Earth Coordinates & Grid Coordinate Systems

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1 Earth Coordinates & Grid Coordinate Systems You will always find parallels and meridians on largescale maps This is done to make the map a very close approximation to the size and shape of the piece of the ellipsoidal earth that it represents Horizontal Reference Datums Datums are the collection of very accurate control points (points of known accuracy) surveyors use to geo-reference all other map data Surveyors determine the precise latitude and longitude of horizontal control points spread across the landscape

2 Horizontal control point monuments are fixed objects established by surveyors when they determine the exact location of a point See Figure 1.15, page 17, Muehrcke Often found on top of a hill or other prominent feature From the 1920s to the early 1980s,these control points were surveyed relative to the surface of the Clarke 1866 ellipsoid, together forming what is called the North American Datum of 1927 (NAD 27) Topographic maps, nautical and aeronautical charts and many other large-scale maps of this time period had graticule lines or ticks based on this datum See Figure 1.16, page 17, Muehrcke

3 The Corvallis, Oregon map was published in 1969 and the bottom-right corner of the map has the NAD 27 latitude and longitude By the early 1980s, better knowledge of the earth s shape and size and far better surveying methods let to the creation of a new horizontal reference datum, the North American Datum of 1983 (NAD 83) The NAD 27 control points were corrected for surveying errors where required, then these were added to thousands of more recently acquired points The latitudes and longitudes of all of these points were determined relative to the Geodetic Reference System of 1980 (GRS 80) ellipsoid, which is essentially identical to the WGS 84 ellipsoid The change of horizontal reference datum meant that the coordinates for control points across the continent changed slightly. This change had to be shown on large-scale maps published earlier but still in use See Figure 1.16, page 17, Muehrcke

4 On topographic maps the new position of the map corner is shown by a dashed plus sign Many times the shift is in the 100 meter range and must be taken into account when plotting on older maps the latitudes and longitudes obtained from GPS receivers Even today, many countries continue to use local datums or ellipsoids resulting in incompatibilities The European Terrestrial Reference System of 1989 (ETRS 89) is a way to standardize mapping coordinates but many countries continue to use locally-derived ellipsoids The Earth as a Geoid When we treat the earth as an oblate ellipsoid, we neglect mountain ranges, ocean trenches and other surface features There is justification for doing this The earth s surface is truly smooth when we compare the surface undulations to the 7,918 mile diameter of the earth

5 The greatest relief variation is the approximately 12.3 mile difference between the summit of Mt. Everest (29,035 feet) and the deepest point in the Marianas Trench (36,192 feet) This vertical difference is immense on our human scale, bit only 1/640 th of the earth s diameter It has been said that is the earth were a bowling ball, it would be smoother than the bowling ball The top of Mt. Everest is located at: o 27 o 59 N, 86 o 56 E, 29,035 feet (8,852 meters) What is this elevation relative to? Vertical Reference Datums Elevations and depths are measured relative to what is called a vertical reference datum An arbitrary surface with an elevation of zero The traditional datum used for land elevation is mean sea level (MSL)

6 Surveyors define mean sea level as the average for all low and high tides at a particular starting location over a metonic cycle (the 19 year cycle of the lunar phases and days of the year) Early surveyors chose this datum because of the measurement technology of the day Surveyors first used the method of leveling, where elevations are determined relative to the point where mean sea level is defined They used horizontally aligned telescopes and vertically aligned leveling rods A small circular monument was placed in the ground at each surveyed benchmark elevation point A benchmark is a permanent monument that establishes the exact elevation of a place Later, surveyors could determine elevation by making gravity measurements at different locations on the landform and relating them to the strength of gravity at the point used to define MSL Gravity differences translate into elevation differences

7 Mean sea level is easy to determine along coastlines, but what about inland locations? What is needed is to extend MSL across the land Imagine that the MSL is extended under the continental land masses This is the same as extending a surface having the same strength of gravity as mean sea level See Figure 1.17, page 18, Muehrcke This imaginary equal gravity surface doesn t form a perfect ellipsoid because differences in topography and earth density affect gravity s pull at different locations The slightly undulating, nearly ellipsoidal surface that best fits mean sea level for all the earth s surface is called a global geoid The global geoid rises and falls approximately 100 meters above and below the oblate ellipsoid surface in an irregular pattern

8 World maps showing land topography and ocean bathymetry use land heights and water depths relative to the global geoid surface The mean sea level datum based on the geoid is so convenient that it is used to determine elevations around the world It is used as the base for the elevation data found on nearly all topographic maps and nautical charts Be aware that the local geoid used in your area is probably slightly above or below (usually within 2 meters) the global geoid elevations used on world maps This difference is caused by mean sea level at one or more locations being used as the vertical reference datum for your nation or continent, not the average sea level for all oceans In the US you may see elevations relative to the National Geodetic Vertical Datum of 1929 (NGVD 29) on older topographic maps

9 This datum was defined by the observed heights of mean sea level at 26 tide gauges, 21 in the US and 5 in Canada It was also defined by the set of elevations of all benchmarks resulting from over 60,000 miles of leveling across the continent totaling over 500,000 vertical control points North American Vertical Datum of 1988 (NAVD 88): In the 1980s surveyors adjusted the 1929 datum with new data Mean sea level for the continent was defined at one tidal station on the St. Lawrence River at Rimouski, Quebec, Canada NAVD 88 was a necessary update of the 1929 datum since about 400,000 miles of leveling was added to the NGVD since 1929 Additionally, numerous benchmarks had been lost over the decades and the elevations of others had been affected by vertical changes caused by rising of land elevations since the retreat of glaciers at the end of the last ice age (isostatic rebound) or

10 subsidence from sedimentation and the extraction of natural resources like oil and water GPS has created a second option for measuring elevation GPS receivers calculate what is called the ellipsoidal height, the distance above or below the surface of the WGS 84 ellipsoid along a line from the surface to the center of the earth See Figure 1.17, page 18, Muehrcke An ellipsoidal height is not an elevation, since it is not measured relative to the mean sea level datum for your local geoid You must convert GPS ellipsoidal height values to mean sea level datum elevations before you can use them with existing maps See Figure 1.18, page 19, Muehrcke

11 Grid Coordinate Systems There are several ways to pinpoint locations on a map The latitude and longitude graticule has been used for over 2000 years as the worldwide locational reference system Geocentric latitude and longitude coordinates on a sphere or geodetic latitudes and longitudes on an oblate ellipsoid, still key to modern position finding, are not as well suited for making measurements of length, direction and area on the earth s surface The basic difficulty is the fact that latitude-longitude is a coordinate system giving positions on a rounded surface It would be much simpler if we could designate location on a flat surface using horizontal and vertical lines spaced at regular intervals to form a square grid We could then simply read coordinates from the square grid of intersecting straight lines

12 Most maps are created by projecting the earth s surface onto a flat surface The advantage of the flat map projection surface is that we can locate something by using a two-axis coordinate reference system This coordinate system is the basis for the square grid of horizontal and vertical lines on a map We call a plane-rectangular coordinate system based upon and mathematically placed on a map projection a grid coordinate system To devise such a system for large areas, we have to deal somehow with the earth s curvature We know that transferring something a spherical to something flat always introduces geometrical distortion We also know that map projection distortion caused by the earth s ellipsoid shape is minimal for fairly small regions

13 If we superimpose a square grid onto flat maps of small areas, we can achieve positional accuracy good enough for many map uses All geographic grid systems are based on Cartesian coordinates, invented in 1637 by the famous French philosopher and mathematician Rene Descartes Cartesian Coordinates If you superimpose a square grid on a map, with divisions on a horizontal x-axis and a vertical y-axis where the axes cross at the system s origin, you have established the Cartesian Coordinate System See Figure 4.1, page 64, Muehrcke You can now pinpoint any location on the map precisely and objectively by giving its two coordinates (x,y)

14 The Cartesian Coordinate System is divided into four quadrants (I-IV) based on whether the values along the x- and y-axes are positive or negative Mapmakers use only quadrant I for grid coordinate systems so that all coordinates will be positive numbers relative to the (0,0) grid origin Grid Coordinates The simplest way of defining map positions based on Cartesian Coordinates has definite advantages over using the spherical graticule to define positions Measuring x- and y-coordinates from horizontal and vertical axes with equally spaced distance increments greatly simplifies locating environmental features because you do not have to deal with the decreasing separation between meridians converging towards the poles Grid coordinates systems based on the Cartesian Coordinate System are especially handy for such map analysis procedures as finding the distance or

15 direction between locations or determining the area of a mapped feature like a lake Two popular grid coordinate systems are: o Universal Transverse Mercator (UTM) o State Plane Coordinate System Universal Transverse Mercator System A grid coordinate system can be used worldwide if enough zones are defined to insure reasonable geometric accuracy The best known is UTM The UTM grid extends around the world from 84 o N to 80 o S 60 north-south zones are used, each 6 o longitude See Figure 4.2, page 65, Muehrcke Each zone has its own central meridian and uses a secant case transverse Mercator projection

16 centered on the zone s central meridian for each of the 60 zones This projection makes it possible to achieve a geometrical accuracy of one part in 2,500 maximum scale error Scale factors ranging from to within each zone Each zone is individually numbered from west to east Each zone has separate origins for the northern and southern hemispheres Easting: the east-west x-coordinate in a grid coordinate system. That is the distance east from the origin In both the northern and southern hemispheres, an easting value of 500,000 meters (written 500,000mE) is assigned to the central meridian of each UTM zone This value, called the false easting, is added to all x- coordinates so that there are no negative eastings in the zone

17 A northing is the north-south y-coordinate in a grid coordinate system In the northern hemisphere a northing value of 0mN is assigned to the equator so all northing values are positive numbers In the southern hemisphere, the equator is given a false northing of 10,000,000mN There are no negative y-values in the southern UTM zone because this false northing values places the origin of the zone very close to the south pole UTM Example See Figure 4.3, page 66, Meuhrcke Zone 10 covers much of the western seaboard The x-axis follows the equator The central meridian for the zone is 123 o W The longitude range is 120 o W to 126 o W The origin lies on the equator 500,000 meters west of the central meridian at 123 o W

18 UTM Coordinates of the 68 th St Entrance to the North Building mE, mN, Zone 18 North The near global extent of the UTM grid makes is a valuable worldwide referencing system The UTM grid is indicated on many foreign maps and on all recent USGS maps All GPS vendors program the UTM specifications into their receivers Note that UTM coordinates will differ when different datums are used You should check the datum information in the GPS receiver to assure that the coordinates are being recorded in the correct system Because meridians and not stat boundaries delimit UTM zones, it usually takes more than one UTM zone to cover a state completely

19 Universal Polar Stereographic System UTM grid zones extend from 80oS to 84oN To complete the global coverage, a complementary rectangle coordinate system called the universal polar stereographic (UPS) system was created UPS grid consists of a north zone and a south zone Each zone is superimposed on a secant case polar stereographic projection that covers a circular region over each pole See Figure 4.4, page 66, Muehckre The north zone extends from 84 o N to the north pole The UPS coordinate at grid center is: o 2,000000mE, 2,000,000mN This assures all coordinates are positive Virtually all large-scale maps of these high latitudes are based on the UPS grid The UTM grid system was not extended to the poles because the 60 zones converge at the poles meaning

20 that a new zone would be encountered every few miles State Plane Coordinate (SPC) System The state plane coordinate system was created in the 1930s by the land surveying profession in the US as a way to define property boundaries that would simplify computation of land parcel perimeters and areas The idea was to completely cover the US and its territories with grids laid over map projection surfaces so that the maximum scale distortion error would not exceed 1 part in 10,000 A distance measured over a 10,000 foot course would be accurate to within a foot of the true distance This level of accuracy could not be achieved if only one grid covered the whole country, because the area is too large The solution was to divide each state into one or more zones and make a separate grid for each zone

21 The US was originally divided into 125 zones Each has its own projection surface based on the Clarke 1866 ellipsoid and NAD 27 geodetic latitudes and longitudes Most states have several zones See Figure 4.5, page 67 Secant case Lambert conformal conic projections are used for states of predominantly east-west extent Secant case transverse Mercator projections are used for states of greater north-south extent States with more than one zone use the names North, South, East, West and Central to identify zones California uses Roman numerals In recent years the following states have combined zones into a single one: o Nebraska o South Carolina

22 o Montana The logic of the SPC system is simple Zone boundaries follow state and county boundaries because surveyors have to register land surveys in a particular county Each zone has its own central meridian that defines the vertical axis for the zone An origin is established to the west and south of the zone Usually 2,000,000 feet west of the central meridian for the Lambert conformal conic zones and 500,000 feet for the transverse Mercator zones This means that the central meridians will usually have an x-coordinate of either 500,000 feet for the transverse Mercator or 2,000,000 for the Lambert conformal conic zones These large numbers for zone centers were selected so that all x-coordinates will be positive numbers

23 Although different for each zone, the origin is always at a parallel to the south of the zone to ensure that all y-coordinates will be positive numbers SPC coordinates are read in the same manner as UTM coordinates first to the east and then to the north of the zone s origin For example the location of the state capitol dome in Madison Wisconsin is: o 2,164,600 ft E, 392,280 ft N, Wisconsin, south zone In 1983 the SPC system was modernized by switching to NAD 83 and the GRS 80 ellipsoid Zones were redefined in metric units The SPC system served the needs of states when it was created and state plane coordinates have been widely used for public works and land surveys

24 The SPC system is now largely obsolete as far as surveyors are concerned One reason is modern surveying methods can exceed the accuracy Also, each SPC zone is a separate entity with its own grid definition a fact that frustrate and discourages uses across zone boundaries Other Grid Systems US state grids combine UTM zones into a single state grid European grid coordinate systems British National Grid coordinate system Swiss coordinate system

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