LI Jiemeihui , SHI Yang , YANG Yixin , , and CHEN Cheng

1)School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710000, China

2)Shaanxi Key Laboratory of Underwater Information Technology, Northwestern Polytechnical University, Xi’an 710000, China

Abstract Traditional acquisition method of sound speed profiles using hydro-acoustic instruments is accurate but time-consuming and costly. To overcome this problem, some inversion methods have been developed over the last few decades. In this study, a comprehensive comparison of two inversion methods – the acoustic inversion method (AIM)and the satellite observation reconstruction method (SOR)– is presented. For AIM, the sound speed profile is first parameterized by the empirical orthogonal function (EOF)and the optimal parameters are searched by simulated annealing algorithm with respect to the cross-correlation function of the receiving signal and the simulation signal. For SOR, remotely sensed data are used to construct sound speed profiles. An experiment was conducted in the northeast of the South China Sea to verify the two methods. Both methods can obtain sound speed profiles quickly and cheaply. Compared with the sound speed profiles obtained by a conductivity-temperature-depth (CTD)instrument, the root-meansquare-error (RMSE)of AIM is 0.55 m s-1 and that of SOR is 1.71 m s-1. It is clear that AIM provides better inversion performance than SOR. Another primary benefit of AIM is that this method has no limitation to the inversion depth. The simulation results of sound propagation in regard to the inversed sound speed profiles show that the transmission losses of AIM and CTD are consistent and that of SOR is adversely affected by the inversion error of the sound speed and the inversion depth. But SOR has particular advantages in the inversion coverage. Together, all of these advantages make the AIM particularly valuable in practice.

Key words sound speed profile; inversion; EOF; cross-correlation; remote sensing; South China Sea

1 Introduction

The sound wave has better propagation performance than the electromagnetic wave and the light in the ocean. The marine environment plays a significant role in the acoustic propagation (Abraham, 2019). Sound speed is one of the most important factors affecting the underwater acoustic propagation and, consequently, the underwater target detection, location, tracking, and recognition are also greatly affected (Miller and Miller, 2018; Xuet al., 2019; Sazontov and Smirnov, 2020). Sound speed is the function of temperature, salinity and depth (Urick, 1983). It varies linearly with depth and non-linearly with temperature and salinity according to an empirical formula derived from numerous experimental measurements (Biancoet al., 2016).

Traditional sound speed profile (SSP)acquisition methods are mainly based on hydro-acoustic instruments carried by vessels, such as conductivity-temperature-depth (CTD)instruments and expendable bathythermographs (XBTs)(Songet al., 2009; Zhanget al., 2016). In addition, many global observation experiments are carried out by moorings, Argo floats, gliders, and other underwater mobile platforms (Jeffreyet al., 2011; Matsumotoet al., 2011; Liet al., 2022).The field has gradually broadened as three-dimensional ocean sensor networks (Wanget al., 2012). These methods have notable benefits in terms of accuracy. Therefore, these methods have been widely adopted in the field of underwater parameter acquisition. However, these direct acquisitions generally require substantial time, manpower and material resources.

There are growing appeals for acquiring underwater SSPs quickly and cheaply. Thus, a number of inversion methods have been developed over the past few decades (Boden and DeSanto, 1987; Chapman, 2008). A common method involves the use of satellite observations. The US Naval Research Laboratory designed the Modular Oceans Data Assimilation System (MODAS)to produce gridded sound speed fields using real-time remotely sensed data from satellite and historical climatology (Foxet al., 2002; Chuet al., 2004;Liet al., 2019). It has been widely used by the US Navy and many universities. Maoet al. (2013)proposed a threedimensional temperature and salinity reconstruction system in the South China Sea. Chenet al. (2016)described a single datum assimilation method for inversing the SSPs in the Philippine Sea.

Another popular approach to SSP acquisition is the acoustic inversion method. This method constitutes a relatively new research area which has emerged from the underwater acoustic propagation and historical acoustic data. Le-Blanc proposed a method of describing SSPs in terms of the empirical orthogonal function (EOF)in 1980, which laid the foundation for the acoustic inversion (LeBlanc and Middleton, 1980; Shenet al., 1999). On this basis, Munket al. (1995)reported an SSP inversion method using the arrival time delay in 1995. In the same year, Makriset al.(1995)investigated the effect of environment parameters on three-dimensional sound speed inversion under the EOF theory.

Since then, various inversion methods of sound speed have emerged. Yu proposed an inversion method of sound speed with the parallel genetic algorithm and estimated empirical orthogonal functions (Yuet al., 2010). Charantonis utilized the hidden Markov model defined by the self-organizing network to inverse the biological and geographical parameters of the marine profiles according to the sea surface data (Charantoniset al., 2015a). Subsequently,they used the self-organizing network to reconstruct the temperature-salinity profiles based on the sparse database of the underwater autonomous glider (Charantoniset al.,2015b).

The South China Sea has a complex environment and an important geographic location. However, the effectiveness of different SSP inversion methods in the South China Sea has rarely been directly discussed.

In this study, a comprehensive comparison of two SSP inversion methods is presented. The acoustic inversion method (AIM)is a new method that uses the cross-correlation function matching, EOF theory and simulated annealing to get sound speed profiles in the ocean. The control group is the satellite observation reconstruction method (SOR)using remotely sensed data. An ocean experiment was carried out to verify their accuracy, validity, and application scope compared with SSPs measured by the CTD instrument. The overall goal of this paper is to explore the inversion method of the sound speed with lower errors and costs. Section 2 describes the principle and procedure of two methods. The experiment is introduced in Section 3.Section 4 discusses the comparison results. Finally, Section 5 is the conclusion.

2 Methods

The main objective of AIM and SOR is to reduce the cost of obtaining SSPs without performance degradation in inversion accuracy. AIM obtains SSPs according to the signals received by a single hydrophone and matching optimization. SOR provides multiple SSPs in a grid using the optimal interpolation technique and remotely sensed data.

2.1 Acoustic Inversion Method via Cross-Correlation Function Matching

AIM is based on EOF representation of the sound speed.Therefore, the main focus of the method is to search for the appropriate EOF parameters. The autocorrelation function of signals received by a hydrophone and that of model simulation are matched to find the optimal parameters according to simulated annealing. These parameters are enough to synthesize a sufficiently accurate SSP.

According to the SSPs synthesized by different EOF parameters, the received signals are simulated by the Bellhop model. The Bellhop, Gaussian beam ray model, is sensitive to the sound speed because the ray transmission trajectory function can be expressed by the sound speed (Jensenet al., 2011).

The implementation steps are as follows.

Step 1. The process begins with the transmitting signals(n). A single hydrophone receives the signalx(n)through acoustic propagation. The autocorrelation function isRx(m).

whereEis the expected value operator, and the asterisk denotes complex conjugation.

Step 2. The first part of parameter simulation is to convert the SSP into an EOF representation (LeBlanc and Middleton, 1980; Shenet al., 1999). Multiple historical SSPs form a matrixSNand its covariance matrix isR. The matrixRis eigen decomposed. The eigenvalues are arranged from maximum to minimum.Fis the eigenfunction matrix ofR, which is also called the empirical orthogonal function matrix.Nis the rank of matrixR.

Hence, the SSP can be expressed as

whereSmeanis the average SSP ofSaveraged by time andanis the EOF parameter. If a few eigenvalues corresponding tofnare much larger than the others, it is feasible to characterize the sound speed profile with fewer parameters of EOF.

The parameter range is obtained by fitting the historical SSPs according to the polynomial fitting. They are close to optimal parameters.

Subsequently, the SSPs synthesized by the EOF parameters are input into Bellhop and the transfer functionh(n)is obtained. It consists of the amplitudes and phases received by the hydrophone. When the transmitting signals(n)convolves with the transfer functionh(n), the simulation signalxb(n)is generated. Its autocorrelation function isRxb(m).

Step 3. Two autocorrelation functions combine a crosscorrelation functionRxxb(m). Each set of EOF parameters corresponds to a cross-correlation function.

Taking the maximum value (MRxxb)of the real part ofRxxb(m)as the cost function, this method utilizes simulated annealing optimization algorithm to find the optimal parameters in parameter domain. The searching model is as follows.

The inputs of simulated annealing procedure are the EOF parameters (an,n= 1, 2, ···,N), the parameter limitations (LB,UB), and the objective function (Fun). Fun accepts the inputanand returns a cross-correlation function value evaluated ata. The step 1 (actual signal process)and the step 2 (simulation signal process)are contained within the Fun.

Thus, the parameters corresponding to the maximum is the best solution. Finally, the SSP synthesized by these parameters is the inversion result and the inversion depth depends on the water depth. This process is shown in Fig.1.

2.2 Satellite Observation Reconstruction Method

SOR combines real-time remotely sensed data and historical measurements to generate a three-dimensional temperature-salinity field and then reconstructs the SSPs.

It is a modular tool that uses optimal interpolation for data assimilation, single empirical orthogonal function regression (sEOF-R)for constructions, and numerous statistical analyses (Maoet al., 2013). The process is shown in Fig.2.Initial remotely sensed data are used to inverse the underwater temperature field. Then a three-dimensional temperature field is produced in combination with historical data.The salinity field is obtained by using the temperature and salinity regression. Finally, this method uses an empirical formula to generate SSPs. It can depict near real-time SSPs on a 0.25? × 0.25? grid in the time-varying area between 0?– 25?N in latitude and 105? – 122?E in longitude of the South China Sea. The inversion depth in this method is 1500 m.

Fig.1 Diagram of the acoustic inversion method.

Fig.2 Diagram of the satellite observation reconstruction method.

In this study, the AIM and SOR inversion methods are used to obtain SSPs in the South China Sea. The results are evaluated by comparison with measured SSPs.

3 Experiment

The experiment was carried out in the deep-sea region of the South China Sea on June 17, 2017. Fig.3 shows the location of the experiment. The red dot indicates the site where the experimental ship was moored, and the red star marks the position of the vertical array. The sediment is silty clay with an average water depth of 2500 m.

Fig.3 Experiment setup showing the location. The dot and the star represent the transmitting and receiving sites respectively.

During the experiment, the acoustic part was accomplished by using a transmitting transducer and a hydrophone, 13.54 km apart. As shown in Fig.4, the source, suspended below the ship at the depth of 45 m, transmitted linear frequency modulation (LFM)signals at two central frequencies: 3000 Hz with bandwidth of 200 Hz and 650 Hz with bandwidth of 100 Hz. Each signal had a duration of 2 s and the signal interval was 10 s. Both types of signals were repeated 25 times. The sound source level was 195.7 dB re 1 μPa @ 1 m. These signals were received by a hydrophone attached to a vertical cable at a depth of 1150 m,which is shown in Fig.4. The sample frequency of the hydrophone was 16 kHz and the sensitivity level was -195 dB.

Fig.4 Layout of experimental equipment and the domain time of receiving signal.

A CTD instrument was used to acquire approximately accurate SSPs for comparison with the inversion results.The CTD recorded SSPs as it was lowered slowly and vertically. Then it was moored under the hydrophone near the bottom of the sea.

To keep the data synchronized, remotely sensed data(Figs.5a, b)were downloaded simultaneously by connecting satellites over the sea. These data mainly included sea surface temperature (SST), which comes from the infrared sensor on two satellites (TERRA and AQUA), and sea surface height (SSH), which was derived from three satellite altimeters (China’s ocean 2 satellite HY-2A, the European Space Agency’s ice detection satellite Cryosat-2, and Jason-2). These remotely sensed data were used to reconstruct SSPs.

Fig.5 Remotely sensed data of (a)SST, and (b)SSH in the South China Sea. The horizontal axis represents longitude and the vertical axis represents latitude.

In addition, historical temperature and salinity data required for inversion are derived from the National Oceanic and Atmospheric Administration (www.nodc.noaa.gov)and the China Argo Real-time Data Center (www.argo.org.cn).The internal ID numbers of Argos available are 0380 and 0382 because they moored near the experimental stations.The instrument type is HM2000 and their observations are transmitted through the BEIDOU Profile Buoy Data Service Center.

4 Results and Discussion

4.1 Order Selection of AIM

The order reduction of AIM is discussed in this subsection. The historical SSPs were collected from NOAA and Argo. There were 57 samples in total, and the sampling time was distributed in each month of the year. The EOF, obtained from all the samples, has 56 orders, because the rank of covariance matrix is 56. The EOF curves corresponding to the largest four eigenvalues is shown in Fig.6. Taking the first one, two, three or four order EOF into AIM for simulation respectively, the output of AIM with different orders are presented in the Table 1.

Fig.7 is a normalized fuzzy surface of matching correlation values changing with the first two EOF parameters.The plus sign corresponds to the maximum value of the fuzzy surface. This shows that AIM can distinguish subtle EOF disturbances.

The performance of AIM in inversion accuracy is measured by the root-mean-square error (RMSE)wheresis the inversed sound speed,crepresents the real sound speed from CTD, andKis the number of samples in depth.

Fig.6 Four orders of EOF.

Table 1 Output of AIM with different orders

Fig.7 Normalized fuzzy surface of the first two EOF parameters.

It can be seen from Table 1 that there is little difference in the maximum values of cross-correlation functions. The runtime rises as the EOF order increases. TheRMSEdecreases with the increasing order and it can converge to a constant level quickly in the third order and third-order EOF has relatively less running time. This means that higherorder EOF have little contribution and it is appropriate to select the first three EOF to representS. Three parameters are needed to be searched to describe the SSP. The following experiments uniformly use the third-order EOF for AIM.

4.2 Inversion Results

The results of AIM and SOR in SSP inversion are compared in this subsection. Fig.8 shows six SSPs from historical average data, SOR, CTD, first-order, second-order,and third-order AIM. These profiles from AIM are inversed with the signals which have a central frequency of 3000 Hz. The measurements from CTD are red and this result can be regarded as reliable. The mean of the 57 samples is black. The historical samples were measured by Argo at different times, around the experimental position.

Fig.8 SSPs from historical data (black), CTD (red), SOR(pink), first-order (grey), second-order (green)and thirdorder (blue)AIM.

AIM uses signals with two central frequencies (3000 Hz and 650 Hz)in the experiment. 25 SSPs were obtained for each signal. These results show no significant differences between repeated experiments. In Fig.8, there is a profile from third-order AIM results by using 650 Hz signals under the blue line. They are consistent with theRMSEof 0.0057 m s-1. It is clear that the frequency of the transmitting signal has little effect on the inversion results and the third-order AIM has the lowest error. Another promising finding is that theRMSEof SSP between the CTD and AIM is 0.55 m s-1and that between the CTD and SOR is 1.71 m s-1. In terms of depth, both inversion methods suffer from more inversion errors in the mixed layer (within the first 100 m)and thermocline (about 900 m).

4.3 Inversion Performance

The following subsection focuses on the error comparison at different depth, as shown in Fig.9. It can be seen that the inversion error of AIM is much lower than that of SOR at different depths. TheRMSEbetween AIM and CTD SSPs (blue bar)is 0.81 m s-1in the first 100 m and then reaches a peak of 1.38 m s-1at 500 m. Next, it drops suddenly to 0.33 m s-1at 700 m. Subsequently, theRMSEremains at a relatively low level until the bottom of the sea. The yellow bar in the Fig.9 shows the RMSE between SOR and CTD SSPs. It is 3.08 m s-1in the first 100 m, which is almost 3.8 times as large as AIM. After peaking of 3.14 m s-1at 200 m, it plunges to the lowest point(0.32 m s-1)at 600 m. The second largestRMSEoccurs between 700 and 1100 m, about 1.75 m s-1. This depth is near the acoustic axial channel. After that, theRMSEgradually decreases until 1500 m. Overall, the present results confirm that the error in shallow water is much larger than that in deep water for both inversion methods because the mixed layer is affected by the season and wind waves.

Fig.9 RMSE in different depths from AIM (blue)and SOR(yellow).

4.4 Comparison of AIM and SOR

Together, the inversion results confirm that both inversion methods are cheap and accurate, but have different characteristics. Table 2 provides a detailed comparison.

Table 2 The inversion performance of AIM and SOR

First, when compared with the SSP measured by only the CTD instrument, theRMSEof SOR is almost three times than that of AIM. Furthermore, AIM has smaller errors in different depths. It is clear that AIM gives better inversion performance than SOR. Second, AIM has substantially better inversion advantages in depth than SOR. This is because the longitudinal range of acoustic inversions is the full depth of the sea in AIM, whereas the depth limit of SOR is 1500 m. Third, one limitation of AIM, however, is that it covers only the distance of acoustic propagation in one measurement. Another limitation is the assumption that the horizontal sound speed does not change with the propagation distance. SOR performs well in the inversion coverage because satellite remote sensing signals cover the whole of the South China Sea. In traditional acquisition method, CTD instruments measure at only one point at a time. Fourth,SOR connects to satellites to download data at little cost,while AIM requires at least a transducer and a hydrophone as the basis of the experiment. But SSPs obtained by AIM can be regarded as a bonus in the process of sound propagation. Lastly, when it comes to acquisition time, both methods are fast. CTD instruments need several hours to measure once. Furthermore, AIM can obtain a SSP at every measurement, while SOR can only reconstruct a SSP in one place per day.

4.5 Comparison of Transmission Loss

The transmission losses in the process of acoustic propagation are compared in this subsection. Taking different SSPs into Bellhop, the influence of inversion results on transmission loss can be observed easily. Since the inversion depth of SOR is only 1500 m and the experimental sea depth is 2500 m, part of the SSP must be continued. The SSP obtained by SOR deeper than 1500 m is extrapolated from pressure. Fig.10 shows the comparison of transmission losses (TL)from three SSPs at 3 kHz. The first surface and bottom reflections are in the similar location. However, the reflection widths are inconsistent because of the leakage of sound energy. It is more severe in the sea surface near the location of source.

The average transmission losses with center frequency of 3000 Hz in the different depths are compared in the Fig.11.According to one-third octave, the left and right frequency points of 3000 Hz are about 2600 and 3300 Hz. A frequency point is selected every 100 Hz for simulation, and the transmission losses of eight simulations are averaged. The thickness of the mixed layer is 50 m, so the TL at a depth of 30 m is calculated and shown in Fig.11a. Apparently, TL vary greatly in the range of 2 – 7 km. The maximum difference of TL between CTD and AIM is 4 dB, and that of SOR is approximately twice. 200 m is the conventional depth of the submarine. In Fig.11b, a downward trend after 17.8 km can be found for AIM, while TL of SOR remains rising.1000 m is usually near the axial channel. Only SOR has a 10 dB of TL at 16.7 km in the Fig.11c. In practice, the real sound propagation process is more complicated. Obviously,the inversion results of AIM are closer to the real values than that of SOR in the South China Sea.

Fig.10 Transmission loss for CTD (a), SOR (b), and AIM (c).

Fig.11 Transmission loss for CTD (blue), SOR (gray), and AIM (pink)at the depth of 30 m (a), 200 m (b)and 1000 m (c).

As discussed above, the two inversion methods have different application scenarios. SOR performs well in the case of getting substantial data at once. AIM is the better choice in a vessel or submarine. The combination of the two methods can improve the inversion effect. AIM can fill the mesh gaps of SOR and the inversion results of SOR can be used as historical data to assist the inversion of AIM.

5 Conclusions

In this study, the SSPs obtained from two inversion methods are compared with CTD measurements in the northeast of the South China Sea. Both methods can obtain SSPs quickly and cost less than traditional methods. They have different characteristics. But one of the key benefits of AIM is the high accuracy in different depths. All of these advantages make it particularly valuable in terms of underwater sound speed acquisition. The next step requires further researches about how to combine the two inversion methods effectively.

Acknowledgements

This work was supported by the project funded by the National Natural Science Foundation of China (Nos. 419 06160, 11974286 and 12174312). We would like to thank Prof. Yiquan Qi for providing the information of satellite observation reconstruction method in this study.