SPCS Computations and Use

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STATE PLANE COORDINATE COMPUTATIONS Lectures 14
15 GISC-3325
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Updates and details

• Required reading assignments due 30 April 2008
• Extra credit due 23 April 2008
• Overdue lab assignments/homework will be given
  credit ONLY if received by 21 April 2008.
• Wednesday class 16 April 2008 will be devoted to
  RTK. Mr. Toby Stock will demonstrate, make
  observations and show results. Meet him at
  Blucher during lecture and lab periods.

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• Datum A set of constants specifying the
  coordinate system used to calculate coordinates
  of points on the Earth.

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• 8 Constants
• 3 to specify the origin.
• 3 to specify the orientation.
• 2 to specify the dimensions of the reference
  ellipsoid.

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a Semi major axis b Semi minor axis f a-b
Flattening a
N
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BESSEL 1841 a 6,377,397.155 m 1/f 299.1528128
CLARKE 1866 a 6,378,206.4 m 1/f
294.97869821
GEODETIC REFERENCE SYSTEM 1980 - (GRS 80) a
6,378,137 m 1/f 298.257222101
WORLD GEODETIC SYSTEM 1984 - (WGS 84) a
6,378,137 m 1/f 298.257223563
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Image on left from Geodesy for Geomatics and GIS
Professionals by Elithorp and Findorff,
OriginalWorks, 2004.
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Map Projections
From UNAVCO site
hosting.soonet.ca/eliris/gpsgis/Lec2Geodesy.html
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Taken from Ghilani, SPC
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Conformal Mapping Projections

• Mapping a curved Earth on a flat map must address
  possible distortions in angles, azimuths,
  distances or area.
• Map projections where angles are preserved after
  projection are called conformal

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• http//www.cnr.colostate.edu/class_info/nr502/lg3/
  datums_coordinates/spcs.html

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• SPCS 27 designed in 1930s to facilitate the
  attachment of surveys to the national system.
• Uses conformal mapping projections.
• Restricts maximum scale distortion to less than 1
  part in 10 000.
• Uses as few zones as possible to cover a state.
• Defines boundaries of zones on county-basis.

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http//www.ngs.noaa.gov/PUBS_LIB/pub_index.html
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Secant cone intersects the surface of the
ellipsoid NOT the earths surface.
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d
Ellipsoid
c
d
c
cd lt cd
b
b
ab gt ab
a
a
Grid
Earth Center
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Bs Southern standard parallel (?s) Bn Northern
standard parallel (?n) Bb Latitude of the grid
origin (?0) L0 Central meridian (?0) Nb false
northing E0 false easting
Constants were copied from NOAA Manual NOS NGS 5
(available on-line)
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Zone constant computations
Latitude of grid origin
Mapping radius at equator.
Equations from NGS manual, SPCS of 1983 NOS NGS 5
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R0 Mapping radius at latitude of true projection
origin. k0 Grid scale factor at CM. N0Northing
value at CM intersection with central parallel.
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Convergence angle
Grid scale factor at point.
Conversion from geodetic coordinates to grid.
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Formulas converted to Matlab script.
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Grid to Geodetic Coordinates
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http//www.ngs.noaa.gov/TOOLS/spc.shtml
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• STARTING COORDINATES
• AZIMUTH
• Convert Astronomic to Geodetic
• Convert Geodetic to Grid (Convergence angle)
• Apply Arc-to-Chord Correction (t-T)
• DISTANCES
• Reduction from Horizontal to Ellipsoidal
• Elevation Sea-Level Reduction Factor
• Grid Scale Factor

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• N 3,078,495.629
• E 924,954.270
• N -25.13
• k 0.99994523
• Convergence angle
• 01-12-19.0
• LAPLACE Corr.
• -4.04 seconds

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Laplace correction

• Used to convert astronomic azimuths to geodetic
  azimuths.
• A simple function of the geodetic latitude and
  the east-west deflection of the vertical at the
  ground surface.
• Corrections to horizontal directions are a
  function of the Laplace correction and the zenith
  angle between stations, and can become
  significant in mountainous areas.

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Astronomic to Geodetic Azimuth

• ? F ?
• ? ? - (? / cos ?)
• a A- ?tan ?
• (?, ?) are geodetic coordinates
• (F, ?) are astronomic coord.
• (?, ?) are the Xi and Eta corrections
• (a, A) are geodetic and astronomic azimuths
  respectively)

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Grid directions (t) are based on north being
parallel to the Central Meridian.
Remember Geodetic and grid north ONLY coincide
along CM.
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Astronomic to Grid (via geodetic)

• ag aA Laplace Correction g
• 253d 26m 14.9s - Observed Astro
  Azimuth
• ( - 1.33s) - Laplace
  Correction
• 253d 26m 13.6s - Geodetic Azimuth
• 1 12m 19.0s - Convergence Angle
  (g)
• 254d 38m 32.6s - Grid azimuth
• The convention of the sign of the convergence
  angle is always from Grid to Geodetic.

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Arc-to-Chord correction d (alias t T)

• Azimuth computed from two plane coordinate pairs
  is a grid azimuth (t).
• Projected geodetic azimuth is (T).
• Geodetic azimuth is (a )
• Convergence angle (?) is the difference between
  geodetic and projected geodetic azimuths.
• Difference between t and T d, the
  arc-to-chord correction, or t-T or
  second-term correction.
• t a-? d

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Arc-to-Chord correction d (alias t T)
Where t is grid azimuth.
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When should it be applied?

• Intended for during precise surveys.
• Recommended for use on lines over 8 kilometers
  long.
• It is always concave toward the Central Parallel
  of the projection.
• Computed as
• d 0.5(sin ?3-sin ?0)(?1- ?2)
• Where ?3 (2 ?1 ?2)/3

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Compute magnitude of the second-term correction
from preliminary coordinates. It is not
significant for short sight distances (lt
8km) but The effect of this correction is
cumulative!
Azimuth of line from N Azimuth of line from N
Sign of N-N0 0 to 180 180 to 360
Positive -
Negative -
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Angle Reductions

• Know the type of azimuth
• Astronomic
• Geodetic
• Grid
• Apply appropriate corrections
• Angles (difference of two directions from a
  single station) do not need to consider
  convergence angle.
• Apply arc-to-chord correction for long sight
  distances or long traverses (cumulative effect).

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• N1 N (Sg x cos ag)
• E1 E (Sg x sin ag)
• Where
• N Starting Northing Coordinate
• E Starting Easting Coordinates
• Sg Grid Distance
• ag Grid Azimuth

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Reduction of Distances

• When working with geodetic coordinates use
  ellipsoidal distances.
• When working with state plane coordinates reduce
  the observations to the grid (mapping surface).

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Re is the radius of the Earth in the azimuth of
the line.
Lm is surface Le is ellipsoid
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For most surveys the approximate radius used in
NAD 27 (6,372,000 m or 20,906,000 ft) can be used
for Re.
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Reduce ellipsoid distance to grid
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Final reduced distance

• Measured distances are first corrected for
  atmospheric refraction and earths curvature.
• Distances reduced to ellipsoid.
• Distances reduced to grid by applying the
  combined factor (scale factor by elevation
  factor).

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EF at a point (numeric example)
Let R 6372000, h 48.98 EF R/(R h)
0.999992313 if we do not have h, compute it via
relationship N H
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Reduction of distances
D
h H N
h
H
S
N

REarth Radius 6,372,161 m
20,906,000 ft.
S D x ___R__ R h
S D x R H N
Earth Center
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D5 is the geodetic distance.
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REDUCTION TO ELLIPSOID

• S D x R / (R h)
• D 1010.387 meters (Measured Horizontal
  Distance)
• R 6,372,162 meters (Mean Radius of the Earth)
• h H N (H 2 m, N - 26 m) - 24 meters
  (Ellipsoidal Height)
• S 1010.387 6,372,162 / 6,372,162 - 24
• S 1010.387 x 1.00000377
• S 1010.391 meters
• If N is ignored
• S 1010.387 6,372,162 / 6,372,162 2
• S 1010.387 x 0.99999969
• S 1010.387 meters -- 0.004 m or about 1
  252,600

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REDUCTION TO GRID

• Sg S (Geodetic Distance) x k (Grid Scale
  Factor)
• Sg 1010.391 x 0.99992585
• 1010.316 meters

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COMBINED FACTOR

• CF Ellipsoidal Reduction x Grid Scale Factor
  (k)
• 1.00000377 x 0.99992585
• 0.99992962
• CF x D Sg
• 0.99992962 x 1010.387 1010.316 meters

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STATE PLANE COORDINATE COMPUTATION

• N1 N (Sg x cos ag)
• E1 E (Sg x sin ag)
• N1 4,103,643.392 (1010.277 x Cos 253o 30
  07.4)
• 4,103,643.392 (1010.277 x -
  0.28398094570069)
• 4,103,643.392 (- 286.899)
• 4,103,356.492 meters
• E1 587,031.437 (1010.277 x Sin 253o 30
  07.4)
• 587,031.437 (1010.277 x -
  0.95882992364597)
• 587,031.437 (- 968.684)
• 586,062.753 meters

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• I WANT STATE PLANE COORDINATES RAISED TO GROUND
  LEVEL

• GROUND LEVEL COORDINATES ARE NOT STATE PLANE
  COORDINATES!!!!!

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PROBLEMS WITH GROUND LEVEL COORDINATES

• RAPID DISTORTIONS
• PROJECTS DIFFICULT TO TIE TOGETHER
• CONFUSION OF COORDINATE SYSTEMS
• LACK OF DOCUMENTATION

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GROUND LEVEL COORDINATESIF YOU DO

• TRUNCATE COORDINATE VALUES SUCH AS
• N 13,750,260.07 ft becomes 50,260.07
• E 2,099,440.89 ft becomes 99,440.89
• AND

DOCUMENT DOCUMENT DOCUMENT !!
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GOOD COORDINATION BEGINS WITH GOOD COORDINATES
GEOGRAPHY WITHOUT GEODESY IS A FELONY
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• The Universal Grids Universal Transverse
  Mercator (UTM) and Universal Polar
  Stereographic (UPS) - TM8358.2
• Transverse Mercator Projection
• Zone width 6o Longitude World-Wide
• Northing Origin (0 meters- Northern Hemisphere)
  at the Equator
• Easting Origin (500,000 meters) at Central
  Meridian of Each Zone