Late For Dinner

Approach

To determine the location of the GPS receiver (U_x, U_y, U_z), the four range equations must be solved simultaneously.

     (X₁ – U_x)² + (Y₁ – U_y)² + (Z₁ – U_z)² = (PR₁ –  τc)²
      (X₂ – U_x)² + (Y₂ – U_y)² + (Z₂ – U_z)² = (PR₂ –  τc)²
      (X₃ – U_x)² + (Y₃ – U_y)² + (Z₃ – U_z)² = (PR₃ –  τc)²
      (X₄ – U_x)² + (Y₄ – U_y)² + (Z₄ – U_z)² = (PR₄ –  τc)²

The Newton-Raphson Method was used to solve these non-linear equations simultaneously.  This is an iterative method described by:

     U = U₀ - H^-1η

Where U is a vector of the four unknowns U_x, U_y, U_z, and τ.  U₀ is an initial guess, which was chosen to be Atlanta.  H is the Jacobean matrix of the range equations.  And η is a vector of the range equations in homogenous form.

A Matlab script was written to solve this matrix problem.  One hundred iterations were run to be certain that the answer had converged.

Results

The GPS receiver is at a location of 33.7727041° Lat and -84.3798970° Lon.  Using Google Earth, it was found that this is the intersection of Myrtle Street and Ponce de Leon Ave.

Comments

The altitude ambiguity in the C/A GPS signal was easily resolved using this technique.  We used an initial altitude guess of 6380 km.  Therefore, the answer converged to an altitude that was on the Earth's surface.

The GPS system has an accuracy of about 30 meters when just one measurement is taken as in this problem.  Some possible sources of error are: satellite clock, ephemeris error, ionosphere delay, troposphere delay, receiver noise, and multiplath.