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1 Introduction

The characterization of an unknown atmospheric pollutant’s source following a release is a special case of inverse atmospheric dispersion problem. Such kind of inverse problems are to be solved in a variety of application areas such as emergency response (e.g. Kovalets et al. 2011; Sharan et al. 2012; Singh et al. 2013) pollution control decisions (Koracin et al. 2011) and indoor air quality (Matsuo et al. 2015).

In the urban or industrial scale, there are few researchers that have combined Computational Fluid Dynamics (CFD) with source estimation techniques (Bady et al. 2009; Chow et al. 2008; Keats et al. 2007; Kovalets et al. 2011; Kumar et al. 2016; Libre et al. 2012).

In this point it should be noticed that in some cases of inverse modeling there are some limitations. According to Dhall et al. (2006) there is a typical problem for non-linear least squares fitting due to the ill-posed minimization problem and the non-convex cost function. This problem is called ‘overfitting’ effect. According to this effect, the calculation errors which are introduced by the wrong source location and lead to significant underestimation of the concentration are compensated by the overestimated source rate. Thus, the resulting quadratic cost function reaches minimum for the wrong combined solution (source location and source rate). In context of data assimilation this problem is especially important when the number of measurements is insufficiently small. This ‘overfitting’ effect was also observed in Tsiouri et al. (2014) where the Source Inversion (SI) algorithm produced unsatisfactory results regarding the distance between the true and the estimated source location and the true to estimated source rate ratio.

Efthimiou et al. (2016) presented an integrated and innovative approach, to eliminate the ‘overfitting’ effect, based on two main improvements. First, we propose a non-simultaneous determination of the source location and rate, based on a two-step segregated approach combining a correlation and cost functions. Second, we suggest a correlation coefficient of measured and calculated concentrations, instead of a cost function (as in Kovalets et al. 2011). Moreover, we investigate the impact of the grid resolution, for the numerical simulations, on the determination of source characteristics. The MUST dataset has been selected for the evaluation of the proposed approach owes to its high quality data and because it has been used extensively by other similar works.

A description of the experiment and the computational simulations can be found in Kovalets et al. (2011). The present grid is slightly different than the one of Kovalets et al. (2011). It consists of 58,500 cells with minimum/maximum cell distances dx = 4.93/9.9, dy = 4.95/6.2 and dz = 0.2/2.06.

2 Method of Validation of the Predicted Source Location and Rate

In order to understand the order of magnitude of the error we have used the horizontal \( r_{H} = \sqrt {\left( {x^{s} - x_{t}^{s} } \right)^{2} + \left( {y^{s} - y_{t}^{s} } \right)^{2} } \) and vertical \( r_{V} = \left| {z^{s} - z_{t}^{s} } \right| \) distances of the estimated source location (x s, y s, z s) from the true source location \( \left( {x_{t}^{s} ,y_{t}^{s} ,z_{t}^{s} } \right) \) where index “t” stays for the true source. We have assumed that the predicted source (x s, y s, z s) is located at the center of the cell.

Concerning the source rate we have calculated the relative source rate ratio \( \delta q = \hbox{max} \left[ {\left( {q^{s} /q_{t}^{s} } \right),\left( {q_{t}^{s} /q^{s} } \right)} \right] \) which is always greater than unity for both underestimated and overestimated source rates.

3 Computational System

The solution was performed in a Laptop with 8 GB RAM using the OpenMP protocol and all the cores (four) of the processor (Intel(R) Core(TM) i7-4700MQ CPU @ 2.40 GHz).

4 Results

The horizontal distance (r H ) between the real and the predicted source was found equal to 10.81 m and the vertical distance (r V ) equal to 0.54 m which are slightly better results than Kovalets et al. (2011) (r H  = 11 m and r V  = 0.8 m).

The relative source rate ratio was found equal to 2.26 which is again better result than Kovalets et al. (2011) (δq = 3.1).

5 Conclusions

A major change in the data assimilation code of Kovalets et al. (2011) was performed in Efthimiou et al. (2016) and included the implementation of a two-step approach:

  • At first only the source coordinates were analyzed using a correlation function of measured and calculated concentrations.

  • In the second step, the source rate was identified by minimizing a quadratic cost function.

The validation of the new algorithm was performed for the source location and rate by simulating a wind tunnel experiment on atmospheric dispersion among buildings of a real urban environment. Good results of source location and rate estimation have been achieved when all available measurements (244) were used to solve the inverse problem.