Subsurface radar location of the putative ocean on Ganymede: Numerical simulation of the surface terrain impact

Planetary and Space Science 92 (2014) 121–126

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Subsurface radar location of the putative ocean on Ganymede: Numerical simulation of the surface terrain impact Ya.A. Ilyushin n Atmospheric Physics Department, Physical Faculty, Moscow State University, 119992 GSP-2 Lengory, Moscow, Russia

art ic l e i nf o

a b s t r a c t

Article history: Received 4 October 2013 Received in revised form 6 January 2014 Accepted 24 January 2014 Available online 5 February 2014

Exploration of subsurface oceans on Jupiter's icy moons is a key issue of the icy moons' geology. Radar is in fact the only sounding technique which is able to penetrate their icy mantles, which can be many kilometers thick. Surface clutter, i.e. scattering of the radio waves on the rough surface, is known to be one of the most important problems of subsurface radar probing. Adequate numerical modeling of this scattering is required on all stages of subsurface radar experiment, including design of an instrument, operational strategy planning and data interpretation. In the present paper, a computer simulation technique for numerical simulations of radar sounding of rough surfaces is formulated in general form. Subsurface radar location of the ocean beneath Ganymedian ice with chirp radar signals has been simulated. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Ground penetrating radar Ganymede Ice Ocean Surface clutter Scattering

1. Introduction Geological models of Jupiter's icy moons predict oceans of liquid water beneath their icy shells, due to radiogenic and tidal heating (Spohn and Schubert, 2003). At present, there are several evidences of these subsurface oceans, e.g. magnetometric data (Zimmer et al., 2000), grooved terrain (Carr et al., 1998), and so on. Immediate proof of the presence of liquid water beneath the ice would very much extend our current knowledge of the icy moons geology. The radar wave absorption in the icy crust, measured in this experiment, might help us to constrain the physical and chemical composition of the bulk of ice (Chyba et al., 1998; Moore, 2000). Besides, since these oceans are thought to be potentially habitable, such proof would greatly support our hope to find live beings somewhere in our Solar system outside Earth. However, because these oceans are covered by thick icy layers, they are inaccessible for immediate observation and can be reached only by seismic or electromagnetic waves. While seismic measurements at these depths require powerful sources of waves and landed equipment, a small orbiter carrying the radar onboard can provide deep subsurface sounding with global coverage of the icy moon surface. So far, this approach has been successfully performed in Lunar (Peeples et al., 1978; Oshigami et al., 2009) and Martian (Picardi et al., 2005) research programs.

n

Tel.: þ 7 495 939 3252. E-mail addresses: [email protected], [email protected]

http://dx.doi.org/10.1016/j.pss.2014.01.019 0032-0633 & 2014 Elsevier Ltd. All rights reserved.

In this paper we attempt to assess feasibility of such experiment on Ganymede. Subsurface radar location has now numerous application in geology, geophysics, engineering, and so on. Working principle of this instrument is the transmission of wave packet, which propagates through the medium under investigation and comes back and is then received and analyzed (see Fig. 1). If the coherence of this wave packet is partially lost due to inhomogeneities of the medium, part of this energy is scattered away and does not reach the receiver, and the rest part of the wave packet is distorted. On the other hand, some part of the wave energy emitted by the radar to off nadir directions can be scattered from the rough surface back to the instrument. These are the so-called side clutter echoes, masking useful informative subsurface echo, as it is shown in Fig. 1. To improve the instrument performance, various signal processing techniques are utilized (focused and unfocused aperture synthesis, migration technique and so on). Accurate theoretical simulation of the experiment includes evaluation of the electromagnetic field, excited in the investigated medium by the transmitter and registered by the receiver, and application of all the data processing procedures implemented in real experiment. Depending on the chosen method of electromagnetic field simulation, this might be really time consuming. Evaluation of mean quantities, such as mean intensity, implies averaging over many realization of the signal. This greatly increases computational costs of the problem. Presentation of numerical technique for simulation of subsurface radar probing through rough surface with demonstration of some practical applications of it is the basic purpose of this paper.

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synthetic aperture

for transiting and reflected waves, respectively. Here Δk is the difference of wave numbers in two media, separated by the ith surface. Between the boundaries all the media are homogeneous, i.e. volume scattering is neglected. For nearly vertical wave propagation, in the paraxial approximation the complex amplitude of the transiting or reflected wave can be expressed in the form Z !0 !0 ! 0 0 Eðx; yÞ ¼ Ai expðiϕi ÞEð r ÞGð r ; r Þ dx dy ; ð3Þ

x3 ,x4

z1

y1 ,y2

x1 ,x2

! where r ¼ ðx; y; zÞ is an arbitrary point within the medium, !0 0 0 0 r ¼ ðx ; y ; z Þ is an integration point on the phase screen plane, Ai is the Fresnel amplitude reflection or transmission coefficient, respectively, ϕi is the random phase shift (1) or (2), introduced by the phase screen. The Green function of the Helmholtz equation in the paraxial approximation equals to ! 0 2 0 2   k z  z0  þikðx  x Þ þ ikðy  y Þ : exp ik Gij ¼ ð4Þ 2πijz z0 j 2jz  z0 j 2jz z0 j

z2 ice ocean

Fig. 1. Spherical surface of Ganymede.

sources

receivers

In the point ðx0 ; y0 ; z0 Þ a concentrated (point) source of the ! ! sounding radiation E0 δð r  r 0 Þ is located. Within the paraxial approximation, adopted in this study, the field generated by the source in the medium is Z !0 ! !0 ! 3 !0 ! ! ! ð5Þ Eð r Þ ¼ E0 δð r  r 0 ÞGð r ; r Þ d r ¼ E0 Gð r 0 ; r Þ: 1

Thus, complex amplitude of the wave field of frequency ω in the ! point r N þ 1 in the medium after N reflections from and transmissions through the rough boundary surfaces equals to Z Z N ! ð6Þ ∏ Ai expðiϕi ÞGi;i þ 1 dxi dyi ; Eω ð r N þ 1 Þ ¼ E 0 ⋯

2

i¼1

i+1

! ! where Gi;i þ 1  Gð r i ; r i þ 1 Þ. Taking an average of two fields (6) at different frequencies 〈Eω1 Enω2 〉, we obtain the two frequency correlation function of the field (Ishimaru, 1978) Z ! ! Γ ω1 ω2 ð r N þ 1 ; r 2N þ 2 Þ ¼ 〈Eω1 Enω2 〉 ¼ E0 En0 ⋯

n1

n2 i

Z

i¼1

where

n4

... Fig. 2. General scheme of phase screen technique applied for the calculations.

2. Phase screen technique for simulation of wave propagation One of the widely used approaches to the numerical simulation of wave propagation in inhomogeneous media is the phase screen technique (Ilyushin, 2009a; van de Kamp et al., 2009). The geometry of general model of multi-layered medium is shown in Fig. 2. Rough boundary surfaces, which separate the layers, are approximately modeled by random phase screens ð1Þ

2N

∏

i ¼ N þ2

N

2N

i¼1

i ¼ N þ2

∑ iϕi ðxi ; yi Þ 

exp

∑

Ani Gni;i þ 1 dxi dyi ;

ð7Þ

!+ iϕi ðxi ; yi Þ

ð8Þ

is the characteristic function of the joint distribution of all the random phases. In the case of multi-variable normal distribution of the phases, the characteristic function is (Davenport and Root, 1958) !      1 M ϕ ¼ exp ∑iϕi ¼ exp  ∑βij ðxi ; yi ; xj ; yj Þ ; ð9Þ 2 i;j where βij are the random phase covariations ϕi ðxi ; yi Þϕj ðxj ; yj Þ, accounting their signs in the formula (8). If aperture synthesis of some other averaging procedure is applied to the observed fields, the expressions (6) and (7) should be integrated over corresponding variables with proper weighting (aperture) functions. Similarly, higher order field correlation functions can be evaluated. After matched filtration, we get the temporal profile of the mean signal intensity 〈jEðtÞ2 j〉 p

Z

1 ð2πÞ

2

þ1 1

Z

þ1 1

Hðω1 ÞHðω2 ÞΓðω1 ; ω2 Þexpðiðω1  ω2 Þt Þ dω1 dω2 :

ð10Þ

and ω ϕi ¼ 2Δkhðxi ; yi Þ ¼ 2 Δnhðxi ; yi Þ; c

*

Mfϕg ¼

n5

ϕi ¼  2khðxi ; yi Þ

N

Mfϕg ∏ Ai expðiϕi ÞGi;i þ 1

n3

ð2Þ

An expression for the characteristic function Mfϕg (9), which appears in formula (7) for the two-frequency correlation function,

Ya.A. Ilyushin / Planetary and Space Science 92 (2014) 121–126

can be expanded in power series of the random phase covariations βij. It the correlation functions of the random phases, i.e. random surface roughness (Ostro, 1993), and the aperture functions (Ilyushin, 2009a,b, 2010) are Gaussian, integration of these series term by term results in the sum of terms of general form sffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! Z BT Aij 1 B πn n expð  Aij xi xj þ Bi xi Þ d x ¼ exp : ð11Þ det Aij 4 Therefore, the approach accounts for diffractional effects and correctly treats aperture synthesis. This technique has been recently applied for computer simulation of the subsurface radar sounding through inhomogeneous ionospheric plasma (Ilyushin, 2009b, 2009a, 2010), as well as sounding of rough subsurface horizon through surface terrain (Ilyushin et al., 2012).

3. Radar sounding of the subsurface ocean on Ganymede: model description and numerical simulations Ganymede is the largest of moons in the Solar system. Like some other satellites of giant planets, Ganymede is globally covered with ice. There is a hypothesis about the oceans beneath the ice on these satellites, which is supported by various observational data and simulation results. Getting an immediate proof of the presence of these oceans is an important task, which is being repeatedly discussed in connection with future planned interplanetary missions (Chyba et al., 1998; Moore, 2000; Berquin et al., 2013; Cecconi et al., 2012). Application of an orbital ground penetrating radar to the subsurface exploration of Ganymede is complicated because of rough surface (Squyres and Veverka, 1981, 1982). Recently, statistical characteristics of the Ganymedian terrain height distribution have been retrieved from the photoclinometry of high resolution images (Berquin et al., 2013). It proves that the Allan variances of the random height of the surface terrain for different geological units are well described by Gaussian correlation functions (Fig. 3). For numerical simulations, two cases were selected from the samples investigated by Berquin et al. (2013): Arbela Sulcus vicinity (sample 2) and the so-called bright terrains. Statistical parameters of the rough surface were estimated by fitting the graphic data from Berquin et al. (2013). Numerical values r 0 ¼ 760 m, hrms ¼ 182 m and r 0 ¼ 1420 m, hrms ¼ 383 m (correlation scale and height of the terrain) have been chosen for these two geological units, respectively. According to Berquin et al. (2013), ice thickness within the range 60–80 km is expected. Radar instruments, proposed for icy moon exploration (Bruzzone et al., 2011, 2013), are not suitable for log Δhrms, km 0.6 0.8 1 1.2

that deep sounding. An instrument capable to reach these depths would probably operate at lower frequencies (e.g. below one megahertz). In addition, there is a minimum of Jovian radio emissions at these frequencies (Bruzzone et al., 2013). In that frequency range, correlation scale of Ganymedian surface roughness is less than Fresnel zone size. Therefore, a technique accounting for diffraction is needed for practical simulation of the scattering on the surface terrain, e.g. the one presented in this paper. For the computer simulation, we accept value z2 ¼ 70 km. A standard value ɛ¼3.15 for the dielectric permittivity of the ice at radio frequencies (Gudmandssen, 1971) was assumed. Main focus of this paper is the loss of signal coherence due to scattering on the rough surface. Therefore, dielectric loss and volume scattering in the ice have not been considered, although loss in the bulk of the ice might be quite significant (Chyba et al., 1998; Moore, 2000). The ionospheric phase distortion of the chirp signal assumed to be well compensated by some correction algorithm (Ilyushin and Kunitsyn, 2004). However, our knowledge about Ganymedian ionosphere is at present somewhat uncertain. Available data analysis (Eviatar et al., 2001) suggests not so high values of electron concentration (plasma frequency about 100 kHz of less). Here the problem of subsurface sounding of hypothetic Ganymede ocean through the surface terrain will be considered with this technique. If the water surface beneath the ice is considered to be flat, the problem is mathematically equivalent to the ionospheric models developed in Ilyushin (2009a, 2010). Accounting for the roughness of the bottom of the icy layer complicates the calculations, but the basic algorithm remains principally the same (Ilyushin et al., 2012). Since we are interested in the assessment of signal coherence loss due to diurnal surface terrain, we consider the flat ocean surface to investigate purely the effect of our interest. Similarly, a surface clutter echo from the rough spherical surface of the celestial body can be evaluated (Fig. 1). Accounting for the curvature of the spherical surface, in Cartesian coordinates it can be approximated with an expression z ¼ hðx; yÞ 

x2 y2  ; 2R 2R

E1 ¼

k1 2πiz1

Z exp 2ik1 z1  2ikhðx1 ; y1 Þ þ ik1

0.25

0

0.25

0.5

0.75

1

1.25

1.5

! dx1 dy1 dx3 ;

where z1 is a height of spacecraft above the surface of Ganymede, k1 ¼ ω1 =c is the wavenumber of the given frequency in vacuum, c is the speed of light, x1 and y1 are Cartesian coordinates in the tangent plane to the spherical surface at a subsatellite point in the middle of the synthetic aperture, x3 is a coordinate of the spacecraft along the flight trajectory (approximated by a horizontal straight line along the x direction). Complex amplitude of the field at another frequency ω2 is expressed by the same formula (13) with another set of parameters. Two-frequency correlation function of the field is therefore

Z

Fig. 3. Model structure functions of random terrain height mimicking photoclinometric reconstructions of real Ganymedian terrain statistics (Berquin et al., 2013). Solid line – bright terrains (Voyager data), dotted line – Arbela sulcus vicinity (sample 2).

x2 y2 ðx1  x3 Þ2 þ ik1 1 þ ik1 1 z1 R R

ð13Þ

Γ ω1 ω2 ¼ 〈E1 En2 〉 ¼ log Δr, km

ð12Þ

where R is the radius of the sphere. Then, complex amplitude of the wave at the frequency ω1, reflected from the rough surface and received by an instrument, is expressed as

1.4 1.6

123

k1 k2 1 〈expð  2ikhðx1 ; y1 Þ þ 2ikhðx2 ; y2 ÞÞ〉 4π 2 z21 πL2

x2 y2 ðx1  x3 Þ2 ðx2  x4 Þ2  ik2 þ ik1 1 þik1 1 z1 z1 R R ! 2 2 2 2 x y ðx1  x3 Þ ðx2  x4 Þ  dx1 dx2 dx3 dx4 dy1 dy2 :  ik2 2 þ ik2 1  R R L2 L2 exp 2iðk1  k2 Þz1 þ ik1

ð14Þ

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Ya.A. Ilyushin / Planetary and Space Science 92 (2014) 121–126

(18), through formula (11), where the matrix Aij is

After averaging, we obtain

0

k1 k2 1 ¼ 〈E1 En2 〉 ¼ 2 2 2 4π z1 πL

Γ ω1 ω2 Z

expð2iðk1  k2 Þz1 þ ik1

x2 y2 ðx1  x3 Þ2 ðx2  x4 Þ2  ik2 þ ik1 1 þ ik1 1 z1 z1 R R

x2 y2 ðx1  x3 Þ2 ðx2  x4 Þ2 〈ϕ21 〉þ 〈ϕ22 〉  2〈ϕ1 ϕ2 〉  ik2 2 þ ik2 1    2 R R L2 L2 dx1 dx2 dx3 dx4 dy1 dy2 ;

2

〈hðx1 ; y1 Þhðx2 ; y2 Þ〉 ¼ 〈h 〉 exp 

ðx1  x2 Þ2 ðy1  y2 Þ2  r 20 r 20

!

an expression for the two frequency correlation function Γ ω1 ω2 can be expanded in the power series

βn n ¼ 0 n! 1

k1 k2 1  β e 4π 2 z21 πL2

Z exp 2iðk1  k2 Þz1 þ ik1

 ik2

x2 y2 ðx2 x4 Þ2 þ ik1 1 þ ik1 1 z1 R R

 ik2

x22 y2 ðx1  x3 Þ2 þ ik2 1  R R L2



ðx2  x4 Þ2 L2

ðx1  x3 Þ2 z1

nðx1  x2 Þ2 nðy1  y2 Þ2   r 20 r 20

! dx1 dx2 dx3 dx4 dy1 dy2 ; ð18Þ

2

where β ¼ 2k1 k2 〈h 〉 is the power series parameter. Integration in (18) yields a compact expression for the general term of the series P,dB

f 200

400 kHz r0 400

 rn2

 rn2

760 m hS 600

800

0

0

0

ik2 ik2 n R þ z1 þ r2

0

 ikz12

0

0

ik1 z1

0

1  ikz11 L2

0

0

0

0

 ikz12

0

ik2 1 z1 þ L2

0

0

0

 ikR1  ikz11 þ rn2

 rn2

 rn2

ik2 ik2 n R þ z1 þ r 2

0

0

0

0

0

0

0

0

0

0

0

0

1200

0

C C C C C C C C: C C C C C C A

0

ð19Þ Domain of validity of the expressions derived here is bounded below by the condition λ o r 0 (wavelength does not exceed a typical size of scatterers) and above by the practical convergence of the series.

4. Numerical results and discussion Numerical results within the validity domain of accepted approach (compressed chirp radar signals, power vs. delay time) are presented in Figs. 4–7 for two Ganymedian geological units discussed in Berquin et al. (2013). Different curves in each figure correspond to different heights. For checking, an undisturbed compressed chirp pulse is shown in all the figures together with the simulated signals. The 0 dB level corresponds to the undisturbed chirp pulse reflection from an ideally reflecting plane at the subsatellite point, without the window loss and other loss due to non-ideal numerical signal processing performed during the simulation (Harris, 1978). Following to Ilyushin (2009a), the Hanning window is used throughout the simulation. An optimal synthetic aperture length for the unfocused aperture synthesis (half size of the Fresnel zone (Klauder and et al., 1960)) is adopted. All the algorithms of chirp pulse processing for the simulation are the same as in the previous papers (Ilyushin et al., 2005; Ilyushin, 2007, 2009a). The subsurface echoes are clearly recognized in all the figures, because the signal coherence is not totally destroyed, while delay time between the surface and subsurface echoes is long due to large ice crust thickness expected for Ganymede (Berquin et al., 2013). The subsurface echo exceeds the echo coming from the front surface because of the great difference of these surfaces in radar wave reflectivity and neglected volume extinction in the bulk of ice. This is not something unexpected, and is not unusual in real radargrams, e.g. of Martian polar ice sheets (Lauro et al., 2012; Mouginot et al., 2010). P,dB

182 m 1000

1

ik1 z1

0

ð16Þ

and therefore Gaussian correlation functions of the random phases ! ðx1  x2 Þ2 ðy1  y2 Þ2 2  〈ϕ1 ϕ2 〉 ¼ 4k1 k2 〈h 〉exp  ð17Þ r 20 r 20

∑

 ikR1  ikz11 þ rn2

ð15Þ

where ϕ1 ¼ 2k1 hðx1 ; y1 Þ and ϕ2 ¼ 2k2 hðx2 ; y2 Þ are random phases, corresponding to local terrain heights in x1 ; y1 and x2 ; y2 respectively. For Gaussian correlation functions of the random heights

Γ ω1 ω2 ¼ 〈E1 En2 〉 ¼

B B B B B B B Aij ¼ B B B B B B B @

t,µs 1400 10

20

20

30

30

40

40

50

50

60

60

70

70

Fig. 4. Surface clutter and subsurface echoes (Arbela sulcus vicinity). Mean frequency of the chirp signal 400 kHz, chirp bandwidth 200 kHz. Thick, thin and dashed curves – the spacecraft height 16, 128 and 1024 km, respectively. Thin dotted curve – undisturbed compressed chirp signal.

800 kHz r0

200

10

80

f

400

760 m hS 600

800

182 m 1000

1200

t,µs 1400

80 Fig. 5. Surface clutter and subsurface echoes (Arbela sulcus vicinity). Mean frequency of the chirp signal 800 kHz, chirp bandwidth 200 kHz. Thick, thin and dashed curves – the spacecraft height 16, 128 and 1024 km, respectively. Thin dotted curve – undisturbed compressed chirp signal.

Ya.A. Ilyushin / Planetary and Space Science 92 (2014) 121–126

P,dB

f

200 kHz r0

200

400

1420 m hS 600

800

383 m 1000

1200

t, µs 1400

10

125

future space missions. The results reported in this paper might give some hope for successful detection of the subsurface echo coming from the ocean beneath Ganymedian ice in properly organized radar experiment.

20 30

Acknowledgements

40 50 60 70 80 Fig. 6. Surface clutter and subsurface echoes (bright terrains). Mean frequency of the chirp signal 200 kHz, chirp bandwidth 200 kHz. Thick, thin and dashed curves – the spacecraft height 16, 128 and 1024 km, respectively. Thin dotted curve – undisturbed compressed chirp signal.

P,dB

f

400 kHz r0

200

400

1420 mh S 600

800

383 m 1000

1200

t, µs 1400

10 20 30 40 50 60 70 80 Fig. 7. Surface clutter and subsurface echoes (bright terrains). Mean frequency of the chirp signal 400 kHz, chirp bandwidth 200 kHz. Thick, thin and dashed curves – the spacecraft height 16, 128 and 1024 km, respectively. Thin dotted curve – undisturbed compressed chirp signal.

Due to great difference between dielectric permittivities of ice and liquid water, the reflection coefficient of the ocean surface is close to unity. Thus, there are two main sources of uncertainty in the mode: roughness of bottom of the ice layer and attenuation of the signal in the bulk of ice. Therefore, the scenario suggested here should be regarded as very optimistic. However, we neglect both these uncertainties because in this study we concentrate at the loss of the surface and subsurface echo coherence due to rough diurnal surface of Ganymede.

5. Conclusions and remarks Practical technique for numerical simulations of subsurface radar location through rough surfaces is presented. Examples of application of proposed technique to deep radar probing of Ganymede are demonstrated. According to the results of the simulations performed here, degradation of the signal coherence at frequencies below 1 MHz is relatively moderate and does not prevent subsurface echo detection. More probably, it might be prevented by the wave absorption in the medium due to its dielectric losses, which also increase at higher frequencies and which have been neglected in this study. It can also be noted that the subsurface echo suffers from the surface scattering loss much less than the echo coming from the front surface. So, the numerical simulation performed here may pose some constraints or suggest some preferences for key parameters of the radar experiment (frequency, pulse duration, synthetic aperture length and so on) during the planning and operation of

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