Second-order design of GNSS networks with different constraints using particle swarm optimization and genetic algorithms

1 Introduction

Geodetic networks play a crucial role in several fields such as surveying, navigation, mapping, and environmental studies. Therefore, the need for accurate geodetic networks is inevitable in most of these fields. The establishment and measurements of the geodetic networks may be designed using several techniques, and the selection of the most suitable approach depends on the purpose, requirements, and type of constraints of the geodetic network. In recent years, satellite positioning has become the most popular technique for measuring geodetic networks, especially those with large baseline lengths. Satellite positioning techniques such as the global navigation satellite system (GNSS) offer several advantages that make them valuable for a variety of applications. These advantages include high accuracy, global coverage, cost and time efficiency, versatility, and needlessness for intervisibility between network points.

One of the most critical stages in establishing geodetic networks is the design stage. This stage involves determining the network configuration by selecting the distribution of geodetic point locations and developing an observing plan for the network measurements [1]. The primary goal of this stage is to guarantee achieving desirable accuracy, high reliability, and minimum cost [2]. Grafarend [3] categorized the geodetic network design problem into four groups:

1. Zero-order design (ZOD): selection of optimum datum definition.

2. First-order design (FOD): design of the network configuration by determining the optimum network point locations.

3. Second-order design (SOD): selection of observation weights.

4. Third-order design (TOD): development of the existing network through optimal network densification.

Additionally, mixture order design (MOD), also known as combined order design (COMD), addresses both the FOD and SOD optimization problems simultaneously [4,5]. The optimization of geodetic networks with different orders has been of paramount importance to geodesists and researchers for an extended period. Recently, it has become even more critical due to the rapid use of global satellite observations. Moreover, the optimization process ensures high accuracy, good strength, reliability of geodetic networks, and cost effectiveness of the observation process. Basically, two primary techniques can be employed to solve the optimal design problems of geodetic networks: traditional methods and global optimization approaches. Traditional techniques include the trial-and-error method and analytical approaches such as linear or quadratic programming [1,6]. However, the rapid development of computer technology in recent years has brought global optimization methods to the forefront. These methods include neural networks, particle swarm optimization (PSO), genetic algorithms (GAs), and simulated annealing (SA) [7,8]. These global optimization approaches have demonstrated their effectiveness in various fields due to their ability to perform global searches, minimize the number of adjustable parameters, and execute calculations at a high speed [9]. In contrast, traditional techniques face several limitations. For instance, the trial-and-error method may fail to yield optimal network results, even after extensive calculations. Analytical methods may lead to undesirable outcomes, such as negative weights or disconnected networks. Moreover, traditional techniques may produce local optimization solutions rather than the desired global optimization solutions [7].

Several recent studies have investigated the optimization methods of geodetic networks using various approaches. The optimal design of geodetic networks was initially investigated using the trial-and-error method by Maksimović et al. [9] and Pelzer [10] and later through analytical approaches by Doma and El Shoney [1], Grafarend and Sanso [11], and Schaffrin [12]. In recent years, different global optimization methods have been analyzed extensively by researchers. For instance, Doma and Elshouny [6], El-Shouny [7], and Sahabi et al. [13] examined the use of the GA method for solving the optimal design of geodetic network problems, while Farhan et al. [2], Yetkin et al. [14], and Said et al. [15] analyzed the use of the PSO method in solving these network problems. The work of Doma [16] addressed using the butterfly optimization algorithm (BOA) method for the optimal baseline selection for global satellite system (GPS) networks. Alluhaybi et al. [17] compared the use of GAs and PSO for second-order problems of genetic horizontal deformation monitoring in geodetic networks. Doma and Sedeek [18] investigated the use of five optimization algorithms, artificial bee colony optimization (ABC), ant colony optimization (ACO), GA, PSO, and SA, to determine the optimal GNSS constellation.

Most of these previous studies primarily focused on either a simulated dataset of geodetic networks or small-scale geodetic networks with a limited number of points and a short baseline length. In contrast, this research addresses the optimal design of a large-scale geodetic network using real geodetic data that includes a significant number of points and long distance baselines. This network contains 30 points and 435 baselines, ranging from 4.8 to 301.7 km with a total baseline length of 95168.2 km. This network is located in the Sinai region in the northeastern area of Egypt, and it was designed, established, observed, and solved using relative positioning techniques. Practically, these large-scale geodetic networks demand significant time, effort, and high cost. Therefore, selecting an optimal design for these networks is essential to eliminate unnecessary observations while meeting the required accuracy criteria. In this study, we addressed the SOD problems of the GNSS. The results obtained from these methods were analyzed and compared with those derived using traditional optimization techniques. This article first introduces the geodetic network optimization formula and then explains GA and PSO methods. The study area, observations, and geodetic network solutions are then presented. Finally, a discussion of the obtained results and the conclusions of this article are presented.

2 Geodetic network observations and constraints

Geodetic observations refer to a set of measurements and data collected using various surveying techniques to determine the accurate positions and elevation data points. These observations are classified into two main types: necessary and redundant observations. Necessary observations represent the minimum number of observations required to achieve the primary survey objectives to determine the unique values of the unknown. In contrast, redundant observations are those collected more than the necessary observations. They play a critical role in applying adjustment methods to improve the accuracy and reliability of network point data. Redundant observations are valuable for network quality control, minimizing errors and consistency checks within geodetic networks.

In the GPS network, the redundant observations refer to baselines that are observed in addition to the minimum number of baselines needed to solve the network data. These redundant baselines can be either an independent re-observation of previous measurements or an observation of a point from another base [1]. The necessary and redundant observations for the GPS network can be calculated using the following equations [19]:

where n _o is the number of necessary observations (baselines), n is the total number of net points, f is the number of knowns, R is the number of redundant observations, and L is the number of total observations.

The number of necessary and redundant observations in a geodetic network depends on the type of network constraints. They can be classified into three main types [7,20] as follows: free or floating, minimum constraint, and over-constrained networks. Free or floating networks are networks that contain no control points, and the coordinates of all points still need to be calculated. Minimum constraint networks contain the minimum number of control points required for solution. While an over-constrained network contains more control points than necessary, ensuring the quality and accuracy of the geodetic measurements and facilitating adjustment procedures [21].

3 Optimization of the GNSS network

The primary goal of the design stage of geodetic networks is to achieve optimal precision, reliability, and cost-efficiency. Network optimization aims to select the best network configuration and observing plan to satisfy quality requirements at the lowest possible cost. An optimal design eliminates unnecessary observations which reduces the effort needed for measurements, minimizes time and cost, and mitigates the impact of gross errors. Identifying the optimal configuration of geodetic networks and determining the optimum weights for observations have received much attention from geodesists. By solving optimization procedures, the desired criteria can be achieved with maximum homogeneity and isotropy conditions while minimizing cost [5].

Recently, GNSS surveys have rapidly become the most popular method for designing new geodetic networks and densifying the existing ones. Unlike classical survey networks, which require inter-visibility between each consecutive station, GPS networks do not have this requirement. In a GPS network survey, at least two receivers are used simultaneously to collect observations over a fixed period of time [17]. One or more receivers are installed on control points with fixed positions to process data using relative positioning techniques and estimate the accurate coordinates of the surveyed points. The result of these observations is a baseline vector between each pair of receivers observed simultaneously. These baselines, classified into independent or dependent and called “trivial” baselines, do not provide a unique solution for stations. The total number of baselines and the number of independent baselines in a network can be computed using the following equations [7]:

where N is the number of receivers deployed on the network.

4 GNSS network problem’s mathematical formulation

The design of GNSS networks is crucial to meet the diverse needs of users across various sectors for achieving the requirements of reliability, accuracy, and global coverage. The main objective of the network design stage is to determine the optimal network configuration and observing plan that satisfies quality criteria at a minimum cost while avoiding unnecessary observations. Dispensing these unnecessary observations saves time, effort, and cost of observations as well as reduces errors in observations.

The FOD problem in the GNSS network, also referred to as network configuration, is most similar to the FOD problem in a terrestrial network. However, differences between GNSS and terrestrial network configuration designs exist, such as the needlessness for intervisibility conditions in the GNSS network. The GNSS network configuration design problem basically deals with two factors: first, determining the appropriate observation session times and second, selecting the optimal ground network point locations. The most accurate GNSS observation results are achieved by observing all satellites for as long as possible and measuring all network baselines for the processing stage. However, this approach is impractical due to the significant time and cost involved, especially for large GPS networks with long baselines. For long periods, the main concern for geodesics and researchers is to find the optimum design of GPS network configuration and observation weights to meet the required criteria at a minimum cost. Several studies have been applied theoretically and practically to investigate the design of geodetic networks [5,6,22,23,24,25]. Many of these recent publications have concluded that the necessary observations should be identified and unnecessary observations should be eliminated according to the required criteria [1].

A GNSS network can be defined as a set of stations with coordinates determined through a series of observation sessions formed by placing receivers on the stations including both new points and fixed control stations [26]. The covariance matrix of the vector between stations is written as follows [2]:

The total weight matrix P in the adjustment model is [1,15]

where Σ L i is the covariance matrix of the observed vector i and m is the number of observed vectors.

During the design stage, both A and P matrices are known, and they are used to compute the prior-covariance matrix of the unknown [27]:

where σ o 2 is the prior variance factor usually taken as 1 in the design stage, A is the configuration matrix, and P is the weight matrix of observations.

Collecting the necessary observations, the precision of the GPS network can be computed using the following equation:

where A ₁ is the configuration matrix for necessary observations and P 1 is the weight matrix of the necessary observations.

After calculating the network precision using the minimum necessary observations, we can improve this precision by measuring redundant baseline observations. The modified variance–covariance matrix “C _f” is given by

The main objective is to select the optimal arrangement for the redundant baseline observations to improve the precision of the network and achieve the desired criteria with minimum cost, time, and effort. The solution to this optimization problem can be found using various methods. In this study, we will solve this problem using both GAs and PSO global methods and compare their results with those obtained from classical techniques such as the trial-and-error method.

5 Optimization approaches

Historically, optimization problems for different types of geodetic networks were solved using traditional methods, such as the trial-and-error method, or analytical approaches, such as quadratic programming and linear programming. Unfortunately, these methods have notable limitations. The trial-and-error method often demands significant effort and computations, yet it may fail to find the optimal network solution. Similarly, analytical methods can produce unrealistic solutions, such as negative weights or disconnected networks. They may also converge to a local optimal solution rather than achieving a global optimum solution [7,27]. Recently, with the rapid advancement of artificial intelligence applications, various intelligence optimization techniques have been increasingly adopted for solving geodetic networks, such as SA, GAs, and PSO.

GAs were first introduced in 1975 by Professor John Holland at the University of Michigan, and later they were developed, analyzed, and explained by one of Holland’s students, Goldberg in 1989 [28]. GAs are general purpose heuristic search algorithms that use evolutionary principles to identify the best fit solutions. The use of these algorithms has increased due to their algorithmic simplicity and ability to reach optimization solutions easily even for complex optimization problems [6]. The GA process requires only an objective or fitness function to define the relationship between the parameters. The GA process starts with creating a random initial population out of the entire range of possible solutions. This initial population may be represented as strings of binary digits or other data structures. Then, each solution in the population is evaluated by a fitness function to assign a fitness value for each solution to measure how close the solution is to the optimal solution.

Genetic operation consists of four main operations to go forward from one population to the next one. The first operation is the selection mechanism of new solutions among the created population for producing the next generation. Crossover is the second operation of genetic steps that merges two chromosomes to produce new chromosomes with better characteristics. The mutation operation deals with the most fit solutions by changing some of the binary digits randomly. It helps prevent the algorithm from converging into the local optimum. The final genetic step is eitism, which selects the fittest solution from any generation and transfers it to the next generation to ensure the fitness value is not less than the fitness values in the previous generation. The process terminates when the optimal fitness value is reached or when no significant improvement occurs over consecutive generations [7]. The mathematical equations of GA operations can be summarized as follows [28]:

First, the initial population P 0 is calculated using the following equation:

where N is the population size and x i is an individual solution.

The selection process of an individual (x _i ) is given by the following equation:

where P ( x i ) is the probability of selecting the individual (x _i ) for reproduction or for the next generation, f ( x i ) is the fitness value of the individual (x _i ), N is the total number of individuals in the population, and ∑ j = 1 N f ( x i ) is the sum of fitness values of all individuals in the population.

Then, crossover and mutation processes are applied using the following equations:

where x 1 p and x 2 p are two parents which combine to produce the offspring.

The algorithm stops when a termination condition is met:

where T max is the maximum number of generations and f target is the desired fitness value.

PSO is a computational optimization technique that solves problems by applying an iterative heuristic search algorithm to converge to global optimization. The inspiration for PSO was from the social behaviors of swarming species, such as birds and fish. PSO was originally developed by Dr. Eberhart and Dr. Kennedy in 1995. The PSO approach is more objective and simpler than other optimization methods, and it has the efficiency in exploring large and complex search spaces. So, it is widely used in various fields such as engineering, data mining, machine learning, and different optimization problems. However, similar to stochastic optimization algorithms, it does not guarantee finding the optimal solution but rather aims to find good solutions in a reasonable amount of time. The PSO technique involves a swarm of candidate solutions, known as particles. Each particle represents a possible solution to the optimization problem, and the position of the particle corresponds to a point in the search space. During the optimization process, each particle adjusts their position and velocity around the space search based on their own experience and the experience of their neighbors. This process is repeated for multiple iterations to find the near optimal solution [2,15,29]. Figure 1 illustrates the flowchart of the GA and PSO processes.

Figure 1

Flowchart showing GA and PSO processes.

The velocity and position equations of each particle, initially developed by Kennedy and Eberhart, can be described as follows [29]:

where V ( t ) denotes the particle’s current velocity, X ( t ) denotes the particle’s current position, c 1 and c 2 are the so-called acceleration coefficients, usually equal to 2, r 1 ( t ) and r 2 ( t ) are the random variables uniformly taken in the range [0, 1], p best ( t ) presents the best previous position of each particle, and g best ( t ) presents the best previous position of all particles.

6 Case study and results

The study area is located in Egypt in the Sinai region along the Red Sea coastal lines. This area has continuous development projects and needs protection works along its coastal zones to mitigate the impact of climate change and global warming. Consequently, a permanent geodetic network with high accuracy is essential for these zones. The investigated geodetic network points were established along the Red Sea coastal zones of both the Suez and Aqaba Gulfs. These points cover both sides of the Suez Gulf extending from Suez city in the north to Ras Gareb city in the south on the east side and to Sharm El-Sheikh city on the west side. The network also extends from Sharm El-Sheikh to Taba city, located in the northern area of the Gulf of Aqaba. The total longitudinal length of these coastal lines is approximately 650 km. The locations of the established points and the network geometry are shown in Figure 2. The establishment and observations of the geodetic network points were carried out by the Survey Research Institute on behalf of the Egyptian Coastal Protection Agency [30]. In the following sub-sections, the different stages of establishment and measurements of the geodetic network points are discussed.

Figure 2

Location of network points and ESA control points.

6.1 Establishment of geodetic network points

The first stage of this project was the planning stage. The main purpose of this stage was to select the most appropriate locations for establishing network points along the coastal zones considering future construction projects throughout the area. Also, it was important that the selected locations should be easy to reach and safe to ensure the long-term safety of the points. For this reason, the points were constructed as a concrete prism of 40 cm × 40 cm for the upper section and 60 cm × 60 cm for the lower section. The prisms were 1 m in height, with 50 cm embedded underground for stability. A total of 28 points were established across the study area, and their approximate coordinates are presented in Table 1. Photos of two of these network points taken during GNSS observations are shown in Figure 3.

Table 1

Approximate coordinates of the network points

Point	X	Y	Z	Point	X	Y	Z
p1*	4652030.523	2698782.915	3417283.38	p16	4674684.457	2595312.49	3465777.944
p2*	4662101.264	2467274.325	3573927.147	p17	4676406.878	2583838.744	3471983.564
p3	4682568.091	2674025.458	3395075.789	p18	4681049.761	2572166.304	3474371.422
p4	4691286.839	2659055.527	3394786.42	p19	4683000.106	2517909.673	3511097.885
p5	4689871.153	2647293.968	3405827.953	p20	4684380.292	2502429.813	3520235.844
p6	4688959.39	2636895.485	3415075.477	p21	4682254.844	2485599.247	3534848.786
p7	4691653.584	2627953.191	3418243.602	p22	4676427.671	2469761.027	3553541.788
p8	4695724.504	2616229.282	3421621.774	p23	4662834.242	2463473.401	3575474.246
p9	4702937.583	2607700.214	3418265.272	p24	4648072.015	2474631.375	3586893.442
p10	4707821.112	2568310.936	3441113.226	p25	4624751.057	2508758.9	3593277.914
p11	4707221.036	2541315.639	3461778.11	p26	4622985.398	2511850.629	3593389.996
p12	4662144.757	2673746.813	3423114.883	p27	4593029.004	2544402.59	3608767.084
p13	4671514.624	2652056.61	3427136.88	p28	4590109.718	2551364.851	3607617.692
p14	4674735.854	2635994.157	3435073.885	p29	4582626.761	2559211.378	3611550.562
p15	4677087.792	2621671.624	3442792.669	p30	4569314.608	2572389.635	3618971.363

*P1, P2: ESA control points.

Figure 3

Representative photos of network points.

7 Results

To evaluate the effectiveness of both PSO and GA methods, the MATLAB program was utilized to determine the optimal design of the geodetic network. This program identified the optimized baseline that needed to be observed to achieve the required accuracy of the network points. The network points were required to have a minimum accuracy of 2 cm or better for horizontal positioning and 5 cm or better for vertical positioning. The program was applied to solve both minimum and over-constrained network cases using both PSO and GA methods.

The PSO parameters utilized in this study are detailed in Table 2. These parameters were selected based on the recommendations in the study of Yavari et al. [31] and further through experimental trials to achieve an optimal balance between the global and local searches of PSO. However, it should be noticed that PSO is rather stable to the mild changes of these parameters [18]. Similarly, the GA parameters are outlined in Table 3. The chromosome length is dynamic, depending on the number of variables, but was standardized to a 32-bit representation to accommodate the expected numerical precision, which includes two integer digits and six decimal places.

Subsequently, the network was processed using TBC to analyze the optimized baselines determined by the program for each network solution type. To fairly assess the achieved results of optimization using PSO and GA methods, their outcomes were compared with those obtained from the solution of the network using the traditional method. Table 4 shows the optimization results, including the number and lengths of baselines, as well as the number and percentage of baselines removed for each solution type using PSO and GA methods.

As is shown in Table 4, PSO achieves the highest reduction percentage (43.45%) in the number of baselines for the minimum constraint solution, with a slightly lower percentage (40.12%) for the over-constrained solution. While GAs show a consistent reduction percentage for both minimum constraints (42.30%) and over-constrained solutions (42.03%). Regarding optimized baseline lengths, both methods yielded approximately equal baseline lengths, with the reduction percentage ranging from 40.03 to 42.52% of the total lengths of all possible network baselines, which was 59168.20 km. Figures illustrating all possible network baselines and the optimized baselines for PSO and GA methods are included in Appendix A. The results indicate that while the outcomes of both methods are closely comparable, GA results are slightly better for the over-constrained solution, whereas PSO results are marginally better for the minimum constraint solution. The estimated horizontal precision of the processed baselines for each solution is shown in Figure 4a–d with a statistical summary provided in Table 5.

Figure 4

(a) Baseline horizontal precision of minimum constraint solution using PSO. (b) Baseline horizontal precision of over-constrained solution using PSO. (c) Baseline horizontal precision of minimum constraint solution using GAs. (d) Baseline horizontal precision of over-constrained solution using GAs.

Table 5 and Figure 4(a–d) indicate that PSO and GAs yield the same minimum and maximum values of 0.005 m and 0.016 m, respectively, for the minimum constraint solution. Similarly, for the over-constrained solution, both methods show the same minimum value of 0.003 m, with maximum values of 0.012 m for GA and 0.013 m for PSO results. Scatter plots demonstrate that over-constrained solutions using both methods have the highest positive correlation (R ² value), making them more suitable for networks with long baseline lengths, where minimizing errors are critical. In contrast, the minimum constraint solutions of both methods show lower correlation results, with a slight advantage for the GA solution. While these solutions may be less ideal for networks with long baselines, they still achieve reasonable predictability within a moderate range. Figure 5 presents a boxplot of differences (∆X, ∆Y, and ∆Z) between the values estimated using each network solution type and those determined by solving the network using traditional methods. A statistic summary of these differences is provided in Table 6. The boxplot reveals that over-constrained optimization results obtained through PSO and GA methods are more accurate with consistent coordinate differences, while minimum constraint results are less accurate with higher variability in differences.

Figure 5

Boxplot showing differences ∆X, ∆Y, and ∆Z for each solution.

8 Discussion

The main objective of this study is to address the optimal design solution for large-scale geodetic networks using artificial intelligence approaches. PSO and GA methods were explored for designing minimum constraints and over-constrained types of GNSS network. These methods were employed to identify the optimal baselines required to achieve the desired network accuracy. These methods were applied to a large-scale geodetic network, containing a large number of points (30) and 435 baselines, ranging from 4.8 to 301.7 km with a total baseline length of 95168.2 km. Both PSO and GA methods were employed to solve the network for both constraint types. The optimization results of all cases were presented, compared, and analyzed. The analysis included horizontal precision values for the optimal baselines and differences in network point coordinates derived from the optimization methods compared to the traditional method.

The optimization results demonstrate that both methods effectively reduced the number and total length of optimal selected baselines by approximately 42.4 and 41.4%, respectively. This reduction significantly decreased the observation time and overall project cost. Among the findings, PSO provided the best result for the minimum constraint network, while GAs provided more consistent performance across both network types. Additionally, the differences between coordinates derived from the optimization methods and those obtained from traditional methods indicated that over-constrained solutions using both PSO and GAs are more accurate, whereas minimum constraint solutions exhibited greater variability in coordinate differences.

9 Conclusions

The selection of an optimal observation plan for large-scale GNSS geodetic networks is essential to reduce the time, effort, and cost while achieving the desired accuracy. Several global intelligence techniques have been explored as alternatives to traditional methods. In this research, PSO and GAs demonstrated their effectiveness, simplicity, and fast computation processes in determining optimal solutions. Both methods reduced the observed baselines by approximately 42% of the total network baselines in terms of number and length. This optimization resulted in saving about 40% of the required fieldwork time and reducing costs by a similar margin, all the while meeting the required network accuracy.

The findings indicate that networks with over-constrained solutions are more accurate than those with minimum constraint solutions, as the latter exhibited greater variability in coordinate differences when using both PSO and GAs. For future research, we recommend investigating alternative global intelligence techniques, such as SA or ACO, as well as exploring machine learning approaches for the optimal design of geodetic networks.