# Spherical Reasoning Mystery Cache

##### This cache has been archived.

bytemobile: The time has come to put this one to sleep. I don't think I remember enough about the calculation to help anyone through it anyway.

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A cache by bytemobile
Hidden : 08/24/2004
Difficulty:
Terrain:

Size:  (micro)

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## Geocache Description:

The coordinates at the top of this page are not the actual coordinates of the cache! You must determine the actual coordinates by solving the puzzle. The cache is within 3 miles of the coordinates above.

Calculate the location of the cache by computing it like your GPS receiver does, only in simplified form.

An excellent introduction to how GPS works is at Trimble's web site.

A GPS receiver uses timing to measure range (or pseudo range) to each satellite it processes. (There is no time difference of arrival involved as in my puzzle cache, Hyperbolic Reasoning.) In addition to providing timing, the GPS signal also sends information which allows the receiver to compute the positions of the satellites. The position of two satellites and the distance from them defines two spheres, which intersect in a circle. A third satellite narrows the possibilities down to two points on the circle, one of which is the location of interest. A fourth satellite is used to improve timing accuracy.

This puzzle uses just two satellites, for a “2D” geolocation. Normally a 2D geolocation requires three satellites, but we are not concerned with correcting timing errors in this puzzle. The cache is on the surface of the earth, so you can use the earth as a third intersecting "sphere" (ellipsoid).

The location of GPS satellites is computed to high accuracy and archived in “SP3” files for 15 minute intervals. This puzzle is set up on an even hour, so you won’t have to interpolate the satellite position, but you will have to go hunting for the SP3 file and learn how to read it.

GPS time is kept in week number (from 6 Jan 1980), modulo 1024, and seconds of week, which starts just after midnight Saturday night / Sunday morning. GPS time differs from UTC by the number of leap seconds, which is currently 13, i.e. GPS = UTC + 13.

The problem: On August 5th, 2004, at 08:59:47 PDT, from the cache location, your GPS receiver would have measured the following distances:

Satellite PRN 09: 23665.109 km
Satellite PRN 21: 23785.440 km