the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A Flexible Algorithm for Network Design Based on Information Theory
- NILU – Norsk Institut for Luftforskning
- NILU – Norsk Institut for Luftforskning
Abstract. 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, therefore, is computationally more efficient than methods that require this. The algorithm is demonstrated in two examples, the first looks at designing a network for monitoring CH4 sources using observations of the stable carbon isotope ratio in CH4 (δ13C), and the second looks at designing a network for monitoring fossil fuel emissions of CO2 using observations of the radiocarbon isotope ratio in CO2 (∆14CO2).
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Rona Thompson and Ignacio Pisso
Status: final response (author comments only)
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RC1: 'Comment on egusphere-2022-213', Peter Rayner, 11 Jun 2022
This paper presents a metric for the evaluation of observing networks.
Previous studies have used some scalar function of the posterior
covariance. Most commonly this is also the uncertainty in some linear
functional of the posterior estimate (such as the total flux over a
given region) but metrics like the trace of the posterior covariance
are also used. The metric proposed by this paper is the information
content defined as the difference in entropy between prior and
posterior. Under the common linear Gaussian assumption this turns out
to be closely related to the log of the generalised variance or
determinant of the posterior covariance.The authors develop the necessary mathematics for their metric then
present two examples, adding isotope measurements for methane or
radiocarbon measurements for estimating fossil fuel CO$_2$ emissions.
The mathematics is presented well and the examples are clear and
pertinent. Furthermore the paper clearly lies within scope for the
journal.My concern with the paper is its lack of evaluation of the metric
itself. There is a comparison with the independence metric but not
with the covariance metrics. It is not clear to me that the
information theory metric does the same job as the covariance metric
or a better or worse job. There are two related problems:First the generalised variance is one metric and probably not a very
flexible one. It must include all potential sources. Normally this is
not what we want. We have some target quantity like national emissions
for which we are designing the network. Normally our target quantity
will be a subset of pixels (e.g.\ pixels in one country). I'm not sure
we can easily calculate the determinant of a submatrix of the Hessian.
Next there is the choice of uncertainty quantity to minimise.
\cite{rayner96} pointed out that the preferred network depended on
details of this quantity, such as the total ocean flux vs the average
uncertainty for each ocean basin. The determinant is the volume of the
hyperellipse described by the posterior covariance. It might be a good
general choice but is likely to obscure these differences.Finally I think the computational advantages of the new metric need a
bit more justification. The authors claim that the covariance metric
requires the inversion of a large matrix. Depending on the uncertainty
metric we choose to minimise this might not be true. In general our
target quantity is a linear functional of the posterior sources.
Examples include the sum over some subregion and average over time.
From what I learned to call Riesz's Representation Theorem (though
there seem to be several of these) for any linear functional $f$ on
$R^n$ there is a vector $\mathbf{v}$ such that $f(\mathbf{x}) =
\mathbf{v} \cdot \mathbf{x}$ for all $\mathbf{x} \in R^n$. Thus for
any target quantity $t$ we can find some vector $\mathbf{v}_t$ such
that $t^{\mathrm{a}} = \mathbf{v}_t \cdot \mathbf{x}^{\mathrm{a}}$.
The superscript $\mathrm{a}$ refers to the analysis or posterior. An
example $\mathbf{v}_t$ contains $1$ for every pixel in a region and
$0$ otherwise. This will sum over the region of interest. By the
Jacobian law of probabilities the uncertainty in $t$ is given by
$\mathbf{v}_t^\mathrm{t} \cdot \mathbf{A} \cdot \mathbf{v}_t$ where
the superscript $\mathrm{T}$ denotes transpose and $\mathbf{A}$ is the
posterior or analysis covariance. $\mathbf{A}$ is the inverse of the
Hessian $\mathbf{G}$ so we need to calculate
$\mathbf{v}_t^{\mathrm{T}} \cdot \mathbf{G}^{-1} \cdot \mathbf{V}_t$.
I believe this calculation can be efficiently accomplished by the
Cholesky decomposition of $\mathbf{G}$. If we write $\mathbf{G} =
\mathbf{L}\mathbf{L}^{\mathrm{T}}$ (Cholesky decomposition) then I
believe $\mathbf{G}^{-1} = \mathbf{L}^{-1,\mathrm{T}}
\mathbf{L}^{-1}$. Substituting this we see $t =\mathbf{y}^\mathrm{T}
\cdot \mathbf{y}$ where $\mathbf{y} = \mathbf{L}^{-1} \cdot
\mathbf{v}_t$. Thus I think the target uncertainty can be performed
with a Cholesky decomposition, a matrix-vector product and a
dot-product. This may even be less costly than the determinant via
the Cholesky decomposition.I may just as easily be wrong here but think the comparison of the
cost and generality of the new metric cf the existing uncertainty
metric does need more consideration than it gets here.I only have two specific comments on the paper:
\begin{description}
\item[L45] When citing early literature it is probably fair to cite
the paper that gave rise to the field, \cite{hardt94}.
\item[L65] Summing over the submatrix does indeed account for the
covariance of uncertainty but that isn't it's most important
property. This is that it calculates the uncertainty on the summed
regional flux rather than the individual pixels.
\end{description}
\bibliographystyle{copernicus}
\bibliography{refs} -
RC2: 'Comment on egusphere-2022-213', Anonymous Referee #2, 17 Jun 2022
Overview:
The manuscript “A Flexible Algorithm for Network Design Based on Information Theory” by Thompson and Pisso describes the development of a novel method for optimising the distribution of a measurement network, with the aim of maximising the information content provided by these measurements for a flux inversion. Previous methods, usually based on quantifying the posterior uncertainty of the inversion, were computationally expensive but the metric presented here should be more efficient. The new method is applied to improving the current European measurement network for CH4 and CO2 through inclusion of isotopic measurements at a subset of locations. The paper is well-written and presented, with thorough explanation of the methodology and clear figures. The new method appears to provide a justifiable technique for network design.
My only significant comment is that the discussion of the results of the new method in the context of previous methods is very brief. The results are compared to those using the clustering-based selection discussed earlier in the text, which is based on discounting sites with similar observed signals. However, there is no comparison of the merits of the new method compared to those based on posterior uncertainty. Whilst, for computational reasons, I understand that the authors might not want to explicitly perform such an analysis for direct comparison, I do think that there needs to be some further discussion of the potential differences, advantages and disadvantages of the new method compared to the the full range of alternative methods.
If the last sections are expanded to include such discussion, I am happy to recommend this manuscript for publication in this journal.
Minor/technical comments:
line 30: brackets around reference year
line 70: slightly unclear. heterogeneity in terms of flux?
figure 1: It would be good to also mark the locations of the sites that were not selected by the algorithm in Figure 1. I appreciate that they are shown in a later figure, but it is easiest for the reader, and would aid comprehension of Fig. 1, if they are noted earlier than later.
Figure 5: Is it possible to say anything in the main text concerning why the two sites located very close to each other in France might have been selected using this method?
Rona Thompson and Ignacio Pisso
Rona Thompson and Ignacio Pisso
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