Modeling the Geoid – local and global

Introduction

This section will introduce the basic principles of how and why an ellipsoid is used to model the Geoid surface of the Earth’s gravity field.  The ellipsoidal surface, and thus by definition the ellipsoidal coordinate system, when ‘tied’ to the Geoid surface will create a geodetic datum.  Two approaches are described in this module, known as:  Single Astronomical Point and Astronomical-Geodetic orientation. These address both the creation of local geodetic datums although the latter could be extended to create a global datum.  

 

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Slide 338

These are the basic components of a coordinate system and in particular an ellipsoidal coordinate system. This is a conceptual mathematical model that as yet has no real world use.  It describes a simple ellipsoidal surface through which three axes are superimposed (X, Y and Z) to create a Cartesian coordinate frame.  Coordinates of any point on the surface of the ellipsoid can be described using two angular coordinates which will be called latitude and longitude. At this point they have no true value until the coordinate system is tied to a body to create a geodetic datum.  A third component will describe the elevation of a point above or below the ellipsoidal surface.

 

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When the ellipsoidal surface (and associated coordinate frame) are tied to a body it will create a geodetic datum. The system now has some real benefits for describing absolute coordinates of points on that body.  However, before the ellipsoid is tied to the Earth it is initially required to determine what size of ellipsoid to use.  This depends on geographical location over which the geodetic datum will be used.  Many different ellipsoids have been computed historically and they are derived for geographical locations spanning the entire globe. 

 

 

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The traditional method of determining the size of an ellipsoid (for use in a specific geographic location) is to compute the length of a baseline connecting two survey stations that both lie on or very close to the same meridian, e.g. line of longitude, passing through the area of use. At each survey station the astronomical latitude of that point is determined.  This will enable the angular separation of the two stations to be determined along with the distance between them.

 

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Using these parameters it is possible to determine the radius of the ellipsoid relating to that section of the ellipsoids circumference. These values will then allow a determination to be made as to which ellipsoid parameters will best describe the curvature of the Earth in that geographic region.  This will become the ellipsoid used to model the Geoid over that same area.

 

What follows are high level descriptions of the two principle methods used to tied the ellipsoid to the geoid for the purposes of creating a geodetic datum 

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The two methods do overlap with some of their requirements but differ in the number of survey stations used to compute the fit between the ellipsoid and geoid.  The salient points for each method will be described in the following slides.  First, the single astronomical point method will be discussed.

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A survey station is selected that should be close the center of the area over which the geodetic datum will cover.  This station will act as the origin of the datum, which will become the fundamental point of the datum.

 

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The survey equipment is aligned within the Earth’s gravity field and thus the plumb bob will be aligned with the vertical, which is perpendicular to the Geoid surface or another equi-potential surface.  Next, astronomical observations are made to determine astronomical latitude and astronomical longitude of that point. Finally, astronomical azimuth is determined between the survey station and another survey station within the network.

  

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For the single astronomical point method the geoid surface and the ellipsoidal surface are considered to be coincident. Therefore, astronomical latitude and longitude (referenced to the geoid) will equal geodetic latitude and longitude (referenced to the ellipsoid).  Additionally, astronomical azimuth will equal geodetic azimuth.   A station with these parameters is known as a Laplace Station.

 

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The astronomical and geodetic coordinates are the same therefore the Vertical and the Normal will be parallel and perpendicular to the geoid and ellipsoid respectively.  As such, the Deviation of the Vertical will be zero at this point.

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Although the Deviation of the Vertical will be zero at the origin it will not be zero at any other point in the network because of the assumptions made at the origin.  As the ellipsoid is not aligned along the mean spin axis of the Earth this contributes to the errors that will accumulate in the network away from the origin.

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The second method is known as the Astronomical-Geodetic method (Astrogeodetic method).  In this method multiple stations are used to determine astronomical coordinates and azimuths rather than just one in the previous method.   

 

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Astronomical coordinates are referenced to the Gravity field of the Earth because the survey equipment used to derive the position of points is aligned in the Gravity field.  When a tripod is set up the plumb bob is aligned to the vertical which is the perpendicular plane to the equi-potential surfaces.  

 

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Slide 53Ellipsoidal coordinates are referenced to the ellipsoidal coordinate system. These are not related to the Earth’s gravity field but instead to the ellipsoid surface whose perpendicular plane to that surface is called the Normal.  Therefore, geodetic latitude is defined as being referenced to that plane with the angle it subtends with the equatorial plane of the ellipsoid coordinate system. Geodetic longitude is referenced to a zero plane called in Prime Meridian, which for most global applications is the Meridian line passing through the Greenwich observatory in London.

 

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Flattening is the amount of ‘squashing’ the Earth experiences as a result of the equatorial bulge created by the Earth’s rotation and resultant centrifugal forces. 

 

 

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A list of some of the common ellipsoid definitions used to create both local and global geodetic datums.  For most local geodetic systems the name of the ellipsoid is accredited to its creator along with the year in which it was published.  For example, George Biddell Airy created the reference Geoid for United Kingdom in 1830 and his ellipsoid definition is still used by the Ordnance Survey of Great Britain (OSGB) today as it fits the Geoid of the UK better than its newer global equivalents.  WGS 84 and WGS 72 belong to the group of ellipsoid definitions known as the World Geodetic System (WGS).  These two long with GRS 80 are used to model the geoid surface over the entire globe and not just for localized areas that are addressed by the local geodetic systems. 

 

 

 

Other considerations

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Slide xx - geoid model

The ellipsoid of the reference ellipsoid is a smooth continuous surface.  Due to sub-surface geological variations the Geoid is not and varies geographically.  Variations in the height of the geoid with respect to the reference ellipsoid are known as Geoid anomalies. The longer wavelength anomalies are accredited to deeper sub-surface Mantle density variations caused by convection currents and geoid highs are often associated with active subduction zones where higher density continental crust over rides lower density oceanic crust.  The cause of geoid lows is still a cause of uncertainty with contrasting opinion but they are thought to be a result of deep seated earth structures (perhaps related to the core-mantle boundary).   

 

 

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The ellipsoid is a mathematical representation of the shape of the Earth used to model the Geoid.  As the radius of curvature differs geographically is will affect what shape of ellipsoid best fits the Earth’s shape locally.  The derivation of the many different ellipsoid models used within local geodetic datums is testimony to this.  See slide 14, previous section.

 

 

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The accuracy with which the ellipsoid surfaces models the Geoid surface is important in adopting it within the definition of a geodetic datum.  Two main considerations are addressed with respect its acceptability, namely:  

  • Firstly, the vertical separation between the two surfaces and
  • Secondly, the angular separation between the Vertical (perpendicular surface to the Geoid) and the Normal (perpendicular surface to the ellipsoid).  

The Geoid is an undulating surface, the ellipsoid is not.  How well the latter models the undulations of the former is key to it being adopted.

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The vertical separation between the Geoid and Ellipsoid will describe the Geoid anomalies occurring within the area of interest. The ellipsoidal surface will be fitted to the Geoid such that all values of N along the profile, when squared and summed together will be a minimum.  This is performed as a least squares adjustment to derive the best fit.  The vertical separation between the two surfaces is known as the Geoid-Ellipsoid separation (N), and maximum values for anywhere on the global is about +/-100 meters. 

 

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Density variation in the Mantle (Upper and Lower) will account for the longer wavelength geoid anomalies.  Shorter wavelength harmonics will be superimposed on the longer fundamental frequencies, which result from near surface geological variations in the top 50 kms. These effects will cause an angular deviation between the perpendicular plane to the Geoid (called the vertical) and the perpendicular plane to the Ellipsoid (called the normal).  The amount the vertical deviates from the normal must also be minimized over the area of interest for the ellipsoid to be considered a ‘best fit’.    

 

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END OF MODULE – RETURN TO – INTRODUCTION TO GEODESY