An Evaluation of the Sensitivity of a Source Term Estimation Methodology of Sensor Configuration in an Urban-like Environment

Abstract

Identifying unknown sources of air pollutants is vital for protecting public health, especially in cases involving the emission of toxic substances. The efficiency of this process depends highly on the accuracy of Source Term Estimation (STE) methods and the availability of robust measurements. Therefore, it is important to examine how sensor network characteristics affect STE accuracy. This study investigates the impact of different sensor configurations on STE results for a stationary point source in a complex, urban-like environment. The STE methodology employs the Metropolis–Hastings Markov Chain Monte Carlo (MCMC) algorithm alongside numerical simulations of a Computational Fluid Dynamics (CFD) model. The STE algorithm is applied across several sensor configurations in three distinct release scenarios and real sensor observations from the Michelstadt wind tunnel experiment, assessing both the number of sensors used and the agreement between measured and modeled concentrations. In general, the results indicate that increasing the number of sensors and the model’s accuracy improves the source parameters estimations. However, there is a specific number of sensors in each release scenario where STE outcomes from randomly selected, high-accuracy, and low-accuracy sensors converge to similar solutions. Overall, the findings provide valuable information for designing sensor configurations in urban areas.

1. Introduction

Air pollution is one of the major environmental stressors, posing serious health risks to populations [1]. Policymaking for air pollution control relies on information from various tools, including observations from in situ measurement networks and predictions from Atmospheric Transport and Dispersion Models (ATDM). The development and application of methods and techniques that combine data from modeled and observed air pollutant concentrations can significantly enhance air pollution control strategies [2,3]. One of the most critical situations involving air pollutants is the release of highly toxic airborne agents or gases into the atmosphere, which can lead to serious and profoundly negative impacts on public health and the environment. Incidents involving nuclear and chemical accidents, as well as terrorist activities, underscore the urgent need for prompt intervention by authorities, particularly in densely populated urban regions. When characteristics of the source, such as its location or release rate, are unknown, Source Term Estimation (STE) methods can offer precise assessments through the integration of atmospheric transport and dispersion models, inverse algorithms, and observational data.

An STE from an unidentified source is a sub-category of inverse problems [4]. The predominant techniques developed in prior studies concerning STE applications rely on in situ concentration measurements collected by corresponding static sensor networks. In such scenarios, the observed concentrations are compared with those modeled using an ATDM. The dispersion of the contaminant can be calculated through both “forward” and “inverse” techniques. In the former approach, the concentration is derived by executing the ATDM multiple times, employing varying values for the source parameters to generate the source–receptor relationships. However, this technique incurs increased computational costs due to multiple iterations of the ATDM. Conversely, the inverse module adopts the adjoint approach, which resolves the adjoint advection and diffusion equations from each sensor location once, considering the sensor as a source to generate the source–receptor relationships. Pudykiewicz [5] first used the adjoint approach to estimate the location and strength of a nuclear contaminant source.

STE methods can also be used at the global [5], regional [6,7], and micro-scales [8,9]. In the urban environment, the interaction of wind with buildings and obstacle surfaces leads to strong turbulence phenomena, adding additional difficulties for achieving accurate STE. In such cases, both steady Reynolds-Averaged Navier Stokes (RANS) and transient Large Eddy Simulation (LES) approaches of Computational Fluid Dynamics (CFD) are utilized to calculate the wind flow and the dispersion of airborne contaminants within the area of interest. LES can provide accurate solutions; however, its high computational resources and cost make it impractical in many cases. Xue et al. [10] utilized the gradient diffusion hypothesis [11] to employ an averaged flow field calculated using LES simulation as a RANS-like field. Jia and Kikumoto [12] calculated the source–receptor relationships from LES simulations, with the results indicating the high accuracy of the method in estimating unknown source parameters but also highlighting the significantly higher computational resources required compared to RANS applications. On the other hand, the applicability of RANS models is the main reason this method is widely used in urban or urban-like domains for STE problems.

Various techniques and algorithms have been developed and investigated in STE applications. Hutchinson et al. [13] categorized these methods into two primary categories: optimization and probabilistic methods.

The optimization technique aims to minimize a cost function that correlates the modeled and measured concentrations [13]. Many different categories of optimization algorithms have been investigated in previous studies, such as least squares [14], Simulated Annealing (SA) [15], Genetic Algorithms (GAs) [16,17], and Particle Swarm Optimization (PSO) [18]. Wang et al. [19] applied a hybrid technique in an STE problem by combining PSO, GA, and SA. Efthimiou et al. [20] introduced a two-step algorithm that identifies the source location using a cost function based on the Pearson correlation coefficient and the release rate using quadratic ones.

The probabilistic approach utilizes algorithms based on Bayesian Inference. These algorithms accept Probability Density Functions (PDFs) as input and provide the final estimate of source parameters as PDFs rather than a single value [13]. Various Bayesian algorithms have been employed in source identification applications, including Markov Chain Monte Carlo (MCMC) [21], Sequential Monte Carlo [22], Differential Evolution Monte Carlo [23], and Adaptive Multiple Importance Sampling [24].

One crucial aspect influencing the efficiency of Bayesian applications is the sampling procedure, wherein the algorithm selects the combination of source parameters (proposal state) to assess its accuracy. Metropolis–Hastings [25,26] is among the most popular algorithms for this sampling procedure [21,27], while several other algorithms have been developed to enhance Metropolis–Hastings in STE applications [24].

The scarcity of datasets of concentration measurements in urban areas mapping the dispersion of pollutants due to accidents or malevolent actions poses a limitation to validating the developed models and methods. Thus, the majority of the developed STE models have been evaluated based on field experiments, wind tunnel experiments, or synthetic concentration datasets. Haupt et al. [16] validated the accuracy of a GA in an STE application based on synthetic data created using a Gaussian plume model. Ryan and Arisman [28] utilized the CFD forward simulation of contaminant dispersion to generate synthetic data for evaluating a model capable of estimating the strength and location of transient source terms.

Datasets from field experiments such as the Fusion Field Trial 2007 (FFT07) [29], the Joint Urban Experiment 2003 [30], and the Mock Urban Setting Test (MUST) [31,32] have been widely utilized to evaluate the developed STE methods and algorithms [33,34,35,36,37,38]. Similar datasets of wind flow variables and pollutant concentration measurements have been derived from wind tunnel experiments, including MUST [39], Michelstadt [40,41], and CUTE [42]. These datasets have been extensively used for validation purposes in CFD models concerning wind field, forward dispersion, and STE [20,43,44,45].

In general, the efficiency of STE techniques relies on three crucial factors: the accuracy of the ATDM, the selection of a suitable optimization or probabilistic algorithm, and the availability of reliable observational data provided by the sensor network. Most of the aforementioned studies have focused on improving ATDM or on the development and selection of the inverse algorithm. Since the quality of measured concentration data is highly important for STE accuracy [46], the impact of sensor networks on STE results must be carefully considered. Additionally, designing a sensor network requires balancing technical requirements with financial constraints. This is a challenging task since sensor costs can range widely depending on the sensor type, as evident in cases like the SmartAQnet network [47], where even a coordinated field deployment has incorporated various observation instruments at differing costs. Therefore, the influence of sensor configuration characteristics such as the number, positioning, and measurement accuracy of sensors are of high importance on STE results. In this direction, several STE applications have been conducted using Gaussian models [46,48]. Recently, some studies have explored the impact of sensor network configuration by utilizing CFD models, which are especially effective for complex urban environments.

Kovalets et al. [43] investigated the probability of obtaining accurate solutions in source parameter estimation for different numbers of sensors using the RANS approach in the MUST field experiment. The MUST field experiment dataset was used by Ngae et al. [49], who combined an adequate cost function based on the renormalization inversion method with CFD modeling to determine the optimal positions of the sensors. They found that reducing the number of sensors from 40 to 13 and even to 10 did not affect network performance. Jia and Kikumoto [50] estimated the optimal sensor network configuration by calculating the joint entropy of adjoint concentration distributions in a block-arrayed city model, using data from the LES model. Liu and Li [51] extended this method by incorporating adjoint concentration data for multiple wind directions in a complex geometry area.

All these studies provide valuable insights and analyses regarding sensor configurations in urban environments. However, some limitations remain, and there are research areas open for further investigation. Most of the previous studies are conducted in structured block arrays [43,49,50] rather than on more complex urban layouts. Additionally, many studies rely on synthetic observations [50,51] instead of real measurements. Furthermore, these studies primarily examine the effects of sensor quantity and positioning within each configuration, often without considering factors related to the alignment between observed and modeled concentration values.

The present study aims to investigate the impact of different sensor configurations on determining the location and release rate of an unknown stationary point source, associated with an emergent release, within the highly complex geometry of an urban environment. Specifically, the research focuses on the influence of varying numbers of sensors on STE outcomes. Additionally, it examines the effect of configurations with different levels of agreement between measured and modeled concentrations. For this purpose, datasets from three distinct release scenarios involving individual sources from the Michelstadt wind tunnel experiment were selected as the case study. The open-source CFD suite, OpenFOAM v2106, is utilized to calculate modeled concentrations by solving the adjoint advection-diffusion equation. Estimations of source location and release rate are obtained as PDFs using the well-established Metropolis–Hastings MCMC algorithm. To assess the accuracy of this study, the wind flow model, the forward dispersion model, and the STE results are evaluated against the Michelstadt experiment dataset and compared with corresponding findings from previous studies.

2. Methodology

The main assumption of this study is that the pollutant is emitted from a stationary point source under steady meteorological conditions. Therefore, the initial step of the methodology involves calculating the wind flow. To achieve this, a RANS simulation is performed to determine the three velocity components and the eddy viscosity fields within the spatial domain, which are used for solving the adjoint advection and diffusion equations. Following this, the source–receptor relationships are stored based on the calculated adjoint concentrations. The final step of the methodology is the application of the Bayesian inference algorithm, whose convergence provides the PDFs of the estimated parameters, including the source’s location coordinates and release rate. Since the wind direction remains constant across scenarios, once the source–receptor relationships are determined, the algorithm is directly applied to each sensor configuration scenario.

2.1. Bayesian Inference

In numerous previous studies, Bayesian inference algorithms are combined with CFD simulations to determine unknown source parameters. Based on Bayes’ theorem, these algorithms can estimate the posterior probability of source parameters by incorporating information about the prior knowledge of the source parameters, estimates of sensor measurement errors, and biases in the model calculations. Let $d$ represent the set of observations of air pollutant concentration levels from the sensor network and $\theta$ denote the source term (including source location coordinates and release rate). According to Bayes’ theorem, the posterior probability distribution $p\left(\theta |d\right)$ describes the probability of the source term $\theta$ given the measurements $d$. The posterior probability is calculated using Equation (1), which combines the likelihood function $p\left(d|\theta \right)$, representing the probability of the sensor measurements $d$ given the source term $\theta$, the prior probability $p\left(\theta \right)$, which includes any prior knowledge about the source term, and $p\left(d\right)$, known as the evidence, is a normalizing factor that is independent of $\theta$ and does not affect the relative probabilities.

(1)$p\left(\theta |d\right)={\displaystyle \frac{p\left(d|\theta \right)p\left(\theta \right)}{p\left(d\right)}}\propto p\left(d|\theta \right)p\left(\theta \right)$

The current study focuses on estimating the unknown source location and release rate. Therefore, the vector of the estimated parameters can be defined as $\theta =\left(x_s, q_s\right)$, where $x_s=\left(x, y, z\right)\in \mathsf{\Omega }$ represents the coordinates of the source location, which can be located anywhere within the space $\mathsf{\Omega }$ of the spatial domain, and $q_s$ denotes the release rate. Assuming that $C^{true}\in {ℝ}^N$ is the vector of the true concentrations at sensor locations, $d\in {ℝ}^N$ is the vector of concentrations observed by the sensors of the measurement network, and $C^c\left(\theta \right)\in {ℝ}^N$ is the vector of concentrations calculated using the ATDM using the source term $\theta$, and $N$ is the total number of sensors; the observations and calculated concentrations can be expressed by the relationships:

(2)$d=C^{true}+{\epsilon }^o$

(3)$C^c\left(\theta \right)=C^{true}+{\epsilon }^c$

Here, ${\epsilon }^o$ represents the vector of the measurement errors, while ${\epsilon }^c$ denotes the vector of model errors. Determining both measurement and model errors is inherently challenging [52]. Previous studies [10,12,21] have suggested that these errors can be modeled as normally distributed with zero mean and variances ${\sigma }_o^2$ and ${\sigma }_c^2$, respectively. Consequently, the measurement and model errors are characterized by the following probabilities:

(4)$p\left(d|C^{true}\right)\propto exp\left[-{\displaystyle \frac{1}{2}}{\displaystyle \sum }{\displaystyle \frac{{\left(d_i-{C^{true}}_i\right)}^2}{{\sigma }_{o,i}^2}}\right]$

(5)$p\left(C^{true}|\theta \right)\propto exp\left[-{\displaystyle \frac{1}{2}}{\displaystyle \sum }{\displaystyle \frac{{\left({C^{true}}_i-C^c{\left(\theta \right)}_i\right)}^2}{{\sigma }_{c,i}^2}}\right]$

where $i\in \left[1, N\right]$ is the index of each sensor. Finally, the likelihood function can be expressed using $p\left(d|C^{true}\right)$ and $p\left(C^{true}|\theta \right)$ probabilities:

(6)$p\left(d|\theta \right)\propto {{\displaystyle \int }}_{C^{true}}p\left(d|C^{true}\right)p\left(C^{true}|\theta \right)d{C}^{true}\propto exp\left[-{\displaystyle \frac{1}{2}}{\displaystyle \sum }{\displaystyle \frac{{\left({C^o}_i-C^c{\left(\theta \right)}_i\right)}^2}{{\sigma }_{o,i}^2+{\sigma }_{c,i}^2}}\right]$

The prior probability $p\left(\theta \right)$ represents the prior knowledge about the unknown source parameters $\theta =\left(x_s, q_s\right)$. In this study, we adopt the assumption from [21] that no prior information is available regarding the source location and release rate before the analysis begins. Consequently, the source could be located anywhere within the bounds of the computational domain, and the release rate is assumed to be positive and less than a meaningful maximum value. Therefore, the prior probability is assigned a uniform distribution at the computational domain and maintains a constant value for all possible sets of source parameters:

(7)$p\left(\theta \right)=constant$

Based on this assumption, the value of the posterior probability (Equation (1)) is directly correlated with the value of the corresponding likelihood such as:

(8)$p\left(\theta |d\right)\propto exp\left[-{\displaystyle \frac{1}{2}}{\displaystyle \sum }{\displaystyle \frac{{\left({C^o}_i-C^c{\left(\theta \right)}_i\right)}^2}{{\sigma }_{o,i}^2+{\sigma }_{c,i}^2}}\right]$

2.2. Source–Receptor Relationship

To calculate the probability in Equation (8), and considering that the observation vector is known, it is necessary to estimate the error variances ${\sigma }_o^2$ and ${\sigma }_c^2$, and to calculate the modeled concentrations $C^c{\left(\theta \right)}_i$ in each sensor position, $i$, for all the investigated sets of source parameters, $\theta =\left(x_s, q_s\right)$. For stationary passive non-reactive contaminant emissions, the modeled concentration is given by:

(9)$C^c{\left(x_s,q_s\right)}_i=q_s{C}^c{\left(x_s\right)}_i$

where $C^{\mathrm{c}}{\left(x_s\right)}_i$ represents the source–receptor relationship at every sensor location $i$. The source–receptor relationship expresses the sensitivity at a given measurement position to the source parameters. The source–receptor relationship can be calculated by the forward model based on the solution of the advection-diffusion equation:

(10)${\displaystyle \frac{\partial C_i^{\mathrm{c}}}{\partial t}}+u_j{\displaystyle \frac{\partial C_i^{\mathrm{c}}}{\partial x_j}}-{\displaystyle \frac{\partial }{\partial x_j}}D{\displaystyle \frac{\partial C_i^{\mathrm{c}}}{\partial x_j}}=q_s\delta \left(x_s\right)$

Here, $u_j=\left(U, V, W\right)$ is the vector of the three velocity components in cartesian coordinated $x_j=\left(x, y,z\right)$, $D$ is the turbulent diffusion coefficient, and $\delta (·)$ is the Dirac delta function. The diffusion coefficient can be expressed in terms of the molecular diffusivity, $D_m$, and the turbulent diffusivity, $D_t$:

(11)$D=D_m+D_t=D_m+{\displaystyle \frac{v_t}{S{c}_t}}$

where $S{c}_t$ is the turbulent Schmidt number and $v_t$ is the eddy viscosity.

Calculating the source–receptor relationship for every possible source term requires significant computational resources, especially when the source could be located anywhere within the domain, making this approach impractical. To address this issue, Pudykiewicz [5] introduced a method that calculates source–receptor relationships using adjoint advection-diffusion equations. This approach involves calculating the adjoint concentration $C^{*}$ for each sensor $i$ from the adjoint equation, setting a unitary source term, $p_i$, at the sensor locations:

(12)${\displaystyle \frac{\partial C_i^{*}}{\partial t}}-u_j{\displaystyle \frac{\partial C_i^{*}}{\partial x_j}}-{\displaystyle \frac{\partial }{\partial x_j}}D{\displaystyle \frac{\partial C_i^{*}}{\partial x_j}}=p_i$

The source–receptor relationships need to be calculated only once for each sensor in the measurement network, reducing the computational costs, based on the following equation:

(13)$C^c{\left(x_s,q_s\right)}_i=q_s{C}^{*}{\left(x_s\right)}_i$

A detailed description of adjoint advection-diffusion equations and the corresponding methodology can be found in previous studies [5,10,20,43,53].

2.3. Sampling Algorithm

Although calculating the posterior probability for every possible set of source parameters is feasible, it can be computationally expensive, especially when the parameter values are completely unknown. To address this challenge, various iterative methods are based on proposing new samples of parameter sets, evaluating their proposal probabilities, and either accepting or rejecting them, ultimately converging to an optimal solution. In this study, the Metropolis–Hastings MCMC algorithm is used for sampling and collecting the PDF samples. A detailed description of the algorithm can be found in [25,26].

2.4. Case Study Description

2.4.1. Michelstadt Wind Tunnel Experiment

The extension of the “Michelstadt” wind tunnel experiment [40,41] was performed at the “WOTAN” atmospheric boundary layer wind tunnel within the Environmental Wind Tunnel Laboratory at the Meteorological Institute of the University of Hamburg. Michelstadt represented the geometric features of a typical idealized Central European city district, including narrow streets, open spaces, and sharp building corners, at a 1:225 scale. During the experiment, both continuous and instantaneous releases of ethane tracer gas were carried out under neutral boundary layer conditions. Velocity component and pollutant concentration measurements were taken at multiple points within the geometry, resulting in a database useful for model evaluation and for applications related to urban wind flow and pollutant dispersion [45,54,55,56].

2.4.2. Release Scenarios

In this study, three continuous release scenarios from the Michelstadt experiment were used to validate the wind flow model, the forward dispersion model, and the STE algorithm. The three release cases, named “S2”, “S4”, and “S5”, were conducted under similar meteorological conditions. By scaling the characteristics of the experiment to full scale, the reference velocity was set to $U_{ref}=6 \mathrm{m}/\mathrm{s}$ at a reference height $z_{ref}=99.9 \mathrm{m}$. In all cases, the pollutant was emitted continuously from the ground level. As shown with red “x” symbols in Figure 1a–c, the source was located in an open space in the S2 case, in a street parallel to the flow direction in the S4 case, and in a street perpendicular to the flow direction in the S5 case. The total number of measurement sensors (blue dots in Figure 1a–c) was 52 in the S2 case, 25 in the S4 case, and 22 in the S5 case, with all sensors positioned at a height of 7.49 m in the full-scale setup.

2.5. Model and Method Evaluation

2.5.1. Wind Flow Evaluation

To assess the accuracy of the wind flow simulation, the modeled results are compared against measured velocity component data obtained from the wind tunnel experiment. Specifically, during the Michelstadt experiment, 2158 observations were recorded for horizontal velocity components ($U, V$) at various locations and heights, including street canyons and above-building rooftops. It should be noted that no observations were available for the vertical velocity component ($W$). In the evaluation process, both simulated and observed velocities are normalized using the reference velocity $U_{ref}$. The numerical results are evaluated against the corresponding experimental data both qualitatively, through scatter plots, and quantitatively, using the Hit Rate ($HR$) validation metric, following the recommendations of COST ES 732 [57]. The $HR$ metric that was used for wind field validation purposes in the Michelstadt case study in previous studies [45,56] is defined by the following relationship:

(14)$HR={\displaystyle \frac{{\sum }_{i=1}^N{n}_i}{N}}$

where N is the total number of observations, and $n_i$ is calculated from:

$n_i=\{\begin{array}{c}1, if {\displaystyle \frac{\left|M_i-O_i\right|}{\left|O_i\right|}}\le D or\\ 1, if \left|M_i-O_i\right|\le W\\ 0, else\end{array}$

Here, $M_i$ represents the modeled value, and $O_i$ is the observed value of the variable at the $i^{th}$ measurement point. The HR metric allows for a relative difference up to a threshold value $D$ or an absolute difference up to a threshold value $W$. According to the recommendations by Hertwig et al. [56], the $D$ value is set to 0.0165 for $U/U_{ref}$ and 0.0286 for $V/U_{ref}$ velocity components. The absolute difference threshold value $W$ is set to 0.25, based on VDI guidelines [58].

The acceptance criterion for the hit rate is set at $HR\ge 0.66$, following the VDI guidelines. The value of the $HR$ metric strongly depends on the values of the parameters $W$ and $D$. In this study, although the same threshold value for the relative difference is used as per the VDI guidelines, the threshold value for the absolute difference follows the recommendation of Hertwig et al. [56] and is much lower than the VDI value, set at 0.05 for all velocity components. Therefore, as noted in COST ES 732 Action [57], to evaluate the accuracy of the simulation, it is important to take into account the different values of $W$ in relation to the acceptance criteria limit ($HR\ge 0.66$ for VDI).

2.5.2. Forward Dispersion Model Evaluation

Following the wind field evaluation, and before applying the STE algorithm, the forward dispersion model was calculated via the advection and diffusion equation (Equation (12)) using the true location and release rate for the three release scenarios. For this purpose, the modified ScalarTransportFOAM solver described in Section 4 was utilized, employing the same simulation setup and boundary conditions as the adjoint equation numerical simulations. The forward dispersion model’s performance was assessed qualitatively through scatter plots and quantitatively using the following metrics, as recommended by Chang and Hanna [59].

• Fraction within a factor of two, $FAC2$:

(15)$FAC2={\displaystyle \frac{{\sum }_{i=1}^N{n}_i}{N}}$

where:

$n_i=\{\begin{array}{c}1, if 0.5\le {\displaystyle \frac{C_p}{C_o}}\le 2\\ 0, else\end{array}$

• Fraction within a factor of four, $FAC4$:

(16)$FAC4={\displaystyle \frac{{\sum }_{i=1}^N{n}_i}{N}}$

where:

$n_i=\{\begin{array}{c}1, if 0.25\le {\displaystyle \frac{C_p}{C_o}}\le 4\\ 0, else\end{array}$

• Normalized mean squared error, $NMSE$:

(17) $NMSE={\displaystyle \frac{\overline{{\left(C_o-C_p\right)}^2}}{\left(\overline{C_o}·\overline{C_p}\right)}}$

• Fractional bias, $FB$:

(18) $FB=2{\displaystyle \frac{\overline{C_o}-\overline{C_p}}{\overline{C_o}+\overline{C_p}}}$

• Geometric mean, $MG$:

(19) $MG=exp\left(\overline{ln{C}_o}-\overline{ln{C}_p}\right)$

• Geometric variance, $VG$:

(20)$VG=exp\left(\overline{{\left(ln{C}_o-ln{C}_p\right)}^2}\right)$

where $C_o$ is the measured wind tunnel concentration and $C_p$ is the modeled concentration.

Hanna et al. (2004) [60] state that each metric indicates different aspects when comparing modeled and observed concentrations. Linear-based metrics, such as $NMSE$ and $FB$, are significantly affected by extremely high modeled or measured concentrations. In contrast, logarithmic-based metrics ($MG$, $VG$) are less sensitive to these high extremes, making $MG$ and $VG$ more suitable for cases where predicted and observed concentrations differ by several orders of magnitude. The ideal model values are 1.0 for $FAC2$, $FAC4, MG$, and $VG$ metrics, and 0.0 for $NMSE$ and $FB$. The acceptance criteria for the validation metrics in local-scale urban environments, according to Hanna and Chang [61], are as follows:

$FAC2>0.3$

$NMSE<6$

$\left|FB\right|<0.67$

According to Hanna et al. [60], an $MG$ value of 0.5 or 2.0 suggests a mean bias of a factor of two, while an $MG$ value of 0.25 or 4.0 indicates a mean bias of a factor of four. The $VG$ metric measures the scatter in a log-normal distribution. For a scatter factor of 2, the $VG$ value would be 1.6, and for a scatter factor of 5, it would be 12.0, as detailed in [59]. In this study and following the notes from the COST ES1006 action [42], no specific acceptance criteria for $VG$ and $MG$ are established.

For the calculation of each metric, all pairs of available concentrations were used with one exception: for $MG$ and $VG$, pairs were excluded if either the observed or modeled concentration fell below a specific threshold. The chosen threshold value is set at 10⁻⁴ ppmv, as recommended by COST ES1006 [42].

2.5.3. STE Evaluation

Regarding the STE results, in order to quantify the quality of the unknown source parameters’ estimation, the horizontal distance ($R_H$), the vertical distance ($R_V$), and the release rate ratio ($\mathsf{\Delta }q$) between the true $(x_t, y_t, z_t,q_t)$ and the mean PDF values of the estimated parameters $(x_e, y_e, z_e,q_e)$ are calculated using the following equations:

(21)$R_H=\sqrt{{\left(x_e-x_t\right)}^2+{\left(y_e-y_t\right)}^2}$

(22)$R_V=\left|z_e-z_t\right|$

(23)$\mathsf{\Delta }q=\max \left[\left(q_e/q_t\right),\left(q_t/q_e\right)\right]$

The target values for the distances $R_H$ and $R_V$ are zero ($R_H=0$ m, $R_V=0$ m); for the release rate ratio $\mathsf{\Delta }q$, it is unity ($\mathsf{\Delta }q=1$).

2.6. Sensors Analysis

After assessing the STE results from the complete dataset (which includes observations from all available sensors in each release scenario), the Metropolis–Hastings MCMC is utilized for different sensor configurations. To examine how the number of sensors affects the STE solution, the configurations are organized into groups of five sensors. The first group contains five sensors, and each subsequent group adds five more sensors compared to the previous one, continuing this pattern until the total number of sensors is reached. In each subgroup, the algorithm is applied to 50 configurations, with the sensors being selected at random, and the values for $R_H$, $R_V$, and $\mathsf{\Delta }q$ are calculated. Additionally, the algorithm is applied to configurations featuring sensors that exhibit the highest and lowest agreement between the concentrations predicted by the forward dispersion model and the measurements for each group. The evaluation metrics from these applications are then compared to those derived from the random sensor selection configurations.

2.7. Numerical Simulation Settings

A 3D geometry model and computational domain were constructed based on the wind tunnel experiment’s full-scale dimensions to be used in the numerical simulations. The maximum building height in full scale is $H_{max}=24 \mathrm{m}$. Therefore, following the guidelines of COST ES 732 action [57], the inlet boundary was positioned at a distance of $5\times H_{max}=120 \mathrm{m}$ from the building area, the outlet boundary at $20\times H_{max}=480 \mathrm{m}$, and the lateral boundaries also at $5\times H_{max}=120 \mathrm{m}$. The top boundary was set at $5\times H_{max}=120 \mathrm{m}$ from the ground surface. In overall, the total domain extent along the $x, y, z$ axes are $2070\times 992\times 144 \mathrm{m}$.

For the discretization of the computational domain, an unstructured mesh of tetrahedral cells was constructed using the commercial software ANSYS Meshing R1 2019. The mesh was selected to have higher resolution near the ground and building surfaces while the total number of computational cells is equal to 12,484,910. Figure 2a–c shows the area of the computational mesh on building surfaces (Figure 2a), at the $x–z$ plane (Figure 2b), and at the x–y plane (Figure 2c).

The open-source CFD software OpenFOAM v2106 was employed for conducting the numerical simulations. Airflow simulations utilized the RANS approach coupled with the standard $k–\epsilon$ turbulence model within the Semi-Implicit Method for Pressure-linked Equations (SIMPLE) solver. For passive scalar transport, the built-in ScalarTransportFOAM solver was modified to achieve steady-state conditions. The simulations that resolved the wind flow, the forward, and the adjoint advection-diffusion equations were performed for a sufficient timespan until reaching a steady state.

In the present study, the molecular diffusivity, $D_m$, as described in Equation (11), is considered negligible. As shown by Henry, Bonifacio, and Glasgow [62] in their investigation of particle dispersion, $D_m$ is orders of magnitude smaller than the turbulent diffusivity, $D_t$. Therefore, it is assumed that neglecting the molecular diffusivity will not significantly affect the dispersion of the passive pollutant examined in this work.

The Schmidt number ($S{c}_t$), as described in Equation (11), can range from 0.2 to 1.3 in CFD-applied pollutant dispersion cases, according to Tominaga and Stathopoulos [63]. Examining the impact of the $S{c}_t$ number on the source parameters estimation is beyond the scope of this study. Hence, a $S{c}_t$ value of 0.7 is chosen, following the recommendations of Wang and McNamara [64]. In this study, consistent with wind tunnel conditions, we assume negligible thermal heat flux effects and neutral stability conditions.

The boundary conditions for the three velocity components $\left(U, V, W\right)$, the turbulence kinetic energy $k$, and the turbulence dissipation rate $\epsilon$ in the airflow simulations are set based on the conditions of the wind tunnel experiment. The reference velocity is set at $U_{ref}=6$ m/s at a height of 99.9 m. The vertical profile of the $U$ velocity component and the turbulent kinetic energy $\left(k\right)$ at the inflow boundary are adjusted to match the wind tunnel experiment measurements, while the $V$ and $W$ components are set to zero. The vertical profile of the turbulence dissipation rate $\epsilon \left(z\right)$ is calculated using the equation provided by Richards and Hoxey [65]:

(24)$\epsilon \left(z\right)={\displaystyle \frac{{\left(u^{*}\right)}^3}{\kappa \left(z+z_0\right)}}$

where $u^{*}$ is the friction velocity, $\kappa$ is the von Karman constant, and $z_0$ is the aerodynamic roughness length.

2.8. Metropolis–Hastings MCMC Settings

In each applied case, the Metropolis–Hastings algorithm begins from a random initial position in the special domain within the wider area of buildings. To determine the variables that indicate the measurement and model bias, ${\sigma }_{o,i}^2+{\sigma }_{c,i}^2$, in the posterior probability calculation (Equation (8)), we follow the recommendations of Jia and Kikumoto [12]. In this study, the overall error is considered as ${\sigma }_{o,i}^2+{\sigma }_{c,i}^2=a ·{C^o}_i$, where $a$ is a constant specified by the user and consistent across all sensors. In this study, due to the significant variation in the agreement between modeled and observed concentrations across different sensor configurations, the Metropolis–Hastings MCMC algorithm is applied 10 times for each configuration. In these applications the values of $a$ range from 0.1 to 1, with a step size of 0.1. For each configuration, the solution corresponding to the value of $a$ that yields the lowest $R_H$ value is chosen as the optimal solution. In MCMC algorithms, the initial states of the target parameters may affect the true posterior distribution. Therefore, a so-called burn-in period is used to discard the number of samples until the algorithm converges toward the stationary distribution. In the present work, the Metropolis–Hastings MCMC algorithm is applied for 100,000 iterations, with a burn-in period of 20,000 iterations and posterior distribution samples collected from the subsequent 80,000 iterations.

3. Results

3.1. Wind Field Results

Figure 3a,b shows a scatter plot comparing the modeled and wind tunnel measured $U/U_{ref}$ and $V/U_{ref}$ velocity components across different altitude-based measurement groups. For the $U/U_{ref}$ component, there is a slight overestimation at heights $z>2\times H_{max}=48 \mathrm{m}$, though the data points are highly correlated and closely align with the $y=x$ line. This strong correlation is also seen from the maximum building height ($H_{max}=24 \mathrm{m}$) up to $2\times H_{max}$. However, as measurements are taken at lower altitudes, particularly within street canyons, the correlation diminishes. This reduction in correlation is most evident near the ground ($z<12 \mathrm{m}$), where many data points fall outside the $1/2$ ($y=2x$ line) or $2/1$ ratios ($y=x/2$ line). For the $V/U_{ref}$ component, the correlation is weaker, especially at lower altitudes and within street canyons, although most data points still lie between the $y=2x$ and $y=x/2$ lines.

The results of the quantitative evaluation of the wind field simulation for all measurement points within the domain and the splatted groups are presented in Table 1. The $HR$ values for both $U/U_{ref}$ and $V/U_{ref}$ at all measurement points exceed 0.5. Although $HR$ values are below the VDI guideline limit, they are close to the threshold. Regarding the model validation criteria, it should be noted that the absolute difference $W$ is almost one-third for $U/U_{ref}$ and nearly half for $V/U_{ref}$ compared to the corresponding value of the VDI guideline. This leads to a drop in the $HR$ value, as a lower $W$ value reduces the allowed difference for a pair of modeled and measured velocity components to contribute to the increase of HR in Equation (15). Additionally, Table 1 shows that a very high HR value is achieved for both velocity components at altitudes above $H_{max}$. As seen in the qualitative presentation (Figure 3), the correlation between modeled and measured velocity decreases within street canyons, leading to a lower $HR$ value, especially at heights near the ground ($z<H_{max}/2$). It should be noted that the presented results for velocity are similar to those in previous studies evaluated using the Michelstadt experiment dataset [45,56], and from this point of view, the results are considered acceptable. An extensive investigation of potential improvements in the wind field simulation is beyond the scope of the current study.

3.2. Forward Dispersion Model Results

Figure 4a–c presents the scatter plots on a logarithmic scale of the modeled ($C_p$) and observed ($C_o$) concentrations for the three release scenarios (S2, S4, S5). Moreover, the $y=x$ line is shown in red, and the boundaries for FAC2 (green dashed lines) and FAC4 (yellow dashed lines) agreements are illustrated. Overall, the S2 release scenario demonstrates the best model performance, with most points aligning closely with the $y=x$ line. The accuracy declines for concentrations below 1 ppmv, where some outliers are present, and the model tends to underestimate these lower concentration levels. A similar pattern is noted for the S5 release scenario. In contrast, the S4 scenario shows the lowest correlation, as illustrated in Figure 4b, where the model significantly underestimates concentrations below 1 ppmv and overestimates concentrations for high values (>100 ppmv).

Table 2 presents the calculated values for all metrics for each of the three release scenarios. Additionally, it indicates the overall accuracy of the model across all scenarios. For the case of the overall concentration pairs, two linear metrics ($FAC2$ and $FB$) meet the acceptance criteria, although the $NMSE$ value, while close to the threshold limit (6), slightly exceeds it. Among the individual scenarios, S2 exhibits the best performance, and the S5 results also meet all the criteria. In contrast, the worst $NMSE$ and $FB$ values are observed for the S4 scenario. Specifically, the S4 $NMSE$ value (11.8) surpasses the threshold, impacting the overall results due to a significantly overestimated outlier, as seen in the scatter plot in Figure 4b (concentration point above 10² ppmv). For logarithmic-based metrics, the best results are also found in the S2 scenario, consistent with the linear metrics. The S4 scenario similarly demonstrates the influence of outliers on the results. The overall $MG$ value ($MG=1.45$) suggests a mean bias of less than a factor of two, which is deemed satisfactory. However, the overall $VG$ value $(VG=4.25$) is notably impacted by the S4 results and is higher than the ideal model value ($VG=1.0$). It is noteworthy that the overall metrics for these concentration pairs are either better or comparable to those of CFD dispersion models validated on the Michelstadt dataset from COST ES 1006 [42].

To visually represent the level of agreement between the measured and modelled concentrations, Figure 5a–c displays the distribution of sensors along with the plume from the forward dispersion model for each release scenario. Sensors with a $FAC2$ value of 1, indicating high agreement between the measured and modelled concentrations, are categorized as high agreement (HA). Sensors with a $FAC2$ value of 0 but a $FAC4$ value of 1 are classified as medium agreement (MA), while those with a $FAC4$ value of 0 are labeled as low agreement (LA). In Figure 5, HA sensors are shown as green dots, MA sensors as blue dots, and LA sensors as red dots, with the true source location marked by a yellow “x” symbol.

3.3. STE Results

The results for the source parameter estimates, including location ($x, y, z$ coordinates) and release rate ($q_s$) for the three release scenarios, are quantitatively illustrated in Figure 6a–c. This figure displays the estimated PDF histograms in green, with the mean value of the distribution shown by a blue line and the true value indicated by a red line. It is observed that there is a high agreement between the average values of the distributions and the true $x$ coordinate, particularly in the S2 and S4 scenarios. However, the accuracy for the $y$ coordinate is lower across all scenarios. This suggests that the algorithm provides better estimations for the $x$ coordinate, which aligns with the direction of the inlet velocity flow, regardless of the source location characteristics (e.g., open space, narrow streets). For the vertical coordinate $z$, high agreement between the mean estimated and true values is achieved in the S2 and S5 scenarios, in contrast to the S4 case, where a greater discrepancy is observed. All release rate ($q_s$) distributions have mean values relatively close to the true rates, with the best agreement observed in the S4 case.

Table 3 presents the true and mean estimated parameters $\left(x, y, z, q\right)$ and the quantities $R_H$, $R_V$, and $\mathsf{\Delta }q$ for the three release scenarios. The highest $R_H$ distance is observed in the S2 scenario, while the lowest is found in S5. Furthermore, the $R_V$ distance and $\mathsf{\Delta }q$ are nearly the same for both S2 and S5. Conversely, although the S4 scenario has a relatively low horizontal distance ($R_H=13.61 \mathrm{m}$), its $R_V$ is over 20 m, much greater than the values seen in S2 ($R_V=3.69 \mathrm{m}$) and S5 ($R_V=2.64 \mathrm{m}$). Additionally, the S4 scenario provides the best estimation for the release rate ratio, with $\mathsf{\Delta }q$ approaching the target value ($\mathsf{\Delta }q=1.06$).

The results reveal noticeable differences in final estimations and performance across the release scenarios, despite using the same model setup, velocity, and eddy viscosity fields for all scenarios. Therefore, it is important to examine the differences in each scenario setup, including the number of sensors and source positions. When comparing the horizontal distance results, similar biases are observed across all release scenarios. However, the highest accuracy in horizontal level estimations is achieved with the lowest number of deployed sensors (22 sensors in the S5 case), while the lowest accuracy is associated with configurations that use a higher number of deployed sensors (52 sensors in the S2 case). This variation may be attributed to the source’s location within the computational domain: S2 is situated in open space, whereas the other sources are located in narrow streets. Regarding $R_V$ and $\mathsf{\Delta }q$, it appears that errors in the STE process (including measurement, wind field, and adjoint concentration calculation errors) have a more significant effect on vertical distance and release rate estimations. In all cases, better vertical plane estimations (lower $R_V$) correspond with increased release rate estimation errors (higher $\mathsf{\Delta }q$), and vice versa. Further research is needed to determine if this pattern is consistent across different cases and datasets.

For a deeper understanding of the quality of the present study STE outcomes, the results are compared with those of the two-step STE algorithm introduced by Efthimiou et al. [20] and evaluated in the Michelstadt case [45] across the three release scenarios (S2, S4, S5), among others. In scenario S2, the present algorithm offers a more accurate estimate for both the location and release rate. Specifically, the estimated source of the current is 47.24 m closer to the true source on the horizontal level and 5.81 m closer to the vertical compared to the Efthimiou et al. [45]. Moreover, the $\mathsf{\Delta }q$ value of 1.35 indicates a better estimation of the release rate compared to the 2.25 provided by Efthimiou et al. [45]. In scenario S4, both methods produce similar results, with Efthimiou et al.‘s method providing a slightly better estimate for the source location (0.49 m closer horizontally and 3.51 m closer vertically) and the release rate ($\mathsf{\Delta }q=1.01$ compared to the current study’s 1.06). Similarly, in scenario S5, Efthimiou et al. [45] method achieved slightly better results for the source location in x–y plane, being 4.89 m closer, while the current study’s estimation is by 0.86 m closer vertically. The release rate estimated by Efthimiou et al. [45] ($\mathsf{\Delta }q=1.05$) was also nearer to the true value compared to the current study’s estimate of $\mathsf{\Delta }q=1.35$. Overall, the present study provided significantly better results in scenario S2, whereas the Efthimiou et al. [45] method produced slightly better estimates in scenarios S4 and S5.

3.4. Sensitivity on Sensors Configuration

Section 2.6 outlines the analysis process for sensor configuration, focusing on the effect of sensor count and the alignment between modelled and observed concentrations on STE outcomes. For each release scenario, subgroups are formed by incrementally increasing the sensor count, beginning with five sensors and adding five more for each subsequent group until reaching the total sensor count. For the S2 case, with 52 sensors, groups range from 5 to 50 in increments of five sensors. Similarly, in the S4 and S5 scenarios, which have 25 and 22 sensors, respectively, subgroups are created from 5 to 20 sensors. Within each subgroup, the Metropolis–Hastings MCMC algorithm runs on 50 randomly selected sensor configurations. Additionally, for each sensors number group, the algorithm is applied to the sensor configurations showing the highest agreement (BA) and lowest agreement (WA) with the modelled concentrations. After generating the estimations as probability density functions (PDFs), the mean of each PDF is calculated for the source location coordinates and release rate, which are then used to determine the evaluation quantities values $R_H$, $R_V$, and $\mathsf{\Delta }q$.

Figure 7, Figure 8 and Figure 9 present the distances $R_H$ (Figure 7a, Figure 8a and Figure 9a) and $R_V$ (Figure 7b, Figure 8b and Figure 9b), along with the release rate ratio $\mathsf{\Delta }q$ (Figure 7c, Figure 8c and Figure 9c), for the application described above. Each box represents the distribution of evaluation quantities across the 50 random configurations within each subgroup. The evaluation values for the BA configuration are marked with green dots, while those for the WA configuration are shown with red dots. The true source parameter values are indicated by a dashed red line.

Observing the outcomes for the randomly selected sensor configurations in the S2 scenario (Figure 7a–c) reveals a distinct pattern across all evaluation quantities. For configurations with 5 sensors, while the majority of estimations approach the true source values, a notable number of configurations exhibit significant discrepancies between the estimated and true source values. Additionally, some low-quality solutions (identified as outliers in the plots) appear. As the number of sensors used in the STE algorithm increases, the range of the boxes (located near the true source values) for $R_H$, $R_V$, and $\mathsf{\Delta }q$ decreases, and fewer outliers are observed, suggesting that different sensor combinations lead to more consistent solutions. Remarkably, after the 20-sensor group, estimations become both consistent and highly accurate, regardless of specific configurations or the number of sensors used. For BA and WA configurations, more accurate solutions are observed with the BA configuration for groups with a low number of sensors (except in the 10-sensor configuration). As the number of sensors increases, BA and WA estimations converge, becoming nearly identical to those of the random configurations. This pattern is particularly evident in the $R_H$ quantity, wherein, after the 30-sensor configuration, BA and WA solutions are almost identical. Additionally, BA solutions tend to align with the highest accuracy solutions among the random configurations, though they are not always the best. In contrast, significantly lower quality solutions are observed from some random configurations than WA, especially in groups with fewer sensors. This finding highlights the possibility that factors other than the agreement between modelled and observed concentrations contribute to the low-accuracy solutions in these groups and should be further investigated.

The results for the S4 release scenario are presented in Figure 8a–c. In general, some differences in the patterns are observed compared to S2. For the 50 random configurations, similar patterns to S2 are observed for $R_H$ and $\mathsf{\Delta }q$, where, starting from 15 sensors, the range in the boxplots decreases, with the majority of estimations lying near the true values. On the other hand, the $R_V$ plots show some differences: the box of estimations for the 20-sensor group reduces its range but converges to a solution with lower accuracy compared to some solutions in groups with fewer sensors. However, the 20-sensor solution is converging toward the estimation value obtained with the full 25-sensor dataset. In examining the BA and WA solutions, it is clear that BA consistently yields high accuracy across all sensor groups, as seen in the S2 scenario. In contrast, WA values show lower accuracy across all configurations with 5 to 15 sensors. The discrepancy between the estimated and true source values is evident in the evaluation metrics for each group. For example, while the horizontal distance is low for the 10-sensor group, the very high values for $R_V$ and especially $\mathsf{\Delta }q$ indicate a less accurate solution. Another noteworthy observation is that, for the 20-sensor group, the WA solution is closer to the BA estimations. A key factor contributing to the differences in results between S2 and S4, particularly in the WA solution, is that S4 has the lowest accuracy in the forward model, a higher number of LA sensors, and more extreme variations in these sensors (Figure 4b). In contrast, S2 features the highest FAC2 and shows less discrepancy between modelled and observed concentrations, even for the LA sensors.

Figure 9a–c illustrates the findings for the evaluation quantities in the S5 release scenario. In this case, similarities with the S2 scenario are apparent. At 10 sensors, the solutions from the random selections, as well as the BA and WA configurations values, appear to converge to estimates that are close to the true source location and release rate. In the S5 scenario, compared to S4, although a lower $FAC2$ value for the forward model was calculated, no extreme discrepancies between modelled and measured concentrations were observed. From this perspective, it appears that the number of sensors needed to reduce the likelihood of obtaining low-accuracy solutions is more closely related to the level of bias in the LA sensors than to the total number of HA sensors.

As previously noted, in several cases, the BA and WA sensor configurations do not always yield the best and worst solutions, respectively, according to the evaluation parameters. This is particularly evident in the five-sensor group, where certain sensor combinations result in much lower accuracy than the WA configuration. Therefore, it is essential to investigate other factors of the sensor network that might contribute to these low-quality estimations. These factors could include the sensor positions, the number of LA sensors, or the distance between the sensors and the true source.

To further explore this, Figure 10, Figure 11 and Figure 12 display the sensor configurations on the x–y plane and the evaluation metrics for the BA (Figure 10a, Figure 11a and Figure 12a) and WA (Figure 10b, Figure 11b and Figure 12b) configurations, as well as the best solution (BS) (Figure 10c, Figure 11c and Figure 12c) and worst solution (WS) (Figure 10d, Figure 11d and Figure 12d) for the five-sensor group in each release scenario. HA sensors are represented by green dots, MA sensors by blue dots, and LA sensors by red dots. The true source location is marked with a yellow “x” symbol, and the plume from the forward dispersion model is also shown.

In the S2 scenario (Figure 10a–d), it is observed that, for the BA, BS, and WS configurations, no specific pattern emerges regarding the position of the sensors, their distances from the source, or the influence of the plume. However, in the WA case, it is noted that all five LA sensors are positioned in a specific part of the urban area with low influence from the plume. In contrast, in the BS configuration, three sensors are HA, one is MA, and one is LA, whereas, in the WS case, only one sensor is MA, and four are HA. Therefore, the very high $R_H$, $R_V$, and $\mathsf{\Delta }q$ values observed in the WS case cannot be explained by the position of the sensors, their agreement with the model, the influence of the plume, or the distance between the sensors and the true source position.

The findings in the S5 scenario (Figure 11a–d) show that, for both the BA and BS configurations, the sensors are positioned in the direction of the plume, detecting higher concentrations compared to the WA and WS configurations. In the WS scenario, four of the five sensors are classified as LA. While these factors may contribute to poor estimations by the algorithm, they do not fully explain the significant discrepancy between the estimated and true source values, particularly in the source location estimation on the x–y plane ($R_H$). This is especially apparent when comparing the results for the WA configuration (with five LA sensors) and the BS configuration (with two LA sensors).

The corresponding results for the S5 scenario (Figure 12a–d) show that the sensor positions in the BA and WA configurations align with those observed in the S2 and S4 scenarios. Additionally, the sensor positions and accuracy in the BS case (three HA, one MA, and one LA) are similar to the corresponding WS case (three HA and two LA).

The analysis presented in Figure 10, Figure 11 and Figure 12 does not provide a definitive explanation for the algorithm’s failure to produce accurate solutions, other than the limited number of available sensor data. Although factors such as sensor positioning, the agreement between sensor measurements and model values, and the plume’s influence do affect the STE results, they do not account for the extreme outlier evaluation quantities observed in the 5-sensor subgroups (Figure 7, Figure 8 and Figure 9). In general, the poor availability of observational data can impact the accuracy of the Metropolis–Hastings MCMC algorithm, potentially leading to convergence issues or incorrect optimal solutions. While this may help explain the poor accuracy of the outliers, exploring alternative MCMC algorithms is beyond the scope of this study.

4. Discussion

The current study focuses on examining the impact of different sensor network configurations on the accuracy of an STE methodology within a complex urban area. To this end, the Metropolis–Hastings MCMC algorithm is used to estimate the location and release rate of an unknown source by combining measured concentrations detected by the sensors with corresponding modelled concentrations calculated by the CFD model OpenFOAM v2106. This analysis involves three release scenarios from the Michelstadt wind tunnel experiment case study. The focus is primarily on how the number of sensors and the degree of alignment between observed and modelled concentrations at sensor locations influence STE accuracy. Additionally, the study assesses the wind flow model, forward dispersion model, and STE to enhance the robustness of the sensor configuration analysis.

The main assumptions of this work are aligned with fitting the application of numerical simulations and the overall investigation to the conditions and characteristics of the wind tunnel experiment. In the numerical simulations, the emitted substance is treated as a passive, non-reactive pollutant, with no chemical processes introduced. Furthermore, neutral stability conditions are assumed, and thermal effects are neglected. These assumptions are commonly applied in pollutant dispersion studies in urban settings using CFD models [9,45,66]. A limitation of this work is that only one wind direction is investigated in all three release scenarios. However, applying the methodology across scenarios with different source locations and sensor networks helps to mitigate this limitation. The Michelstadt experiment provides a comprehensive and valuable dataset that has frequently been used to evaluate wind field and ATD models in urban settings. Many previous studies [49,50,51] have used synthetic measurements to analyze sensor configurations by applying CFD models. Typically, synthetic measurements are created by adding noise to modelled data to represent the discrepancy between modelled and observed concentrations. In contrast, real measurements allow for assessing the validity of methods under actual conditions, where the agreement between modelled and observed data is influenced by numerous parameters, potentially leading to varying outcomes.

The evaluation of the wind flow model shows that acceptable performance is achieved. Specifically, the hit rate metric, which compares observed and modelled velocity components, is 0.535 for the $U/U_{ref}$ component and 0.501 for the $V/U_{ref}$ component across all measurement points. These results are comparable to those from previous CFD wind flow models validated using the Michelstadt dataset [45,56]. It is noted that the model’s accuracy is higher above the buildings and decreases within street canyons and near the ground surface.

The validation metrics for the forward dispersion model show very high performance in the S2 case. In the S5 scenario, although accuracy decreases compared to S2, the $FAC2$, $NMSE$, and $FB$ metrics indicate that the model still meets the acceptance criteria. However, extreme concentration outliers in the S4 scenario (Figure 4b) negatively impact the model’s performance, particularly affecting the $NMSE$, $MG$, and $VG$ values. Overall, despite these outliers in the S4 scenario, the forward dispersion model’s performance is considered satisfactory and is comparable to or slightly better than other CFD dispersion models evaluated in Michelstadt [42]. Notably, the variations in agreement levels between the modelled and observed concentrations across the three release scenarios proved extremely useful for analyzing the impact of sensor configuration on the STE, as cases with differing accuracy levels were utilized.

The outcomes from the application of STE methodology in the full measurement data of 52, 25, and 22 sensors in the corresponding S2, S4, and S5 scenarios, indicate that the algorithm is capable of providing high-quality solutions in estimating both the unknown source location and release rate. Specifically, the horizontal distances between the average values of the estimated PDFs and the true source parameter values are below $1.5\times H_{max}=36 \mathrm{m}$, the vertical distances are under $H_{max}=24 \mathrm{m}$, and the estimated release rates are within a factor of two of the true rates across all investigated cases. The best results for source location estimation occur in the S5 release scenario, with a horizontal distance of 8.44 m and a vertical distance of 5.23 m, although fewer sensors are used. On the other hand, in the S2 case, although high accuracy is achieved in the vertical dimension ($R_V=5.51 \mathrm{m}$), the horizontal distance of 32.61 m is the highest compared to the other two scenarios. In the S4 scenario, a vertical distance of more than 20 m is observed between the true and the estimated mean PDF values. The release rate ratio values indicate an almost perfect prediction in the S4 scenario, with similar accuracy in the S2 and S5 cases. It was also observed that a higher error in vertical distance correlates with a lower bias in release rate estimation, and vice versa. However, this observation requires further investigation across different test cases and scenarios. Comparing the findings of the present study with the corresponding estimations from a two-step algorithm in the previous work [45], the current study achieves much higher accuracy in the S2 scenario, while Efthimiou et al. [45] work showed slightly better predictions for the S4 and S5 cases.

The main outcomes of this study relate to the different sensor configurations used by the STE algorithm. As shown in Figure 7, Figure 8 and Figure 9, each release scenario includes a specific number of sensors for which the vast majority of randomly selected sensor configurations yield similar solutions. In the S2 scenario, 20 out of 52 sensors are sufficient, while 20 out of 25 are needed in the S4 scenario, and 10 out of 22 in the S5 scenario. Reducing the number of sensors below these values generally still produces high-quality solutions for most configurations, however, there is a risk of lower accuracy estimation. This effect becomes more pronounced when using a network of only five sensors, where some configurations lead to a significant discrepancy between the estimated and true source parameter values. The evaluation quantities values suggest that increasing the number of sensors generally improves the convergence of the BA and WA estimations. Specifically, when 25 sensors are used in S2, 20 in S4, and 10 in S5, the predictions for source parameters by BA and WA configurations closely align. BA typically provides better accuracy at a lower number of sensors used, as expected. The greater discrepancy between BA and WA estimates in the S4 scenario may be due to substantial mismatches between modelled and observed concentrations at certain sensors (Figure 4b). For configurations that resulted in very low accuracy, particularly those using only five sensors, no clear pattern related to sensor positioning, distance from the source, or agreement between observed and modelled concentrations was identified (Figure 10, Figure 11 and Figure 12). Although using LA sensors or sensors that does not significantly affected by the plume tends to decrease STE accuracy, these factors alone do not account for the extreme discrepancies between estimated and true source parameters. Further investigation is needed to explore potential convergence or false optima issues in the Metropolis–Hastings MCMC algorithm in these limited cases.

5. Conclusions

In the current work, it was generally found that there is a certain threshold in the number of sensors above which all sensor configurations and different levels of model accuracy converge to similar solutions. Below this threshold, STE results are more sensitive to sensor configuration, the accuracy of dispersion models, and algorithmic issues. This study is based on the following assumptions:

• A stationary source,

• A non-reactive, passive pollutant,

• Steady meteorological conditions with a single wind direction,

• Neutral stability conditions without thermal effects.

Despite these limiting assumptions, the study investigates several characteristics that, in combination, are not widely explored in previous sensor configuration analyses for STE. These include:

• Complex geometries that represent real urban environments,

• Highly accurate pollutant concentration measurements obtained from a robust wind tunnel experiment,

• Various release scenarios with different source location characteristics and sensor networks,

• Realistic scenarios with varying levels of agreement between forward-modelled and observed concentrations,

• A sensor configuration analysis based on STE algorithm solutions.

Based on the above the present study provide useful insights for designing sensor configurations in urban environments for STE purposes. However, further research is needed to investigate more complex real urban areas. In previous work by Gkirmpas et al. (2024) [9], the current methodology was applied to the real geometry of Augsburg city center using synthetic measurements, providing accurate estimations for source location and release rate. In that work, it was observed that the accuracy of the estimations is strongly correlated to the number of sensors detecting concentrations above a specified threshold. Further investigations are needed to explore the sensors’ configuration and the sensitivity of the STE in such a real world case. Moreover, future research could investigate different aspects of the current study such as varying meteorological conditions, multiple wind directions, non-stationary sources, and different Bayesian algorithms.

Author Contributions

Conceptualization, P.G. and N.M.; methodology, P.G., F.B., G.T. and N.M.; software, P.G. and P.T.; validation, P.G., F.B., G.T. and N.M.; formal analysis, P.G.; investigation, P.G., F.B., G.T., G.E. and N.M.; resources, C.V., T.R. and N.M.; data curation, P.G.; writing—original draft preparation, P.G.; writing—review and editing, P.G., F.B., G.T., G.E., P.T., T.R., C.V. and N.M.; visualization, P.G.; supervision, N.M.; project administration, N.M.; funding acquisition, N.M. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The experimental data used in this study were made available in the framework of the COST Action ES1006.

Acknowledgments

The authors acknowledge support by the state of Baden-Württemberg through bwHPC. OPENFOAM^® is a registered trademark of OpenCFD Limited, producer and distributor of the OpenFOAM software v2106 via www.openfoam.com (accessed on 15 November 2024).

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Release scenarios S2 (a), S4 (b), and S5 (c). The source is indicated with red “x” symbols and the sensors of the measurement network with blue dots.

Figure 1. Release scenarios S2 (a), S4 (b), and S5 (c). The source is indicated with red “x” symbols and the sensors of the measurement network with blue dots.

Figure 2. Computational mesh on the buildings and ground (a), at the x–z plane (b), and at the x–y plane (c).

Figure 2. Computational mesh on the buildings and ground (a), at the x–z plane (b), and at the x–y plane (c).

Figure 3. Scatter plots of modeled and measured velocity components [Forumla omitted. See PDF.] (a) and [Forumla omitted. See PDF.] (b) for all available velocity observation points. The velocity pairs are divided into four groups based on the altitude of the measurement. The [Forumla omitted. See PDF.] line is indicated in red, while [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] lines are shown with dashed lines.

Figure 4. Scatter plots of forward dispersion modeled [Forumla omitted. See PDF.] and measured [Forumla omitted. See PDF.] concentrations for the three release scenarios S2 (a), S4 (b), and S5 (c). The [Forumla omitted. See PDF.] line is indicated in red.

Figure 5. Spatial distribution of sensors and the plume of the forward dispersion model in the S2 (a), S4 (b), and S5 (c) release scenarios. The HA sensors are depicted with green dots, the MA sensors with blue dots, and the LA sensors with red dots. The true source is shown with a yellow “x” symbol in black circle.

Figure 6. PDF distributions of the estimated parameters [Forumla omitted. See PDF.] for the release scenarios S2 (a), S4 (b), and S5 (c) are indicated with green histograms. The mean value of the distributions are shown with blue lines, while the true source parameter values are indicated with red lines.

Figure 7. The horizontal distance [Forumla omitted. See PDF.] (a), vertical distance [Forumla omitted. See PDF.] (b), and release rate ratio [Forumla omitted. See PDF.] (c) between the mean estimated and true source parameters across all cases in each sensor group for the S2 release scenario. Each box illustrates results from 50 random cases, with green dots representing the BA configuration and red dots the WA configuration. The red line marks the true source values.

Figure 8. The horizontal distance [Forumla omitted. See PDF.] (a), vertical distance [Forumla omitted. See PDF.] (b), and release rate ratio [Forumla omitted. See PDF.] (c) between the mean estimated and true source parameters across all cases in each sensor group for the S4 release scenario. Each box illustrates results from 50 random cases, with green dots representing the BA configuration and red dots the WA configuration. The red line marks the true source values.

Figure 9. The horizontal distance [Forumla omitted. See PDF.] (a), vertical distance [Forumla omitted. See PDF.] (b), and release rate ratio [Forumla omitted. See PDF.] (c) between the mean estimated and true source parameters across all cases in each sensor group for the S5 release scenario. Each box illustrates results from 50 random cases, with green dots rep-resenting the BA configuration and red dots the WA configuration. The red line marks the true source values.

Figure 10. Sensor configurations on the x–y plane and the evaluation quantity values ([Forumla omitted. See PDF.], [Forumla omitted. See PDF.], [Forumla omitted. See PDF.]) for BA (a), WA (b), BS (c), and WS (d) in the 5-sensor group for release scenario S2. HA sensors are shown with green dots, MA sensors with blue dots, and LA sensors with red dots. The true source position is marked with a yellow “x” symbol in black circle.

Figure 11. Sensor configurations on the x–y plane and the evaluation quantity values ([Forumla omitted. See PDF.], [Forumla omitted. See PDF.], [Forumla omitted. See PDF.]) for BA (a), WA (b), BS (c), and WS (d) in the 5-sensor group for release scenario S4. HA sensors are shown with green dots and LA sensors with red dots. The true source position is marked with a yellow “x” symbol in black circle.

Figure 12. Sensor configurations on the x–y plane and the evaluation quantity values ([Forumla omitted. See PDF.], [Forumla omitted. See PDF.], [Forumla omitted. See PDF.]) for BA (a), WA (b), BS (c), and WS (d) in the 5-sensor group for release scenario S5. HA sensors are shown with green dots, MA sensors with blue dots, and LA sensors with red dots. The true source position is marked with a yellow “x” symbol in black circle.

Table 1

Hit rate of the wind field simulation for the velocity components $U/U_{ref}$ and $V/U_{ref}$ across all measurement points, with groups divided based on the altitude of the observation points.

Hit Rate
Measurement Points	$\mathit{U}/{\mathit{U}}_{\mathit{r}\mathit{e}\mathit{f}}$	$\mathit{V}/{\mathit{U}}_{\mathit{r}\mathit{e}\mathit{f}}$
$z<12 \mathrm{m}$	0.173	0.274
$12 \mathrm{m}<z<24 \mathrm{m}$	0.489	0.450
$24 \mathrm{m}<z<48 \mathrm{m}$	0.984	0.770
$z>48 \mathrm{m}$	1.000	0.890
all locations	0.535	0.501

Table 2

Validation metrics values provided separately for each release scenario (S2, S4, S5), as well as for the overall concentration measurement point across the three scenarios.

Release Scenario	$\mathit{F}\mathit{A}\mathit{C}2$	$\mathit{F}\mathit{A}\mathit{C}4$	$\mathit{N}\mathit{M}\mathit{S}\mathit{E}$	$\left|\mathit{F}\mathit{B}\right|$	$\mathit{M}\mathit{G}$	$\mathit{V}\mathit{G}$
S2	0.73	0.90	0.53	0.16	1.05	1.79
S4	0.48	0.64	11.80	0.62	2.45	27.33
S5	0.45	0.77	3.26	0.19	1.70	4.14
All	0.63	0.81	7.11	0.34	1.45	4.25

Table 3

True source parameters, mean values of the distributions of the estimated parameters, horizontal distance, vertical distance, and release rate ratio for the three release scenarios (S2, S4, and S5).

Case	Source	$\mathit{x}$ (m)	$\mathit{y}$ (m)	$\mathit{z}$ (m)	$\mathit{q}$ (kg/s)	${\mathit{R}}_{\mathit{H}}$ (m)	${\mathit{R}}_{\mathit{V}}$ (m)	$\mathit{\Delta }\mathit{q}$ (-)
S2	True Estimated	453.28	659.56	0.0	0.5	15.12	3.69	1.35
		443.29	670.91	3.69	0.37
S4	True Estimated	750.78	572.77	0.0	0.5	13.61	21.01	1.06
		749.93	559.19	21.01	0.47
S5	True Estimated	815.18	534.47	0.0	0.5	9.39	2.64	1.35
		822.93	529.16	2.64	0.37

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