Research PaperNo Access

Accurate Broadband Gradient Estimates Enable Local Sensitivity Analysis of Ocean Acoustic Models

Michael C. Mortenson

Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA

Search for more papers by this author

Tracianne B. Neilsen

Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA

E-mail Address: tbn@byu.edu

Search for more papers by this author

Mark K. Transtrum

Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA

E-mail Address: mktranstrum@byu.edu

Search for more papers by this author

, and

David P. Knobles

Knobles Scientific and Analysis, LLC, Austin, TX 78731, USA

Search for more papers by this author

https://doi.org/10.1142/S2591728522500153Cited by:5 (Source: Crossref)

Abstract

Sensitivity analysis is a powerful tool for analyzing multi-parameter models. For example, the Fisher information matrix (FIM) and the Cramér–Rao bound (CRB) involve derivatives of a forward model with respect to parameters. However, these derivatives are difficult to estimate in ocean acoustic models. This work presents a frequency-agnostic methodology for accurately estimating numerical derivatives using physics-based parameter preconditioning and Richardson extrapolation. The methodology is validated on a case study of transmission loss in the 50–400Hz band from a range-independent normal mode model for parameters of the sediment. Results demonstrate the utility of this methodology for obtaining Cramér–Rao bound (CRB) related to both model sensitivities and parameter uncertainties, which reveal parameter correlation in the model. This methodology is a general tool that can inform model selection and experimental design for inverse problems in different applications.

Keywords:

References

1. A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (Society for Industrial and Applied Mathematics, Philadelphia, 2005). Crossref, Google Scholar
2. B. B. Machta, R. Chachra, M. K. Transtrum and J. P. Sethna, Parameter space compression underlies emergent theories and predictive models, Science 342 (2013) 604–607. Crossref, Google Scholar
3. S. E. Dosso and M. J. Wilmut, Quantifying data information content in geoacoustic inversion, IEEE J. Ocean. Eng. 27 (2002) 296–304. Crossref, Google Scholar
4. S. E. Dosso, P. L. Nielsen and M. J. Wilmut, Data error covariance in matched-field geoacoustic inversion, J. Acoust. Soc. Am. 119 (2006) 208–219. Crossref, Google Scholar
5. J. Dettmer, S. E. Dosso and C. W. Holland, Model selection and Bayesian inference for high-resolution seabed reflection inversion, J. Acoust. Soc. Am. 125 (2009) 706–716. Crossref, Google Scholar
6. J. Dettmer, S. E. Dosso and J. C. Osler, Bayesian evidence computation for model selection in non-linear geoacoustic inference problems, J. Acoust. Soc. Am. 128 (2010) 3406–3415. Crossref, Google Scholar
7. M. K. Transtrum, B. B. Machta and J. P. Sethna, Geometry of nonlinear least squares with applications to sloppy models and optimization, Phys. Rev. E 036701 (2011) 1–35. Google Scholar
8. M. K. Transtrum, B. B. Machta, K. S. Brown, B. C. Daniels, C. R. Myers and J. P. Sethna, Perspective: Sloppiness and emergent theories in physics, biology, and beyond, J. Chem. Phys. 143(1) (2015) 07B201. Crossref, Google Scholar
9. D. Lill, J. Timmer and D. Kaschek, Local Riemannian geometry of model manifolds and its implications for practical parameter identifiability, PloS one 14 (2019) e0217837. Crossref, Google Scholar
10. K. N. Quinn, M. C. Abbott, M. K. Transtrum, B. B. Machta and J. P. Sethna, Information geometry for multiparameter models: New perspectives on the origin of simplicity, arXiv:2111.07176. Google Scholar
11. M. K. Transtrum, B. B. MacHta and J. P. Sethna, Why are nonlinear fits to data so challenging?, Phys. Rev. Lett. 104 (2010) 2–5. Crossref, Google Scholar
12. M. Girolami and B. Calderhead, Riemann manifold langevin and hamiltonian monte carlo methods, J. R. Stat. Soc. Series B Stat. Methodol. 73 (2011) 123–214. Crossref, Google Scholar
13. H. H. Mattingly, M. K. Transtrum, M. C. Abbott and B. B. Machta, Maximizing the information learned from finite data selects a simple model, Proc. Natl. Acad. Sci. 115 (2018) 1760–1765. Crossref, Google Scholar
14. C. H. LaMont and P. A. Wiggins, Correspondence between thermodynamics and inference, Phys. Rev. E 99 (2019) 052140. Crossref, Google Scholar
15. K. N. Quinn, C. B. Clement, F. De Bernardis, M. D. Niemack and J. P. Sethna, Visualizing probabilistic models and data with intensive principal component analysis, Proc. Natl. Acad. Sci. 116 (2019) 13762–13767. Crossref, Google Scholar
16. H. K. Teoh, K. N. Quinn, J. Kent-Dobias, C. B. Clement, Q. Xu and J. P. Sethna, Visualizing probabilistic models in Minkowski space with intensive symmetrized Kullback–Leibler embedding, Phys. Rev. Res. 2 (2020) 033221. Crossref, Google Scholar
17. M. K. Transtrum and P. Qiu, Model reduction by manifold boundaries, Phys. Rev. Lett. 113 (2014) 098701. Crossref, Google Scholar
18. F. Casey, R. Gutenkunst, C. Myers, D. Baird, K. Brown, J. Waterfall, Q. Feng, R. Cerione and J. Sethna, Optimal experimental design in an epidermal growth factor receptor signalling and down-regulation model, IET Syst. Biol. 1 (2007) 190–202. Crossref, Google Scholar
19. K. S. Brown and J. P. Sethna, Statistical mechanical approaches to models with many poorly known parameters, Phys. Rev. E 68 (2003) 9. Crossref, Google Scholar
20. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007). Google Scholar
21. P. Gerstoft and C. F. Mecklenbräuker, Ocean acoustic inversion with estimation of a posteriori probability distributions, J. Acoust. Soc. Am. 104 (1998) 808–819. Crossref, Google Scholar
22. S. E. Dosso, P. M. Giles, G. H. Brooke, D. F. McCammon, S. Pecknold and P. C. Hines, Linear and nonlinear measures of ocean acoustic environmental sensitivity, J. Acoust. Soc. Am. 121 (2007) 42–45. Crossref, Google Scholar
23. M. D. Collins and L. Fishman, Efficient navigation of parameter landscapes, J. Acoust. Soc. Am. 98 (1995) 1637–1644. Crossref, Google Scholar
24. T. B. Neilsen, An iterative implementation of rotated coordinates for inverse problems, J. Acoust. Soc. Am. 113 (2003) 2574–2586. Crossref, Google Scholar
25. J.-P. Hermand, M. Meyer, M. Asch and M. Berrada, Adjoint-based acoustic inversion for the physical characterization of a shallow water environment, J. Acoust. Soc. Am. 119 (2006) 3860–3871. Crossref, Google Scholar
26. H. C. Song, Performance bounds for passively locating a moving source of a known frequency in oceanic waveguide using a vertical array, IEEE J. Ocean. Eng. 18 (1993) 189–198. Crossref, Google Scholar
27. A. Baggeroer and H. Schmidt, Cramér–Rao bounds for matched field tomography and ocean acoustic tomography, in 1995 Int. Conf. Acoustics, Speech, and Signal Processing (IEEE, 1995), Vol. 5, pp. 2763–2766. Crossref, Google Scholar
28. J. R. Buck, Information theoretic bounds on source localization performance, in Sensor Array and Multichannel Signal Processing Workshop Proceedings, 2002 (IEEE, 2002), pp. 184–188. Crossref, Google Scholar
29. C.-F. Huang, P. Gerstoft and W. S. Hodgkiss, Uncertainty analysis in matched-field geoacoustic inversions, J. Acoust. Soc. Am. 119 (2006) 197–207. Crossref, Google Scholar
30. Z.-H. Michalopoulou and M. Picarelli, Gibbs sampling for time-delay-and amplitude estimation in underwater acoustics, J. Acoust. Soc. Am. 117 (2005) 799. Crossref, Google Scholar
31. A. B. Baggeroer, W. A. Kuperman, H. Schmidt and A. B. Baggeroer, Matched field processing: Source localization in correlated noise as an optimum parameter estimation problem, J. Acoust. Soc. Am. 83 (1988) 571–587. Crossref, Google Scholar
32. B. Hochwald and A. Nehorai, Concentrated Cramér–Rao bound expressions, IEEE Trans. Inf. theory 40 (1994) 363–371. Crossref, Google Scholar
33. E. Naftali and N. C. Makris, Necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the Cramér–Rao Lower Bound. Part I. General approach with an application to time-delay and Doppler shift estimation, J. Acoust. Soc. Am. 110 (2001) 1917–1930. Crossref, Google Scholar
34. A. F. Brouwer and M. C. Eisenberg, The underlying connections between identifiability, active subspaces, and parameter space dimension reduction, arXiv:1802.05641. Google Scholar
35. F. Nielsen, Cramér–Rao lower bound and information geometry, in Connected at Infinity II (Hindustan Book Agency, Gurgaon, 2013), pp. 18–37. Crossref, Google Scholar
36. E. W. Weisstein, Statistical correlation, Mathworld — A Wolfram Web Resource. Google Scholar
37. E. K. Westwood, C. T. Tindle and N. R. Chapman, A normal mode model for acousto-elastic ocean environments, J. Acoust. Soc. Am. 100 (1996) 3631–3645. Crossref, Google Scholar
38. L. F. Richardson and J. A. Gaunt, VIII. The deferred approach to the limit, Philos. Trans. Royal Soc. Lond. Series A 226 (1927) 299–361. Crossref, Google Scholar
39. J. Perktold and A. Verheyleweghen, numdifftools, 2021, https://pypi.org/project/numdifftools/. Google Scholar
40. M. Siderius and J. Gebbie, Environmental information content of ocean ambient noise, J. Acoust. Soc. Am. 146 (2019) 1824–1833. Crossref, Google Scholar
41. M. F. P. Tolsma, A. van den Bos and G. Blacquière, Experimental design for geoacoustic parameter inversion, J. Acoust. Soc. Am. 109 (2001) 2423–2423. Crossref, Google Scholar
42. M. Hawkes, S. Member and A. Nehorai, Effects of sensor placement on acoustic vector-sensor array performance, IEEE J. Ocean. Eng. 24 (1999) 33–40. Crossref, Google Scholar
43. H. Lai and K. Bell, Cramér-Rao lower bound for DOA estimation using vector and higher-order sensor arrays, in 2007 Conference Record of the Forty-First Asilomar Conference on Signals, Systems and Computers (IEEE, 2007), pp. 1262–1266. Crossref, Google Scholar
44. P. K. Tam, K. T. Wong and Y. Song, An hybrid Cramér-Rao bound in closed form for direction-of-arrival estimation by an ’acoustic vector sensor’ with gain-phase uncertainties, IEEE Trans. Signal Process. 62 (2014) 2504–2516. Crossref, Google Scholar
45. A. B. Baggeroer, The stochastic Cramér-Rao bound for source localization and medium tomography using vector sensors, The Journal of the Acoustical Society of America 141(5) (2017) 3430–3449. Crossref, Google Scholar
46. P. M. Daly, Cramér-Rao bounds for matched field tomograph and ocean acoustic tomography, Thesis, Massachusetts Institute of Technology (1997). Google Scholar
47. S. Narasimhan and J. Krolik, A Cramér–Rao bound for source range estimation in a random ocean waveguide, in IEEE Seventh SP Workshop on Statistical Signal and Array Processing (IEEE, 1994), pp. 309–312. Crossref, Google Scholar
48. J. Gebbie and M. Siderius, Optimal environmental estimation with ocean ambient noise, J. Acoust. Soc. Am. 149 (2021) 825–834. Crossref, Google Scholar
49. A. Thode, M. Zanolin, E. Naftali, I. Ingram, P. Ratilal and N. C. Makris, Necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the Cramér–Rao lower bound. II. Range and depth localization of a sound source in an ocean waveguide, J. Acoust. Soc. Am. 112 (2002) 1890–1910. Crossref, Google Scholar
50. M. Zanolin, I. Ingram, A. Thode and N. C. Makris, Asymptotic accuracy of geoacoustic inversions, J. Acoust. Soc. Am. 116 (2004) 2031–2042. Crossref, Google Scholar
51. F. B. Jensen, W. A. Kuperman, M. B. Porter and H. Schmidt, Computational Ocean Acoustics (Springer, New York, 2011). Crossref, Google Scholar
52. G. R. Potty and J. H. Miller, Effect of shear on modal arrival times, IEEE J. Ocean. Eng. 45 (2020) 103–115. Crossref, Google Scholar
53. F. A. Bowles, Observations on attenuation and shear-wave velocity in fine-grained, marine sediments, J. Acoust. Soc. Am. 101 (1997) 3385–3397. Crossref, Google Scholar