An Average Speed of Sound Method for Underwater Positioning Based on Time of Arrival

This article uses the distance and sound spread time of each monitoring base station in the underwater positioning system, calculates the effective sound speed of the sound wave to the base station, and proposes a TOA underwater positioning method based on the average speed. The influence of small underwater sound changes the system positioning accuracy, reducing system positioning errors. Simulation experiments show that compared with the traditional fixed sound velocity method, this method has higher positioning accuracy and lower error. The positioning effect in a long baseline underwater positioning system is less dependent on the number of monitoring base stations. The cumulative probability of positioning error within 5 m can reach 70%, and the cumulative probability of error within 10 m can reach over 95%.


Introduction
In recent years, there were more challenges and opportunities for underwater positioning systems because of the stronger demands, such as marine resource exploration and underwater dynamic monitoring [1].Exploring underwater positioning methods of different scenarios was significant for the economic and social value of ocean utilization.Underwater acoustic positioning gained more attention and emerged numerous achievements.Among the various achievements of underwater acoustic positioning, there are three types of positioning systems based on long baseline positioning systems (LBL), short baseline positioning systems (SBL), and ultra-short baseline positioning systems (USBL) [2].In LBL, located estimation with high accuracy of the underwater target could be gained even without prior information [3], which provided underwater positioning services in vast and deep-sea areas generally [4].In SBL and USBL, the information of the located target is measured by using transducers that are installed on surface buoys, docking stations, or mobile platforms.The located estimation of the target is calculated by using array positioning methods.
Generally, SBL and USBL provided services for underwater robot location and navigation, and autonomous underwater vehicle recovery operations [5].
In [6], the uncertainty of sound speed and reduced errors were overcome by using ray tracing technology and Gauss-Newton iteration.In [7], sound speed was estimated based on acoustic propagation parameters in an unknown scenario and optimized the effectiveness of sound speed using particle swarm optimization.
In [8], the cluster feature field of sound signals was established and target positioning was obtained through the parameter feature matching search.In [9], an iterative constrained weighted algorithm was used to obtain higher positioning accuracy.In [10], the particle swarm algorithm coordination method was introduced to reduce the increasing errors of the Chan algorithm with distance.Although there are many methods for target positioning under the water, high accuracy often means computational complexity.How to reduce computational complexity, avoid local optimization and achieve both positioning accuracy and computational efficiency [11] remains one of the research hotspots in underwater positioning systems.

Problem Description
The number of n monitoring base stations with homogeneous characteristics and locations is assumed to deploy under the water, which is denoted as a 1 ,a 2 ……a n.And time synchronization is between base stations.The coordinates of base station a 1 are represented as (x 1 , y 2 , z 3 ), a i is represented as (x i , y i , z i ), and the target T is represented as (x t , y t, z t ).

2.1.TOA Localization Principle
The method is based on TOA, which obtained the position of the target by capturing the arrival times of the acoustic waves from the source to each monitoring base station and calculating the distances between the source and each base station.
By drawing circles with the base stations as centers and the propagation distances of the acoustic waves as radii, the intersection points or overlapping regions of multiple circles provide an estimation of the target's position.
If c denotes the propagation speed of the acoustic waves in the water, t i is the time taken for the acoustic waves to propagate from the target point T to the monitoring base station a i , and According to the TOA positioning principle, the system of equations is established, as shown in Equation 1.

2.2.Localization Model
From the previous assumption, considering the deployment of n monitoring base stations a 1 , a 2 ……a n , in the three-dimensional underwater area, the distance between base stations a i and a j is denoted as r ij , and the time taken for the acoustic wave propagated from a i to a j is represented as . And the coordinates of monitoring base stations are considered known and reliable conditions.As a result, it is possible to calculate the distances r ij between each monitoring base station accurately.When using base station a i as the sound source and remaining base stations as monitoring points, the speed of sound c ij from a j to a i could be calculated by using the relationship between the distance r ij and the propagation time taken t ij of the acoustic wave.The average value of c ij of all base stations is viewed as the effective speed of sound c i , , which represents the speed of any acoustic wave to reach monitoring base station a i within the area underwater.The formula for calculating c i is shown as Equation 2.

(
) To iterative calculate the effective speed of sound c 1 ,……c i ……c n which represents the speed of any acoustic wave to reach monitoring base station a i within the area underwater, that could use these effective speed c i to correct the fixed speed of sound c in Equation 1.To improve the accuracy of the distance between the target T and the monitoring base station a i , . In addition, according to the Chan algorithm [12], ) , ( , then Equation (3) could be transformed into the following form: ( ) ) By rewriting Equation (4) into the following form: , where T', and K 0 variables are independent of each other, and Q represents the covariance matrix of the measurement values.Equation ( 5) is obtained as:

2.3.Method Description Diagram
In long baseline positioning systems, the results of the TOA method based on the intersection areas of spherical surfaces primarily.Therefore, TOA requires 3 monitoring base stations at least, which are not in the same straight line.Additionally, to reduce the impact of the layout on the positioning accuracy of the system, the layout of the monitoring base stations should be symmetrically distributed as much as possible.Therefore, when the monitoring station base is deployed, the positions are fixed, and the coordinates are viewed as known in long baseline positioning systems.As mentioned earlier, assuming the coordinates of any base station a i is (x i , y i , z i ), the distance


, could be calculated between any two base stations a i and a j by using the Euler formula.
Usually, the times of sound waves to propagate back and forth between two points are equal.However, in underwater environments, the speed of sound is influenced by depth, salinity, temperature, and noise disturbances of underwater.Therefore, the time tokens of round-trip may differ between any two base stations a i and a j .Assuming a small constant  , and sound waves propagate from a i to a j are denoted as t ij , and propagate from a j to a i is denoted as t ji , if , it would be considered a large error between the two base stations, then propagation time tokens t ij and t ji would be recalculated until Using Equation 2, the average speed of sound c i for each base station could be calculated, and the time t i which propagated from target T to each base station could be obtained also.Finally, the estimated location (x t , y t, z t ) of the target T would be calculated by using Equation 5 and Equation 6.The algorithm flowchart was shown in Figure 1.Calculate the distance r ij between any two base stations Record propagation delay of waves round-trip between any two base stations: According to Formula 2, the average speed c i be calculated for waves reaching each base station Are the round-trip delays consistent between any two base stations?
The propagation delay of waves between target point T and each base station be recorded, t i The position estimation of target point T be calculated according to equations 5 and 6,(x t , y t , z t )

Simulation Verification
In underwater positioning systems, 1500 m/s is a fixed sound speed that is used commonly.This paper compared the performance of the average sound speed method with the traditional fixed sound speed method by using an acoustic ray model -Bellhop with Munk sound speed profile and Indian Ocean sound speed profile.The Munk sound speed profile is shown in Figure 2

3.1.Munk Sound Speed Profile Verification
The Munk sound speed profile is one of the typical deep-sea environment profiles used in underwater acoustic research.In the Munk sound speed profile, we consider an underwater area with a depth of 2 km and a propagation distance of 3 km in the east-west and north-south directions.Five monitoring stations are deployed in non-co-planar positions with coordinates (0,0,500), (3000,0,500), (3000,3000,1000), (0,3000,1000), and (1500,1500,2000), as shown in Figure 3.Under the Munk sound speed profile, the target is generated randomly and 100 times independent experiments are conducted by using both the average sound speed and fixed sound speed method.Through repeated verification, the trend of error range using average sound speed is consistent with the fixed sound speed method.However, the average sound speed method is easier to get high-accuracy results than the traditional method with the fixed sound speed in an underwater positioning system.Figure 4 shows the comparison of results under the Munk sound speed profile.
In addition, to examine the effect on the positioning accuracy of the average sound speed method when monitoring stations' depth changes, the paper adjusts the depth of monitoring stations with a step of 100 m, and without changing their horizontal coordinates.Starting with stations 1 and 2 descending from 500 m to 1000 m, stations 3 and 4 then ascending from 1000 m to 500 m, after each iteration of adjusting the depth of stations, 100 times independent random experiments are conducted, and the Root Mean Square Error (RMSE) values for both the fixed sound speed and average sound speed method are calculated.The simulation shows that, with the same number of monitoring stations, the trend of error range using average sound speed remains consistent with the fixed sound speed method.However, the average sound speed method yields smaller RMSE values and better positioning accuracy.Moreover, when stations 1, 2, 3, and 4 are at approximately the same depth, the deployment as a cube-shaped network makes the system easier to obtain smaller positioning errors.The RMSE of underwater positioning with station depth change is shown in Figure 5.

3.2.Indian Ocean Sound Speed Profile Verification
In the complexity of the marine environment, although Munk sound speed profile is used widely, some ASMA-2023 Journal of Physics: Conference Series 2638 (2023) 012010 IOP Publishing doi:10.1088/1742-6596/2638/1/0120107 methods such as actual measurement and inversion calculation are needed for obtaining the sound speed profile of specific application systems frequently.Therefore, to compare with the Munk sound speed profile, we select the Indian Ocean sound speed profile on April 28, 2013, which is located at 9°59'797"N, 86°59'706"E, as shown in Figure 2 (b).
Under the Indian Ocean sound speed profile, 100 times independent random experiments with the 5-station deployment shown in Figure 3 are repeated, and the results are shown in Figure 6.The analysis of positioning errors shows average sound speed method reduces errors significantly and obtains better effects.However, in experiments 37, 51, 57, 76, and 94, the target was located in a depth range of 0 m to 300 m and near the edge of the sound propagation radius, resulting in larger positioning errors by the relatively drastic sound speed variations.In addition, compared with the target results using the Munk sound speed profile in Figure 4, under the Indian Ocean sound speed profile, there is a better performance of the average sound speed method.The reason may be sound speed variation becomes more gentle than Munk when depth is below 300 m in the Indian Ocean sound speed profile, which is more suitable for the average sound speed method.
Typically, increasing the number of monitoring stations is one of the most effective methods to improve positional accuracy and reduce errors in underwater positioning systems.There are a few larger errors that stay in the 5-station experiments under the Indian Ocean sound speed profile, we reconstruct a 7-station underwater positioning system, and the deployment of monitoring stations is shown in Figure 7.The coordinates of the stations are as follows: (0, 0, 1000), (3000, 0, 1000), (3000, 3000, 1000), (0, 3000, 1000), (1500, 1500, 2000), (1500, 0, 1000), and (1500, 3000, 1000).Through 100 times independent random experiments, the accumulation probability curves of errors for both the average sound speed and fixed sound speed method are calculated with the deployment of 7-station and 5-station, as shown in Figure 8.As shown in Figure 8, whether it is a 5-base station or 7-base station deployment in an underwater positioning system, error accumulation probability curves of the average sound speed method are both over 70% within 5 m, and it increases to approximately 95% within 10 m.However, the error accumulation probability curves of the fixed sound speed method are only about 20% within 10 m, and it decreases to 5% within 5 m.Therefore, compared to the traditional method with the fixed sound speed in long baseline underwater positioning systems, the average sound speed method could obtain higher accuracy and reduce positioning errors of underwater positioning systems significantly.In addition, Figure 8 shows the fixed sound speed method performance better obviously in the 7-base station than in the 5-base station deployment.However, the average sound speed method's positioning accuracy in the 7-base station and the 5-base station do not much different, and the error cumulative probability curve is overlapping approximately.Therefore, it indicates that the positioning accuracy of the average sound speed method is affected less by the number of base stations, and is more suitable for applications in long baseline with weak base station deployments.

3.3.Analysis of Algorithm Complexity
The method proposed in this paper calculated the average sound speed between any two base stations and the sound propagation time token mainly.The complexity of the algorithm depends on the number of sound propagation between base stations, which formed a directed complete graph.If the number of base stations was n, the algorithm complexity of the average sound speed method was O(n*n-1).However, in the experimental verification of this paper, the average sound speed method was less affected by the number of base stations, indicating its suitability for monitoring weakly deployed long baseline underwater positioning systems.Therefore, when the value of n was small, the algorithm complexity of the average sound speed method would be lower relatively.

Conclusion
In underwater positioning systems, sound speed is one of the most important factors affecting accuracy.However, it is difficult to find a perfect sound speed optimization method for every practical application in reality, The average sound speed method could obtain better positioning accuracy in long baseline underwater positioning systems with base stations deployed weakly, especially in areas with gradual sound speed profiles, and it is a new reference for the diversity of underwater positioning system applications.

InitializeInput:
Number of base stations n, Coordinates (x i , y i , z i )

Figure 2 .
(a), and the Indian Ocean sound speed profile is shown in Figure 2 (b).Sound Speed Profile Diagram

Figure 3 .
Figure 3. Deployment diagram of the 5 base stations

Figure 4 .
Figure 4. Comparison of positioning results for targets under the Munk sound speed profile

Figure 5 .
Figure 5. RMSE Comparison of Base Station Depth Changes.

Figure 6 .
Figure 6.Comparison of target localization results under the Indian Ocean sound speed profile.