# Strategies for Deploying a Sensor Network to Explore Planetary Lava Tubes
## 5 Communication  
With multiple sensors laid down along the cave, a direct line-of-sight connection link is maintained from a robot at the surface of the pit to the farthest robot inside the cave; these robots can then be used to relay data out of the cave and vice versa. This section provides an analysis of the performance of such a multi-hop communication channel. The data rate, \(B_{r}\) between a transmitter and a receiver for an Additive with Gaussian Noise (AWGN) channel is bounded by Equation (1) [39].  
\[B_{r}=W\log\left[1+\frac{P_{t}G_{t}G_{r}}{N_{0}d^{2}}\right] \tag{1}\]  
where \(W\) is the channel bandwidth in Hz, \(P_{t}\) is the transmit power, \(N_{0}\) is the noise floor level, \(G_{t}\) and \(G_{r}\) are the transmitter and receiver antenna gain, and \(d\) is the distance between the transmitter and the receiver. Moreover, the bit error rate (BER) describes the number of bit errors per unit time. For an AWGN channel with quadrature phase-shift keying (QPSK)  
Figure 14: Placement of sensors for maintaining a direct line of sight link inside a cave represented by a non-convex polygon. The red polygons are the obstacles; the black squares are the sensors, and the lines connecting them are the direct line of sight connection link. (**a**) Cave model 1, (**b**) Cave model 2, and (**c**) Cave model 3.  
Figure 15: Placement of sensors inside an unknown cave over 5 successive Explore\(\rightarrow\)Place Sensor cycles.  
modulation, the BER is a function of the normalized carrier-to-noise ratio \(E_{b}/N_{0}\) (energy per bit to noise power spectral density ratio) and is defined as Equation (2) [39].  
\[BER=\frac{1}{2}erfc\Bigg{(}\sqrt{\frac{E_{b}}{N_{0}}}\Bigg{)} \tag{2}\]  
where \(erfc\) is the error function, multiple sensors are placed between the first transmitter and the last transmitter for a multi-hop communication link inside a cave. With the transmit power amplified at each sensor, the noise floor level \(N_{0}\) also gets amplified, and hence the data rate for the \(n^{th}\) sensor is defined in Equation (3).  
\[B_{r}=W\log\left[1+\frac{P_{b}G_{t}G_{r}}{nN_{0}d^{2}}\right] \tag{3}\]  
The BER for the multi-hop communication link over N hops and each channel assumed to be independent is defined as Equation (4).  
\[BER_{multi-hop}=1-((1-BER_{1})(1-BER_{2})\dots(1-BER_{N})) \tag{4}\]  
Assuming each BER \(<<1\), the above expression is approximated as Equation (5).  
\[BER_{multi-hop}=\sum_{i=1}^{N}BER_{i} \tag{5}\]  
However, with the noise amplified, the BER for the \(n^{th}\) sensor is defined as Equation (6).  
\[BER_{n}=\frac{1}{2}erfc\Bigg{(}\sqrt{\frac{E_{b}}{nN_{0}}}\Bigg{)} \tag{6}\]  
The BER for the multi-hop channel is then calculated as Equation (5). For our simulations, we considered three frequencies (a) 500 MHz RF (radio frequency), (b) 2 GHz RF, and (c) 200 THz optical. The 500 MHz and 2 GHz channels were simulated for both 10% and 25% bandwidth, and the 200 THz channel for 1% bandwidth. Figure 16a shows the variation of data rate over a distance of 1000 m for each combination of frequency and bandwidth when the sensors are placed at a distance of 100 m. It can be seen that a data rate of 18 kbps is possible for 500 MHz, 10% bandwidth RF communication channel, while the data rate increases to 0.7 Gbps for 200 THz, 1% bandwidth optical communication channel. Although the data rate does not vary much as the distance between the sensors varies in the orders of 10 s of meters, the BER has a significant impact. Figure 16b shows the variation of BER over \(E_{b}/N_{0}\) in dB for different numbers of hops over a distance of 1000 m.