International Glaciological Society. Acoustic impedance measurement of snow density o. MARCO,I O. BusER,2 P. VILLEMAIN,3 F. TOUVIER,'I PH. REVOL1.
AnnaLs qfGLacioLogy 26 1998
© International Glaciological Society
Acoustic impedance measurement of snow density o.
MARCO,I
O.
BusER,2
P. VILLEMAIN,3 F. TOUVIER,'I PH.
REVOL 1
Division Torrents, Neige et Avalanches, CEMAGREF, Domaine llniversitaire, BP 76, 38402 Saint-Martin-d'Heres Cedex, France 2Swiss FederaL InstituteJor Snow and AvaLanche Research, CH-7260 Davos Doif, Switzerland 3 Laboratoire d'Instrllmentation de Microinformatique et d'ELectroniqlle, UniversitiJoseph Fow·ier, BP 76, 38041 GrenobLe Cedex, France 4Centre ditudes de La Neige, Centre NationaL de Recherches MeteoroLogiqlleS, Meteo-France, 38406 Saint-Martin-d'Heres Cedex, France I
ABSTRACT. Today, it is possible to estimate the density of snow in many different
ways (gravimetry, attenuation of gamma rays and velocity of electromagnetic waves). In this paper, we suggest using the acoustic properties of snow. First, we recall the main studies on the acoustic properties of snow. Then, we show, from laboratory measurements, the correlation between density and the acoustic parameters of snow. We found a very good correlation between density and the modulus of the normalized characteristic impedance. We also show that use ofa three-parameter model simulating the propagation of an acoustic wave in porous media allows one to calculate the density of snow from its acoustic porosity.
LIST OF SYMBOLS
Real(A) Rth
Rv
A
e
f
cp
"t "tm
h hg Im(A) j
Ji k K -
-
l A Aapp
Ah Amin
PP+ Pi
r P Pe 92
Cv Cl'
Complex conjugate value of A Coefficient of attenuation of the propagation of an acoustic wave through the snow Prandtl number Sound velocity in free air Specific heats Density Backing length Frequency Phase of the characteristic impedance W of snow Propagation constant Propagation constant measured with impedance tube Acoustic porosity Gravimetric porosity Imaginary part of A Imaginary unit Bessel's function of order i Coefficient of structure Ratio of specific heats Sample length Wavelength in free air Apparent thermal conductivity Thermal conductivity Minimal wavelength in air according to a planewave condition in a tube Reflected component of Pl Incident component of PI Fourier transform of acoustic pressure at microphone i Coefficient of tortuosity Pore radius Coefficient of determination Complex reflection coefficient at sample surface Volumetric mass Equivalent complex volumetric mass
Bp
v W w
Will
Z
Real part of A Pore radius taking into account thermal effects Pore radius taking into account viscous effects Pore-shape ratio factor Cinematic viscosity of air Characteristic impedance ratio with respect to the free-air wave impedance Radian frequency Characteristic impedance ratio measured with the impedance tube Sample normal acoustic impedance ratio p, = R.jW/v
INTRODUCTION In this paper, the notation of Zwikker and Kosten (1949) will be used. No derivations of the formulae will be given, because they can be found in Zwikker and Kosten (1949). The experiments were performed to seek phenomenological relationships between acoustic parameters and snow density. For light snow (p < 300 kg m \ Oura (1952) indicated that the air contained in the snow propagates the acoustic wave. His observations were conlirmed by Ishida's (1965) and Buser's (1986) experiments. For the dense snow (400 kg m 3 ::; P ::; 900 kg m 3) of Greenland, Smith (1965) found a linear correlation between density and the longitudinal acoustic wave velocity in the ice frame. According to Bogorodsky and others (1974), Sommerfeld (1982) and Lee and Rogers (1985), it seems clear that below 200 kg m 3, the pore air is more important in sound propagation, whereas above 300 kg m - 3 propagation in the ice frame is more important. Johnson (1982) tried to explain this phenomenon by using the theory of Biot (1962) which predicts the existence of two longitudinal acoustic waves. They involve coupled motion between the interstitial air and the ice frame. Buser (1986) and Attenborough and Buser (1988) suggested the rigid-frame model of porous media for snow when only the movement of pore air is of interest, which has been validated up to 400 kg m 3.
Alareo and others: Acoustic impedance measurement qfsnow density To measure the density of all sorts of snow, the method used by Smith (1965) would be the best if measurements of the longitudinal wave pressure in the frame did not require a perfect contact between the sensors and the ice grains of the skeleton. H owever, Zephoris a nd others (1975) showed that this experimental constraint is very difficult to meet with light snow (p ::; 300 kg m 3). For these densities, Buser a nd Good (1987) found that the acoustic porosity, h, a parameter of Zwikker's and Kosten's (1949, p.18- 21) rigid-frame model is identical to the gravimetric porosity up to a density of400 kg m 3. Our final goal is to find a method of measuring the density inside the body ofa flowing dense avalanche where the density of the snow can reach 700 kg m :l. In this paper, experiments were performed, first to seek a correlation between the density of dense snow (up to 680 kg m 3) and one of its acoustic parameters, and second ly to verify the validity of the rigid-frame model to describe acoustic waves in the pore air for this dense snow.
EXPERIMENTAL PROCEDURE The selection of snow samples and the method used to measure their acoustic parameters are the main aspects of the experimental procedure to be described.
SaIUpling Naturally deposited snow density varies from 200 to 500 kg m 3. Snow is a porous medium in which the ice grains form a skeleton. The shape and the size of these grains vary accordi ng to the state of snow metamorphism a nd deposited snow density is not entirely independent of the ice-grain shape. Usually, light snow is a compound of fresh or branched snow and dense snow is composed of more rounded particles produced by metamorphosis. Sound propagation through snow pores may be influenced by the frame structure. In this study, snow samples of each kind were compacted mechanically to obtain 55 snow samples with p varying from 142 to 682 kg m- 3 (Fig. I). This sampling includes and extends that used by Buser (1986). The snow samples have a diameter of 3 cm. Their thickness is between 6 and 4 cm when p < 400 kg m 3; and between 2 andl cm for dense snow. To a ll ow the snow-sample temperature to reach equi liCompacted snow
:(~\ • •
700 I600 500 -
E 400 Cl
0
:1=
a. 300 -
200 i100 l-
III
0
+
:
t
Before
After
Fig. 2. The eX/Jerimental proeedurejor preparing the comjJac/ing snow sample. brium with the cold-room temperature, the snow samples were prepared not less than I day before they were used for acous tic measurements. They were stored in a cold room at - 10 C. An hydraulic press was used to compact the snow in the co ld room (1 mm min I). This procedure (Fig. 2) prO\·ided homogeneous snow samples. Acoustic IUeasureIUents
The characteristic acoustic pa rameters of a material are W, the characteristic impedance ratio with respect to the free-air wave impedance, and ,,(, the propagation constant (see li st of symbols). A two-microphone impedance-measurement tube was used. This method was developed by Chung and Blaser (1980a, b), and the associated error analysis was studied by Seybert and Soenarko (1980) and Boden and Abom (1986). The basic principle of the two-microphone method is to measure the acoustic pressure of a broad-band stationary random signal by microphones at two locations on the wall of the tube. After a Fourier transform , the acoustic pressure spectra at microphone 1 and microphone 2 (Fig. 3) may be written as:
Pl (f)
P2(f)
= p+(w) + p_ (w),
(1) w
w
= p+(w) exp( - j C d) + p_ (w) exp(j C"ir d)
where f = w/27r is a fi-equency for which there is a plane wave in the tube, Pi is the Fourier transform of acoustic pressure at microphone i, P+ and p_ arc respectiyely the incident and reflected component of PI and Cnir is the speed of sound. Using the notation of Figure 3, the complex reflection coefficient at the su rface of the sample is:
T(f) = p+(f) p- (f)
x
l>.
a
T(f) =
p\(I) P2( f)
Ca;, Cm
1 + 7'
Z=-
Rounded particfes
Fig. 1. Snow samples.
Small round grains
Fresh or branched snow
(4)
7' \ (1)
1'2(1 )
Then the normal acoustic impedance ratio Z is:
"
D Buser's (1986) samples
(3)
exp(-j....eLd)
exp(j--""-d) _
I
+
(2)
air
A
0
:;:
~
I. , . - - - - Snow - - -___ I sample holder
Using Equations (I) and (2), one obtains:
~
Q
Homogeneous sieved snow
••
~
0
"!-
•
Hydraulic press
Depth hoar crystal
Snow of avalanche deposit. rounded grains
(5)
1-1' So, with on ly one measurement, [he two-microphone method can gi\·e Z for each frequency of the plane-wave domain. To calcu late Wand ,,(, one needs two values of Z for different backing distances e. With the two-microphone method, it is impossible, for a random signa l, to work with 93
MaTco and others: Acoustic impedance measurement qfsnow density Mic. 1 Mic. 2
that the snow sample was in the sample holder was never longer than 10 minutes. This is importa nt when investigating snow, even in a cold room at - 10 e. According to Buser 's (1986) work, the frequency range of interest is 500- 4096 H z. Zc a nd Zo were calculated for 25 different frequencies with a step of 150 ± 2 Hz and e (Fig. 3) was ta ken equal to 0.01 m. 0
Acoustic source
p-
RELATIONSHIP BETWEEN DENSITY AND THE ACOUSTIC PARAM ETERS OF SNOW
Fig. 3. The basic configuration qfthe two-microphone method. only one wavelength. To obtain Wand, one uses the basic eq uations:
Zc = W Z'(e = 0) coshbl) + W sinhbl) Z'(e = 0) sinhbl) + W coshbl) Z'(e) coshbl) + Wsinhbl) Z o = W -=:-:--:--:-~~-=--:--:....:..-:: Z'(e) sinh b l) + W coshbl)
= W cotbl) (6) (7)
=
a
=
+ j(w/C).
,=
Four real numbers can be distinguished: IWI and cP, resp ectively the modulus and the phase of the characteri stic impedance W of the snow.
a and C, respectively the coeffi cient of attenuation and
with
Z'(e
The aco ustic properties of snow are defi ned by the two IWI (cos rj; + j sin rj;) and complex numbers, W
0) "# Z'(e).
(8)
Or,
the velocity of propagation of a n aco ustic wave through snow. R elation s hip b etween d and resp ective ly a a n d cP
w
Z'(e) = -j cot (C . e)
a nd
Z'(e = 0) =
(9)
00.
all'
So, Equation (8) implies
A e "# n"2
(n: integer "# 0).
(10)
If one chooses 0 < e < Amin/2 (Amin is the smallest wavelength in air satisfying the condition [or plane waves in the tube), one obtains from Equations (6) and (7):
Zc cotbl) = W W2
(11 )
= Zo(Z'(e) + Zc) - Z'(e)Zc.
(12 )
The real par t of W is positive a nd there is only one pair of complex solutions (W, , ). For the experiments presented in this paper, we used Eruel a nd Kj Rv). For a medium consisting of spheres, situated on a cubic lattice, Aa ",,/ Ah is constant and equal to 0.45 and independent of frequency, radius of the spheres and porosity (Zwikker ancl Kosten, 1949, p.43). In the fi rst step, we obtain simultaneously the three parameters by fitting the frequency dependent Wand T Before each experiment, air pressure and temperature in the cold room are measured to calculate the speed of sound and characteristic impedance of the air. Then, functions W(q = 1, h = 1, R , 1) and "I( q = 1, h = 1, R , 1) are determined with B equal to 0.86. Values issued from experiments, 95
Marco and others: Acoustic impedance measurement !ifsnow densiry Wm(f ) a nd , p,(f) are obtained for 25 frequencies. Accord-
1.0
ing to the least-squares fitting method, a nd assuming th at the three pa rameters a re independent of the frequency, th e soluti on (h, q, R) must minimize the function S: 25
S=
L
(I I ~ W (q = l , h = I , R , ji) -
,= 1
+ Il q,(q =
.c'"
2
Wm(MII 2
1, h = 1, R , M -'m(fi)11)
0.8
0,6
.
We obtain the functions q(R) a nd h(R) such that as j ah = 0 a nd aSjaq = o.
0,4
25
L
Real(,( h = 1, q = 1, R , j i) '1m(fi)) q(R )=-i= -1--- - - - - - - - - - - - - - - - - - - - - - 25
L h'(h = 1, q = 1, R, 1;)1
2
Fig. 8, Relation between acoustic porosiry h and gravimetric porosiry hg. 0 this study; • Buser (1986); l:,. ice grains in fro nt !if the surface !if the sample; 0 sample with a chipped surface.
i= l
25
""' Lt i= l
1
h(R) = q(R )
IW(h =
2 1, q = 1, R , M 1
25
L
Real(W (h
= 1, q = 1, R , M . W m(f;))
;= 1
(16) Real (A ): real pa rt of complex A. A: compl ex conjugate value of A. So, we need only one parameter R to minimize the function S(h(R) , q(R) , R) . Th e mathematical software MATLAB of Scientific Software Group (Mathworks, 1994), has been used to fit R for each snow sampl e. Then, hand q a re calcul ated easily.
RESULTS In a nother paper (Marco a nd others, 1996), we show th a t the r igid-fra me model proposed by Zwikker and Kosten (1949) for porous media cannot expla in the acoustic wave propagation through all snow types, especially not for dense snow. Certainly, one or more of its three pa rameters ar e not consta nt in the frequency ra nge used. The onl y interesting res ult for our probl em is the very good correlation between the acoustic porosity h determined by minimi sing S and the gravimetric porosity hg (Fig. 8). With the phenomenological relationship obtained between d and IWI, the use of the three-parameter model seems not to be the most convenient way to determine the density of snow from measurements of its acoustic pa rameters.
CONCLUSION From experiments in a cold labora tory, we fo und a good relationship between the density of snow a nd the modulus of its cha racteristic imped a nce. Thi s estimate is much more convenient than the one obta ined by using the existing three-pa ra meter model describing the propagation of an acoustic wave through not very dense snow. The m ain difficulties for conducting fi eld measurem ents to determine density are the uncertainti es associated with the experimental param eters. For exa mple, the exact di sta nce between the microphones a nd th e surface of the snow sample has to be known. Unfortun ately, thi s condition will be very difficult to fulfill for measurements inside th e body of a fl owing avalanche and even in a natura l snow cover. 96
h
(15)
REFERENCES Attenborough, K. 1982. Acoustica l cha rac teristics of porous materia ls, Physics Lelt., 82 (3). 182- 190. Att enborough, K. a nd O. Buser. 1988. On the applicati on of rigid- poro us models to impeda nce data for snow.] S01lnd Vib., 124 (2), 315-327. Bi ot, l\1. A. 1962. Mecha nics of deform ation and acoustic propagati on in porous media. ] Appl. Ph},s" 33, 1482- 1498, Boden, H. a nd M . Abom. 1986. InOuence of errors on the two-microphone method for measuring aco ustic properties in ducts. ] Acoust. Soc, Am" 79 (2),541- 549. Bogo rodski y, V. v. , V. P. G avrilo a nd V. A. Nitikin. 1974. C haracteristics of sound propagati on in snow. Sov. Phys. Aconst. , 20 (2), 121- 122. Bruel a nd Kja:r. 1983. Two -micro/)/lOllf impedance measurement tube type 4206 Lyon, Bruel a nd Kj a:r. Technical Documentation. Buse r, O. 1986. A rigid frame model of porous med ia for the acoustic impedence of snow. Journal ofSowld and Vibration, 111 (I), 71- 92. Buse r, O. a nd \>\'. Good. 1987. Acoustic, geometric a nd mechanical parameters of snow. {ntemational Association oJ Jlydrological Sciences Publica/ion 162 (Symposi um at Davos 1986 - Avalanche Formation, Movement and Effects ), 61 71. C hung, J Y. a nd D. A. Bl ase r. 1980a. Tra nsfe r function method of measuring in-duct aco usti c proper ties. I. Theory. ] Acoust. Soc. Am. , 68 (3), 907- 913. C hung, J. Y. a nd D. A. Bl aser. 1980b. Tra nsfer fun ction method of measuring in-duct aco ustic properti es. 2, Ex periment.] Acoust. Sac, Am., 68 (3), 914- 921. Ishida, T. 1965. Aco ustic properti es of snow. COil/rib. fnsl. Low Temp. Sci., Ser. A 20,23- 63. J ohnson, J. B. 1982. On the applicati on of Bi ot's theory to acoustic wave propagati on in snow. Cold Reg. Sci. Technol., 6 (1), 49- 60. Lee, S. l\1. and J. C. R ogers. 1985. C ha rac terisation of snow by acoustic so unding: a feasibilit y study. ] SoundVib. , 99 (2), 247- 266. Ma rco, O . 1994. Instrumentati on d'un site avala ncheux: de l'utilisation des propri etes acoustiques de la neige et des techniques d'im age ri es pour la mesure de pa ra metres phys iques d'une ava la nche dense. (These de doctora t, Uni versiteJoseph Fouri er, G renobl e.) Ma rco, 0 ., 0. Buser a nd P. Vill ema in. 1996, Ana lysis of a rigid fra me model of poro us medi a for th e aco ust ic pro perties of dense snow. ] Sound Vib., 196 (4),439- 451. Ma thworks. 1994. M ATLAB 4.2Clfor Macintosh, neural network tootbox, versioll 2.0a. Nati ck, l\IA , The M athworks Inc. O ura, H . 1952. [Sound velocit y in th e snow cover] Low Temp. Sci. , 9, 171- 178. II nJapanese with English summ ary.] Scybe rt, A. F. a nd B. Soena rko. 1980. Error a na lysis of spectra l estimates with a pplicati on to the measurements of acoustic pa ra me ters using ra ndom so und fi elds in ducts. ] Acoust. Soc. Am., 69 (4), 1190- 1199. Smith, J. L. 1965. The el astic constants, strength a nd density of Greenla nd snow as determined from meas urements of sonic wave velocity, CRREL Teck Rep. 167. Sommcrfcld, R .A. 1982. A review of snow acousti cs. Rev. Geo/)hys. Space Pltys. , 20 (1), 62- 66. Ze phoris, l\ L, D. Ma rtin a nd C. Pontiki s. 1975. Propagation des ondes 1IlIrasonores au sein du manleau neigfux. Toulouse, Meteorologic Nationale. (Etude AR 3162.) Z wikker, C. a nd C. W. Kosten. 1949. Sound absorbing materials. New York, e tc. , Elsevier.
