Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 2, No. 3, 2022 

 

59 

Three-dimensional positioning based on TOA 
combining acoustic signal and optical signal 

Run Zhang 

Key Laboratory of Electronic and Information Engineering (Southwest Minzu University), State Ethnic Affairs Commission, College of 

Electronic and Information, Southwest Minzu University, Chengdu, China. 

 

Abstract: In recent decades, the massive depletion of the earth's terrestrial resources has turned people's attention to the ocean, 

so many underwater technologies have emerged and developed rapidly with the comprehensive progress of human science. Since 

the TOA positioning algorithm only needs a synchronous time source, the application of the TOA positioning algorithm by the 

traditional underwater wireless sensor system network is very common. Underwater positioning technology uses underwater 

sensor network (UWSN) to locate unknown nodes, which can be divided into underwater acoustic communication positioning 

and water optical communication positioning according to different transmission channels. The characteristics of underwater 

acoustic communication and water optical communication determine that the former has better accuracy in long-distance 

transmission, and the latter has better accuracy in short-range positioning. In this paper, the weight coefficients of sound and 

light are set separately to obtain the distance cut-off values of sound and light channels. The distance demarcation value can be 

changed according to the actual environment and needs. We will set the threshold of the impact factor of the underwater 

environment to ensure that the accuracy error caused by environmental losses is minimized. The reasonable arrangement of 

UWSN's anchor nodes will maximize energy savings. Simulation results show that the proposed method makes reasonable use 

of the advantages of acoustic signal and optical signal, and obtains more accurate positioning results after setting the threshold. 

Keywords: UWSN; TOA; LOS; Acoustic signal; Optical signal; Anchor node arrangement. 

 

1. Introduction 

Underwater wireless localization has been drawing much 

attention for many years because of the security of ocean 

environment and the popularity of wireless smart sensor 

networks (WSSN). Due to the localization accuracy plays a 

key role in underwater wireless localization,so many practical 

localization methods was put forward,such as Time of Arrival 

(TOA)–, Time Difference of Arrival (TDOA),, Received 

Signal Strength (RSS)–,Angle of Arrival (AOA), ,and hybrid 

localization–. 

In an outdoor environment,the unknown node (UN) can be 

located with highly accuracy from the Global Positioning 

System (GPS),or from BeiDou Navigation Satellite 

System.In the underwater environment, GPS cannot be 

applied to these land-based devices due to the shadow effect 

of water. Therefore, underwater wireless sensor networks can 

be well used for locating unknown nodes. 

The specific positioning method needs to be based on the 

hardware facilities and positioning environment of the 

wireless sensor network, for example, using the angle of 

arrival A0A method, the node needs to be equipped with a 

special antenna array; The arrival time TOA or arrival time 

difference TDOA method only requires the node to be 

equipped with a clock system with a certain accuracy. This 

will help improve the positioning accuracy of unknown 

nodes.In this paper, we will study the underwater positioning 

of TOA, and the main innovations are as follows: 

1) Set a threshold for the dynamic factors of the underwater 

environment, the anchor node is detected, and when the 

environmental impact factor is greater than this threshold, the 

positioning process is not carried out. 

2)Take advantage of the advantages and disadvantages of 

acoustic signals and optical signals, complement them, and 

obtain higher accuracy beyond this distance value. 

3)According to the nature of underwater influencing 

factors, reasonably set the position of the anchor node. 

2. Problem overview 

In this section, we propose a problem to improve the 

positioning accuracy of the TOA positioning algorithm 

combining acoustic and optical signals in underwater wireless 

sensor networks, and to rationally arrange nodes in the 

underwater environment where LOS dynamically changes. 

Given that existing wireless sensor network (WSN) 

localization methods will be based on the TOA method, the 

following discussion will first focus on TOA optimization, 

and then will discuss localization in LOS environments. 

 

Fig.1 Schematic diagram of spatial positioning 

First, given four non-coplanar anchor nodes in three-

dimensional space, an unknown node to be located. Use a 

matrix to calculate the coordinates of unknown nodes. 

Supposing that four points A, B, C and D are known in the 

space, and their coordinates are (X1, Y1, Z1), (X2, Y2, Z3), 

(X3, Y3, Z3), (X4, Y4, Z4). The coordinates of unknown node 

P are: (X, Y, Z), and the distances from P to the four points are 

d1, d2, d3, d4, as shown in figure 1. According to the distance 



 

60 

formula, the following equations can be obtained:  

{

(X1 − X)2 + (Y1 − Y)2 + (Z1 − Z)2 = d12

(X2 − X)2 + (Y2 − Y)2 + (Z2 − Z)2 = d22
(X3 − X)2 + (Y3 − Y)2 + (Z3 − Z)2 = d32

(X3 − X)2 + (Y3 − Y)2 + (Z3 − Z)2 = d32

    (1) 

Arranging (1)and can get: 

 

{

X2 − 2X1 ∗ X + Y2 − 2Y1 ∗ Y + Z2 − 2Z1 ∗ Z = d12 − X12 − Y12 − Z12
X2 − 2X2 ∗ X + Y2 − 2Y2 ∗ Y + Z2 − 2Z2 ∗ Z = d22 − X22 − Y22 − Z22
X2 − 2X3 ∗ X + Y2 − 2Y3 ∗ Y + Z2 − 2Z3 ∗ Z = d32 − X32 − Y32 − Z32
X2 − 2X4 ∗ X + Y2 − 2Y4 ∗ Y + Z2 − 2Z4 ∗ Z = d42 − X42 − Y42 − Z42

    (2) 

 

The third-order difference of the equation is used to 

eliminate the higher-order term to obtain the following first-

order equation:  

 

{

2X(X2 − x1) + 2Y(Y2 − Y1) + 2Z(Z2 − Z1) = d12 − d22 + X22 − X12 + Y22 − Y12 + Z22 − Z12
2X(X3 − x1) + 2Y(Y3 − Y1) + 2Z(Z3 − Z1) = d12 − d32 + X32 − X12 + Y32 − Y12 + Z32 − Z12
2X(X4 − x1) + 2Y(Y4 − Y1) + 2Z(Z4 − Z1) = d12 − d42 + X42 − X12 + Y42 − Y12 + Z42 − Z12

  (3) 

The equation is expressed in matrix form: AC=B 

A =[

2(X2 − X1) 2(X3 − X1) 2(X4 − X1)
2(Y2 − Y1) 2(Y3 − Y1) 2(Y4 − Y1)
2(Z2 −  Z1) 2(Z3 −  Z1) 2(Z4 − Z1)

]        (4) 

C =
𝑋
𝑌
𝑍

          (5) 

B =
d12 − d22 + X22 − X12 + Y22 − Y12 + Z22 − Z1

d12 − d32 + X32 − X12 + Y32 − Y12 + Z32 − Z12
d12 − d32 + X32 − X12 + Y32 − Y12 + Z32 − Z12

       (6) 

The equation can be written as: 

C = A-1B                 (7) 

3. TOA Based Distance Measurements 

TOA(time-of-arrival) positioning is a distance-based 

positioning method, which can be used to locate unknown 

nodes by multiplying the arrival time of the signal by the 

propagation speed to obtain the distance. Without considering 

the error, only four non-coplanar anchor nodes are needed in 

the three-dimensional space to calculate the target position. 

Without losing generality, it is assumed that the start time 

of each source ranging is 0, and the time signal from the ith 

anchor node is received at time ti, which means that ti is a 

TOA measurement. The relationship between distance and 

time is as follows-: 

 

di = ti * c, i=1,2,3,…n              (8) 

  

Where ti is the time of the signal from the anchor node to 

the UN,di is the calculated distance from the UN to the ith 

anchor node,c is the speed of the signal under water.The 

calculation process of UN coordinates is shown in step A. 

4. Acoustic And Optical Hybrid Based 
Distance Measurements 

4.1. Acoustic  

We set the received signal of the underwater acoustic 

channel as:  

 

xri(t) = A sin(2πfAct + φi + φwi + φ0) + wi(t)         (9) 

 

Where i represents four non coplanar anchor nodes that 

have been successfully positioned, a represents the amplitude 

of the received signal, φi represents the phase shift due to 

interference noise, φwi represents the phase uncertainty, and 

wi(t) is a Gaussian zero mean random noise. 

The attenuation of underwater acoustic signal strength is 

related to various factors, which will affect the amplitude and 

frequency of the acoustic signal. The acoustic signal strength 

is written as: 

10 log10 ( 
A0

A
 )2= k · 10 log10 d + d · 10 log10α    (10) 

Where k represents the geometric diffusion factor, which is 

usually between 1 and 2, d represents the distance between 

the anchor node and the UN, α represents the underwater 

acoustic absorption coefficient, and the value of α is usually 

calculated by Thorp's formula, :  

10 log10 α = 
0.1fAc2

1+fAc2
+

40fAc2

4100+fAc2
+2.75×10-4×fAc2+0.003  (11) 

4.2. Optical 

Underwater optical communication systems employ 

directly detectable intensity modulation (IM/DD), which uses 

an ON–OFF keying (OOK) in the physical layer. The initial 

simulations will be done in the LOS environment, which will 

save us from having to talk about light in the presence of 

obstacles.The radiant intensity of light underwater can be 

expressed as,: 

Is(d, φ)=P
m+1

2𝜋d2cos m φ        (12) 

Where d is the angle between the optical path direction of 

the anchor node and the vertical direction, d is the distance 

between the anchor node and the UN, m is the Lambert’s 

modulus of the beam effect of the anchor node, p is the 

average optical power, where m is expressed as,: 

  m=
− 𝑙𝑛 2

ln(cosφ1/2)
                   (13) 

 

Fig.2 Main parameters of the optical link model 

Let Aeff be the effective area of the UN receiving the 

incident light radiation,and which is expressed as:  

A eff(ψ) = f(ψ)Ar cos ψ, |ψ| ≤ ψc           (14) 

where Ar is the UN area, ψ is the incident angle and f(ψ) is 

the light concentrator gain given by:  

f(ψ) = {
n2

sin2ψc
, if |ψ|  ≤  c

0        , if|ψ|  > c   
               (15) 

Where ψc is the half-angle Field of View (FOV) of the 

optical detector, and n is the refractive index of the seawater. 

Since the attenuation of light underwater is mainly 

absorption, in order to accurately determine the position of 

the UN, we set the absorption coefficient as:  

Lch = exp(−Kad)               (16) 

where Ka is the diffuse attenuation coefficient. 

Using (15), (17), and (19), the received optical signal 

strength in dB at the UN side can be calculated as: 

POW = 10 log10(IsA effLch) + ξ        (17) 

Where ξ is the average value of noise (in dB) generated by 

natural light underwater. 

5. Improvement of toa 

In underwater UN positioning, acoustic positioning is often 

used due to the limitations of the scale of the simulated 



 

61 

environment and energy saving. However, considering the 

advantages of underwater optical communication such as 

high speed, high bandwidth, and high security, its advantages 

in short-range communication are very significant. Therefore, 

considering the underwater situation of sound and light, we 

evaluate the advantages of sound and light in distance dd, that 

is, when the positioning distance d is greater than dd, the 

redundant sound part d (d-dd) uses acoustic communication 

data, and the range from 0 to dd uses optical communication 

data, combined with sound and light to locate unknown nodes. 

In the following simulation, we will carry out the process of 

obtaining dd by this method several times, and extend the data 

to the general case, that is, save the dd data obtained by each 

anchor node using optical communication, which greatly 

saves energy consumption. 

Table 1. Hybrid Acoustic/Optical Algorithm 

1: Locate an UN and use optical communication first 

2: If localization distance d<=dd 
3: Notify that the acoustic communication does not need 

positioning, obtain the optical communication distance d, 
and the positioning of the anchor node is completed   

4: end if 
5: If d>dd 
6: Notification acoustic communication positioning, 

then d=dd+dacoustic; 
7: end if 

6. Simulation 

In this section, we will simulate the underwater positioning 

algorithm on the Matlab platform and simulate the 

localization problem of the TOA algorithm in the underwater 

LOS environment, as shown in Figure 3: 

 

Fig.3 LOS simulation environment 

Figure 3 shows the anchor node arrangement in an 

underwater LOS environment. Place four anchor nodes at the 

four corners of the water surface in three-dimensional space, 

and one anchor node in the center of the bottom. Such a sensor 

network system arrangement will effectively overcome the 

characteristics of underwater environment loss, that is, the 

underwater environment due to salinity, temperature and 

other factors, the deeper the loss of the signal, the greater the 

loss. Therefore, such an anchor node arrangement will also be 

a major innovation in this paper. 

First, we will simulate the distance critical values of 

acoustic and optical signals during positioning, and the 

simulation results are shown in Figure 4: 

 

Fig.4 Determination of acousto-optic distance 

We believe that the opposite properties of underwater 

sound and light will create a distance equilibrium point. We 

believe that the cost and benefit of acousto-optic positioning 

are almost the same in the distance balance point after the 

judgment is completed, and this balance point will also be 

regarded as the critical point of acousto-optic combination. As 

can be seen in the simulation results shown in Figure 4, the 

simulation result obtained is 98m, and this distance value can 

be changed according to the actual environmental 

requirements. Taking advantage of the combination of 

acoustic signal and optical signal, the positioning simulation 

of TOA algorithm is carried out, and the results are shown in 

Figure5: 

 

Fig.5 Hybrid acoustic/optical localization 

It can be seen from Figure 5 that the positioning accuracy 

of the TOA positioning algorithm after the combined acousto-

optic signal is higher than that of a single communication 

channel after the obtained distance value. 

7. Conclusion 

Underwater wireless sensor network positioning system 

using TOA positioning algorithm mostly uses acoustic signals 

for signal transmission, and acoustic and optical signals have 

their own disadvantages, so the advantages and disadvantages 

of the two complement each other, and better positioning 

accuracy can be obtained than a single signal. At the same 

time, in the setting of the underwater environment, this paper 

also considers the influence factor of the underwater 

environment, that is, if the impact factor is too large, the 

signal loss is large, which will seriously affect the positioning 

accuracy, and such influence parameters need to be monitored. 

We have made reasonable settings for the positioning system 

in three-dimensional space, which greatly reduces the use of 

anchor nodes and energy consumption. Simulation proves 

that the proposed algorithm has good positioning accuracy. In 

the future, we will consider anchor node placement and 

simulation in the NLOS environment. 



 

62 

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