This study presents an optimization methodology for reducing the size of an existing monitoring network of the sensors measuring polluting substances in an urban-like environment in order to estimate an unknown emission source. The methodology is presented by coupling the simulated annealing (SA) algorithm with the renormalization inversion technique and the computational fluid dynamics (CFD) modeling approach. This study presents an application of the renormalization data-assimilation theory for optimally reducing the size of an existing monitoring network in an urban-like environment. The performance of the obtained reduced optimal sensor networks is analyzed by reconstructing the unknown continuous point emission using the concentration measurements from the sensors in that optimized network. This approach is successfully applied and validated with 20 trials of the Mock Urban Setting Test (MUST) tracer field experiment in an urban-like environment. The main results consist of reducing the size of a fixed network of 40 sensors deployed in the MUST experiment. The optimal networks in the MUST urban region are determined, which makes it possible to reduce the size of the original network (40 sensors) to

In the case of an accidental or deliberate release of a hazardous contaminant in densely populated urban or industrial regions, it is important to accurately retrieve the location and the intensity of that unknown emission source for risk assessment, emergency response, and mitigation strategies by the concerned authorities. This retrieval of an unknown source in various source reconstruction methodologies is completely dependent on the contaminant's concentrations detected by some pre-deployed sensors in the affected area or a nearby region. However, the pre-deployment of a limited number of sensors in that region required an optimal strategy for the establishment of an optimized monitoring network to achieve maximum a priori information regarding the state of emission. It is also required to correctly capture the data while extracting and utilizing information from a limited and noisy set of concentration measurements. Establishing optimal monitoring networks for the characterization of unknown emission sources in complex urban or industrial regions is a challenging problem.

The problem of monitoring network optimization is complex and may consist of a first deployment of the sensors, updating an existing network, reducing the size of an existing network, or increasing the size of an existing network. These problems are independent and each one has its own requirements. The degree of complexity also depends on (i) the network type (mobile network deployed only in an emergency, permanent mobile network, permanent static network), (ii) the scale (local, regional, etc.), and (iii) the topography of the area of interest (flat terrain without obstacles, complex terrain, cities, urban, industrial regions, etc.). It is important to note that the optimization also depends on the objective of a network design, such as the reconstruction of an emitting source, analysis of the air quality, and/or the triggering of an alert. This study is focused on reducing the size of an existing network at a local scale in an urban-like terrain for source reconstruction.

This study presents an optimization methodology for reducing the size of an existing monitoring network in a geometrically complex urban environment. The measurements from a reduced optimal network can be used for the source term estimation (STE) of an unknown source in an urban region with almost the same level of source detection ability as the original network of a larger number of samplers. The establishment of an optimal network required sensor concentration measurements, along with the availability of meteorological data, an atmospheric dispersion model, the choice of an STE procedure, and an optimization algorithm. These types of networks can have great applications in the oil and gas industries for the estimation of emissions of greenhouse gases (GHGs) like methane. In order to utilize an inversion method to estimate methane emissions, accurate measurements of methane in a network of high-precision sensors downwind of a possible source are a prerequisite. However, these sensors may not be deployed in large numbers due to their high cost. Alternatively, low-cost sensors (which may not be as high precision) can rapidly be deployed specifically for collecting the initial measurements. Using these less accurate measurements and the proposed optimization methodology, a reduced optimal network can quickly be designed to provide the “best” positions for the deployment of high-precision sensors to obtain accurate methane measurements. These high-precision measurements can be utilized in an inversion method to estimate accurate methane emissions. A similar and very useful application of the method proposed here can be applied for the estimation of methane emissions from landfills.

In this study, a simulated annealing (SA) stochastic optimization algorithm

The reduced optimal networks are validated using an STE technique to estimate the unknown parameters of a continuous point source. The STE problem for atmospheric dispersion events has been an important topic of much consideration as reviewed in

This study deals with a case of optimally reducing the size of an existing monitoring network. For this purpose, a predefined network of sensors deployed in an area of interest is considered to determine an optimized network with a smaller number of sensors but with comparable information. This work explores two requirements of the optimal networks that modify the spatial configuration of an existing network by moving the sensors and also reducing the number of sensors of an existing large network. In real situations, this methodology can be applied for the optimization of mobile networks deployed in an emergency situation. The methodological approach to optimally reduce the size of an existing monitoring network in an urban environment is presented by coupling the SA stochastic algorithm with the renormalization inversion technique and the CFD modeling approach. The concentration measurements obtained from the optimally reduced sensor networks in 20 trials of the Mock Urban Setting Test (MUST) field tracer experiment are utilized to validate the methodology by estimating an unknown continuous point source in an urban-like environment.

In the context of an inversion approach, source parameters are often determined using concentration measurements at the sensor locations and a source–receptor relationship.
The release is considered continuous from a point source located at the ground or at a horizontal plane corresponding to an altitude of a known source height.
Since the optimization methodology presented in the next section utilizes some concepts from the renormalization inversion methodology

A source–receptor relationship is an important concept in the source reconstruction process and it can be linear or nonlinear. This study deals with the linear relationship because except for nonlinear chemical reactions, most of the other processes occurring during the atmospheric transport of trace substances are linear: advection, diffusion, convective mixing, dry and wet deposition, and radioactive decay

For a given set of concentration measurements

This process involves a weight function

The weight function in the above-discussed renormalization process is computed by using an iterative algorithm proposed by

A predefined large network of

A cost function is defined (based on renormalization theory) as a function that minimizes the quadratic distance between the observed and simulated measurements according to the

The problem of optimization of a network is solved using a simulated annealing algorithm.
The SA optimization algorithm is utilized here for the determination of the optimal networks by comparing its performance with the genetic algorithm (GA)

SA is a random optimization technique based on an analogy with thermodynamics. The technique was introduced to computational physics over 60 years ago in the classic paper by

For SA, each network is considered as a state of a virtual physical system, and the objective function is interpreted as the internal energy of this system in a given state. According to statistical thermodynamics, the probability of a physical system to be in the same state follows the Boltzmann distribution and depends on its internal energy and the temperature level. By analogy, the physical quantities (temperature, energy, etc.) become pseudo-quantities. And during the minimization process, the probabilistic treatment consists of accepting a new network selected in the neighborhood of the current network following the same Boltzmann distribution and depending on both the cost difference between the new and current
networks and on the pseudo-temperature (simply called temperature). To find the solution, SA incorporates the temperature into a minimization procedure. So at high temperature (i.e., starting temperature), the space of solution is widely explored, while at lower temperature the exploration is restricted. The algorithm is stopped when the cold temperature is reached. It is necessary to choose the law of decreasing temperature, called the cooling schedule. Different approaches to parameterize SA are explored in

This step involves

Given a sensor location

set the

set rows of matrix

determine

compute the source term

compute the cost function

If

When

In this step,

If the maximum number of iterations of a bearing

Temperature is cooled using the cooling schedule and the iteration variable is reset to zero.

The cold temperature

At this step, the last best network

In stochastic optimization algorithms, especially in SA, it was observed that there is no guarantee for the convergence of the algorithm with such a strong cooling

The MUST field experiment was conducted by the Defense Threat Reduction Agency (DTRA) in 2001. It aimed to help develop and validate numerical models for flow and dispersion in an idealized urban environment. The experimental design and observations are described in detail in

The meteorological values (wind speed

A schematic diagram of the MUST geometry showing 120 containers and source (stars) and receptor (black filled circles) locations. In a given trial only one source was operational.

Flow diagram of the simulated annealing procedures to determine an optimized monitoring network.

The flow field in atmospheric dispersion models in geometrically complex urban or industrial environments cannot be considered homogeneous throughout the computational domain. This is because the buildings and other structures in that region influence and divert the flow into unexpected directions. Consequently, the dispersion of a pollutant and computations of the adjoint functions (retroplumes) are affected by the flow field induced by these structures in an urban region.
Recently,

The simulation results with fluidyn-PANACHE in each MUST trial were obtained with inflow boundary conditions from vertical profiles of the wind

In order to compute the retroplumes in each MUST trial, first the CFD simulations were performed to compute the converged flow field in the computational domain. Second, the flow field is reversed and used in the standard advection–diffusion equation to compute the adjoint functions

The calculations were performed by coupling the SA algorithm to a deterministic renormalization inversion algorithm and the CFD adjoint fields to optimally reduce the size of an existing monitoring network in an urban-like environment of the MUST field experiment.
The network optimization process consists of finding the best set of sensors that leads to the lowest cost function.
In this study, the validation is realized following two separate steps. The first step consists of forming two optimal monitoring networks by using the presented optimization methodology, which makes it possible to reduce the size of an original network of 40 sensors to approximately one-third (13 sensors) and one-fourth (10 sensors). The second step consists of comparing the a posteriori performance of the obtained reduced-size optimal networks with the MUST original network of 40 sensors at 1.6 m above the ground surface.
In first step, a comparison (based on a cost function) with networks of the same size (e.g., 10 sensors) was implicitly performed during the optimization process. As the SA is an iterative algorithm, during the optimization process networks of the same size are compared at each iteration and the best one is retained. The networks have also been generated randomly like in

The size of the MUST predefined (original) network is 40 sensors, and the sizes of the optimized networks are fixed after performing a first optimization with the number of sensors from 4 to 16

The optimization calculations were performed using MATLAB on a computer with the configuration Intel^{®} Core^{™} i7-4790 CPU @ 3.60 GHz and 16 GB of RAM. The averaged computational time for the optimization of one 10-sensor network was

The optimal networks of 10

Figure

In order to analyze the performance of the optimal monitoring configurations of smaller sizes, source reconstructions were performed to estimate the unknown location and the intensity of a continuous point release. These source reconstruction results were obtained using the information from the optimal monitoring networks formed by 10 and 13 sensors in each MUST trial.
In this performance evaluation process, the retroplumes and the concentration measurements were utilized from the sensors corresponding to these optimal networks. The retroplumes were computed using CFD simulations, considering the dispersion in a complex terrain. The source reconstruction results from both the optimal monitoring networks were also compared with results computed from the initial MUST network formed by 40 sensors

Source estimation results from the different monitoring networks for each selected trial of the MUST field experiment.

Source estimation results from the different monitoring networks are shown in Table

For a given trial, the skeleton parameter represents the common sensors between two optimal networks of different sizes (with 10 and 13 sensors). These results show that the SA algorithm coupled with renormalization inversion theory and the CFD modeling approach succeeded in reducing the size of an existing larger network to estimate unknown emissions with similar accuracy in an urban environment.

Isopleths of the renormalized weight function

Figure

Statistics for the source reconstruction results from each monitoring network. Here,

From the distribution of the optimized sensors in the networks in Fig. 3 for trials 5, 11, and 19 as well as in Figs. S2.1 and S2.2 for all selected trials, it was noted that a larger number of sensors are close to the source position in the optimal networks in most of the trials.
The source reconstruction results from the optimal monitoring networks formed by 10 sensors have an averaged location error

For all 20 trials, the averaged location error

In some trials, it was also noted that the distance of an estimated source to a real source decreases with a decrease in sensor number and also increases with the number of sensors in some other cases. It may be because the information added by a new sensor was not necessarily beneficial. It is noticeable that in a particular meteorological condition (i.e., wind direction, wind speed, and atmospheric stability), some of the sensors in a network may have little contribution to the STE. So, increasing the number of sensors may not always provide the best estimation because with the addition of more sensors, we also add more model and measurement errors in the estimation process. These errors can affect the source estimation results in some trials. In some cases, it may also depend on the sensitiveness of the added sensor's position in an extended optimal network to the source estimation. It is also noted that for a monitoring network, not only the number of sensors but also the sensor distribution (or sensor position) affects the information captured from the network.

In fact, both optimal networks for each trial show a diversity of structures independently of the number of sensors considered. For this, the skeleton was used to analyze the heterogeneity of the structures of different optimal networks.
A skeleton with seven sensors is considered a strong common base for the networks. This is the case for trials 3, 6, 14, 15, and 20 (Table

Considering the networks of intermediate structures with skeletons varying from four to six sensors, for trials 2, 4, 5, 7, 8, 9, 12, 13, 16, 17, 18, and 19, no obvious trend is noticed. These results tend to show that for a given trial, one or more optimal networks can satisfy the conditions of a nearly overall optimum (to be minimized). The obtained optimal networks may have a more or less common structure (having a greater or lesser number of skeletons).

Moreover, uncertainties calculated for different network sizes do not show an obvious trend. Indeed, a general relationship between the number of samplers and the uncertainties is not obvious. One may notice that changing the size of the network (increasing or decreasing the number of sensors) can lead to the growth or diminution of the uncertainties in the source parameter estimations. As an example, for trial 7 uncertainties grow, while for trial 17 uncertainties diminish (Table

It should be noted that this study deals with the case of reducing the number of sensors in order to obtain an optimal network from an existing large network. This optimization was carried out under the constraints of an existing network of the original 40 sensors in the MUST field experiment.
If one compares the performances of the obtained optimal monitoring networks of smaller sizes with the initial (original) network of 40 sensors in the MUST environment, both optimal networks provide satisfactory estimations of unknown source parameters. The 40-sensor network gives an averaged location error of 14.62 m for all trials, and the release rate was estimated within a factor of 2 in 75 % of the trials. However, reducing the number of sensors to

It should also be noted that the optimization evaluation in this study is performed using the MUST set of measurements, and this makes it more likely that the resulting sensor configuration performs well in reconstructing the source (in other words, the same measurements should not be used for the optimization and for the reconstructions). However, this does not limit the application of the proposed methodology for some important practical applications like accurate emission estimations. In fact, this can be considered a limitation of the data used for this application because for a complete process of optimization and then evaluation one requires a sufficiently long set of measurements so that all the data can be divided into two parts, (i) one part for designing an optimal sensor network and (ii) another part for the evaluation of the designed optimal network. However, the durations of the releases in the MUST field experiment were not sufficiently long to divide all the data from a test release separately into two parts for designing the optimal sensor network and then its evaluation. In further evaluations of the resulting optimal sensor configuration, a different set of concentration measurements can be constructed by adding some noise to the measurements. For a continuous release in steady atmospheric conditions, the average value of the steady concentration in a test release is not expected to deviate drastically from the mean values in each segment of the complete data. So this new set of concentration measurements with added noise can partially fulfill the purpose of evaluating a designed optimal network. As shown in Table 2, the errors in the estimated source parameters are small even with the new sets of concentration measurements constructed by adding 10 % Gaussian noise. This exercise shows that even if we have utilized a partially different set of the measurements for the evaluation of the optimal networks, the optimal networks have almost the same level of source detection ability in an urban-like environment. However, realistic data are required for further evaluation of the optimization methodology.

Although the MUST field experiment has been widely utilized for the validation of atmospheric dispersion models and inversion methodologies for unknown source reconstruction in an urban-like environment, its experimental domain was only approximately 200 m

This study describes an approach for optimally reducing the size of an existing monitoring network of sensors in a geometrically complex urban environment. It is a matter of reducing the size of networks while retaining the capabilities of estimating an unknown source in an urban region. Given an urban-like environment of the MUST field experiment, the renormalization inversion method was chosen for the source term estimation. It was coupled with the CFD model fluidyn-PANACHE for the generation of the adjoint fields. Combinatorial optimization by simulated annealing consisted of choosing a set of sensors that leads to an optimal monitoring network and allows for an accurate unknown source estimation. This study demonstrates how the renormalization inversion technique can be applied to optimally reducing the size of an existing large network of concentration samplers for quantifying a continuous point source in an urban-like environment with almost the same level of source detection ability as the original network with a larger number of samplers.

The numerical calculations were performed by coupling the simulated annealing stochastic algorithm to the renormalization inversion technique and the CFD modeling approach to optimally reduce the size of an existing monitoring network in urban-like environment of the MUST field experiment. The optimal networks were constructed to reduce the size of the original networks (40 sensors) to approximately one-third (13 sensors) and one-fourth (10 sensors). The 10- and 13-sensor optimal networks have estimated average location errors of 19.20 and 17.43 m, respectively, and have comparable source estimation performances with an averaged location error of 14.62 m from the original 40-sensor network. In 80 % of trials with optimal networks of 10 and 13 sensors, the emission rates are estimated within a factor of 2 of the actual release rates. These are also comparable to the performance of the original 40-sensor network, whereby in 75 % of the trials the releases were estimated within a factor of 2 of the actual release.

It was shown that in most of the MUST trials, the number of sensors in optimal networks slightly influences the location error of an estimated source, and this error tends to increase as the number of sensors decreases. In 20 MUST trials, an analysis of the networks formed by 10 and 13 sensors revealed the heterogeneity of their structures in an urban domain. It was observed that for some trials, optimal networks had a strong common structure. This tends to prove that a certain number of sensors have a primordial role in reconstructing an unknown source. It would reflect the fact that disjoint sets of sensors can lead to the best estimate of an unknown source in an urban region. This opens enormous prospects for assessing the relative importance of each sensor in a source reconstruction process in an urban environment. Defining a global optimal network for all meteorological conditions is a complex problem, but it is of greater importance that one may realize. This challenge consists of defining an optimal static network able to reconstruct the sources in all varied meteorological conditions. This information can be of great importance to determine an optimal monitoring network by reducing the number of sensors for the characterization of unknown emissions in complex urban or industrial environments.

The authors received access to the MUST field experiment dataset from Marcel König of the Leibniz Institute for Tropospheric Research. The MUST database was officially available from the Defense Threat Reduction Agency (DTRA). Code developed and utilized for this work is accessible from

Following

A cost function is defined (based on the renormalization theory) as a function that minimizes the quadratic distance between the observed and the simulated measurements according to the

When considering a point source,

The supplement related to this article is available online at:

HK and PN conceived the idea of the optimization of the sensor networks and developed the algorithm. PK conducted the CFD calculations for the MUST field experiment and analyzed the results to utilize in the optimization algorithm. HK, PN, PK, and AAF analyzed the results and prepared the paper with contributions from NB. All authors reviewed the paper.

The authors declare that they have no conflict of interest.

The authors would like to thank Marcel König of the Leibniz Institute for Tropospheric Research and the Defense Threat Reduction Agency (DTRA) for providing access to the MUST field experiment dataset. The authors gratefully acknowledge Fluidyn France for use of the CFD model fluidyn-PANACHE. We also thank Claude Souprayen from Fluidyn France for useful discussions. Finally, we thank the reviewers Janusz Pudykiewicz, George Efthimiou, two anonymous reviewer, and the topical editor Ignacio Pisso for their detailed and technical comments that helped to improve this study.

This paper was edited by Ignacio Pisso and reviewed by George Efthimiou, Janusz Pudykiewicz, and two anonymous referees.