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Ellipsoids and geoids are crucial concepts in geospatial engineering. They provide the foundation for accurate mapping, positioning, and height measurements on Earth's complex surface.

Understanding these models is key to working with GPS data, creating maps, and performing surveying tasks. The relationship between ellipsoids and geoids impacts how we measure and interpret heights in various applications.

Ellipsoids as Earth models

  • Ellipsoids are simplified mathematical models used to represent the Earth's shape and size in geospatial applications
  • They provide a reference surface for geodetic calculations, mapping, and positioning

Ellipsoid definition and parameters

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  • An ellipsoid is a three-dimensional surface formed by rotating an ellipse about its minor axis
  • Key parameters include (a), (b), and (f) which is the difference between a and b divided by a
  • (e) is another important parameter that describes the deviation of the ellipsoid from a perfect sphere
  • Ellipsoid parameters are chosen to best fit the Earth's shape and size based on various geodetic measurements and observations

Commonly used reference ellipsoids

  • Different ellipsoids have been developed over time to better represent the Earth's shape for specific regions or globally
  • Examples include the WGS84 ellipsoid used for GPS, the GRS80 ellipsoid used for many national datums, and the Clarke 1866 ellipsoid used for the North American Datum of 1927 (NAD27)
  • The choice of reference ellipsoid depends on the geographic area, the desired level of accuracy, and the compatibility with existing datasets and systems

Ellipsoid vs sphere comparison

  • While a sphere is a simple and intuitive model of the Earth, it does not accurately represent the Earth's true shape
  • The Earth is an oblate spheroid, slightly flattened at the poles due to its rotation, which is better approximated by an ellipsoid
  • Using an ellipsoid instead of a sphere reduces errors in distance, area, and direction calculations, especially over large distances and near the poles

Geoid overview

  • The geoid is a complex surface that represents the Earth's shape under the influence of gravity and rotation
  • It is a crucial concept in geospatial engineering, geodesy, and related fields

Geoid definition and properties

  • The geoid is defined as the equipotential surface of the Earth's gravity field that best fits
  • It is an irregular surface that accounts for variations in the Earth's density, mass distribution, and gravitational acceleration
  • The geoid is perpendicular to the direction of gravity at every point and extends through the continents
  • It is a more accurate representation of the Earth's shape than an ellipsoid, but it is more complex and difficult to model mathematically

Geoid undulations and height

  • Geoid undulations, also known as geoid heights, represent the separation between the geoid and a reference ellipsoid
  • Undulations can be positive or negative, depending on whether the geoid is above or below the ellipsoid at a given location
  • Geoid heights are important for converting between ellipsoidal heights (obtained from GNSS) and orthometric heights (elevations above the geoid)
  • Undulations typically range from -100m to +100m globally, with larger variations in mountainous regions and areas with significant gravitational anomalies

Geoid models and data sources

  • Geoid models are developed using various data sources, including gravity measurements, satellite observations, and topographic data
  • Global geoid models, such as and , provide geoid heights on a global scale with varying spatial resolutions
  • Regional and local geoid models are created for specific areas to provide higher accuracy and resolution for surveying and engineering applications
  • Geoid models are continually refined and updated as new data and techniques become available, improving the accuracy of height measurements and conversions

Ellipsoid vs geoid

  • Understanding the differences between ellipsoids and geoids is essential for accurate positioning, mapping, and height determination in geospatial applications

Ellipsoid as mathematical surface

  • An ellipsoid is a simple mathematical surface that approximates the Earth's shape and size
  • It is defined by a few parameters (semi-major axis, semi-minor axis, and flattening) and is smooth and regular
  • Ellipsoids are used as a reference surface for geodetic calculations, map projections, and GNSS positioning
  • They provide a consistent and easily computable framework for geospatial data

Geoid as physical surface

  • The geoid is a complex physical surface that represents the Earth's shape under the influence of gravity and rotation
  • It is an equipotential surface of the Earth's gravity field that coincides with mean sea level and extends through the continents
  • The geoid accounts for variations in the Earth's density, mass distribution, and gravitational acceleration
  • It is a more accurate representation of the Earth's true shape but is irregular and difficult to model mathematically

Ellipsoid-geoid separation

  • The separation between the ellipsoid and the geoid is known as the or geoid height
  • Undulations vary from place to place, depending on the local gravity field and the choice of reference ellipsoid
  • Ellipsoid-geoid separation is important for converting between ellipsoidal heights (from GNSS) and orthometric heights (elevations above the geoid)
  • Accurate knowledge of the geoid and its relationship to the ellipsoid is crucial for precise surveying, engineering, and geospatial applications

Geodetic vs orthometric height

  • Geodetic and orthometric heights are two different ways of expressing the vertical position of a point on the Earth's surface
  • Understanding the differences between these is essential for surveying, mapping, and engineering applications

Geodetic height definition

  • , also known as ellipsoidal height, is the vertical distance between a point on the Earth's surface and the reference ellipsoid
  • It is measured along the normal to the ellipsoid and is obtained directly from GNSS observations
  • Geodetic heights are purely geometric and do not account for variations in the Earth's gravity field
  • They are used in GNSS positioning, satellite orbit determination, and some surveying applications

Orthometric height definition

  • is the vertical distance between a point on the Earth's surface and the geoid
  • It is measured along the plumb line (the direction of gravity) and represents the height above mean sea level
  • Orthometric heights are physically meaningful and are used for most surveying, engineering, and mapping applications
  • They are obtained by combining geodetic heights with geoid undulations (N) using the equation: H = h - N, where H is orthometric height, h is geodetic height, and N is geoid undulation

Height differences and applications

  • The difference between geodetic and orthometric heights varies from place to place, depending on the geoid undulations
  • In areas where the geoid coincides with the ellipsoid, the difference is minimal, while in areas with significant undulations, the difference can be several meters
  • Accurate conversion between geodetic and orthometric heights is crucial for integrating GNSS observations with traditional surveying techniques
  • Orthometric heights are used for building and infrastructure design, floodplain mapping, and topographic surveys, while geodetic heights are used for GNSS positioning and geodetic studies

Vertical datums and reference surfaces

  • Vertical datums provide a consistent reference for measuring and expressing heights in surveying, mapping, and engineering applications
  • The choice of vertical datum depends on the geographic area, the available data, and the specific requirements of the project

Local vs global vertical datums

  • Local vertical datums are established for specific regions or countries based on local mean sea level observations and leveling networks
  • Examples include the North American Vertical Datum of 1988 () for the United States and the European Vertical 2000 () for Europe
  • Global vertical datums, such as the International Terrestrial Reference Frame (), provide a consistent reference for global geospatial applications
  • Global datums are based on a combination of GNSS observations, satellite altimetry, and other geodetic techniques

Mean sea level as vertical datum

  • Mean sea level (MSL) is often used as a vertical datum for local and regional applications
  • MSL is determined by averaging tidal observations over a specific time period (usually 19 years) at one or more tide gauges
  • MSL-based datums, such as NAVD88, are used for most surveying and engineering projects in coastal areas
  • However, MSL is not a level surface due to variations in ocean currents, temperature, salinity, and atmospheric pressure, which can cause differences in MSL between locations

Ellipsoid and geoid as vertical datums

  • Ellipsoids and geoids can also serve as vertical datums for specific applications
  • Ellipsoidal heights from GNSS observations can be used directly as a vertical reference for some surveying and mapping projects
  • Geoid models, such as for the United States, can be used to convert between ellipsoidal and orthometric heights
  • The choice of ellipsoid or geoid as a vertical datum depends on the accuracy requirements, the available data, and the compatibility with existing datasets and systems

Coordinate conversions

  • Coordinate conversions are essential for integrating geospatial data from different sources and reference systems
  • Understanding the various height conversions and datum transformations is crucial for accurate surveying, mapping, and engineering applications

Ellipsoidal to geoidal height conversion

  • Converting ellipsoidal heights (h) to geoidal heights (N) involves subtracting the geoid undulation from the ellipsoidal height
  • The equation for this conversion is: N = h - H, where H is the orthometric height
  • Geoid undulations are obtained from geoid models, such as EGM2008 or GEOID12B, which provide undulation values on a grid or as a set of coefficients
  • Accurate ellipsoidal to geoidal height conversions are essential for combining GNSS observations with orthometric heights from leveling or other sources

Geoidal to orthometric height conversion

  • Converting geoidal heights (N) to orthometric heights (H) involves adding the geoid undulation to the geoidal height
  • The equation for this conversion is: H = h - N, where h is the ellipsoidal height
  • This conversion is used to obtain physically meaningful heights (elevations above the geoid) from GNSS-derived ellipsoidal heights
  • Accurate geoidal to orthometric height conversions are crucial for applications such as floodplain mapping, infrastructure design, and topographic surveys

Datum transformations and considerations

  • Datum transformations are necessary when working with geospatial data referenced to different horizontal or vertical datums
  • Horizontal datum transformations, such as between NAD27 and NAD83, involve shifts, rotations, and scale changes to account for differences in the underlying reference frames
  • Vertical datum transformations, such as between NGVD29 and NAVD88, involve applying a set of corrections based on leveling observations and gravity data
  • When performing coordinate conversions and datum transformations, it is important to consider the accuracy of the input data, the choice of transformation method, and the potential propagation of errors
  • Proper documentation and metadata are essential to ensure the transparency and reproducibility of coordinate conversions and datum transformations

Gravity and geoid modeling

  • Understanding the Earth's gravity field and its variations is fundamental to and accurate height determination in geospatial applications

Earth's gravity field and variations

  • The Earth's gravity field is a complex and dynamic phenomenon influenced by the distribution of mass within the Earth, its rotation, and external gravitational forces
  • Gravity varies with latitude, elevation, and the local density and topography of the Earth's surface
  • Temporal variations in gravity occur due to tidal forces, changes in groundwater and ice mass, and other geodynamic processes
  • Accurate modeling of the Earth's gravity field is essential for geoid determination, orbit determination, and other geodetic applications

Gravity measurements and techniques

  • Gravity measurements are obtained using various instruments and techniques, including gravimeters, gravity gradiometers, and satellite observations
  • Terrestrial gravimeters measure the absolute or relative acceleration due to gravity at a specific location on the Earth's surface
  • Airborne and shipborne gravimeters measure gravity variations over larger areas, while satellite gravimetry provides global coverage of the Earth's gravity field
  • Satellite missions, such as GRACE and GOCE, have revolutionized our understanding of the Earth's gravity field and its temporal variations

Geoid determination methods

  • Geoid determination involves combining gravity measurements, satellite observations, and other geodetic data to model the shape and undulations of the geoid
  • The Stokes integral is a classical method for computing geoid undulations from gravity anomalies, which are the differences between observed and theoretical gravity values
  • Modern geoid determination methods, such as the remove-compute-restore (RCR) technique, combine global models, local gravity data, and terrain information to improve the accuracy and resolution of geoid models
  • Least-squares collocation (LSC) is another method that uses statistical interpolation to estimate geoid undulations from heterogeneous data sources
  • Ongoing research and advancements in geoid modeling aim to improve the accuracy and resolution of geoid models for various applications in geospatial engineering and Earth sciences

Applications in geospatial engineering

  • The concepts of ellipsoids, geoids, and height systems have numerous applications in geospatial engineering, surveying, and related fields

Surveying and leveling

  • Ellipsoids and geoids provide the reference surfaces for surveying and leveling operations
  • Geodetic surveying techniques, such as GNSS and total station measurements, rely on accurate knowledge of the reference ellipsoid and its relationship to the local vertical datum
  • Leveling is used to establish orthometric heights and vertical control networks based on the geoid or a local vertical datum
  • Combining GNSS observations with leveling data requires accurate geoid models and height conversions to ensure consistency and accuracy

GNSS height measurements

  • GNSS (Global Navigation Satellite Systems) provide ellipsoidal heights based on the reference ellipsoid used by the system (e.g., WGS84 for GPS)
  • Converting GNSS-derived ellipsoidal heights to orthometric heights requires accurate geoid undulation information
  • Geoid models, such as GEOID12B or EGM2008, are used to interpolate geoid undulations at the location of GNSS measurements
  • Accurate GNSS height measurements and conversions are essential for applications such as land surveying, construction, and monitoring of vertical deformation

Terrain modeling and analysis

  • Digital Elevation Models (DEMs) and Digital Terrain Models (DTMs) are used to represent the Earth's surface for various applications, such as hydrology, land use planning, and infrastructure design
  • DEMs can be referenced to either the ellipsoid or the geoid, depending on the data source and the intended application
  • Geoid undulations are used to convert between ellipsoidal and orthometric heights in DEM generation and analysis
  • Accurate terrain modeling and analysis require consistent vertical reference systems and proper handling of height conversions and datum transformations
  • Applications such as floodplain mapping, watershed analysis, and infrastructure planning rely on accurate and properly referenced terrain data
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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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