ESDEarth System DynamicsESDEarth Syst. Dynam.2190-4987Copernicus PublicationsGöttingen, Germany10.5194/esd-9-627-2018Bias correction of surface downwelling longwave and shortwave radiation for the EWEMBI datasetBias correction of surface downwelling radiation for the EWEMBI datasetLangeStefanslange@pik-potsdam.dehttps://orcid.org/0000-0003-2102-8873Potsdam Institute for Climate Impact Research, Telegraphenberg A 31, 14473 Potsdam, GermanyStefan Lange (slange@pik-potsdam.de)24May2018926276456September201725September201722March20183May2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://esd.copernicus.org/articles/9/627/2018/esd-9-627-2018.htmlThe full text article is available as a PDF file from https://esd.copernicus.org/articles/9/627/2018/esd-9-627-2018.pdf
Many meteorological forcing datasets include bias-corrected surface
downwelling longwave and shortwave radiation (rlds and rsds). Methods used
for such bias corrections range from multi-year monthly mean value scaling to
quantile mapping at the daily timescale. An additional downscaling is
necessary if the data to be corrected have a higher spatial resolution than
the observational data used to determine the biases. This was the case when
EartH2Observe E2OBS; rlds and rsds were bias-corrected
using more coarsely resolved Surface Radiation Budget
SRB; data for the production of the meteorological
forcing dataset EWEMBI . This article systematically
compares various parametric quantile mapping methods designed specifically
for this purpose, including those used for the production of EWEMBI rlds and
rsds. The methods vary in the timescale at which they operate, in their way
of accounting for physical upper radiation limits, and in their approach to
bridging the spatial resolution gap between E2OBS and SRB. It is shown how
temporal and spatial variability deflation related to bilinear interpolation
and other deterministic downscaling approaches can be overcome by downscaling
the target statistics of quantile mapping from the SRB to the E2OBS grid such
that the sub-SRB-grid-scale spatial variability present in the original E2OBS
data is retained. Cross validations at the
daily and monthly timescales reveal that it is worthwhile to take empirical
estimates of physical upper limits into account when adjusting either
radiation component and that, overall, bias correction at the daily timescale
is more effective than bias correction at the monthly timescale if sampling
errors are taken into account.
Introduction
High-quality observational datasets of surface downwelling radiation are of
interest in many fields of climate science, including energy budget
estimation and climate model
evaluation . As part of so-called climate
or meteorological forcing datasets such as those generated within the Global
Soil Wetness Project GSWP;, at Princeton University
, and within the Integrated Project Water and Global
Change
WATCH;, the longwave and shortwave components of
surface downwelling radiation (abbreviated as rlds and rsds or just longwave
and shortwave radiation in the following) are used to correct model
biases in climate model output and
drive simulations of climate impacts, for example
.
These meteorological forcing datasets are global, long-term meteorological
reanalysis datasets such as those produced by the National Centers for
Environmental Prediction – National Center for Atmospheric Research
NCEP–NCAR; and the European Centre for
Medium-Range Weather Forecasts ECMWF;, refined
by bias correction using global, gridded observational data. For the
components of surface downwelling radiation, such a bias correction is often
necessary because observations of these variables are not assimilated in the
reanalyses, which makes them subject to modelling biases of
land–atmosphere interactions and cloud processes, for example
.
Different approaches are adopted in order to carry out these bias
corrections. apply indirect corrections at the
monthly timescale using near-surface air temperature observations for rlds
and observations of atmospheric aerosol loadings and cloudiness for rsds.
directly rescale rlds and rsds to match observed
multi-year monthly mean values. directly adjust
distributions of daily mean rsds. The observational dataset commonly used for
such direct adjustments of rlds and rsds is the Surface Radiation Budget
(SRB) dataset assembled by the National Aeronautics and Space Administration
(NASA) and the Global Energy and Water Exchanges project
GEWEX;.
Another meteorological forcing dataset, the EartH2Observe, WFDEI and
ERA-Interim data Merged and Bias-corrected for ISIMIP
EWEMBI;, was recently assembled to be used as the
reference dataset for bias correction of global climate model output within
the Inter-Sectoral Impact Model Intercomparison Project phase 2b
ISIMIP2b;. The surface downwelling longwave and
shortwave radiation data included in EWEMBI are based on daily rlds and rsds
from the climate forcing dataset compiled for the EartH2Observe project
E2OBS;. In order to reduce deviations of E2OBS rlds and
rsds statistics from the corresponding SRB estimates in particular over
tropical land , for EWEMBI, the former were bias-adjusted to
the latter at the daily timescale using two newly developed parametric
quantile mapping methods.
These methods are conceptually similar to the method, which
fits beta distributions to reanalysed and observed daily mean rsds for every
calendar month, thereby accounting for upper and lower physical limits of
rsds using the multi-year monthly maximum value as the upper and zero as the
lower limit of the distribution, and then uses quantile mapping to adjust the
distributions. In contrast to , the methods developed to
adjust E2OBS rlds and rsds for EWEMBI applies moving windows to estimate beta
distribution parameters for every day of the year. This precludes
discontinuities at the turn of the month and
promises a better bias correction where the seasonality of radiation is very
pronounced such as for rsds at high latitudes. Also, the new methods estimate
the physical upper limits of rlds and rsds differently, acknowledging that
these limits are necessarily greater than or equal to the greatest value
observed during any fixed period. Lastly, while linearly
interpolate SRB rsds from its natural horizontal resolution of
1.0∘ to the 0.5∘ reanalysis grid prior to bias
correction, the new methods aggregate the E2OBS data from their original
0.5∘ grid to the 1.0∘ SRB grid, where the bias
correction is then carried out, and disaggregates these aggregated and
bias-corrected data back to the E2OBS grid. Depending on the disaggregation
method, this approach promises to generate bias-corrected data with more
realistic temporal as well as spatial variability.
The new methods are comprehensively described and cross validated in this
article. Moreover, several modifications of the new methods are tested here
that differ in how they handle the spatial resolution gap between the E2OBS
and SRB grids, and how they account for the physical upper limits of rlds and
rsds. Also included are bias correction methods that operate at the monthly
timescale in order to test if bias correction of daily or monthly mean
values yields better overall cross-validation results. The lessons learned
from these analyses shall benefit bias corrections of surface downwelling
radiation to be carried out in future generations of climate forcing
datasets.
DataE2OBS
The EartH2Observe E2OBS; daily mean rlds and
rsds data bias-corrected for EWEMBI cover the whole globe on a regular
0.5∘×0.5∘ latitude–longitude grid and
span the 1979–2014 time period. Over the ocean, E2OBS rlds and rsds are
identical to bilinearly interpolated ERA-Interim ERAI;
rlds and rsds. Over land, they are identical to WATCH Forcing Data
methodology applied to ERA-Interim reanalysis data
WFDEI; rlds and rsds. WFDEI rlds, in turn, is identical
to bilinearly interpolated ERAI rlds, adjusted for elevation differences
between the ERAI and Climatic Research Unit CRU; grids.
WFDEI rsds is identical to bilinearly interpolated ERAI rsds bias-corrected
at the monthly timescale using CRU TS3.1/3.21 mean cloud cover and
considering effects of interannual changes in atmospheric aerosol optical
depths .
SRB
The observational data used for the bias correction of E2OBS daily mean rlds
and rsds for EWEMBI were the NASA–GEWEX Surface Radiation Budget
SRB; primary-algorithm estimates of daily mean rlds
and rsds from the latest SRB releases available at the time, which were
release 3.1 for rlds and release 3.0 for rsds. These data cover the whole
globe on a regular 1.0∘×1.0∘
latitude–longitude grid and span the July 1983–December 2007 time period. For bias
correction and cross validation, a 24-year subsample of these data that spans the December 1983–November 2007 time period was used
and is used here. Additional data
from the adjacent months November 1983 and December 2007 are employed for computations of
running mean values. The SRB estimates of rlds and rsds are based on
satellite-derived cloud parameters and ozone fields, reanalysis meteorology,
and a few other ancillary datasets. Due to a lack of satellite coverage
during most of the July 1983–June 1998 time period over an area centred at
70∘ E, SRB data artefacts are present over the Indian Ocean
(https://gewex-srb.larc.nasa.gov/common/php/SRB_known_issues.php, last access: 18 May 2018;
see
Figs. –,
). Every SRB grid cell contains
exactly four E2OBS grid cells.
Methods
For the reader who is is not familiar with the concepts of quantile mapping
and/or statistical downscaling, a short introduction including definitions of
relevant terms is given in Appendix .
The parametric quantile mapping methods introduced in the following are named
according to the scheme BCvtpx, where v,t,p are used to
distinguish between methods for longwave and shortwave radiation (v=l,s) operating at the daily and monthly timescales (t=d,m) using basic and advanced distribution types or parameter
estimation techniques (p=b,a). Index x=0,1,2 is used for
variants of these methods that differ in how they handle the spatial
resolution gap between the SRB and E2OBS grids. For the
BCvtp0 methods, the SRB data are spatially bilinearly
interpolated to the E2OBS grid and the E2OBS data are then bias-corrected
using these interpolated SRB data; this is to mimic the
approach. For bias correction with the BCvtp1 methods,
E2OBS data are spatially aggregated to the SRB grid, and the aggregated data are
then bias-corrected and the resulting data disaggregated back to the E2OBS
grid; this approach was used to produce the EWEMBI radiation data. Lastly,
the BCvtp2 methods adjust mean values and variances at
the E2OBS grid such that mean values and variances of spatial aggregates to
the SRB grid match the corresponding SRB estimates while the sub-SRB-grid-scale spatial structure of mean values and variances present in the original
E2OBS data is retained; this is to overcome the variability deflation induced
by the other two approaches. Since the BCvtp0 and
BCvtp2 methods are based on the
BCvtp1 methods, the latter are introduced first. Readers
who are merely interested in how the EWEMBI radiation data were produced are
informed that methods BClda1 and BCsda1 were used for
that purpose.
Bias correction at the SRB grid scale
For the BCvtp1 methods, daily mean E2OBS rlds and rsds
are first aggregated to the SRB grid using a first-order conservative
remapping scheme . The conservative remapping ensures that
each aggregated value is the grid-cell area-weighted mean of the underlying
four E2OBS values. The methods of bias correction of these aggregated values
are described in the following. The method used for the subsequent
disaggregation to the E2OBS grid is described in Sect. .
Estimation of parameters of quantile mapping methods used for the
bias correction of longwave (a, b) and shortwave (c, d)
radiation at the daily (a, c) and monthly (b, d) timescales. This example is based on SRB daily mean rlds and rsds data from
79.5∘ N, 12.5∘ E and the December 1983–November 2007 time
period. Climatological distribution parameters are estimated based on
empirical 24-year mean values (dark grey), standard deviations (light grey
range around mean values), and minimum and maximum values (black) of daily
mean (a, c) and 31-day running mean (b, d) radiation
computed for every day of the year. The distribution parameters estimated for
the basic (red) and advanced (blue) bias correction methods (see Table ) include mean values and standard deviations (dotted
red, dashed blue), and upper bounds (solid red, solid blue) where beta
distributions are used. Note that the basic and advanced estimates of mean
values and standard deviations only differ in panel (c) near the
beginning and end of polar night (see Table ). The green
line in panel (a) represents 25-day running mean values of 25-day
running maximum values of 24-year maximum values of daily mean rlds, which
are used to estimate the upper bounds of the climatological beta
distributions used by the BClda1 method (solid blue line in panel a). The lower bounds of all climatological beta distributions are
set to zero.
The BCvtp1 methods use parametric transfer functions of
the form FvtpSRB-1(FvtpE2OBS(⋅)),
where FvtpE2OBS and FvtpSRB are climatological
cumulative distribution functions (CDFs) of aggregated E2OBS and SRB data,
respectively. The CDFs are estimated individually for every SRB grid cell and
day of the year (Fig. ).
In order to quantify the extent to which bias correction results benefit from
explicitly accounting for physical radiation limits, the basic and advanced
methods BCltb1 and BClta1 for
longwave radiation use normal and beta distributions, respectively. For
shortwave radiation, the relevance of physical limits is less questionable,
given that the lower limit of zero matters at least during polar night, and
that the solar radiation incident upon land and ocean surfaces is limited by
the solar radiation incident upon the top of the atmosphere (see
Fig. ). Therefore, all
BCstp1 methods use beta distributions and the basic and
advanced methods only differ in how they estimate the beta distribution
parameters (see Fig. ,
Table ).
Distribution types and parameter estimation methods of bias
correction methods BCvtp1 for day d of the year (see
Fig. ). Please note that
the lower bounds of all climatological beta distributions are set to zero and
that 24-year statistics are replaced by 12-year statistics for
cross validation.
MethodDistribution typeMean value µdVariance σd2Upper bound bdBCldb1normal〈〈xij〉i24〉j25d〈{xij}i24〉j25d–BClda1beta〈〈xij〉i24〉j25d〈{xij}i24〉j25dA〈〈xij〉i24〉j25d+BBClmb1normal〈〈xij〉j31d〉i24{〈xij〉j31d}i24–BClma1beta〈〈xij〉j31d〉i24{〈xij〉j31d}i24〈bjlda1〉j31dBCsdb1beta〈〈xij〉i24〉j25d〈{xij}i24〉j25d〈[[xij]i24]j25k〉k25dBCsda1beta〈〈xij〉i24〉j25d*〈{xij}i24〉j25d*CrsdtdBCsmb1beta〈〈xij〉j31d〉i24{〈xij〉j31d}i24〈bjsdb1〉j31dBCsma1beta〈〈xij〉j31d〉i24{〈xij〉j31d}i24〈bjsda1〉j31d
xij is the daily mean rlds (for BCltp1) or rsds (for
BCstp1) on day j of year i. Brackets
〈⋅〉, {⋅}, and [⋅] denote the
calculation of sample mean values, variances, and maximum values,
respectively. Bracket subscripts i24, j31d, j25d, and
j25d* indicate that these sample statistics are calculated over
years i∈{1,…,24}, over days
j∈{d-15,…,d+15}, over days
j∈{d-12,…,d+12}, and over days
j∈{d-n,…,d+n} with n=min{12,max{n≥0:∀j∈{d-n,…,d+n}:rsdtj>0}}, respectively. Constants A, B, and C are
determined by argminA,B′∑l=1365(〈[[xij]i24]j25k〉k25l-A〈〈xij〉i24〉j25l+B′)2, min{B>0:∀l∈{1,…,365}:A〈〈xij〉i24〉j25l+B≥〈[[xij]i24]j25k〉k25l}, and min{C>0:∀j∈{1,…,365}:Crsdtj≥[xij]i24}, respectively.
Bias correction at the daily timescale
The parameters of the climatological CDFs FvdpE2OBS
and FvdpSRB are estimated based on empirical
multi-year mean values, variances, and maximum values of daily mean radiation
from the December 1983–November 2007 time period. Data from the whole period were used
for the production of EWEMBI rlds and rsds. Data from some half of the period
(see Sect. ) are used for cross validation in this
study.
For shortwave radiation, the basic daily bias correction method is designed
to resemble the method outlined by Sect. 3.4.
BCsdb1 estimates mean values and variances of climatological beta
distributions by 25-day running mean values of multi-year daily mean values
and variances, respectively, and their upper bounds by 25-day running mean
values of 25-day running maximum values of multi-year maximum values of daily
mean rsds (solid red line in
Fig. c). The idea behind
this upper-bound estimate is that 25-day running maximum values of multi-year
maximum values of daily mean rsds resemble the multi-year monthly maximum
values of daily mean rsds used by . Please note that using
the same window length for the running maximum calculation and the additional
smoothing ensures that the resulting upper bounds are always greater than or
equal to the multi-year maximum values of daily mean rsds.
The BCsda1 method employs the climatology of daily mean shortwave
insolation at the top of the atmosphere (rsdt; see Appendix for how rsdt is calculated in
this study) for the upper-bound estimation. This is motivated by rsds being
limited by rsdt in most locations and seasons, which suggests that the annual
cycle of the upper bound of daily mean rsds has a similar shape as the
climatology of daily mean rsdt. Therefore, method BCsda1 uses a
rescaled daily mean rsdt climatology as the upper-bound climatology of daily
mean rsds (solid blue line in
Fig. c). The rescaling
is performed with the smallest possible factor that guarantees that the resulting
upper bounds are greater than or equal to the multi-year maximum values of
daily mean rsds on all days of the year with rsdt ≥ 50 W m-2.
An extension of this guarantee to days of the year with lower rsdt would
inflate the rescaling factor because during dusk and dawn of polar night,
rsds can exceed rsdt due to diffuse radiation coming in from lower latitudes.
Therefore, on days of the year with rsdt < 50 W m-2, the
maximum of the rescaled rsdt and the empirical multi-year maximum daily mean
rsds is used as the upper rsds bound. Mean values and variances of the
climatological beta distributions of the BCsda1 method are
estimated by running mean values of multi-year daily mean values and
variances, respectively. The window length used for these running mean
calculations is 25 days by default. On days that are fewer than 13 days away
from the beginning or end of polar night (as defined by daily mean rsdt going
to zero), the window length is shortened to 2n+1, where n is the number
of days between the day in question and the beginning or end of polar night.
For longwave radiation, both the basic and the advanced daily bias correction
methods use 25-day running mean values of multi-year daily mean values and
variances to estimate climatological mean values and variances, respectively.
The upper bounds used by BClda1 are not estimated by the often
rather un-smooth 25-day running mean values of 25-day running maximum values
of 24-year maximum values of daily mean rlds (solid green line in
Fig. a) but by a
suitably shifted and rescaled mean value climatology (solid blue line in
Fig. a; formulas in
Table ).
Since the choice of the window length used for all the running mean and
maximum value calculations mentioned above is somewhat arbitrary, the window
length dependence of the overall performances of the
BCvda1 methods is investigated in Appendix . Sensitivities are found to be very low
for window lengths between 10 and 40 days.
Bias correction at the monthly timescale
In order to mimic a bias correction at the monthly timescale as was performed
by Sect. 3.d.3, for example, the
BCvmp1 methods bias-correct 31-day running mean
values and then rescale each daily value by the corrected-to-uncorrected
ratio of the respective 31-day running mean value.
Mean values and variances of the climatological CDFs
FvmpE2OBS and FvmpSRB of
31-day running mean values are simply estimated by 24-year (or 12-year for
cross validation) daily mean values and variances of 31-day running mean
values, respectively, with 29 February values replaced by averages of
28 February and 1 March values.
Upper bounds of beta distributions are estimated by 31-day running mean
values of the upper bounds of the corresponding CDFs
FvdpE2OBS and FvdpSRB of
daily mean radiation (see
Fig. , Table ) because 31-day running mean values of multi-year
maximum values of daily mean radiation are mathematically always greater than
or equal to multi-year maximum values of 31-day running mean radiation. The
resulting upper bounds are typically much larger than observed 24-year
maximum monthly mean radiation (see
Fig. d) because 31
consecutive days of daily mean radiation at the respective physical upper
limit are very unlikely to occur in reality.
Disaggregation to the E2OBS grid
In principle, the disaggregation of aggregated and bias-corrected E2OBS data
from the SRB to the E2OBS grid can be carried out in various ways. The simplest
approach would arguably be a mere interpolation, which is disadvantageous
since it ignores the sub-SRB-grid-scale spatial variability present in the
original E2OBS data. Probabilistic disaggregation methods
that are designed to retain that variability
seeSect. 3.b.1, are impractical if, as in the
present case, the purpose of the disaggregation is the production and
publication of a dataset because all variants of the dataset that can
potentially be generated by a probabilistic algorithm are, as long as all
conceivable constraints have been incorporated in the algorithm, equally
plausible candidates for the one dataset to be published. Therefore, not a
probabilistic but the following deterministic disaggregation approach was
used for the production of EWEMBI rlds and rsds and is adopted here for all
BCvtp1 methods.
First, E2OBS-grid-scale upper bounds of daily mean radiation are estimated by
bilinearly interpolated maximum values of the climatological upper bounds of
SRB all-sky and clear-sky radiation, which in turn are estimated using the
BClda1 method for rlds and the BCsda1 methods for rsds
(see Table and blue lines in
Fig. a, c). The clear-sky
radiation data are included in order to prevent the E2OBS-grid-scale upper
bounds from being much lower than the real physical limits of daily mean
radiation at that spatial scale, given that due to sub-SRB-grid-scale spatial
variability, upper radiation bounds at the E2OBS grid scale may exceed those
at the SRB grid scale.
The original daily E2OBS data are then clamped between zero and these upper
bounds, and the resulting values (or their distances to their upper bounds)
are rescaled day by day and SRB grid cell by SRB grid cell such that their
SRB-grid-scale aggregates match the bias-corrected values. More precisely,
for a fixed but arbitrary SRB grid cell and a fixed but arbitrary day, let
Y denote the bias-corrected value at the SRB grid scale, wk with
∑k=14wk=1 the area weights of the four E2OBS grid cells
k=1,2,3,4 contained in the SRB grid cell, Xk the clamped original E2OBS
data values with upper bounds bk, and Yk the bias-corrected values at
the E2OBS grid scale to be computed. If Y≤∑k=14wkXk, then
Yk is computed according to Yk=fXk with f=Y/∑k=14wkXk. Otherwise Yk is computed according to Yk=bk-f(bk-Xk)
with f=(Y-∑k=14wkbk)/∑k=14wk(Xk-bk). This
rescaling procedure ensures that 0≤Yk≤bk and ∑k=14wkYk=Y.
Bias correction at the E2OBS grid scaleThe BCvtp2 methods
The disaggregation method introduced above corrects the original E2OBS values
from the four E2OBS grid cells contained in one SRB grid cell as if they must
all be too low (high) if their area-weighted average is too low (high). This
implicit assumption is questionable since it rules out the possibility that
the area-weighted average is too low because one of the four values is much
too low while the others are slightly too high, to give just one example. A
statistical manifestation of this problem is illustrated and discussed in
Sect. .
The assumption does not need to be made if the bias correction is carried out
directly at the E2OBS grid. With target distributions fixed at the SRB grid,
target distributions at the E2OBS grid can be defined such that the
bias-corrected data have the SRB-grid-scale target distributions and the
sub-SRB-grid-scale structure of the original E2OBS data. For parametric bias
correction methods such as those introduced above, this can be achieved via
suitable definitions of the parameters of the E2OBS-grid-scale target
distributions. Here, for every BCvtp1 method, a
corresponding BCvtp2 method is defined to operate at the
same temporal scale and to use the same source (at the E2OBS grid) and target
(at the SRB grid) distribution type and parameter estimation technique (see
Table ). E2OBS-grid-scale target climatologies of mean
values, variances, and (where necessary) upper bounds are defined as follows.
The mean value estimates of the original E2OBS data are shifted by a common
offset per SRB grid cell and day of the year to obtain the E2OBS-grid-scale
target mean values. The offsets are chosen such that the E2OBS-grid-scale
target mean values aggregated to the SRB grid match the corresponding SRB
mean value estimates. E2OBS data bias-corrected using these E2OBS-grid-scale
target mean values have SRB-grid-scale aggregates that match the SRB-grid-scale target mean values because (i) the aggregation is a linear
operation and (ii) the mean value of a linear combination of random variables
is equal to the same linear combination of the mean values of these random
variables.
To obtain the E2OBS-grid-scale target variances, the variance estimates of
the original E2OBS data are rescaled by a common (to all four E2OBS grid
cells contained in one SRB grid cell) factor fij per day i of the year
and SRB grid cell j. For the derivation of the formula for fij let
Yijk (and Xijk) denote random variables representing bias-corrected
(and original) E2OBS data from day i of the year and E2OBS grid cells
k=1,2,3,4 contained in SRB grid cell j. Then the estimated variance of
the SRB-grid-scale aggregate of Yijk can be expanded to
Var∑k=14wjkYijk=∑k,l=14wjkwjlCov(Yijk,Yijl)=∑k,l=14wjkwjlCor(Yijk,Yijl)Var(Yijk)Var(Yijl),
where wjk is the area weight of E2OBS grid cell jk with ∑k=14wjk=1 for all j, Cov(Yijk,Yijl) is the estimated
covariance of Yijk and Yijl, Cor(Yijk,Yijl) is
the estimated Pearson correlation of Yijk and Yijl, and
Var(Yijk) is the estimated variance of Yijk. A bias
correction would be deemed successful if the left-hand side of
Eq. () was equal to the estimated variance of Zij, the
SRB data from day i of the year and grid cell j. On the right-hand side
of Eq. (), fijVar(Xijk) can be
substituted for Var(Yijk) by definition of the scaling
factors, and Cor(Yijk,Yijl) can be approximated by
Cor(Xijk,Xijl) since quantile mapping preserves ranks and
therefore rank correlations and therefore approximately Pearson correlations.
The variance scaling factors fij for method BCvtp2
are therefore calculated based on
Var(Zij)=fij××∑k,l=14wjkwjlCor(Xijk,Xijl)Var(Xijk)Var(Xijl),
where the variances are estimated using the respective
BCvtp1 approach (see Table ), and the
Pearson correlations are estimated by inversely Fisher-transformed 25-day
running mean values of Fisher-transformed 24-year daily Pearson correlations
of daily (for BCvdp2) or 31-day running mean
(for BCvmp2) radiation data. The Fisher
transformations are invoked here in order to approximately account for
correlation value-dependent sampling error intervals
.
The E2OBS-grid-scale target upper bounds are calculated in the same way as
the E2OBS-grid-scale target mean values. This way, the latter rarely exceed
the former. Where they do, the latter are reduced to 99 % of the former.
For longwave (shortwave) radiation, such reductions are necessary in four
(11 % of all) E2OBS grid cells on an average of 15 % (5 %) of all days
of the year.
Furthermore, in order to obtain realistic E2OBS-grid-scale target beta
distributions, the E2OBS-grid-scale target variances calculated using
Eq. () are limited to 40 % of µ(b-µ), where µ
and b are the E2OBS-grid-scale target mean values and upper bounds,
respectively. This limit is imposed because (i) the variance σ2 of a
random variable taking values from within the interval [a,b] can generally
not be greater than (µ-a)(b-µ) if µ is the random variable's mean
value; (ii) if that random variable is beta distributed and σ2>(µ-a)(b-µ)/2 then the probability density function is U shaped
, which is considered unrealistic for climatological
distributions of rlds and rsds; and (iii) σ2/(µ(b-µ)) has an
empirical upper limit of about 40 % in the original E2OBS radiation data.
The 40 % condition is never met for longwave radiation whereas for
shortwave radiation it is met in 14 % of all E2OBS grid cells on
an average of 2 % of all days of the year.
The BCvtp0 methods
For the BCvtp0 methods, daily SRB data are first
bilinearly interpolated to the E2OBS grid. The E2OBS data are then
bias-corrected directly at the E2OBS grid using the interpolated SRB data and
transfer functions defined exactly as for the respective
BCvtp1 method.
Biases relative to SRB in mean values (a–f) and
standard deviations (g–l) of spatially aggregated (to the
SRB grid) daily mean longwave (a–c,
g–i) and shortwave (d–f,
j–l) radiation after bias correction with methods
BCvda0 (left), BCvda1 (middle), and
BCvdb1 (right). The biases are calculated individually
for each calendar month (January to December) and calibration data sample
(every1st, every2nd) pooling SRB and corrected E2OBS data from all years of
the corresponding validation data sample (every2nd, every1st, respectively)
and omitting shortwave radiation data from months with monthly mean rsdt less
than 1 W m-2 (see
Appendix and
Fig. c).
Depicted are median and agreement in direction (sign of bias) of these
individual biases, represented by hue and saturation of a grid cell's colour,
respectively. Categories of agreement in bias direction are defined based on
one-sided p values obtained from modelling
underestimations and overestimations for individual calendar months and
validation data samples as outcomes of independent 50/50 Bernoulli trials.
More saturated colours indicate higher statistical significance of biases
remaining after bias correction.
Same as
Fig. but for
biases in skewness (a–f) and 12-year maximum values (g–l).
Same as
Fig. but for
relative biases in interannual standard deviations of monthly mean radiation
remaining after bias correction with methods BCvda1
(left) and BCvma1 (right).
Results
In the following, the bias correction methods introduced above are
cross validated at the SRB grid scale (Sect. ), and
their disaggregation performance is assessed by comparing sub-SRB-grid-scale
spatial variability before and after bias correction
(Sect. ).
Cross validation at the SRB grid scale
For the cross validation against
SRB data, 24 years worth of overlapping E2OBS and SRB data are divided into
two 12-year samples of which the first one is used to calibrate and the
second one to validate the method. Common practice would be to use data from
the first and second half of the 24-year period to define these samples.
However, due to climate change this definition may yield calibration and
validation data samples that differ statistically. These differences in turn,
which are essentially climate change signals, may differ in extent between
the E2OBS and SRB data. have shown that such differences
in climate change signals may then dominate cross-validation metrics and
thereby distort the comparative validation of bias correction methods. In
order to minimise this climate change impact on cross-validation results,
here, calibration and validation data samples are composed of data from all odd years and all even years or
vice versa, respectively. The samples are accordingly labelled every1st and
every2nd.
Please note that results for BCvtp2 are not shown or
discussed in this section because BCvtp1 and
BCvtp2 produce virtually identical data at the SRB grid
scale.
BCvtp0 vs. BCvtp1
The first question addressed here is how the bilinear spatial interpolation
of SRB data to the E2OBS grid before bias correction with the
BCvtp0 methods impacts the distribution of bias-corrected
rlds and rsds values at the SRB grid scale. To quantify these impacts, biases
in multi-year daily mean values, standard deviations, and maximum values
remaining after bias correction with methods BCvda0 and
BCvda1 are compared in the left and middle columns of
Figs. and
.
Since linear interpolation always yields values that are intermediate to the
values at the interpolation knots it is expected that daily SRB data
bilinearly interpolated to the E2OBS grid and then aggregated back up to the
SRB grid will be more smooth overall both in space and time than the original
SRB data. Manifestations of the increased smoothness in time are the more
negative biases of standard deviations
(Fig. ) and
maximum values
(Fig. )
remaining after bias correction with BCvda0 than with
BCvda1. Standard deviations after bias correction with
BCvda0 in particular are negatively biased by more than
4 % (median over calendar months × validation data samples) in most
regions. In mountainous and therefore spatially heterogeneous regions, multi-year monthly mean radiation is also
changed significantly by the
interpolation, with median biases over calendar months × validation
data samples remaining after bias correction with BCvda0
exceeding 2 W m-2 in many such places
(Fig. ).
BCvtax vs. BCvtbx
Next is an assessment of how the treatment of the upper bound of the
distributions estimated by the BCvdp1 methods
impacts the distribution of bias-corrected rlds and rsds values at the
SRB grid scale. To quantify these impacts, biases in multi-year daily mean
values, standard deviations, and maximum values remaining after bias
correction with methods BCvda1 and
BCvdb1 are compared in the middle and right columns of
Figs. and
.
For longwave radiation, the basic method BCldb1 assumes
normally distributed values and therefore does not account for any upper
physical limit of rlds whereas the advanced method BClda1
assumes the existence of such a limit and estimates it empirically. Figure shows that
the advanced method generally yields a better correction of 12-year maximum
values. In contrast, standard deviations are slightly better corrected by the
basic method and mean values are equally well corrected by both methods
(Fig. ).
For shortwave radiation, both the basic and the advanced method empirically
estimate upper physical limits of rsds and take these into account in the
form of upper bounds of beta distributions. The limit estimates are based on
downwelling shortwave radiation at the surface and at the top of the
atmosphere for BCsda1, and on rsds only for
BCsdb1. Figure shows that
the basic method generally yields a better correction of 12-year maximum
values. Standard deviations and mean values are also slightly better
corrected by BCsda1 than by BCsdb1
(Fig. ).
BCvdpx vs. BCvmpx
Overall performance of bias correction methods
BCvda1, BCvda0,
BCvdb1, and BCvma1 for longwave
(top) and shortwave (bottom) radiation at the daily
(left) and monthly (right) timescales as quantified by
p values of two-sample Kolmogorov–Smirnov test statistics of the respective
E2OBS and SRB data before (black) and after (colours) bias correction (see Appendix ; greater p values indicate stronger
agreement of E2OBS and SRB distributions). The p values are determined
individually for each grid cell, season, and calibration data sample, with
all corresponding values pooled into one distribution and omitting shortwave
radiation data from months with average rsdt less than 1 W m-2.
The horizontal lines of each box–whisker plot represent the 90th, 75th, 50th,
25th, and 10th (from top to bottom) grid-cell area-weighted percentiles of the
natural logarithms of these p values over calibration data sample (1sthalf,
2ndhalf), latitude, and longitude. The grey horizontal line marks the p=10 % significance level.
Same as Fig. but
based on p values of Kuiper's two-sample test statistic.
Relative change by bias correction with methods
BCvda0 (left), BCvda1
(middle), and BCvda2 (right) of the
root-mean-square deviation (RMSD) of daily mean E2OBS-grid-scale longwave
(a–f) and shortwave (g–l) radiation
from the aggregated SRB-grid-scale values based on 1∘ grid cells of
the SRB grid (a–c, g–i) and the
staggered SRB grid (d–f, j–l; see
text). For every 1∘ grid cell and calendar month, the RMSDs are
calculated using original or bias-corrected E2OBS data from the four
0.5∘ grid cells contained in the 1∘ grid cell, pooling data
from the entire December 1983–November 2007 time period and omitting
shortwave radiation data from months with average rsdt less than
1 W m-2. Depicted are median and agreement in the direction of monthly
RMSD changes by bias correction (same colouring scheme as in
Fig. ). Very
similar results are obtained for the corresponding basic bias correction
methods.
Next is a comparative cross validation of methods BCvdpx
and BCvmpx operating at the daily and monthly timescales,
respectively. The cross validation itself is also performed at the daily and
monthly timescales based on statistics of daily and monthly mean radiation,
respectively. A joint assessment of these cross validations shall reveal
whether bias correction at the daily or monthly timescale is better overall.
By design, the BCvdpx and BCvmpx
methods are equally good at correcting multi-year mean values of daily mean
radiation. However, both day-to-day and year-to-year variability are expected
to be differently well corrected by the methods operating at different timescales. Since day-to-day variability is (not) explicitly adjusted by the
methods operating at the daily (monthly) timescale, the
BCvdpx methods are expected to perform better at the
daily timescale than the BCvmpx methods. The
year-to-year variability, however, is explicitly corrected by the
BCvmpx methods and it is not by the
BCvdpx methods because daily data from different years
are pooled before quantile mapping is carried out at the daily timescale.
Consequently, biases in interannual standard deviations of monthly mean
radiation are much larger after bias correction with
BCvda1 than with BCvma1
(Fig. ),
and the BCvmpx methods are generally expected to perform
better at the monthly timescale than the BCvdpx methods.
In order to assess whether bias correction at the daily or monthly timescale
is more effective overall, a performance measure is needed that is comparable
across timescales. Common performance measures of distribution adjustments
at individual timescales are the two-sample Kolmogorov–Smirnov (KS) and
Kuiper's two-sample test statistics. While Kuiper's test is equally sensitive
to CDF differences at all quantiles, the KS test is more sensitive at the
median than in the tails. A straightforward comparison of these test
statistics across timescales is not very meaningful because sample sizes at
the daily and monthly timescales differ by a factor of 30, which implies
that the same value of a test statistic has different statistical
significance at the daily and monthly timescales. A better comparability can
be achieved by comparing the test statistic's p value, which represents the
statistical significance of CDF differences. In the present cross validation,
the CDFs compared are based on bias-corrected E2OBS and the corresponding SRB
data, and a higher p value indicates more similar CDFs and therefore a
better bias correction. For details of the calculation of p values of the
two-sample KS and Kuiper's two-sample test statistics see Appendix .
Global distributions of p values of two-sample test statistics for seasonal
distributions of daily and monthly mean rlds and rsds are shown in
Fig. for the KS test and
Fig. for Kuiper's test. In accordance
with expectations, both tests indicate that CDFs are generally better
adjusted by BCvdpx than by BCvmpx at
the daily timescale and vice versa at the monthly timescale. However,
performance differences between BCvdpx and
BCvmpx are clearly more significant at the daily than at
the monthly timescale. This suggests that bias-correcting at the daily
instead of at the monthly timescale yields bias decrements at the daily timescale that exceed bias increments at the monthly timescale. Therefore, bias
correction at the daily timescale is deemed more effective overall than bias
correction at the monthly timescale.
To elaborate on this further, the p=10 % significance level is marked by a
grey horizontal line in all panels of
Figs. and
and is to be compared with the 10th
percentiles of the global distributions of p values of the two-sample test
statistics. Any coincidence of such a 10th percentile with the 10 %
significance level suggests that the corresponding p-value distribution is
in agreement with the null hypothesis of the respective test. Since the null
hypothesis of both tests is that the samples compared are from the same
underlying distribution, such a coincidence suggests that the bias correction
which produced one of the samples compared worked perfectly within the limits
of sampling uncertainty. Similarly, 10th percentiles of p-value
distributions above (below) the 10 % significance level suggest
overcorrections (undercorrections) in terms of sampling uncertainty. In that
sense, the BCvtpx methods generally overcorrect at the
monthly timescale and undercorrect at the daily timescale.
The KS and Kuiper's test statistics also confirm the finding of
Sect. that at the daily timescale, the
BCvda1 methods outperform the BCvdb1
methods for longwave radiation and vice versa for shortwave radiation. This
holds true for all seasons and irrespective of CDF differences being
generally greater in summer and winter (DJF and JJA) than in the transition
seasons (MAM and SON) both before and after bias correction. Moreover, the
test statistics find both BCvda1 and
BCvdb1 to outperform BCvda0 at the
daily timescale, which is in line with the finding of
Sect. that the BCvda0 methods
deflate day-to-day variability.
The fact that all BCvdp1 methods undercorrect at
the daily timescale demonstrates the imperfections of these parametric
quantile mapping methods. The remaining CDF differences must be linked to
imperfect bias corrections of moments of higher-than-second order since
multi-year mean values and standard deviations are well adjusted by design.
To illustrate this, relative skewness biases remaining after bias correction
with BCvdp1 are shown to exceed 50 % (median over
calendar months × validation data samples) in many regions
(Fig. ).
Another manifestation of the imperfections are remaining biases in the tails
of the distribution of daily mean rlds and rsds. These must be larger than
the remaining median biases because p values of Kuiper's test statistics
for these distributions are generally larger than those of the corresponding
KS test statistics.
Spatial disaggregation and sub-SRB-grid-scale spatial variability
As outlined in Sect. , the BCvtp1
approach to the disaggregation of bias-corrected daily mean rlds and rsds
values from the SRB to the E2OBS grid scale is based on the implicit
assumption that the original E2OBS values of daily mean radiation onto the
four E2OBS grid cells contained in one SRB grid cell must all be too low
(high) if their area-weighted average is too low (high). The four original
values are then all increased (decreased) by the BCvtp1 method. In
order to account for their upper (lower) physical bounds, the increases
(decreases) are performed by a common scaling factor applied to the distances to
these bounds. This leads to a reduction of the differences among the four
values (this is necessarily true if the four bounds are equal; it is true in most cases if they are similar), i.e., to a deflation of sub-SRB-grid-scale spatial variability.
In order to illustrate and quantify the extent of this variability deflation
and compare the BCvtp0, BCvtp1, and BCvtp2
methods in terms of their impact on sub-SRB-grid-scale spatial variability,
the root-mean-square deviation (RMSD) of the four E2OBS-grid-scale values of
daily mean radiation per SRB grid cell from their area-weighted average is
calculated over all days of a given calendar month both before and after bias
correction with either method. Median relative bias-correction-induced
changes of these RMSDs are depicted in
Fig. and demonstrate that
BCvda1 indeed generally deflates them, in some regions by
more than 20 % (median over calendar months) for both longwave and
shortwave radiation. In contrast, BCvtp0 and BCvtp2
deflate or inflate them depending on variable and region.
In an analogous manner, such RMSDs can be computed based on data from the
four E2OBS grid cells contained in one staggered SRB grid cell, where the
staggered SRB grid is a regular
1.0∘×1.0∘ latitude–longitude grid
shifted by 0.5∘ latitude and 0.5∘ longitude
relative to the SRB grid, i.e., every staggered SRB grid cell contains
E2OBS grid cells contained in four different SRB grid cells. Median relative
bias-correction-induced changes of these RMSDs are also depicted in
Fig. . Ideally, bias-correction-induced changes of RMSDs from SRB and staggered SRB grid cell mean
values would be equal. It would then be impossible to tell from their
comparison whether the bias correction's target distributions were defined on
the SRB or on the staggered SRB grid.
The BCvdp1 methods do not fulfil this criterion as they
deflate RMSDs from SRB grid cell mean values everywhere while inflating RMSDs
from staggered SRB grid cell mean values in many regions, in particular over
the tropical oceans. The criterion is much better fulfilled by the
BCvdp2 and BCvdp0 methods. The RMSDs
are generally greater after bias correction with BCvdp2
than with BCvdp0, i.e., BCvdp2
produces data with greater sub-SRB-grid-scale spatial variability than
BCvdp0. This difference is most visible for longwave
radiation, for which BCvdp0 produces a stark land–sea
contrast of RMSD changes with strong RMSD reductions over land whereas
BCvdp0 does so to a much lesser extent. This strong
deflation of sub-SRB-grid-scale spatial variability by
BCvtp0 is believed to be another artefact caused by the
bilinear interpolation of SRB data to the E2OBS grid.
Summary and conclusions
This article introduces various parametric quantile mapping methods for the
bias correction of E2OBS daily mean surface downwelling longwave and
shortwave radiation using the corresponding SRB data. The quantile mapping
methods differ in (i) the timescale at which they operate, (ii) if and how
they take physical upper radiation bounds into account, and (iii) how they
handle the spatial resolution gap between E2OBS and SRB.
A cross validation at the SRB grid scale demonstrates that statistics of
daily mean radiation are better corrected by methods operating at the daily
timescale than by methods operating at the monthly timescale, and vice
versa for statistics of monthly mean radiation. Since these performance
differences are statistically more significant at the daily than at the
monthly timescale, overall, bias correction at the daily timescale is
deemed more effective then bias correction at the monthly timescale.
The cross validation further suggests that it is generally worthwhile to
explicitly take physical upper radiation bounds into account during quantile
mapping. For shortwave radiation, different approaches to their estimation
are tested. A simple approach using running maximum values is found to
outperform a more complicated one based on daily mean insolation at the top
of the atmosphere (rsdt). This must be due to other factors besides rsdt that
influence the upper physical bounds of rsds. Atmospheric humidity is an
example for such a factor: The highest rsds values usually occur under
clear-sky conditions and they are higher the drier the atmosphere.
Atmospheric humidity in turn is limited by the water vapour holding capacity
of the atmosphere, which is controlled by atmospheric temperature. The
climatology of atmospheric temperature lags behind that of rsdt. Hence, the
climatology of the upper physical bounds of rsds can be expected to deviate
from the rsdt climatology.
The cross validation also reveals to what extent the bilinear spatial
interpolation of SRB data to the E2OBS grid prior to bias correction with the
BCvtp0 methods deflates day-to-day variability. This variability
deflation has a greater effect on bias correction performance than a change
of if and how physical upper radiation bounds are taken into account during
quantile mapping, but a much smaller effect than a change of the timescale
at which the quantile mapping is carried out.
Lastly, the cross validation at the daily timescale shows that none of the
quantile mapping methods tested here are perfect, concerning in particular the
adjustment of distribution tails and moments of higher-than-second order.
This indicates that the true distribution of rlds and rsds is not always
exactly normal or beta, as assumed by the parametric quantile mapping methods
tested here. Potentially, non-parametric quantile mapping methods (that do
not rely on such assumptions) could yield better cross-validation results as
long as overfitting is avoided e.g.,. However, an
introduction of and comparison to such methods is beyond the scope of this
article.
To bridge the spatial resolution gap between E2OBS and SRB, the methods used
for the production of EWEMBI rlds and rsds deterministically disaggregate the
E2OBS data previously aggregated to and bias-corrected at the SRB grid. It is
shown that the method used for that disaggregation introduces artefacts in
the sub-SRB-grid-scale spatial variability, which can be overcome by applying
quantile mapping directly at the E2OBS grid using either bilinearly
interpolated SRB data or target distribution parameters that are based on the
more coarsely resolved SRB data as well as on sub-SRB-grid-scale spatial
variability present in the original E2OBS data. This latter approach yields
both good cross-validation results at the SRB grid scale and suitable
adjustments of the sub-SRB-grid-scale spatial variability.
The best methods identified here are therefore BClda2 for rlds and
BCsdb2 for rsds. In comparison to BClda1 and
BCsda1 used for the production of EWEMBI rlds and rsds, bias
correction with these methods yields more natural sub-SRB-grid-scale spatial
variability and, in the case of rsds, slightly better cross-validation
results at the SRB grid scale.
The EWEMBI dataset is publicly available via https://doi.org/10.5880/pik.2016.004 (Lange, 2016).
Quantile mapping and statistical downscaling
Quantile mapping is used to adjust the distribution of values from a data
sample. In the context of bias correction, the distribution to be adjusted –
the source distribution – is believed or known to be more biased than the
distribution the source distribution is adjusted to – the target
distribution. In practise, source and target distributions are empirically
estimated from the respective samples, in the present case of E2OBS and SRB
radiation data, in the form of cumulative distribution functions (CDFs)
FE2OBS and FSRB, respectively. Quantile mapping is
then defined by
x↦FSRB-1(FE2OBS(x)),
where FSRB-1(FE2OBS(⋅)) is called the transfer function.
Quantile mapping is called parametric if the CDFs are assumed to take certain
functional forms. Their estimation then reduces to the estimation of the
parameters of these functions. Otherwise, quantile mapping is called
non-parametric and CDFs are estimated by estimating selected quantiles,
between and beyond which quantiles are interpolated and extrapolated,
respectively e.g.,.
In the present study, source and target distributions are assumed to be
normal or beta distributions. Mean values and variances of normal
distributions are estimated by running mean values of multi-year daily sample
mean values and variances. Lower and upper bounds of beta distributions are
set to zero and estimated by physical upper limits of daily mean radiation,
respectively. Shape parameters of beta distributions are estimated with the
method of moments using running mean values of multi-year
daily sample mean values and variances.
Bias correction includes a spatial disaggregation or downscaling step if the
data behind source and target distributions have different spatial
resolution, as in the present case, or represent area mean values and point
values, as in the case of quantile mapping between gridded and station data.
If the data behind the target distribution have higher resolution or represent
finer spatial scales than the data behind the source distribution, then
quantile mapping may lead to both temporal and spatial variability inflation
. For the reverse case, the present study shows how
quantile mapping may lead to both temporal and spatial variability deflation.
suggests solving the inflation issue with stochastic
downscaling. It is shown here that the deflation issue of the reverse case
can also be overcome with deterministic downscaling at the transfer function
level.
Daily mean insolation at the top of the atmosphere
Over the course of a year, the total solar irradiance, S, varies according
to S=S0(1+ecos(Θ))2, where S0=1360.8 W m-2 is
the solar constant , e=0.0167086 is the Earth's current
orbital eccentricity, and Θ is the angle to the Earth's position from
its perihelion, as seen from the Sun. If the orbital angular velocity of the
Earth is approximated to vary sinusoidally in time then the total solar
irradiance on day n after 1 January of the first year of a 4-year cycle
including one leap year is approximately given by
S=S01+ecos2πn-2365.25+2esin2πn-2365.252,
since S is at its maximum when the Earth is at its perihelion, which on average occurs on 3 January.
The daily mean insolation at the top of the atmosphere, rsdt, at some fixed
geolocation depends on the location's latitude, ϕ, and on the
declination of the Sun, δ, which varies over the course of a year. On
day n after 1 January of the first year of a 4-year cycle including one
leap year, the declination of the Sun is approximately given by
sinδ=cos2πn+10365.25+2esin2πn-2365.25sinδmin,
since δ is at its minimum value δmin=-23.4392811∘ at the December solstice, which on average occurs
on 22 December. Latitude and declination of the Sun determine the hour angle
at sunrise, h, according to
cosh=min{1,max{-1,-tanϕtanδ}}.
The daily mean insolation at the top of the atmosphere at latitude ϕ on day n is then given by
rsdt=Sπ(hsinϕsinδ+sinhcosϕcosδ).
For a given latitude, the rsdt climatology used to estimate the upper bounds
of the climatological beta distribution of rsds in the BCsdax
methods is derived using Eqs. ()–() to compute rsdt
over a 4-year cycle including one leap year and then averaging calendar
day values over the four cases of leap year occurrence in the 4-year
cycle.
Two-sample Kolmogorov–Smirnov test and Kuiper's two-sample test
The overall effectivity of the bias correction methods introduced in this
study is measured by similarities of empirical CDFs of SRB and E2OBS data
before and after bias correction using the two-sample Kolmogorov–Smirnov (KS)
test and Kuiper's two-sample test
. Let F1 be the empirical CDF of
uncorrected or corrected daily or monthly mean longwave or shortwave E2OBS
data for one particular grid cell, calendar month, and validation data sample,
with all corresponding values pooled into one distribution, and let F2 be
the empirical CDF of the corresponding SRB data. Then the two-sample KS test
statistic, D, and Kuiper's two-sample test statistic, V, of these CDFs
are given by
D=suprF1(r)-F2(r),V=suprF1(r)-F2(r)+suprF2(r)-F1(r).
The null hypothesis of both the KS test and Kuiper's test is that the two
data samples whose empirical CDFs are compared have the same underlying
distribution. According to Sect. 14.3, the
probability p of incorrectly rejecting this null hypothesis can be
approximated by
p=1-Fn+0.12+0.11/nDandp=1-Gn+0.155+0.24/nV
for the KS test and Kuiper's test, respectively, where F and G are the
CDFs of the asymptotic distributions of nD and nV,
respectively, n=n1n2/(n1+n2) is the effective sample size, and
n1 and n2 are the sizes of the samples behind F1 and F2,
respectively. This approximation of the true p value is not only
asymptotically accurate but already quite good for n≥4see.
In order to adjust these p values for potential autocorrelations in the
samples compared here, which are in fact time series, n1 and n2 in the
formula for n are replaced by n1(1-ρ1) and n2(1-ρ2),
respectively, as proposed by , where the autoregression
coefficients ρ1 and ρ2 of first-order autoregressive processes
fitted to the time series are estimated by the respective sample
autocorrelation at lag one.
Window length for running mean and maximum calculations
The climatologies of mean values, variances, and upper bounds of daily mean
radiation estimated by the BCvdpx methods are based on
running mean values of empirical multi-year daily mean values, variances, and
running maximum values, respectively. A common window length of 25 days is
used for these running mean and maximum value calculations (see Table
). An obvious question is how sensitive the bias
correction results are to the choice of this window length.
The question is addressed here via variants of the BCvda1
methods that use uneven window lengths between 10 and 40 days for their
running mean and maximum value calculations and are otherwise identical to
the BCvda1 method introduced in
Sect. . The performance of these
BCvda1 variants is then quantified by p values of
two-sample KS statistics of bias-corrected E2OBS data cross validated against
SRB data (see Sect. and Appendix
). The window lengths that maximise these p values
vary considerably with location, calendar month, and calibration data sample
(Fig. ). The
reason for this high variability is illustrated in
Fig. , where the overall
performance of the BCvda1 variants, quantified by
p values of two-sample KS statistics aggregated over time (calendar months)
and space (grid cells), is shown to only weakly depend on the chosen window
length.
The optimal window length is thus highly uncertain. For longwave (shortwave)
radiation, the overall performance of the BCvda1 variants
is slightly higher for window lengths from the upper (lower) end of the
investigated range (Fig. ). For
practical matters, one can apply the methods using any window length between
10 and 40 days and expect similarly well adjusted radiation biases. The
choice of 25-day running windows made here for both longwave and shortwave
radiation ensures a close-to-optimal performance of the
BCvda1 methods for both variables.
Optimal window length for running mean and maximum calculations that
precede the estimation of parameters of the climatological distributions of
longwave (v=l; top) and shortwave (v=s; bottom) radiation that are used for bias correction with
BCvda1 (see Table ). Window lengths
are varied between 10 and 40 days. Optimal window lengths maximise the
p value of the two-sample KS statistic of bias-corrected E2OBS data
cross validated against SRB data (see Sect. and Appendix
) and are determined individually for every grid
cell, calendar month (with all corresponding values pooled into one
distribution), and calibration data sample (every1st, every2nd). Zonal medians
of optimal window lengths for each month and calibration data sample are
shown in panels (a) and (c). Results are masked in
(c) where and when the monthly mean rsdt
(Eqs. –) is less than 1 W m-2.
Panels (b) and (d) show medians of optimal window lengths
over months and calibration data samples.
Dependence of two-sample KS statistic p values on window length
for different radiation types and calibration data samples (see text and
Fig. ). Plotted
are the grid-cell area-weighted 50th (a) and second (b)
percentiles of the natural logarithms of the p values over months,
latitudes, and longitudes.
The author declares that no competing interests are present.
Acknowledgements
The author is grateful to Katja Frieler, Jan Volkholz, and Alex Cannon for
various helpful discussions at different stages of this work, to Paul
W. Stackhouse Jr. for his guidance with SRB data products and the provision
of SRB elevation data, to Graham Weedon and Emanuel Dutra for their guidance
during the initial stage of assembling the EWEMBI dataset, and to the two
anonymous referees, who provided highly valuable comments on the discussion
paper version of this paper. This work has received funding from the
European Union's Horizon 2020 research and innovation programme under grant
agreement no. 641816 Coordinated Research in Earth Systems and Climate:
Experiments, kNowledge, Dissemination and Outreach
(CRESCENDO). The article processing charges
for this open-access publication were covered by the Potsdam
Institute for Climate Impact Research (PIK). Edited by:
Valerio Lucarini Reviewed by: two anonymous referees
ReferencesCalton, B., Schellekens, J., and Martinez-de la Torre, A.: Water Resource
Reanalysis v1: Data Access and Model Verification Results,
10.5281/zenodo.57760, 2016.Cannon, A. J.: Multivariate quantile mapping bias correction: an N-dimensional
probability density function transform for climate model simulations of
multiple variables, Clim. Dynam., 50, 1–19, 10.1007/s00382-017-3580-6,
2017.Chang, J., Ciais, P., Wang, X., Piao, S., Asrar, G., Betts, R., Chevallier, F.,
Dury, M., François, L., Frieler, K., Ros, A. G. C., Henrot, A.-J.,
Hickler, T., Ito, A., Morfopoulos, C., Munhoven, G., Nishina, K., Ostberg,
S., Pan, S., Peng, S., Rafique, R., Reyer, C., Rödenbeck, C., Schaphoff,
S., Steinkamp, J., Tian, H., Viovy, N., Yang, J., Zeng, N., and Zhao, F.:
Benchmarking carbon fluxes of the ISIMIP2a biome models, Environ. Res. Lett., 12, 045002, 10.1088/1748-9326/aa63fa, 2017.Dee, D. P., Uppala, S. M., Simmons, A. J., Berrisford, P., Poli, P., Kobayashi,
S., Andrae, U., Balmaseda, M. A., Balsamo, G., Bauer, P., Bechtold, P.,
Beljaars, A. C. M., van de Berg, L., Bidlot, J., Bormann, N., Delsol, C.,
Dragani, R., Fuentes, M., Geer, A. J., Haimberger, L., Healy, S. B.,
Hersbach, H., Hólm, E. V., Isaksen, L., Kållberg, P., Köhler, M.,
Matricardi, M., McNally, A. P., Monge-Sanz, B. M., Morcrette, J.-J., Park,
B.-K., Peubey, C., de Rosnay, P., Tavolato, C., Thépaut, J.-N., and Vitart,
F.: The ERA-Interim reanalysis: configuration and performance of the data
assimilation system, Q. J. Roy. Meteor. Soc., 137, 553–597, 10.1002/qj.828, 2011.Dutra, E.: Report on the current state-of-the-art Water Resources Reanalysis,
Earth2observe deliverable no. d.5.1, available at: http://earth2observe.eu/files/Public Deliverables (last access: 18 May 2018), 2015.
Fisher, R. A.: Frequency Distribution of the Values of the Correlation
Coefficient in Samples from an Indefinitely Large Population, Biometrika, 10,
507–521, 1915.
Fisher, R. A.: On the “probable error” of a coefficient of correlation
deduced from a small sample, Metron., 1, 3–32, 1921.Frieler, K., Lange, S., Piontek, F., Reyer, C. P. O., Schewe, J., Warszawski,
L., Zhao, F., Chini, L., Denvil, S., Emanuel, K., Geiger, T., Halladay, K.,
Hurtt, G., Mengel, M., Murakami, D., Ostberg, S., Popp, A., Riva, R.,
Stevanovic, M., Suzuki, T., Volkholz, J., Burke, E., Ciais, P., Ebi, K.,
Eddy, T. D., Elliott, J., Galbraith, E., Gosling, S. N., Hattermann, F.,
Hickler, T., Hinkel, J., Hof, C., Huber, V., Jägermeyr, J., Krysanova, V.,
Marcé, R., Müller Schmied, H., Mouratiadou, I., Pierson, D., Tittensor, D.
P., Vautard, R., van Vliet, M., Biber, M. F., Betts, R. A., Bodirsky, B. L.,
Deryng, D., Frolking, S., Jones, C. D., Lotze, H. K., Lotze-Campen, H.,
Sahajpal, R., Thonicke, K., Tian, H., and Yamagata, Y.: Assessing the impacts
of 1.5 ∘C global warming – simulation protocol of the Inter-Sectoral Impact
Model Intercomparison Project (ISIMIP2b), Geosci. Model Dev., 10, 4321–4345,
10.5194/gmd-10-4321-2017, 2017.Garratt, J. R.: Incoming Shortwave Fluxes at the Surface – A Comparison of GCM
Results with Observations, J. Climate, 7, 72–80,
10.1175/1520-0442(1994)007<0072:ISFATS>2.0.CO;2, 1994.Gennaretti, F., Sangelantoni, L., and Grenier, P.: Toward daily climate
scenarios for Canadian Arctic coastal zones with more realistic
temperature-precipitation interdependence, J. Geophys. Res.-Atmos., 120, 11862–11877, 10.1002/2015JD023890, 2015.Gudmundsson, L., Bremnes, J. B., Haugen, J. E., and Engen-Skaugen, T.:
Technical Note: Downscaling RCM precipitation to the station scale using
statistical transformations – a comparison of methods, Hydrol. Earth Syst.
Sci., 16, 3383–3390, 10.5194/hess-16-3383-2012, 2012.Harris, I., Jones, P. D., Osborn, T. J., and Lister, D. H.: Updated
high-resolution grids of monthly climatic observations – the CRU TS3.10
Dataset, Int. J. Climatol., 10.1002/joc.3711, 2013.Hempel, S., Frieler, K., Warszawski, L., Schewe, J., and Piontek, F.: A
trend-preserving bias correction – the ISI-MIP approach, Earth Syst. Dynam.,
4, 219-236, 10.5194/esd-4-219-2013, 2013.Iizumi, T., Takikawa, H., Hirabayashi, Y., Hanasaki, N., and Nishimori, M.:
Contributions of different bias-correction methods and reference
meteorological forcing data sets to uncertainty in projected temperature and
precipitation extremes, J. Geophys. Res.-Atmos., 122, 2017JD026613, 10.1002/2017JD026613,
2017.Ito, A., Nishina, K., Reyer, C. P. O., François, L., Henrot, A.-J.,
Munhoven, G., Jacquemin, I., Tian, H., Yang, J., Pan, S., Morfopoulos, C.,
Betts, R., Hickler, T., Steinkamp, J., Ostberg, S., Schaphoff, S., Ciais, P.,
Chang, J., Rafique, R., Zeng, N., and Zhao, F.: Photosynthetic productivity
and its efficiencies in ISIMIP2a biome models: benchmarking for impact
assessment studies, Environ. Res. Lett., 12, 085001, 10.1088/1748-9326/aa7a19, 2017.Jones, P. W.: First- and Second-Order Conservative Remapping Schemes for Grids
in Spherical Coordinates, Mon. Weather Rev., 127, 2204–2210,
10.1175/1520-0493(1999)127<2204:FASOCR>2.0.CO;2, 1999.Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L.,
Iredell, M., Saha, S., White, G., Woollen, J., Zhu, Y., Leetmaa, A.,
Reynolds, R., Chelliah, M., Ebisuzaki, W., Higgins, W., Janowiak, J., Mo,
K. C., Ropelewski, C., Wang, J., Jenne, R., and Joseph, D.: The NCEP/NCAR
40-Year Reanalysis Project, B. Am. Meteorol. Soc.,
77, 437–471, 10.1175/1520-0477(1996)077<0437:TNYRP>2.0.CO;2, 1996.Kiehl, J. T. and Trenberth, K. E.: Earth's Annual Global Mean Energy Budget,
B. Am. Meteorol. Soc., 78, 197–208,
10.1175/1520-0477(1997)078<0197:EAGMEB>2.0.CO;2, 1997.Kistler, R., Collins, W., Saha, S., White, G., Woollen, J., Kalnay, E.,
Chelliah, M., Ebisuzaki, W., Kanamitsu, M., Kousky, V., van den Dool, H.,
Jenne, R., and Fiorino, M.: The NCEP-NCAR 50-Year Reanalysis: Monthly Means
CD-ROM and Documentation, B. Am. Meteorol. Soc.,
82, 247–267, 10.1175/1520-0477(2001)082<0247:TNNYRM>2.3.CO;2, 2001.
Kolmogorov, A.: Sulla determinazione empirica di una leggi di distribuzione,
Giornale dell' Istituto Italiano degli Attuari, 4, 83–91, 1933.Kopp, G. and Lean, J. L.: A new, lower value of total solar irradiance:
Evidence and climate significance, Geophys. Res. Lett., 38, l01706, 10.1029/2010GL045777, 2011.Krysanova, V. and Hattermann, F. F.: Intercomparison of climate change impacts
in 12 large river basins: overview of methods and summary of results,
Climatic Change, 141, 363–379, 10.1007/s10584-017-1919-y, 2017.
Kuiper, N. H.: Tests concerning random points on a circle, in: Koninklijke
Nederlandse Akademie van Wetenschappen, vol. 63 of A, 38–47, 1962.Lange, S.: EartH2Observe, WFDEI and ERA-Interim data Merged and Bias-corrected
for ISIMIP (EWEMBI), 10.5880/pik.2016.004, 2016.Ma, Q., Wang, K., and Wild, M.: Evaluations of atmospheric downward longwave
radiation from 44 coupled general circulation models of CMIP5, J. Geophys. Res.-Atmos., 119, 4486–4497, 10.1002/2013JD021427, , 2014.Maraun, D.: Bias Correction, Quantile Mapping, and Downscaling: Revisiting the
Inflation Issue, J. Climate, 26, 2137–2143,
10.1175/JCLI-D-12-00821.1, 2013.Müller Schmied, H., Müller, R., Sanchez-Lorenzo, A., Ahrens, B., and Wild,
M.: Evaluation of Radiation Components in a Global Freshwater Model with
Station-Based Observations, Water, 8, 450, 10.3390/w8100450,
2016.Ruane, A. C., Goldberg, R., and Chryssanthacopoulos, J.: Climate forcing
datasets for agricultural modeling: Merged products for gap-filling and
historical climate series estimation, Agr. Forest Meteorol.,
200, 233–248, 10.1016/j.agrformet.2014.09.016, 2015.Rust, H. W., Kruschke, T., Dobler, A., Fischer, M., and Ulbrich, U.:
Discontinuous Daily Temperatures in the WATCH Forcing Datasets, J. Hydrometeorol., 16, 465–472, 10.1175/JHM-D-14-0123.1, 2015.Sheffield, J., Goteti, G., and Wood, E. F.: Development of a 50-Year
High-Resolution Global Dataset of Meteorological Forcings for Land Surface
Modeling, J. Climate, 19, 3088–3111, 10.1175/JCLI3790.1, 2006.Smirnov, N.: Table for Estimating the Goodness of Fit of Empirical
Distributions, Ann. Math. Stat., 19, 279–281,
10.1214/aoms/1177730256, 1948.
Stackhouse Jr., P. W., Gupta, S. K., Cox, S. J., Mikovitz, C., Zhang, T., and
Hinkelman, L. M.: The NASA/GEWEX surface radiation budget release 3.0:
24.5-year dataset, Gewex news, 21, 10–12, 2011.Stephens, M. A.: The Goodness-Of-Fit Statistic Vn: Distribution and
Significance Points, Biometrika, 52, 309–321, 10.2307/2333685, 1965.
Stephens, M. A.: Use of the Kolmogorov-Smirnov, Cramer-Von Mises and Related
Statistics Without Extensive Tables, J. R. Stat. Soc. Series B. Met., 32, 115–122, 1970.Switanek, M. B., Troch, P. A., Castro, C. L., Leuprecht, A., Chang, H.-I.,
Mukherjee, R., and Demaria, E. M. C.: Scaled distribution mapping: a bias
correction method that preserves raw climate model projected changes, Hydrol.
Earth Syst. Sci., 21, 2649–2666, 10.5194/hess-21-2649-2017,
2017.Trenberth, K. E., Fasullo, J. T., and Kiehl, J.: Earth's Global Energy Budget,
B. Am. Meteorol. Soc., 90, 311–323, 10.1175/2008BAMS2634.1, 2009.Uppala, S. M., Kållberg, P. W., Simmons, A. J., Andrae, U., Bechtold, V.
D. C., Fiorino, M., Gibson, J. K., Haseler, J., Hernandez, A., Kelly, G. A.,
Li, X., Onogi, K., Saarinen, S., Sokka, N., Allan, R. P., Andersson, E.,
Arpe, K., Balmaseda, M. A., Beljaars, A. C. M., Berg, L. V. D., Bidlot, J.,
Bormann, N., Caires, S., Chevallier, F., Dethof, A., Dragosavac, M., Fisher,
M., Fuentes, M., Hagemann, S., Hólm, E., Hoskins, B. J., Isaksen, L.,
Janssen, P. A. E. M., Jenne, R., Mcnally, A. P., Mahfouf, J.-F., Morcrette,
J.-J., Rayner, N. A., Saunders, R. W., Simon, P., Sterl, A., Trenberth,
K. E., Untch, A., Vasiljevic, D., Viterbo, P., and Woollen, J.: The ERA-40
re-analysis, Q. J. Roy. Meteor. Soc., 131,
2961–3012, 10.1256/qj.04.176, 2005.Veldkamp, T. I. E., Wada, Y., Aerts, J. C. J. H., Döll, P., Gosling, S. N.,
Liu, J., Masaki, Y., Oki, T., Ostberg, S., Pokhrel, Y., Satoh, Y., Kim, H.,
and Ward, P. J.: Water scarcity hotspots travel downstream due to human
interventions in the 20th and 21st century, Nat. Commun., 8, 15697, 10.1038/ncomms15697, 2017.
Vetterling, W. T., Press, W. H., Teukolsky, S. A., and Flannery, B. P.:
Numerical Recipes in C (The Art of Scientific Computing), Cambridge
University Press, 2nd edn., 620–628, 1992.Weedon, G. P., Gomes, S., Viterbo, P., Österle, H., Adam, J. C., Bellouin,
N., Boucher, O., and Best, M.: The WATCH forcing data 1958–2001: A
meteorological forcing dataset for land surface and hydrological models,
Technical report no. 22, available at: http://www.eu-watch.org/publications/technical-reports (last access: 18 May 2018), 2010.Weedon, G. P., Gomes, S., Viterbo, P., Shuttleworth, W. J., Blyth, E.,
Österle, H., Adam, J. C., Bellouin, N., Boucher, O., and Best, M.: Creation
of the WATCH Forcing Data and Its Use to Assess Global and Regional Reference
Crop Evaporation over Land during the Twentieth Century, J. Hydrometeorol., 12, 823–848, 10.1175/2011JHM1369.1, 2011.Weedon, G. P., Balsamo, G., Bellouin, N., Gomes, S., Best, M. J., and Viterbo,
P.: The WFDEI meteorological forcing data set: WATCH Forcing Data methodology
applied to ERA-Interim reanalysis data, Water Resour. Res., 50,
7505–7514, 10.1002/2014WR015638, 2014.Wild, M., Folini, D., Schär, C., Loeb, N., Dutton, E. G., and
König-Langlo, G.: The global energy balance from a surface perspective,
Clim. Dynam., 40, 3107–3134, 10.1007/s00382-012-1569-8, 2013.Wild, M., Folini, D., Hakuba, M. Z., Schär, C., Seneviratne, S. I., Kato,
S., Rutan, D. A., Ammann, C., Wood, E. F., and König-Langlo, G.: The
energy balance over land and oceans: an assessment based on direct
observations and CMIP5 climate models, Clim. Dynam., 44, 3393–3429,
10.1007/s00382-014-2430-z, 2015.
Wilks, D. S.: Statistical Methods in the Atmospheric Sciences, Academic Press,
San Diego, CA, 1995.Xu, X.: Methods in Hypothesis Testing, Markov Chain Monte Carlo and
Neuroimaging Data Analysis, PhD thesis, Harvard University, available at: http://nrs.harvard.edu/urn-3:HUL.InstRepos:11108711 (last access: 18 May 2018), 2013.Zhao, M. and Dirmeyer, P. A.: Production and analysis of GSWP-2 near-surface
meteorology data sets, Center for Ocean-Land-Atmosphere Studies
Calverton, available at: http://ww.w.monsoondata.org/gswp/gswp2data.pdf (last access: 18 May 2018), 2003.