OBJECTIVES This exercise involves understanding map scale and coordinate systems. You will learn to read geographic coordinates in latitude/longitude, the State Plane Coordinate System, and the Universal Transverse Mercator system. You will learn to work with these coordinate systems by examining a recent surveying controversy that claims the Four Corners Monument is actually misplaced. Make your own measurements to decide it if it s off the mark. MATERIALS Ruler Pencil USGS Yellow Rock Point East Quadrangle 7.5 GeoPDF (available for free download at http://store.usgs.gov) TerraGo Desktop (available for free download at www.terragotech.com) BACKGROUND Key Terms Latitude Longitude Coordinate System Prime Meridian Datum Angular measurement north and south of the equator. Angular measurement east and west of the Prime Meridian. A method of representing points in a space of given dimensions by coordinates from an origin Meridian, line of longitude, passing through Greenwich, England at which longitude is defined to be 0 Reference points on the earth s surface against which measurements are made Common units and conversions Linear unit conversions are important to understand, especially the more common ones which will be used throughout this series of labs. Get a grasp of the magnitude of some units relative to other units. If you are not familiar with unit conversions, you may need some practice setting up the calculations so that the units cancel. 1 km = 1000 meters 1 meter = 100 centimeters = 1000 millimeters 1 mile = 5280 feet 1 foot = 12 inches 1 meter = 3.28 feet 1 square meter = 10.76 square feet Introduction A geographic coordinate system is used to describe the position of a location in two or three dimensional space with respect to an origin and a specified unit of distance. Standard coordinate systems are in use to assign locational information to places in unambiguous, accurate terms. A map coordinate uses two numbers, an easting and a northing, to specify a location with respect to a defined origin. Most coordinate systems assume a particular earth model, ellipsoid, datum, and central meridian to establish an origin. We will discuss the three coordinate systems included on your quadrangle. Page 1 of 10
Latitude/Longitude Geographical coordinates in latitude and longitude are angular measurements given with respect to an arbitrarily selected origin at the intersection of the equator and the Prime Meridian, which was standardized by the International Meridian Conference of 1884 when a line passing through Greenwich Observatory in England was adopted as the zero-longitude reference line. Latitude ranges from 90 degrees north or south of the equator while longitude ranges from 180 degrees east or west of the Prime Meridian. To precisely locate points on the earth's surface, degrees longitude and latitude have been subdivided into minutes (') and seconds (") of arc. There are 60 minutes in each degree. Each minute is divided into 60 seconds. Seconds can be further divided into tenths, hundredths, or even thousandths. At any point on earth, the ground distance covered by one degree of latitude is about 111 km. The ground distance covered by one degree of longitude at the equator is also about 111 km, but decreases until it becomes zero at the poles. Latitude and longitude is the basis for USGS map tiling that is, the edges of maps follow lines of equal latitude and longitude. Universal Transverse Mercator (UTM) The UTM coordinate system is included on most USGS topographic maps. This grid-based coordinate system divides the earth into 60 pole-to-pole zones, each 6 degrees of longitude wide, starting at 180 W (the International Date Line), and increasing east. Each zone is divided by its hemisphere into N or S. For the coterminous United States, California falls into zones 10N and 11N. A false origin for each zone is established. In the northern hemisphere, this false origin is at the equator beyond the westerly limit of each zone. In the southern hemisphere, the zero northing is at the South Pole. This numbering scheme allows all coordinates to be given in positive northings and eastings, a location s distance in meters from the false origin. The central meridian for each zone is given the value of 500,000 meters. State Plane Coordinate System (SPCS) SPCS divides the United States into a number of smaller zones following state and county boundaries. Each state contains one or more zones that use a simple Cartesian coordinate system, thus reducing map projection distortion because such issues are minimal for fairly small regions. Outside a specific state plane zone, accuracy rapidly declines, thus the system is not useful across zone boundaries. Most state plane zones are based on either a Transverse Mercator projection or a Lambert conformal conic projection depending upon the shape of the state and its zones. States that are elongated in the east-west direction typically use the Lambert conformal conic projection because it maintains accuracy along an east-west axis. Zones that are elongated in the north-south direction use the Transverse Mercator projection because it is better at maintaining accuracy along a north-south axis. SPCS coordinates are given in meters or feet north and east of the false origin, which is arbitrarily defined beyond the southwesternmost point of each zone. Page 2 of 10
EXERCISE Four Corners Monument Controversy The Four Corners Monument, which adjoins southwest Colorado, northwest New Mexico, northeast Arizona and southeast Utah, is the only location in the US where four states meet. The area was first surveyed by the U.S. Government in 1868 as part of an effort to form the state of Colorado. In 1875, surveyor Chandler Robbins was contracted by the US government to survey the entire boundary between Arizona and New Mexico. He was instructed that the boundary which separated these two states should be coincident with the 37 th parallel and the 32 nd meridian of longitude west of the Washington DC Meridian (77 03 W), what would be equivalent to 109 03 W as referenced to the Greenwich Meridian. The Four Corners monument remains today in the location established by Robbins. UTAH ARIZONA COLORADO NEW MEXICO Controversy arose in April 2009 when several media reports claimed that recent surveys revealed the monument to be mislocated by 2.5 miles east due to surveying errors and a datum change. Your task today is to determine if there is any weight to these claims. Using a USGS quadrangle for the area, you will determine if the marker is misplaced, and if so, calculate the amount of error in its displacement. Part 1. Understanding Coordinate Systems Open the USGS Yellow Rock Point East Quadrangle GeoPDF which was downloaded from the Map Locator at http://store.usgs.gov. If you do not have the TerraGo toolbar installed on your computer, it can be downloaded at www.terragotech.com. This free toolbar provides an easy way to access spatially referenced information stored in USGS topographic maps. a. Geographic coordinates appear at the four corners of the USGS map. What are the values of latitude and longitude lines that enclose the Yellow Rock East map? 1. 2. 3. 4. b. UTM Zone 1 runs from 180 W to 174 W. The Four Corners region lies in UTM Zone 12N. What is the range of longitude degrees covered by this projection zone? c. How many SPCS zones are in your state? Which map projection does your state use? Part 2. Using map coordinates This section focuses on using the lat/lon coordinate system, but UTM and SPCS have the advantage of being easier to read on a map because their coordinates correspond to distances on the ground (i.e. The UTM coordinate 242000 meters North, 3819000 meters East describes a point that is a distance north and east of the zone s false origin). The distance between two points given in UTM and SPCS coordinates can easily be computed with the Pythagorean Theorem. Page 3 of 10
Look for the symbol that marks the current Four Corners monument at the bottom of the map. (Hint: Follow the state boundary lines to find the square marker symbol). Click on the first button on the TerraGo toolbar. Read the information in the dialog box that appears which provides instructions on displaying coordinates. Zoom into the Four Corners monument location. Position your mouse cursor over the marker as precisely as possible, then single click on the marker symbol to freeze the coordinate display on the bottom right. a. What are the lat/lon coordinates of the marker as it appears on the map? The Four Corners monument was intended to be established at 37 N, 109 2 59.25 W. You will locate this point on the USGS Yellow Rock Point East Quadrangle using the TerraGo toolbar. Click on the second button in the toolbar, with an icon of arrow superimposed on a globe. A window should appear that will enable you to input search coordinates. b. What is the current datum and projection setting? What is the coordinate system and unit? Click on the arrow on the bottom right of this window Click on Set Projection Parameters in the drop down menu. In the window that appears, click on the Predefined tab and scroll through the list of coordinate systems to find NAD83 (NSRS2007). Select this projection and click OK. The Locate window should update to the selected coordinate system. Page 4 of 10
c. First, convert the coordinates for the monument s intended location from degrees minutes seconds (DMS) to decimal degrees (DD). Example: 20 10 15 Leave the degree, the first number, unchanged: 20 Multiply the minutes by 1/60 to convert it to degrees: 10 * (1/60) = 0.167 Multiply the seconds by 1/60 to convert it to minutes. Then multiply by 1/60 again to convert it to degrees: 15 * (1/60) * (1/60) = 0.004 Sum all three numbers: 20 + 0.167 + 0.004 = 20.171 Show your work: What are the coordinates in DD? Enter the coordinates in the Locate window. d. Does the coordinate you entered coincide with the marker for the monument printed on the map? Following the above procedure for setting projection parameters, change the coordinate system to SPCS and then UTM. e. What are the coordinates for the intended monument location in SPCS and UTM Zone 12N? Be sure to include units. SPCS: UTM: Part 3. Measuring error a. According to the National Geodetic Survey datasheet, the Four Corners monument is currently located at 36 59 56.31570 N, 109 02 42.62076 W in reference to NAD 83(2007). Convert this coordinate to DD, record it here, and enter the coordinate into the Locate window. Page 5 of 10
Use the third button on the TerraGo Toolbar to measure the length of any discrepancy between the marker s current location and its intended location. (In the next lab Land Navigation, you will learn how to use the Pythagorean Theorem to calculate the Euclidean distance between two points. For now, you will approximate the distance using this digital tool.) b. Is the actual marker misplaced from its intended location? If so, how far away is it in feet? In meters? Describe the marker s directional location in relation to its intended location. c. News articles publicized that the marker was misplaced by 2.5 miles east. Is this correct? Compute their accuracy. A way to compute the percent accuracy, given an estimate E (reported displacement) and an observation V (actual displacement) is: V E A 1 100 V Example: Suppose the measured distance is 2650 and the observed value is 2686, then: A = [1-2686-2650 /2686] x 100 (Note the use of absolute values ) = [1 (36/2686)] x 100 = 98.65% Show your work: The accuracy is % d. According to the NGS, the Four Corners Monument is exactly in the right place. Read the following article on the controversy at http://www.ngs.noaa.gov/info/fourcorners.shtml and summarize why monuments rule. Page 6 of 10
Part 4. Map interpolation To complete this portion of the lab, you will need to print out the attached map which displays a portion of a USGS quad for the Four Corners region. Or click HERE to download it. Although using GeoPDFs simplify the coordinate reading process, map interpolation is an important skill to acquire so that coordinates can be read without the use of a computer. Reading lat/lon coordinates is straightforward if a location falls on the lines of latitude and longitude printed on the map. However, if the place of interest lies somewhere between the graticule lines, the task becomes more difficult. Reading Latitude/Longitude Coordinates We will use a proportional method and paper strip technique to determine the geographic coordinates for three locations on the attached topomap. Map interpolation requires a graticule as a starting point. First, we will determine the parallels and meridians that enclose this map. The attached map is bounded: Longitude 109 07 30 W and 109 02 30 W (an interval of 5 = 300 ) Latitude 36 57 30 N and 37 N (an interval of 2.5 = 150 ) As mentioned above, the distance covered by a degree of longitude varies by latitude because the meridians converge at the poles. On the attached map, at latitude 37 N the length of a degree of longitude is 89011.64 meters. The length of a degree of latitude is 110,977.62 meters (approximately 111 km). a. Recall that there are 60 minutes in a degree. What is the length of one minute of longitude at 37 N? Using the lengths of degree given above, the Total Longitude Distance covered by this map can be calculated as 7417.636 meters. The Total Latitude Distance covered by this map is 4624.068 meters (110,977.62 m divided by 60, multiplied by 2.5). (If these lengths were unknown, one could use the graphic scale on the bottom of the map to make these measurements.) Locate Point 1 on the map. You will use the paper edge technique to measure the straight-line distance between Point 1 and the southern boundary of the map, which we will call the Partial Latitude Distance. Step I. Lay a straight-edged piece of paper on the map so that the edge of the paper touches both points and extends past them. Step II. Make tick marks on the edge of the paper at each point. Page 7 of 10
Step III. The earth distance is determined by checking the measurement against the graphic bar scale. If the measured distance exceeds the length of the graphic scale, several partial measurements should be combined. Step IV. The partial distance is in direct proportion to the total distance, and will be used to set up a ratio. Partial Distance Total Distance x arcseconds Example: Suppose we determine the partial latitude distance is 4 160 meters. Taken as a proportion of the known total latitude distance and arc seconds then: 4160 x 4624 150" 4624 * x = 150 * 4160 use cross multiplication to solve for x x = 134.9 Convert seconds to minutes by dividing by 60. The integer quotient will provide the minutes, while the remainder will be the seconds: 134.9 /60 = 2 14.9 Now add this number to the southern latitude coordinate that bounds the map, and this will be the latitude of Point 1. 2 14.9 + 36 57 30 N = 36 59 44.9 N b. Repeat the above paper strip technique to measure the partial longitude distance. Then use this number to set up a proportional ratio to find the longitude of this point. Partial Longitude Distance: Longitude coordinate: Page 8 of 10
c. Repeat the paper strip and proportional method to find the coordinates of Point 2. Latitude: Longitude: Reading UTM Coordinates UTM coordinates are easily interpreted if a point falls on gridlines; however, when the point of interest lies between the lines, some interpolation is required. Again, reading coordinates is simpler in UTM and SPCS because we can use the provided tick marks and map scales to compute distances. To measure a coordinate: 1. First find the nearest grid lines to the south and west of your point of interest. 2. Next, measure the distance from those grid lines to the point using the paper edge trick, then use the graphic scale to determine this distance on the ground. 3. Add the gridline values to the measured distance. (Note: UTM lines are aligned with grid north instead of true north. It would be helpful to align corresponding tick marks at the top and bottom of the map to superimpose a grid and read coordinates.) d. Interpolate the UTM coordinates for Point 3 on the attached map. Northing: Easting: Page 9 of 10
e. Interpolate the UTM coordinates for Point 4 on the attached map. Northing: Easting: RELATED LINKS Dr. Map article that appeared in ACSM Bulletin. April 2009. http://www.webmazine.org/issues/bull238/documents/askdrmap238.pdf NGS Four Corners Marker Datasheet. http://www.ngs.noaa.gov/cgi-bin/ds_mark.prl?pidbox=ad9256 NGS National Spatial Reference System http://www.ngs.noaa.gov/info/onepagers/nsrs.html REVIEW 1. What are the intervals and units of the three coordinate systems that appear on your quadrangle map? How are they symbolized? 2. How are the UTM and SPCS reference systems similar? 3. How are UTM zones divided? What type of projection does it use? Conformal or equivalent? This lab was developed by Susan Tran UCSB Geography September 2009 Page 10 of 10