Process by/in which
A process by which” describes the effect of the process. “A process in which” describes what happens in the process.
For example:
Steeping is a process by which tea is made.In other words:
Tea is made by the process of steeping.On the other hand:
Steeping is a process in which water absorbs flavor from tea leaves.Means:
In the process of steeping, water absorbs flavor from tea leaves.There is often some overlap between these uses, i.e. there are cases in which either is grammatically correct. This is because what happens in a process and what the process does are not clearly distinct. The choice of one or the other can change the connotation of the sentence.
approximation to/of
X to Y makes Y a target or destination that X wants to reach.
X of Y makes Y relate to X such that you can say “X belongs to Y”.
This model is a good approximation to the data.This sounds like you are trying to make the model fit the data. You had an idea of what the model should be before making it.
This model is a good approximation of the data.This sounds like you made the model, then compared with the data, and found that it “belongs”. You didn’t necessarily have an idea of what the model should be before making.
