Archaeology is not an "exact science" but a scientific doctrine. In the beginning it was like early psychology: a typical "human science", now [after evolving, like psychology] it's a scientific doctrine.
The limit of these disciplines is obvious: they cannot be totally experimental.
1. Because of reality. You cannot recreate today the historical context of the battle at Alesia to make an experiment to find out if a theory is correct or not. Don't think to the group that reenact: they don't live like legionaries, they don't have their limited education, they don't reach the battle field marching for weeks, they don't eat what legionaries ate, present Gauls are not the warriors or Vergingetorix ... and so on.
2. Ethical reasons. You cannot make an experiment in psychology to find out how many persons would prefer the suicide to burn alive in a oven. [If you ignore ethic, it would be a mere scientific behavioral experiment!].
Anyway, becoming scientific doctrines they have adopted [in the possible measure] the scientific method. This has improved Economy, Psychology, Archaeology, History, Sociology ...
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May be we should enter details and try and check the datation of a Monarch of the early dynasties ...
I've never heard the word doctrine used in a philosophy of science context before, but I think I know what you mean. As for not being totally experimental, that's not really a problem, lots of sciences are like that. Evolutionary biology, or even Astronomy/Astrophysics are in that regard. But if you can't do rigorous controlled experiments, the least the scientific method demands is that you try and make the experiments you can perform as "blind" as possible. Asking someone what answer they expect beforehand flies in the face of that! Imagine the outrage if medical trials were done like that.
My father started a PhD in archaeology/palaeontology back in his youth, and has stories of people massaging or exaggerating finds to fit their personal theories. Not much seems to have changed in the last few decades.