As a signal processing novice, I would appreciate your help with this common
situation, for which I imagine there are standard answers.

I have some time-series recordings as vectors V, each of which consists of a
main signal plus some noise.
I have reason to think that one component of the noise has a specific time
profile: ie, it can be represented as a vector N, which is like a 'basis
vector' in that it has to be scaled and shifted in amplitude to match the
component present in any data vector V. However we can assume that N is not
scaled or shifted in the time domain. Therefore the noise component is (a*N +
b), where the scale and shift parameters a,b need to be estimated from the
data.

Obviously I want to subtract this component from the raw signal:
D = V - (a*N + b), where a,b are the optimal values of noise amplitude and
offset with respect to some 'merit condition'.

So far I have been finding a & b by brute search of the amplitude/offset space.

Are there analytical ways of finding the optimum values of a and b ?
What if i) if the 'merit condition' is min( sum( (D)^2 ) ) ? (the minimum
least-squares form)
ii) if the 'merit condition' is min( sum( abs(D) ) ) ? (less
tractable but less sensitive to outliers)

For an arbitrary vector N, what choices of method are there? Can you give me
some names (or preferably, explicit algorithms) that I can look up?

Thanks

Ross