Simulations of regional or global climate models are often used
for climate change impact assessment. To eliminate systematic errors, which
are inherent to all climate model simulations, a number of post-processing
(statistical downscaling) methods have been proposed recently. In addition
to basic statistical properties of simulated variables, some of these
methods also consider a dependence structure between or within variables. In
the present paper we assess the changes in cross- and auto-correlation
structures of daily precipitation in six regional climate model simulations.
In addition the effect of outliers is explored making a distinction between
ordinary outliers (i.e. values exceptionally small or large) and dependence
outliers (values deviating from dependence structures). It is demonstrated
that correlation estimates can be strongly influenced by a few outliers even
in large datasets. In turn, any statistical downscaling method relying on
sample correlation can therefore provide misleading results. An exploratory
procedure is proposed to detect the dependence outliers in multivariate
data and to quantify their impact on correlation structures.
Introduction
The investigation of climate change impact on the hydrological cycle is one
of the crucial topics in the field of water resources management and
planning (Mehrotra and Sharma, 2015). Simulations of regional and global
climate models (RCMs and GCMs) represent a fundamental data source for
climate change impact studies. It is well known that raw climate model
outputs cannot be used directly in impact studies due to inherent biases
which are found even for basic statistical properties (Chen et al., 2015).
The bias is caused primarily by a simplified representation of important
physical processes (Solomon et al., 2007), which often results from low
spatial resolution of the RCMs.
Therefore, many methods have been developed to post-process the climate
model outputs in order to move their statistical indicators closer to
observations. An overview of these methods is presented, e.g. by Maraun et
al. (2010). Precipitation is a key input into hydrological climate change
impact studies and at the same time it belongs to meteorological variables
that are most affected by bias. The comparison of correction methods
commonly used for precipitation data is provided by Teutschbein and Seibert
(2012). Nevertheless, these standard methods correct only the bias in
statistical indicators (mean, variance, distribution function) of individual
variables. The bias in persistence parameters of time series as well as the
bias in cross-dependence structures between variables is often neglected.
However, the dependence structures of the meteorological variables affect
the hydrological response of a catchment (Bárdossy and Pegram, 2012), and
thus their inadequate representation in the data can impair hydrological
impact studies (Teng et al., 2015; Hanel et al., 2017).
In recent years several studies attempted to overcome this limitation.
Hoffmann and Rath (2012) and Piani and Haerter (2012) focused on the
relationship between precipitation and temperature data from a single
location. Bárdossy and Pegram (2012) developed two procedures correcting
a spatial correlation structure of RCM precipitation. Mao et al. (2015)
proposed a stochastic multivariate procedure based on copulas. Johnson and
Sharma (2012) developed a procedure correcting common statistics (mean,
variance) together with lag-1 autocorrelation in multiple timescales. The
procedure was later extended with a recursive approach by Mehrotra and
Sharma (2015) and subsequently with a non-parametric quantile mapping by
Mehrotra and Sharma (2016) to correct the bias in auto- and cross-dependence
structures across multiple timescales. An approach based on the principal
components was presented by Hnilica et al. (2017), correcting bias in
cross-covariance and cross-correlation structures.
This study is focused on a temporal stability of dependence structures. We
evaluate the temporal changes in cross-and auto-correlation structures in
multivariate precipitation data simulated by an ensemble of climate models.
We further investigate whether the magnitude of the changes considerably exceeds the natural variability. Attention is finally paid to the
effect of outlying values, which can significantly affect the correlations
and can thus lead to artefacts in bias-corrected time series.
The paper is organised as follows. In Sect. 2 the data used in this study
are presented and Sect. 3 describes the methodology. In Sect. 4 the results
are reported and in Sect. 5 their consequences for climate changes impact
studies are discussed.
Data and study area
The daily precipitation data from six EURO-CORDEX (Giorgi et al., 2009)
regional climate models were considered. The ensemble of models was composed
of two RCMs (CCLM,RCA) driven by three GCMs (EC_EARTH, HadGEM2-ES and MPI-ESM-LR); see Table 1 for the overview. The simulations
with 0.11 ∘ spatial resolution forced by the representative concentration pathway 8.5 (RCP8.5) were used. The data
from 12 model grid boxes located in the western part of the Czech
Republic were analysed; see Fig. 1 for the details of the area. The control
period spans the years from 1971 to 2000, the future period the years from
2051 to 2080.
Global and regional climate models used in the present study.
Location of the considered grid boxes in the Czech Republic.
The numbering of individual pairs of grid boxes. The figure depicts
the correlation matrix, the orders of rows and columns correspond to the grid box
labels from Fig. 1. The sub-diagonal part of the (symmetrical) matrix was
used for the numbering of individual pairs of grid boxes – the numbers inside of
the matrix represent the identifiers used in Fig. 4.
Overview of the changes in correlation structures for all models:
(a) the changes in binary cross-correlations, (b) the
changes in cross-correlations of overlapping wet periods, (c) the changes in lag-1
auto-correlations.
Methods
The wet and dry periods were treated separately in this study. The
cross-correlations were calculated in two stages. Firstly the binary
cross-correlations were calculated to assess the correspondence of wet and dry
periods, using the time series with the values replaced by 0 (dry day) or by
1 (wet day). In the second stage the cross-correlations of overlapping wet
periods were calculated. The auto correlations were analysed through the
lag-1 auto-correlation coefficient, where only the non-zero pairs of
neighbouring values xi and xi+1 were considered.
The individual grid boxes were labelled by numbers 1–12, as shown by labels
in Fig. 1. The cross-correlation between the grid boxes i and j is denoted as
ri,j. The symbol R denotes the correlation
matrix (i.e. the square matrix with elements ri,j). The lag-1
auto-correlation from grid box i is denoted as ri1. If
appropriate, the subscripts denoting the grid boxes are omitted for clarity.
The changes in correlation coefficients were calculated as
Δr=rF-rC,
where Δr denotes the change in r (cross- or auto-correlation), and
subscripts F and C denote the future and control periods, respectively. Note
that the first-order moments needed for calculation of rF and
rC are calculated independently for the future and control periods.
Another option, leading potentially to larger changes, is to consider a fixed reference, e.g. the mean for the control period. However, the changes in the
mean are relatively small; the average change is 0.14 mm across all
considered simulations, which represents approximately 5 % of the value
from the control period. Therefore the differences in the calculated Δr and their significance are not expected to be large.
The sampling variability of individual cross- and auto-correlation was
investigated to assess the statistical significance of their changes. The
confidence intervals were derived using the block bootstrap approach
(Davison and Hinkley, 1997). Specifically, the confidence interval around
the correlation ri,j was obtained as follows:
One-year blocks from the time series for basins i and j were randomly
selected with replacement (30 times to obtain the same sample size as the
original data). Subsequently the correlation of the 30 year sample was
calculated.
Step 1 was repeated 1000 times.
The 95 % confidence interval was derived as a range between the 0.025 and
0.975 quantiles of the resampled correlations.
The block approach was chosen to preserve seasonal variability in the
bootstrap samples. For the presentation of confidence intervals, the unique
identifier (ID) was assigned to each pair of grid boxes, and the numbering was
done according to rows of correlation matrix; the scheme is depicted in Fig. 2. The confidence intervals for auto-correlation were derived in the same
way using 1-year blocks of time series. Due to random selection of the
blocks, the beginning part of the blocks is independent on the end of the
previous block. To minimise bias introduced by block resampling, data that
are potentially influenced (joints of the adjacent blocks) were not
considered for the calculation of the serial correlation.
The confidence intervals around the correlations from control and future
period were used to visually assess their overlap. In addition, the real
bootstrap-based tests of significance of individual changes were performed.
In each of the thousand steps the correlation of resampled control data was
subtracted from the correlation of resampled future data. The change was
found to be insignificant if the confidence interval of these differences contained
zero.
The 95 % confidence intervals of the individual
cross-correlation coefficients for overlapping wet periods for all models.
The identifiers of grid-box pairs (ID) are explained in Fig. 2. The blue
lines separate identifiers located in successive rows in the correlation
matrix (see Fig. 2). The arrow marks the confidence intervals around the
r5,11 (ID 50) of the model 2A, discussed in more detail in Sect. 4.2.
ResultsChanges in correlation structures
In the case of 12-dimensional data, the change in the cross-correlation
structure consists of changes in ri,j coefficients for 66 pairs of
grid boxes (corresponding to the sub-diagonal part of the correlation
matrix). For clarity, these 66 changes are presented in the form of
box plots for individual models.
Figure 3a and b present the changes in the binary cross-correlations and
in the cross-correlations of wet periods, respectively. As seen from the
figures, the binary correlations are relatively stable; their changes range
approximately from -0.02 to 0.03. Therefore, the correspondence of
wet and dry periods between individual grid boxes remains similar in the control and
future periods. The correlations of wet periods change more substantially;
the changes range from -0.16 to 0.05. Nevertheless, there are strong
differences between individual models; the models 1A and 2A reach noticeably
higher changes than other models. In addition, the changes in fractions of
dry days were calculated. In general, the fraction of zeroes fluctuates
around 0.25 in time series across all models and it was found that it
slightly increases in most cases. The difference can be found between the
regional models A and B. While in the simulations of the model A the
fractions of zeroes increase on average by 0.05; for the model B the
average increment is only 0.009.
The 95 % confidence intervals of the individual lag-1
auto-correlation coefficients for all models.
Figure 3c presents the changes in lag-1 auto-correlations; the box plots for
individual models are compiled from 12 changes in time series from
individual grid boxes. The changes range from -0.1 to 0.025; the widest
range of changes is reached by the model 2A. The maximal relative changes in
cross-correlation reach up to 18 % of the value from the control period,
and
in the case of auto-correlation it is almost 45 %. This is because the
auto-correlations are in general markedly lower than cross-correlations (the
mean cross-correlation of individual models exceeds 0.8; the mean lag-1
auto-correlation is around 0.23). In addition, it is worth noting that the
reported changes do not completely characterise the changes in the
auto-dependence structure. Substantial changes in dependence at longer
temporal scales have been reported in several studies (Mehrotra and Sharma,
2015, 2016; Hanel et al., 2017).
The significance of the changes in wet-period correlations was assessed
using a block bootstrap. Figure 4 presents the 95 % confidence intervals
of individual cross-correlations for all models. The blue dividers identify
the successive rows below the diagonal in the correlation matrix. In
general, the majority of changes show little significance; the intervals
from control and future periods overlap considerably (except models 1A and
2A, which in many cases show exceptionally wide intervals for the future period). Figure 5 shows the same for lag-1 auto-correlations of individual
grid boxes. Also in this case the majority of changes do not exceed the
sampling variability; the most significant changes are reached by the model
3B, but the overall trend is a drop in the future.
To verify these results, the significance of individual changes was tested
using the bootstrap approach. The results of tests correspond well with the
visual assessment presented in Figs. 4 and 5. In the case of
cross-correlations, only four changes were found significant for the model
1A, no changes for the model 1B, two for the model 2A, eight for the model 2B,
two for the model 3A and no changes for the model 3B. In case of
auto-correlations, the significant changes were found only for the model 3B.
Note, however, that the fraction of significant changes might be larger in
the case of a fixed reference being used for calculating correlations and
auto-correlations (see Sect. 3).
Effect of outliers
The previous section demonstrated that in some cases the changes in
cross-correlation show little significance despite their high absolute
values, which is particularly related to the models 1A and 2A. At the same time,
it can be seen in Fig. 4 that some confidence intervals for these models are
exceptionally wide. Further analyses showed that this instability of
correlation estimates is introduced by outlying values, which cause seeming
changes in the correlation structures.
In the simulation of the model 2A, the sample correlation
r5,11 decreased from 0.90 in the control period to 0.73 in the
future period. Figure 6a depicts the data from the future period (values
from the grid box 5 plotted against values from the grid box 11; the data
with any zero values are excluded). The decrease is in large part caused by
one outlying point, which is circled in the plot. Its removal from the data
increases the correlation in the future period to 0.86, which markedly
reduces the change. On the other hand, high values do not necessarily affect
the correlation, as seen in Fig. 6b, where the data from grid boxes 11 and
12 are plotted (again the model 2A, future period). The circled outlier does
not affect the correlation in this case, since the location of the point is
in accordance with the configuration of the data – the point lies approximately
in a direction of a potential regression line.
The effect of outliers on correlation structures of model 2A in
the future period (the outliers are circled): (a) the outlier strongly
affecting the cross-correlation, (b) the outlier with no effect on the
correlation, (c) the outlier affecting the calculation of the serial
correlation r121.
The difference between the ordinary and dependence outliers. The
dashed lines define the standard coordinate system; the solid lines define
an alternative coordinate system. The outlying points in the standard
coordinate system are ordinary outliers (point A); the outlying points in the
alternative coordinate system are denoted as dependence outliers (point deviating
from the dependence structure, point B). The construction of alternative
coordinates is explained in the text.
Outlying values affect also the auto-correlation. The largest change in the
auto-correlation was achieved by the model 2A, where r121 decreased
from 0.23 in the control period to 0.12 in the future period. This decrease
is caused by the outlier 349.4 mm in the future data; this extraordinary
value was simulated by the model 2A for 8 May 2080. Figure 6c
depicts the data for the calculation of r121, i.e. the values
xi plotted against the values xi+1, where i denotes the order of
the value x in the time series. The outlier is employed twice within the
calculation (as xi and as xi+1, circled values in Fig. 6c),
which markedly affects the result. If the outlier is removed from the time
series, r121 increases from 0.12 to 0.22, which reduces the change
almost to zero. The calculation of other members of the auto-correlation
function is affected by the outlier in the same way. We note that the
effect of an outlier on the auto-correlation strongly depends on the
values which the outlier is surrounded by in the time series. The presence of a
noticeable outlier thus makes the calculation of the auto-correlation very
unstable.
Detection of outliers
The examples showed that outliers can distort cross- and auto-correlation
structures of a large dataset comprising many thousands of values.
Nevertheless, it should be realised that not each extreme value necessarily
affects the correlation (as seen in Fig. 6b). Therefore, a more specific
concept of outliers is presented in this study. Values deviating from the
correlation structure are denoted as dependence outliers. As well as ordinary
outliers, the dependence outliers are values which are a long distance from the
origin; nevertheless, the difference between them and ordinary outliers
depends on the coordinate system in which the distance is measured. Figure 7
illustrates this by an example of synthetic two-dimensional data. The dashed
lines and coordinates in square brackets define the standard (canonical)
coordinate system. The ordinary outliers are points a long distance from
the origin [0, 0], measured in standard coordinates; the point A
represents an example. The solid lines and coordinates in round brackets
define an alternative coordinate system, which reflects the intensity of
linear dependence between the variables X and Y. The dependence outliers are
points a long distance from the origin (0, 0), measured in alternative coordinates; the point B represents an example. Let us remark
that the point B is an extreme value neither in X nor Y
data, which is in contrast to the point A. Nevertheless, the point B deviates from
the dependence structure, which becomes apparent when its distance from (0, 0) in
alternative coordinates is calculated. The alternative coordinate system is
constructed through the covariance matrix of the data. The directions of the
axes are given by the eigenvectors of the matrix, the lengths of unit
vectors are given by the square root of the corresponding eigenvalues and
the origin is located in the mean of the data. The construction of the
system is related to the principal component analysis; see for example Wilks (2011) for details.
The demonstration of the exploratory procedure: (a) the detection of
dependence outliers for two-dimensional data from Fig. 6a (the plot of
δRm indicates a
noticeable outlier in the data). (b) the same for the data from Fig. 6b – a
gradual evolution of δRm indicates that data do not contain
dependence outliers.
The problem is that the presence of outliers is not easily detected from
the changes in dependence structures. It can be indicated indirectly from
the analysis of sampling variability; nevertheless, the wide confidence
intervals do not necessarily imply the presence of outliers. Or
alternatively, it can be found when the individual pairs of datasets are
visually checked. We propose a procedure allowing for identification of
significant dependence outliers and assessment of their effect on
correlation structure. The procedure consists of three steps:
The most outlying (multivariate) value is found in the data (in alternative
coordinates).
The value is removed from the data and a new correlation matrix is
calculated.
A difference between the new and the previous correlation matrix is
calculated and recorded.
These three steps are repeated. The difference in step 3 is quantified
through
δRm=Rm-Rm-1,
where Rm denotes the correlation matrix of the data
from which m largest outliers were removed; the notation ⋅ denotes the Frobenius matrix norm. The most outlying value in the
step 1 is simply defined as the value with the highest distance from origin
(measured in alternative coordinates). A result of the proposed
exploratory procedure is a sequence of δRm, which clearly indicates the presence of
noticeable outliers. We note that the alternative coordinate system in which
the dependence outliers are searched is data-dependent (in contrast to the
standard coordinates). This means that after each outlier removal the
alternative coordinates change slightly and must be recalculated to
correspond to the remaining data.
The procedure is demonstrated on two simple two-dimensional examples. Figure
8a depicts the sequence of δRm for the
data from Fig. 6a. A massive impact of the first outlier is clearly visible;
the removal of the next outliers does not affect the correlation matrix
substantially (the first member δR1
corresponds to the circled outlier in Fig. 6a). Figure 8b depicts the same
for the data from Fig. 6b; a gradual evolution of δRm indicates that the data do not contain noticeable
(dependence) outliers.
This procedure is very useful as it allows a large set of multivariate
data to be explored as a whole. The n-dimensional outliers can be searched for in the
same way as the two-dimensional outliers in the examples presented above. A
result of the procedure is always a one-dimensional plot of δRm, regardless of the dimension of the input dataset.
Figure 9 shows the plots of δRm for
the complete 12-dimensional data from the future period for all models. The
strong outliers in data from 1A and 2A are easily detected from the plots.
Generally, a plot of δRm enables a
simple assessment of the internal structure of the data and a direct
evaluation of the importance of individual outliers.
The detection of dependence outliers for complete 12-dimensional
data from all models in the future period. The strong outliers in data from
the models 1A and 2A are clearly distinguishable.
Conclusions
The examples presented demonstrate that outliers can strongly affect the cross- and
auto-correlation structures of the data comprising many thousands of values.
In general, it must be stressed that the presence of outliers cannot be
considered as a bias. The extreme precipitation values as well as the
dependence outliers naturally occur. Nevertheless, although the dependence
structures are markedly influenced by a small number of outliers, they
characterise the data as a whole. Therefore a substantial bias can arise
when data with noticeable outliers are used to assess the dependence
structures, or when their dependence structures are used, e.g. for calibration
of the bias correction functions. The cross- and auto-correlation structures
are the key ingredients in several multivariate bias correction methods;
for examples see Mehrotra and Sharma (2015) and Mehrotra and Sharma (2016).
The results based on these methods can be devalued by outliers; see the
Supplement to this paper.
From this point of view there is no need to distinguish between real
extremes and “genuine” outliers (for example measurement errors). The real
extremes as well as genuine outliers affect the correlation structures in
the same way, which subsequently affects the bias corrections (or stochastic
generators). Therefore the dependence outliers, regardless of their origin,
should be removed from the calibration data. The appropriate tool for
testing the presence of outliers is the analysis of the difference
δRm between the new and previous
correlation matrix (Eq. 2) presented above; the exploratory procedure can be
automatised and included in the modelling chain as a pre-processing step to
automatically remove at least the most noticeable outliers.
The analysis of significance showed that in most cases the correlations are
stable in time; their changes are insignificant and are caused by outlying
values. Therefore the climate projection can be interpreted as a linear
transformation of an initial state, because a nonlinear transformation would
change the correlations substantially. From this point of view a reasonable
scenario of future precipitation can be obtained by the corresponding linear
transformation of observations, i.e. by the multiplicative delta method
(Déqué, 2007). Such an approach avoids the problems of complex bias correction methods (e.g. their increasing complexity and unclear effect on
climate change signal), which have recently been the subjects of serious
criticism, for example by Ehret et al. (2012) or Maraun et al. (2017).
Data availability
The RCM data, the source codes and the plot data
are available online at 10.5281/zenodo.1407992 (Hnilica, 2018), which allows the
generation of all results and reproduction of all plots.
The supplement related to this article is available online at: https://doi.org/10.5194/hess-23-1741-2019-supplement.
Author contributions
JH an MH designed the study and wrote the paper. JH
wrote the source codes. VP provided the theoretical background for the principal
component analysis and for the bootstrap. All authors participated in the
interpretation of the results.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
This study was supported by the Czech Science Foundation
(Jan Hnilica from grant no. 16-05665S and Martin Hanel from grant no. 16-16549S). Moreover, the financial
support from RVO: 67985874 is greatly acknowledged. We acknowledge the World
Climate Research Programme's Working Group on Regional Climate, and the
Working Group on Coupled Modelling (former coordinating body of CORDEX and
responsible panel for CMIP5). We also thank the climate modelling groups
of the CLM community and the Rossby Centre (Swedish Meteorological and Hydrological
Institute) for producing and making available their model outputs.
Review statement
This paper was edited by András Bárdossy and reviewed by Geoff Pegram and Ashish Sharma.
ReferencesBárdossy, A. and Pegram, G.: Multiscale spatial recorrelation of RCM
precipitation to produce unbiased climate change scenarios over large areas
and small, Water Resour. Res., 48, W09502, 10.1029/2011WR011524, 2012.Chen, J., Brissette, F. P., and Lucas-Picher, P.: Assessing the limits of
bias-correcting climate model outputs for climate change impact studies, J.
Geophys. Res.-Atmos., 120, 1123–1136, 10.1002/2014JD022635, 2015.
Davison, A. C. and Hinkley, D. V.: Bootstrap methods and their application,
Cambridge University Press, Cambridge, United Kingdom, 1997.Déqué, M.: Frequency of precipitation and temperature extremes over
France in an anthropogenic scenario: Model results and statistical
correction according to observed values, Global Planet. Change, 57, 16–26,
10.1016/j.gloplacha.2006.11.030, 2007.Ehret, U., Zehe, E., Wulfmeyer, V., Warrach-Sagi, K., and Liebert, J.: HESS
Opinions “Should we apply bias correction to global and regional climate
model data?”, Hydrol. Earth Syst. Sci., 16, 3391–3404,
10.5194/hess-16-3391-2012, 2012.
Giorgi, F., Jones, C., and Ghassem, R.: Addressing climate information needs
at the regional level: the CORDEX framework, World Meteorological
Organization (WMO) Bulletin, 58, 175–183, 2009.
Hanel, M., Kožín, R., Heřmanovský, M., and Roub, R.: An R
package for assessment of statistical downscaling methods for hydrological
climate change impact studies, Environ. Modell. Softw., 95, 22–28, 2017.Hnilica, J.: Supporting data and source codes, Data set,
Zenodo, 10.5281/zenodo.1407993, 2018.Hnilica, J., Hanel, M., and Puš, V.: Multisite bias correction of
precipitation data from regional climate models, Int. J. Climatol., 37,
2934–2946, 10.1002/joc.4890, 2017.Hoffmann, H. and Rath, T.: Meteorologically consistent bias correction of
climate time series for agricultural models, Theor. Appl. Climatol., 110,
129–141, 10.1007/s00704-012-0618-x, 2012.Johnson, F. and Sharma A.: A nesting model for bias correction of
variability at multiple time scales in general circulation model
precipitation simulations, Water Resour. Res., 48, W01504,
10.1029/2011WR010464, 2012.Mao, G., Vogl, S., Laux, P., Wagner, S., and Kunstmann, H.: Stochastic bias
correction of dynamically downscaled precipitation fields for Germany through
Copula-based integration of gridded observation data, Hydrol. Earth Syst.
Sci., 19, 1787–1806, 10.5194/hess-19-1787-2015, 2015.Maraun, D., Wetterhall, F., Ireson, A. M., Chandler, R. E., Kendon, E.
J.,Widmann, M., Brienen, S., Rust, H. W., Sauter, T., Themeßl, M.,
Venema, V. K. C., Chun, K. P., Goodess, C. M., Jones, R. G., Onof, C., Vrac,
M., and Thiele-Eich, I.: Precipitation downscaling under climate change:
recent developments to bridge the gap between dynamical models and the end
user, Rev. Geophys., 48, RG3003, 10.1029/2009RG000314, 2010.
Maraun, D., Shepherd, T. G., Widmann, M., Zappa, G., Walton, D.,
Gutiérrez, J. M., Hagemann, S., Richter, I., Soares, P. M. M., Hall, A.,
and Mearns, L. O.: Towards process-informed bias correction of climate change
simulations, Nat. Clim. Change, 7, 764–773, 10.1038/NCLIMATE3418, 2017.Mehrotra, R. and Sharma, A.: Correcting for systematic biases in multiple
raw GCM variables across a range of timescales, J. Hydrol., 520, 214–223,
10.1016/j.jhydrol.2014.11.037, 2015.Mehrotra, R. and Sharma, A.: A multivariate quantile-matching bias
correction approach with auto-and cross-dependence across multiple time
scales: Implications for downscaling. J. Climate, 29, 3519–3539,
10.1175/JCLI-D-15-0356.1, 2016.Piani, C. and Haerter, J. O.: Two dimensional bias correction of temperature
and precipitation copulas in climate models, Geophys. Res. Lett., 39,
L20401, 10.1029/2012GL053839, 2012.
Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, M., Averyt, K. B.,
Tignor, M., and Miller, H. L. (Eds.): Climate Change 2007: The Physical
Science Basis Contribution of Working Group I to the Fourth Assessment Report
of the Intergovernmental Panel on Climate Change, Cambridge University Press,
Cambridge, UK and New York, NY, USA, 2007.Teng, J., Potter, N. J., Chiew, F. H. S., Zhang, L., Wang, B., Vaze, J., and
Evans, J. P.: How does bias correction of regional climate model
precipitation affect modelled runoff?, Hydrol. Earth Syst. Sci., 19,
711–728, 10.5194/hess-19-711-2015, 2015.Teutschbein, C. and Seibert, J.: Bias correction of regional climatemodel
simulations for hydrological climate-change impact studies: Review and
evaluation of different methods, J. Hydrol., 456, 12–29,
10.1016/j.jhydrol.2012.05.052, 2012.
Wilks, D. S.: Statistical methods in the atmospheric science, 3rd edn., Academic, Amsterdam, 2011.