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Let's assume, hidden in a forest, there's a beacon. I walk in the forest and, at random intervals, ping the beacon. For each ping I get a lat/lng pair and the signal strength of the ping at that point (but no direction). At the end of the day I have n readings where n is a random number.

  1. How would I compute the location (lat/lng) of the beacon?

  2. If, at a later time, I would get another reading, how would I add this to my current estimate of the beacon's location?

Note: we do not know how the forest is affecting the signal strength, but we can assume the effect is uniform - the beacon could be in a clearing, or buried beneath a stone.

I am assuming that all my n readings can be drawn on a map as circles where the lat/lng is the center of the circle and the signal strength is the radius. If the readings are accurate then the beacon should be in that area where all the circles overlap. I have however no clue how to find the center of the overlapping areas of n circles. Can somebody help?

Edit: clarifications
1. Each reading indicates the position I am currently standing in (random location)
2. The signal strength is inversely proportional to the distance between the reading and the beacon although I do not know what is the coefficient. I do however know that this coefficient is the same for all the readings
3. While from a theoretical point of view only two readings are necessary I would like to be able to incorporate as many of the readings as possible in the calculation as all these readings might not be 100% accurate

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    So lat/lng will always be the same number unless the beacon is moving, right?2011-12-28
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    @Raskolnikov: no, I think he gets lat/long of his current position and signal strength.2011-12-28
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    @Thomas: Yes, that would definitely make more sense. But it's not very clear from the question.2011-12-28
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    Is signal strength proportional to a negative power (or is it at least a decreasing function) of distance?2011-12-28
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    note: if my interpretation of your description is correct, and if signal strength is an exact indicator of your distance to the beacon (i.e. if you can derive your distance to the beacon from signal strength -- you did not say this) then you need only two measurements from different locations to deduce two points one of which will be the location of the beacon, since two circles in the plane will intersect in at most two points (there will be at least one since the beacon is somewhere). A third measurement might leave only one solution.2011-12-28
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    The comments to your question indicate you should possibly make it more precise :-)2011-12-28
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    Thanks for all the comments. I have included the unclear information2011-12-28
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    As a practical point, signal strength is a poor predictor of distance, because it is physically impossible for an RF beacon to radiate isotropically in all directions, and your the sensitivity of your receiver will probably also have an unknown directional dependence. It's much more reliable to use a directional antenna to triangulate the location.2011-12-28
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    @Henning, you are perfectly correct in what you say, but the parameters and data are what I have indicated.2011-12-28

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