Proceedings of the Institute of Acoustics

 

Gaussian processes for merging water column parameters

 

F. Boulton, University of Liverpool, Liverpool, L69 7ZL, UK
D.J. Colquitt, University of Liverpool, Liverpool, L69 7ZL, UK
S. Timme, University of Liverpool, Liverpool, L69 7ZL, UK
D.P. Williams, Dstl, Porton Down, Salisbury, SP4 0JQ, UK

 

 

1 INTRODUCTION

 

Accurate sound speed profiles are essential for effectively modelling the propagation of acoustic waves. However, in-situ, point-in-time measurements are often limited to XBT and XCTD probes, at a discrete location and time, providing environmental parameters for the top of the water column. Often historical climatological, or very occasionally forecast, data for a given time-period is used as a “best estimate”. In cases where measured in-situ data exists, invariably these measurements either does not match the climatological/forecast data and/or does not cover the entire water-column. To address this problem we use a Gaussian Process (GP) data fusion method called Multi-Fidelity Data Fusion (MFDF) to extend in-situ measurements, in depth, using historical/forecast climatological data, yielding potentially more accurate, up to date, estimates of the chosen water column parameter. We apply this method to in-situ temperature data from XBT probes and historical climatological data. The merged data is then used to calculate the sound speed profile. We examine the effect of using MFDF to water column parameters with acoustic propagation modelling. 

 

In-situ measurements of temperature and depth can be obtained with instruments such as XBT probes, while simultaneous in-situ estimates of temperature, salinity and depth can be performed by Expendable Conductivity Temperature-Depth (XCTD). These instruments are typically rated up to depths of approximately 460m to 1000m, depending on the type of instrument. As such, these instruments measure parameters in the upper section of the water column which is the most variable in temporal changes. The deep water section of the profile remains relatively constant over time in many deep water areas, hence we can use pre-existing climatology to estimate the parameter profile in this section of the water without resorting to extrapolation. These merged parameter profiles can then be used to calculate the sound speed using equations such as the United Nations Educational, Scientific and Cultural Organization (UNESCO)⁵, or the Leroy et al¹⁴ sound speed equations. 

 

Various factors can affect the speed of sound and how sound waves propagate through water, including temperature, pressure, and salinity. Small changes in these parameters can have a significant impact on sound speed profiles, introducing features such as ducts and sound channels (a region within the water column where sound is trapped), which dramatically affect acoustic propagation. These profiles are essential for sonar systems, aiding in target localization, range estimation, and detection performance. In underwater communication, reliable data transmission relies on understanding sound speed variations. Additionally, sound speed profiles are vital for environmental monitoring, helping to track oceanic phenomena and interpret oceanographic data. Sound speed profiles also naturally play an important role in underwater imaging techniques like sonar imaging. Indeed, accurate sound speed profiles underpin many aspects of underwater acoustics, enabling better decision-making and more effective use of acous tic technologies. 

 

There are several existing methods to merge in-situ measurements with historical/forecast data. The simplest approach is to append historical/forecast data to the in-situ data, such that the merged profile covers the whole water column. This method is simple and easy to implement but often results in discontinuities in the merged profile, which can have a significant impact on acoustic propagation modelling. Another method developed by McHugh¹⁸ uses a decaying shift equation based on the idea that the temperature profile from the historic climatology is shifted by a small amount at the maximum depth of the in-situ profile such that the in-situ and historic profiles coincide, then a decaying shift is applied at subsequent depths to smoothly merge the two profiles. Anand et al.² also used this approach, developing software to merge water column parameters collected from in-situ measurements and historical climatology from the 2009 World Ocean Atlas¹⁵. A piece of open source software called Sound Speed Manager was developed by Masetti et al.¹⁶, the main functionality of which, is to read data collected in various formats, which can then be enhanced by in-situ profiles using oceanographic atlases before exporting this output for use in other applications. A more recent effort by Huang et al.¹⁰ combines in-situ data with the 2018 world ocean atlas⁴to reconstruct sound speed profiles using the UNSECO sound speed equation⁵. 

 

In this paper, we develop an approach to merging water column parameters using GPs, specifically MFDF. A GP is a collection of random variables, any finite number of which have a joint Gaussian distribution²¹. In essence, GPs provide a principled, non-parametric approach to regression and classification tasks, offering a natural way to incorporate prior knowledge and uncertainty into predictions²¹. GP have found applications in underwater acoustics for a range of tasks including, for example, Direction-of-Arrival Estimation¹¹ and underwater communication¹². MFDF is a developing methodology that focuses on integrating information from multiple data sources, each with varying levels of fidelity or accuracy, to improve predictive modelling and decision-making processes⁹. This approach is particularly relevant in scenarios where data from different sources have different costs, resolutions, or reliability levels. For this application, less expensive data sources (such as climatological data) might provide lower resolution or less reliable information, while more expensive sources (in-situ measurements) could offer higher resolution or more dependable data. By combining information from these sources, MFDF aims to exploit complementary strengths and mitigate individual weaknesses, leading to more accurate and robust predictions. It has been applied in aerodynamics^{3, 8, 13} for aircraft wing pressure distributions, fusing data from computational fluid dynamics simulations, wind tunnel experiments, and flight tests. GPs provide a natural framework for MFDF due to their ability to encapsulate complex relationships and quantify uncertainty. 

 

We apply the MFDF method to in-situ temperature XBT data and historical climatological temperature data. The merged data and salinity data from climatology is then used to calculate the sound speed profile using the Leroy et al¹⁴ equation, which is used as it enables accurate sound speed calculations from a single equation in a wide range of oceanographic conditions (up to salinity of 42 ppt). We illustrate the effects of merged water column parameters on acoustic wave propagation, comparing the MFDF method with two methods that have previously used in this application: (i) the append method and (ii) the decaying shift method. Using a realistic example, with real-world data, we demonstrate how small changes or uncertainty in temperature can lead to significant variations in acoustic propagation paths. Indeed, we demonstrate the choice of data fusion method can affect the presence of strongly localised propagation ducts in the merged environment. In section 2, we describe the GP-based MFDF method for merging water column parameters. In section 3, we present the results of our merged profiles and sound speed calculations. Finally, in section 4, we present our conclusions and future work. 

 

2 MERGING WATER COLUMN PARAMETERS

 

In this section we describe the append, decaying shift, and MFDF merging processes. As described in section 1, the simplest approach is to append historical/forecast data to the in-situ data, such that the merged profile covers the whole water column.

 

2.1 Decaying Shift Method

 

The method developed by McHugh¹⁸ uses a decaying shift equation

 

 

where PM_i is the merged profile, P_i is the in-situ profile, D_i is the depth of the in-situ profile, ∆ is the difference between the in-situ parameter P_j and historical parameter θ_j at the last data collection point j. C is a constant equal to 0.835, D_i is the i^th climatology depth, d_j is the deepest in-situ depth. That is, the temperature profile from the historic climatology is shifted by a small amount at the maximum depth of the in-situ profile, such that the in-situ and historic profiles coincide, then a decaying shift is applied at subsequent depths to smoothly merge the two profiles.

 

2.2 Gaussian Processes

 

We consider noisy measurements of the form y = f(x) + ϵ, where ϵ ∼ N (0, τ ²). That is, we have some x values and corresponding observed y values, and we predict the value of f(x^∗). Formally a GP is a collection of random variables, any finite number of which have a joint Gaussian distribution²¹. A GP is completely specified by its mean function m(x) : D → R and covariance function k(x, x^′) : D × D → R, where D is the input space and a vector in this space is x ∈ D. The GP is written as

 

 

where x and x′ are two points in the input space D . It can be shown 9 that the prediction at a test point x^∗ is given by

 

 

 

where µ_GP(x) is the posterior mean, σ_GP(x) is the posterior standard deviation, X = [x₁, x₂, ..., x_N ] is a vector of N training data points with x_i ∈ D, f = f(x) = [f₁, f₂, ..., f_N ]^T is the corresponding vector of function values, k(x^∗, X) = [k(x^∗, x₁), k(x^∗, x₂), ..., k(x^∗, x_N )] = k(X, x^∗)^T is the kernel, k(X, X) is a square symmetric matrix of the covariance function evaluated at the training points X, and y = y(X) = [y₁, y₂, ..., y_N ]^T is the observed data.

 

Common kernels include the squared exponential, Matern, and periodic kernels⁶ . The problem presented here is best suited to the squared exponential or Matern kernels as they are smooth and differentiable. The squared exponential kernel is defined as

 

 

where σ² and ℓ are the variance and length-scale hyperparemeters respectively, and we write θ = [σ, ℓ_k]. The hyperparameters can be found by maximising the log marginal likelihood⁹

 

 

i.e.,

 

 

A common concern with GPs is the computational cost of inverting the covariance matrix [ K(X, X)+τ²I ], which is O(N³), where N is the number of training points. This can severely limit the scalability of the GP especially in high dimensional spaces. However, there are several methods to mitigate this cost, such as inducing points, sparse GPs, and variational inference²¹.

 

2.3 Multi-Fidelity Data Fusion

 

In simple terms, the approach of MFDF is to describe multiple data sources, each with varying levels of fidelity or the confidence in the data. For each information source we construct an intermediate GP model µ_GP,i, σ_GP,i  as described in section 2.2. Here the standard deviation σ_GP,i(x) is the uncertainty in the prediction away from a training point. To model the uncertainty associated with the information sources themselves, we introduce a fidelity function for each information source¹³ σ_f,i (x), the total variance is then given by

 

 

A key challenge with using MFDF is that of how to appropriately define the fidelity functions σ_f,i (x). The choice of fidelity function is subjective and depends on the problem at hand. We will discuss our choice of fidelity functions in section 3. The multifidelity fused estimate of f(x) is a weighted sum of the individual GP models, where more importance is given to high confidence models⁹ as follows

 

 

It should be emphasised that, whilst more importance is given to high confidence models, this means that if a particular model has a high fidelity but is poorly represented by the intermediate GP it will nevertheless have a small contribution to the fused estimate.

 

3 RESULTS

 

We apply our MFDF approach to the problem of merging in-situ temperature data with historical climatological temperature data. The goal is to create a smooth, physically realistic, and representative merged temperature profile. We use the XBT data from the World Ocean Database⁴ as the in-situ measurement, and the Copernicus monthly mean Global Ocean Physics Analysis and Forecast⁷ as the historical clima tological data. We use GPflow¹⁷ — a Python package for GP which uses Tensorflow¹ — for the MFDF framework. The fidelity function for the in-situ data is defined as the accuracy of the sensor, which is typically ±0.1^◦C depending on the model, the fidelity function for the historical data is defined as the monthly temperature variance observed in the data. We compare the MFDF method to the decaying shift and append methods. With the merged temperature profile in hand, we use salinity data from the Copernicus monthly mean Global Ocean Physics Analysis and Forecast⁷ to calculate the sound speed profile using the Leroy et al sound speed equation¹⁴, which has the inputs temperature, salinity, latitude and depth. From here we perform acoustic propagation modelling using Bellhop²⁰, specifically bellhopcxx¹⁹. Finally, we compare the results of the three merging methods with the sound speed profile generated from only climatology data with no in-situ update.

 

 

Figure 1: Temperature, Salinity, and Sound Speed profiles for the North Atlantic ocean.

 

Our example XBT probe was dropped on 2023-01-01 in the North Atlantic ocean, specifically located at 36.807^◦N, 48.46^◦W. The XBT probe used in this example has a maximum rated depth of 760m and the XBT data extends past this point to around 960m where it then fails. However, we cannot be confident that the XBT probe is accurate past 760m and as such this data is excluded. The merged profiles are shown in Figure 1. We observe that the append method results in a discontinuity at the point where the in-situ data ends, whereas both the decaying shift and MFDF methods result in smooth transitions. We also notice that including the in-situ data results in a temperature profile which is very close to the climatology profile, however slight differences in curvature can have a significant impact on the sound speed profile in this case causing a duct from 200m to 400m in depth. 

 

To illustrate the effect of these merging methods on the propagation of acoustic waves, we model a 100Hz source positioned at 300m in depth with launch angles of ±10^◦from the horizontal. The results are plotted in Figures 2 and 3. We see that the append and decaying shift methods produce less physically reasonable acoustic propagation paths, due to the noise from the XBT data. The MFDF method accounts for this noise producing a smooth profile which results in smoother acoustic propagation paths. Using the MFDF method, we also observe the duct from 200m to 400m in depth. This duct is not readily apparent in the climatology only model, nor the append and decaying shift approaches. Intuitively, it is expected that the averaging processes used for the climatological data typically reduces variable, but acoustically important sound-speed features. Never the less, these results demonstrate the importance of fusing multiple sources of environmental data and using appropriate algorithms when modelling acoustic wave propagation. 

 

4 CONCLUSION

 

We have presented a method for merging in-situ temperature data with historical climatological temperature data using the MFDF approach. We applied this method to XBT data from the World Ocean Database and the Copernicus monthly mean Global Ocean Physics Analysis and Forecast. The merged temperature profile was then used to create a sound speed profile.

 

 

Figure 2: Acoustic propagation with merged profiles using different methods.

 

 

Figure 3: Averaged acoustic propagation in range and depth.

 

We demonstrated the importance of merged water column parameters in acoustic propagation modelling, showing that small differences in temperature profiles can lead to significant variations in sound speed profiles and thus acoustic propagation. We have shown that the MFDF method produces a smooth temperature profile that accounts for in-situ updates of water column parameters, resulting in a more accurate acoustic propagation model compared to the append and decaying shift methods. Another advantage of the MFDF method is that it can be extended to higher dimensional spaces, allowing for the merging of multiple water column parameters over a wide area. This is ongoing work with future development focusing on uncertainty quantification in the merged profiles and the application to higher dimensional spaces.

 

ACKNOWLEDGMENTS

 

The authors are grateful for the comments of the technical reviewer from the Defence Science and Technology Laboratory (Dstl), which improved the manuscript immensely. FB gratefully acknowledges the financial support by the Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training in Distributed Algorithms (grant number 2447391 as part of EP/S023445/1) in partnership with Dstl. DJC acknowledges funding from The Leverhulme Trust through Research Project Grant RPG- 2022-261. Part of this work was commissioned by Dstl as part of the FASTER programme. The authors would like to thank the FASTER team for their support and guidance. The contents include material subject to © Crown copyright (2024), Dstl. This material is licensed under the terms of the Open Government Licence except where otherwise stated. To view this licence, visit http://www.nationalarchives.gov.uk/doc/open-government-licence/version/3 or write to the Information Policy Team, The National Archives, Kew, London TW9 4DU, or email: psi@nationalarchives.gov.uk

 

REFERENCES

 

1. Mart´ın Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. TensorFlow: a system for Large-Scale machine learning. In 12th USENIX symposium on operating systems design and implementation (OSDI 16), pages 265–283, 2016.

2. P. Anand, P.V. Hareesh Kumar, and K.G. Radhakrishnan. A software to merge temperature and salinity profiles with climatology. Computers & Geosciences, 52:356–360, March 2013. ISSN 00983004. doi: 10.1016/j.cageo.2012.10.018. URL: https://linkinghub.elsevier.com/retrieve/pii/S0098300412003676.

3. Mehdi Anhichem, Sebastian Timme, Jony Castagna, Andrew J Peace, and Moira Maina. Data fusion of wing pressure distributions using scalable gaussian processes. AIAA Journal, 62(5):1946–1961, 2024. doi: 10.2514/1.J063317. URL: https://arc.aiaa.org/doi/10.2514/1.J063317.

4. Tim P Boyer, Hern´an E Garc´ıa, Ricardo A Locarnini, Melissa M Zweng, Alexey V Mishonov, James R Reagan, Katharine A Weathers, Olga K Baranova, Christopher R Paver, Dan Seidov, et al. World ocean atlas. 2018.

5. Chen-Tung Chen and Frank J. Millero. Speed of sound in seawater at high pressures. The Journal of the Acoustical Society of America, 62(5):1129–1135, November 1977. ISSN 0001-4966. doi: 10.1121/1.381646. URL: https://doi.org/10.1121/1.381646.

6. David Duvenaud. Automatic model construction with Gaussian processes. PhD thesis, 2014.

7. E.U. Copernicus Marine Service Information (CMEMS). Global ocean physics analysis and forecast. E.U. Copernicus Marine Service Information (CMEMS). Marine Data Store (MDS). DOI: 10.48670/moi-00016 (Accessed on 01-May-2024), 2024.

8. Alex Feldstein, David Lazzara, Norman Princen, and Karen Willcox. Multifidelity Data Fusion: Application to Blended-Wing-Body Multidisciplinary Analysis Under Uncertainty. AIAA Journal, 58(2):889–906, February 2020. ISSN 0001-1452. doi: 10.2514/1.J058388. URL: https://arc.aiaa.org/doi/10.2514/1.J058388. Publisher: American Institute of Aeronautics and Astronautics.

9. Alexander W Feldstein. Multi-fidelity data fusion for the design of multidisciplinary systems under uncertainty. PhD thesis, Massachusetts Institute of Technology, 2018.

10. Chenhu Huang, Meiping Wu, Xianyuan Huang, Juliang Cao, Jianlun He, Cheng Chen, Guojun Zhai, Kailiang Deng, and Xiuping Lu. Reconstruction and evaluation of the full-depth sound speed profile with world ocean atlas 2018 for the hydrographic surveying in the deep sea waters. Applied Ocean Research, 101:102201, August 2020. ISSN 01411187. doi: 10.1016/j.apor.2020.102201. URL: https://linkinghub.elsevier.com/retrieve/pii/S0141118719306297.

11. Ishan D. Khurjekar, Peter Gerstoft, Christoph F. Mecklenbr¨auker, and Zoi-Heleni Michalopoulou. Direction-of-arrival estimation using gaussian process interpolation. In ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1–5, 2023. doi: 10.1109/ICASSP49357.2023.10094761.

12. George P Kontoudis and Daniel J Stilwell. A comparison of kriging and cokriging for estimation of underwater acoustic communication performance. In Proceedings of the 14th International Conference on Underwater Networks & Systems, pages 1–8, 2019.

13. R´emi Lam, Douglas L Allaire, and Karen E Willcox. Multifidelity optimization using statistical surrogate modeling for non-hierarchical information sources. In 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, page 0143, 2015.

14. Claude C. Leroy, Stephen P. Robinson, and Mike J. Goldsmith. A new equation for the accurate calculation of sound speed in all oceans. The Journal of the Acoustical Society of America, 124(5):2774–2782, November 2008. ISSN 0001-4966, 1520-8524. doi: 10.1121/1.2988296. URL: https://pubs.aip.org/jasa/article/124/5/2774/910920/A-new-equation-for-the-accurate-calculation-of.

15. Sydney Levitus, Ricardo A Locarnini, Timothy P Boyer, Alexey V Mishonov, John I Antonov, Hernan E Garcia, Olga K Baranova, Melissa M Zweng, Daphne R Johnson, and Dan Seidov. World ocean atlas 2009. 2010.

16. G Masetti, B Gallagher, B R Calder, C Zhang, and M Wilson. An open-source application to manage sound speed profiles. INTERNATIONAL HYDROGRAPHIC REVIEW, 2017.

17. Alexander G de G Matthews, Mark Van Der Wilk, Tom Nickson, Keisuke Fujii, Alexis Boukouvalas, Pablo Le, Zoubin Ghahramani, James Hensman, et al. Gpflow: A gaussian process library using tensorflow. Journal of Machine Learning Research, 18(40):1–6, 2017.

18. G. F. Mchugh. An improved method for matching and merging shallow in situ with deep-ocean climatological temperature profiles. Marine Technology Society Journal, 29:32–45, 1995. URL: https://api.semanticscholar.org/CorpusID:114634750.

19. Louis A Pisha, Joseph Snider, Khalil Jackson, and Jules S Jaffe. Introducing bellhopcxx/bellhopcuda: Modern, parallel bellhop (3d). The Journal of the Acoustical Society of America, 153(3 supplement):A218–A218, 2023.

20. Michael B Porter. The bellhop manual and user’s guide: Preliminary draft. Heat, Light, and Sound Research, Inc., La Jolla, CA, USA, Tech. Rep, 260, 2011.

21. Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian processes for machine learning. Adaptive computation and machine learning. MIT Press, Cambridge, Mass, 2006. ISBN 978-0-262-18253-9. OCLC: ocm61285753.