Ray tracing based on a closed-form expression for the sound propagation path in a stratified moving medium such as a stratified atmosphere is suggested. The approach involves a one-dimensional (1D) integration over the vertical coordinate and can complement existing ray tracing codes based on a set of differential equations. It can also be used as a benchmark solution, because 1D integration is usually computationally simpler than these codes. Using this approach, a closed-form expression for the azimuthal deviation between the true and apparent source location is obtained. The result enables estimation of the azimuthal deviation given a crosswind or remote sensing of the crosswind provided the azimuthal deviation is measured. Based on these formulations, sound propagation in the atmospheric surface layer characterized by the friction velocity and surface heat flux is studied.

1.
R. M.
Jones
,
J. P.
Riley
, and
T. M.
Georges
, “
HARPA: A versatile three-dimensional Hamiltonian ray-tracing program for acoustic waves in the atmosphere above irregular terrain
,” National Technical Reports Library (
U.S. Department of Commerce
,
Washington, DC
,
1986
).
2.
K.
Attenboroughand
and
T.
Van Renterghem
,
Predicting Outdoor Sound
, 2nd ed. (
CRC Press
,
Boca Raton, FL
,
2021
).
3.
E.
Salomons
,
Computational Atmospheric Acoustics
(
Kluwer Academic
,
Dordrecht, the Netherlands
,
2001
).
4.
V. E.
Ostashev
and
D. K.
Wilson
,
Acoustics in Moving Inhomogeneous Media
, 2nd ed. (
CRC Press
,
Boca Raton, FL
,
2015
).
5.
V. E.
Ostashev
, “
On the sound field of a point source in a stratified moving two-component medium
,”
Izv. Acad. Scienc. USSR. Atmos. Ocean. Phys.
21
(
9
),
731
735
(
1985
).
6.
M.
Hornikx
,
M.
Kaltenbacher
, and
S.
Marburg
, “
A platform for benchmark cases in computational acoustics
,”
Acta Acust. united Ac.
101
,
811
820
(
2015
).
7.
P.
Schäfer
and
M.
Vorlaender
, “
Ray tracing for efficient simulation of curved sound propagation paths: Towards real-time auralization of aircraft noise
,”
J. Acoust. Soc. Am.
148
,
2524
(
2020
).
8.
V. E.
Ostashev
,
M. V.
Scanlon
,
D. K.
Wilson
, and
S. N.
Vecherin
, “
Source localization from an elevated acoustic sensor array in a refractive atmosphere
,”
J. Acoust. Soc. Am.
124
(
6
),
3413
3420
(
2008
).
9.
V. E.
Ostashev
and
D. K.
Wilson
, “
Relative contributions from temperature and wind velocity fluctuations to the statistical moments of a sound field in a turbulent atmosphere
,”
Acta Acust. Acust.
86
(
2
),
260
268
(
2000
).
10.
V. E.
Ostashev
,
T. M.
Georges
,
S. F.
Clifford
, and
G. H.
Goedecke
, “
Acoustic sounding of wind velocity profiles in a stratified moving atmosphere
,”
J. Acoust. Soc. Am.
109
,
2682
2692
(
2001
).
11.
P.
Mialle
,
D.
Brown
, and
N.
Arora
, “
Advances in operational processing at the International Data Centre
,” in
Infrasound Monitoring for Atmospheric Studies
, edited by
E.
Le Pichon
,
A.
Blanc
, and
A.
Hauchecorne
(
Springer
,
Cham
,
2019
).
12.
T. M.
Georges
and
W. H.
Beasley
, “
Refraction of infrasound by upper-atmospheric winds
,”
J. Acoust. Soc. Am.
61
(
1
),
28
34
(
1977
).
13.
S.
Arrowsmith
,
D.
Norris
,
R.
Whitaker
, and
D.
Anderson
, “
Sources of error model and progress metrics for acoustic/infrasonic analysis: Location estimation
,”
Pure Appl. Geophys.
171
,
587
597
(
2014
).
14.
L. G.
Evers
and
H. W.
Haak
, “
Infrasonic forerunners: Exceptionally fast acoustic phases
,”
Geophys. Res. Lett.
34
(
10
),
L10806
, (
2007
).
15.
D.
Norris
, “
Ray methods and infrasound
,”
J. Acoust. Soc. Am.
148
,
2523
(
2020
).
16.
E. T.
Kornhauser
, “
Ray theory for moving fluids
,”
J. Acoust. Soc. Am.
25
(
5
),
945
949
(
1953
).
17.
P.
Ugincius
, “
Ray acoustics and Fermat's principle in a moving inhomogeneous medium
,”
J. Acoust. Soc. Am.
51
(
5
),
1759
1763
(
1972
).
18.
C. I.
Chessel
, “
Three-dimensional acoustic-ray tracing in an inhomogeneous anisotropic atmosphere using Hamilton's equations
,”
J. Acoust. Soc. Am.
53
(
1
),
83
87
(
1973
).
19.
K. M.
Li
, “
A high-frequency approximation of sound propagation in a stratified moving atmosphere above a porous ground surface
,”
J. Acoust. Soc. Am.
95
(
4
),
1840
1852
(
1994
).
20.
G. A.
Korn
and
T. M.
Korn
,
Mathematical Handbook for Scientists and Engineers
, 2nd ed. (
McGraw-Hill
,
New York
,
1968
).
21.
A. S.
Monin
and
A. M.
Obukhov
, “
Basic laws of turbulent mixing in the surface layer of the atmosphere
,”
Tr. Akad. Nauk SSSR Geophiz. Inst.
24
(
151
),
163
187
(
1954
).
22.
R. B.
Stull
,
An Introduction to Boundary Layer Meteorology
(
Kluwer Academic
,
Dordrecht, the Netherlands
,
1988
), pp.
347
404
.
23.
J. R.
Garratt
,
The Atmospheric Boundary Layer
(
Cambridge University Press
,
Cambridge, UK
,
1992
).
24.
D. K.
Wilson
, “
An alternative function for the wind and temperature gradients in unstable surface layer
,”
Boundary Layer Meteorol.
99
,
151
158
(
2001
).
25.
V.
Červený
,
Seismic Ray Theory
(
Cambridge University Press
,
Cambridge, UK
,
2001
).
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