Is it possible to model a system for Kalman filter with spatial (not time) derivatives? In my application the x_n can be expressed as
x_n+1 = x_n + x'_n * d
x'_n+1 = x'_n
where x' is the spatial derivative of x, i.e differentiation with respect to distance. Can these make a valid model for Kalman filter? The d is an input/parameter without any noise, like a control input.
I can't think of any problems that you would encounter, and I don't think that the filtering process would change at all. The dynamics would just be taking you to the next step, rather than the next time step. The measurements aren't dependent on time, so there will be no changes there.