2.1 Mathematical model of geometric position of average sound speed

GNSS-A technology links GNSS and acoustic positioning methods to co-locate to accurately locate the position of the seafloor datum point. As shown in Figure 1, the transducer is installed under the ship and performs active ranging and positioning during the voyage, forming N sea-level dynamic control points within a period of time. Using GNSS technology can accurately determine their three-dimensional (3D) coordinates. When there are multiple distance observations, the 3D position information of the target point can be obtained through the distance intersection method (Sun, Reference Sun, W.2007). The underwater positioning observation model based on the average sound speed is:

(1)\begin{equation}{L^s} = {v^s}{t^s} = {||{{\bf x} - {{\bf x}^s}} ||_2} + {\varepsilon _L}\quad s \in \{{1,2,\ldots , N} \}\end{equation}

where ${L^s}$ is the propagation distance of the acoustic signal from the sea surface transducer to the transponder; ${v^s}$ is average sound speed; ${t^s}$ is the propagation time; ${||{{\bf x} - {{\bf x}^s}} ||_2}$ is the Euclidean distance between two points; ${\varepsilon _L}$ is the equivalent ranging error; N represents the number of sea surface movement control points.

Figure 1.

GNSS/acoustic positioning local coordinate system

Carry out the Taylor series expansion of Equation (1) at the initial value ${{\mathbf x}_0}$ to obtain the linearised equation:

(2)\begin{equation}d{L^s} = {{\mathbf e}_i}{\mathbf dx} + {r_i} + \varepsilon\end{equation}

where ${\mathbf dx} = {\mathbf x} - {{\mathbf x}_0}$ is coordinate correction number; ${r_i}$ is the linearisation residual term, which can generally be ignored;

(3)\begin{equation}{{\mathbf e}_i} = ({{\mathbf x}_0} - {{\mathbf x}_i})/{d_i}({{{\mathbf x}_0}} )\end{equation}

${{\mathbf e}_i}$ is the direction cosine. When there are three or more observations, the adjustment criterion is introduced, and the least square solution can be expressed as:

(4)\begin{equation}{\bf d\hat{x}} = {({{{\bf J}^T}{\bf PJ}} )^{ - 1}}{{\bf J}^T}{\bf Pb}\end{equation}

where ${\bf P}=\left[ {\begin{array}{*{20}{c}} {{{\bf P}_1}} & \ldots & {{{\bf P}_{\rm n}}} \end{array}} \right]$ is the observation weight matrix; ${\bf b = }\left[ {\begin{array}{*{20}{c}} {d{\boldsymbol{L}_1}} & {d{\boldsymbol{L}_2}} & \cdots & {d{\boldsymbol{L}_n}} \end{array}} \right]$; ${\bf J}=\left[ {\begin{array}{*{20}{c}} {{\bf J^{\prime}}} & {{{\bf k}_n}} \end{array}} \right]$ is called Jacobi matrix:

(5)\begin{equation}{\bf J}=\left[ {\begin{matrix} {{\bf e}_1^T } & {{\bf e}_2^T } & \cdots & {{\bf e}_n^T } \end{matrix}} \right] \quad {{\bf k}_n} = {\left[ {\begin{matrix} 1 & \cdots & 1 \end{matrix}} \right]^T }\end{equation}

${\bf k}$ is the column matrix with a content of 1. Based on the initial value ${{\mathbf x}_0}$, the coordinates of the underwater point to be measured can be calculated iteratively:

(6)\begin{equation} \hat{\bf x} = {{\bf x}_{\bf 0}}+ {\bf d}\hat{\bf x}\end{equation}

From the range-finding underwater positioning model, we can see that the sound speed is the most important parameter, and the selection of the sound speed directly determines whether the positioning result is reliable. Therefore, a lot of research and analysis have been done on the sound speed correction method in order to improve the accuracy of sonar underwater positioning.

2.2 Description of average sound speed

In the underwater positioning model, the range is determined by the sound speed and the propagation time. Currently, the time measurement error is generally within 0·1 ms (Zhao et al., Reference Zhao, S., Wang, Z. and Liu, H.2018). Therefore, the key to improving the positioning accuracy lies in the determination of the sound speed. The equation of state of sea water (Cai et al., Reference Cai, Y. and Cheng, P.2012) is:

(7)\begin{equation}dp = {v^2}d\rho\end{equation}

where $\rho$ is the density at a certain point i in seawater; p is the instantaneous sound pressure at the same point; v is the sound speed which can be expressed as:

(8)\begin{equation}v = \sqrt {dp/d\rho } = {1 / {\sqrt {\rho \beta } }}\end{equation}

The three influencing factors mainly change with the depth direction, so the sound speed can be roughly expressed as a function of water depth, and the relationship between them is expressed by the SSP. The SSP can be measured by CTD (conductivity, temperature, depth) or calculated according to the sound speed experience model. The water environment in the ocean is relatively complex, and temperature is the first factor that affects the sound speed. Generally, the sound speed of the surface layer and thermocline in the ocean changes significantly, while the temperature of the isothermal layer is relatively stable (Zhao et al., Reference Zhao, J and Liang, W2019). Seasonal temperature and flow field changes can cause instability of the SSP. Therefore, the SSP has significant temporal and spatial characteristics, and its changes are irregular. Studies have shown that the sensitivity of SSP mismatch to source positioning at different depths is different (Su et al., Reference Su, Ma, Song, Guo and Lu2015), so it is necessary to measure the SSP of this area at the same time during underwater positioning measurement (Li, Reference Li, S.2015).

The classic ideal Munk-SSP (Figure 2) model is a parametric model of SSP structure. In the past few decades, it has been widely used in related research. It is a classic SSP model suitable for deep-sea areas based on the measured layered characteristics of deep-sea density structure. It is usually expressed by the function v(z) of sound speed v and water depth z (Munk, Reference Munk1974):

(9)\begin{equation}v(z )= 1500[{1 \cdot 0 + \kappa ({\tilde{z} - 1 + {e^{ - z}}} )} ]\end{equation}

where $\kappa$ = 0·00737, conversion depth $\tilde{z} = {{2({z - 1300} )} / {1300}}$. When positioning without SSP, the Munk-SSP can be used as a simple replacement, or the global SSP can be constructed based on Argo data (Cai et al., Reference Cai, Cheng, Wen, Yang and Zhang2014).

Figure 2.

Classic Munk sound speed profile

The premise of the positioning model based on distance measurement is that the sound speed is a constant, and the trajectory of acoustic rays in the water is a straight line. In order to improve the efficiency of underwater positioning in the project, based on experience, sometimes the sound speed value is directly taken as 1500 m/s. Obviously, this will bring a large ranging error. When the SSP is measured, the average sound speed based on the SSP is expressed as:

(10)\begin{equation}{V_A} = \frac{1}{H}\sum\limits_{i = 0}^{N - 1} {\frac{{({{V_{i + 1}} + {V_i}} )({{z_{i + 1}} - {z_i}} )}}{2}}\end{equation}

where H is the water depth, ${V_i}$ and ${V_{i + 1}}$ are the sound speed values of the i and i+1 layer respectively, ${z_i}$ and ${z_{i + 1}}$ are the depths of the i and i+1 layer respectively. The positioning model of the average sound speed method is simple, the calculation method is convenient, and the positioning efficiency is high, but there is still a large residual error.

2.3 The effect of average sound speed on positioning

The acoustic ray, in practice, will bend due to the change of sound speed. Studies have shown that the sound speed determines the accuracy of positioning by affecting the accuracy of ranging. Based on the acoustic ray tracing algorithm, the relationship between ranging and surface sound speed error is derived (Zhao et al., Reference Zhao, J and Liang, W2019):

(11)\begin{equation}dL = \frac{{k\sin {\theta _0}}}{{{V_0}}}\sum\limits_{i = 1}^N {\frac{{{V_i}\Delta {z_i}}}{{\sqrt {{{[1 - {{(k{V_i})}^2}]}^3}} }}d{V_0}}\end{equation}

where k is Snell's constant, L is ranging, ${\theta _0}$ is the incident angle of surface acoustic rays, ${V_0}$ is surface sound speed. It can be seen from Equation (11) that there is a non-linear relationship between the incident angle and the sound speed on the ranging, and the error of the sound speed and the incident angle are positively correlated with the ranging error. The acoustic ray tracing algorithm is a high-precision underwater positioning method commonly used to reduce the acoustic ray bending error (Sakic et al., Reference Sakic, Ballu, Crawford and Woeppelmann2018). When the observed incident angle is large, the average sound speed positioning model and acoustic ray tracing algorithms are used to calculate the horizontal distance of acoustic ray propagation at an incident angle of 85°, the average sound speed is 1470 m/s.

It can be seen from Figure 3 that, when the acoustic signal propagates to the horizontal distance of about 30 m, the difference between the two algorithms begins to appear. When the horizontal distance between the transponder and the source is close to 350 m, under the average sound speed model, the acoustic ray travels to 30 m underwater. While the acoustic ray travels to 70 m under the acoustic ray tracing algorithms, it has a significant impact. In addition, the influence of the average sound speed error on the ranging error under different incident angles is also calculated.

Figure 3.

Influence of average sound speed error

In Figure 4, the sound source depth is 1 m, the incident angle is 30°, 50° and 70°, the average sound speed is 1,470 m/s, and the propagation time is 0·477 s. It can be seen from the experimental results that the horizontal distance reached by the signal propagation under different incident angles is different; The ranging error is affected by the acoustic ray bending error caused by the sound speed error, and it is positively correlated with the incident angle. When the transponder is directly below the sound source, there is no influence of acoustic ray bending error. In addition, the longer the sound signal propagation time, the greater the impact of the sound speed error. Assuming the propagation time is 1 s, the sound speed error of 1 m/s will result in a 1 m ranging error.

Figure 4.

Effect of average sound speed error on ranging error

The acoustic ray tracing algorithm can effectively reduce the influence of the acoustic ray bending error and restore the footing of the beam more realistically. However, the calculation model of the algorithm is relatively complicated, and the calculation efficiency is lower than that of the average sound speed positioning model, and rapid positioning cannot be achieved. Therefore, it is necessary to consider sound speed correction to reduce the influence of the sound speed error. At present, the main sound speed correction methods include: empirical sound speed method, weighted average sound speed method, Taylor series expansion method, effective sound speed method, equivalent SSP method and acoustic ray tracing algorithm. Due to the limitation of the algorithm, the above method has many applicable conditions, and each has its advantages and disadvantages (Vincent and Hu, Reference Vincent and Hu2000; Nie, Reference Nie, Wang and Li2015; Zhang, Reference Zhang, B.2019).

It is easy to obtain from ray theory that when the depth of the target point to be measured is certain, the larger the incident angle, the longer the propagation distance. Therefore, an improved model of incident angle sound speed is considered. On the basis of the traditional average sound speed, the influence of incident angle error is considered, and the calculation complexity is simplified, so as to improve the positioning accuracy while ensuring the calculation efficiency.

4.1 Simulation experiment and analysis

The simulated trajectory of the ship on the sea surface is a circle with radius R = 100 m, the plane position of the unknown target point is at the centre of the circle, and the vertical position is 100 m underwater. Based on Munk-SSP, the ray tracing algorithm is used to simulate the propagation time of the acoustic signal. The simulated sea surface is a cosine wave with a wave height of 2  m, the measured ship speed is 3 kn/h, the sampling interval is 4 s, and the ranging error is 0·3% times the slope distance (Zhao et al., Reference Zhao, S., Wang, Z. and Liu, H.2018). The positions of the seafloor target in simulation test are known, so it is possible to compare the accuracy of positioning results of different models.

Three different sound speed determination methods are used for positioning experiments. The first model is the average sound speed of Munk-SSP, that is, ${\textrm{V}_{\rm 1}}\textrm{ = mean}[{\textrm{SSP(}\textrm{v}\textrm{)}} ]$. The second model is based on the positioning results of the ray tracing algorithm, that is, ${\textrm{V}_2} = \textrm{tracing}(SSP)$. The core idea of the ray tracing algorithm is given in the form of a program in the Appendix. The third model is to use the new sound speed positioning model related to the incident angle ${\textrm{V}_3}({\theta } )$ proposed in this paper; its calculation flow is shown in Figure 7. The positioning experiment was repeated 500 times, and the deviation sequence and variance of different positioning models and true values were counted to evaluate the reliability of the results.

Several conclusions can be seen from the experimental results in Figure 8 and Table 1:

1. 1. When the average sound speed model ${\textrm{V}_1}$ is used for positioning, since the sea surface control graphic is a symmetrical circle, the positioning accuracy in the plane direction is high, and the error with the real position of the target is small. The positioning error of the average sound speed model is mainly reflected in the vertical direction, because the water surface control points are approximately coplanar, resulting in an ill state in the vertical direction, so the vertical positioning accuracy is poor.

2. 2. When the ray tracing model ${\textrm{V}_2}$ is used for positioning, it can be seen that the positioning result is very close to the real position of the target. The reason is that the observation configuration is symmetrical, the incident angle is approximately equal, and the incident angle error affects the positioning accuracy through the ranging error.

3. 3. When the new positioning model ${\textrm{V}_3}({\theta } )$ is used for positioning, it can be seen that the deviation between the positioning result and the real position in the three directions is very small, and it is similar to the ray tracing model. The reason is that the sound speed used to calculate the distance measurement is no longer a single sound speed value, but a function related to the incident angle.

4. 4. Compared with the average sound speed positioning model, the new positioning model related to the incident angle improves the vertical positioning accuracy. Compared with the ray tracing model, the positioning accuracy is equivalent under the symmetrical circle control graph; when the sea surface control graphics is not a circularly symmetrical graph, the advantages of the new model will be reflected, and the authors will verify this conclusion in subsequent experiments.

Figure 8.

Underwater geometric positioning errors caused by three sound velocity models under the same incident angle

Table 1.

Variance of positioning results under different models

In addition, a simulation experiment was conducted with different incident angles. The surveying track of the simulated ship on the sea is a rectangle, and the error simulation method is the same as the above simulation experiment. The surveying track of the ship on the sea is shown in Figure 9, ‘Target’ indicates the position of the underwater target point.

Figure 9.

Rectangular surveying track

Correspondingly, according to the change of the incident angle, the corresponding sound speeds are calculated by Equation (14), and the relationship between them is shown in Figure 10.

Figure 10.

Relationship between sound speed and incident angle

Similar to the above simulation experiment, three different positioning models are used for calculation. ${\textrm{V}_1}$ is the average sound speed positioning model, ${\textrm{V}_2}$ is the ray tracing positioning model, and ${\textrm{V}_3}$ is the new sound speed positioning model. The positioning error is shown in Figure 11.

Figure 11.

Underwater geometric positioning errors caused by three sound velocity models under different incident angles

Through the experimental results, we can see that the ${\textrm{V}_1}$ model has poor positioning accuracy in three directions for the observations with the unequal incident angles. The reason is that a fixed sound speed is used for positioning. The positioning accuracy of the ${\textrm{V}_2}$ and ${\textrm{V}_3}$ models perform better than ${\textrm{V}_1}$, and the overall conclusion is the same as the above simulation test with the equivalently incident angles.

In summary, whether it is a circular calibration experiment or a positioning experiment with irregular observation patterns, the new positioning model related to the incident angle can quickly achieve high-precision underwater positioning.

4.2 Real experiment and analysis

In order to verify the validity of the new positioning model related to the incident angle proposed in this paper, relevant actual measurement experiments were carried out. The positioning experiment was carried out in October 2012 on a lake in northern China, with a total area of about 554 km² and maximum water depth of about 77 m. The target transponder was about 60 m underwater. The main equipment required for the experiment were: a measurement ship equipped with Leica GPS1200 dual-frequency real-time kinematic (RTK), vertical reference unit (VRU), Gyroscope (Gyro), long-base line (LBL) positioning system, sound velocity profile (SVP), data processing centre, etc. The LBL system used Sonardyne's 6G LBL acoustic positioning system. The ship's attitude sensor model was Seatex MRU-05, with roll/pitch measurement accuracy of ±0·05° and heave accuracy of ±5 cm. The underwater sound speed meter used AML SVP to measure the SSP of the water area Figure 12.

Figure 12.

Sound speed profile of the survey area

The experiment used track navigation and positioning, that is, the measuring ship did not follow a regular shape on the water surface, and the observation values at different incident angles were obtained. The sailing track is shown in Figure 13.

Figure 13.

Survey ship's trajectory and the position of the receiver

According to the initial value of the transponder position and the coordinates of the dynamic control point of the measuring ship, the incident angles of different observations can be inversely calculated, and then the new sound speed positioning model ${\textrm{V}_3}({\theta } )$ can be calculated by Equation (14); at the same time we give the average sound speed ${\textrm{V}_1}$. Figure 14 shows the comparison of the relationship between the two sound speed values.

Figure 14.

Comparison of two sound speeds

Positioning results were performed under two different models. The residual information contains important information about the basic assumptions of the model, and the residual relationship under the two positioning results is given (Figure 15).

Figure 15.

Comparison of residuals of two positioning results under two models

It can be seen from the positioning residuals of the two positioning models that the residuals of the new sound speed positioning model ${\textrm{V}_3}({\theta } )$ are generally smaller than the average sound speed positioning model ${\textrm{V}_1}$. The reason is that the ${\textrm{V}_3}({\theta } )$ is used instead of a single sound speed value ${\textrm{V}_1}$, which releases the residual information, reduces the influence of the sound speed error on the positioning, and improves the accuracy of parameter estimation.

In addition, it was noticed that there is a significant mutation for positioning residuals approximately at the 210th epoch. Combined with the navigation route in Figure 12, and through the position relationship between the transducer and the underwater transponder, the authors believe that at the 210th epoch the ship sailed to the farthest point, and then the route and direction of sailing changed, where the incident angle and other related errors have the greatest impact.

Taking the acoustic ray tracing positioning result as the true value, Table 2 shows the deviation of the positioning results in three directions under different models. From the results, the results of the ${\textrm{V}_3}({\theta } )$ model in the three directions are closer to the results of acoustic ray tracing, and the positioning accuracy has been greatly improved. The three directions of X, Y, and Z are raised by about 3 m, 0·8 m and 2·4 m, respectively. From the positional error (PE) value, $\textrm{PE = }\sqrt {{{\mathrm{(\Delta X)}}^2} + {{\mathrm{(\Delta Y)}}^2} + {{\mathrm{(\Delta Z)}}^2}}$, the PE values under the two models are 4·9848 and 1·1233, respectively. It is obvious that the ${\textrm{V}_3}({\theta } )$ model is effective.

Table 2.

Deviation of positioning results under different models: unit (m)