Sensitivity Operator Framework for Analyzing Heterogeneous Air Quality Monitoring Systems
Abstract
:1. Introduction
2. Materials and Methods
2.1. Chemical Transport Model
 We suppose that only a given set of species ${L}_{src}$ is emitted. For the rest of species ${r}_{l}(\mathbf{x},t)=0$, $l\notin {L}_{src}$.
 The emission sources are supposed to be constant in time (${r}_{l}(\mathbf{x},t)={r}_{l}\left(\mathbf{x}\right)$).
 We do not require the emission sources to be positive since variables unconsidered in the model, chemical transformations, various land types, and meteorological conditions, such as rains and snowfalls, can act as sinks for the specific chemicals.
 In the Direct problem, $\mathbf{v}$ and $\mathbf{q}\in Q$ are given, and we find $\mathbf{\phi}$ from (1)–(4). The solution of the direct problem is denoted by $\mathbf{\phi}\left[\mathbf{q}\right]$. Let there be an “exact” uncertainty function value ${\mathbf{q}}^{(*)}={\mathbf{r}}^{(*)}$ to be found and ${\mathbf{\phi}}^{(*)}=\phi \left[{\mathbf{q}}^{(*)}\right]$ be the corresponding solution of the direct problem with the source function ${\mathbf{r}}^{(*)}$.
 In the Inverse problem, the uncertainty function ${\mathbf{q}}^{(*)}$ has to be identified from the partial information (“measurement data”) about ${\mathbf{\phi}}^{(*)}$, described in Section 2.3.
2.2. SensitivityOperator Based Representation of Measurement Data
2.3. Measurement Data Types
 “Timeseries”: ${N}_{Timeseries}$ time series of concentrations of the specific species in the specific points. In the state function terms:$$\left\{{\phi}_{{l}^{\left(m\right)}}({x}^{\left(m\right)},t),t\in \left[0,T\right],\left({x}^{\left(m\right)},{l}^{\left(m\right)}\right)\in {\left(\Omega \times L\right)}_{meas},m=1,\dots ,{N}_{Timeseries}\right\}.$$Projection system:$${h}^{\left(\xi \right)}=C(T,{\theta}^{\left(\xi \right)},t)\delta (x{x}^{\left(\xi \right)})\delta (l{l}^{\left(\xi \right)}),\phantom{\rule{1.em}{0ex}}\xi =1,\dots ,{\Xi}_{Timeseries}.$$For any element of $\left({x}^{\left(m\right)},{l}^{\left(m\right)}\right)$, the parameter $\theta $ ranges from 0 to ${\Theta}_{Timeseries}1$. The number of the frequencies ${\Theta}_{Timeseries}$ is the parameter of the projection system. This parameter is responsible for the temporal resolution of the considered data. Hence the total number of projection functions corresponding to the Timeseries is$${\Xi}_{Timeseries}={\Theta}_{Timeseries}\times {N}_{Timeseries}.$$
 “Pointwise”: ${N}_{Pointwise}$ Pointwise concentration measurements of the specific species at specific moments and specific points. In the state function terms:$$\left\{{\phi}_{{l}^{\left(m\right)}}({x}^{\left(m\right)},{t}^{\left(m\right)}),\left({x}^{\left(m\right)},{t}^{\left(m\right)},{l}^{\left(m\right)}\right)\in {\left({\Omega}_{T}\times L\right)}_{meas},m=1,\dots ,{N}_{Pointwise}\right\}.$$Projection system:$${h}^{\left(\xi \right)}=\delta (x{x}^{\left(\xi \right)})\delta (t{t}^{\left(\xi \right)})\delta (l{l}^{\left(\xi \right)}),\phantom{\rule{1.em}{0ex}}\xi =1,\dots ,{\Xi}_{Pointwise}.$$The projection system is naturally defined by the measurement points. Hence the total number of the projection functions is ${\Xi}_{Pointwise}={N}_{Pointwise}$. In the case of a large number of points, these data can be aggregated.
 “Integral”: ${N}_{Integral}$ Integrals of concentrations over the time interval of the specific species in the specific points. In the state function terms:$$\left\{{\int}_{0}^{T}{\phi}_{{l}^{\left(m\right)}}({x}^{\left(m\right)},t)dt,\left({x}^{\left(m\right)},{l}^{\left(m\right)}\right)\in {\left(\Omega \times L\right)}_{meas},m=1,\dots ,{N}_{Integral}\right\}.$$Projection system:$${h}^{\left(\xi \right)}=\delta (x{x}^{\left(\xi \right)})\delta (l{l}^{\left(\xi \right)}),\phantom{\rule{1.em}{0ex}}\xi =1,\dots ,{\Xi}_{Integral}.$$Here ${\Xi}_{Integral}={N}_{Integral}$. Integral measurements are equivalent to “Timeseries” measurements with ${\theta}^{\left(\xi \right)}=0$.
 “Snapshot”: ${N}_{Snapshot}$ specific species concentration fields images at specific moments in time. In the state function terms:$$\left\{{\phi}_{{l}^{\left(m\right)}}(x,{t}^{\left(m\right)}),x\in \Omega ,\left({t}^{\left(m\right)},{l}^{\left(m\right)}\right)\in {\left(\left[0,T\right]\times L\right)}_{meas},m=1,\dots ,{N}_{Snapshot}\right\}.$$Projection system:$${h}^{\left(\xi \right)}=C(X,{\theta}_{x}^{\left(\xi \right)},x)C(Y,{\theta}_{y}^{\left(\xi \right)},y)\delta (t{t}^{\left(\xi \right)})\delta (l{l}^{\left(\xi \right)}),\phantom{\rule{1.em}{0ex}}\xi =1,\dots ,{\Xi}_{Snapshot}.$$The projection system has two parameters: ${\Theta}_{Snapshot}^{\left(x\right)}$ and ${\Theta}_{Snapshot}^{\left(y\right)}$, which define the spatial resolution of the considered data. For any image, ${\theta}_{x}$ and ${\theta}_{y}$ range in $0,\dots ,{\Theta}_{Snapshot}^{\left(x\right)}1$ and $0,\dots ,{\Theta}_{Snapshot}^{\left(y\right)}1$, respectively. Hence ${\Xi}_{Snapshot}=$${N}_{Snapshot}\times {\Theta}_{Snapshot}^{\left(x\right)}\times {\Theta}_{Snapshot}^{\left(y\right)}$.
2.4. SensitivityOperatorBased Analysis of Measurement System
2.4.1. Inverse Problem Solution
 (1)
 The “exact” solution ${\mathbf{q}}^{(*)}$ is given. In our case, this is the location and capacity of the emission sources.
 (2)
 The “exact” solution ${\mathbf{q}}^{(*)}$ is then used to simulate the “measurement data“. This “measurement data” is used in the algorithm to solve the inverse problem.
 (3)
 The result ${\mathbf{q}}^{(\infty )}$ of the algorithm is compared with the “exact” solution. In this case, both the reconstruction of the source is estimated, and the convergence parameters of the algorithm are analyzed.
2.4.2. Sensitivity Operator Properties Analysis
2.5. Inverse Modeling Scenario
 Geographical domain.
 Monitoring system characteristics: locations and accuracy.
 Main emission sources to construct the “exact” solution.
 Chemical transformation mechanism, initial, and boundary conditions.
 Meteorological conditions, determining CTM model coefficients.
3. Results
3.1. Heterogeneous Measurements
3.2. Specific Measurement Types
3.3. Accuracy of the Reconstruction’s Prediction
 Red: Section 3.1, “realistic” source case;
 Green: Section 3.1, “single” source case;
 Blue: Section 3.1, “unified” source case;
 Black: Section 3.2, Timeseries experiment;
 Cyan: Section 3.2, Pointwise experiment;
 Magenta: Section 3.2, Snapshot experiment.
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
BNT  Baikal Natural Territory 
SVD  Singular value decomposition 
COSMO  Consortium for Smallscale Modeling 
CTM  Chemical transport model 
Appendix A. Newton–KantorovichType Algorithm
Algorithm A1 Newton–Kantorovichtype Algorithm 

Appendix B. Projection Equivalence
Appendix C. Chemical Transformation Model (Leighton RelationshipBased)
Appendix D. The Description of the Meteorological Scenario
 23.07 Rain zone in the foothills of the Altai, in the Kuzbass. The cold front from the west offset to the east. There is practically no leading stream. Weak variable wind in the west of Lake Baikal.
 24.07 Rain zone in the foothills of AltaiSayan (Khakassia), Western Sayan (Daily precipitation HMS 29698 Nizhneudinsk57mm). With the approach of a cold front from the west, the wind is mainly southeasterly.
 25.07 The rain zone encircles the Western Sayans from the north. Cold front, offset to the east, the wind weakens and changes direction to mainly western.
 26.07 The cold front approaches the Hangar from the west. Baikal, in the warm sector orographically cut off in the south (baric depression, thunderstorms south and north of Lake Baikal).
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Type  ${\Xi}_{\mathbf{Type}}$  ${\mathit{N}}_{\mathbf{Type}}$  Description 

Pointwise  60  60  $5\phantom{\rule{4.pt}{0ex}}\mathrm{sites}\times 12\phantom{\rule{4.pt}{0ex}}\mathrm{measurement}\phantom{\rule{4.pt}{0ex}}\mathrm{moments}$ 
Timeseries  60  6  $6\phantom{\rule{4.pt}{0ex}}\mathrm{sites}\times 10,\phantom{\rule{1.em}{0ex}}{\Theta}_{Timeseries}=10$ 
Integral  5  5  $5\phantom{\rule{4.pt}{0ex}}\mathrm{sites}$ 
Snapshot  625  1  $1\phantom{\rule{4.pt}{0ex}}\mathrm{image}\times 25\times 25,\phantom{\rule{1.em}{0ex}}{\Theta}_{Snapshot}^{\left(x\right)}={\Theta}_{Snapshot}^{\left(y\right)}=25$ 
Composite  750  Sum of the above 
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Penenko, A.; Penenko, V.; Tsvetova, E.; Gochakov, A.; Pyanova, E.; Konopleva, V. Sensitivity Operator Framework for Analyzing Heterogeneous Air Quality Monitoring Systems. Atmosphere 2021, 12, 1697. https://doi.org/10.3390/atmos12121697
Penenko A, Penenko V, Tsvetova E, Gochakov A, Pyanova E, Konopleva V. Sensitivity Operator Framework for Analyzing Heterogeneous Air Quality Monitoring Systems. Atmosphere. 2021; 12(12):1697. https://doi.org/10.3390/atmos12121697
Chicago/Turabian StylePenenko, Alexey, Vladimir Penenko, Elena Tsvetova, Alexander Gochakov, Elza Pyanova, and Viktoriia Konopleva. 2021. "Sensitivity Operator Framework for Analyzing Heterogeneous Air Quality Monitoring Systems" Atmosphere 12, no. 12: 1697. https://doi.org/10.3390/atmos12121697