GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus GmbHGöttingen, Germany10.5194/gmd-8-1525-2015Objectified quantification of uncertainties in Bayesian atmospheric inversionsBerchetA.antoine.berchet@empa.chPisonI.ChevallierF.BousquetP.BonneJ.-L.ParisJ.-D.Laboratoire des Sciences du Climat et de l'Environnement, CEA-CNRS-UVSQ, IPSL, Gif-sur-Yvette, Francenow at: Laboratory for Air Pollution/Environmental Technology, Swiss Federal Laboratories for Materials Science and Technology, Empa, Dübendorf, SwitzerlandA. Berchet (antoine.berchet@empa.ch)26May2015851525154628May201429July20145March20155May2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://www.geosci-model-dev.net/8/1525/2015/gmd-8-1525-2015.htmlThe full text article is available as a PDF file from https://www.geosci-model-dev.net/8/1525/2015/gmd-8-1525-2015.pdf
Classical Bayesian atmospheric inversions process atmospheric
observations and prior emissions, the two being connected by an
observation operator picturing mainly the atmospheric transport.
These inversions rely on prescribed errors in the observations, the
prior emissions and the observation operator. When data pieces are
sparse, inversion results are very sensitive to the prescribed error
distributions, which are not accurately known. The classical
Bayesian framework experiences difficulties in quantifying the
impact of mis-specified error distributions on the optimized fluxes.
In order to cope with this issue, we rely on recent research results
to enhance the classical Bayesian inversion framework through a
marginalization on a large set of plausible errors that can be
prescribed in the system. The marginalization consists in computing
inversions for all possible error distributions weighted by the
probability of occurrence of the error distributions. The posterior
distribution of the fluxes calculated by the marginalization is not
explicitly describable. As a consequence, we carry out a Monte Carlo
sampling based on an approximation of the probability of occurrence
of the error distributions. This approximation is deduced from the
well-tested method of the maximum likelihood estimation. Thus, the
marginalized inversion relies on an automatic objectified diagnosis
of the error statistics, without any prior knowledge about the
matrices. It robustly accounts for the uncertainties on the error
distributions, contrary to what is classically done with frozen
expert-knowledge error statistics. Some expert knowledge is still
used in the method for the choice of an emission aggregation pattern
and of a sampling protocol in order to reduce the computation cost.
The relevance and the robustness of the method is tested on a case
study: the inversion of methane surface fluxes at the mesoscale
with virtual observations on a realistic network in Eurasia.
Observing system simulation experiments are carried out with
different transport patterns, flux distributions and total prior
amounts of emitted methane. The method proves to consistently
reproduce the known “truth” in most cases, with satisfactory
tolerance intervals. Additionally, the method explicitly provides
influence scores and posterior correlation matrices. An in-depth
interpretation of the inversion results is then possible. The more
objective quantification of the influence of the observations on the
fluxes proposed here allows us to evaluate the impact of the
observation network on the characterization of the surface fluxes.
The explicit correlations between emission aggregates reveal the
mis-separated regions, hence the typical temporal and spatial scales
the inversion can analyse. These scales are consistent with the
chosen aggregation patterns.
Introduction
Characterizing the global biogeochemical cycles of greenhouse gases
requires a reliable understanding of the exchanges at the
surface–atmosphere interface. The description of these exchanges
must encompass the absolute amounts of gas released to and removed
from the atmosphere at the surface interface, the spatial
distribution and the temporal variability of the fluxes, and the
determination of the underlying physical processes of emissions and
sinks. Such an integral depiction is still missing for most
greenhouse gases . One of the possible
approaches to inquire into the surface fluxes is the analysis of the
atmospheric signal. The drivers of the spatial and temporal
variability of the atmospheric composition are atmospheric
transport, chemistry and surface fluxes. Therefore, monitoring the
atmospheric composition and using a representation of the
atmospheric transport and chemistry with global circulation models
(GCMs) or chemistry-transport models (CTMs) can help in inferring
information on the fluxes
. This
approach, called atmospheric inversion, suffers from two practical
issues in its implementation. First, the atmospheric composition is
still laconically documented, though the number of global monitoring
projects with extensive surface observation networks and satellite
platforms has been increasing in the last decades e.g.. Indeed,
the satellite platforms have a global coverage but the observed
atmospheric composition is integrated over the vertical column,
while the surface sites can provide continuous observations but only
at fixed point locations. Second, the atmosphere behaves as an
integrator and the air masses are mixed ambivalently through the
transport . Thus, the inverse problem
of tracking back the fluxes from the variability of the atmospheric
composition cannot be solved univocally. The Bayesian formalism
allows for statistical analyses of the atmospheric signal, so that one
can identify confidence intervals of fluxes compatible with the
atmospheric composition .
Bayesian inversions have been extensively used at the global scale,
providing insights on the greenhouse gas budgets e.g.. However, non-compatible discrepancies
in the results appear between the possible configurations of
atmospheric inversion systems . The
various configurations include the choice of the atmospheric
transport, its spatial and temporal resolutions, the meteorological
driving fields, the type and density of the observations, etc. In
the Bayesian formalism, some assumptions also have to be made on the
transport model error statistics, on the errors made when comparing
a discretized model to observations and
on the confidence we have on the prior maps and time profiles of
emissions . All these choices are based
on technical considerations and on the expert perception of the
problem to solve. Comparing results based on different choices that
are physically adequate, but subjective, is difficult, especially to
track inconsistencies, which enlarge the range of flux estimates.
In the following, we focus on the development of an enhanced
Bayesian method that objectifies the assumptions on the statistics
of the errors and that takes the unavoidable uncertainties generated
by our lack of knowledge on these error statistics into account. In
this approach, the confidence ranges of the optimized surface fluxes
are computed by a Monte Carlo marginalization on all the possible
error statistics, which is more general than the usual Bayesian
approach deducing posterior uncertainties from a single error
statistic combination only. The weight function for the
marginalization is inferred from an already-tested maximum
likelihood approach e.g., processing the pieces of information carried
by the differences between the measurements and the prior simulated
concentrations. The potential and consistency of the method is
tested through observing system simulation experiments (OSSEs) on a
realistic configuration of atmospheric inversion.
The case study is the quantification of methane fluxes in the
Siberian Lowlands with a network of surface observation sites that
have been operated for a few years by the Japanese National
Institute for Environmental Studies
and the German Max Planck Institute
. The characterization of the
region is challenging, with co-located massive methane emissions
from anthropogenic activity (oil and gas extraction) and from
wetlands in summer. Moreover, the wetland emissions have a very high
temporal variability due to their sensitivity to the water
table depth and to the temperature; e.g.. Their
quantification is then difficult. In order to catch the influence of
the sampling bias due to non-regularly distributed observation sites
and non-continuous measurements, we produce virtual observations
from a known “truth” at locations where real observations are
carried out and at dates when the logistical issues do not prevent
the acquisition of measurements. We then check the capability of our
method to reproduce consistent flux variability and distribution
with seven degraded inversion configurations (perturbed transport,
flat flux distributions, etc.).
In Sect. , we describe the theoretical framework of our method of marginalization.
The enhancements on the general theoretical framework need a cautious definition of the problem to be
implementable in terms of computational costs and memory limits.
In Sect. , guidelines for a suitable definition of the problem are developed.
The whole structure of the method is summarized in Sect. .
In Sect. , we present the particular set-up of the OSSE carried out for proving the robustness of the method.
The specific Siberian configuration we test our method on is detailed in Sect. .
The OSSEs are evaluated along defined objective statistical scores in Sect. .
Marginalized Bayesian inversion
We first describe the motivations for using a marginalized inversion in Sect. .
In Sect. , we describe the marginalization itself and the Monte Carlo approach chosen in order to compute it.
Context and motivation for the marginalizationBayesian inversion framework
The surface–atmosphere fluxes, through transport, cause a
variability in the atmospheric mixing ratios of the species we are
interested in. The atmospheric inversion relies on the processing of
the atmospheric variability in order to infer the surface–atmosphere
fluxes. Since the atmosphere is diffusive and irreversibly mixes air
masses from different origins, it is physically impossible to infer
univocal information on the fluxes from the integrated atmospheric
signal alone . We
then pursue a thorough characterization of the pdf (probability density function) of the state of
the system x (e.g. the spatial and temporal distribution of
the surface fluxes, but also background concentrations and
baselines in some cases), assuming some prior knowledge on the
system and having some observations of the atmospheric physical
variables related to our problem. That is to say, we want to
calculate the pdf p(x|yo-H(xb),xb) for all
possible states x; yo is a vector gathering
all the available observations, xb is the
background vector including the prior knowledge on the state of the
system and H is the observation operator converting the
information in the state vector to the observation space. Typically,
H embraces the atmospheric transport and the discretization of the
physical problem. In the scope of applications of the atmospheric
inversions, the observation vector yo gathers
measurements of dry air mole fraction. As for the observation
operator, it is computed with a model which simulates mixing ratios.
As we are interested in trace gases, we will consider that the dry
air mole fractions can be treated as mixing ratios. In all the
following, we also consider that H is linear; hence, H is
represented by its Jacobian matrix H and
H(xb)=Hxb. This approximation
is valid for all non-reactive atmospheric species at scales large
enough, so that the treatment of the local-scale turbulence by the
model does not generate numerical non-linearity. When the
atmospheric chemistry must be taken into account (for instance with
methane), either the window of inversion must be short compared with
the typical lifetime in the atmosphere for the linear assumption to
be valid, or the concentration fields of the reactant species (e.g.
OH radicals for methane) must be accurately known.
In general, the characterization of the pdf is built within the
Bayesian formalism with the assumption that all the involved pdfs
are normal distributions . The pdfs
are then explicitly described through their mode and their matrix of
covariance. In this case, the pdf p(x|yo-Hxb,xb)∝N(xa,Pa) is
defined by its mode, xa, the posterior state, and
its matrix of covariance, Pa. In addition to the
linear assumption, we also consider that the uncertainties are
unbiased. That is to say, p(x-xb)∝N(0,B) and
p(yo-Hxt)∝N(0,R) where xt is the
true state of the system. The uncertainty matrix B (resp. R) encompasses the uncertainties on the background
xb (resp. on the measurements and on the model,
including representation errors, i.e. the errors made when
approximating the real world by a numerical gridded model). Under
these assumptions, we can explicitly write the posterior vector and
the posterior matrix of covariance:
p(x|yo-Hxb,xb)∝N(xa,Pa):xa=xb+K(yo-Hxb)Pa=B-KHB,
with K=BHT(R+HBHT)-1 the Kalman gain matrix.
Ambivalent uncertainty set-up
Atmospheric inversion is straightforward (apart from technical
issues in the numerical implementation of the theory) as long as the
uncertainty matrices R and B are defined.
Some of their components can be calculated unambiguously, such as
measurement errors in matrix R. Other errors are derived,
in most cases, following expert knowledge on, e.g. the behaviour of
the atmospheric transport and of the surface fluxes. This expert
knowledge is acquired, for example, through extensive studies on
the sensitivity of the transport model to its parametrization and
forcing inputs e.g.,
or by comparing prior fluxes to measured local fluxes
e.g.. Some studies also rely
on pure physical considerations e.g..
However, the complex and unpredictable structure of the
uncertainties is hard to reproduce accurately from the expert
knowledge alone and an ill-designed couple of uncertainty matrices
(R,B) can have a dramatic impact on the
inversion results
e.g.. The
discrepancies between the possible configurations of inversion can
also reveal some biases, η, in the models: in that
case p(yo-Hxt)∝N(η,R) instead of
p(yo-Hxt)∝N(0,R), which would require a different
handling of Eq. (). For example, the
horizontal wind fields can be biased or the vertical mixing in the
planetary boundary layer systematically erroneous. That makes it
difficult to compare simulated concentrations in the boundary layer
to measurements e.g.. Biases can have critical impacts on
inversion results and must be inquired into independently
e.g.. Nevertheless, for our study,
we decide to neglect the biases in the inversion. We discuss in
Sect. the potential impacts of biases
that are not significant in our specific application. We then focus
only on the mis-specification of the uncertainty matrices
R and B.
Possible uncertainty handling
In order to address the uncertainty issue in atmospheric inversions,
efforts are carried out towards objectifying the way the error
statistics are chosen e.g.. These efforts focus
on specific algebraic properties of the uncertainty matrices
e.g. or more generally on understanding the
likelihood of the prior innovation vector,
yo-Hxb, as a function of the
uncertainty matrices . Under Gaussian
assumptions, the likelihood of the innovation vector can be written
p(yo-Hxb|R,B,xb)=e-12(yo-Hxb)T(R+HBHT)-1(yo-Hxb)(2π)d|R+HBHT|,
with d the dimension of the observation space and |⋅| the determinant operator.
Distribution of one component of the Monte Carlo posterior
ensemble. The histogram displays the raw posterior distribution. The
dark hatched part of the histogram depicts the proportion of the
ensemble within the tolerance interval TI68,
[xlow,xhigh] (as defined in
Sect. ). The red curve represents
the normal distribution with the same mode and tolerance interval;
the green one stands for a normal distribution with the same mode
and the same standard deviation; the black one is the posterior
distribution computed with the maximum likelihood couple of
uncertainty matrices, presenting under-estimated skewness compared
with the Monte Carlo distribution.
In the likelihood framework, the couple of uncertainty matrices
(R,B) that maximizes Eq. () is
considered as optimal and will be hereafter referred to as the
maximum likelihood. This maximum likelihood optimally balances the
observation and prior state error variances and covariances
according to the prior innovation vector
yo-Hxb. A direct algorithm computing the
maximum likelihood applied to atmospheric inversion in,
e.g. is then
supposed to provide a good approximation of the couple of optimal
matrices (Rmax,Bmax) which can
be used forward in the inversion (Eq. ). In
order to dampen the computation cost of the maximum likelihood
estimation, most studies just maximize the likelihood on
hyperparameters (e.g. correlation lengths), describing the couple
of matrices (R,B) in a more simple way.
Though general, the estimation of the innovation vector maximum
likelihood relies on strong assumptions, it can suffer from strong
numerical errors and it is not necessarily univocal. More
explicitly, as showed in previous works, the pdf of the
uncertainty matrices p(R,B) behaves as a
χ2 distribution with d degrees of freedom, d being the
dimension of the observation space. Thus, the likelihood is highly
dominated by the mode of p(R,B), co-located with
the maximum likelihood. However, the peaked likelihood argument may
be too rough in some cases. As the number of observations decreases
compared to the number of state dimensions, this optimal case
becomes less univocal. In the frameworks where observations are too
scarce, the maximum likelihood may lead to flawed results. To assess
the validity of the peak assumption, estimations of the Hessian
matrix of the likelihood at its maximum have been used e.g.. Hessian matrices
give the magnitude of the uncertainties on the computation of the
uncertainty matrices. Nevertheless, to our knowledge, no atmospheric
inversion accounts for the impact of the Hessian matrix of the
likelihood on the inversion results.
In addition, even when the pdf p(R,B) is
intensely peaked at its maximum, the inferred inversion results from
Eq. () with a direct maximum likelihood
algorithm would erroneously under-estimate uncertainties on the
result see Fig. and,
e.g.. Indeed, at the maximum likelihood,
all the pieces of information in the system are considered perfectly
usable by the inversion, which then gives too optimistic posterior
uncertainties in this case.
Marginalization of the inversionTheoretical formulation
Here, we compute the pdf p(x|yo-Hxb,xb) by
a marginalization on the uncertainty matrices to comprehensively
account for the uncertainties in the characterization of the
uncertainties and to quantify the impact of ill-specified
uncertainty matrices. In statistics, marginalizing a pdf p(x) consists in rewriting it as a
sum of conditional probabilities p(x|z) weighted by
p(z).
Thus, the complete pdf p(x|yo-Hxb,xb)
classically described by Eq. () is separated
into a sum of the contribution of each possible couple of covariance
matrices (R,B) weighted by the probability of
occurrence of the couple (R,B):
p(x|yo-Hxb,xb)=∫(R,B)p(x|yo-Hxb,xb,R,B)×p(R,B|yo-Hxb,xb)d(R,B)∝∫(R,B)N(xã,Pã)×p(R,B|yo-Hxb,xb)d(R,B).
In Eq. (), (.)̃ depicts a
dependency to the couple (R,B). The complete
pdf p(x|yo-Hxb,xb) then has the shape of an infinite sum of weighted
normal distributions. This infinite sum could be described as a
multi-variate T-distribution .
The general expression of Eq. () encompasses
the classical case with only one couple of matrices
(R,B) which considers p(R,B|yo-Hxb,xb) as a
Dirac-like distribution (centred at the maximum likelihood or at
any expert-based couple of uncertainty matrices). More generally,
p(R,B|yo-Hxb,xb) is
not so well known as discussed in
Sect. above.
Monte Carlo sampling
Hereafter, a direct Monte Carlo characterization of
Eq. () is carried out to deduce p(x|yo-Hxb,xb).
The Monte Carlo ensemble is to be defined along the pdf
p(R,B), but the exact distribution of the error
statistics is intricate. In all the following, we then approximate
the pdf p(R,B) by a multi-variate χ2
distribution with d (the number of observations) degrees of
freedom, centred at the maximum likelihood of the prior innovation
vector following. The Monte Carlo
marginalization is consequently a direct extension of the maximum
likelihood estimation now classically used in the atmospheric
inversion framework.
The maximum likelihood can be estimated first by a
quasi-Newtonian descent method. However, descent methods have high
computation costs and thus require a reduced number of
hyperparameters (variances, correlation lengths, etc.) to describe
the full uncertainty matrices. From here, we decide to reduce the
distribution of the matrices (R,B) to the
subspace of the diagonal positive matrices. Using a subspace of the
possible error statistics can dampen the generality of the method.
In particular, error correlations will be excluded with diagonal
uncertainty matrices. Correlations can be used in some frameworks to
detect the biases in the system . But,
more importantly, correlations of observation or background errors
can indicate redundant pieces of information in the inversion
system. For instance, an inversion computed with no observation
correlation tries to use too much information and is expected to
give too optimistic a reduction of uncertainties on the fluxes.
Nevertheless, in Sect. , we reduce the
observation and state spaces in order to numerically compute the
Monte Carlo marginalization. The reduction of the observation and
state spaces indirectly depicts correlations in the full-resolution
system. In this configuration, the correlation issue is then
attenuated and the diagonal assumption is valid.
At the end, for each diagonal term of the uncertainty matrices
(R,B), we prescribe a χ2 distribution with
d (i.e. the dimension of the observation space) degrees of
freedom, rescaled so that its average equals the associated term in
the computed maximum likelihood couple
(Rmax,Bmax). That is to say,
for each diagonal element ri,i of the matrix
R (equivalently of the matrix B):
pri,irmaxi,i×d∝χ2(d)
as the mean of the χ2 distribution with d degrees of freedom, χ2(d), is d.
The χ2 distributions are then sampled on a large ensemble –
the Monte Carlo approach stabilizes after tens of thousands
of draws in our case study – to characterize the final output pdf
p(x|yo-Hxb,xb). Each samples of the ensemble must take into
account the spread of
N(xã,Pã)
in Eq. (). To do so, we describe the pdf
p(x|yo-Hxb,xb) not from the ensemble of posterior fluxes xã, but from a perturbed ensemble
of x̃, with each
x̃ a random sample of
N(xã,Pã).
Processing the Monte Carlo posterior ensemble
In Fig. , we draw an example of the distribution
of the Monte Carlo posterior vector ensemble along one component of
the state space. The black curve depicts the posterior distribution
inferred from the maximum likelihood, with under-estimated spread
compared to the Monte Carlo distribution. On the opposite, as
illustrated by the green curve, a normal distribution with the same
mode and the same standard deviation gives a misleading flat shape.
As for a Gaussian, we then define the symmetric tolerance interval,
so that 68.27% of the samples are in the range (the hatched
portion of the histogram in Fig. ). This interval
is equivalent to the Gaussian ±σ interval, with σ
the standard deviation. One must remember that the computed tolerance
interval does not depict a normal distribution. A normal
distribution with the same tolerance interval (the red curve in
Fig. ) is still misleadingly flat. In all the
following, we will write the tolerance interval TI68,
[xlow,xhigh].
Block diagram of the method. Green boxes represent the raw
inputs of the system. The blue ones are intermediary results and red
ones the outputs to be interpreted. The yellow ones depict the
algorithms to compute. Details in Sects.
and . Insights for output analyses are given
in Sect. .
To summarize (as represented in the block diagram of
Fig. ), the maximum likelihood is first
estimated using a quasi-Newtonian algorithm, similarly to what has
been done in the literature e.g.. We deduce from
this maximum likelihood a plausible distribution of the uncertainty
matrices (R,B). Through a Monte Carlo sampling
of uncertainty matrices (R,B) along the deduced
distribution, we compute an ensemble of possible posterior vectors
(xã(R,B)). We can
then define the tolerance intervals TI68 and a posterior
covariance matrix filled by the covariances of the ensembles of
state components with each other.
Posterior covariance matrices are not always easy to compute in
the atmospheric inversion framework. Here, the posterior covariance
matrix is computed explicitly and objectively. The explicit
definition of this matrix can give valuable information on the
ability of the inversion to separate co-located emissions and
emissions at different periods and locations. This capacity is used
for the evaluation of the OSSEs in
Sects. and .
Informed definition of the problem
The general approach defined in Sect.
applies a Monte Carlo method on tens of thousands of individual
inversions after the completion of a maximum likelihood algorithm.
This procedure requires extensive amounts of memory and computation
power that cannot be afforded in most real cases. For instance, the
explicit computation of H with a CTM closely depends on the dimension of the state space:
every column of the observation operator needs one model simulation
. Additionally, each step of the
algorithm to compute the maximum likelihood of the prior innovation
vector and each step of the Monte Carlo method relies on matrix
products, matrix determinants and matrix inverses. At first sight,
all these operations are as many technical issues in high-dimension
problems.
As a consequence, the application of the theoretically simple
framework developed in Sect. relies
closely on an informed definition of the problem. The dimensions of
the observation and state spaces should be reduced to dampen the
numerical obstacles, but one shall keep resolutions physically
relevant for the system we are analysing. By synthesizing the
recent literature on the subject, we show in the following that
approximations can be reasonably applied to the full-resolution
problem while not impacting the quality of the marginalized
inversion results. Applying the Monte Carlo marginalized inversion
is then technically feasible in a problem defined with a reduced
dimension from the full-dimension problem.
Principle for problem reductionMotivations and definition
In the observation space, more and more surface observation sites
nowadays provide quasi-continuous measurements at least a
few data points per minute in the data set we use;. For long
windows of inversion at the regional scale (of a few weeks or
months), such a frequency of acquisition generates a number of data
points technically impossible to assimilate all together in our
framework. Concerning the fluxes, one shall aim at a
characterization of the fluxes on very fine pixels and at a high
temporal resolution. As the window of inversion lengthens and the
domain widens, the number of flux unknowns grows dramatically.
In the inversion framework, the most straightforward way of
minimizing the dimension of a problem is to reduce the dimensions of
the observation and state spaces. Aggregating components of the
state space and sampling observations are classically used for this
purpose. In most studies, the reduction of the problem is carried
out arbitrarily. However, aggregation can generate large errors
, which
would mitigate the benefits of the Monte Carlo marginalized approach
compared to more classical ones applied in other atmospheric
inversion studies with no aggregation e.g. variational
inversions;. Here, we propose a more objective way to
do so following recent literature.
Using the formalism from , we aim at
defining a representation ω that encompasses the horizontal
and temporal resolution of the fluxes, the choice of the regions of
aggregation and the temporal sampling of the observations. The
representation ω is defined through two operators
Γω and Λω, which
projects respectively the full-resolution state and observation
space to smaller ones. After the state space “projection” with
Γω, the inversion applies corrections on
regions of aggregation with fixed emission distributions, instead of
on single pixels. The adjoint of this operator,
ΓωT, then redistributes total
emissions on finer scales with the same fixed emission distribution.
The choice of Γω impacts both the state vector
x and the observation operator H. The observation
sampling Λω can consist in averaging or
picking one value per time step (chosen accordingly to the physical
resolution inquired into). For instance, one can decide to average
the observations by day in order to study the synoptic variability
of the atmosphere, related to the fluxes at the mesoscale. The
observation sampling applies to both the observation vector
yo and the observation operator H. The
observation operator H computes the contribution from
single sources to single observations. The adjoint of the
observation sampling, ΛωT, will then
redistribute an average or a sample for each chosen time step along
this same time step. The redistribution will follow the raw observed
temporal profile within the processed time step.
Mathematical formulation
At first glance, choosing the aggregation pattern and the sampling
protocol can be considered as two independent physical problems.
However, as they both influence the dimension of the observation
operator H, they cannot be fixed separately. More
explicitly, we can derive a formula which links
Γω and Λω. Indeed, our
final objective is to compute total posterior fluxes for each
aggregated region that are as close as possible to the posterior
fluxes from a full-resolution inversion aggregated afterwards. That
is to say, we want to confine the norm of
xωa-Γωxta
with xωa, the posterior state vector resolved
in the representation ω and xta the
posterior state vector computed with a full-resolution
representation of the problem. Algebraic manipulations lead to
xωa-Γωxta=ΓωBEω(yo-Hxb),
where
Eω=PωHTΛωTSω-1Λω-HTS-1,S=R+HBHT,Sω=ΛωR+H(Aω+PωBPω)HTΛωT,Pω=(Γω)TΓω,Aω=(I-Pω)xtxtT(I-Pω),xt is the true state of the system,I is the identity matrix.
In Eq. (), R and B are the full-resolution matrices of the error statistics.
For the aggregation errors to be limited, Eω
(Eq. ) must tend towards 0. Then,
S (Eq. ) and Sω
(Eq. ) must be as close as possible to each other
and the impact of Pω (Eq. ) and
of the sandwich product with Λω,
ΛωT(⋅)Λω, must be as small as possible. ΓωT
extrapolates the fluxes from the aggregated regions to a finer
resolution following an a priori repartition. The matrix
Pω then redistributes the fluxes over a region
with respect to the prior repartition, but keeping the same total
emissions on the region.
In Sect. below, we explain how
to reduce these terms. The exact estimation of
Eq. () is complicated and requires extensive
numerical resources e.g.. In the
following, we use physical considerations towards minimizing
Eq. (). The errors that are intrinsic to the
aggregation process and that are unavoidable are controlled so that
the benefit from the general marginalization is not wasted. We show
in Sect. that the physical
considerations for choosing the representation ω in our case
do not depreciate the inversion results compared to what would have
been obtained with the exact resolution of Eq. ().
Considering the computer resources we use, all the principles we
define are applied in order to limit the size of the observation
space (resp. the state space) to a dimension of roughly 2000 (resp. 1500). For instance, in the mesoscale Eurasian case study described
in Sect. , these considerations lead to the
aggregation patterns displayed in Figs. and
. With these problem dimensions, the
ensemble used in the Monte Carlo sampling consists of 60 000 draws.
When the observation and the state space aggregation are chosen, the
operator H can be computed with the so-called “response
functions”, based on forward simulations of the transport for each
state component .
Representation choiceObservation space sampling
The sandwich product with Λω,
ΛωT(⋅)Λω,
aggregates the errors in the observation space and diffuses them
back within each aggregate along a prescribed temporal profile. For
example, it can typically compute the average error per day; then, it
allocates for each subdaily dimension an error proportional to the
contribution of the related component of yo to the
daily mean. However, a daily averaging would be dominated by the
outliers (e.g. singular spikes or night-time observations when the
emissions remain confined close to the surface due to weak vertical
mixing) that are generally associated with very high observation
errors (due to fine-scale misrepresentations of the transport and
erroneous night vertical mixing). For this reason, we decide to
define Λω as the sampling operator, which,
for each day and observation site, picks the component of the
observation vector when the daily minimum of concentrations within a
planetary boundary layer higher than 500 m is observed. Below this
threshold, the vertical mixing by the model is known to be possibly
erroneous e.g.. The daily
resolution is chosen in order to keep a representation of the
transport relevant to the mesoscale expectations on flux
characterization. Higher time resolution would not improve the
inversion efficiency due to strong within-day temporal correlations
of errors .
Observational constraints
One can notice that far from the observational constraints, the
atmospheric dispersion (depicted by the sandwich product with
H, H(⋅)HT) makes the
potential errors negligible compared to the errors generated in the
areas close to the stations. Indeed, gathering two close hotspots
of emissions thousands of kilometres away from the observation sites is not
problematic since the air masses coming from the two spots will be
well mixed. On the contrary, two hotspots that are as distant from
each other as the first two, but close to an observation site, will
generate plume-like air masses with a very high sensitivity to the
errors of mixing and transport in the model. We use an estimation of
the observation network footprints (approximating HT) in order to fix the typical regions constrained by the network
and avoid unfortunate grouping. At this step, approximate
footprints are preferred to the heavy computation of the complete
HT and are sufficient for our physical
considerations. Within the constrained regions, we use a small
spatial resolution for the fluxes and the transport and fine
aggregation patterns; outside of them, we choose a coarse resolution
and large aggregation patterns. These guidelines for using
footprints prior to an inversion can be applied more systematically,
as what is done in . An
illustration of aggregation patterns in our case study can be looked
at in Fig. .
Flux aggregation
Some terms in Eq. () are directly related to the
aggregation of the fluxes. The term HAωHT in Eq. () depicts the
aggregation errors coming from the uncertain distribution and
temporal profile of the fluxes within each aggregation region, then
transported to the observation sites. It must be close to
0. In our application below, this is particularly
important for hotspots of emissions, the locations of which are poorly
known. The term HPωBPωHT in Eq. ()
must be as close as possible to HBHT. The factors of divergence between these two
terms come from the areas that are not well constrained by the
observations. If, within a region of aggregation, a part is upwind
the observation sites, while the other is not seen, then the
aggregation errors will scatter on the unseen part of the region.
The main sources of errors can then be separated into two different
types: (1) the resolution/representation type, and (2) the constraint
type.
The type-1 errors are mainly related to the resolution of the
observation operator. The models consider that the fluxes and the
simulated atmospheric mixing ratios are uniform on a subgrid basis
and neglects subgrid processes. This discretization contributes to
type-1 errors, as “representation” errors
. Additionally, the distribution within
each aggregation region is fixed and subregion rescaling is
forbidden. The fine resolution close to the observation network is
bound to confine type-1 errors e.g..
Additionally, the representation error is critical for co-located
emissions, especially when the typical temporal and spatial scales
of these emissions are different. For instance, grouping hotspots
from oil extraction emissions with widespread wetland emissions that
quickly vary in time is hazardous. We then aggregate the emissions
along their typical time and space scale, hence according to the
underlying physical process. An in-depth analysis of the footprints
and the small patterns of aggregation close to the observation sites
should limit the type-2 constraint errors. Areas under high
observational constraints should not be grouped with
under-constrained areas.
The resolution and aggregation choices can be computed objectively,
but at a very high cost and only within a framework of prescribed
frozen error matrices . For our purpose, we cannot afford such
computation costs and rely on heuristic choices: small resolution
and aggregation patterns close to the observation sites, aggregation
by type of emission, separation of constrained/under-constrained
areas by analysing the footprints. These non-optimal subjective
choices may damp the efficiency of our method and must be carried
out cautiously. Nevertheless, in our case, checking our choices after the
computation shows that they did not have a critical impact on the inversion
results.
Numerical artefacts
In addition to the need of defining a well-sized problem, smart
adaptations can be applied to the computation of the method in order
to enhance the efficiency of the algorithm. We face several sources
of numerical artefacts in the computation of the method. In the
quasi-Newtonian maximum likelihood algorithm, numerical artefacts
are generated by the under-constrained regions. After a few steps,
the computed gradient of the likelihood is dominated by these
regions and the algorithm stays stationary. This issue could be
partly related to the under-optimality of the chosen representation
ω as suggested by the optimality criteria described in
. The stagnation of the maximum
likelihood algorithm could then be used to detect too small regions
of aggregation.
The under-constrained regions perturbing the maximum likelihood
algorithm can be diagnosed using the diagonal terms of the influence
matrix KHwith K defined in
Eq. () and following. This matrix represents the
sensitivity of the inversion to elementary changes in the
observations. Strong observation constraints are related to high
sensitivity. After stagnation, the regions with a diagnosed
KH<0.5 are flagged out and the algorithm is carried on.
This way, only the sufficiently constrained components of the state
vector are processed until the algorithm converges. A third to half
of the regions are flagged out this way in our case study.
The detection of the misrepresentation of hotspot plumes should
also be enhanced. Despite the minimum daily sampling and the fine
resolution close to the observation network, the plume issue can
still generate strong temporal and spatial mismatches. For example,
a point source can influence a station in the real world but not in
the model because it has been mis-located, and vice versa. This
creates significant differences between the simulated and the
observed concentrations. The maximum likelihood algorithm attributes
such mismatches to prior errors and/or observation errors. High
diagnosed errors in the maximum likelihood algorithm are then a
criterion for plausible mismatches. We know such plumes must be
flagged out from the inversion to avoid irrelevant high influence
from very local sources hard to represent. Since we notice that the
observation and prior computed errors seem to follow a
Fischer–Snedecor distribution, we choose to flag out the
observations that are within the 95 % tail of the distribution.
Validation experiments
In Sect. , we described our modified
atmospheric inversion by marginalization. In
Sect. , we proposed some essential rules to
follow in order to properly define the problem, so that the rather
simple theoretical framework is not hindered by finite numerical
resources. The marginalization method has to be validated along
objective criteria. In the following, we summarize the general
structure of the method in order to identify the critical points to
test in the method (Sect. ). We deduce
from these points some OSSEs to carry out. In
Sect. , we define the scores
according to which the method will be evaluated.
Required testsMethod summary
The method described in Sects.
and is condensed in the block diagram in
Fig. . The marginalized inversion takes the
same input as any other atmospheric inversion: some atmospheric
measurements and prior maps of fluxes with specified resolution and
temporal profiles. In Sect. , we gave
recommendations on the processing of the “raw” inputs, so we get an
observation vector yo, a prior state vector
xb and an observation operator H that are
small enough to be computable by the method. These highlights are
mainly the sampling of the observations per day (in accordance with
our objective of characterizing mesoscale fluxes in our case study)
and the aggregation of the fluxes by regions (based on physical
considerations and footprint analysis). The maximum likelihood
algorithm processes yo, xb and
H in order to find a couple of optimal diagonal error
matrices (Rmax,Bmax). This
maximum likelihood is found by a quasi-Newtonian descent method. We
then infer from (Rmax,Bmax) the
approximate χ2 shape of the distribution of all the possible
error matrices (R,B). We carry out a Monte Carlo
sampling on these distributions of errors and get an ensemble of
posterior state vectors (xa^). The
processing of this ensemble provides the final output of the method:
a tolerance interval TI68 of the posterior state and the
posterior correlations between the components of the state space.
The method also allows for the explicit computation of the influence
matrix KmaxH in order to analyse the
constrained regions of emissions only.
To summarize, the marginalized inversion processes two vectors and
one operator: yo, xb, and
H, as any other atmospheric inversion. The main
difference with most other atmospheric inversions resides in the
objective and automatic computation of the influence of
ill-specified error statistics, in contrast with the traditional
assigning of frozen error matrices based on expert knowledge and
with the more recent online computations of error hyperparameters.
Thus, we do not have to inquire into the sensitivity of our method
to the prescribed spatial correlations of flux errors, or to the
error variances. Such a sensitivity is transposed to the choice of
the aggregation patterns and the sampling protocol, as defined in
Sect. . The chosen configuration of
aggregation and the sampling protocol are checked afterwards to be
relevant in our case study. OSSEs are then to be carried out to
evaluate the sensitivity of the method to yo,
xb, H.
Test strategy
We assume that, in our case, the method is not sensitive to errors
in yo. Indeed, in all the following, we consider
that the measurement errors are negligible compared to transport
errors; this is true for surface sites that fulfil the World
Meteorological Organization's strict recommendations for accuracy and
precision . This approximation does not
hold for satellite total column measurements, for which the
transport errors are smoothed over the vertical atmospheric column,
and the instrument errors are larger. In addition,
representativeness errors may also impact yo. OSSEs
should account for these errors. However, OSSEs may face
difficulties in explicitly highlighting these errors. Therefore, we
do not perturb yo in order to represent the
instrumental uncertainties and representativeness errors in the
OSSEs.
The OSSEs are then based on perturbations of xb and
H. The discrepancies between the background
xb and the “truth” xt are of two
types: (1) the erroneous distribution and temporal profile of the
fluxes within aggregation regions, and (2) incorrect total emissions
by region. For example, in Eurasia, the maps of the distribution of
the wetlands differ drastically from one database to another
. Apart from the distribution, the amount of
gas emitted by each process is uncertain, due to
mis-parametrizations or, for anthropogenic emissions, mis-specified
activity maps e.g.. The
transport H differs from the “true” transport mainly
because of the resolution of the model, the parametrization of
subgrid processes (such as vertical turbulent mixing in the
planetary boundary layer or deep convection), and the meteorological
forcing fields (which are not necessarily optimized for transport
applications).
The main sources of errors in the inversion are then (1) a wrong
representation of the transport (highly dependent of the transport
model used, its resolution, its parametrization and the exactitude
of forcing wind fields), (2) an erroneous distribution of the fluxes
within aggregation regions (each inventory and database has
different statistical methods and parameters to reproduce surface
fluxes), and (3) incorrect total emissions by regions. In order to
evaluate the impact of each point on the inversion result, we carry
out OSSEs with perfect synthetic observations from a nature run
(i.e. with “true” emissions and “true” transport, as defined in the
set-up in Sect. ). We test the ability of the
marginalized inversion to reproduce the “true” fluxes or, at least,
to consistently include the “truth” within the tolerance intervals.
There are eight possible combinations of correct or perturbed phases
of the three parameters. The “all true” combination is not relevant:
yo-Hxb=0 and the maximum
likelihood algorithm is stationary. Seven combinations remain,
detailed in Table . We run the marginalized
inversion for the seven OSSEs and evaluate them along the scores
defined in Sect. below.
OSSEs summary.
Three parameters of the inversion (subtotal masses emitted per regions, emission distribution and transport) can be perturbed compared with the “truth”.
The seven possible combinations are depicted by = and ≠ signs for each parameter and each OSSE.
Every OSSE is evaluated along the scores defined in Sect. .
The scores are given in percentages for the best correlation threshold for grouping the state space components as presented in Sect. .
The influence score must be as close to 100% as possible.
The other two scores must be as small as possible.
The regions are grouped along a correlation criterion rmax (see Sect. );
we present the scores only for rmax with the best results.
For OSSE 7, the scores are zeros for the fossil fuel regions because most of these regions were filtered out.
The few remaining ones are very well constrained.
We expect an atmospheric inversion to provide reliable ranges of
uncertainties for surface fluxes. That is to say, for as many
components of the state vector xi as possible, the “truth”
xit should be within the tolerance interval
TI68, [xilow,xihigh] (defined
in Sect. ). In order to evaluate the
ability of producing consistent fluxes, we define a relative score
zrel for each component of the state vector:
(zrel)i=2|xia-xit|xihigh-xilow.
Hereafter, all the scores will be expressed in percentages for better
readability. Statistically, zrel has no upper bound.
Relative scores bigger than 100% are not statistically
inconsistent, but, for the method to be validated, we expect that
the proportion of state components with zrel<100%
is dominant.
Furthermore, the atmospheric inversion is supposed to reveal pieces
of information to the understanding of the system. Then, we also
expect that a correct relative score below 100% is not reached by
specifying huge tolerance intervals. To evaluate the ability of the
marginalization of getting close to the reality, i.e. providing
valuable information on the state of the system, we define an
absolute score zabs:
(zabs)i=xiaxit-1. The smaller the absolute score, the more accurate the
marginalized inversion.
An inversion also must be able to evaluate the observation
constraints on the regions. An objective estimator of the
constraints on the regions is the influence matrix KH
defined in Sect. . The Kalman gain matrix
depends on the couple (R,B). Amongst all the
Monte Carlo draws, we compute the influence matrix
KmaxH for the couple associated with the maximum
likelihood. The diagonal terms of this matrix range from 0 to 1.
They give for all components of the state space the constraint given
by the observations. We then define the influence score:
(zinfl)i=(KmaxH)i. The
closer these terms are to 100%, the more constraints the inversion
provides. We can then deduce the typical range of influence of the
observation sites and detect the blind spots of the used network.
For each component i of the state space, we then have defined three indicators:
(zrel)i=2|xia-xit|xihigh-xilow,(zabs)i=xiaxit-1,(zinfl)i=(KmaxH)i.
Posterior correlation processing
Another point most inversions do not compute explicitly and
objectively is the typical temporal and spatial scales the inversion
can differentiate in the fluxes, considering the atmospheric
transport and the density of the observations. Our marginalized
inversion gives access to an explicit matrix of correlations as
defined in Sect. . Strong positive and
negative correlations between two components of the state space
indicate that the inversion cannot separate the contributions from
the two components. For example, air masses observed at a station
and coming from two regions upwind the station will have a mixed
atmospheric signal difficult to analyse. Co-located emissions are
also not necessarily differentiated in the atmospheric signal.
Moreover, in a regional framework, when a model of limited area is
coupled to lateral boundary conditions (LBC), the inversion must
explicitly alert on the regions that cannot be separated from the
boundary conditions, i.e. from the baseline signal.
In the case of strong correlations between two components of the
state space in the posterior covariance matrix, we consider that it
is not relevant to try to infer specific information for the two
separate components. Then, we group the state space components
according to their posterior correlations. We define a threshold of
correlation rmax and associate couples of regions (i,j)
within groups such that |ri,j|>rmax. If we prescribe
rmax=0, all the regions will be grouped; conversely,
if rmax=1, no group will be formed. The optimal
correlation threshold is not evident. We test the grouping for all
possible rmax values. We flag out from the processing of the
results all the groups, which include some contributions from the
LBC. Thus, from this post-processing, we only keep the regions that
are clearly constrained by the observation sites, with no
interference from the LBC. Moreover, we can infer the spatial and
temporal scale that the inversion can resolve from the grouping
patterns.
In Table , the three scores defined in
Eq. () are averaged on the whole domain of interest for
the optimal correlation threshold rmax (as discussed in
Sect. ).
Set-up of the OSSEs
We compute the OSSEs that we described in
Sect. in a realistic mesoscale case.
We focus on a domain spanning over Eurasia, from Scandinavia to
Korea. At this scale, the air masses' residence time is typically of
days to a few weeks. This timescale is small compared to the
8–10-year lifetime of methane in the atmosphere mainly
due to oxidation by OH radicals;.
Hence, the observation operator can be considered linear. We apply the
method on a region characterized by significant fluxes, with
collocation of different sources with different emission
timescales: Siberia. We describe the region of interest and the
chosen “truth” for the experiments in
Sect. . We use two transport models in
order to simulate atmospheric transport. The technical details on
these models are summarized in Sect. . In
Sect. , we explain how we choose and
compute the synthetic observations for our experiments.
Topographic map of the domain of interest. The colour bar
shows the altitude above sea level from ETOPO1 database;. Red dots (resp. orange triangle) depict
hotspots of CH4 emissions (based on EDGAR v4.2 inventory; see
Sect. ) related to oil welling and
refineries (resp. gas extraction and leaks during distribution in
population centres). Purple squares highlight the observation site
localization. Blueish shaded areas represent average inundated
regions, wetlands and peatlands from the Global Lakes and
Wetlands Database;
Virtual true state xtState space components
In the region of interest, the emissions of methane are dominated
by wetland, anthropogenic (here, mainly related to the oil and gas
industry) and wildfire emissions. In Fig. ,
the distributions of the wetlands and of the oil and gas industry in
the region are displayed. Anthropogenic emissions of methane in the
region are mainly hotspots related to the intense oil and gas
industry in the Siberian Lowlands and to the leaks in the
distribution system in population centres in the south part of
Siberia. Wetland emissions are mainly confined in the lower part of
Siberia in the west Siberian plain, half of which is lower than
100 m above sea level.
The spatial distribution of the associated fluxes is deduced from the
(1) EDGAR database v4.2 (http://edgar.jrc.ec.europa.eu) for
year 2008 for anthropogenic emissions, (2) LPX-Bern v1.2 process
model at a monthly scale for wetland emissions
, and (3) GFED database at daily scale
for wildfires . The EDGAR inventory
uses economic activity maps by sectors and convolves them with
emission factors estimated in laboratories or with statistical
studies . LPX-Bern is an update of
process model LPJ-Bern . It includes
a dynamical simulation of inundated wetland areas, dynamic nitrogen
cycle, and dynamic evolution of peatlands
. The model uses
CRU TS 3.21 input data (temperature, precipitation rates, cloud
cover, wet days) and observed atmospheric CO2 for each year for
plant fertilization. GFED v4 is built from the burnt-area satellite
product (MCD64A1). CH4 emissions at monthly and daily scales are
deduced from the burnt areas using the
Carnegie–Ames–Stanford Approach CASA model; and emission factors
. Wildfire emissions can be very
strong and are punctual in time and space; they are then difficult
to analyse by the inversion. Wildfires are included as inputs to the
marginalized inversion but are automatically filtered out during
the computation. In all the following, we evaluate the OSSEs only in
terms of anthropogenic and wetland emissions.
In addition, at the mesoscale, we use a CTM (see
Sect. ) with a limited area domain.
Initial and lateral boundary conditions (IC and LBC) are then also
to be optimized in the system to correct the atmospheric inflow in
the domain. Lateral concentrations are deduced from simulations at
the global scale by the general circulation model LMDZ with the
assimilation of surface observations outside the domain of interest
. We aggregate the LBC along four
horizontal components and two vertical ones (1013–600 and
600–300 hPa).
Generation of a perturbed reference state xt
The EDGAR fluxes are given at the yearly scale and the LPX fluxes
are calculated at a monthly scale. Additionally, LPX monthly fluxes
exhibit smoothed patterns while wetland emissions can vary
drastically from a point to another. We want the nature run for
OSSEs to reproduce the potential spatial and temporal variability of
the emissions. To do so, we intensify the spatial and temporal
contrasts from the databases to the nature run. We then compute the
“true” state vector xt by perturbing EDGAR emissions
on a monthly basis and LPX on a 10-day basis. That is to say
xt=α⊗xdata, with
the vector α depicting the scaling factors by state
space component, ⊗ the point-wise multiplication operator
and xdata the emissions from the databases. The
perturbations in α from original EDGAR and LPX
databases applied to get the “truth” are scaling factors of up to 10.
These scaling factors could have been chosen randomly, but we prefer
inferring them with a raw expert-knowledge-based inversion using
real data. The purpose of using real data for computing
xt is to generate potentially realistic variations
within the state space.
Distribution of the scaling factors applied to the
emission databases in order to compute the “truth”. All the emission
components of the state vector have been included in the histogram.
The selection of the scaling factor distribution is detailed in
Sect. .
For both anthropogenic and wetland emissions, the scaling factors can significantly differ from a period of inversion to another.
We can then evaluate the ability of the marginalized inversion to catch quick variations.
The distribution of the scaling factors α is shown in Fig. .
These factors are plausible, knowing the uncertainties on the wetland emissions and gas leakage e.g..
Such target scaling factors are at the edge of the validity of the Gaussian assumption and of the positivity of methane fluxes.
The ability of the marginalization to recover such correction factors will prove its robustness.
As for anthropogenic and wetland emissions, we apply the scaling factors α on the components of xt related to LBC by periods of 10 days.
The OSSEs rely on xb perturbed from xt, or not.
We apply two types of perturbations as summarized in Table .
In OSSE 1, 4, 5 and 7, we only implement prior fluxes with different total emissions on the regions of aggregation.
We take the emissions of the raw inventories as background to test the ability of recovering “true” fluxes from realistic background fluxes without assigning frozen prior errors.
In OSSE 2, 4, 6 and 7, the distribution of the prior fluxes is modified from the “truth”.
We choose all flat flux distributions for each region of aggregation as prior fluxes.
Simulation of the observation operator H
The observation operator H is deduced from simulations of atmospheric
transport.
We use two different transport models in order to evaluate the impact of the transport on the inversion.
We define HFLEXPART with the Lagrangian dispersion model FLEXPART and HCHIMERE
with the Eulerian chemistry-transport model CHIMERE.
Any transport model can be considered at some point biased compared with the reality.
Confronting the results from FLEXPART to those from CHIMERE will allow us to test the robustness of our method to the biases.
The Lagrangian model: FLEXPART
With the Lagrangian dispersion model FLEXPART
, we can compute the footprints of the
observations, hence HFLEXPARTT. We use
FLEXPART version 8.2.3 to compute numerous back trajectories of
virtual particles from the observation sites. The model is forced by
the European Centre for Medium-range Weather Forecast (ECMWF)
ERA-Interim data at a horizontal resolution of
1∘×1∘, with 60 vertical levels and 3 h
temporal resolution . Virtual particles
are released in a 3-D box centred around each observation site with a
10-day lifetime backwards in time. The footprints are computed on a
0.5∘×0.5∘ horizontal grid, following the method
of , taking the boundary layer height at
each particle location into account. The footprints only have to be
convolved with the emission fields in order to get simulated
concentrations at the observation sites. The method for computing
the footprints considers that only the particles within the boundary
layer are influenced by surface emissions and that the boundary
layer is well-enough mixed to be considered as uniform. The uniform
vertical mixing of the mixing layer can generate a bias on the
surface-simulated concentrations. Such a bias is critical in the
classical inversion framework and consequently in the one we
describe since all the uncertainties are considered unbiased.
FLEXPART can easily compute an estimation of the adjoint of the
full-resolution observation operator before choosing the
representation ω. Hence, despite the expectable biases, we
use this model to estimate the footprints of the network and deduce
the aggregation patterns needed to compute
HCHIMERE. This same model FLEXPART may also be
used to compute explicitly and rigorously the representation
ω according to objective criteria
.
The Eulerian model: CHIMERE
Using the Eulerian mesoscale chemistry transport model CHIMERE
constrained by
non-hydrostatic meteorological fields, we explicitly define the
observation operator HCHIMERE by computing the
forward atmospheric transport from the emission aggregated regions
(defined according to Sect. criteria) to
the observation sites. This model was developed in a framework of
air quality simulations
but is also used
for greenhouse gas studies
. We use a
quasi-regular horizontal grid zoomed near the observation sites
after the considerations of Sect. . The
domain of interest is of limited area and spans over the mainland of
the Eurasian continent (see Fig. ). The
average side length of the grid cells near the stations is 25 km,
while it spans over 150 km away from the observation sites. The
3-D-domain roughly embraces all the troposphere, from the surface to
300 hPa (∼9000m), with 29 layers geometrically spaced. The
model time step varies dynamically from 4 to 6 min depending on the
maximum wind speed in the domain. The model is an offline model
which needs meteorological fields as forcing. The forcing fields are
deduced from interpolated meteorological fields from ECMWF with a
horizontal resolution of 0.5∘×0.5∘ every 3 h.
Eurasian site characteristics (Sect. ). The altitudes of the sites are given
as metres above sea level (a.s.l.) and the inlet height is in metres above ground level
(a.g.l.).
StationIDLocation InletLon.Lat.Alt.height(∘ E)(∘ N)(m a.s.l)(m a.g.l.)AzovoAZV73.0354.7110050BerezorechkaBRZ84.3356.1515080DemyanskoeDEM70.8759.797563IgrimIGR64.4263.192547KarasevoeKRS82.4258.255067NoyabrskNOY75.7863.4310043PallasPAL24.1267.975605ShangdianziSDZ117.1240.652870Tae-ahn PeninsulaTAP126.1236.72200Ulaan UulUUM11.0844.459140VaganovoVGN62.3254.5020085YakutskYAK129.3662.0921077ZotinoZOT89.3560.80104301Synthetic observations yo
We compute the nature run, i.e. the synthetic observations, from
the “true” state vector, with the CTM CHIMERE. That is to say, in
all the following, we consider that
yo=HCHIMERExt. The
site and date of available observations are chosen according to the
operated observation sites in the region. Thirteen Eurasian surface
sites have been selected. These sites are maintained by NIES
Tsukuba, Japan;, IAO (Tomsk,
Russian Federation), MPI Iena, Germany;, NOAA-ESRL Boulder, United
States of America;, and KMA
(Seoul, Korea). The description of the sites is given in
Table . The observation sites do not carry out
measurements all year-round due to logistical issues and
instrument dysfunctions. In order to reproduce this sampling bias,
we generate synthetic observations only when real measurements are
available from January to December 2010.
Results and discussion
After the description of the set-up in Sect. ,
we now have a “true” state xt and some reference
observations yo. We also have two observation
operators HCHIMERE and
HFLEXPART and several possible prior fluxes
xb as inputs for the marginalized inversion
developed in Sect. . In order to evaluate
the method, we now carry out the OSSEs described in
Table following the complete procedure in
Fig. . In Sect. ,
we examine the average robustness of the method. Then, in
Sect. , we explore the spatial
efficiency of the marginalized inversion in our case study. In
Sect. , we discuss the enhancement
provided by our method compared to the classical Bayesian framework,
despite some limitations.
Score comparison on fossil fuel (up) and wetland (bottom)
regions for all OSSEs along correlation thresholds rmax of
region grouping (see details in
Sect. ). Left: influence correlation
zinfl profile. Centre: relative score
zrel correlation profile. Right: absolute score
zabs correlation profile. The red arrows depict the
direction from lowest scores to best ones. The blue arrows denote
the direction of grouping, from all grouped (“G”, rmax=0)
to all separated (“S”, rmax=1). The OSSEs are indexed
along Table numbering. Thin (resp. thick) lines
stand for correct (resp. perturbed) subtotal emissions. Green
(resp. brown) lines depict correct (resp. perturbed) emission
distributions. Solid (resp. dotted) lines represent correct (resp. perturbed) transport. As in Sect. ,
the scores are noted in percentages.
Robustness of the methodImpact of the correlation processing
The marginalization should consistently reproduce the nature run in
the OSSEs or, at least, it should detect its inability in
characterizing the fluxes from the given atmospheric constraints. As
detailed in Sect. , the aggregation
regions may have strong posterior correlations after the
marginalized inversions. This denotes the difficulties that the
inversion encounters in separating some emissions. The aggregation
regions can be grouped along correlation thresholds rmax
arbitrarily chosen in order to explicitly take the emission dipoles
into account. In Fig. , we plot the profiles of
the scores defined in Sect. along
the possible correlation thresholds rmax for grouping the
regions. Specifying a correlation threshold rmax allows
identifying the typical temporal and spatial scales that the
inversion can separate. In the case of a limited domain CTM, the
influence of the LBC and of the inside fluxes can be mis-separated.
The correlations take this issue into account and the correlation
threshold specifies the tolerance to such mis-separations.
For all OSSEs, the influence score zinfl increases
with rmax. In the correlation processing after the
computation of the marginalized inversion, the threshold
rmax depicts the degree of tolerance to mis-separation
between inside fluxes and LBC. The higher the threshold of tolerance
rmax, the fewer inside fluxes are considered inseparable
from the LBC. Hence, fewer aggregation regions are eliminated from
the inversion and more fluxes are corrected by the inversion. As the
number of constraints increases, we notice that the absolute and
relative scores, zabs and zrel, also
tend to increase with rmax. That is to say, if we only try
to get average information on big under-resolved regions, the
posterior fluxes can be expected to be closer to the “truth”. On the
contrary, if we try to process too much spatial information from the
inversion, we must expect more discrepancies with the “truth”.
In particular, in Fig. , one can notice some outlier peaks for low rmax.
For low rmax, very few regions are computed in the inversion.
The peaks are created by the regions that are not any more considered as mis-separated from the LBC when rmax increases.
For some OSSEs, these newly computed regions have very wrong scores and dominate upon the other few, computed regions.
For this reason, one should be very careful in the chosen correlation threshold.
In order to avoid the score instability, the optimum threshold should be chosen higher than 0.4.
Above 0.5, in our mesoscale case study, as described above, the inversion is limited
by the temporal and spacial variability of the fluxes to optimize and by the transport biases.
Then, it cannot reach the requirement of consistent reproduced fluxes.
One should find a balance between the physical scales one wants to separate and the consistency of the results.
In Table , we summarize the scores of every OSSE for a chosen correlation threshold with respect to result consistency.
Hotspots and large-area emissions
Both in Table and Fig. ,
looking at a given correlation threshold rmax, one would
expect influence, relative and absolute scores that get worse
when the inversion condition degrades.
The fossil fuel influence score follows this trend: the more
perturbed the transport and the prior fluxes are, the more state
space components are considered un-invertible. The hotspot regions
of emissions are broadly filtered out and the remaining regions can
be well characterized by the inversion even with wrong distribution
and transport patterns. Some effects in the degrading conditions of
the inversion can however compensate each other. For example, the
absolute scores of OSSEs 5 and 7 are better than the one of OSSEs 3
and 6.
The situation for wetland emissions is different.
These emissions are smoother than oil and gas emissions and are then not excluded because of wrong transport or distributions.
For this reason, the influence score does not exhibit a clear trend with degrading inversion conditions.
For wetland regions, transport seems to be the predominant factor of errors.
OSSEs 3, 5, 6 and 7 do not consistently reproduce the “truth” with relative scores higher than 100 % when rmax≥0.4.
These discrepancies can be attributed to the very high variability prescribed in the “true” wetland emissions.
An erroneous transport will fail in detecting brutal changes of emissions at the synoptic scale.
The wetland emissions should then be grouped temporally and spatially in order to average the point releases of methane.
The erroneous tolerance intervals can also be attributed to the
biased transport in FLEXPART compared with CHIMERE. Since we
filtered out most of the plumes with spatial and temporal mismatches
with the observations, the horizontal biases in the transport are
confined. Concerning the vertical bias, a wrong simulated vertical
mixing in the planetary boundary will affect all the fluxes. This
bias will then have an impact on the atmospheric concentrations that
is relatively smoothed, uniform and constant. Therefore, an accurate
detection of such a bias is very difficult. Any inversion relies on
the unbiased assumption of the errors. The inversion will attribute
the biases to the flux for wetland regions, impacting the result of
the inversion. As other inversions, despite the marginalization, it
appears that the results on wetland regions may be sensitive to
vertical transport biases in the models (see discussion in
Sect. ).
Thus, the marginalized inversion seems to be sensitive to transport
biases and to fluxes varying too quickly, as any other inversions.
Nevertheless, post-processing is made possible by the explicit and
objective computation of the posterior covariances and of the
influence matrix. This post-processing proves that the atmospheric
inversion is not able to inquire into very fine scales in our case
study. The correlation grouping of indifferentiable regions allows
for an accurate analysis of the best possible signal detectable by the
inversion. In the following, we take a correlation threshold of 0.5
as a good balance between sufficient constraints on the system and
consistent posterior fluxes.
Map of the average scores as defined in
Sect. for OSSE 1 (see
Table ) projected on the aggregation grid defined in
Sect. . Top: influence score
zinfl. Middle: relative score zrel.
Bottom: absolute score zabs. The colour maps have
been chosen so that redder regions correspond to better scores
(denoted by ⊖ and ⊕ symbols). The resolution and
physical projection of the maps are the same as in Fig. .
Spatial evaluation
We have chosen a threshold of correlation grouping the regions so
that the averaged scores on the whole domain of interest are
optimal. The scores are not uniformly distributed. In
Fig. , the distributions of the three scores
are displayed for fossil fuel regions and wetlands for OSSE 1
(transport and distribution of the fluxes same as the “truth”,
perturbed masses by regions; see Table ). We choose
the “easiest” OSSE configuration in order to evaluate the behaviour
of the marginalized inversion in the best configuration possible,
thus getting the upper bound for the expectable quality of the
results. Any more realistic set-up likely gives results that are not as good.
In the figure, the scores are projected on the aggregation grid
built on the considerations in Sect. . Most
of the observation sites are located in the centre of the domain
(see Fig. ). Then, the influence score is on
average better close to the core of the network for the wetlands.
For the fossil fuel regions, the influence score is relatively high
also upwind the monitoring network (dominant winds blow from west to east
in the region). In addition to the network density, the inversion
suffers from mis-separation of side regions and LBC. For this
reason, side regions tend to be less constrained than centre ones.
However, one can notice in both wetland and fossil fuel maps that
some centre regions are, in general, significantly less constrained than the core
of the domain. These are regions of very high and dense
emissions close to the observation sites (<500km). The air masses
coming from these regions to the observation sites are plume-shaped
air masses. The inversion has troubles in assimilating single
plumes. In Sect. , filters have been
implemented in order to detect these problematic regions. The
marginalized inversion effectively filtered out these regions.
The absolute and relative scores also show unexpected patterns.
The regions of Scandinavia and China own some of the best absolute and
relative scores. These two side regions are filtered out most of the
time because of strong correlations with the LBC components of the
state space (confirmed by their low influence score). Consequently,
when not filtered out, these regions are very well and
unambiguously constrained, thus the good relative and absolute scores.
For the rest of the domain, the scores are mostly the better, the
closer to the observation network.
Limitations and benefitsPromising computation of the uncertainties
The marginalized inversion provides an objectified quantification of
the errors in the inversion system. With the Monte Carlo approach we
implemented, we are able to consistently take the sources of
uncertainties in the inversion process into account, especially
those from the prescribed error covariance matrices. As evaluated
through OSSEs, the method proved to consistently catch “true” fluxes
on average in the particular Siberian set-up. Moreover, the Siberian
set-up is a difficult case study for atmospheric inversions, with
co-located intense fluxes that vary at temporal and spatial scales
smaller than the mesoscale. The processing of hotspots, critical
in most inversion configurations, is consistently managed through
filters on the plume-shaped air masses. An in-depth analysis of the
temporal variability of the fluxes is carried out in a sister
publication with the Siberian set-up and real observations
. Additionally, as a comparison, we
carried out the same OSSEs on the same particular Siberian set-up,
but with expert-knowledge frozen error matrices (diagonal matrices
with the same representation ω as for the other OSSEs). The
correlation profiles and the spatial structures of the scores with
the expert-knowledge matrices are not shown because the general
patterns are very similar to what is described for the marginalized
inversion. Though similar in patterns, the values of the scores are
significantly depreciated from the marginalized inversion to the
expert-knowledge one. The expert-knowledge relative and absolute
scores are several times bigger than the ones from the marginalized
inversion, thus statistically incompatible with the “truth”.
The marginalized inversion explicitly and objectively computes the posterior covariance matrix and the influence matrix.
The physical interpretation of the inversion results are then enhanced by a clear analysis of the observation constraints to the fluxes.
The processing of the posterior correlations makes the detection of the dipoles and of indistinguishable regions possible.
The influence of the lateral boundary conditions, specific to the mesoscale and to the use of limited area CTMs, is estimated.
Thus, the regions upwind the observation sites and mixed with lateral air masses can be excluded from the inversion.
From the correlations, the grouping of regions gives an estimate of the typical spatial and temporal scale the method can compute.
In our case, with few and distant observation sites, the groups of regions cover very large areas.
Thus, a grid-point high-resolution inversion would not have given deep insights into the fluxes we are looking at.
The reduced problem approach described in Sect. is then relevant when computed cautiously.
Subjective choices and biases
Despite all these benefits compared with the classical Bayesian
framework, our method still has limitations. The technical
implementation of the method needs extensive computation power and
memory requirements. For this reason, we have to drastically reduce
the size of the problem to solve. The size reduction relies on
rigorous considerations that are difficult to formulate
analytically. Therefore, we applied heuristic principles in order to
choose the aggregation patterns of the observations and the fluxes.
This subjective procedure can modify the results of the inversion
and must be carried out very cautiously. The way we group the
regions after the marginalized inversion in order to physically
interpret the results is also subjective. We choose a correlation
threshold of 0.5 in order to counterbalance the need of useful
constraints from the inversion and the requirements of consistently
reproducing the “true” fluxes. Other thresholds could have been
chosen and the typical distinguishable temporal and spatial scales
would slightly differ from one threshold to another. But, in any
chosen correlation threshold, we notice that most aggregation
regions are grouped within bigger ensembles, suggesting that the
chosen aggregation patterns are small enough to have reduced impact
on the inversion post-processed results.
The marginalized inversion suffers from transport biases as any
other inversion. However, the maximum likelihood algorithm considers
the biases as random errors and includes them into the error matrix
Rmax. The biases are then taken into account in
the marginalized inversion, though as random errors. Biases can be
represented, or at least detected, with non-diagonal matrices as
suggested by , but a non-diagonal
framework would make the computation of the marginalized inversion
critically complicated. Despite the implicit inclusion of the biases
as random error in Rmax, we reduced the impact of
the horizontal transport biases through filters on the plume-shaped
air masses. The vertical biases are smoother and more difficult to
detect. This issue must be inquired into in further works. Biases
can be studied through marginalizations on the input vectors
e.g.. Coupled marginalizations on
the input vectors and on the error statistics would provide a more
complete view on atmospheric inversion uncertainties.
Conclusions
At the mesoscale, inconsistencies between inversion configurations
appear in the classical Bayesian framework. One of the main sources
of inconsistencies is the specification of the error matrices and
the non-inclusion of the tenacious uncertainties on these matrices.
Synthesizing the recent literature, we developed an updated Bayesian
method of inversion from the classical Bayesian framework based on a
marginalization on the error matrices and on an objectified
specification of the probability density function of the error
matrices. This new method makes the comprehensive inclusion of the
impact of ill-specified uncertainty matrices possible for the first
time, to our knowledge, in atmospheric inversion. In principle, this
method needs very high computation power and memory resources. To
avoid technical limitations, we reduce the size of the problem by
aggregating the fluxes by region, following objective principles for
reducing aggregation errors. We test this method through OSSEs on
methane in a domain of interest spanning over Eurasia with
significant emissions of different types and different time and
space scales. The OSSEs are based on synthetic observations
generated from a nature run. We evaluate the consistency and
robustness of the method on OSSEs with inversion configurations from
the more favourable to the most disadvantageous one (perturbed
atmospheric transport, flat flux distribution and wrong total
masses). The method produces very consistent and satisfactory
results. In most cases, the tolerance intervals given by the
inversion include the “true” fluxes and the results remain close to
the “truth”. The method also provides an explicit computation of the
constraints on the regions and allows flagging out regions
critically mis-separated from the lateral boundary condition. We
hence have developed a robust and objectified method able to
consistently catch “true” greenhouse gas emissions at the mesoscale
and to explicitly group the regions that are physically
un-distinguishable with the atmospheric signal only. In addition, we
developed a method that explicitly produces posterior tolerance
intervals on the optimal distinguishable time and space flux scales
and that computes the observation network influence on the fluxes.
The robustness of our method on the Siberian case with a biased
transport proves that it can be generically applied to other
mesoscale frameworks. The high spatial and temporal variability of
the fluxes in Siberia ensures the possibility of using the system in
an “easier” inversion set-up. Actual observations from the sites we
used for the validation of the method are exploited in further steps
of our work in order to quantify the “real” methane fluxes in the
Siberian Lowlands .
Acknowledgements
We thank all the PIs from the sites we used for providing us with
information on their data. We especially thank Jost Lavrič and
Jan Winderlich (Max Planck Institute, Jena, Germany), Motoki Sasakawa
(Center for Global Environmental Research, NIES, Tsukuba,
Japan), and Michael Yu. Arshinov (V. E. Zuev Institute of
Atmospheric Optics, SB-RAS, Tomsk, Russia) for the information on
the Siberian sites. We are thankful to the two reviewers, including
Marc Bocquet (CEREA, Champs-sur-Marne, France), for their
constructive criticism of the first version of this manuscript and
for their fruitful comments. We thank the François Marabelle (LSCE)
IT support team for the maintenance of computing resources. This
study extensively relies on the meteorological data provided by the
European Centre for Medium-range Weather Forecast.
This research was supported by the Commissariat à l'Énergie Atomique et aux Énergies Alternatives.
Edited by: V. Grewe
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