# Mars ionosphere TEC estimation from MARSIS data: A new approach

M. Cartacci, B. Sánchez-Cano, A. Cicchetti, R. Noschese, B. Langlais, R.

Orosei

► To cite this version:

M. Cartacci, B. Sánchez-Cano, A. Cicchetti, R. Noschese, B. Langlais, et al. Mars ionosphere TEC estimation from MARSIS data: A new approach. Icarus, 2025, 434, pp.116545. 10.1016/j.icarus.2025.116545. hal-05364399

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# Mars ionosphere TEC estimation from MARSIS data: A new approach

M. Cartacci<sup>a,*</sup>, B. Sánchez-Cano<sup>b</sup>, A. Cicchetti<sup>a</sup>, R. Noschese<sup>a</sup>, B. Langlais<sup>c</sup>, R. Orosei<sup>d</sup>

## ARTICLE INFO

**Keywords:**

Mars
Radar observations
Ionospheres

## ABSTRACT

We describe a new implementation of the Contrast Method (CM) algorithm (Picardi and Sorge, 2000; Ilyushin Ya and Kunitsyn, 2004; Cartacci et al., 2013) used to correct the phase distortion of the echoes recorded by the Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) (Picardi et al., 2005; Jordan et al., 2009) in its subsurface mode.

The New CM is designed following an improved approach which allows not only to perform its original task but also to estimate the Total Electron Content (TEC) of the Mars ionosphere with a better accuracy respect to the past (Cartacci et al., 2013; Cartacci et al., 2017), in the nightside when the transmitted frequency is close to the ionosphere plasma frequency as well as in the dayside, thanks to a lesser use of theoretical approximations in the method development.

Moreover, taking advantage of almost 16 years of MARSIS operations, corresponding to about 8700 orbits, we present an improved global map of the night side TEC variations due to the interaction with the Mars crustal magnetic field (Safaeinili et al., 2007; Cartacci et al., 2013).

## 1. Introduction

The total electron content (TEC) of the ionosphere is a very useful parameter that indicates the number of free electrons that are present in a column with the cross section of 1 m² between two points in space. On Earth, it is used as a monitor for the state of the ionosphere, which is known to have several degrading effects on satellite communications and position determination. At Mars, it is also a very useful parameter to determine the state of Mars' ionosphere, its variability with seasons, solar activity, crustal magnetic field anomalies (e.g., Safaeinili et al., 2007; Mendillo et al., 2013; Sánchez-Cano et al., 2016; Cartacci et al., 2013; Campbell et al., 2024), or even low atmospheric cycles (Sánchez-Cano et al., 2018) or FUV aurora occurrence (Chirakkil et al., 2024).

There are several methods used to retrieve the TEC at Mars, most of which are based on resolving the integral $TEC = \int_{h1}^{h2} N_e dh$ if the electron density profile as a function of altitude ($N_e$) is the measured parameter, where h1 and h2 are minimum and maximum altitudes above the surface of Mars where electron density is present. However, a more complex method is necessary when using radars to obtain this parameter. In this case, the TEC is retrieved as a by-product from the

analysis of MARSIS signal distortion caused by the dispersive ionosphere (e.g., Safaeinili et al., 2007; Mouginot et al., 2008; Cartacci et al., 2013). There are several algorithms designed using slightly different methodologies to correct the signal distortion as will be described in this work. However, historically, there have been significant differences in their retrievals, mainly at lower solar zenith angles (i.e., dayside) when the transmitted frequency is closer to the critical frequency of the ionosphere (more details can be found in Sánchez-Cano et al., 2015). Although the methodologies have a robust physics basis, it is critical to understand the origin of these differences and if they are due to the uncertainties introduced by the approximations necessary to develop each technique as the TEC is a parameter that now more than ever is essential for the characterization of radio signal propagation at Mars given the amount of robotic exploration planned for the near future (e. g., Sánchez-Cano et al., 2022; Lillis et al., 2021). After a first attempt was made by Sánchez-Cano et al., 2015, new improved retrievals have emerged such as Cartacci et al. (2017) or Plaut (2023).

Therefore, the goal of this study is to do a critical and objective assessment of the retrieval method, including the assumptions made regarding the shape of the ionosphere and how it contributes to the

<footnote><sup>a</sup> Istituto di Astrofisica e Planetologia Spaziali (IAPS), Istituto Nazionale di Astrofisica (INAF), Rome, Italy</footnote>

<footnote><sup>b</sup> Planetary Science Group, School of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom</footnote>

<footnote><sup>c</sup> Nantes Université, Univ Angers, Le Mans Université, CNRS, Laboratoire de Planétologie et Géosciences, LPG UMR 6112, 44000 Nantes, France</footnote>

<footnote><sup>d</sup> Istituto di Radioastronomia (IRA), Istituto Nazionale di Astrofisica (INAF), Bologna, Italy</footnote>

<footnote>* Corresponding author.
E-mail address: marco.cartacci@inaf.it (M. Cartacci).</footnote>
retrieval errors per se, in order to understand how Mars' ionosphere contributes to the propagation of radio signals and the error of the TEC observations. Moreover, we use this knowledge to produce the most updated map of the nightside TEC of Mars, which is of high interest for the scientific community.

We note that similar studies about the TEC estimation from a radar signal (Campbell et al., 2013; Campbell and Watters, 2016) were conducted using data collected from SHARAD (Seu et al., 2004), another radar sounder that works at higher frequencies than MARSIS. Anyway, in the present study we considered only the MARSIS data.

## 2. MARSIS instrument

The Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) (Picardi et al., 2005; Jordan et al., 2009), is a nadir-looking pulse limited radar sounder and it is carried by ESA's Mars Express (MEX) spacecraft, the first European interplanetary mission. The main goals of MARSIS are to find and map the distribution of water (Orosei et al., 2018; Lauro et al., 2021), both liquid and solid, as well as other materials on Mars, and to characterize the structure of the Martian ionosphere. In order to perform at best the defined tasks, MARSIS is able to work in different operative modes, the Sub-Surface (SS) Mode and the Active Ionosphere Sounding (AIS) Mode, each characterized by a particular radar configuration.

The AIS Mode (Gurnett et al., 2007) is an operational configuration dedicated to the study of Mars' ionosphere.

When sets in AIS Mode, MARSIS transmits a swept of 160 radio signals in a frequency range between 0.1 and 5.5 MHz which propagates through the Ionosphere from the topside to lower regions with increasing electron density.

In this way, when a wave pulse of a particular frequency reaches a layer with the same plasma frequency, the wave is reflected characterizing the upper profile of the ionosphere and estimating the topside electron density Ne.

The second configuration is represented by the SS Mode that is used to investigate the sub-surface of Mars, and the one we use in this study.

In this configuration, the radar transmits a signal linearly modulated in frequency over a bandwidth $B = 1$ MHz ("chirps"), with four different central frequencies (1.8 MHz, 3 MHz, 4 MHz, 5 MHz), alternating the transmission of two different frequencies with a Pulse Repetition Frequency (PRF) of 127.27 Hz.

Another important characteristic of MARSIS is its use of synthetic aperture (SAR) techniques to improve the signal-to-noise ratio and the along-track resolution.

SAR processing produces synthetic apertures (frames) adjacent to each other, with a ground resolution of 5.5–10 km along the track and of 17–30 km across the track, where lower and higher resolutions pertain to higher and lower spacecraft altitudes, respectively. During an orbit, MARSIS performs hundreds of synthetic apertures where the echoes received for each transmitted pulse are reduced to a single radar return, through an azimuth and range compression.

Since the ionosphere is known to cause significant radio propagation signal distortion (Armand et al., 2003; Safaeinili et al., 2003; Cartacci et al., 2013), in order to optimize the deep penetration of the surface of Mars, MARSIS operates at the lowest possible frequencies capable of propagating through the Martian ionosphere, i.e., well above the critical plasma frequency, $f_p$, of the ionosphere.

For this reason, the two frequencies are chosen according to the predicted Solar Zenith Angle (SZA), as this is the main parameter determining the number of electrons found in the ionosphere (e.g., Sánchez - Cano et al., 2013; Sánchez-Cano et al., 2015). The main reason is to increase the possibility that the transmitted bandwidth is always higher than the local plasma frequency, and therefore, be able to penetrate the full atmosphere with the smallest possible degradation.

Considering that the ionospheric electron density is lower in the night side and increases considerably during the day side, a proper

selection of the transmitted frequencies is very important.

Normally, the typical configuration during the SS Mode is to use the carrier frequencies 4 MHz and 3 MHz during the night side (SZA ≥ 90°), while during the day side (SZA < 90°) the carrier frequencies used are 5 MHz and 4 MHz. The lowest frequency of 1.8 MHz was used mainly in the earlier years of the mission. However, it was observed that it can work properly only during the deep night side (SZA > 115°) and if there is no particle precipitation from the Martian tail into the atmosphere, for this reason it was rarely adopted in the following years. Despite this operational efforts, it is well known that a careful selection of the frequencies is not sufficient to completely preserve the radar signal from the ionosphere effects (Safaeinili et al., 2003; Cartacci et al., 2013), because many other factors play a role in the ionospheric variability, such as the crustal fields, space weather, auroras, etc. (e.g., Sánchez - Cano et al., 2019; Lester et al., 2022; Harada et al., 2024), so a dedicated algorithm, the so-called Contrast Method (CM) (Cartacci et al., 2013) is implemented in the processing pipeline of MARSIS SS data to remove the signal distortion.

The CM works on both frequencies separately, producing independent output parameters.

## 3. Mars's ionosphere and TEC estimation

As described in previous publications (Safaeinili et al., 2007; Mouginot et al., 2008; Cartacci et al., 2013; Cartacci et al., 2017; Conroy et al., 2019), the phase shift induced by the ionosphere in a radar signal of frequency $f$ can be written as

$$ \Delta\varphi(f) = \frac{4\pi}{c} f \int_{0}^{L} [n(z) - 1] \quad (1) $$

where $L$ is the ionospheric thickness, $c$ is the speed of light in vacuum, and $n(z)$ is the refraction index:

$$ n(z) = \sqrt{1 - \frac{f_p^2(z)}{f^2 - jf\nu} \approx \sqrt{1 - \frac{f_p^2(z)}{f^2}}} \quad (2) $$

where $f_p$ is the plasma frequency, $\nu$ the electron-neutral collision frequency and $z$ is the altitude above ground. If we consider a typical MARSIS operation frequency (i.e. in the 1.8–5 MHz range), the imaginary term in the denominator of Eq. (2) can be neglected, because generally $\nu \sim 10–60$ kHz. In this way, the refractive index $n$ is reduced to a real number, thus the computations are simplified. Even if there is the possibility to use Eq. (2) in its complex form (Zhang et al., 2009), this approach is the most common one (Safaeinili et al., 2007; Mouginot et al., 2008; Cartacci et al., 2013).

The plasma frequency, in Hz, can be written as

$$ f_p(z) = 8.98 \sqrt{N_e(z)}, \quad (3) $$

where $N_e$ is the electron density in $m^{-3}$. The maximum value of $f_p$ corresponds to the maximum value of the electron density $N_{emax}$.

According to Eq. (2), Eq. (1) can be rewritten as

$$ \Delta\varphi(f) = \frac{4\pi}{c} f \int_{0}^{L} \left[ \sqrt{1 - \left(\frac{f_p(z)}{f}\right)^2} - 1 \right] dz \quad (4) $$

At this point, all the studies previously introduced, applied a Taylor expansion that is valid for small values of the ratio $f_p/f$ (Safaeinili et al., 2003).

This approach allows a simpler implementation of the respective distortion compensation algorithms.

In detail, Safaeinili et al., 2007 and Mouginot et al., 2008 introduced the same expansion of Eq. (2).

Safaeinili et al., 2007 Eq. (2)
$$n(z) = \sqrt{1 - \frac{f_p^2(z)}{f^2}} \approx 1 - \frac{1}{2} \frac{f_p^2(z)}{f^2} - \frac{1}{8} \left( \frac{f_p^2(z)}{f^2} \right)^2 - \frac{1}{16} \left( \frac{f_p^2(z)}{f^2} \right)^3 \quad (5a)$$

Mouginot et al. (2008) Eq. (2)

$$n(z) = \sqrt{1 - \frac{\omega_p^2}{\omega^2}} \approx 1 - \frac{1}{2} \frac{\omega_p^2}{\omega^2} - \frac{1}{8} \left( \frac{\omega_p^2}{\omega^2} \right)^2 - \frac{1}{16} \left( \frac{\omega_p^2}{\omega^2} \right)^3 \quad (5b)$$

obtaining the following similar expressions.

Safaeinili et al. (2007) Eq. (4)

$$\Delta\varphi(f) = -\frac{2}{c} \left[ \frac{8.98^2\pi}{f} \int N_e(z) dz + \frac{8.98^4\pi}{3f^3} \int N_e^2(z) dz + \frac{8.98^6\pi}{8f^5} \int N_e^3(z) dz \right] \quad (6a)$$

Mouginot et al., 2008 Eq. (4)

$$\Delta\varphi(f) = \frac{4\pi}{c} \left[ \frac{40.32}{f} \int_{h_1}^{h_2} n_e dh + \frac{812.851}{f^3} \int_{h_1}^{h_2} n_e^2 dh + \frac{32774.2}{f^5} \int_{h_1}^{h_2} n_e^3 dh \right] \quad (6b)$$

While Cartacci et al. (2013) introduced an expansion of Eq. (4) producing the following expression:

$$\Delta\varphi(f) \approx a_0 + a_1(f - f_0) + a_2(f - f_0)^2 + a_3(f - f_0)^3 + a_4(f - f_0)^4 + \dots, \quad (7)$$

where:

$$\begin{align}
a_0 &= \frac{4\pi}{c} \int_0^L \left( \sqrt{f_0^2 - f_p^2} - f_0 \right) dz \quad [\text{rad}] \nonumber \\
a_1 &= \frac{4\pi}{c} \int_0^L \left( \frac{f_0}{\sqrt{f_0^2 - f_p^2}} - 1 \right) dz \quad [\text{rad}/\text{Hz}] \nonumber \\
a_2 &= -\frac{4\pi}{c} \int_0^L \left( \frac{f_p^2}{2(f_0^2 - f_p^2)^{\frac{3}{2}}} \right) dz \quad [\text{rad}/\text{Hz}^2] \tag{8} \\
a_3 &= \frac{4\pi}{c} \int_0^L \left( \frac{f_0 f_p^2}{2(f_0^2 - f_p^2)^{\frac{5}{2}}} \right) dz \quad [\text{rad}/\text{Hz}^3] \nonumber \\
a_4 &= \frac{4\pi}{c} \int_0^L \left( \frac{4f_0^2 f_p^2 + f_p^4}{8(f_0^2 - f_p^2)^{\frac{7}{2}}} \right) dz \quad [\text{rad}/\text{Hz}^4] \nonumber
\end{align}$$

$$\begin{align*}
a_3 &= \frac{4\pi}{c} \int_0^L \left( \frac{f_0 f_p^2}{2(f_0^2 - f_p^2)^{\frac{5}{2}}} \right) dz \quad [\text{rad}/\text{Hz}^3] \\
a_4 &= \frac{4\pi}{c} \int_0^L \left( \frac{4f_0^2 f_p^2 + f_p^4}{8(f_0^2 - f_p^2)^{\frac{7}{2}}} \right) dz \quad [\text{rad}/\text{Hz}^4]
\end{align*}$$

The use of the previous expressions leads to the implementation of dedicated algorithms able to compensate for the ionosphere distortion, and secondarily, to estimate the Total Electron Content (TEC) of Mars's ionosphere, which is a by-product of this compensation.

As is well known, the TEC can be expressed as follows:

$$\text{TEC} = \int N_e(z) dz \quad (9)$$

Considering the studies of Safaeinili et al., 2007 and Mouginot et al., 2008, it can be seen that the first terms on the right side of Eqs. (6a) and (6b) represent exactly the TEC value.

Otherwise, the studies of Cartacci et al. (2013) and Cartacci et al. (2017) estimated the TEC using the terms of Eq. (8) through the following equations:

Cartacci et al. (2013) Eq. (7)

$$\text{TEC} \approx - \frac{a_2 c f_0^3}{2\pi(8.98)^2} \quad (10)$$

Cartacci et al. (2017) Eq. (10)

$$\text{TEC} \approx \frac{(2a_1 + a_2 f_0) c f_0^2}{2\pi(8.98)^2} \quad (11)$$

Since Eq. (11) represents a more accurate version of Eq. (10), mainly during the day side (see Cartacci et al., 2017 for more details), it will be the only one considered in the present study. We note that although a priori both techniques should provide similar retrievals, only the TEC values obtained on the nightside with both algorithms are very similar, while, on the dayside, they significantly differ (e.g., Sánchez-Cano et al., 2015). Thus, the algorithms developed by Safaeinili et al. (2007) and Mouginot et al. (2008) provide similar results, while they differ from the ones estimated using the Eq. (11) from Cartacci et al. (2017).

Actually, the comparison shows a good accordance for SZA ≥ 95° while the differences increase mainly during the dayside (Figs. 3–4 Cartacci et al., 2013, Figs. 6–7 Conroy et al., 2019).

This discrepancy is very important, because the TEC is a key scientific parameter to characterize the behavior of Mars' ionosphere and could be a clue about the reason why the methods have different behaviors.

A possible explanation can rely on the expansion approximation adopted to obtain Eqs. (6a), (6b) and (7) that are valid only when $f_p/f < <1$.

It is worth noting that, the constraint about the ratio $f_p/f$ should be satisfied not only by the central frequencies listed before (1.8 MHz, 3 MHz, 4 MHz, 5 MHz) but also within the signal bandwidth of 1 MHz, so the effective range of frequencies are 1.3÷2.3 MHz, 2.5÷3.5 MHz, 3.5÷4.5 MHz, 4.5÷5.5 MHz respectively.

If this constraint is not completely satisfied, it is difficult to foresee the effects on the results obtained by the various algorithms, and indeed, the discrepancies between the estimations arise approaching the day side when the plasma frequency increases.

The discussion about this topic is still ongoing between the scientific community and given the importance of an accurate retrieval of TEC estimates on the dayside, in this study we provide a new approach to estimate the TEC to overcome the described criticism and obtain the most accurate TEC estimates of the Martian ionosphere to date.

## 4. The new contrast method (CM) and Mars's ionosphere models

As described in the previous section, the TEC estimation is a byproduct of algorithms dedicated to correcting ionospheric distortion. Thus, its accuracy is strictly dependent on the ones of the related correction methods.

Focusing our attention on the TEC estimation from Eq. (11), introduced in Cartacci et al., 2017, this is the result of various approximations (see Eqs. (8), (9), (10), Cartacci et al., 2017) each of one could introduce some inaccuracy.

Considering the criticism due to the fact that Eq. (7) might lose its validity, with unpredictable results, for transmitted frequencies too close to the plasma frequency, we decided to directly use the terms of Eq. (8), avoiding further approximations (Eq. A.11, Cartacci et al., 2013).

This decision implies the necessity to modify the CM (see appendix A, Cartacci et al., 2013) allowing it to correct the ionosphere distortion directly using the plasma frequency profile ($f_p$) of Eq. (8).

This is not an easy task because the plasma frequency profile along the path of the propagating radar signal is unknown.

For this reason, we introduce the use of two ionospheric models to obtain the plasma frequency profiles: the NeMars Model (Sánchez-Cano et al., 2013; Sánchez-Cano et al., 2016) during the day side and a Gamma distribution function (hereafter Gamma Model) during the night side.

Furthermore, a series of tests were conducted using also a simple Gaussian distribution to verify the impact of the model used in the TEC estimation.
## 4.1. Comparison of the CM different versions

At this point, it is important to recall with some details how the CM works, and which are the modifications introduced in the new version.

Firstly, it is worth noting that the main peculiarity of Eq. (7) is to split the ionosphere distortion that affects the radar signal in different effects.

In other words, every parameter of Eq. (8) has its own effect on the radar signal.

As it is clearly visible in Fig. 1 from Cartacci et al., 2013, term a1 represents the entire group delay, the term a2 is responsible for the signal defocusing, while the terms a3 and a4 affect the signal shape and its side lobes in different ways.

This fact it is very important because allowed to design the original CM optimizing its functioning in correcting the term a2 which is the most important one, accepting a lower accuracy for the terms a3 and a4 that are obtained from a2 using formulas derived through a very simple model (see Eq. A.12, Appendix Cartacci et al., 2013), and ignoring completely the a1 term, that was estimated only in the following version (Cartacci et al., 2017) with the introduction of a signal simulator.

Actually, the original CM estimates the a2 term empirically testing hundreds of correction phases (see Eq. A.1, Appendix Cartacci et al., 2013) within a loop until it achieves the best range compression.

Obviously, this approach is not optimal because the less accuracy of the terms a3 and a4 in Eq. A.1 could still influence negatively the correct estimation of the a2 term.

With the introduction of the New CM it is possible to use directly Eq. (8), theoretically improving the terms a2, a3 and a4, and secondarily estimates the term a1 without the signal simulator.

This is very important, because the main goal of the CM, in all its versions, is to estimate the parameters a1, a2, a3 and a4 as close as possible to the real ones, in order to correct each effect of the ionosphere distortion at best.

In the following sub-sections, we'll introduce some models that will be used to estimate the terms of Eq. (8), both during the day side and the night side.

The aim is to verify if, with the proper settings, the various models are able to reproduce correctly the terms a1, a2, a3 and a4, eventually improving the CM performances or, on the contrary, if they are not suitable for the task due to their bad modeling.

Finally, we'll compare the related TEC values verifying similarities and differences.

## 4.2. The NeMars model

For the dayside, we use the NeMars Model. The NeMars Model is an empirical model for the dayside electron density of the Martian ionosphere that includes both primary and secondary layer and is mainly based on MARSIS AIS data (Sánchez - Cano et al., 2013).

Considering Eqs. (7) and (8) from Sánchez - Cano et al., 2013, the ionosphere electron density profile is characterized by two main layers:

Main layer peak.

$$N_{mo1} = \left( \left( \frac{1}{d} \right)^* \left( (4.5*10^{8}*F) - \left( 1.9*10^{10}*\exp\left(\frac{SZA}{37.2}\right) \right) + 2.3*10^{11} \right) \right) \quad (12)$$

Second layer peak.

$$N_{mo2} = \left( \left( \frac{1}{d} \right)^* \left( (1.2*10^{8}*F) - \left( 3.5*10^{9}*\exp\left(\frac{SZA}{32.9}\right) \right) + 5.9*10^{10} \right) \right) \quad (13)$$

With respective neutral scale height, H, expressed in km.

H of the main layer:

$$H_{sc1} = 2.6*\exp\left(\frac{SZA}{100.5}\right) + 9.4 + ((SZA*(-3.5*10^{-4})) + 0.08) * (h-h_{mo1}) \quad (14)$$

H of the secondary layer:

$$H_{sc2} = 12 \quad (15)$$

Where d is the Mars-Sun distance, F is the $F_{10.7cm}$ index that describes the solar activity, and h is the ionospheric altitude ranging from 50 km to 400 km and $h_{mo1}$ is the main layer peak altitude and can be expressed by the following equation:

$$h_{mo1} = \left( (0.001*\exp^{\frac{SZA}{8.3}}) + 132.4 \right) \quad (16)$$

Following studies (Sánchez-Cano et al., 2016) introduced some modifications in H to allow the NeMars Model to consider the Solar cycle effects.

This implies that four different versions of the H scale height of the main layer were defined while the H scale height of the second layer did not change.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Comparison between Electron Density Profiles, for Mars Sun Distance d = 1.45 AU, $F_{10.7} = 70$, $SZA = 80°$, obtained with the General Model (blue line), in case of Solar Activity Period A (red line), Period B (black line). Period C (magenta dotted line) and Period C (green dotted line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
For moderate solar activity in the descending phase of the solar cycle (called hereafter Period A), H is:

$$H_{sc1} = 11.222 - (0.0019603*SZA) + ((0.099034 - (0.00023124*SZA))* (h - h_{mo1})) \quad (17)$$

For the low solar activity phase of the solar cycle (called hereafter Period B), H is:

$$H_{sc1} = 11.656 - (0.0014153*SZA) + ((0.087487 - (0.000212*SZA))* (h - h_{mo1})) \quad (18)$$

For moderate solar activity of the ascending phase of the solar cycle (called hereafter Period C), H is:

$$H_{sc1} = 16.557 - (0.092701*SZA) + ((0.10735 - (0.00032849*SZA))* (h - h_{mo1})) \quad (19)$$

For the high solar activity phase of the solar cycle (called hereafter Period D), H is:

$$H_{sc1} = 16.105 - (0.042347*SZA) + ((0.16212 - (0.0013043*SZA))* (h - h_{mo1})) \quad (20)$$

The previous equations need some basic input to work properly. The Mars-Sun distance “d” and the SZA values are auxiliary data that can be downloaded from the ESA planetary archive together with the scientific data. The F10.7cm index, is not available simultaneously with the MARSIS data but can be obtained from the NASA website (https://cdaweb.gsfc.nasa.gov/index.html) afterwards.

Fig. 1 shows a comparison between the electron density profiles obtained with the General NeMars Model (Eq. (14), blue line) and the ones obtained with the model adapted to Solar Cycle (Eqs. (17)–(19)) for the same input parameters.

We note that Eqs. (12)–(20) cannot be directly used in the new CM, it is necessary to introduce some flexibility in order to use them. We also note that the NeMars Model describes an average behavior of the ionosphere, while the actual radar signals can be affected by local variations due to different causes (e.g., the solar cycle, solar rotations, solar flares, galactic cosmic rays, gamma ray bursts, etc. (Lillis et al., 2010, Lillis et al., 2011)).

One of these is the presence of local crustal magnetic fields (Acuna et al., 1999; Langlais et al., 2019) that can produce heavy variation in the ionosphere density, even modulating its behavior. These variations produce fluctuations in the signal distortions that are easily recorded by MARSIS (Safaeinili et al., 2007; Cartacci et al., 2013).

For this reason, the physical properties that define the main layer in the NeMars Model, which is responsible for the largest part of the distortion, in detail the main layer peak Nmo1 (Eq. (12)) and the H scale height Hsc1 (Eq. (14)), are modified in order to allow them to vary around theoretical value, thus matching every local variation.

We wish to underline that for this study, the NeMars Model adopted was the General one only, for the following reasons:

* The classification of the Solar activity phases requires a dedicated study that needs some processing after observations are made, while the main goal of the New CM is to process data almost in real time. The use of two different Models for the night side and the day side introduces a transition phase when the algorithm switches between one to another. This transition must be optimized through a refining of the various parameters. Thus, the use of NeMars Model for Solar activity phases increases the complexity of algorithm optimization.

* The general version of the model provides an overall reduction of the algorithm complexity and criticism to increase its reliability during this testing phase.

The decision to use the NeMars General Model at this phase of the study does not exclude the possibility to develop, in the future, an improved version of the algorithm able to use even the NeMars Model with Solar Activity Phases.

### 4.3. The gamma model

For the nightside, we use the Gamma Model because unfortunately, the NeMars model validity is limited to the day side. Therefore, we have the necessity to adopt something else for the night side.

Considering that during the night side the ionosphere profile is simpler, typically consisting of only one layer, we decided to use a Gamma distribution (Picardi et al., 2008) function to directly model the plasma frequency behavior.

In particular, we use:

$$f_{P}(z) = f_{pmax} \frac{z - h_{0}}{b}^{1 - \frac{z - h_{0}}{b}} \quad (21)$$

Where f_pmax is the maximum plasma frequency of the ionosphere, h_0 is the altitude at the beginning of the ionospheric layer, and “b” is the shape factor that depends on the theoretical h_0 and the main peak altitude h_max, according to the following equation:

$$b = h_{max} - h_{0} \quad (22)$$

Fig. 2 shows the overall behavior of the main peak altitude on both the day sides and night sides. The blue circles represent the NeMars model outputs from Eq. (16), while the red ones, used the Gamma Model outputs obtained through the interpolation method.

A particular aspect to consider is that the f_pmax parameter, while for the day side is well defined by Eq. (12), for the night side there is not a similar equation to predict its theoretical value according to the SZA. Therefore, this parameter can vary within a loop in the Gamma Model in a range of possible values defined taking into account all possible variations that could affected its behavior (Lillis et al., 2009; Nemec et al., 2010a; Nemec et al., 2010b), through almost twenty years of operations.

In order to verify whether the Gamma Model can simulate with a good approximation the plasma frequency profile of the ionosphere for the night side, Fig. 3 shows a comparison of the Models used in this paper.

It can be seen that, a less sophisticated model than the NeMars Model, the Gamma Model reproduces the ionosphere shape with sufficient accuracy even during the day side, even if with larger H scale height.

Nevertheless, the Model was tested with the New CM using it in two different ways:

* As a complement to the NeMars Model during the night side

* As a substitute of the NeMars Model even during the day side

The aim of the latest test was to verify the reliability of the Gamma Model during the day side in terms of ionosphere distortion compensation and TEC estimation. In particular, the main goal was to investigate how the lack of the ionospheric secondary layer could deteriorate its performance in both tasks.

### 4.4. The Gaussian model

Finally, in order to verify the impact of the models used in the algorithm implemented, we conducted a series of tests, both during the day and night sides, with an even more simple Gaussian Model to describe the plasma frequency profile similarly to Mouginot et al., 2008:
<figure><img src="image_3.png" /><figcaption>Fig. 2. Main peak altitude behavior vs SZA.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Example of plasma frequency profile from NeMars Model (black line) using SZA = 86.8°, d = 1.41 AU, F₁₀.₇ = 69, Gamma Model (blue line) using b = 50 km, h₀ = 133 km and fₚmax = 1.83 MHz and Gaussian Model (red line) using b₁ = 50 km, hₘₐₓ = 183 km and fₚmax = 1.83 MHz. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

In particular, we use:

$$f_{p}(z) = f_{pmax} \exp \left( \frac{(z-h_{max})^2}{2b_{1}^{2}} \right) \quad (23)$$

Where fₚmax is the maximum plasma frequency, hₘₐₓ is the main peak altitude of the ionosphere, and “b1” is peak width (or standard deviation). An example of plasma frequency profile obtained with a Gaussian distribution is reported in Fig. 3 (blue line), where it can be seen how it differs from the other models implemented.

## 5. The new CM algorithm description

The structure of the CM is always the same (Appendix Cartacci et al., 2013), the main differences are related to the estimation of the

parameters a1, a2, a3 and a4.

In the previous implementation (hereafter Old CM), the a2 term was estimated empirically while the terms a3 and a4 were obtained from a2 (Eq. A.12 Cartacci et al., 2013) and the term a1 through signal simulation (Cartacci et al., 2017).

In the proposed new version (hereafter New CM) the estimation of the parameters a1, a2, a3 and a4 is obtained through different models and the CM algorithm is dedicated to select which is the best quartet.

Fig. 4 shows a schematic flow chart of the tested algorithm, that describes the various steps in which the process is divided, revealing the complexity of the New CM structure.

As introduced in the previous section, each Model needs defined inputs to produce the ionosphere profile necessary to evaluate the
<figure><img src="image_5.png" /><figcaption>Fig. 4. Flow chart describing the new CM functioning.</figcaption></figure>

parameters of Eq. (8), and the introduction of some flexibility in key
parameters is necessary to match the local variation of the ionosphere,
which leads to the introduction of more steps of elaboration to reduce
the computation time.

Actually, the functions that implemented the Models generated grids
of values for each term a1, a2, a3 and a4 (Fig. 4 light blue boxes). These
values are estimated around the theoretical ones in order to consider all
the predictable cases due to the ionosphere conditions. Obviously, this
approach can increase considerably the computing time, so, to reduce it
to reasonable values, it is necessary to perform a selection between the
original sets of parameters.

For this reason, the first step of the algorithm (green box in Fig. 4) is
to use the Old CM (Cartacci et al., 2013) version that performs a pre-
liminary range compression that allows to obtain a starting value for the
a2 term, called a2cm.

The a₂cm term can be used as a reference value to the next step,
represented by the blue boxes.

In this phase, the ionospheric Model is selected according to the SZA
value, and produces the sets of parameters a1, a2, a3, a4 with the related
values of fpmax and TEC. Using the a2cm value it is possible to reduce the
sets of parameters to a small group "n", by selecting only the ones that
are characterized by a term a2 close (±20 % max) to the a2cm (dark blue
box).

This approach allows us to reduce the computing time of the
following step that consists in the CM algorithm (yellow boxes) that
performs a signal range compression using each set of parameters a1n,
a2n, a3n and a4n selecting the one that allows to minimize the signal
energy within the receiving window (see Appendix A Cartacci et al.,
2013) that corresponds to the best ionosphere distortion correction. The
output of this step, represented by the optimal value of a1, a2, a3 and a4
(and the related fpmax and TEC values) will be used to correct the signal
in the final range compression phase (red box).

It is worth noting that Fig. 4 represents the architecture of the final
implementation of the New CM that will be integrated in the MARSIS
ground segment pipeline, while the version used to test both the Gamma
and Gaussian model during the day side was simpler, since the models

are used regardless of the SZA values.

6. The new CM algorithm performances

The main goal of developing the New CM algorithm is to obtain
better accuracy in the TEC estimation to improve the ionospheric
correction and have scientific reliable TEC estimates. For that, the first
step in the algorithm validation is to verify the signal processing per-
formances comparing the signal obtained with the Old CM with the new
one. The task is not easy since the comparison involved a huge number
of synthetic apertures (more than 17.5\*10⁶ frames). Because of this, we
focused our attention on three key parameters: the a1, the a2 value and
the signal energy that is obtained summing the signal samples intensity.

Checking the a2 values obtained by the two different algorithms
allows us to verify not only if the decision to use the Old CM as a filter in
the new algorithm works well, but also to quantify the differences be-
tween the two methods.

The comparison of the energies that characterize each frame helps to
understand if the signal behavior was effectively improved or worsened
by changing the processing algorithm.

Finally, checking the a1 values allows us to verify if the ionosphere
delay is correctly estimated and the radargram signal is well positioned
inside the receiving window.

6.1. a2 checking

Despite both algorithms using three parameters to compensate the
ionosphere distortion (a2, a3 and a4 of Eq. (8)), the a2 parameter is the
most important one (see Fig. 1 Cartacci et al., 2013) in both of them, as it
produces the signal main lobe widening.

In order to compare correctly the a2 values obtained for each frame
of the data set using the old CM and the new one, we need to define a
procedure.

Considering the CM theory exposed in the Appendix of Cartacci et al.,
2013, we fixed a threshold of Δa₂ = 6.28 10⁻¹² rad/Hz² (see Eq. A.10
Cartacci et al., 2013) that represents the necessary accuracy to perform
the ionospheric correction correctly.

After that we calculate the difference between the a2 values obtained
by the Old CM (a2OCM) and the ones produced by the Models (a2NCM),
verifying if the module of such value is lower than the defined threshold.

Table 1 shows that even with signal of poor quality, with a low SNR,
the differences between the a2 values of the two CMs are below the Δa₂
threshold for at least the 93 % of the cases considering all the operative
frequencies.

If we consider a signal of very good quality with a SNR of 20 dB the percentage rises to more than 98 %. Actually, the comparison shows that only for a very small percentage the two algorithms produce values substantially differently (see row 3 of Table 1).

It is important to underline that this result means that the two CM,
the Old and the New one, compensate for the ionosphere distortion
practically with the same effectiveness.

This is a very important result because it means that the data pro-
cessed, and delivered, till now are well compressed and perfectly suit-
able for the scientific analysis.

Unfortunately, Table 1 does not allow us to discern whether the
differences between the a₂ terms means that there is a signal improve-
ment or not. In other words, the value of 0.63 % estimated with a SNR ≥
20 dB reported in the third row includes both cases.

To avoid such ambiguity, it is necessary to verify the signal energy
variation.

6.2. Signal energy checking

As introduced in the previous section and described in detail in Ap-
pendix of Cartacci et al., 2013, both the Old and the New CM use the
signal energy to check if the signal is well compressed. As it is possible to
<table><caption>**Table 1**
Old CM vs New CM comparison considering all frequencies.</caption><thead><tr><th></th><th></th><th>SNR 5 dB</th><th>SNR10dB</th><th>SNR 15 dB</th><th>SNR 20 dB</th><th>SNR 25 dB</th></tr></thead><tbody><tr><td>1</td><td>|a2OCM - a2NCM| ≤ Δa<sub>2</sub></td><td>93,3 (%)</td><td>95,23 (%)</td><td>96,58 (%)</td><td>97,8 (%)</td><td>98,76 (%)</td></tr><tr><td>2</td><td>Δa<sub>2</sub> &lt; |a2OCM - a2NCM| ≤ 2Δa<sub>2</sub></td><td>2.83 (%)</td><td>2.62 (%)</td><td>2.18 (%)</td><td>1.57 (%)</td><td>0.96 (%)</td></tr><tr><td>3</td><td>|a2OCM - a2NCM| &gt; 2Δa<sub>2</sub></td><td>3.87 (%)</td><td>2.15 (%)</td><td>1.24 (%)</td><td>0.63 (%)</td><td>0.28 (%)</td></tr></tbody></table>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Night Side signals energy comparison for all operative frequencies. The black and yellow lines represent the thresholds E<sub>WR</sub> and E<sub>IR</sub> introduced successively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

see from Fig. 1 of Cartacci et al. (2013), the signal distorted is much wider than the compensated one. This means that also the overall energy is higher. The energies comparison is more accurate with respect to the a2 term to verify the performances of the new algorithm with respect to the old one.

Fig. 5 shows the ratio between the signal's energy obtained with the two algorithms considering the frames collected during the night side for all the operative frequencies used, while Fig. 6 shows the ratio for the frames collected during the day side only for 4 and 5 MHz that are the only two frequencies used for SZA < 90°.

The analysis was performed using only the frames characterized by a SNR ≥ 20 dB.

This requirement is important because the performance of both algorithms can be reduced if the signal has a low SNR due to various effects such as the presence of surface clutter or the ionosphere attenuation. Moreover, for signals with a good SNR, it is easier to verify visually the obtained results and check for which ratio value the signal is effectively worsened or improved in a way that can affect its scientific relevance.

The samples with values greater than 1 represent frames where the old algorithm produces a signal with a higher energy respect to the new one, and then with a worse range compression, while the values lower than 1 represent frames where the opposite occurs. In this case, it is the new CM that is performing worse.

Actually, the performances analysis is more complex because a ratio greater or lower than one does not automatically mean an effective improvement or a worsening of the signal respectively, because the

difference can be so small to be negligible from the scientific point of view.

For this reason, it was necessary to directly compare enough frames processed with both algorithms in order to identify the best thresholds that allow the correct classifications of the results.

In Fig. 7 there are few examples of frames that show how the new algorithm produces a worse range compressed signal respect to the old CM.

As can be seen, a worse energy ratio (E<sub>WR</sub>) does not mean automatically loss of information from a scientific point of view. Fig. 7a-b show that with E<sub>WR</sub> = 0.982 and E<sub>WR</sub> = 0.98 the signal shape and power are almost identical for both algorithms. These E<sub>WR</sub> values do not produce a loss of information on the frames. On the contrary, Fig. 7c shows that the frame processed with the New CM is wider and with a significant loss of power. Fig. 7d shows an increasing deterioration of the signal performances respect to the frames processed with the Old CM. According to these examples and to many other that show similar results, a threshold of E<sub>WR</sub> ≤ 0.98 was selected to classify whether a frame processed with the New CM is worse from the scientific point of view.

The same approach was followed to determine the upper threshold (E<sub>IR</sub>) that identifies an improvement of the signal.

Fig. 8 shows a selected quartet of frames as an example.

As in the previous case, it is clear that only for certain values of E<sub>IR</sub> the differences assume a relevance capable of interfering with a correct interpretation of the signal. While in Fig. 8a and b the differences are almost negligible, the signals represented in Fig. 8c and d show a visible improvement produced by the New CM. A reasonable threshold appears
<figure><img src="image_10.png" /><figcaption>Fig. 6. Day Side signals energy comparison for operative frequencies 4 and 5 MHz. The black and yellow lines represent the thresholds $E_{WR}$ and $E_{IR}$ introduced successively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7. Comparison between a quartet of frames, collected at 4 MHz, characterized by different values of $E_{WR}$.</figcaption></figure>

$E_{IR} = 1.018$, so the signals with an $E_{IR} \ge 1.018$ could be considered as improved respect to the past.

In Table 2 it is possible to verify the correlation between the results reported in Figs. 7 and 8 with the related $a_2$ values obtained by the different algorithms.

Checking the table, it is easy to see that all the frames characterized by an evident variation of the energy, for example, orbits 13,536 and 11,342 for $E_{WR}$ or orbits 4877 and 18,299 for $E_{IR}$, are also characterized by a $|a_{2OCM} - a_{2NCM}| > 2\Delta a_2$.

As a consequence of the previous analysis, it is possible to insert both the upper and the lower threshold in Figs. 5 and 6 as two dashed yellow and black lines allowing to visualize better the comparison between the

two algorithms. Moreover, it is possible to determine quantitatively how many frames improve and how many perform worse. It is worth nothing that, fortunately, the largest part of the frames has a ratio very close to one and within the thresholds, both in the day side than during the night side. This means that the two algorithms produce the same performances. This result matches the one related to the $a_2$ values reported in Table 1.

From the Figs. 5 and 6, it seems that the New CM performs a bit better than the old one in particular for the lower frequencies. This result is due to a drawback of the Old CM.

As described in Cartacci et al., 2013 (see Appendix), when the Old CM starts the range compression, it has no information about the signal,
<figure><img src="image_12.png" /><figcaption>Fig. 8. Comparison between a quartet of frames, collected at 4 MHz, characterized by different values of E<sub>IR</sub>.</figcaption></figure>

<table><caption>**Table 2**

Correlation between the energy differences and the a<sub>2</sub> values of the different algorithms.</caption><thead><tr><th>Orbit number</th><th>Frame number</th><th>E<sub>WR</sub></th><th>E<sub>IR</sub></th><th>|a2OCM - a2NCM|</th></tr></thead><tbody><tr><td>12,669</td><td>640</td><td>0.982</td><td>-</td><td>≈ 4•10<sup>-12</sup> rad/Hz<sup>2</sup></td></tr><tr><td>11,431</td><td>795</td><td>0.98</td><td>-</td><td>≈ 1•10<sup>-12</sup> rad/Hz<sup>2</sup></td></tr><tr><td>13,536</td><td>1140</td><td>0.978</td><td>-</td><td>≈ 13•10<sup>-12</sup> rad/Hz<sup>2</sup></td></tr><tr><td>11,342</td><td>1103</td><td>0.976</td><td>-</td><td>≈ 16•10<sup>-12</sup> rad/Hz<sup>2</sup></td></tr><tr><td>17,666</td><td>271</td><td>-</td><td>1.0175</td><td>≈ 11•10<sup>-12</sup> rad/Hz<sup>2</sup></td></tr><tr><td>7029</td><td>918</td><td>-</td><td>1.018</td><td>≈ 10•10<sup>-12</sup> rad/Hz<sup>2</sup></td></tr><tr><td>4877</td><td>1113</td><td>-</td><td>1.02</td><td>≈ 17•10<sup>-12</sup> rad/Hz<sup>2</sup></td></tr><tr><td>18,299</td><td>338</td><td>-</td><td>1.025</td><td>≈ 23•10<sup>-12</sup> rad/Hz<sup>2</sup></td></tr></tbody></table>

so it needs few frames to converge to an optimal ionosphere correction. This happens for the first frames of an orbit and when there is a switch between two transmitted frequencies. Also, a sudden fading of the signal can produce a wrong estimation of the a2, a3 and a4 parameters, that needs few frames to be corrected adequately.

Moreover, for increasing plasma frequency values the approximation used in the Old CM (see Eq. A.11 Cartacci et al., 2013) starts to fail, producing a compensation that does not completely remove the distortion.

These drawbacks are overcome by the new algorithm that, thanks to the use of the models as a reference, is more robust.

It is clearly visible from Figs. 5 and 6 that the higher concentration of samples that show an improvement of the performances is located mainly around the terminator (SZA = 90°) and, for 1.8-3 MHz, also for SZA = 105° due to the switch between 3 and 4 MHz or between 1.8 and 3 MHz.

Actually, the Figs. 5 and 6 could lead to a misleading deduction that the New CM is a huge improvement respect to the old one. Indeed, considering the data set collected with the frequency of 4 MHz, the only one used intensively for whatever SZA (both day and night sides), that satisfies the constraint of a SNR ≥ 20 dB (about 5•10<sup>6</sup> frames), only the 0.22 % (11•10<sup>3</sup> frames) of the signals show a visible variation that could modify the scientific information and lesser than two hundreds are worsened respect to the older processing. The last cases appear to be concentrated for SZA a bit lower than 90° (see Figs. 6 and 7) suggesting that there is probably residual work to do for optimizing the transition phase between the two implemented models.

This is good news because it demonstrates that the data collected

until now and processed with the Old CM are still completely reliable for scientific purposes. We wish to underline again that the main goal of the New CM is to provide a TEC estimation with better accuracy and only secondarily to improve the range compression performances.

### 6.3. a1 validation

Last but not least, we verified if the new algorithm correctly estimates the ionosphere delay, comparing the signal position within the receiving window with its related signal simulation as introduced in Cartacci et al., 2017.

This task is very tricky due to the radar signal characteristics.

First of all, MARSIS signal is sampled with a frequency fs = 1.4 MHz, this means that each sample corresponds to a time bin of 0.714 µs in the range direction. As is clearly visible in Figs. 7 and 8 the effect of this sampling rate is a poor reconstruction of the signal shape. As obvious, the same issue affects the simulated signal.

Moreover, the Mars's surface can degrade the signal behavior introducing various effects due to surface roughness, clutter from craters, gullies and other features.

The combination between surface and ionosphere effects can add so many variables that it is not easy to compare the simulated signal position with respect to the real one. Actually, is virtually impossible for the two signals to match completely, so it is more reasonable to verify if the estimated discrepancies between the two signals are within an acceptable margin of error.

From Table 3 it is possible to check that the ionosphere delay compensation achieved by the New CM is very good, since the differences with the related simulations are always of few samples corresponding to an error smaller than 3 µsec for almost all the frames.

In conclusion, the New CM, working with the parameters of Eq. (8) directly estimated through the NeMars Model (day side) and the Gamma Model (night side), proved to be effective in removing the ionosphere distortion, slightly improving the overall performances respect to the Old CM. Only a very small number of frames (a few hundred) show results that were not completely satisfactory. This means that both models, and the implemented settings, are very reliable in approximating the real electron density profiles of the ionosphere.

## 7. TEC estimation and comparison

As introduced in the previous sections, the TEC estimation is no
<table><caption>**Table 3**
Estimated error between simulated and real signal position considering all frequencies and all SZA.</caption><thead><tr><th></th><th>SNR 5 dB</th><th>SNR10dB</th><th>SNR 15 dB</th><th>SNR 20 dB</th><th>SNR 25 dB</th></tr></thead><tbody><tr><td>|a1SIM - a1NCM| ≤ 2 samples (1.42µs)</td><td>71,31 (%)</td><td>73,97 (%)</td><td>75,68 (%)</td><td>76,91 (%)</td><td>78,13 (%)</td></tr><tr><td>|a1SIM - a1NCM| ≤ 3 samples (2.14µs)</td><td>82,38 (%)</td><td>85,37 (%)</td><td>87,25 (%)</td><td>88,71 (%)</td><td>90,19 (%)</td></tr><tr><td>|a1SIM - a1NCM| ≤ 4 samples (2.85µs)</td><td>87,74 (%)</td><td>90,82 (%)</td><td>92,63 (%)</td><td>93,91 (%)</td><td>95,08 (%)</td></tr></tbody></table>

longer a secondary product obtained from the output parameter of the processing algorithm (see Eqs. (10) and (11)) but it is a main product of the NeMars Model or the Gamma Model for the day side and the night side respectively. Considering the NeMars Model (Eqs. (12)–(16)), when a frame is processed, the main parameters (h, d, F and SZA) are well known, this means that both the electron density profile (Ne) and the TEC can be easily evaluated.

The New CM only performs a finer tuning of the main layer peak Nmol and of the H scale height Hsc1 to match at best the distortion that affects the radar signal according to the local Ionosphere conditions. In this way, for each frame, it defines the optimal electron density profile and then the related TEC.

For the night side, the same result is obtained through the Gamma distribution. In this case, the Gamma Model parameters have a variation range defined empirically but the final result is similar, allowing the estimation of the plasma frequency profile and then the related TEC.

As a preliminary analysis of the New CM with the NeMars Model during the dayside (SZA < 90°) and the Gamma Model during the night-side (SZA ≥ 90°), Fig. 9 shows the average values of the estimated TEC for three cases:

* High Solar Activity (2011–2016)

* Low Solar Activity (2007–2009)

* Complete available Data Set (Period 2005–2021)

The average values are computed for SZA bins of 0.25°.

It is worth noting that the TEC values are estimated using data collected at 4 MHz for SZA ≥ 90° and 5 MHz for SZA < 90°.

Moreover, Fig. 10 shows the behavior of the related peak plasma frequencies, evaluated in the same way again using data collected at 4 MHz for SZA ≥ 90° and 5 MHz for SZA < 90°.

An interesting observation from Figs. 9–10 is related to the behavior

of both TEC and Peak plasma frequency for 90° ≥ SZA ≥ 100°. Within this range the average values of the three periods diverge from each other. This effect is probably due to the fact that the constraint $f_p/f \ll 1$ is no more satisfied, in particular for high solar activity (blue circle).

Indeed, for SZA < 90°, the different values for both TEC and $f_{pmax}$ are closer to each other because the frequency switched from 4 to 5 MHz, and the differences are due to the transition between the two adopted models.

The next step is a comparison of the obtained TEC to verify its behavior and the possible variation respect to the previous estimates.

For this aim we have decided to compare the TEC values obtained through the following algorithms:

* New CM with NeMars Model (day side) and Gamma Model (night side).

* Old CM (Cartacci et al., 2017).

* TEC delivered from Grenoble University based on the Mouginot et al., 2008 algorithm.

* NeMars Model (Sánchez – Cano et al., 2013).

* New CM with Gamma Model (day side and night side) (only for test purposes).

* New CM with Gaussian Model (day side and night side) (only for test purposes).

In Figs. 11 and 12 the comparison between the various TEC summarized before is reported for the orbits 4710 and 4711.

As it is clearly visible, during the night side, for SZA ≥ 95° the TEC values are quite similar, while between 90° ≥ SZA ≥ 95° they start to differ since the approximation used and the models used to initialize the respective parameters start to influence the results differently.

During the day side, the behavior differs considerably. In particular, the TEC from Grenoble (black line) represents the lower value among

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Comparison between average TEC values obtained for the entire period under analysis (red circles), for a range time of High Solar Activity (blue circles) and Low Solar Activity (black circles). The frequencies used are 4 MHz for SZA ≥ 90° and 5 MHz for SZA < 90°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Comparison between average maximum plasma frequency values obtained for the entire period under analysis (red circles), for a range time of High Solar Activity (blue circles) and Low Solar Activity (black circles). The frequencies used are 4 MHz for SZA ≥ 90° and 5 MHz for SZA < 90°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** TEC comparison from various algorithms. In case of TECs derived from the CM algorithm (both new and old version), the values are obtained from the higher frequency (5 MHz for SZA < 90°, 4 MHz for SZA ≥ 90°).</figcaption></figure>

the others as noticed also in previous work (Cartacci et al., 2017).

The TEC obtained from the New CM with the NeMars Model (blue line) is the highest one and obviously is set around the TEC estimated directly from the Model (grey line). Anyway, a range of variability of 10–15 % around the theoretical values is sufficient to take into account local variations of the ionosphere respect to the model.

As expected, the TEC estimated using the New CM with only the Gamma Model (magenta line) is lower than the one obtained from the NeMars Model (blue line), considering the lack of the low altitude secondary layer.

Interestingly, the TEC from the Old CM (red line) and the ones estimated using the New CM with both the Gamma Model (magenta line) and the Gaussian Model (blue dotted line) are very close to each other.

These results are particularly important because it demonstrates that two algorithms (Old and New CM) developed in different ways and estimating the TEC with completely different approximations and models still can obtain values very similar. This fact is clear evidence that the TEC estimation from an algorithm designed to correct to ionosphere distortion is affected by a high level of ambiguity.

Furthermore, another interesting result is represented by the fact that the TEC estimated using the New CM and the Gaussian Model is the closest to the Grenoble one.

In order to verify the obtained results for orbits 4710 and 4711, the same comparison was performed considering almost 1000 orbits collected in 2007.

In Fig. 13 there is a comparison of the TEC values obtained through the different models considered in this work evaluated for the 2007 data.
<figure><img src="image_16.png" /><figcaption>Fig. 12. TEC comparison from various algorithms. In case of TECs derived from the CM algorithm (both new and old version), the values are obtained from the higher frequency (5 MHz for SZA < 90°, 4 MHz for SZA ≥ 90°).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 13. Comparison between the TEC values obtained through different algorithms averaged for SZA bins of 0.25° and related to the year 2007.</figcaption></figure>

Therefore, Fig. 13 confirms the results shown in Figs. 11 and 12.

To complete the analysis, and verify the correctness of using the Gamma Model during the day side and the Gaussian Model for every SZA, we checked also the performances related to the estimation of the a2 term of the ionosphere correction phase in comparison with the one of the Old CM.

In particular, Figs. 14 and 15 show how the differences between the a2 values estimated for orbit 4710 and 4711 respectively, are well below the threshold of Δa2 = 6.28 10⁻¹² rad/Hz². This result, in correlation with Table 2, demonstrates that the ionosphere correction is performed with the same results with both the New CM and the Old CM, confirming that this phase is not influenced by the approximations used to develop the algorithm and the models used to initialize its parameters.

Furthermore, we conducted an analysis on the a2 parameters produced by the different algorithms, considering all available frequencies

with the constraint of an SNR ≥ 20 dB, focusing our attention only on the frames where the a2 values overcome the threshold identified by the following expression:

$$|a2OCM - a2NCM| > 2\Delta a_2 \qquad (24)$$

The result was that whatever models we used to initialize the New CM, the percentage of frames that satisfy the Eq. (24), was around 1 % during the day side (SZA < 90°) and around 0.29 % during the night side (SZA ≥ 90°).

In other words, all the algorithms used to compensate for the ionosphere distortion converge and result in similar output parameters, in particular the a2 value, virtually identical.

This is evidence that correct compensation of the ionosphere distortion is not sufficient to guarantee a univocally estimation of the TEC.
<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Comparison between the $a_2$ values obtained by the New CM through different models with the $a_2$ values obtained with the Old CM for the same orbit 4710. The values considered are related to the higher frequency (5 MHz for SZA < 90°, 4 MHz for SZA ≥ 90°).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Fig. 15.** Comparison between the $a_2$ values obtained by the New CM through different models with the $a_2$ values obtained with the Old CM for the same orbit 4710. The values considered are related to the higher frequency (5 MHz for SZA < 90°, 4 MHz for SZA ≥ 90°).</figcaption></figure>

Finally, the main results represented in Figs. 11–15 can be summarized as follows:

### Night Side

* For SZA ≥ 95°, when the ratio $f_p/f \ll 1$ is well satisfied, all the considered methods estimate very similar TEC values.

* For 90° ≤ SZA ≤ 95° the increasing influence of a higher plasma frequency spreads out the TEC values on a wider range.

### Day Side

* The same algorithm (New CM) can correctly compensate for the ionosphere distortion using three different models to initialize Eq.

(8) (NeMars, Gamma and Gaussian), even if the obtained TEC values are completely different.

* Two algorithms (Old CM and New CM), developed in different ways and estimating the TEC with completely different approximations and model (Gamma and Gaussian), not only correct the ionosphere distortion in the same way but still obtain very similar TEC values.

* A good correction of the ionosphere distortion is not sufficient evidence to verify if an algorithm is able to estimate the TEC values correctly.

In conclusion, during the night side, both the Gamma and the Gaussian Model seem to reproduce sufficiently well the ionosphere profile, allowing an estimation of the related TEC values that are very
close to each other and very similar to the ones obtained from Mouginot et al., 2008 and Cartacci et al., 2017. For SZA ≥ 95° the various TEC start to differ, in particular the one obtained from Cartacci et al., 2017 seems a bit overestimated, and this is probably because the constraint $f_p/f \ll 1$ is not completely satisfied.

During the day side, the interpretation of the obtained results is more complicated. As introduced in Section 4, it was well known that the Gamma and Gaussian models were not reliable to reproduce correctly the ionosphere behavior for SZA < 90°. Indeed, the aim of the test was to verify how the lack of the ionosphere low altitude layer could affect the performance of the New CM.

Nevertheless, the algorithm performs well in correcting the ionosphere distortion and thus introducing a big question about the correctness to use this criterion as a quality check for the related TECs.

Finally, the NeMars Model deserves few more words.

With respect to the Gamma and the Gaussian models, the NeMars Model was specifically designed to represent at best the day side Mars' Ionosphere behavior, and it was successfully tested and verified in many previous studies. The good result obtained by this model in combination with the New CM is further evidence of its reliability.

Cross-checking the results reported in Sections 6 and 7, the TEC derived from the NeMars Model seems to be the best approximation of the real one.

## 8. TEC from gamma model, a further validation

The last step consists in a thorough checking of the Gamma Model and its performances, because, while the NeMars Model is well known and a long-tested model (e.g., Sánchez-Cano et al., 2013, Sánchez-Cano et al., 2016; Krishnan and Karatekin, 2024; Parrott et al., 2024), the use of the Gamma Model during the night side is not so utilized. As

introduced in the previous section, the first evidence of its good performance relies in the behaviors reported in Figs. 14 and 15.

Indeed, during the night side the differences between the $a_2$ values estimated by the Old CM and the ones estimated by the New CM initialized through the Gamma Model are negligible.

This means that, the ionosphere distortion corrections are very close to each other.

Considering the TEC values estimated by the two algorithms, Figs. 11 and 12 show that the behavior is, again, very close to each other.

In order to verify better the validity of the Gamma Model in reproducing the plasma frequency profile, we decided to estimate the map (see Fig. 9a Cartacci et al., 2013) that shows the TEC variations in relation to the Martian crustal field (Langlais et al., 2004; Langlais et al., 2010) during the night side (Safaeinili et al., 2007; Cartacci et al., 2013).

The aim of this test is to verify that the Gamma Model is able, not only to estimate with a good approximation the TEC intensity, but also to reproduce well its fluctuations due to the crustal magnetic fields. With respect to the past work, we could add more than 4000 orbits to the previously used data set, for a total exceeding 8700 orbits. In this way, it was possible to improve the coverage and the level of detail of the map. Similarly, the magnetic field model has been updated, and it now used Mars Global Surveyor and MAVEN magnetic field measurements (Langlais et al., 2019).

Using the same approach described in Section 5.3 of Cartacci et al., 2013, the TEC variation (in percentage) estimated respect to a best fitting behavior, $\Delta\text{TEC}_{\text{pc}}$, was binned in a grid $0.25^\circ \times 0.25^\circ$. Fig. 16 shows the latitude-longitude map of the final $\Delta\text{TEC}_{\text{pc}}$. Moreover, it was used the same grey scale colour coding of Cartacci et al., 2013. Thus, it is possible to appreciate better the improvement achieved and verify that the Gamma Model works very fine.

To simplify the analysis, Fig. 17 shows an improved version of the

<figure><img src="image_20.png" /><figcaption>Fig. 16. Latitude-longitude maps of $\Delta\text{TEC}_{\text{pc}}$ plotted over a shaded relief Martian map background. The thin yellow and red dashed lines allow us to identify some features that are clearly correlated to the crustal magnetic fields. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Fig. 17.** Latitude-longitude maps of α is the angle between the ambient magnetic field vector of internal origin, predicted by the model of *Langlais et al., 2019*, and the local vertical direction. The data set is plotted over a shaded relief Martian map background. This figure shows the same thin yellow and red dashed lines reported in Fig. 16, for an easier comparison of the maps. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

latitude-longitude map (see Fig. 9b, Cartacci et al., 2013) of the angle between the local vertical direction and the ambient magnetic field vector of internal origin, predicted by a spherical harmonics model up to degree 134 (Langlais et al., 2019).

Thin yellow and red dashed lines were drawn to underline some of the features that are clearly identified on both maps.

In conclusion, the obtained results not only agree with the ones reported in past studies (Safaeinili et al., 2007, Cartacci et al., 2013) achieving a great improvement in terms of coverage and accuracy, but it also demonstrates that the map is a good representation of the night side ionosphere as more data have been added with the same results.

## 9. Discussion and summary

In this paper we have introduced a new version of the Contrast Method, a tool developed to compensate for the ionospheric distortion effects on radar signals.

We have demonstrated that the use of two Ionosphere Models, the NeMars Model for the day side and the Gamma Model for the night side, to initialize the CM parameters, allows to reduce at minimum the approximation adopted in the TEC estimation improving its accuracy. Moreover, the new algorithm slightly improves the ionosphere distortion compensation performances mainly when the MARSIS radar switches between two operative frequencies.

We have also demonstrated that a right ionosphere distortion compensation is not sufficient, alone, to demonstrate the correctness of the TEC estimation.

Finally, we have estimated a complete map of the TEC variations related to its interaction with the Mars crustal magnetic field (Langlais et al., 2019), finalizing a previous work (Cartacci et al., 2013) exploiting

the availability of the data collected from more than 8700 orbits.

The TEC estimation, as a by-product from the algorithms dedicated to correct the signal distortions introduced by the Mars ionosphere, started at the beginning of the MARSIS operations. For almost twenty years, various methods and studies (Safaeinili et al., 2007; Mouginot et al., 2008; Cartacci et al., 2013; Cartacci et al., 2017; Conroy et al., 2019) have been discussing this topic but yet failed to converge towards a shared interpretation.

This was mainly due to the significant differences between the TEC estimated by each proposed algorithm.

As described in Section 2, every method is characterized by a series of approximations that can vary considerably from each other, influencing the final result. Moreover, for transmitted frequencies $f_0$ that not completely satisfied the constraint $f_p < <f_0$, the possible negative effects in the obtained TEC are unpredictable.

Probably, the main issue is related to the assumption that a correct compensation of the ionosphere distortion is a reliable indicator of a good TEC estimation.

The comparison between the values of term a2 obtained by the Old CM and the New CM (see Figs. 5, 6, 14, 15) shows that, with the right settings, every proposed model (NeMars, Gamma and Gaussian) has sufficient flexibility to reproduce the effects of the ionosphere distortion, regardless their theoretical validity, without significative variation of the range compression performances.

This result demonstrates that a good ionosphere correction is not sufficient validation of the related TEC values, introducing the necessity to carefully analyze their theoretical validity.

Actually, as Figs. 11, 12 and 13 clearly show, all the TEC values are close to each other only during the night side for $SZA \ge 95^\circ$, when the higher frequency transmitted (4 MHz) satisfies the constraint $f_p < <f_0$.
For lower SZA angles, the various TECs start to diverge from each other according to the approximations adopted.

Nevertheless, for SZA ≥ 90°, all methods considered show a good consistency that combined with the related good performances in correcting the ionosphere distortion, allow us to assert that the estimated TEC values are reasonably close to the real ones.

The analysis of the results obtained during the day side is far more complex, because the TEC values obtained through the New CM with the models (both Gamma and Gaussian) and the Old CM during the day side (see Fig. 13) are surprisingly very similar, considering that these methods share only the generic architecture of the algorithms and the starting Eq. (7).

In particular, while the New CM TEC uses directly Eq. (8) and the plasma frequency profile given by the models, the Old CM TEC is implemented using a simplified version of Eq. (8) (Eq. A.11 Appendix, Cartacci et al., 2013) and Eq. (11).

Moreover, as already highlighted in Section 7 conclusions, both the Gamma and the Gaussian model are not suitable for a good representation of the ionosphere for SZA < 90°, due to the lack of the second layer at low altitude (see Fig. 3), thus introducing severe doubts about the correctness of the related TEC and also about the one estimated using the method from Cartacci et al., 2017.

Finally, the TEC estimated by the New CM in combination with the NeMars Model seems to be the most reliable since it is obtained using a model that reproduce the ionosphere more accurately.

The use of three different models during the day side (NeMars, Gamma and Gaussian), demonstrates how the accuracy of the model adopted can greatly modify the final results.

In particular, the same New CM algorithm correctly performs the ionosphere distortion correction with all models, but the TEC values differ considerably.

The present study demonstrates that the validation of an ionosphere model through its use as part of an algorithm designed to correct the signal distortion is not completely correct.

This approach can be heavily influenced by the approximation used to define the algorithm itself and the tuning of its parameters. For these reasons, only during the night side all the methods produced TEC values very similar to each other.

It is worth noting that, considering MARSIS radar, the operative mode designed for the ionosphere analysis was AIS, as described in Section 1.

SS3 data processing can produce indirect information about the ionosphere but with some limitations since the operative frequencies are very close to the plasma frequency, mainly during the day side.

If the constraint $f_p/f \ll 1$ it is not completely satisfied, Eqs. (6a), (6b), (7) and (8) are barely valid and consequently there is also the possibility that the ionosphere distortion is not completely removed from the signal. In this condition every TEC estimation lost validity.

A more reasonable approach seems to be the use of a model obtained directly from more suitable data produced by instruments, such as MARSIS AIS and MAVEN, dedicated exclusively to ionosphere analysis.

In this way, the ionosphere correction that affects the SS3 data can be used to test the Models verifying their correctness to reproduce the ionosphere distortion.

Following this reasoning, the New CM, described in this paper, is designed with a high level of modularity in order to be easily updated according to the most recent studies related to the Mars ionosphere.

The final result of this work is related to the upgrade of the map of the TEC variations related to interaction of the ionosphere, characterizing the night side, with the Mars crustal magnetic field (Langlais et al., 2019).

This final map (see Fig. 16) is obtained collecting information of more than 8700 orbits, achieving a global coverage of the entire planet.

Moreover, this improved map demonstrates the validity of the Gamma Model to correctly describe the plasma frequency profile, matching very well with the older version (Cartacci et al., 2013).

To conclude this section, we wish to outline the directions along which we intend to further develop our activities: 1) investigate the possibility to introduce the NeMars Model that takes into account the impact of the Solar Cycle; 2) improve the transition for SZA = 90° between the models; 3) further extend the data base including the observations from 2022 till the present day with the final goal to achieve a full regime real time processing of the future data; 4) update the study of the TEC variations related to interaction of the ionosphere with the Mars crustal magnetic field introducing a map related to the day side data.

### CRediT authorship contribution statement

M. Cartacci: Investigation, Formal analysis, Data curation. B. Sánchez-Cano: Investigation, Formal analysis. A. Cicchetti: Investigation. R. Noschese: Investigation. B. Langlais: Investigation. R. Orosei: Investigation.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The author is not an Editorial Board Member/Editor-in-Chief/ Associate Editor/Guest Editor for any Journal and was not involved in the editorial review or the decision to publish this article.

The authors declare that there are no financial interests/personal relationships which may be considered as potential competing interests.

### Acknowledgements

The authors gratefully acknowledge support from the Italian Space Agency (ASI) through contract 2024-40-HH.0. Operations of the Mars Express spacecraft by the European Space Agency (ESA) are gratefully acknowledged. B.S.-C. acknowledges support through STFC Ernest Rutherford Fellowship ST/V004115/1.

### Data availability

Data will be made available on request.

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