## 2 Evolutional Computational Methodology for Radiation Mapping
### Simplified Radioactivity Model of the Shelter Object  
The 3D geometric model of the SO is simplified for the purposes of this study, compounded by the limited amount of detailed information that exists relating to the Matryoshka doll-like configuration of the structure (the original concrete building and other materials, enclosed by the outer shell) and the unknown source-term contained within. As such, the SO was modeled as a suite of isotropic gamma point sources with a step of 1 m, located on its outer shell, as shown in Figure 1. No internal sources or other available information is hence used.  
As a true model of the SO would possess considerable uncertainties, predominantly because of the unknown \"media\" between the real, multiple sources and sensors, this evolutional algorithm approach simplifies the scenario by representing the radioactivity as external \"contamination\" of the SO shell via point sources with a consistent step spacing of 1 m. Owing to workers and equipment operating outside the SO exterior during decommissioning activities, it is of limited significance if the radioactive source is on the shell or behind the shell in 30 m, under 15 cm of concrete or debris. Therefore, such a surface-based model can be considered applicable and appropriate.  
The model does not consider the existing but poorly studied effect of gamma-ray re-reflection from the shell of the NSC since the real detector is shielded from such radiation by the above-mounted MCS.  
Here, the photon flux reaching the omnidirectional detector from each point source, accounting for the decrease with increasing distance (inverse square law), is modeled. In this simplified model, \(S=F(a_{1},\ldots,a_{n\times m};\ x_{1},\ldots,x_{n})\), the activity measured by the sensor, \(S_{j}\) within coordinates \((x,\ y)\), is expressed in (1).  
\[S_{j}=\sum\limits_{i=1}^{n}a_{ij}\times x_{i}+\varphi_{j} \tag{1}\]  
where \(x_{i}\) is the activity in the point \(i\), \(a_{ij}\) is the coefficient of decrease that depends on the distance between sensor \(S_{j}\), and point \(x_{i}\), and \(\varphi_{j}\) represent the uncertainty associated with measurement error and other random factors. This is represented graphically in Figure 2.  
Then, for all \"virtual detectors\" and all point sources, modeling SO as an underdetermined system of linear Equation (2) can be constructed.  
\[\left\{\begin{array}{l}S_{1}=\sum\limits_{i=1}^{n}a_{i1}\times x_{i}+\varphi _{1}\\ \ldots\\ S_{j}=\sum\limits_{i=1}^{n}a_{ij}\times x_{i}+\varphi_{j}\\ \ldots\\ S_{m}=\sum\limits_{i=1}^{n}a_{im}\times x_{i}+\varphi_{m}\end{array}\right. \tag{2}\]  
However, some constraints in (3) are linked to the fact that the photon flux cannot not be negative and practically could not exceed a reasonable value. It should be noted that the set of equations in (3) does not comprise an Objective Function, instead comprising a constraint only, with the Objective Function that which is being optimized, while the fitness function is used to guide to such an optimization. The Objective Function is the value of the approximation error.  
\[\forall i\in(1,n),j\in(1,m)\big{|}V_{min}<x_{i}<V_{max},a_{ij}>0 \tag{3}\]  
Here \(V_{max}\) and \(V_{min}\) are a maximum and a minimum of the dose-rate in the actual system. Thus, the construction of the gamma-ray map is reduced to finding some approximation to the solution of this system of equations with practical accuracy.  
Figure 1: Simplified 3D geometric model of the shelter object, which houses the Unit 4 reactor.