# Analytical Modeling for Underground Risk Assessment in Smart Cities
## 3 Problem Formulation  
For underground risk analysis and assessment, we need to collect certain parameters of interest via sensors deployed at selected locations. Risk at any location and at any point in time depends on various associated factors, e.g., ground status, pipeline leakage, etc. These individual risk factors are referred to _risk contributing factors (RCFs)_ here and denoted as \(s_{i}\). Each _RCF_ has its contribution towards _final risk index (FRI)_ at that particular location. We can express set of risk contributing factors as  
\[S=\{s_{1},s_{2},\ldots s_{m}\}\]  
For the sake of generality, we assume that \(m\) is the total number of _RCFs_. The risk index value for \(k^{th}\) individual factor is expressed as \(M_{k}\) and is calculated by fusion and aggregation of relevant sensor data after necessary processing. In the real world, we cannot have sensors at every location; rather sensors are deployed at selected locations. Assume we have \(n\) selected locations in our area of interest, i.e., \((x_{1},y_{1}),(x_{2},y_{2})\ldots(x_{n},y_{n})\). We are interested in collecting information of about each \(s_{i}\in S\) for selected locations where value of each _RCF_ is calculated by fusion and aggregation of various related sensors data, e.g., water supply pipeline leakage is calculated by fusing camera sensor data with water leak sensors along with flow and pressure sensors. For the sake of simplicity, we express set of sensors required for estimation of \(k^{th}\) RCF as \(Sens_{k}\). As we are considering \(m\)_RCFs_, therefore, at each selected location, we can have \(m\) different set of sensors, i.e., \(Sens_{1},Sens_{2},\ldots Sens_{m}\) collecting information about every individual _RCF_. Thus, we need at most \(n\times m\) set of sensors in our area of interest. Each set of sensor gives us reading of a particular \(k^{th}\) parameter on a selected \(i^{th}\) location \((x_{i},y_{i})\) at particular time \(t\) expressed as \(Sens_{k}(x_{i},y_{i},t)\). Sensor  
Figure 2: Typical \(5\times 5\) risk matrix. reading defines risk probability at particular location where impact of the risk is estimated by experts depending upon the location sensitivity. Risk at any location and at any point in time depends on various associated _RCF_, e.g., ground status, pipeline leakage, etc., contributing towards the final risk index at that particular location. Thus, first we need to estimate risk index value for individual _RCF_ at every selected location. As mentioned previously, we assume _m risk contributing factors (RCF)_ where risk index of \(k^{th}\) factor at time \(t\) at location \((x_{i},y_{i})\), i.e., \(M_{k}(x_{i},y_{i},t)\) depends on two main factors: (a) risk probability; and (b) risk severity. Mathematically, we can express it as below  
\[M_{k}(x_{i},y_{i},t)=H(p_{k}(x_{i},y_{i},t),r_{k}(x_{i},y_{i}))\in[0,1] \tag{3}\]  
In Equation (3), \(H\) is used to express an unknown function (because the actual relationship is unknown) taking two parameters, i.e., \(p_{k}\) and \(r_{k}\), to compute \(M_{k}\). Here, \(p_{k}\) is the risk probability calculated from sensors readings for \(k^{th}\) factor at time \(t\) and \(r_{k}\) is the impact severity of the \(k^{th}\) risk factor at location \((x_{i},y_{i})\). Risk probability \(p_{k}\) of \(k^{th}\) factor depends on the sensors reading at time \(t\) deployed at \((x_{i},y_{i})\), i.e.,  
\[p_{k}(x_{i},y_{i},t)=P(x=Sens_{k}(x_{i},y_{i},t)) \tag{4}\]  
In other words, risk probability of \(k^{th}\) risk contributing factor depends on the sensor reading installed for that purpose. Various sensor readings are considered for concluding risk index of a particular risk contributing factor and individual sensors reading can be combined in several ways, for instance simple or weighted averaging scheme. Actual sensors reading and experts knowledge will be required to formulate the relationship among sensors readings and corresponding risk index value.