a porosity-based biot model for acoustic waves in snow · 2.1. porous material properties for snow...

1

A porosity-based Biot model for acoustic waves in snow

Rolf Sidler

Department of Earth Sciences, Simon Fraser University, 8888 University Drive, BC V5A1S6 Burnaby, Canada

ABSTRACT. Phase velocities and attenuation in snow can not be explained by the widely

used elastic or viscoelastic models for acoustic wave propagation. Instead, Biot’s model of

wave propagation in porous materials should be used. However, the application of Biot’s

model is complicated by the large property space of the underlying porous material. Here

the properties of ice and air as well as empirical relationships are used to define the prop-

erties of snow as a function of porosity. Based on these relations, phase velocities and plane

wave attenuation of shear- and compressional-waves as functions of porosity or density are

predicted. For light snow the peculiarity was found that the velocity of the first compres-

sional wave is lower than the second compressional wave that is commonly referred to as

the “slow” wave. The reversal of the velocities comes with an increase of attenuation for the

first compressional wave. This is in line with the common observation that sound is strongly

absorbed in light snow. The results have important implications for the use of acoustic waves

to evaluate snow properties and to numerically simulate wave propagation in snow.

1. INTRODUCTION

The use of acoustics to investigate snow is complicated by

the fact that sometimes lower wave velocities can be observed

with increasing density of the snow (Oura, 1952). This obser-

vation is at odds with elastic or visco-elastic wave propagation

theory, for which higher velocities are expected for the consid-

erably higher bulk and shear moduli of denser snow. Yet, the

observed wave velocities can be explained with wave propa-

gation theory for porous materials, where a second compres-

sional wave, also known as “slow” wave, is predicted (John-

son, 1982; Smeulders, 2005).

Oura (1952); Smith (1969); Yamada and others (1974) mea-

sured acoustic wave velocities and attenuation in field and

laboratory environments. Gubler (1977) measured accelera-

tion in the snowpack and air pressure above the snowpack for

explosives used in avalanche mitigation operations. Johnson

(1982) successfully used Biot’s model for wave propagation

in porous materials to predict wave velocities in snow. Som-

merfeld and Gubler (1983) observed increased acoustic emis-

sions from unstable snowpacks compared to acoustic emis-

sions of stable snowpacks. Mellor (1975) and Shapiro and

others (1997) published extensive reviews on snow mechanics

including acoustic wave propagation and proposed wave ve-

locity as a potential index property for snow. Amongst others,

Buser (1986); Attenborough and Buser (1988); Marco and

others (1996, 1998); Maysenholder and others (2012) investi-

gated acoustic impedance and attenuation of snow based on

the so called “rigid-frame” model (Terzaghi, 1923; Zwikker

and Kosten, 1947) in which the wave traveling in the pore

space is completely decoupled from the wave traveling in the

frame of the porous material. Recently acoustic methods have

been used to monitor and spatially locate avalanches (Suri-

nach and others, 2000; van Herwijnen and Schweizer, 2011;

Lacroix and others, 2012), to estimate the height and sound

absorption of snow covering ground (Albert, 2001; Albert and

others, 2009, 2013) and to estimate the snow water equiva-

lent of dry snowpacks (Kinar and Pomeroy, 2009). Kapil and

others (2014) used metallic waveguides to measure acoustic

emissions from deforming snowpacks.

The advantage of the rigid frame model is that it is rela-

tively straight forward to extract tortuosity of the pore space

and pore fluid properties from the phase velocities of the

slow wave. Applications are widespread and range from non-

destructive testing, medical applications and soil characteri-

zation to sound absorption (Fellah and others, 2004; Jocker

and Smeulders, 2009; Shin and others, 2013; Attenborough

and others, 2013).

The rigid-frame model can be deduced from Biot’s (1956a;

1962) theory under the assumption that the stiffness of the

porous frame is considerably higher than the stiffness of the

pore fluid. Consequently, the rigid-frame model does not ac-

count for the interaction between the pore fluid and the porous

frame as does Biot’s theory. Also the viscous effects of the

pore fluid are approximated with complex moduli in the rigid-

frame model and consequently accounted for with a phe-

nomenological instead of a physical model as in Biot’s theory

where the viscous friction of the fluid moving relative to the

solid frame is causing the observed attenuation. Especially

in light snow and in wet snow, where the frame and the

stiffness of the pore fluid are of comparable order of mag-

nitude Biot theory is expected to provide superior results

than the rigid-frame model (Hoffman and others, 2012). Also

a physical model is to be preferred over a phenomenological

as the results can be compared to complementary measure-

ments and consequently has more predictive power. However,

a disadvantage for the application of Biot’s model is the large

number of properties that have to be specified.

While phase velocities obtained with plane wave solutions

for Biot’s theory tend to correspond relatively well with its

measured counterparts, the plane wave attenuation can gen-

erally not be readily compared. The wave attenuation is com-

plicated by the superposition of effects that all lead to a de-

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2 Sidler: Acoustic waves in snow

crease of wave amplitude and are difficult to separate. Plane

wave attenuation does, for example, not account for geomet-

rical spreading, that strongly depends on the geometry of the

experiment and is present in virtually all physical measure-

ments.

Here, we propose a porous snow model as a function of

porosity and use it to estimate the wave velocities and atten-

uation of compressional and shear wave modes using plane

wave solutions for Biot’s (1956a) differential equation of wave

propagation in porous materials. We compare the results to

measurements from the literature and investigate the sensi-

tivity of the fast and slow compressional waves to individual

parameters of the porous model such as, for example, specific

surface area (SSA).

2. METHODS

2.1. Porous material properties for snow

An inherent problem when working with Biot-type porous

models is the large number of material properties involved.

To address this problem empirical relationships and a priori

information is gathered in this section to express the porous

material properties as a function of porosity.

A Biot-type porous material is characterized by ten prop-

erties with porosity arguably being the most significant. For

the stress strain relations the fluid bulk modulus Kf , the

bulk modulus of the frame material Ks, the bulk modulus of

the matrix Km, and the shear modulus µs have to be known.

The equations of motion require the densities of the solid and

fluid materials ρs and ρf , the porosity φ, and the tortuosity

T . The energy dissipation due to the relative motion of the

fluid to the solid are based on Darcy’s (1856) law and requires

the knowledge of the permeability κ and the viscosity η of the

pore fluid.

Typical values for Young’s modulus of ice, the frame ma-

terial of snow, are between 9.0 GPa and 9.5 GPa with a

Poisson’s ratio of ± 0.3 (Hobbs, 1974; Mellor, 1983; Schul-

son, 1999). The Young’s modulus E can be converted to bulk

modulus K as (Mavko and others, 2009)

K =E

3(1− 2ν), (1)

where ν is the Poisson’s ratio. The resulting frame bulk mod-

ulus Ks for snow is approximately 10 GPa.

For the snow matrix bulk modulus Km, the Krief equation

(Garat and others, 1990; Mavko and others, 1998)

Km = Ks(1− φ)4

(1−φ) , (2)

can be parameterized as

Km = Ks(1− φ)a

(b−φ) , (3)

and values for a = 30.85 and b = 7.76 can be obtained by

a least square inversion on the measurements presented by

Johnson (1982). In Figure 1 the Young’s moduli resulting

from Equation (3) are shown in comparison with measure-

ments and theoretical estimates of Young’s moduli (John-

son, 1982; Smith, 1969; Schneebeli, 2004; Reuter and others,

2013).

In combination with the Poisson’s ratio of snow, Equation

(3) can also be used to estimate shear moduli of snow as a

0.0 0.2 0.4 0.6 0.8 1.0Porosity

10-1

100

101

102

103

104

Young s

modulu

s (M

Pa)

0200400600800Density (kg/m3 )

Fig. 1. Krief equation fitted to dynamic measurements (solid line)

and compared to dynamic measurements from Johnson (1982) and

Smith (1969) indicated with circles and stars, respectively. Theo-

retical values obtained from numerical modeling of microtomogra-

phy snow structures are indicated with diamonds and triangles for

Schneebeli (2004) and Reuter and others (2013), respectively.

function of porosity by using the relationship (Mavko and

others, 2009)

µs =3

2

Km(1− 2ν)

1 + ν. (4)

For this purpose the linear relationship

ν = 0.38− 0.36φ, (5)

is used to express the Poisson’s ratio of snow as a function

of porosity. Figure 2 shows how this function relates to mea-

surements from Bader (1952), Roch (1948), and Smith (1969).

The shear moduli resulting from Equations (3), (4) and, (5)

are shown in Figure 3 and compared to measurements from

Johnson (1982) and Smith (1969).

The tortuosity T describing the ‘twisting’ of the actual

flow path of the pore fluid compared to a straight line can be

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Porosity

0.0

0.1

0.2

0.3

0.4

0.5

Pois

son's

rati

o

200300400500600700800900Density (kg/m3 )

Fig. 2. Poisson’s ratio as a function of porosity according to Equa-

tion (5) compared to measurements from Bader (1952) (dashed

lines), Roch (1948) (dotted lines), and Smith (1969) (circles).

Sidler: Acoustic waves in snow 3

0.3 0.4 0.5 0.6 0.7 0.8Porosity

104

105

106

107

108

109

1010

Shear

modulu

s (P

a)

200300400500600Density (kg/m3 )

Fig. 3. Shear moduli for snow as a function of porosity by using

Equations (3) and (5) compared to measurements presented by

Johnson (1982) (diamonds) and Smith (1969) (stars).

estimated based on geometrical considerations as

T = 1− s(

1− 1

φ

), (6)

where s is the so called shape factor (Berryman, 1980). For

a packing of sphere the shape factor is 0.5 and Equation (6)

reduces to

T =1

2

(1 +

1

φ

). (7)

The permeability is estimated using the Kozeny-Carman

relation

κ = Cr2φ3

(1− φ)2, (8)

where κ is the permeability, φ the porosity, and C is an em-

pirical constant (Mavko and others, 1998). For sediments the

constant is Cs = 0.003 (Mavko and Nur, 1997; Carcione and

Picotti, 2006), but one order of magnitude larger for snow

CB = 0.022 (Calonne and others, 2012; Bear, 1972). The

grain diameter r can be related to the specific surface area

(SSA) of snow with the relation

r =3

SSAρi, (9)

where ρi is the density of ice. Substituting Equation (9) into

Equation (8) leads to

κ = 0.2φ3

(SSA)2(1− φ)2. (10)

Due to the compaction and metamorphosis processes inher-

ent to snow it can be assumed that the specific surface area by

itself is a function of porosity (Legagneux and others, 2002;

Herbert and others, 2005). Domine and others (2007) use the

equation

SSA = −308.2 ln(ρS)− 206.0, (11)

to relate specific surface area to snow density. For dry snow

the density ρs can be obtained from snow with porosity φ as

ρs = (1− φ) · 916.7kg

m3. (12)

Table 1. Pore fluid properties of air (Lide, 2005).

density, ρf 1.30 kg/m3

viscosity, η 1.7 · 10−5 Pa s

bulk modulus, Kf 1.42 · 105 Pa

1

The equation (11) yields negative values for the specific sur-

face area for porosities smaller than φ = 0.44. Therefore

the relationship is used here only for porosities larger than

φ = 0.65 and a constant value for specific surface area of

15 m2/kg is used for lower porosities. In this study Equation

(11) is intended to reflect an average trend and is shown to-

gether with high and low constant values for specific surface

area to illustrate the possible variability.

The density ρf , viscosity η, and bulk modulus Kf of air

as the pore fluid of snow are assumed to be constant, which

means independent of temperature and altitude and are given

in Table 1 (Lide, 2005).

2.2. Phase velocities and plane waveattenuation

A convenient way to get closed form solutions for Biot’s (1956b)

differential equations is to assume plane waves solutions and

substitute those into the differential equations. The complex

plane wave modulus is then obtained by solving the result-

ing dispersion relation (Johnson, 1982; Pride, 2005; Carcione,

2007). As in the poroelastic case the dispersion relation is a

quadratic equation, there are two roots that correspond to

the first and second compressional waves. The phase veloc-

ity V and attenuation in terms of the dimensionless quality

factor Q can then be obtained from the complex plane wave

modulus Vc as (O’connell and Budiansky, 1978)

V (ω) =[Re(Vc(ω)−1)

]−1, (13)

Qp(ω)−1 = 2Im(Vc(ω))

Re(Vc(ω)), (14)

where ω is the angular frequency.

The solutions are generally computed for individual fre-

quencies and porosities. However, under the assumption that

any dissipative effects are ignored and that the bulk mod-

ulus of the fluid is much smaller than the bulk modulus of

the solid matrix it can be shown that the first compressional

wave velocity V∞1 can be expressed as

V∞1 =

√Em

ρ− φρf/T, (15)

and the second compressional wave velocity V∞2 as

V∞2 =

√Kf

ρfT, (16)

where Em = Km+(4/3)µs (Bourbie and others, 1987, p. 81).

While these expressions may deviate from the Biot phase ve-

locities, they illustrate that the first compressional wave trav-

els mainly in the skeleton and the second compressional wave

mainly in the fluid. The first compressional wave is therefore


also most sensitive to the properties of the ice matrix. The

second compressional wave is sensitive mostly to properties

of the pore space and the pore fluid.

2.3. Dynamic viscous effects

The fluid flow in the pores of the material has a different

character for lower and higher frequencies (Biot, 1956b). For

lower frequencies the flow is of Poisson-type where the flow

is fastest in the center of a pore and reduces gradually in a

parabolic shape towards the outside of the pores. For higher

frequencies the importance of inertial forces increases. The

fluid in the center of the pores flows all with the same ve-

locity like an ideal fluid, while the fluid at the outside of the

pores remains attached to the pore walls. In between the two

forms the so called viscous boundary layer (Pride, 2005). The

transition between the low and high frequency flow behavior

occurs when the viscous boundary layers are smaller than the

pore diameter. The frequency at which this transition occurs

is called the Biot frequency fBiot and can be computed as

fBiot =ηφ

2πT κρf, (17)

where η and ρf are the viscosity and density of the pore fluid,

and T and κ are the tortuosity and the permeability of the

pore space, respectively (Carcione, 2007, p. 270).

Biot’s (1956a) theory is considered valid for frequencies up

to the Biot frequency. For higher frequencies Johnson and

others (1987) introduced a frequency dependent permeability

that accounts for the different flow behavior in the low and

high frequency limit and is often referred to as a frequency

correction or the JDK model. Figure 4 shows the Biot fre-

quency for snow based on the relationships between porosity

and the involved material properties as presented in Section

2.1, where the permeability further depends on the specific

surface area. To illustrate the variability of Biot’s frequency

due to specific surface area the Biot frequency is plotted for

Equation (11) which depends on porosity and for constant

values of SSA = 15 m2/kg and SSA = 90 m2/kg.

The results shown in this study are evaluated using the fre-

quency correction (Johnson and others, 1987). However, for

the first compressional wave there are virtually no differences

when the frequency correction is neglected. For the second

compressional wave the differences are rather low, except for

frequencies in the range of the Biot frequency, where moder-

ate differences can be observed.

3. RESULTS

In this section phase velocities and plane wave attenuation

for snow are presented as a function of porosity based on

the relationships presented in Section 2.1. Figure 5 shows the

predicted phase velocity for the first compressional wave and

a frequency of 1 kHz as a function of porosity. The predicted

phase velocities are compared to measurements from Smith

(1969) and Johnson (1982). In addition, the predicted velocity

for an individual and a combined variation of 25 % in bulk

and shear modulus are shown. The velocity strongly decreases

with increasing porosity and the variation of bulk and shear

modulus for snow of the same porosity are small compared

to the change of velocity over the porosity range.

0.3 0.4 0.5 0.6 0.7 0.8 0.9Porosity

101

102

103

104

105

Bio

t fr

equency

(H

z)

SSA (φ)

SSA = 15 m2 /kg

SSA = 90 m2 /kg

100200300400500600Density (kg/m3 )

Fig. 4. Biot’s characteristic frequency as a function of porosity

based on the relations presented in Section 2.1. The solid line cor-

responds to frequency dependent variation of specific surface area

according to Equation 11 and the dashed and dotted line corre-

sponds to constant values of SSA = 15 m2/kg and SSA = 90

m2/kg, respectively

The predicted shear velocities at 1 kHz are compared to

measurements by Johnson (1982) and Yamada and others

(1974) in Figure 6. Similar to the first compressional wave,

the shear velocity strongly decreases with increasing porosity.

The predicted phase velocities of the second compressional

wave at 500 Hz as a function of porosity are shown in Figure

7 and are compared to measurements from Oura (1952) and

Johnson (1982). As the pore fluid properties are assumed to

be constant, the phase velocity of the second compressional

wave depends almost exclusively on variations in permeabil-

ity and tortuosity. Variation in frame bulk modulus and shear

modulus have virtually no influence on phase velocity and at-

0.3 0.4 0.5 0.6 0.7 0.8 0.9Porosity

0

500

1000

1500

2000

2500

Velo

city

(m

/s)

25 % variation in Km

25 % variation in µs

25 % variation in both

100200300400500600Density (kg/m3 )

Fig. 5. Predicted phase velocities for the compressional wave of

the first kind at 1 kHz compared to measurements from Johnson

(1982) (diamonds) and Smith (1969) (crosses). The dashed and

dotted lines correspond to predicted velocities for a 25 % variation

of matrix bulk modulus and shear modulus, respectively. The dash-

dot line corresponds to a 25 % variation in both.


0.2 0.3 0.4 0.5 0.6 0.7 0.8Porosity

0

200

400

600

800

1000

1200

1400

1600

Velo

city

(m

/s)

200300400500600700Density (kg/m3 )

Fig. 6. Predicted shear velocities at 1kHz. Squares and crosses cor-

respond to shear wave velocity measurements from Johnson (1982)

and Yamada and others (1974), respectively.

tenuation of the second compressional wave. The tortuosity

has a stronger lever on the phase velocity than the permeabil-

ity and 30 % variation in tortuosity leads to larger changes

in phase velocity than a 50 % variation in permeability. The

phase velocity of the second compressional wave shows only

little variation with porosity and is mainly sensitive to the

geometrical structure of the pore space.

The plane wave attenuation for the first compressional wave

as a function of porosity is shown for three different frequen-

cies in Figure 8. The plane wave attenuation of the first com-

pressional wave is orders of magnitude higher for light snow

with a porosity φ & 0.8 than for denser snow. Figure 8b)

therefore shows the porosity range between φ = 0.55 and φ

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95Porosity

0

50

100

150

200

250

300

350

400

Velo

city

(m

/s)

30 % variation in tortuosity

50 % variation in permeability

50100150200250300350400Density (kg/m3 )

Fig. 7. Predicted phase velocities for the second compressional

wave at 500 Hz. The dashed and dotted lines correspond to phase

velocities for 30 % variation in tortuosity and 50 % variation in

permeability, respectively. Squares represent wave velocity mea-

surements from Johnson (1982). Crosses correspond to measure-

ments from Oura (1952). Please note that an increasing tortuosity

decreases the velocity while an increase of permeability increases

the velocity of the second compressional wave.

a)

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95Porosity

0.00

0.01

0.02

0.03

0.04

0.05

Att

enuati

on (

1/Q

)

100 Hz

1 kHz

10 kHz

50100150200250300350400Density (kg/m3 )

b)

0.55 0.60 0.65 0.70 0.75 0.80Porosity

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

Att

enuati

on (

1/Q

)

100 Hz

1 kHz

10 kHz

200250300350400Density (kg/m3 )

Fig. 8. Predicted attenuation for the first compressional wave as a

function of porosity. Figure b) shows a fragment of a) for porosities

below φ = 0.8. The attenuation of the first compressional wave is

orders of magnitudes higher for light snow than for denser snow.

= 0.8 where variations in attenuation can not be resolved in

Figure 8a).

Homogeneous Biot-type porous materials are known to have

a characteristic peak of attenuation (Geertsma and Smit,

1961; Carcione and Picotti, 2006). In Figure 9 these attenu-

ation peaks are shown for snow of different densities. Again,

the figure is split into two subfigures to account for the sig-

nificant difference of attenuation levels for light and dense

snow that was already apparent in Figure 8. Peak attenua-

tion shifts towards lower frequencies and the attenuation level

increases with increasing porosity. The same is true for light

snow but with considerably higher attenuation levels. Also

the peak attenuation frequency overlap for a porosity range

around φ = 0.8.

Phase velocity and attenuation for the second compres-

sional wave obtained with and without using the frequency

correction discussed in Section 2.3 are shown in Figure 10.

The effect of the dynamic viscous effects are relatively small

except in the range of Biot’ frequency, where the phase veloc-

ity shows a moderate difference between the solutions includ-

ing and neglecting a frequency correction (Johnson and oth-

ers, 1987). In contrast to the first compressional wave there

is no distinctive difference in attenuation for dense and light


a)

100 101 102 103 104 105 106

Frequency (Hz)

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006A

ttenuati

on (

1/Q

)φ = 0.4

φ = 0.5

φ = 0.6

φ = 0.7

b)

100 101 102 103 104 105 106

Frequency (Hz)

0.00

0.02

0.04

0.06

0.08

0.10

Att

enuati

on (

1/Q

)

φ = 0.82

φ = 0.85

φ = 0.87

φ = 0.9

Fig. 9. Frequency dependent attenuation for for the first com-

pressional wave in (a) medium to dense and (b) light snow. The

peak of the attenuation shifts toward higher frequencies for denser

snow. Please note that the amplitude of the attenuation is orders

of magnitude larger for light snow with a porosity φ & 0.8.

snow for the second compressional wave. The sharp bend in

phase velocity and attenuation is due to the relationship be-

tween porosity and the specific surface area that was chosen

to be constant for φ < 0.65 to avoid the negative values re-

sulting from Equation (11).

The variation of the phase velocity of the second compres-

sional wave due to changes in specific surface area are shown

in Figure 11. For fixed values of specific surface area of SSA =

15 m2/kg and SSA = 90 m2/kg the phase velocity at 500 Hz is

plotted with dashed and dotted lines, respectively. The solid

line represents the phase velocities resulting from Equation

(11). The variation is larger for denser snow than for light

snow where permeability is less affected by specific surface

area.

In Figure 12 attenuation for both compressional waves is

shown for a constant specific surface area of SSA = 15 m2/kg,

and SSA = 90 m2/kg, as well as for specific surface area as

a function of porosity by the use of Equation 11. The at-

tenuation levels of both compressional waves increase with

an increase of specific surface area. When comparing Figure

12a) to Figure 8a) and Figure 12b) to Figure 10b) it becomes

obvious that the effects of the specific surface area has a sim-

a)

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95Porosity

0

50

100

150

200

250

300

350

400

Velo

city

(m

/s)

100 Hz

1 kHz

10 kHz

50100150200250300350400Density (kg/m3 )

b)

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95Porosity

0.0

0.5

1.0

1.5

2.0

2.5

Att

enuati

on (

1/Q

)

100 Hz

1 kHz

10 kHz

50100150200250300350400Density (kg/m3 )

Fig. 10. a) Phase velocity and b) attenuation for the second

compressional wave for 100 Hz, 1 kHz and 10 kHz. The black lines

correspond to solutions including dynamic viscous effects consid-

ered by Johnson and others (1987) while the red lines corresponds

to solutions of Biot’s (1956a) differential equations without fre-

quency correction. The signs denote velocity measurements from

Oura (1952) and Johnson (1982).

0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95Porosity

0

50

100

150

200

250

300

350

400

Velo

city

(m

/s)

SSA (φ)

SSA = 15 m2 /kg

SSA = 90 m2 /kg

50100150200250300350400Density (kg/m3 )

Fig. 11. Predicted phase velocities for the compressional wave of

the second kind at 500 Hz for a specific surface area as a function

of porosity (solid line) and constant values of SSA = 15 m2/kg

(dashed line) and SSA = 90 m2/kg (dotted line). Squares and

crosses correspond to measurements from Johnson (1982) and Oura

(1952), respectively.


a)

0.3 0.4 0.5 0.6 0.7 0.8 0.9Porosity

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Att

enuati

on (

1/Q

)

SSA (φ)

SSA = 15 m2 /kg

SSA = 90 m2 /kg

100200300400500600Density (kg/m3 )

b)

0.3 0.4 0.5 0.6 0.7 0.8 0.9Porosity

0.0

0.5

1.0

1.5

2.0

Att

enuati

on (

1/Q

)

SSA (φ)

SSA = 15 m2 /kg

SSA = 90 m2 /kg

100200300400500600Density (kg/m3 )

Fig. 12. Predicted attenuation at 500Hz for the compressional

wave of a) the first and b) second kind as a function of porosity.

The dashed and dotted lines correspond to fixed values for specific

surface area of SSA = 15 m2/kg and SSA = 90 m2/kg, respectively.

The solid line corresponds to equation (11) and a constant value

of SSA = 15 m2/kg for densities above 315 kg/m3.

ilar effect on attenuation as a change of frequency. This is an

effect of scale. A higher specific surface area goes along with

lower permeability and therefore has a similar effect as if the

wavelength would be increased.

4. DISCUSSION

4.1. Slow first compressional phase velocity

Compressional wave velocities as a function of porosity com-

pared to measurements presented by Johnson (1982) and

Sommerfeld (1982) are shown in Figure 13. The relations

between porosity and the properties of the porous material,

especially the strong decrease of matrix bulk modulus with

increasing porosity, lead to the peculiarity that the predicted

first compressional wave becomes slower than the second com-

pressional wave for light snow with a porosity φ & 0.8. In

most materials, the second compressional wave is consider-

ably slower than the first compressional wave and is there-

fore sometimes also called the ‘slow’ wave. No measurements

of first compressional wave with a lower phase velocity than

the second compressional wave have been reported for snow.

0.3 0.4 0.5 0.6 0.7 0.8 0.9Porosity

0

500

1000

1500

2000

2500

Velo

city

(m

/s)

P1

P2

100200300400500600Density (kg/m3 )

Fig. 13. Predicted velocities for compressional waves of the first

(solid line) and second kind (dashed line) as a function of porosity

based on empirical relationships for frame bulk and shear mod-

ulus, tortuosity and permeability. The dashed lines identify mea-

surements of first compressional waves compiled by (Sommerfeld,

1982) and the diamonds and squares represent wave velocity mea-

surements compiled by Johnson (1982) for compressional waves of

the first and second kind, respectively.

However, a first compressional wave with lower phase veloc-

ity than the second compressional wave has been observed

in high porosity reticulated foam (Attenborough and others,

2012).

From the plane wave solutions it is not immediately clear

that the lower compressional velocity in light snow corre-

sponds to the first compressional wave mode as there are no

explicit rules to choose the signs of the square roots. To illus-

trate that it is indeed the velocity of the first compressional

wave that is slower than the phase velocity of the second

compressional wave, two numerical simulations of wave prop-

agation in poroelastic materials were performed. In the first

simulation the homogeneous poroelastic material corresponds

to snow with a porosity φ = 0.7, where the first compressional

wave is expected to be faster than the second compressional

wave. In the second simulation the porosity of the snow is

chosen to be φ = 0.9 and the first compressional wave is

expected to be slower than the second compressional wave.

The relations from Section 2.1 are used to characterize the

remaining porous material properties.

For the simulation a pseudo spectral modeling code was

used (Sidler and others, 2010). To avoid differences due to the

source characteristics a pressure source with a waveform of a

Ricker wavelet with a central frequency of 500 Hz is placed in

the air above the poroelastic material and the boundary con-

ditions are assumed to be of the ’open pore’ type (Deresiewicz

and Skalak, 1963). The field variables of the simulation are

the velocity of the solid frame, the relative velocity of the pore

fluid to the solid frame, the stress tensor, and the pore pres-

sure. Please note that these are particle velocities and should

not be confused with the phase velocities discussed before.

Even though the wave modes do not travel independently

of each other in the numerical simulation, in a homogeneous

material, the motion of the solid frame corresponds roughly


to the first compressional wave that travels mainly in the

solid frame. Likewise, the relative motion of the pore fluid

corresponds roughly to the second compressional wave (see

also equations (15) and (16)).

In Figure 14 snapshots of horizontal components of the

velocity of the porous frame and the relative velocities of

the pore fluid to the porous frame are shown 15.6 ms after

triggering the acoustic source for the two simulations. The

velocity of the second compressional wave is almost the same

in both simulations (Figure 14 b) and d) ). However, it is

evident that the first compressional wave is faster than the

second compressional wave in the first simulation (Figure 14

a) ) and slower than the second compressional wave in the

second simulation (Figure 14 c) ).

4.2. Increased sound absorption of lightsnow

The attenuation levels of the first compressional wave differs

significantly for light snow with a porosity φ & 0.8 and snow

with a lower porosity. This separation corresponds roughly

to a separation between freshly fallen and aged snow (Judson

and Doesken, 2000). In between the two porosity ranges the

attenuation vanishes completely as the two wave modes have

the same velocity and the viscous effects leading to attenu-

ation are not effective. The sound absorption above ground

is a complex combination of effects involving amongst others

the interference of incident and reflected waves, the reflection

coefficient, geometrical spreading as well as surface and non

geometrical waves (Embleton, 1996). However, it is clear that

if the reflection coefficient of the ground decreased also the

sound level above the ground decreases (Watson, 1948; Nico-

las and others, 1985). Due to the high porosity of snow and

the open pore boundary conditions, the pressure of the air

above the snow pack interacts mainly with the air in the pore

space and little energy is transmitted into the ice frame. As

the velocity of the second compressional wave is almost equal

to the velocity of the air above the the snow, there is almost

no impedance contrast that would lead to a reflection. The

low velocities of the first compressional wave in snow with

porosity φ > 0.8 and the corresponding higher attenuation

further decrease the impedance contrast and also reduce the

contribution of refracted waves.

5. CONCLUSIONS

A method to predict phase velocities and plane wave atten-

uation of acoustic waves as a function of snow porosity is

presented. The method is based on Biot’s (1956a) model of

wave propagation in porous materials and uses empirical re-

lationships to assess tortuosity, permeability, bulk, and shear

moduli as a function of porosity. The properties of the ice

frame of the snow and air as the pore fluid are assumed to be

constant. The method is not restricted to porosity as a sin-

gle degree of freedom and additional information on specific

surface area or any other of the properties characterizing a

Biot-type porous medium can be readily incorporated.

For light snow with a porosity φ & 0.8 the particularity

is found that the velocities of the compressional wave of the

first kind is slower than the phase velocities of the compres-

sional wave of the second kind which is commonly referred

to as the ‘slow’ wave. Such a reversal of the velocities of the

compressional waves has been observed in reticulated foam

before and is due to the week structure of the ice matrix in

fresh and light snow. The wave velocity reversal is a rela-

tively sharp boundary for the attenuation level of the first

compressional wave which is orders of magnitudes larger for

highly porous snow. This finding is in accordance with the

well known observation that freshly fallen snow absorbs most

of the ambient noise, while after a relatively short time this

absorbing behavior vanishes.

The first compressional wave is sensitive mainly to ma-

trix and shear bulk modulus. A variation of ∼25 % in both,

shear and matrix bulk modulus can characterize the variabil-

ity in the measured velocities. The attenuation of the second

compressional wave decreases with increasing porosity and

is considerably higher than for the first compressional wave.

Also frequency dependence of the attenuation is considerably

more distinct than for the first compressional wave. The ve-

locity of the second compressional wave depends strongly on

tortuosity, permeability, and the related specific surface area.

The variation of measured wave velocities for the second com-

pressional wave can be obtained by altering the tortuosity by

∼30 % or by altering the permeability by ∼50 %.

This method is a viable prerequisite for numerical modeling

of acoustic wave propagation in snow, which allows, for exam-

ple, to assess the design of acoustic experiments to probe for

snow properties or to assess the role of acoustic wave propa-

gation in artificial or skier triggered snow avalanche releases.

Further research will address the presence of liquid water in

the pore space, a more complete analysis of sound absorption

above snow of different porosity, and numerical simulations

of explosive avalanche mitigation experiments.

6. ACKNOWLEDGEMENTS

This research was founded by a fellowship of the Swiss Na-

tional Science Foundation.

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