There are base stations in the basement of some buildings. this can happen if there is a base station in a nearby basement.
Why would anyone put GPS "base stations in the basement of some buildings"?
Ahahah!
GSM base stations are used instead of GPS stations. GSM stations and GPS stations are generally same thing. GSM stations of different brands serve as GPS for all phones.
GPS operate in the same frequency band, but not in the same frequency. While Smartphones do use GSM stations (and WIFI) in addition to GPS data to provide better and faster positioning, they still have built-in GPS modules and GPS-chips (
https://www.lifewire.com/iphone-gps-set-up-1683393). And GSM is a completely different standard and communication protocoll than GPS - so, this is not possible to use the same station to emit GPS and GSM signal, you need separate anntennas, some hardware (atomic clocks, which is base for calculations) and software
Most standalone GPS does not use GSM chips. That's why they don't work inside the interiors while cellular network is, and that's why they work in the middle of nowwhere (forest), while smartphone does not. Once again, i can provide you some videos.
I don't know, you claim, that some GSM stations emit GPS signals and some not?
But that's not what we observe. We observe, that GPS signal is available much more widely in open areas than GSM signal, very often available in difficult areas like mountains or forests, where there is no GSM reception. At the same time, GSM signal tends to penetrate buildings interiors much better than GPS signal (especially in cities).
That's extremely obvious if we accept, that gps is provided by the satellites, but it's ridiculous otherwise.
If they would install according to your claims (?) separate towers for GSM and GPS, then you have a lot of additional problems here. First, you would need to double the costs of all GSM towers in the world. This is huge. And GPS is provided without costs, so who would pay for all these GPS stations in Poland, Hungary, France, Russia (!)? NASA ?!
Then there is a next problem. Such huge investition requires a lot of technicians who are able to install and test these towers. I know personally few people who do this for GSM and they would all know this "secret", they would also need to be paid. That's an absurd and needs a proof.
So, what's really your theory about this?
You clearly know nothing about radio signal propagation because the basement of a building would an extremely ineffective place for any transmitter.
Nope. I know more than you about it. A station on the basement floor is only for service to the people in basement floor. If you still want to talk to your phone 20 meters below the ground and you have enough money, you will call your GSm company and they will install a small station in your basement. this is actually a small wireless network connection connected by cable. it is not very expensive. I know that because I had it made. they want 1 square meter of place in basement floor and some money. 
That's called repeater, or amplifier. In Poland it's illegal to buy and install them by yourself, you have to prove that there is no reception, provider must check if there is not something wrong with their towers and then you could get a positive response, but they will deliver you their own device. We tried to get something like this few years ago, and we failed.
So, in Poland most buildings does not have something like this. Anyway, these are very small devices, and mostly capable to provide a range inside one building. The power is small, and the anntennas are also small. They are not capable to trasmit between many buildings in the area.