Appendix B Calculating Pipe Length and Mean Slope from GPS Data B.1 THE BASICS: NORTHING AND EASTING Dimensions of the potential site for a water network can be measured with landbased surveying equipment, such as a transit or an Abney level and measuring tape. Alternately, more sophisticated electronic equipment may be used, such as a global positioning system or GPS. A GPS uses signals transmitted between itself and satellites to accurately determine the position of the device. The position is given in terms of three coordinates, latitude, longitude, and elevation from sea level. In principle, these are sufficient to allow us to determine everything we need for the survey. However, latitude and longitude are not immediately useful because they are angles measured from the equatorial plane for latitude (between 0 and 90 ; positive for the northern hemisphere, negative for the southern) and the prime meridian for longitude Gravity-Driven Water Flow in Networks. By Gerard F. Jones Copyright 2010 John Wiley & Sons, Inc. 495
496 APPENDIX B JV N, E vnl N E 2.N 2, K E Figure B.l Easting-Northing coordinates for two points on the earth's surface. (between 1 and 180, positive for the eastern hemisphere, negative for the western) and we require distances. A conversion is needed. Universal Transverse Mercator (UTM 1 ) coordinates are Cartesian coordinates that have been converted from latitude and longitude angles. The two coordinates are referred to as Easting and Northing, where Easting is the distance coordinate in the east-west direction (larger numbers are east of smaller numbers) and Northing is that in the north-south direction (larger numbers are north of smaller numbers). In engineering, we normally refer to Easting and Northing as the x and y coordinates, respectively; see Figs. B.l and B.2 for a comparison between the two. The conversion is accomplished by applying formulas from geometry relating angular measurements and distances at the earth's surface. For example, 1 of longitude at the equator (0 latitude) is 111.3 km to better than 0.1% depending on longitude. This distance gets smaller with increases in either north or south latitude. Fortunately, we do not have to program these formulas ourselves since others were kind enough to do this for us. A Microsoft Excel spreadsheet written by Dutch (2009) is available for the conversions. The inputs include the model for the shape of the earth, latitude and longitude values entered in either decimal or degree-minutes-seconds (DMS) format, and specifications of north or south for latitude and east or west for longitude. For the DMS format there are 60 minutes in a degree and 60 seconds in a minute. The conversion between the DMS and decimal format is thus, Θ = 0+Μ + Λ_ (B.D 60 3600 'The Transverse Mercator Projection that is used on many world maps is a cylindrical projection. In this projection, the earth is contained within an imaginary cylinder that contacts the globe along its equator. The earth is then projected on the cylinder to produce the Mercator Projection. To visualize this, imagine a flashlight positioned on the north-south axis (and normal to it) running through the globe, pointing outward toward the cylinder. Note that landmasses at the larger north and south latitudes will be compressed in the north-south direction relative to those at the equator once project onto the cylinder. This is a characteristic of the Mercator Projection. v '
THE BASICS: NORTHING AND EASTING 497 y, x ivi Χ 2^9. X, 2 X Figure B.2 Easting-Northing coordinates of Fig. B. 1 written in conventional x-y form. where Θ is the latitude or longitude angle in decimal format. For example, a latitude of 19, 45', 33" converts to 19.759167. Note the designations of a single prime for minutes and a double prime for seconds in the DMS format. The model for the earth's shape essentially means the assumed diameters of the earth at the equator and poles; it is generally understood that the earth is larger at the equator than at the poles. Different models, referred to as "datums", use very slightly different diameters. The most recent datums are NAD83/WGS84 (World Geodetic System 1984) and GRS80 (the foundation for the North American Datum of 1983 or NAD83) which agree with each other to within 1 part in 10,000. Use the NAD83/WGS84 datum unless information is available to dictate otherwise. A word about accuracy of the results from these calculations is in order. Since there are at most 111,300 m in 1 of longitude, 1 m of resolution in the Easting coordinate will require certainty in longitude to five decimal places. The same accuracy in latitude is required for the Northing coordinate. Longitude or latitude data to only four decimal places will produce, at best, certainty to ±10 m. The accuracy of readings from a standard GPS receiver is also unlikely to be greater than ±10 m. 2. Thus, the understanding is that calculations involving dimensions from GPS data will be accurate only to this order of magnitude. To obtain the most meaningful results at any accuracy level, multiple latitude and longitude readings, say 10 or more, are recommended before converting the average of these readings to UTM coordinates. More accurate horizontal and elevation measurements, at the cost of additional time and perhaps expense, are obtained by surveying the site with a transit as described in Chapter 13. 2 One popular brand of GPS in the United States, Garmin, states that its GPS receivers are accurate to within 15 m on average (Anon., 2009) This is probably a worst-case estimate.
498 APPENDIX B B.2 AN EXAMPLE Consider the calculation of Northing and Easting coordinates for two locations listed in Table B. 1. 3 Download the Excel spreadsheet (Dutch, 2009) and launch Excel to run the sheet. Input the data in DMS format and obtain the results as shown in Table B.2. Table B.l Data for a Northing-Easting Example Name Latitude Longitude Treasure Island Diamond Island 43 33' 0" 43 34' 25" -71 17' 0" -71 19' 20" Table B.2 Northing-Easting Results Location Treasure Island Diamond Island Northing Coordinate (m) Ni = 4,824,427.26 N 2 = 4,827,136.64 Easting Coordinate (m) i = 315, E 2 = 312, 555.29 487.09 The Pythagorean theorem is used to calculate the distance between the two locations. Obtain I = [(ivx - iv 2 ) 2 + (E 1 - E 2 ) 2 } 1 ' 2 (B.2) The distance between the two locations is calculated as I = 4093.24 m. If this is the final result to be reported, it would be rounded to the nearest 10 m. We get ί = 4090 m = 4.09 km. To calculate the mean slope between these two locations, assuming a source at Treasure Island Az = 10 m higher than the delivery location on Diamond Island, from the definition of slope, we use.- from which we get s = 0.244%. From the Pythagorean theorem we can also calculate the pipe length if run directly from source to delivery or tank location. Thus, L = {i 2 + Az 2 ) 1 / 2 = [(ivj - N 2 ) 2 + (E 1 - E 2 f + Az 2 } 1 ' 2 (B.4) We get L = 4093.25 m. Because of the relatively small value of Az, L and I are essentially identical. 3 The locations are on Lake Winnipesaukee, New Hampshire. This is the location of the author while completing this appendix.
REFERENCES 499 References Anon. How Accurate is GPS? http://www8.garmin.com/aboutgps/, 2009. [Online; accessed 3-November-2009]. S. Dutch. Converting UTM to Latitude and Longitude (or vice versa). Technical report, University of Wisconsin - Green Bay, Green Bay, WI, 2009. URL \url{http://www.uwgb.edu/dutchs/usefuldata/ UTMFormulas.HTM}. [Online; accessed 18-November-2009].