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While cumbersome, is it permissible when asking for a pop-up IFR clearance to state, at the intersection of WWW radial XXX and YYY radial ZZZ?
Obviously, during VMC, you have your pick of visual checkpoints, but in IMC, are there any better suggestions?
Additionally, if you are near a navaid or published intersection, and take a guess at your distance from it, how far off would your guess have to be before ATC wouldn't really even consider it helpful?
I'm going on the assumption you're operating in IMC (or perhaps VMC above a cloud deck) and have no reference to the ground. In a situation if you need a pop-up IFR clearance or ATC is requiring you to provide position reports for some other reason (non-radar, or your transponder is dead and they want to identify you but don't have time to issue you confidence maneuvers).
I'm going to further constrain the scenario with the assumption that all you have are "basic" IFR instruments (6-pack, clock, compass, and VOR -- no VFR/Handheld GPS, DME, or other fancy stuff).
The AIM has some guidance on this subject in 5-3-2, but the general idea is your report should be "as accurate as possible". For routine position reporting like you'd use in non-radar environments (where you're giving an estimate of when you'll reach a reporting point) they're looking for "within 2 minutes flying time" on your estimates in most situations, so you'll want to give your position with the same sort of accuracy.
In the case you describe, with no DME, GPS, or other precision tool for reporting your distance and bearing from a fix your the most accurate option would be to report crossing a published intersection, VOR, or other reporting point noted on your chart - it's the most definite "I am HERE" report you can give, and ATC will certainly know the location.
If that's not an option (ATC needs to know where you are right now, and you're not over a VOR or charted intersection) I would probably start by computing the distance using the intersection of two VOR radials (whether this is a charted intersection doesn't matter, though it would help if your report is relative to one): Break out your chart and mark the location, then take your ruler and measure the distance from one of the stations (or a nearby intersection along the airway you're using) and you've got your position "as accurately as possible".
If you don't have a chart or can't get signal from a second VOR for some reason it's time to work out a time/speed/distance problem in the air:
If you flew over ABC VOR 10 minutes ago and are now flying from the VOR on the 090 radial at 120 knots, and you've got a 20 knot headwind (which you know from the Winds Aloft forecast, updated with data from previous position and time/speed/distance computations) then you know you've traveled about 17 miles, so you're "17 miles from ABC VOR, on the 090 radial".
The time/speed/distance computation has a lot of potential error (winds not as forecast, for one), but sometimes it's your only option.
Press the NRST button on your GPS, and give them the distance/direction from the fix:
"5-4-XRay is 7 miles southeast of K-A-S-H"
Just because you are /U, it doesn't mean you can't use a VFR or handheld GPS for non-navigational position reporting. If you're IFR and don't yet, in 2015, have a an IFR or VFR GPS, or a handheld GPS running an aviation app in the cockpit, you should turn in your ticket.
You don't say whether you talking about center or a tower. I would definitely never give a tower such a position. If I was talking to an ATCSCC, maybe in wierd scenario (broken transponder for example), but in all honesty it is the pilot's job to have charts and report position properly. Trying to get a controller to pull out a chart and compute your position is not a good idea, and I doubt they would cooperate unless it was an emergency.
In real life, what happens when you don't know your position, is that you would have the controller find you with radar, so you just say "Boston Center, Cessna November 54321", and center will answer "Go ahead November 54321", then you say "November 54321, what is my position?" and control will say "Ident", then you push a button on your radar transponder. This causes a "blossom" on the controller's radar screen and he can see exactly where you are. The controller will then give your position to you.
Note that this often happens even when you accurately report your position. So, for example, you might call in "Nashua Tower, Cessna one seven two November five four three two one, six miles west of water tower, maintaining three thousand five hundred, with information Romeo", then tower might say, "November five four three two one, ident", then you push the button. Then, they will know exactly where you are.
