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I have asked the local police department several times in different cities.
They all ways say no problem, don't leave a mess.
Get the officers name that says yes in case your questioned later.
Then you can focus on the hunt and not have to look over your back.
NY-waterhunter, while:
a) what you are saying might/would deflect *some* busy-bodies, and
b) it's certainly an improvement over asking city park's people (since cops have much bigger fish-to-fry, and will be more inclined to see md'ing as innocuous),
Yet it's also been shown to be toothless and fail in multiple other occasions. The oft-cited scenario goes like this: The md'r goes to the police (or city people or whomever) to "get permission". Then they gleefully get the name of the person, as you say (or even to the extent of "getting it in writing!

) Then the md'r gets accosted in the field by a busy-body. But no problem: The proudly whip out their permission or name-to-cite. The busy-body then merely gets on their cell-phone, calls down to the police or city hall person and says:
"But he's tearing the place up!" (which isn't true, of course, but guess who'll be on the loosing end of the semantics of that?). And then guess what happens to your "permission"? It is promptly revoked. This scenario has been played out many times in various posts.
But sure, sometimes a busy-body slinks away embarassed for having stopped you, of course. So you might be inclined to say:
"well, it deflects some busy-bodies, so why not still do it?". My answer to that is, that by asking, you risk a "no", where there's no real rule that says that, nor would anyone have ever cared or noticed (till you asked). Not sure which is the greater of evils, but they both happen. Glad your success rate is 100% so far, don't get me wrong. But I have also heard of persons going to the police like you do, and getting a "no". Or something strange like
"yes, but you have to turn in everything you find, because it belongs to the city", or
"yes but you can't dig" (even though you never mentioned digging. Or
"go ask permission from the park's dept as that's their domain", and so forth.