How does one measure the Fourier components of a signal?
Is the signal split by many narrow band pass filters in paralell? Are these filters RLC circuits? Are there chips that contains many such filters in paralell?
No, these are typically collected as time signal, and then transformed digitally (by a DFT implemented by an FFT) to discrete Fourier domain for processing reasons.
In the sense that yes, there's many hardware implementations of FFTs, and the FFT can be considered as polyphase boxcar filterbank, there's chips containing a filterbanks – but they're processing a digitized time signal, not an analogue one.
(The reasons being pulse compression, mainly, i.e. you need to do a huge correlation with the known signal to even see your signal from the noise – there isn't exactly much signal coming back to the satellite after being radiated from lower earth orbit to ground, through the ground, scattered there, and back. Correllation is very computationally intense, as it goes with the square of the length of the signal, but with processing in frequency domain, you can significantly reduce that amount, and bonus, the data might become way easier to compress, store and transmit to the ground station.)
(By the way, I wrote a top-of-my-head list of devices that do a Fourier transform during operation here, maybe you'll find it interesting.)
Generally, anything that can be done after digitization digitally or before in analogue is usually done digitally — unless doing it in analogue reduces the digitization effort by magnitudes. Analogue electronics mostly have undesirable features for signal process – tolerances, non-linear in/out relationship that is hard to model, non-linear phases, temperature dependency, …
So, if you're in the business of processing signals, you typically want your signals to be digitized at the quality necessary, and afterwards do the math you want to do with the digital signal (which is actually numbers) instead of approximating the math you want to do with analogue components. You'll find that the same
Maybe a place for some explanations.
There's no such thing as recording a radio signal in frequency domain. In receiver circuit only voltage as a function of time is available. There exists nothing else which could be amplified, stored or processed.
I guess you want to send frequency sweep bursts. Echos of them really can contain perfectly usable information, as good as a pulse radar could give with 1000 or 10000 times bigger transmitting power. It isn't only a guess. Because only quite low frequencies penetrate long distances into the soil, the used power must be limited to avoid problems with telecommunication companies. I bet pulse radars are out of the question for that reason.
Think a transmitted which sends with linearly growing frequency. The burst has constant power and it lasts until there's got echo signal from the most distant point under interest.
Receiving and detection: Transmitter output signal is splitted, about one milliwatt of it is directed to a mixer which gets the received signal simultaneously. Mixer output signal is amplified, stored and it's Fourier transform is calculated. The result IS the same as a pulse radar would give. That's easy to see, why. The mixer produces frequency components at differerence frequencies (the sum frequency components are filtered out). The frequency of an echo is the higher the further the echo had came from.
The actual circuitry: I have no ground radar circuit to give, so you must design it by yourself or buy it. I'm afraid to be able to do the design from scratch, you should have combined radio, analog electronics and digital signal processing knowledge total worth of 4 years of full time studies + excellent electronics development lab work skills. And of course, you need the lab.
You should get some realistic estimates how strong echo signals are available. Do not forget that the echos get weaker as the distance grows - that's because the soil attenuates and the signal spreads to wider area. If you are lucky, the transmitter can be weak enough to make possible simultaneous receiving without saturation and the most distant echos still are detectable under the noise. You need both measurements and math to find those basic limiting facts.
Probably you can also find them from published academic works. To be able to find and understand them the already mentioned knowledge is a must.
You can divide the design effort into few major areas:
1) finding the basic limits
2) overall system design
3) radio design (= finding numerical specs for transmitter, antenna, receiver and signal processing)
5) signal processing programming
6) system control software programming
You were wondering, if RLC filter bank could do the signal processing job. In theory it can separate frequencies, but collecting the outputs is complex. Radar technology books contain practical solutions. Today Digital signal processing is the main route, but in the past various other methods were in use (=no fast enough computers available). Frequency dispersive surface acoustic filtering was one way to construct a pulse radar type output signal. The methods in general radar vocabulary were under term "pulse compression".
Get a ground penetrating radar technology book, find one which is understandable with your existing math, radio and electronics knowledge. If you collect more of that infrastructure knowledge, get higher level GPR technology book, too.