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HF (high frequency) radar sounder technology has been developed for several missions of Mars surface/subsurface exploration. This paper presents a model of rough surface and stratified sub-surfaces to describe the multi-layer structure of Mars polar deposits. Based on numerical simulation of radar echoes from rough surface/stratified interfaces, an inversion approach is developed to obtain the parameters of Polar Layered Deposits,
*i.e.* layers thickness and dielectric constants. As a validation example, the SHARAD radar sounder data of the Promethei Lingula of Mars South Polar region is adopted for parameters inversion. The result of stratification is also analyzed and compared with the optical photo of the deep cliff of Chasma Australe canyon. Dielectric inversions show that the deposit media are not uniform, and the dielectric constants of the Promethei Lingula surfaces are large, and become reduced around the depth of 20 m - 30 m, below where most of the deposits are nearly pure ice, except a few thin layers with a lot of dust.

The physical properties of Mars polar deposits have been studied for several decades. Some studies show that North Polar Layered Deposit (NPLD) and South Polar Layered Deposit (SPLD) might be rich in water ice [

HF radar waves can penetrate through the Mars regolith media several kilometers. The MARSIS (Mars Advance Radar for Subsurface and Ionospheric Sounding) onboard the Mars Express operates in 4 bands centered at 1.8, 3, 4, 5 MHz with a bandwidth of 1 MHz, and can penetrate through the media as deep as 4 km [

To study the radar sounder data, Mouginot et al. [

Based on numerical simulations of the radar sounder echoes from the one-layer model with rough surface/subsurface media, Ye and Jin [

Based on [

1) A Model of Parallel Stratified Media

The SHARAD radar sounder data has shown that there are multi-layer structures in Mars Polar regions [

Suppose that multiple reflection and transmission between interfaces are neglected. It means that the echo from the n-th interface experienced one-reflection, round trip of 2(n − 1) transmissions (i.e. including round trip attenuation) through previous (n − 1) layer media. This assumption is based on small difference on final surface reflectivity caused by underlying multi-layer structures. Thus, as the incident radar power through ionosphere is directly on the top surface P 0 = P 0 ( a ) e − A , the echo from the n-th interface is written as

P n = P 0 r n ∏ m = 1 n − 1 [ t m 2 exp ( − 4 k ″ m d m ) ] (1)

where P 0 ( a ) is denoted as the transmitted power of dipole antenna (with notation (a)), and similarly, P n ( a ) = P n e − A denotes the n-th reflected peak power received by the radar antenna, i.e. observation. In Equation (1), t m is the transmittivity between the (m − 1)-th and the m-th media, r n is the reflectivity of the n-th layer, d m is the thickness of the m-th layer, k ″ m is the imaginary part of the wave number of the m-th layer, exp ( − 4 k ″ m d m ) is the round-trip attenuation in the m-th layer [

k m = k 0 ε m = k 0 ε ′ m + i ε ″ m ≈ k 0 ε ′ m + i 1 2 k 0 ε ″ m ε ′ m (2)

Here k 0 is the wave number of free space. ε ′ m and ε ″ m are the real and imaginary parts of the m-th layer dielectric constant, respectively. k ″ m is written as

k ″ m = 1 2 k 0 ε ″ m ε ′ m (3)

The transmittivity satisfies

t m = 1 − r m (4)

and the layer thickness can be expressed as

d m = c ε m τ m 2 (5)

where c is the light speed in free space, and τ m is the time delay as EM wave propagates through the m-th medium twice, i.e. the time delay between two echoes from two successive interfaces.

Substituting Equations (3)-(5) into Equation (1), the reflected power from the n-th interface is derived as

P n = P 0 r n ∏ m = 1 n − 1 [ ( 1 − r m ) 2 exp ( − k 0 c τ m ε ″ m ε ′ m ) ] = P 0 r n ∏ m = 1 n − 1 [ ( 1 − r m ) 2 exp ( − k 0 c τ m tan δ m ) ] (6)

where tan δ m = ε ″ m / ε ′ m is the (attenuation) loss-tangent of the m-th layer.

2) Calculation of Loss Tangent

Since roughness of underlying interfaces is totally unknown, the model makes all underlying interfaces as plane-stratified, as shown in

It has been known from Mars studies that the loss tangents of NPLD and SPLD are actually very small, as usually 0.001 - 0.005 [

Certainly, the reflectivity r n ( n = 1 , ⋯ , n − 1 ) is a main factor to affect P n , comparing with τ m , tan δ m and exp ( − k 0 c τ m tan δ m ) . To reduce the number of unknowns and reach final inversion, all small loss tangents of the n-layers are trivial and seen as the same within one illuminated area (e.g. 1.81 km in the next example). The low value of the loss tangent makes such approximation reasonable. Thus, Equation (6) is simplified as

P n = P 0 r n exp ( − k 0 c tan δ ∑ m = 1 n − 1 τ m ) ∏ m = 1 n − 1 ( 1 − r m ) 2 (7)

Taking natural log of both sides of Equation (7), it gives

ln P n = − k 0 c tan δ ∑ m = 1 n − 1 τ m + ln P 0 + ln r n + 2 ∑ m = 1 n − 1 ln ( 1 − r m ) (8)

It means that the echo from each interface is a linear function of the time delay

ln P n = − ( k 0 c tan δ ) τ t o t a l , n + b + ξ (9)

where τ t o t a l , n = ∑ m = 1 n − 1 τ m is the time delay from the n-th interface echo to the top

surface echo, b is an unknown constant, ξ is a random variable to take account of different reflectivities, r n , of the interfaces.

Equation (9) can be seen as a linear regression model with the regress or τ . Using the radar range echoes from all interfaces and their respective ranges, the linear fitting is obtained with the least square method. Then the loss tangent can be calculated by the slope of linear function in Equation (9).

3) Solution of Dielectric Constant of Each Layer

Since the loss tangent is obtained, the number of unknowns now becomes n + 1. The set of Equation (7) can be directly solved, and the reflectivity is written as

r n = P n P 0 exp ( − k 0 c tan δ ∑ m = 1 n − 1 τ m ) ∏ m = 1 n − 1 ( 1 − r m ) 2 (10)

Equation (10) can be solved, iteratively.

Reflectivity is a function of the dielectric constants of layering media. Because the roughness of all sub-interfaces is totally unknown, it is practicable to model all sub-interfaces between layers as flatly stratified within a limited area. This is a good and workable assumption within a limited area illuminated by the radar waves, even if a small error might be caused due to small scale roughness of the interfaces.

It is also noted that in derivation of Equation (1), multiple-reflection and transmission are neglected. Thus, the reflectivity from the (n − 1)-th layer to the n-th layer is derived based on a half-space model, i.e. the reflectivity of this interface is written as

r n = ( ε n − ε n − 1 ε n + ε n − 1 ) 2 (11)

Since the loss tangent is very small, it yields ε m ≈ ε ′ m , Equation (11) becomes

ε ′ n = ( 2 1 ± r n − 1 ) 2 ε ′ n − 1 (12)

But, there would be two solutions from Equation (12) due to the term ± r n . Using the phase change of the echoes, an unique solution may be obtained.

The echo phase from the n-th interface is written as

φ n = φ 0 + k 0 c ∑ m = 1 n − 1 τ m + 2 ∑ m = 1 n − 1 φ t , m + φ r , n (13)

where φ 0 is the phase of EM incidence, φ t , m denotes the phase from each transmission, and φ r , n the phase of each reflection.

As EM wave is vertically incident from the (n − 1)-th layer to the n-th layer, the reflection coefficient and transmission coefficient are, respectively, written as

R n = ε n − ε n − 1 ε n + ε n − 1 (14)

T n = 2 ε n ε n + ε n − 1 (15)

Since the loss tangent is very small, it can be seen that if ε ′ n > ε ′ n − 1 , it makes R n > 0 and φ r , n = 0 ; otherwise, if ε ′ n < ε ′ n − 1 , it makes R n < 0 and φ r , n = π . In transmission, the phase keeps unchanged, i.e. φ t , m = 0 .

As incident upon the top surface, φ r , 1 = 0 . Equation (13) gives φ 0 = φ 1 . Thus, all phases due to reflections from all interfaces can be calculated from the data of radar range echoes as

φ r , n = φ n − φ 1 − k 0 c ∑ m = 1 n − 1 τ m (16)

Based on these approximations, it yields the dielectric constant of each layer as

ε ′ n = { ( 2 1 − r n − 1 ) 2 ε ′ n − 1 φ r , n = 0 ( 2 1 + r n − 1 ) 2 ε ′ n − 1 φ r , n = π (17)

4) Calculation of Ionospheric Attenuation and Dielectric Constant of the Surface Medium

Propagation through the ionosphere causes phase distortions and attenuation. The SHARAD Reduced Data Record (RDR) data has already corrected the phase distortion using Phase Gradient Autofocus (PGA) method [^{ }

Since the SHARAD data, as available, have not been absolutely calibrated [

P n ( a ) = P n e − A ≡ C P n ( n = 1 , ⋯ ) (18)

where calibration constant C ≡ e − A can be seen to take account the one-way ionospheric attenuation.

It has been studied [_{2} ice covering an area of the South Pole of Mars, and its dielectric constant is known as about 2.2. Thus, based on the observation data P 1 ( a ) ( ε 1 = 2.2 ) , as available, and our simulated data P 1 ( ε 1 = 2.2 ) at this South Pole location, C of Equation (18) is obtained and applied to the whole inversion region.^{ }

The top surface is modeled as a rough surface, described by the known DEM data. From the radar equation [

P 1 ( ε ) = γ ( ε ) P 0 (19)

where γ is the backscattering coefficient of rough surface.

The Kirchhoff approximation (KA) of rough surface scattering requires the curvature radius of the surface much larger than the radar wavelength [

Thus, it is derived as [

γ ( ε ) = k 0 2 | ∬ S e i 2 k i ⋅ r ″ d S ′ | 2 π A r 1 ( ε ) (20)

where k i is the incident wave vector, r ″ is the distance vector from the pixel center of integral to the nadir point.

Suppose that the top surface has a test value ε 1 0 , the simulation gives P 1 ( ε 1 0 ) . The ratio of the observation P 1 ( a ) ( ε 1 ) with an unknown ε 1 over the simulation C P 1 ( ε 1 0 ) with assumed ε 1 0 gives [

P 1 ( a ) ( ε 1 ) P 1 ( a ) ( ε 1 0 ) = P 1 ( a ) ( ε 1 ) P 1 ( ε 1 0 ) C = P 0 γ ( ε 1 ) C P 0 γ ( ε 1 0 ) C = r 1 ( ε 1 ) r 1 ( ε 1 0 ) . (21)

where C was defined in Equation (18), and actually can be evaluated in the next approach. Substituting Equation (11) into Equation (21), it gives the inverted ε ′ 1 , as follows

ε ′ 1 = ( 2 1 − P 1 ( a ) ( ε 1 ) C P 1 ( ε 1 0 ) r 1 ( ε 1 0 ) − 1 ) 2 . (22)

It is noted that this inversion is applicable for surfaces with gentle roughness and zero mean slope. Indeed, there are large areas on the SPLD to fit this description as shown from DEM data. Our approach focuses on those cases and ignores those cases with steep slopes or highly varying roughness.

Using the inverted ε ′ 1 and Mars Orbiter Laser Altimeter (MOLA) elevation data, the backscattering coefficient γ ( ε 1 ) can be calculated in our numerical simulation [

P 0 = P 1 ( ε 1 ) γ ( ε 1 ) (23)

Substituting the inverted ε ′ 1 , P 0 and observation P 2 ( a ) into Equations (10), it gives r 2 . Then, Equation (17) gives ε ′ 2 . Sequentially, it yields ε ′ n ( n = 3 , ⋯ ) , etc. The thickness of each layer can be then calculated by Equation (5).

1) SHARAD Radar Echoes Data from the Promethei Lingula

As shown in

We choose the track 17485_01 of SHARADRDR data on PDS Geosciences node (filename: r_1748501_001_ss11_700_a.dat on website http://pds-geosciences.wustl.edu/), which passes by Promethei Lingulanear the Chasma Australecanyon during the night (SZA is about 112˚), for inverting the parameters of multi-layer media. The vertical resolution of the data is 15m in vacuum and about 8.5 m in pure ice. The latitude and longitude of each frame are indicated in the SHARAD RDR data, and the along-track distance between each frame can be calculated. Especially, the distance between each frame in

2) Echoes from the Surface/Sub-Surfaces and Interface Locations

In the radargram, the nadir surface echo, off-nadir clutters of rough surface, and echoes from layering interfaces must be identified and treated, separately. While the strongest peak is often from the nadir echo, the surface roughness and DEM geometry can make bright off-nadir returns [

S ( i , j ) = 1 2 n ∑ q = − m m ∑ p = − n n s ( i + q , j + p ) (24)

where S ( i , j ) = { 1 the i -thsample offrame j islocalmaximum 0 the i -thsample offrame j isnotlocalmaximum .

For example, let n = 25 and m = 1, and hence S indicates the ratio of local maximum from totally nearby 50 frames with the similar time delay (not exceeding 1 sampling interval). If S > 0.7 , it means that more than 70% of adjacent frames have reflector with the same time delay, and the reflector is judged to be from the interface. Otherwise, the isolated reflector is seen as the surface clutter. In this way, the surface echoes and interface echoes are distinguished from frame 69,301 - 69,350 of the track 17485_01, which extends about 1.81 km.

Finally, the stratification of multi-interfaces is shown in

Thus, the surface echoes, the echoes from underlying interfaces, i.e. P n , n = 2 , ⋯ , can be obtained. Sometimes, the echoes from different locations of the same interface might be quite different. It might be caused by different interface- topography, or happens to be mixed by the surface clutters. Dimmer or brighter radar echoes may also be caused by the change of interface reflectivity or the change of the interface time delay, for the time delay changes lead to different interference in the radar signal. To avoid such fluctuations of the interface echoes to affect final inversion, the echoes from the same interface is taken as an averaged value.

3) Dielectric Constant of the Surface Medium

Using the MOLA (Mars Orbiter Laser Altimeter) elevation data, the surface echoes from this rough surface can be numerically simulated [

The ratio of

shown in _{2} ice cap covering the South Pole, which can be also seen from blue colors of _{2} is much lower than water ice and rocks. So the echo power of CO_{2} cap is much less than other places. The constant of C, Equation (18), is actually obtained from these figures.

Using the algorithm aforementioned, the dielectric constant of Mars surface over the South polar region, ε ′ 1 , can be inverted, as shown in _{2} ice, because the data used in the inversion are all acquired during autumn and winter, and the Mars polar regions are covered by a thin CO_{2} ice layer less than 1 m thick at that time [_{2} ice covers on the top player [_{2} ice is variable and uncertain for data acquired in different time, which is much thinner in terms of SHARAD resolution, the inverted dielectric property of the first layer can be seen as the effective dielectric constant including thin CO_{2} ice layer as available. The inverted results are the equivalent dielectric constants of a cluster of thin layers comprised of CO_{2} ice and water ice with dust. The ε ′ 1 of most areas of SPLD is between 3 and 4. These results show that SPLD is mainly comprised of water ice, which is consistent with previous studies [

4) Calculation of Loss Tangent

To reduce the fluctuation of different reflectivities of the interfaces ( ξ in Equation (9)), the echoes powers of the interfaces with the same time delay are averaged. Using the least square method to make a linear fitting for the average echoes power with different time delay, a linear equation of ln P n = − 1.11 × 10 5 τ t o t a l , n + 4.3 is obtained, as shown in

The hypothesis of linear regression model to well fit the data is tested using F distribution. It takes F = 13.1 for total 42 data points in the linear fitting. Setting the significant level α = 0.01 , it gives F α ( 1 , 40 ) = 7.31 and F > F α ( 1 , 40 ) . Thus, the linear regression is good to fit the data. The loss tangent is much smaller than previous result [

5) Echoes Phase Estimation and Unique Solution Determination

The sampling interval of the SHARAD data is 0.075 μs [

φ ¯ n = Im { ln ( ∑ f = 69301 69350 exp ( i φ n , f ) ) } (25)

where φ n , f is the phase of frame f and n-th interface.

Substituting Equation (25) into Equation (16), the phase of reflection is calculated. Considering that φ r , n usually is not exactly equal to 0 or π due to the phase accuracy, Equation (17) is changed as

ε ′ n = { ( 2 1 − r n − 1 ) 2 ε ′ n − 1 φ r , n ∈ [ − π 2 , π 2 ] ( 2 1 + r n − 1 ) 2 ε ′ n − 1 φ r , n = [ − π , − π 2 ) ∪ ( π 2 , π ] (26)

6) Inversions and Validation

The dielectric constant of surface medium ε ′ 1 inverted from the frame 69,301 - 69,350 (83˚S 102˚E in

Substituting #Math_130# into Equation (25), and changing the loss tangent from 0.0004 to 0.0014, the varying range of the layered dielectric constants are iteratively calculated, as shown in

Using Equation (5), the thickness of each layer is calculated, as shown in

The inversion accuracy depends on the evaluation of interface locations and echo powers. The phase accuracy might be also interfered by surface clutter.

Suppose the depth of each interface do not change across the track, as shown in

It can be seen that the inverted multi-layering structure is well described on the cliff of Chasma Australe canyon near track 17485_01, which was indicated in a HiRISE optical image, even not exactly the same matching. The optical layers are much finer than the radar resolution, and they cannot be a 1:1 correlation. The layers in radargram correspond to the packets of thinner layers in optical image. Many layers can be seen on optical image but cannot be detected by radar, because this HF radar technology is capable only to detect the interfaces with a significant change of dielectric media. The inversion model presented is based on the implicit assumption that there is no more than one such interface within a SHARAD vertical resolution (10 - 15 m). If SHARAD reflections are caused by merged reflections of packets of thinner layers with different dielectric constants, the inversed layer-depths and dielectric properties are understood on the average or effective sense for radar sounder echoes. Moreover, details of the technology parameters and measurements, such as SNR for each echo and high resolution elevation data might further improve the inversions.

The inversion results show that the dielectric deposits in Promethei Lingula are not uniformly stratified, seen along the dashed line indicated in

ε = [ ( 1 − c dust ) ε ice 1 3 + c dust ε dust 1 3 ] 3 (27)

where c dust is the dust fraction, and can be calculated as

c dust = ε 1 3 − ε ice 1 3 ε dust 1 3 − ε ice 1 3 (28)

Taking the dust basalt ( ε dust = 8 ), the dust fraction is calculated using Equation (28), as shown in

Making a dielectric average of top 8 layers media, it yields

〈 ε 〉 = ∑ i = 1 8 d i ε i / ∑ i = 1 8 d i (29)

It gives the dielectric constant of regolith impurity, 〈 ε 〉 = 3.6 , which corresponds to the dust fraction about 12%.

This paper presents a model of rough surface and stratified interfaces to describe the multi-layer structure of Mars Polar Layered Deposit. The range echoes of HF radar sounder from rough surface/subsurface is numerically calculated. And under the Kirchhoff approximation with a mean zero slope, the received echo at nadir direction preserves the functional dependence of the surface reflectivity. In radargram to show the radar range echoes, the nadir surface echo and the echoes from layering interfaces are separated from off-nadir surface clutters. As the surface dielectric constant is derived from the ratio of the received echo peak power, the inversion approach is designed to obtain the dielectric constant and layer thickness of next layers, sequentially.

As a validation example, the SHARAD data are adopted to inversions of the Promethei Lingula stratified media of Mars South Polar region. The vertical profile of the dielectric constants of layering media is obtained. Correspondingly, the layered structure is obtained, which is visually similar to the optical image on the cliff of Chasma Australe canyon. The inversion results show that the surface media of Promethei Lingula has larger dielectric constant with impurity, while as the media below the surface might contain much less impurity and more pure water ice. As more details of the technology parameters and measurements can be taken into account, the inversion accuracy can be further improved.

The inversion model is based on the assumption that there is no more than one interface within a SHARAD vertical resolution. As SHARAD reflections might be caused by packets of thinner layers, multi-layer model under this resolution won’t be able to see them, individually, which are merged with other stronger adjacent reflections. The inversed layer-depths and dielectric properties are understood on the average or effective sense for radar sounder echoes.

This work was supported by the National Key Research and Development Program of China 2017YFB0502703.

The MOLA elevation data and SHARAD Reduced Data Record (RDR) data are all from PDS Geosciences Node (website: http://pds-geosciences.wustl.edu/). The HiRISE optical data are from the HiRISE website (http://hirise.lpl.arizona.edu/).

The authors declare no conflicts of interest regarding the publication of this paper.

Liu, C. and Jin, Y.Q. (2019) Parameter Inversions of Multi-Layer Media of Mars Polar Region with Validation of SHARAD Data. International Journal of Astronomy and Astrophysics, 9, 335-353. https://doi.org/10.4236/ijaa.2019.93024