Inverse models use observations of a system (observation vector) to quantify the variables driving that system (state vector) by statistical optimization. When the observation vector is large, such as with satellite data, selecting a suitable dimension for the state vector is a challenge. A state vector that is too large cannot be effectively constrained by the observations, leading to smoothing error. However, reducing the dimension of the state vector leads to aggregation error as prior relationships between state vector elements are imposed rather than optimized. Here we present a method for quantifying aggregation and smoothing errors as a function of state vector dimension, so that a suitable dimension can be selected by minimizing the combined error. Reducing the state vector within the aggregation error constraints can have the added advantage of enabling analytical solution to the inverse problem with full error characterization. We compare three methods for reducing the dimension of the state vector from its native resolution: (1) merging adjacent elements (grid coarsening), (2) clustering with principal component analysis (PCA), and (3) applying a Gaussian mixture model (GMM) with Gaussian pdfs as state vector elements on which the native-resolution state vector elements are projected using radial basis functions (RBFs). The GMM method leads to somewhat lower aggregation error than the other methods, but more importantly it retains resolution of major local features in the state vector while smoothing weak and broad features.

Inverse models quantify the state variables driving the evolution of
a physical system by using observations of that system. This requires
a physical model

A critical step in solving the inverse problem is determining the amount of
information contained in the observations and choosing the state vector
accordingly. This is a non-trivial problem when using large observational
data sets with large errors. An example that will guide our discussion is the
inversion of methane emissions on the basis of satellite observations of
atmospheric methane concentrations

The simplest approach would be to use the native resolution of the CTM in
order to extract the maximum information from the observations. However, the
observations may not be sufficiently dense or precise to optimize emissions
at that level of detail, resulting in an underdetermined problem.

An additional drawback of using a large state vector is that analytical
solution to the inverse problem may not be computationally tractable.
Analytical solution requires calculation of the Jacobian matrix,

Reducing the dimensionality of the state vector in the inverse problem thus
has two advantages. It improves the observational constraints on individual
state vector elements and it facilitates analytical solution. Reduction can
be achieved by aggregating state vector elements. For a state vector of
gridded time-dependent emissions, the state vector can be reduced by
aggregating grid cells and time periods. However, this introduces error in
the inversion as the underlying spatial and temporal patterns of the
aggregated emissions are now imposed from prior knowledge and not allowed to
be optimized as part of the inversion. The resulting error is called the
aggregation error

Previous work by

Here we present a method for optimizing the selection of the state vector in
the solution of the inverse problem for a given ensemble of observations
without requiring an accurate specification of the native-resolution prior
error covariance matrix. Instead, we use the expected error correlations
between native-resolution state vector elements as criteria in the
aggregation process. Relative to

Inverse problems are commonly solved using Bayes' theorem,

The analytical solution to the inverse problem thus provides full error
characterization as part of the solution. It does require that the forward
model be linear. The Jacobian matrix must generally be constructed
numerically, requiring

The limitation on the state vector size can be lifted by finding the solution
to

The resolution of the forward model (e.g., grid resolution of the CTM) places an upper limit on the dimension for the state vector, which we call the native dimension. As we reduce the dimension of the state vector from this native resolution, the smoothing error decreases while the aggregation error increases. Here we present analytical expressions for the aggregation and smoothing error covariance matrices and show how they can be used to select an optimal state vector dimension.

As in

Aggregation error is the error introduced by aggregating state vector elements in the inversion. The relationship between the aggregated elements is not optimized as part of the inversion anymore and instead becomes an unoptimized parameter in the forward model, effectively increasing the forward model error and inhibiting the ability of the model to fit the observations. The aggregation error is thus a component of the observational error.

The aggregation error can be quantified by comparing the observational error
incurred by using the native-resolution state vector,

Following

From Eq. (

Additional consideration of aggregation error for a reduced-dimension
state vector

From these relationships we derive the total error covariance matrix
as

Each of the three error terms above depends on state vector dimension.
Because the smoothing error increases with state vector dimension while the
aggregation error decreases, analysis of the error budget can potentially
point to the optimal dimension where the total error is minimum. It can also
point to the minimum state vector dimension needed for the aggregation error
to be below a certain tolerance, e.g., smaller than the observation error. We
give an example in Sect.

A caveat in the above expressions for the aggregation and smoothing error
covariance matrices is that they are valid only if the prior

Illustration of different approaches for aggregating a state
vector. Here the native-resolution state vector is a field of
gridded methane emissions at

Aggregation of state vector elements to reduce the state vector dimension
introduces aggregation error, as described in Sect.

Previous work by

The simplest method for reducing the dimension of the state vector is to merge adjacent elements, i.e., neighboring grid cells. This method considers only spatial proximity as a source of error correlation. It may induce large aggregation errors if proximal, but otherwise dissimilar regions are aggregated together. In the case of methane emissions, aggregating neighboring wetlands and farmland would induce large errors because different processes drive methane emissions from these two source types.

The other two methods enable consideration of additional similarity factors
besides spatial proximity when aggregating state vector elements. These
similarity factors are expressed by vectors of dimension

Table

Similarity vectors for inverting methane emissions in North
America

Let

This approach of using a similarity matrix

In this method we cluster state vector elements following the principal
components of the similarity matrix. It is generally not practical to derive
the principal components in state vector space because the

Here we use a Gaussian mixture model

The first step in constructing the GMM is to define a

Projection of the native-resolution state vector onto the GMM involves four
unknowns:

Gaussian mixture model (GMM) representation of methane
emissions in Southern California with Gaussian pdfs as state vector
elements. The Gaussians are constructed from a similarity matrix
for methane emissions on the

The GMM allows each native-resolution state vector element to be represented
by a unique linear combination of the Gaussians through the RBFs. For a state
vector of a given dimension, defined by the number of Gaussian pdfs, we can
achieve high resolution for large localized sources by sacrificing resolution
for weak or uniform source regions where resolution is not needed. This is
illustrated in Fig.

We apply the aggregation methods described above to our example problem of
estimating methane emissions from satellite observations of methane
concentrations, focusing on selecting a reduced-dimension state vector that
minimizes aggregation and smoothing errors. The inversion is described in
detail in

For the purpose of selecting an aggregated state vector for the inversion, we
consider a subset of observations for May 2010 (

Aggregation and smoothing error dependences on the
aggregation of state vector elements in an inverse model. The
application here is to an inversion of methane emissions over North
America using satellite methane data with 7366 native-resolution
state vector elements

Figure

Figure

Total error budget from the aggregation of state vector
elements in an inverse model. The application here is to an
inversion of methane emissions over North America using satellite
methane data with 7366 native-resolution state vector
elements

From Fig.

Previous work by

We presented a method for optimizing the selection of the state vector in the solution of the inverse problem for a given ensemble of observations. The optimization involves minimizing the total error in the inversion by balancing the aggregation error (which increases as the state vector dimension decreases), the smoothing error (which increases as the state vector dimension increases), and the observational error. We further showed how one can reduce the state vector dimension within the constraints from the aggregation error in order to facilitate an analytical solution to the inverse problem with full error characterization.

We explored different methods for aggregating state vector elements as a means of reducing the dimension of the state vector. Aggregation error can be minimized by grouping state vector elements with the strongest correlated prior errors. We showed that a Gaussian mixture model (GMM), where the state vector elements are multi-dimensional Gaussian pdfs constructed from prior error correlation patterns, is a powerful aggregation tool. Reduction of the state vector dimension using the GMM retains fine-scale resolution of important features in the native-resolution state vector while merging weak or uniform features.

For advice and discussions, we thank K. Wecht (Harvard University). Special thanks to R. Parker and H. Boesch (University of Leicester) for providing the GOSAT observations. This work was supported by the NASA Carbon Monitoring System and by a Department of Energy (DOE) Computational Science Graduate Fellowship (CSGF) to A. J Turner. We thank the Harvard SEAS Academic Computing center for access to computing resources. We also thank M. Bocquet and an anonymous reviewer for their thorough comments. Edited by: R. Harley