## 3 General Concept and Passive Detection
### Practical Issues in Passive Detection
#### 3.2.1 Estimation of Transmission Cost on a Specific Route  
On any specific route \(i\) consisting of a total number of \(J_{i}\) nodes, the expected total energy cost in transmission \(E(i)\) is determined by the following [30]:\[E(i)=E(i,1)+E(i,2)+...+E(i,J_{i}-1) \tag{3}\]  
where \(E(i,m)\) is the estimated transmission cost from the \(m\)-th node on this route to its next hop (\(m\) is an integer and satisfies \(1\leq m\leq J_{i}-1\)). Energy cost depends on a successful packet transmission, which may need a number of retries. To be more specific, the transmission cost is written as follows:  
\[E(i,m)=K(i,m)[P(i,m)+P_{c}+P_{r}]t \tag{4}\]  
where \(K(i,m)\) is the predicted average number of retries needed for a successful packet delivery from node \(m\) to its next hop node \(m+1\); \(P(i,m)\) is the minimum required radio transmission power level at node \(m\) to transmit a data packet to the next hop successfully; \(P_{c}\) is the processing power at node \(m\) (consumed by circuits on this node at the stage of preparation of radio transmission, such as coding and modulation); \(P_{r}\) is the receiving power at next hop \(m+1\) (used for data receiving process, such as demodulation and decoding); and \(t\) is the transmission time needed to transmit a packet (\(t=\frac{packsize}{datarate}\)). The energy model adopted in this paper mainly refers to previous studies given in [36] that focuses on the energy efficiency issue of WSN. Therefore, \(P(i,m)\) can be defined by the following formula:  
\[P(i,m)=\frac{\beta N_{0}[d(i,m)]^{\gamma}}{\ln[1-\mathcal{P}_{out}(i,m)]} \tag{5}\]  
where \(\beta\) is the signal to noise ratio (SNR) threshold, \(N_{0}\) is the variance of white Gaussian noise (AWGN) since the noise components in this paper are modeled as AWGN), \(d(i,m)\) is the distance between node \(m\) and its next hop, \(\gamma\) is the path-loss exponent and \(\mathcal{P}_{out}(i,m)\) is the probability that the packet has not been delivered (in other words, outage probability) from node \(m\) to node \(m+1\) on any attempt. Some of the nodes are assumed to have energy harvesting capability. The harvested energy from the surrounding environment is considered as free and can partially offset \(E(i,m)\) as follows:  
\[E(i,m)=K(i,m)[P(i,m)+P_{c}+P_{r}-\alpha(i,m)R]t \tag{6}\]  
where \(R\) is the maximum output power of the photo-voltaic power generator, and \(\alpha(i,m)=0\) if node \(m\) is not capable for energy harvesting or \(\alpha(i,m)\) is a random number defined over \([0,1]\) if this node has energy harvesting capability. As mentioned earlier in Section 1, for those applications under consideration, solar cells are more practical for sensor nodes considering their acceptable size (by contrast, wind driven generator is too bulky) or energy source accessibility (by comparison, motion power is almost not available since nodes are deployed in severe environment where human or animal activities are relatively rare). For these nodes, \(\alpha(i,m)=R^{{}^{\prime}}/R\) where \(R^{{}^{\prime}}\) is the active power level of the photo-voltaic power generator. \(R^{{}^{\prime}}\) is assumed to follow a \(\beta-\)distribution defined by the following probability density function [37]:  
\[F(R^{{}^{\prime}})=\frac{\Gamma(p_{sh}+q_{sh})}{\Gamma(p_{sh})\Gamma(q_{sh})} \left(\frac{R^{{}^{\prime}}}{R}\right)^{p_{sh}-1}\left(1-\frac{R^{{}^{\prime} }}{R}\right)^{q_{sh}-1} \tag{7}\]  
where \(p_{sh}\) and \(q_{sh}\) are the shape parameters of \(\beta-\)distribution, and \(\Gamma\) is the Gamma function. \(\beta\) distributions suit the past record of sunlight data using the algorithm that minimizes the K-S statistic [38], and its shape parameters \(p_{sh}\) and \(q_{sh}\) depend on the specific geographic regions where these data are recorded. This assumption is also based on the past records of sunlight data and statistical correlation analysis of solar radiance together with consumer load.