How should we estimate at (x,y)?

Detective Story #4: "What's wrong with these estimates?"

You want to estimate the value of the dreaded "glarp" at location $(x,y)$, and you have data for glarp on points at locations $(x_i,y_i)$. How do you proceed?
If you answered "it depends", why?
Unsupervised Inverse Distance weighting is bad: why?
But we need to weight the data somehow -- so what do we do? Hey, I've got an idea!;)