Source localization with a geoacoustic model requires optimizing the model over a parameter space of range and depth with the objective of matching a predicted sound field to a field measured on an array. We propose a sample-efficient sequential Bayesian optimization strategy that models the objective function as a Gaussian process (GP) surrogate model conditioned on observed data. Using the mean and covariance functions of the GP, a heuristic acquisition function proposes a candidate in parameter space to sample, balancing exploitation (sampling around the best observed objective function value) and exploration (sampling in regions of high variance in the GP). The candidate sample is evaluated, and the GP conditioned on the updated data. Optimization proceeds sequentially until a fixed budget of evaluations is expended. We demonstrate source localization for a shallow-water waveguide using Monte Carlo simulations and experimental data from an acoustic source tow. Compared to grid search and quasi-random sampling strategies, simulations and experimental results indicate the Bayesian optimization strategy converges on optimal solutions rapidly.

1.
C. E.
Rasmussen
and
C. K. I.
Williams
,
Gaussian Processes for Machine Learning
(
MIT Press
,
Cambridge, MA
,
2006
).
2.
D.
Caviedes-Nozal
,
N. A. B.
Riis
,
F. M.
Heuchel
,
J.
Brunskog
,
P.
Gerstoft
, and
E.
Fernandez-Grande
, “
Gaussian processes for sound field reconstruction
,”
J. Acoust. Soc. Am.
149
(
2
),
1107
1119
(
2021
).
3.
Z.-H.
Michalopoulou
,
P.
Gerstoft
, and
D.
Caviedes-Nozal
, “
Matched field source localization with Gaussian processes
,”
JASA Express Lett.
1
(
6
),
064801
(
2021
).
4.
Z.-H.
Michalopoulou
and
P.
Gerstoft
, “
Inversion in an uncertain ocean using Gaussian processes
,”
J. Acoust. Soc. Am.
153
(
3
),
1600
1611
(
2023
).
5.
I. D.
Khurjekar
,
P.
Gerstoft
,
C. F.
Mecklenbräuker
, and
Z.-H.
Michalopoulou
, “
Direction-of-arrival estimation using Gaussian process interpolation
,” in
Proceedings of International Conference on Acoustics, Speech and Signal Processing
(
2023
), pp.
1
5
.
6.
D.
Zimmerman
,
C.
Pavlik
,
A.
Ruggles
, and
M. P.
Armstrong
, “
An experimental comparison of ordinary and universal kriging and inverse distance weighting
,”
Math. Geol.
31
(
4
),
375
390
(
1999
).
7.
K. P.
Murphy
,
Probabilistic Machine Learning: An Introduction
(
MIT Press
,
Cambridge, MA
,
2022
).
8.
B.
Shahriari
,
K.
Swersky
,
Z.
Wang
,
R. P.
Adams
, and
N.
de Freitas
, “
Taking the human out of the loop: A review of Bayesian optimization
,”
Proc. IEEE
104
(
1
),
148
175
(
2016
).
9.
J. J.
Beland
and
P. B.
Nair
, “
Bayesian optimization under uncertainty
,” in
NIPS BayesOpt 2017 Workshop
,
Long Beach, CA
(
2017
), Vol.
3
.
10.
D. R.
Jones
, “
A taxonomy of global optimization methods based on response surfaces
,”
J. Global Optim.
21
(
4
),
345
383
(
2001
).
11.
N.
Srinivas
,
A.
Krause
,
S. M.
Kakade
, and
M. W.
Seeger
, “
Information-theoretic regret bounds for Gaussian process optimization in the bandit setting
,”
IEEE Trans. Inf. Theory
58
(
5
),
3250
3265
(
2012
).
12.
D.
Ginsbourger
,
R.
Le Riche
, and
L.
Carraro
, “
Kriging is well-suited to parallelize optimization
,” in
Computational Intelligence in Expensive Optimization Problems
(
Springer
,
Berlin, Heidelberg
) (
2010
), Vol.
2
, pp.
131
162
.
13.
J. T.
Wilson
,
F.
Hutter
, and
M. P.
Deisenroth
, “
Maximizing acquisition functions for Bayesian optimization
,”
Adv. Neural Inf. Process. Syst.
(published online
2018
), arXiv:1805.10196.
14.
J.
Wang
,
S. C.
Clark
,
E.
Liu
, and
P. I.
Frazier
, “
Parallel Bayesian global optimization of expensive functions
,”
Oper. Res
68
(
6
),
1850
1865
(
2020
).
15.
M.
Balandat
,
B.
Karrer
,
D. R.
Jiang
,
S.
Daulton
,
B.
Letham
,
A. G.
Wilson
, and
E.
Bakshy
, “
BoTorch: A framework for efficient Monte-Carlo Bayesian optimization
,”
Adv. Neural Inf. Process. Syst.
33
,
21524
21538
(
2020
).
16.
W.
Jenkins
,
P.
Gerstoft
, and
Y.
Park
, “
Bayesian optimization with Gaussi
an process
surrogate model for geoacoustic inversion and parameter estimation
,” https://github.com/NeptuneProjects/BOGP (
2023
) (Last viewed July 25, 2023).
17.
A.
Baggeroer
,
W.
Kuperman
, and
P.
Mikhalevsky
, “
An overview of matched field methods in ocean acoustics
,”
J. Ocean. Eng
18
(
4
),
401
424
(
1993
).
18.
J.
Bergstra
and
Y.
Bengio
, “
Random search for hyper-parameter optimization
,”
J. Mach. Learn. Res.
13
(
10
),
281
305
(
2012
).
19.
I.
Sobol'
, “
On the distribution of points in a cube and the approximate evaluation of integrals
,”
USSR Comp. Math. Math. Phys.
7
(
4
),
86
112
(
1967
).
20.
A.
Saltelli
,
M.
Ratto
,
T.
Andres
,
F.
Campolongo
,
J.
Cariboni
,
D.
Gatelli
,
M.
Saisana
, and
S.
Tarantola
,
Global Sensitivity Analysis. The Primer
,
1st ed.
(
Wiley
,
West Sussex, UK
,
2007
).
21.
S.
Joe
and
F. Y.
Kuo
, “
Constructing Sobol sequences with better two-dimensional projections
,”
SIAM J. Sci. Comput.
30
(
5
),
2635
2654
(
2008
).
22.
L.
Dumaz
,
J.
Garnier
, and
G.
Lepoultier
, “
Acoustic and geoacoustic inverse problems in randomly perturbed shallow-water environments
,”
J. Acoust. Soc. Am.
146
(
1
),
458
469
(
2019
).
23.
B.
Kayser
,
B.
Gauvreau
,
D.
Écotière
, and
V.
Mallet
, “
Wind turbine noise uncertainty quantification for downwind conditions using metamodeling
,”
J. Acoust. Soc. Am.
151
(
1
),
390
401
(
2022
).
24.
J.
Nocedal
and
S. J.
Wright
,
Numerical Optimization, Springer Series in Operations Research
,
2nd ed.
(
Springer
,
New York
,
2006
).
25.
R.
Schmidt
, “
Multiple emitter location and signal parameter estimation
,”
IEEE Trans. Antennas Propag.
34
(
3
),
276
280
(
1986
).
26.
M. E.
Tipping
, “
Sparse Bayesian learning and the relevance vector machine
,”
J. Mach. Learn. Res.
1
,
211
244
(
2001
).
27.
D.
Wipf
and
B.
Rao
, “
Sparse Bayesian learning for basis selection
,”
IEEE Trans. Signal Process.
52
(
8
),
2153
2164
(
2004
).
28.
K. L.
Gemba
,
S.
Nannuru
,
P.
Gerstoft
, and
W. S.
Hodgkiss
, “
Multi-frequency sparse Bayesian learning for robust matched field processing
,”
J. Acoust. Soc. Am.
141
(
5
),
3411
3420
(
2017
).
29.
Y.
Park
,
F.
Meyer
, and
P.
Gerstoft
, “
Sequential sparse Bayesian learning for time-varying direction of arrival
,”
J. Acoust. Soc. Am.
149
(
3
),
2089
2099
(
2021
).
30.
Y.
Chi
,
Y. M.
Lu
, and
Y.
Chen
, “
Nonconvex optimization meets low-rank matrix factorization: An overview
,”
IEEE Trans. Signal Process.
67
(
20
),
5239
5269
(
2019
).
31.
S.
Li
,
L.
Cheng
,
T.
Zhang
,
H.
Zhao
, and
J.
Li
, “
Graph-guided Bayesian matrix completion for ocean sound speed field reconstruction
,”
J. Acoust. Soc. Am.
153
(
1
),
689
710
(
2023
).
32.
M. J.
Bianco
,
P.
Gerstoft
,
J.
Traer
,
E.
Ozanich
,
M. A.
Roch
,
S.
Gannot
, and
C.-A.
Deledalle
, “
Machine learning in acoustics: Theory and applications
,”
J. Acoust. Soc. Am.
146
(
5
),
3590
3628
(
2019
).
33.
Y.
Liu
,
H.
Niu
, and
Z.
Li
, “
A multi-task learning convolutional neural network for source localization in deep ocean
,”
J. Acoust. Soc. Am.
148
(
2
),
873
883
(
2020
).
34.
P.-A.
Grumiaux
,
S.
Kitić
,
L.
Girin
, and
A.
Guérin
, “
A survey of sound source localization with deep learning methods
,”
J. Acoust. Soc. Am.
152
(
1
),
107
151
(
2022
).
35.
A.
Varon
,
J.
Mars
, and
J.
Bonnel
, “
Approximation of modal wavenumbers and group speeds in an oceanic waveguide using a neural network
,”
JASA Express Lett.
3
(
6
),
066003
(
2023
).
36.
L.
Zhang
,
T.
Zhang
,
H.-S.
Shin
, and
X.
Xu
, “
Efficient underwater acoustical localization method based on time difference and bearing measurements
,”
IEEE Trans. Instrum. Meas.
70
,
1
16
(
2021
).
37.
P.
Gerstoft
, “
Inversion of seismoacoustic data using genetic algorithms and a posteriori probability distributions
,”
J. Acoust. Soc. Am.
95
(
2
),
770
782
(
1994
).
38.
P.
Gerstoft
and
C. F.
Mecklenbräuker
, “
Ocean acoustic inversion with estimation of a posteriori probability distributions
,”
J. Acoust. Soc. Am.
104
(
2
),
808
819
(
1998
).
39.
S. E.
Dosso
, “
Quantifying uncertainty in geoacoustic inversion. I. A fast Gibbs sampler approach
,”
J. Acoust. Soc. Am.
111
(
1
),
129
142
(
2002
).
40.
S. E.
Dosso
and
P. L.
Nielsen
, “
Quantifying uncertainty in geoacoustic inversion. II. Application to broadband, shallow-water data
,”
J. Acoust. Soc. Am.
111
(
1
),
143
159
(
2002
).
41.
S. E.
Dosso
and
M. J.
Wilmut
, “
Uncertainty estimation in simultaneous Bayesian tracking and environmental inversion
,”
J. Acoust. Soc. Am.
124
(
1
),
82
97
(
2008
).
42.
M. D.
Collins
,
W. A.
Kuperman
, and
H.
Schmidt
, “
Nonlinear inversion for ocean-bottom properties
,”
J. Acoust. Soc. Am.
92
(
5
),
2770
2783
(
1992
).
43.
M. D.
Collins
,
W. A.
Kuperman
, and
W. L.
Siegmann
, “
Propagation and inversion in complex ocean environments
,” in
Full Field Inversion Methods in Ocean and Seismo-Acoustics
(
Kluwer Academic
,
The Netherlands
,
1995
), pp.
15
20
.
44.
M. D.
Collins
and
L.
Fishman
, “
Efficient navigation of parameter landscapes
,”
J. Acoust. Soc. Am.
98
(
3
),
1637
1644
(
1995
).
45.
J.
Dettmer
,
S. E.
Dosso
, and
C. W.
Holland
, “
Trans-dimensional geoacoustic inversion
,”
J. Acoust. Soc. Am.
128
(
6
),
3393
3405
(
2010
).
46.
J.
Dettmer
and
S. E.
Dosso
, “
Trans-dimensional matched-field geoacoustic inversion with hierarchical error models and interacting Markov chains
,”
J. Acoust. Soc. Am.
132
(
4
),
2239
2250
(
2012
).
47.
S. E.
Dosso
and
J.
Bonnel
, “
Joint trans-dimensional inversion for water-column sound speed and seabed geoacoustic models
,”
JASA Express Lett
3
(
6
),
060801
(
2023
).
48.
P.
Gerstoft
, “
Inversion of acoustic data using a combination of genetic algorithms and the Gauss–Newton approach
,”
J. Acoust. Soc. Am.
97
(
4
),
2181
2190
(
1995
).
49.
M. R.
Fallat
and
S. E.
Dosso
, “
Geoacoustic inversion via local, global, and hybrid algorithms
,”
J. Acoust. Soc. Am.
105
(
6
),
3219
3230
(
1999
).
50.
N.
Booth
,
P.
Baxley
,
J.
Rice
,
P.
Schey
,
W.
Hodgkiss
,
G.
D'Spain
, and
J.
Murray
, “
Source localization with broad-band matched-field processing in shallow water
,”
J. Ocean. Eng.
21
(
4
),
402
412
(
1996
).
51.
Z.-H.
Michalopoulou
and
M.
Porter
, “
Matched-field processing for broad-band source localization
,”
J. Ocean. Eng.
21
(
4
),
384
392
(
1996
).
52.
R. M.
Neal
,
Bayesian Learning for Neural Networks
, Lecture Notes in Statistics Vol. 118 (
Springer
,
New York
,
1996
).
53.
K. P.
Murphy
,
Probabilistic Machine Learning: Advanced Topics
(
MIT Press
,
Cambridge, MA
,
2023
).
54.
R. H.
Byrd
,
P.
Lu
,
J.
Nocedal
, and
C.
Zhu
, “
A limited memory algorithm for bound constrained optimization
,”
SIAM J. Sci. Comput.
16
(
5
),
1190
1208
(
1995
).
55.
C.
Zhu
,
R. H.
Byrd
,
P.
Lu
, and
J.
Nocedal
, “
Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization
,”
ACM Trans. Math. Software
23
(
4
),
550
560
(
1997
).
56.
S.
Kim
,
G. F.
Edelmann
,
W. A.
Kuperman
,
W. S.
Hodgkiss
,
H. C.
Song
, and
T.
Akal
, “
Spatial resolution of time-reversal arrays in shallow water
,”
J. Acoust. Soc. Am.
110
(
2
),
820
829
(
2001
).
57.
B.
Letham
,
B.
Karrer
,
G.
Ottoni
, and
E.
Bakshy
, “
Constrained Bayesian optimization with noisy experiments
,” arXiv:1706.07094 (
2018
).
58.
R. T.
Bachman
,
P. W.
Schey
,
N. O.
Booth
, and
F. J.
Ryan
, “
Geoacoustic databases for matched-field processing: Preliminary results in shallow water off San Diego, California
,”
J. Acoust. Soc. Am.
99
(
4
),
2077
2085
(
1996
).
59.
Marine Physical Laboratory
, “
SWellEx-96 Experiment
,” http://swellex96.ucsd.edu/ (
2003
) (Last viewed May 30, 2023).
60.
M. B.
Porter
, “
The KRAKEN normal mode program
,” SACLANT Undersea Research Centre Memorandum (SM-245)/Naval Research Laboratory Memorandum Report 6920 (
1991
).
61.
G. L.
D'Spain
,
J. J.
Murray
, and
W. S.
Hodgkiss
, “
Mirages in shallow water matched field processing
,”
J. Acoust. Soc. Am.
105
(
6
),
3245
3265
(
1999
).
62.
Y.
Park
,
S.
Nannuru
,
K.
Gemba
, and
P.
Gerstoft
, “
SBL4 from NoiseLab
,” https://github.com/gerstoft/SBL (
2020
) (Last viewed May 30, 2023).
63.
J. R.
Gardner
,
G.
Pleiss
,
D.
Bindel
,
K. Q.
Weinberger
, and
A. G.
Wilson
, “
GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration
,”
Adv. Neural Inf. Process. Syst.
(published online
2018
), arXiv:1809.11165.
64.
Meta Platforms, Inc.
, “
Adaptive Experimentation Platform
,” https://ax.dev (
2023
) (Last viewed May 30, 2023).
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