Acoustic ray tracing in the atmosphere: with gravitational

ANNALS OF GEOPHYSICS, 57, 5, 2014, A0542; doi:10.4401/ag-6474

A0542

Acoustic ray tracing in the atmosphere: with gravitational effectand attenuation considered

Yang Song*, Yuannong Zhang, Chen Zhou, Zhengyu Zhao

Wuhan University, Department of Space Physics, School of Electronic Information, Wuhan, China

ABSTRACT

An acoustic ray tracing model is developed to take into account the im-pacts of gravitational field and realistic atmospheric attenuation. Raytracing equations are deduced from the real part of the dissipative dis-persion relation, while the acoustic attenuation coefficient and growthrate in a stratified moving atmosphere are deduced from the imaginarypart of the dispersion relation. To account for the non-isothermal effectand realistic attenuation, the buoyancy frequency and the cut-off fre-quency are substituted by the values in the slowly varying atmosphere,and the attenuation coefficient is corrected by the realistic absorption. Inthe validation by numerical experiment, the ray trajectory obtained bythis ray tracing model agrees well with the result calculated by the FDTDmethod. It is shown that the acoustic trajectory can be accurately pre-dicted by this ray tracing model. The numerical results for 5 Hz acousticwaves show that in the stratospheric ducting the gravitational effect playsa leading role while the attenuation effect could be neglected. But for thethermospheric ducting, the contribution of the absorption becomes moreimportant.

1. IntroductionIt is widely considered that the acoustic wave is

one of the essential forms of energy transportation inthe atmosphere. Depending on the profiles of the tem-perature and wind, waves could be ducted through thetroposphere, stratosphere, and thermosphere [Drob etal. 2003]. This phenomenon, called long range soundpropagation (LRSP) [Kulichkov 2004], is frequently ob-served at infrasonic frequencies because of the low at-tenuation in the lower atmosphere [Bedard and Georges2000]. Consequently, natural and artificial sources couldbe detected and monitored through the network of in-frasonic stations, such as the International MonitoringSystem [Le Pichon et al. 2009]. In addition, as the prop-agation is controlled by the background profiles, it isalso possible to study the properties of the atmosphereby analyzing infrasonic signals, such as probing the highaltitude wind [Le Pichon et al. 2005]. Therefore, it is im-

portant to comprehensively understand the propaga-tion properties of acoustic waves.

Ray tracing is an efficient way to study the propa-gation of acoustic waves in the atmosphere. It can beused to calculate the trajectories of acoustic waves aswell as a number of propagation parameters includingtravel time and arrival angle. With the ray path avail-able, the location and intensity of the fluctuations canbe predicted [Pilger and Bittner 2009]. Since the earlymodel developed by Barry [1963], enormous progresseshave been achieved in ray tracing studies. There arethree classical ray tracing models, namely HARPA,WASP-3D and Tau-P, all of which have been widely ap-plied to study the sound propagation [e.g., Le Pichon etal. 2010]. HARPA is derived from the dispersion rela-tion in a stratified moving atmosphere on the basis ofHamilton equations [Jones et al. 1986]. WASP-3D [Dessaet al. 2005] is also a Hamiltonian solution extendedfrom 2D Cartesian ones [Virieux et al. 2004], but it isdeduced from the Eikonal equation. Tau-P model [Gar-cés et al. 1998] is developed for computation of traveltimes as a reformulation of the original method pre-sented by Buland and Chapman [1983]. All of these threemodels are developed to study the atmospheric wavepropagation through a windy, non-isothermal, non-dis-sipative atmosphere.

Theoretically, the high frequency approximationadopted by the above three models confines the ray trac-ing modeling to acoustic signals above 5 Hz for whichthe gravity influence is neglected. Moreover, the at-tenuation of acoustic wave is not considered well in thethree models. However, in the atmosphere acousticwaves can extend down to much lower frequencies~0.02 Hz [Drob et al. 2003]. In addition, there are manytypes of signals in the 0.02-5 Hz frequency band, dueto the radiation by some natural or artificial sources.For instance, signals with frequencies from 0.5 to 5 Hz

Article historyReceived December 10, 2013; accepted September 2, 2014.Subject classification:Acoustic ray tracing, Dissipative dispersion relation, Gravity effect, Attenuation.

were observed in the cases of large bolides [Hedlin etal. 2010], meteorites [Le Pichon et al. 2013], volcano[Kanamori et al. 1994, Evers and Haak 2005] and explo-sions [Stevens et al. 2002, McKenna et al. 2007]. Underthe influence of gravity, the propagation of acousticwaves becomes more dispersive and anisotropic as fre-quency decreases. For modeling the propagation ofacoustic waves below 5 Hz, the high frequency ap-proximation is no longer valid and the effect of gravityneeds to be taken into account. Damping of acousticwaves in the realistic atmosphere is another importantingredient of acoustic propagation modeling [de Groot-Hedlin et al. 2011], which however is often ignored inray tracing modeling. Based on the calculation of thespeed and attenuation at altitudes up to 160 km devel-oped by Sutherland and Bass [2004], the attenuation ef-fects on the propagation of infrasound is studied withoutconsidering gravitational effect [Bass et al. 2007]. Thedamping effect of low frequency waves is not significantnear the ground, but it increases remarkably at higheraltitudes. In addition, the accumulation of attenuationeffects along the long-range ray path is nonnegligible.Therefore, in order to model the ray path of acousticwaves precisely, the wave attenuation also needs to beincluded.

Ray theory for lossy media has been discussed inradio wave propagation by Jones [1970]. To take lossesinto account, the complex phase refractive index in-stead of the real index defined in the ideal media isused, so that the ray path can be calculated using theextended Snell’s law or Haselgrove’s equations. The ex-pression of such a path is in complex space, so onlythose rays ended in real space are physically significant.Another way is developed by extending Hamilton’sequations for lossy media under the constraint that thespace, time and group velocity should be real [Sonnen-schein et al. 1997]. When studying the acoustic wavepropagation, especially the infrasonic propagation,both the effects of attenuation and the gravitational dis-persion should be taken into account. Other than usingthe complex index, it could be a better way to obtainthe ray equation through the Hamilton’s equations byemploying the dissipative dispersion relation as an ex-tension.

In the present study, an acoustic ray tracing modelin the realistic atmosphere is developed on the basis ofanalyzing both imaginary part and real part of the dis-sipative dispersion relation. In Section 2.1, the dissipa-tive dispersion relation is derived from the governingequations of an isothermal dissipative atmosphere. InSections 2.2~2.4, the acoustic attenuation coefficientand growth rate in the moving atmosphere are deducedfrom the imaginary part of the dispersion relation,

while ray tracing equations are obtained from the realpart of the dissipative dispersion relation. Modificationsfor the parameters are presented in Section 3. Thebuoyancy frequency and the cut-off frequency are sub-stituted by the values in the slowly varying atmosphere.The acoustic attenuation coefficient is corrected by thetheory of absorption in the atmosphere up to 160 km[Sutherland and Bass 2004, 2006], aiming to include thenon-isothermal effect and the realistic atmospheric at-tenuation. This ray tracing model is validated by thecomparison with FDTD solution in Section 4. Numer-ical examples are shown in Section 5 to study the influ-ences of the gravitational effect and attenuation,followed by the conclusion in Section 6.

2. Inclusion of gravity and absorption into ray trac-ing equations

In order to introduce gravity influence and dissi-pation effects into acoustic ray tracing theory, we startwith the compressible fluid equations with molecularviscosity and thermal diffusivity in three dimensionalCartesian systems [Beer 1974].

(1)

where t, p, s and T are particle density, pressure, en-tropy, and temperature respectively, V = (u,v,w) is thewind velocity, is the Stokes derivative, gis the gravitational acceleration, h and p are shear andbulk viscosity. For the convenience of formula deriva-tion, the viscous term is neglected in following calcula-tions.

With the knowledge of the thermodynamic rela-tionship, the entropy and temperature in the atmos-phere can be given by

(2)

In Equations (2), c= cp/cv is the ratio between spe-cific heat at constant pressure and constant volume, andR is the universal gas constant.

It is assumed that the basic quantities such as t, pand V can be expressed as the sum of the ambient value(with subscript 0) and the perturbation caused by thepassage of a sound wave (with subscript 1), i.e.

(3)

As the acoustic wave in the atmosphere is an irro-tational wave, the variable V satisfies ∇∇·V1=∇2V1.

VDtD

t $22 d= +

V

V g VV

t

DtD p

T DtD s

30

0

2

$

$

22

d

d dd d

tt

t t ph

t

h

+

- + = +

=

=

+

^ hZ

[

\

]]]

]]]

` j

const ,lns cp

T Rp

v yt t= + =c m

, , V V Vp p p0 1 0 1 0 1t t t= + = + = +

SONG ET AL.

2

3

With the approximation of a stratified atmosphere,Equation (1) can be simplified and linearized to yield

where V0 = (u0,v0,0) is the wind profileof the atmosphere, is the adiabatic soundspeed with T0 as the ambient temperature.

2.1. Dissipative dispersion relationAssume that the acoustic wave has a plane wave

solution in the form

(5)

where P, R, X, Y and Z are polarization terms, ~ andk=(kx,ky,kz) are angular frequency and wave numbercomponents, respectively. Substitution into Equation(4) yields a set of fifth-order algebraic equations for(u1,v1,w1,t1/t0,p1/p0), which can be expressed in thematrix form as

(6)In the coefficient matrix of Equation (6), X = ~ −

V0 · k, the Doppler-shifted frequency, is the coefficient of kinematic viscosity of air, H = c2/cgis the scale height. According to the Cramer’s rule, fora non-trivial solution of Equation (6), the determinantof the coefficient matrix must vanish. Therefore, thedissipative dispersion relation becomes

(7)

Equation (7) is complex because the inhomogene-ity and dissipation of the atmosphere results in complexwave number components. Obviously, if the absorptionis neglected, Equation (7) reduces to the dispersion re-

lation of acoustic gravity waves [Yeh and Liu 1974].Since the wave vector k is complex, it can be written as

(8)

where k' = (k'x,k'y,k'z) is the real part of the wave vector,known as the wave vector in an ideal medium, and k" =(k"x,k"y,k"z) is the imaginary part of the wave vector thatcan characterize the attenuation and growth of waveamplitude in the inhomogeneous atmosphere. Subse-quently, the dispersion relation Equation (7) can be de-composed into two equations, i.e., the imaginary part,Equation (9) and the real part, Equation (10).

(9)

(10)

The first term on the left side of the dissipative dis-persion relation, g2 (c − 1)(k2

x + k2y), plays an important

role in the real part of the dispersion relation, as it rep-resents the influence of the gravity that generates grav-ity dispersion. This term is discarded in the derivationof Equation (9), because the imaginary part of the dis-sipative dispersion relation is mainly associated with theattenuation relation between frequency and imaginarywave vector k". However imaginary wave vector k"represents the changes of wave amplitude, we keep it inthe derivation of Equation (10).

The imaginary wave vector k" can be derived fromEquation (9) in order to calculate the amplitude varia-tions along a ray path. By substituting k" into Equation(10), the ray equations can be determined on the basisof Hamilton equations.

2.2. The attenuation and growth of acoustic wavesAs acoustic waves propagate in the atmosphere,

the wave amplitude usually drops due to the attenua-tion effect. Growth in the wave amplitude may occurbecause of the atmospheric density gradient in the ver-tical direction. Thus, in the dissipative stratified atmos-phere, the imaginary wave vector can be considered asa combination of omnidirectional attenuation andgrowth in the vertical direction. Assuming the attenu-ation efficient and growth factor as a and b respectively,we obtain that

(11)

By substituting Equation (11) into Equation (9) and(10), the imaginary part and real part of the dissipativedispersion relation can be rewritten as

0

DtD

u xp

DtD p

DtD p

g

DtD

x z w

DtD

p c DtD

zp

c z w

u

v y v

w z w

uyv

zw

34

34

34

0

00

11

00

11 2

1

00

11

12

1

01 0

10

10

1 01

01 0

2 01 0

2 01

0

212

2

22

d

22

d

22

22

22

22

22

22

dt p h

t p h

t t p h

t t t tt

tt

+ = +

+ = +

+ + = +

+ + + + =

- + - =

Z

[

\

]]]]]]

]]]]]]

``

`

c

jj

j

m

expp Pp

R Xu

Yv

Zw

A i t k y k y k zx y z

0

1

0

1 1 1 1tt

~

= = = = =

= - - -^ h6 @

,VDtD

t0

0 $2 d= +

34

0g p h t= +` j

1

0

g k ik i k

c k ik i gkyx

z

2 22 2

2 2 2 2

c g

g c

X X

X X

- - + +

- + - =

^ ^ ^

^

h h h

h

k k ki= +l m

2 0k k kc gkz02 2$ g cX+ - =l m l l

yx1

2

2

0k k

k k

k k

g k

gk

k

c z

2 22

02 2 2 2 $

$c g

g c

X X

X

-

+

+ - +

- - - =

l l l m

l m l m m

^ ^ ^

^

h h h

h6 @

, ,k kk

k k k kk

xx

y zza a a b= = = +

ykm

ll

mll

mll

c RT0 0c=

ACOUSTIC RAY TRACING IN THE ATMOSPHERE

(4)

1

i k

ik

i k

ik

i k

ik H

H

g

i

i

igHkigHk

igHk g

i

uvw

pp

00

0

0

0

0

00

1

00

0

0

x

x

y

y z

z

2

2

2

2

1

0

1

g

g

g

cc

tt

X

X

X

X

X X

+

-

+

-

+

- -

-

-

-

- - =

p

r

qqqqqqqq

p

r

qqqqqqqq

t

v

uuuuuuuu

t

v

uuuuuuuu

(12)

The attenuation coefficient and growth rate canthen be solved from Equation (12) as

(14)

(15)

In Equation (14), minus sign means the decrease ofwave amplitude; Doppler-shifted frequency X representsthe impact of wind on the attenuation by changing thepropagation distance in a unit time. In the windless at-mosphere, X reduces to ~, and this coefficient becomesthe classical absorption coefficient [Beer 1974]. Equation(15) indicates that an exponential amplification or decaywith height can be caused by the inhomogeneity of theatmosphere in the vertical direction.

2.3. Final form of dissipative dispersion relationSubstituting the solved attenuation coefficient a

and growth factor b into Equation (13), the real part ofthe dissipative dispersion relation becomes the finalform of dispersion, written as

(16)

the term , which is thecontribution of the gravitational effect, where ~2

g =(c− 1)g2/c2, ~g is the isothermal Vaisala-Brunt frequency;and ~a = c/(2H) is the acoustic cut-off frequency; theterm is the con-tribution of attenuation. If both gravitational effect andattenuation effect are neglected, the dispersion will be-come the classic acoustic dispersion relation

(17)

According to the theory of atmospheric waves, thewaves whose frequency is greater than acoustic cut-offfrequency are identified as acoustic waves, and thewaves whose frequency is less than Vaisala-Brunt fre-quency are the internal gravity waves. The dispersionrelation Equation (16) can be employed to describeboth acoustic waves and internal gravity waves, but thiswork mainly focuses on the propagation of the acoustic

wave, the internal gravity waves will not be discussed. Actually, when the dispersion is employed to ob-

tain the ray tracing equations, the accuracy of the raytracing equations depend on the dispersion relationequation. So the gravitational term Gr and attenuationterm At are of great importance to the dispersion rela-tion. It can be discussed in two limiting cases that arelow frequency case and high frequency case.

For the acoustic wave with low frequency which isclose to the acoustic cut-off frequency (~→ ~a), thedissipative effect is so small that could be neglected,meanwhile the gravitational effect is enhanced, so thedispersion relation is transformed as

(18)

Equation (18) is the classic dispersion relation ofacoustic gravity wave.

As to the acoustic waves with high frequency(~ >> ~a), the dispersion equation becomes

(19)

Different from the classic acoustic dispersion rela-tion, Equation (17), the attenuation term At is takeninto consideration, because as the frequency becomeshigher, the attenuation becomes stronger. In the highfrequency approximation, the gravitational dispersionterm , but the scale height of amean density variation still exists. The term ~2

a can betaken as the contribution of the inhomogeneity of at-mosphere density.

Both limiting cases demonstrate that the inclusionof the gravity and attenuation are reasonable enrich-ment for the classic acoustic dispersion relation. The ef-fect caused by gravity and attenuation can be studiedby the group velocity which can be derived as

where u0 and v0 represent zonal wind and meridionalwind respectively. If both gravitational effect and at-tenuation are neglected, Equation (20) would be sim-

2c k k k gk 0z02

22a b g cX+ + - =l l l l^ h

x y

2

g k k

c k kk

g kk

c k k k

1

4 2 0

z

zz

2 2 2 4

02 2 2 2 2

2 3 2 2

c

a b ab

c a b a b

X

X

X X X

- + + -

- - - -

+ + - + =-

-

l l

lll

ll

l l

^ ^

^ ^

h h

h h

cc

mm

ck2 0

2ag X

=-l

cg

H2 21

02bc

= =

c k Gr At 04 2 2 2X X- + + =l

x yGr c k kg a2 2 2 2 2 2~ ~X= + -l l^ h

7 2At k k c c kHk

4z2 2 2

2

2

22 23g

gXX

X X= - -+l ll

l^ h` j

0c k2 2 2X - =l

c k Gr 02 2 2X - + =l

0c k Ata2 2 2 2~X - - + =l

x yc k k 0g2 2 2 2 2 "~ X+l l^ h

x y

x y

x y

k ck k At

c kc

k Atu

k ck k At

c kc

k Atv

k ck k At

c k At

4 21

21

4 21

21

4 21

21

x g

xg

x k

y g

yg

y k

z g

z k

3

2 22 2

22

2 2

0

3

2 22 2

22

2 2

0

3

2 22 2

2

x

y

z

22

2

2

22

2

2

22

2

2

~~

~

~~

~

~~

XX

X

XX

X

XX

=-

- + +

- + ++

=-

- + +

- + ++

=-

- + +

- +

X

X

X

l l

l l

l l

l l

l l

l

l

l

l

^

^

^

h

h

h

Z

[

\

]]]]]]]

]]]]]]]

SONG ET AL.

4

(13)

(20)

5

plified as the group velocity derived from the classicacoustic dispersion relation. In Equation (20), we canfind that the terms which mainly con-trolled by attenuation modify the valve of the groupvelocity. The gravity term Gr brings in the gravitationaldispersion and makes the sound propagation becomeinhomogeneous. Taking the gravity and attenuationinto consideration, the profile of group velocity in theatmosphere can be predicted more accurately.

2.4. Ray tracing equationsEquation (16) is a real variable equation. On the

basis of Equation (16), ray equations can be derivedfrom the Hamilton equations, expressed as

(21)

where r is a point on the ray path, ∇r and ∇k are the gra-dient operators in r and k space, respectively, and H isthe dispersion relation [Yeh and Liu 1974].By substitut-ing Equation (16) into (21), the ray equations can be de-rived as

(22.a)

(22.b)

For the components of the wave number,

(23)

In Equations (22) and (23),

(24.a)

(24.b)

(24.c)

From Equations (22-24), we can observe that theparameter g and play important roles in the raytracing equation. As mentioned in Section 2.3, with at-tenuation considered, the group velocity would bemodified. What’s more, the gradient of kinematic vis-cosity also can enhance the gradient of the groupvelocity. Both terms demonstrate that the acoustic tra-jectory is influenced by the effect due to the attenua-tion in a dissipative atmosphere.

For given atmospheric parameters including thewind profile, temperature field and the profile of vis-cosity g, the ray trajectory in the dissipative atmospherecan be determined by numerically solving Equations(22-24).

3. Parameter modifications in the realistic atmosphereDerivation of the dissipative dispersion is very com-

plex, because a fifth-order determinant has to be solved.In order to simplify the derivation, two assumptions areemployed. Firstly, the atmosphere is assumed isother-mal so that the gradients of pressure and temperaturecan be described in a simple form in terms of the scaleheight. Secondly, only the viscous dissipation in the mo-mentum equation is considered. In the realistic atmos-phere, the temperature varies with height, and otherdissipation mechanisms, such as heat conduction andmolecular relaxation losses, also play a role.

Provided that the atmospheric temperature variesvery slowly and the high order derivatives of scaleheight can be ignored, modifications of ~a and ~g canbe obtained as

(25)

In Equation (25), ~an and ~B are the acoustic cut-off frequency and Vaisala-Brunt frequency in thenon-isothermal atmosphere, respectively [Beer 1974,Matsumura et al. 2012]. By using the temperature andwind profiles given by the MSISE-00 [Picone et al. 2002]and HWM-93 [Hedin et al. 1996] empirical referencemodels, the typical values of acoustic cut-off frequencyand Vaisala-Brunt frequency in the realistic atmosphereare shown in Figure 1. Varying with altitude, the acousticcut-off frequency reaches the extreme value of about0.0039 Hz and 0.005 Hz at the height of the strato-

andAt Atk2 2Xl

,dtdr H dt

d Hkk rd d= =

zdd

zx

Ck H g k k

Ak Bu2

2z

x3

2 2

0

gc g

XX

=+ - +

+

l l l

l

^ h

zdd

zy

Ck H g k k

Ak Bv2

2z

y3

2 2

0

gc g

XX

=+ - +

+

l l l

l

^ h

z

d dk kd d 0, dz z z

Ck H g k kD E

2 2

x y x

z3 2 2g c gX X

= = =

=- + +

+

dk

l l l^ h

2

4 2

A c ck g k

27

g2 2 2

22 4

2 2 22

~ g

g gc

X X

X X

= - - +

+ -l l

^ h

4 2 2 7

2

c k ck

kk

c k

B

H 6

a2 2 2

2

2 2

2 2 2 2

33

4

~g

gg

X X XX

X X

= - - - +

+ + -

ll

ll

l^ h

2 27

4C c c k2 22

22 2 2

4ggX

XX=- - + l

yx

4

D zc

k z z kc

k

k u k v c k2

a

z a

g

x z y

22

22

02

22 2

2

02 2 2

22

22

2

2~ ~

~

X

X X

= +

-

+ + -

+ - +

l l l

l

^ ^ ^^

^ ^

h h hh

h h

;6E

@

7

7 2

2

E c zc k z k k c

kz z

c k

k u k ck

k

Hk

Hk c k

z Hk

c k

Hk

Hc H

v

87

62

4

x z z

z z z

y

z

32 4 2

22 2

2

2 22

0 0 2

2 2 32 4

2 2 23 2 2

22

2 2

22

22

22

22

22

gg

g

gg

g g g

X XX

X X

XX

XX X

= + -

+ - + +

+ - + -

-

- - + -

-

l l l

l l l

ll

l l l l

^

^

h

h

``

c

jj

m

z2 g

z2 g

,Tg

zT

Tg

zT

an a B g2 2

22

22

~ ~ ~ ~= + = +


(24.d)

(24.e)

sphere and thermosphere, respectively. The prediction of acoustic attenuation in the real-

istic atmosphere should consider the contribution ofdifferent dissipation mechanisms including viscosity,diffusion and molecular relaxation [Brown and Hall1978]. The coefficients for different kinds of dissipationmechanism can be combined as one net attenuation co-efficient in terms of linear superposition. In this study,kinematic viscosity g is modified by the value of thenet attenuation coefficient, so that we can account forthe realistic attenuation effects into dissipative disper-sion relation.

According to the algorithms developed by Suther-land and Bass [2004, 2006], which can precisely estimatethe acoustic absorption up to 160 km, the net attenua-tion coefficient at (Np/m), is written in the form of lin-ear summation

(26)

where acr is the attenuation coefficient associated withthe combination of classical and rotational relaxationlosses, adiff is the attenuation coefficient due to diffusionloss, and Amax,i is the maximum loss per wavelength forthe ith molecular vibration relaxation component witha relaxation frequency fvib,i. In the realistic atmosphere,four primary gas components, O2, N2, CO2, and O3 areconsidered. Typical profiles of attenuation coefficientat frequencies of 0.05, 0.5, 5 and 8Hz are shown in Fig-ure 2. Clearly, the total attenuation coefficients for allconsidered frequencies increase significantly in the ther-mosphere.

From Equation (14), in the windless conditions,the parameter g that describes the acoustic attenuationcan be given by

(27)

Combining Equation (26) with Equation (27), weunderstand the parameter g as an artificial viscosity,which is determined as a function of frequency, temper-ature, pressure and relative humidity.

Using the modified values of ~a, ~g, and g, the raytracing model can compute the trajectory of acousticwaves more accurately, since the contributions of bothtemperature gradient and acoustic attenuation aretaken into account.

4. Comparison with FDTD solutionTo validate this ray tracing model, we simulate the

propagation of a Gaussian infrasound wave packet byFDTD method, and compare the trajectory obtainedby FDTD solution with the result calculated by the raytracing model. By using a finite-difference time-domainmethod [Wochner et al. 2005] consisting a dispersionrelation preserving scheme in space and a Runge-Kuttascheme in time, this FDTD model is built on the basis ofthe governing equations of acoustic waves, Equation (4).

In the FDTD experiment, the Gaussian infrasoundwave packet at the frequency of 0.5 Hz is set to be emit-ted eastward with shooting angle of 15° at the heightof 10 km. The pressure perturbation of the packet hasthe following form:

(28)

where pc = 100 Pa; xc = 8 km; zc = 10 km; kx and kz arethe horizontal and vertical component of the wave num-ber vector respectively; and vx and vz are the half widthof the wave packet in the horizontal and vertical direc-tion respectively. In the simulation, vx = vz = 1.6 km.According to the background profiles, the parameter kxand kz are initially set by using the dispersion Equation(16). The other initial quantities such as u1, w1 and t1

/

/ /

A c

f f f f12,maxt cr diff ii

vib vib22

#

#

a a a= + +

+^ ^^h h h

6

6

@

@

"

,

/

2.k

c t2

g~

a=

-

,p x z

k x x k z zc

x x z z

x c z c2 2x

c

z

c2

2

2

2

=

- + -v v

---

sinp e e

^

^ ^^ ^

h

h hh h

6 @

SONG ET AL.

6

Figure 1. Profiles of acoustic cut-off frequency (solid) and Vaisala-Brunt frequency (dashed), calculated using the MSISE-00 andHWM-93 empirical model.

Figure 2. Total attenuation coefficient as a function of altitude forfrequencies of 0.05, 0.5, 5 and 8Hz (from left to right).

7

are derived from the polarization equations of acousticwaves in the atmosphere [Beer 1974].

The background profiles of density, temperatureshown by Figure 3A are provided by the MSISE-00 [Pi-cone et al. 2002], and the wind profile shown by Figure3B is provided by HWM-93 [Hedin et al. 1996]. The at-tenuation coefficient profile is derived by the algorithmsfounded by Sutherland and Bass [2004, 2006] which isalready exhibited in Figure 2. For the MSISE and HWMmodels, the time we set is the 300th day of the year at9:00 UT; while the geographic location of the source is(39.92°, 116.46°) and the parameter F10.7 index is 150.

The simulation results are represented by theacoustic intensity of the packet. Continuous capture ofthe infrasound packet every 30s during the propagationare shown by Figure 3C. As seen in Figure 3C, the po-sition and altitude of the packet are changed due to therefraction controlled by the atmospheric profile. Becauseof the effects of attenuation and geometric spreading, theintensity of the infrasound packet becomes weaker asthe packet gets higher. In this case, the trajectory of theinfrasound is the wave energy propagation path whichcan be represented by the positions of the wave packet’s

energy center. The propagation path of the packet ob-tained from the FDTD solution and the ray path pre-dicted by the ray tracing model are compared in Figure4. The comparison shows that the ray path predictedby the ray tracing model is equal to the propagationpath of the infrasound packet obtained from the FDTDsolution within the computational precision. Based onthe results shown in Figure 4, we can conclude that thisray tracing can be proven accurate and reliable.

5. Numerical resultsTo investigate the impacts of gravity dispersion

and acoustic dissipation, based on the backgroundmodels of the MSISE-00 and HWM-93, three cases of


Figure 4. Comparison between the trajectory obtained by theFDTD solution and ray solution. The FDTD solution is denoted by‘+’, while the ray solution is represented by the solid line.

Figure 3. (A) background profiles of density and temperature; (B)background profiles of zonal wind and meridional wind; (C) theacoustic intensity of the infrasound packet in the form of continu-ous capture every 30s.

A B

C

numerical results are presented. The first is calculatedusing the acoustic ray tracing model for the realistic at-mosphere, as described in Section 2, in which bothgravity dispersion and acoustic dissipation are consid-ered. The second is calculated using the lossless raytracing model that considers only the gravity disper-sion, which can also be called “the acoustic gravitywave ray tracing model” [Yeh and Liu 1974]. The last isobtained from the classical ray tracing model in whicheither gravity dispersion or acoustic dissipation is neg-lected [Jones et al. 1986]. The numerical results for theabove three cases are compared with each other underthe condition of long range sound propagation (LRSP).

5.1. TrajectoriesIn this calculation, the input parameters of MSISE-

00 and HWM-93 are set as same as the validation ex-periment in Section 4. Under the background profilesof density, temperature and wind shown by Figure 3Aand Figure 3B, We assume that the acoustic wave witha frequency of 5 Hz is emitted eastward by an infra-sonic source on the ground. Shooting angles of the raysvary from 3° to 43°, with an increment of 8°.

Figure 5 shows the wave trajectories for the threecases. The solid, dotted, and dashed lines denote thewave trajectories obtained using the acoustic ray trac-ing model for the realistic atmosphere, the acousticgravity wave ray tracing model, and the classical raytracing model, respectively. Two channels can be iden-tified from these modeled trajectories. One is the re-flection from the height less than 50 km, which is called“stratospheric ducting”, and the other is “thermos-pheric ducting” that occurs at higher altitudes from theheight of thermosphere. The effect of acoustic dissipa-

tion can be registered by comparing the solid and dot-ted curves, and the effect due to the gravity can be eval-uated by comparing the dotted and dashed curves. Wefirst discuss the effect of acoustic dissipation and thenthe gravitational effect.

For the rays with shooting angles of 3°-19°, whichare in the channel of stratospheric ducting, the two setsof trajectories coincide well. But for the other rays withshooting angles of 27°-43° that are in the channel ofthermospheric ducting, there are obvious differencesbetween the two sets of trajectories. It indicates thatthe dissipation of acoustic waves has an effect on thetrajectory of 5 Hz acoustic waves. The influence dueto the dissipation is quite significant for thermosphericducting but less important to stratospheric ducting. It ismainly because the attenuation coefficient varies withaltitude. As shown in Figure 2, the net attenuation co-efficient for 5 Hz waves reaches a peak valve (~0.03dB/km) at the altitude of 10km in the troposphere andstratosphere. As expressed by Equation (18), the dis-persion relation would be reduced to the traditional dis-persion of acoustic gravity wave when the attenuationcould be neglected. As the altitude increases, the atten-uation coefficient increases, reaching ~400 dB/km inthe thermosphere. Because the attenuation coefficientis very small (< 0.03 dB/km), the effect of dissipationhas little influence on the rays in the stratospheric duct-ing. For thermospheric ducting, in the height range100 km < z <140 km, the turning points of ray trajec-tories with the realistic attenuation are at an altitudelower than that of rays in the lossless atmosphere. Ob-viously, the refraction is enhanced by the strongacoustic attenuation in the thermosphere.

By comparing the dotted and dashed lines, dis-crepancies occur in both stratospheric ducting and ther-mospheric ducting. For the trajectory with shootingangle 3°, due to the gravitational effect, the dotted linesare below the dashed line in the region z <160 km.After the intersection of the two types of lines, the dot-ted lines are above the dashed lines. However, this phe-nomenon does not occur for all rays. Take the trajectorywith shooting angle 19° for an example. The dottedlines are always above the dashed lines. Since compar-isons between the trajectories are complicated, we willdescribe the discrepancies of the three kinds of trajec-tories via some concrete parameters such as turningpoint and propagation distance.

5.2. Turning point and propagation distanceTurning point and propagation distance of the

three kinds of trajectories as a function of elevation areshown in Figure 6. With the same range of shootingangle but a smaller step of 1°, a clearer view of the in-

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Figure 5. Ray paths estimated based on three ray tracing models. Thesolid and dotted lines show the results calculated using our acousticray tracing model and the acoustic gravity wave ray tracing modelrespectively, which both consider the effect of gravity dispersion. Thedashed lines represent the results calculated using the classical rayracing model that ignores both gravity dispersion and acoustic dis-sipation. Shooting angles of 5 Hz acoustic waves vary from 3 to 43°with a step of 8°.

9

fluences associated with the gravity dispersion and dis-sipation can be obtained.

A turning point is an altitude that characterizes thetransformation of wave rays between the two channels,i.e., stratospheric ducting and thermospheric ducting.In Figure 6A, for the solid and dotted lines, the eleva-tion range of stratosphere ducting is from 3° to 19°,while the elevation range of stratosphere ducting is20°~43°. For the dashed lines, the elevation range ofstratospheric ducting and thermospheric ducting are3°~21° and 22°~43°, respectively. Due to the gravita-tional effect, the rays with shoot angles 20° and 21° nolonger belong to the stratospheric ducting but the ther-mospheric ducting, suggesting that the gravitational ef-fect play an important role in determining the channelof acoustic ray trace.

By comparing the dotted line and the dashed line inFigure 6A, the calculation deviation of the turning pointis about 1.5 km in stratospheric ducting, and it will riseto about 8 km (at the initial elevation 31°) in the propa-gation channel of thermospheric ducting. As indicatedin Figure 5, little difference is observed between solidline and dotted lines in stratospheric ducting in both Fig-ure 6A and Figure 6B. This difference illustrates that thedeviation between the solid line and dashed line instratospheric ducting is caused by the gravity.

In thermospheric ducting, deviation of the turningpoint between solid line and dotted lines could reachabout 15 km, as indicate in Figure 6A. Controlled by thedeviation of the turning point, deviation of the propa-gation distance between solid line and dotted lines in-creases along with the initial elevation, and reaches thepeak value of about 160 km, then decreases to the levelof approximately 60 km (Figure 6B). These great devia-tions between solid line and the other lines shown inFigure 5 indicate that the strong attenuation is a domi-nant factor for the ducts in the thermosphere. As seen in

Figure 2, the profile of acoustic attenuation varies withfrequency. The attenuation decreases when frequencygoes down, so for the propagation with the frequencysmaller than 5 Hz, the influence due to attenuation inthe thermospheric ducting will become weaker.

6. ConclusionsAn acoustic ray tracing model including the effects

of gravity and attenuation is developed in the presentstudy. The propagation of acoustic waves in non-isothermal lossy atmosphere can be described by thismodel. By comparing the ray path estimated by thismodel with the FDTD solution, the ray tracing modelhas been proven accurate and reliable. The influencesdue to absorption and gravity are investigated by com-paring the numerical results obtained using thisacoustic ray tracing model with the other two models,i.e., acoustic gravity wave ray tracing model and theclassic ray tracing model that ignores the two effects.

The results indicate that in stratospheric ducting,the gravitational effect plays a leading role while the ef-fect associated with the dissipation is weak. Because ofthe gravity, some rays ducted in the stratosphere as alossless medium can merge into the channel of ther-mospheric ducting in the realistic atmosphere. For ther-mospheric ducting, the refraction is enhanced by thestrong absorption. This results in the dominance of dis-sipative effect over gravitational effect in thermosphericducting.

To conclude, by taking the gravity into considera-tion, the acoustic frequency in our model is no longerconstrained by the high frequency approximation. In-stead, it is available to accurately predict the propaga-tion of infrasound with frequencies lower than 5 Hz.Consideration of the dissipation effect in our model canalso improve the computation accuracy of ray trajec-tories and the efficiency of wave amplitude evaluation.


Figure 6. Comparison of the turning height (A) and propagation distance (B) as a function of elevation angle. The solid and dotted lines showthe results calculated using our acoustic ray tracing model and the acoustic gravity wave ray tracing model respectively, which both considerthe effect of gravity dispersion. The dashed lines represent the results calculated using the classical ray racing model that ignores both grav-ity dispersion and acoustic dissipation. Shooting angles vary from 3~43° with a step of 1°.

A B

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*Corresponding author: Yang Song,Wuhan University, Department of Space Physics,School of Electronic Information, Wuhan, China;email: [email protected].

© 2014 by the Istituto Nazionale di Geofisica e Vulcanologia. Allrights reserved.


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Acoustic ray tracing in the atmosphere: with gravitational