I'd be VERY cautious about this sort of mathematical inversion technique. We do it routinely in my field, so I'm familiar with the method and its drawbacks.

What it does is to take the pattern of reflectivity (here - radar, in my field - sound), and convert that to the "steps" in physical property (permittivity for radar and accoustic impedence for sound) that would produce the same reflection series.

The method is "blind" to any changes at higher wavelength than the scale of the measurements and also tends to "wander" at long wavelengths. So it might be that deep sections appear to be quite high or low in their absolute values of permittivity or accoustic impedence, even when they are really quite average.

In my field, we calibrate the accoustic inversion technique by 2 methods:-

1) We have a good idea of average rock velocities and densities with depth, and density*velocity = accoustic impedence. This is essentially "common sense"

2) When we have well data available, that gives us hard data to constrain the long wavelength behaviour.

In my experience, unconstrained inversions are of little value - particularly if the results exceed the expectations of "common sense".

Now, there's a long story about the MARSIS investigators. The Italians are very enthusiastic, but very naieve. They have demonstrated this to me personally on many occasions. They are very "mathematical" geophysicists, rather than practical people, and they are prone to beleive the numbers they calculate, even when the answers are plumb crazy. (Actually, this isn't just true of the Italians - most of ESA has this problem).

For Mars, obviously we have no drill data from the base of the polecap, so it comes down to "common sense"

Unfortunately, the prevailing ideas about temperature gradients and thermal state of the crust and icecap of Mars are based on a long history of wishful thinking. I've bashed my head against this problem for the better part of a decade now, and made limited progress.

Let's look at a VERY simple analogue for the south polar cap of Mars - the south polar icecap of the Earth. ANY model must be able to explain both of these at once, or it isn't valid.

On Earth, the polar icecap is about 3 km thick, and it reaches melting point at its base.

On Mars, the polar icecap is 2-3 km thick. The question is - DOES it reach melting point at its base?

There are five factors that strongly affect the melting point at the base of an icecap:-

1) Thickness
2) Surface Temperature
3) Thermal Conductivity
4) Heat flow
5) Composition

Let's take them in turn:

1) The thickness of the pole cap on Mars is a little less than that on Earth, so it is a little less likely to melt at its base

2) The surface temperature on Mars is a LOT colder than Earth (the maximum low ever recorded on Earth's south pole is -88C. Average temperatures are around -40 to -60C. Mars temperature is constrained by the condensation and sublimation of solid CO2 to around -129C. As a result, the base of the icecap on Mars is MUCH less likely to melt at its base

3) Thermal conductivity of ice INCREASES at lower temperatures, so the base is LESS likely to melt

4) Heat flow on Mars is a LOT less than on Earth, so the base is MUCH less likely to melt

5) Ice is ice, so maybe this effect is neutral, but let's be open-minded. We know that Mars has a lot of sulphate minerals around, originally formed from strong acidic and salt-laden solutions. Maybe some of this stuff is still buried under the southern polecap and acting as a "flux" to melt some of the ice. On Earth, we don't see this because there is a conveyor-belt of ice through the polecap, with fresh snow on top and melting at the base. Any salts get flushed away. On Mars, a much more static situation exists and melting may be minor, gradual, or occasional.

Items 1-4, taken together, would push the melting depth of ice at the poles from the ~3 km on Earth to 15-20 km. Only item (5) is working to raise it.

Phew! Pause for breath...

OK, so where are we?

The theory says that Mars' southern polecap is VERY unlikely to have any appreciable amount of liquid water in it. (and any that it does will be deep, acidic, and salty).

My experience with Inversion is that it is intrinsicly unreliable without calibration and often suggests "false positives"

My experience of the investigating team is that they have a history of making facile assumptions

The evidence has not been presented in any open and scientific way, and is merely a hint that values equivalent to water are "possible"

My conclusion is that I'm not eating my hat yet!