Page 1
Geosci. Model Dev., 12, 3055–3070, 2019
https://doi.org/10.5194/gmd-12-3055-2019
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.
Trend-preserving bias adjustment and statistical downscaling
with ISIMIP3BASD (v1.0)
Stefan Lange
Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association,
P.O. Box 60 12 03, 14412 Potsdam, Germany
Correspondence:
Stefan Lange (slange@pik-potsdam.de)
Received: 7 February 2019 – Discussion started: 7 March 2019
Revised: 13 June 2019 – Accepted: 24 June 2019 – Published: 17 July 2019
Abstract.
In this paper I present new methods for bias adjust-
ment and statistical downscaling that are tailored to the re-
quirements of the Inter-Sectoral Impact Model Intercompar-
ison Project (ISIMIP). In comparison to their predecessors,
the new methods allow for a more robust bias adjustment
of extreme values, preserve trends more accurately across
quantiles, and facilitate a clearer separation of bias adjust-
ment and statistical downscaling. The new statistical down-
scaling method is stochastic and better at adjusting spatial
variability than the old interpolation method. Improvements
in bias adjustment and trend preservation are demonstrated
in a cross-validation framework.
1
Introduction
Bias adjustment in climate research is the adjustment of
statistics of climate simulation data for the purpose of mak-
ing them more similar to climate observation data. In many
application cases, these climate simulation and observation
data have different spatial resolution. In most of these cases,
the climate observation data are more highly resolved. In any
of these cases, bias adjustment requires bridging the resolu-
tion gap.
In previous phases of the Inter-Sectoral Impact Model
Intercomparison Project (ISIMIP; Warszawski et al., 2014;
Frieler et al., 2017), climate simulation data were always
more coarsely resolved than the climate observation data
used for their bias adjustment, and the goal of this bias ad-
justment was not just to remove systematic biases from the
simulation data but also to increase their spatial resolution to
that of the observation data. In application cases like these,
bias adjustment as it is commonly understood involves two
distinct problems: (i) the actual bias adjustment at the spatial
resolution of the simulation data and (ii) a statistical down-
scaling to the spatial resolution of the observation data.
Commonly, the bulk of resources for the development of
solutions to these problems is allocated to problem (i), and
problem (ii) is solved by a mere spatial interpolation of the
simulation data to the spatial resolution of the observation
data prior to bias adjustment. For example, this approach
was adopted in the ISIMIP Fast Track (Hempel et al., 2013),
in ISIMIP2b (Frieler et al., 2017) and for the generation of
the NASA Earth Exchange Global Daily Downscaled Pro-
jections data set (NEX-GDDP; Thrasher et al., 2012). The
simplicity of this approach comes at a price if, as usual, the
same univariate bias adjustment method is independently ap-
plied in every cell of the observation data grid. The bias ad-
justment then retains the spatial coherence of the interpolated
simulation data and inflates temporal variability at their orig-
inal spatial resolution (Maraun, 2013).
These issues can be overcome by spatially multivariate
bias adjustment or, as suggested by Maraun (2013), using a
statistical downscaling method which is able to add the spa-
tiotemporal variability that is missing at the simulation data
resolution. He argues that such a method should be stochas-
tic, given the multivalued nature of statistical downscaling
(there are infinitely many high-resolution fields compatible
with the same low-resolution field) and the multifaceted in-
flation issues caused by deterministic methods such as spatial
interpolation.
In this paper, I present the bias adjustment and statisti-
cal downscaling methods to be used in phase 3 of ISIMIP.
These methods have been developed following the paradigm
Published by Copernicus Publications on behalf of the European Geosciences Union.
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S. Lange: ISIMIP3BASD
of a clear separation of bias adjustment and statistical down-
scaling. In ISIMIP3, climate simulation data shall first be
bias-adjusted at their original spatial resolution using spa-
tially aggregated climate observation data. In a second step,
their spatial resolution shall be increased using the original
climate observation data and a stochastic statistical down-
scaling method.
In addition to this paradigm shift, the new bias adjust-
ment method has been developed to work better than its
predecessor in several respects. The following design deci-
sions were taken in this context. While structurally differ-
ent bias adjustment methods (including but not restricted to
quantile mapping methods) were used for different climate
variables in ISIMIP2b (Frieler et al., 2017), the ISIMIP3
method applies a newly developed quantile mapping method
to all climate variables since this allows for the controlled
adjustment of biases in all quantiles. The new method is
approximately trend-preserving in all quantiles and there-
fore features a more comprehensive trend preservation than
the ISIMIP2b method. The new quantile mapping method is
parametric because this promises a more robust adjustment
of biases in extreme quantiles than nonparametric quantile
mapping (Switanek et al., 2017). The new bias adjustment
method also includes a modified version of the event likeli-
hood adjustment introduced by Switanek et al. (2017). This
new feature facilitates a confinement of extreme values to the
physically plausible range, which had to be enforced using
cap values in ISIMIP2b.
The remainder of this paper is organized as follows. Cli-
mate simulation and observation data used in this study are
described in Sect. 2. Details of the ISIMIP2b and ISIMIP3
bias adjustment and statistical downscaling methods are pre-
sented in Sect. 3. Also in Sect. 3 I explain how the new and
old methods are tested in the following. Test results are pre-
sented and compared in Sect. 4. Conclusions are drawn in
Sect. 5.
2
Data
2.1
Climate simulation data
Climate simulation data are taken from the fifth phase of
the Coupled Model Intercomparison Project (CMIP5; Tay-
lor et al., 2011). I use data produced with the four climate
models that were also used in ISIMIP2b (GFDL-ESM2M,
HadGEM2-ES, IPSL-CM5A-LR, MIROC5; Frieler et al.,
2017). For bias adjustment at 2
◦
spatial resolution, daily data
for 10 variables (see Table 1) are conservatively interpolated
(Jones, 1999) to a global 2
◦
×
2
◦
latitude–longitude grid. I
concatenate output data of the historical CMIP5 experiment
with output data of the rcp85 CMIP5 experiment to obtain
climate simulation data representing the historical time pe-
riod 1980–2015. Only output data of the rcp85 CMIP5 exper-
iment are used to obtain climate simulation data representing
the future time period 2064–2099.
2.2
Climate observation data
As observational reference data for bias adjustment and sta-
tistical downscaling I use the EartH2Observe, WFDEI and
ERA-Interim data Merged and Bias-corrected for ISIMIP
(EWEMBI; Lange, 2019a), which cover the entire globe at
0
.
5
◦
spatial and daily temporal resolution from 1979 to 2016.
For a description of the EWEMBI data set including vari-
ables covered, data sources used, and bias adjustments ap-
plied, see Frieler et al. (2017, Sect. 3.1 and Table 1). For
statistical downscaling from 2 to 1
◦
and for bias adjustment
at 2
◦
spatial resolution, these data are conservatively aggre-
gated to global 1
◦
×
1
◦
and 2
◦
×
2
◦
latitude–longitude grids,
respectively.
3
Methods
3.1
ISIMIP2b method
The ISIMIP2b bias adjustment and statistical downscaling
method is comprehensively described in Frieler et al. (2017),
Lange (2018), and Hempel et al. (2013). For statistical down-
scaling, simulation data are bilinearly interpolated to the ob-
servation data grid. These interpolated data are then bias-
adjusted in different ways for different climate variables.
For pr, psl, rlds, sfcWind, and tas, monthly mean values
are adjusted for the purpose of removing the bias in their his-
torical multiyear mean value (for abbreviations for variables,
see Table 1). This adjustment is done multiplicatively for pr,
rlds, and sfcWind and additively for psl and tas. In order to
preserve trends in multiyear monthly mean values, the same
scaling factor or offset is used in all application periods. In a
second step, day-to-day variability around the monthly mean
value is adjusted using transfer functions derived for every
calendar month from historical simulations and observations.
An indirect bias adjustment of tasmax and tasmin is done
by adjusting tasmax
−
tas and tas
−
tasmin using monthly scal-
ing factors which remove the bias in the mean value of all
historical daily values of these nonnegative variables from a
given calendar month. The adjusted values are then added to
and subtracted from bias-adjusted tas values in order to ob-
tain bias-adjusted tasmax and tasmin values, respectively.
Bias adjustment of rsds is done by parametric quantile
mapping using beta distributions with lower bounds of zero
and upper bounds estimated by rescaled climatologies of
downwelling shortwave radiation at the top of the atmo-
sphere. For trend preservation, upper bounds, mean values,
and variances of historical observations are modified using
simulated trends prior to quantile mapping. Using beta dis-
tributions with fixed lower and upper bounds of 0 and 100 %,
respectively, this method is also used to bias-adjust hurs.
Bias-adjusted prsn values are obtained by multiplying bias-
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Table 1.
Climate variables considered in this study.
Variable
Short name
Unit
Daily mean near-surface relative humidity
hurs
%
Daily mean precipitation
pr
kg m
−
2
s
−
1
Daily mean snowfall flux
prsn
kg m
−
2
s
−
1
Daily mean sea-level pressure
psl
Pa
Daily mean surface downwelling longwave radiation
rlds
W m
−
2
Daily mean surface downwelling shortwave radiation
rsds
W m
−
2
Daily mean near-surface wind speed
sfcWind
m s
−
1
Daily mean near-surface air temperature
tas
K
Daily maximum near-surface air temperature
tasmax
K
Daily minimum near-surface air temperature
tasmin
K
adjusted pr values with the original prsn-over-pr ratio. This
ratio is therefore not bias-adjusted.
3.2
ISIMIP3 method
The newly developed ISIMIP3 bias adjustment and statistical
downscaling method is comprehensively described in the fol-
lowing. It consists of a bias adjustment method that is applied
at the spatial resolution of the climate simulation data and
a statistical downscaling method that is applied to the bias-
adjusted climate simulation data for the purpose of increasing
its spatial resolution to that of the climate observation data.
These two new methods are presented in the following two
subsections.
3.2.1
ISIMIP3 bias adjustment method
This section describes the new bias adjustment method. Be-
fore diving into details I will explain the concept of the
method and outline how the new unified bias adjustment
framework can be customized for an application to the cli-
mate variables listed in Table 1.
The ISIMIP3 bias adjustment method is a parametric
quantile mapping method that has been designed to (i) ro-
bustly adjust biases in all percentiles of a distribution and
(ii) preserve trends in these percentiles. It is applicable for
bias adjustment of different kinds of climate variables includ-
ing those listed in Table 1. Like the ISIMIP2b bias adjust-
ment method, it is independently applied to every variable,
grid cell, and calendar month.
In order to overcome the multitude of approaches to bias
adjustment used for different variables in ISIMIP2b, the new
method features a unified framework, which can be cus-
tomized for an application to one particular climate variable.
Customization specifications for the variables considered
here are listed in Table 2. Note that biases in prsn, tasmax,
and tasmin are not adjusted directly. Instead, the ISIMIP3
method adjusts biases in pr and prsnratio
=
prsn
/
pr, and mul-
tiplies the resulting values to arrive at bias-adjusted prsn val-
ues. This is done in order to (i) ensure 0
≤
prsnratio
≤
1
and (ii) preserve trends in prsnratio. For tasmax and tas-
min, Piani et al. (2010) point out that an independent bias
adjustment of tas, tasmax, and tasmin may result in large
relative errors in the daily temperature range, tasrange
=
tasmax
−
tasmin, and the skewness of the daily temperature
cycle, tasskew
=
(
tas
−
tasmin
)/
tasrange. They also demon-
strate that these errors can be minimized by a direct and inde-
pendent bias adjustment of tas, tasrange, and tasskew. There-
fore, the ISIMIP3 method derives bias-adjusted tasmax and
tasmin values from bias-adjusted tas, tasrange, and tasskew
values.
In the following, I will describe the unified framework of
the ISIMIP3 bias adjustment method in detail. In this con-
text, let
x
obs
hist
be the time series of historical observations for
one climate variable, grid cell, and calendar month. Further,
let
x
sim
hist
and
x
sim
fut
be the simulated time series for the histori-
cal and future time period, respectively, and the same climate
variable, grid cell, and calendar month. Since the bias adjust-
ment method is trained on
x
obs
hist
and
x
sim
hist
, the historical time
period is also called the training period. Since it is applied
to
x
sim
fut
the future time period is also called the application
period.
The bias adjustment algorithm with inputs
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
and output
y
sim
fut
proceeds in the following steps, which
are explained in more detail below.
1. (For rsds only.) Scale values in
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
to
the interval
[
0
,
1
]
.
2. (For prsnratio only.) Replace missing values in
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
by random sampling from available val-
ues.
3. (For psl, rlds, and tas only.) Detrend
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
.
4. (For bounded variables only.) Randomize values be-
yond thresholds in
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
.
5. (For all variables.) Transfer the simulated climate
change signal for every distribution quantile from
x
sim
hist
and
x
sim
fut
to
x
obs
hist
. Let
x
obs
fut
be the resulting time series of
pseudo future observations.
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S. Lange: ISIMIP3BASD
Table 2.
Specification of the ISIMIP3 bias adjustment method for all climate variables considered in this study. Where a lower (upper)
bound is specified, no values smaller (greater) than this bound will occur in the bias-adjusted data. For every lower (upper) bound, a lower
(upper) threshold is defined, which is only slightly greater (smaller) than the bound. The lower (upper) threshold is used to adjust the
relative frequency of values smaller (greater) than the threshold. Note that the units of prsnratio
=
prsn
/
pr, tasrange
=
tasmax
−
tasmin, and
tasskew
=
(
tas
−
tasmin
)/
tasrange are 1, kelvin, and 1, respectively. For units of the other climate variables, see Table 1. The lower threshold
of pr is equivalent to 0.1 mm d
−
1
. For a description of the different kinds of trend preservation, see Eqs. (1)–(8) and the text around them.
Variable
Lower
Lower
Upper
Upper
Distribution
Trend
Detrending
Other
short name
bound
threshold
bound
threshold
preservation
hurs
0
0.01
100
99.99
beta
bounded
no
–
pr
0
0.1/86400
–
–
gamma
mixed
no
–
prsnratio
0
0.0001
1
0.9999
beta
bounded
no
sampling of missing values
psl
–
–
–
–
normal
additive
yes
–
rlds
–
–
–
–
normal
additive
yes
–
rsds
0
0.0001
1
0.9999
beta
bounded
no
upper bound scaling
sfcWind
0
0.01
–
–
Weibull
mixed
no
–
tas
–
–
–
–
normal
additive
yes
event likelihood not adjusted
tasrange
0
0.01
–
–
Rice
mixed
no
–
tasskew
0
0.0001
1
0.9999
beta
bounded
no
–
6. (For all variables.) Use parametric quantile mapping to
adjust the distribution of values in
x
sim
fut
to the distribu-
tion of values in
x
obs
fut
. For bounded variables, also bias-
adjust the frequency of values beyond thresholds. Let
y
sim
fut
be the resulting time series.
7. (For psl, rlds, and tas only.) Add trend subtracted from
x
sim
fut
in step 3 to
y
sim
fut
.
8. (For rsds only.) Scale values in
y
sim
fut
back to their actual
range.
Steps 1 and 8 are only applied to rsds and reflect that this
climate variable has a physical upper bound which varies
over the annual cycle. In order to fit this case into the uni-
fied framework, which at its core assumes constant bounds
and thresholds, rsds values are scaled to the interval
[
0
,
1
]
in
step 1 and back to their actual range in step 8. These scal-
ings are done using annual cycles of upper bounds that are
estimated from the rsds values in
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
. Fol-
lowing Lange (2018), annual cycles of upper bounds at daily
temporal resolution are estimated as running mean values of
running maximum values of multiyear daily maximum val-
ues. Here, a window length of 31 d is used for the running
window calculations. Let
b
obs
hist
,
b
sim
hist
, and
b
sim
fut
be these annual
cycles estimated for time series
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
, respec-
tively. Further, let
x
ij
be the value of one of these time series
on day
j
of year
i
, and let
b
j
be the upper bound for that day
of the year according to the corresponding annual cycle, then
x
ij
≤
b
j
holds true for all years
i
and
j
=
1
, . . .,
366. The
scaling in step 1 is done according to
x
ij
7−→
x
ij
/b
j
. The
scaling in step 8 requires an annual cycle of upper bounds
to the bias-adjusted rsds values. Let
b
obs
fut
denote this annual
cycle. Following Frieler et al. (2017, Eq. 2), it is estimated
according to
b
obs
fut
=
b
obs
hist
b
sim
fut
/b
sim
hist
. The scaling in step 8 is
then done according to
y
ij
7−→
y
ij
b
j
, where
y
ij
is the value
of
y
sim
fut
on day
j
of year
i
, and
b
j
is the upper bound for that
day of the year according to
b
obs
fut
.
Step 2 is only applied to prsnratio and reflects that val-
ues of this variable are missing on days of zero precipita-
tion because on these days the ratio prsn
/
pr is not defined.
In order to fit this case into the unified framework, which at
its core assumes gap-less time series, missing prsnratio val-
ues are replaced by random sampling from available values.
More precisely, for every missing value in
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
, an independent random number
p
is drawn from the in-
terval
[
0
,
100
] ⊂
R
with uniform probability, and the missing
value is replaced by the
p
th empirical percentile of all avail-
able values in
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
, respectively. This proce-
dure approximately preserves the distribution of values in the
time series.
Steps 3 and 7 are only applied to psl, rlds, and tas and re-
flect that these variables can have significant trends not only
between but also within training period and application pe-
riod. In order to prevent a confusion of these trends with in-
terannual variability during quantile mapping (steps 5 and
6), linear trends within
x
obs
hist
,
x
sim
hist
, and
x
sim
fut
are removed in
step 3 and restored in step 7. Trend lines
t
obs
hist
,
t
sim
hist
, and
t
sim
fut
are estimated at annual temporal resolution, i.e., by linear
regression of annual mean values of the daily values of the
respective time series. Let
x
ij
be the value of one of these
time series on day
j
of year
i
, and let
t
i
be the value for
year
i
of the corresponding trend line, which is shifted such
that
P
i
t
i
=
0. Then detrending in step 3 is done according to
x
ij
7−→
x
ij
−
t
i
, and the trend simulated within the applica-
tion period is restored in step 7 according to
y
ij
7−→
y
ij
+
t
i
,
where
y
ij
is the value of
y
sim
fut
on day
j
of year
i
, and
t
i
is the
value for that year of trend line
t
sim
fut
.
Step 4 is only applied to bounded variables, i.e., vari-
ables which have either a lower bound (and threshold) or
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S. Lange: ISIMIP3BASD
3059
an upper bound (and threshold) or both (see Table 2). These
bounds reflect physical limits to values these variables can
take. Thresholds located slightly above the lower bound and
slightly below the upper bound are used in step 6 to bias-
adjust the frequencies of occurrence of values close to the
bounds. In particular, the lower threshold of pr is used to
bias-adjust the dry-day frequency, i.e., the frequency of oc-
currence of pr
<
0
.
1 mm d
−
1
. In most cases, the simulated
dry-day frequency will be lower than the observed one (driz-
zle effect), and its bias adjustment is easily done by setting pr
values on some initially wet days to 0. Conversely, if the sim-
ulated dry-day frequency is too high, its bias adjustment re-
quires turning initially dry days into wet days. These days are
randomly selected following Cannon et al. (2015): all values
below the lower threshold,
α
, and above the upper threshold,
β
, are replaced by random numbers drawn from the open in-
terval
(a, α)
and
(β, b)
, where
a
and
b
are the lower bound
and the upper bound, respectively. These new values can then
be moved across the respective threshold by quantile map-
ping in step 6. In contrast to Cannon et al. (2015), random
numbers from
(a, α)
and
(β, b)
are not drawn with uniform
probability but with power-law probability that increases to-
wards the respective bound, as this approach is found to al-
leviate kinks in the distribution of wet-day precipitation after
bias adjustment.
Step 5 generates pseudo future observations, which are
needed for parametric quantile mapping in step 6. These
pseudo future observations are generated such that trends in
all quantiles between any two application periods are approx-
imately the same before and after quantile mapping. This
makes the bias adjustment method trend-preserving in all
quantiles. Different kinds of trends are preserved for different
climate variables (Table 2).
Pseudo future observations for one specific future time pe-
riod are generated by transferring simulated climate change
signals between the historical and the future time period to
the historical observations. This transfer is done quantile by
quantile using a nonparametric kind of quantile mapping. In
the following, I will describe the transfer for additive, multi-
plicative, mixed, and bounded trend preservation. Figure 1
provides an illustration for the former three of these four
cases.
In what follows, let
F
obs
hist
,
F
sim
hist
, and
F
sim
fut
be the empirical
cumulative distribution function of all values in
x
obs
hist
,
x
sim
hist
,
and
x
sim
fut
, respectively. Let
Q
obs
hist
,
Q
sim
hist
, and
Q
sim
fut
be the corre-
sponding quantile functions. Let
x
be one of the many values
of
x
obs
hist
, let
p
=
F
obs
hist
(x)
be the cumulative probability of
x
,
and let
y
be the pseudo future observation corresponding to
x
. Additive trend preservation is achieved by an additive cli-
mate change signal transfer, i.e., in this case,
y
is generated
according to
y
=
x
+
1
additive
(p),
where
(1)
1
additive
(p)
=
Q
sim
fut
(p)
−
Q
sim
hist
(p).
(2)
Additive trend preservation is the goal here for climate vari-
ables psl, rlds, and tas.
Multiplicative trend preservation is achieved by a multi-
plicative climate change signal transfer, i.e., in this case,
y
is
generated according to
y
=
x 1
multiplicative
(p),
where
(3)
1
multiplicative
(p)
=
max
0
.
01
,
min
100
, 1
∗
multiplicative
(p)
,
and
(4)
1
∗
multiplicative
(p)
=
(
1
if
Q
sim
hist
(p)
=
0
,
Q
sim
fut
(p)/Q
sim
hist
(p)
otherwise.
(5)
Note that the limits imposed in Eq. (4) are usually only
reached for very small values of
x
. Multiplicative trend
preservation is in most cases but not always the goal here
for climate variables pr, sfcWind, and tasrange. It is
not
the
goal here if
x
=
Q
obs
hist
(p)
is much larger than the correspond-
ing quantile of the historical simulations
Q
sim
hist
(p)
(this cor-
responds to a large negative bias in the historical time pe-
riod) because in this case even moderate multiplicative cli-
mate change signals
Q
sim
fut
(p)/Q
sim
hist
(p)
can result in unreal-
istically large
y
values, as illustrated in Fig. 1.
In order to prevent generating such unrealistically large
y
values, pseudo future observations for pr, sfcWind, and tas-
range are generated by a mixed (multiplicative and additive)
climate change signal transfer; i.e., for these climate vari-
ables,
y
is generated according to
y
=
γ (p) x 1
multiplicative
(p)
+
(
1
−
γ (p))(x
+
1
additive
(p)),
where
(6)
γ (p)
=
1
if
Q
sim
hist
(p)
≥
Q
obs
hist
(p),
0
.
5 1
+
cos
Q
obs
hist
(p)
/Q
sim
hist
(p)
−
1
π/
8
if
Q
sim
hist
(p) < Q
obs
hist
(p) <
9
Q
sim
hist
(p),
0
otherwise.
(7)
This translates to a multiplicative trend preservation for pos-
itive biases, an additive trend preservation for large negative
biases, and a mixed trend preservation for moderate negative
biases in the historical time period. A smooth transition from
multiplicative to additive trend preservation is facilitated by
the function
γ (p)
(Eq. 7 and Fig. 2).
For climate variables with both lower bound
a
and upper
bound
b
, climate change signals are transferred respecting
these bounds, i.e., for these climate variables,
y
is generated
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3060
S. Lange: ISIMIP3BASD
Figure 1.
Schematic of climate change signal transfer from simulations to observations for wet-day precipitation. Empirical cumulative
distribution functions of historical and future simulations and observations are displayed using a linear precipitation scale in
(a)
and a
logarithmic precipitation scale in
(b)
. Pseudo future observations generated preserving different kinds of trends are shown in red with
different line styles. For the 90th percentile, black double-headed arrows indicate additive trend preservation in
(a)
and multiplicative trend
preservation in
(b)
. Mixed trend preservation is explained in the text.
Figure 2.
Function
γ (p)
used to transition from multiplicative to
additive trend preservation in mixed trend preservation (Eqs. 6–7).
according to
y
=
a
+
(x
−
a) Q
sim
fut
(p)
−
a
/
Q
sim
hist
(p)
−
a
if
Q
sim
hist
(p) > Q
sim
fut
(p),
x
if
Q
sim
hist
(p)
=
Q
sim
fut
(p),
b
−
(b
−
x) b
−
Q
sim
fut
(p)
/
b
−
Q
sim
hist
(p)
otherwise.
(8)
Bounded trend preservation is the goal here for climate vari-
ables hurs, prsnratio, scaled rsds, and tasskew.
Step 6 is the core of the new unified bias adjustment frame-
work. For unbounded climate variables, it consists of a para-
metric quantile mapping of
x
sim
fut
to the pseudo future obser-
vations generated in step 5. For climate variables with at least
one bound, it consists of a bias adjustment of the frequency
of values beyond thresholds and a parametric quantile map-
ping of all other values in
x
sim
fut
.
Frequencies of values beyond thresholds (see Table 2) are
bias-adjusted as follows. For climate variables with a lower
bound
a
and lower threshold
α
, let
P
obs
hist
,
P
sim
hist
, and
P
sim
fut
be
the relative frequency of values smaller than
α
in
x
obs
hist
,
x
sim
hist
,
and
x
sim
fut
, respectively. Similar to step 5, a pseudo future ob-
servation of this frequency,
P
obs
fut
, is generated by transfer-
ring the simulated climate change signal to the historically
observed value,
P
obs
fut
=
P
obs
hist
P
sim
fut
/P
sim
hist
if
P
sim
hist
> P
sim
fut
,
P
obs
hist
if
P
sim
hist
=
P
sim
fut
,
1
−
1
−
P
obs
hist
1
−
P
sim
fut
/
1
−
P
sim
hist
otherwise.
(9)
Then, if
x
sim
fut
is of length
n
, the
nP
obs
fut
lowest values of
x
sim
fut
are set to
a
. Similarly, for climate variables with an upper
bound
b
and upper threshold
β
, the relative frequency of val-
ues greater than
β
is bias-adjusted by setting the
nP
obs
fut
high-
est values of
x
sim
fut
to
b
, where
P
obs
fut
is generated using Eq. (9)
with relative frequencies of values smaller than
α
replaced
by relative frequencies of values greater than
β
.
All other values in
x
sim
fut
(or all values in the case of an un-
bounded climate variable) are bias-adjusted using parametric
quantile mapping, the pseudo observations
x
obs
fut
generated in
step 5, as well as the historical observations and simulations
x
obs
hist
and
x
sim
hist
, respectively. Distributions used for paramet-
ric quantile mapping are the beta distribution for bounded
climate variables (hurs, prsnratio, scaled rsds, tasskew), the
gamma distribution for pr, the normal distribution for un-
bounded climate variables (psl, rlds, tas), the Weibull dis-
tribution for sfcWind, and the Rice distribution for tasrange.
For unbounded climate variables, distributions are fitted to
all values in
x
obs
hist
,
x
obs
fut
,
x
sim
hist
, and
x
sim
fut
. For climate variables
with a lower and/or upper bound, distributions are only fit-
Geosci. Model Dev., 12, 3055–3070, 2019
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S. Lange: ISIMIP3BASD
3061
ted to values greater than
α
and/or smaller than
β
in
x
obs
hist
,
x
obs
fut
, and
x
sim
hist
, and to all values in
x
sim
fut
that where not set to
a
or
b
in the first part of step 6. Let
ˆ
F
obs
hist
,
ˆ
F
obs
fut
,
ˆ
F
sim
hist
, and
ˆ
F
sim
fut
be the cumulative distribution functions of these fitted
distributions.
The parametric quantile mapping method used in ISIMIP3
is inspired by the scaled distribution mapping method intro-
duced by Switanek et al. (2017), which in addition to biases
in quantiles also adjusts biases in the likelihood of individ-
ual events. For the sake of argument, let me assume that the
number of values
ˆ
F
obs
hist
,
ˆ
F
obs
fut
,
ˆ
F
sim
hist
, and
ˆ
F
sim
fut
were fitted to
is the same for all four cumulative distribution functions, and
let
ˆ
x
sim
hist
,
ˆ
x
sim
hist
,
ˆ
x
sim
fut
, and
ˆ
x
sim
fut
be the lowest of these values,
respectively. Then
ˆ
x
sim
fut
is quantile-mapped according to (see
Eq. (15) for the definition of the logit function)
ˆ
x
sim
fut
7−→ ˆ
F
obs
fut
−
1
logit
−
1
L
obs
hist
+
1
log
−
odds
,
where
(10)
1
log
−
odds
=
max
−
log 10
,
min
log 10
, L
sim
fut
−
L
sim
hist
,
(11)
and
L
obs
hist
=
logit
ˆ
F
obs
hist
ˆ
x
obs
hist
,
(12)
L
sim
hist
=
logit
ˆ
F
sim
hist
ˆ
x
sim
hist
,
(13)
L
sim
fut
=
logit
ˆ
F
sim
fut
ˆ
x
sim
fut
.
(14)
Values of higher rank are quantile-mapped in the same way,
i.e., using Eqs. (10)–(14) and
ˆ
x
sim
hist
,
ˆ
x
sim
hist
,
ˆ
x
sim
fut
, and
ˆ
x
sim
fut
of
equal rank. Additional interpolations need to be introduced
in Eqs. (10)–(14) to make them work in the general case of
unequal sample sizes, as explained by Switanek et al. (2017).
Equations (10)–(14) result in a perfect match in distribu-
tion if training and application period are identical. In this
case,
1
log
−
odds
=
0 and likelihoods of events are mapped
from
ˆ
F
sim
fut
ˆ
x
sim
fut
= ˆ
F
sim
hist
ˆ
x
sim
hist
to
ˆ
F
obs
hist
ˆ
x
obs
hist
. In all other
cases, the simulated climate change signal in event likelihood
is transferred to the historically observed event likelihood
such that changes in odds are multiplicatively preserved. To
see that this is true, note that
logit
p
1
+
logit
p
2
−
logit
p
3
=
log
p
1
1
−
p
1
+
log
p
2
1
−
p
2
−
log
p
3
1
−
p
3
=
log
p
1
1
−
p
1
p
2
1
−
p
2
p
3
1
−
p
3
.
(15)
Asymptotically, i.e., for extreme values, the odds scaling
used here is equivalent to the return interval scaling used by
Switanek et al. (2017). The limits imposed in Eq. (11) are to
prevent the generation of unrealistic event likelihoods.
Note that in contrast to all other climate variables, the
likelihood of individual events is not adjusted for tas. In-
stead, in this case, Eqs. (10)–(14) are replaced by
ˆ
x
sim
fut
7−→
ˆ
F
obs
fut
−
1
ˆ
F
sim
fut
ˆ
x
sim
fut
. The reason for this exception is that
the event likelihood adjustment can produce artifacts if large
nonlinear trends are present within the training or applica-
tion period. Examples of such cases have (only) been found
for tas.
3.2.2
ISIMIP3 statistical downscaling method
This section describes the new statistical downscaling
method. Before diving into details I will explain the concept
of the method, reveal its algorithmic origin, and elaborate on
prerequisites and best practices of its application.
As described in the introduction, the ISIMIP3 bias adjust-
ment method shall be applied at the spatial resolution of
the climate simulation data using spatially aggregated cli-
mate observation data. Since the resulting data can be con-
sidered to have unbiased distributions of daily values per
climate variable, grid cell, and calendar month, their subse-
quent statistical downscaling should be done using a method
which preserves values at the aggregated spatial resolution.
The ISIMIP3 statistical downscaling method has this prop-
erty. Since the new method is based on the MBCn algorithm
by Cannon (2017) it is abbreviated to MBCnSD in the fol-
lowing.
The MBCnSD algorithm is independently applied to every
climate variable and calendar month. It requires the coarse
grid of the climate simulation data and the fine grid of the
climate observation data to be compatible in the sense that
every fine grid cell is entirely contained in one coarse grid
cell. For example, that is the case if the coarse and fine grid
are the global 2
◦
×
2
◦
and 1
◦
×
1
◦
latitude–longitude grid, re-
spectively, since then every coarse grid cell contains exactly
K
=
4 fine grid cells. In order to make MBCnSD applicable
in cases with originally incompatible coarse and fine grids as
well, the original climate simulation data need to be interpo-
lated (before bias adjustment) to a grid that is both compati-
ble with the fine grid and similar in resolution to the original
coarse grid.
The MBCn algorithm by Cannon (2017) is a multivari-
ate quantile mapping bias adjustment method. It is employed
here in the context of statistical downscaling because the
downscaling problem at hand can be regarded as yet another
bias adjustment problem: once the climate data to be down-
scaled have been broadcasted to the fine grid, their statistical
downscaling can be achieved by an adjustment of the multi-
variate distribution of all time series contained in one coarse
grid cell.
The MBCn algorithm applies a series of univariate non-
parametric quantile mappings along randomly chosen axes.
Mathematically, this is achieved by repeatedly rotating the
climate simulation and observation data using random
K
×
K
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3062
S. Lange: ISIMIP3BASD
Figure 3.
Two-dimensional illustration of one iteration of the mod-
ified MBCn algorithm used for statistical downscaling in ISIMIP3
(MBCnSD), which at its core consists of two steps. In the first step,
data point
A
is quantile-mapped to data point
B
like in the original
MBCn algorithm. In the second step, MBCnSD projects data point
B
onto the weighted sum-preserving hyperplane of data point
A
,
here with equal weights on all axes. The result is data point
C
.
orthogonal matrices, each time followed by
K
univariate
quantile mappings. This use of random rotation matrices
makes the ISIMIP3 statistical downscaling method stochas-
tic.
The MBCn algorithm cannot be used as it is to solve the
downscaling problem at hand because it does not have the
required preservation property. The preservation of values
at the aggregated spatial resolution translates to a preserva-
tion of the weighted sum of all time series contained in one
coarse grid cell. With MBCnSD, this is achieved by an ad-
ditional conservation step following the
K
univariate quan-
tile mappings in every iteration of the algorithm. For
K
=
2,
this is illustrated in Fig. 3. A corner case of what can hap-
pen without this additional step is shown in Fig. 4: for cer-
tain axes rotation sequences, the MBCn algorithm almost re-
verses the ranks of values along one axis, which results in
strongly changed aggregated values. Figure 4 also exempli-
fies that this is prevented by the MBCnSD algorithm.
If the resolution gap between climate simulation and ob-
servation data is large, then statistical downscaling can be
done in one big step or in multiple small steps. Vandal et al.
(2018) have shown that statistical downscaling with neural
networks works better in multiple small steps. For statistical
downscaling with the MBCnSD algorithm, both approaches
yield similar results. But as in the neural network case, down-
scaling in multiple small steps yields slightly smoother fields
than downscaling in one big step (Fig. 5) and is therefore
deemed the preferred approach.
In the following, I will describe the MBCnSD algorithm in
detail. In this context, let
x
sim
ij
be the previously bias-adjusted
Figure 4.
Statistical downscaling of artificial two-dimensional cli-
mate data with the original MBCn algorithm by Cannon (2017) and
the modified MBCn algorithm used for statistical downscaling in
ISIMIP3 (MBCnSD). Panel
(a)
shows an example of observation
and simulation data drawn from a bivariate standard normal distri-
bution with cross-correlation 0 and 1, respectively. Panels
(c)
–
(f)
show the result of statistical downscaling after 4
(c, d)
and 32
(e,
f)
iterations of the MBCn
(c, e)
and MBCnSD
(d, f)
algorithm. To
track changes, the color of a data point in panels
(c)
–
(f)
is the same
as the color of the corresponding original data point in panel
(a)
.
Gray lines in panels
(c)
–
(f)
represent the axes along which univari-
ate quantile mappings are applied in the respective iteration. Note
that MBCn and MBCnSD use the same sequence of axes rotations
here. Panel
(b)
shows the energy distance (Székely and Rizzo, 2013)
between observation and simulation data over iterations for 30 ran-
dom data samples (thin dashed lines) and on average over these 30
samples (thick solid lines).
climate simulation data to be statistically downscaled, with
i
being the time index and
j
being the coarse grid cell index.
Further, let
x
obs
ij k
be the historical climate observation data on
the fine grid, with
i
,
j
as for
x
sim
ij
, and
k
=
1
, . . ., K
being the
index for the fine grid cells contained in one coarse grid cell.
Finally, let
w
j k
be proportional to the area of fine grid cell
k
in coarse grid cell
j
. Then the MBCnSD algorithm works as
follows.
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S. Lange: ISIMIP3BASD
3063
Figure 5.
Statistical downscaling of precipitation from 2 to 0
.
5
◦
spatial resolution in two small steps (2 to 1
◦
and 1 to 0
.
5
◦
) versus
in one big step. In this example, the MBCnSD algorithm is applied
for statistical downscaling of spatially aggregated historical obser-
vation data. Shown are precipitation fields over Europe for one par-
ticular day: the original precipitation field in panel
(a)
and the result
of statistical downscaling in two small steps and in one big step in
panels
(b)
and
(c)
, respectively.
1. (For all variables.) Bilinearly interpolate
x
sim
ij
to the fine
grid. Let
x
sim
ij k
be the result.
2. (For bounded variables only.) Randomize values be-
yond thresholds in
x
sim
ij
,
x
sim
ij k
, and
x
obs
ij k
.
3. (For all variables.) Apply the core of the MBCnSD al-
gorithm independently to every coarse grid cell
j
. Let
y
sim
ij k
be the result.
4. (For bounded variables only.) De-randomize values be-
yond thresholds in
y
sim
ij k
.
Step 1 broadcasts the previously bias-adjusted climate
simulation data to the fine grid. This is done using bilinear in-
stead of conservative interpolation, which in this case would
Figure 6.
Statistical downscaling of precipitation from 1 to 0
.
5
◦
spatial resolution using different broadcasting methods in step 1 of
the MBCnSD algorithm. In this example, the algorithm is applied
for statistical downscaling of spatially aggregated historical obser-
vation data. Shown are downscaled precipitation fields over Europe
for one particular day, using bilinear and conservative interpolation
for broadcasting in panels
(a)
and
(b)
, respectively. The correspond-
ing original precipitation field is shown in panel
(a)
of Fig. 5.
be equivalent to setting
x
sim
ij k
=
x
sim
ij
for all
k
. Broadcasting
with bilinear interpolation is preferred because it results in
smoother fields than broadcasting with conservative interpo-
lation, as exemplified in Fig. 6. There are two reasons for
this. First, bilinear interpolation already generates some of
the spatial variability within each coarse grid cell that statis-
tical downscaling has to add, whereas conservative interpola-
tion does not. Therefore, the MBCnSD algorithm would have
to add more variability after conservative than bilinear inter-
polation, with the result of more noisy fields. Secondly, bi-
linear interpolation transfers spatial gradients between coarse
grid cells to the fine grid, whereas conservative interpolation
does not. The MBCnsD algorithm can then preserve these
gradients to the degree that they are meaningful, which re-
sults in smoother fields.
Steps 2 and 4 are only applied to variables which have
either a lower bound (and threshold) or an upper bound (and
threshold) or both. The bounds and thresholds used for statis-
tical downscaling are identical to those used for bias adjust-
ment (Table 2) for all climate variables except rsds. For sta-
tistical downscaling, rsds values are not scaled, have a lower
bound of 0, a lower threshold of 0.01 W m
−
2
, and no upper
bound (or threshold). The randomization itself works exactly
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S. Lange: ISIMIP3BASD
as in step 4 of the ISIMIP3 bias adjustment method. For de-
randomization, all values below the lower threshold are set to
the lower bound, and all values above the upper threshold are
set to the upper bound. Note that the MBCnSD algorithm is
applied for statistical downscaling of hurs, pr, prsnratio, psl,
rlds, rsds, sfcWind, tas, tasrange, and tasskew. Bias-adjusted
and statistically downscaled prsn, tasmax, and tasmin values
are then derived as described in Sect. 3.2.1.
Step 3 is the core of the MBCnSD algorithm and is ap-
plied independently to every coarse grid cell
j
. Therefore,
in the following, let
j
be arbitrary but fixed. Let
X
=
(X
i
)
be the vector with components
X
i
=
x
sim
ij
, and let
Y
=
(Y
ik
)
and
Z
=
(Z
ik
)
be matrices with components
Y
ik
=
x
sim
ij k
and
Z
ik
=
x
obs
ij k
, respectively. Further, let
w
=
P
k
w
j k
be the sum
of all fine grid cell area weights and let
e
w
=
q
P
k
w
2
j k
be
their root sum square. Let
W
=
(W
k
)
be the vector with com-
ponents
W
k
=
w
j k
/
e
w
.
The core of the MBCnSD algorithm proceeds in three sub-
steps, which I will refer to as 3a, 3b, and 3c in the following.
Sub-step 3a adjusts
Y
for the purpose of restoring the spa-
tially aggregated values which shall be preserved by the al-
gorithm but have been altered by bilinear interpolation in step
1. In addition, sub-step 3a adjusts
Z
for the purpose of trans-
ferring the simulated climate change signal to the historical
climate observation data on the fine grid. This signal is incor-
porated in
X
, given that
X
is the result of quantile mapping to
the pseudo future climate observation data generated in step
5 of the ISIMIP3 bias adjustment algorithm. Since the sim-
ulated climate change signal is only available at the coarse
resolution, it is transferred to
Z
at that resolution, and all sta-
tistical dependencies at higher resolution are left unchanged.
Mathematically, sub-step 3a proceeds as follows.
1. Generate a
K
×
K
orthogonal matrix
O
whose first col-
umn is equal to
W
(all other columns can be chosen at
will). Set
O
total
=
O
. Rotate
Y
,
Z
,
W
using
O
.
2. Set
Y
i
1
to
X
i
w/
e
w
to restore the spatially aggregated
values.
3. Do a nonparametric quantile mapping of
(Z
i
1
)
to
(X
i
w/
e
w)
to transfer the simulated climate change sig-
nal.
Here and in the following, to rotate
Y
,
Z
,
W
using
O
means
to apply the matrix multiplications
Y
7−→
YO
,
(16)
Z
7−→
ZO
,
(17)
W
7−→
(
W
T
O
)
T
=
O
T
W
,
(18)
and to set
Y
,
Z
,
W
to the respective result. To do a non-
parametric quantile mapping of
(A
i
)
to
(B
i
)
means to use
empirically estimated quantiles
a
p
and
b
p
of
(A
i
)
and
(B
i
)
,
respectively, corresponding to cumulative probabilities
p
∈
{
0 %
,
2 %
,
4 %
, . . .,
100 %
}
, with
a
0
=
min
i
A
i
,
a
1
=
max
i
A
i
,
and the same for
b
0
,
b
1
, to define a transfer function
f
using
the linearly interpolated quantile–quantile pairs
(a
p
, b
p
)
, to
then map
A
i
7−→
f (A
i
)
, and to set
A
i
to the result for all
i
.
In sub-step 3b, the following three steps are repeated ei-
ther a fixed number of times or until
Y
has converged to
Z
in distribution. The last two of these steps are illustrated in
Fig. 3.
1. Generate a random
K
×
K
orthogonal matrix
O
. Rotate
O
total
,
Y
,
Z
,
W
using
O
. Set
e
Y
=
Y
.
2. For all
k
, do a nonparametric quantile mapping of
(Y
ik
)
to
(Z
ik
)
.
3. Project
Y
onto the weighted sum-preserving hyperplane
of
e
Y
by subtracting
((
Y
−
e
Y
)
W
)
⊗
W
from
Y
.
In the first of these steps, random orthogonal matrices are
drawn from the circular real random matrix ensemble using
the algorithm by Mezzadri (2007). In the last step,
⊗
denotes
the outer product of two vectors. Note that the results pre-
sented in Sect. 4 are obtained using a fixed number of 20
iterations in sub-step 3b of the MBCnSD algorithm, as this
was deemed sufficient for the MBCn algorithm by Cannon
(2017).
In sub-step 3c, all data are rotated back to the original axes.
A last quantile mapping along these axes ensures that there
are no values out of bounds in the resulting data. Mathemat-
ically, sub-step 3c proceeds as follows.
1. Set
O
=
O
T
total
. Rotate
Y
,
Z
,
W
using
O
.
2. For all
k
, do a nonparametric quantile mapping of
(Y
ik
)
to
(Z
ik
)
.
For the arbitrary but fixed coarse grid cell
j
, the result
y
sim
ij k
of
step 3 of the MBCnSD algorithm is then given by
y
sim
ij k
=
Y
ik
.
3.3
Comparison
Results obtained with the ISIMIP2b and ISIMIP3 bias ad-
justment and statistical downscaling methods will be com-
pared in Sect. 4. This section describes how that comparison
will be done.
I will begin with results of bias adjustment applied at 2
◦
spatial resolution, i.e., using climate simulation and observa-
tion data both on the global 2
◦
×
2
◦
latitude–longitude grid.
In particular, I will compare the methods’ ability to (i) ad-
just biases in percentiles of distributions of daily values and
(ii) preserve trends in these percentiles. Percentiles chosen
for this comparison are the 5th, 50th, and 95th, representing
the lower tail, the center, and the upper tail of a distribution,
respectively. An exception is made for pr and prsn, for which
instead of the 5th percentile I consider the dry-day frequency,
i.e., the frequency of precipitation or snowfall flux, to be less
than 0.1 mm d
−
1
, and instead of the 50th and 95th percentile
Geosci. Model Dev., 12, 3055–3070, 2019
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S. Lange: ISIMIP3BASD
3065
of all values I consider the 50th and 95th percentile of all
values which exceed 0.1 mm d
−
1
, i.e., the 50th and 95th per-
centile of wet-day precipitation.
I will then compare results obtained with the ISIMIP2b
and ISIMIP3 statistical downscaling methods. To that end,
both downscaling methods are combined with the ISIMIP3
bias adjustment method. These bias adjustment–statistical
downscaling method combinations are abbreviated LI
+
BA
and BA
+
SD in the following. The combination LI
+
BA rep-
resents the ISIMIP2b approach to statistical downscaling and
therefore consists of a bilinear interpolation of the climate
simulation data from their original spatial resolution of 2 to
0
.
5
◦
followed by a bias adjustment using the climate observa-
tion data at their original spatial resolution of 0
.
5
◦
. The com-
bination BA
+
SD represents the ISIMIP3 approach to statis-
tical downscaling and therefore consists of a bias adjustment
at 2
◦
spatial resolution using the climate observation data ag-
gregated to that resolution followed by statistical downscal-
ing from 2 to 0
.
5
◦
in two steps (2 to 1
◦
and 1 to 0
.
5
◦
) using
the ISIMIP3 statistical downscaling method.
Results at 2 and 0
.
5
◦
spatial resolution will first be as-
sessed based on the same metrics used to compare the
ISIMIP2b and ISIMIP3 bias adjustment methods at 2
◦
spa-
tial resolution. This is done to demonstrate that the BA
+
SD
approach does not impair data quality with regard to bias
adjustment or trend preservation relative to the LI
+
BA ap-
proach. For the comparison at 2
◦
spatial resolution, the bias-
adjusted and statistically downscaled climate simulation data
are conservatively aggregated back to that resolution.
Secondly, I will compare results at 0
.
5
◦
spatial resolution
with regard to spatial variability within 2
◦
×
2
◦
grid cells. Two
ways of placing these grid cells will be considered. The first
way is to place their centers at odd-numbered latitudes and
longitudes (measured in degrees). These grid cells constitute
the regular 2
◦
×
2
◦
latitude–longitude grid of the original cli-
mate simulation data. The second way is to place their cen-
ters at even-numbered latitudes and longitudes (measured in
degrees). These grid cells form a grid that is staggered by 1
◦
latitude and 1
◦
longitude relative to the regular one.
For 2
◦
×
2
◦
grid cells placed in the first way it is ex-
pected that spatial variability within them is better adjusted
by BA
+
SD than by LI
+
BA by design. It is less clear if
this also holds true for spatial variability within staggered
2
◦
×
2
◦
grid cells since these contain time series whose sta-
tistical dependence is not adjusted by the ISIMIP3 statistical
downscaling method. In both cases, spatial variability within
a 2
◦
×
2
◦
grid cell is measured by the root-mean-square de-
viation (RMSD) of the 16 time series contained in that grid
cell from their spatial average: let
x
ij
be the value on day
i
in
0
.
5
◦
×
0
.
5
◦
grid cell
j
. Then, for time series of length
n
, the
RMSD is calculated according to
RMSD
=
v
u
u
u
t
1
n
n
X
i
=
1
1
16
16
X
j
=
1
x
ij
−
1
16
16
X
k
=
1
x
ik
!
2
.
(19)
The comparison of methods with regard to their ability
to adjust biases and spatial variability is done in a cross-
validation framework. This is done to prevent different ex-
tents of overfitting by different methods to dominate dif-
ferences in results. I first use odd-numbered years from the
time period 1980–2015 for training and even-numbered years
from the same time period for application. Secondly, I swap
these training and application years. Finally, I merge the
results of application to odd-numbered and even-numbered
years to arrive at bias-adjusted and statistically downscaled
data for cross-validation which fully cover the 1980–2015
time period. Compared to the more common use of con-
secutive and nonoverlapping time periods (here 1980–1997
and 1998–2015) for training and validation, the division into
even-numbered and odd-numbered years reduces the influ-
ence of climate trends on cross-validation results (Switanek
et al., 2017). The downside of the cross-validation framework
used here is that the resulting validation and training data sets
are rather similar in terms of decadal climate variability. For
the comparison of methods with regard to trend preservation,
I use the full 1980–2015 time period for training and the full
time periods 1980–2015 and 2064–2099 for application.
The metrics introduced above (RMSD, dry-day frequency,
percentiles) are calculated independently for every data
set (climate observations, original climate simulations, cli-
mate simulations bias-adjusted with the ISIMIP2b/ISIMIP3
method, climate simulations bias-adjusted and statistically
downscaled with the LI
+
BA/BA
+
SD method combination),
climate variable, calendar month, and grid cell. The good-
ness of spatial variability adjustment, trend preservation, and
bias adjustment is then quantified using absolute errors: for a
fixed metric, adjustment method, climate variable, calendar
month, and grid cell, let
x
obs
hist
, x
sim
hist
, x
sim
fut
, y
sim
hist
, and
y
sim
fut
rep-
resent values of the metric calculated for historical observa-
tions, historical simulations, future simulations, adjusted his-
torical simulations, and adjusted future simulations, respec-
tively. Then, in the case of spatial variability adjustment and
bias adjustment, the absolute error
e
is calculated according
to
e
=
y
sim
hist
−
x
obs
hist
.
(20)
In the case of trend preservation, the absolute error
e
is cal-
culated according to
e
=
y
sim
fut
−
y
sim
hist
−
x
sim
fut
−
x
sim
hist
.
(21)
Values of these errors are then aggregated over all calendar
months and grid cells using the grid cell area-weighted me-
dian. For prsn I only aggregate errors from higher than 60
◦
latitude. The aggregated values are then used to compara-
tively assess method performance.
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Geosci. Model Dev., 12, 3055–3070, 2019
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3066
S. Lange: ISIMIP3BASD
4
Results
In the following I will first present results obtained with
the ISIMIP2b and ISIMIP3 bias adjustment methods ap-
plied at 2
◦
spatial resolution. I will then compare results ob-
tained with the ISIMIP2b and ISIMIP3 statistical downscal-
ing methods applied for downscaling from 2 to 0
.
5
◦
spatial
resolution.
4.1
Comparison of bias adjustment methods
The goodness of bias adjustment and trend preservation by
the ISIMIP2b and ISIMIP3 bias adjustment methods is as-
sessed based on Fig. 7, which shows how well these methods
adjust biases and preserve trends in the 5th, 50th, and 95th
percentile of daily values of the 10 climate variables listed in
Table 1 (note the special treatment of pr and prsn described in
Sect. 3.3). Results suggest that in most calendar months and
grid cells, biases are better adjusted by the ISIMIP3 method
than by the ISIMIP2b method for all 10 climate variables.
The greatest gains are found for hurs, rlds, rsds, and sfcWind.
The least yet still considerable gains are found for tas, tas-
max, and tasmin. Intermediate gains are found for pr, prsn,
and psl.
Results further suggest that in most calendar months and
grid cells, trends in psl, rlds, and tas are considerably bet-
ter preserved by the ISIMIP3 method than by the ISIMIP2b
method. Trends in hurs, rsds, sfcWind, tasmax, and tasmin
are mostly better preserved by the ISIMIP3 method than
by the ISIMIP2b method, yet there are a few exceptions of
slightly better trend preservation by the ISIMIP2b method
for these climate variables. For pr, results suggest that the
ISIMIP3 method is much better at preserving trends in dry-
day frequency, while both methods are similarly good at pre-
serving trends in the 50th percentile of wet-day precipitation,
and the ISIMIP2b method is a bit better at preserving trends
in the 95th percentile of wet-day precipitation. Trends in prsn
are generally better preserved by the ISIMIP2b method, pre-
sumably because the prsn
/
pr ratio is left unchanged by this
method, whereas it is adjusted and therefore changed by the
ISIMIP3 method.
4.2
Comparison of statistical downscaling methods
The goodness of bias adjustment and trend preservation by
the LI
+
BA and BA
+
SD method combinations applied for
bias adjustment and statistical downscaling from 2 to 0
.
5
◦
spatial resolution is assessed based on Figs. 8 and 9, which
show how well they adjust biases and preserve trends in the
5th, 50th, and 95th percentile of daily values of the 10 climate
variables listed in Table 1 (note again the special treatment of
pr and prsn described in Sect. 3.3). The goodness of bias ad-
justment is assessed at 2 and 0
.
5
◦
spatial resolution in Figs. 8
and 9, respectively, while the goodness of trend preservation
is only assessed at 2
◦
spatial resolution (Fig. 8) because sim-
ulated trends are only available at that resolution.
Differences between LI
+
BA and BA
+
SD in their ability
to adjust biases at 2
◦
spatial resolution reflect structural dif-
ferences between the ISIMIP2b and ISIMIP3 approaches to
statistical downscaling. Bias adjustment in BA
+
SD is car-
ried out at 2
◦
spatial resolution and followed by a statis-
tical downscaling which approximately preserves values at
that resolution. Therefore, biases can be expected to be well
adjusted at 2
◦
spatial resolution by BA
+
SD. In contrast,
bias adjustment in LI
+
BA follows a bilinear interpolation
to 0
.
5
◦
spatial resolution and is independently applied to ev-
ery 0
.
5
◦
×
0
.
5
◦
grid cell. Spatial dependencies between time
series within 2
◦
×
2
◦
grid cells are not adjusted. Therefore, bi-
ases at 2
◦
spatial resolution are expected to be better adjusted
by BA
+
SD than LI
+
BA. Results shown in Fig. 8 are largely
in line with this expectation. The greatest gains are found
for pr. Only biases in the center of the distribution of tasmax
and tasmin are slightly better adjusted by LI
+
BA than by
BA
+
SD in most calendar months and 2
◦
×
2
◦
grid cells.
At 2
◦
spatial resolution, BA
+
SD is expected to outper-
form LI
+
BA with regard to trend preservation for the same
structural reason as with regard to bias adjustment. Results
(Fig. 8) are in line with this expectation. Only for pr, prsn,
and psl are there cases in which LI
+
BA preserves trends
slightly better than BA
+
SD. Otherwise, trends are better pre-
served by BA
+
SD, by considerable margins in particular for
sfcWind and tas.
Biases at 0
.
5
◦
spatial resolution (Fig. 9) are slightly bet-
ter adjusted by BA
+
SD than by LI
+
BA in most calendar
months and grid cells for all climate variables except pr and
prsn. Dry-day frequency biases are better adjusted by LI
+
BA
than by BA
+
SD, arguably because the parametric bias ad-
justment of pr that is done following a bilinear interpola-
tion in LI
+
BA adjusts it explicitly and therefore precisely,
whereas the nonparametric quantile mapping that is used for
statistical downscaling after bias adjustment in BA
+
SD ad-
justs it implicitly and only approximately (see Sect. 3.2).
Biases in the 50th percentile of wet-day precipitation are
slightly better adjusted by LI
+
BA than by BA
+
SD in most
calendar months and grid cells. The opposite is true for the
95th percentile of wet-day precipitation.
In order to assess the goodness of bias adjustment across
spatial scales, the
y
=
1
/x
line in Fig. 9 is considered to sep-
arate cases in which BA
+
SD outperforms LI
+
BA (below
the line) from cases in which LI
+
BA outperforms BA
+
SD
(above the line). Results suggest that BA
+
SD adjusts biases
better than LI
+
BA in the vast majority of cases.
The goodness of spatial variability adjustment by LI
+
BA
and BA
+
SD is assessed based on Fig. 10, which shows
how well these method combinations adjust spatial variabil-
ity within regular and staggered 2
◦
×
2
◦
grid cells. Results
suggest that, as expected, spatial variability within regular
2
◦
×
2
◦
grid cells is better adjusted by BA
+
SD than by
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S. Lange: ISIMIP3BASD
3067
Figure 7.
Goodness of bias adjustment (
x
axis) and trend preservation (
y
axis) by the ISIMIP2b and ISIMIP3 bias adjustment methods for
the lower tail
(a)
, center
(b)
, and upper tail
(c)
of the distribution of daily values of 10 different climate variables (color, Table 1) simulated by
four different climate models (symbols) at 2
◦
spatial resolution. Values on both axes represent ratios of spatiotemporally aggregated absolute
errors after bias adjustment with the two methods (see Sect. 3.3). Values greater than 1 indicate better bias adjustment or trend preservation
by the ISIMIP2b method than by the ISIMIP3 method and vice versa for values smaller than 1.
Figure 8.
Same as Fig. 7 but for LI
+
BA and BA
+
SD applied for bias adjustment and statistical downscaling from 2 to 0
.
5
◦
spatial resolution.
Like in Fig. 7, absolute errors are calculated at 2
◦
spatial resolution based on conservatively aggregated bias-adjusted and statistically
downscaled climate simulation data.
LI
+
BA for most calendar months and grid cells in all cases
but one (prsn simulated by HadGEM2-ES).
Spatial variability within staggered 2
◦
×
2
◦
grid cells is
better adjusted by BA
+
SD than by LI
+
BA in most cases
for hurs, tas, tasmax, and tasmin and vice versa for prsn
and psl. Results are mixed for pr, rlds, rsds, and sfcWind.
In order to assess how spatial variability is adjusted over-
all, the
y
=
1
/x
line in Fig. 10 is considered to separate
cases in which BA
+
SD outperforms LI
+
BA (below the line)
from cases in which LI
+
BA outperforms BA
+
SD (above the
line). Results suggest that BA
+
SD adjusts spatial variability
better than LI
+
BA in the vast majority of cases.
5
Conclusions
The ISIMIP3 bias adjustment and statistical downscal-
ing methods outperform their predecessors in several re-
spects. The new trend-preserving parametric quantile map-
ping method used for bias adjustment preserves trends and
adjusts biases in distribution quantiles more accurately than
the ISIMIP2b bias adjustment method. The new stochastic
method used for statistical downscaling prevents the variabil-
ity inflation caused by spatial interpolation in ISIMIP2b.
A major fraction of the bias adjustment gains can be at-
tributed to the newly introduced adjustment of the likelihood
of individual events. This new feature effectively corrects
for the imperfections of the distribution fits that are the ba-
sis of parametric quantile mapping. In addition, it simplifies
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Geosci. Model Dev., 12, 3055–3070, 2019
Page 14
3068
S. Lange: ISIMIP3BASD
Figure 9.
Same as Fig. 8 but with goodness of bias adjustment at 0
.
5
◦
spatial resolution on the
y
axis.
Figure 10.
Goodness of adjustment of spatial variability within regular (
x
axis) and staggered (
y
axis) 2
◦
×
2
◦
grid cells by the LI
+
BA and
BA
+
SD bias adjustment and statistical downscaling methods for 10 different climate variables (color, Table 1) simulated by four different
climate models (symbols). Values on both axes represent ratios of spatiotemporally aggregated absolute errors after bias adjustment and
statistical downscaling with the two methods (see Sect. 3.3). Values greater than 1 indicate better spatial variability adjustment by LI
+
BA
than by BA
+
SD and vice versa for values smaller than 1.
the confinement of extreme values to the physically plausible
range.
Trend preservation works better with the new methods be-
cause they apply it to all distribution quantiles compared an
application to only distribution mean values for most climate
variables in ISIMIP2b. In addition, the new approach of bias
adjustment at the spatial resolution of the climate simulation
data followed by statistical downscaling to the spatial reso-
lution of the climate observation data ensures that trends are
preserved at the spatial resolution at which they were simu-
lated.
The new approach also better adjusts spatial variability at
the spatial resolution of the climate observation data than the
old approach of a bilinear interpolation of climate simulation
data to the spatial resolution of the climate observation data
followed by bias adjustment of these interpolated data. Over-
all, the results presented in this paper can be regarded as a
proof of concept of the new paradigm of a clear separation of
bias adjustment and statistical downscaling.
The next version of the ISIMIP3 bias adjustment and sta-
tistical downscaling method (ISIMIP3BASD) is already un-
der development. In order to improve inter-variable consis-
tency, bias adjustment in ISIMIP3BASD v2.0 will be done
in a multivariate manner. In particular, the MBCn algorithm
will be employed for an adjustment of the inter-variable
copula. This additional adjustment step will be inserted be-
tween steps 4 and 5 of the bias adjustment algorithm pre-
sented herein. Apart from that, ISIMIP3BASD v2.0 and
ISIMIP3BASD v1.0 will be identical: the bias adjustment
method for the marginal distributions of all climate variables
as well as the statistical downscaling method will remain un-
changed.
Geosci. Model Dev., 12, 3055–3070, 2019
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S. Lange: ISIMIP3BASD
3069
Code and data availability.
The
ISIMIP3
bias
adjustment
and
statistical
downscaling
code
is
publicly
available
at
https://doi.org/10.5281/zenodo.2549631
(Lange,
2019b).
The
ISIMIP2b
bias
adjustment
code
is
publicly
avail-
able
at
https://doi.org/10.5281/zenodo.1069050
(Lange,
2017).
The
EWEMBI
data
set
is
publicly
available
via
https://doi.org/10.5880/pik.2019.004
(Lange,
2019a).
The
CMIP5
multi-model
ensemble
output
is
publicly
avail-
able
via
https://doi.org/10.1594/WDCC/CMIP5.NGEMhi
(Dunne
et
al.,
2014a)
for
GFDL-ESM2M
histori-
cal,
https://doi.org/10.1594/WDCC/CMIP5.NGEMr8
(Dunne
et
al.,
2014b)
for
GFDL-ESM2M
rcp85,
https://doi.org/10.1594/WDCC/CMIP5.MOGEhi
(Jones
et
al.,
2014)
for
HadGEM2-ES
histori-
cal,
https://doi.org/10.1594/WDCC/CMIP5.MOGEr8
(Sanderson
et
al.,
2014)
for
HadGEM2-ES
rcp85,
https://doi.org/10.1594/WDCC/CMIP5.IPILhi
(Denvil
et
al.,
2016a)
for
IPSL-CM5A-LR
histor-
ical,
https://doi.org/10.1594/WDCC/CMIP5.IPILr8
(Denvil
et
al.,
2016b)
for
IPSL-CM5A-LR
rcp85,
https://doi.org/10.1594/WDCC/CMIP5.MIM5hi
(AORI,
NIES, and JAMSTEC, 2015a) for MIROC5 historical, and
https://doi.org/10.1594/WDCC/CMIP5.MIM5r8
(AORI,
NIES,
and JAMSTEC, 2015b) for MIROC5 rcp85.
Competing interests.
The author declares that no competing inter-
ests are present.
Acknowledgements.
The author is grateful to Alex Cannon, Matt
Switanek, Douglas Maraun, Simon Treu, Jens Heinke, and Katja
Frieler for various helpful discussions at different stages of this
work. He appreciates the constructive comments on the discussion
paper version of this paper by Stefan Hagemann and one anony-
mous reviewer. He acknowledges the World Climate Research Pro-
gramme’s Working Group on Coupled Modelling, which is respon-
sible for CMIP, and he thanks the climate modeling groups for pro-
ducing and making available their model output.
Financial support.
This work has received funding from the Eu-
ropean 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 publica-
tion were covered by the Potsdam Institute for Climate Impact
Research (PIK).
Review statement.
This paper was edited by Heiko Goelzer and re-
viewed by Stefan Hagemann and one anonymous referee.
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