GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus GmbHGöttingen, Germany10.5194/gmd-8-1259-2015A sparse reconstruction method for the estimation of
multi-resolution emission fields via atmospheric inversionRayJ.jairay@sandia.govLeeJ.YadavV.LefantziS.MichalakA. M.van Bloemen WaandersB.Sandia National Laboratories, P.O. Box 969, Livermore, CA 94551, USACarnegie Institution for Science, Stanford, CA 94305, USASandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0751, USAJ. Ray (jairay@sandia.gov)29April2015841259127323July201420August20147March20157April2015This 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/8/1259/2015/gmd-8-1259-2015.htmlThe full text article is available as a PDF file from https://www.geosci-model-dev.net/8/1259/2015/gmd-8-1259-2015.pdf
Atmospheric inversions are frequently used to estimate fluxes of atmospheric
greenhouse gases (e.g., biospheric CO2 flux fields) at Earth's surface.
These inversions typically assume that flux departures from a prior model are
spatially smoothly varying, which are then modeled using a multi-variate
Gaussian. When the field being estimated is spatially rough, multi-variate
Gaussian models are difficult to construct and a wavelet-based field model
may be more suitable. Unfortunately, such models are very high dimensional
and are most conveniently used when the estimation method can simultaneously
perform data-driven model simplification (removal of model parameters that
cannot be reliably estimated) and fitting. Such sparse reconstruction methods
are typically not used in atmospheric inversions. In this work, we devise a sparse reconstruction method, and illustrate it in an idealized atmospheric inversion problem for the
estimation of fossil fuel CO2 (ffCO2) emissions in the lower 48 states
of the USA.
Our new method is based on stagewise orthogonal matching pursuit (StOMP), a
method used to reconstruct compressively sensed images. Our adaptations
bestow three properties to the sparse reconstruction procedure which are
useful in atmospheric inversions. We have modified StOMP to incorporate prior
information on the emission field being estimated and to enforce
non-negativity on the estimated field. Finally, though based on wavelets, our
method allows for the estimation of fields in non-rectangular geometries, e.g.,
emission fields inside geographical and political boundaries.
Our idealized inversions use a recently developed multi-resolution (i.e.,
wavelet-based) random field model developed for ffCO2 emissions and
synthetic observations of ffCO2 concentrations from a limited set of
measurement sites. We find that our method for limiting the estimated field
within an irregularly shaped region is about a factor of 10 faster than
conventional approaches. It also reduces the overall computational cost by
a factor of 2. Further, the sparse reconstruction scheme imposes
non-negativity without introducing strong nonlinearities, such as those
introduced by employing log-transformed fields, and thus reaps the benefits
of simplicity and computational speed that are characteristic of linear
inverse problems.
Introduction
The estimation of spatially resolved fields, e.g., permeability fields in
aquifers or CO2 fluxes in the biosphere, from limited observations,
are required for many scientific or engineering analyses. These fields are
generally represented on a grid whose spatial resolution is dictated by the
analyses. The observations are usually too scarce to allow the estimation of
the field's values in each grid cell independently. If the field is known to
be smooth, one can impose a spatial correlation between the grid cells (e.g.,
model the field as a realization from a stationary multi-variate Gaussian
distribution) and reduce the effective dimensionality of the estimation
problem so that the limited observations suffice. In contrast, if the field
is complex, i.e., non-smooth or non-stationary (in the statistical sense,
implying different characteristic length scales at different locations),
multi-variate Gaussian models are difficult to construct
and an alternative parameterization may be preferable. The
parameterization has to be low dimensional, i.e., have few independent
parameters, so that they can be estimated from limited observations.
The construction of the spatial parameterization for complex fields poses
a stiff challenge. The parameterization is usually problem dependent and
sometimes based on heuristics. One may use an easily observed covariate (or
predictor) of the field being estimated to construct such a model; for
example, see for a description on how images of lights at
night were used to create a spatial parameterization for fossil fuel
CO2 (ffCO2) emissions. However, one is never quite sure if
the resultant parameterization is too simple or too complex – in the former
case, the estimates will be needlessly inaccurate, while in the latter case,
one may not obtain a unique solution or the estimates may reproduce the noise
in the observations (overfitting). Consequently, one uses some method, for
example, Akaike information criterion, to devise models of suitable
complexity. However, if the quality of the observations changes with time,
then, ideally, a different parameterization is constructed for each
time instant. In practice, often the simplest model that can be used with all
the observations is employed. This degrades estimation accuracy.
Sparse reconstruction methods can allow one to circumvent these problem which
arise from the dimensionality of spatial parameterization (also called the
random field model). Sparse reconstruction methods such as matching
pursuit MP;, orthogonal matching pursuit OMP; and Stagewise OMP StOMP; are
optimization methods that are used to fit high-dimensional models to limited
observations. Unlike other optimization methods, these methods enforce
sparsity, i.e., they identify the model parameters that are not informed by
the observations and set them to zero. This is accomplished by augmenting the
objective function (usually a ℓ2 norm of the data – model discrepancy
or residuals) with a penalty formulated as a ℓ1 norm over the
parameters being estimated. (The ℓ2 norm of a vector x is
defined as ||x||2=∑ixi2, while the ℓ1 norm is
||x||1=∑i|xi|.) An optimizer is used to manipulate model
parameters to minimize the objective function. The parameters that do not
impact the residual appreciably are quickly driven to zero, as it minimizes
the ℓ1 penalty, i.e., the optimizer performs dimensionality reduction
while it fits the model to data. This model simplification characteristic
of sparse reconstruction methods allows one to dispense with the offline
construction of a spatial parameterization and postulate a general,
high-dimensional random field model instead; thereafter, the optimization
method simplifies (reduce the dimensionality of) the random field model in
a data-driven manner. In the case of observations with time-variant quality,
sparse reconstruction methods have the potential to be particularly useful.
Our interest in sparse reconstruction methods arises from a need to
develop accurate spatially resolved estimates of emissions that
are not smoothly distributed in space; ffCO2 emissions are
one such example. Estimates of ffCO2 emissions are used to
assess regional contributions to greenhouse gas emissions and to drive
climate change simulations . Currently,
spatially resolved estimates of ffCO2 emissions are typically
derived from national-level emissions inventories, and are mapped
spatially using population density or some other proxy of human
activity; examples of such spatially resolved inventories are
described in , , and . Their shortcomings
arise from errors in national/provincial reporting and the choice of
the proxy used in spatial disaggregation . Recently,
the possibility of using atmospheric observations to constrain fossil
fuel emissions, and thereby improve inventories, has been
explored . Such applications involve the
solution of an inverse problem driven by ffCO2 concentration
measurements . Note that such improvements would
be contingent on a good representation of an estimation problem within
the context of an inversion, including the use of a suitable
parameterization for the emissions, the characterization of transport
model errors, and the availability of an observational network that is
sufficient to provide an adequate constraint on ffCO2
emissions; ffCO2 emissions for individual urban domes have
been estimated using atmospheric
measurements , i.e., without solving an
inverse problem, but existing methods do not offer a scalable approach
to updating entire inventories in this manner.
As a step towards enabling such applications, we constructed a wavelet-based
spatial parameterization, called the multiscale random
field MsRF;, to represent ffCO2 emission
fields. The MsRF was used to model ffCO2 emissions in the lower 48
states of the USA at 1∘× 1∘ spatial resolution. The
MsRF covers a rectangular region described by the corners 24.5∘ N,
63.5∘ W and 87.5∘ N, 126.5∘ W. The emissions
are modeled using Haar wavelets, which provide the sparsest representation of
ffCO2 emissions in the relevant region. The model has O(103)
independent model parameters which were selected using images of lights at
night. Due to its high dimensionality, the MsRF model cannot be used directly
given realistic in situ observational limitations. However, a data-driven
dimensionality reduction of the MsRF model, using a sparse reconstruction
method, could help constrain the inverse problem and make it possible to
capture coarse spatial patterns of ffCO2 emissions (and, perhaps,
finer details in the vicinity of the sensors), conditioned on atmospheric
measurements.
The use of sparse reconstruction methods poses certain methodological
challenges. First, these reconstruction methods do not provide
a mechanism for imposing non-negativity, which is a requirement when
estimating emission fields. Second, sparse reconstruction methods
have, to date, been used with wavelet-based random field models which
can only model rectangular domains; in contrast, the geometry of
emission fields could be decided by geographical or political
boundaries. (A random field model is a spatial
parameterization for a field defined on a grid. It can be
constructed using orthogonal bases such as wavelets; the wavelets'
weights are the model parameters and are treated as random
variables. Realizations of these random variables produce a
realization of the field. Depending upon the choice of the basis set,
e.g., if it contains only a subset of wavelets that can be supported
by the grid, the random field model may be able to produce only a
subset of the infinite number of fields that the grid can support).
Finally, sparse reconstruction methods do not provide a simple
mechanism to incorporate prior information or guesses of the field
being estimated, a common technique to ensure a unique solution to an
inverse problem. This is because methods such as OMP and StOMP were
largely developed for the reconstruction of compressively sensed
images where prior information is weak. In contrast,
many emission fields of an anthropogenic nature have
inventories that can serve as very informative priors and
reconstruction methods could profitably use them.
Our previous work focused on the spatial
parameterization (the MsRF described above) for
estimating ffCO2 emission fields via atmospheric inversion. In
this paper, we describe the methodological innovations in sparse
reconstruction techniques that allowed us to perform the inversion,
despite the high dimensionality of the parameterization. These
innovations result in an extension of StOMP which can address the
peculiarities of reconstructing an emission field. The StOMP extension
will be demonstrated in a top-down inversion, using synthetic
observations generated from a known, ground-truth emission field
so that we may examine certain algorithmic and numerical
aspects of the estimation technique, as described
below. The novel algorithmic developments addressed in this paper
are
Incorporation of a prior model of spatially rough emissions:
we demonstrate a novel and simple method to introduce
prior information on spatially rough emission fields (in the form of
an approximate field fpr) into StOMP. Currently,
sparse reconstruction methods employ no other prior information
beyond the phenomenological observation that most fields can be
represented quite accurately with a sparse set of judiciously chosen
wavelet bases .
Note, that the term prior model or prior information is used
somewhat loosely here since our method is not strictly
Bayesian. However, fpr serves a similar function
by providing regularization in the inverse problem.
Estimating fields in irregularly shaped regions: the
MsRF model, being based on wavelets, can only model fields in
rectangular domains, whereas our emission field is distributed over
an irregular region R, the lower 48 states of the USA. We
demonstrate how this geometrical constraint can be imposed
efficiently using random projections, a technique that underlies
much of compressive sensing. The reconstruction of fields in
non-rectangular geometries has no parallel in the compressive
sensing of images and the method discussed in this paper is the
first of its kind.
Imposition of non-negativity: the estimation of
the emission field is posed as a linear inverse
problem (see Sect. ). Non-negativity of emissions
can be enforced by log transforming the field, but converts the
problem into a nonlinear one, requiring computationally expensive,
iterative sparse reconstruction methods, like the one developed
in . We develop a simple, iterative post-processing
method to enforce non-negativity on the estimated ffCO2
emissions. The non-negativity enforcement mechanism uses StOMP but
does not use the MsRF model. The imposition of non-negativity in the sparse reconstruction of an emission field has never been explored before;
for example, in , the non-negativity constraint was
not applied to CH4 emissions from landfills.
In this study, we demonstrate our method on an idealized
atmospheric inversion of a spatially rough emission field in R.
The method is general, but we use ffCO2 as the test case.
The idealizations are enumerated below.
We assume that ffCO2 can be measured independently
without interference from biospheric CO2 fluxes. As
described in , this could be performed using
Δ14CO2 (radiocarbon) or other non-CO2
tracers, but the measurement technology is expensive and far from
being widely deployable.
Inversions require us to adopt a statistical error model for
the mismatch between observations and model predictions using the
estimated emission field. This error quantifies the aggregate of
measurement uncertainties and errors introduced by the
approximations in the transport model, among others. It varies
between measurement locations. In this study, we model this
mismatch as i.i.d. (independent and identically distributed)
Gaussian random variables. We assume a value
for the standard deviation of the distribution that is too small
compared to what is possible using existing transport models and
measurement technologies; further, we use the same error model for
all the measurement locations (details in Sect. ). The
small error allowed us to investigate the numerical aspects of our
formulation and solution algorithms without being substantially
affected by observational noise. The small error was also required
due to the nature of the measurement network employed in the
synthetic data test (see below).
The synthetic measurements are obtained at a set of 35 towers, a network that
existed in 2008 (see , for their locations). This
network, sited with biospheric CO2 measurements in mind, has towers
which tend to be far from urban areas and thus sources of ffCO2
emissions. Consequently, the modeled ffCO2 concentrations at these
towers tend to be low, forcing us to employ an error model that is
unrealistically small. These idealizations lead to limits on the inferences
that can be drawn regarding the use of our method in a real-data inversion
for ffCO2 emission fields; they are discussed in
Sect. .
We will estimate the emission field at 1∘× 1∘
resolution. Emission fields are averaged over 8 days, and estimated over
360 days, i.e., we estimate 360/8=45 fields. ffCO2 emission from
the Vulcan inventory (version 1)
http://vulcan.project.asu.edu/index.php; serve as
the ground truth, to generate the synthetic or pseudo-observations
yobs of time-variant ffCO2 concentrations. (The
Vulcan inventory provides hourly ffCO2 emissions at 10 km resolution
for the lower 48 states of the USA for 2002; it can also be downloaded at
0.1∘ resolution.) The prior model fpr will be
constructed using the Emission Database for Global Atmospheric
Research source: European Comission, Joint Research Centre/Netherlands Environmental Assessment Agency; Emission Database
for Global Atmospheric Research (EDGAR), release version
4.0, http://edgar.jrc.ec.europa.eu, 2009;, which provides
a single emission field at 1∘ resolution for 2005. These choices were
driven solely by the easy availability of data.
We evaluate our inversion method using the following
metrics. First, we check whether the incorporation of prior
information into our modification of the StOMP algorithm improves
estimates. Second, we investigate the sparsifying nature of
our algorithm. The aim of sparse reconstruction is to estimate
parameters supported by data (usually large-scale spatial patterns
in the rough emission field) and remove details that are not. We
check whether this property of StOMP is retained after our
modifications. Finally, we check the efficiency with which our
method reconstructs emission fields inside an irregular
R. Our use of a wavelet-based spatial parameterization
incurs a computational cost which can be limited by a user-defined
setting. We check if there is a principled way of computing this
setting, e.g., if improvements in results follow a diminishing
returns behavior with the computational cost.
Note that in this study, we do not use the accuracy of the estimated
field as a metric for evaluating our method; we only use estimation
accuracy to select between competing formulations of the inverse
problem. The estimation accuracy depends on (1) the spatial
parameterization (the MsRF) and (2) the information content of the
data set, and was explored in detail in our previous
paper . There, we fixed the observational data and
used the accuracy of the estimated emission field to gauge the
quality of the MsRF. The converse problem – fixing the MsRF and
varying the quantity of data – is not very useful for our
StOMP-based algorithm, since StOMP's sensitivity to data was
addressed in .
The paper is structured as follows. In Sect. , we
review sparse reconstruction techniques, their use with wavelet models
of fields and the tenets of compressive sensing that establish the
necessary conditions for successful sparse reconstructions. In
Sect. , we pose the inverse problem and describe the
numerical method used to solve it. Three formulations, differing in
the manner in which they incorporate fpr are
examined. In Sect. we perform inversion tests with
synthetic data to select the best formulation. We also explain, using
the properties required for sparse reconstruction, why the selected
formulation performed better than the others. The efficacy of
limiting the estimated field within R using random
projections is also investigated. Conclusions are in
Sect. .
Background
In this section, we review techniques used to estimate CO2
fluxes, compressive sensing and the use of sparse reconstruction in
inverse problems.
Estimation ofCO2fluxes: let the vector f
be the CO2 flux defined on a grid with NR grid cells.
Let f be of size KNR, representing a flux field defined
over K time periods. The flux is assumed to be time invariant during a
given time period. The transport of CO2 is modeled as that of
a passive scalar, i.e., the concentration of CO2 due to f at
an arbitrary set of sites, is given by y=Hf. Here,
y is a vector KsNs long, Ns being
the number of locations where measurements are collected Ks times
over the K time periods. The matrix H (KsNs×KNR) contains the sensitivity of
measurements to a CO2 source in each grid cell and is computed using
an atmospheric transport model such as the Stochastic Time-Inverted
Lagrangian Transport Model STILT;. In an atmospheric
inversion, CO2 concentration yobs are measured at
a limited set of locations, usually a set of measurement towers (as in our
case) or as column-averaged satellite soundings. The measurements are too few
or too uninformative to estimate f, with each grid cell treated
independently. In case of biospheric fluxes, a prior flux fpr
(with the same dimensions as f) can be obtained from a biogeochemical
process-based model such as CASA Carnegie–Ames–Stanford
Approach;. The discrepancy (f-fpr) is
usually modeled as a multi-variate Gaussian field with covariance
Q (a KNR×KNR matrix), and the
estimation of f is typically performed by minimizing the objective
function
J=(yobs-Hf)TRe-1(yobs-Hf)︸Observation term+(f-fpr)TQ-1(f-fpr),︸Prior term
where Re is a diagonal matrix with the data – model variances
and includes many sources of errors including measurement errors, aggregation
errors and transport model inaccuracies. Methods to solve this linear inverse
problem are reviewed in . A comparison of biogenic CO2
fluxes and ffCO2 emissions Fig. 1 in shows that
ffCO2 are multiscale in nature and a multi-variate Gaussian field
approximation of (f-fpr) is unlikely to be accurate.
This motivated us to construct the MsRF model for ffCO2 emission
fields . The solution of an inverse problem using MsRF
requires the use of a sparse reconstruction method that, to date, has been
used in the reconstruction of compressively sensed images.
Compressive sensing of images: compressive
sensing is a very efficient means of representing
images using wavelets. Wavelets are a family of orthogonal bases with compact
support that are routinely used to model complex fields, including
ffCO2 emissions . Compact support refers to the
fact that a wavelet is defined over a finite region (compact support).
This is in contrast to other commonly used bases, e.g., sin(kπx),-∞≤x≤∞,k∈Z, which have infinite support. However,
like Fourier bases, wavelets are orthogonal; the scalar product of two
different wavelets of the same type and order, e.g., Daubechies wavelets of
order 4, is 0. Compressive sensing (CS) is based on two key tenets:
compressible representation and encoding via random projections. CS assumes that an image, projected onto
a suitable wavelet basis set, will yield wavelet weights (represented
by a vector w) that are mostly very small (i.e., a
compressible representation) and can be set to zero. Removing
the small wavelets results in a sparse approximation of
the image. Encoding via random projections is more involved and
determines the necessary conditions for successful sampling. Random
encoding is central to our method for applying boundary conditions,
viz., limiting ffCO2 emissions within complex, non-rectangular
boundaries.
Consider an image g of size N, that can be represented sparsely
using L≪N wavelets. Random encoding, as used in CS, asserts that the
image may be sampled by projecting it onto a set of random vectors
ψj, to obtain compressive measurements g′,
of size Nm, L<Nm≪N:
g′=Ψg=ΨΦw=Aw,
where the rows of the sampling matrix Ψ consist of the
random vectors ψj, the columns of Φ
consist of the orthonormal basis vectors (the wavelets) ϕi
and w are the weights (or coefficients) of the wavelets.
Φ is a N×N matrix while Ψ is
Nm×N. The bulk of the theory was established
in , , and .
In order that one may recover the original image g from
g′ using sparse reconstruction, Ψ and
Φ must satisfy incoherence and a
restricted isometry property. Incoherence implies
that no row ψk in Ψ is co-aligned with
column ϕl in Φ and thus collects
information on all bases. It is ensured by choosing some well-known wavelets
bases (e.g., Haars or Daubechies 4 and 8) for Φ and random
vectors for Ψ. This is formally
quantified by the mutual coherenceμ(Ψ,Φ) of Ψ and Φ:
μ(Ψ,Φ)=Nmax1≤(k,l)≤N|<ψk,ϕl>|=Nmax(|Akl|),
where Akl are elements of A. Each row of Ψ
is normalized to a unit vector. The term |<ψk,ϕl>| is the projection of row ψk on a wavelet basis
ϕl. Co-alignment of a row ψk′ with a
wavelet ϕl′ would lead to
Ak′l′=1 and Ak′l=0 for all l≠l′, indicating that a random vector ψk′
collects information on only one wavelet. The scaling by N is
conventional. A small mutual coherence ensures that all projections of
Φ on the rows of Ψ are of moderate
magnitude (O(10-1)–O(10-3)). A small mutual
coherence aids accurate reconstruction. When μ(Ψ,Φ)≪N, we loosely refer to Ψ and
Φ as being incoherent. The restricted isometry property
(RIP) is a condition imposed on A which ensures that w can
be recovered from g′ uniquely without the use of priors
(except sparsity). We did not pursue this thread since the use of a prior –
making the inventory that supplies fpr consistent with
observations – is the motivation behind this investigation.
Sparse reconstruction of images from compressive measurements: the aims of reconstruction in CS are to (1) recover
the sparsity pattern (alternatively, identify the components of w
that can be estimated from g′) and (2) estimate those
elements of w that are informed by g′ while setting
the rest to zero. The former can be realized by minimizing the ℓ0 norm
of w while the latter is typically achieved by minimizing the
ℓ2 norm of the measurement – model discrepancy. However, an objective
function that contains a ℓ0 norm is discontinuous, and consequently
ℓ0 is replaced by an ℓ1 norm, which is more
tractable . Thus, the optimization problem is posed as
minimizew∈RN||w||1,subject to||g′-Aw||2<ϵ2.
This optimization problem can be solved using methods like MP, OMP and StOMP.
Bayesian equivalents also exist , where Laplace priors
are used to enforce sparseness in the inferred w. Algorithms based on
convex optimization that serve the same purpose are reviewed
in . All these algorithms are general and do not exploit any
particular structure in g except sparsity. However, one may also
create a prior model for wavelet distributions, e.g., by using a database of
similar images, for higher quality
reconstructions . In order to do so,
sparse reconstruction methods have to be modified to incorporate prior
information.
Sparsity is sometimes used to solve inverse problems in physics, with
the Ψ operator representing the physical
process. Most of these inverse problems have been in the estimation of
log-transformed permeability fields , seismic
tomography and estimation of point and
distributed emissions . A more detailed review of
the sparse reconstruction methods can be found in our previous
paper . Most of these inverse problems involved
nonlinear models, i.e., y=a(w), rather
than y=Aw, for which incoherence (and
RIP) are not well defined and consequently were not investigated.
To summarize, sparse reconstruction techniques and wavelet-based random field
models have been used in nonlinear inverse problems. In contrast, the problem
of estimation of spatially rough emission fields is linear, raising the
possibilities that (1) the same approach may offer a solution to the emission
estimation problem and (2) mutual incoherence may provide analytical metrics
for the quality of observations and, consequently, solutions. We build on the
principles of compressive sensing and sparse reconstruction methods to design
an inversion scheme for rough emission fields. In particular, we show (using
coherence metrics) why the use of fpr was necessary. We also
show the degree of computational saving achieved when we use random
projections to limit ffCO2 emissions within R.
Formulation of the estimation problem
developed a MsRF model
for ffCO2 emissions in the USA. The MsRF model allows
ffCO2 emissions to be represented as f=Φw, where Φ is
a collection of Haar wavelets. Consequently, the observational term in
Eq. () can be written as ||yobs-HΦw||22. Compared with
Eq. (), we see that the transport model H
serves as the sampling matrix Ψ. Since we seek to
estimate the wavelet weights w from
yobs, an optimization problem like
Eq. () could be posed with the constraint ||yobs-Aw||2<ϵ2,
A=HΦ. In order to solve this
problem via sparse reconstruction, one requires that H and
Φ be incoherent. As we will show in
Sect. , the incoherence requirement is not met, and
sparsity (solely) is not sufficient to solve the problem accurately
(as tested in Sect. ). Consequently, we modify StOMP to
incorporate a prior emission field fpr. We also
adapt it to accommodate fields defined over irregularly shaped domains
as well as to ensure non-negativity of the estimated field.
Let f be a time-variant, non-negative field defined in an
irregular region R, gridded with NR
grid cells. In our case f models ffCO2 emission
fields. The field is averaged over a time period T and covers K
time periods, i.e., it is a vector NRK long. f
drives a linear model of observations of ffCO2 concentrations:
yobs=y+ϵ=Hf+ϵ,
where H is the sensitivity matrix obtained from an atmospheric
transport model (see Sect. ), ϵ is the model–data
mismatch due to measurement and transport model errors and
yobs is a vector of time-variant measurements collected at
Ns measurement towers. Each tower collects Ks
measurements over the K time periods, i.e., yobs is a vector
KsNs long. The H matrix is (KsNs)×(NRK).
Prior models
We employ two prior models in our work – the MsRF model for ffCO2
emissions and a time-invariant approximation of ffCO2 emissions
fpr. The MsRF is a collection of wavelets and model emissions
in the logically rectangular domain given by the corners 24.5∘ N,
63.5∘ W and 87.5∘ N, 126.5∘ W. The MsRF
discretizes the domain using a dyadic 2M×2M mesh. Haar
wavelets are defined on all M levels of this dyadic grid, but not all of
them are retained in the MsRF. Wavelets constituting the MsRF model are
chosen using radiance-calibrated images of lights at
night http://ngdc.noaa.gov/eog/data/web_data/v4composites/F152002.v4.tar;,
which serve as a proxy for human activity and thus capture the spatial
patterns of ffCO2 emissions. The emission field is allowed to assume
non-zero values only within R, the lower 48 states of the USA. We
denote the field during the kth time period as fk and model it
as
fk=wk′ϕ′+∑s=1M∑i,jws,i,j,kϕs,i,j,{s,i,j}∈W(s)=Φwk,
where W(s) contains the L wavelets that constitute the MsRF
model, and L is a fraction of the 4M wavelets that can be supported
by a 2M×2M mesh.
The MsRF is also the starting point for developing the second prior
model fpr. The MsRF provides a sparse
representation of the radiances X(s):
X(s)=w(X)′ϕ′+∑l,i,jw(X),l,i,jϕl,i,j,{l,i,j}∈W(s).X(s) is used to calculate a time-invariant prior model
for ffCO2 emissions as fpr=cX(s). c is computed such that
∫RfV‾dA=∫RfprdA=c∫RX(s)dA=c∫R(w(X)′ϕ′+∑l,i,jw(X),s,i,jϕl,i,j)dA,{l,i,j}∈W(s).
Equation () implies that c is calculated such that both
fV‾ and fpr provide the same value for the
total emissions in R. fV‾ in our case is the
annually averaged 2005 emission field obtained from EDGAR. The leftmost term
∫RfV‾dA quantifies the total
emissions as predicted by EDGAR over R. The term c∫RX(s)dA is an estimate of emissions
over the same region but modeled using fpr. The rightmost
term simply replaces X(s) with its wavelet model, in accordance
with
Eq. (). The details of how the MsRF and fpr were
constructed are in . fpr differs from the
ground truth (Vulcan emissions aggregated over the lower 48 states) by
5–25 % see Fig. 9 in.
Posing and solving the inverse problem
We seek emissions over an entire year (360 days), i.e., we seek F={f1,f2,…fK}={Φw1,Φw2,…ΦwK}=Φ̃w. F models the field in R∪R′, where R′ models the region
outside R (but inside the rectangular domain modeled by the MsRF)
with zero ffCO2 emissions. We separate out the fluxes in
R and R′ by permuting the rows of
Φ̃F=FRFR′=Φ̃RΦ̃R′w,
where Φ̃R and
Φ̃R′ are (NRK)×(LK) and (NR′K)×(LK) matrices,
respectively. Here NR′ is the number of grid cells in
R′. The modeled concentrations at the measurement towers,
caused by FR, can be written as y=HFR. For arbitrary w, FR′
(the emissions in R′) are not zero and
FR′=0 will have to be imposed as a constraint in
the inverse problem.
Specifying the constraint in individual grid cells is not very efficient
since it leads to NR′K constraints. This can get very
large in a global inversion at high spatial resolutions. Instead, we adapt an
approach from compressive sensing to enforce this constraint approximately.
Consider a Mcs×(NR′K) matrix
R, whose rows are direction cosines of random points on the
surface of NR′K-dimensional unit sphere. This matrix
is called a uniform spherical ensemble . The Mcs
projections of the emission field FR′ on
R, i.e., RFR′, compressively
samples FR′ and setting them to zero during
inversion allows us to enforce zero emissions outside R. In
Sect. , we will investigate the degree of computational saving
afforded by imposing the FR′=0 constraint in
this manner. The problem is now modeled as
Y=yobs0≈HΦ̃RRΦ̃R′w=Gw.
In this equation, G is akin to A in Eq. ().
The left hand side Y is approximately equal to Gw
since the observations yobs contain measurement errors that
cannot be modeled with H.
The case where R′ contains non-zero
emissions requires the use of boundary fluxes and is discussed in Ray et al. (2014).
The wavelet coefficients w in Eq. () are not
normalized and usually display a large range of magnitudes. The
wavelets in W(s) at finer scales, i.e., those with a small
support, tend to have coefficients with a large magnitude. Their small
support cause the fine-scale wavelets to impact only neighboring
measurement towers. In contrast, wavelets at the coarser scales have
large footprints that span multiple measurement locations. Total
emissions in R, as well as yobs, are
very sensitive to their coefficients. Solving Eq. ()
as it is incorporates no information from fpr beyond
the selection of wavelets to be included in
Φ̃. We explore the incorporation of
fpr in the estimation of w using three
different approaches:
Approach A: this is the baseline approach and solves
Eq. () as it is. The lack of normalization of w, in
conjunction with the sparse reconstruction procedure described below, leads
to artifacts that will be described in Sect. .
Approach B: in this formulation, we include fpr as a
prior. We write the emissions as F=fpr+ΔF. Substituting into Eq. (), we get Y≈Hfpr+GΔw, where Δw=w-w(X). Here, w(X)=c{w(X)′,w(X),s,i,j},{s,i,j}∈W(s), where c
is obtained from Eq. (). Simplifying, we get
ΔY=Y-Hfpr≈GΔw.
Approach C: in approach B we expressed the true flux
F as an additive correction over fpr, thus
incorporating the prior information in fpr. In approach C, we
use the spatial pattern of fpr, as captured by its wavelet
coefficients w(X), to normalize w. We rewrite
Eq. () as
Y≈GBB-1w=G′w′=HΦ̃R′RΦ̃R′′w′,B=cdiag(w(X)),
where w′={ws,i,j/(cw(X),s,i,j)},{s,i,j}∈W(s) are the wavelet coefficients
normalized by those of fpr,
Φ̃R′=Φ̃Rdiag(w(X)) and
Φ̃R′′=Φ̃R′diag(w(X)). If fpr is close to
F, the elements of w′ will be O(1). If
fpr is a gross underestimate, the elements of
w′ will still be of the same order of magnitude, but
not O(1). Thus, normalization with w(X) removes the large
differences that exist between the wavelet coefficients at different scales.
In all the three cases, we obtain an underdetermined set of linear
equations of the form
Υ≈Γζ.
Here Υ represents Y in approaches A and C, and
ΔY in approach B. Γ represents G
in approaches A and B and G′ in approach C.
ζ represents w in approach A, Δw in
approach B and w′ in approach C.
Since yobs is obtained from a set of locations
sited with an eye towards biospheric CO2 fluxes
see , it is unlikely that it will allow the
estimation of all the elements of ζ. Further,
a priori, we do not know the identity of these un-estimateable
elements and so we use sparse reconstruction to find and compute them.
Equation () is recast similar to Eq. ():
minimizeζ∈RN||ζ||1,subject to||Υ-Γζ||22<ϵ2.
We solve Eq. () using StOMP. ||ζ||1 is
minimized by setting to zero as many elements of ζ as
possible, thus enforcing sparsity. Meanwhile, the constraint ||Υ-Γζ||2 ensures
that the solutions being proposed by the optimization procedure provide
a good reproduction of the observations. Note ζ contains
only the wavelets in W(s). The StOMP algorithm is detailed in
. We will refer to this step in the estimation procedure as
step I.
Enforcing non-negativity on FR
Estimates of w calculated by StOMP do not necessarily provide
FR=Φ̃Rw that
are non-negative. In practice, negative values of FR occur
in only a few grid cells and are usually small in magnitude. A large fraction
of elements of w are set to zero by StOMP. Having identified the
sparsity pattern, i.e., the spatial scales that can be estimated from
yobs, we devise an iterative procedure for enforcing
non-negativity on FR. We discard
FR′ and manipulate the field (the emissions) in
R directly, rather than via the wavelet coefficients.
We seek the non-negative vector E={Ei},i=1…Q,Q=(NRK) such that
||yobs-HE||2||yobs||2≤ϵ3.E is constructed iteratively through a sequence E1,E2,…EN. E0 is initialized by using the absolute values
of FR calculated by solving Eq. ().
At each iteration m, we seek a correction ξ={ξi},i=1…Q, where |ξi|≤1, such that
E(m)=diag(exp(ξ1),exp(ξ2),…,exp(ξQ))E(m-1)≈diag(1+ξ1,1+ξ2,…,1+ξQ)E(m-1)=E(m-1)+ΔE(m-1),whereΔE(m-1)=ξTE(m-1).
Since the field must satisfy yobs≈HE(m), we get
yobs-HE(m-1)=Δy≈HΔE(m-1).
This is an underconstrained problem, and we seek the sparsest set of updates
ΔE(m-1) using StOMP. The corrections are calculated, and the
field updated as
ξi=sgnΔEi(m-1)Ei(m)max1,ΔEi(m-1)Ei(m),Ei(m)=Ei(m-1)exp(ξi)
to obtain E(m). The convergence requirement in
Eq. () is checked with E(m), and if not met, the
iteration count is updated m:=m+1 and Eq. () is solved
again.
We will refer to this step in the estimation procedure as step II.
Numerical results
In this section, we test the sparse estimation technique in
Sect. , using synthetic observations. The time period T over
which the ffCO2 emissions are averaged is 8 days. K=45, i.e., we
estimate emissions over 8×45=360 days. Ns=35 towers,
which are a subset of NOAA's Earth System Research Laboratory (ESRL) Global
Monitoring Division's cooperative air sampling network ; their
locations are in . These towers provide continuous
observations of CO2 concentrations (in parts per million by volume,
ppmv), and 3-hourly averaged synthetic observations are used here (i.e.,
Ks=24/3× 8 ×45=2880). We discretize the
domain covered by the MsRF using 1∘× 1∘ grid cells
i.e., M=6. The number of grid cells in the entire domain (the rectangle
with the corners 24.5∘ N, 63.5∘ W and 87.5∘ N,
126.5∘ W), N, is 4M=4096, which is also equal to the
number of wavelets that can be defined on the mesh. The number of wavelets
retained in the MsRF, L, is 1031. R denotes the lower 48 states
of the USA. They are covered with NR=816 grid cells. The number
of grid cells outside R, NR′=N-NR=3280.
The H matrix in Eq. () is calculated per the
description in . We use the Stochastic Time-Inverted
Lagrangian Transport Model , with wind fields from the Weather
Research & Forecasting model , version 2.2, driven by 2008
meteorology to compute H. Concentration sensitivities are
calculated at 3 h intervals over a North American grid, at a resolution of
1∘× 1∘. The sensitivity of the CO2
concentration at each observation location due to the flux at each grid cell
is calculated in units of ppmvµmol-1m2s1. The
sensitivity of y to the 8-day-averaged emissions were obtained from
the 3 h sensitivities by simply adding the 8×24/3=64
sensitivities that span the 8-day period.
The true ffCO2 emissions in R are obtained, for 2002,
from the Vulcan inventory. Hourly Vulcan fluxes are coarsened from
0.1∘ resolution to 1∘, and averaged to 8-day periods. These
8-day-averaged fluxes at 1∘ resolution are multiplied by H
to obtain ffCO2 concentrations at the measurement towers. Note that
averaging over 8 days removes the diurnal variations of ffCO2
emissions in Vulcan. Observations are generated every 3 h and span a full
year. A measurement error ϵ∼N(0,σ2) is added to the
concentrations to obtain yobs (see Eq. ), as
used in Eq. (). The same σ is used for all towers. We
use σ=0.1ppmv, which is too small and represents an
idealized inversion scenario that is used here to test the quality of the
proposed numerical method. Realistic values of transport model errors for
some of the towers used in this study are in . Radiocarbon
measurement errors can be found in .
Plot of ffCO2 emissions (micromoles of C m-2 s-1)
as reported by EDGAR for 2005. Emissions below 0.02 micromoles of C
m-2 s-1 are grayed out.
Comparison of optimization formulations
We choose between approaches A, B and C by solving the inverse problem for
the ffCO2 emission field. The inversion is performed for the
emissions F={fk},k=1…K, for the entire year. The
following parameters are used in the inversion process: ϵ2=10-5,ϵ3=5.0×10-4,Mcs=13 500, i.e., 300
random projections for each 8-day period. The rationale for these values can
be found in our previous paper .
In Fig. we plot the estimated emissions during the 31st
8-day period, as calculated using approaches A, B and C. The true emissions
are also plotted for reference. Four quadrants are also plotted for easier
comparison and reference. The distribution of measurement towers is very
uneven, with most of the towers being concentrated in the northeast (NE) quadrant,
where we expect the reconstruction to be most accurate. We see that
approach A (Fig. , top right) provides estimates that
have large areas in the northwest (NW) and southwest (SW) quadrants with
moderate levels of ffCO2 emissions. In contrast, the true emissions
(Fig. , top left) are mostly empty. Thus, we see that the
minimization of ||ζ||1 (alternatively ||w||1)
drives the wavelet coefficients to small values, but not identically to zero.
In Fig. (bottom left), approach B provides estimates
that show much structure in the eastern quadrants, and the patterns seen in
fprsee are reproduced. The reason is
as follows. While fpr captures the broad, coarse-scale
patterns of ffCO2 emissions, it incurs significant errors at the
finer scales. Equation () seeks to rectify the discrepancy
between fpr and true emissions using observations. However,
as mentioned in Sect. , fine-scale wavelets tend to have large
wavelet coefficients and the minimization of ||ζ||1
(alternatively ||Δw||1) removes them since the constraint ||Υ-Γζ||22<ϵ2 is not very sensitive to individual wavelets at the fine scale.
(See for a discussion on the largely local impact of a
CO2 flux source.) The inability to rectify the fine-scale
discrepancies led to a final ffCO2 estimate that resembles
fpr in the finer details. Figure
(bottom right) plots the estimates obtained using approach C, which uses
normalized wavelet coefficients w′. The estimates from
approach C show large areas of little or no emissions in the western
quadrants, similar to the true emissions in the top-left figure. In the
eastern quadrants, the emissions show less spatial structure than the true
emissions as well as those obtained using approach A.
Plots of ffCO2 emissions during the 31st 8-day
period. The units are micromoles of C m-2 s-1.
Emissions below 0.02 micromoles of C m-2 s-1 are grayed out.
Top left, we plot true emissions from the Vulcan inventory.
Top right, the estimates from approach A. Bottom left
and right figures contain the estimates obtained from approaches B
and C, respectively. Each figure contains the measurement towers as
white diamonds. Each figure is also divided into quadrants. We see
that approach A, unconstrained by fpr provides low levels of
(erroneous) emissions in large swathes of the western
quadrants. Approach B reflects fpr very strongly. Approach C
provides a balance between the influence of fpr and the
information in yobs.
The quality of the estimate is due to both the MsRF model and the new sparse
reconstruction scheme. The limited observations are sufficient to allow the
estimation of the coarse MsRF wavelets, and in certain areas, e.g., the NE
quadrant, finer details. The MsRF model is sufficiently flexible to
accommodate the spatial heterogeneity in detail, but requires a sparse
reconstruction method to address the high dimensionality that such
flexibility entails. Further, the multi-resolution nature of MsRF model
allows for
the accurate estimation of coarse-scale patterns of ffCO2 emissions,
i.e., we expect that aggregate measures of emission quality, such as
integrated emissions in R, will be accurate. It will incur larger
errors as the domain of integration has shrunk.
In Fig. (top) we evaluate the accuracy of the
reconstruction quantitatively. We integrate the emissions in R to
obtain the country-level ffCO2 emissions and compare that with the
emissions from Vulcan. We plot a time series of errors defined as
a percentage of total, country-level Vulcan emissions
Errork(%)=100K∑k=1KEk-EV,kEV,k,where Ek=∫REkdA and EV,k=∫RfV,kdA.
Here, fV,k are Vulcan emissions averaged over the kth 8-day
period and Ek are the non-negativity enforced emission estimates in
the same time period. A positive error denotes an overestimation by the
inverse problem. In Fig. (bottom) we plot the Pearson
correlation coefficient between the true and reconstructed emissions in
R over the same duration. We define the Pearson correlation
coefficient between Ek and fV,k as
CEk,fV,k=cov(Ek,fV,k)σEkσfV,k,
where σEk2 and σfV,k2 are the variances
of the true and reconstructed fluxes and cov(Z1,Z2) is the
covariance between two random variables Z1 and Z2. It is clear that
approach B provides the worst reconstructions, with the largest errors and
smallest correlations. Approach C tends to over-predict emissions a little
more than approach A, but has better spatial correlation with the Vulcan
emissions.
Comparison of estimation error (top) and the correlation
between true and estimated emissions (bottom) using approaches A, B
and C. It is clear that approach B is inferior to the
others.
In Fig. we see the essential difference between
approach A and C. We plot the reconstruction error (top) and correlation
between true and reconstructed emissions (bottom) in the northeast (NE) and northwest (NW) quadrants.
Errors in the emissions are represented as a percentage of the total (true)
emissions in that quadrant. We see the approach C has smaller errors in both
the quadrants. It also provides higher correlation in the NW quadrant, which
does not have many measurement towers (white diamonds in
Fig. ). Both the approaches have errors of opposite
signs in the quadrants which largely cancel out when errors are assessed over
R as a whole, leading to approximately similar estimation
accuracies by both the approaches in Fig. . However, the
estimates produced by approach A (without the use of fpr)
show larger spatial variability and error than approach C. This is because
normalization using w(X) and minimization of
||ζ||1 (alternatively ||w′||1) prevents
large departures from fpr and also rectifies the tendency to
remove large wavelet coefficients belonging to the finer wavelets. Approach C
therefore provides a formulation that is more accurate and robust at the
quadrant scale, even though both have similar fidelity at the scale of
R.
Evaluating formulation using compressive sensing metrics
Having established empirically that approach A is less accurate than
approach C, we can explain why this is the case. We employ coherence metrics for this
purpose.
In compressive sensing, random matrices such as Gaussians, Hadamard,
Circulant/Toeplitz or functions such as
noiselets serve as
Ψ. In Fig. , we plot the distribution of
log10(|Ai,j|), the elements of
AΨ=ΨΦ
for these standard sampling matrices. Φ contains
only the wavelets in W(s). Note that max(|Aij|) specifies
the mutual coherence, and small values of max(|Aij|) indicate
informative measurements. We see that log10(|Ai,j|) may
assume continuous (Gaussian and circulant sampling matrices) or
discrete (Hadamard, scrambled-block Hadamard and noiselets)
distributions, and generally lie between -3 and -1. This provides
a range for the level of coherence observed in theoretical CS
analyses.
Reconstruction error (top) and correlation between the
true and estimated emissions, using approaches A and C, for the
northeast (NE) and northwest (NW) quadrants. We see that approach
C, which includes information from fpr, leads
to lower errors in both the quadrants and better correlations in
the less instrumented NW quadrant.
In Eq. (), H serves a similar sampling purpose, and
the efficiency of sampling depends on the incoherence between H
and Φ. We construct a new H′ by picking
the rows of H corresponding to two towers and for the 21st and 22nd
8-day periods. We compute AH′=H′Φ, and in Fig. , plot the log-transformed
magnitudes of the elements of AH′. The
distributions for the two towers are almost identical. We clearly see that,
unlike AΨ, AH′
contains a significant number of elements that are close to 1, and a large
number of elements that are close to 0 (e.g., near 10-6). This is
a consequence of the rows of H′ being approximately
aligned to some of the columns of Φ and, consequently,
nearly orthogonal to others. The small values in
AH′ indicate that the CO2
concentration prediction y at the two selected towers are insensitive
to many of the wavelets, i.e., to many scales and locations, as observed in
Sect. . Further, the coherence μ(H′,Φ) is larger than μ(Ψ,Φ), indicating a sampling efficiency a few orders of
magnitude inferior to those achieved in the CS of images. Consequently,
approach A, based solely on sparsity, and identical to the method adopted in
CS, would not work well. Thus, approach C, which employed both sparsity and
fpr, proved superior to approach A.
Numerical consistency and computational efficiency
We now address some of the numerical aspects of the solution. The results
presented here are not tests of accuracy of the estimated emission
field; estimation accuracy also depends on the MsRF and was investigated
in . Here we empirically verify that certain necessary
conditions of our sparse reconstruction are satisfied.
Comparison of the distribution of the elements of
AΨ and
AΦ. We see that Gaussian and circulant
random matrices lead to continuous distributions, whereas Hadamard,
scrambled-block Hadamard (sbHadamard) and noiselets serving as
sampling matrices lead to AΨ where the
elements assume discrete values. In contrast, the elements of
AH′ assume values which are spread
over a far larger range, some of which are quite close to 1 while
others are very close to 0.
In Fig. (top), we plot y predicted by the
reconstructed emissions at two towers, BAO (Boulder Atmospheric Observatory, Colorado)
and MAP (Mary's Peak, Oregon). These towers were included in the inversion
and are not being used as an out-of-sample test of the accuracy of
the estimated emission field. Rather, the MsRF for rough fields allows the
estimation of local sources which can help reproduce a tower's measurements
very closely, unless neighboring towers provide a constraint; in a sparse
network, this is not always possible. Thus, an accurate reproduction of a
tower's observations is not necessarily a sign of an accurately estimated
emission field, but a bad reproduction can be a sign of a malfunctioning
sparse reconstruction method. We see that the ffCO2 concentrations
are well reproduced by the estimated emissions. In
Fig. (bottom) we plot the wavelet coefficients
obtained by projecting the emissions (both the true and reconstructed) on the
wavelet bases. The wavelet coefficient values have been subjected to
a hyperbolic tangent transformation for ease of plotting. The true wavelet
coefficients with a magnitude above 0.01 are plotted with red symbols. The
true (Vulcan) emissions have a large number of coefficients with small
magnitude; these are usually for small-scale features, i.e., have coefficient
indices in the right half of the range (red symbols in Fig. , bottom). During sparse reconstruction, these coefficients are set to
zero (blue symbols in Fig. , bottom). The low-index
coefficients, which represent large structures, are estimated accurately. The
explicit separation of scales is thus leveraged into omitting fine-scale
details which are difficult to inform with data and focusing model-fitting
effort on the large scales instead. Sparse reconstruction achieves this in an
automatic, purely data-driven manner, rather than via a pre-processing,
scale-selection step.
(Top) predictions of ffCO2 concentrations at two
measurement locations, using the true (Vulcan) and reconstructed
emissions (blue lines) over an 8-day period (period
no. 31). Observations occur every 3 h. We see that the
concentrations are accurately reproduced by the estimated
emissions. (Bottom) Projection of the true and estimated emissions
on the wavelet bases for the same period. Coarse wavelets have
lower indices, and they progressively get finer with the index
number. We see that the true emissions have a large number of
wavelets with small, but not zero, coefficients. In the
reconstruction (plotted in blue), a number of wavelet coefficients
are set to very small values (almost zero) by the sparse
reconstruction. The larger scales are estimated accurately.
Finally, we address the issue of enforcing the FR′=0 constraint via random Mcs projections. Naively, the constraint
can be enforced for every individual grid cell, resulting in
NR′=3280 linear equations per 8-day period in
Eqs. () and (). Considering that
yobs=HΦ̃R
results in 64×35=3240 linear equations per 8-day period, we see
that enforcing the constraint is as expensive as computing
FR. Instead, we set Mcs≪NR′ random projections of
FR′ to zero in Eqs. () and
(), exploiting the basic efficiency-via-random-sampling tenet
of CS. Since Eq. () is solved approximately, and due to the
small number of wavelets in W(s) that span R′, the
constraint FR′=0 is not satisfied exactly. This
error varies with Mcs; a larger Mcs results in a closer
realization of the constraint. Errors in the enforcement of the
FR′=0 constraint lead to commensurate errors in
FR. Here we check the trade-off between Mcs
(computational efficiency) and accuracy of the estimated emissions
(FR and FR′). In practice, this
affects only step I of the procedure, where an approximation of ffCO2
emissions is calculated, and thereafter it is used as a guess in step II.
However, a good estimate of the emission field accelerates the second step.
The quality of the solution from step I, quantified as the cumulative
distribution function of the fluxes can be found
in . There are only a few grid cells with
negative emissions and their magnitudes are small.
The impact of the number of compressive samples
Mcs on the reconstruction of FR
(ηR) and FR′
(ηR′). ηR and
ηR′ are plotted on the Y1 and Y2 axes,
respectively. Results are plotted for the 31st 8-day period. We
see that Mcs>103 does not result in an appreciable
increase in reconstruction quality. Also, Mcs<102
shows a marked degradation in ηR′.
In Fig. , we plot the impact of Mcs on the
reconstruction. We perform sparse reconstruction of the emission field, for
the 31st 8-day period and compute the ratios
ηR=||fk,R||2||fV,k||2andηR′=||fk,R′||2||fV,k||2fork=31,
where fk,R and fk,R′ are
the emissions over R and R′ from step I.
fV,k is the true (Vulcan) emission field during the same period.
These ratios are plotted as a function of log10(Mcs) per 8-day
period. We see that 10 projections per 8-day period is too few, leading to
around 20 % errors in fk,R′
(ηR′≈0.2). Beyond about 100 projections per
8-day period, ηR′ oscillates around 0.1. The
corresponding errors in fk,R are about 5 %
(ηR≈1.05). In our study we used 300 random
projections for each 8-day period. This is about 10 % of the 3280 linear
constraints that we would have enforced under a naive implementation of the
FR′=0 constraint. It also halves the
computational cost of step I.
Conclusions
In this study, we have developed a sparse reconstruction scheme that could be
used for solving physics-based linear inverse problems. Our method is an
extension of stagewise orthogonal matching pursuit (StOMP) and borrows
many concepts from the compressive sensing (CS) and sparse reconstruction of
images . This scheme is useful for estimating non-stationary
fields, e.g., permeability or flux fields, provided their random field model
consists of independent parameters. This is typically achieved by
representing the fields in terms of orthogonal bases, e.g., wavelets or
Karhunen–Loève modes, if a prior covariance is available. The
dimensionality of the resultant representation is not an issue; the sparse
reconstruction method estimates only those parameters that are informed by
the observations while setting the rest to zero.
Our new method has three novel characteristics. First, it can impose
non-negativity on the estimated field, without resorting to
log transformations. This retains the linear nature of the inverse
problem and consequently, its computational efficiency. Second, it
allows one to estimate geometrically irregular fields while using
a random field model designed for rectangular domains. Third, it
allows us to incorporate a prior model of the field being estimated
into the sparse reconstruction procedure. While other model-based
sparse reconstruction methods exist , our
method is simple and is seen empirically to recover the correct
solution.
We have demonstrated our method in an atmospheric inverse
problem for the estimation of a spatially rough emission field. It is an
idealization of the estimation of ffCO2 emissions in
R, the lower 48 states of the USA. The
emissions were modeled in a square domain, with a 64×64 grid,
using a recently developed multiscale random field (MsRF)
model . It uses Haar wavelets and images of lights
at night to capture the spatial patterns of ffCO2 emission
fields. The observational data consists of ffCO2 measurements
at a limited set of towers, which are linked to the emission field via
a CO2 transport model (the forward model). We draw parallels
between our physics-based inverse problem and the sparse
reconstruction of images in CS, and show that a fundamental CS tenet
– incoherence – holds only approximately. Consequently, such inverse
problems may not bear an accurate solution if they are regularized
solely using sparsity. We demonstrate this in our study and show how
incorporation of prior information, in the form of spatial patterns in
images of lights at night, and a prior model of ffCO2
emissions can enable a solution. We also demonstrate how CS concepts
can be used to restrict the estimated field to an irregular region (in
our case, R) with a factor-of-ten less computational
effort than a naive approach. Finally, we show how non-negativity of
ffCO2 emissions can be imposed using a simple post-processing
step.
We also tested whether step I (Sect. ) was necessary by bypassing
it completely, and starting step II (Sect. ) with E0
initialized using an inventory. We do not present results of these tests in
this paper, but find that the iterative scheme converges only when
E0 is very close to the true results. For example, initializing
using perturbed Vulcan emissions led a converged solution, whereas
fpr did not. Thus, step I is required for robustness and
generality. This is particularly relevant for developing countries where
inventories contain larger errors.
Our sparse reconstruction scheme suffers from one serious drawback – it does
not provide uncertainty bounds on the estimated field due to the paucity of
data, and/or the shortcomings of the models. While this can be rectified
using a Kalman filter, it does not provide any mechanism for reducing the
dimensionality of the random field model, should the observational data prove
inadequate. This is currently being investigated. Also, we assumed that there
were no emissions outside R, but in reality, there are. See our
previous paper on how they could be accommodated as
boundary fluxes. Our use of the MsRF in the inversion is a second source of
error; in the limit of a very informative measurement network, the accuracy
of the inversion is limited by the ability of the MsRF to represent
ffCO2 fields accurately.
Due to the lack of a good tracer for ffCO2 emissions, we
demonstrated our method in an idealized inversion problem. The
idealizations include a very small model – data mismatch ϵ
(much smaller than what can be supported by contemporary transport
models and radiocarbon measurement technology) and an ability to
measure ffCO2 accurately, without interference from
biospheric CO2 fluxes (i.e., we treated ffCO2 like a
radiocarbon tracer). In order that our method could be
used in a real-data inversion for ffCO2 emissions, our
method would need to be extended in a number of ways. First, we
would require a measurement network better suited for ffCO2
measurements, with sensors near large sources; one could be designed
by conducting an observation system simulation
experiment, perhaps using the method described here. Second,
we would have to extend the method to perform
a joint ffCO2–biospheric CO2 inversion, by including
a spatial parameterization and priors for biospheric
CO2. Finally, we would have to devise a separate ϵ
for each tower to reflect transport model errors.
In conjunction with this paper, we are also providing, at our
website , the MATLAB® code
required to construct the MsRF model for ffCO2 emissions and perform
the inversion using synthetic observations. The website also contains links
to the (free) MATLAB® toolkits that our
code depends on, along with a user's manual.
Acknowledgements
This work was supported by Sandia National Laboratories' LDRD
(Laboratory Directed Research and Development) funds, sponsored by
the Geosciences Investment Area. Sandia National Laboratories is
a multi-program laboratory managed and operated by Sandia
Corporation, a wholly owned subsidiary of Lockheed Martin
Corporation, for the US Department of Energy's National Nuclear
Security Administration under contract DE-AC04-94AL85000. Edited by: A. Stenke
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