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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The great variability usually found in underwater media makes modeling a challenging task, but helpful for better understanding or predicting the performance of future deployed systems. In this work, an underwater acoustic propagation model is presented. This model obtains the multipath structure by means of the ray tracing technique. Using this model, the behavior of a relative positioning system is presented. One of the main advantages of relative positioning systems is that only the distances between all the buoys are needed to obtain their positions. In order to obtain the distances, the propagation times of acoustic signals coded by Complementary Set of Sequences (CSS) are used. In this case, the arrival instants are obtained by means of correlation processes. The distances are then used to obtain the position of the buoys by means of the Multidimensional Scaling Technique (MDS). As an early example of an application using this relative positioning system, a tracking of the position of the buoys at different times is performed. With this tracking, the surface current of a particular region could be studied. The performance of the system is evaluated in terms of the distance from the real position to the estimated one.

All systems need a good understanding of the medium in which they are deployed. Particularly, underwater medium is highly dynamic and difficult to model due to some effects as swell, turbulences or the irregular spatial distribution. Nevertheless, a propagation model is needed to test the behavior of any system prior its actual deployment. Until now, a large number of models have been proposed, based on different mathematical approaches such as ray tracing, normal mode or the parabolic equation, to name a few, where their use is suggested for different scenarios or purposes.

Ray tracing provides an intuitive approach to acoustic propagation, assuming that the energy of the wave is confined in different paths, allowing to think in rays rather than waves. This is a good assumption if the amplitude of the wave and sound speed do not change noticeably in a wavelength. Through all these years, several ray tracing codes have been developed [

Normal mode models are based on the integral representation of the wave equation, and provide the sound field as a sum of normal modes. They can compute transmission loss easily for a given combination of frequency, source depth, receiver depth and ranges, but they are range independent and the number of modes to compute depend on frequency, so they are recommended for frequencies below 500

Fast field theory is very similar to normal mode, but it uses an asymptotic expansion on the equation for the acoustic pressure field. The resulting infinite integral is evaluated by means of a fast Fourier transform and includes a branch line integral term that is usually neglected in normal modes [

Finite elements models divide the medium into a mesh. The length of the sides of this mesh is usually one tenth of the wavelength, and they intersect at nodal points. The wave equation is replaced by a system of algebraic equations, that can be solved to obtain the field at each node in the mesh. Because of that, a large computation time is required for long range and high frequency configurations [

Other approach is the finite-difference time-domain, where a discretization of the time dependent curl equations of Maxwell is performed, and the wave propagation is simulated in the time domain. The feasibility on underwater acoustics problems was introduced in [

The last main technique for modeling underwater acoustic propagation is the parabolic equation. This is an approximation of the elliptic Helmholtz equation, introduced in underwater acoustics in 1973 [

Whereas underwater modeling has been an active field since decades ago, positioning systems are a relatively new research field in underwater environments that is being very active recently. In the last decade, some systems appeared based on GPS [

On the other hand, applications like sonars or positioning systems need accuracy in the estimation of times of flight. To achieve this accuracy, coding the emitted signal is a good solution, obtaining these times of flight by means of correlation processes. This technique has been widely used in airborne environments [

As the bottom interaction is not the main purpose of this work, a high frequency signal has been used (20

The model has been developed in Matlab. A cluster of computers has been used to perform a statistical study of the behavior of the relative positioning system. This system uses acoustic signals coded with CSS to obtain the times of flight. With these times of flight and knowing the sound speed value, the distances between the buoys are obtained and fed to the MDS technique, which obtains the relative positions of the buoys knowing only the distances between them.

This system does not need GPS measurements, nor prior information regarding the position of the buoys. Every buoy is also capable to locate itself and the others. Additionally, it would be an inexpensive solution as well as a system easy to deploy and use. As an application example, the relative positioning system is used to track the movement of surface buoys due to a surface current, although its main advantage would be in the positioning of submerged objects. The distance between the real position and the estimated position is used as a performance criteria, which allows to study the feasibility of this kind of system.

The rest of the paper is organized as follows. In Section 2 the fundamentals of the propagation model are presented. Section 3 describes the relative positioning system, the CSS coding scheme and the MDS relative positioning algorithm. Section 4 shows some simulated results for the behaviour of the system, and Section 5 outlines the conclusions and future work.

In this section, the underwater acoustic propagation model is presented. First, some fundamentals about the ray tracing technique are given. Then, the main parameters involved in acoustic propagation are described in detail.

Although this model is based on the ray tracing technique, it does not solve the differential equations, but rather uses a geometrical approach. As stated before, ray tracing assumes that the energy of the wave is confined in different paths or rays. This assumption is valid for high frequencies, so it is a good choice for modeling kilohertz signals and above.

The water column is assumed to be stratified, obtaining a large number of layers. In each layer, sound speed is considered to be constant, but it can change from one layer to the next one. Thus, a ray path will follow a straight line within each layer [

The rays will propagate through the medium, and they will lose energy due to different processes. This model considers the energy loss caused by geometrical spreading, absorption and rebound losses at the surface and the bottom. This transmission loss is computed for each ray that arrives at the receiver, which is called an eigenray. Both the sea surface and the bottom are considered flat for computing the ray paths. Nevertheless, wind speed is included in the model, which will cause a Doppler spread in the surface-reflected signal due to the swell.

The eigenrays are obtained following an intensive search. First, a small number of rays (typically between 20 and 40 rays) are launched. Then, the number of rebounds of two adjacent rays are compared. If they have the same number of rebounds, the final positions in the water column at the receiver end of these rays are compared with the receiver depth. If the receiver is placed between the two rays, it can be assumed that there will be a ray between them that will hit the receiver. The properties of this eigenray are obtained interpolating the values of the other two. However, if the receiver is not placed between them, there will be no eigenray. Another possible situation is when two adjacent rays do not have the same number of rebounds. If that is the case, another 10 rays are launched between them, searching for the edge rays with the same properties than the other two. Then two beams will be obtained out of one, and the process above is repeated. This intensive search is performed for each two adjacent rays.

The block diagram of the model is shown in

During the development of the model, its results were compared with the ones provided by the ray tracing code Bellhop [

Sound speed in water _{T}_{0} is the equilibrium density. However, the variations of these parameters with temperature and depth are not easy to predict, so in the last decades several empirical formulas have been given. In this propagation model, the equation by Chen and Millero has been used, _{ij}_{ij}_{ij}_{ij}

Sound speed is computed by

There are three main contributions to energy loss of an acoustic wave in water: geometrical spreading, absorption and rebounds at the surface and the bottom. This energy loss must be computed for each ray.

Geometrical spreading _{geo}_{abs}_{i}_{i}_{1} and _{2} are the relaxation frequencies of boric acid and magnesium sulfate, respectively;

The model can compute the bottom loss _{bot}_{b}/ρ_{w}_{w}/c_{b}_{b}_{w}_{w}_{b}_{bot}

As for the surface rebounds, one of the simplest equations to obtain the surface loss _{sur}_{s}_{2} = 378^{−2}.

Adding together all the terms introduced before, the total transmission loss for each eigenray can be obtained by _{bot}_{sur}

It has been previously stated that the surface is flat for computing the ray paths. However, the swell will cause a moving surface and a motion of the reflection point. This motion will cause a Doppler spread ^{−1} and _{s}

One of the main advantages of relative positioning systems is that only the distances between all the buoys are needed to obtain their positions. In this work, the relative positioning system consists of four buoys. Two of them are anchored at a fixed position: one of them is considered the origin of the coordinate system, so its position in the horizontal plane will always be (0, 0)

At 1

For this simulated example, the buoys are supposed to be perfectly synchronized, e.g., by a RF link. As the errors in the system are in the order of a meter, as it is shown in Section 4, the system will be robust to synchronization errors up to approximately two hundred microseconds. These synchronization errors would cause a positioning error of the order of few decimeters. This tolerable synchronization error is greater than those found on the literature [

In order to obtain the times of flight more accurately, the acoustic signals are coded with CSS [

At each buoy, the received signal consists of the signals coming from the other three, which have suffered a fading obtained with the propagation model. Every buoy correlates this received signal with the CSS codes of the other three buoys, obtaining several correlation peaks, as can be seen in

The measure of the time-of-flight (TOF) is done from the maximum amplitude peak from the correlation function. As the correlation peak is obtained when the entire coded signal has passed through the correlator, the TOF between two buoys (_{ij}_{code}_{ij}

The buoys position are obtained with the MDS positioning algorithm [_{ij}_{ji}

To obtain the buoys positions, it is necessary to build another matrix _{ij}_{c}

In this section, some simulated results using the relative positioning system are presented. A preliminary version of these results were presented at the OCEANS 2011 Conference [

The positioning system is simulated to be deployed in the coast of Comodoro Rivadavia, Argentina. The latitude of this city is −45.8647°, and it has a bottom depth of 6 ^{−3} and a sound speed of 1, 749 ^{−1}. A value of 1, 024 ^{−3} has been considered for the water density, as well as a salinity of 34.1‰ and a pH of 7. All these values are between the most common that one can encounter in the medium [

The value for the water temperature has been obtained by means of the Levitus Atlas [

Due to the statistical nature of the dynamic transfer function, a statistical study has been conducted. For each value of wind speed and SNR, a hundred simulations have been performed and the average error for each buoy has been obtained.

The positioning system consist of four buoys. The statistical study has been made considering only one position for all the buoys. The first buoy is fixed at (0, 0)

In this work, the SNR is defined as _{b}_{0}, where _{b}_{0} is the noise power spectral density, assuming an additive white Gaussian noise. The SNR values used were 12, 0 and −6 ^{−1}, considering different situations ranging from almost no wind to a remarkable wind speed between them, where these values are easily found in Comodoro Rivadavia. The effect of the wind in the impulse response and the time spread can be seen in

The results for the average error in each buoy, considering different values of SNR and wind speed are shown in

The first conclusion that can be drawn from these results is that outliers mask the behavior of the system. These outliers are due to the near-far effect, when a cross-correlation peak has a greater amplitude than an autocorrelation peak, causing greater errors in value, of tens or hundreds of meters, as the peak detected is from another buoy. As can be seen in

More interesting conclusions can be drawn by removing the outliers. First of all, the errors are now lower, as can be seen in

In the statistical study, all the buoys were considered to be in a constant position. In this case, the two free-moving buoys will vary their positions due to a surface current. The objective is to track the buoys and obtain their positions at each time. Knowing the distance traveled and the time difference between emissions, the velocity and direction of the surface current could be determined.

Both the SNR and the wind speed can vary from one position to the next one, as well as the value and direction of the surface current, to represent a realistic as possible situation. The fixed buoys were placed at (0, 0)

As shown in

An underwater acoustic propagation model has been proposed in this work. This model takes into account a sound speed profile, so it can be used both in shallow waters and deep waters. Additionally, the equations that are used to compute the sound speed and transmission loss are valid in a wide range of input values, so it is not restricted to very specific environmental conditions. It also considers the dynamic effect of swell that worsens the properties of the surface-reflected signal and uses a dynamic transfer function.

Also, a relative positioning system has been presented. The performance of this system has been studied using the propagation model described before. For a particular position, the average error has been obtained for each buoy varying the SNR and the wind speed. Multipath caused by low wind speeds has been identified as the most damaging effect, and some outliers have been detected, due to the near-far effect, which disguise the behavior of the system.

As an early example of application, the relative positioning system has been used to track the position of the buoys, where two of them can freely move through the sea surface due to surface currents. The system is capable to track the positions in different times, so the value and direction of the surface current could be computed. Using the MDS algorithm, the error in these measurements is about 1

As future work, some improvements can be made into the model, like adding a bathymetry profile or a range-dependent option, so the performance of this system could be studied in more complex environments. The tracking of the surface buoys must be estimated continuously, so a fast algorithm with low computational complexity has been used (MDS). The outliers could be reduced by implementing refinement algorithms, which will increase the complexity of the system to the detriment of the time between measurements. Finally, it is important to remark that although the tracking of moving buoys in the surface is interesting, this is not the final application of this system. That would be the simultaneous positioning of buoys in the surface and submerged objects in the sea, where acoustic systems can work in more complex environments than other wireless techniques, whose use is severely limited underwater.

This work has been possible thanks to the support of the Spanish Ministry of Science and Innovation (TIN2009-14114-C04-01/04), the University of Alcalá (FPI fellowship) and the Regional Government of Extremadura through the European Regional Development Funds (FEDER - GR10097).

Block diagram of the propagation model.

Positioning system configuration.

Correlation peaks at fixed buoy 1.

Autocorrelation peaks at buoy 1 with the code from buoy 4.

Dependence of the relative impulse response on wind speed.

Average error of estimated positions for different values of SNR and wind speed:

Buoys 3 and 4 movement due to a surface current.

Absolute error for each buoy at each measurement.

Parameters for calculating the Chen–Millero equation [

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

| |||||

_{00} |
1402.388 | _{01} |
5.03830 | _{02} |
−5.81090E-2 |

_{03} |
3.3432E-4 | _{04} |
−1.47797E-6 | _{05} |
3.1419E-9 |

_{10} |
0.153563 | _{11} |
6.8999E-4 | _{12} |
−8.1829E-6 |

_{13} |
1.3632E-7 | _{14} |
−6.1260E-10 | _{20} |
3.1260E-5 |

_{21} |
−1.7111E-6 | _{22} |
2.5986E-8 | _{23} |
−2.5353E-10 |

_{24} |
1.0415E-12 | _{30} |
−9.7729E-9 | _{31} |
3.8513E-10 |

_{32} |
−2.3654E-12 | _{00} |
1.389 | _{01} |
−1.262E-2 |

_{02} |
7.166E-5 | _{03} |
2.008E-6 | _{04} |
−3.21E-8 |

_{10} |
9.4742E-5 | _{11} |
−1.2583E-5 | _{12} |
−6.4928E-8 |

_{13} |
1.0515E-8 | _{14} |
−2.0142E-10 | _{20} |
−3.9064E-7 |

_{21} |
9.1061E-9 | _{22} |
−1.6009E-10 | _{23} |
7.994E-12 |

_{30} |
1.100E-10 | _{31} |
6.651E-12 | _{32} |
−3.391E-13 |

_{00} |
−1.922E-2 | _{01} |
−4.42E-5 | _{10} |
7.3637E-5 |

_{11} |
1.7950E-7 | _{00} |
1.727E-3 | _{10} |
−7.9836E-6 |

Parameters used in the simulation.

Measurement | SNR (dB) | w (^{−1}) |
Surface current (^{−1}) |
---|---|---|---|

| |||

1 | 6 | 2.5 | (0.11,0.06) |

2 | 6 | 2.5 | (0.11,0.06) |

3 | −3 | 2.5 | (0.11,0.06) |

4 | 6 | 0.6 | (0.11,0.06) |

5 | −6 | 0.6 | (0.06,0.08) |

6 | 3 | 4 | (0.03,0.08) |

7 | 3 | 4 | (0.06,0.08) |

8 | 3 | 1 | (0.06,0.03) |

9 | 6 | 2 | (0.06,0.03) |

10 | 6 | 2 | (0,0) |