Atmospheric inversions are frequently used to estimate fluxes of atmospheric
greenhouse gases (e.g., biospheric CO

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 ffCO

The estimation of spatially resolved fields, e.g., permeability fields in
aquifers or

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

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

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;

As a step towards enabling such applications, we constructed a wavelet-based
spatial parameterization, called the multiscale random
field

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

Our previous work

Note, that the term

We assume that

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.

We will estimate the emission field at 1

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

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

The paper is structured as follows. In Sect.

In this section, we review techniques used to estimate

Consider an image

In order that one may recover the original image

Sparsity is sometimes used to solve inverse problems in physics, with
the

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

Let

We employ two prior models in our work – the MsRF model for

The MsRF is also the starting point for developing the second prior
model

We seek emissions over an entire year (360 days), i.e., we seek

Specifying the constraint in individual grid cells is not very efficient
since it leads to

The case where

The wavelet coefficients

In all the three cases, we obtain an underdetermined set of linear
equations of the form

Since

Estimates of

We seek the non-negative vector

We will refer to this step in the estimation procedure as step II.

In this section, we test the sparse estimation technique in
Sect.

The

The true

Plot of

We choose between approaches A, B and C by solving the inverse problem for
the

In Fig.

Plots of

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

In Fig.

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.

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

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

In Eq. (

We now address some of the numerical aspects of the solution. The results
presented here are

Comparison of the distribution of the elements of

In Fig.

(Top) predictions of

Finally, we address the issue of enforcing the

The impact of the number of compressive samples

In Fig.

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)

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

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

We also tested whether step I (Sect.

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

Due to the lack of a good tracer for

In conjunction with this paper, we are also providing, at our
website ^{®} code
required to construct the MsRF model for ^{®} toolkits that our
code depends on, along with a user's manual.

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