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NYSAPLS 2013 Conference

January 2013

Joseph V.R. Paiva, PhD, PS, PE

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n  Why do we have it anyway? n  Is it those darn software

manufacturers? n  Why can’t we have the good old

“ground” days? n  Dealing with the grid is so expensive

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n  Surveys: limited scope vs. large extent n  Plane surveys vs. geodetic n  Making large surveys easier n  Solving the GPS and total station raw

data mixing problem n  SPCS Calculations n  Strategies for dealing with the grid/

ground “thing”

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n  Most small-area surveys: assume the earth is flat (plane surveys)

n  For large areas, Earth’s curvature has to be considered

n  This usually involves determining geodetic positions (latitude and longitude) of survey stations

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n  SPCS was designed in the early 1930s by Coast and Geodetic survey to solve the problem of surveys of large extents for the “local” surveyor

n  In addition to allowing plane survey concepts to be used for surveys of large extent, it delivers several additional benefits

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n  Simplifies calculations for surveys over large distances

n  Provides common datum of reference for all surveys (if tied in)

n  Well suited for engineering projects of large extent, i.e. highways, but also photogrammetry, large scale cadastral surveys, etc.

n  Supplies vital grid for GIS (other options exist)

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n  When surveys are tied into the SPCS, their locations become (potentially) indestructible

n  With GPS, the problem of what coordinates to use once geocentric coordinates of GPS have been transformed in geodetic coordinates makes SPCs a natural choice

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n  The basic problem with plane surveying is that it assumes the earth is flat

n  Some problems…

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n  meridians converge

Parallel is curved

except at equator

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n  On the Earth, “straight lines” are not straight except for meridians (or the equator) and the difference gets larger as you extend them

N

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n  Changes in elevation cannot be ignored, that is why all geodetic distances are at “sea level”

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n  To have a plane coordinate system, it is necessary to distort the curved surface of the earth to a fit on a plane

n  Orange peel analogy n  This process of flattening

must be systematic in order to have accuracy

n  In surveying this process is called a projection

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n  Systematic way to portray (curved) surface of the earth on a flat surface

n  Different projections are used because each minimizes distortion in some properties at the expense of others

n  Again, distortions inevitable

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n  Different mathematical treatments are given to projections depending on the result desired

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n  A shape that can be made into a plane • Cone • Cylinder • Plane (of course)

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n  Cylindrical

Tangent

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A

B

C

D

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Cylinder

Sphere

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n  Conic

Secant

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n  Varying central apex angle of cone changes section of ellipsoid that is intersected

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n  Planar n  This type of projection

is created when surveyor sets up arbitrary coordinate system for a survey

n  Plus…many miscellaneous

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n  Once developable surface parameters are picked, plane is created

n  Because a developable surface is used, while there are distortions in converting coordinates on the earth to the developable surface, there is no further distortion of shape or size when it is unrolled or “developed”

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n  Lambert conformal (conic) n  Transverse Mercator (cylinder) n  Also…skewed (or oblique or Hotine)

Mercator where axis of cylinder is not east-west

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n  System for specifying geodetic stations using plane rectangular coordinates

n  Over 120 zones for U.S. n  Long N-S states use

Transverse Mercator n  Long E-W states use Lambert n  If square, use either

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n  Alaska, Florida and New York use both types of projections

n  In addition Alaska has an oblique projection for the southeastern part of the state

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n  Coordinate values changed (N and E) n  Meters n  Types of projections changed for

some states n  Zones different in some n  Numbers of zones per state changed

in some

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n  U.S. Survey foot = [m] x 3937/1200

n  International foot = [m] / 0.3048

n  2 PPM! n  [0.01 ft in a mile] n  [but with a coord value of

500,000 m, difference is 1 m!]

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n  NOAA Manual NOS NGS 5 State Plane Coordinate System of 1983

n  http://www.ngs.noaa.gov/ n  [www.ngs.noaa.gov/PUBS_LIB/ManualNOSNGS5.pdf]

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n  Scale is exact where cone or cylinder intersects ellipsoid surface

n  Scale is less than one between lines of true scale (i.e. length on ellipsoid is greater than length on plane)

n  Scale is more than one outside lines of true scale (i.e. length on ellipsoid is smaller than length on plane)

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n  Where the zone intersects the Earth, and whether it is tangent or secant controls the distortions

n  By strategic placement, distortions are minimized, scale differences can be kept to 1:10,000 or less

n  Done by keeping zone size to <158 mi and keeping zone width such that two-thirds of the zone is between lines of true scale (secant lines)

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n  Zones are designed to overlap each other considerably

n  Thus a survey done near a zone boundary can be done in either zone

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n  Also conformal n  Scale varies east to west but not

north to south n  Scale is true at the secant line n  All geodetic meridians are curved,

converging at the pole

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n  All parallels (of latitude) are curved n  CM is assigned to a meridian line n  All lines on the plane parallel to the

CM are grid north n  East-west lines on the plane are

perpendicular to the CM

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A

B

C

D

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Cylinder

Sphere

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A

B

C

D

Scale greater

than true

Scale greater

than true

Scale less than true

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Cylinder

View perpendicular to axis of cylinder

Scale greater than true

Scale less than true

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n  Also called grid declination or variation

n  Greek letter - γ [gamma]

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CM (Central Meridian) E0

Item Value Zone New York E (3101) also NJ Type Transverse Mercator Central Meridian 74° 30’ W * Grid origin latitude 38° 50’ N * Grid origin longitude 74° 30’ W * Grid origin X coordinate -easting 150,000 m Grid origin Y coordinate- northing 0 m Scale at central meridian 1:10,000 *

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Item Value Zone New York C (3102) Type Transverse Mercator Central Meridian 76° 35’ W Grid origin latitude 40° 00’ N Grid origin longitude 76° 35’ W Grid origin X coordinate -easting 250,000 m Grid origin Y coordinate- northing 0 m Scale at central meridian 1:16,000

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Item Value Zone New York W (3103) Type Transverse Mercator Central Meridian 78° 35’ W Grid origin latitude 40° 00’ N Grid origin longitude 78° 35’ W Grid origin X coordinate -easting 350,000 m Grid origin Y coordinate- northing 0 m Scale at central meridian 1:16,000

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Item Value Zone Lambert (3104) Type Lambert Conformal Central Meridian 74° 00’ W Standard parallel N 41° 02’ N Standard parallel S 40° 40’ W Grid origin latitude 40° 10’N * Grid origin longitude 74° 00’W Grid origin X coordinate -easting 300,000 m Grid origin Y coordinate- northing 0 m Scale at central meridian 1:16,000

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N

E

E0

EP’

EP

Distance from Central Meridian

CM

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Local Topography

Geoid

Ellipsoid

Mass Deficiency

Mass Excess

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Rm = 20,906,000 ft or 6,372,000 m n  Approximate SLF can be

calculated for project where relief is small

n  In high relief areas need to calculate individually using average elevation of the line

....

...

..

DistSurfSLFDistGeod

DistGrndElevRRDistGeod

m

m

×=

×+

=

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n  k is sometimes called SF (scale factor)

n  k is calculated from equations or interpolated from tables in state or NOAA documents

kDistGeodDistGrid ×= .

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n  “k” based on longitude (EP’) n  A single Scale Factor (SF), can be

picked for projects that are not large (under ~8 km)

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Grid Dist = Surf Dist x SLF x SF n  If average elevation and E for the

project are being used, multiply SLF and SF and use it as the Grid Factor (GF)

n  Grid factor also sometimes called “Combined Scale Factor” (CSF)

n  SF converts from geodetic to grid n  GF converts from ground to grid

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Grid Az = Geod Az - γ + Second Term

n  For most surveys Second Term can be ignored (lines under 8 km)

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«

γ Geodetic azimuth

Grid azimuth

«

γ Geodetic azimuth

Grid azimuth

Central Meridian

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n  Varies with longitude but can use same γ for many surveys

LaPlace correction may need to be added if using astro-azimuths

).sin()..( StaLatStaLongCMLong ×−=γ

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n  adjust traverse n  determine SLF using elevation

(either for project or dist. by dist.)

n  determine SF using dist. from CM (either for project or dist. by dist.)

n  calc. GF if desired n  convert all distances to grid

distances using GF n  convert all azimuths to grid

azimuths

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n  Assuming one of the traverse points has a known SPC, calc of the coordinates (SPC) of the other points is straightforward

n  Always multiply distances n  NEVER multiply coordinates!

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n  Conformal: true angular relationships are maintained around all points in small regions

n  Scale varies north to south but not east to west

n  Secant lines, where scale is true, are called standard parallels

n  All geodetic meridians are straight, converging at the pole

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n  All parallels (of latitude) are arcs of concentric circles have their center at the apex

n  CM is assigned to a meridian line n  All lines on the plane parallel to the

CM are grid north n  East-west lines on the plane are

perpendicular to the CM

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n  Varying central apex angle of cone changes section of ellipsoid that is intersected

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Standard parallels Pr

ojec

tion

limits

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n  Also called grid declination or variation

n  Greek letter - θ [theta]

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n  Same as for Transverse Mercator except…

n  Tables for the zone have the value

of the long. of the CM and l n  General pattern for calcs is the

same

lStaLongCMLongAzGeodAzGrid

×−=

−=

.)..(....

θθ

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n  Elevation 0 m; ground dist = 1000 m

n  Elevation 1,000 m; ground dist = 1000 m

....

...

..

DistSurfSLFDistGeod

DistGrndElevRRDistGeod

m

m

×=

×+

=

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n  Assume distance from CM is 30,000 m (doesn’t matter whether east or west)

n  Enter table and pick off value for 30,000 m: 0.9999444

n  If not a round number, will have to interpolate!

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n  Dist from CM = 31,457 m n  Table value for 30,000: 0.9999444 n  Table value for 31,500: 0.9999455 n  Difference (sometimes tabulated):

0.0000011 n  SF?

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n  GF = SLF x SF n  Also called Combined Scale Factor

(CSF)

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n  Sta. Long. = 93°00’00” n  CM = 92°30’00” n  Sta. Lat = 38°00’00”

).sin()..( StaLatStaLongCMLong ×−=γ

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n  Tie in to monuments with SPCs, therefore don’t need to calculate mapping angle

n  Project coordinates sometime used—be careful!

n  On plats show SPCs. If you must show ground distances, show grid distances also!

n  Meta data!

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n  How to use? n  My suggestion: use all grid (coordinates)

or all ground (distances) n  If all ground distances, publish a table of

grid coordinates of all the points n  If all grid coordinates, publish a table of all

ground distances and, if desired, azimuths/bearings on non-grid basis

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n  DO NOT publish “ground coordinates” unless X and Y values are readily differentiable

n  On the plat if you show ground values and grid values use a suffix or prefix (GRID & ground)

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n  If you have to, use different fonts or different styles (regular vs. italics)

n  But make sure they can be easily differentiated

n  Do NOT use different colors to differentiate!

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n  A point is a point n  It doesn’t matter whether it is on the

plane (grid), ellipsoid or surface n  Do some work on a survey nearby…hand

calculate grid or ground values…then see if your data collector and PC software handle correctly

n  Need to have fairly long distances to see differences between grid and ground (figure PPM to know how long)

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n  Using a data collector do the math is OK, as long as it does it correctly

n  Do NOT multiply or divide coordinates by scale factor!

n  Remember: GIGO

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