In spacecraft borne radar investigations of planets such as Venus and Mars the waves must penetrate an ionospheric layer which causes absorption and dispersive phase delay if the waves reach the surface at all. If the purpose of the radar system is to explore the planetary subsurface, the radar frequency should be as low as possible for maximum skin depth, yet high enough to reach the surface. In order to resolve subsurface discontinuities the radar must use short pulses and because of the low frequency the dispersive pulse distortion in the ionosphere becomes a problem. In this paper we discuss ways to avoid the pulse distortion and to recover the original pulse shape. As the ionosphere is unknown and changes with time and position, the necessary ionospheric data must be derived from the radar observations themselves. The scheme described is designed to do this. We also discuss the effect of additive noise on the accuracy of this observation scheme. The investigation was motivated by the MARSIS experiment on Mars Express and by a similar experiment on the Russian spacecraft Mars96. The implementation of the scheme in MARSIS will depend on the computational resources which can be allocated to the task. In future space missions with the proper planning it is thought that the scheme presented will be even more attractive.
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 Ground-based radar has been developed into an important research tool in planetary investigations [Ostro, 1993]. Initial studies of the moon provided the first crude information on the small scale topography of the lunar surface, and lead to estimates of the dielectric properties and the surface material, the rms slope of the surface on scales much smaller than can be obtained from high resolution photographs [Hagfors and Evans, 1964; Margot et al., 1999]. Further development of the delay Doppler technique, identical to the side-looking radar applied in airplanes and satellites for Earth observations has also found applications in planetary research both on the Moon and on Venus, in the latter case in the projects Venera and Magellan, with spectacular results [Pettengill et al., 1991].
 Most interpretations of the data have been in terms of a quasi-specular reflection with diffuse scattering from small scale structure superimposed. Attempts have also been made to interpret the data with a model that involves the penetration of a low loss surface layer, and with scattering and reflection inside this layer.
 Ground penetrating radar has been used on Earth, particularly on glaciers, or in dry desert areas [Evans and Smith, 1969; Davis and Annan, 1989]. Ground penetrating radar to explore the planet Mars is, therefore, an obvious extension. The reason for the interest in seeing below the surface stems from the supposition that the water, which has been flowing over the surface in the past, and which has left its imprint on the surface, must have been trapped underneath the surface in the form of ice mixed with regolith material, and even in the form of liquid water or brine. It is thought that there is no mechanism which can have made all the water escape into space, hence the expectation.
 On the ESA space mission Mars Express launched in 2003 a long wavelength radar designed to penetrate the Martian surface has been installed. It is a linearly chirped radar with an instantaneous bandwidth of 1 MHz which can operate at four different center frequencies. For a description of the radar system, termed MARSIS (MArs Radar for Subsurface and Ionospheric Sounding), see Picardi et al. .
 The penetration into the surface of Mars requires as low a frequency as possible. However, the MARSIS signals are subject to distortion due to the dispersion in the ionospheric plasma. This distortion is determined by two factors. One is the maximum electron concentration determining the transparency of the ionosphere to radio waves. The other is the Total Electron Content (henceforth TEC) which is the main ionospheric characteristic defining the level of distortion and the signal shape after penetration of the ionosphere.
 The dispersion of the signals caused by the frequency dependence of the plasma permittivity leads to distortion of the radio pulse shape. This is fully the case of the pulses radiated by the MARSIS radar of the Mars Express mission. Therefore for studies of the surface the operation on the night side of Mars is desirable as both TEC and the maximum plasma frequency are at a minimum. The need to form very short pulses (broadband signals) makes the dispersion problem particularly severe.
 Our task in this paper is to research the character of these distortion, its action on the quality of the radar operation and the consideration of possible ways of distortion correction. The radio wave absorption in the Martian ionosphere will not be taken into account (plasma is supposed to be collisionless). Taking absorption into consideration with frequency dependent attenuation will not lead to principal difficulties and will not qualitatively change the results. Therefore attenuation is not taking into account at this step of our investigation.
2. Characteristics of the MARSIS Pulses
 We first briefly characterize the pulses used in the MARSIS system. The multi-frequency radar has four bands centered on frequencies = 1.8, 3.0., 4.0 and 5.0 MHz. In future, we shall number the bands in the order of increasing center frequencies (first, second, etc.). The chirp with a frequency band of 1.0 MHz is used for the transmitted signals. An important parameter is the ratio of the signal frequency half bandwidth and the center frequency,
where B is the spectral band of the signal. In our case, B = 1 MHz. Parameter p will be referred in future as the broadband coefficient. The main characteristics of the MARSIS signals are collected in Table 1. All bands are broadband with 2p ≥ 0.2.
Table 1. Main Characteristics of the MARSIS Signals
Center frequency, MHz
 The signals themselves, as functions of time t, can be described in the first approximation by the formula of the form:
where T0 is the duration of the pulse transmitted. The signal amplitude is set to unity. This is not significant for future considerations. The signal spectrum is represented as:
where b = BT0 is the coefficient of signal compression (base) and
The value of the module of the spectrum with base b = 100 normalized to unity is shown in the Figure 1. One can see the spectrum is practically uniform within the spectral band B. Only the spectral behavior close to the band edges disturbs this uniformity slightly. The property of the spectrum can be obtained from (3) by asymptotic integration. Because the coefficient of compression is large the integration in (3) can be extended to infinity (stationary phase method) and we obtain
One must remember that this representation of the signal spectrum is valid only within the spectral interval −ΔΩ/2 + ΔΩ/2. The received signal is obtained by means of optimal filtration. The frequency characteristic of the optimal (matched) filter is the complex-conjugate of the transmitted signal spectrum, i.e.,
After the matched filter the received signal will appear with the spectrum
The received pulse is described by
 In what follows we consider the voltage at the optimal filter (convolver) output as the received signal. An example of this voltage is shown in Figure 2 for the case of the coefficient of compression equal to 10 and the broadband coefficient p equal to 0.1. The variable ξ is the relative time ξ = t/T0. In following, we shall be interested in the envelope of the signal defined in such a way that its shape is subject to distortion while the radio wave propagates in a dispersive medium. In the case considered, the “undistorted” pulse power is described by the formula
 Here the relative (dimensionless) time q is expressed in units of the compressed pulse timescale. This timescale is defined as
3. Pulse Propagation in the Plasma
 We will assume in this part pulse propagation in a uniform plasma with electron density N. As was mentioned above, the radio wave absorption in plasma is ignored and, therefore, its refractive index is represented as
Here, e = 1.6 · 10−19 Coulomb is the electron charge, m = 9.1 · 10−31 kg − its mass and ε0 = 8.859 · 10−12 Coulomb/(Volt.meter). Let us introduce the wave number
Then, in the case of the propagation along the z axis, the general expression for the signal can be written as
Below we shall consider the radio propagation in the plasma under the condition ωp2/ω2 ≪ 1. Strictly speaking, the neglect of absorption is valid only when this is the case. The expansion for small ωp/ω is given by
Under the above assumption the group velocity, as a frequency function, can be determined from the relation
Let us also define the extra group delay,
which determines the delay time of the pulse propagating with group velocity relative to the arrival time of a pulse traveling at light speed.
 Turning again to considerations of the pulse shape after the optimal filter, the compressed pulse spectrum after passing the plasma layer is of the form
in the given spectral band. However in what follows, it will be more convenient to regard this spectrum as
Then we have
instead of (8). As described we have separated the carrier exp[ik()z − it] which has a propagation speed equal to the phase velocity c/n(), and the envelope
The real part of the signal is the object of our interest. It is represented by
As a result, the phase modulated signal with variable amplitude (amplitude modulation)
will appear at the matched filter output. In the case of small shape distortion of the signal, Φ(Ω, z) ≅ k′()z, AS(z, t) = 0, and
The last result represents pulse propagation at group velocity without distortion, as it should.
 After these comments setting the stage for the following discussion, we turn to the analysis of propagation of the MARSIS pulses trough the ionosphere of Mars. As a first step, we provide some information about the Martian ionosphere.
4. Characteristics of the Martian Night Ionosphere
 The information about the Martian night side ionosphere is poor. The common way of determining the parameters characterizing the Martian ionosphere is the radio occultation technique. Due to the mutual position of Earth and Mars the observation by this method of the night side Martian ionosphere, i.e., at solar zenith angles much greater than 90° is not possible. Therefore, there is the opportunity for the observation only of the crepuscular ionosphere. Here, there are definite difficulties too. As a matter of fact, the main data on the Martian ionosphere were obtained from the American spacecraft Viking. However, the data mainly apply to the day side of the planet. As for the night side ionosphere, or more exactly the crepuscular section, the observations were hampered by the high level of the solar activity and by electron concentration spikes in the solar wind and in the Earth ionosphere. These variation of the electron density caused problems in the interpretation of the observed data.
 In addition, the frequencies used in the communication channels in the Viking spacecraft are so high (about 2.3 GHz) that the sensitivity of the occultation technique suffers. Lower frequency was used in the Russian spacecraft (about 0.94 GHz). This provided sufficient sensitivity for the detection of plasma with concentration 109 m−3 and lower, which is important for the study of the Martian night side ionosphere. The two best profiles obtained by the missions Mars-4 and Mars-5 were taken in 1974. One of them was obtained 10.02.74 at a solar zenith angle of 127°. The data of this profile are represented in Table 2 and are also plotted in Figure 4. The data of the horizontal axis reflect the electron concentration N(z) in thousands per cubic centimeter. The second profile, obtained during these missions on 18.02.74, is similar. These data give us the background for numerical estimation of different parameters of the Martian ionosphere. The most important ones are the maximal plasma frequency
and the total electron content (TEC)
Table 2. Electron Concentration of Martian Nightside Ionosphere
El. Concentr., cm−3
El. Concentr., cm−3
El. Concentr., cm−3
 In the specific case considered, the computations lead to the results fm = 6.17 · 105 = 0.617 MHz and Nt = 4.31 · 1014 m−2. From this, one obtains the estimate of the thickness of the night side ionosphere
In what follows we shall also be interested in the integral
The effective thickness values z1 and z2 given here are specifically related to the ionospheric profile of Figure 4.
5. Pulse Distortion
 The Martian night-side ionosphere parameters given open the way for an estimation of the character of the MARSIS signals distortion. At that, one must take into account the radio double pass through the ionosphere, and make the changes:
 The maximum bandwidth for acceptable distortion can be shown to be given by
See Figure 5. The phase of a spectral component is determined by the product φ(ω, z) = k(ω)z in the case of radio propagation in a uniform plasma. In the case of the ionosphere, the plasma frequency is a function of altitude, and the wave number becomes a function of the coordinate, i.e.,
Hence, the phase is determined by the relation
in the geometrical optics approximation. We have
replacing (18). In the case of narrow - band signals, when the parameter p = ΔΩ/2 = B/2 is small, the expansion k(ω, ζ) ≅ k(, ζ) + k′(, ζ)Ω is sufficient. It leads to the result
This corresponds to the description of the pulse propagation with a changing group velocity. Proceeding in the same way, we will have
We have multiplied by a factor of 2 to account for the double pass of the ionosphere, and, where it does not cause misunderstanding, we have extended the integration to infinity.
 The expansion over Ω assuming small values of p = Ω/ is not appropriate for broadband signals. More exact is the expansion on the basis of the approximation
not assuming p = Ω/ to be small, but keeping the plasma frequency smaller than the signal frequencies. Let us carry out the expansion and add and subtract the expression τg()/ + Ω for reasons of simplification to follow below. In the result, we will then have
The last summand in this expression is relatively small compared to two first ones. Even for the first band where this summand is the largest, it is approximately ten times less than the second term. Therefore, it may be omitted and if so:
The expressions (22) in this approximation become:
Here τ = t − 2z/c. In this representation, the time reckoning is carried out relatively to the signal arrival moment at light speed. It is useful for the following to make the substitution for the integration variable Ω = ΔΩs/2. Then
were introduced. The parameter q is dimensionless time reckoned relatively the pulse moment of arrival with light speed. The parameter qg defines the dimensionless time of group time delay. Sometimes, it is more convenient to reckon the time relatively to the pulse arrival at the group velocity. Then, the main energy of the signal is concentrated close to the “zero” moment of this timescale. Such presentation is especially effective for the comparison of the shape distortion of pulses of different center (carrier) frequencies. The function introduced are described, in this presentation, by the formulae
where qe = q − qg(compare with (25)). Note that the last expressions reduce to the Fresnel approximation at small broadband coefficients. Let us note also, that functions introduced are the real and imagery part of complex functions
As an example for the following computation, the model of the ionosphere described in section 4 will be taken. Table 3 gives the parameter values for the MARSIS bands.
Table 3. Ionospheric Parameters for the MARSIS Signals
Group delay, μ s
Dimensionless group delay
Product of the coefficient of broadbandedness and dimensionless group delay
 The pulse shapes represented by
are shown for the different bands in Figure 6. The pulse concentrated about zero corresponds to propagation in the ionosphere without distortion and extra delay, i.e., to a pulse with very high center frequency. One can see, from the computational results, the distortion of the MARSIS pulses when propagating in the night side Martian ionosphere. Only the pulse of the fourth band keeps its shape.
 Let us define the moment of the pulse “arrival.” One possible definition is the following:
It is easy to show that without distortion = qg corresponding to the “arrival” of the signal at group speed. This speed differs from the group speed in the presence of distortion. In greater detail:
The spectral presentation of the delta function
was used. The integral in the numerator of (42) is calculated similarly:
In the process, the following relations
were used. In the result
The “arrival” time as defined above is practically the same as the group time in the case of narrow - band signals but differs essentially for broadband signals.
 Turning now to the definition of the signal duration as defined by
From this, the unknown value
may be defined as the coefficient of dispersion broadening. Table 4 gives the arrival times and pulse broadening for the MARSIS bands for the assumed nighttime ionospheric model.
Table 4. Parameters of Distorted Pulses
Dimensionless arrival time
Coefficient of the dispersion broadening
6. Signal Shape Restoration
 The strong distortion of the MARSIS pulse shape requires the signal shape to be restored. This means we must find a filter capable of signal compression. In other words, it has to be a reverse filter. This means that the filter in equation (6) must be replaced by the more complicated:
If the filter parameters are chosen correctly, the pulse shape is restored to the form given in equation (9). Thereby however, it is necessary to know with sufficient accuracy the parameters of the medium of propagation, i.e., parameters of the Martian ionosphere, at the time of radar operation. This knowledge a priory is not realistic and, therefore, one can only talk about approximate knowledge of these parameters and following its more precise definition during the processing using some optimization criteria. The main parameter relating to the ionosphere is the group time delay determined by the formula (34), which leads to the conclusion that it is necessary, in the first place, to know the TEC. As a matter of fact, we shall have
neglecting the insignificant second term in (34). The TEC cannot be known for any particular time of the Martian ground remote probing. The assumed value of TEC Nt = 4.31 · 1014 m−2 can be considered a tentative one, following from the data on an individual profile of the electron concentration on the night side Martian ionosphere.
 Let us suppose that the preliminary TEC value is known with the accuracy δNt = 1 · 1014 m−2. It means that the signal shape is still distorted after the corrective filtering. However the distortion effect will be similar to that when the radio wave is passing the ionosphere with a TEC of 1 · 1014 m−2. Since the group time delays are determined by the relation, τg = 2.69 · 107/2, the corresponding values of the dimensionless group time delay at the operating frequencies chosen will be 26, 9.4, 5.3 and 3.4. The results of the correction are represented in Figure 7.
 After this preliminary step of correction one can observe the progress in the pulse properties, relatively to those shown in Figure 6. The pulses of the second and third bands are becoming normal in shape. Only the pulse of the first channel is still strongly disturbed. The well- defined shape of the higher frequency pulses permits us to fix their “arrival” time and estimate the relative group time delay. For example qg = 5.354, 3.421 for the third and fourth bands respectively. The difference between these values gives TEC (see equation (49)). The corresponding value is δNt = 1.02 · 1014 m−2, which differs little from the one chosen preliminarily. Now, we can correct TEC by adding this value. However, the correction will not be fully due to the remainder of 2 · 1012 m−2 in TEC. In our calculation, this remainder is the result of approximations, but will be considered as an inaccuracy in the second step approach. Now, if we assume the unknown value of TEC being δNt = 2.0 · 1012 m−2 the corresponding values of the relative time becomes: qg = 0.522, 0.216, 0.117, 0.073 for the four MARSIS channels. The pulses corresponding to the remaining TEC correction are displayed in Figure 8.
 One can see that pulses in all channels are practically restored, and the procedure of dedispersion of the signals can be finished. The uncompensated part of the group time delay is only a small fraction of the of the pulse duration. One may continue the procedure in order to determine these values more exactly. However, this is probably not worthwhile because of the role of additive noise and the reflection by the rough Martian surface. This back scattering is the sum of pulses from surface inhomogeneity with a small additional delay leading to pulse broadening of the initial pulse. This broadening can differ for the different bands and must be considered in the procedure of pulse restoration.
 The correction procedure described is based on a preliminary initial estimate of the electron content. This estimate can be “bad” in the sense that errors can cause the ineffective operation of the inverse filter.
 This estimate is necessary only in the case that it fails precisely to determine an arrival time at least for one signal. The initial value of the electron contents is not critical for iterative procedure.
 The preliminary initial estimate of the electron content is not required, if the arrival times of one or two signals can be determined. In this case a preliminary initial estimate of the electron content can be determined under the formula (49), using meanings of the arrival times for two signals. It is possible to determine a preliminary initial estimate of the electron content on measurements of one signal.
 In this case, one has to look for ways to improve the preliminary TEC estimate based on the signal itself. One of the ways is the half-and-half split of the signal spectra and the formation of two similar pulses with small enter frequency difference. The spectral band of the signals thus formed will be half as wide. One may expect that, after passing the ionosphere, these pulses will be weakly distorted compared to the original pulses. The further procedure of the preliminary initial estimate of the electron content is similar to the procedure at reception of two signals.
 An example of shapes of such split signal pulses of the third channel is demonstrated on Figure 9. The center frequencies, in this case, were 3.75 and 4.25 MHz and the spectral bands 0.5 MHz for each of the pulses. TEC = 4.31 · 1014 m−2 was assumed for the calculation. The difference of center frequencies leads to a difference of the “arrival” time due to different group speed. The difference in the peak positions allows us to determine the electron content by the procedure described above. In this case the initial TEC is 4.28 · 1014 m−2 which is not far from the real one. The preliminary estimation of the electron content can be obtained from the analysis of the quantities of the pulses “arrival” time on the formula (34, 49) base. The difference of the mean “arrival” time for two pulses of different center frequencies provides the opportunity to estimate TEC as was done above. This way, the operation of differentiation, which is needed for the peak position definition, is interchanged with the process of integration which is more stable against noise.
 Finally, we shall consider one more way to estimate the TEC. As mentioned above the amplitudes AC and AS are the real and imaginary parts of a function being the complex amplitude, written in the form:
taking the signal spectra into consideration. The time reckoning is relative to the instant of the signal “arrival” with group speed. Generally speaking, this moment is not known beforehand, and the signal, as a time function, is known at the receiving point relative to the local time τl. If we introduce a preliminary unknown time τ0, and put τe = τl + τ0, then the spectrum of the function is given by the relation
This spectrum can be obtained by Fourier transform relative to the received signal. The spectrum obtained allows us to define the phase
as a function of frequency Ω. Then, one can choose this form of frequency dependence for the experimental data approximation. In the process, the unknown preliminary values τ0 and τg can be selected on the basis of some criteria, for example, the least squares method. However this cannot be simply realized because the phase is defined modulo 2π. In practice there is the opportunity to obtain only a trigonometric function of the phase. The real and imagery parts of the expression (51) give the signal spectral density of power multiplied by cosine and sine of the phase. Their ratio does not depend on the spectral density and gives tan (ϕ(τ0, Ω)). A graph of the tangent behavior as a frequency function is represented in Figure 10 for the case of the MARSIS first band and with the assumed ionospheric profile. In this way the phase discussed is considered versus relative frequency s = 2Ω/ΔΩ = 2F/B. The graphs correspond to the cases q0 = ΔΩτ0/2 = πBτ0 = 0, and 2.5 which are reflected by the values of the phase ϕ first argument. One way to determine the unknown parameters is to use specific points of the tan-function. For example, putting to ±π/2 the tan argument at the points s1 and −s2 close to zero, where tan becomes infinite, we will obtain two equations
from which q0 can be eliminated and the dimensionless group delay time can be determined
The knowledge of this value allows us to make the TEC estimation needed to start the correction algorithm.
7. Effect of Noise on the Signal Shape Restoration
 So far the distorted signal has been recovered from a received signal with no additive noise. We have applied the deconvolution procedure described to a particular example of a noise input. The noise added to the distorted signal was derived from a random number generator and the noise level in relation to the ideally deconvolved signal was chosen, in the particular case to be considered, at −5.7 up to −6.3 dB (0.23–0.27) depending on channel number. At this noise level only the fourth (5. MHz) has an appropriate shape. The first (1.8 MHz) and the second (3.0 MHz) channels are destroyed completely. The curves shown in Figure 11 describe the radar signals shape after their double passage through the night ionosphere of Mars with additive noise (thin line) and without (bold line). Using the iterative procedure allows us to restore successfully the pulses from all the channels. In this case the correction proceeds as follows: at the first stage we determine the envelope peak the for the third (4. MHz) and fourth (5. MHz) channels. From the formula
we determine as a first approximation the total electron content. This value we use to account for the envelope function as a corrective addition. Then we determine the envelope peak for the second and third channels and regard it as the total electron content. Taking account of this value we correct again the envelope functions for all channels. At the third stage a similar procedure is applied for the second and first channels. Further procedure of correction is then applied to the first and fourth channels. The final results of correction after the fourth stage are given in Figure 12. Thus the iterative correction procedure allows us to restore the initial pulse shapes even in the presence of high noise level with good accuracy.
 We have described a method of constructing an adaptive matched filter for planetary observations eliminating the effect of an ionosphere and restoring the pulse shape. The method depends on the simultaneous, or near simultaneous, observation of the echoes from the planetary surface at several adjacent low frequencies near but above the maximum plasma frequency in the ionosphere. We have shown that an iterative scheme can be established which gradually restores the distorted pulse shape. The scheme depends on a fairly undistorted pulse shape at the highest frequency. When this is not the case one can start by splitting the signal spectrum and start the procedure from a wider pulse and work the iteration procedure from there.
 In this paper we worked out the details, at least in terms of a specific example, taking additive noise into account, and showed that the adaptive filtering scheme is robust, at least as long as the reflecting surface is smooth. Future investigations must also consider the effects of a rough reflecting surface on the ability to eliminate the ionospheric effects. With a rough surface there will not be a clean surface return equivalent to a delta function to establish the pulse response of the propagation circuit, and the clutter from the irregularities of the rough surface will produce what amounts to extra noise. Detailed investigations of these effects must be undertaken in future work.
 We have only considered a very simple ionospheric model without absorption and without a magnetic field. In particular the effect of the magnetic field may be of the greatest interest in the case of Mars, where there is a crustal magnetic field which influences the distribution of the ionospheric plasma, as well as the propagation of electromagnetic waves in it [Acuña et al., 1999].
 This work which started out to eliminate the effects of the ionosphere on the short radar pulses required to identify abrupt transitions of subsurface electrical properties associated with the presence of subsurface ice or water in the Marsis project may well lead to new insights into the irregular surface structure, the horizontal distribution of ionospheric plasma and the strength and configuration of the magnetic field in Marsis and in future long wavelength investigations of planets and satellites.
 N. Armand and V. Smirnov are indebted to the DLR under contract 50 QM 0004 for financial support and to the Max Planck Institute for Aeronomy for providing office space and computer support during the course of the work described.