Independent studies of attenuation of X-band (8.5 GHz) and Ka-band (32 Ghz) radio signals when traversing Martian dust were carried out. These analyses turned out remarkably similar. The computational method is essentially that of T. S. Chu but uses observed optical depth at 0.67 microns rather than visibility as the measure of optical attenuation from which to derive the microwave attenuation. An awkwardness in the approach is that the size distribution of Martian dust particles is not well known, but the mean is probably around 4 microns, whereas in the terrestrial case it is nearer 10 microns. As a consequence, there will be a larger tail of particles still in the Mie regime in the Martian case as compared to the terrestrial one. The computational error will, therefore, be somewhat larger for Martian than Earth-bound dust. Fortunately, the indicated attenuations are small enough for the worst case (1.3 dB at 32 GHz) that the error is academic

Flock, W. L.

Smith, E. K.

English

NASA Technical Reports Server

TDA Prqress Report 42-87 July-September 1986 
Propagation Through Martian Dust at 8.5 and 32 GHz 
E. K. Smith 
Telecommunications Systems Section 
W. L. Flock 
University of Colorado 
Independent studies o f  attenuation o f  X-band (8.5 GHz) and Ka-band (32 GHz) 
radio signals when traversing Martian dust were carried out by the authors. These analy- 
ses turned out remarkably similar. The computational method is essentially that o f  
T. S. Chu but uses observed optical depth a t  0.67 microns rather than “visibility” as 
the measure of optical attenuation from which to derive the microwave attenuation. 
A n  awkwardness in the approach is that the size distribution of Martim dttst particles 
is not well known, but the mean is probably around 4 microns, whereas in the terres- 
trial case it is nearer 10 microns. As a consequence, there will be a larger tail o f  particles 
still in the Mie regime in the Martian case as compared to the terrestrial one. The com- 
putational error will, therefore, be somewhat larger for Martian than Earth-bound dust. 
Fortunately, the indicated attenuations are small enough for the worst case (1.3 dB a t  
32 GHz) that the error is academic. 
1. Introduction 
This article makes use of the approaches and results of 
studies of terrestrial sand and dust storms for the rather 
analogous problem on Mars. Dust storms on Mars are, if any- 
thing, more impressive than on Earth. Typically, in spring in 
the southern hemisphere, dust storms will start, grow, move 
northward, and frequently encircle the globe. In 1971, there 
was a huge storm which obliterated the surface features of 
the entire planet. 
Il. Terrestrial Approach 
In the case of the Earth, the most convenient yardstick for 
a dust storm is “visibility,” which is defined as the distance 
at which a mark disappears against the background. According 
to Middleton (Ref. 12), at the distance of disappearance, 
the luminance L of the object has decreased to 0.031 of its 
translates to: L/Lo = 10 log 0.031 = 15.086 dB, which is 
usually shortened to 15 dB. An approach developed by Chu 
(Ref. 6), and used by Ansari and Evans (Ref. I )  and Gold- 
hirsh (Ref. s) ,  incorporates the visibility V into the expres- 
sion for microwave attenuation coefficient A(2):  
Lo. As !umin2nce is 2 me2s”:P cf pc!‘?!Pr, this :..:+:-1 ..-l.,o l l l L C l U l  “( l lU” 
r = particle radius, m 
E = er - je,. = complex dielectric constant 
A = wavelength, m 
291 
https://ntrs.nasa.gov/search.jsp?R=19870005926 2020-03-20T12:10:13+00:00Z
As shown above, particles are assumed to  be uniform in 
size and to have known permittivity. The development assumes 
single scat€&. Thisxis fljt bad g t  microwave, but the optical 
‘.visibilitf liMt kwell  into the multiple scatter regime. 
Particle size distributions for four dust storms in Khartoum 
have been published by Ghobrial (Ref. 7), who finds parti- 
cles with sizes ranging from 0.1 to 300 microns (0.0001 to 
0.3 mm). Wind tunnel measurements (Ref. 9) indicate that 
0.08 mm is the most likely size of particle to  be picked up by 
wind from the Earth’s surface. 
111. Mars 
Mars has long captured the public fancy as possibly the 
most hospitable planet, next to Earth, for life. This is not so 
unreasonable, as can be seen in Table 1. However, the tempera- 
ture is considerably colder, the atmospheric pressure at the 
surface less than 1% that of the Earth, and the atmosphere 
almost totally lacking in oxygen (95.3% CO,, 2.7% N, , 1.6% 
argon and perhaps 0.13% 0,; Ref. 13). Mars’ orbit around the 
sun is considerably more eccentric than that of the Earth 
(206.7 X lo6 km at perihelion and 249.1 X I O 6  km at aphe- 
lion) with the result that the insolation is some 45% greater 
during perihelion (late spring in the southern hemisphere) than 
at aphelion (late spring in the northern hemisphere). 
Mars has been visited by spacecraft from both the US and 
USSR as is seen in Table 2. The most recent visits were by the 
two US Viking missions, and the data presented here came 
largely from that source. 
Martian dust storms tend to originate in the late spring or 
summer in the southern hemisphere when the Martian surface 
temperatures and the temperature gradients are highest. The 
greatest storm on record started in September of 1971 and 
became planetwide (Ref. 11). Great storms (planetary scale) 
occur once or twice each Martian year (Ref. 19). Quite typi- 
cally, a storm will move northward and reach planet-encircling 
proportions and last several weeks. The dust has been reported 
as high as 50 km above the surface. Optical depths as great 
as 6 at 0.67 microns have been measured and a dust density 
scale height of 10 km proposed (Refs. 15 and 16). The size 
and permittivity of Martian dust remain controversial. Fig- 
ure 1 portrays a particle density distribution (Ref. 10) typical 
of those found in papers published in the late 1970s. The 
mean of the particle radius is 0.4 microns. Also shown is the 
geometric cross-section distribution, GN(r), which is more 
directly pertinent to extinction calculations. Recent workers 
have favored somewhat larger particles, around 2 microns 
(Ref. 19 and Footnote 1) or greater (Ref. 5). The favored 
substances for the dust are basalt and montmorillonitic clays. 
Tabulated dielectric constant at 50 microns (the longest wave- 
length available) for the latter is 2.18 - jO.14.’ 
Water was detected in the soils at the two Viking Lander 
sites, but the integrated water in the Martian atmosphere 
does not exceed 100 precipitable microns (Ref. 4). 
IV. Attenuation by Dust 
A simple slab model of a Martian dust storm is shown in 
Fig. 2. The assumption is made that the reflected components 
of the electric field E, and E,, are much less than the trans- 
mitted component Et (E, >> Er, Et >> E,,). We define a 
complex propagation constant K,: 
where 
2 m r  
0 = Kon, = - - phase constant (2) 
X O  
2 m i  
Q = Koni = - = attenuationconstant (3) 
A0 
and 
ni = n, - ini  = complex refractive index 
2n K = -  
O b  
where 
Xo = free-space wavelength 
One can then write for the electric field (Ref. 2) 
E = Er(O, 1,O) exp {j(wt - Kz)} (4) 
where E,(O, 1,O) is the complex amplitude of the transmitted 
plane wave with electric vector polarized in t h e y  direction and 
travelling downward (positive z direction). Substituting (1) 
into (4) yields 
R. A. West, Jet Propulsion Laboratory, private communication, 
1986. 
292 
which may also be written 
E = E ,  (0,1,0) exp - -7  exp j wc - - ( ; ) 1 ( 2T)/ (6) 
where 7 is the optical depth. The attenuation coefficient A in 
decibels is then given by: 
A (dB) = 8.686 (a (nepers) (7) 
A (dB) = 8.686 Koni dB/m (8 1 
Attenuation from dust = 4.3437 dB (9) 
A. Chu’s Development 
We now turn to the work of Chu (Ref. 6) who wrote a 
much quoted article on “Effects of Sandstorms on Microwave 
Propagation.” Chu considered sand of radius 0.01 to 0.1 mm 
(10 to 100 microns) with dielectric constant 2.5 (1 - j O . O l )  
to 10 (1 - jO.01). He invoked the Rayleigh approximation at 
microwave and the very large sphere approximation in the ~. .. 
visual range. The attenuation coefficient (a is defined as 
(a = l - N ( a ) C ( a )  da 
where 
C(a) = extinction coefficient 
N(a) = number density 
a = particle radius 
For sand of radius a 
wh-ere Q,,, = C/ra2 is the extinction efficiency, and S = (4/3) 
~ Q ’ N  is the fraction of the volume actually made up of sand. 
As the size of particles become large with respect to wave- 
length, Qext becomes asymtotic to 2 (Refs. 17 and 18): 
Qext = 2 , a > >  X (12) 
As can be seen in Fig. 3, if the sphere is at all conducting, the 
extinction becomes very close to  2 by the time 2ra /h= 15 or 
a z 2 . 4 X .  Hence, in the case treated by Chu, if we take 
0.6 microns for the wavelength of white light, any particle 
larger than 1.5 microns would meet the requirements of (1 2). 
The question of multiple scatter needs to be raised. Van de 
Hulst (Ref. 18) suggests that whenever the optical depth 
exceeds 0.3 multiple scattering becomes a factor. In this 
case, the limit of visibility occurs at 7 = 3.47, well into the 
multiple scatter regime. However, this limit occurs at the 
point where a mark disappears into the background, and it 
can be argued that this is the coherent component that one is 
tracking and that single scatter still applies. 
The introduction of (12) is done by combining (1 1) and 
(1 2) and solving for N :  
(13) 
N = &Oa 
6.5 [(4/3)ra3] 
where (ao is the optical attenuation coefficient in dB/m. 
The next step is to make use of an expression presented, 
but not derived, by Van de Hulst (Ref. 18) for the effective 
refractive index m of a scattering medium: 
where S(0) is the forward scattering function, and K is the free 
space phase constant. In the Rayleigh regime S(0) is given by 
The second term is negligible for X >> a as is the case at 
radio frequencies. Substituting (13) and (15) into (14) and 
using the terminology of (1) through (9), the phase shift is 
given by 
K(R,  6 - 1) = 3K j-j (ao“ [ Re ( ‘ - ‘ ) I ( ~ ) d e g / m  ~ + 2  (16) 
and the attenuation coefficient becomes 
K(1mZ) = 3K -13 (a 0 a [ Im (:;:)1(8.68)dB/rn - (17) 
B. Application to the Martian Problem 
For a uniform slab the total attenuation and total phase 
shift for the wave of relations (4) through (6) is obtained by 
multiplying (16) and (17) by z ,  the distance traversed. For 
total attenuation (17) becomes 
293 
and, substituting 4.347 f o r a o  Z, we may write 
r 1 
K(1m E) = 3K -ra 13 1 Im (:;:)1(3’7.’7)dB/m - (18b) 
This relation (18b) now gives the microwave attenuation 
for uniform spheres of known radius a and dielectric constant 
E for a slab of constant particle density N and agrees with 
Eq. (13) of Goldhirsh (Ref. 8). Martian dust undoubtedly 
follows some particle distribution such as shown in Fig. 1 ,  
but the uncertainty as to the mean size is such that the as- 
sumption of a single particle radius is as good as any. Similarly, 
the dielectric constant can only be guessed at, but can rea- 
sonably be bounded. The dust density probably follows an 
experimental decay with height, so the equivalent slab will be 
used. 
C. Equivalent Slab for a Martian Dust Storm 
Pollack (Ref. 15) has suggested a scale height for dust 
particle density, H d ,  of 10 km. Assuming Hd is a constant 
over the region of interest, we may write 
where the right-hand side is obtained by performing the indi- 
cated integration; and No is the surface density of the dust. 
Thus, it is seen that the Martian dust storm may be repre- 
sented by a slab of height Hd = 10 km and density N o .  A 
spherical model of such a slab is shown in Fig. 4.  
D. Calculation of Microwave Attenuation 
leads to 
Performing the indicated complex arithmetic on (1 8b) 
where E = E~ - jei and a, the particle radius, and h are in 
meters. 
For zenith angles of 0” to 80” we use: 
7 = r2 sec 0, 8 < 80” (21) 
For zenith angles greater than 80” we need to consider refrac- 
tion effects. The Martian atmosphere is 95% CO,; CO, at 
0°C and 1 atmosphere and for white light has an index of 
refraction of 1.000450. In terms of refractivityN= (n - 1)106; 
this may be written 
N(C0 , )  = 12 1.3 p(mb)/T(K) (22) 
If p = 6 mb and T = 280 K on the Martian surface, the 
surface refractivity is 2.6N units as compared to about 300 
on the surface of Earth, and refractive bending will be a 
second-order effect and can be ignored. Using the geometry 
in Fig. 4 for zenith angles 8 greater than 80” or elevation 
angles 41 between 0” and IO”, the optical depth r is 
for @ = 90” - 8 < 10” (23) 
Shown in Fig. 5 is the zenith angle dependence for the 
“best guess” and “worst case” calculations using relations 
(20) and (23). Dust storms under the worst case scenario only 
become significant for ray paths nearly tangential to the 
surface of Mars. Shown in Fig. 5 is the dependence of atten- 
uation on choice of particle radius and dielectric constant. 
V. Discussion 
Attenuation through Martian dust storms has been com- 
puted for 8.5 and 3 2  GHz one-way radio transmission for the 
maximum expected optical depth (r=6).  These are first- 
order calculations based on single scatter. Due to  uncertain- 
ties in the Martian dust parameters - particle size distribution 
and dielectric constant - “best guess’’ and “worst case” 
situations have been selected. Attenuation, even at 32 GHz, 
will not be significant, except for rays almost tangential to  
the Martian surface. 
References 
1. Ansari, A. J.,  and Evans, B. G., “Microwave Propagation in Sand and Dust Storms,” 
Proc IEE, Vol. 129, Part F, pp. 31 5-322, Oct. 1982. 
294 
2.  Booker, H. G., “Energy in Electromagnetism,”ZEE Electromagnetic Waves, Series 13, 
3. Carr, M. H., The Surface ofMars, Yale University Press, 1981. 
4. Carr, M. H., The Geology of the Terrestrial Planets, NASA SP-469, Supt. of Docu- 
ments, 1984. 
5. Christensen, P. R., “Regional Dust Deposits on Mars: Physical Properties, Age and 
History,”J. Geophy. Res., Vol. 91, No. B3, pp. 3533-3545, 1986. 
6 .  Chu, T. S., “Effects of Sandstorms on Microwave Propagation,” Bell Syst. Tech. J . ,  
Vol. 58, pp. 549-555, Feb. 1979. 
7 .  Ghobrial, S. F.,  “The Effect of Sandstorms on Microwave Propagation,” Proc. Nut. 
Telecomm. Conf., Houston, TX,Vol. 2, pp. 43.5.1-43.5.4, 1980. 
8. Goldhirsh, J., “A Parameter Review and Assessment of Attenuation and Backscatter 
Properties Associated with Dust Storms Over Desert Regions in the Frequency 
Range of 1 to 10 GHz,” IEEE Trans. Ant. Propag., Vol. AP-30, pp. 1121-1127, 
Nov. 1982. 
9 .  Greeley, R., Leach, R., White, R., Iverson, J., and Pollack, J., “Threshold Wind 
Speeds for Sands on Mars: Wind Tunnel Simulations,” Geophy. Res. Lett., Vol. 7, 
10. Hunt, G. E., “On the Opacity of Martian Dust Storms Derived by Viking IRTM 
11. Martin, L. J., “Clearing the Martian Air: The Troubled History of Dust Storms,” 
12. Middleton, W. E. K., Vision Through the Atmosphere, University of Toronto Press, 
13. Owen, T., Biemann, K., Rushneck, D. R., Biller, J. E., Howarth, D. W., and Lafleur, 
A. L., “The Composition of the Atmosphere at the Surface of Mars,” J. Geophys. 
Res., Vol. 82, pp. 4635-4639, 1977. 
Peter Peregrinus, Ltd., 1982. 
pp. 121-124,1980. 
Spectral Measurements,” J. Geophys. Res., Vol. 84, pp. 8301-8310, 1979. 
Icarus, Vol. 57, pp. 317-321, 1984. 
Canada, 1952. 
14. Pollack, J .  B., The Solar System, Chap. 8 ,  Scientific American, 1975. 
15. Pollack, J. B., Colburn, D., Kahn, R., Hunter, J., Van Camp, W., Carlston, C. E., 
and Wolf, M. R., “Properties of Aerosols in the Martian Atmosphere, as Inferred 
from Viking Lander Imaging Data,” J. Geophys. Res., Vol. 82,  pp. 4479-4496, 
1977. 
16. Pollack, J .  B., Colburn, D., Flaser, M., Kahn, R., Carlston, C. E., and Pidek, D., 
“Properties and Effects of Dust Particles Suspended in the Martian Atmosphere,” 
J. Geophys. Res., Vol. 84, pp. 2929-2945, 1979. 
17. Ulaby, F. ‘T., Moore, R. K., and Fung, A. K., Microwave Remote Sensing: Active 
and Passive, Volume I, Fig. 5.1 1 ,  p. 292, Addison-Wesley Publishing Co., Reading, 
MA, 1981. 
18. Van de Hulst, H. C., Light Scattering by Small Particles, John Wiley and Sons, 
New York, Chapman and Hall, Ltd., London, 1957. 
19. Zurek, R. W., “Martian Great Dust Storms: An Update,” Icarus, Vol. 30, pp. 288- 
310,1982. 
295 
Bibliography 
Bagnold, R. A,, The Physics o f  Blown Sand and Desert Dunes, Methuen & Co., Ltd., 
Flock, W. L., Telecommunications in Planetary Environments, JPL Publication 81-84, 
Flock, W. L., Propagation Effects on Satellite Systems at  Frequencies Below 10  GHz, 
Ishimaru, A., Wave Propagation and Scattering in Random Media, Vol. I: Single Scatter- 
London, 1941. 
Jet Propulsion Laboratory, Pasadena, CA, 1981. 
NASA Reference Publication 1108, NTIS, 1983. 
ingand Transport Theory, Academic Press, New York, 1978. 
296 
Table 1. Some physical properties of Earth and Mars 
Property Earth Mars 
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
Mean distance from sun, km X lo6 
Period of revolution, days 
Rotation period, hours 
Equatorial diameter, km 
Mass (Earth = 1) 
Density (Water = 1) 
Atmosphere 
Water vapor (integrated column), cm2 
Water content in surface 
Mean atmospheric pressure, m b  
Mean surface temperature, "C 
Mean particle size in dust storms (diam), 
Real part refractive index 
Imaginary part refractive index 
Real part refractive index 
Imaginary part refractive index 
microns 
(clay 8.5 GHz) 
(clay 8.5 GHz) 
(clay 32 GHz) 
(clay 32 GHz) 
149.6 
365.28 
23.9344 
12,756 
1 
5.5 
Nitrogen 
Oxygen 
2.2 
0 - 30% 
1013.25 
22 
0.5 to 100 
206.7 
227'9 1249.1 
687 
24.6321 
6,787 
0.108 
3.9 
Carbon dioxide 
Nitrogen, Argon 
0.001 
-1% 
6 
-23 
0.1 to 10 
[2.5 + 0.5 (% HZO)] 
10.02 + 0.15 (% H20)] 
[2.5 + 0.3 (% H20)] 
[0.06 + 0.1 (% HzO)] 
Notes: 1. Data for 1 to 8 from Ref. 14. 
2. Values for 13 to 16 derived for data for clay from Table 1, Ref. 1. 
Table 2. Spacecraft missions to Marsa 
Mission Arrival date Type 
IJnited Statec 
Mariner 4 
Mariner 6 
Mariner 7 
Mariner 9 
Viking 1 
l"r&Lyg 2 
Soviet 
Mars 2 
Mars 3 
Mars 4 
Mars 5 
M a i s  6 
Mars 7 
July 15, 1965 
July 30, 1969 
August 4 ,  1969 
November 14,1971 
June 19, 1976 
A,.-. ?, 1976 
November 27, 1971 
December 2,1971 
March 1974 
March 1974 
March 1974 
Mn*rh 1974 
2.1u.d.. I 
Orbiter/Lander 
Orbiter/Lander 
Orbiter 
Orbiter 
Lander 
1 .,"Am? -....--.. 
aFrom Ref. 3. 
297 
PARTICLE RADIUS, 1 
101 102 1 03 104 105 
10-2- ' ' " I  ' ' " I  ' " ' I  ' I " -  10-1 
- N(r) = araexp[- (a/r)(r/rm)yl 
- (Ref. 17) 
f o r a  = 2, y = 1/2, rm = 0.4 p n  
10-3 - 
I 
L 
I z 
w 
V z 
z 
a n 
a 
0 
2 10-4 - 
w 1
+ LT 
6 
10-5 - 
(0.67pm) 
A 
10-6, , , 1 , I , I I ,  I , I I ,  I , 1 , 1 .  10-5 
10-3 10-2 10-1 1 00 101 
PARTICLE RADIUS, prr 
Fig. 1. Martian dust particle distribution 
Y TX 
z 
0 0" 
t 
0 z 
w 100 u 
U 
U 
W 
- 
- 
Q, = Q,+Q, - 
% = EXTINCTION - 
COE F F IC I ENT 
FOR ABSORPTION - 
n = 1.29 - j0.047 - 
I-1 I 0. = EXTINCTION 4 
a COEFFICIENT 
FOR SCATTERING 4 1 / n =  1.29 
10-21 I I  I I I I I I I I 1 I I I I I 
0 2 4 6 8 10 12 14 
X = 2 r r h  
Fig. 3. The extinction efficiency Q,for various types of spheres, 
shown in the range 0 < x < 5 (Ref. 7) 
SLAB OF DENSITY No, HEIGHT Hd = 10 km 
I 
/ \ 
\ 
I 
f 
I 
I 
Fig. 4. Model of equivalent Martian atmosphere 
Fig. 2. Simple slab model of a Martian 
dust storm 
298 
# - *  
Y 
.I . 
I I I I I I I 1 
ZENITH OPTICAL DEPTH T~ = 6, SCALE HEIGHT = 10 km 
1 00 
10-1 
m 
2 
2 
s 
5 
t 10-2 
V 
+ 
+ 
U 
$ w 
z 
0 
10-4 I I I 1 1 1 I I 1 
0 10 20 30 40 50 60 70 80 90 
L C l Y l  I "  r \ l Y U L C ,  "e9 -?,-*IITII A L l r l  r >-. 
OPTICAL DEPTH T = 6 
n WORSTCASE 
WORST CASE 
MONTMORILLONITE 
- - 
BEST GUESS CASE 
Fig. 6. Dependence of attenuation on choice of particle radius and 
dielectric constant 
Fig. 5. Zenith angle dependence of dust attenuation in Martian 
299 


Propagation through Martian dust at 8.5 and 32 GHz

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