Yes and no. The radar isn’t only looking down into the ground. The antenna pattern has side lobes which can potentially generate large echos in the radargram, e.g. from rocks on the surface etc. you only know that there is something in some distance (or rather time delay).
The useful signal is extremely weak anyway and the clutter from the surface hides the useful signal in many cases unless you habe really strong scatterers (large and highly reflective) buried in the ground.
Plenty of commercial GPR devices operate on Earth just fine with the explicit goal to detect changes in the subsurface's dielectric properties. It doesn't matter if you're on Mars or here, GPR works in the same way and I'm pretty sure that the antenna and the signal processing has been designed for the purpose, possibly even more meticulously than the antennas of commercial GPR pushcarts. Your comment makes something simple sound highly involved and problematic.
The useful signal is extremely weak anyway and the clutter from the surface hides the useful signal in many cases unless you habe really strong scatterers (large and highly reflective) buried in the ground.