GPS Coordinate Measurement

I wanted to record the exact latitude and longitude coordinates of the center of my bedroom, seeing as we bought a GPS unit for Christmas. I had so much fun in the process that I decided to post my methods here. There were two problems I had to overcome, first was that the GPS unit won't operate indoors, so I could not obtain a direct reading of the coordinates. Second, the unit is only accurate to within 15 feet.

I took the GPS receiver outside, and found a good spot close to my bedroom window, in which I could have the best view of the sky, to ensure maximum accuracy. This entailed climbing up on a garbage pail and leaning against the top of the block fence, where I also entertained myself by looking into the neighbor's yard briefly. Because of the nature of the GPS signals and the accuracy, on a small scale the readings are constantly fluctuating. Statistically, the most accurate reading can be obtained by recording a number of coordinates over a short period of time and averaging them. Thus, I created 28 waypoints (which store the coordinate data as well as elevation), approximately every 5 seconds. I present them below (scroll past the table to continue reading):

With that completed, I averaged the values, and we have: 33.31831429° N, 112.0109256° W, and and elevation is 1328.54 feet (rounding will be done at the end). The next step is to adjust these values for the actual center of my room. Currently they represent a point outside.

In the diagram, my bedroom window center is the X on the left, and my measurement location was the X on the right. Because of the short distance from the measurement position to the window, I was unable to obtain a reliable GPS direction heading. Thus, I walked parallel to the house (line on the left), in the direction shown by the arrow, and recorded a heading of 325°. Latitude and longitude are along the 90-270° and 0-180° lines, respectively. I measured along the house for the distance from the window to the point that would be on the line of 59° that intersects the measurement point. This distance was 84 inches. I also measured the distance from the measurement point to the house along that line of 59°. It was 154 inches. Based on the Pythagorean Theorem for the triangle shown in the diagram, the true vector from the measurement point to the window was of magnitude 175.42 inches. The direction was measured with trigonometry; the angle opposite the 154" side was tan^-1(154/84), or 61.39°. Thus the direction from the window to the measurement point was 325+61.39, modulo 360. This is 26.39°, but because we want the other path along this line, the inversed value is taken, which is 206.39°.

As I mentioned the direction of the latitude and longitude coordinates, we need to break apart that vector of 206.39° into vectors in the N-S and W-E directions.

In the second diagram above, you can see that with two basic trig operations we can calculate the W-E and N-S components of the vector. The angle alpha is 26.39°, so the value x (the west-east component) is 175.42*sin(26.39°). The value of y is 175.42*cos(26.39°). These values work out to 77.97" and 157.14", respectively. It is actually -157.14", because that resolved vector points south when the latitude is expressed in north. There is one further adjustment to be made, in that those values are only to the outside center of the window. I wanted it to be in the center of my room. I will spare the calculations to resolve the vectors here, but the results are an adjustment of 51.61" longitude, and -36.14" latitude. Adding these with the offsets for outside, there is a total positional difference of 129.58" for longitude, and -193.28" for latitude.

In order to correct the original latitude and longitude, we must convert our inch differences into the appropriate coordinate values. Because the earth's circumference is 131,481,141.7 feet, dividing by 180 for longitude and converting into inches, then dividing 129.58" by that produces a difference of .0000147831°. Similarly for latitude (only 360 degrees), we have a difference of -.0000441°. Finally, we obtain our finished coordinate values of 33.31827019° N, and 112.0109404° W. Now I'll detour briefly for the elevation measurement. The average value was 1328.54 feet, however I was standing 46 inches above the ground, with the GPS unit held 12" above the top of my head, and I am 67" tall. Thus I was 125" above the outside ground, which measured as 21" below the bottom of the window frame. The window frame provides a constant point of reference for both indoor and outdoor elevation. On the other side of it, the rug in the house is 18" below the bottom. Thus the overall elevation change from the measurement point to the floor of my room is 122 inches. The final elevation number is then 1319 feet (this is feet above sea level).

Back to the coordinates, and converting into a degree-minute format, these are the values: 33°19.0962' N, 112°00.6564' W. Overall, this was a fun and nerdy task to occupy myself for a short time.