I find the subject line a bit misleading because it makes it look as if the preposition changes the meaning. But you can use both with both:

This is based on/upon the figures...

= dies basiert auf den .... Zahlen

We based our assumption on/upon the fact that...

and here it gets a bit complicated because you can' t use 'auf etw. basiesen' here:

* Wir basieren die Annahme auf der Tatsache, dass* is incorrect IMO.

You'd have to say, again: die Annahme basiert auf...

or rephrase with a different verb.

I personally usually use simply 'on', too.

Thank you very much!

I was confused, because I did not get the same result (= preposition) for search ENG->GER and GER->ENG