Calculating a gps coordinate using satellites position

Question

I was wondering if there is a way to figure out an exact GPS coordinates using the information from Satellites. I have this information:

Satellite Name / NORAD Number / Tx Time (GMT) / Lat. / Long. / Altitude (km)

GPS BIIR-3 (PRN 11) / 25933 / 16:02:30.0829041384 / 40.63 / -52.99 / 19808.1

GPS BIIF-2 (PRN 01) / 37753 / 16:02:30.0813097068 / 27.58 / -54.84 / 20072.3

GPS BIIF-5 (PRN 30) / 39533 / 16:02:30.0809520507 / 27.01 / -93.49 / 20153.9

GPS BIIA-23 (PRN 04) / 22877 / 16:02:30.0801554947 / 43.43 / -33.86 / 19884.1

GPS BIIR-5 (PRN 28) / 26407 / 16:02:30.0772779343 / 56.07 / -114.62 / 20715.9

And with the following assumptions: •All four signals reached your GPS receiver simultaneously •The waypoint is at sea level •The Earth's atmosphere has no effect on the GPS signal •The Earth is a perfect sphere, with a radius of 6370 kilometers •The speed of light is 3.00E+08 meters per second •Your GPS receiver has no time differential error

I also know the waypoint was taken around 12pm eastern time zone. I am sure all I need is the actual time to determine the GPS coordinate but I am not sure how to figure out the time. Does anyone have any ideas?

Answer 1

Yes, you have enough information with four satellites. You are trying to determine four numbers, your $x,y,z$ position and the time. You have four pieces of data, which are the times the satellites sent out each signal. If you are willing to assume sea level, you are only trying to measure three numbers, so three satellites would suffice. Having a fourth can improve your accuracy through a least squares fit. Now that your edit has come it and the data is readable, it looks like you have data from five satellites.

What you need to do is to translate the Lat/Lon/altitude to $x,y,z$ coordinates for each satellite. For each satellite you write an equation based on distance to satellite=(time now-time signal was sent)*speed of light. The distance to satellite uses the satellite coordinates you computed. This gives you five equations in four unknowns. As you have assumed your altitude is zero, you really have only three unknowns-your Lat/Lon and the time of observation. Normally you would compute the error in each of the observations as a function of your variables, then use a minimizer to find the best fit values of your variables. You don't solve for the time first, you solve for time and your position simultaneously.

Your assumptions and data are quite coarse. Reporting satellite Lat/Lon to only two decimals of a degree gives a position error of the order of a mile. The earth radius varies by $13$ miles depending on longitude. This will not make the measurement fail, it will just render it inaccurate.