JOURNAL OF RESEARCH of the National Bureau of Standards- D. Radio Propagation

Vol. 64D, No. 4, July- August 1960

Measured Electrical Properties of Snow and Glacial Ice 1, 2

A . D. Watt and E. L. Maxwell

(J anuary 22, 1960)

The electrical proper tie;:; of s now and glacial ice near 0° C have been observed over the frequency range from 20 cycles per seco nd to 200 kil ocycles per second . In general , the condu ctiv ity of s now an d glacial icc is found to be much hig her than t hat fOI' purc ice. T his is particularly so at frequencies below 2 kilocycles per second.

The magnitude of t he complex con du ctivity for glacia l icc appears Lo in crease wit h temperature at frequ encies below 200 cycles per seco nd a nd Lo deCl'case with temperaLure a bove this frequency.

1. Introduction

The electrical properties (i.e., conductivity and permittiviLy) of snow and ice can have a n appreciable effeet upon the propagation of electromagnetic waves over such m edia. Similarly, the char acter­istics of ground-based anLe nnas may be severely modifi.ed [1,2).3 Although the proper Lies of pure icc as m easured in the laboratory are relatively well Imown, the phys ical characteristics and formation of snow and glacial ice are d ifferent, and thus it is important thaL their properties be measured under actual field condiLions. Considerable dispersion exists in the electrical properties of ice throughou t the audiofrequency range, and in order to permit measuremenLs of the elecLri cal properties of snow and glacial ice in their natural state , two independent methods have been employed over the frequeney range from 20 cps to 200 kc. These are described in the follo lving secLion and the resulting electrical proper ties presented.

2. Measurement Methods and Nomenclature

The bridge and substitu tion method shown in figm'e 1 permits observations to be made on the conductivity and permittivity (dieleetric constant) of a lossy dielectric. Numerous forms of electrodes have been employed including parallel plates and the parallel rods shown. The parallel rods have the great advantage of permitting their insertion into the medium by the simple expedient of drillin g~:two holes with an ice auger. This method produees a minimum amount of disturbance in the medium, and when the experiment is carefully condu cted the effects of fringe capacity etc. , are greatly minimized. This method suffers from one primary deficiency in that the eleetrodes must fit very closely in the ice, and there is always the question of possible errors in

I Contribut ion from Central R.adio Propaga tion Laboratory, National Bureau of Standards, Boulder, Colo.

' The studi es conta iJl e<l in this paper werc sponsored by the Omce of l aval Research und er Contract No. NR 371- 291.

3 Figures io brackets indicate the literature rcfcrences at tbe end of this paper.

Signal Generator

GR Bridge Type 1650A

VTYM

Precision Capacitor

Pr e cision Resistance

~~ ;O, " ~~ / snow

Electrodes

F r OU RI" 1. Bridge and substitution method for measming conducl?:vity and dielectric constant.

Lhe deLerminaLion of conduct ivit~T and permitLivity due to in terface 01' surface-layer effecLs on the electrodes themselves.

The other method employs two pairs of electrodes. It is described generally by Wait [3] and has been used in measuring ground conductiviLy in the audio­frequency region. The use of four electrodes permits a determination of the magnitude of the complex conductivity 10:1 to be made in such a manner that surface-layer effects [4] at the electrodes are elimi­nated. At least three basic arrangements of elec­trodes were employed during thcse measurements including the yVenner , Eltran, and right-angle arrays. The Eltran arrangement shown in figure 2 proved to be the convenient form to use since it permits a ready changing of electrode spacings, and accurate re ults can be obtained over a relatively wide range of electrode spacings provided that the precaution of elevating the wire several feet above the surface of the snow or ice is taken . If thc wires connecting the electrodes were not elevated, the large dielectric constant of the snow and the rather high terminal resistance resulted in a nonuniform distribution of Cllrren t along the wire. This, of course, invalidates the relations employed in calculating 10:1 based upon a model where the current and potential electrodes

357

are discrete points. At very short spacings, the results with all three methods were similar.

It is important that the vacuum-tube voltmeter across the potential electrodes have a very high input impedance and that the voltmeter stake im­pedances be made as low as possible by pouring salt or salt water around electrodes placed in the ice so that the voltage indicated is equal to the true potential difference between the potential electrodes. These precautions are necessary since the stake im­pedances are effectively in series with the po ten tial source whose voltage is desired. In some cases, it may be difficult to reduce the stake impedances below 0.5 meg and correc tions of the voltmeter readings may be necessary [4).

Signal Generator VTV M VTVM

-+ q 'e ~--tt---O ~ a I a I a ~

FIGURE 2.

I ~

mhos /meter

F'o w' electrode [ELTRAN] army f01' measuring the magnitude of the com plex conductivity.

The signal generator should be capable of pro­ducing relatively high vol tages so that sufficient current can be induced into the current elec trodes. This makes the vol tage produced at the poten tial elec trodes appreciably greater than the background noise caused by atmospheric noise and earth-current fields. Signal generators with outputs of 1 v have been employed for spacings up to 20 ft while beyond this it has been found necessary to employ an addi­tional power amplifier with a capability of supplying 300 v into 6,000 ohms. This arrangement has per­mitted electrode spacings of up to 1,000 ft provided that some filtering is employed at the vacuum-tube voltmeLer to eliminate the fields produced by VLF transmitting stations.

The four-electrode method using the instrumenta­tion shown eliminates the effects of surface phenom­ena at the electrodes but does not give the magnitude of the conductivity and permittivity independently .4 The magnitude of the complex conductivity em­ployed by this means can, however, be compared with that obtained by the bridge substitution method as a check to see whether or not surface phenomena at the electrodes in the bridge measurements was great enough to cause an error in the observations.

The physical characteristics of the electrodes, used in the bridge method, and the relation between the equivalent circuit and the basic proper ties of the dielectric material are shown in figure 3. The two basic physical properties to be measured are the conductivi ty (J" expressed in (mhos/meter), and the permittivity (dielectric constant), E, expressed in

• This call be accJ:npli sbed if the relative phase bat.veen current I and voltage V is measured [4J.

A/I dimensions in meters C

r~'A./'~'\,-1 r~.Vv'~1 ;' / I I I 1 // ~' I~

~;~ ~ .~ /" ~ ;; , ~ tn/e~/ace V .......... LoSSy melectrlc mo/enol

LIST OF SYMBOLS

Conductiv ity (tnhos/meter)

Permittivity. i. e" dielectric constant (farads/me t er)

A* wh, plate area (rneters Z)

R d/ A* rr = T ' re sistance (ohms)

G l/R = (T A/d , conductance (mhos)

C E A/d, capacit ance (farads)

A Effective area ~ A* when d «w and h

x c

B

Y

- ..!. , reactance (ohms) w e

- ~ = w c , susceptance (mhos}

G + jB . admittance (mhos)

u AID +jw e AID

(T + jWE complex conductivit y" admittivit y" (mhos / m e t er)

oJ 2 Z magnitude of complex conductivit y II admittivity" (T + (w E) (mhos/meter)

El _ j e " complex dielectric constant

e' E/e = C/C real part of relative dielectric constant o 0

e" rr /w E = ~ ilnaginary part of relativ e dielectric constant o RwC

o 5.56 x 10-11 E" X f( cJ s) (mhos/m e ter)

F I C1.:RE 3. Lossy dielecl1'ic capacitor and equivalent circuit.

(farads/meter ). ·When the plates or rods are im­mersed in an effectively infinite medium, it is neces­sary to use the effective area to separation ratio A /d rather than the ratio of physical values A */d shown in the illustration .

It is eviden t tha t

1 (1) (J"=---

RA/d and that

C (2) E= A /d

where (J" is the conductivity in mhos/meter, R is the equivalent parallel resistance in ohms, A is the effective area in square meters, d is the effective separation in meters, E is the permittivity of the sub­stance in farads/meter, and C is the capacity in farads. Since the ratio A /el may influence the inter­face effect, the value of this ratio is given for each electrode arrangement employed. Although A /el can be calculated with fairly good accuracy, the value determined experimentally from the air capacity of the electrodes is used in all the calculations of this paper. Most of the nomenclature employed is now well standardized [4). The complex conductivity, u, may not be well known and the possibility of employing "admittivity" in future work is suggested. It should be emphasized that the quantity obtained by the four-electrode method is lui the magnitude of

358

the "admittivity" rather than the conductivity (J

which is the dominant term in most ground meaSU1'ements.

3 . Properties of Pure Ice

The electrical and physical properties of ice have been discussed in considerablc dctail by Dorsey [5]. H e presents tabulations of permittivity and con­ductivity as reported by numerous observers. A more recent paper by Auty and Cole [6] has analyzed the lack of close agreement between measm ed di­electric constants of ice and has found tha t in gen­eral they appear to be attributable to the formation of voids or cracks in the samplcs being mcasm ed.

For a material with a single relaxation frequency the simple theory of dielectric relaxation predicts a complex dielectric constant €

(3)

where the time factor is eiwl , f dc , and f a> arc the so­called equilibrium and limiting high-frequency valuC's of relative dielectric cons tan t, f is the frequency at which the electrical properties are being measured, and f e is defined as the relaxa tion frequency which is equal to l /T, where T is the relaxation time. The complex plane presen LaLion of € forms semicircles as shown in figure 4. This type of presentation is known as the Argand diagram . Auty and Cole [6]

80.----r~------------~------------_,

. 7.2 kC/S} RELAXATION

60

20

FIGURE 4. From Auty and Cole.

- 280c/s FREOUENCIES (f,) I 63 c/s ' C . , , , . . .

_ "T""7"-_ 200' TEMP °C - 56.8 0

-32 0

- 0./ 0

40 60 80 120

E ' Electrical pToperties oj pure ice.

140

with carefully designed equipment in experimental procedures found that pme ice does in fact adhere very closely to the form predicted by this simple theory, and they were able to obtain results with a high degree of repeatability. From their data we have obtained the relaxation frequency f c as a function of temperatme as shown in figure 5. The high-frequency dielectric constant f a> was found to be very elose to 3 for all temperatmes ranging from - 0.1 ° to - 65.8° C; and fde, also shown in figme 5, has values ranging from approximately 90 to 114 over the temperatme range indicated . The value of f corresponding to the relaxation frequency, f e ,

permits a rapid construction of Argand diagrams since this point represents the center of the circular diagram. It is obvious that the frequency at the

'" "-u

10000

100

500

0

0

lOOO

2000

1000

0 10

500

300

200

100

10

50

30

20

10 -70

-

V /

/ -60 -50

= I-

'/ -

/

V / =

/ -

/ -

/ /",(fc)RELAXATION FREOUENCY

ie.f WHEN Ion ¢ = /

I -""CdC WHEN f-O /

~ ., ¢ (cc)l.e.c WHEN Ion =1

-40 - 30 -20

TEMPERATURE,OC

FIGURE 5. P ure ice--Jrequency and Telative permi ttivity when tan q,= 1 versus temperature.

From Au ty and Colc.

top of the semicircular diagram will be the r elaxation frequency. Frequencies at other points on this d iagram can r eadily be obtained by the relationship

tan cf> fife (4)

where cf> is the angle indicated in figme 4. The rela­tive dielectric constant f' = f/ eo can be readily ob­tained from diagrams such as figme 4 and the results presented as a function of frequency for various temperatures as shown in figure 6.

\JJO

;;; 100 t~~~~=::::;~;;:j~-----f-"

:'i 80 1----- \c--"rt-'\ Wf­>Ul i=z <l: 0 601---'- -1,::---\-- -\; ...Ju

~~ g: 40 1------\+-- \ U W ...J W 101-----o

10 10 100,000

FREQUENCY, cis

FLO VRE 6. Relative dielectric constant of pW'e ice. Calculat ion based on Auty and Colo.

359

It is interesting to note that the equilibrium relative dielectric constant Edc increases with decrease in temperat1U'e. This same general increase in Edc

has been predicted theoretically by Powles [7]. The actual m agnitude, however, is either below or somewhat above the values obtained experimentally depending upon the type of theoretical model em­ployed in the calculations.

The conductivity of pure ice can be obtained from the Argand diagram of figure 4 by means of the relationship

(5)

which can also be written as

(6)

where f is th e frequency in cycles per second and u is the conductivity in mhos/meter. The results of these calculations, shown in figure 7, indicate that for a given temperature and at frequencies below the relaxatio~l frequency that the conductivity is d irectl~~ proportional to thc square of the frequ ency.

E 10 "-V> o L;

E :l...

b -

>­f--

I_Ie

:> f= u => o z 8 0.1 1----- --1'----/-,,-----, +-- 0.1 TEMPERATURE DC

I II 111111

100 1000 10000

FR EQUENCY, cis

F I GURE 7. Conductivity of pure ice. Calculations based on Au ty and Cole.

4. Experimental Results

4.1. Snow Drifts in Colorado

I I

100000

The properties of drifted sno w were mea"ured at two locations in the moun tains of Colorado. The Argand diagrams resulting are shown in figure 8 where it is obvious that the characteristics are dif­feren t from those of pure ice . Four electrode methods of measuring the absolute magnitude of the eomplex conductivi ty are also employed and a com­parison of results is shown in figure 9, wh ere it is seen tbat relatively good agreemen t exists between the two types of measurements. The type of

140 ,---r--~~-------'----'-"""T--'

50 K 80 K

, 200K

lOO K

110

100

60

ELEC TRODES, PARALLEL RODS, d/ A = 0.356 ALTITUDE ~ 9,000 FEET

.... - L To;r ::::: 12°C , Tsnow ~ aoc I j-: : 'l ' 1'2~0 I I I .

REO ROCK LAKE, COLORADO r----.,-- . ,1 6-25-59 DENSITy= o.6

FR~OU~4!C/+ '4100 I I' I I I

-T----4 'I I IK

,-, ~- ~ , 1-r ' - I 2 I I I ,IK

I I" I I ~TRA/L RIDGE ROAD, COLORADO

6 -12-59

+ 40 60 80 100 110

E'

F I G UHE 8. Eleclrical pTOpeTlies of spring dnf/ snow. Compact and wet.

E "-V> 0 L;

E :l... - -~

liil (ELTRAN ARRAY;

0

~ w~

10 10 100 IK lOOK

FREOUENCY, cis

F I GU HE 9, Elecll'ical properties of wet snow. T rail Ridge Road, Colorado, June 12, 1959,

140

departures from pure icc are ill general agreemen t with observations mad e by Kuroiwa [8] and D e­vaney [9]. It should be pointed out that the rather large values of E" which result from an increase in conductivity are probably due to the presence of impurities in the snow and possible interface (over­voltage) effects at the part icl es [4]. The effect of impurities was verified by Kuroiwa by means of chemical analysis of t he snow and by an experiment in which he produced hoarfrosL crystals from chem­ieaHy pure water and found that the Argaud diagram was a semicircle as observed wi th pure ice although the dielectric constant was, of course, much lower because of the deer'eased densi ty.

4 .2 . Athabasca Glacier, Columbia Icefields, Canada

Observations made on the Athabasca glacier near Jasper in Alberta, Canada, produced the Argand diagram shown in figure 10 . It is seen from this

360

200

f . ~ 5JC .1/'".Ic / ~o·1 f.-- aIr 10 . Ice

l80

f- .-J60

ROD~ ON RIDGE 8/5/59",-

d i A - r631 II /

140

/ II

FREDUENCtfS I~ c/) !zoo / f l 120

00

/ / 250

100

II / RODS IN CREVASSE,,, IfF / r- 8/1//59, d/A =0..693 1\ 200

v-tciK ~4K roo

80

60

40 20Kl// Vl9 K V5K~ N / j

~i/zOK b4 ~---I ~K)K

~kJ/IIK -I- 2r ~/400 4K x2K IK-I---

\-;jRALLEL PLATES IN SURFACE ICE, LlA 20

40K~ 80 K :::::6f

lOOK 0 rr40K 8't7' I I d(A 10.5(, I -I---

10 40 00 100 110 140

E'

FIG lJRE 10. Eleclr·ical properties oj (Jlacial ice. Ath abasea glacier, Alberta, Canada.

figure that the general circular form expected for pure icc having a simple dielectric relaxation is ob­tained for frequencies down to approximately 2 kc. Below this the e" values increase quite rapidly indicating an apprcciable amount of conductivity.

It should be noted tha t the radius of these Argand diagrams is greater than that predicted for 0° C pure ice from figure 4. This general type of departure was also observed by Auty and Cole [6] who ex­plained the discrepancy on the basis of electrode interface effects. They indicate that this effect disappears at larger electrode spacings which, for the equipment they employed, would be a dlA ratio of 29. The smallest ratio of dlA which they employed was in the order of 3.3. In our experiments ellA varied from 0.36 to 0.69. A comparison of the magnitude of the complex conduc tivity as obtained by bridge and four-electrode Eltran measurements, figure 11, shows close agreement. This may be due to the fact tha t the rods were surrounded by a very thin in ter­face of water. These sets of observations were obtained in the same general area on the glacier and would indicate that the large observed values of e' and e" are probably real. It should be pointed out that some dispersion was found between various observations in different areas on the glacier due pos­sibly to cracks and crevasses.

The parallel plate observations of fiO'ure 10 were made near the surface of the ice which was rather porous and wet at the time of measurement. The rods on the ridge were placed in the same general area as the parallel plate measurements, but being 4 ft long are expected to give results typical for ice several

E ....

I§ o z <t

(MAGNITUDE OF Co.MPLEX CONDUCTIVITY)

/0=/-v&-2 +(WE)2~

x/ /

./

x

~ ____ ~x _____ ~ ____ ~ ___ ~

_ x....... BRIDGE MEASUREMENTS (4 FOOT RODS, liNCH IN DIAMETER)

10 100 I!

FREQUENCY, cis

Pre LJ RI~ 11. Electri ral properties oj glacial ice . Athabusea glacier, Alberta, Canada, Au g. 5, 1959.

feet below the surface which was observed to be much more solid Lhan the surface icc. The third curve for rods in a crevasse represents measurements made some 30 ft below the surface of the glacicr where the icc was less disturbf'd by wind action, and in addition, it was expected to be slightly colder than the surfacfl icc . This reduction in temperature with depth is indicatcd by the fact t hat rods in or near the surface caused mel ting around them while rods a hundred feet or so below the surface would free~e in place.

F igure 12 shows 10:1 as a function of frequency for three diffcrent electrode spacings on the Athabasca

ELTRAN ARRAY WITH WIRES

ELEVATED ABOUT 3 FEE T.

100 Fool spoeilg

I-------I--------,)F-A,L- 20 Fool spoe",

10 100 1000 10000

Frequency I C /s

FIGURE 12. l\IJa(Jnitude oj complex cond1tctivily vs. frequency at three electrode spaC£n(Js.

Athabasca glacier, Alberta, Canada, Aug. 1959.

361

glacier. It is interesting to note the reversal in magnitude at the 100-cps region. Similar data is shown in figure 13 where liT l is plotted as a function of electrode spacing at different frequencies. It is obvious from these results that the properties of the glacier change with depth and that this change has an inverse effect at frequencies above and below the 100-cps region.

10 0

I I I+~ I I = I l- I I 1- -!----- ~- POINTS TAKEN 8-9-59

I-(A80UT 300 FEET FROM MARGINAL MORAINEl

-0- POINTS TAKEN 8-1/-59 1- x - POINTS TAKEN 8 -12-59 -

( NEAR THE CE~TER OF THE GLACIER J. , x '--

~~~i ~ x 2000 c/$

~ 0

-

= -I--

- -I f-- !

-

~ - - t- f--

x 0

I~ r I x I J 200c/s

l-n 1 01-" r 10F-

-- -

f~ J -

1-'\ r -r

I 4 0I

c/$ 1-

-

"'-

~ II

Ib

"'-- J_ 1_ 1

I r- Illltt' f---

I o o 100 400 600 1000 1100

ELECTRODE SPACI NG , FEET

F I GUR E 13. VaI'iations of magnitude of complex cond1,ctivity with electrode spacing.

Athabasca glaCier, Alberta, Canada.

The magnitude of the complex conductivity, 1;;:1, obtained by means of the bridge measurements is shown in figure 14, and it is interesting to observe that on the surface lui is greater below 200 cps and less above this frequency than is found for the subsurface measurements. This general behavior is in agreement with the observations of figures 12 and 13 when the variation of effective penetration with stake spacing is considered. This effective physical depth for the various electrode spacings is discussed in considerable detail by Wait [3].

It is instructive to observe the manner in which the conductivity, admittivity, and loss tangent vary as a function of frequency. A typical example taken from the Athabasca glacier for 0° C glacial ice is shown in figure 15 where it can be seen that the loss tangent increases below 1 kc rather than continuing to decrease as it does for pure ice.

>- 100 l­ 1 S> i= o :::l o Z o o E x :;; W 0 -' .c 0- E I ~ :t o _

~C§: W o :::l I­Z ~ :2

1 ~

0

1---'

Ilzl , 100

? V

~

, 1 ' I 1_ ------= -~bl r---rr /0

- - ;;1 - -I--1---

tl I T~O°C -/'

,,/

~ 1 /?,s, ~ , I I I

-~ RODS ON RIDGE 8-5-59 -

~ RODS INCREVASSE"8-11-59

• 30 FEET BELOW SURFACE AND SLIGHTLY COOLER THAN SURFACE.

~ -.l I 11Jl.ul 1 l~~ ~ I~

1000 ~OOO KlI000

FREOUENCY, cis

FIGURE 14. Jl![agnitude oj complex conductivity f or s1uJace and subsurface ice.

Athabasea glaCier, Alberta, Canada.

100,--------,----,-----,--;-,--------,--, 10

ICE TEMPERATURE O°C

E _1- \V

';;; 3 o ~ 1 -=--CONDuc7;; r-~--- ~

;,: 10i-="'---=:----'---+------,""'---r--z"--t---------j LO 15 ~ LOSS TANGENT ~

C If i I~ §

II LOO---""--------'-laoo-'-----'-----~IO~aoo--- ' 00000 O.L

FREQUENCY, cis

FIGURE 15, Electrical propeTties of glacial ice. Atbabasea glaCier, Alberta, Can ada, Aug. 5, 1959.

5. Conclusions

It is apparent from our limited observations- that the electrical properties of snow and glacial ice are appreciably different than those of pure ice. The greatest difference appears in the region below 2 kc. The conductivity, IJ , of glacial ice appears to be approaching a constant in the vicinity of 0.5 to 3 f.I. mhos/m for frequencies of 200 cps and lower. For pure ice, IJ appears to be continuing to decrease with decreasing frequency and at 100 cps, 1J~0.01 f.I. mhos/m at - 0.1° C. Typical values of IJ at 20 kc are 30 !1- mhos/m for pme ice, and 36 !1- mhos/m for the ice of the Athabasca glacier.

The close agreement between the values of com­plex conductivity magnitude obtained with the bridge measurements and four electrode methods would appear to indicate that electrode smface effects have not greatly influenced the results re­ported here. For future bridge measurements it

362

I I

r

l

:

>

l ,

r \

l

does, howev er, appeal' desirable to employ several rod spacings including d/A ratios up to 10 or more. For general propagation studies and wave-tilt calcu­lations, the values of (J and E given for ice should in most cases b e used in the s tratified earth formulas of Wait [10] rather than in the simple r elation for an infinitely deep layer . Typical values of (J for the material below the ice may vary from 0.1 to 10 milli­mhos/m. Sm'face measUTemen ts near the Athabasca glacier yielded values esse ntially independent of frequency of (J,,-, 2 millimho /m for soil in the creek beds, and ",0.3 mill imhos/m for a rocky ridge.

The authms aclmowledge the assistance obtained from: J. R . VVait , for many valuable suggestions dUTing the COUTse of this work; E . C. Bolton and R. W . Plush , for assistance in preparation of equip­ment and obtaining data ; H enri Vetter and D . n,. ' Va t t, for assis ting in the meaSUTemen ts on the Athabasca Glacier: G. D . Garland, whose in terest made the m eaSUTemen ts on the glacier possible; F. J. Weyl, for his in teres t and encouragemen t dUTing the eOUTse of this work ; and Mrs. Winifred M au , for her assistance dUTing the prepar ation of this manu­script .

363

-- ------

6. References

[1] J . R . \Ya it and A. M . Cond a, Pat t ern of an ante nna on a cur ved lossy surface, IRE Trans. PGAP AP- 6, 348 (1958) .

[2] A. D . Watt, E. L. Maxwell , and E. H . Whela n, Low­frequen cy propagation paths in Arctic areas, J . R e­search N BS 63D , 99 (1959).

[3] J. R . \Yait a nd A. M . Conda, On t he measurcment of groun d condu ctivity at VLF, IRE Trans. P GAP AP-6, 273 (1958) .

[4J J. R. Wait (Ed .), Overvol tage r esearch and gcophysical appli cations (Pcrgamon Press, New York, N .Y. , 1959) .

[5] N . E. D orsey, ProperLie of ordinary water-substance (Reinhold P ublish ing Co rp. , N cw York, N.Y., 1940) .

[6] R. P . Aut~r and R . I-I . Cole, DielecLric prop erties of ice a nd solid D 20, J . Chem . Ph ys. 20, 1309 (1952).

[7] J . G. P owles, A calculation of t he static d ielectric constant of ice, J . Chcm . P h ys. 20, 1302 (1952) .

[8] D . Kuroiwa, The dielectric proper ty of snow (Inst. Low T emp. Sci. , Hokk aido Un iv., Soygcoro, J ap an) Proc. I GU, R ome (1 954).

[9] T. E . D evaney (private communication). [10] J . R. "Vait, P ropagation of radi o waves over sLraWied

gr'ou nd, Gcoph ysics 18, 416 (1953).

(Paper 64D4- 69)