While budgeting always involves estimating, that's not our only consideration. We draw from a lot of sources.
Time & Materials
Why budget at all? Just have your team start working, and see where you end up. (Not recommended.)
Fixed-Budget, Scope-Controlled
Carefully adjust the scope, in either breadth or depth, incorporating new knowledge and ideas.
Fixed Price
You can ignore the surprises and learning, since you're locked into your original, detailed plan. Or you can renegotiate your job with a series of change orders.