the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# A flexible algorithm for network design based on information theory

### Ignacio Pisso

A novel method for atmospheric network design is presented, which is based on information theory. The method does not require
calculation of the posterior uncertainty (or uncertainty reduction) and is
therefore computationally more efficient than methods that require this. The algorithm is demonstrated in two examples: the first looks at designing a network for monitoring CH_{4} sources using observations of the stable carbon isotope ratio in CH_{4} (*δ*^{13}C), and the second looks at designing a network for monitoring fossil fuel emissions of CO_{2} using observations of the radiocarbon isotope ratio in CO_{2} (Δ^{14}CO_{2}).

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The optimal design of any observing network is an important problem in order to maximize the information obtained with minimal cost. In atmospheric sciences, observing networks include those for weather prediction, as well as for air quality and the monitoring of greenhouse gases (GHGs). For air quality and GHGs, one essential purpose of the observation network is to learn about the underlying sources and, where relevant, the sinks. This application is based on inverse methodology in which knowledge about some unknown variables, in this case the sources (and sinks), can be determined by indirect observations, that is, the atmospheric concentrations or mixing ratios, if there is a model or function that relates the unknown variables to the observations. Inverse methodology provides a means to relate the observations to the unknown variables and provides an optimal estimate of these (Tarantola, 2005).

In atmospheric sciences, the methodology is most often derived from Bayes'
theorem, which describes the conditional probability of the state variables,
*x*, given the observations, *y*.

Assuming a Gaussian probability density function (PDF), the following cost function can be derived (Rodgers, 2000).

The ** x** for which

*J*(

**) is the minimum is the state vector that minimizes the sum of two distances: one in the observation space, between the modelled,**

*x***(**

*H***), and observed,**

*x***, variables, and the other in the state space, between**

*y***and a prior estimate of state variables,**

*x*

*x*_{b}. These two distances are weighted by the matrices

**R**and

**B**, which are, respectively, the observation error covariance and prior error covariance. Expressions for the centre and variance of the posterior PDF of

**are given by e.g. Tarantola (2005).**

*x*The choice of the locations for the observations has important consequences
for how well the state variables can be constrained. Increasing the number
of observations will decrease the dependence of the solution on
*x*_{b}, but where those observations are made is also a
critical consideration and depends on how they relate to the state variables,
as described by the transport operator, ** H**(

**). Here only the linear transport case is considered in which this operator can be defined as the matrix**

*x***H**.

In practical applications of network design, there is usually a predefined budget that would allow the establishment of a given number of sites, either to create a new network or to add to an existing one. The possible locations of sites are usually a predefined set, since these need to fulfil certain criteria, e.g. access to the electrical grid, internet connection, road access, an existing building on site to house instruments, and the agreement of the property owner, and may include having an existing tower if measurements are to be made above the surface layer. Thus, the question is often the following: which potential sites should be chosen to provide the most information about the sources and sinks?

The founding work on network design was actually in the field of seismology
(Hardt and Scherbaum, 1994), but there are already a number of examples of
network design in the framework of atmospheric monitoring in the scientific
literature. An early example is the optimization of a global network for
CO_{2} observations to improve knowledge of the terrestrial CO_{2}
fluxes (Gloor et al., 2000; Patra and Maksyutov, 2002; Rayner et al., 1996).
These studies dealt only with small dimensional problems, i.e. with few
state variables, relatively low-frequency observations, and thus small
**B** and **H** matrices, and the criteria by which the network
was chosen involved minimizing the posterior uncertainty. Gloor et al. (2000) solved the
problem using a Monte Carlo method (specifically simulated annealing), but
they found this method took considerable time to converge and up to
5 × 10^{5} iterations were needed. Patra and Makysutov (2002) used a less
computationally demanding approach, the incremental optimization method,
which is based on the “divide and conquer” algorithm principle. In this
method, the problem to solve is broken down into steps, i.e. sequentially
choosing the best site from the set of potential sites and correspondingly
depleting this set by one with each step. In the incremental optimization
approach only $\sum _{i=\mathrm{1}}^{k}(p-i+\mathrm{1})$ calculations are
needed, where *k* is the number of sites to select and *p* the number of potential sites to choose from. The incremental optimization approach, however, may lead to a different selection of sites compared to testing all possible combinations of sites, which would involve $p\mathrm{!}/\left(k\mathrm{!}\right(p-k\left)\mathrm{!}\right)$ calculations, but
this in many cases may be a prohibitively large number.

More recently, the problem of network design has been addressed in the context of regional networks for GHG observations (Lucas et al., 2015; Nickless et al., 2015). Again, in both these studies the metric for selecting the network was the posterior uncertainty, either by using the trace of the posterior error covariance matrix, which is equivalent to minimizing the mean square uncertainty for all grid cells (Lucas et al., 2015), or by minimizing the sum of the posterior error covariance matrix (or submatrix for a particular region), which also accounts for the covariance of uncertainty between grid cells (Nickless et al., 2015). These studies both used Monte Carlo approaches (specifically, genetic algorithms) to find the network minimizing the selected metric.

However, for large problems any metric involving the posterior uncertainty
becomes a bottleneck, if not unworkable, since the calculation of the
posterior error covariance matrix, **A**, requires inverting the matrix
${\mathbf{H}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+{\mathbf{B}}^{-\mathrm{1}}$ which has
dimensions of *n*×*n* where *n* is the number of state variables. For this reason, methods were proposed based on criteria considering how well a
network resolves the atmospheric variability or “signal” or, in other
words, how well they sample regions of significant atmospheric heterogeneity
(Shiga et al., 2013). In this approach, the atmospheric signal (e.g. mixing
ratio) is modelled using an atmospheric transport model and a prior flux
estimate, and sites are sequentially added to the network so that the
distance of any grid cell from an observation site is within some
predetermined correlation scale length. For this method, the number of
calculation steps is equal to the sites to be selected (Shiga et al., 2013).
Although computationally very efficient, this method does not consider the
information gained about the state variables but only the optimal sampling
of atmospheric variability.

An alternative method, but also based on the consideration of atmospheric variability, is to consider how “similar” the atmospheric signal is between potential sites in a network and to reduce the number of sites, leaving only those with significantly different signals (Risch et al., 2014). Risch et al. (2014) applied a clustering method to cluster sites with similar signals (i.e. strongly correlated sites), and individual sites were removed from each cluster based on the premise that they did not contribute any significant new information, whereas sites in clusters of one member were all retained. However, as in the method of Shiga et al. (2013), this approach does not consider the information gained about the state variables and how atmospheric transport alone may influence the variability at each site.

Here a method for network design is proposed based on information theory.
This method requires precomputed transport operators for each potential
site, so-called site “footprints” or “source–receptor relationships” (SRRs), which can be calculated directly using a Lagrangian atmospheric
transport model (Seibert and Frank, 2004) or from forward calculations of an
Eulerian transport model for each source (Rayner et al., 1999; Enting,
2002). The method can be applied to the problem of creating a new network or
expanding an existing one and can be applied to observations of mixing
ratios, isotopic ratios, column measurements, or a combination of these. It
provides an alternative criterion to the posterior uncertainty (or
uncertainty reduction) to assess a potential network and can be used with
either incremental optimization or Monte Carlo approaches. It has a number
of advantages compared to previous methods: (i) it does not require the
inversion of any large matrix, except for **B**, but this is needed only once, making it computationally efficient; (ii) it accounts for spatial
correlations in the state variables; and (iii) it allows for an exact
formulation of the problem to be solved, i.e. what is the improvement in
knowledge about the unknown variables. On the other hand, it requires
linearity of the operator from the state space to the observation space,
which is not the case for methods examining only atmospheric variability.

Two example applications are presented, which are based on real-life network
design problems. The first considers adding measurements of the stable
isotope ratio of CH_{4}, i.e. *δ*^{13}C, to a subset of existing
sites measuring CH_{4} mixing ratios in order to maximize the information
about CH_{4} sources. The second considers designing a network for Δ^{14}CO_{2} measurements to maximize the information about fossil fuel emissions of CO_{2}.

In information theory, the information content of a measurement can be thought of as the amount by which knowledge of some variable is improved by making the measurement, and the entropy is the level of information contained in the measurement (Rodgers, 2000). In this case, one can consider the PDF a measure of knowledge about the state variables, and the information provided by a measurement can be found by comparing the entropy of the PDFs before and after measurement was made. Furthermore, the information content of the measurement is equal to the reduction in entropy. In the application of network design, all observations within the potential network are considered to be one “measurement”.

The entropy, *S*, of the PDF given by *P*(*x*) is

And the information content, *I*, is the reduction in entropy after a
measurement is made:

where *P*(*x*) is the prior PDF (before measurement) and *P*(*x*|*y*) is the posterior
PDF (after the measurement, *y*). The entropy is given by integrating Eq. (3)
over the bounds −∞ to +∞ (Rodgers, 2000), which for a
Gaussian PDF of a scalar variable is

where *σ* is the standard deviation. In the multivariate case with *m*
variables the entropy is given by

where *λ*_{i} is an eigenvalue of the error covariance matrix. By
rearrangement one can write the following.

In Eq. 9 $\left|\mathbf{B}\right|$ is the determinant of the prior error covariance matrix, using the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues. Similarly, the entropy for the posterior PDF can be derived, giving the information content as

where **A** is the posterior error covariance matrix. In this case the
determinant can be thought of as defining the volume in state space occupied by
the PDF, which describes the knowledge about the state; thus, *I* is the change in the log of the volume when observation is made. From Eq. (10) one can derive the following.

And given that the inverse of **A** is equal to the Hessian matrix of
*J*(** x**) (Eq. 2)

one obtains

where **R** is the observation error covariance matrix, **H** is the model operator (for atmospheric observations it is the atmospheric
transport operator), and **I** is the identity matrix.

The principle of this network design method is to choose the sites that maximize the information, and this criterion can be used in either the incremental optimization or Monte Carlo approach. The incremental optimization approach is computationally efficient, requiring only $\sum _{i=\mathrm{1}}^{k}(p-i+\mathrm{1})$ calculations and delivers, if not the same, at least similar results as testing all possible combinations of sites (Patra and Maksyutov, 2002).

The calculation of the matrix ${\mathbf{BH}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+\mathbf{I}$ can be quite fast, since **H** and **R** can be made quite small. **H** does not need to represent all observations for each
site, but only the average observation corresponding to different levels of
uncertainty or “characteristic observations”. In the case that
observations at each site have only one characteristic uncertainty, then
**H** will have dimension *k*×*n* where *n* is the number of state variables, and **R** will be *k*×*k*; in practice **R**
is most often diagonal. In the case that the uncertainty of an observation
at a given site varies depending on when it was made, e.g. daytime or
nighttime, then the dimension of **H** will be 2*k*×*n*. The
computationally demanding step is the calculation of the matrix determinant.
However, this calculation can be made very efficient if the matrix
${\mathbf{BH}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+\mathbf{I}$ is decomposed into **B** and
(${\mathbf{H}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+{\mathbf{B}}^{-\mathrm{1}}$), which
are both symmetric positive definite matrices, and using the fact that the
log of the determinant of a symmetric positive definite matrix can be
calculated as the trace of the log of the lower triangular matrix of the
Cholesky decomposition:

where **B**=**LL**^{T} and
${\mathbf{H}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+{\mathbf{B}}^{-\mathrm{1}}={\mathbf{MM}}^{\mathrm{T}}$ where **L** and **M** are the lower triangular
matrices. Note that if temporal correlations in **B** can be ignored,
then **B** only needs to be formulated for a single time step, i.e.
**B**_{t}, which is a considerably smaller matrix than **B**,
and ${\mathbf{H}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+{\mathbf{B}}^{-\mathrm{1}}$ can be
calculated stepwise adding ${\mathbf{B}}_{t}^{-\mathrm{1}}$ for each time step.
Furthermore, ${\mathbf{B}}_{t}^{-\mathrm{1}}$ (or **B**^{−1}) only needs to be
calculated once, since it does not change with choice of sites. In this case
the information content is simply

where *q* is the number of time steps and **L** in this case is the lower triangular matrix of **B**_{t}.

The computational complexity of the whole algorithm can be estimated
considering that the Cholesky decomposition of a symmetric matrix has a
complexity of $O\left({n}^{\mathrm{3}}\right)/\mathrm{3}$. The calculation of **B**^{−1} from
**B** requires $O\left({n}^{\mathrm{3}}\right)/\mathrm{3}$ operations. The calculation of
**R**^{−1} from **R** requires $O\left({k}^{\mathrm{3}}\right)/\mathrm{3}$ operations. The calculation of **H**^{T}**R**^{−1}**H**
requires $O(n{k}^{\mathrm{2}}+{n}^{\mathrm{2}}k)\sim O\left({n}^{\mathrm{2}}\right)$
operations if *k*≪*n*. Then only the calculation of the
determinant of the matrix $\left|{\mathbf{H}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+{\mathbf{B}}^{-\mathrm{1}}\right|$ remains, which given that it is symmetric and positive definite
also takes $O\left({n}^{\mathrm{3}}\right)/\mathrm{3}$ operations (Aho et al., 1974). The subsequent
logarithm and the trace operations are linear with respect to *n*, i.e.
*O*(*n*). The total complexity yields $O\left({n}^{\mathrm{3}}\right)/\mathrm{3}+O\left({k}^{\mathrm{3}}\right)/\mathrm{3}+O\left({n}^{\mathrm{2}}\right)+O\left({n}^{\mathrm{3}}\right)/\mathrm{3}+O\left(n\right)\approx \mathrm{2}O\left({n}^{\mathrm{3}}\right)/\mathrm{3}+O\left({k}^{\mathrm{3}}\right)/\mathrm{3}$,
which is comparable to e.g. one *n*×*n* LU factorization if *k*≪*n*.

## 3.1 Enhancing a network for estimating sources of CH_{4}

This example considers the enhancement of a network of atmospheric
measurements of CH_{4} mixing ratios by adding observations of stable
isotopic ratios, *δ*^{13}C, at a selected number of sites within the
existing network in order to improve knowledge of the different CH_{4}
sources. For the example, the case of the Integrated Carbon Observation System
(ICOS) network (https://www.icos-cp.eu, last access: 12 January 2023) in Europe is used,
which consists of 24 operational sites in geographical Europe measuring
CH_{4} mixing ratios (Table 1). In this hypothetical case, the budget is
available to equip 5 of the 24 sites with in situ instruments measuring
*δ*^{13}C at hourly frequency, as is now possible with modern
instrumentation (Menoud et al., 2020). The problem can thus be formulated
as follows: given the existing information provided by 24 sites measuring CH_{4}
mixing ratios, which sites are the best to choose for the additional *δ*^{13}C observations?

The *δ*^{13}C value is the ratio of ^{13}C to ^{12}C in a sample
relative to a reference value measured in per mil (‰).

The *δ*^{13}C value in the atmosphere varies as a result of
variations in the *δ*^{13}C value of the sources, the oxidation of
CH_{4} in the atmosphere and in soils, and atmospheric transport. Sources
of CH_{4} can be grouped according to their characteristic *δ*^{13}C value, with microbial sources being the most depleted in ^{13}C,
while thermogenic sources such as from oil, gas, and coal are less
depleted, and pyrogenic sources, such as biomass burning, are the least
depleted (Fisher et al., 2011; Dlugokencky et al., 2011; Brownlow et al.,
2017) (Table 2). In this example, CH_{4} sources were grouped into anthropogenic
microbial sources, namely agriculture and waste (agw); fossil sources,
namely fossil fuel and geological emissions (fos); biomass burning sources
(bbg); and natural microbial sources, principally wetlands (wet) and the ocean
source (oce). The change in CH_{4} mixing ratios from all sources can thus
be written as

where Δ*c* is the change in CH_{4} mixing ratios, ** x** is the
vector of fluxes, and

**H**is the transport operator. Analogously, the change in

*δ*

^{13}C can be defined as

where *δ*_{x} is the isotopic signature for each source type.
Therefore, the transport operator for an observation of the change in
*δ*^{13}C is just the transport operator **H** but scaled
by *δ*_{x} for each source.

For this example, SRRs were calculated for all 24 sites in the ICOS network
using the Lagrangian particle dispersion model, FLEXPART (Pisso et al.,
2019), driven with ERA-Interim reanalysis wind fields. Retro-plumes were
calculated for 10 d backwards in time from each site at an hourly frequency.
The SRRs were saved at 0.5^{∘} × 0.5^{∘} resolution
over the European domain of 12^{∘} W to 32^{∘} E and
35 to 72^{∘} N and averaged over all observations within
a month to give a monthly mean SRR for each site.

The uncertainty in the *δ*^{13}C measurements was set to the same
value for each site, that is, at 0.07 ‰ based on
experimental values (Menoud et al., 2020). Similarly, the uncertainty in
CH_{4} mixing ratio measurements was also set to the same value at all
sites, at 5 ppb (WMO, 2009). The prior uncertainty *σ* for each grid
cell was calculated as 0.5 times the prior flux, with a lower threshold
equal to the first percentile value of all grid cells with non-zero flux for the
smallest flux source. The spatial correlation between grid cells was
calculated based on exponential decay over distance with a correlation scale
length of 250 km over land. The prior error covariance matrix was then
calculated as

where **C** is the spatial correlation matrix and **Σ** is
a diagonal matrix with the diagonal terms equal to the prior uncertainties
for each grid cell.

For this example, the optimal network was found for three different
scenarios: (1) monitoring all sources in EU27 countries plus the UK, Norway, and Switzerland (EU27 + 3); (2) monitoring only anthropogenic sources in
EU27 + 3; and (3) as in scenario 1 but ignoring the existing information
provided by CH_{4} mixing ratios at all sites.

For these scenarios the influence of the fluxes that are not the target of
the network needs to be projected into the observation space and included in
the **R** matrix. For example, in scenario 1 this is the influence of
fluxes outside EU27 + 3, and in scenario 2 it is the influence of all
non-anthropogenic sources plus the influence of fluxes outside EU27 + 3.
This is calculated as

where **R**_{meas} is simply the prior measurement uncertainty and **B**_{other} is the prior error covariance matrix for the other (i.e. non-target) fluxes.

For all scenarios the choice of the first four optimal sites was the same,
that is, IPR, SAC, KIT, and LIN, while the last site chosen was KRE in
scenarios 1 and 2 (Fig. 1) and LUT in scenario 3. All chosen sites are
strongly sensitive to anthropogenic emissions, and the choice to optimize
all sources or only anthropogenic sources made no difference in this
example, likely because the natural sources (predominantly wetlands) are a
relatively small contribution to the total CH_{4} source in Europe (only
12 %). On the other hand, ignoring existing information provided by
CH_{4} mixing ratios led to LUT being chosen over KRE, likely because LUT
provides a stronger constraint on the region with the largest emissions and
diverse sources, i.e. Benelux (Fig. 2), which is more important in the
absence of CH_{4} mixing ratio data.

## 3.2 Network of ^{14}CO_{2} measurements for fossil fuel emissions

This example concerns the establishment of a network for measurements of
radiocarbon dioxide, ^{14}CO_{2}, which can be used as a tracer for
fossil fuel CO_{2} emissions; since fossil fuel contains no ^{14}C, its
combustion depletes the atmospheric background value of ^{14}CO_{2}
(Turnbull et al., 2009). Similar to the previous example, the ICOS network
is used, which also has CO_{2} measurements at 24 sites in Europe. The
hypothetical problem can be formulated as follows: if there is a budget to
equip 10 sites in the ICOS network with weekly flask samples for
^{14}CO_{2} analysis, which sites should be chosen to gain the most
knowledge of fossil fuel emissions? In this case, only weekly measurement
frequency is examined as ^{14}CO_{2} measurements cannot be made
continuously, and the measurement method, either via counting radioactive
decay or by accelerator mass spectrometry, is costly and time-consuming. The
optimization problem needs to consider the information already brought by
the CO_{2} measurements at all sites (in this example hourly measurements)
and, in addition, the influence on the atmospheric signal from other
sources, which may change the sensitivity of a site to fossil fuel
emissions.

Measurements of ^{14}CO_{2} are reported as the ratio of ^{14}CO_{2}
to CO_{2} relative to a reference ratio and given in units of per mil
(‰).

Since fossil fuels contain no ^{14}C, its isotopic ratio is
−1000 ‰. Other than fossil fuels, atmospheric values of
Δ^{14}CO_{2} are determined by the natural production of
^{14}CO_{2} in the stratosphere, nuclear power and spent fuel-processing
plants, ocean and land biosphere fluxes, and atmospheric
transport. Ocean fluxes affect ^{14}CO_{2}, since the ocean is not in
isotopic equilibrium with the atmosphere owing to higher values of
atmospheric ^{14}CO_{2} in the past due to nuclear bomb testing and
similarly for plant respiration fluxes of CO_{2} (Bozhinova et al., 2014).

The change in CO_{2} mixing ratios can be described as follows:

where *x*_{fos} is the fossil fuel emission, *x*_{pho} the land biosphere photosynthesis flux, *x*_{res} the land biosphere respiration flux, and *x*_{oce} the net ocean flux. A similar expression for the change in Δ^{14}CO_{2} can be derived following Bozhinova et al. (2014) as

where Δ^{14}*c* is the change in Δ^{14}CO_{2}, the term Δ_{x} is the isotopic signature of the corresponding source, and
**H**Δ_{nuc}*x*_{nuc} is the production of ^{14}CO_{2} from nuclear facilities. There is a term missing from Eqs. (23) and (24), namely
the stratospheric production of CO_{2} and ^{14}CO_{2}. This term is
ignored as the direct stratospheric contribution is negligible for the time
and space domain considered by the Lagrangian model, since the observations
are close to the surface. Equation (24) can be further simplified by removing
the term **H**Δ_{pho}*x*_{pho}, since photosynthesis,
although it affects the ^{14}CO_{2} mixing ratio, does not affect Δ^{14}CO_{2} (Turnbull et al., 2009). Furthermore, the ocean and
respiration fluxes can be split into a background term and a disequilibrium
term, Δ_{bg}+Δ_{ocedis} and Δ_{bg}+Δ_{resdis}, respectively. As for photosynthesis, the background terms for ocean and respiration fluxes do not change Δ^{14}CO_{2}, but only the disequilibrium terms change Δ^{14}CO_{2}. For the domain in
consideration, these terms are much smaller than that of fossil fuels and
are ignored as in Bozhinova et al. (2014). With these simplifications, Eq. (24) becomes the following.

Since *x*_{nuc} is pure ^{14}CO_{2}, Δ_{nuc} would be infinite; therefore, the approach of Bozhinova et al. (2014) is used and Δ_{nuc} is approximated as the ratio of the activity of the sample and the referenced standard, giving ${\mathrm{\Delta}}_{\mathrm{nuc}}\approx \mathrm{0.7}\times {\mathrm{10}}^{\mathrm{15}}$ ‰.

Because, in this example, only the fossil fuel emissions are the unknown
variables and the target of the network, the matrix **B** corresponds
only to the uncertainty in the fossil fuel emissions and is resolved
monthly. The other terms influencing CO_{2} and Δ^{14}CO_{2}
are projected into the observation space and included in the **R**
matrix using Eq. (21). For the Δ^{14}CO_{2} observations,
**B**_{other} is only the nuclear source, and for CO_{2}
observations, **B**_{other} includes photosynthesis and respiration,
the sum of which is net ecosystem exchange (NEE) and the ocean flux, for
which the effect on the observed CO_{2} signal is very small and is thus
ignored here. For both NEE and nuclear emissions, an uncertainty of 0.5 times the value in each grid cell was used to calculate **B**_{other}
with a spatial correlation length of 250 km. Since NEE fluxes have large
diurnal and seasonal cycles which co-vary with atmospheric transport, for
the CO_{2} observations, **R** was calculated using **H** and
**B**, resolved for day, night, and monthly. Note, only one
uncertainty value was calculated for each site, which represents the annual
mean uncertainty for a daytime observation. Each site has a different
uncertainty for CO_{2} mixing ratios and Δ^{14}CO_{2} depending
on the influence of NEE fluxes and nuclear emissions, respectively. This can
be simply interpreted in terms of a signal-to-noise ratio. For example, for
CO_{2} mixing ratios where there is a large influence of NEE, the
time series becomes noisier, similarly for the influence of nuclear
emissions on Δ^{14}CO_{2} observations. The measurement
uncertainty, **R**_{meas}, was set to the same value for each site, that is, at 2 ‰ for Δ^{14}CO_{2} (Turnbull et
al., 2007) and 0.05 ppm for CO_{2} mixing ratio measurements (WMO, 2018).

For this example, SRRs were calculated for all 24 sites in the ICOS network
using FLEXPART with retro-plumes calculated for 5 d backwards in time
from each site at an hourly frequency. The SRRs were saved at 0.5^{∘} × 0.5^{∘} and 3-hourly resolutions over the European domain of 15^{∘} W to 35^{∘} E and 30 to 75^{∘} N
and were averaged to give mean day and night SRRs for each month for each
site. Estimates of NEE fluxes were used from the Simple Biosphere Model –
Carnegie Ames Stanford Approach (SiBCASA) and were resolved 3-hourly
(Schaefer et al., 2008), estimates of nuclear emissions were used from the *C**O*_{2} human emissions (CHE)
project (Potier et al., 2022) and were an annual climatology, and estimates
of fossil fuel emissions were from GridFED at monthly resolution (Jones et al., 2020a).

Figure 3 shows the uncertainty in the observation space at each site due to
the influence of uncertainties in NEE and nuclear emissions on CO_{2}
mixing ratios and Δ^{14}CO_{2} values, respectively. For
CO_{2}, sites in western Europe have the largest uncertainties, while
sites in northern Scandinavia and southern Europe have smaller uncertainties
following the pattern of NEE amplitude. For Δ^{14}CO_{2}, most
sites have only small uncertainties owing to nuclear emissions, but two
notable exceptions are NOR and KIT, and both are close to large nuclear
sources.

The optimal sites in the order selected are SAC, KIT, LUT, KRE, STE, LIN,
GAT, IPR, TRN, and TOH (Fig. 4). Two of the sites, SAC and TRN, are
relatively close to one another (approximately 95 km apart); however, they
have somewhat different footprints with SAC sampling more of the Paris
region and TRN sampling more of the south and east. If the prior error
covariance matrix, **B**, and the transport operator, **H**, are not resolved monthly but only annually, the optimal sites differ by only one site, namely HPB instead of TRN. If the existing information
provided by CO_{2} mixing ratios is ignored (i.e. the network is designed
only considering information from Δ^{14}CO_{2}), then the choice
of optimal sites differs slightly, and TRN and TOH are no longer selected but
OPE and LMP. The choice of LMP may seem unexpected at first, but it is close
to an emission hotspot in Tunis, Tunisia (Fig. 5). The reason this site is
not selected when the information from CO_{2} mixing ratios is included is
presumably because the CO_{2} mixing ratio already provides a reasonable
constraint on the fossil fuel emissions with the NEE signal being relatively
small.

An obvious question is how does the criterion of information content compare
to criteria based on the posterior uncertainty? The information content
describes the change in probability space from before an observation is made
(prior probability) compared to after an observation is made (posterior
probability) and is thus more closely linked to the observations themselves
than to the exact definition of the posterior uncertainty metric. The
performance of the two metrics, i.e. information content versus the sum of
the posterior error covariance matrix, was examined using the CH_{4}
example in scenario 1 (described in Sect. 3.1). For this example, a second
network was selected using the criterion of the sum of the posterior error
covariance matrix and consisted of the sites HPB, HTM, KRE, PUY, and TRN
(only KRE was also selected using the information criterion). Two inversions
were performed using pseudo-observations generated by applying the transport
operator, **H** (with rows corresponding to daily afternoon means for
each site and columns corresponding to the six source types resolved
annually), to the annual mean fluxes for each source type, ** x**, and
adding random noise according to the error characteristics of

**R**.

In these inversions, the prior was generated by randomly perturbing the
fluxes according to the error characteristics of **B**.

Both inversions used the same prior fluxes and uncertainties and differed
only in the set of sites used. The performance of the inversions was tested
using the so-called gain metric, *G*, based on the ratio of the distance of the posterior from the true fluxes to the distance of the prior from the true
fluxes:

where *x*_{a} is the posterior flux vector. The larger the
value of *G*, the closer the posterior is to the true flux. Using the optimal
sites according to the information content *G*=0.6996, while using the
optimal sites according to the posterior error covariance *G*=0.6988. (A
comparison of the prior and posterior compared to the true fluxes is shown
in Fig. 6.) Thus, the information content performs at least as well as the posterior error covariance metric for determining a network.

Another question that arises is how does this method compare to methods
based on the analysis of the variability in the time series at the different
sites? To answer this question, a clustering method was applied to the
example of designing a network for fossil fuel CO_{2} emissions. For this,
a time series of Δ^{14}CO_{2} was generated for each of the 24
sites using Eq. (25) (see the Supplement for plots of the
time series). The values were generated hourly, but, since generally only
daytime values are used in inverse modelling, data were selected for the
time interval 12:00 to 15:00 LT. A dissimilarity matrix was calculated for the
24 time series (using the R function proxy::dist with the dynamic time warp
(DTW) method; Giorgino, 2009). The divisive hierarchical clustering method
(R function cluster::diana) was applied to the dissimilarity matrix, stopping
at 10 clusters. The first cluster contained 13 sites, that is, those with
little signal (e.g. JFJ, CMN, and ZEP). Two clusters contained two sites,
namely IPR and KRE, as well as OPE and TRN, while the remaining clusters contained
only one site. Based on the principle of choosing sites that display
different signals, one would choose the sites which are in a cluster of one.
This would lead to the choice of GAT, KIT, LIN, LUT, SAC, STE, and TOH. These
seven sites are also chosen by the method based on information content. However,
the question is how to choose the remaining three sites from clusters with more
than one site. For this there is no single answer. Moreover, the sites that
are the most dissimilar are not necessarily those that will provide the most
information about the target fluxes of the network, since the reasons for
dissimilarity are various, e.g. having little signal, being sensitive to
sources that are not the target of the network, or owing to distinct
atmospheric circulation patterns. Sites with high degrees of
similarity may both offer a strong constraint and both be valuable to a
network (in this example IPR and KRE were in the same cluster, but both sites
are chosen in the method based on information content).

In the examples presented, the atmospheric transport matrix, **H**, and the matrix, **B**, were resolved at 0.5^{∘} × 0.5^{∘} (and considered only land grid cells) and monthly. The size of the matrix **B** (and the matrix
${\mathbf{H}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+{\mathbf{B}}^{-\mathrm{1}}$) for the
example of a network for fossil fuel CO_{2} emissions was ∼ 11 Gb. However, in the case of finer spatial resolution or a larger domain,
which means the size of the matrices exceeds the available memory, it is
still possible to use this method as long as **B** and
${\mathbf{H}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+{\mathbf{B}}^{-\mathrm{1}}$ defined
for one time step do not exceed the memory. In this case, the problem can be
solved by summing the information content calculated separately for each
time step. Disaggregating the problem in this way does not lead to the same
value of information content as when all time steps are considered together;
however, the choice of sites is nearly the same. For the example of a
network for fossil fuel CO_{2} emissions, the two methods (i.e.
disaggregating versus not disaggregating) differed by only one site. For the
example of a fossil fuel network, the total computation time was
∼ 3 h using multi-threaded parallelization on eight cores.

In addition to the memory requirements, there is the question of the
computational cost determined by the complexity of the algorithm, in
particular compared to the more established method using a metric based on
the posterior error covariance. Such analysis can be performed putting aside
the practical considerations related to particular software and/or hardware.
It has been established that the algorithmic complexity, and hence the
computational cost, of the calculation of the determinant is the same as
that of matrix multiplication (Strassen, 1969; Aho et al., 1974). Ignoring
the particularities of the algorithm used and its hardware implementation,
the analysis can be simplified by counting the number of matrix
multiplications: *O*(*m**n**k*) for two generic rectangular matrices, Cholesky
decompositions ($O\left({n}^{\mathrm{3}}\right)/\mathrm{3}$), matrix inversions, and determinant
calculations (both obtained e.g. from the Cholesky decomposition). Both the
error covariance metric and the information metric require the inversion of
the matrix **B**. The covariance metric requires the Hessian matrix
$\mathbf{G}={\mathbf{H}}^{\mathrm{T}}{\mathbf{R}}^{-\mathrm{1}}\mathbf{H}+{\mathbf{B}}^{-\mathrm{1}}$that
takes one inversion of **B** ($\sim O\left({n}^{\mathrm{3}}\right)/\mathrm{3}$), one
inversion of **R** ($\sim O\left({k}^{\mathrm{3}}\right)/\mathrm{3}$), and the product of
three matrices ($O(n{k}^{\mathrm{2}}+{n}^{\mathrm{2}}k)\sim O\left({n}^{\mathrm{2}}\right)$ if
*k*≪*n*). This yields **G** in $O\left({n}^{\mathrm{3}}\right)/\mathrm{3}+O\left({k}^{\mathrm{3}}\right)/\mathrm{3}+O(n{k}^{\mathrm{2}}+{n}^{\mathrm{2}}k)$ operations. The inverse of
**G** yields the posterior covariance in $O\left({k}^{\mathrm{3}}\right)/\mathrm{3}$ operations via
the Cholesky decomposition. Subsequent steps are of lower computational
order. Even if some simplifications are possible, its complexity is bounded
below by $\mathrm{2}O\left({n}^{\mathrm{3}}\right)/\mathrm{3}$. Therefore, the information metric is not
computationally more expensive than the covariance metric. The algorithm
presented here is faster than the naive computation of the information
content from its formal definition.

A method for designing atmospheric observation networks is presented based
on information theory. This method can be applied to any type of atmospheric
data: mixing ratios, aerosols, isotopic ratios, and total column
measurements. In addition, the method allows the network to be designed with
or without considering existing information, which may also be of a
different type, e.g. mixing ratios of a different species or isotopic
ratios. Since the method does not require inverting any large matrices
(e.g. for the calculation of posterior uncertainties), and the calculation
of **B**^{−1} only needs to be performed once, it is very efficient
and can also be used on large problems. The only constraint is that the
matrices **B** and **H**^{T}**R**^{−1}**H** +
**B**^{−1} defined for one time step do not exceed the available
memory. The algorithm allows the exact problem to be defined, that is, to
target specific emission sources or regions. Two examples are presented: the
first is to select sites from an existing network of CH_{4} mixing ratios
for additional measurements of *δ*^{13}C to constrain emissions in
EU countries (plus Norway, Switzerland, and the UK), and the second is to select
sites from an existing network of CO_{2} mixing ratios for additional
measurements of Δ^{14}CO_{2} to monitor fossil fuel CO_{2}
emissions. These examples demonstrated that the optimal network differs
depending on its exact purpose, e.g. is the network targeting emissions
over the whole domain or for a specific region? And should existing
information be considered or not? Thus it is important that the method
of network design is able to account for these considerations.

The R code for the network design algorithm presented in this paper is available from Zenodo: https://doi.org/10.5281/zenodo.7070622 (Thompson and Pisso, 2022).

The CH_{4} emissions data for anthropogenic sources are available from the EDGAR website (http://data.europa.eu/89h/488dc3de-f072-4810-ab83-47185158ce2a, last access: 12 January 2023; Crippa et al., 2019). The biomass burning sources are available from the GFED website (https://www.geo.vu.nl/~gwerf/GFED/GFED4/, last access: 12 January 2023; van der Werf et al., 2017). The wetland sources and soil sinks from the LPX-Bern model are available on request to Jurek Müller (jurek.mueller@unibe.ch), and the ocean sources are available from the website (https://doi.org/10.6084/m9.figshare.9034451.v1, Weber, 2019). The CO_{2} emissions data for anthropogenic sources are available from Zenodo (https://doi.org/10.5281/zenodo.3958283, Jones et al., 2020b). The NEE fluxes from the SiBCASA model are available on request to Ingrid Van der Laan (ingrid.vanderlaan@wur.nl), and the nuclear emissions estimates of ^{14}CO_{2} are available on request to Elise Potier (elise.potier@lsce.ipsl.fr).

The supplement related to this article is available online at: https://doi.org/10.5194/amt-16-235-2023-supplement.

RLT developed the algorithm, wrote the code, and carried out the examples. IP contributed to the modelling of Δ^{14}CO_{2} and to the algorithm, performed the algorithm complexity analysis, and provided feedback on the paper.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the European Commission, Horizon 2020 Framework
Programme (VERIFY, grant no. 776810). We would like to acknowledge Jurek Müller for preparing the LPX–Bern simulations of wetland CH_{4} fluxes, as well as Elise Potier and Yilong Wang for providing the estimates of nuclear emissions of ^{14}CO_{2}.

This research has been supported by the European Commission, Horizon 2020 Framework Programme (VERIFY, grant no. 776810).

This paper was edited by Thomas Röckmann and reviewed by Peter Rayner and one anonymous referee.

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