The coherence of rough sea-surface-scattered acoustic fields decreases with increasing frequency. The frequency-difference autoproduct, a quadratic product of acoustic fields at nearby frequencies, mimics a genuine field at the difference frequency. In rough-surface scattering, the autoproduct's lower effective frequency decreases the apparent surface roughness, restoring coherent reflection. Herein, the recovery of coherent reflection in sea surface scattering via the frequency-difference autoproduct is examined for data collected off the coast of New Jersey during the Shallow Water '06 (SW06) experiment. An acoustic source at depth 40 m and receiver at depth 24.3 m and range 200 m interrogated 160 independent realizations of the ocean surface. The root mean square surface height h was 0.167 m, and broadcast frequencies were 14–20 kHz, so that 2.5  k h cos θ  3.7 for acoustic wavenumber k and incidence angle θ. Measured autoproducts, constructed from scattered constituent fields, show significant coherent reflection at sufficiently low difference frequencies. Theoretical results, using the Kirchhoff approximation and a non-analytic surface autocorrelation function, agree with experimental findings. The match is improved using a numerical strategy, exploiting the relationship between autoproduct-based coherence recovery, the ocean-surface autocorrelation function, and the ocean-surface height spectrum. Error bars computed from Monte Carlo scattering simulations support the validity of the measured coherence recovery.

1.
M.
Darmon
,
V.
Dorval
, and
F.
Baqué
, “
Acoustic scattering models from rough surfaces: A brief review and recent advances
,”
Appl. Sci.
10
,
1
27
(
2020
).
2.
T. M.
Elfouhaily
and
C. A.
Guérin
, “
A critical survey of approximate scattering wave theories from random rough surfaces
,”
Wav. Rand. Med.
14
,
R1
R40
(
2004
).
3.
L.
Fortuin
, “
Survey of literature on reflection and scattering of sound waves at the sea surface
,”
J. Acoust. Soc. Am.
47
,
1209
1228
(
1970
).
4.
F. G.
Bass
and
I. M.
Fuks
,
Wave Scattering from Statistically Rough Surfaces
(
Pergamon
,
Oxford, UK
,
1979
).
5.
J. A.
Oglivy
,
Theory of Wave Scattering from Random Rough Surfaces
(
IOP Publishing, Ltd
.,
Bristol, UK
,
1991
).
6.
P.
Beckmann
and
A.
Spizzichino
,
The Scattering of Electromagnetic Waves from Rough Surfaces
(
MacMillan
,
New York
,
1963
).
7.
H.
Medwin
and
C. S.
Clay
,
Fundamentals of Acoustical Oceanography
(
Academic Press
,
New York
,
1998
).
8.
L. M.
Brekhovskikh
and
Y. P.
Lysanov
,
Fundamentals of Ocean Acoustics
(
Springer-Verlag
,
New York
,
1991
).
9.
P. H.
Dahl
, “
On the spatial coherence and angular spreading of sound forward scattered from the sea surface: Measurements and interpretive model
,”
J. Acoust. Soc. Am.
100
,
748
758
(
1996
).
10.
D. R.
Dowling
and
D. R.
Jackson
, “
Coherence of acoustic scattering from a dynamic rough surface
,”
J. Acoust. Soc. Am.
93
,
3149
3157
(
1993
).
11.
E. Y.
Gorodetskaya
,
A. I.
Malekhanov
,
A. G.
Sazontov
, and
N. K.
Vdovicheva
, “
Deep-water acoustic coherence at long ranges: Theoretical prediction and effects on large-array signal processing
,”
IEEE J. Ocean. Eng.
24
,
156
171
(
1999
).
12.
R. J.
Urick
,
Principles of Underwater Sound
, 3rd ed. (
McGraw-Hill
,
New York
,
1983
).
13.
B. M.
Worthmann
and
D. R.
Dowling
, “
The frequency-difference and frequency-sum acoustic-field autoproducts
,”
J. Acoust. Soc. Am.
141
,
4579
4590
(
2017
).
14.
D.
Tang
,
J. N.
Moum
,
J. F.
Lynch
,
P.
Abbot
,
R.
Chapman
,
P. H.
Dahl
,
T. F.
Duda
,
G.
Gawarkiewicz
,
S.
Glenn
,
J. A.
Goff
,
H.
Graber
,
J.
Kemp
,
A.
Maffei
,
J. D.
Nash
, and
A.
Newhall
, “
Shallow Water '06: A joint acoustic propagation/nonlinear internal wave physics experiment
,”
Oceanography
20
,
156
167
(
2007
).
15.
N. J.
Joslyn
and
D. R.
Dowling
, “
Recovery of coherent reflection from rough-surface scattered acoustic fields via the frequency-difference autoproduct
,”
J. Acoust. Soc. Am.
151
,
620
633
(
2022
).
16.
P. H.
Dahl
, “
Observations and modeling of angular compression and vertical spatial coherence in sea surface forward scattering
,”
J. Acoust. Soc. Am.
127
,
96
103
(
2010
).
17.
P. H.
Dahl
,
W. J.
Plant
, and
D. R.
Dall'Osto
, “
Vertical coherence and forward scattering from the sea surface and the relation to the directional wave spectrum
,”
J. Acoust. Soc. Am.
134
,
1843
1853
(
2013
).
18.
P. J.
Welton
, “
Cross correlation of omnidirectional, broadband signals scattered by a random pressure-release surface
,”
IEEE J. Ocean. Eng.
40
,
485
494
(
2015
).
19.
C.
Eckart
, “
The scattering of sound from the sea surface
,”
J. Acoust. Soc. Am.
25
,
566
570
(
1953
).
20.
W. J.
Plant
, “
A stochastic, multiscale model of microwave backscatter from the ocean
,”
J. Geophys. Res., [Oceans]
107
,
3-1
3-21
, https://doi.org/10.1029/2001JC000909 (
2002
).
21.
C. S.
Clay
, “
Coherent reflection of sound from the ocean bottom
,”
J. Geophys. Res.
71
,
2037
2046
, https://doi.org/10.1029/JZ071i008p02037 (
1966
).
22.
J. M.
Berkson
, “
Measurements of coherence of sound reflected from ocean sediments
,”
J. Acoust. Soc. Am.
68
,
1436
1441
(
1980
).
23.
D. R.
Dowling
, “
Revealing hidden information with quadratic products of acoustic field amplitudes
,”
Phys. Rev. Fluids
3
,
110506
(
2018
).
24.
B. M.
Worthmann
and
D. R.
Dowling
, “
The effects of refraction and caustics on autoproducts
,”
J. Acoust. Soc. Am.
147
,
3959
3968
(
2020
).
25.
B. M.
Worthmann
and
D. R.
Dowling
, “
Autoproducts in and near acoustic shadow zones created by barriers
,”
J. Acoust. Soc. Am.
147
,
1863
1873
(
2020
).
26.
D. J.
Geroski
,
M. A.
Dzieciuch
, and
D. R.
Dowling
, “
Measurements of the correlation of the frequency-difference autoproduct with acoustic and predicted-autoproduct fields in the deep ocean
,”
J. Acoust. Soc. Am.
149
,
853
865
(
2021
).
27.
J. E.
Lipa
,
B. M.
Worthmann
, and
D. R.
Dowling
, “
Measurement of autoproduct fields in a Lloyd's mirror environment
,”
J. Acoust. Soc. Am.
143
,
2419
2427
(
2018
).
28.
S. H.
Abadi
,
H. C.
Song
, and
D. R.
Dowling
, “
Broadband sparse-array blind deconvolution using frequency-difference beamforming
,”
J. Acoust. Soc. Am.
132
,
3018
3029
(
2012
).
29.
A. S.
Douglass
,
H. C.
Song
, and
D. R.
Dowling
, “
Performance comparisons of frequency-difference and conventional beamforming
,”
J. Acoust. Soc. Am.
142
,
1663
1673
(
2017
).
30.
A. S.
Douglass
and
D. R.
Dowling
, “
Frequency-difference beamforming in the presence of strong random scattering
,”
J. Acoust. Soc. Am.
146
,
122
134
(
2019
).
31.
B. M.
Worthmann
,
H. C.
Song
, and
D. R.
Dowling
, “
High frequency source localization in a shallow ocean sound channel using frequency difference matched field processing
,”
J. Acoust. Soc. Am.
138
,
3549
3562
(
2015
).
32.
D. J.
Geroski
and
D. R.
Dowling
, “
Long-range frequency-difference source localization in the Philippine Sea
,”
J. Acoust. Soc. Am.
146
,
4727
4739
(
2019
).
33.
D. J.
Geroski
and
D. R.
Dowling
, “
Robust long-range source localization in the deep ocean using phase-only matched autoproduct processing
,”
J. Acoust. Soc. Am.
150
,
171
182
(
2021
).
34.
D. J.
Geroski
,
J. R.
Johnson
, and
D. R.
Dowling
, “
Localization of a remote source in a noisy deep ocean sound channel using phase-only matched autoproduct processing
,”
J. Acoust. Soc. Am.
153
,
2223
2237
(
2023
).
35.
B. M.
Worthmann
,
H. C.
Song
, and
D. R.
Dowling
, “
Adaptive frequency-difference matched field processing for high frequency source localization in a noisy shallow ocean
,”
J. Acoust. Soc. Am.
141
,
543
556
(
2017
).
36.
L.
Yang
,
Y.
Wang
, and
Y.
Yang
, “
Aliasing-free broadband direction of arrival estimation using a frequency-difference technique
,”
J. Acoust. Soc. Am.
150
,
4256
4267
(
2021
).
37.
Y.
Park
,
P.
Gerstoft
, and
J. H.
Lee
, “
Difference-frequency MUSIC for DOAs
,”
IEEE Signal Process. Lett.
29
,
2612
2616
(
2022
).
38.
N. J.
Joslyn
,
A. S.
Douglass
, and
D. R.
Dowling
, “
Spatial coherence comparisons between the acoustic field and its frequency-difference and frequency-sum autoproducts in the ocean
,”
Acoustics
4
,
764
782
(
2022
).
39.
T.
Hagfors
, “
Some properties of radio waves reflected from the Moon and their relation to the lunar surface
,”
J. Geophys. Res.
66
,
777
785
, https://doi.org/10.1029/JZ066i003p00777 (
1961
).
40.
D. E.
Weissman
, “
Two frequency radar interferometry applied to the measurement of ocean wave height
,”
IEEE Trans. Antennas Propagat.
21
,
649
656
(
1973
).
41.
G. T.
Ruck
,
D. E.
Barrick
, and
T.
Kaliszewski
, “
Bistatic radar sea state monitoring
,”
NASA Technical Report No. NASA-CR-137469
, Battelle Columbus Laboratories (
1972
).
42.
W. J.
Plant
and
D. L.
Schuler
, “
Remote sensing of the sea surface using one- and two-frequency microwave techniques
,”
Radio Sci.
15
,
605
615
, https://doi.org/10.1029/RS015i003p00605 (
1980
).
43.
W.
Alpers
and
K.
Hasselman
, “
The two-frequency microwave technique for measuring ocean-wave spectra from an airplane or satellite
,”
Boundary-Layer Meteorol.
13
,
215
230
(
1978
).
44.
D. J.
Geroski
and
B. M.
Worthmann
, “
Frequency-difference autoproduct cross-term analysis and cancellation for improved ambiguity surface robustness
,”
J. Acoust. Soc. Am.
149
,
868
884
(
2021
).
45.
E.
Thorsos
, “
Surface forward scattering and reflection
,”
Technical Report 7-83
,
Applied Physics Lab—University of Washington
(
1984
).
46.
J. F.
McDonald
, “
Fresnel-corrected second-order interfrequency correlations for a surface-scatter channel
,”
IEEE Trans. Commun.
22
,
138
145
(
1974
).
47.
Wolfram Research, Inc.
, “
Mathematica
,” https://www.wolfram.com/mathematica/. (Last viewed May 11, 2022).
48.
D. R.
Olson
, “
A series approximation to the Kirchhoff integral for Gaussian and exponential roughness covariance functions
,”
J. Acoust. Soc. Am.
149
,
4239
4247
(
2021
).
49.
A.
Darawankul
and
J. T.
Johnson
, “
Band-limited exponential correlation function for rough-surface scattering
,”
IEEE Trans. Geosci. Remote Sens.
45
,
1198
1206
(
2007
).
50.
D. M. F.
Chapman
, “
The directional nature of attenuation of sound due to scattering at a rough ocean surface
,”
J. Acoust. Soc. Am.
68
,
1475
1481
(
1980
).
51.
P. H.
Dahl
, “
On bistatic sea surface scattering: Field measurements and modeling
,”
J. Acoust. Soc. Am.
105
,
2155
2169
(
1999
).
52.
P. H.
Dahl
,
J. W.
Choi
,
N. J.
Williams
, and
H. C.
Graber
, “
Field measurements and modeling of attenuation from near-surface bubbles for frequencies 1–20 kHz
,”
J. Acoust. Soc. Am.
124
,
EL163
EL169
(
2008
).
53.
M. B.
Porter
and
H. P.
Bucker
, “
Gaussian beam tracing for computing ocean acoustic fields
,”
J. Acoust. Soc. Am.
82
,
1349
1359
(
1987
).
54.
H.
Li
and
J. T.
Johnson
, “
On the amplitude distributions of bistatic scattered fields from rough surfaces
,”
IEEE Trans. Geosci. Remote Sens.
55
,
6883
6892
(
2017
).
55.
A.
Rohatgi
, “
WebPlotDigitizer
,” https://automeris.io/WebPlotDigitizer (Last viewed June 14, 2023).
You do not currently have access to this content.