the radar system for the exploration of titan

IL NUOVO CIMENTO VOL. 15 C, N. 6 Novembre-Dicembre 1992

The Radar System for the Exploration of Titan (*).

G. PICARDI (1), R. SEU (1), A. CORADINI (2), E. ZAMPOLINI (1) and A. CIAFFONE (1)

(1) INFO-COM, Universit& di Roma ,La Sapienza,,, Italia (2) IAS, Reparto Planetologia, CNR, Roma, Italia

(ricevuto il 14 Luglio 1992)

S u m m a r y . - In the field of the exploration of the Solar System NASA and ESA have jointly planned two cornerstone missions whose aim is the close observation of two largely unknown planetary bodies: Titan (Cassini mission) and comet nuclei (Rosetta mission). Our attention in this paper will be devoted to the first one, however some results of this study can be applied also to the second one. Titan is one of Saturn's moons and is characterized by the presence of a dense atmosphere, opaque for optical sensors. Therefore radar data are the only ones at least theoretically capable of giving relatively detailed informations about the geomorphologic structure of Titan. As a matter of fact the radar cross-section depends on both the dielectric constant of the target by means of the Fresnel reflectivity and on its surface roughness. Moreover if the attenuation is sufficiently low, the radar in principle could be able to detect subsurface discontinuities as well. The above considerations hold true only if a well-suited analytical model accounting for surface backscattering is known. All throughout this paper the radar cross-section of planetary bodies will be evaluated in the case of high-resolution radars.

PACS 96.30 - Planets and satellites (excluding the Moon). PACS 95.85.Ek - Radar, microwave.

1 . - I n t r o d u c t i o n .

The task of the radar system which is a par t of the payload of the Cassini miss ion is to provide the scientific c o m m u n i t y with as m u c h detailed as possible informat ion about the geomorphologic s t ruc ture of Titan. In par t icular the following scientific objectives have been established:

i) est imate the composi t ion of the surface and of the subsurface where

possible,

(*) Paper presented at the V Cosmic Physics National Conference, S. Miniato, November 27-30, 1990.

1149

1150 G. PICARDI, R. SEU, A. CORADINI, E. ZAMPOLINI and ~_ CIAFFONE

ii) estimate the elevation profile (that is provide with a topographic map) of the surface,

iii) estimate the ocean depth (if any).

A radar system is in principle able to afford this task since the radar wavelengths allow a good penetration depth (either in atmospheres or other materials) provided that the dielectric constant is not too high. Anyway in order to define the main radar parameters and the optimum signal processing to work out the required estimates it is quite important to have some knowledge of the target (Titan) and of the scattering mechanism which generates the radar echo.

In the paper, first of all, some hypotheses about the composition of the Titan atmosphere and surface will be made, then a well-suited analytical model accounting for surface backscattering valid for a Beam Limited Radar Altimeter will be discussed considering the case of smooth or rough surface. The above models have been obtained by evaluating the interaction between the propagating electric field and the surface. Once the related radar cross-section has been evaluated, the possibility of estimating the Fresnel reflectivity of the surface (subsurface if detectable) can be investigated.

It is worth noting that the obtained models could also justify the apparent disagreement existing between the ground-based radar observations of Titan carried out by Mulhemann [1] at the Arecibo facility and the available Titan's models (including the one proposed herebelow). It would also clarify the capabili- ties of the Titan Radar Mapper to those scientists who have had some concern about the usefulness of the radar instrument in the observation of asteroids and icy satellites.

2. - T i t a n m o d e l .

The Titan satellite is characterized by the presence of an atmosphere and a surface likely composed by oceans a n d / o r dry lands and sediments.

The atmosphere is characterized by a dense layer of clouds (therefore it is opaque for optical sensors) mainly composed by N2, CH4 and argon (see table I).

The size of the aerosol droplets should be very small in the troposphere (micron size), of the order of (10 -- 100) pm (composition: nitriles and acetylene core, C~Hs, C2H6, CH4 envelope) close to the surface and larger than 50 microns in the clouds.

Given the above information we expect a very small electromagnetic extinction; therefore the Titan atmosphere can be considered transparent to the radar frequencies. Anyway such extinction (if unknown) could worsen the accuracy in radar backscattering measurement.

The pressure and temperature profiles have been obtained by the Voyager observations: the pressure is ,~ 1.5 bar and the temperature is ,~ 94 K at the surface. In particular the presence of zonal winds in high troposphere and stratosphere can be hypothesized just on the basis of the temperature profile. On the contrary, the wind on the surface should be absent.

The o c e a n composition is the result of chemical reactions (photolysis) and of ocean-atmosphere interactions.

THE RADAR SYSTEM FOR THE EXPLORATION OF TITAN 1151

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1152 G. PICARDI, R. SEU, A. CORADINI, E. ZAMPOLINI and & CIAFFONE

Some theoretical models [2-4] seem to indicate that the ocean is a mixture of CH4, C2H~, C8H8, N2 and Ar in solution. The relative fraction of the various components is really uncertain and table I shows only an estimated percentage.

The uncertainty of the order of at least a factor 6 in the gaseous methane in the atmosphere and the uncertainty of the temperature at the interface atmosphere-ocean [4] lead to two models of the ocean composition and conse- quently of the ocean depth [2]. Therefore the knowledge of the ocean depth is particularly important and its determination is one of the main requirements for the radar instrument.

The photolysis and radiation chemical processes also produce insoluble com- pounds, these products can be gravitationally differentiated and sedimented on the bottom of the ocean. The thickness of this layer can be of the order of (100 -- 500) m. Due to the high density of the ocean components in solid phase, the ice, if exists, sediments in the bottom, therefore the ocean should be very smooth (the wind is supposed absent at the surface level)[2-5].

The dry lands could be composed by water ice or rock [6, 7]. Water ices and rocks could be covered by sediments but the methane rains can remove them periodically.

Table II shows the horizontal and vertical scale in order to define the spatial and vertical resolution required by the radar.

By means of the above-defined models, a general agreement with the requirements to obtain images results

low resolution ( ( 2 - 10) km) and

high resolution ( ~ 300 m).

The elevation profile could be obtained with a resolution of ~ 30 m.

TABLE II

Horizontal scale:

smooth surface ocean and atmospherical phenomena tectonic surface crater: large size

medium size small size

Vertical scale:

topography craters on icy bodies

depth rim height

ocean tide coast line and bottom topography

100 km 10 km 5 km 100 km 25 km 5 km

(0.1 § 1) km

l k m 100 m (10 -- 100) m (10 + 100) m


Figure 1 shows the permit t iv i ty vs. the in t rus ion vo lume f rac t ion or vs. the vo lume f rac t ion of the m a i n c o m p o n e n t s of hos t mater ia l . By analysing this f igure we can conc lude tha t it is theore t ica l ly poss ib le to dis t inguish be tween ocean regions and dry lands.

7.0

6.0

5.0

:~ 4.0

3.0 X

2.0

1.0

10)

8)

6 4)

0 .0 , , vi(%) 0 ~ 1'0 15 2'o 25 30 1-v2(~)

Fig. 1 . - Permittivity vs. intrusion volume fraction for Titan's materials. 1) Dry land-rock (v2 = 50%, ~ ' = 3.1). 2) Dry land-water ice (v2 = 50%, lunar fines). 3) Dry land-water ice (v2 = 50%, meteoritic material). 4) Sediments (simple organic e'----2.2, polymers e ' = 5.2). 5) Dry land-rock (v2 = 80%, s ' = 3.1). 6) Dry land-water ice (v 2 = 80%, lunar fines). 7) Dry land-water ice (v2 -- 80%, meteoritic material). 8) Dry land-rock (v2 = 50%, e' ---- 8.6). 9) Dry land-rock (v2 = 80%, ~ ' = 8.6). 10) a) Ocean s ' = 1.8. b) Ocean ~'= 1.6.

3 . - S u r f a c e b a c k s c a t t e r i n g .

In o rde r to ana lyse the sca t te r ing of an e.m. wave f rom an i r regu la r surface [8, 9], the sur face i tself m u s t be cha rac te r i zed in t e r m s of

i) p robabi l i ty dens i ty func t ion of the sur face height z(x , y): we will suppose a Gauss i an model ;

ii) s t anda rd deviat ion of the sur face height: 6h = ~ / ( - ~ - _ ~ ) 2 ( . ) ;

L:J2 k J2

1 ~ ~ Z(X, y) dxdy (*) -5 = LxLy

-~/2 -L~/2 L~/2 Ly/2

-5"~_ 1 ~ ~ z2(x, y)dxdy. L x Ly

- L : , / ~ - ~ / 2

1154

(1)

iv) spatial cor re la t ion length: lx, l~ = ly = 1;

v) surface r.m.s, slope m [9] (**)

(2)

(3)

G. PICARDI, R. SEU, A. CORADINI, E. ZAMPOLINI and ~ CIAFFONE

iii) spatial cor re la t ion func t ion (* ) :

p ( x -- x ' , y -- y ' ) ----- p (iv);

l~; in the following we will suppose

1-- _ m----X/2ah/l for a Gaussian corre la t ion func t ion ,

m----ah/l for an exponent ia l cor re la t ion func t ion .

Firs t of all we would like to r e m e m b e r some intuitive evaluations of the backsca t te r f rom flat and spher ical surfaces. As it is known, a flat surface of area S when i l luminated by an e.m. radia t ion propagat ing in the normal d i rec t ion with respec t to it, behaves like an an tenna , the radar cross-sect ion is t hen

4u 2 (4) a ---- ~ - S F ( 0 ) ,

where ~ is the wavelength and F (0) is the Fresne l reflectivity.

On the con t r a ry if we have a spher ical surface of radius RT ent i re ly involved by the e.m. field and 2~R/2 >> 1, t hen it is well known that

(5) a ---- gR~ F (0)

i ndependen t ly of the surface roughness . Moreover, by cons ider ing a flat wavefront of the e.m. field impinging on a

smooth spher ical surface (see fig. 2), the greates t cont r ibut ion to the backscat tering is due only to the flat region opposite to the e.m. wave ( cohe ren t backscat ter) .

(*) P(P) = ~z ( r ) z (p + r)dr ~z 2 (r) dr

(**) The surface slope is defined by

z ( x + Ax) - z ( x ) s = lim

Ax

and r.m.s, slope is given by

being X 12

p (x ' ) = 1 + p" (0) �9 - - + ... 2


1 I

Fig. 2. - Geometry of scattering from a spherical body.

The spatial region X, of the spherical body involved in this coherent backscattering can be evaluated by letting y _< ~ /4 in fig. 2; therefore X1 becomes:

(6) X, _< ~ f ~ -- X L .

On the other hand, it is possible to demonstrate that the power backscattered by a fiat sphere is effectively backscattered by a portion of it defined by XL (see fig. 2), therefore ff our body is for instance an ocean in a region bigger in size than XL, the backscattering is always given by eq. (5).

Moreover it is possible to define (by means of eqs. (4) and (5)) a fiat surface, as a disc of radius Xeq, equivalent from the backscattering point of view to the smooth spherical surface

(7) ---~" ( ~ 2 e q ) 2 ---- 7cR2w.....~ Zeq = ~ ' ~ ;

XL results larger than Xeq by a factor n//~: this can be intuitively explained by thinking that the phase relationship is different in the case of a flat surface with respect to a curved surface.

The case of plane surface and spherical wave is the dual of the previous one so that in eq. (5) R~ becomes the distance ( H ) between the e.m. source and the surface.

If we consider finally the case of spherical wave and spherical surface we can demonstrate that RT in eq. (5) becomes:

R T �9 H

(8) Rp -- RT +-----H"

In the following the coherent and non-coherent baekscattering will be evaluated by considering a beam-limited radar altimeter.

1156 G. PICARDI, R. SEU, A~ CORADINI, E. ZAMPOLINI and A. CIAFFONE

3"1. Coherent-scattering models. - It is well known in the literature that the scattering from an irregular surface consists in general of a coherent and a non-coherent component. The coherent one is important at small-incidence angles and if the roughness can be considered negligible while the non-coherent scattering becomes the most important at larger incidence angles.

If the correlation length and the curvature radius in each point of the surface are larger than the wavelength (Kirchhoff approximation), the scattered electric field due to a spherical incident wave (exp [jkR~]/R1) from an undulating surface is given by [8] (see fig. 3)

(9) E s -~-- R(01) (r q) exp [jk(2R1)]/(R~ ) dS

S

in the case of monostatic radar, where

S is the area defined by the altimeter operative mode,

R(01) is the Fresnel reflection coefficient in P: F ( O ) - - I R ( O ) I 2, r is the unit vector normal to the surface S in P,

is the unit vector in the direction from P to the radar position (R),

k----2~/~ where ~ is the wavelength,

R1 is the path length from P to the radar.

Under the following hypotheses:

i) r = ~ (unit vector in the z-direction - see fig. 3 - small surface slope),

ii) the surface S is limited by Gaussian beam (one-side one-way beam- width ---- fl0),

iii) the size of the region under investigation (S) is very small with respect

Z

Oo

01

R1

Ro

x

/ y

Fig. 3. - Geometry for the evaluation of surface backscattering.


to R0 (R0 is the distance of the central point of S from the radar) and the surface height is small compared to the size of the same S,

we can write

(10) R 1 = x / ( R o c o S 0 o - - z) 2 + y2 + (x + Rosin0o) 2

1 2 ~-- Ro -- zcosO o + OoxsinOo +-~oo(X + y2).

In the following we will consider a conventional radar al t imeter operative mode so that 00 = 0 and eq. (10) becomes

1 x 2 + y2 (1 I ) R 1 ~ R 0 -- z +

2 R 0

We can also suppose

R (0) ~ R (0o) = R (0)

and eq. (9) can be rewrit ten as

(12) ff }] Es ~ 2---~ exp [ j 2kRo] exp Ro k -- 2kz G (x 2 + y2) dx dy ,

s

where

exp [ j k ( x 2 + y2)/Ro] is the contr ibut ion due to the spherical wave,

e x p [ - - j 2 k z ] is the contr ibut ion due to the surface height d isplacement ( roughness) ,

G is the normalized an t enna gain. In the following, according to the conventional a l t imeter operat ion we will suppose an an t enna with circular symmetry, so, the gain depends only on the distance of the considered point from the nadir point (0, see fig. 3).

For a s tat ionary Gaussian-distr ibuted random surface, with variance given by a~ and spatial correlat ion p ( p ) (see eq. (1)) , the average square magni tude of the scattered field is given by [8]

k 2 F (0) f f f f _ x ' 2 (13) (EsE*~) 4~2R 4 e x p [ j k ( x 2 + y 2 --y '2)/Ro].

s

�9 exp [ -- 4 k 2 a ~ [ 1 -- p (x -- x ' , y -- y ' ) ] ]- G (x 2 + y 2). G (x'2 + y ,2) dx dx ' dy d y ' ,

where F (0) = I R (0) 12.

1158 G. PICARDI, R. SEU, A_ CORADINI, E. ZAMPOLINI and ~ CIAFFONE

The radar cross-sect ion a is def ined [9] by

(14) a - { E - - ~ > 4~R~ = < EsE* >4~R~, IEi I

where E i is the inc ident field on the surface. In o rder to evaluate the c o h e r e n t contr ibut ion, according to the convent ional

approach we use a series representa t ion:

(15) exp [4k2 a~ p] = (4k2 a~p) n

n=o n!

Tha t r ep resen ta t ion is useful if 4k 2 a 2 p < 1, and the n -- 0 t e rm inser ted in eqs. (13) and (14) gives the c o h e r e n t scat ter ing and the cross-sect ion is given

by IS l

F (0) ~ 2 (16) a ---- ( ( l / k 2 R~ 2 fl02) + (f10~/4)) exp [-- 4k 2 a 21 �9 ~ (floRo)

which cor responds to the backscat ter ing cont r ibut ion evaluated for beam-limited altimeter (BLA), in the case where the roughness is smal ler t han the wavelength.

F r o m eq. (16) we can obtain

(17) IT

F (0)" uR 2 exp [-- 4kay]

'

1 + o Ro/ /

flo Ro / x /~ can be cons ide red as the spatial resolu t ion (Rxz) and the normal iza t ion was pe r fo rmed according to eq. (7).

The behaviour of eq. (17) is shown in fig. 4.

o

- 2

h~ o

- 6

- 8

- 2 I I I I I I I

-1 0 1 2 log (x)

Fig. 4. - Normalized backscattering coefficient x = R~z/x/~oo/2~.


By neglecting in eq. (16) the term ( k R o r io) -2 with respect to the term f l ~ / 4 we can obtain

(18) a ---- F (0) ~R2o exp [-- 4ka2h].

By considering ah----0 and the dual condition (spherical surface and flat wavefront) we have a full agreement with eq. (5) (see eq. (7)).

3"2. N o n - c o h e r e n t s c a t t e r i n g . - When the surface roughness is not negligible with respect to the incident wavelength, a non-coherent scattering process arises.

Let us consider a rectangular antenna pattern and neglect the spherical term, so that eq. (13) can be rewritten. We can suppose G-- 1 in the footprint and the cross-section (see eq. (14)) is then given by

2 RAg

(19) a - - - - - - exp [--4k 2 a~ [1 --p (u, v)]] (2RAz-- u) (2RAz--IV I) du dv

2 RAT,

by letting in eq. (13) x ' = x - - u , y ' = y - - v and integrating on (x', y ')-plane. In the following we shall assume a Gaussian correlation coefficient

[ (20) p = exp ~ .

By considering that the integrand term is negligible for large values of I u l and I vl for a suitable value of kah , and that the spatial resolution can be supposed larger than correlation length ( l ) we can write ]9]:

(21) 2R~ >> (u ) , 2R~ >> (v) ;

therefore in polar coordinates (r, tp) eq. (19) can be rewritten as

2RAz

(22) a----k 2 F ( 0 ) ' ( 2 R ~ ) 2 exp - - 4 k 2a~ 1 - - e x p -- dr 2.

0

By expanding the correlation coefficient in Taylor's series and stopping it at the first two terms eq. (22) becomes:

l 2 F (0) (2RAz) 2 (23) a ----- k 2 F (0) (2RAz) 24k 2 a2 h -- 2m 2

in the same hypothesis(*) of eq. (21) and geometric optics model comes out [9].

(*) Moreover it easy to show that a more exact form of eq. (41) is given by

a ~ 2m 2 (2RAz) 2 ld- h 2exp[ 4k ea ;

therefore, by considering (e.g.) 2RAz/1 N 25 and 4k2ah = 10 [3], the use of geometric optics model causes an error of the order of 1 dB.

1160 G. PICARDI, R. SEU, A_ CORADINI, E. ZAMPOLINI and & CIAFFONE

In the following we want to generalize the previous conclusion by considering the spherical term of eq. (13) and Taylor's series development. The cross-section can be rewritten as

F u 1 (24) a - - - exp[-- 2k2m2u2]2(2R~)sin[(k/R~ d

[_.- ~

where m is given by eq. (2) and setting x = u k x / ~ o eq. (23) becomes

(25) a - - -

F 2RAg k ~

k2F-(O)/ f exp[ - - 2km2 R0x] �9 J L-2R~

�9 2 (2RAz) 2 k x / ~ o x 2 R A z

k ~ F (0) z Ro [ 2i"z'~ -- - - r 4R .~-~--

[-- 2RAzn

i 2R.4Zn ~ r (o) = ~ exp [- 2km2Rox] �9

n

therefore

(26)

[ -- 2km 2 R 0 x] sin x (2R .-~ _-- [ x [ ) |dxl 2 exp X 2 R Azh J

sin x (2RAzn -- Ix[ )

X dx~,

Equation (26) is shown in fig. 5, where it is included coherent and non- coherent backscattering.

- 1 0 . 0 . =10

~ -20"01 .=1oo

-50.0~

-2.0 010 2.0 log RAzn

Fig. 5. - Backscattering coefficient vs. ground resolution: fl = 0 coherent scattering.


Conclusions.

Notwithstanding the relatively close observations of Voyager, mainly due to the impossibility of making observations with optical sensors, Titan remains a largely unknown planetary body.

In spite of the lack of information, the scientists have worked out some quite exciting models discussed in the first part of the paper: the surface of Titan could be covered by a large hydrocarbon ocean and /o r icy dry lands while the atmosphere wrapping the planet is made up of dense clouds.

Only recently Earth-based radar observations have been made, but unfortuna- tely the results seem to be in disagreement with the above models. A deeper analysis of the backscattering models could, however, demonstrate that the disegreement is only apparent. As a matter of fact the models discussed in the paper show that the radar backscattering might have a rippled behaviour with peaks up to 6 dB over the mean value so that a very high backscattering level could be justified also by a relatively small fiat region like the Titan ocean.

The authors are indebted to P. T. Melacci of the Dipartimento di Fisica, Univesit~ di Perugia, for his valuable help. This work has been partially supported by ASI, the Italian Space Agency, contract n. 1991 RS23.

R E F E R E N C E S

[1] D. O. MUHLEMAN, A. W. GROSSMAN, B. J. BUTLER and M. A. SLADE: Radar Reflectivity of Titan, Science, Vol. 248, May 1990, pp. 975-980.

[2] W. R. THOMPSON and S. W. SQUYRES: Icar?~s, 86, 336 (1990). [3] J. I. LUNINE and B. RIz~:: Icarus, 80, 370 (1989). [4] J. I. LUNINE: Icarus, 81, 1 (1989). [5] N. DUBOULOZ et al.: Icarus, 82, 81 (1989). [6] D. M. HUNTEN et al.: Titan, in Saturn, edited by T. GEHRELS and M. S. MATTEWS

(University of Arizona Press, Tucson, 1984). [7] J. L. LUNINE and D. J. STEVENSON: Evolution of Titan's coupled ocean-atmosphere

system and interaction of ocean with bedrock, in Ices in the Solar System, edited by KLINGER (1984).

[8] A. K. FUNG and M. J. EOM: IEEE Trans. Antennas Propag., 31, 68 (1983). [9] F. T. ULABY, R. K. MOORE and A. K. FUNG: Microwave Remote Sensing (Addison-

Wesley Publishing Company, 1981).

the radar system for the exploration of titan

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