direct electromagnetic wave propagation in planetary sounding radar 

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Introduction

Radar sounding has shown great success in the study of terrestrial and extra-terrestrial surfaces. Today, both Mars and the Moon have been probed by mean of ground-penetrating radars, which is one of the very few techniques that can remotely probe subsurfaces. On the Moon, deep interfaces have been observed in the mare which might correspond to the basement of these thick magmatic flows (Ono et al., 2009; Pommerol et al., 2010). On Mars, sounding of the polar caps was achieved with great success by both Marsis and Sharad radars, providing unique information on the structures and formation mechanisms of these deposits (Grima et al., 2011; Plaut et al., 2007, 2009). In the framework of planetary radars, Galilean satellites appear as highly interesting targets given the suspected presence of a superficial water ocean (McCord et al., 2001), with a high habitability potential. Oceans are expected to be as deep as 3–40 km for Europa and 60–80 km for Ganymede (Spohn and Schubert, 2003; Zimmer et al., 2000). Although radar penetra-tion can be quite important in water ice (down to few kilometers in the case of Marsis radar), surface topography can have first order effects on the instrument performance. An important work has already been conducted for Europa and Ganymede to some extent (Schenk, 2009). Our goal in this paper is to characterize Ganymede’s surface topography to better understand its surface properties from a radar point of view. These results should help to put constraints on the design of a possible future radar. We use topographic data derived from the Voyager and Galileo missions images to try to constrain the surface structure and to quantify its geometry (in terms of slopes and RMS heights). Scale dependency and its implication on wave propagation are also discussed therein as well as comparison to analog terrains with available radar data.

Topographic data

The Ganymedian surface is often described as a mix of two types of terrain: older, moderately cratered to highly cratered, dark regions and somewhat younger, brighter regions marked with an extensive array of grooves and ridges (sulcii) (Head et al., 2002; Oberst et al., 1999; Patterson et al., 2010; Prockter et al., 1998, 2010; Squyres, 1981). The dark terrain covers about one-third of the surface. Analyses of Galileo images have shown that locally, terrains can have surface characteristics which differ noticeably (Pappalardo et al., 2004). In order to investigate the different terrain types observed on Ganymede’s surface, three typical examples were selected (Fig. 1). They include both bright and dark regions and should give a fairly good insight of the topography that would be encountered. Vertical resolutions in DEMs range from less than 10–50 m.

Arbela Sulcus region

A first DEM derived from Galileo images using stereo image analysis techniques (Giese et al., 1998, 2001; Schenk, 2003) was obtained in Arbela Sulcus region (bottom left of Fig. 1).
Y. Berquin et al. / Planetary and Space Science 77 (2013) 40–44 41
Fig. 1. Top left: raw Ganymede Digital Elevation Model (DEM) data. Note: scale and topography are in kilometers. Bottom left: base Digital Elevation Model (DEM) in the vicinity of Arbela Sulcus. Numbers (1, 2 and 3) correspond to the three data subsets studied. Right: base DEM in the vicinity of Harpagia Sulcus. The small black dots correspond to corrupted data (interpolation was carried out for the study).
The effective resolution of the elevation model (i.e. horizontal resolution of the topography map which is different from the one of the base images) is around 350 m at best. Arbela Sulcus is a prominent SSW–NNE trending smooth, 20 km-wide band (Pappalardo et al., 1998). Most parts of the band are topographi-cally lower than the near surroundings, primarily dark terrain. The band is not smooth but there is lineated topography within. The eastern boundary consists of a ridgelike feature with a top-to-bottom elevation of up to 200 m on its western edge. It stands higher than the surrounding, older dark terrain to the east but does not embay it. The western boundary of the band is not elevated and stands lower than or at about equal topographic level as the surrounding terrain to the west. Grooved terrain is characterized by sub-parallel ridges and troughs at different scales. Such type of terrain is featured by the SW–NE trending band cut by Arbela Sulcus (visible to the west of Arbela in Fig. 1). It has an undulatory topography with a characteristic length of about 6 km and amplitudes reaching 400 m. Rifted terrains can be observed in many places (Pappalardo et al., 1998) but most of these rifts are too small to be resolved by the DEM. Craters with different sizes and degrees of structural deformation are distrib-uted across the study area.

Harpagia Sulcus region

A second DEM derived from high resolution Galileo images using stereo image analysis techniques as well (Giese et al., 1998; Schenk, 2003) was obtained in Harpagia Sulcus region (right of Fig. 1). The effective resolution of the elevation model is around 350 m at best. This specific bright region appeared as a surpris-ingly smooth surface on Voyager data (Head et al., 2001). How-ever, observations from Galileo proved this area to be quite rough and heavily pitted by small craters and to contain relatively common but degraded linear elements. This smooth terrain is clearly cut by younger grooved terrain along its eastern margin.

Bright terrain (Voyager data)

A third and last low resolution DEM (top left of Fig. 1) was derived from Voyager images using photoclinometric techniques (Squyres, 1981). This data set covers a considerably larger area and has an effective resolution of approximately 630 m. It dis-plays typical features (ridges mostly here) observed within bright terrain areas as well as numerous impact craters of different sizes which significantly alter the topography.

Methodology

In order to fully characterize terrains (for radar sounding purpose), different parameters were studied. We mainly focused on surface slopes, correlation lengths and height standard devia-tions. Each of these parameters was computed over DEM square samples of dimension L L (L being called hereafter window length). Graphics usually display parameters mean for a set of window lengths L for the available DEMs.
At a given point on a surface z ¼ f ðx, yÞ, the slope S is defined as a function of gradients at x and y (i.e. WE and NS) directions.
where fx and fy are the gradients at NS and WE directions, respectively. From the previous equation, it is clear that the key for slope computation is the estimation of fx and fy. Since we are interested in surface properties for radar sounding design, slopes are considered over a surface and not over a profile. Furthermore, since we are interested in slope variations with lateral scale L, we subdivide the surface into L L planes. We determine the slope and aspect of each plane by fitting it to the surface height point using a least squares method.
Correlation length was defined as the minimum threshold value at which the normalized 2D correlation function of a given DEM sample equals 0.37 (i.e. 37% of its maximum). It is a simplistic definition. However, as mentioned above, we are interested in the possible hampering arising from the surface in radar sounding. Hence, rough estimations are sufficient at the moment. DEM samples are detrended through removing a plan fitted in the least square sense.
We paid extra attention to scale dependency behaviours in surface statistics. Indeed, surface characteristics are functions of scales at which they are observed. One way to deal with it is to assume self-affine behaviour of the surface within a certain range of scales. This topic has been extensively discussed over the last decades and such behaviours have been shown to be well suited to describe natural surfaces (Orosei et al., 2003; Power and Tullis, 1991; Picardi et al., 2004). This approach is particularly convenient since it allows an explicit roughness scale dependent formulation (Shepard et al., 1995; Shepard and Campbell, 1999). For a self-affine profile or surface, RMS height variations, RMS slopes are a function of the sample length over which they are measured (Shepard et al., 1995). The surface is fully described through a standardized reference length and its Hurst exponent (Shepard and Campbell, 1999). We report here surface RMS deviations n in Fig. 2 (also referred to as structure function, variogram or Allan deviation). In essence, this parameter is a measure of the difference in height between points separated by a distance Dx.
Unfortunately, as seen in Fig. 2, available DEM cover too limited spatial ranges to build a scale dependent model. Figs. 2 and 4 provide an insight on the scale dependency of surface parameters. Small scales close to DEM resolution are likely affected by DEM resolutions which smooth the surface (attenuation of height variations). This effect probably accounts for slope breaks observed in Fig. 2. At large scales, height variations are usually bounded and reach a plateau. Overall, no typical self-affine behaviour was observed so far over a sufficient range of scales to extract robust Hurst exponents.

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Results

We mainly differentiated between available data sets and additionally we subdivided the data set in the Arbela region into three sets containing terrains to the west of the sulcus, the sulcus (Arbela band) and terrains to the east of the sulcus due to their obvious morphologic differences as presented in Fig. 1. Each data set obtained was then considered to be spatially stationary (note: this hypothesis solely relies on geological considerations at scales we are interested in). Main results are presented in Figs. 2–4. It is worth noticing that DEM in the Harpagia Sulcus region does not cover sufficient scale ranges to perform scale dependent analysis.
In addition, a comparison was conducted with two sets of surfaces on Mars observed with MOLA (Mars Orbiter Laser
Fig. 3. Slope histograms for major observed terrains. Data are for sites observed at 630 m resolutions roughly (except for MOLA data observed at 463 m). Bright terrains from Voyager images are plotted with a thin plain line, terrains in Arbela Sulcus vicinity are plotted with dotted lines and numbered according to areas defined in Fig. 1. Histogram with large circles corresponds to Harpagia Sulcus area and remaining histograms with large plain lines correspond to MOLA data on Mars (rough area in the Olympus Mons vicinity). Histograms from Mars data are within the range of values observed on Ganymede, expect for terrain in the sulcus (although Mars histograms have larger tails, i.e. more extreme topographic events). Area below the curves are normalized to one.
Fig. 4. Mean slopes as a function of window width L. Deg–log(km) scales. Dotted lines correspond to terrains within Arbela Sulcus vicinity numbered according to areas defined in Fig. 1. Large plain lines correspond to MOLA data on Mars and plain line corresponds to the DEM obtained with Voyager images. Mean slopes from MOLA data are comparable to those observed on Ganymede except for terrains located in the sulcus which are noticeably smoother.
Altimeter) at 463 m resolution (Kreslavsky and Head, 1999, 2000). These surfaces are located in the vicinity of Olympus Mons. They display, at these scales similar behaviours to those observed on Ganymede in terms of slopes, correlation lengths and height variations (Figs. 2–4). Radargrams obtained in these areas are available (Fig. 5). This area on Mars is considered as very rough in comparison to the rest of the planet. Materials on Ganymede differ noticeably from those on Mars. In terms of scattering this will primarily affect the electric permittivity in Eq. (3) (see Discussion).
Terrains on the edges of Arbela Sulcus and terrains observed within Harpagia Sulcus region display surprisingly similar slope histograms (Fig. 3) with large mean values around 7.51 to 81 at 630 m. Such terrains will likely produce important lateral radar echoes when performing radar sounding. In comparison, terrain located in Arbela Sulcus shows gentle slopes averaging 3.51 at 630 m. Slopes from DEM obtained from Voyager images sit in the middle, which may be explained by the presence of both sulcii and cratered/ridged terrains. These observations are very similar to those conducted on Europa (Schenk, 2009).
Variations of mean slopes with window lengths display inter-esting features (Fig. 4). When window lengths are below 1 km roughly, the behaviour is mainly dictated by DEM limited resolu-tions (smoothing effect). However, at larger scales, slope varia-tions are probably due to natural terrains roughness. This is true only over a limited range of scales. Once again, terrain in Arbela Sulcus shows much smaller average slopes in comparison to other terrains. Considering the limited range of scales available for each DEM (one order of magnitude roughly), we shall only derive parameters at window lengths of the order of magnitude of DEMs sizes. Through such description, we somehow make the assump-tion that topography within each set can be considered as stationary (topography could be modelled for instance by Gaus-sian or exponential correlated surfaces).
Typical correlation lengths observed in terrains to the East and West of Arbela Sulcus are around 1.5 km. However this value does not fully describe correlation functions since terrains are strongly anisotropic. Large structures can be clearly seen on the DEM image (Fig. 1). Correlation lengths associated to the DEM obtained with Voyager data are clearly larger reaching few kilometres (3–4 km) which may indicate larger structures in the area, although the resolution might be too low to resolve smaller features. Structures within Arbela Sulcus are smaller with typical correlation lengths around 800 m. Terrains in Harpagia Sulcus region have correlation lengths ranging from 450 m to 1000 m, increasing towards the east. Smaller structures are most likely present within these terrains but are not resolved by available DEMs.
RMS heights within Harpagia Sulcus region are ranging from 40 m to 75 m increasing towards the east. Terrains to the east and west of Arbela Sulcus have RMS heights around 120 m which corresponds to a value of 0.9 in Fig. 2. Whereas Arbela Sulcus terrain RMS height is much smaller around 30 m ( 1.5 in Fig. 2). DEM obtained from Voyager images has an RMS height of 150 m ( 0.82 in Fig. 2).
Overall, terrains on Ganymede could be qualified as rough in comparison to what has been observed on Mars (see slope values in Fig. 3). Important lateral surface echoes and surface diffusion of the radar signal are very likely to occur during radar sounding experiments. Smoothest areas are located within sulcii which display obvious topographic differences from the rest. These narrow bands (10–100 s of kilometers wide) highlight the pre-sence of relatively smooth terrains on Ganymede that might allow good radar sounding performances. Galileo and Voyager observa-tions have permitted to build models for these grooved terrains. These models mainly induce rift-like processes with a significant role for tilt-block style normal faulting, high thermal gradient, locally high extensional strain, the potential for tectonism alone to cause resurfacing in some regions, and a generally less prominent role for icy volcanism (Pappalardo et al., 2004).

Table of contents :

i introduction 
1 planetary sounding radar and the juice mission: an overview
2 a study on ganymede’s surface topography: perspectives for sounding radar
3 a foreword on the work
ii direct electromagnetic wave propagation in planetary sounding radar 
4 implementing huygens-fresnel’s principle using a meshed boundary surface
4.1 Discretization of the boundary problem
4.2 Application to planetary sounding radar
4.3 Implementation
4.4 Examples
iii an inverse problem formulation using huygens-fresnel’s principle 
5 recovering reflectivity parameters
5.1 Towards a linear inverse problem
5.2 Tackling non-linear reflectivity parameters with an iterative scheme
5.3 A statistical perspective
6 discussing the other parameters
6.1 Sounding the upper medium
6.2 Reconstruction of surface obstacle using far field radar measurements
iv discussions 
7 conclusion and perspectives
7.1 A quick overview of the manuscript
7.2 A foreword on subsurface imaging
7.3 Beyond the manuscript
v appendix 
a maxwell’s equations and the constitutive relations
b direct problem formulation
b.1 Electromagnetic wave scattering
b.1.1 The Stratton-Chu formulation
b.2 Instrument characteristics and on-board processing
b.3 Huygens-Fresnel in the far field approximation
b.4 Fields on the boundary surface
c a modern formulation of the problem
c.1 Maxwell’s equations using differential geometry
c.2 Huygens-Fresnel’s principle
bibliography 

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