# Surveying on NAD83 State Plane Coordinate System

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1 Surveying on NAD83 State Plane Coordinate System By Dr. Joshua Greenfeld Surveying Program Coordinator NJ Institute of Technology Objective Modern surveying operations which involves working with GIS/LIS and GPS requires the utilization of a nationwide or at least a statewide reference coordinate system. The most common reference system in the US is The NAD83 State Plane Coordinate System (SPCS). 1

2 Objective The purpose of the seminar is to provide the Professional Land Surveyor with the necessary information for working with the NAD83 SPCS. It will include a discussion on projecting the earth onto a plane, definitions of NAD83 and computational aspects of SPCS. Practical examples such as traverse computations will be practiced by the seminar participants. OUTLINE Background on State Plane Coordinate Systems - The need - The law Definition of Terms - Geodetic Reference System - Geodetic Coordinate Systems - Datum 2

3 OUTLINE Introduction to map projections From the curved earth to a flat map Classes of map projections Map projections for surveying NAD83 Why replace NAD27? Characteristics of the new datum NAD83/9x OUTLINE State plane coordinate system General New Jersey s system Pennsylvania s system Coordinate computations and conversions Geodetic to Plane Plane to Geodetic Meridian convergence and Scale Factor 3

4 OUTLINE Traverse computation on SPCS (t-t) correction Scale factor correction Final coordinate values for traverse points GPS applications The National Spatial Reference System NSRS serves as the basis for: mapping GIS navigation military activities 4

5 The nature of NSRS changed over the years Name Objective 1. New England Datum Local 2. NAD 27 Adjustment of North America 3. NAD 83 Development of EDM 4. NAD 2000 (ITRF) GPS Some of the benefits for surveyors to work with SPCS All surveys correlate to a single reference framework. This means that all surveys, old and new, can be combined seamlessly to a consistent and contiguous mapping project. Points from old and new surveys can be used without the need to recompute the old measurements. Surveyors having numerous projects in a certain area could, theoretically, cut and paste different projects to produce a map without compromising the accuracy of the new product. 5

6 Some of the benefits for surveyors to work with SPCS Large projects can be surveyed in parallel as independent sections. Although during the time of the execution of the project the different sections are not yet connected physically, they are connected computationally because they all share a common reference framework. As the work progresses, all sections will be connected and the accuracy of the entire project will be maintained throughout. Some of the benefits for surveyors to work with SPCS Data sharing among surveyors is simplified if everyone is working on the same reference system. This means that surveys can be repackaged and sold for additional profit. Data is a precious commodity in the GIS world. Surveyors have an abundance of spatial information; if it is in a useable form (such as SPCS) it has a market value. 6

7 Some of the benefits for surveyors to work with SPCS No point can be considered lost because it can be recovered by its coordinates. If you have the State Plane Coordinates of a point, you can use, for example, GPS to recover it. There is no need to recover points using ties that may have been also been destroyed. (Unless there is a legal issue involved.) Some of the benefits for surveyors to work with SPCS Using SPCS, the earth can be viewed mathematically as a plane. This means that plane geometry and trigonometry mathematics can be used in our computations. One needs only to apply a small, well defined, correction to compensate for the plane approximation. This manual explains what corrections have to be made and provides an example of how to apply them. 7

8 Some of the benefits for surveyors to work with SPCS Working with SPCS provides an extra external computation check for our surveys. Loop closures such as a closed traverse check only the inner consistency of the survey. If, for example, there is a systematic scale error in the traverse, it will not be detected by summing up the latitudes and the departures. Only if we tie the traverse to two or more points with given State Plane Coordinate values can this error be realized and corrected. A similar argument holds for the orientation of the traverse. To maintain proper orientation of a traverse we need to tie it to at least two control points with State Plane Coordinates. NJ SPCS Law Assembly, No. 21-L.1989, c. 218 This bill changes the system of plane coordinates which comprises the official survey base for the State. The plane coordinates system currently in use is a transverse Mercator projection of Clarke's spheroid of 1866, with a central meridian located 74 40' west from Greenwich. Under the bill, the system will be a transverse Mercator projection of the Geodetic Reference System of 1980, with a central meridian located 74 30' west from Greenwich. The bill also provides for the use of' the North American Datum of 1983 or the most recently published adjustment by the National Geodetic Survey. 8

9 NJ SPCS Law (cont ) All coordinates of the system shall be expressed in meters rather than in feet. The bill revises the scale of the coordinate system from 1:40,000 (expressed in feet) to 1:10,000 (expressed in meters). It also provides that standard conversions from meters to feet shall be the adopted standards of the National Oceanic and Atmospheric Administration. COMMITTEE AMENDMENTS The committee amended the bill to correct the reference to the Geodetic Reference System of 1980 and to provide that the New Jersey coordinate system defined by the North American Datum of 1927 may be used concurrently with or in lieu of the system defined by the North American Datum of 1983 for a period of 36 months after the effective date of this act. Some of the benefits for surveyors to work with SPCS In many states such as New Jersey there is a SPCS law. 9

10 GEODESY Geodesy is one of the oldest sciences Definition: The science of determining the size and shape of the earth including its gravity field, in four-dimensional space-time Different Surfaces In Geodesy The topography - the physical surface of the earth The geoid - the level (equipotential) surface at mean sea level. In simple terms a level surface is a surface on which water will stand still. The geoid is a level surface at elevation zero. The ellipsoid - the mathematical surface which approximates the shape and size of the earth, and is used as a reference frame for position computations. 10

11 Surfaces in Geodesy The Figure of the Earth Minor Axis a b f = a Major Axis b a 11

12 Common Ellipsoids Name Use a - (meters) 1/f Clarke 1866 NAD GRS 1967 (Old GRS) GRS 1980 NAD WGS-72 Previous GPS WGS-84 Current GPS Geodetic Coordinate System 12

13 Geodetic Coordinate System Pole Equator b ϕ Normal to ellipsoid a H Greenwich λ Map projection A Map projection is a systematic representation of a round body such as the earth on a flat (plane) surface. Each map projection has specific properties that make it useful for specific purpose. 13

14 The Problem of map projection The earth is an irregular surface that has to be represented in terms of a simple but consistent graph. Computation of Distances and Directions (Az) are very complicated on spheroid but very easy using X,Y coordinates. Solution 1. Approximate the earth by a sphere (or an ellipsoid) 2. Reduce the size of the sphere (globe) 3. Project the globe into another surface (developable surface) that can be flattened out 4. develop a method for calculating the resultant distortions. 14

15 Distance distortion Shape distortion 15

16 Distortions Mercator Cylindrical Equal Area An ideal map projection must satisfy the following: 1. All distances and areas on the map should have correct relative magnitude. 2. All Azimuths and angles should be correctly shown on the map 3. All Great circles on the earth should appear as straight lines. 4. Geographic latitudes and longitudes of all points should be shown correctly on the map 16

17 An ideal map projection? It is impossible to satisfy all of these conditions. So, we devise different map projections for specific purposes. The best map projection is the one that best serves a particular application. The most common classification of map projections: Azimuthal projection: Shows correct directions and AZ from one central point Equidistance projection: shows correct distances from one central point. Conformal projection: Shows correct angles from all points, if a small portion of the earth is projected. Equal area projection: Shows correct areas. Shape is usually very distorted. 17

18 Choice of Datum Ellipsoid Sphere Scale Reduction Globe Cylindrical Projection Conic Projection Azimuthal Projection Map Projection Systematic (mathematical) construction of No actual projection graduate 18

19 Classification of map projections Extrinsic Problem (Projection Surface) Class Varieties Nature Plane Conical Cylindrical Coincidence Tangent Secant Poly superficial Position Normal Transverse Oblique Classification of map projections Intrinsic Problem (Projection Characteristic) Class Varieties Properties Equidistance Equivalent Conformal Generation Geometric Semi- Geometric conventional Mapping extent World Continent Region Use Topographic Thematic Navigation 19

20 Glossary Map projection - An orderly system of lines on a plane representing a corresponding system of imaginary lines on an adopted datum surface, and its mathematical concept. Developable surface- A developable surface is a simple geometric form capable of being flattened without stretching Datum - Any numerical or geometrical quantity or set of such quantities which may serve as reference or base for other quantities. Glossary Latitude - The angle which the normal to ellipsoid at a point makes with the plane of the equator. Longitude - The angle between the plane of the meridian and the plane of an initial meridian, arbitrarily chosen (Greenwich) Great Circle - A great circle is formed on the surface of a sphere by a plane that passes through the center of the sphere. The arc of a great circle is the shortest distance between points on the surface of the earth. 20

21 Glossary Meridian - An arc formed by the intersection of the plane containing a point on the earth and its two poles. Central meridian - A meridian that is located in the center of the projected area (used only in some map projection) Parallel - An arc formed by connecting all the points that have the same latitude Standard Parallel - A specific parallel at which the developable surface intersect the surface of the earth Glossary Rhumb Line (Loxodromes) - A rhumb line is a line on the surface on the earth cutting all meridians at the same angle. (not necessarily the shortest distance!) Linear Scale - Linear scale is the relationship between a distance on a map and the corresponding distance on the earth. 21

22 Glossary Mapping Equations - Mapping equations are the mathematical relationship between the coordinates of the earth and the coordinates of the map x = f 1 (ϕ,λ) y = f 2 (ϕ,λ) The simplest mapping equations are: x = λ y = ϕ Glossary For T.M. x = k o N [A+ (1-T+C)A 3 /6 + (5-18T+T 2 +72C-58e 2 )A 5 /120] Where: e 2 = (a 2 - b 2 ) / b 2 N = a /(1-e 2 sin 2 ϕ) 1/2 T = tan 2 ϕ C = e 2 cos 2 ϕ A = cosϕ (λ-λ o ) y =... 22

23 Some Map Projections 23

24 Orthographic Polar Mapmaker selects North or South Pole Plane of projection Oblique Mapmaker selects any point of tangency except along Equator or at Poles Equator Equatorial Mapmaker selects central meridian 24

25 Orthographic Used for perspective views of the Earth, Moon, and other planets. The Earth appears as it would on a photograph from deep space. Used by USGS in the National Atlas of the United States of America. The Orthographic projection was known to Egyptians and Greeks 2,000 years ago. Orthographic Directions are true only from center point of projection. Scale decreases along all lines radiating from center point of projection. Any straight line through center point is a great circle. Areas and shapes are distorted by perspective; distortion increases away from center point. Map is perspective but not conformal or equal area. In the polar aspect, distances are true along the Equator and all other parallels. Azimuthal-Geometrically projected onto a plane. Point of projection is at infinity. 25

26 Stereographic Polar Mapmaker selects North or South Pole Oblique Mapmaker selects any point of tangency except along Equator or at Poles Equator Point of projection Plane of projection Equatorial Mapmaker selects central meridian Stereographic Used by the USGS for maps of Antarctica and American Geographical Society for Arctic and Antarctic maps. May be used to map large continent-sized areas of similar extent in all directions Used in geophysics to solve spherical geometry problems. Polar aspects used for topographic maps and charts for navigating in latitudes above 80º. Dates from 2nd century B.C. Ascribed to Hipparchus. 26

27 Stereographic Directions true only from center point of projection Scale increases away from center point. Any straight line through center point is a great circle. Distortion of areas and large shapes increases away from center point. Map is conformal and perspective but not equal area or equidistant. Azimuthal-Geometrically projected on a plane. Point of projection is at surface of globe opposite the point of tangency. Gnomonic Polar Mapmaker selects North or South Pole Oblique Mapmaker selects any point of tangency except along Equator or at Poles Plane of projection Equator Equatorial Mapmaker selects central meridian 27

28 Gnomonic Used along with the Mercator by some navigators to find the shortest path between two points. Used in seismic work because seismic waves tend to travel along great circles. Considered to be the oldest projection. Ascribed to Thales, the father of abstract geometry, who lived in the 6th century B.C. Gnomonic Any straight line drawn on the map is on a great circle, but directions are true only from center point of projection. Scale increases very rapidly away from center point. Distortion of shapes and areas increases away from center point. Map is perspective (from the center of the Earth onto a tan- gent plane) but not conformal, equal area, or equidistant. Azimuthal-Geometrically projected on a plane. Point of projection is the center of a globe. 28

29 Central Meridian (selected by mapmaker) Rhumbs lines (direction true between any two points Equator touches cylinder if cylinder is tangent Great distortions In high latitudes Mercator Reasonably true shapes and distances within 15 o of equator Mercator Used for navigation or maps of equatorial regions. Any straight line on map is a rhumb line (line of constant direction). Presented by Mercator in Directions along a rhumb line are true between any two points on map, but a rhumb line usually is not the shortest distance between points. (Sometimes used with Gnomonic map on which any straight line is on a great circle and shows shortest path between two points.) 29

30 Mercator Distances are true only along Equator, but are reasonably correct within 15º of Equator; special scales can be used to measure distances along other parallels. Two particular parallels can be made correct in scale instead of the Equator. Areas and shapes of large areas are distorted. Distortion increases away from Equator and is extreme in polar regions. Map, however, is conformal in that angles and shapes within any small area (such as that shown by a USGS topographic map) are essentially true. Mercator The map is not perspective, equal area, or equidistant. Equator and other parallels are straight lines (spacing increases toward poles) and meet meridians (equally spaced straight lines) at right angles. Poles are not shown. Cylindrical-Mathematically projected on a cylinder tangent to the Equator. (Cylinder may also be secant.) 30

31 Transverse Mercator Central Meridian selected by mapmaker touches cylinder if cylinder is tangent Equator Can show whole Earth, but directions, distances and areas are reasonably accurate only within 15 o of the central meridian No straight rhumb lines Transverse Mercator Used by USGS for many quadrangle maps at scales from 1:24,000 to 1:250,000; such maps can be joined at their edges only if they are in the same zone with one central meridian. Also used for mapping large areas that are mainly northsouth in extent. Presented by Lambert in Distances are true only along the central meridian selected by the mapmaker or else along two lines parallel to it, but all distances, directions, shapes, and areas are reasonably accurate within 15º of the central meridian. 31

32 Transverse Mercator Distortion of distances, directions, and size of areas increases rapidly outside the 15º band. Because the map is conformal, however, shapes and angles within any small area (such as that shown by a USGS topographic map) are essentially true. Graticule spacing increases away from central meridian. Equator is straight. Other parallels are complex curves concave toward nearest pole. Central meridian and each meridian 90' from it are straight. Other meridians are complex curves concave toward central meridian. Cylindrical-Mathematically projected on cylinder tangent to a meridian. (Cylinder may also be secant.) Oblique Mercator Line of tangency the great circle that touches cylinder if cylinder is tangent In this projection, shortest distance between points along line of tangency are straight lines. No straight rhumb lines Equator 32

33 Oblique Mercator Used to show regions along a great circle other than the Equator or a meridian, that is, having their general extent oblique to the Equator. This kind of map can be made to show as a straight line the shortest distance between any two preselected points along the selected great circle. Developed by Rosenmund, Laborde, Hotine et al. Oblique Mercator Distances are true only along the great circle (the line of tangency for this projection), or along two lines parallel to it. Distances, directions, areas, and shapes are fairly accurate within 15º of the great circle. Distortion of areas, distances, and shapes increases away from the great circle. It is excessive toward the edges of a world map except near the path of the great circle. The map is conformal, but not perspective, equal area, or equidistant. Rhumb lines are curved. 33

34 Oblique Mercator Graticule spacing increases away from the great circle but conformality is retained. Both poles can be shown. Equator and other parallels are complex curves concave toward nearest pole. Two meridians 180º apart are straight lines; all others are complex curves concave toward the great circle. Cylindrical-Mathematically projected on a cylinder tangent, (or secant) along any great circle but the Equator or a meridian. Directions, distances, and areas reasonably accurate only within 15º of the line of tangency. Robinson Pseudocylindrical or orthophanic ("right appearing") projection Central Meridian Equator Concave meridian are equally spaced Straight equator, parallels, central meridian central meridian is 0.53 as long as equator 34

35 Robinson Uses tabular coordinates rather than mathematical formulas to make the world "look right." Better balance of size and shape of high-latitude lands than in Mercator, Van der Grinten, or Mollweide. Soviet Union, Canada, and Greenland truer to size, but Greenland compressed. Presented by Arthur H. Robin- son in Used in Goode's Atlas, adopted for National Geographic's world maps in 1988, appears in growing number of other publications, may replace Mercator in many classrooms. Robinson Directions true along all parallels and along central meridian. Distances constant along Equator and other parallels, but scales vary. Scale true along 38º N & S, constant along any given parallel, same alone N & S parallels same distance from Equator. Distortion: All points have some. Very low along Equator and within 45º of center. Greatest near the poles. Not conformal, equal area, equidistant, or perspective. Pseudocylindrical or orthophanic ("right appearing") projection 35

36 Albers Equal Area Conic Two standard parallels (selected by mapmaker) Equal area. Deformation of shapes increases away from standard parallels Albers Equal Area Conic Used by USGS for maps show-ing the conterminous United States (48 states) or large areas of the United States. Well suited for large countries or other areas that are mainly east-west in extent and that require equal area representation. Used for many thematic maps. Maps showing adjacent areas can be joined at their edges only if they have the same standard parallels (parallels of no distortion) and the same scale. Presented by H. C. Albers in

37 Albers Equal Area Conic All areas on the map are proportional to the same areas on the Earth. Directions are reasonably accurate in limited regions. Distances are true on both standard parallels. Maximum scale error is 1.25% on map of conterminous States with standard parallels of 29.5 N and 45.5'N. Scale true only along standard parallels. USGS maps of the conterminous 48 States, if based on this projection, have standard parallels 29.5'N and 45.5'N. Such maps of Alaska use standard parallels 55'N and 65'N, and maps of Hawaii use standard parallels 8'N and 18'N. Albers Equal Area Conic Map is not conformal, perspective, or equidistant. Conic-Mathematically projected on a cone conceptually secant at two standard parallels. 37

38 Lambert Conformal Conic Two standard parallels (selected by mapmaker) Large-scale map sheets can be joined at edges if they have same standard parallels and scales Lambert Conformal Conic Used by USGS for many 7.5 and 15-minute topographic maps and for the State Base Map series. Also used to show a country or region that is mainly east-west in extent. Presented by Lambert in One of the most widely used map projections in the United States today. Looks like the Albers Equal Area Conic, but graticule spacings differ. 38

39 Lambert Conformal Conic Retains conformality. Distances true only along standard parallels; reasonably accurate elsewhere in limited regions. Directions reasonably accurate. Distortion of shapes and areas minimal at, but increases away from standard parallels. Shapes on large-scale maps of small areas essentially true. Lambert Conformal Conic Map is conformal but not perspective, equal area, or equidistant. For USGS Base Map series for the 48 conterminous States, standard parallels are 33'N and 45'N (maximum scale error formap of 48 States is 21/2%). For USGS Topographic Map series (7.5- and 15-minute), standard parallels vary. For aeronautical charts of Alaska, they are 55'N and 65'N; for the National Atlas of Canada, they are 49'N and 77'N. Conic-Mathematically projected on a cone conceptually secant at two standard parallels. 39

40 Geographic Gnomonic 40

41 Stereographic Mercator 41

42 Transverse Mercator (TM) Robinson 42

43 Albert Equal Area Conic Lambert Conformal Conic 43

44 The Universal Transverse Mercator (UTM) Projection The Universal Transverse Mercator (UTM) Projection 44

45 Characteristics of NAD'83 Redefinition - Define a new datum that: Has a worldwide fit (not localized to best fit North America) The center of the ellipsoid is defined at the mass center of the earth. (moved from Meades Ranch KS to be consistent with satellite systems) The size (a) and the shape (1/f) are defined by GRS'80. a = 6,378, meters 1/f = Characteristics of NAD'83 Readjustment - Compute a new Least Square solution for old and new observations. Simultaneous adjustment of 1,785,772 observations for 266,436 stations. Combine classical terrestrial and satellite (space) data Mathematical modelling to remove unpredictable distortions in the network. 45

46 Characteristics of NAD'83 Plane Coordinates - For various states (or zones) are derived from a conformal mapping projections that: Maximum scale distortion is less than 1:10,000 Cover an entire state with as few zones of a projection as possible 46

47 Characteristics of NAD'83 Define boundaries of projection zones as an aggregation of counties TM - Transverse Mercator (N-S zones) or LCC - Lambert Conformal Conic (E-W zones) OTM - Oblique Transverse Mercator (Alaska) Coordinate values are different from NAD'27 due to: Change in datum Changes as a result of a new adjustment New mapping (projection) equations (derived mathematically not empirically) to support 1mm accuracy 47

48 Coordinate values are different from NAD'27 due to: Changes in numerical grid value of the origin of each zone Changes in mapping constants in some zones (new standard parallel or meridian) Azimuth orientation is due North The use of Metric units rather then Feet Plane coordinates North-South Direction Plane Straight up or parallel to the direction of the North Arrow Ellipsoid Slanted (not uniformly) towards the North Pole. All lines pointing to North, converge at the North Pole. 48

49 Plane coordinates C S D C D Distances Plane Straight lines (line C-D) Ellipsoid Curved lines (curve C-S -D) Plane coordinates Sum of angles in a quadrilateral Even coordinate differences correspond to: Plane 360 Even (same length) distances Ellipsoid spherical excess Uneven distances, i.e. the length of an arc of 2 of longitude near the pole is much shorter than 2 of arc near the equator. 49

50 Most Common Projection for SPCS Projecting part of an ellipsoid onto a plane Scale >1 Scale <1 Scale=1 Projection Plane Ellipsoid 50

51 Y LCC Projection θ Standard Parallels Scale =1 Scale >1 R cos q R R b Miles Miles or or less less Scale =1 Scale <1 X P R sin q P Scale >1 Y P X E o TM Grid Scale >1 Scale =1 Standard Lines Central Meridian Scale <1 Scale =1 Scale >1 g Standard Lines 51

52 Computations with SPCS 1. Direct Problem (ϕ, λ) (N, E ) 2. Inverse Problem (N, E ) (ϕ, λ) 3. Meridian convergence 4. Scale factor Manual computation Software NAD83 SPCS Zone Constants for New Jersey Zone Code 2900 Projection TM Central Meridian Scale Factor 1:10,000 Grid Origin: j l E 150,000 N 0 52

53 NAD83 SPCS Zone Constants for Pennsylvania Zone N S Code Projection LCC LCC Standard Parallels Grid Origin: j l E 600, ,000 N 0 0 INTERPOLATION OF LCC NAD 83 (Pennsylvania North) Input Zone Constants ϕ = 41 25' 23.85" ϕ b = λ = 78 15' 12.12" λ cm = 77 45' R b = E o = N o = 0. sin ϕ o =

54 SPC from LCC NAD 83 (Tables) (Pennsylvania North) Interpolation. R 41 26' R 41 25' R= R R = R " = ' 60" Computation. γ = ( λ cm - λ ) sinϕ o = -0-19' " E = R sinγ + E o = N = R b - R cosγ + N o = SPC from LCC NAD 83 (Tables) (Pennsylvania North) Input: E = N = Computation. E E0 tan γ = = R ( N N ) b Rb ( N N 0) R = = cosγ γ λ = λcm - = 78 15' 12.12" sinϕ 0 To compute ϕ perform a backward interpolation using R. 0 54

55 Direct Problem (ϕ, λ) (N, E ) L = λ d - λ 0 N = S' + A 2 L 2 + A 4 L 4 - * 2nd order correction E = E 0 + A 1 L + A 3 L 3 - * 2nd order correction Where: λ d - λ 0 - E 0 - Geodetic Longitude in D.ddddd (Decimal Degrees) 74.5 (NJ s Geodetic Longitude of C. Meridian) 150,000m (NJ s Grid Easting Origin Shift) S',A 2,A 4,A 1,A 3,-Tabulated coefficients as a function of ϕ ϕ- Geodetic Latitude in D M' S N,E- Grid Northing and Easting (m) * 2nd order corrections for accuracies better than ±0.001m. 55

56 56

57 Inverse Problem (N, E ) (ϕ, λ) Q E = 150,000. 1,000,000. ϕ = ϕ' + B 2 Q 2 + B 4 Q 4 - * 2nd order correction λ = λ 0 + B 1 Q + B 3 Q 3 + B 5 Q 5 - * 2nd order correction Where: λ- Geodetic Longitude in D.ddddd (Decimal Degrees) ϕ- Geodetic Latitude in D.ddddd (Decimal Degrees) λ (NJ s Geodetic Longitude of C. Meridian) ϕ'- Footpoint Latitude. Tabulated as a function of N. E 0-150,000m (NJ s Grid Easting Origin Shift) E- Grid Easting (m) B 2,B 4,B 1,B 3,B 5 -Tabulated coefficients as a function of N (Northing) * 2nd order corrections accuracies better than ±0.0001". 57

58 Data Sheet National Geodetic Survey, *********************************************************************** KV2920 DESIGNATION - E 78 KV2920 PID - KV2920 KV2920 STATE/COUNTY- NJ/SOMERSET KV2920 USGS QUAD - RARITAN (1981) KV2920 KV2920 *CURRENT SURVEY CONTROL KV2920 KV2920* NAD 83(1996) (N) (W) ADJUSTED KV2920* NAVD (meters) (feet) ADJUSTED KV2920 KV2920 X - 1,283, (meters) COMP KV2920 Y - -4,677, (meters) COMP KV2920 Z - 4,127, (meters) COMP KV2920 LAPLACE CORR (seconds) DEFLEC99 KV2920 ELLIP HEIGHT (meters) GPS OBS KV2920 GEOID HEIGHT (meters) GEOID99 KV2920 DYNAMIC HT (meters) (feet) COMP 58

59 Data Sheet KV2920 HORZ ORDER - SECOND KV2920 VERT ORDER - FIRST KV2920 ELLP ORDER - FOURTH CLASS II CLASS I KV2920 KV2920. The horizontal coordinates were established by GPS observations KV2920. and adjusted by the National Geodetic Survey in May KV2920 KV2920. The orthometric height was determined by differential leveling KV2920. and adjusted by the National Geodetic Survey in June KV2920 KV2920. The X, Y, and Z were computed from the position and the ellipsoidal ht. KV2920 KV2920. The Laplace correction was computed from DEFLEC99 derived deflections. KV2920 KV2920. The ellipsoidal height was determined by GPS observations KV2920. and is referenced to NAD 83. KV2920 KV2920. The geoid height was determined by GEOID99. KV2920 KV2920. The dynamic height is computed by dividing the NAVD 88 KV2920. geopotential number by the normal gravity value computed on the KV2920. Geodetic Reference System of 1980 (GRS 80) ellipsoid at 45 KV2920. degrees latitude (g = gals.). KV2920 KV2920. The modeled gravity was interpolated from observed gravity values. Data Sheet KV2920; North East Units Scale Converg. KV2920;SPC NJ - 194, , MT KV2920;UTM 18-4,493, , MT KV2920 KV2920 STATION DESCRIPTION KV2920 KV2920'DESCRIBED BY NATIONAL GEODETIC SURVEY 1978 KV2920'2.6 MI NW FROM SOMERVILLE. KV2920'2.6 MILES NORTHWEST ALONG STATE ROUTE 28 FROM THE COUNTY COURT HOUSE KV2920'AT SOMERVILLE, SET 40.0 FEET NORTHWEST OF THE ENTRANCE TO THE VILLA, KV2920'48.5 FEET SOUTHWEST OF THE CENTER LINE OF STATE ROUTE 28, 50.1 FEET KV2920'SOUTHEAST OF A FIRE HYDRANT AND 4.7 FEET EAST OF THE SOUTHWEST CORNER KV2920'OF THE BRICK FOUNDATION OF THE SIGN WITH THE VILLA ON IT. 59

60 USEFUL FORMULAS FOR COMPUTATION OF DISTANCES AND DIRECTIONS ON SPCS The Grid Scale Factor correction for projecting a curved line onto a plane Measured Distance Topography EllipsoidDistance h H Ellipsoid Geoid N 60

61 Grid Scale Factor K 12 for a Line from Point 1 to Point 2 Purpose: To correct for scale distortion due to the projection of the ellipsoid onto a plane. Formula: K 12 = K + 4K + K 1 m 2 6 Where: K 12 -Grid Scale factor of a line between points 1 and 2. K 1 -Grid scale factor at point 1 K 2 -Grid scale factor at point 2 K m -Grid scale factor at the line's mid- point. Grid Scale Factor K 12 for a Line from Point 1 to Point 2 Usage: A reasonable approximation for the above formula is to compute a simple average of K 1 and K 2. A further approximation is to compute a single K value for the entire line or for the entire survey area. This will be demonstrated later in our traverse example. 61

62 Scale Factor for LCC (Aproximate) Where: k - N - k = k 0 ( N + 2 The Scale Factor 2 N0) 2 r0 Northing of the point N 0, r 0, k 0, ϕ 0 - Zone Constants ( N N + 3 0) 3 6r0 tan ϕ PA North PA South N 0 = N 0 = r 0 = r 0 = k 0 = k 0 = ϕ 0 = ϕ 0 = k = k0 + ( E Where: k - The Scale Factor E - Easting of the point E 0, r 0, k 0 - Zone Constants For New Jersey E 0 = k 0 = = x10-14 or Scale Factor for TM (Aproximate) 2 0 K = (E - E o ) E 0 ) 2 1 2r 2 0 2r 62

63 Reducing Measured Horizontal Distance to Grid Distance Purpose: To reduce a measured horizontal distance to the projection plane. As mentioned earlier, field measurements are carried out on the physical surface of the earth, while office computation are performed on the projection of the earth onto a plane. This reduction is in essence the bridge between field measurements and the computations on the state plane coordinate system. Reducing Measured Horizontal Distance to Ellispoid h H N D S Topography Sea Level - Geoid Ellipsoid S' D = R R + h R R S' = D ( R + ) h R S' = D ( R + H + ) N 63

64 Reducing Measured Horizontal Distance to Grid Distance Formula: S = R D ( R + H + ) N K 12 Where: S- Grid Distance D- Horizontal (Measured) Distance H- Mean Elevation (Above Mean Sea Level) N- Mean Geoid Height (About -32m in NJ) R- Mean Radius of the Earth (About 6,372,000m) K 12 -Grid Scale factor of the Line. Reducing Measured Horizontal Distance to Grid Distance Usage: Obtain the elevations of the terminal points of the line, the Geoid height of the region and compute. 64

65 Relationship Between Geodetic and Grid Azimuths Purpose: To account for the convergence of the north direction towards the pole vs. parallel north direction on a plane Formula: AZ Grid = AZ Geodetic - γ + (t-t) Where: AZ Grid - Grid Azimuth AZ Geodetic - Geodetic Azimuth γ- Meridian Convergence (t-t)- Arc-to-chord correction Relationship Between Geodetic and Grid Azimuths Geodetic North Grid North N 2 Projected meridian g t T a d Projected Geodetic line Line of sight 1 E 65

66 Relationship Between Geodetic and Grid Azimuths Usage: This above formula (except for the t-t correction) has to be used only if we are given with Geodetic Azimuth and want to perform our computations on plane coordinates. Grid Azimuth is computed from an inverse Geodetic Azimuth from NGS, GPS or Astronomical Az plus Laplace Correcetion Arc-to-Chord Correction (t-t) for line 1-2 Purpose: The (t-t) correction is due to the fact that the measured direction between two points is actually a curved line, on the surface on a body such as an ellipsoid, that passes through these points. When projected onto a plane, the geodetic direction looks like an arc not a straight line. The angle that we compute from field notes is defined by the difference between two measured directions. Thus, the computed angle differs from the plane angle that we have to use when working with the State Plane Coordinate System. This difference is expressed by (t-t). 66

67 Arc-to-Chord Correction (t-t) for line 1-2 Formula: (t-t)" = 25.4 N E Where: N = N 2 - N 1 E2 E1 E = E0 2 (t-t)" -Arc-to-chord correction in seconds of arc. N 1 -Northing of point 1 N 2 -Northing of point 2 E 1 -Easting of point 1 E 2 -Easting of point 2 Arc-to-Chord Correction (t-t) for line 1-2 Usage: The size of the correction is rather small and can be neglected for most ordinary work (not high accuracy). The sign of the correction is dependent on the direction of the line with respect to the north (Azimuth dependent). Table 3.1 presents the size (magnitude) of the (t-t) correction and figure 6, describes the sign convention for (t-t), as it applied for the NJ State Plane Coordinate System. One should note that the following example does not apply to States with Lambert Conformal Conic projection (e. g. PA) or states that their central meridian is not 150,

68 Table 3.1 Size of the correction in seconds of arc ( ): (for NJ, E o = 150,000) Average Easting of the line N 150, , ,000 50, , ,000 2 km 0" 0.3" 0.5" 0.8" 5 km 0" 0.6" 1.3" 1.9" 10 km 0" 1.3" 2.5" 3.8" 20 km 0" 2.5" 5.1" 7.6" Sign of the correction for States using Transverse Mercator projection WEST DE <E o EAST DE >E o Central Meridian

69 Sign of the correction for States using Transverse Mercator projection WEST DE <E o EAST DE >E o Central Meridian Observed Direction Plane Direction Table 3.1 Size of the correction in seconds of arc ( ): (for LCC) N-No E 50, , , , ,000 2 km 0.3" 0.5" 0.8" 1.0" 1.3" 5 km 0.6" 1.3" 1.9" 2.5" 3.2" 10 km 1.3" 2.5" 3.8" 5.1" 6.4" 20 km 2.5" 5.1" 7.6" 10.2" 12.7" 69

70 Sign of the correction for States using LCC projection NORTH DN >N o Observed Direction Plane Direction Central Parallel SOUTH DN <N o Sign of the correction for States using LCC projection NORTH DN >N o 180 Central Parallel 0 SOUTH DN <N o Observed Direction Plane Direction

71 10 steps for computing a traverse on NAD 83 SPCS Computing a traverse on NAD 83 State Plane Coordinate System may include the following steps: 1. Obtain starting and ending coordinates and grid azimuth. 2. Compute preliminary Azimuth for each line. 3. Compute preliminary coordinates for each traverse point. These three steps are essentially computing an open traverse without applying corrections for closure errors. 10 steps for computing a traverse on NAD 83 SPCS 4. Obtain or compute approximate elevations (mean or for each point) of the traverse. Step 4 is necessary in order to reduce the distances from the topography to mean sea level. Unless high precision results are sought or there are substantial elevation differences between the various traverse points, a mean elevation of the area will be sufficient. 5. Compute grid scale factor (mean or individual for each line). Again, the choice between computing a mean value or individual scale factors for each line depends on the accuracy objectives of the project and the magnitude of elevation differences between the traverse points. 71

72 10 steps for computing a traverse on NAD 83 SPCS 6. Reduce horizontal distances to grid distances. 7. Compute (t-t) correction for each line (if necessary). For most traverses, (t-t) correction is practically negligible. If the Northing component (usually called the Latitude) of a side of the traverse is less than 1 mile long, the maximum (t-t) correction will not exceed 1 (second of arc). Thus, the correction is equal or smaller than the accuracy with which we are able to carry out our measurements. (t-t) becomes significant only for long Latitudes (larger than 10 km) and at the east/west most parts of the State. One can assume that the type of work for which (t-t) is significant will be done with GPS. 10 steps for computing a traverse on NAD 83 SPCS 8. Apply (t-t) correction to each Azimuth (if necessary). 9. Balance (or adjust) the traverse. 10. Compute final State Plane Coordinates for the traverse points. Steps 9 and 10 are the standard procedures for balancing the traverse with the Compass Rule or adjusting it with Least Squares. 72

73 D Traversing on SPCS C 2 B 1 A Step 1: Obtain starting and ending coordinates and grid azimuth and place the information in a table to facilitate the computations. Traversing on SPCS Point Angle Distance AZIMUTH N E A (m) (m) (m) ' 35.0" B 60 50' 37" ' 32" ' 40" C ' 52" ' 18.6" D Also given: Average Elevation of the traverse is: 200m above MSL 73

74 Traversing on SPCS Steps 2 and 3: Compute preliminary Azimuth for each line and compute preliminary coordinates for each traverse points. Point Angle Distance AZIMUTH N E A LAT DEP B C Comp D Given Closure Traversing on SPCS Step 4: Obtain or compute approximate elevations (mean or for each point) of the traverse. The average elevation is given as 200m above MSL. It is assumed that there are no substantial height differences among the traverse points. Thus, the average value will be used. 74

75 Traversing on SPCS Step 5: Compute Grid and Elevation scale factors (mean or individual for each line) The Grid scale factor (GSF) is computed based on the approximate Easting of the points. The results of the interpolation are: Point GSF B C Mean Traversing on SPCS One should note that the largest error committed by using the mean GSF instead of GSF for individual points will occur on side B-1 (the longest). The magnitude of this error is about 3mm (0.01 ft). This error is smaller than our ability to measure the distance B-1 with a total station. Thus, it is justifiable to use a mean GSF for our example traverse. 75

76 Traversing on SPCS The Elevation scale factor (ESF) is required in order to reduce the traverse from the topography to the ellipsoid. The computation is based on the elevation of traverse points above MSL (actually the Geoid) and the height of the ellipsoid above the Geoid (N). Data: R = H = 200 N = -32 Using the ratio: Mean ESF is = Finally, the combined scale factor = GSF ESF = R R + H + N Traversing on SPCS Step 6: Reduce horizontal distances to grid distances. The reduction of the distances is performed by multiplying each measured distance with the combined scale factor from step 5. Line Dist Grid Dist B C

77 Traversing on SPCS Step 7: Compute (t-t) correction for each line. Line DN DE (t-t)" B C Traversing on SPCS Step 8: Apply (t-t) correction to each Azimuth (if necessary). It can be seen from step 7 that the magnitude of this correction is negligible for most projects. Only very precise traverses performed with first order theodolites may be subject to such a small correction. Nevertheless, let s see how to apply the correction for reference purposes only. 77

78 Traversing on SPCS Sincethe central meridian in NJ is 150,000 and our traverse points have smaller Eastings (between 124,000. and 131,000.), we have to use the west (left) side of the sign diagram NW 0 NE SW 180 SE Traversing on SPCS From the preliminary traverse computation (steps 2-3) we have the following information: Point Backsight AZ Quad./Sign Foresight AZ Quad./Sign B SE / SW / NE / NW / SE / NW / + C SE / NW / + 78

79 Traversing on SPCS Combining the sign information with the computed values for (t-t) the corrections for each observed angle are: Observed BS FS FS-BS Corrected Point Angle Corr Corr Total Angle B C Note that the Backsight at point B and the foresight at point C are control points with State Plane coordinates. Thus, the Azimuths to these points are Grid Azimuths and thus already corrected for (t-t). Traversing on SPCS Step 9: Balance (or adjust) the traverse. In this example we use the Compass Rule. Balanced N E Point Angle Distance Azimuth LAT DEP A B C Closure 2.3 Comput D Given Closure Relative precision = 1: 23,698 79

80 Traversing on SPCS Step 10: Compute final State Plane Coordinates for the traverse points. Point N E B C GPSing on SPCS A large tract of land has to be surveyed and delivered with State Plane Coordinates. The monuments of the two west corners of the tract were recovered and documents revealed that the Azimuth of the line is (determined from sun shots ). In order to tie the traverse to SPC, four GPS points were ordered from a GPS service firm. GPS4 TR1 TR2 TR3 GPS2 GPS3 TR4 TR5 TR6 GPS1 80

81 GPSing on SPCS The following data was obtained from the GPS observations: Point E N Elev GPS GPS GPS GPS GPSing on SPCS You decided to check if the GPS data is correct by performing the following computations: 1. Compute the Bearing of the line from GPS3 to GPS4 and compare it with the given Azimuth. You discover that the computed Bearing is: while the given Azimuth is What is wrong? (Answer: compute meridian convergence) 81

82 GPSing on SPCS 2. Now that you found a problem you decided to measure the distance between GPS3 and GPS4 with your EDM. The measured distance was The computed distance from GPS data is problem? You remember that NAD83 coordinates are in Meters, so you convert the measured distance to Meters. Now the computed distance is Why is there a 0.1m (0.3 ) difference? GPSing on SPCS You realize that the discrepancy could be because of the Grid Scale Factor. Computing the corrected distance for GSF yields This improves the situation somewhat, but still the distances are about 1/4 ft off. How can you account for this difference? (Answer: compute sea level SF) 82

83 Compute ground level distances from State Plane Coordinates. In some cases it is necessary to compute actual ground level distances between points that have coordinates in SPCS. The actual distance is computed from: Distance actual = Distance grid Comb. S. F. Compute ground level distances from State Plane Coordinates. Thus, in order to compute the actual distance from a computed distance from coordinates we need to reverse the computation procedure we discussed earlier. The reversed procedure is to compute GSF (Grid scale factor), ESF (elevation scale factor) and the combined scale factor as described in step 5. Subsequently, apply the correction by dividing the grid distance by the combined scale factor. 83

84 Compute ground level distances from State Plane Coordinates. For example, the actual ground distance between point 1 and 2 in our example is computed as follows: Grid Distance = m Combined scale factor = Actual ground distance = m (computed from coordinates) Computing Stakeout Distances from SPCS 84

85 Example E N Ground Distance Computation Point 1 2 E N Distance Elevation

86 Ground Distance Computation Point 1 2 E N Distance Elevation Ground SF Ground Distance Computation Point 1 2 E N Distance Elevation Ground SF Elevation SF

87 Ground Distance Computation Point 1 2 E N Distance Elevation Ground SF Elevation SF Comb SF Ground Distance Computation Point 1 2 E N Distance Elevation Ground SF Elevation SF Comb SF SF Correction SP Distance Distance ft

88 Ground Distance Computation Pt. E N Distance Elevation SP Distance Ground Coordinates Comp Dist Actual Dist

89 Shifted Ground Coordinates Shift to Lower Left Corner Comp Dist Actual Dist Relative Ground Coordinates Shift to Project Center Comp Dist Actual Dist

90 Shifted+Relative Ground Coordinates (Project Center +50,000, 80,000) Comp Dist Actual Dist The End 90

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