I presume robinh means why not set all those additional params when the search form is posted rather than the initial get.

EDIT: Ok the follow up comment doesn't indicate that... but it would be just as valid to do it that way if they are indeed search modifiers.

That was what I was trying to ask: can you do both, or would that be absurdly impractical/impossible? Alternatively, can't they do only a POST and somehow generate URLs dynamically based on the search query?

It's perfectly possible to POST to a URL with GET query parameters.

For Google though I'm not sure you'd want POST requests at all; unless it's to set cookies and do a redirect on POST.