Introduction
Global climate models (GCMs) and regional climate models (RCMs)
are used to study the climate system. However, their outputs often appear
biased compared to observational references e.g.,. For
example, the temperature means can be shifted. Thus, removing this bias is
often necessary to drive impact studies such as those based on crop or
hydrological models . The main goal of bias correction (BC)
is to match the statistical features of climate model outputs with
observations see, e.g.,. The
most used method is the quantile mapping
, which adjusts the quantiles of
the variables of interest in the stationary case . The
importance of the stationarity hypothesis has been discussed by a few studies
. Some extensions, like CDF-t
Cumulative Distribution Function transfer,,
can take into account some of the non-stationarity in GCMs or RCMs.
Most of those methods are univariate, and do not take into account the
spatial and inter-variable correlations, which may alter the quality of the
corrections e.g.,. have
pointed out that correcting model output could induce biases of physical
processes and that such procedures require an understanding of the nature of
the biases. In particular it is crucial to investigate the way key climate
variables co-vary.
This shortcoming has led to the recent development of multivariate
techniques. As mentioned by , two kinds of methods are
currently available. The first type corrects separately each marginal and
applies afterwards a correction of the dependence structure
e.g.,. The second kind
performs recursive corrections: each variable is corrected conditionally on
the previously already corrected variables .
These last methods have two main limitations. First, the correction depends
on the ordering of the marginals. Second, each marginal is adjusted
conditionally on previously corrected marginals, which reduces the number of
data at each step. Furthermore, the variability of observations is generally
greater than that of the climate models. To increase the variability,
, and suggested
introducing a stochastic component into the bias correction procedure. In
this paper, we develop a multivariate and stochastic bias correction method,
different from the two categories presented, based on elements from optimal
transport theory.
Optimal transport theory is a natural way to measure the dissimilarity
between multivariate probability distributions
, especially in a
multivariate case. For example, this has already been successfully applied in
image processing to transfer colors between images
. Here, our goal is to apply optimal
transport techniques to perform bias correction in estimating a particular
joint law (called a transport plan) that links the
probability distributions of a biased random variable and its correction.
This joint law minimizes a cost function, representing the energy
needed to transform a multivariate probability distribution to another. In
this optimal transport context, any realization of the biased random variable
induces a conditional law of the transport plan, associating the realization
and its correction. As the corrections are randomly drawn from these
conditional laws, the suggested method is stochastic by construction.
Moreover, also stressed that BC methods do not correct the
physical processes of the model, and errors can propagate into the
corrections. However, one key aspect of the present work is to highlight
that, in a climate change context (or more generally, in a framework where
corrections are performed in conditions different from the calibration
dataset), a proper BC method should provide changes – from calibration to
projection periods – in agreement with the modeled data to be corrected. Knowing the quality of the raw modeled data
(and of the underlying processes) is therefore an important a priori step.
Nevertheless, this is beyond the scope of bias correction per se.
This paper is organized as follows. In Sect. ,
the developed theoretical framework to perform bias correction is presented.
In particular, the classical definition of bias correction as transfer
function is generalized with optimal transport theory. Two methods are
presented: optimal transport correction (OTC, stationary case) and dynamical
optimal transport correction (dOTC, non-stationary case). In
Sect. , the proposed methodology is
tested on an idealized non-stationary case based on chaotic attractors. In
Sect. , a multivariate bias correction is performed
on a regional climate model (RCM) simulation of temperatures and
precipitation, in a cross-validation case. Section
provides conclusions and perspectives.
Theoretical framework
The general goal of this paper is the correction of a random variable,
denoted X (e.g., a biased climate model output) with respect to a
reference random variable, denoted Y. The random variables
X and Y live in dimension d. If d=1, we denote them
X and Y. The probability law of X (or Y) is denoted
PX (or PY).
Following , a bias correction method of X
with respect to Y is a map
T:Rd→Rd, called a
transfer function, such that the random variable
T(X) (the correction) follows the same law as
Y,
i.e., PT(X)=PY. This
definition covers most of the practical cases, but we can construct random
variables where no deterministic transfer function exists, e.g., if
X is constant and Y is not. Thus, beyond a multivariate
transfer function, it is necessary to extend the definition of bias
correction.
In the first part, we highlight our method of bias correction with a
univariate example starting from quantile mapping. In the second part, the
mathematical theory is explained. Finally, an extension of our method in a
non-stationary context is presented.
From quantile mapping to optimal transport
We start with the construction of a quantile mapping method in the univariate
case, i.e., with d=1. In this context, the biased and reference random
variables are denoted X and Y, respectively. A transfer function
T between X and Y is constructed on the cumulative
distribution functions (CDFs) of X and Y, defined by
FX(x):=PX(X≤x),FY(y):=PY(Y≤y).
A realization y of Y is the correction of a realization x of X if and
only if FX(x)=FY(y). Under the assumption that FY is invertible,
the correction y of x is given by
y=(FY-1∘FX)︸T(x).
Thus the transfer function is written T=FY-1∘FX.
This method is called quantile mapping. Indeed, the quantiles of X
and Y are matched through the relation FX(x)=FY(y).
Histogram of two Gaussian laws X and Y in blue and red. (a) The
x axis indicates the edges of each bar. The black arrows indicate how the
quantile mapping matches an element of X with its correction. (b) The
x axis gives the center of each bar. The black arrows indicate the
possibilities for how the probability of obtaining the value x1 for X
can be distributed among the possible values yj of Y. The γ1j
correspond to the number of realizations moved. These arrows can be
generalized to each xi. (c) The x axis gives the center of each bar.
The black arrows indicate the non-zero γij estimated by the OTC
method. (d) Bivariate histogram of two Gaussian laws. The black arrows
represent how the OTC method fits each xi with its
correction. To facilitate readability, only 30 arrows are represented.
![]()
We illustrate the quantile mapping method with an example in
Fig. a. In this example, the random variables X and Y
are two Gaussian laws centered, respectively, on 0 and 10, with a standard
deviation of 1. We cut R into cells of length 1 and estimate
the histograms. Fig. a shows the two histograms of X and
Y in red and blue, respectively. The x axis gives the empirical quantiles of the edges of each cell. The black arrows indicate how the
quantile mapping connects a cell of X to a cell of Y.
For example, the realizations of X in the first blue cell are corrected and
transferred to realizations in the first three red cells of Y.
The main point here is the following: in the univariate context, we can
perform a bias correction with only the black arrows. A realization in a cell
of X is corrected to a realization into a cell of Y connected by a black
arrow. Because in a multivariate context the quantile mapping can not be used
to estimate these arrows (CDFs are not invertible), our problem is the
following: how to construct these black arrows in a multivariate context.
For this, let xi (yj) be the centers of each cell of the histogram of
X (Y). Let pxi be the number of realizations of X in the interval
xi, and let pyj be the number of realizations of Y in the
interval yj. We represent all possible black arrows by a collection of
coefficients γij. A γij value corresponds to the number
of realizations in the cell xi that are transferred to realizations in the
cell yj. We obtain the following two equalities:
pxi=∑jγij,
representing how the cell xj is split into each cell yj; and
pyj=∑iγij,
representing how cell yj received the realizations from each cell xi.
We depict the γ1j coefficients in Fig. b. The
black arrows represent the number of realizations γ1j that are
transferred to each yj.
The problem is to calculate the coefficients γij. For each
displacement γij, we can associate a cost, which is the square of
the length of the displacement, |xi-yj|2. This choice comes from
the optimal transport theory, and will be highlighted in the next section. To
correct γij realizations, we have a cost of
γij|xi-yj|2. We thus obtain a global cost associated with
the γij coefficients:
C(γ):=∑ij|xi-yj|2γij.
Our bias correction method is defined by the γij coefficients
minimizing the functional C. The γij obtained by minimizing C
for our example are shown in Fig. c. Comparing with the
quantile mapping in Fig. a, we can see that the obtained
coefficients (the black arrows) are similar. Indeed, the coefficients induced
by the quantile mapping are precisely those minimizing the functional C.
Proofs of this statement can be found in Appendix A and
chap. 2. In other words, even in the absence of
CDF, a bias correction can be carried out by calculating the minimum of the
function C.
The advantage of this approach is that the functional C can be written in
the multivariate case by replacing |xi-yj| by
‖xi-yj‖, where xi and
yj are the center of multivariate cells, and ‖⋅‖ the
Euclidean norm. We illustrate in Fig. d how the
displacements are carried out in the case of two bivariate Gaussian
distributions. The black arrows again represent the non-zero coefficients
estimated by the OTC method (we only represent 30 arrows).
In the next section we present the mathematical theory behind this example
with probability measures of X and Y. If we normalize the number of
realizations of X and Y in each bin by the total number of realizations
of X and Y, we obtain pxi and pyj. Therefore, the transport can
be written as a transport of a fraction of mass, instead of a transport of
the number of realizations.
Bias correction as a joint distribution
In the multivariate context we assume the existence of a transfer function
T between X and Y. By construction, the random variables
X and T(X) are dependent, and their
associated joint law can be summarized by the function
κ:Rd→Rd×Rd
κ(x):=(x,T(x))∈Rd×Rd.
The map κ connects the random variable X with its
correction T(X) on the space
Rd×Rd. Furthermore, the map κ induces a
probability law on Rd×Rd, denoted
PT, and given for all measurable sets
A⊂Rd×Rd by
PT(A):=PX(κ-1(A))=PX({x∈Rdsuch thatκ(x)∈A})=P((x,T(x))∈A).
The critical property here concerns the margins of
PT: the first (second) margin of
PT is PX
(PY). To understand why it is critical, let
Γ(PX,PY) be the set of
probability measures on Rd×Rd for which
PX is the first margin and PY
the second one. By definition,
PT∈Γ(PX,PY).
Thus, any bias correction method defined by a transfer function is an element
of Γ(PX,PY).
We argue that any probability distribution in
Γ(PX,PY) induces a bias correction method. For
γ∈Γ(PX,PY),
γ(x,y) can be interpreted as the probability that y is the correction of x. Formally, the
Jirina theorem see, e.g.,chap. 5 states that there
exists a collection of probability laws γx,
x∈Rd, such that γx are the
conditional laws of Y given X. In other words,
for B⊂Rd, γx(B) is the probability
that the correction y∈B, given X=x. The
correction of x is then sampled from the law
γx. Thus, any
γ∈Γ(PX,PY) defines a
bias correction method, through the conditional laws γx.
This highlights the stochastic part of this approach: all corrections are
sampled from the laws γx, and the corrected values follow
the law PY (by definition of a conditional law).
We note that the problem where X is constant is easily solved with
this approach. The set
Γ(PX,PY) is reduced to one
element: the independent law
δx×PY, where
δx is the Dirac mass in x. Thus,
γx=PY, and the correction of
X is given by sampling each correction with the law
PY.
We have defined a bias correction method as an element of
Γ(PX,PY). However, this set
can be very large. The goal of the next section is to present a
criterion to select an element of
Γ(PX,PY).
Selection of a joint law with optimal transport theory
To select a probability law
γ∈Γ(PX,PY), we
propose using a cost function on this set. The minimum of this cost function
corresponds to an optimal bias correction method. We propose minimizing the
energy needed to transform a realization x of X to its
correction y, i.e., minimizing ‖x-y‖2,
weighted by γ(x,y). Thus, the cost function C is
given by
C:Γ(PX,PY)⟶R+,γ↦∫Rd×Rd‖x-y‖2⋅dγ(x,y).
This cost function minimizes the square of the distance between x
and its correction y. Our bias correction method is associated
with the law γ that minimizes C. This cost function stems from
optimal transport theory . The choice of the square
in Eq. () guarantees the uniqueness of the solution. In
the univariate case, it can be shown that the joint law defined by the
quantile mapping minimizes the cost function C of
Eq. (). Proofs of this statement can be found in
Appendix A and chap. 2.
Our next step is to explain how this minimization strategy can be extended in
the multivariate case.
Multivariate bias correction with optimal transport selection: the stationary case
We assume that (X1,…,Xn) and
(Y1,…,Yn) are two independent and identically
distributed (i.i.d.) samples of the random variables X and
Y. A first step is to estimate the empirical distributions,
P^X and P^Y. We
denote ci a collection of regularly spaced cells that partition
Rd and cover (X1,…,Xn) and
(Y1,…,Yn). The center of each cell is also
denoted ci. With this notation, P^X
and P^Y can be written as a sum of I and J
Dirac masses:
P^X(A)=∑i=1IpX,iδci(A),wherepX,i=1n∑l=1n1(Xl∈ci),andA⊂Rd,P^Y(B)=∑j=1JpY,jδcj(B),B⊂Rd.
The scalar pX,i (or pY,j) is the empirical weight
around ci (or cj) and induced from the sampling of
X (or Y). A natural estimator of
γ∈Γ(PX,PY) can be
written as
γ^(A×B)=∑i,j=1I,Jγi,jδ(ci,cj)(A×B).
The coefficients γij are the probabilities to transform
ci (i.e., a x∈ci) to cj (i.e., a
y∈cj). They are unknown, and they have to obey the
marginal properties:
∑j=1Jγij=pX,i,∑i=1Iγij=pY,j.
Finally, the cost function defined in Eq. () can be
approximated by
C^(γ^)=∑i,j=1I,J‖ci-cj‖2γij.
Finding γij, i.e., solving the problem defined by constraints of
Eqs. ()–() and
minimization of Eq. (), is called a
linear programming problem. It can be solved (for example) by the
network simplex algorithm see, e.g.,. We use the
python implementation of . To correct X, we
have to follow the plan of γij. For a realization Xl
of X, we take the cell ci that contains
Xl. Following γ^, ci is moved to
cj with probability γij/pX,i (applying
Eq. (), the sum over j is 1). To determine
cj, we randomly draw it according to the conditional law
γ^Xl=(γi1,…,γiJ)/pX,i.
Finally, we draw uniformly y in cj. This methodology is
summarized in Algorithm A1, and we refer to it as optimal transport
correction (OTC).
Note that the traditional one-dimensional quantile mapping preserves the
ordering of quantiles. In the multivariate case, this type of property can be
viewed as the Monge– shortening principle
see, e.g.,chap. 8. The idea is that the
extremes of a multivariate distribution are moved to extremes, the boundary
to the boundary, the level lines to level lines, etc.
Representation of bias correction in the context of climate change.
Present
Future
Numerical model
X0
X1
Observations
Y0
unknown (Y1)
Non-stationary bias correction
Climate models offer a valuable tool to study future realistic climate
trajectories. Climate model outputs of the present period need to be bias
corrected with respect to current observations. Future climate simulations
also need to be adjusted. However, no observation is available for the future
and clear assumptions have to be made to correct simulations for future
periods. Table displays the basic framework of bias
correction. Future unobserved data, say Y1, should be inferred
from the current reference vector, Y0, and two numerical runs,
one in the present, say X0, and one in the future, say
X1. Period 0 is called the calibration period, and
period 1 the projection period. In the univariate case, denoting
Fi (Gi) the CDF of Xi (Yi), the CDF-t (CDF transform)
method of assumes that
(G1)-1∘G0=TY0,Y1=TX0,X1=(F1)-1∘F0.
Recombining Eq. (), the CDF of Y1 is given by
G1=G0∘(F0)-1∘F1, and can be used to perform a
quantile mapping correction. Here, the fundamental hypothesis
TY0,Y1=TX0,X1 means that the transfer
functions to capture the temporal changes are identical in the model and
observational worlds.
CDF-t learns the change between X0 and X1, and transfers it to
Y0 to estimate Y1. In the multivariate case, following CDF-t, we
want to learn the evolution (i.e., the change or the temporal evolution)
between X0 and X1, and apply it to
Y0. This generates Y1, and OTC can then be
applied between X1 and Y1. Note also that the
reverse hypothesis TY1,X1=TY0,X0
could be considered, meaning that the bias is learned, and transferred along
the dynamic. In this case, the correction of example given in
Sect. does not correspond to the
reference (not shown), so we rejected this assumption. Thus, our definition
of non-stationary bias correction assumes a transfer of the evolution of the
model to the observational world. Indeed, climate change is one of the main
signals that we want to account for in the projected corrections. However,
the change in the observations can be different, and therefore the resulting
corrections can also be different from observations. Nevertheless, this
methodology is justified because different simulations can have different
evolutions; e.g., the four RCP scenarios provide four different simulations,
giving four different corrections. This is also true for different climate
models, which can show different changes. This information is therefore kept
in the corrections.
Estimation of the unobserved random variable Y1.
The random variables X0, X1 and Y0
are known. Plans γ and φ are the optimal joint laws in the
sense of Eqs. ()–().
φ̃ is the evolution of Y0 estimated from γ
and φ. OTC is used to correct X1 with respect to the
estimation of Y1.
![]()
Using OTC, we define two optimal plans: the optimal plan γ, between
X0 and Y0, and the optimal plan φ,
between X0 and X1. The law γ is the
bias between X0 and Y0, whereas
φ is the evolution between X0 and
X1. Our goal is to move φ along γ,
defining a plan φ̃, to estimate Y1 as the
evolution of Y0,
i.e., Y1=φ̃(Y0). Then, we correct
X1 with respect to
Y1=φ̃(Y0), with the OTC method. This
is summarized in Fig. .
Bivariate histogram with bin size equal to 0.1. In each panel we
have a Gaussian law centered on (0,0) with covariance 4Id2
(P^X0), a Gaussian law centered on (10,0)
with covariance 1/4Id2 (P^X1)
and a Gaussian law centered on (0,10) with covariance 1/4Id2
(P^X1). The red arrow is the local evolution
between P^X0 and
P^X1. (a) The probability
distribution P^Y1 is the correction with
OTC-t and D=Id2. The grey arrow is the estimation of
the evolution of P^Y0. (b) The
probability distribution P^Y1 is the
correction with dOTC and D given by
Eq. (). The grey arrow is the estimation of the
evolution of P^Y0.
![]()
The estimation of φ̃ is performed in three steps:
transformation of φ into a collection
of vectors,
transferral of these vectors along γ and
adaptation of these vectors to Y0.
To illustrate our methodology, Fig. shows an example
where the random variables X0, X1 and
Y0 follow a bivariate Gaussian law. They are, respectively,
centered at (0,0), (10,0) and (0,10), with covariance matrices
4×Id2, Id2/4 and Id2/4 (the
matrix Idd is the d-dimensional identity matrix). Without
loss of generality, we write the empirical distribution of X0,
X1 and Y0 as a sum of Dirac masses,
P^X0=∑i=1IpX0,iδci,
P^Y0=∑j=1JpY0,jδcj,
P^X1=∑k=1KpX1,kδck.
Random variables generated by the
model, OTC, dOTC, quantile mapping and CDF-t. (a) Biased random
variable X0 (red) and references Y0 (blue) for
time period 0. (b) Biased random variable X0 (red)
and correction Z0 with OTC (green). (c) Biased
random variable X0 (red) and correction Q0 with
quantile mapping (green). (d) Biased random variable
X1 (red) and references Y1 (blue) for time period
1. (e) Biased random variable X1 (red) and
correction Z1 with dOTC (green). (f) Biased random
variable X1 (red) and correction Q1 with
CDF-t (green).
![]()
Step 1. Transformation of φ. Using the OTC method, φ
moves the bin ci of P^X0 to the bin
ck of P^X1. The vector
vik:=ck-ci represents the evolution from
ci to ck (i.e., the local evolution between
X0 and X1). The collection of vectors
vik is an estimation of the process between X0 and
X1. In Fig. , the red arrow is an example
of vector vik.
Step 2. Transfer along γ. Using the OTC method, γ moves
the bin ci of P^X0 to the bin
cj of P^Y0. Thus, the estimation of
φ̃ could be defined by the vector vik applied to
cj; i.e., a realization of Y1 is given by
cj+vik. The grey arrow in Fig. a
depicts this operation. But the vik can cross, and the correction
is not coherent. This is due to normalizing issues and because the collection
of vectors vik applied to Y0 does not define an
optimal transport plan. The standard deviation decreases between
X0 and X1, whereas it increases between
Y0 and Y1 in our example. Furthermore, the
quantiles are inverted in this example (low values are moved to high values).
Consequently, we have to adapt the vectors vik to
P^Y0.
Step 3. Adaptation of vik. To solve this problem, we
introduce a matrix factor D, which rescales the collection of
vectors vik. In the univariate case, proposed a
factor σY0σX0-1, where σ
is the standard deviation. The idea is to remove the scale of
X0 and to replace it by the scale of Y0.
and proposed a multivariate
equivalent that uses the Cholesky decomposition of the covariance matrix.
Denoting Σ the covariance matrix, and Cho(Σ) its
Cholesky decomposition, we multiply (in a matrix sense) vik by the
following matrix:
D:=Cho(ΣY0)⋅Cho(ΣX0)-1.
The Cholesky decomposition only exists if Σ is symmetric and
positive-definite. Some covariance matrices do not have this property, e.g.,
highly correlated random variables. In such a case, Σ must be slightly
perturbed to be positive-definite see, e.g.,.
Furthermore, the Cholesky decomposition can be poorly estimated if the number
of available data is too small compared to the dimension. Indeed, the inverse
of a covariance matrix is highly biased. In this case, a pragmatic solution
is to replace the matrix D by the diagonal matrix of the standard
deviation, i.e., D=diag(σY0σX0-1).
Finally, a realization of Y1 is given by
cj+D⋅vik. Figure b
shows an estimation of Y1. Visually, the shape of
Y1 appears coherent with the evolution between X0
and X1. The mean of Y1 is (2.53,10). The
standard deviation between X0 and X1 is divided
by 4. The mean shift between X0 and X1 is
(10,0). This shift of 10 units is correctly taken into account in the
rescaling of Y0 by the standard deviation (equal to 4)
between X0 and X1:
(2.53,10)︸Y1mean=(10,0)︸meanshiftbetweenX0andX1/4︸Rescaling+(0,10)︸Y0mean.
The value of the covariance matrix of Y1 is
ΣY1≃0.018×Id2. It is close to the
expected value
(1/4)/16×Id2≃0.015×Id2. The shift of
10 units of the model is not
followed. It is interpreted as a correction of the bias into the evolution
of the model. However, depending on the hypotheses desired by the user, the
dOTC method can easily provide corrections whose mean evolutions and trends
are in agreement with those given by the simulations to be corrected, like in
the EDQM bias correction method . The complete method of
correction is summarized in Algorithm A2. We refer to it as dOTC (dynamical
optimal transport correction).
We first propose evaluating OTC and dOTC on an idealized case.
Bias correction on an idealized case
Model and methodology
To evaluate our bias correction method, we construct an idealized biased
case, based on the model. This
three-dimensional system is generated by the differential equations
dxdt=-x22-x32-(x1-ψ(t))/4x1x2-4x1x3-x2+1x1x3+4x1x2-x3.
The function ψ(t) is a linear forcing proposed by .
Classically, ψ also contains a seasonal cycle , where
the length of a “year” is fixed at t=73 time units. Here we integrate
this equation for the following forcing between 0 and 7×73
(i.e., 7 “years” of integration):
ψ(t)=9.5-20t-TT1{t>T},T=6×73.
The integration is performed with a Runge–Kutta (order 4) scheme with a time
step of size 0.005. All trajectories of the
model converge on a unique subset of R3 (called an attractor),
and remain trapped on it. According to , the first 5
“years” correspond to the time required to trap the trajectories.
One realization of random variable Y0 (Y1) is
year 6 (year 7). Each year contains 14600(=73/0.005) elements.
According to Eq. (), the linear forcing is applied during
year 7. The non-stationarity is induced by the change between the two time
periods.
We introduce a bias by multiplying each point of the trajectories by a
triangular matrix S, and add a vector m,
i.e., X=SY+m. The addition changes the
mean, whereas the multiplication alters the covariances. The matrix
S is chosen empirically such that the covariance matrices of
X0, X1, Y0 and Y1
differ. We fix
S=1.2200-0.411.040-0.410.560.52,m=123.
(a) Map of the southeast of France. The 12 black squares
are the locations where corrections are performed. (b–h) The x
axis of the panels is the evolution of the correction with dOTC. The y axis
of panels (b)–(h) is the evolution of WRF in red and the
evolution of SAFRAN in blue. The red line is the linear regression between
the evolution of correction and the evolution of WRF. The black cross markers
are the scatterplots between the evolution of correction with CDF-t and
evolution of WRF. (b) Evolution of mean precipitation,
i.e., difference between the projection period and the calibration period.
(c) Evolution of variance of precipitation. (d) Evolution
of spatial covariance of precipitation. (e) Evolution of covariance
between precipitation and temperatures. (f) Evolution of mean
temperatures. (g) Evolution of variance of temperatures.
(h) Evolution of spatial covariance of temperatures.
![]()
The random variables X and Y are plotted in
Fig. a, d. The blue (red) curve of
Fig. a is the trajectory of Y0
(X0). The mean is largely altered. We estimate the covariance
matrices as
Cov^Y0=0.43-0.37-0.24-0.370.930.17-0.240.170.69,Cov^X0=0.64-0.68-0.62-0.681.391.0-0.621.00.92.
Similarly to Fig. a, d depicts in blue
Y1 and in red X1. The forcing of
Eq. () has changed the properties of the trajectories, and
they became chaotic. It is worthwhile noticing that the dynamic of
Y is comparable to the one of X. The covariance
matrices are largely affected:
Cov^Y1=0.27-0.09-0.14-0.090.810.08-0.140.080.73,Cov^X1=0.4-0.25-0.29-0.251.00.65-0.290.650.64.
We estimate the empirical distributions PY0,
PY1, PX0 and
PX1 with a three-dimensional histogram. We cut a
large cube around the trajectories into cells of size
0.2×0.2×0.2. Then we count the number of points in each cell.
Finally, we evaluate the quality of the correction by comparing the
covariance matrices of Y0 and X0, and the
covariance matrices of Y1 and X1.
Correction of the biased Lorenz (1984) model
We apply our method to correct X0 and X1. The
random variable X0 is corrected with respect to
Y0 and using the OTC method. The random variable
X1 is corrected with respect to the estimation of
Y1, coming from the dOTC method. The resulting random variables
Z0 and Z1 are given in
Fig. b, e. We show in Fig. 4c, f a univariate
correction with quantile mapping for the period 0, generating the random
variable Q0. The same is shown for CDFt, period 1 and the random
variable Q1.
The correction Z0 is visually very similar to the reference
in blue in Fig. a. The covariance matrix is almost
perfectly reproduced:
Cov^Z0=0.42-0.36-0.24-0.360.930.17-0.240.170.69,supCov^Z0-Cov^Y0=0.004.
The correction Z1 is depicted in green in
Fig. d. It is visually hard to compare to
Fig. b, but we recognize Y1. The
covariance matrix is correctly rectified:
Cov^Z1=0.26-0.11-0.11-0.110.820.08-0.110.080.71,supCov^Z1-Cov^Y1=0.03.
Finally, the cost of transformation, given by Eq. (),
of Z1 into Y1 is 93 % smaller than the cost between
Y1 and X1; i.e., PZ1 is more
similar to PY1 than PX1.
Furthermore, if we replace the Cholesky matrix of dOTC by the matrix of
standard deviation, the maximum difference between covariance matrices
increases to 0.22, but the cost is 85 % smaller. Thus, using the standard
deviation slightly degrades the correction. However, visually, it is very
hard to distinguish the corrections with the Cholesky matrix or the standard
deviation matrix. The figure corresponding to Fig.
with the standard deviation matrix is given in the Supplement.
By contrast, Q0 and Q1, depicted,
respectively, in Fig. c, f, do not reproduce
Y0 and Y1. Thus, the multivariate correction is
largely better than the univariate correction. This is confirmed by the
covariance matrices, which reproduce exactly the covariances of
X0 and X1:
Cov^Q0=0.42-0.42-0.42-0.420.950.68-0.420.680.69,supCov^Q0-Cov^Y0=0.51,
Cov^Q1=0.13-0.1-0.14-0.10.590.39-0.140.390.46,supCov^Q1-Cov^Y1=0.31.
We have performed a tri-variate correction on a nonlinear system exhibiting
non-standard probability measures (i.e., non-Gaussian, non-exponential). In
the stationary case, the OTC method works almost perfectly. In the
non-stationary case, the dOTC method produces a probability distribution
closed to the expected result. We propose now to apply OTC and dOTC to
climate model simulations.
Bias correction of an RCM simulation
Data
The dataset used as a reference for the bias correction (BC) is the Systeme
d'Analyse Fournissant des Renseignements Atmospheriques a la Neige
SAFRAN, reanalysis. SAFRAN is a hourly reanalysis
over France between 1958 and present, with a horizontal resolution of
8 km × 8 km. claimed that the daily
mean of the surface atmospheric temperature (tas) and precipitation (pr)
presents no bias compared to observations from the climatological database of
Météo-France. This justifies the use of SAFRAN as a reference.
We test our multivariate BC method on a simulation of the Weather Research
and Forecast (WRF) atmospheric model performed within
the EURO-CORDEX initiative with a
0.11∘ × 0.11∘ horizontal resolution. The boundaries
of the simulation were forced by a historical simulation of the Institut
Pierre-Simon Laplace (IPSL) coupled model .
This EURO-CORDEX historical simulation will be called “WRF” in the
following.
SAFRAN and WRF data are re-mapped onto the same grid, with a spatial
resolution of 0.11∘ × 0.11∘
(i.e., ∼ 12 km × 12 km). The nearest neighbor interpolation
is used. We only keep the land region comprised in
1.8–7.85∘ E × 41.8–45.2∘ N, i.e., covering the
southeast of France. This region is characterized by a complex topography,
which creates a strong spatial heterogeneity, especially for precipitation.
For the present application, we extract 12 grid points regularly spaced
(see Fig. a), with a one-to-one spatial
correspondence between SAFRAN and WRF.
In both datasets, we will consider daily surface air temperatures and
precipitation. The goal of this section is to correct the bias in tas and pr
in the WRF data with respect to SAFRAN.
Cross-validation protocol
We focus on the daily timescale over the 1970–2000 period. We correct the
warm season (May–September). The analysis and conclusions are available for
the cold season, and the corresponding figure
(i.e., Fig. ) is given in the Supplement. We
split that period into two sub-periods, 1970–1985 (2295 days) and 1985–2000
(2295 days), to perform a cross-validation. The SAFRAN (WRF) values over the
first time period correspond to the random variable Y0
(X0), and are called the calibration period. The
SAFRAN (WRF) values over the second time period correspond to
Y1 (X1), and are called the projection period. SAFRAN during 1985–2000 (i.e., Y1) is assumed to be
unknown, and is used for cross-validation.
We perform two bias corrections: univariate and 24-variate (12 grid
points and 2 variables).
For univariate correction, quantile mapping is used for the calibration
period, and CDF-t for the projection period.
For 24-variate correction, OTC is used for the calibration period,
and dOTC for the projection period. The spatial structure and the
dependence between the two variables are used. Due to the dimension,
the Cholesky matrix is poorly estimated. We replace it by the matrix
of standard deviation in the rescaling step.
We estimate the empirical distributions by computing histograms with bins of
size 0.1 in each dimension. Furthermore, CDF-t and dOTC can shift close
to 0 values to negative values for precipitation. Thus, negative
precipitation values are replaced by 0 after correction. We test the
quality of the correction by plotting the evolution of the mean, the standard
deviation, and the spatial and inter-variable covariance, i.e., the
difference between the projection and calibration periods. These indicators
are summarized in Fig. . During the
calibration period, the goal is that the probability distribution of
correction of the WRF simulation will be the probability distribution of
SAFRAN. By construction of OTC, the correction is almost perfect, and we
focus on the projection period. In the projection period, the goal is that
the evolution of corrections will be close to the evolution of the WRF
simulation.
Evolution analysis
As we have seen in the previous section, the corrections of X1
and Y1 are identical only if the evolution of SAFRAN is
identical to the evolution of WRF. To analyze the evolution of WRF, SAFRAN
and the corrections, we compute the difference of statistical indicators
between the projection and the calibration period at each grid point. The
indicators are the mean (Fig. b, f), the
variance (Fig. c, g), the covariance between
pr and tas (Fig. e) and the spatial covariance
for each variable (Fig. d, h).
r-value, p-value and standard error (SE) of linear regression
between the evolution of correction and evolution of WRF.
r-value
p-value
SE
Mean evolution pr
0.98
10-8
0.08
Mean evolution tas
0.99
10-9
0.05
Variance evolution pr
0.71
10-2
0.37
Variance evolution tas
0.57
5×10-2
0.25
Covariance pr/tas evolution
0.81
2×10-3
0.24
Spatial covariance pr
0.59
10-14
0.08
Spatial covariance tas
0.76
10-29
0.05
The x axis of Fig. a–h is the evolution of
the correction
(i.e., E(Z1)-E(Z0),…).
The y axis of Fig. a–h is the evolution of
WRF in red
(i.e., E(X1)-E(X0),…),
and the evolution of SAFRAN in blue
(i.e., E(Y1)-E(Y0),…).
Furthermore, the red line is the linear regression between the evolution of
the 24-variate correction and the evolution of WRF. The correlation
(r-value), p-value and standard error of each linear regression are
summarized in Table .
The linear regression between evolution of 24-variate correction and
evolution of WRF (red line) shows a strong statistical link for all
statistical indicators. The evolution of the mean is almost perfectly
reproduced for the two variables (r-values is at least equal to 0.98,
with a maximal p-value at 10-9). The evolution of variance of WRF is
also reproduced, the linear regression being significant (maximal p-value
is 5×10-2).
The evolution of dependence structure is given by the evolution of spatial
and inter-variables covariance. The minimal r-value for linear regression
is equal to 0.59 with a maximal p-value equal to 2×10-3. This
means that dOTC reproduces the evolution of WRF between calibration and
projection period. Because the calibration period is perfectly corrected, the
correction during projection period appears as the evolution of WRF, applied
to SAFRAN.
A linear regression, the Spearman rank correlation between the evolution of
SAFRAN, and the evolution of the correction with WRF do not show a
significant statistical link (not shown). We conclude that the evolution of
WRF is different of the evolution of SAFRAN. This indicates it is not
possible to reproduce SAFRAN during projection period using dOTC and WRF. For
example, WRF predicts an increase between 0.2 and 0.4 K of the mean
temperature, whereas SAFRAN gives an increase between 0.2 and 1 K.
The correction with CDF-t appears to be satisfactory for the temperatures,
and very similar to the correction with dOTC. But for the precipitation, the
structure is not coherent with WRF or SAFRAN. This dissimilarity is due to
the difference between the probability distribution of temperatures
(quasi-Gaussian) and precipitations (exponential/Gamma laws).
We conclude that the evolution of the 24-variate correction with dOTC
between calibration and projection periods is close to the evolution of WRF.
Furthermore, the evolution of SAFRAN is very different from the evolution of
WRF. In particular, this example illustrates how the classical
cross-validation methodology does not differentiate the variations of SAFRAN
and WRF, and that the correction can not be compared to the reference during
the projection period.
Conclusions
We have developed a new method for multivariate bias correction, generalizing
the quantile mapping in the multivariate case. To do so, we have developed a
new theoretical framework to understand any bias correction (BC) method: any
BC method is here characterized by a joint law between the biased dataset and
the correction. This joint probability distribution is estimated based on
optimal transport techniques, and the BC method is then referred to as
optimal transport correction (OTC). A definition of non-stationary bias
correction is also proposed: the evolution of the model is learned and
transferred to the reference world. An extension of OTC called dynamical OTC
(dOTC) has been developed to account for temporal non-stationarities.
OTC and dOTC methods have been tested on an idealized three-dimensional case
based on time-dependent attractors, which
induced changes in the correlation between variables. The bias correction
appeared to perform very well in those idealized experiments.
Then, 12 grid points of a WRF simulation have been corrected with respect
to SAFRAN reanalyses for precipitation and temperature in Southern France. A
24-variate correction was performed. The correction in stationary context
was almost perfect. In the non-stationary case, the evolutions of WRF and
SAFRAN were different, and, as expected, the correction with dOTC differed
from SAFRAN. However, the correction presented a multidimensional evolution
similar to that of WRF. We can therefore conclude that the correction is
consistent with the definition proposed for the non-stationary case.
This is consistent with the results of : the fundamental
errors of a model are not corrected, but transferred to the world of
observations. The dOTC method preserves the signal of climate change inferred
from the model simulations. As suggested by , our
cross-validation method does not compare the correction to the observations
on the validation period, which can produce false positive or true negative
due to internal variability of model or observations, but assesses whether
the statistical evolution of the model is kept.
Furthermore, although the number of available data is very small compared to
the dimensions (2295 days and 24 dimensions), the OTC and dOTC performed a
correction without numerical problems, and, moreover, only in a few minutes
on a personal computer.
The methods OTC and dOTC are able to correct the dependence structure
(i.e., the joint law), and not only the inter-variable and spatial
correlations. In particular, the copula function (which contains the
information about dependence) is corrected. In addition, dOTC proposes a
definition of non-stationarity, and explicitly gives what the correction
corresponds to (the evolution of the model applied to observations). In the
particular case of the temperatures/precipitation correction, compared to,
e.g., and , the correction is at least as
good during the calibration period, although the comparison is not done over
the projection period, because the indicators are different.
As a perspective of improvement of the method, we note that the optimal plan
can only be used to correct data points that are already known. If a new data
point is obtained, and alters the estimate of the probability density
function, then the plan needs to be recomputed. However, such a situation is
relatively rare in bias correction. Indeed, the corrections usually have
to be performed on climate model simulations that cover many
years and decades. This means that the
whole time series are available at once and are not continuously updated.
One possibility would be to “smooth” the optimal plan that,
thus, could be applied to new points without recalculating the plan.
Finally, a promising application of this method is the post-processing of
operational forecasts.
In such a case, the question of internal variability
would not affect the bias correction procedure as climate
dynamics is consistently represented between the model and observations.