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