Introduction to GPS: Theory and Applications

Introduction to GPS: Theory and Applications

Homework No. 1

GS 608: Introduction to GPS: Theory and Applications

Autumn 2000

Assigned: October 5, 2000

Due: October 19, 2000

  1. Reference systems.
  2. Visit the following web page (if you have not done it already): print the section: Geodetic Datum Overview including the figures. We have covered most of this material in class, but please, use this write-up as further reference.
  3. Define reference system and explain what is the difference between the local and global datum (reference system).
  4. What is the difference between geographical and geodetic latitude, longitude and height? For what types of applications (say in mapping) would you recommend using geodetic vs. geographical coordinates?
  1. Cartesian to (from) geographical (geodetic) coordinate conversion.
  2. Using the class notes distributed last week, do the following coordinate conversion:
  3. Given geographical coordinates: Latitude = 40N, Longitude = 83W, h= 210m, convert them to 3-D Cartesian coordinates X, Y and Z. Use the radius of the reference sphere r = 6371 km.
  4. Now perform the backward conversion, going from your XYZ triplet back to Latitude, Longitude and height. Have you obtained your original coordinates?
  5. Now assume that the coordinates given in (i) are actually geodetic coordinates, and thus refer to the ellipsoidal surface with the semi major axis a = 6378137 m and 1/f = 298.257223563.
  6. Perform the coordinate conversion from Latitude, Longitude and height given in (a) to XYZ.
  7. What is the difference between your current results and the results obtained in (ai) above? Why?
  8. Perform the coordinate conversion going from the XYZ obtained in (i) to ellipsoidal Latitude, Longitude and Height using two different methods:
  9. Direct solution presented in class, according to Bowring, 1976.
  10. Closed formulas presented in the hand out (Borkowski, 1989).
  11. Discuss the difference between the results from (iii 1 and 2).
  1. NAD83 to SPCS83 conversion.
  2. What is the SPCS?
  3. Follow the instructions given below to perform coordinate conversion between NAD83 Latitude and Longitude and SPCS83.

SPCS83 is datum conversion software developed by NGS (National Geodetic Survey). It converts geodetic coordinates from NAD 83 to NAD 83 SPCS (State Plane Coordinates System) and vice versa. It allows the users to input data in the interactive mode or as a batch file. The software is available in the NGS anonymous ftp station on ftp://ftp.ngs.noaa.gov/pub/pcsoft/spcs83/spcs83.exe, and its documentation (spsc83.doc) can also be found in the same directory.

  • Type ftp://ftp.ngs.noaa.gov/pub/pcsoft/spcs83/ in your Internet browser and save spcs83.exe and spcs83.doc to your local directory. Note: this is not the only way to get access to this anonymous ftp location. If you are familiar with ftp client software, you can use that as well.
  • Spcs83.doc contains a list of the menus and all other submenus. In Appendix D, the State Plane Zone Codes can be found.
  • In your local directory: double click on spcs83.exe. A DOS-prompt menu will pop up. Type 1 and run the program interactively with a selected output file name (you need to type an output file name).
  • Now input the coordinates of the following station in North Ohio:
  • Columbus (station name), 83W, 40N. The input format is (DD MM SS.SSSS). For example 83 could be (83 00 00.0000). The system will prompt you for the east or west longitude.
  • Check the Zone Code from the documentation spcs83.doc and type it in.
  • Type N to end this program.
  • Go to Windows Explorer and check your local directory for your output file. Open it with Notepad to see the result. What are the Northing and Easting you’ve got?
  • Now, double click spcs83.exe again and choose option 2 to convert your Northing and Easting back to latitude and longitude. Use the Northing and Easting you’ve got in the previous practice and follow the prompt.
  • What are your latitude and longitude? Are they the same as your original coordinates?
  • What would happen if you selected the wrong Zone Code? Why? Check it out!

This is the equation form of Borkowski’s FORTRAN codes (c.f., IERS Technical Note 21, pp12-13). It transforms Cartesian coordinates to geodetic ones. Given: X, Y, Z Cartesian coordinates and the semi-major axis (a) and the inverse flattening (fr) of the reference ellipsoid, it calculates the geodetic latitude and ellipsoidal height. Variable v and variable s will be calculated recursively. For example, using the following quantities,

a = 6378137

fr = 298.2572221

X = 596261.727 m

Y = -4856162.066 m

Z = -4077921.293 m,

you will get your result:

 = -40h = -100 m.

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