Application of capacitance instrumentation to the measurement of density and velocity of flowing snow

cold regions science

and technology ELSEVIER

Cold Regions Science and Technology 25 (1997) 47-63

Application of capacitance instrumentation to the measurement of density and velocity of flowing snow Michel Y. Louge a,*, Roland Steiner a, Stephen C. Keast a, Rand Decker b, James Dent c, Martin Schneebeli d a Sibley School of Mechanical and Aerospace Engineering, Cornell Unit,ersity, Upson Hall, Ithaca, NY 14853, USA b Department of Cieil Engineering, University of Utah, Salt Lake City, UT 84112, USA c Department of Civil Engineering, Montana State Universit3; Bozeman, MT 59717, USA d Eidgeni~ssisches lnstitutfiir Schnee- und Lawinenforschung, CH-7260 Davos-Dorf Switzerland Received 7 November 1995; accepted l July 1996

Abstract

We describe capacitance instrumentation suitable for the measurement of density and velocity of flowing snow with moderate liquid-phase water content. A wand, consisting of two adjacent sensors protected by guard circuits, produces signals that are related to snow density through calibration. Cross-correlation of the signals permits velocity measurements. Calibration is accomplished using a capacitance device that records the dielectric properties of a snow sample while subjecting it to controlled levels of compaction and volume change. Non-invasive probe geometries are also presented. The instrumentation is tested in artificial and natural avalanches. Keywords: Snow; Avalanche; Dielectric properties; Electrical methods; Measurement

1. Introduction

Because in situ measurements of density and velocity of snow are challenging, models of snow avalanches have yet to be verified on the geological scale. Laboratory experiments with surrogate materials permit fine control of flow conditions, see for example Hungr and Morgenstern (1984) or Hutter et al. (1995). However, it is likely that the scale of actual avalanches promotes phenomena that are difficult to reproduce in the laboratory. Direct measurements of avalanche flow parame-

* Corresponding author. Michel.Louge @cornell.edu

Fax:

+ 1(607)255-1222.

E-mail:

ters are limited. Gubler and Hiller (1984), Salm and Gubler (1985) and Gubler (1987) used frequencymodulated, continuous wave (FMCW) radar to resolve snow density profiles and doppler radar to record avalanche velocity. Dent et al. (1994) recorded the velocity of snow grains by cross-correlating the backscattering signals from adjacent pairs of optical sensors protected by a shed. The corresponding velocity profiles suggested the existence of a relatively thin granular shear layer of order a few cms at the base. Because radiation backscattering is not a function of density alone (Van de Hulst, 1957), the optical instrument of Dent et al. does not produce quantitative measurements of snow density without calibration. The relatively well-established dielectric proper-

0165-232X/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S01 6 5 - 2 3 2 X ( 9 6 ) 0 0 0 1 6 - X

48

M. K Louge et al. / Cold Re,ictus Science and Technology 25 (1997) 47-63

ties of snow (Mellor, 1964; Kuroiwa, 1967) suggest an alternative. The idea is to record the effective dielectric permittivity of snow suspensions with capacitance probes and to infer snow density through calibration. Denoth et al. (1984) reviewed capacitance instruments that deduce the water equivalent of snow from a record of its dielectric properties and an independent measurement of snow density. Common instruments employ radio frequencies in the range 106 to 109 Hz and exploit differences in the dielectric properties of liquid water and ice. At these high frequencies, the real part of the dielectric permittivity is typically a linear function of density and a quadratic function of liquid-phase water content. However, because these instruments are generally designed for stationary assays of the snow pack, their electronics are often unsuitable for rapid measurements of density time histories, while their geometries are unfit for insertion in an avalanche. in flows of gas-solid suspensions and dielectric granular materials, capacitance probes have produced rapid record of solid volume fraction. A conventional technique is to balance the probe capacitance in a bridge excited by an oscillator at a few kHz (e.g., Lanneau, 1960; Geldart and Kelsey, 1972; Chandran and Chen, 1982; Abed, 1984). However, this method is often corrupted by the influence of other conductive surfaces in the vicinity (stray capacitance) and by the capacitance of the cable connecting the probe to the bridge. To eliminate such stray and cable capacitances, Acree Riley and Louge (1989) added a new "guard" conductor to their probe geometries. In this way, they resolved excursions in the volume fraction as small as _+0.5%. Using a similar principle, Louge (1995) and Louge et al. (1996) designed noninvasive probes that record solid volume fraction and velocity near the grounded wall of an inclined chute flow. The objective of the present work is to develop a capacitance technique suitable for rapid records of snow density and velocity in a snow avalanche of moderate liquid-phase water content. The assumption is that the density dependence of the effective dielectric properties can be established shortly before or after the event through calibration. The challenge is to design a probe geometry that minimizes disturbances to the flow while confining the measurement volume near the probe.

This paper begins with a summary of the probe electronics and the corresponding rapid determination of the complex dielectric permittivity. After outlining a calibration procedure suitable for field work and discussing stability of the electronics, we present two wall-mounted, non-invasive probe geometries. We then describe a twin probe in the shape of a wand designed to record velocity from signal cross-correlation. We present static and dynamic tests of the wand and discuss its performance. Finally, we report data from artificial avalanches produced in Davos, Switzerland and from natural avalanches triggered in Bridger Bowl near Bozeman, Montana. 2. Guarded capacitance measurement This section summarizes the principle of the processing electronics, which are available from MTI Instruments or Capacitec, two competing firms that apply this technology to the accurate measurement of distances. Dorman (1979) provides a more exhaustive description of the electronics. As Fig. 1 illustrates, generic probes consist of three conductors called the "sensor", "ground" and "'guard" electrodes. The electronics records the impedance Z between sensor and ground, while a buffer amplifier maintains the guard at precisely the same sinusoidal voltage as the sensor's. Because it absorbs distortions of the electric field caused by external interferences, the guard protects the sensor from stray capacitances. In addition, because the sensor is connected to the processing circuits using a guarded coaxial cable, the cable capacitance does not participate in the measurement. By eliminating stray and cable capacitances, the technique can detect capacitances three to six orders of magnitude smaller than conventional bridges. To ensure proper operation of the guard electrode, the electronics must maintain the guard voltage ~,~ equal to the sensor voltage ~,~. This is accomplished by sampling ~:~ through a buffer amplifier of unity gain. This buffer has a suitably high input impedance to avoid disturbing the sensor circuit. Further protection of the sensor circuit from external interferences is accomplished by surrounding it with a guarded shield. The output of the system is the rectified guard voltage V, which the buffer keeps proportional to the amplitude of the sensor voltage.

49

M. E Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63

Another objective of the electronics is to pass a sinusoidal current of constant amplitude through the test impedance Z. To this end, the carrier oscillator generates a voltage t,r of constant, albeit user-adjustable, amplitude. The amplitude of tile sensor current i is kept constant by controlling the voltage (v~ - c l) across the reference impedance Z r. This is achieved by sampling the sensor voltage t,~ through the buffer, adding or subtracting the difference (L'~ ~"L) to the oscillator voltage, and feeding the result to the reference impedance through an amplifier of high gain H (Fig. 1). Assuming negligible current input into the amplifiers, the voltage t,'l into the reference impedance is L'I = H(/.' r -I- a/;g - a v I) = i ( Z r + Z )

(1)

(2)

L'~ = Zi

In Eq. (1), 6 is - 1 for the Capacitec amplifier and + 1 for the MTI system. Eliminating t, l from Eqs. (1) and (2) yields t;r -

t:~

1 = -

Zr(I+6H] +

H

Z~

H

J

(3)

(7)

V = Qs/gC

where Q~ is a system constant and g is an adjustable parameter. For the MTI amplifier, Q~ = 3.44 × 10-12 Coulomb and 1 < g _< 6.5. Electronics with different ranges are also available. From Eq. (7), the effective dielectric constant e e of a homogeneous suspension covering the probe is obtained by forming the ratio of V0, the rectified probe output in air, and V, the output in the presence of the dielectric powder of interest, - C/Co

while the guard voltage is

Cg =

quency of the oscillator and j2 = _ 1. In this case, the system produces a rectified output voltage that is related to C, the capacitance between the ground and sensor surfaces, through the empirical relation

(8)

= Vo/V

where C O is the probe capacitance in air. For a dielectric powder, it remains to infer the volume fraction from the measurement of 6e. This is achieved by analyzing or calibrating the dielectric behavior of the suspension (Louge and Opie, 1990). For powders with a significant imaginary part of the permittivity, the voltage ratio V o / V yields the modulus of Z,

and -Y =

vr

1

i=--

Z

Because H is large and real, inspection of Eq. (4) indicates that the current is nearly independent of Z, i ,-~ 3 b ' r / Z r

(5)

and the modulus of Eq. (3) is, to a good approximation, I 'rl - -

I%1

Izrl =

- -

Iz[

(6)

Because rectification produces a voltage proportional to Vg, the output V is itself proportional to the modulus of Z. With a carrier oscillator operating at 16 kHz, the rectified output can monitor variations of the suspension density as rapid as 3.5 kHz. For powders with negligible imaginary part of the permittivity, Z = 1 / j 2 " r r f C , where f is the fie-

-

2 fCo

(9)

z

where Z 0 is the probe's impedance in air. For isotropic media, the electric displacement D and the electric intensity E are colinear vectors; however, because they are not generally in phase, the effective dielectric constant 6~ may exhibit real and imaginary parts, Ee - D / w o E =

6'-j6"

(10)

where 6' and 6" are both functions of frequency and o-0 = 8.854 × 10 -12 F / m is the dielectric permittivity of free space and, to a good approximation, of air. The corresponding impedance between sensor and ground is 1

Z=

f 2 7rfo-o( #' + j 6 ' )

(ll)

In this expression, / is a characteristic length of the capacitance probe geometry, /==- c o ~ < ,

(

50

M.Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63

High gain amplifier / Reference IV Ix" -/ impedance ~

I

~'r ~1~ Vl ± o,o,,,=

,,-," Coaxial

~ x , cable

,'" ~_ _~.~ VS ISensor/"',Z ',

Vg

V

Fig. 1. Schematic of the electronic system. The dashed lines represent the physical boundary of the processing circuits. Z is the impedance between sensor and ground.

In the presence of a homogeneous, isotropic suspension, f is independent of e e. Thus, from Eqs. (9)(12), the voltage ratio yields the modulus of e e, V(...~)= 1/e. 2 + et 2 V

(13)

In order to resolve both components of ee, we exploit the phase lag ~b between vr and t,g, which are both readily available for measurement. From Eq. (3), m E' tan~b =

e' m --+--+ e" n

tz E"

1

(14)

2 ~fcr0 e " f n ( 1 + H )

where m and n are the real and imaginary parts of the conjugate of Z r, Zr=-m-n j

(15)

For the MTI amplifier, the constants m / n and n(1 + H ) are found by recording ~ for a series of known capacitances. For the Capacitec amplifier, Z,. is purely capacitive, so m = 0. Note that, in order to permit successful measurements of capacitance in air, the condition 127rf~r0fn(l + 6 H ) I >> 1 must be satisfied to let Eq. (6) be a valid

(16)

approxima-

tion of Eq. (3). Consequently, for the Capacitec amplifier, Eq. (14)reduces to E t!

Itan6] -~ e'

(17)

so the phase lag between vr and t,,~ is a direct measure of the "loss tangent" ( e " / e ' ) . If both real and imaginary parts of e~ are required, the Capacitec measurement yields e' and e" through Eqs. (13) and (17). For the MTI amplifier, the determination of e' and e" requires a more complicated solution involving Eqs. (13) and (14) and the knowledge of f for each probe. For the present technique, we follow others (e.g., Kuroiwa, 1967; Denoth et al., 1984) in assuming that the snow sample contained in the probe's measurement volume is sufficiently isotropic and homogeneous to possess an effective dielectric constant satisfying Eq. (10). Then it remains to evaluate this constant through calibration.

3. Calibrations At radio- and microwave frequencies, the real part of the effective dielectric constant of snow approaches the optical dielectric constant e~ which, to

M.Y. Louge et al. / Cold Regions Science and Technology" 25 (1997) 47-63

a good approximation, grows linearly with snow density and quadratically with volumetric liquidphase water content (Denoth et al., 1984). Because is nearly independent of temperature and snow type and texture, frequencies in the range 10 MHz to 1 GHz would permit robust capacitance measurements that only require the independent determination of liquid-phase water content. Unfortunately, it is difficult to implement a guarded circuit under these conditions. As Fig. 1 suggests, proper operation of the buffer requires that the capacitive impedance 1 / 2 rr.fjCsg between sensor and guard be large enough not to interfere with the buffer's operations. Frequencies above 10 MHz generally fail to satisfy this criterion with cables of reasonable lengths. Consequently, measurements with guarded capacitance probes are limited to lower frequencies, unless one forgoes the convenience of the coaxial cable and locates the processing electronics near the probe head. In the present study, we employ standard commercial electronics operating at 16 kHz, where snow exhibits a significant imaginary part of ee. Because at this frequency I%1 is considerably larger than e~, the capacitance measurement is more sensitive to the volume fraction of snow than at radio frequencies. The presence of liquid-phase water or impurities further raises the values of e", and to a lesser extent, e' (Kuroiwa, 1967).

However, because our present measuring frequency is close to a resonance of ice, Ee also depends upon snow texture and temperature, in addition to snow density, liquid-phase water content, and impurities. The role of temperature is underscored by the simple calculation that follows. Unless snow is wet or laden with impurities, it often behaves as a Debye dielectric near 16 kHz and thus its components of ee fall on a " C o l e - C o l e " circle in the complex plane (Kuroiwa, 1967), whence ~e = e ' - j e "

ier

Fig. 2. The "snow press."

= ~ +

1 +j2'rrfT

(18)

In this expression, e~ is the limit of e' at vanishing frequencies and r is the relaxation time of D subject to a step change in the electric field. As a relatively crude estimate, we adopt for ~- the value for pure ice, which depends upon the absolute temperature T through ~"(sec) ~ 5.3 X l O - ' 6 e x p [ O / r ]

(19)

where 0 ~ 6640 K is an activation temperature. Thus, a manipulation of Eq. (18) shows that a temperature excursion AT corresponds to a relative change in the modulus of Ee

Al~el J~el 1 +

tic der

51

2(2 f )2 T AT (20)

At 16 kHz, for typical dry snows, the value of #' is nearly maximum (Kuroiwa, 1967) i.e., the product (27rf~') is of order 1; further, ¢~ ~ 20 and ~ ~ 2. In this case, a 1 K temperature increase around T ~ 270K corresponds to a 4% increase in levi. Consequently, it is crucial to record the dielectric properties of a particular snow near the actual temperature of its avalanche. We evaluate these properties using the capacitance instrument sketched in Fig. 2. This "snow press" includes a grounded piston traveling in a plastic cylinder of 102 mm diameter. The base features a circular sensor surface of 25 mm diameter surrounded by a guard plate of 197 mm diameter. Gaps in the range 12 to 76 mm between piston and

M. Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63

52

e-

4

2 1

0

0

0.1

0.2

0,3

0.4

p (g/cm 3)

Fig. 3. Density dependence of the dielectric properties of Montana snow at - 8.8°C. Solid lines are empirical least-squares fits of the form E ' = l + ( p / p ' o )L3 and e"=(p/p'[)) 16 with p'0=0.136 g / c m 3 and p{] = 0.217 g / c m 3,

base are recorded using a depth micrometer, Because these gaps are smaller than the lateral width of the guard surface, the electric field lines shed by the sensor are straight and parallel to the axis of the cylinder. For this capacitor, the characteristic length f is AJd, where d is the gap width and A~ is the surface area of the sensor. In a typical calibration test, the amplitude of the oscillator is adjusted to produce an output voltage V0 = 10 V with a gap d o = 51 mm. After removing the piston assembly, snow is gently introduced in the press to nearly the same depth. The piston is then slowly brought to touch the upper surface of the sample. It is progressively lowered to bring snow to denser compactions. At each new position of the piston, snow is allowed to equilibrate volumetrically for a few seconds before the output V and depth d are recorded. Because the press capacitance in air is inversely proportional to depth, the modulus of the dielectric constant is I~1

avo

= ----

do V

G-1

%-1 c -

3G

-

(22)

(23)

ep + 2 G

where ep is a complex dielectric constant to be fitted empirically and t, = P/Pitt is the solid volume fraction ~ of the grain assembly with Pice = 0.92 g / c m 3. As expected, Fig. 4 also indicates that both the real and imaginary parts of the dielectric constant increase with temperature. in experiments with flowing snow, the present

I 5

I

I

0.3

0.4

.7oC

4O

o

3-

_o

2

-~_

I

-

(21)

After the test, the sample is extracted and weighed in order to calculate its overall density at each position. Assuming homogeneous snow distribution in the press, the average density of the compressed sample is p = M/dAp

where M is the sample mass a n d Ap is the crosssectional area of the piston. In our calibration tests of various snow samples, we have often distinguished two broad categories of snow response. The first is exhibited by relatively light, dry, crystalline snows. A typical example is shown in Fig, 3, where we plot both components of the dielectric constant of relatively fresh snow collected on the Montana State University campus. For these snows, it is our experience that both real and imaginary parts of the dielectric constant increase with density. However, the increase of one or the other part may be small on occasion. Irregular faceted ice grains represent the second broad category of snow behavior that we have observed. As Fig. 4 illustrates, the density dependence of G for these ice grains is well represented by a complex generalization of the self-consistent model of B~Sttcher (1952),

Y I

0

0.1

0.2

0.5

p (g/cm 3)

Fig. 4. Dielectric properties of ice grains of 1 mm mean size observed in Davos. The solid lines are least-squares fits to Eq. (23) using ep = 18.1 - 8 . 9 j at - 9 ° C and ep -- 2 3 . 0 - 2 5 . 4 j at - 7oc.

53

M.Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63 6

I

I

I

I

0.1

0.2

0.3

0.4

4 I£el 3

,i 1

0

0.5

p (g/cm 3)

Fig. 5. Modulus of the effective dielectric constant versus density for a variety of crystalline snows. The circles, squares and diamonds represent, respectively, fresh snow, old snow, and "graupel" with fresh snow collected in Alta at -6.5°C; the downward triangles are fresh snow from Bridger Bowl at - 11°C; the upward triangles are Cornell snow at -6.9°C; the crosses are other, likely purer, Cornell snows at -4.2°C.

amplifier design makes it inconvenient to record the time-history o f the phase shift between oscillator and guard, unless both voltages are continuously acquired throughout the event at a rate at least twice the oscillator frequency. In contrast, because they involve the acquisition of a single voltage o f relatively small bandwidth, measurements of the modulus IEe ] are straightforward. Fortunately, since our objective is to record the evolution of snow density with time, the monotonic increase o f l eel with p makes it superfluous to resolve both components of Ee. Therefore, in most practical situations, it suffices to calibrate the dependence of l eel on snow density. Fig. 5 shows the dielectric modulus for a variety of relatively dry, crystalline snows collected in the community of Alta near Salt Lake City, Utah, as well as in Bozeman, Montana, and on the Cornell campus. For these snows, the variations of [eel with density are nearly linear. Our observations are that the modulus o f e e is generally larger for Cornell snows than for Utah or Montana snows. The greater values betray the presence of impurities that may result, for example, from the systematic salting of roads in the Northeastern US, or perhaps from the passage of our storm tracks above the Ohio River industrial basin.

In other calibrations of Jeel, we dug deep into the Alta snow pack to extract "temperature-gradient" (TG) snow formed of irregular faceted ice grains of approximately 2 mm diameter. Cylindrical samples were cut using a metal cylinder and introduced directly in the snow press. Particular care was taken not to upset their structural integrity while doing so. Fig. 6 shows the resulting values of lee] along with other results from Davos calibrations. The squares and triangles represent samples extracted along the vertical and horizontal directions of the Alta snow pack, respectively. The observed density ranges revealed different mechanical behaviors of the two samples. In particular, the " h o r i z o n t a l " sample yielded immediately under compression to produce a relatively compact aggregate with p = 0.32 g / c m 3, while the " v e r t i c a l " sample produced stable compacts below this density. Despite the anisotropic structure that these observations suggest, both samples exhibited a similar dependence of l eel on snow density. As Fig. 6 illustrates, this dependence was well represented by a simpler version of the B~ttcher model,

3]eel

~

I @ + 2 (-eeI

(24)

which we have adopted to infer density from direct measurements of leel in an avalanche of faceted ice grains.

8

m

6

f~e I 4 2

0

0.1

0.2

0.3

0.4

9 (g/cm3)

Fig. 6. Modulus of the effective dielectric constant versus density for faceted ice grains. The solid lines are best least-squares fits through the data based upon Eq. (24). The squares and triangles are, respectively, the "vertical" and "horizontal" Alta TG snow samples at -6.5°C. The diamonds are Davos ice grains at -9°C.

M. E Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63

54

The snow press thus permits accurate determination of both components of the effective dielectric constant. However, a limitation of the calibration is that it cannot capture densities below those of loose snow packing. In the dilute suspensions expected around powdered snow avalanches, for example, we must therefore infer density by interpolating le~l between air and the loosest available packing. For snows like those shown in Figs. 3 and 5, a fit of the form IE~i= 1 + ( p / p o ) ~ is reasonable. However, for ice grains, an interpolation based on the Bt~ttcher model may be invoked (Fig. 6). We also employ the variable capacitance produced by the press to evaluate the processing electronics in the absence of snow. For example, a record of the phase lag between vr and eg at different values of d provides information on the reference impedance Z r. In air where the impedance is purely capacitive, the phase is w for the Capacitec amplifier. In contrast, for the MTI electronics, it is given by cot~

n

d

m

m(1 + H)2~rf~roa ~

i + j2 ~ f R C

(26)

In this case, the phase lag between ~'r and vg is

Itan~h[

1

2 ~rfRC

0

I

I ~

1

2

3

4

5

d (mm)

Fig. 7. Phase recorded by the Capacitec amplifier while connected to the snow press in parallel with a known resistor R = 5.56 MI). The abscissa is the distance between ground and sensor. The circles are the measurements, while the line is the slope predicted by Eq. (27). Agreement is better than 3%.

where the capacitance in air is C = o-0 A J d . Thus, Itan~bl is proportional to d, with a slope of (2~fRo-0A~) -L As Fig. 7 shows, the Capacitec amplifier captures the expected slope quite accurately.

4. Amplifier stability

()J), R

0

I

(25)

For example, a linear regression of cot~b versus depth yields m(l + H ) = 139 M I ) and n(l + H ) 164 M 1] for the MTI amplifier. Another use of the snow press is to demonstrate the accuracy of our technique. While capacitance probes are known to provide accurate dielectric constant magnitudes for many different materials (e.g., Louge and Opie, 1990), exploiting the phase to resolve both components of complex dielectric constants requires an independent verification. A convenient way to do so is to record the phase lag produced by the variable capacitance C of the snow press connected in parallel with a known resistor R to ground. This method simulates the behavior of a dielectric with first-order impedance resembling Eq.

Z-

2

(27)

A common difficulty with this technique is the ability of the control system of Fig. 1 to supply the sensor surface with stable currents of constant amplitude. In this section, we identify the origin of amplifier instabilities and prescribe a way to avoid them. The control system may become unstable for three principal reasons. The first arises when capacitances are too small. In this case, the impedance Z may become so large as to require a voltage v~ beyond the saturation limit of the high-gain amplifier (Eq. (l)). The second arises when the impedance is too small. Here, the guard voltage falls below detectable limits. Generally, such instabilities are avoided through careful design. For example, our experience is that the MTI amplifier requires capacitances in air in the range 50 _< Co ~ 350 fF, while the Capacitec electronics are somewhat more forgiving. The third source of instabilities is specific to the dielectric behavior of snow, and thus it merits further discussion here. This behavior manifests itself as sudden jumps of the output voltage V to incoherent values (e.g., V < 0 or V > Vo) that cannot correspond

M.Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63

to natural snow density signals. Their onset is preceded by a gradual increase of the guard frequency. In this section, we explain the origin of these instabilities by analyzing the electronics in Fig. 1 in the context of the Capacitec amplifier. The Capacitec system may be regarded as a control loop with oscillator voltage L,r as input and current i as output. The corresponding closed-loop transfer function is i --

H =

(28)

Z+Zr(I + 6H)

For the Capacitec amplifier, 6 - - 1 and Z r = 1/sC~, where s is the complex Laplace frequency. The

impedance produced by a snow sample is derived from Eq. (11), 1

Z =

I

(29)

sfo" 0( e' - j e " )

while for a typical snow the dependence of e" and e' on the real frequency f = s / 2 7 r j may be plotted as sketched in Fig. 8. Thus, the denominator of Eq. (28) is a complicated function of s that depends upon the dielectric characteristics of the snow sample. A necessary condition for stability of the control system is that this denominator should have complex zeros with a negative real part. To evaluate these zeros, we expand the complex dielectric constant to first order about the amplifier frequency f0, Et

12,

55

t

= eo + ( f - L )

~

+ o(f-fo)

z

(30a)

I

e " = eO ' + ( f - f O ) - ~ f + O ( f - f O )

2

(30b)

8

4

I I 10,000 100,000 1,000,000 1 (Hz)

o 1 ,ooo

W e define a ' -= Oe'/Of and a " - Oe"/Of and note that a ' is always negative, while a " may be of either sign. W e also define the intercepts E~ - e~ a'fo and E~ - e~ - a"fo and note that E~ is always positive. From these definitions, we calculate the denominator Z -~- Zr( 1 -~- ~ a )

10 '

'

d (b)

1

~" ~"<0

I(X">OI 0 0~.'0 I

cC'<0

o" ' ""J ,

l+6H

\

t

+ - -

t

o

,~

0

I

I

4

8

I I,

12

(31)

Crs

I I

16

E'

Fig. 8. Dielectric properties of a snow sample of density p = 0.45 g/cm 3 at a temperature of -3°C (after Kuroiwa, 1967). (a) The abscissa is frequency in Hz. The circles and triangles are the real and imaginary parts of the dielectric constant, respectively. The thick lines are best visual fits through the data. The solid symbols locate the Capacitec amplifier frequency. (b) Imaginary versus real parts of the dielectric constant. The semi-circle is the best fit "Cole-Cole" circle assuming e0 = 13 and e~ = 2. a" is positive in the region between the two vertical lines.

The real part of its zeros have the sign of the quantity n-

e;a" - ~;a' +

a" C r

(1 +

I4)/(r 0

(32)

Because of inequality (16), and unless a" is unrealistically large, the last term in Eq. (32) is negligible and the stability condition reduces to n=E~a"-~a'

<0

(33)

M.Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47 63

56

For most snows, two cases arise depending on the sign of a". When it is positive, c~" is often small enough that 0
4

Stable [ ~ amplifier operations

0 .I

I

0

0.1

(34)

The introduction of a parallel capacitor is therefore equivalent to increasing the real part of the dielectric

I

I

I

constant by an amount / J / . stability criterion becomes

ul

0

0

Stable I I 0.1 0.2

I 0.3

- ~' ( e0"- ~".ti,) < 0.

(35)

+ (/+/,)

(36)

As this expression shows, the introduction of a parallel capacitance makes it necessary to know the density dependence of both e' and e" in order to infer p from Vo/V.

=~o0.5 ii

The corresponding

Thus, when a " < 0, one can always increase /'~ until the amplifier recovers stable operations (Fig. 10). With this method, the voltage ratio in Eq. (13) now becomes

v -

I 70kHz

0.2

Fig. 10. Dielectric properties of fresh Cornell snow at - 6 . 9 ° C versus density. The solid symbols represent conditions where the Capacitec amplifier was originally unstable with U = 33.2 ram, but could be re-stabilized by connecting a capacitance o f / ] = 22.8 mm in parallel. Dashed lines are least-squares fits of the fi3rm le~l= 1 + ( p ~ po )1 I and e" = ( p ~ p'('l)12 with Pl) = 0.058 g / e r a ~ and P'~'I= 0.068 g / c m 3.

(p) 1,0

,"

""~"

P (g/cm 3)

~"( e0' - ~'f0 + / , / / ) Z = f 2 w f 0 o . 0 [ e,, +j(e' + / J / ) ]

I I

=

I 0.4

0.5

p (g/em a)

Fig. 9. Calculated density dependence of El') / E ' o for the snow sample of Fig. 8 at 70 kHz. To construct this figure, we assume that the density dependence of e' and e" is captured by the complex B~ttcher model of Eq. (23) and that c~' and a " are independent of density. Because it is approximate, this figure is only meant as an illustration of stability limits.

5. Probe geometries This section describes three probe geometries which we tested in snow avalanches. The first is the non-invasive "wall probe" sketched in Fig. 11. It belongs to a family of probe geometries with peripheral ground surface and central guard (Louge, 1995). It was inserted flush with the base of the Davos

M.Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63

Guard~sensor

14

57

b

~J

Gr( Flow -~,- - -

Fig. l 1. Top view of the " w a l l " probe face. The flow passes over the probe in the direction shown. Its outer diameter is 50 mm; its characteristic length is f = 22 m m and its measurement volume extends approximately 6.7 mm from the base of the chute.

chute. To a good approximation, its measurement volume may be regarded as the region swept by electric field lines emanating from the edges of the sensor surface. The relatively large height of this region contained a sufficient number of the coarse 1 mm ice grains available in Davos. Louge et al. (1996) discuss this configuration in detail and prescribe optimum dimensions for its manufacture. For the tests in Bridger Bowl, we designed a "strip probe" with a smaller measurement volume better suited to the dense avalanches of small grains generally available there (Fig. 12). Another objective of this probe was to calibrate the optical density sensors of Dent et al. (1994). The probe was mounted flush with a vertical surface naturally grazed by the avalanche. Its principle was similar to the "wall probe" in that its sensors shed electric field lines

"~

20

~ ~

20

10 :- ~

Optical sensor

' Flow 5O I

Guard

Sensor

/

I ~ow

base

Fig. 12. Side view of the " s t r i p " probe face. Fhe flow passes over the probe in the direction shown. Its characteristic length is f = 5.5 mm and its measurement volume extends approximately 2.7 m m from the wall of the chute.

sensorguard Front guard Backsensor Fig. 13. Active electrical surfaces of the " s n o w w a n d " . For clarity, internal connections of the respective coaxial cables are not shown. For this probe, a = 21.2, b = 17.3 and c = 10.4 mm.

outward to the ground surface. Electrical insulation was achieved by anodizing the aluminum guard surfaces. Because of its small thickness, the anodized guard layer sandwiched between brass sensor and aluminum ground acted as an additional dielectric between these conductors. At - 1 5 ° C , this layer effectively added to each probe a parallel capacitance of characteristic length f t ~ 49 mm, which ensured stability of the amplifier. Because the dielectric permittivity of aluminum oxide grows with temperature, a drawback was that the value of f t for the anodized layer depended upon ambient temperature. In Davos, we tested the "snow wand", a probe suitable for simultaneous measurements of snow density and velocity (Fig. 13). Its geometry is an axisymmetric extension of the twin capacitance probes described by Louge et al. (1996). It is meant to protrude from a test structure erected in an avalanche corridor. To minimize flow disruption, the cylindrical probe head of 12.7 mm diameter features a bluff extremity, which is pointed toward the direction of mean flow. Unlike the generic one-channel probe sketched in Fig. 1, this capacitance instrument includes two sensor surfaces, each of which is surrounded by its own guard. The " t w i n " guard/sensor systems are driven by two identical circuits sharing the same voltage oscillator, thus having synchronous sinusoidal currents of constant and equal amplitudes. In addition to recording the local snow density as before, each sensor can now detect separately the passage of a disturbance in the local dielectric constant caused by a density fluctuation.

M, Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47 63

58

If such a disturbance travels at a finite speed above the probe, the normalized cross-correlation integral 4h*2(~-d) of the signals Vl(t) and Vz(t) from the two output channels peaks at a value % that is a measure of the mean time delay between the two sensors. An estimate of the integral is

have toroidal shapes (Fig. 13). Their extent is of order the radius R' of the outermost field lines, a 2 _ C2

R'

The characteristic length / either sensor to ground is /=2Rln

1 (37) where s~ and s 2 are the standard deviations of the two signals, V1 and V2 are their averages, and Td is the duration of signal acquisition. The normalization of integral (37) is appropriate for all signals. In addition, unlike conventional cross-correlation, this equation also produces a sharper peak in response to density steps likely to arise at the leading edge of an avalanche. Then, a measure of the mean velocity of the particles is

(40)

2c

-

of the capacitances of

- X - -

a+c

(41)

a-b

where the dimensions a, b and c are shown in Fig. 13 and R = 6.4 mm is the wand radius. With these values, R ' = 16.4 mm and / = 15.6 ram. In the presence of an inhomogeneous suspension, the twin guard/sensor systems do not generally exhibit the same voltage and, consequently, the electric field is not symmetrical about the mid plane separating them. Louge et al. (1996) analyze the resulting effects on the instrument. Their calculations prescribe probe dimensions that ensure confinement of the measurement volume as well as insensitivity to machining tolerances or deterioration of the probe

(38)

surface. They recommend c / a = ~"~- - 2 and ( b / a ) ~ 0.82, values which we have adopted here.

where L is the separation between the two sensors. In this case, unlike the measurement of snow density, cross-correlation velocimetry does not require an absolute calibration. However, in the presence of density waves, cross-correlation may yield wave speed rather than actual snow velocity. Because the resolution in the measurement of % is inversely proportional to the sampling frequency fro, the relative uncertainty in v m is

Our prototype of the snow wand has aluminum guard, ground and sensor surfaces arranged in adjacent layers. Anodization ensures adequate resistive insulation of order 1 Mf~ among these. A grounded axial stainless steel " s p i n e " holds the assembly together and provides structural rigidity. Through the spine, the probe head is attached to a longer stainless steel tube of 21 mm diameter that may be fastened to the avalanche pylon. Anodized holes are drilled in each metal layer to guide the two coaxial cables and the spine. Particular care is paid to avoid capacitive coupling between the back sensor piece and the coaxial cable carrying the guard voltage of the front sensor. Failure to do so would disturb considerably the circuit of the back sensor. Similarly, the grounded spine is surrounded everywhere by guard surfaces to avoid capacitive coupling with either sensors.

= L/

,n

{)In

AUra

- -

Cm

= +

-

(39)

-- 2.fm L

where f,n cannot exceed the maximum output frequency of the processing electronics. With L = b + c ~-, 27.7 mm and fm = 3.5 kHz, the present design can detect velocities as large as 19 m / s e c with an uncertainty no greater than +_ 10%. If greater velocities are expected, other probes may be designed with greater L. If the twin probes are covered by a homogeneous dielectric medium, then both sensors exhibit the same voltage. In this case, the measurement volumes

6. S t a t i c c r o s s - c o r r e l a t i o n

tests

For convenience and reproducibility, we first evaluated the performance of the snow wand in tests

M.Y. Louge et a L / Cold Regions Science and Technology 25 (1997) 47-63 I

0

I

- 50

0

j ;

59

1.3

I

l

1.2-

1.1 1.0

50

I

°, I

I i

0.8 -50

1 O0

0

Fig. 14. Immersion of the snow wand in a plastic powder of effective dielectric constant ee = 2.7. The abscissa is the location of the probe tip with origin at the bed surface and positive direction along the downward vertical. The solid line is the voltage ratio V0 / V of the sensor nearest to the tip; the dashed line is the ratio of the other sensor. The vertical dashed lines indicate locations of their respective centers.

where the probe was exposed to surrogate dielectric materials in various stationary configurations. Fig. 14 shows the response of the two sensor channels to the progressive immersion of the wand into a stationary bed of plastic powder. This test simulates, for example, the onslaught of a snow avalanche onto the probe. In this experiment, the free surface of the bed is horizontal and the probe is vertical. As this figure indicates, rather than exhibiting a monotonic growth of the apparent dielectric constant with probe depth, the signals from the two sensors show some degree of correlation. This effect is due to the voltage imbalance that accompanies the exposure of the two sensors to different media. Louge et al. (1996) discuss how the imbalance produces an electric field that influences both sensor simultaneously. It is greatest when the two sensors are immersed in media of widely different densities. In this light, the test in Fig. 14 may be interpreted as a worst case scenario. In another stationary test, the wand traversed a 1.6 mm thick plate of PVC plastic through a hole of 14.3 mm diameter. The plate was held perpendicular to the wand's axis while translating along the probe. The resulting signals from both sensors are shown in Fig. 15. We distinguish sharp peaks as the plate passes directly above each sensor. Because the signals are symmetrical about the boundary between the

50

100

x (ram)

x (ram)

Fig. 15. Translation of the wand through a plastic plate. For lines and symbols, see Fig. 14.

two guard/sensor systems, the geometrical differences between the ground surface at the tip and its counterpart near the post appear inconsequential. However, as before, the passage over one sensor induces a small spurious signal from the other sensor. The spatial cross-correlations of the static test signals are shown in Fig. 16. By analogy with Eq. (37), it is defined as

4,?.(Ax)

,

X

s=

s~

..).x (42)

where X is the span of the displacement along x. As I

1,0

,i,,o. -''°'" 0.5

.~ 0.0

-0.5

-1.0

-25

0

25

50

75

(ram) Fig. 16. Spatial cross-correlation (42) of the signals from Fig. 14 (dashed line) and from Fig. 15 (solid line).

M.Y. Louge et al. / Cold Regions Science and Technolog3 25 (1997) 47 63

60 0.5

I

I

I

400

600

0.4

g"

0.3

o.2 0.1

0

200

time (msec)

Fig. 17. Typical density time-history recorded by the " w a l l " probe with a snow temperature of - 9 ° C . Densities were calcu lated from Eq. (24) using levi = 19.9 from a calibration with snow deposited after the event.

expected, the peaks in this figure correspond to the distance between the two sensors. Note that the " r a m p " signals of Fig. 14 lead to a wider correlation function. In contrast, the " p u l s e " signals of Fig. 15 produce a smaller secondary peak at zero separation, which arises from the interference between the two sensors.

7. Artificial avalanches The "wall probe" and " s n o w wand" were tested at the Weissfluhjoch facility of the Swiss Federal Institute for Snow and Avalanche Research in Davos. There, at temperatures ranging from - 9°C to - 3°C, artificial avalanches of faceted grains of lmm mean size were produced in a chute 20 m long, 2.5 m wide, 1 m high and inclined at a 35 ° angle from the horizontal. Fig. 17 illustrates signals obtained with the " w a l l " probe within 6.7 mm of the base of the chute. The smaller densities recorded by the probe are consistent with the presence of a more dilute, agitated layer of grains at the bottom of the flow. Because of relatively low densities, few instabilities were observed in the signal. Fig. 18 is the record of another avalanche passing over the snow wand. For this experiment, the wand was held near the chute axis parallel to the base at a distance of 105 ram. Its tip protruded 88 mm from a

relatively massive aluminum holder. Because ambient air was relatively warm, the snow dumped in the chute reached a temperature of - 3 ° C , while in the pack it was as low as - 9°C. Perhaps because of the relatively high temperatures and the occasional melting that resulted, the signal exhibited amplifier instabilities, which we removed from Fig. 18 for clarity. Unfortunately, because at that time we had not yet conducted the analysis of Section 4, we did not consider stabilizing the signals by connecting an additional capacitor in parallel with the wand. Fig. 19 shows signals near the onslaught of the avalanche. Fig. 20 illustrates later details, where the density was smaller and, as expected from the analysis in Section 4, the amplifier was more stable. Shortly after the front, snow density was always higher over the downstream probe. This phenomenon probably resulted from the intrusion of the probe's holding piece, which was located too close to the tip. It is likely that, as the flow passed over this obstruction, it slowed down and, by virtue of mass and momentum conservation, its density increased. A shortcoming of our current calibration technique is evident from an inspection of Figs. 4 and 18. At present, the snow press can only provide limited isostatic compression of the snow sample, while avalanches produce shearing leading to greater compaction. Our suggestion for a better snow press is to allow a rougher piston to rotate and impart tangential stresses to the sample while compressing it.

0.6

~-E

0.4

I

, 0

e_

~ 0.2 o

0.0

o

o

0

o

_

~.....,~,==.=m~ 500

1000

time (msec)

Fig. 18. Density time-history recorded by the upstream sensor of the "'snow w a n d " with a snow temperature of - 3 ° C . Densities were calculated from Eq. (24) using le~,l = 32.6.

M. Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63 0.6

(a)

I

0.5

I

T

q bottom sensor

o ~,

-

o _

~" 0.2

• •

o

•

E ~ 0.3

•

I ~° -5

0

15

10

,. 1 ._o

.

0

-

10

20

200

O.2

~ 0.1 400101~ 2000 3000

Fig. 21. Density time-histories at two elevations above the base of the Revolving Door avalanche at - 1 5 ° C . Triggering failed to capture the avalanche front occurring approximately 35 msec before signal acquisition. Progressive expansion of the time axis underlines signal features.

I

-

0,1 ~ 0

~

time (msec)

time (msec)

I

0.3

0.2

I

5

(b)

0.5

"

o

0.0

~

O'4

o~O~, °~,oo

0.4

6!

~

-

m 0

~

0.5

0.0

--

-5

0

5

10

time (msec) Fig. 19. Detail of the event of Fig. 18 near the front of the artificial avalanche. (a) The open and filled circles represent the w a n d ' s upstream and downstream sensors, respectively. (b) The corresponding cross-correlation using Eq. (37) peaks at r,, = 2.29 + 0 . 1 4 msec, yielding v m = 12.1 + 0 . 7 m / s e e .

0.15 0.10

"~ o.o5 I 40

20

60

time (msec)

_

0.8

~" 0.6 =

0.4

e

o

0.2

-5

0 5 time (msec)

10

Fig. 20. (a) Detail near t = 150 msec in Fig. 18. The top trace is from the downstream sensor. (b) ~'m = 3.71 + 0 . 1 4 msec i.e., L,m = 7.4_+0.3 m / s e e .

The signals featured sufficient fluctuations to permit evaluation of velocities from cross-correlation during most of the event. Overall, we found that velocity decreased from 12.1 + 0.7 m / s e e at the front (Fig. 19) down to 7.4 + 0.3 m / s e e at 150 msec (Fig. 20), and 6.0 + 0.2 m / s e e at 450 msec after the front. For the occasional snow trailing behind the bulk of the avalanche, we measured 4.8 _+ 0.1 m / s e e at 750 msec. Independent visual measurements produced an average velocity of 7.5 m / s e e . This progressive flow deceleration behind the front is typical of such avalanches. The "strip probe" of Fig. 12 was tested in a dense snow avalanche of approximately 1.5 m depth triggered at the Revolving Door site near Bridger Bowl, Montana. The long axes of the two sensors were parallel to the slope; the bottom and top sensors were respectively located at distances of approximately 10 mm and 60 mm above the undisrupted snow cover. As Fig. 21 shows, the resulting density time-histories exhibited flow-induced fluctuations that disappeared as soon as snow was deposited in front of the sensors. As expected from a gradual snow deposition starting at the base, the fluctuations recorded by the bottom sensor disappeared sooner. The signals also revealed progressive snow compaction as the avalanche continued to flow high above the sensors, Although they both registered similar densities while snow was flowing, the bottom sensor recorded greater compaction of deposited snow than the top. This observation was consistent

62

M.Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63

with independent density measurements of 0.4 g / c m 3 and 0.25 g / c m 3 from samples dug alter the event around the bottom and top sensors, respectively.

compaction) will be exploited to design better probes and procedures in future.

Acknowledgements 8. Conclusions In this paper, we have described guarded capacitance instruments that yield the density and velocity of snow from measurements of its effective dielectric constant. This new method has advantages over conventional capacitance techniques. The first is its ability to capture rapid and wide density changes. The second is the virtual elimination of problems associated with cable capacitance, which permits placement of sensitive processing electronics away from the probe. Another advantage is the availability of low-cost, dedicated commercial electronics. While careful probe sizing is necessary to confine measurement volume and minimize stray capacitances, the guarding principle allows many possible geometries, which can be optimized for specific apparatus and flow conditions. Using an independent test involving a known resistor and a variable guarded capacitor, we have shown that the processing electronics yields both components of a complex dielectric constant with accuracy. At the 16 kHz frequency of commercial oscillators, we have found that the dielectric constant of snows with moderate moisture correlates well with density. However, because dielectric properties also depend upon temperature, liquid-phase water content, impurities and texture, accurate measurements in the field require calibration. To this end, we have presented a method that can produce rapid assays of avalanche debris. We have also observed that the frequency dependence of the complex dielectric constant of snow can produce instabilities of the processing electronics. In our experience with commercial 16 kHz amplifiers, these instabilities are avoided by connecting another guarded capacitance probe in parallel. Finally, we have shown density time-histories and velocities obtained from artificial avalanches of ice grains in Davos and from natural avalanches triggered in Bridget Bowl. The lessons learned and phenomena observed during these trials (e.g., probe intrusion, sensor separation, electronic stability, snow

The lead author is indebted to the entire staff of the Federal Institute for Snow and Avalanche Research for their advice and support during the Davos test campaign. He is particularly grateful to Christian Camponovo, Dieter Issler, Martin Hiller and Gunter Klausegger for their active participation, hospitality and assistance during the experiments. He is equally grateful to Walter Ammann for suggesting the avalanche tests and securing the generous financial support of the Swiss Federal Government. We are also indebted to Edward E. Adams, D. Scott Schmidt and K. Jay Burrell for assisting in the experiments at Bridger Bowl and for helping the lead author endure extreme climatic and skiing conditions. We also wish to thank Daniel A. Howlett, Thor Femenias, the Center for Snow Science at Alta and the Alta Ski Lifts Company for their assistance and hospitality during the 1994 field tests of the snow press in Utah. Finally, we are grateful to Robert E. Davis and Hansueli Gubler for helpful discussions and advice.

References Abed, R., 1984. The characterization of turbulent fluid bed hydrodynamics. In: D. Kunii and E. Toei (Editors), Fluidization. Engineering Foundation, New York, pp. 137-144. Acree Riley, C. and Louge, M.Y., 1989. Quantitative capacitive measurements of voidage in dense gas-solid flows. Partic. Sci. Tech.. 7: 51-59. BiSttcher, C.J.F.. 1952. Theory of Electric Polarization, Elsevier, New York. Chandran, R. and Chert, J.C., 1982, Bed-surface contact dynamics for horizontal tubes in fluidized beds. AIChE J., 28(6): 907914. Denoth, A., Foglar, A., Weiland, P., M~itzler, C., Aebischer, H., Tiuri, M. and Sihvola, A., 1984. A comparative study of instruments for measuring the liquid water content of snow, J. Appl. Phys., 56(7): 2154-2160. Dent, J.D., Adams, E.E., Schmidt, D.S., Jazbutis, T.G. and Bailey, 1.J., 1994. Measurements of vertical velocity profiles in a snow avalanche. In: Twelfth U.S. National Congress on Applied Mechanics, Seattle, WA, paper ICB6 5.

M.Y. Louge et al. / Cold Regions Science and Technology 25 (1997) 47-63 Dorman, R.A., 1979. Signal amplifier system for controlled carrier signal measuring sensor/transducer of the variable impedance type. US Patent 4,176,555. Geldart, D. and Kelsey, J.R., 1972. The use of capacitance probes in gas fluidized beds. Powder Tech, 6: 45-60. Gubler, H. and Hiller, M., 1984. The use of microwave FMCW radar in snow and avalanche research. Cold Reg. Sci. Technol., 9(2): 109-119. Gubler, H., 1987. Measurements and modeling of snow avalanche speeds. In: B. Salm and H. Gubler (Editors), Avalanche Formation, Movement and Effect. IAHS Publ. 162, pp. 4 0 5 420. Hungr, O. and Morgenstern, N.R., 1984. Experiments on the flow behavior of granular materials at high velocity in an open channel flow. G6otechnique, 34: 405-413. Hutter, K., Koch, T., Pliiss, C. and Savage, S.B., 1995. The dynamics of avalanches of granular materials from initiation to runout. Part II. Experiments. Acta Mech., 109: 127-165. Kuroiwa, D., 1967. Snow as a material. Chapter J. Electrical properties of snow. In: Cold Regions Science and Engineering, Part II, Section B: Physical Sciences. US Army Materiel Command, Cold Regions Research and Engineering Laboratory, Hanover, NH, pp. 63-79.

63

Lanneau, K.P., 1960 Gas-solids contacting in fluidized beds. Trans. Inst. Chem. Eng., 38: 125-143. Louge, M.Y., 1995. Guarded Capacitance Probes for Measuring Particle Concentration and Flow. US Patent 5,459,406. Louge, M. and Opie, M., 1990. Measurements of the effective dielectric permittivity of suspensions. Powder Tech., 62: 8 5 94. Louge, M., Tuccio, M., Lander, E. and Connors, P., 1996. Capacitance measurements of the volume fraction and velocity of dielectric solids near a grounded wall. Rev. Sci. Instrum., 67: 1869-1877. Mellor, M., 1964. Properties of snow. Chapter VII. Electrical properties. In: F.J. Sanger (Editor), Cold Regions Science and Engineering, Part III, Section A: Snow Engineering. US Army Materiel Command, Cold Regions Research and Engineering Laboratory, Hanover, NH, pp. 86-105. Salm, B. and Gubler, H., 1985. Measurement and analysis of the motion of dense flow avalanches. Ann. Glaciol., 6: 26-34. Van de Hulst, J., 1957. Light Scattering by Small Particles. Dover, New York.