I didn’t see it as a woosh. I watched one of the Sunday political shows (ABC?) in 2003, and distinctly remembered hearing a Bush flunky gave an estimate that the total cost of the war to the US would be around three billion (however, the above cite by Johnny LA is superior to my memory.). Iraq oil would be used to offset the cost (which seemed horribly amoral–why should the Iraqis pay for a war we started?). I think this estimate occurred before the war started.
The segment was surreal. Instead of giving out a non-answer, or low-balling the cost of with a range of numbers, the guy gave out this hard number that no believed. Was the guy being sincere, but misinformed, or lying through his teeth? Per Colin Powell’s experiences, you can never be sure.