Control Surveying

1
Control Surveying

• CE 363
• Roy Frank Jr.
• Assistant Professor

2
Geodetic Surveying

• Geodesy is the science of measuring the size and
  shape of the earth
• Geodetic Surveying is when all distances and
  horizontal angles are projected onto the surface
  of the spheroid which represents mean sea level
  on the earth.
• Geodetic Position referenced to geodetic latitude
  and longitude
• Latitude angle between the direction normal to
  spheroid and at station and the plane of the
  equator.
• Longitude angle between the meridian of origin
  (Greenwich) and the meridian through the station.

3
Direction Geodetic Azimuths

• NGS expresses azimuths in terms of clockwise
  angle from the south.
• Geodetic azimuth astronomic azimuth and laplace
  correction normally only a few seconds and this
  covers a large area
• Laplace correction is supplied by NGS upon
  request for any area
• When requesting you must supply approximate
  longitude and latitude

4
Direction (contd)

• Available Data 2 major sources
• IDOT
• Most major highways have permanent monuments with
  locations identified these can be used.
• NGS (USGS) - Rockville, Maryland
• Nationwide system of stations, normally within 10
  miles of each other -TRIANGULATION STATIONS
• Latitude, longitude and elevation are given with
  a description
• Azimuth-azimuth between stations given
• Azimuth between station and existing structure

5
Definitions

• Ellipsoid Flattened earth at the poles that
  approximates mean sea level
• Geoid undulating surface based upon earths
  magnetic field
• Heights
• Ellipsoid height (h) distance along plumb line
  from earths surface to ellipsoid
• Geoid height (N) distance along plumb line from
  ellipsoid to geoid
• Orthometric height (H) distance along plumb
  line from earths surface to geoid

6
NGS Minimum Standards

• Old NGS Standards for error in surveys are
  distance based
• First Order - 1100,000
• Second Order
• Class I 1 50,000
• Class II 1 20,000
• Third Order
• Class I 1 10,000
• Class II 1 5,000

7
1988 Geometric Accuracy Standards
Survey Category Order Base
Error Dependent cm ppm Error
Global Regional Geodynamics AA 0.3 0.01 1100,000,000
National Geodetic Reference Network (Primary) A 0.5 0.10 110,000,000
National Geodetic Reference Network (Secondary) B 0.8 1.00 11,000,000
National Geodetic Reference Network Terrestrial Based C below below
First Order Second Order 1.0 10.00 1100,000
Class I Class II Third Order 2.0 3.0 5.0 20.00 50.00 100.00 1 50,000 1 20,000 1 10,000
8
National Map Accuracy Standards

• Horizontal Specification
• Map produced at scales larger than 120,000
• On smaller scale maps, the limit of horizontal
  error is 1/50 inch (0.5 mm).
• Vertical Specification
• Not more than 10 of elevations tested shall be
  in error by more than one-half the contour
  interval and none can exceed the interval

9
Vertical Accuracy Classes

• First Order
• Class I 4 mm error
• Class II 5 mm error
• Second Order
• Class I 6 mm error
• Class II 8 mm error
• Third Order 12 mm error

mm Error C/v(K) C circuit error in mm K
length in KM
10
Specification for Local Horizontal Control Surveys

• Control Surveys are needed for most accurate
  mapping projects
• Most are tied to the national geodetic network
• Advantages
• Monuments that are destroyed can be easily
  relocated
• Adjacent, but unconnected survey will be in
  correct position
• Provides excellent check to work
• This has become much more feasible since the
  advent of EDM

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• Control Surveys have two categories Horizontal
  and Vertical
• Triangulation and Trilateration
• Traverse (Precise)
• Trilateration
• Vertical is then divided by method
• Spirit Leveling most accurate
• Trigonometric leveling

12
Reconnaissance

• When planning a control survey, a field recon is
  needed
• Can show simpler solutions
• Which monuments are there

13
Monumentation Description

• Control monument set to be in place permanently
• Must be thoroughly described and referenced
• Do not use same type of monument for reference
  point as control monument
• Prior to using a previously established monument
  check its location prior to starting based on
  references

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Control survey minimum standards
First and Second order horizontal and vertical
control Purpose To prescribe standards and
specifications that will provide accurate
horizontal and vertical positioning
Definitions

• Positional Accuracy of a station is the accuracy
  of the station related to the reference stations
  that are held fixed National Geodetic Survey or
  other higher order stations in the process of the
  adjustment.
• Relative Accuracy is the relative position of one
  station with respect to another station.
• Both are computed from the constrained correctly
  weighted, least squares adjustment at the 95
  confidence level.

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Accuracy of Horizontal Control

• Acceptable accuracy of 1st and 2nd order control
• The accuracy of a horizontal control station is
  classified according to constrained and
  unconstrained relative accuracy of the distance
  between the stations and the positional accuracy
  of the station relative to the known stations.
• FIRST ORDER HORIZONTAL CONTROL
• The relative accuracy of the distance between
  directly connected adjacent points shall be equal
  to or less then 25 mm for distances equal to or
  less than 1 km and 10 ppm for distances greater
  then 1 km
• The positional accuracy of a station shall be 30
  mm in urban areas and 60 mm in rural areas.

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• SECOND ORDER HORIZONTAL CONTROL
• The relative accuracy of the distance between
  directly connected adjacent points shall be equal
  to or less than 25 mm for distances equal to or
  less than 1 km and 20 ppm for distances greater
  than 1 km.
• The positional accuracy of a station shall be 60
  mm in urban areas and 100 mm in rural areas.

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GPS Survey Guidelines

1. Direct connections must be made to any adjacent
   observable National Geodetic Reference System
   station located 5 km or less from any new station
2. At least three existing higher or equal order
   control points must be included in any proposed
   GPS survey. Whenever possible, these should be
   three 3D control points. Otherwise two sets of 3
   points (three 2D horizontal points and three
   vertical control points) must be used. These
   control points should be chosen to be roughly
   equidistance on the periphery of the network so
   that they enclose as much of the proposed network
   as possible.

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1. Each new point to be established by the proposed
   GPS survey must be occupied as least 2 separate
   times to enable proper checking of blunders (ie,
   incorrect path, setup errors, incorrect antenna
   heights). A separate occupation is one where the
   antenna has been taken down and set up again and
   the receiver restarted.
2. Each point must be connected by simultaneous
   occupations to at least three other points in the
   network after outer base lines have been rejected
   from the adjustment. Because it is generally
   easier to resolve the integer phase ambiguities
   over shorter base line, adjacent points should be
   connected wherever possible.
3. At least two receivers must be used for relative
   positioning, although three or more may be used
   for more efficient operation and increased
   station reoccupation and base line repeatability.

19

• A pre-analysis should be performed to determine
  the minimum occupation time required to achieve
  the required standard of accuracy. In addition,
  the most appropriate satellites to observe at
  each site should also be selected for receivers
  unable to track all of the visible satellites.
  The pre-analysis should be specific for carrier
  phase relative positioning.
• In order to meet second order accuracies, the
  carrier beat phase must be observed together with
  a time tag for each observation. Pseudo-range
  observations are not precise enough for control
  surveys and cannot be used.

20

• A detailed field log must be kept during
  observations taken at each station. At a minimum,
  the following must be recorded
• Universal Time Correction (UTC) date of
  observations
• Station identification (name and number)
• Session identification
• Serial numbers of receiver, antenna and data
  logger
• Receiver operator
• Antenna height and offset from monument, if any
  to 1 mm
• Diagram illustrating stamping on monument
• Most common error
• Other stations observed during session
• Starting and ending time (UTC) of observations
• Satellites observed

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1. The raw data files for all station occupations
   must be kept. Each file called an R-file, will
   consist of one set of raw observations for each
   station occupation session.
2. The unadjusted base line vector solution files
   for all observed base lines, non-trivial and
   trivial, will be kept
3. Station descriptions must include station name,
   county, township, range, sections, USGS 7.5 quad
   name, date monumented, date of observations,
   complete descriptions of the station, azimuth,
   and all reference monuments, a current to reach
   description, property owners name, address
   phone number. A sketch depicting the station and
   reference marks with dimensions and directions
   shown should accompany all narrative data.
4. If the GPS survey project includes any surveys
   using conventional horizontal surveying
   techniques, copies of all field notes, and
   associated data must be kept. Also, when the
   survey includes conventional differential
   leveling, copies of the field notes and
   associated data must be kept.

22

1. When the GPS survey project includes surveys
   using conventional or terrestrial horizontal
   surveying techniques, copies of all field notes
   and associated data must be submitted. This
   would include eccentric point establishment and
   reduction. Polaris, solar, or direct
   observational data to establish azimuth marks
   shall also be submitted.
2. A tabulation of the results of the repeat
   baseline comparisons will be included in the
   project report.
3. A minimally constrained (free) least squares, 3D
   adjustment will be submitted in the form of the
   input and output files.

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Traverse Survey Guidelines

• First Order Traverse Procedure
• The location of first order traverse lines and
  monumented stations shall be determined by a
  thorough field reconnaissance. Traverse point
  spacing shall not be less than 600 meters.
• All first order traverse lines shall start from,
  and close upon first order stations of the
  National Geodetic Reference System in accordance
  with these procedures.

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1. Properly maintained theodolites with a least
   count of one second shall be used to ovserve
   directions and azimuths. At least four positions
   or repetitions of the angles shall be observed.
   The theodolite and targets should be centered to
   within two mm over the survey station or traverse
   point.
2. Electronic distance measuring instruments shall
   be used to measure all distances. EDM instruments
   shall be tested on a certified baseline at the
   start and on the completion of any first or
   second order traverse. Barometric pressure to
   plus or minus five mm and temperature to one
   degree celsius shall be recorded for each
   measurement.

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• Each traverse shall be tied to a minimum of two
  benchmarks. Trig or spirit leveling will be
  observed along all traverse lines. All HIs,
  HOs and zenith angles will be recorded and
  submitted.
• The traverse shall be controlled by an astronomic
  azimuth at each end of the traverse line and at
  not more than every six segments along the line.
  Astronomic azimuths shall have a standard
  deviation of one and one-half second or better.
• All field data shall be submitted to DNR in a
  format acceptable to the Department. This shall
  include directions, distances, azimuth and
  elevations

26

• Second Order Traverse Procedure
• The location of second order traverse lines and
  monumented stations shall be determined by a
  thorough field reconnaissance. The traverse
  point spacing shall not be less than 300 meters.
• A second order traverse line shall start and
  close upon second order or higher stations of the
  NGRS in accordance with these procedures.
• Properly maintained theodolites with a least
  count of one second shall be used to observe
  directions and azimuths. At least 4 positions or
  repetitions of the angles shall be observed. The
  theodolite and targets shall be centered to
  within 2 mm over the station.

27

1. EDM instruments shall be used to measure all
   distances and shall be tested on a certified
   baseline at the start and completion of any
   traverse. Barometric pressure to /- 5 mm and
   temperature to one degree celsius shall be
   recorded for each measurement
2. Sections E and F from First Order Traverse
   Procedure shall be met.
3. The Traverse shall be controlled by an astronomic
   azimuth at each end of the traverse line and at
   not more than every eight segments along the
   line. Astronomic azimuth shall have a standard
   deviation of 2 seconds or better.

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Horizontal Control Nets

• Original Technique Triangulation
• Precise measurement of one side of triangle as a
  baseline and determine each angle of the triangle
• Addition of EDMs Trilateration
• Solution of triangle by measuring all sides
• Combination of two techniques provides the option
  for the most accurate figures

29
Several Options

• Chain of single Triangles
• Chain of Double triangles (quadrilaterals)

30
Control Points should be Located Based On

• Good visibility to other control points and an
  optimal number of layout points
• Visibility must exist for exiting conditions but
  also potential visibility during all phases of
  construction.
• Minimum of 2 (prefer 3) reference ties for each
  control point
• Control Points placed at locations not to be
  affected by primary or secondary construction
  activity
• Control points must be established on solid
  ground.
• Once locations are selected, plot to determine if
  solid geometrics exist.

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Positional Accuracies

• Primary Control Stations
• Permissible deviations when measuring the
  position of primary points and those calculated
  from adjusted coordinates can not exceed
• Dist /- 0.75(vL) in mm
• Ldistance between station in meters
• Angle /- 0.045/(vL) in degrees
• L shorter side of angle
• Permissible deviations when checking the
  positions of primary points
• Distance /- 2(vL) in mm
• Angle /- 0.135/(vL)

32

• Secondary Control Stations
• Permissible deviation for checked distance from
  given or calculated distance between primary and
  secondary point shall not exceed
• Distance /- 2(vL) in mm
• Permissible deviation for checked distance from
  given or calculated distance between two
  secondary points in same system
• Distance /- 2(vL) in mm
• Permissible deviations for checked distance from
  given or calculated distance between 2 points in
  different secondary systems for same project
• Dist /- K/(vL) in mm
• LDistance in meters
• K Constant

33
Control Survey Markers

• Used for
• State or Provincial Coordinate Grids
• Property or Boundary Delineation
• Project Control

34
Control Survey Markers (contd)

• Type of marker varies dependent on
• Type of soil or material at site
• Degree of permanence required
• Cost of replacement
• Precision required
• Key is horizontal vertical Stability
• Most popular marker types
• Property Markers iron rods (1/2 1 diameter)
• Construction Control Rebar (1/2 5/8) with
  or without caps and concrete monuments with brass
  caps
• Control Surveys bronze tablet markers, sleeve
  type survey markers and aluminum break-off markers

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Control Surveying

• Major network to which smaller surveys tied
• Network must be permanent and higher level of
  precision at least one level above project
  needs
• Generally networks are based upon North American
  Datum

37
Horizontal

• NAD 1927 based on Clarkes Spheroid of 1866
• Uses equatorial radius of 6378206.4 M
• NAD 1983 based on GRS 80 ellipsoid
  (Earth-mass-centered ellipsoid)
• uses an equatorial radius of 6,378,17 M
• Most NAD 83 stations are 1st Order
• 1200,000 or better

38

• Old Standards
• First Order 1 100,000
• Second Order
• Class I 1 50,000
• Class II 1 20,000
• Third Order
• Class I 1 10,000
• Class II 1 5,000
• Worker well until GPS electronic systems

39

• New Standards
• 1988 Geometric Geodetic Accuracy Standards
• The new primary secondary comprise the HARN
  (High Accuracy Reference Network)
• Monuments spaced at 25 km 125 km
• Established by Static GPS w/ multiple 5-8 hr
  session
• Data available NGS Information Center/ U.S.
  Coast Guard GPS Information Center
• 300,000 control monuments
• Indiana completed system in 1997 with 16,000
  stations
• Accuracy emphasis changed from distance to
  position with absolute positioning

40
Vertical

• Old NAVD 29
• New North American Vertical Datum 1988
• 600,000 benchmarks in U.S. and Canada

41

• Original Triangulation - because basic
  measurement and angles could be done more
  precisely than distances
• Since EDMs
• Combined triangulation trilateration
• Precise Traverse
• GPS

42
First and Second Order Horizontal and Vertical
Control

• Positional Accuracy of a station is the accuracy
  of the station related to the reference stations
  that are held fixed NGS or other higher order
  stations in the process of the adjustment.
• Computed from the constrained, correctly
  weighted, least squares adjustment at the 95
  confidence level.
• Relative Accuracy is the relative position of one
  station with respect to another station.
• Rural Area any 2nd, 3rd, or 4th class county
  according to 48.020 RSM
• Urban Area

43
Accuracy of Horizontal Control

• Accuracy of a horizontal control station is
  classified according to constrained and
  unconstrained relative accuracy of the distance
  between the stations and the positional accuracy
  of the station relative to the known stations
• First Order Horizontal Control
• Equal to or less than 12 mm or 10 ppm
• Pos Accuracy 30 mm urban 60 mm - rural
• Second Order Horizontal Control
• Equal to or less than 25 mm or 20 ppm
• Pos Accuracy 60 mm urban 100 mm - rural

44

• Standard deviation provides an indication of
  the precision of a simple value with respect to
  other values of the same origin
• Formula v(sum or squares of residuals/(n-1))
• Mean average of all values
• Angles Set of 8

45
Information on Stations available on CD
(www.ngs.noaa.gov)

• Systems provide information in both latitude and
  longitude and coordinate systems
• Latitude and longitude angles must be expressed
  to 4 decimals of a second to give position to the
  closest 0.01 feet.
• Lat angle of equator to point N or S (f, phi)
• Long angle E or W of Greenwich (?, lambda)
• At 44 degrees lat (1 latitude 101 and 1
  longitude 73 feet
• Carbondale at lat 37-43-45 N long 89-12-30 E

46
Coordinate Systems

• Universal Transverse Mercator GRID System (UTM)
• Projection placing cylinder around earth with
  its circumference tangent to the earth along a
  central meridian
• With projections, scale is exact at central
  meridian and becomes more distorted as distance
  increases from central meridian
• Distortion is minimized 2 ways
• By keeping the zone width relatively narrow
• By reducing the radius of the projections cylinder

47

• Characteristics of UTM Grid System
• Zone is 6 wide. Zone overlap of 0 30
• Latitude of Origin is the Equator - 0
• Easting Value of Central Meridian 500,000 M
• Northing Value of Equator 0.00 M Northern
  Hemisphere 10,000,000 M Southern
  Hemisphere
• Scale factor at the central meridian is 0.9996
• Zone numbering commences with 1 in the zone 180W
  to 174W and increases eastward to zone 60 at
  zone 174E to 180E.
• Projection limits latitude 80S to 80N

48

• Utilized extensively for large mapping and by
  Geographers
• Proponents say advantages of uniform worldwide
  grid are
• Eliminates confusion resulting from different
  grid coordinate axes in the same area.
• Permits quick ground data correlation between
  neighboring or distant government agencies
• Permits a uniformity in maps and map referencing
• Facilities worldwide or continental data base
  sharing

49
State Plane Coordinate Systems

• Devised by U.S. Coast and Geodetic Survey in 1933
• Each state has its own system
• Transverse Mercator Projection (cylindrical)
• Relatively distortion free North-South
• Lambert Projection conical
• Relatively distortion free East-West

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• Basis of system is a mathematical projection of
  the earths surface like that of a cylinder or
  cone which is then developed into a plane
• NGS computes And publishes the plane coordinate
  on the appropriate state system for all points it
  determines positions
• Only one point to a pair of coordinates for a
  zone of any state
• An engineer of surveyor who ties a well executed
  survey to the national network can perform the
  surveying calculations utilizing the normal
  office procedures of plane surveying

53
Advantages of State Plane Coordinate Systems

• Positive checks can be applied to all surveys to
  prevent accumulation of errors in the measurement
  of angle and direction.
• Tie back into control Station networks
• Surveys that are done at widely separate
  locations within a zone will have the same
  directional and coordinate systems, thus tied
  together
• ANY STATION WHOSE STATE PLANE COORDINATES HAVE
  BEEN DETERMINED IS PERMANENTLY LOCATED
• Destroyed monuments can be relocated
• THERE IS NO FORM OF FIELD SURVEYING CALLED STATE
  COORDINATE SURVEYING AND NO STATE COORDINATE
  STATION.

54
To have maximum utility, a state plane coordinate
system must

1. X Y coordinates of a survey point should be
   readily obtainable from latitude and longitude
   and vise versa
2. Forward and back azimuths must differ by 180
3. Length of survey line calculated from grid coord.
   Must be equal to ground distance or some method
   to convert must exist
4. Angular relationship should be retained angle on
   grid must angles on ground
5. Should cover as large an area as possible

55
Lambert Projection(Lambert Conformal Proj.)

• Lambert best suited for areas narrow in N-S
  direction and long in E-W direction
• Uses a cone with axis coinciding with axis of
  Earth
• Cone slices earths surface 2 lines are
  standard parallels
• Section of the cone as developed into a plane
• Central meridian has known longitude and
  longitude at any point given with respect to it
• Latitude of standard parallels known and points
  developed from these

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• Characteristics of Lambert Projection
• Scale exact along standard parallels
• Scale to large outside standard parallels
• Scale to small between standard parallels
• Parallels of latitude are curved lines and
  perpendicular to meridian at all points
• Meridians appear as straight lines converging
  toward central meridian at all points
• Projection can be extended E-W indefinitely with
  out problems with accuracy
• The closer together standard parallels, the more
  closely plane and Earths surface coincide and
  grid lengths are closer to ground lengths

58
Transverse Mercator Projection

• Consists of cylinder perpendicular to earths
  axis which cuts earths surface along 2 parallels

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• Characteristics
• Scale exact along 2 lines of intersection
• Scale too large outside lines, too small inside
• Straight lines (not latitude) which are
  perpendicular to central meridian of sphere and
  perpendicular to central meridian of grid
• Neither lines of latitude or longitude appear as
  straight lines except at central meridian
• Projection can be extended N-S indefinitely with
  out loss of accuracy
• Closer circles of intersect the less difference
  in surface and projection

62
Transverse Mercator Proj (contd)

• Best suited for areas narrow in E-W and long in
  the N-S
• Grid width maximum 158 miles, with most being
  narrower to minimize scale error
• Most systems have an X value of central meridian
  of 500,000 ft and a Y value at X axis of 0 ft.

63

• Formulas
• XP H??/-ab
• XP XP XC
• YP YO V(? ?/100)2 /- C
• YO, H, V and a are based on latitude state
  tables
• ?? longitude central meridian longitude
  point in seconds
• b and c related to ? state tables

64
Comp of Coord. Transverse Mercator

• Illinois has 2 zones
• East Central Meridian _at_ 88o20 W long.
• (Y axis) West Central Meridian _at_ 90o10 W
• X axis for both _at_ 36o40 N lat.
• East zone 88o20 W, 36o40 N
• 1927 y0.00 x500,000
• 1983 y0 M x300,000 M

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• EAST ZONE 88o 20W 36o 40N
• 1927 y0.00 x 500,000
• 1983 y0.00M x300,000M
• WEST ZONE 90o 10 W 36o 40 N
• 1927 y0.00 x500,000
• 1983 y0.00M x300,000M

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Station King

• Lat. 40o4337.202 IL East Zone
• Long. 88o4135.208
• ?? 88o4135.208 88o20 - 0o2135.208
• Value if longitude less than CM, - if greater
• ?? -0o2135.208 -1295.208
• (?/100)2 (-1295.208/100)2 167.756
• From lat. 40-43-37.202
• 37.202/60 .620033(Percent of 1)

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• H Page 12 Tables
• 40-4377.004565
• 40-4476.985354
• .019211 Absolute Difference
• .620033 X .019211 .011911
• 77.004565 - .011911 76.992654 H

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Calculate V Page 12 Tables

• 40-43 1.217865
• 40-44 1.217973
• 0.000108 Absolute Difference
• .000108X.620033 .000067
• 1.217865.000067 1.217932 V

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Calculate a Page 12 Tables

• 40-43 - .493
• 40-44 - .491
• -.002 Absolute Difference
• 0.002X.620033 .001240
• -.493 .00124 -0.492 a

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Calculate Yo Page 12 Tables

• 40-43 1474960.92
• 40-44 1481032.87
• 6071.95 Absolute Difference
• 6071.95X.620033 3764.8094
• 1474960.923764.80941478725.73 Yo

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Calculate b c Page 22 Tables

• Dl 1295.208
• 95.208/100 .95208 ( of 100)
• b 1200 2.384
• 1300 2.553
• 0.169 X .95208 .160902
• 2.384 .1609 2.545 b
• c 1200 - .032
• 1300 - .038
• .006 X .95208 .006
• -.032 .006 -.038 c

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Calculate Y YoV(Dl/100)2 /- c

• Yo 1,478,725.73
• V(Dl/100)2 204.32
  (167.756X1.217932) 204.32
• 1,478,930.05
• /- c - .04
• Y 1,478,930.01

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Calculate X X X H x Dl /- ab X X
500,000

• H x Dl 76.992654 x (-1295.208) - 99,721.50
• ab -.492 x 2.545
  - 1.25
• X
  - 99,720.25
• X 500,000 (-99,720.25)
  400,279.75
• If ab is negative, decrease H x Dl numerically
• Y1,478930.01 X400,279.75

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Uses of State Plane Coordinate Systems

• SPC provide all of the advantages of geodetic
  positioning without difficulties of geodetic
  computations
• Long. Centerline projects on SPC allows several
  crews to work in separate areas and data is still
  correlated.
• County and city mapping separate surveys are all
  tied together and provide checks
• Provide for accurate recovery of lost or
  destroyed monuments

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Grid Azimuths

• Projection lines from both systems (TMP LP) are
  grid because all north-south lines are parallel
  to central meridian and perpendicular to all E-W
  lines
• Only place grid and geodetic azimuth are the same
  is _at_ central meridian
• Difference between grid and geodetic azimuth
  becomes greater as distance from central meridian
  increases

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• Points west of central meridian, grid azimuth is
  greater than geodetic azimuth points east of
  central meridian, grid azimuth is less than
  geodetic azimuth
• This directional difference called mapping or
  convergence angle
• Expressed as ?, theta, in lambert and ?a, alpha,
  in transverse mercator
• For transverse mercator
• ??dif in long. from central meridian and P
• ?a ??sinFP
• FPlatitude point P

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• If distance from central meridian is known,
• ? 32.392 dk tan F
• ? 52.13 d tan F
• ?convergence angle in seconds
• ddeparture distance from central meridian in
  miles
• dKdeparture distance in kilometers
• F average latitude of the line

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Grid Distance

• Must convert ground distance to mean sea level
  (geodetic distance) or spheroid then convert
  geodetic distance to grid distance (plane
  projection)
• Step 1 Ground to geodetic multiply by elevation
  factor
• For areas 500-750 elev. Correction is minimal,
  also error of 500 in elevation will cause
  cumputational error of 1/41,800

80

• Elevation Factor
• EF x ground distance Geodetic distance
• EF 1- (elevation(avg)/20,906,000
• Scale Factor
• SF x Geodetic distance Grid Distance
• Scale factor based upon X value (Distance from
  Central Meridian)
• X Coordinate 500,000 X (Page 28 Tables)
• Surface Distance X EF Geodetic Distance X SF
  Grid Distance
• Grid Distance / SF Geodetic Distance / EF
  Surface Distance
• Elevation Factor X Scale Factor Grid Factor,
  thus
• Surface Distance X Grid Factor Grid Distance
• Grid Distance / Grid Factor Surface Distance

81
EXAMPLE Grid - Surface

• Point 101Z Y 403116.25 X 788232.55 EL
  410.35
• Point 109A Y 405316.19 X 787858.20 EL
  624.86
• Elevation Factor
• (410.35 624.86) / 2 517.605 Mean
  Elevation
• 1 (517.605 / 20906000) .999975241
  Elevation Factor
• Scale Factor
• (288232.55 287858.20) / 2 288045.375
  Average X
• From Page 28 285000 1.0000340
  3045.375 / 5000 .609075
• 290000 1.0000373
• 
  .0000033 X .609075 .0000020
• 1.0000340 .0000020 1.0000360 Scale
  Factor
• Inverse Grid Distance 2231.5631
• Grid Factor .999975241 X 1.0000360 1.00001124
  
• Surface Distance 2231.5631 / 1.00001124
  2231.5380

82

• Reduce field distance to grid distance (elevation
  factor based on elev. 750)
• Therefore X 500,000 468148 31852
• Elev Factor .9999642 Scale factor .9999424
• Grid factor .9999066

GROUND DISTANCE A-1 754.25 x .9999066 754.18 1-2
517.12 517.07 2-3 808.11 808.03 3-4 1617.63
1617.48 4-5 982.63 982.52 5-6 3165.07 316
4.77 6-7 2354.55 2354.33 7-8 3296.43 3296.12
8-B 1241.74 1241.62 Now Compute Coordinates
by Standard Methods
83
Astronomical Observations

• Measuring positions of sun or certain stars (most
  often polaris) to determine the direction of
  astronomic meridian
• Can be done to determine latitude and longitude
  of points
• Seldom used because field procedure and
  computations are time consuming and GPS allows
  quick longitude, latitude two point direction.

84

• Astronomic Meridian at any point it is a line
  tangent to and in the plane of the great circle
  which passes through the point, the earths north
  and south geographic poles
• Latitude angle between equatorial plane and
  ellipsoid normal to the point
• Longitude angle between Greenwich meridian and
  meridian through point

85
Ursa Minor
Polaris
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Simple methods of Determining the Meridian

• 2 methods that require no computations
• Shadow Method need straight pole and string
• Establish pole in smooth, level surface
• Mark end of shadow at 30 min intervals between 9
  am and 3 pm
• Marks are joined by smooth line
• Using string, scribe a radius about pole, such
  that radius intersects shadow arc twice
• Locate midpoint of intersects
• Line from pole through midpoint provides meridian
  /- 30

94

• Equal Altitude of Sun
• Requires use of instrument with sun lens and
  knowledge of time of sun path
• Instrument over point and bisect sun with both
  vertical and horizontal crosshair at 9 am
• Vertical angle read
• Lower scope and set point at about 500
• Shortly before 3 pm (6 hrs), set vertical angle
  in gun and follow sun until both cross hairs
  bisect (only remove horizontal)
• Again depress scope and set point at 500
• Split horizontal angle between two set points to
  get astronomic meridian
• /- 30 accuracy

95
Usual Procedure for Astronomic Azimuth
Determination

• Field
• Total station set up at one end of a line where
  azimuth is to be determined
• Back sight station at other end of line and 0 set
  gun
• Rotate instrument clockwise and sight celestial
  body
• Read and record horizontal and vertical angles to
  body
• Precise time of sighting recorded

96

• Office procedure
• Obtain precise location of celestial body at
  instant of sighting from an ephemeris (almanac of
  celestial body positions)
• Compute celestial bodies azimuth (Z, angle
  between body and north) based on observed and
  ephemeris data
• Calculate lines azimuth by applying measured
  horizontal angle to computed azimuth

97

• Accuracies
• Variables
• Precision of instrument
• Ability and experience of observer
• Weather conditions
• Quality of clock or chronometer used to measure
  time
• Celestial body sighted and its position when
  observed
• Accuracy of ephemeris and other data available
• Polaris capable of /- 1, reality 5
• Sun /- 10, reality 15-20

98
Ephemeris

• Almanacs containing data on the positions of the
  sun and various stars versus time
• U.S. Bureau of Land Management _at_
  http//www.cadastral.com has data on Sun and
  Polaris
• Sokkia Celestial Observation Handbook and
  Ephemeris
• Published annually
• 800-255-3913 (Sokkia Corp.)

99

• The Apparent Place of Polaris and Apparent
  Sidereal Time
• U.S. Department of Commerce
• The Nautical Almanac U.S. Naval Observatory
• All values given for universal time (UT)
  (Greenwich Civil Time)

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101
Definitions

• Celestial Sphere infinite radius sphere with the
  earth as the center.
• Due to earths rotation on the axis, all
  celestial bodies rotate about the axis
• Zenith (Z) point where a plumb line projected
  upward meets the celestial sphere (overhead)
• Point on celestial sphere vertically above the
  observer
• Nadir point on celestial sphere vertically
  beneath the observer and exactly opposite zenith
• North Celestial Pole (P) point where the
  earths rotational axis extended from north
  geographic opine intersects celestial sphere

102

• South Celestial Pole (P) point where earths
  rotational axis extended from the south
  geographic pole intersects celestial sphere
• Great Circle any circle on the celestial sphere
  whose plane passes through the center of the
  sphere
• Vertical Circle any great circle of celestial
  sphere passing through zenith and nadir
• Celestial Equator the great circle of the
  celestial sphere whose plane is perpendicular to
  the axis of rotation of earth
• Earths equator enlarged in diameter

103

• Hour Circle any great circle on celestial
  sphere which passes through the north and south
  celestial poles
• Perpendicular to plane of celestial equator
• Meridians (longitudinal lines)
• Used to measure hour angles
• Horizon a great circle on the celestial sphere
  whose plane is perpendicular to the direction of
  the plumb line
• Celestial (Local) Meridian the hour circle
  containing the observers zenith

104

• Diurnal Circle the complete path of travel of
  the sun or a star in its apparent daily orbit
  about the earth
• Lower Culmination bodys position when it is
  exactly on lower branch of celestial meridian
• Eastern Elongation where body is farthest east
  of the celestial meridian with its hour circle
  and vertical circle perpendicular
• Upper Culmination
• Hour Angle exists between meridian of reference
  and hour circle passing through celestial body
• Measured westward from meridian of reference

105

• Greenwich Hour Angle of a heavenly body at any
  instant of time is the angle measured westward
  from meridian of Greenwich to meridian over which
  the body is located (GHA)
• Local Hour Angle angle measured westward from
  observers celestial meridian to meridian of
  heavenly body
• Meridian Angle like local hour angle, except
  measured either east or west from observers
  meridian (value always between 0 and 180)
• Declination angular distance measured along the
  hour circle between the body and the equator
• Denoted by d (delta)
• Positive when north of equator and negative when
  south
• Polar Distance (Co-declination) of a body is
  90o minus declination

106

• Altitude angular distance measured along
  vertical circle between celestial body and
  horizon
• Denoted h
• Astronomical Triangle (P25) spherical triangle
  whose vertices are the pole (P), zenith (Z), and
  astronomic body (S)
• Azimuth (of a heavenly body) angle measured in
  the horizontal plane clockwise from either the
  north or south point to the vertical circle
  through the body.
• Equal to Z angle on P25 triangle
• Latitude (of observer) angular distance
  measured along the meridian from the equator to
  observers position
• Denoted by F in formulas
• Vernal Equinox intersection point of the
  celestial equator and hour circle through the sun
  at the instant it reaches 0 declination (March 21
  /-)

107

• Refraction causes angular increase in the
  apparent (observed) altitude of the celestial
  body
• Caused by bending of light rays that pass through
  the earths atmosphere at an angle
• Varies from 0o for altitude of 90o to maximum of
  35 horizontal
• Value in minutes of arc is roughly equal to the
  cotangent of the observed altitude
• Exact value also depends on barometric pressure
  and temperature
• Higher the celestial body, lower refraction

108

• Compute using
• Cr 16.38b/(460 F)tanV
• Cr refraction correction in minutes
• b barometric pressure (inches mercury)
• F temperature in Fahrenheit
• V observed altitude
• CORRECTION IS ALWAYS SUBTRACTED FROM OBSERVED
  ALTITUDE!!

109

• Parallax results from observations being made
  from earths surface rather than center
• Causes a small angular decrease in apparent
  altitude the CORRECTION IS ALWAYS ADDED
• Insignificant when star is target, but must be
  taken into account when sun is used.
• Compute using
• CP 8.79cosV
• Due to inversions and non-uniformities in air
  pressure and temp. these corrections are
  inaccurate
• Astronomic observations for azimuth generally do
  not require vertical or zenith angles which
  eliminates used for correction

110

• Time 4 kinds can be used
• Sidereal Time Sidereal Day is interval of time
  between two successive upper culminations at same
  meridian
• Star time
• Apparent Solar Time Apparent solar day is time
  interval between two successive lower
  culminations of the sun
• There is one less day of solar time/year than
  sidereal time due to earths rotation about sun
• Sidereal day is shorter than solar day by about 3
  min 56 sec.

111

• Mean Solar or Civil Time Related to fictitious
  mean sun which is assumed to move at a uniform
  rate
• Basis of time we use with 24 hour day
• Standard Time mean time at 15o meridians appart
  measured east and west of Greenwich
• 15o meridian 1 hour
• Universal Time (UT) Greenwich Civil Time (GCT)
• Eastern Standard time at 75th meridian differs
  from GCT by 5 hours earlier.
• Central Standard time by 6 hours

112

• Daylight Savings Time in a zone is equal to
  standard time in adjacent zone to the east

360o of longitude 24 hours 15o of longitude
1 hour 1o of longitude 4 minutes Time
Zones Longitude of Correction in
Hours Central time To Add to Obtain UT
(Standard) Atlantic 60o 4 Eastern
75o 5 Central 90o 6 Mountain
105o 7 Pacific 120o 8
113
Star Positions

• Polaris brightest star nearest to north pole
• part of Ursa Minor (Little Dipper)
• Circumpolar (never moves below horizon) for all
  of U.S.
• Southern Cross used in southern hemisphere

114
Azimuth from Polaris Observation

• 3 Methods
• Polaris by hour angle most often used because
  it can be done anytime
• Polaris at culmination
• Advantage direction is due north, thus no
  subsequent comps. But comps must be made to
  determine exact time of culmination
• Disadvantage star speed fastest
• Polaris at elongation
• Advantage apparent movement is vertical (easier
  observation) and computations are simple
• Lasts 15-20 minutes, but time must be computed

115
Hour Angle Method (Polaris)

• Only the horizontal angle and precise time are
  needed
• To make sure the correct star is sighted, zenith
  angle can be turned
• Equal numbers of direct and reverse observation
  taken to allow averaging

116

• tan Z - sin t /(cos F x tan d) (sin F x cos
  t)
• F latitude
• d declination
• t hour angle of polaris
• Latitude (F) of observers position is arc HP
  thus arc PZ is 90o F
• Declination (d) of star is arc SS thus SP 90o
  d (Polar distance)
• Angle ZPS is t (meridian angle)

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Meridian Angle (t)
119

• North celestial Pole (P) is at the center of the
  stars diurnal circle (viewed from observers
  position within sphere)
• West is left and apparent rotation is counter
  clockwise
• Angle ? between Greenwich (G) and Local meridian
  (L) through the observers position is the
  longitude of observers position

120

• The stars Greenwich Hour Angle (GHA) for an
  observation time is taken from the ephemeris
• If GHA and ? are plotted, the stars position is
  known, thus LHA GHA ?
• If star is west, LHA is between 0o 180o
• If star is East, LHA is between 180o 360o
• If star is west, t LHA
• If star is east, t 360o LHA
• Latitude of observers Position is needed to
  compute Z
• Longitude of observers Position is needed to
  compute either t or LHA
• BOTH CAN BE SCALE FROM 7.5 USGS TO DO OR USE
  HAND HELD GPS

121
Relationship between LHA, sign of Z and Azimuth
of star

• LHA Zlt0 Zgt0
• 0o 180o Azimuth 360oZ Azimuth 180oZ
• 180o-360o Azimuth 180o Z Azimuth Z

122

• Example
• Observation on Polaris, December 3, 2000
• Determine Azimuth of Line A-B.
• Instrument_at_ A(Latitude 43o0524N Longitude
  89o 26 00W)
• OBS INST STA. OBS. TIME HORZ.
  CIRCLE
• D/R SIGHTED (pm
  cst)
• 1 D B
  0 00
  00
• Polaris
  83049 51
  09 15
• 2 R B
  
  180 00 01
• Polaris
  83939 231
  07 14
• 3 D B
  
  0 00 00
• Polaris
  84433 51 05 35
• 4 R B
  
  180 00 00
• Polaris
  84946 231
  04 20
• Zenith angle verification of Polaris 47 26 -
  17
• When observations are made over a short period of
  time, average values for time and horizontal
  angle can be used and one reduction made
• Time span must be under 20 minutes
• For most accurate results, calculate each and
  average results

123

• Computation for Observation 1
• 1. UT of Obs. 8 30 49 PM CST 12/3/00
• 12 For PM
• 6
  Correction for Greenwich
• 26 30 49
• 263049 240000 23049 UT
  4 December, 2000
• 2. GHA of Polaris (Ephemeris)
• GHA 0h UT 4 Dec. 34 34 44.4
• GHA 0h UT 5 Dec. 35 34 06.4
• Change in GHA for 24 hr
• (360 353406.4) (34-34-44.4)
  360-59-22.0
• This accounts for star making a
  complete diurnal circle in 24 hours.
• 23049 2.51361 hr.
• (2.51361 / 24) X 360-59-22.0
  37-48-28.1
• Compute GHA of Polaris_at_ Observation
• 34-34-44.4 37-48-28.1 72-23-12.5
  GHA Polaris

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• 3. Declination of Polaris (Ephemeris)
• d _at_ 0 hr UT, 4 Dec 89 16 10.04
• d _at_ 0 hr UT, 5 Dec 89 16 10.36
• compute change in 24 hours 0.32
• change for 2 h 30 m 49 s
• (2.51361/ 24) X 0.32 0.03
• Declination 89-16-10.04 0.03 89
  16 10.07
• 4. Local Hour Angle of Polaris
• GHA 72 23 12.5
• l - 89 - 26 - 00
• - 17 02 - 47.5
• 360 (To
  normalize direction)
• LHA 342 57 12.5 (Polaris is E. of
  North)
• 5. Azimuth of Polaris by Equation
• Z tan -1 -sin 342-57-12.5/(cos 43-05-24
  tan 89-16-10.07) (sin 43-05-24 cos
  342-57-12.5)
• Z tan -1 (.0051775)
• Z 0 17 47.9

125

• 6. Azimuth line A-B (Eq a 360 Z q)
• Az. Polaris 0 17 47.9
• 360
  To normalize
• 360 17 47.9
• - 51 09 15 Horz.
  Angle to B
• 309 08 33 Azimuth
  A-B
• Computation of other observations
• 2. 309-08-30
• 3. 309-08-42 Mean of 4 Angles
  309-08-35.2
• 4. 309-08-36
• Have BS _at_ least 1000 away to minimize
  refocusing.
• Level instrument carefully, 10 out of level
  10 Az. Error _at_ 45oN
• Make sure crosshairs BS are illuninated.

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Solar Observation

• NEVER POINT AT SUN WITHOUT A SOLAR LENS
• Will cause damage to total station EDM and cause
  permanent eye damage
• Reduction uses same equations as Polaris with 2
  exceptions
• Because sun is close, linear interpolation for
  declination is not adequate, must use
• d sun do (d24 do)xUT1/240.0000395 dox
  sin(7.5UT1)
• do tabulated declination of sun at 0 h UT, on
  day of observation
• d24 tabulated declination of sun at 24 h UT1 on
  day of observation (0 h UT1 following day)
• UT1 Universal Time of Observation

127

• Depends on filter used
• Standard Filter the most accurate sight is the
  suns trailing edge, therefore, must correct for
  the horizontal angle of suns semi-diameter which
  varies, from ephemeris
• CSD suns semi-diameter/cos(h) (altitude)
• The need for this is eliminated using a Roelof
  solar prism
• Provides 4 overlapping images of the sun and
  crosshair is placed in center

128
Sources of Error in Astronomic Observations

1. Instrument not level
2. Instrument out of adjustment
3. Error in pointing
4. Time
5. Parallax in readings

129
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