Geodesy#

Geodesy is the science of measuring and understanding Earth’s shape, size, orientation in space, and gravity field.

A reference ellipsoid is a simple, smooth shape used to approximate the Earth’s true form (or that of another planet). It is basically a mathematical definition of a planet’s surface defined as an oblate spheroid, ignoring the bumps and dips caused by mountains, valleys, and other features.

The geoid is a version of Earth’s shape, smoothed out and stripped of mountains, valleys, and other surface features. It represents the average sea level if the oceans were calm, with no tides, currents, or air pressure changes.

The geoid is uneven because of differences in the planet’s gravity caused by variations in its interior composition and density.

For example, its surface sits higher than the reference ellipsoid in areas where there’s extra mass, causing stronger gravity (a positive gravity anomaly.)

On the other hand, it dips below the reference ellipsoid in places where there’s less mass, leading to weaker gravity (a negative gravity anomaly.)

Such changes are, however, very small and are usually measured in milligals. Thus, they are not considered in the reference ellipsoid, and neither in this package.

Since reference ellipsoids are easier to work with than the irregular geoid, they are used for mapping and calculations. They provide the surface on which coordinates like latitude, longitude, and elevation are defined.

World Models#

The two most important geospatial systems used in mapping, navigation, and Earth science applications are the World Geodetic System (WGS84) and the World Magnetic Model.

These systems often work together in navigation devices to provide both positional (WGS84) and directional (WMM) information.

The WGS84 is a standard reference coordinate system for Earth widely used for global positioning systems (GPS) and serves as the underlying geodetic system for latitude, longitude, and altitude coordinates.

WGS84 approximates Earth’s shape using a Reference Ellipsoid model to simplify the calculation of other geodetic parameters, such as the Earth’s radius, circumference, and surface area.

The WMM (World Magnetic Model) is a mathematical model that represents the Earth’s magnetic field. It is used to calculate magnetic declination (the angle difference between true north and magnetic north.)

It provides magnetic field data for navigation systems, including compasses, aircraft, and smartphones.

This model is updated every 5 years (e.g., 2020-2025 is the current model) to account for changes in the Earth’s magnetic field.

These three systems (reference ellipsoid, WGS84, and WMM) are implemented in this package.

The WGS84 model is enough for most applications including the navigation of aircraft, ships, and cars, as well as for surveying and mapping on Earth. Therefore, we will use the WGS84 as the default reference ellipsoid throughout this library for all our implementations.

Classes and Functions#

Additionally, the library provides a set of methods that have been used historically in geodesy, but are not part of the above impllementations.

Reference Frames#

A frame of reference specifies the position of an object in relation to a reference within a coordinate system.

In the context of global navigation, the reference frame is used to define the position of an object in relation to the Earth. The Earth is a non-inertial frame, and the object is usually a sensor or a vehicle.

In this case, the sensor frame is usually attached to the object, and the navigation frame is attached to the Earth.

There main frames are:

  • Earth-Centered Inertial Frame (ECI), also noted as i-frame, has its origin at the center of mass of the Earth. The X-axis points towards the vernal equinox in the equatorial plane. The Z-axis is along the rotation axis of the Earth. The Y-axis completes with a right-hand system. This is also sometimes known as the Celestial Reference System (CRS.)

  • Earth-Centered Earth-Fixed Frame (ECEF), also noted as e-frame, has its origin and Z-axis aligned to the i-frame, but rotates along with the Earth. Therefore, is Earth-Fixed.

  • The North-East-Down (NED) system has its origin fixed at the center of gravity of the aircraft. It is defined relative to a specific location on Earth, described by latitude, longitude and altitude.

  • The local East-North-Up (ENU) coordinates are formed from a plane tangent to the Earth’s surface fixed to a specific location and hence it is sometimes known as a Local Tangent or “Local Geodetic” plane.

  • An Azimuth-Elevation-Range (AER) system uses the spherical coordinates (az, elev, range) to represent position relative to a local origin. The local origin is described by the geodetic coordinates (latitude, longitude, height). Azimuth, elevation, and slant range are dependent on a local Cartesian system, for example, an ENU system.

The most common reference frames transformations used in geodesy are also implemented in the library.