Experimental Results of Underwater Sound Speed Profile Inversion by Few-Shot Multi-Task Learning

Abstract

Underwater Sound Speed Profile (SSP) distribution is crucial for the propagation mode of acoustic signals, so fast and accurate estimation of SSP is of great importance in building underwater observation systems. The state-of-the-art SSP inversion methods include frameworks of matched field processing (MFP), compressive sensing (CS), and feed-forward neural networks (FNNs), among which the FNN shows better real-time performance while maintaining the same level of accuracy. However, the training of FNN needs quite a lot historical SSP samples, which is difficult to satisfy in many ocean areas. This situation is called few-shot learning. To tackle this issue, we propose a multi-task learning (MTL) model with partial parameter sharing among different training tasks. By MTL, common features could be extracted, which accelerates the learning process on given tasks, and reduces the demand for reference samples, enhancing the generalization ability in few-shot learning. To verify the feasibility and effectiveness of MTL, a deep-ocean experiment was held in April 2023 in the South China Sea. Results show that MTL outperforms the other mainstream methods in terms of accuracy for SSP inversion, while inheriting the real-time advantage of FNN during the inversion stage.

1. Introduction

Underwater observation systems have become a good way to provide positioning, navigation, and timing (PNT) services in recent years [1,2,3]. Communication and localization are the two most important technological bases for underwater observation systems, and because of the attenuation problems, the sound wave becomes the main carrier for long-distance signal transmission in underwater environment. However, the non-uniform distribution of sound speed will cause a significant Snell effect, which means the signal will propagate non-straightly. So, the sound field information will be dynamically changed, such as signal propagation time or received signal strength [4]. Fortunately, when obtaining the distribution of sound velocity, the sound field distribution can be accurately estimated according to ray theory [5] or normal mode theory [6,7], thus the accuracy of ranging and positioning could be improved [8,9,10].

With the same range scale, sound speed distribution varies greater with depth than the horizontal direction, so sound speed profiles (SSPs) are usually used to represent the distribution of sound speed [4]. Recently, many approaches leveraging sound field information have been proposed for inverting SSPs [11,12]. For underwater positioning, navigation and timing systems, the real-time performance is very important. To ensure service quality, it is necessary to first obtain real-time regional sound velocity distribution. However, traditional measurement methods by instruments such as conductivity, temperature and depth profilers (CTDs) or sound velocity profilers (SVPs) require a long period of time, and existing sound velocity inversion methods based on Monte Carlo ideas have inadequate computational efficiency, which poses a great challenge for the rapid acquisition of sound velocity distribution. For example, the navigation of autonomous underwater vehicles or submarines requires periodic and rapid estimation of sound velocity distribution, followed by positioning and navigation correction. With the help of pre-deployed underwater sensor networks, such as Seaweb [13], the PNT service could be effectively achieved.

The state-of-the-art SSP inversion framework contains matched field processing (MFP) [14], compressive sensing (CS) [15,16] and feed-forward neural networks (FNNs) [12,17]. The estimation of real-time SSP is difficult work because, to the best of our knowledge, it could not be directly calculated through any empirical formula with measured sound field information. Nevertheless, based on ray theory, Tolstoy et al. [14] proposed a MFP framework combining empirical orthogonal function (EOF) decomposition for SSP inversion. MFP avoids establishing the mapping relationship from sound field data to SSP distribution, so that it has become the mainstream framework for SSP inversion for a long time.

The optimal solution of [14] is searched through a traversing way, so it is quite time consuming. To accelerate the SSP inversion process, many researchers have adopted heuristic algorithms into the search of optimal solutions, such as the simulated annealing algorithm [18], the particle swarm optimization (PSO) algorithm [19,20], and the genetic algorithm [21,22]. However, the core idea of the heuristic algorithm is based on Monte Carlo test, so they still require a lot of iterative searches to determine the matching items. To further improve the efficiency for SSP inversion, refs. [15,16] proposed a CS framework, which establishes a dictionary to directly map the sound field information to SSPs. Compared with MFP, the CS framework only requires a few iterations to train the dictionary, so that the computational complexity can be reduced. However, the mapping relationship is linearly simplified through the first-order Taylor expansion, which sacrifices the accuracy performance.

Recently, Bianco et al. [23] performed a comprehensive survey that machine learning has gained broad application prospects in the field of underwater acoustics, such as seafloor characterization [24], range estimation [25], and SSP inversion [12,17,26]. In our early work [12], we proposed a FNN structure for SSP inversion with the assistance of ray theory. The model training can be completed offline in advance. After the model converging, only one round of forward propagation is required to invert the SSP when feeding the measured sound field data into the network. Thus, the FNN model shows better efficiency performance than the MFP and CS models. To improve the robustness of FNN under noise interference, we proposed an auto-encoder feature-mapping neural network (AEFMNN) in [26]. By denoising and reconstructing sound field information, hidden features with stronger anti-interference ability are extracted, thereby improving the accuracy of SSP inversion.

The work of MFP, CS, or FNN models all rely on a large amount of historical SSPs as references. However, due to the high economic and labor costs of historical SSPs measured by CTD or SVP, the reference SSPs available in many spatio-temporal ocean areas are so limited that there may be insufficient reference data for model training, making the model prone to be over-fitting. Many approaches for few-shot learning surveyed in [27,28] have been proposed to weaken the over-fitting effect, such as regularization [29], training data set expanding with generative adversarial networks [30], multi-task learning (MTL) [31,32], transfer learning (TL) [33,34], and meta-learning approaches [27,28,35].

For solving the few-shot learning problem in SSP inversion, we propose a multi-task learning (MTL) approach to accurately estimate the regional SSP distribution. We aim to extract the common features from SSP clusters with different distributions via a partial parameter shared neural network, forming a good set of initialization parameters for inversion task. When training on the few-shot samples of the task, the learning rate is dynamically adjusted according to the distance of spatio-temporal information between the reference SSP sample and the task mission. MTL could accelerate the training process and maintain the model’s sensibility to input data, which enhances the generalization ability of the model. To demonstrate the effectiveness of MTL, we conducted a deep-ocean experiment in the South China Sea in April 2023. The contributions of our paper can be concluded as:

• To achieve good accuracy performance of SSP inversion with insufficient training data, we propose the MTL approach. Through learning on multiple tasks (different kinds of SSPs), common features of SSPs are extracted to shorten the training process of the model on any given task, so as to enhance the generalization ability.
• To verify the feasibility of MTL, a deep-ocean experiment was conducted. The accuracy performance of SSP inversion is evaluated based on measured data and is compared with other mainstream approaches.

The rest of this paper is organized as follows. In Section 2, we propose the MTL model for SSP inversion. In Section 3, we introduce the scenarios and processes of the deep-ocean experiment for SSP inversion. Experimental results are discussed in Section 4, and conclusions are given in Section 5.

2. MTL Model for SSP Inversion

To reduce the SSP estimation error under few-shot learning situation, we build a MTL model, in which different types of SSPs sampled in different spatio-temporal regions will be used for pre-training. While for training on the task group, the learning rate will be dynamically adjusted according to the distance of spatio-temporal information between the reference SSP and the inversion task. The illustration of MTL training and SSP inversion is shown in Figure 1.

The MTL model is a three-layer FNN proposed in our previous work [12], as shown in Figure 2. There is a multi-task learner and a task learner in out model. The input layer is signal propagation time sequence, and the number of input layer neurons is consistent with the sonar observation data collected by the transmitting and receiving nodes, and the detailed sonar observation data collection process will be given in the ocean experiment section. The output layer is the inverted SSP, while the label data are the training SSP. For pre-training, weight coefficients between the input layer and the hidden layer are shared, while the weights between the hidden layer and the output layer are unique related to the training task. During each epoch, N kinds of SSPs with V-shot SSPs for each are selected for the multi-task model training, which aims to learn a good set of initialization parameters that could be shared with the task model. Thus, the task learner would quickly converge with only a little training time.

2.1. Label SSP Data Preparation

The training historical SSP data for the multi-task learner could be either clustered manually or intelligently. In our previous work [36], we proposed a SSP clustering method based on Euclidean distance, which can effectively group SSPs with similar distribution, even through they are sampled in different spatio-temporal ocean regions.

For a given SSP inversion task, a small number of reference SSPs are still necessary for training the task learner. However, the distribution characteristics of the target task is not known as prior information. In our previous work [36], we propose a method to find proper training data for a given task according to its spatio-temporal information. These methods will be adopted in this paper.

2.2. Simulation of Signal Propagation Times

Both the training of the multi-task learner and the task learner require sound field information as model’s input under a given SSP distribution. However, the real measured sound field data are usually collected at the inversion stage, and those sound field data used for model training need to be simulated.

For ocean sensing networks, underwater anchor nodes can be located by a long baseline positioning system composed of several buoys, while the coordinates of buoys can be obtained via the global positioning system (GPS). Let a buoy node be the signal sender, and M underwater anchor nodes be the receivers, the horizontal propagation distance of the signal with a given SSP $S=[(s^1,1),\dots ,(s^d,d)]$ can be calculated according to our previous derivation in [26] as:

$$\begin{array}{cc}\hfill h^m& =\frac{s^1}{cos{\theta }^{1,m}}\sum _{d=1}^{D-1}\left|\frac{\Delta z_d}{s^{d+1}-s^d}\left(\sqrt{{\Gamma }_d^m}-\sqrt{{\Gamma }_{d+1}^m}\right)\right|,\hfill \\ \hfill {\Gamma }_d^m& =1-{\left(\frac{cos{\theta }^{1,m}}{s^1}\right)}^2{\left(s^d\right)}^2,\hfill \end{array}$$

(1)

where $h^m$ is the horizontal signal propagation distance from the buoy to the mth anchor node, d is the index of depth layers, ${\theta }^{1,m}$ is the initial grazing angle from source to the mth receiver, $s^d$ is the sound speed value at depth with index d, and $\Delta z_d$ is the depth difference.

With known horizontal distance $h^m$, the grazing angle of signal from source to each receiver can be searched by Equation (1). Then, the ideal signal propagation time from source to the mth receiver can be simulated as:

$$t^m=\sum _{d=1}^{D-1}\left|\frac{\Delta z_d}{s^{d+1}-s^d}ln\left(\frac{s^d\left(1+\sqrt{{\Gamma }_{d+1}^m}\right)}{s^{d+1}\left(1+\sqrt{{\Gamma }_d^m}\right)}\right)\right|,$$

(2)

from which it can be noticed that $t^m$ is a function of ${\theta }^{1,m}$.

2.3. Training of the Multi-Task Learner

The first training phase of MTL that can be conducted offline is the training of multi-task learner. In this stage, the weight parameter of multi-task learner is initialized as $W_{mt,h}^1$ and $W_{mt,o}^1$. $W_{mt,h}^1$ is the weight matrix between the input layer and the hidden layer, and $W_{mt,o}^1$ is the weight matrix between the hidden layer and the output layer.

At the beginning of the pth epoch (an epoch indicates a round of parameter updating), N SSP clusters are chosen from the K available SSP clusters ($N\le K$) for training the multi-task learner. In each selected SSP cluster, there are V trainings ($S_v,v=1,2,\dots ,V$). For SSP cluster n, the V SSPs are utilized for one epoch of training with the cost function:

$$l^{\left(n\right)}\left(W_{mt,n}^p\right)=\sum _{v=1}^V\left(\frac{1}{2}\sum _{d=1}^D{\left(s_v^d-{\tilde{s}}_v^d\right)}^2+{∥W_{mt,n}^p∥}_1\right),W_{mt,n}^p=\left[W_{mt,h,n}^p,W_{mt,o,n}^p\right],$$

(3)

where $s_v^d$ is the sound speed value, ${\tilde{s}}_v^d$ is the corresponding inverted sound speed, and ${∥W_{mt,n}^p∥}_1$ is the regularization item. Then, local parameters are updated by back propagation (BP) algorithm [37]:

$${\dot{W}}_{mt,n}^p=W_{mt,n}^p-\frac{\xi }{N}{\nabla }_{W_{mt,n}^p}{l}^{\left(n\right)}\left(W_{mt,n}^p\right),$$

(4)

where $\xi$ is the learning rate of the multi-task learner.

For a specified SSP inversion task, the parameters of multi-task learner ${\dot{W}}_{mt,h}^P$ is transferred as the initialization for the task learner, thus $W_{tk,h}^1={\dot{W}}_{mt,h}^P$, where $W_{tk,h}^1$ is the weight matrix between the input layer and the hidden layer. The weight matrix $W_{tk,o}^1$ between the hidden layer and the output layer will be initialized randomly. The task learner will be trained on a few reference SSPs to form the final model ${\dot{W}}_{tk}$.

Let the ith reference SSP be $S_{tk,ri}=\left[\left(s_{tk,ri}^1,1\right),\dots ,\left(s_{tk,ri}^d,d\right)\right]$, $d=1,2,\dots ,D$ with coded sampling location as $\left({\beta }_{tk,ri}^x,{\beta }_{tk,ri}^y\right)$ and time information as ${\alpha }_{tk,ri}$, so the spatio-temporal distance ${\varphi }_{tk,ri}$ between the ith task reference SSP and the inversion task (with coded location as $\left({\beta }_{tk}^x,{\beta }_{tk}^y\right)$ and time information as (${\alpha }_{tk}$) can be calculated according to Equations (7)–(10) in [38].

To reduce the negative transfer effect of MTL, we propose a dynamic learning rate adjustment strategy based on the spatio-temporal distance between samples and the task during model training. For the ith training SSP, the learning rate of task learner ${\eta }_{tk,ri}$ will be:

$${\eta }_{tk,ri}=\xi \frac{\frac{1}{{\varphi }_{tk,ri}}}{{\displaystyle \sum _{i=1}^I}\frac{1}{{\varphi }_{tk,ri}}}.$$

(5)

where ${\varphi }_{tk,ri}$ is calculated by Equation (7) in [38] with ${\lambda }_{tk,ri}=0.9$.

If there are total J epochs for task learner training, the cost function $l_{tk}\left(W_{tk}^j\right)$ of the jth ($j=1,2,\dots ,J$) batch will be:

$$l_{tk}\left(W_{tk}^j\right)=\frac{1}{2}\sum _{d=1}^D{\left(s_{tk,ri}^d-{\tilde{s}}_{tk,ri}^d\right)}^2,$$

(6)

where ${\tilde{s}}_{tk,ri}^d$ is the inverted SSP related to reference SSP $S_{tk,ri}$. Then, the parameter updating of task learner is conducted according to:

$${\dot{W}}_{tk}^j=W_{tk}^j-{\eta }_{tk,ri}{\nabla }_{W_{tk}^j}{l}_{tk}\left(W_{tk}^j\right).$$

(7)

At last, the converged model will be ${\dot{W}}_{tk}={\dot{W}}_{tk}^J$.

When feeding measured sound field information into model ${\dot{W}}_{tk}$, the output SSP ${\tilde{S}}_{tk}$ can be quickly calculated through once forward propagation.

3. Deep-Ocean Experiments

To evaluate the feasibility and effectiveness of proposed MTL method for SSP inversion under few-shot learning situation, we conducted deep-ocean experiments in the South Sea of China with areas of 10 km × 10 km in middle April 2023, where the depth is about 3500 m. The relevant data collection corresponding to SSP inversion lasted for a total of 3 days.

The system composition is shown in Figure 3, including four anchor nodes, a ship unit that containing a CTD, a set of expendable CTDs (XCTDs), and an ultra short baseline (USBL) system fixed to the right side of the ship. The difference between CTD and XCTD is that the data collected and transmitted back through disposable probes by XCTD, and XCTD can measure sound speed during slow ship movement, while CTD requires fixed-point measurement. Since the XCTD probe does not need to be recovered, it can save time in measuring the distribution of sound velocity. The anchor nodes and USBL system were developed by Harbin Engineering University, China [39,40], and used for collecting sonar observation data, while SSP samples were collected by CTD and XCTD. (When applied in practice, the on-site CTD is not needed, and the historical data collected by Argos or during shared voyages could be used for model fine-tuning.) The influence of sound speed causes the signal bending. The propagation time of the signal needs to be measured from the sea surface to the bottom, so that its path can cover the entire depth profile, meaning that the sound line contains the sound velocity information of the entire profile. By this means, the distribution of sound speed between this depth range can be inverted.

At the beginning, four anchor nodes were sunk in turns to the seabed and their positions were calibrated using signal round-trip propagation time measured by USBL in a circular trajectory. These four anchor nodes formed a diamond topology. The real–time position of USBL was located through a ship-borne GPS, which was installed near the central axis of the ship. In order to improve the position accuracy of anchor nodes, the lever–arm error between USBL and GPS needs to be corrected. The distance measurement of lever–arm in the horizontal direction between the GPS and USBL is shown in Figure 4. The horizontal distance was measured in three sections, and the vertical distance was about 18.86 m ($7.5$ m under the water surface) measured under a basically wave-free environment in a harbor. Let the GPS receiver of on the ship be the coordinate origin, then the relative coordinates of USBL would be $\left(6.774, 8.392, -18.8603\right)$ in meters in X, Y, and Z (depth) directions.

For deploying each anchor node, a full depth of SSP was measured through ship-borne CTD, the product model of which is SBE911 produced by Sea-bird Scientific [41], and the anchor node was located with the measured SSP. Thus, during sound velocity inversion, the location of the seabed node is known, and the measured SSP will be used as historical reference data for model training. Considering the high time costs of SSP measurement by CTD (almost 3 h for once measurement with no ship movement), the XCTDs were adopted to collect the other nine SSPs, the model of which is HYLMT-2000 produced by [42]. XCTD provides a fast way for SSP measurement that can be performed during ship navigation, and the time cost is related to the measurement depth. For HYLMT-2000 used in this experiment, it takes only about 20 min to measure an SSP with maximum depth of 2000 m. The CTD and XCTD were arranged at the stern of the ship, and the water entry coordinates of CTD and XCTD were measured by the real–time ship borne GPS. These 13 SSPs were collected as reference SSPs, and the time interval between adjacent SSP measurements of these 13 SSPs was about 2 h.

For testing, the last full depth of SSP was measured by SVP as an SSP inversion task at the topology center among the four anchor nodes. At the same time, the USBL interacted with four anchor nodes to collect sonar observation data at a period of 8 s. Specifically, USBL broadcasts a ranging message every 8 s, and each seabed anchor node receives a response message with a timestamp. After receiving the message from each node, USBL stores and preprocesses the data, outputting node number and signal round-trip propagation delay information. The measurement of sonar observation time lasted for a total of 3 h until the measurement of the last full depth of SSP (by SVP). The location of anchor nodes and sampled SSPs are shown in Figure 5.

Although XCTD has obvious advantages of time efficiency compared with the SBE911 CTD, the depth coverage of XCTD is limited due to the pressure resistance characteristics of sensors. To tackle this issue, we adopt the SSP extending method proposed in our previous work [43] with the workflow in Figure 6.

4. Results and Discussions

4.1. Pre-Processing of SSP Data

For data-driven SSP inversion methods such as EOF–MFP and deep learning, the amount of experimental data collected is relatively small, so these models are prone to be over–fitting. To improve the accuracy of SSP estimation in few-shot learning situations, we propose the MTL model and dynamically adjusting the learning rate during task learner training stage.

As mentioned above, the SSP measured at the center of four anchor nodes is used for testing, while the other 13 SSPs are used for task learner training. However, there is still a lack of reference data for multi-task learners to extract common features of sound speed distribution. To solve this problem, 300 historical SSPs (covering at least 3500 m depth) sampled from the Pacific, Atlantic, and Indian Oceans in April of the last 10 years are adopted, which come from the world ocean database 2018 (WOD’18) [44] and are clustered into 10 groups.

For most SSPs, especially those sampled by XCTD, the depth scale can not cover the propagation depth of sonar signal, so the SSPs need to be extended. In this paper, SSP samples are extended through two steps, namely EOF matching extension for SSPs less than 3200 m depth and linear extension for SSPs more than 3200 m depth, but less than 3500 m depth. The core idea of EOF matching extension is to maintain the measured results and extend the unmeasured depth portion according to the regional empirical SSP distribution. In the EOF matching extension, the full-depth and partial-depth principal components of the empirical SSPs are first extracted separately, and then the projection coefficients are obtained by projecting the measured partial-depth SSP onto the partial-depth principal components, and finally the full-depth SSP is composed of the projection coefficients and the full-depth principal components. A detailed algorithm for EOF matching extension can be referred to in our work [43]. For linear extension, we set the gradient computation window to be 50 m, and the SSP is linearly extended based on the gradient of the last 50 m of the measured SSP up to the specified depth. An example of EOF matching extending and linear extending is given in Figure 7.

During the pre-processing of SSPs, SSPs are all standardized and interpolated with a depth spacing of 1 m, and are finally evenly divided into 50 depth layers to form down-sampled data for being reference data of neural networks.

4.2. Performance of MTL

To evaluate the accuracy performance of MTL, the root mean squared error (RMSE) of inverted SSP at the center location of four anchors is compared with the real measured SSP. The parameter settings of MTL is given in

Table 1. Parameter settings of MTL.

Training SSP clusters	10
SSP Clusters per epoch K	3
SSPs for multi-task learner training (per cluster) $S_v$	10
Multi-task learner training epochs *	20
Task learner training epochs	20
Task training SSPs per epoch	5
Maximum SSP depth	3500 m
Points of simplified SSPs	50
Learning rate $xi$	0.000002
Task learning rate	0.01
Input layer neurons	120
Hidden layer neurons	300
Output layer neurons	50
Factor for task classification ${\lambda }_{tk}$	0.02
Factor for learning rate adjustment ${\lambda }_{tk,ri}$	0.9

* One epoch corresponds to a round of parameter updating, using 3 SSP clusters.

, and data are processed by “Matlab 2023a”.

For further comparison of accuracy performance, three state-of-the-art baseline methods are also adopted: spatial interpolation (SIP), EOF-MFP, and FNN. For SIP, the proportional weight follows the principle of inverse proportion of spatial distance. For EOF-MFP, the PSO is utilized for searching matching coefficients with 20 particles and 30 iterations, and the order of the principal component is 3. For FNN, the network structure is the same as the task learner of MTL, and the learning rate is $0.01$, which is equal to that of task learner in MTL.

Table 2. RMSE of inverted SSP by different methods.

Methods	SIP (m/s)	EOF-MFP (m/s)	FNN (m/s)	MTL (m/s)
Average RMSE	0.3895	0.3341	0.2653	0.2007
0–200 (m)	0.6875	0.5578	0.3366	0.2077
200–800 (m)	0.7529	0.6675	0.2086	0.1394
800–1300 (m)	0.4144	0.3743	0.3232	0.2801
1300–3500 (m)	0.0694	0.0617	0.2567	0.1917

gives the average accuracy performance of inverted SSPs with 100 repeated results. It shows that the accuracy of MTL outperforms other state-of-the-art methods under a few-shot learning situation, implying that the mapping relationship from sound field data to SSP distribution can be better and faster captured through MTL. An example of inverted SSP is shown in Figure 8. Due to the fact that MTL performs sound velocity inversion based on sonar observation data, the main reason of error in MTL algorithm is the sonar observation error. When the error in sonar observation data are too large, the model will find it difficult to distinguish whether the changes in sonar observation data are caused by changes in sound velocity. Thus, the anti-noise interference performance of the basic model is worth researching. The sound speed on the sea surface is mainly determined by the influence of temperature changes. Due to the small proportion of the depth scale relative to the entire profile, the influence of shallow water on sonar observation data are difficult to be captured by MTL. However, the sound speed in shallow water is more chaotic due to the irregular temperature changes, so the inversion accuracy of shallow water is relatively low.

MTL relies on sonar observation data for inversion of SSPs, so the quality of sonar observation data has an important impact on inversion accuracy. In order to further test the impact of sonar observation data errors on inversion accuracy performance, we added sonar observation data noise through simulation to test the accuracy performance of the model. We add Gaussian noise with 0 means and different standard deviations, which is the noise level coefficient multiplied by the mean of the measured sonar observation data.

Table 3. Average RMSE of inverted SSP with different noise level of sonar observation data.

Noise Coefficient	0	0.001	0.002	0.005	0.01	0.05	0.1
RMSE (m/s)	0.2007	0.2031	0.2117	0.2233	0.2338	0.2494	0.2725
ratio	1.000	1.012	1.055	1.113	1.165	1.243	1.358

gives the simulation results under 100 times of testing with each noise level. The ratio is the RMSE under a certain level of noise divided by the RMSE without noise. It shows that the inversion error begins to increase significantly when the coefficient is greater than 0.05, and with a noise coefficient of 0.1, the RMSE increases about 35.8%. These results indicate the necessity of improving the model’s noise resistance performance.

To illustrate the reason for the fast convergence of MTL, the convergence performance of MTL is compared with that of FNN in Figure 9. After learning different types of SSPs during the meta learning stage, the initialization parameters of the task learner in the MTL are closer to the converged parameters, which is beneficial for faster convergence. Moreover, due to prior knowledge of sound speed distribution, the model learning process has less fluctuations, which is conducive to reaching the convergence state faster.

Since the training of neural networks for SSP inversion can be performed offline, the time consumption during SSP inversion stage are more noteworthy. The average time consumption of different methods are given in

Table 4. Time efficiency of inverted SSP by different methods.

Methods	SIP	EOF-MFP	FNN	MTL
Inversion stage (s)	0.0033	38.1980	0.0005	0.0008

. MTL inherits the time efficiency advantages of FNN during the inversion stage, because it only needs once forward propagation when feeding signal propagation time into the model, and this process can be performed by matrix operation.

5. Conclusions

In this paper, we propose an MTL method for SSP inversion to improve the accuracy under few-shot learning situations. For verifying the feasibility and effectiveness of the proposed model, a deep-ocean experiment at the South China Sea was conducted in April 2023. Through verification on real sampled SSP data and sonar observation data, it is shown that the proposed MTL has better accuracy performance in few-shot learning SSP construction issues, the RMSE of which is less than 0.25 m/s within the area of 5 km × 5 km. For SIP and EOF-MFP, the construction error of sound velocity mainly occurs in the shallow water range from 0 to 800 m, while for FNN and MTL, the construction error of sound velocity mainly occurs in the deep sea range from 1300 to 3500 m. This phenomenon indicates that the neural network model has stronger learning ability for the sound velocity distribution in shallow waters, but this strong learning ability can lead to a certain degree of over-fitting in deep sea areas with small changes in sound velocity. Moreover, due to the fact that MTL is based on neural networks, after model convergence, only one turn of forward propagation is required to obtain the sound velocity distribution, thus inheriting the high real-time performance of neural network models in sound velocity inversion.

Though MTL has good performance in accuracy and efficiency compared with the state-of-the-art methods in few-shot learning, it still faces with some challenges that limit the performance. (1) The real measured signal propagation time is coupled with strong environmental noise, and high noise level will affect the SSP inversion accuracy. (2) The determination of the distribution type of SSPs which the inversion task belongs to has a significant impact on the training of the model. To ensure the accuracy of sound velocity inversion, it is necessary to first make an accurate judgment of the distribution type of the SSPs which it belongs to.

In our future work, we are going to further study the anti-noise interference performance of the basic model to enhance the whole robustness of SSP inversion.