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Origin of Geographic Coordinates

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					GIS in Water Resources

  Review for Midterm Exam
                Latitude and Longitude
                  in North America

Austin:
(30°N, 98°W)

Logan:
(42°N, 112°W)
                         90 W
 Length on Meridians and Parallels
(Lat, Long) = (f, l)


Length on a Meridian:
AB = Re Df                             R
(same for all latitudes)        R Dl       D
                                    C
                           Re    Df B
                                Re
Length on a Parallel:
                                     A
CD = R Dl = Re Dl Cos f
(varies with latitude)
Example: What is the length of a 1º increment along
on a meridian and on a parallel at 30N, 90W?
Radius of the earth = 6370 km.


Solution:
• A 1º angle has first to be converted to radians
p radians = 180 º, so 1º = p/180 = 3.1416/180 = 0.0175 radians

• For the meridian, DL = Re Df = 6370 * 0.0175 = 111 km

• For the parallel, DL = Re Dl Cos f
                      = 6370 * 0.0175 * Cos 30
                      = 96.5 km
• Parallels converge as poles are approached
     Horizontal Earth Datums
• An earth datum is defined by an ellipse and
  an axis of rotation
• NAD27 (North American Datum of 1927)
  uses the Clarke (1866) ellipsoid on a non
  geocentric axis of rotation
• NAD83 (NAD,1983) uses the GRS80
  ellipsoid on a geocentric axis of rotation
• WGS84 (World Geodetic System of 1984)
  uses GRS80, almost the same as NAD83
       Vertical Earth Datums
• A vertical datum defines elevation, z
• NGVD29 (National Geodetic Vertical
  Datum of 1929)
• NAVD88 (North American Vertical Datum
  of 1988)
• takes into account a map of gravity
  anomalies between the ellipsoid and the
  geoid
          Coordinate System
     A planar coordinate system is defined by a pair
     of orthogonal (x,y) axes drawn through an origin

                                      Y




                     Origin                             X

                                       (xo,yo)
(fo,lo)
Universal Transverse
Mercator
• Uses the Transverse Mercator projection
• Each zone has a Central Meridian (lo),
  zones are 6° wide, and go from pole to pole
• 60 zones cover the earth from East to West
• Reference Latitude (fo), is the equator
• (Xshift, Yshift) = (xo,yo) = (500000, 0) in
  the Northern Hemisphere, units are meters
UTM Zone 14
          -99°
     -102°   -96°




     6°




   Origin
                            Equator
  -120°          -90 °   -60 °
   ArcInfo 8 Reference Frames
• Defined for a feature
  dataset in ArcCatalog
• Coordinate System
   – Projected
   – Geographic
• X/Y Domain
• Z Domain
• M Domain
                 X/Y Domain
                 (Max X, Max Y)     Long integer max value
                                    of 231 = 2,147,483,645




(Min X, Min Y)

       Maximum resolution of a point = Map Units / Precision
       e.g. map units = meters, precision = 1000, then
       maximum resolution = 1 meter/1000 = 1 mm on the ground
                  Network Definition
• A network is a set of edges and junctions
  that are topologically connected to each
  other.
           Edges and Junctions
• Simple feature classes: points and lines
• Network feature classes: junctions and edges
• Edges can be
   – Simple: one attribute record for a single edge
   – Complex: one attribute record for several edges in a
     linear sequence


• A single edge cannot be branched

                                          No!!
Polylines and Edges
                 Junctions
• Junctions exist at all points where edges
  join
  – If necessary they are added during network
    building (generic junctions)
• Junctions can be placed on the interior of an
  edge e.g. stream gage
• Any number of point feature classes can be
  built into junctions on a single network
              Connectivity Table
              p. 132 of Modeling our World
                                                     J125
Junction Adjacent Junction and Edge
J123    J124, E1
                                                    E2
                                             J124
J124    J123, E1 J125, E2 J126, E3
                                              E1     E3
J125    J124, E2
                                      J123
J126    J124, E3                                            J126

                   This is the “Logical Network”
Flow to a sink
   Eight Direction Pour Point Model

               32     64     128


               16             1


               8       4      2


Water flows in the direction of steepest descent
             Flow Direction Grid



32 64 128

16       1

8    4   2
Cell to Cell Grid Network
 Through the Landscape




                     Stream cell
         Contributing Area Grid
1    1   1    1    1   1   1       1             1   1

1   4    3    3    1   1       4        3        3   1

1   1    12   1    2                             1
                       1   1                         2
                                   12
1   1    2    16   1   1                2            1
                           1                 16
1   3    6    25   2   1       3        6
                                                     2
                                            25


    Drainage area threshold > 5 Cells
Delineation of Streams and
 Watersheds on a DEM
Stream Segments
0   0       0            0    0

0       3       2        2    0

                         0
0   0                         1
            11
0             1               0
    0
                         15
0       2       5
                              1
                    24
Stream Segments in a Cell Network

                    1
                        1           2
                            1   2
                                                  5
                                3                 5
                                    3         5
                        4 4 4           3 5
                                4 4
                                         6 6
                                           6
Watershed Outlet
Watershed Draining to This Outlet
      Watershed and Drainage Paths
       Delineated from 30m DEM




Automated method is more consistent than hand delineation
1000 Cell Threshold Exceeded at Stream Junction




                        510    989
                                1504 (>1000)
Subwatersheds for Stream Segments




           Same Cell Value
Vectorized Streams Linked Using
 Grid Code to Cell Equivalents



Vector
Streams
                          Grid
                          Streams
Delineated Subwatersheds and Stream Networks
A Mesh of Triangles

                Triangle is the only
               polygon that is always
                   planar in 3-D




  Points   Lines      Surfaces
    Tin Triangles in 3-D
                       (x3, y3, z3)


    (x1, y1, z1)
                          (x2, y2, z2)

       z

y                           Projection in (x,y) plane

                   x
 Delauney Triangulation
Maximize the minimum interior angle of triangles
No point lies within the circumcircle of a triangle

  Yes                           No
Flow On a Triangle
13.0

       10.0
                      9.0


          5.0



                2.0
Flow On a TIN

                                 120
                           110
                     100
                90
           80
      70




      60




     50

				
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