subsurface radar sounding of the martian polar cap: radiative transfer approach

10
Planetary and Space Science 53 (2005) 1427–1436 Subsurface radar sounding of the martian polar cap: radiative transfer approach Ya.A. Ilyushin a, , R. Seu b , R.J. Phillips c a Atmospheric Physics Department, Physical Faculty, Moscow State University, Moscow 119992, Russia b Infocom Department, ‘‘La Sapienza’’ University of Rome, Via Eudossiana, 18, 00184 Rome, Italy c Department of Earth and Planetary Sciences, Washington University, Campus Box 1169, One Brookings Drive, St.Louis, MO 63130, USA Received 19 January 2005; received in revised form 3 August 2005; accepted 22 August 2005 Available online 11 October 2005 Abstract The problem of subsurface radar sounding of the martian polar caps [Ilyushin, 2004. Martian northern polar cap: layering and possible implications for radar sounding. Planet. Space Sci. 52, 1195–1207] is considered from the point of view of incoherent radiative transfer theory. Since it has been previously shown that the radar signal field within the polar cap has diffuse structure, there is a need for a statistical approach to the problem. Radiative transfer theory, which is now well developed, seems to be the most appropriate formalism for this approach. Several physical models of polar caps have been formulated. The asymptotic solutions for all proposed models are derived here. In the present paper only the case of orbital ground penetrating radar is considered, because it is of great interest in relationship to currently developed radar experiments. In principle, the approach is believed to be applicable to a wide class of short pulse and compressed chirp radar experiments, including both orbital and landed instruments and media more complicated than a simple plane parallel geometry. This work, however, is postponed to future papers. Techniques for retrieval of physical properties of polar caps from the radar measurements are proposed. From the observational data, the macroscopic parameters of the medium appearing in radiative transfer theory, i.e. the single scattering albedo and volume extinction coefficient can be estimated. These estimates put certain constraints on the physical parameters of the medium model introduced in the paper. With some additional information, known a priori or from other observations, these estimates can be used to retrieve physically meaningful information, for example, the average content of impurities in the ice. r 2005 Elsevier Ltd. All rights reserved. Keywords: Mars; Polar ice sheets; Ground penetrating radar; Climate record; Radiative transfer 1. Introduction Ground penetrating radar (GPR) sounding is now the primary instrumental approach for exploration of the interior of celestial bodies, including planets, comets, asteroids, etc. A number of GPR experiments have been proposed for subsurface exploration of Mars (Seu et al., 2004; Picardi et al., 2004; Barbin et al., 1995; Berthelier et al., 2000; Oya and Ono, 1998; Vannaroni et al., 2004). Investigation of the structure of martian polar caps with GPR is of particular interest, because martian polar deposits are the source of key information on the climate of Mars and its water cycle (Clifford et al., 2000). The basic idea of this part of the study is to apply the non-coherent theory of radiative transfer to the problem of subsurface radar sounding of martian polar layered deposits, which are the random layered media (Ilyushin, 2004). The general view of the experimental geometry is shown in Fig. 1. The common approach is coherent treatment of the problem (Nicollin and Kofman, 1994; Shchuko et al., 2003), often leading to results intractable in terms of the initial model of the medium, such as mean thickness of individual layers. Within the approach applied in the present paper, the compressed radar pulse is ARTICLE IN PRESS www.elsevier.com/locate/pss 0032-0633/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2005.08.002 Corresponding author. Tel.: +7 095 939 3252. E-mail addresses: [email protected], [email protected] (Ya.A. Ilyushin).

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ARTICLE IN PRESS

0032-0633/$ - se

doi:10.1016/j.ps

�CorrespondE-mail addr

(Ya.A. Ilyushin

Planetary and Space Science 53 (2005) 1427–1436

www.elsevier.com/locate/pss

Subsurface radar sounding of the martian polar cap: radiativetransfer approach

Ya.A. Ilyushina,�, R. Seub, R.J. Phillipsc

aAtmospheric Physics Department, Physical Faculty, Moscow State University, Moscow 119992, RussiabInfocom Department, ‘‘La Sapienza’’ University of Rome, Via Eudossiana, 18, 00184 Rome, Italy

cDepartment of Earth and Planetary Sciences, Washington University, Campus Box 1169, One Brookings Drive, St.Louis, MO 63130, USA

Received 19 January 2005; received in revised form 3 August 2005; accepted 22 August 2005

Available online 11 October 2005

Abstract

The problem of subsurface radar sounding of the martian polar caps [Ilyushin, 2004. Martian northern polar cap: layering and

possible implications for radar sounding. Planet. Space Sci. 52, 1195–1207] is considered from the point of view of incoherent radiative

transfer theory. Since it has been previously shown that the radar signal field within the polar cap has diffuse structure, there is a need for

a statistical approach to the problem. Radiative transfer theory, which is now well developed, seems to be the most appropriate

formalism for this approach.

Several physical models of polar caps have been formulated. The asymptotic solutions for all proposed models are derived here. In the

present paper only the case of orbital ground penetrating radar is considered, because it is of great interest in relationship to currently

developed radar experiments. In principle, the approach is believed to be applicable to a wide class of short pulse and compressed chirp

radar experiments, including both orbital and landed instruments and media more complicated than a simple plane parallel geometry.

This work, however, is postponed to future papers.

Techniques for retrieval of physical properties of polar caps from the radar measurements are proposed. From the observational data,

the macroscopic parameters of the medium appearing in radiative transfer theory, i.e. the single scattering albedo and volume extinction

coefficient can be estimated. These estimates put certain constraints on the physical parameters of the medium model introduced in the

paper. With some additional information, known a priori or from other observations, these estimates can be used to retrieve physically

meaningful information, for example, the average content of impurities in the ice.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Mars; Polar ice sheets; Ground penetrating radar; Climate record; Radiative transfer

1. Introduction

Ground penetrating radar (GPR) sounding is now theprimary instrumental approach for exploration of theinterior of celestial bodies, including planets, comets,asteroids, etc. A number of GPR experiments have beenproposed for subsurface exploration of Mars (Seu et al.,2004; Picardi et al., 2004; Barbin et al., 1995; Berthelier etal., 2000; Oya and Ono, 1998; Vannaroni et al., 2004).Investigation of the structure of martian polar caps with

e front matter r 2005 Elsevier Ltd. All rights reserved.

s.2005.08.002

ing author. Tel.: +7095 939 3252.

esses: [email protected], [email protected]

).

GPR is of particular interest, because martian polardeposits are the source of key information on the climateof Mars and its water cycle (Clifford et al., 2000).The basic idea of this part of the study is to apply the

non-coherent theory of radiative transfer to the problem ofsubsurface radar sounding of martian polar layereddeposits, which are the random layered media (Ilyushin,2004). The general view of the experimental geometry isshown in Fig. 1. The common approach is coherenttreatment of the problem (Nicollin and Kofman, 1994;Shchuko et al., 2003), often leading to results intractable interms of the initial model of the medium, such as meanthickness of individual layers. Within the approach appliedin the present paper, the compressed radar pulse is

ARTICLE IN PRESS

Martian crust H2O

Martian polar cap

Orbiting spacecraft

Layered deposits

Fig. 1. The general view of the experimental geometry.

Ya.A. Ilyushin et al. / Planetary and Space Science 53 (2005) 1427–14361428

regarded as a short plane wave packet, reflecting from, andpenetrating through, the interfaces between layers. It iscorrect from the mathematical point of view because, forthe simulation purposes, the compression of the pulse andwave propagation through medium can be interchanged.The martian polar deposits are characterized by acomplicated random layered structure (Clifford et al.,2000). It has been shown by Milkovich and Head (2005)that these layers are nearly horizontal (a systematictilt about 0.5 degree has been found) and wide-spread,because the layer patterns on bordering scarps ofdifferent polar troughs correlate very well. Assuming theselayered structures to have many internal parallel interfacesreflecting the sounding wave, one can consider the limitingcase of continuously scattering medium. The theory ofradiative transfer is a well developed tool for treating suchproblems.

2. The scattering medium model

The macroscopic parameters of the medium, appearingin radiative transfer theory, are the volume extinctioncoefficient k, the single scattering albedo l, and the energypropagation velocity c. These parameters for real polarlayered deposits of Mars should be derived from thephysical and chemical composition of the deposits and thegeometrical parameters of their structures. We essentiallyexploit the model of the medium, as introduced in therecent paper by Ilyushin (2004). The medium of ice,perhaps with some small amount of dust, contains manythin parallel horizontal layers, rich in dust compared withthe surrounding ice. In fact, the medium is a stack ofinterlaced parallel layers of two types, which we denote as‘‘icy’’ and ‘‘dusty’’, or ‘‘type 1’’ and ‘‘type 2’’, respectively.All the quantities, indexed as ‘‘1’’ or ‘‘2’’, are related to thelayers of the corresponding type.

The problems of propagation of waves and particles inquasi-periodical structures are well known, for example, insolid state theory (Ashkroft and Mermin, 1976). However,relatively few cases allow analytical solution; in general,

numerical calculations are required. For our purposes, wehave to make several additional assumptions concerningthe geometry of a layered medium, justifying our simplifiedtreatment of the problem. All the propositions made hereshould be verified by calculations. The dusty layersare expected to be thin compared to the width of thecompressed radar pulse dT2 ¼ 1=Bn02, where B is the radarlinear FM (LFM) pulse bandwidth and n2 ¼ n02 þ in002 isthe refractive index of the dusty layer. According tothis requirement, the reflection and transmission coeffi-cients of every individual layer can be regarded as constantwithin the frequency band LFM pulse. All the dusty layersin our model have the same thickness l2. The thickness oficy layers, on the contrary, are believed to be largecompared to dT1 ¼ 1=Bn1 on average, and distributedrandomly with variance also greater then dT1. Thisassumption should guarantee that all the dusty layerswithin an icy medium scatter the radar pulses indepen-dently and incoherently. Coherent effects in scatteringin layered media lead to a periodic structure of thefrequency response of the medium, as has been discussedby Ilyushin (2004).Now consider the process of scattering of electromag-

netic waves by an individual dusty layer embedded in theicy medium with complex refractive index n1 ¼ n01 þ in001.The normal waves, propagating in the icy medium, aree�ik1z, where the complex wave number k1 ¼ k01 þ ik001 ¼2pfn1=c0, c0 ¼ 2:9979� 108 m/s—light speed in a vacuum.The normal waves within the dusty layers are e�ik2z,k2 ¼ k02 þ ik002 ¼ 2pfn2=c0, n2 ¼ n02 þ in002—complex refrac-tive index of the dusty layers. Consider the problem of theincidence of a normal wave in an icy medium of unityamplitude onto the dusty layer, centered at z ¼ 0. Thesolution of this problem is well known (Brekhovskikh,1980). However, it is not invariant to the position ofthe embedded layer, because the normal waves in thesurrounding medium are attenuating. It is worth, therefore,deriving a solution with normal waves normalized to unityat the middle of the layer.Above the dusty layer both incident and reflected waves

eþik1z þ A�e�ik1z are present, while beneath the layer onlythe propagated wave survives Cþeþik1z. Within the layerthere are both waves Bþeþik2z þ B�e�ik2z going in oppositedirections. The wave solution should be continuous with itsfirst derivative, so we can write down the system ofequations, which require this continuity at the boundariesz ¼ �l2=2:

e�i2 k1l2 þ A�e

i2k1l2 ¼ Bþe�

i2k2l2 þ B�e

i2k2l2 , (1)

ik1e� i

2k1l2 � iA�ei2k1l2k1 ¼ iBþk2e

�i2 k2l2 � iB�e

i2k2l2k2, (2)

Cþei2k1l2 ¼ B�e�

i2k2l2 þ Bþe

i2k2l2 , (3)

iCþei2k1l2k1 ¼ �iB

�k2e� i

2k2l2 þ iBþei2k2l2k2. (4)

The amplitude of the normal wave, reflected from the layer,is A�; the amplitude of the wave penetrated through the

ARTICLE IN PRESS

10 20 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

4

8

12

16

f, MHz

t, µs

Fig. 4. Group delay of the wave penetrated through a particular layer.

0.03

0.04

0.05

0.06

0.07

8

12

16

t, µs

Ya.A. Ilyushin et al. / Planetary and Space Science 53 (2005) 1427–1436 1429

layer is Cþ. The solution to the system of Eqs. (1)–(4) is

A� ¼e�il2k1 ð�1þ e2il2k2 Þðk2

1 � k22Þ

ð�1þ e2ik2l2 Þðk21 þ k2

2Þ � 2ð1þ e2ik2l2Þk1k2

, (5)

Cþ ¼�4k1k2e

�iðk1�k2Þl2

ð�1þ e2ik2l2 Þðk21 þ k2

2Þ � 2ð1þ e2ik2l2Þk1k2

. (6)

The plots of jCþj2 and jA�j2, i.e. energy of the reflectedand penetrated waves, are shown in Figs. 2 and 3,respectively. Each group of curves is labeled by the valuesof e02 and has five values of tan d2 ranging from 0 to 0.05ðtan d1 ¼ 0:1 tan d2Þ. The group delays of these waves dueto the dusty layer, i.e. d argðCþÞ=do and d argðA�Þ=do areshown in Figs. 4 and 5. One can see that the amplitudes donot abruptly change within the LFM bands, so thedegradation of the radar pulse shape due to behavior ofthe amplitude would hardly be expected. The group delay,due to interaction of the wave with the dusty layer can beassumed to be small compared to travel time between twointeractions. Henceforth we shall assume the energytransfer velocity to equal c ¼ c0=n01. Dispersion coefficientsof the wave d2 argðCþÞ=do2 and d2 argðA�Þ=do2 are shownin Figs. 6 and 7. It is clearly seen that the dispersion

10 20 30

0.6

0.7

0.8

0.9

1 4

8

12

16

|C+|2

f, MHz

Fig. 2. Absolute value of jCþj2.

10 20 30

0.1

0.2

0.3

0.4

0.5

4

8

12

16

f, MHz

|A- |2

Fig. 3. Absolute value of jA�j2.

10 20 30

0.01

0.02

4

f, MHz

Fig. 5. Group delay of the wave reflected from a particular layer.

10 20 30

f, MHz

-0.004

-0.002

0.002

0.004

4 8

12

16

d2 |A

- |/dω

2 , M

Hz-2

Fig. 6. Dispersion of the wave reflected from a particular layer.

10 20 30

-0.004

-0.002

0.002

0.004

48

12

16

f, MHz

d2 |C

+|/d

ω2 ,

MH

z-2

Fig. 7. Dispersion of the wave penetrated through a particular layer.

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0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

λ

4

5

6

10

15

l2, m

Fig. 8. The single scattering albedo l.

0.2 0.4 0.6 0.8 1l2, m

0.005

0.01

0.015

0.02

κ, d

B/m

45

67

89

1011

1213

1415

16

Fig. 9. The volume extinction coefficient k.

Ya.A. Ilyushin et al. / Planetary and Space Science 53 (2005) 1427–14361430

coefficients, even accumulated after hundreds of interac-tions, are small compared to the values necessary fornotable phase distortion of LFM chirp pulse. Thus, thedegradation of the LFM pulse in the single scattering eventcan be neglected. This means that the signal propagationthrough this medium is dominated by the spatial distribu-tion of scattering (dusty) layers, rather than by the spectralfeatures of a single scattering event.

By definition, the volume extinction coefficient k is anexponent in the Beer’s law

I ¼ I0e�kz (7)

regulating the decrease of the incident stationary intensitywith depth. In particular, for a homogeneous lossy medium(ice) characterized by the complex wave number k1, it isk ¼ 2 Imðn1=c0Þ ¼ 2k001. For another idealized medium(ice without loss with embedded thin lossy layers ofdust), it can be defined as k ¼ 2 lnCþ=hli, where hli isthe characteristic mean period of the dusty layers andCþ is the amplitude transmission coefficient defined byexpression (6). The single scattering albedo l should bedefined as the ratio of the backscattered energy to thefull energy removed from the flux due to scattering on asingle layer:

l ¼jA�j2

1� jCþj2. (8)

To define l for lossy ice with imbedded dusty layers, wehave to calculate the portion of energy removed from theflux by the dusty layers, and then calculate a portion ofscattered energy within it. Keeping in mind that the normalwaves appearing in Eqs. (1)–(4) are attenuating waves, theeffective optical thickness of the dusty layer, i.e. thelogarithm of attenuation imposed by it, can be defined ast2eff ¼ 2k001l2 � 2 ln jCþj. We assume the relative contribu-tion of the dusty layer in the attenuation over the wholeperiod l1 þ l2 is to equal the ratio of effective opticalthicknesses

t2efft1eff þ t2eff

¼2k001 l2 � 2 ln jCþj

2k001ðl1 þ l2Þ � 2 ln jCþj. (9)

In the martian polar layered deposits typically l1bl2.Combining expressions (8) and (9), we get an approximatedefinition of the effective single scattering albedo for ourmedium:

l �2k001 l2 � 2 ln jCþj

ð2k001hl1i � 2 ln jCþjÞ

jA�j2

ð1� jCþj2Þ. (10)

The effective volume attenuation coefficient can be definedas

k ¼t1eff þ t2effhl1i þ l2

� 2k001 �2 ln jCþj

hl1i. (11)

The families of plots of single scattering albedoand volume extinction coefficients at the SHARADcenter frequency f ¼ 20MHz are shown in Figs. 8 and 9,respectively. The following parameters were chosen:

e01 ¼ 3:15, e02 ¼ 4 . . . 16, tan d1 ¼ 0:0005, tan d2 ¼ 0:005,l1 ¼ 30m.

3. Semi-infinite medium

Due to the geometry of the nadir-looking radar soundingexperiment (Fig. 1), it is commonly assumed that thesounding wavefront in the medium is locally plane andparallel to the surface. The radar pulse with a planewavefront travels in the medium in the vertical direction.The only inhomogeneities in the medium are horizontalplane-parallel interfaces, so the scattering process in themedium conserves the vertical direction of the pulsepropagation. Thus, the medium is exactly described by atrue one-dimensional model, and there is no need for theangular variable in the radiative transfer equations. It isargued above that the scattering process can be regarded tobe instantaneous, and always backwards, because in theone-dimensional model it is not necessary to account forthe forward scattering explicitly.

ARTICLE IN PRESS

Fig. 10. The inverse Laplace contour.

Ya.A. Ilyushin et al. / Planetary and Space Science 53 (2005) 1427–1436 1431

Thus we start from the system of one-dimensional non-stationary radiative transfer equations (Nagirner, 1974):

qIþðy; tÞqy

þqIþðy; tÞ

qt¼ �Iþðy; tÞ þ lI�ðy; tÞ, (12)

qI�ðy; tÞqy

�qI�ðy; tÞ

qt¼ �I�ðy; tÞ þ lIþðy; tÞ, (13)

where dt ¼ kdz is the optical thickness, dy ¼ kcdt is thenormalized (dimensionless) time, z is the depth, k is thevolume extinction coefficient, l is the single scatteringalbedo, Iþ is the incoming intensity (going in the positivedirection of z), I� is the outgoing intensity (in the negativedirection of z), and c is the propagation velocity of theradiative energy in the medium. We assume all theproperties of the medium to be constant, i.e. k ¼ const,c ¼ const, l ¼ const. The initial conditions for bothintensities are

I�ð0; tÞ ¼ 0. (14)

Both solutions are bounded at infinity

I�ðy;1Þo1, (15)

that is sufficient for existence of their Laplace transform inthe right complex half-plane. The short sounding pulsecomes to the front surface at the moment t ¼ t0. Neglectingthe reflection of radiation from the surface, we can writedown

Iþðy; 0Þ ¼ dðt� t0Þ ¼ kcdðy� y0Þ, (16)

where dð�Þ is the Dirac delta-function. The case of thereflecting boundary will be treated later. Applying theLaplace transform

F ðsÞ ¼

Z 10

f ðyÞe�sy dy (17)

with respect to y in the system of Eqs. (12) and (13) withthe initial conditions (14) and boundary conditions(15, 16), we obtain a system of ordinary differentialequations for the Laplace transforms of I�:

ð1þ sÞ ~Iþðs; tÞ þ

qqt

~Iþðs; tÞ ¼ l ~I

�ðs; tÞ, (18)

ð1þ sÞ ~I�ðs; tÞ �

qqt

~I�ðs; tÞ ¼ l ~I

þðs; tÞ, (19)

~Iþðs; 0Þ ¼ kce�sy0 , (20)

~I�ðs;1Þo1. (21)

The time of arrival of the sounding pulse y0 can beset equal to 0 without loss of generality. The solutionof Eqs. (18)–(19), satisfying the boundary conditions(20–21), is

~Iþðs; tÞ ¼ kce�Qt, (22)

~I�ðs; tÞ ¼ kc

l1þ sþQ

e�Qt, (23)

where

Q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2 þ 2sþ s2

p. (24)

To get the outgoing intensity I�ðy; 0Þ, we have to performan inversion of the Laplace transform

I�ðy; 0Þ ¼1

2pi

I~I�ðs; tÞesy ds (25)

where the integral is calculated over the straight line ðg�i1; gþ i1Þ as shown in Fig. 10, g is an arbitrary positiveconstant chosen so that the contour of integration lies tothe right of all the singularities of ~Iðs; tÞ. The onlysingularities of the integrand are its branch pointss ¼ �1� l, so we must isolate the branch cut. Thus theinverse Laplace contour is equivalent to the loop aroundthe branch cut with its endpoints, as shown in Fig. 10. Thesquare root Q gets different signs on different sides of thebranch cut. It is not possible to evaluate this integralexactly in closed form; however, one can calculate itsasymptotic values at small and large times:I

~I�ðs; 0Þesy ds � kc

4

3yl2; y! 0, (26)

I~I�ðs; 0Þesy ds � kc

y�3=2ffiffiffiffiffiffiffiffi2plp e�ð1�lÞy; y!1. (27)

One can construct a simple approximate expression

I�ðy; 0Þ � kcyeð�1þlÞy

3

4l2þ

ffiffiffiffiffiffiffiffi2plp

y52

(28)

whose asymptotic behavior is given by (26) and (27) atsmall and large times, respectively. One should keep inmind that the asymptotic solution (26) at small times,strictly speaking, is not valid, because the radar pulse has

ARTICLE IN PRESS

Re

Im

Fig. 11. The poles of expression (31).

Ya.A. Ilyushin et al. / Planetary and Space Science 53 (2005) 1427–14361432

not yet been scattered enough and the field structure is stillcoherent. However, the intermediate approximation (28)between small and large times provides reasonable agree-ment with coherent simulations of the signal.

4. A medium with a reflecting boundary

Now consider a medium with deep reflecting boundary,say the base of the polar cap, at the optical depth t0. Theintensities I� obey Eqs. (12)–(13) with the boundaryconditions (16) at the outer surface, initial conditions (14)and the condition

I�ðy; t0Þ ¼ RIþðy; t0Þ, (29)

at the base instead of the requirement of boundedness (15)at infinity. The ground is supposed to be a perfect specularscatterer, whose reflectivity R here is the intensity reflec-tion coefficient, which is the square of the amplitudereflection coefficient for normal wave incidence. Respec-tively, ~I

�ðs; t0Þ satisfies the system of Eqs. (18)– (20) and

~I�ðs; t0Þ ¼ R ~I

þðs; t0Þ. (30)

The solution of interest, satisfying all these equations, atthe surface is

~I�ðs; 0Þ ¼ kc

�RðQþ e2Qt0 ðQ� sÞ þ sÞ þ l� le2Qt0

s�Q� Rl� ðQþ s� RlÞe2Qt0,

(31)

where s ¼ sþ 1 and Q is defined in (24). Letting t0approach infinity and putting R ¼ 0 in (31), one retrievesthe solution for the semi-infinite medium (23). One can seethat the expression (31) is invariant to our choice of thebranch of the square root Q! ð�QÞ, so it has no branchpoints. However, there are poles obeying the transcenden-tal equation

expð2Qt0Þ ¼s�Q� RlsþQ� Rl

. (32)

The poles are all in the left part of the complex planeas is shown in Fig. 11 . Note that both axes are scaledarbitrarily. All the poles s�n are located near the twoexponential asymptotes:1

Im s�n � �Rl2

expð2t0ð�1� Re s�nÞÞ. (33)

The rightmost pole s0 lies on the real axis. By moving theLaplace contour in (25) leftwards, one can move it behindall the poles. It can be shown that the series of residues atall the poles converge. In this case asymptotic behavior ofI�ðy; 0Þ at large times ybt0 is determined by the residue atthe rightmost pole s0.

I�ðy; 0Þ � kc Re s½esyI�ðs; 0Þ�js¼s0 . (34)

One can show that this asymptotic expression is valid forthe times large enough, namely y42t0. From the physical

1See Appendix A.

point of view, this means that the expression is not validuntil the incident pulse, reflected from the deep boundary,comes back to the surface. At shorter times, yo2t0, thesolution for semi-infinite medium (22,23) is valid. When themedium is optically thick enough, i.e. t0 is on the order of 1or more, the pole s0 approaches �1� l. In this case, thesolution also approaches (22,23).

5. A medium with two reflecting boundaries

Let us now consider a layer of optical thickness t0 withthe intensity reflection coefficients R1 and R2 at the outerand inner boundaries, respectively. The intensity transmis-sion coefficients are T1 ¼ 1� R1 and T2 ¼ 1� R2, regard-less of the direction of the flux. At the time y ¼ 0, theincoming pulse reaches the outer boundary. To treat theproblem, we explicitly extract an incident radiation fromthe solution, as is commonly done in radiative transfertheory. The incident d-pulse, traveling back and forth afterreflection from the inner and outer boundaries, is

Iþ0 ðy; tÞ ¼ T1ðR1R2Þne�t�2nt0dðtþ 2nt0 � yÞ,

2nt0oyoð2nþ 1Þt0; 0 otherwise, ð35Þ

I�0 ðy; tÞ ¼ T1R2ðR1R2Þnet�ð2þ2nÞt0dðt� ð2þ 2nÞt0 þ yÞ,

ð2nþ 1Þt0oyoð2nþ 2Þt0; 0 otherwise. ð36Þ

The initial conditions for I�ðy; tÞ are (14). The correspond-ing inhomogeneous equations for the Laplace transformsof the diffuse radiation intensities I� are

ð1þ sÞ ~Iþðs; tÞ þ

qqt

~Iþðs; tÞ ¼ l ~I

�ðs; tÞ þ l ~I

0 ðs; tÞ, (37)

ð1þ sÞ ~I�ðs; tÞ �

qqt

~I�ðs; tÞ ¼ l ~I

þðs; tÞ þ l ~I

þ

0 ðs; tÞ, (38)

with the boundary conditions

~I�ðs; t0Þ ¼ R2

~Iþðs; t0Þ, (39)

~Iþðs; 0Þ ¼ R1

~I�ðs; 0Þ. (40)

ARTICLE IN PRESSYa.A. Ilyushin et al. / Planetary and Space Science 53 (2005) 1427–1436 1433

The Laplace transforms of (35) and (36) are

~Iþ

0 ðs; tÞ ¼T1e

�st

1� R1R2e�2st0, (41)

~I�

0 ðs; tÞ ¼R2T1e

sðt�t0Þ

1� R1R2e�2st0, (42)

where s ¼ 1þ s. The solution of these equations at theouter boundary is

0.00002 0.00004 0.00006 0.00008 0.0001t, µs

-100

-60

-40

-20

dB

MARSIS Band II 2.5 -3.5 MHz

0

0.02

0.04

noise levels

-80

Fig. 13. Exact and asymptotic intensity plots for MARSIS band II.

~I�ðs; 0Þ ¼

est0 ð1þ eQt0ÞT1ðð1þ eQt0 Þð�1þ est0 ÞQR2 þ ð�1þ eQt0Þðest0ð�R2sþ lÞ � R2sþ R22lÞÞ

ðe2st0 � R1R2Þð�ðð1þ e2Qt0 ÞQð�1þ R1R2ÞÞ þ ð�1þ e2Qt0Þðsþ R1R2s� ðR1 þ R2ÞlÞÞ. (43)

The poles of this solution are shown in Fig. 12. At largetimes y42t0, the asymptotic behavior of I�ðy; 0Þ isdetermined by the rightmost pole of (43), which lies onthe real axis and can be easily found numerically. When t0is large enough, the solution at large times approaches onefor semi-infinite medium (27).

The Eqs. (37)–(38) can be also applied to the semi-infinitemedium with a reflecting boundary. The Laplace transformof the incident pulse Iþ0 ðy; tÞ ¼ ð1� R1Þe

�tdðy� tÞ is

~Iþ

0 ðs; tÞ ¼ ð1� R1Þe�st, (44)

while ~I�

0 ðs; tÞ ¼ 0. A corresponding solution of theseequations, satisfying (40) at the surface and limited atinfinity (15), at the surface is

~I�ðs; 0Þ ¼

ð1� R1Þ2l

s� R1lþQ. (45)

The only pole of (45) is s ¼ ð�2R1 þ ð1þ R21ÞlÞ=ð2R1Þ, and

it lies on the branch Qo0, which we do not address. Thusthe right end of the branch cut (Fig. 10) makes the principalcontribution to the integral (25). One can show that itleads to the asymptotic solution (27) at large times, notdepending on R1.

Re

Im

Fig. 12. The poles of expression (43).

6. Numerical simulations and discussion

For several of the most expected situations, the radarsignals were simulated numerically and compared to theasymptotic solutions derived above. If the total opticalthickness of the whole ice sheet is great enough, theasymptotic solutions for the semi-infinite medium shouldbe expected to work well. The signals for Band II of theMARSIS instrument (Picardi et al., 2004) and the

SHARAD instrument (Seu et al., 2004) for the northernpolar cap are shown in Figs. 13 and 14, respectively.The following parameters of the model were chosen(Ilyushin, 2004): l1 ¼ 14 . . . 45m (distributed uniformly,provided hl1i � 30m), l2 ¼ 1m, e1 ¼ 3:15 ð1þ i tan d1Þ,e2 ¼ 9 ð1þ i tan d2Þ. The loss tangents are varied, providedtan d2 ¼ 10 tan d1. The thickness values of the icy layers ofthe northern cap, estimated early on to range between 14and 45m (Blasius et al., 1982), have recently beenconfirmed by Milkovich and Head (2005). A value of 1mhas previously been suggested for the thickness of the dusty

2 4 6 8 10 12t, µs

-20

-40

-60

-80

dB

SHARAD 15-25 MHz

0

0.025

0.05

noise level

Fig. 14. Exact and asymptotic intensity plots for SHARAD.

ARTICLE IN PRESS

20 40 60 80 100 120t, µs

-140

-120

-100

-80

-60

-40

-20

dB

MARSIS Band III 3.5-4.5 MHz South Pole

ice

water

noise levels

Fig. 15. Exact and asymptotic intensity plots for south pole (MARSIS).

Ya.A. Ilyushin et al. / Planetary and Space Science 53 (2005) 1427–14361434

layers (Malin, 1986). Recently, Skorov et al. (2001) hasshown that a layer of fine dust as thin as 5mm cancompletely stop further sublimation of ice beneath it undermartian polar conditions. This would imply that the dustylayers could be very thin. However, MOC high resolutionimages (Thomas et al., 2000) show some surface roughnessof the northern polar cap at the meter scale. The physicalprocesses responsible for this roughness might cause somelocal resurfacing processes at these scales, leading to theformation of a meter-thick layer, enriched with dust. Forthese reasons, the value l2 ¼ 1mhas been preserved in thepresent simulations. The whole thickness of the model capis about 3600m, which is about the maximum thickness atthe center of the northern polar cap (Clifford et al., 2000).Solid curves are numerically simulated signals (Ilyushin,2004), dashed curves are radiative transfer asymptoticsolutions. Each pair of curves is labeled by the value of losstangent of dusty layers tan d2. The results of the twoapproaches are in good agreement, even at large timesy42t0, i.e. after the expected basal echo ðt � 40msÞ,because the cap is optically thick enough. The dashedgridlines show the estimated noise levels (Ilyushin, 2004)for minimal and maximal operating height of MARSIS.For SHARAD, the characteristic noise level has beenestimated in the same manner for the following values ofparameters (Seu et al., 2004): spacecraft height z ¼ 300 km,transmitted power P ¼ 10W, radar antenna gain G ¼

2:1 dB (dipole antenna), B ¼ 10MHz, number of radarpulses in the synthetic aperture NA ¼ 300.2

For finite slabs of moderate optical thickness, as is thecase for southern polar cap at low frequencies, theasymptotic solutions for the finite slabs should work.Simulations for MARSIS Band III (3.5–4.5MHz) for thesouthern polar cap are shown in Fig. 15. Two types ofsubstrate were tested: pure ice ðe ¼ e1Þ and pure fresh watere � 80. The following parameters were chosen: l1 distrib-uted uniformly in the range ¼ 80 . . . 120m (Herkenhoffand Murray, 1990), total thickness about 1900m (typicalvalue for peripheral areas of the southern polar cap), l2; e1

2Estimated from PRF � 700Hz.

and e2 are the same as for the northern cap (providedtan d2 ¼ 0:005). Power reflection coefficients R1 and R2 atthe boundaries are the squares of the amplitude reflectioncoefficients, which are in turn calculated by the classicalimpedance formula RA ¼ ðZ � 1Þ=ðZ þ 1Þ. The asymptoticsolutions of radiative transfer equations are shown in thesame figure: the one (34) for a single reflecting boundary(thin dashed line) and another one (43) accounting forreflection from both boundaries (thick dotted line). Despitethe systematic discrepancy in absolute power level due tothe neglected residues at the poles, one can see goodagreement in the rate of the temporal decrease of thesimulated signals and the asymptotic solutions. Thesolution (43), accounting for both boundary reflections, isin better agreement with the simulated signal.

7. Conclusions and remarks

The incoherent radiative transfer theory provides aneffective tool for estimating unknown electromagneticproperties of such a complicated medium as the martianpolar layered deposits. The trend of a temporal decrease ofthe signal power provides independent estimates of volumeextinction coefficient k and single scattering albedo l, fromwhich the physical properties of the medium can beretrieved by means of formulae (10) and (11). Formoderate values of total optical thickness of the medium,the nature of underlying substrate can be found from thesignal decrease through the asymptotic solutions (34) and(43). The theory applied in this paper may not work wellfor very thin dusty layers. However, in this case the capshould be very transparent to radar waves, as has beenshown by Ilyushin (2004), so the basal echo should be quitedistinctive and informative.The approach presented in this paper is, after minor

modifications, also applicable for qualitative interpretationof sounding with landed radar instruments (Barbin et al.,1995; Berthelier et al., 2000; Vannaroni et al., 2004). Thiswill be done in a separate paper.We have not yet considered the ionospheric phase

distortion and side clutter, which are two of the mostimportant factors limiting the capabilities of orbital radarinstruments. Techniques of compensation of ionosphericphase shift have been extensively studied in a number ofpapers (Armand et al., 2003; Safaeinili et al., 2003; Ilyushinand Kunitsyn, 2004). For the SHARAD instrument thisproblem should be relatively simple, since the operatingfrequency well exceeds the plasma frequency of the martianionosphere (Armand et al., 2003).The side clutter problem has also been extensively

studied (Picardi et al., 2004; Seu et al., 2004). For thenorthern martian polar cap, the clutter situation should notbe very bad, as the surface is very smooth at hundred meterand larger scales (Kreslavsky and Head, 2000; Malamudand Turcotte, 2001). The latest MOC high resolutionimages show some roughness (Thomas et al., 2000) at themeter scale. The characteristic height variation of 2m,

ARTICLE IN PRESSYa.A. Ilyushin et al. / Planetary and Space Science 53 (2005) 1427–1436 1435

estimated from the images by these authors, should notcause too much clutter, at least at MARSIS frequencies.The clutter produced by large-scale features of the northerncap has been recently estimated (Ilyushin, 2004).

Acknowledgements

One of the authors (Ya.I) is grateful to D.V. Kartashovand O.B. Shchuko for valuable comments on the manu-script and Prof. V.P. Boudak and Dr. S. Byrne for helpfuldiscussions. The authors also thank the anonymousreviewer for the constructive review.

Appendix A

When jsj ! 1, Eq. (32) takes the asymptotic form

e2t0s � �Rl2s

, (46)

where s ¼ sþ 1. Substituting s ¼ xþ iy, one obtains theequation for the absolute value:

~I�

I ðy; 0Þ ¼QIIðð�1þ e2QIt0ÞsðlI � lIIÞ þ ð1þ e2QIt0ÞQIlIIÞ þ ð�1þ e2QIt0 ÞlIðs2 � l2IIÞ

ð1þ e2QIt0ÞQIðQIIsþ s2 � l2IIÞ þ ð�1þ e2QIt0Þðs3 � sl2II þQIIðs2 � lIlIIÞÞ, (51)

e2t0x �Rl

2jxþ iyj. (47)

Formula (33) is the approximate solution of Eq. (47) forlarge jyj. Making use of (33) and (47), one can show thatthe residues rn ¼ Re s½I�ðsn; 0Þ� of expression (31) at thepoles sn at large jsj are asymptotically

jrnj / esny sn

lt0

��������. (48)

Thus, the series of the residues rn up to an arbitraryconstant factor is uniformly exceeded by the expression

S ¼X

n

jsnesnyj. (49)

Substituting (33) into (49), one can prove convergence ofthe latter according to D’Alembert ratio test (Bromwichand MacRobert, 1991) at large times y42t0.

Appendix B. A slab lying on a semi-infinite medium

For the general case of arbitrary depth profiles ofmedium parameters lðtÞ;kðtÞ, and cðtÞ the radiativetransfer equations should be solved numerically. Only afew particular cases allow analytical treatment. Thus,reconstruction of depth profiles of these parameters is ingeneral very complicated. However, analysis of high-resolution images of the northern polar cap (Milkovichand Head, 2005) suggests a model of piecewise constantparameters of the observed layer periodicity is adequate.

For this reason, we present results of the simulations for amedium of such type, thus showing a way for interpreta-tion of sounding data. Consider a slab of the layeredmedium, lying on the semi-infinite medium, both char-acterized by different effective parameters lI;II; kI;II, andcI;II, indexed as ‘‘I’’ and ‘‘II’’ for the slab and the mediumbeneath, respectively. For the sake of simplicity, we putkI ¼ kII ¼ k and cI ¼ cII ¼ c. In this case yI ¼ yII ¼ y. Theoptical thickness of the whole slab is t0. The intensitiesI�I;IIðy; tÞ in both media satisfy the system of Eqs. (12) and(13) with l ¼ lI and l ¼ lII, respectively. At the surface,IþI ðy; tÞ satisfies the boundary condition (16). At infinitedepth, I�IIðy; tÞ are finite. At the interface between twomedia I and II, the intensities should be continuous:

I�I ðy; t0Þ ¼ I�IIðy; t0Þ. (50)

All the intensities satisfy the zero initial condition (14).Applying the Laplace transform to all the equationsmentioned above, one gets the system of ordinaryequations for ~I

I;IIðs; tÞ. The solution of interest to us is

where s ¼ 1þ s, QI ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 � l2I

q, and QII ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 � l2II

q. We

have already seen that for an infinite medium the structureof the solution is determined by the branch points ofI�ðs; tÞ, while for a finite medium it is determined by itspoles. This solution is invariant with respect to changeQI ! ð�QIÞ, but it has branch points s ¼ �1� lII due toQII. The branch cut is drawn between these two branchpoints. The analysis of the structure of the poles of (51) iscumbersome. However, from the physical point of view it isobvious that at times much greater in comparison withtime of propagation through the slab, the temporaldecrease of the intensity should be determined by thesemi-infinite medium. This in turn means that there are nopoles rightwards from s ¼ �1þ lII. This can be easilychecked numerically. Integrating along the path around thebranch points, such as shown in Fig. 10, one gets theasymptotic contribution from the point s ¼ �1þ lII atgreat times y:

I�I ðy; 0Þ ¼kcy�

32eð�1þlIIÞyðlI þ lIIÞffiffiffiffiffiffiffiffiffiffiffi

2plIIp

ðlI þ lII coshð2t0LÞ þ L sinhð2t0LÞÞ,

(52)

where L ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�l2I þ l2II

q. At small times yo2t0, the

asymptotic solution for semi-infinite medium I is valid.

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