#### 5.1. WSN Planning Preliminaries and Lunar Target Area Selection

Among the different candidate lunar areas, we propose a closer study of the Dionysius region [

56] by means of a WSN. The Dionysius region is located near the western edge of Mare Tranquillitatis (see

Figure 6a) and is centered on the Dionysius crater (

$2.{8}^{\circ}$N,

$17.{3}^{\circ}$E), which has a diameter of 18 km. This region is known to have a high concentration of ilmenite material (FeOTiO

${}_{3}$), which is thought to contain

${}^{3}$He (Zheng

et al. [

15]).

Our aim is to deploy the WSN at the points where models have predicted a greater abundance of

${}^{3}$He. We assume that trustworthy points correspond to zones with TiO

${}_{2}$ abundance (see

Section 1), a good tracer of ilmenite. Therefore, a digital map of TiO

${}_{2}$ content represents an indicator function corresponding to the presence of

${}^{3}$He and can potentially be used for our proposal.

**Figure 6.**
Dionysius region of interest with coordinate values S 1.6 N4.2 and W15 E 19 in degrees. (

**a**) NASA’s Lunar Reconnaissance Orbiter (LRO) Wide Angle Camera (WAC) relief image in orthographic projection of the lunar near side and the Dionysius region in the center. Source:

http://wms.lroc.asu.edu/lroc; (

**b**) percentage of TiO

_{2} weight (wt%) using the Lucey

et al. [

57] method. Source:

http://www.lpi.usra.edu/lunar/tools/clementine/; (

**c**) the Dionysius region of interest with an overlay of TiO

_{2} percentage in black tones.

**Figure 6.**
Dionysius region of interest with coordinate values S 1.6 N4.2 and W15 E 19 in degrees. (

**a**) NASA’s Lunar Reconnaissance Orbiter (LRO) Wide Angle Camera (WAC) relief image in orthographic projection of the lunar near side and the Dionysius region in the center. Source:

http://wms.lroc.asu.edu/lroc; (

**b**) percentage of TiO

_{2} weight (wt%) using the Lucey

et al. [

57] method. Source:

http://www.lpi.usra.edu/lunar/tools/clementine/; (

**c**) the Dionysius region of interest with an overlay of TiO

_{2} percentage in black tones.

We used an image map of TiO

${}_{2}$ abundance in our area of interest, delimited by latitude S

$\phantom{\rule{0.166667em}{0ex}}1.{6}^{\circ}$ N

$\phantom{\rule{0.166667em}{0ex}}4.{2}^{\circ}$ and longitude W

$\phantom{\rule{0.166667em}{0ex}}{15}^{\circ}$ E

$\phantom{\rule{0.166667em}{0ex}}{19}^{\circ}$ and generated through the Clementine Mapping Project of the Lunar and Planetary Institute (LPI) [

58] web service. The image size is

$1213\times 789$ pixels, and its scale is

$0.1$ km/pixel in single cylindrical projection (

plate carrée) [

59] corresponding to an approximate surface of

$120\times 78$ km

${}^{2}$. The weight percent (wt%) of TiO

${}_{2}$ is computed based on the method described by Lucey

et al. [

57], as shown in

Figure 6b, where brighter tones indicates higher Ti content (

i.e., higher importance or estimated

${}^{3}$He). For the convenience of this work, the original RGB

Figure 6b was color inverted and indexed, such that each pixel in the image has an associated value, ranging from zero for the whitest areas (minimum TiO

${}_{2}$ abundance) to 255 for the darkest areas (maximum TiO

${}_{2}$ abundance). It represents our importance function, as defined in

Section 3, indicating the expected

${}^{3}$He distribution at each location.

Figure 6c shows the resulting indexed image overlying a relief image of the same coordinates in simple cylindrical projection obtained from the Lunar Reconnaissance Orbiter Camera [

60].

Without loss of generality, we can make the following practical considerations for the deployment:

Excluding the centered Dionysius crater, the region of deployment is smooth enough to be considered a flat surface (i.e., it is not rugged). Although there may be some ${}^{3}$He inside the crater, the amounts are small and distant from other parts of the scenario and, thus, can be ignored.

The maximum number of sensor nodes has been restricted to

$N\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}150$, because spacecraft payload capacity is always limited [

61]. In order to scatter these nodes in our huge target area, parameter

${r}_{t}$ needs to be adjusted. In our tests, we have set a long transmission range

${r}_{t}=6$ km.

Antennas are assumed to be omnidirectional dipoles placed at sufficient height above the Moon surface to ensure that signal propagation (reflection, diffraction, penetration,

etc.) is not affected by ground effects. Under these conditions, the propagation model on the lunar surface could be approximated to the

free-space model, even for long-range distances [

62,

63].

We assume that the transmission power of our nodes may be adjustable between $0\phantom{\rule{0.166667em}{0ex}}$dBm and $20\phantom{\rule{0.166667em}{0ex}}$dBm; we also assume a carrier frequency of $900\phantom{\rule{0.166667em}{0ex}}$MHz. This frequency allows reduced antenna dimensions of $8.32\phantom{\rule{0.166667em}{0ex}}$cm, which are suitable and easy to manage in space applications and also require less energy consumption than higher operation frequencies.

An estimation of the received power at a 6-km distance can be computed using the well-known Friis equation [

64].

For instance, if we select a transmitting power of

$10\phantom{\rule{0.166667em}{0ex}}$dBm (assuming typical dipole gains

${G}_{r}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{G}_{t}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}2.15$ and a system loss factor of

$L\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1$), then we obtain a received power of

$-92.8\phantom{\rule{0.166667em}{0ex}}$dBm. Commercial transceivers of these characteristics are easily available [

65].

The sensing range is set to

${r}_{s}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1.5\phantom{\rule{0.166667em}{0ex}}$ km (15 pixels in

Figure 6c).

The deployment of nodes on the lunar surface could be achieved using a rover, navigating the lunar surface. This scheme would allow controlled positioning of the nodes, although it might take a long time to put all of the nodes in place. Possible alternative methods include dropping the nodes from a spacecraft or launching them from a rover (Sanz

et al. [

66]).

#### 5.2. Validation Tests

Based on adjustment tests in the deployment region, the following

${\text{ACO}}_{\mathbb{R}}$ parameters were selected:

Number of solutions within archive T: $K\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}300$.

Number of ants: $p\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}8$.

$q=0.025$ and $\xi \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0.65$.

$maxiter\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}70,000$.

The better the tuning of

${\text{ACO}}_{\mathbb{R}}$, the better the algorithm will perform (higher objective function and lower computation time). For instance, the size

K of the solutions table is critical because it determines the complexity of the pdfs that the ants have to sample to generate new solutions. This parameter has been tested for different network sizes (

$N\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}10,30,50,70,90,110,130,150$).

Figure 7 shows the relative importance for

$N\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}30$ and

$N\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}50$. The relative importance is defined as the ratio of the importance map covered Equation (

3) to the total importance (

${\Gamma}_{\text{max}}$) contained in the map,

**Figure 7.**
Relative importance sensed and computing time.

**Figure 7.**
Relative importance sensed and computing time.

Results for

$K\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}300$ have the highest relative importance and the smallest computing time of the algorithm. All computations were performed on an 8-CPU Xeon E5 computer with 128 Gb of RAM.

Table 1 and

Table 2 collect WSN deployment and

${\text{ACO}}_{\mathbb{R}}$ operating parameters, respectively.

**Table 1.**
Technical parameters.

**Table 1.**
Technical parameters.
number of nodes | $N\le 150$ |

transmission range | ${r}_{t}=6$ km |

sensing range | ${r}_{s}=1.5$ km |

transmitter power | ${P}_{t}=10$ dBm |

**Table 2.**
${\text{ACO}}_{\mathbb{R}}$ initialization parameters.

**Table 2.**
${\text{ACO}}_{\mathbb{R}}$ initialization parameters.
T size | $K=300$ |

number of ants | $p=8$ |

heuristic parameter | $q=0.025$ |

pheromone evaporation rate | $\xi =0.65$ |

termination condition | $maxiter=70000$ |

Next, we contrast our results with those obtained using a reference heuristic used previously in Rebai

et al. [

42], which for convenience we call

four-directional placement (FDP). For this FDP heuristic, a grid is considered over the target area, with an

${r}_{t}/2$ space lattice. The FDP is an iterative algorithm, which starts at a random position. At each step, it selects the adjacent, previously unselected point of the grid with the highest importance, such that the network remains connected. Following the up, down, right and left directions, the adjacent points are evaluated at

${r}_{t}/2$ and

${r}_{t}$ distances from the current position. If several points have the same value, FDP chooses one at random. During this process, the points evaluated are kept in a sorted table (observed points table) in descending order of relevance. The top one is selected as the node position, and the process continues from this point. Note that this algorithm guarantees full connection of the network.

In our target area of

Figure 6b, we considered the deployment of

$N=10,30,...,150$ sensors and

$\theta \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1$, evaluating the relative importance. These experiments were repeated 20 times (

$8\times 20$ deployment simulations) for each of these algorithms,

${\text{ACO}}_{\mathbb{R}}$ and FDP. The relative importance is displayed in

Figure 8a and the efficiency in

Figure 8b, with a confidence level of

$95\%$.

**Figure 8.**
Performance comparison of ${\text{ACO}}_{\mathbb{R}}$-based versus the four-directional placement (FDP) heuristic.

**Figure 8.**
Performance comparison of ${\text{ACO}}_{\mathbb{R}}$-based versus the four-directional placement (FDP) heuristic.

Efficiency

ρ is computed as joint-coverage

$\left({f}_{1}\right)$ divided by the maximal information that can be sensed by

N nodes. That is,

$\rho ={f}_{1}/\left(N\pi {r}_{s}^{2}{v}_{max}\right)$, where

${v}_{max}=255$ is the maximum value of importance assigned to a point on the map. Efficiency provides insight into the quality of the deployment. Clearly, efficiency decreases with the number of nodes, since as network size increases, more nodes are used to gather less important data or simply to convey information from distant zones.

Table 3 shows the maximum joint-coverage and the efficiency for different deployment instances, some of which (

$N\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}30,90,150$) are depicted in

Figure 9. The sensing coverage zone of each node is represented by a semi-transparent yellow circle; yellow lines are the shortest paths between the sink node (marked in white) and each node. Note that with

${\text{ACO}}_{\mathbb{R}}$, sensing zones do not overlap in order to maximize

${f}_{1}$.

**Table 3.**
Maximum joint-coverage in deployments.

**Table 3.**
Maximum joint-coverage in deployments.
| Relative Importance (${f}_{1}/{\Gamma}_{\text{max}}$) | Efficiency (ρ) |
---|

N nodes | ${\text{ACO}}_{\mathbb{R}}$ | FDP | ${\text{ACO}}_{\mathbb{R}}$ | FDP |

10 | 0.0110 | 0.0110 | 0.9919 | 0.9883 |

30 | 0.0321 | 0.0312 | 0.9568 | 0.9328 |

50 | 0.0535 | 0.0489 | 0.9581 | 0.8762 |

70 | 0.0741 | 0.0600 | 0.9478 | 0.7669 |

90 | 0.0947 | 0.0742 | 0.9415 | 0.7377 |

110 | 0.1150 | 0.0847 | 0.9356 | 0.6892 |

130 | 0.1346 | 0.0919 | 0.9265 | 0.6323 |

150 | 0.1533 | 0.0972 | 0.9142 | 0.5800 |

**Figure 9.**
Node placement examples for different numbers of nodes in the Dionysius region.

${\text{ACO}}_{\mathbb{R}}$ (left) and FDP (right). Numerical results in

Table 3.

**Figure 9.**
Node placement examples for different numbers of nodes in the Dionysius region.

${\text{ACO}}_{\mathbb{R}}$ (left) and FDP (right). Numerical results in

Table 3.

The results of

Figure 8 demonstrate that our

${\text{ACO}}_{\mathbb{R}}$-based algorithm outperforms the FDP heuristic, even in scenarios with few nodes. These figures show how with

${\text{ACO}}_{\mathbb{R}}$, joint-coverage

$\left({f}_{1}\right)$ grows steadily as the number of nodes increases. When the network is small (

$N\le 20$), the efficiency of

${\text{ACO}}_{\mathbb{R}}$ and FDP is comparable. However, as the network size increases,

${\text{ACO}}_{\mathbb{R}}$ maintains a high

ρ, even for complex networks (e.g.,

$85\%$ at

$N\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}150$), but FDP efficiency decreases steadily.

Figure 10 depicts the evolution of the algorithm convergence time

versus the number of nodes and map size in both scenarios. We performed 50 tests with three different scenario sizes:

$625\times 407$,

$950\times 618$ and

$1213\times 789$ pixels (reducing the original map size). As expected, increasing the number of nodes raises the convergence time, which grows almost exponentially for

$N\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}90$, but linearly for larger network sizes. Furthermore, results for

$N\phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.166667em}{0ex}}80$ have a higher variance in both scenarios.

**Figure 10.**
Convergence time versus number of nodes (N) and scenario size.

**Figure 10.**
Convergence time versus number of nodes (N) and scenario size.

Figure 11 shows the results of the Pareto front of functions

${f}_{1}$ and

${f}_{2}$.

**Figure 11.**
Pareto front (in blue) of

${f}_{1}$ (network sensing coverage)

vs.

${f}_{2}$ (network cost). Deployments of

Figure 12 in red.

**Figure 11.**
Pareto front (in blue) of

${f}_{1}$ (network sensing coverage)

vs.

${f}_{2}$ (network cost). Deployments of

Figure 12 in red.

The solver was executed 2040 times, from different initial positions, selected at random.

Figure 11 shows a subset of representative solutions of varying parameter

θ. The blue line shows the “best solutions”, in the sense that it is impossible to improve one of the goals without worsening the other.

**Figure 12.**
Sensing coverage of TiO_{2} content in the Dionysius region of interest. (**a**) θ = 0.0025, f_{1}/Γ_{max} = 0.1608 and f_{2} = 4067; (**b**) θ = 0.05, f_{1}/Γ_{max} = 0.5282 and f_{2} = 14179; (**c**) θ = 0.36, f_{1}/Γ_{max} = 0.8176 and f_{2} = 22607.

**Figure 12.**
Sensing coverage of TiO_{2} content in the Dionysius region of interest. (**a**) θ = 0.0025, f_{1}/Γ_{max} = 0.1608 and f_{2} = 4067; (**b**) θ = 0.05, f_{1}/Γ_{max} = 0.5282 and f_{2} = 14179; (**c**) θ = 0.36, f_{1}/Γ_{max} = 0.8176 and f_{2} = 22607.

The result reveals that the Pareto front approach is useful. The solution shows the tradeoff between cost and joint-coverage of the network. Cost is related to the energy consumed by the network during lunar nights (periods without direct sun exposure). Therefore, mission planners could compute battery consumption during lunar nights, combine this with other costs (variables of the mission), such as battery weight, performance, durability, and so on, and obtain optimal positions in terms of expected

${}^{3}$He abundance. Besides, the Pareto front results are linear, showing that there is an inverse proportional relationship between both optima magnitudes. Finally,

Figure 12 also displays three sensor deployments in our region of interest for three choices of

$\theta \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}(0.0025,0.05,0.36)$ matching the red points on the Pareto frontier represented in

Figure 11. This shows how several “optimal” solutions may behave distinctly, depending on the prioritized variable in the tradeoff balance.