete precipitation time
series remains too wet, whereas in the worst example of ME (Dresden;
Appendix ) the bias correction performs well for most values and
only fails in the highest quantile. In JJA the CvM score, and hence the cdf
bias, is very low, and no pronounced differences between the subdomains can
be identified (Fig. a).
The representation of heavy precipitation by the selected predictor is
evaluated by the percentage of simulated values that are higher than the 95th
percentile of the observations on wet days
(%sim > perc95obs, Figs. c, d and c, d).
Thus, in a “perfect” model this would be exactly 5 % (yellow). In many
locations there are slightly too many “extremes”; i.e., the occurrence of
heavy precipitation (> perc95obs) is overestimated,
particularly in DJF. Consistent with the CvM score, the overestimation in
heavy-precipitation occurrence increases in DJF from west to east
(FR → ME → EA) and is again highest in EA,
followed by ME and SC (Fig. c). In JJA the occurrence of heavy
precipitation is quite well represented in AL and BI (Fig. d); it is,
however, underestimated in some locations (Fig. d). In the other
subregions the occurrence of heavy precipitation is also slightly
overestimated in JJA (Fig. d).
Evaluation of Step 2: downscaling vs. station
Here we present some examples to illustrate the performance of the VGLM gamma
for different climates, calibrated between gridded (E-OBS) and point-scale
(station) observations. All results that are shown for the evaluation of the
downscaling step (step 2, Figs. – and Appendix )
are calibrated over the complete time period and then predicted by E-OBS as a
predictor for the same time period. As we do not use the cross-validated time
series here, the best possible relationship is presented. This allows us to
evaluate the goodness-of-fit and is a necessary step before evaluating the
model in a cross-validation setup. For a detailed evaluation of the VGLM
gamma for the relationship between nudged RCM/GCM simulations and station
observations over the British Isles, refer to and
.
Step 2: downscaling. QQ plots for example stations in DJF. VGLM
gamma standardized to the stationary gamma distribution fitted to observed
wet day intensities between gridded and point-scale precipitation
observations (mm day-1). (a) Karasjok, (b) Stornoway,
(c) Brocken, (d) Dresden, (e) Sibiu,
(f) Sonnblick, (g) Sion, (h) San Sebastian, and
(i) Malaga.
Step 2: downscaling. Estimated relation between gridded and
point-scale precipitation observations for example stations in DJF. VGLM
gamma where both parameters depend on the predictor fitted to observed wet
day intensities. The predictor is E-OBS. Circles: observed precipitation
intensities (mm day-1); lines: 0.1, 0.25, 0.5, 0.75, 0.9, and 0.95
modeled quantiles (mm day-1). (a) Karasjok,
(b) Stornoway, (c) Brocken, (d) Dresden,
(e) Sibiu, (f) Sonnblick, (g) Sion,
(h) San Sebastian, and (i) Malaga.
To evaluate the goodness-of-fit, we use residual QQ plots (Fig. for
DJF and Appendix for JJA). As a QQ plot requires quantiles of an
unconditional distribution, we standardized the day-to-day varying
distribution to a stationary gamma distribution
Standardization is
performed as (1) compute probabilities for reference values (here: station
observations) from an estimated non-stationary gamma distribution (i.e.,
gamma parameters depend on the predictor and, thus, vary from day to day);
(2) compute quantiles of a gamma distribution with stationary parameters for
these probabilities of a non-stationary distribution; (3) plot these
quantiles against quantiles of stationary gamma distribution for theoretical
probabilities: (1 : n)/(n+ 1).
. This
stationary distribution no longer has the predictor-dependent day-to-day
variations; i.e., the effect of the predictor is approximately removed. Due
to this procedure the goodness-of-fit of the regression model can be
evaluated separately, instead of evaluating only the combined effect of
predictor and regression model which is present in the time-varying gamma
parameters, and, thus, also in realizations drawn from these varying
distributions. Therefore, deficiencies that are indicated by these
standardized QQ plots are either due to inappropriate model structure or
ill-fitting parameters. Note that the values of model and observation are
shifted due to the standardization, depending on the strength of the
predictor.
Improvements by the VGLM gamma compared to the predictor can be seen in most
examples ranging from Scandinavia to the Mediterranean and from the Atlantic
coast to eastern Europe in both seasons. However, in some locations the
quantiles modeled by the VGLM gamma compare well to station observations (at
least in Malaga, better than the predictor) up to a certain quantile (e.g.,
Sibiu, ∼ 12 mm day-1, and Malaga, ∼ 42 mm day-1, in
DJF), while there is a wet bias for intensities of the higher quantiles. It
has been verified that precipitation at these locations is gamma-distributed
(not shown). To understand this model behavior we analyze the
predictor–predictand relationship of both observations and VGLM in
Fig. for DJF and Appendix for JJA. Circles are the
observed gridded against point-scale precipitation intensities, showing the
spread of point-scale predictands for a given grid-scale predictor. The lines
represent the 10, 25, 50 (median), 75, 90, and 95 % quantiles of the VGLM
gamma model as a function of the predictor. This function of course fits best
in the range where most of the values used to estimate the relationship are.
For instance, in Sibiu (Malaga) for higher predictor values (Sibiu:
> 15 mm day-1; Malaga: > 42 mm day-1) the predictands are
around or below the 25 % (50 %) quantile of the model, and, thus,
simulated systematically too high by the VGLM. In both cases the bulk of the
distribution is well captured however. This problem is also visible at other
stations, e.g., Dresden or Karasjok. In JJA it is even more pronounced
(Appendix ), particularly in Dresden and Sibiu, where the high
predictor values are even below the modeled 10 % quantile. These examples
indicate that the VGLM basically allows for three different generalized
linear relationships between the predictor and the parameters of the gamma
distribution: concave (i.e., Brocken DJF), straight (i.e., San Sebastian
DJF), or convex (i.e., Malaga DJF). No changes from lower to higher quantiles
between these three types are possible. In some locations this appears to be
not flexible enough to capture the true relationship, which can be nonlinear.
A more flexible relationship that allows for a changed model behavior for
higher values could improve the results but comes with the risk of
overfitting. Additionally, in eastern Europe the station density included in
E-OBS is low
For station density of actual E-OBS versions refer to
the ECA&D website:
http://www.ecad.eu/dailydata/datadictionary.php.
. Hence, in the E-OBS
gridbox closest to Sibiu, there may be only very few (one or two) stations
included, implying most likely a misrepresentation of gridbox precipitation.
This problem affects the calibration of the model where E-OBS is used as a
reference as well as simulations employing E-OBS or precipitation that is
corrected to E-OBS as a predictor. We do not show results of the
cross-validation here as the described problems with the VGLM in some
locations are already present when repredicting the calibration period where
the skill should be higher than in a cross-validation setup where a period is
predicted that is not part of the calibration period. This clearly highlights
deficiencies in the model for these locations.
In both DJF (Fig. ) and JJA (Appendix ) Sonnblick and
Brocken show a concave function, whereas the function in the other example
stations is generally convex. The rain gauges at Sonnblick and Brocken are on
top of the respective mountain. Although their climate is quite different as
Sonnblick is a high mountain in the Alps (altitude: 3106 m), whereas the
Brocken is the highest mountain in the northern German Harz low mountain
range (altitude: 1142 m), they have an exposed position, along with high
variability, in common. These results show that the VGLM gamma is capable of
modeling the scale relationship for such exposed places of high variability
quite well.
Evaluation of the combination of steps 1 and 2: bias correction and downscaling vs. station
In the combined model the VGLM gamma, calibrated against E-OBS, is applied to
the predictor selected in Sect. (Fig. ). Here we
evaluate precipitation simulated by the combined model (predictor and VGLM) with station
observations as a reference, and compare it to the uncorrected RCM-simulated
precipitation and to the QMgrid-corrected precipitation. The
cross-validated time series are evaluated. For the VGLM the evaluation
criteria were computed for 100 realizations and then averaged.
Steps 1 and 2: combined model. (a, b) CvM values for the
selected cross-validated model. The threshold for values under which the
model cdf is not statistically significantly different at the 95 % level
from reference cdf: 0.461. (c, d) Percentage of cross-validated
model values exceeding the 95th percentile of station observations
(%sim > perc95obs) for the cross-validated CvM-selected
model. For the VGLM the criteria were computed for 100 realizations and then
averaged. Selected model: squares: combined model (predictor and VGLM); circles:
QMgrid-corrected RCM; triangles: uncorrected RCM. Note the
different color scales than in Fig. .
To evaluate the predictor and VGLM combined model, we apply the same criteria
as for the first step (bias correction, Sect. ), but with station
observations (i.e., point scale) as a reference. The CvM scores (a, b) and
the percentage of simulated values that are higher than the 95th percentile
of the observations on wet days (%sim > perc95obs, c, d)
for the selected best model based on the CvM criterion are shown in
Fig. and summarized by boxplots for the European subdomains in
Fig. . QQ plots for example stations are provided in Fig.
for DJF and in Appendix for JJA. Precipitation is improved in
most cases by (parts of) our method. The uncorrected RCM (Fig. ,
triangles) is only selected at eight (seven) stations in DJF (JJA). However,
even if the RCM is selected, the other models do not necessarily perform much
worse, such as in Stornoway in DJF (Fig. ) or in Malaga in JJA
(Appendix ). The combined predictor and VGLM model (plotted as squares)
is selected by CvM 25 times (45 times) in DJF (JJA). The more frequent
selection of the VGLM in JJA compared to DJF is likely related to the
dominant underlying mechanism; i.e., in summer there are many small-scale
convective precipitation events, whereas in winter precipitation is mainly
caused by large-scale weather systems.
Steps 1 and 2: combined model. Boxplots of (a, b) CvM score
and (c, d) percentage of simulated wet days exceeding the observed
95th percentile (%sim > perc95obs) for the cross-validated
CvM-selected model. Regions: British Isles (BI), Iberian Peninsula (IP),
France (FR), Mid-Europe (ME), Scandinavia (SC), Alps (AL),
Mediterranean (MD), and eastern Europe (EA). Note the different scales of the
y axes than in Fig. . Outliers out of range in (a) ME and
all: 22.19; SC and all: 27.12 and 31.35.
The CvM values of the selected model (Fig. a and b) indicate that the
cdf bias is again generally lower in JJA than in DJF, and for DJF lowest in
the Mediterranean region. In eastern Europe and Scandinavia in DJF the VGLM
is only rarely selected – in these regions the QMgrid-corrected
time series which is on grid scale is mostly selected, although the reference
cdf is on point scale (Fig. a). This might be due to problems with
the VGLM gamma as explained in Sect. . The rather large cdf bias
in ME, SC, and EA in DJF could hence be related to the remaining scale gap as
the QMgrid-corrected time series is not expected to correctly
represent the point scale. The QQ plot of Sibiu in DJF (Fig. )
illustrates this problem. The higher QMgrid-corrected quantiles
are as expected too low and the VGLM fails at this station in DJF (see also
Sect. ). Finding an adequate stochastic model to bridge the scale
gap might improve the representation of precipitation in such cases. Also in
JJA there are examples where the VGLM has not been selected, but a suitable
VGLM would likely further improve the results (Appendix , San
Sebastian, Dresden, and Karasjok). For, e.g., Brocken JJA and Sion DJF an
improved VGLM may likely even improve the result although the VGLM has been
selected. However, finding the optimal model for all 86 stations is beyond
the scope of our study. The boxplots confirm again the good performance for
DJF in the Mediterranean region (MD and IP), and also in AL (Fig. a).
The CvM score and, thus, the cdf bias, are again very low in JJA, indicating
good performance of our method with no pronounced difference between the
European subregions (Fig. b). However, the sensitivity of the CvM
score is illustrated by Stornoway in JJA (Appendix ), as this
example still yields suitable results despite the relatively high CvM score.
The occurrence of heavy precipitation in the CvM-selected model is slightly
overestimated in most subregions in DJF (Figs. c and c),
though quite well represented in IP and FR (Fig. c). In JJA heavy
precipitation occurrence is quite well estimated (Figs. d
and d). The median of most subregions is very close to 5 % (the
“perfect” model would have exactly 5 %). However, some stations,
particularly in EA, underestimate the occurrence of heavy precipitation.
These are in most cases stations where the VGLM has not been selected, likely
indicating problems with the VGLM and the remaining scale gap (see the
section before and Sect. ).
Ideal performance of our combined model is illustrated in the example QQ plot
of Malaga in DJF (Fig. ); i.e., QMgrid corrects the
RCM-simulated precipitation on the same scale and the VGLM bridges the
remaining scale gap, resulting in a good match of the observations. Sonnblick
in DJF (Fig. ) and JJA (Appendix ) and Brocken in DJF
(Fig. ) are also well-performing examples. The QQ plot of San
Sebastian in DJF (Fig. ) shows the benefit of selecting the
predictor by CvM as in this case the RCM is used as a predictor for the VGLM.
Here using the QMgrid-corrected time series may result in overly
high extremes. Sion in JJA (Appendix ) is another good example
of the benefit of model selection where the RCM has been selected as a
predictor. Here the high VGLM-simulated quantiles are already overestimated
in this setting and would likely be even higher should the
QMgrid-corrected predictor be employed.
QQ plots for example stations of different models (cross-validated)
against station observations for DJF (mm day-1). (a) Karasjok,
(b) Stornoway, (c) Brocken, (d) Dresden,
(e) Sibiu, (f) Sonnblick, (g) Sion,
(h) San Sebastian, and (i) Malaga. For the VGLM the
quantiles (i.e., sorted time series) of 100 realizations are averaged.
Predictor for VGLM as selected by the CvM criterion: (red circles)
QMgrid bias-corrected RCM; (brown triangles) uncorrected RCM. For
examples to illustrate model performance and predictor selection (San
Sebastian and Malaga), the VGLM is plotted for both predictors. Selected
predictor: San Sebastian: RCM; Malaga: QMgrid.
Intercomparison of all cross-validated models (not only the selected
best model). Models: uncorrected RCM, QMgrid-corrected RCM to
grid scale, QMpoint-corrected RCM to point scale, and predictor
and VGLM-downscaled RCM. Boxplots of the CvM score for all models in
different subregions: (a) British Isles (BI), (b) Iberian
Peninsula (IP), (c) France (FR), (d) Mid-Europe (ME),
(e) Scandinavia (SC), (f) Alps (AL),
(g) Mediterranean (MD), (h) eastern Europe (EA), and
(i) all locations. For the VGLM the CvM score was computed for
100 realizations and then averaged. Outlier out of range in (d, i)
RCM DJF: 54.19.
Intercomparison of all models
In this section an intercomparison of all models (not only the selected best
model from Sect. ) for all subregions is presented and compared
to the classical application of QMpoint. Figure shows
boxplots for the CvM score. Generally the cdf bias is lower in JJA than in
DJF for all models, already for the uncorrected RCM (apart from BI). In the
Mediterranean region (MD and IP) there is a very low cdf bias in all models,
indicating general good performance. The QM improves the cdf bias in many
regions, with QMgrid and QMpoint being similar in
many cases. The effect of the VGLM depends on region and season. The
representation of precipitation is generally improved by the VGLM in BI, IP,
AL, and MD in both seasons. However, in FR, ME, SC, and EA in DJF the VGLM
introduces biases. The bias increases from west to east
(FR → ME → EA) with the largest spatial
variability in EA, extending to high CvM values. For continental winter
climate the used VGLM gamma model thus appears not to be the ideal model,
which suggests that in these regions it may be better to only correct the
bias. This raises the question why the results become worse when statistical
post-processing is applied. One potential reason for these problems with the
VGLM in some regions is that the VGLM gamma is not flexible enough to capture
the true predictor–predictand relationship if this relationship is nonlinear
as discussed in Sects. and . The final downscaled
marginal distribution may thus be wrong even though it was properly adjusted
by the bias-correction step. As the predictor–predictand relationship is
always estimated such that it follows well the bulk of the distribution, this
problem occurs for predictand values at the very low ends of the VGLM
conditional distribution. Furthermore, particularly in EA and FR, E-OBS may
be an inappropriate reference for calibration in both QMgrid and
VGLM due to low station density. However, in SC stations in E-OBS are
relatively dense and, thus, the bias introduced by the VGLM is in that case
not attributable to E-OBS quality. In DJF SC has the highest RCM bias among
all subregions. This suggests a detailed evaluation of this high bias which
is beyond the scope of our study however.
To infer the performance of all studied models in estimating the occurrence
of heavy precipitation, boxplots for the percentage of simulated values that
are higher than the 95th percentile of the observations on wet days
(%sim > perc95obs) for all models are provided in
Fig. . Particularly in JJA the QMgrid improves the
occurrence of heavy precipitation but remains slightly too dry, which is
expected due to the remaining scale gap. The estimated occurrence of heavy
precipitation is improved by the VGLM in many cases, although generally
slightly overestimated. The results of the VGLM and QMpoint are
generally similar, with the QMpoint often being slightly closer
to the 5 % line and the VGLM slightly too wet. In AL the VGLM considerably
improves the cdf bias (Fig. f) and the occurrence of heavy
precipitation (Fig. f) in both DJF and JJA compared to the
uncorrected and QMgrid-corrected RCM. In SC in DJF one should be
careful, as although the occurrence of heavy precipitation is considerably
improved by the VGLM (Fig. e), it introduces biases when the whole
cdf is evaluated (Fig. e), and is thus not recommended. Concerning
heavy precipitation occurrence our model shows a similar behavior for all
subregions in JJA and for IP also in DJF (Fig. ). The
QMgrid bias correction improves the representation but remains
too dry. The dry bias is then eliminated by the VGLM though to slightly too
many “extremes”. This model behavior as exhibited in JJA is exactly what
would be expected due to the scale gap between gridded and point scales. Due
to more small-scale convective extremes this scale gap has a larger impact in
summer, whereas in winter most extremes are caused by large-scale weather
systems that are generally better represented by the gridbox scale, also at
coarser resolutions. While the cdf bias and the occurrence of heavy
precipitation reveal how well properties of the precipitation distribution
are represented, they do not allow us to draw conclusions about the
predictive power of the model.
As in Fig. , but for percentage of simulated wet days
exceeding the observed 95th percentile (%sim > perc95obs).
Outlier out of range in (g, i) QMgrid DJF:
41.07 %.
To infer whether our model has predictive power, we cannot assess temporal
correspondence compared to observations as in and
because we use an RCM that is not nudged, and even though
driven with perfect boundary conditions (reanalysis) this is not a clean
pairwise setup. Instead, we evaluate spatial autocorrelation, which is the
correlation of a variable with itself in geographical space. This allows us
to evaluate whether the model correctly reproduces daily spatial
autocorrelations and, thus, the spatial extent of precipitation patterns,
including its variability in time compared to observed precipitation. In
Fig. correlograms of the cross-validated time series of all models
(RCM, QMgrid, QMpoint, 100 VGLM realizations) and
station observations as a reference are provided. The spatial autocorrelation
of QM-bias-corrected precipitation decays very similarly to uncorrected RCM
precipitation and thus shows only little improvement of spatial
autocorrelation compared to point-scale observations. Differences between
QMgrid and QMpoint are negligible. This confirms that
the QM approach is not capable of modeling small-scale variability, and a
stochastic model is thus needed to bridge the scale gap. The spatial
autocorrelation of VGLM-downscaled precipitation decays more similarly to the
station observations than the QM corrected or uncorrected RCM, particularly
in JJA. The spatial dependence is thus improved by the stochastic downscaling
step. The long decorrelation length in DJF is underestimated by our
stochastic, single-site model, which indicates a slightly too strong noise
component. A spatial model considering more than one station or including
more physically based predictors (i.e., sea level pressure) might improve the
predictive power of our model in DJF.
Spatial autocorrelation (cross-validated). Correlogram (circles) and
smoothed spline fitted to the correlogram (lines) for (a) DJF and
(b) JJA. The correlogram is estimated by the centered Mantel
statistic using R package ncf . For the VGLM
100 realizations of the stochastic model for each station were used to
estimate the correlogram.
Conclusions
We introduced the concept of a combined statistical bias correction and
stochastic downscaling method for precipitation. We thereby extend the
stochastic model output statistics (MOS) approach developed by
beyond nudged simulations to free-running GCM/RCM
simulations. We applied our method to precipitation simulated by the RCM
KNMI-RACMO2 driven with ERA-Interim boundary conditions within the
EURO-CORDEX framework. As the RCM is driven with reanalysis we only correct
RCM biases. Our method corrects the “drizzle effect” (i.e., too many wet
days), overly low precipitation values in the rain shadows caused by not
enough windward air masses crossing the mountain range “location
bias”,, and precipitation intensity. To correct the
“drizzle effect” we increased the wet day threshold such that the number of
wet days (closely) matches the gridded observations with a threshold of
0.1 mm day-1. To overcome the “location bias” we
selected the RCM gridbox that best represents the climate in the respective
gridbox of the gridded observations . Note that when
transferring the approach to free-running simulations this gridbox selection
step has to be calibrated with a reanalysis-driven simulation of the RCM to
ensure temporal correspondence. Consequently, only the location bias caused
by the RCM is corrected. How a potential location bias of the driving GCM may
affect the results should be analyzed in future work. Precipitation
intensities were corrected by a parametric quantile mapping (QM) approach
between RCM and gridded observations on the same spatial scale. As
precipitation is highly variable in space and time, not all variability can
be explained by the gridbox scale . To bridge the gap
between gridbox and point scale we applied a stochastic regression-based
model. For evaluation we adopted the experimental framework of VALUE
. In this context, we applied our method to 86 example rain
gauges across Europe representing different climates, and carried out a
5-fold cross-validation for the time period 1979–2008. Both steps of the
combined method were evaluated individually and combined. A comparison to
classical QM between RCM and point scale is also provided.
The proposed parametric model structure appears not to be the optimal choice
for all considered stations. Yet given that the aim of our study is a proof
of concept, the identification of an optimal model for all individual cases
would be beyond the scope of this work. Nevertheless, where our
implementation is not adequate we provide suggestions for improvements within
the presented framework. Our specific implementation for the QM bias
correction (first step) of wet day intensities employs the mixture
distribution of a gamma distribution for the precipitation mass and a
generalized Pareto (GP) distribution for the extreme tail
. The stochastic regression-based model for
downscaling (second step) was calibrated between observations on gridded and
point scales, and then transferred to bias-corrected RCM-simulated
precipitation. This corresponds to a perfect prog (PP) approach. The
regression model consists of a logistic regression to model wet day
probabilities and a vector generalized linear model (VGLM) predicting the
parameters of a gamma probability distribution for precipitation intensities.
The QM-corrected time series (first step) was used as a predictor for
downscaling (second step) if it improves the representation of precipitation
compared to the uncorrected RCM. Thus, we selected the predictor based on the
lower cdf bias by applying the CvM criterion with the gridded E-OBS dataset
as a reference.
Precipitation was in most cases improved by (parts of) our combined method
across different European climates; to what extent depends on region and
season though. The method generally performs better in JJA than in DJF and in
DJF best in the Mediterranean region, with a mild winter climate, and worst
for the continental winter climate in Mid- and eastern Europe or
Scandinavia. Seasonal and regional differences depending on the underlying
mechanism have already been reported for resolution dependence of extreme
precipitation in GCMs and RCMs
. Hence, for a good representation of
precipitation extremes, the complexity of the model can be chosen at each
step of the modeling cascade based on the underlying mechanism in order to
use computational resources efficiently.
Although our bias correction (first step) improved simulated precipitation
for many locations in both seasons, wet biases may remain even after bias
correction, particularly for continental winter. In agreement with our
results, large improvements by bias correction over the Alps, Spain, and
France have been reported by . However, in contrast to our
results these authors also obtain good results for Mid- and eastern Europe,
where we find persisting biases even after bias correction. In the cases
where the quantile mapping approach does not improve RCM-simulated
precipitation, another transfer function might be more suitable. Choosing
between different parametric transfer functions as proposed by
could improve the results. By employing a quantile mapping
approach we presumed both a stationary statistical relationship and
stationary cdfs that also apply in a changed future climate. However, in a
climate change context RCM-simulated trends in the cdf are modified by
applying such statistical post-processing. For cases where the GCM/RCM
simulates plausible climate change trends the CDF-t concept suggested by
and might be an appropriate
framework. In their concept the correction function explicitly accounts for
future trends in the RCM-simulated distribution. Thereby simulated trends in
all moments are approximately preserved after bias correction. For instance,
regions where an increase in extreme precipitation accompanied by a decrease
in mean precipitation is projected e.g., in central European
summer,, these trends might be better
represented by employing a CDF-t method. However, in this study we have not
employed this variant as in our setting the validation period is too short to
achieve an appropriate fit of the future mixture distribution. Quantifying
the differences between the quantile mapping approach we employed here and a
CDF-t approach is left for future work when our combined method will be
applied to climate change scenarios.
The stochastic downscaling (second step) improves the estimated occurrence of
heavy precipitation in many regions, but introduces biases in continental
winter climate. Furthermore, spatial autocorrelation in JJA is improved by
the VGLM, showing the importance of randomization in the framework of
downscaling as already pointed out by, e.g., and
. Moreover, when downscaling climate change scenarios the
randomization component of the VGLM that adds small-scale unexplained
variability does not modify trends, in contrast to purely deterministic
methods, e.g., QM . However, the deterministic part of the
VGLM that corrects systematic local effects (e.g., lee/windward side of a
mountain) alters the pdf, and may thus also change trends. The stochastic
downscaling step is more important in JJA than in DJF for both estimation of
heavy precipitation occurrence and spatial autocorrelation. This can be
attributed to the different underlying main mechanism for heavy
precipitation. In summer heavy precipitation is often caused by small-scale
convective events, whereas in winter large-scale weather systems dominate.
Hence, there is less small-scale variability unexplained by the gridbox in
DJF. In DJF spatial autocorrelation is slightly underestimated by the VGLM,
which is likely related to the long decorrelation length of precipitation in
winter that is not correctly represented in our single-site model, indicating
a slightly too strong noise component. An extension of our method to a
multi-site model and/or including more physically based predictors (i.e., sea
level pressure) would likely improve this feature and can be the subject of
future work. A possible extension to multi-variate or full fields might be
based on copulas e.g., or
random cascade models . A good representation of the mild
climate in the British Isles is consistent with and
. In France, Mid-Europe, eastern Europe, and Scandinavia in
DJF the VGLM introduces biases, raising the question why the results become
worse when statistical post-processing is applied. Particularly in France and
eastern Europe the E-OBS gridded observational dataset may be an unreliable
reference for model calibration for both the QM and the VGLM due to low
station density. The “true” resolution of E-OBS in these regions might be
coarser than the resolution it is gridded to. This highlights the fact that
the applicability of our method is limited to regions where high-quality
gridded datasets are available. However, a detailed evaluation of the
sensitivity of our method to station density in the gridded dataset is beyond
the scope of this study. The bias introduced by the VGLM generally increases
from west to east, and, thus, from maritime to continental winter climate.
However, in Scandinavia the VGLM also introduces biases even though station
density is high. This indicates that although the quality of the E-OBS data
may contribute to these problems, it can not be identified as the main source
of error. It is rather one potential reason among others. For instance, in
some cases the generalized linear relationship between the predictor and the
parameters of the gamma distribution appears to be not flexible enough to
capture the true predictor–predictand relationship, which can be nonlinear,
particularly in but not restricted to continental winter climate. In these
regions there may be a more adequate parametric relationship than our
specific implementation. Problems with the current implementation may be
related to, e.g., the linear structure of the model or the choice of the link
function. For instance, another distribution in the VGLM (e.g., mixture
model), splines as applied in or a vector generalized
additive model VGAM,, are potential approaches. However,
employing a more complex model also comes with the risk of overfitting.
Finding the optimal model for each of the analyzed stations is beyond the
scope of this study however.
The varying performance of our specific implementation clearly shows that
bias correction and downscaling methods should be reevaluated when
transferring them to locations with different climatic conditions. In some
regions a specific implementation different from the one we used is required.
We recommend our model in summer for all studied regions. However, in winter
it should only be used for the British Isles, the Alps, the Mediterranean
region, and the Iberian Peninsula, but not for continental winter climates
(Scandinavia, Mid-Europe, and eastern Europe) and France. While the
stochastic downscaling step (VGLM) is very important for representing spatial
autocorrelation in summer, it is less important in winter, where the
application of solely the bias-correction step might be sufficient. The
concept can generally be extended to a wide range of method combinations.
Transferring this concept to other climate variables should in principle be
possible. Our specific implementation should be applicable to any
gamma-distributed variable. However, our approach has so far only been
evaluated for precipitation. Thus, users need to evaluate the model for the
particular variable at the chosen location when transferring it.
We developed our model in the present-day climate. In a climate change
context the model does not explicitly modify climate trends on a physical
basis. Our model is thus only applicable where changes are correctly
simulated by the GCM/RCM. For instance, changes in the dynamics of local
extreme convective events in summer that need even higher resolution up to
convection-permitting simulations
e.g., will also not be
represented after statistical post-processing is applied. Bias correction and
(dynamical and statistical) downscaling of precipitation is only applicable
if the large-scale patterns and changes therein are simulated reasonably by
the driving GCM . Therefore, when transferring our
method to a GCM or GCM-driven RCM the relevant processes for precipitation in
the studied region need to be correctly simulated. For instance, biases in
simulated precipitation related to biases in the storm track
, El Niño–Southern Oscillation
ENSO;, the monsoon , or persistent
weather regimes cannot be statistically
corrected in a physically sensible way.
The general concept of combining two methods and thereby separating bias
correction (MOS) and downscaling (PP) into two steps is a powerful approach
as it benefits from the respective methodological advantages. Additionally,
the strength of this two-step method is that the best combination of methods
can be selected. This implies that the concept can be extended to a wide
range of method combinations.
The RCM output from the KNMI that was used in this study is
available within the CORDEX framework from the Earth system grid federation
(e.g., https://esgf-data.dkrz.de/projects/esgf-dkrz/) upon registration.
The E-OBS gridded dataset is available at the ECA&D website (http://www.ecad.eu/download/ensembles/download.php).
The ECA&D station data are provided by the KNMI on the ECA&D website
(http://www.ecad.eu/dailydata/customquery.php). The specific selection
for the VALUE experiment can be downloaded from the VALUE website
(http://www.value-cost.eu/data). Bias-corrected and downscaled data as
well as the source code from this study are available from the authors upon request.
Technical details for bias-correction implementation and model selectionTechnical details for model implementation
A non-zero wet day threshold assigns zero probability density to all
intensities between zero and the threshold, resulting in a misfit of the
gamma distribution . To avoid this we shift precipitation on
all wet days by subtracting the threshold for calibration. The estimated
distribution is subsequently shifted back by the threshold.
Numerical instabilities in the estimation of the mixture cdf may in rare
cases result in a discontinuous cdf (Fig. a). In these cases we
interpolate linearly between the continuous probabilities surrounding the
discontinuity. The example cdf in Fig. a illustrates that this
procedure is a reasonable estimation for these quantiles. If the cdf does not
“jump back” as in Fig. a but continues as illustrated in
Fig. b, the model has to be sorted out as there is no
straightforward possibility of handling this artifact caused by numerical
instability. However, the latter case only occurs extremely rarely.
Technical details for model selection
The AIC performs best for the part of the distribution where most of the
values are. Hence, a good fit for the bulk of the distribution might include
large biases in the extremes and still have the lowest AIC (example:
Fig. c). To avoid such a model choice with unreasonable high
extremes, we introduce a criterion based on the extremes to sort out mixture
model fits yielding overly high extremes before AIC-model selection is
applied. This criterion is based on a comparison between the 100 season
return level estimated by the mixture model (RL100Smixture) and
the 95 % confidence interval of the RL100SGP estimated by the
GP distribution only. The RL100SGP and the corresponding 95 %
confidence interval are estimated according to . This
criterion is applied differently for Fobs and FRCM
considering the respective relevant quantity for the correction function. For
Fobs this criterion is based on the return level itself, whereas
for FRCM the probability for the return level is considered. In
particular, for Fobs the RL100Smixture must not
exceed the 95 % confidence interval of the RL100SGP. For
FRCM the mixture model probability (pmixture) for the
RL100SGP must not exceed pGP for the 95 %
confidence interval of RL100SGP. Furthermore,
pmixture for the 95 % confidence interval of the
RL100SGP must not be very close to 1 (i.e.,
> 1–1 × 10-15) as a reasonable extrapolation to potentially higher values under
climate change would not be possible in that case.
Examples of problems with the mixture model. (a) Numerical
instability: discontinuous cdf; (b) numerical instability: cdf that
jumps to the upper bound of 1000 mm day-1 and does not jump back as
in (a), and (c) problematic model selection: QQ plot of a
selected mixture model that fits well for most quantiles but corrects the
extremes to too wet.
Additional resultsStep 1: bias correction
Step 1: bias correction to grid scale. QQ plots of RCM-simulated and
QMgrid-corrected (cross-validated) precipitation (mm day-1)
against E-OBS for stations with a high CvM score. (a) Graz JJA,
(b) Leba DJF, (c) Siedlce DJF, and (d) Dresden
DJF.
Step 2: downscaling
Step 2: downscaling. QQ plots for example stations in JJA. VGLM
gamma standardized to the stationary gamma distribution fitted to observed
wet day intensities between gridded and point-scale precipitation
observations (mm day-1). (a) Karasjok, (b) Stornoway,
(c) Brocken, (d) Dresden, (e) Sibiu,
(f) Sonnblick, (g) Sion, (h) San Sebastian, and
(i) Malaga.
Step 2: downscaling. Estimated relation between gridded and
point-scale precipitation observations for example stations in JJA. VGLM
gamma where both parameters depend on the predictor fitted to observed wet
day intensities. The predictor is E-OBS. Circles: observed precipitation
intensities (mm day-1); lines: 0.1, 0.25, 0.5, 0.75, 0.9, and 0.95
modeled quantiles (mm day-1). (a) Karasjok,
(b) Stornoway, (c) Brocken, (d) Dresden,
(e) Sibiu, (f) Sonnblick, (g) Sion,
(h) San Sebastian, and (i) Malaga.
Combination of steps 1 and 2: bias correction and downscaling
QQ plots for example stations of different models (cross-validated)
against station observations for JJA (mm day-1). (a) Karasjok,
(b) Stornoway, (c) Brocken, (d) Dresden,
(e) Sibiu, (f) Sonnblick, (g) Sion,
(h) San Sebastian, and (i) Malaga. For the VGLM the
quantiles (i.e., sorted time series) of 100 realizations are averaged. Predictor for VGLM as selected by the CvM
criterion: (red circles) QMgrid bias-corrected RCM, (brown
triangles) uncorrected RCM. Highest VGLM-modeled quantile in Dresden out of
range: 3609 mm day-1.
Douglas Maraun had the initial idea for this combined method. Claudia Volosciuk implemented the method
and performed the evaluation with help from Mathieu Vrac and Douglas Maraun.
All authors discussed details of the implementation and the results.
Claudia Volosciuk prepared the manuscript with contributions from all
co-authors.
The authors declare that they have no conflict of interest.
Acknowledgements
We thank the KNMI for producing and making available their model output. We
acknowledge the E-OBS dataset from EU-FP6 project ENSEMBLES
(http://ensembles-eu.metoffice.com) and the data providers in the
ECA&D project (http://eca.knmi.nl). We thank S. Kotlarksi,
S. Hagemann, and one anonymous
reviewer for comments on the manuscript. The analysis was carried out with R,
using the packages evir, ncdf, MASS, stats, stats4, fields, aspace, ncf, and
rworldmap. This study was funded by the EUREX project of the Helmholtz
Association (HRJRG-308) and the PLEIADES project of the Volkswagen Foundation
(grants 85423 and 85425). Claudia Volosciuk has received a Short-Term
Scientific Mission Grant from EU COST Action ES1102 VALUE. The article processing charges for this open-access
publication were covered by a Research Centre
of the Helmholtz Association. Edited by:
L. Samaniego Reviewed by: S. Kotlarski, S. Hagemann, and one
anonymous referee
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