first of all, sorry if my question is trivial or silly, but I'm definitely not much into statistics.

I have some spatially extended data spanning a long time, let's call it F(x,y,t). Imagine that I can describe my data as this:

F(x,y,t) = A(x,y)*a(t) + B(x,y)*b(t).

Now, if I knew a(t) and b(t) I could extract A(x,y) and B(x,y) with a simple linear bivariate regression. But, instead, I know a(t) and B(x,y). Is there a technique to extract A(x,y) and b(t) simultaneously (knowing F(x,y,t), a(t), B(x,y))? If not, is there any strong reason to believe this is not possible?

Any suggestion is welcome, thanks,

Andrea