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Section V
Control Surveys
- A. Geodesy .................................................................................................................V- Table of Contents
- General ...............................................................................................................V-
- Control Datums ...................................................................................................V-
- a. Horizontal Datum ...........................................................................................V-
- b. Vertical Datum ...............................................................................................V-
- Coordinate Systems..........................................................................................V-
- State Plane Zones ............................................................................................V-
- a. State Plane Coordinates ..............................................................................V-
- b. Surface Coordinates ....................................................................................V-
- Azimuths ...........................................................................................................V-
- a. Azimuth References ....................................................................................V-
- b. Forward and Back Azimuths ........................................................................V-
- B. GPS Surveying .....................................................................................................V-
- The Global Positioning System .........................................................................V-
- A Brief History of GPS ......................................................................................V-
- Global Navigation Satellite Systems .................................................................V-
- How GPS Works ...............................................................................................V-
- a. Measuring Distance .....................................................................................V-
- b. Signal Timing ...............................................................................................V-
- The GPS Signal ................................................................................................V-
- Satellite Geometry ............................................................................................V-
- Error Sources in GPS .......................................................................................V-
- a. Atmospheric Errors ......................................................................................V-
- b. Obstructed Signals and Multipath Errors .....................................................V-
- c. Satellite Errors .............................................................................................V-
- d. GPS Equipment Errors ................................................................................V-
- V-2 Revised February, Control Surveys - e. Human Errors ..............................................................................................V-
- GPS Accuracy...................................................................................................V-
- GPS Surveying Procedures ..............................................................................V-
- a. GPS Methods ..............................................................................................V-
- a. Equipment ...................................................................................................V-
- b. Weather Conditions .....................................................................................V-
- C. Differential Leveling ............................................................................................V-
- General .............................................................................................................V-
- Bench marks .....................................................................................................V-
- Procedures .......................................................................................................V-
- Instrument Person’s Duties ...............................................................................V-
- Rod Person’s Duties .........................................................................................V-
- D. Extendible Control Surveys ................................................................................V-
- Extendible Control Coordinates ........................................................................V-
- a. Method 1......................................................................................................V-
- b. Method 2......................................................................................................V-
- c. Method 3 ......................................................................................................V-
- Traverse adjustment .........................................................................................V-
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While the ellipsoid gives a common reference, it is still only a mathematical concept. Geodesists often need to account for the undulating surface of the Earth. To meet this need, the geoid was created. A geoid is a theoretical surface perpendicular at every point to the direction of gravity. It is also commonly associated with mean sea level. Since the Earth’s mass is unevenly distributed, certain areas of the planet experience more gravitational “pull” than others. Figure V-1 is an illustration of the ellipsoid and geoid.
Latitude is measured in a north-south direction and is expressed as degrees of departure parallel to the equator. The equator is defined to be 0° latitude and is the intersection of the Earth’s surface with the plane perpendicular to its axis of rotation. It is nearly equidistant from the North Pole and South Pole and divides the Earth into northern and southern hemispheres.
Longitude is measured in an east-west direction and is expressed as degrees of departure from the prime meridian. The longitude of the prime meridian is arbitrarily set as 0° and passes through the Royal Observatory in Greenwich, England. The prime meridian and its opposite meridian (at 180° longitude) divide the Earth into the eastern and western hemispheres. The International Date Line closely follows the 180° longitudinal meridian, occasionally deviating around land masses and island groups. Figure V-2 is an illustration of latitudes and longitudes.
Figure V-2. Latitudes and longitudes.
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Gravity is the force that pulls all objects in the universe toward each other. On Earth, gravity pulls all objects downward, toward the center of the planet. According to Newton’s Universal Law of Gravitation, the attraction between two bodies is stronger when their masses are larger and closer together. This rule applies to the Earth’s gravitational field as well. Because the Earth rotates and its mass and density vary at different locations on the planet, gravity also varies.
The variation in Earth’s gravity is measured because it plays a major role in determining mean sea level. Elevations on the Earth’s surface are based on mean sea level. Knowing how gravity affects sea level helps geodesists make more accurate measurements. Generally, areas of the planet where gravitational forces are stronger, the mean sea level will be higher because the water will be “pulled” to these locations. Conversely, areas where the gravitational forces are weaker, the mean sea level will be lower.
To measure the Earth’s gravity field, geodesists use instruments located in space and on land. In space, satellites gather data on gravitational changes as they pass over points on the Earth’s surface. On land, devices called gravimeters measure the gravitational pull on a suspended mass. With this data, geodesists can create detailed maps of gravitational fields and adjust existing elevations.
Because of the variations in gravitational force, the geoidal surface is irregular, but considerably smoother than the actual surface. The geoid varies from 350 ft (107 m) below to 280 ft (85 m) above the reference ellipsoid. As shown in Figure V-3, areas in red and yellow indicate regions where the Earth’s gravitational pull is stronger. In these areas, the geoid is above the reference ellipsoid. Areas in green and blue indicate regions where the Earth’s gravity is weaker and the geoid is below the reference ellipsoid.
Figure V-3. Global geoid undulations.
Every topographic point on the Earth’s surface has an orthometric elevation defined as the height above mean sea level. Near coastal areas, mean sea level is determined with by tidal gauges. In areas far from the coast, mean sea level is determined by the geoid. The geoid is a theoretical surface used to closely approximate mean sea level. The orthometric elevation is the distance or height from the geoid to a point on the Earth’s surface, measured along the plumb line normal to the geoid. Each point also has an ellipsoid elevation which is the height of the surface above the reference ellipsoid. Geoid separation is defined as the distance
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local control monuments are used as a reference for the collection of preliminary, cadastral, and construction surveys.
a. Horizontal Datum A horizontal datum is a network of survey monuments that have been assigned precise latitude and longitude measurements. Survey stations in the datum were typically marked with a brass, bronze, or aluminum disk set in concrete or rock. These markers were placed so that surveyors could see one marked position from another. To maximize the line-of-sight between monuments, they were usually set on hilltops or other areas of high elevation. Monuments placed in areas with little vertical relief had towers built to aid surveyors in locating them.
Figure V-5. USC&GS Brass cap.
The datum is then used as a reference for the development of new control networks. Surveyors have historically used a procedure referred to as triangulation to “connect” the horizontal monuments into a unified network. Using this procedure, the location of a point is determined by measuring angles to it from other known points. The new point is fixed as the third point of a triangle with one known side and two known angles. Another procedure used by surveyors is the traverse method.
A traverse starts from two known points to provide a beginning azimuth (or direction) and position. Angles and distances are measured throughout the traverse at intermediate points. The traverse is then completed at two known points to check the ending azimuth and position. Today, surveyors rely almost exclusively on the Global Positioning System (GPS) to determine monument positions. Regardless of the method used to determine monument positions, the observations are adjusted to correct misclosure errors.
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(1) History In 1807, the U.S. Coast Survey was established to chart the country’s coast in the New York Bay area. Shortly thereafter, its mission changed to include surveys of the interior as the nation grew westward. In 1878 the agency was reorganized into the United States Coast and Geodetic Survey (USC&GS).
The first coordinate reference system was established from geodetic surveys performed in 1816 and 1817. The reference system has evolved from the original 11 local markers to more than 250,000 monuments around the country. These stations support various activities such as: Topographic mapping Nautical and aeronautical charting Engineering and construction Public utility management Tectonic motion studies Environmental hazard analysis Geographic information systems Early surveys were often based on a local datum or reference system that was determined by astronomical observations. These surveys were performed to develop nautical charts of small areas. Many other local surveys were used to develop maps as the country expanded westward. It soon became apparent that a common set of reference points were needed. Without a common reference, maps and charts produced from these surveys would not be compatible.
By 1900, a sufficient amount of observations were obtained to complete a national geodetic datum. The datum, containing approximately 2,500 monuments, was based on the Clarke 1866 reference ellipsoid. The datum became known as the U.S. Standard Datum of 1901.
In 1913, the U.S. Standard Datum became known as the North American Datum (NAD) when the governments of Canada and Mexico adopted it. The geodetic center of the datum is a survey station named Meades Ranch. The monument is located in Kansas near the geographic center of the contiguous United States.
In the 1920’s, the USC&GS expanded the national network to more than 25, survey monuments. This network established limited geodetic control in many areas that were not involved in the 1901 datum. These new observations were incorporated into an adjustment known as the North American Datum of 1927 (NAD 27).
An increase in economic and scientific growth after World War II resulted in a need for accurate coordinate information. Development of distance measuring equipment and aerial photography enhanced the capabilities of geodesists, surveyors, and cartographers to provide more precise positional data. Satellite and remote sensing
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the horizontal datum. Beginning in 1978, the NGS introduced an improved bench mark into the National Vertical Control Network. The reference point for the elevation is the top of a stainless steel rod which is protected inside an aluminum casement. The rod, driven to refusal, is accessed by lifting a hinged cover. The bench mark is designed to prevent near-surface soil disturbances such as frost heave, soil shrinkage, and soil swelling. This is accomplished by encasing the rod in a lubricated sleeve to the depth of expected soil movement.
Figure V-6. Modern NGS bench mark.
The traditional method for establishing new elevations is differential leveling. This method uses a known elevation at one location to determine the elevation at another location. For further information on differential leveling, see part C in this Section.
(1) History The U.S. Coast Survey established the first geodetic quality leveling route in the United States in 1856. The leveling survey was required for tide and current studies in the New York Bay and Hudson River. The USC&GS began the transcontinental level line in 1887 at bench mark ‘A’ in Hagerstown, Maryland. The survey followed the 39th^ parallel and reached the Pacific by 1904.
By 1900, the vertical control network in the U.S. included 4,200 bench marks and more than 13,000 miles of geodetic leveling. Because the vertical networks in each area were usually fixed to a local reference, most of the data was not compatible. A single vertical datum was needed to link the level elevations. A vertical datum was created and referenced to local mean sea level. Mean sea level is the average (or mean) height of the ocean’s surface measured by tidal stations over a 19-year period. This time period, known as a tidal epoch, is a complete sun and moon cycle and
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accounts for the effects on ocean levels. Subsequent adjustments of the leveling network were performed by the USC&GS in 1903, 1907, and 1912.
By the late 1920’s, over 60,000 miles of leveling data had been collected. Mean sea level was being measured at 26 tide gauges in the United States and Canada. The gauges were connected through tidal bench marks to an extensive leveling network throughout the United States. However, the height of mean sea level was found to vary slightly from one tidal gauge to another.
In 1929, the USC&GS began a least-squares adjustment of all geodetic leveling data completed in the United States and Canada. Because of the variations in mean sea level, the network was adjusted to set the elevation of mean sea level at each tidal gauge to zero. This adjustment established the 1929 Sea Level Datum, to reference each bench mark elevation to mean sea level. The datum was later renamed the National Geodetic Vertical Datum of 1929 (NGVD 29).
Since 1929, approximately 385,000 miles of leveling has been added to the National Geodetic Reference System (NGRS). Periodic discussions were held to determine the proper time for the inevitable adjustment. In the early 1970’s, NGS conducted an extensive inventory of the vertical control network. The search identified thousands of bench marks that had been destroyed. Many existing bench mark elevations were affected by: Changes in sea level Movement of the Earth’s crust Uplift due to postglacial rebound Ground subsidence resulting from the withdrawal of underground water and oil Beginning in 1977, the NGVD 29 datum was adjusted to remove inaccuracies and to correct distortions in the network adjustment. Much of the first-order NGS vertical control network had to be re-leveled. Damaged or destroyed monuments were replaced with newer, more stable deep-rod bench marks. Due to the local variations at each tidal station, mean sea level was based on a single tidal gauge located in Quebec. In 1991, the result of the vertical adjustment of new and old leveling data was released. This adjustment also included level runs completed in Mexico and Canada. This new datum, called the North American Vertical Datum of 1988 (NAVD 88), provides a more accurate vertical reference system.
Similar to the horizontal datums, there isn’t an exact correlation or translation between vertical datums. VERTCOM is an NGS conversion program that computes orthometric height differences between the NGVD 29 and NAVD 88 datums. The conversion is determined for any location specified by latitude and longitude. This conversion tool is available on the NGS website and can be accessed through the following link: http://www.ngs.noaa.gov/TOOLS.
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The polar coordinate system is another coordinate system used to define points located in a single plane. The location of a point is determined by the angle and distance from the origin. The angular coordinate ( ) is the angle between the polar (X) axis and a line to the point. The radial coordinate (r) is the distance from the origin to the point.
Polar coordinates can be converted to Cartesian coordinates using the sine and cosine trigonometric functions. Conversely, Cartesian coordinates can be converted to polar coordinates using the Pythagorean Theorem and the inverse of the tangent trigonometric function. ;
;
Figure V-8. Polar coordinate system.
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A three-dimensional (3-D) Cartesian coordinate system adds a third (Z) axis to provide another dimension of measurement. The perpendicular distance from the Z axis defines the elevation (z) of a point. The position of a point within the coordinate system is expressed with easting, northing, and elevation (x, y, z) values.
Figure V-9. Three-dimensional coordinate system.
4. State Plane Zones The state plane coordinate system (SPCS) was established in 1933 by the United States Coast and Geodetic Survey (USC&GS). The USC&GS, now known as the National Geodetic Survey (NGS), developed the system to simplify geodetic calculations. Prior to the development of the SPCS, geodetic positions were given in latitudes and longitudes and involved complex computations on the surface of an ellipsoid. By ignoring the curvature of the Earth, the SPCS allows surveyors to use a rectangular coordinate system to define specific locations.
The SPCS is a network of individual state plane zones designed for specific regions throughout the United States. Each zone has an independent rectangular (or Cartesian) coordinate system with its own point of origin. The zones were created by using map projections to transform geodetic coordinates on a curved surface to rectangular coordinates on a flat plane. Distortions between the curved surface and the plane are not evident for small areas. However, as the projection area becomes larger, the distortions become more apparent.
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Most state plane zones are based on either a transverse Mercator map projection or a Lambert conformal conic map projection. The map projection is centered about a longitudinal line referred to as the central meridian. The specific map projection is dependent on the shape and size of the state. States that are longer in the east-west direction are divided into similar shaped zones that are also longer in the east-west direction. These zones use the Lambert projection to superimpose an imaginary cone over the ellipsoid. The apex of the cone is aligned with the Earth’s rotational axis. Figure V-11 is an illustration of a Lambert conformal conic projection.
Figure V-11. Lambert conformal conic projection.
States that are longer in the north-south direction are divided into zones that are also longer in the north-south direction. These zones use the transverse Mercator projection to superimpose an imaginary cylinder over the ellipsoid. The axis of the cylinder lies in the Earth’s equatorial plane. Figure V-12 is an illustration of a transverse Mercator map projection. All four Wyoming state plane zones are transverse Mercator projections.
Either map projection intersects the ellipsoid along two lines, called secants. Along the secant lines, distortions between the curved surface and the plane are essentially zero. However, distortions increase as the distance from the secant lines increase. To maximize the accuracy of each zone, the width of either projection is limited to 158 miles (254 km).
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Also, the secant lines are positioned such that 2/3 of the zone lies between them and 1/6 of the zone lies outside.
Figure V-12. Transverse Mercator projection.
a. State Plane Coordinates To convert geodetic positions from the ellipsoid to a plane, points are first mathematically projected onto an imaginary surface. This surface is then laid out flat without further distortion in shape or size. A rectangular grid is superimposed over the flat surface to establish x and y state plane coordinates. Easting coordinates increase from west to east and are measured as the distance from the origin. The northing coordinates increase from south to north and are measured as the distance from the origin. The x and y coordinate values assigned to the grid’s origin are termed “false easting” and “false northing”.
The grid origin is located south of each state plane zone to assure that the northing coordinates are positive. The easting coordinate at the origin is assigned a sufficiently large number to assure that these values remain positive. As mentioned earlier, each state plane zone has its own independent coordinate system. The easting and northing coordinates in adjacent zones are sufficiently different in magnitude to avoid confusing the coordinates.
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One component of the adjustment involves the projection of positions from a curved surface to a flat plane. A grid scale factor is used to convert positions located on the ellipsoid to positions on the state plane grid. It is a dimensionless scale factor that reflects the difference between distances on the ellipsoid and distances on a plane. The grid scale factor varies across the state plane zone and is dependent on the distance from the central meridian in the east-west direction. It is less than 1.0 at the central meridian, equal to 1.0 at the secant lines, and greater than 1.0 when the state plane is above the ellipsoid. For each Wyoming state plane zone, the grid scale factor is 0.9999375 at the central meridian. This equates to a scale factor equal to 1 part in 16,000.
Figure V-14. Grid scale factor.
The other component of the conversion is a function of elevation. An elevation factor is another dimensionless scale factor used to convert distances. This scale factor is used to convert a distance on the ground to an equivalent distance projected onto the ellipsoid. The elevation factor varies as the elevation of the Earth’s surface changes.
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As the ground elevation increases, the distance from the center of the Earth to its surface increases. This distance is equal to the radius of the Earth. As the radius increases, the corresponding arc length also increases. Thus, a distance measured on the ellipsoid is shorter than a corresponding distance measured on the ground due to the longer radius. A distance measured on the surface must be reduced in proportion to the change in radius between the ellipsoid and the surface.
The grid and elevation factors for each project are determined from the adjusted project control positions by GPS post-processing software. The combined datum adjustment factor (DAF) is a product of the grid scale factor multiplied by the elevation factor. State plane coordinates are multiplied by the reciprocal of the DAF to determine corresponding surface coordinates.
Each project control monument is “occupied” by GPS receivers in a series of static and rapid-static networks. The raw GPS data is then adjusted with proprietary post- processing GPS software. Although combined factors are computed for each control monument, a single DAF is used for the entire project. This DAF is an average of the individual DAF values of each project control point. The single adjustment factor does not cause an appreciable loss in accuracy and will eliminate confusion caused by multiple factors. The DAF value is carried out to nine decimal places so that surface coordinates can be accurately calculated to the nearest ten-thousandth of a meter.
Figure V-15. Scale factors.
The purpose of the DAF is to keep surface coordinate computation errors less than 1:50,000 for the entire project. This equates to a linear error of less than 0.02 ft (0. m) in a 1000 ft (305 m) distance. Occasionally, the DAF for an individual control point will differ from the project DAF by more than 0.00002. When this happens, errors greater than 1:50,000 will occur. These situations typically take place on projects that are extremely long, have a considerable elevation difference, or run in a predominantly east- west direction. The project may need to be broken into shorter lengths with a separate
