5.1.1.  International Organization for Standardization

The International Organisation for Standardization (ISO) is a global network of 167 national Standards bodies with one member per country. It is an independent, non-governmental organization that provides a platform for developing practical tools through common understanding and cooperation with all stakeholders. ISO develops, approves and publishes Standards for everything except electrical and electronic engineering and telecommunication.

5.1.2.  Open Geospatial Consortium

The Open Geospatial Consortium (OGC) is an open-membership, worldwide voluntary consensus Standards organization that defines, documents, approves, and maintains geoprocessing Standards. OGC geospatial Standards can be used in geographic information systems (GIS) and systems for Earth imaging, Web mapping, location-based services, surveying and mapping, CAD-based facility management, webs of geolocated sensors, navigation, cartography, automated mapping, etc. The OGC Standards define open interfaces, protocols, schemas, and other components. Implementation of these Standards enable different systems and applications to exchange geospatial data and instructions and enable the complete integration of these capabilities into a range of information systems. The OGC collaborates with other Standards organizations such as the W3C and ISO. Since several Standards were jointly developed by ISO and the OGC they will be discussed together.

5.1.3.  Consultative Committee for Space Data Systems

Another organization that often cooperates with ISO is the Consultative Committee for Space Data Systems (CCSDS). CCSDS was formed by the major space agencies of the world to provide a forum for discussion of common problems in the development and operation of space data systems. CCSDS is currently composed of 11 member agencies, 32 observer agencies, and over 119 industrial associates. CCSDS Standards are widely used in aerospace. The publications of the CCSDS can be divided into different categories, color coded. The most important are Recommended Standards (Blue), Recommended Practices (Magenta), and Informational Reports (Green). Therefore, the CCSDS does not necessarily provide Standards but rather conventions and practices.

5.1.4.  International Earth Rotation and Reference Systems Service

The main objective of the International Earth Rotation and Reference Systems Service (IERS) is the suggested conventions regarding various transformations and reference system definitions that are widely used in geodesy and of great importance for the work defined in this ER. The IERS provides definitions of various coordinate reference systems and their realizations as well as the Earth orientation parameters required to study Earth orientation variations to transform between the defined CRSs. For all systems and frames, the associated Standards, constants and models are given.
A detailed discussion of the IERS observations, conventions, and more is made in Clause 6 — Clause 8.

5.1.5.  NASA’s Navigation and Ancillary Information Facility

NASA’s Navigation and Ancillary Information Facility is responsible for developing a tool that will be discussed later in this ER. NASA’s Navigation and Ancillary Information Facility (NAIF) was established at the Jet Propulsion Laboratory to lead the design and implementation of the “SPICE” ancillary information system. The NAIF is dedicated to the issues of producing high precision, clearly documented and readily used “ancillary information” required by space scientists and engineers.

5.1.6.  International Astronomical Union

The International Astronomical Union (IAU) was founded in 1919 to promote and safeguard the science of astronomy in all its aspects, including research, communication, education, and development, through international cooperation. Its members — structured into Divisions, Commissions, and Working Groups — are 12113 astronomers active in professional research and education in astronomy from 93 countries worldwide. Among other activities, it acts as the recognized authority for assigning designations and names to celestial bodies (stars, planets, asteroids, etc.) and any surface features on them.
The Working Group on Cartographic Coordinates and Rotational Elements was established with the purpose to avoid a proliferation of inconsistent cartographic and rotational systems, and therefore to define the cartographic and rotational elements of the planets and satellites on a systematic basis and to relate the new cartographic coordinates rigorously to the rotational elements.

5.2.1.  Referencing by coordinates

ISO 19111:2019 Geographic information — Referencing by coordinates is arguably the most important standard addressed by this initiative. TC211 and OGC strive to maintain a separation between coordinate values and the coordinate reference system where those values apply. This allows new coordinate reference systems (projections, geoids, etc.) to be defined without impacting existing standards and implementations. However, this requires that there be a standard way to describe a coordinate reference system. ISO 19111 fills this need by defining “the conceptual schema for the description of referencing by coordinates” and the data required to define a coordinate reference system. ISO 19111 compliant registries allow applications to retrieve, interpret, and apply these coordinate reference system definitions at run time. This is a key capability enabling geospatial technologies to be applied to non-Earth locations.

A detailed description of this ISO 19111 can be found in the OGC Testbed 18 D025 — Reference Frame Transformation Engineering Report (OGC 22-038) and is therefore not discussed here.

5.2.2.  Temporal Schema

ISO 19108:2002 Geographic information — Temporal schema defines concepts for describing temporal characteristics of geographic information. The majority of this Standard addresses the representation of Earth-centric systems (calendars) for dates. However, it also provides a framework for representing time. This framework allows for the association of spatial and temporal reference systems. These associations make it possible to define a spatial-temporal reference system; a coordinate reference system for 4D space-time. This concept is explored in greater detail within this Engineering Report and in D025 — Reference Frame Transformation Engineering Report.

5.2.3.  A Feature Model for Space Objects

Before we can describe non-terrestrial objects, we must first have a conceptual model of the universe. The OGC and ISO TC211 provide that model in ISO 19109:2015 Geographic information — Rules for application schema. 19109 defines the General Feature Model (GFM). The GFM defines the concept of a Feature, its components, and behaviors.

The OGC and ISO define a Feature as an “Abstraction of real-world phenomena” (ISO 19101-1:2014).

A Feature, then, is a high-level abstraction for anything that does or could exist in the universe.

The General Feature Model further refines the concept of a Feature. The following principles are most relevant for this ER.

1. A Feature can be a FeatureType or an Instance of a FeatureType (AnyFeature).

2. FeatureTypes can form a taxonomy (inheritance).

3. Features possess characteristics (Properties).

4. A Property of a Feature can be an Operation, Attribute, or Association.

Figure 1 — General Feature Model

The resulting model is sufficient to describe a Feature’s identity (IdentifiedType), what it is (FeatureType), what it can do (Operations), the Feature’s observable characteristics (Attributes), and any associations with other Feature instances.

5.2.4.  Geometry in 3 Dimensions

The applicable OGC/ISO standard for geometries is ISO 19107:2003 Geographic information — Spatial schema. While a new version was approved in 2019, it hasn’t propagated through the rest of the ISO standards baseline. As a result, 19107:2003 is the most recent implemented version.

5.2.4.1.  Features and Geometry

The General Feature Model treats geometry as an attribute of the Feature. In addition, it defines the following three types of attributes which are useful for associating geometry with a Feature in a standard manner.

• SpatialAttributeType: Geometries (GM_Object) and Topologies (TP_Object)

• LocationalAttributeType: Named locations, extents, and points

• TemporalAttributeType: Temporal objects (TM_Object)

Figure 2 — General Feature Model Attribute Types

Figure 3 — General Feature Model Spatial Attribute Types

Figure 4 — General Feature Model Locational Attribute Types

Figure 5 — General Feature Model Temporal Attribute Types

5.2.4.2.  The Geometry Model

An important feature of the ISO Geometry Model is that it does not specify or assume a coordinate reference system. The SC_CRS class, defined in ISO 19111, is used to define a coordinate reference system. The “Coordinate Reference System” association is used to associate a GM_Object instance with the appropriate SC_CRS instance. Since all geometry classes are descended from GM_Object, any geometry object can have its own unique coordinate reference system.

Figure 6 — GM_Object UML Model

While the ISO Geometry Model is very complex, at its core is the DirectPosition class. This is the fundamental specification of a position within a coordinate reference system. Its purpose is simply to hold the coordinates for a position within the specified coordinate reference system.

Figure 7 — Direct Position UML Model

The DirectPosition class contains two attributes, “dimension” and “coordinate”. The “coordinate” attribute is a sequence of numbers. Each number represents the location of the DirectPosition on a coordinate axis. There are no constraints on the number of numbers in the sequence. Therefore, a DirectPosition can represent an unlimited number of dimensions. The “dimension” attribute specifies the number of axis in the applicable coordinate reference system. This corresponds to the number of values in the “coordinate” sequence.

Like GM_Object, DirectPositions are associated with an SC_CRS through the “coordinateReferenceSystem” association. This association is typically not used since DirectPositions, as data types, will usually be included in larger objects (such as GM_Objects) that have their own references to SC_CRS. When this association is left NULL, the coordinate reference system of the DirectPosition instance will take on the value of the containing object’s SC_CRS.

One limitation of the DirectPosition class is that it does not support complex numbers. However, since DirectPosition does have a “coordinateReferencesystem” association with SC_CRS, it should be possible to model complex numbers in the CRS definition as two orthogonal axis.

5.2.4.4.  3D Geometries

ISO 19107 makes a distinction between a geometric object and the surface which contains that object. One advantage of this approach is that there can be multiple surfaces associated with one object. For example, an island located in a lake would be represented by an interior surface (the island) of a polygon (the lake) bounded by the exterior surface (the shoreline).

ISO 19107 uses the GM_Object class to define an object and the GM_Boundary class to define a containing surface. Both GM_Object and GM_Boundary are defined as root level geometry classes. The association between GM_Object and GM_Boundary is achieved through the “boundary()” operation on the GM_Object class. This operation is inherited by all subclasses of GM_Object.

In the case of a 3D Feature, GM_Solid is the subclass of GM_Object while GM_SolidBoundary is the subclass of GM_Boundary. GM_Solid describes the volume while GM_SolidBoundary describes the shape.

Figure 8 — 3D Geometry UML Model

5.2.4.6.  Shapes

A 3D volume is delineated by a bounding surface. GM_Boundary is the root class for boundaries. The subclass GM_PrimitiveBoundary provides the boundary for GM_Primitives. The GM_PrimitiveBoundary subclass GM_SolidBoundary is defined as the boundary for a GM_Solid.

Figure 9 — 3D Boundaries UML Model

ISO 19107 goes even farther. A GM_SolidBoundary is composed of both interior and exterior boundaries. These boundaries are defined by the GM_Shell class. Therefore, the following can be observed.

• A GM_Shell is a GM_CompositeSurface

• A GM_CompositeSurface is composed of GM_OrientablePrimitives

• A GM_Surface is a type of GM_OrientablePrimitive

• A GM_PolyhedralSurface is a type of GM_Surface

• A GM_PolyhedralSurface can be composed of GM_Polygons

A GM_PolyhedralSurface which is composed of GM_Polygons is an example of Boundary Representation (B-Rep) of a surface. This approach is fundamental to rendering 3D computer graphics. (ref Adam Powers 1981)

5.2.4.6.1.  Closure Surfaces

Some structures, such as a tunnel or overpass, pose difficulties for this geometry model. The boundary surface can be constructed so that it continues into the interior of the structure. That would make the interior of a tunnel external to the tunnel object. This is not always a desireable result. CityGML 3.0 addresses this issue thought the concept of a “Closure Surface”.

A Closure Surface is a surface which is a logical part of the object but does not correspond to a physical part of the object. For example, the entrance to a tunnel can have a closure surface. This surface allows the tunnel to be treated as a three-dimension solid, even though there is a hole in the bounding surface.

Figure 10 — Closure Surface UML Model

As implemented in CityGML 3.0, the ClosureSurface class an impressive history. This concept might need to be generalized for use outside of CityGML. However, the capabilities provided by the ancestor classes do provide value and may be worth incorporating into a general 3D model.

5.2.4.6.2.  B-Rep

The polyhedral surfaces which bound volumetric shapes are similar to the Boundary Representation (B-Rep) approach used in CAD and computer graphics. B-Rep defines a 3-dimensional surface which serves as the interface between the interior of the volumetric shape and the exterior. This surface is usually defined by a collection of shape elements (polygons) which together form a closed surface.

https://en.wikipedia.org/wiki/Boundary_representation

5.2.4.6.3.  Point Clouds

Boundary surfaces can also be defined using 3D point clouds. This allows the spatial representation of a bounding surface by a set of points located on that surface. In this way, the geometry of a Feature could, for instance, be modeled directly from the result of a mobile laser scanning campaign.

5.2.5.  Moving Features

The ISO Standard for Moving Features is ISO 19141:2008 Geographic information — Schema for moving features. This Standard extends the geometry model from ISO 19107 and, by association, the Feature Model from ISO 19109.

Figure 11 — Moving Features

A high-level view of ISO 19141 is provided in Figure 11. The classes identified in this figure are described below. But first, a discussion of coordinate reference systems (CRS) is in-order.

5.2.5.1.  Coordinate systems for Moving Features

Figure 12 — External, Global, and Local CRS

Moving Features deal with three spatial coordinate systems as well at one temporal coordinate system. The spatial coordinate systems are referred to as the External, Global, and Local CRS (see Figure 12).

When dealing with Moving Features, converting coordinates between the three spatial CRS is frequently required. The GM_Object class provides the transform() operation which can be used for this purpose.

5.2.5.1.1.  External CRS

The External coordinate system is the coordinate system within which the Moving Feature exists. Typically, this is an Earth-centric geographic CRS such as WGS 84. In the ISO 19111 model this would be a geodetic coordinate reference system.

5.2.5.1.2.  Local CRS

The local coordinate system is internal to the Feature. This is usually a cartesian coordinate system with the origin at a prominent point in the Feature such as the center of mass. In the ISO 19111 model this would be an engineering coordinate reference system.

For rigid bodies, the local coordinate system is fixed over time. It does not change regardless of any motion by the associated Feature.

5.2.5.1.3.  Global CRS

The Global coordinate system provides a transition between the External CRS and Local CRS. Global CRS is a moving CRS which follows the trajectory of the Moving Feature. As such, it provides the External CRS a time variant local reference system, while providing a time invariant context for the Local CRS.

The origin of the Global CRS is the location of the Moving Feature on the trajectory curve at a specific time. From the perspective of the External CRS, the origin translates and rotates as a function of time. From the perspective of the Internal CRS, however, the origin of the Global CRS is static.

The axis of the Global CRS can be defined using two techniques.

The first approach treats the Moving Feature as a black box viewed by an external observer. From this perspective, the Global CRS is defined in terms of the trajectory alone. No knowledge of the properties or even the shape of the Moving Feature are required. This CRS starts by defining x and y as two orthogonal axis which define a plane tangential to the Trajectory curve at the origin. Positive x is in the direction of motion. The positive y axis is perpendicular to the x axis and forms either a right or left handed CRS. The z axis is perpendicular to the tangent plane. Positive z can be either up or down. The result is a cartesian reference system which is always tangential to the trajectory of the Moving Feature.

The second approach defines the Global CRS in terms of motion properties of the Moving Feature. The x axis, for example, could be defined as the heading of the Moving Feature. This is the direction the Feature is pointing, but not necessarily the direction it is moving. The y and z axis can be defined using similar properties (for example; pitch, yaw, and roll).

In general, the black box approach is appropriate for ballistic Moving Features such as in cases where the Feature is not anticipated to take any actions which would modify the trajectory. The motion properties approach is appropriate for navigated Moving Features such as in cases where a Feature is expected to take actions which would modify the trajectory.

{
    "type": "Feature",
    "id": "A",
    "geometry": {
        "type": "LineString",
        "coordinates": [[11.0,2.0,50.0], [12.0,3.0,52.0], [10.0,3.0,56.0]]
    },
    "properties": {
        "datetimes": ["2012-01-17T12:33:51Z", "2012-01-17T12:33:56Z", "2012-01-17T12:34:00Z"],
        "state": ["walking", "walking"],
        "typecode": [1, 2]
    }
},

5.2.5.5.  Temporal Properties

The OGC Moving Features JSON encoding standard introduces the concept of temporal properties.

“A TemporalProperties object is a JSON array of ParametricValues objects that groups a collection of dynamic non-spatial attributes and its parametric values with time.”

Logically TemporalProperties should be a subclass of MF_OneParamProperties. Since Geometry is a property, then MF_TemporalGeometry should be a subclass of TemporalProperties, which results in the following UML.

Figure 14 — Temporal Properties

Temporal properties are particularly useful for capturing state change. For example, the fuel load of an aircraft will change over time. The leafproperty() operation on a temporal fuel_load object would return the amount of fuel onboard at the specified time.

5.2.5.6.  Location

ISO 19141 represents the location of a Moving Feature using two classes: MF_Trajectory and MF_TemporalTrajectory.

Figure 15 — Trajectory

A MF_Trajectory is a curve (GM_Curve) representing every position that the Feature has occupied during it’s journey and does not necessarily represent the time when each location was reached.

MF_TemporalTrajectory makes the MF_Trajectory a MF_TemporalGeometry which represents location along the trajectory as a function of time. Therefore, each location is fully defined in both space and time.

A Temporal Trajectory has two operations of particular interest; leaf() and leafgeometry(). The input parameter for these operations is always time (TM_Coordinate). The leaf() operation returns the spatial location (Direct_Position) that the Moving Feature passes at the time (TM_Coordinate) specified by the input parameter which is a point on the trajectory GM_Curve geometry. The leaf() operation also serves as the origin of the Global CRS at that location on the trajectory.

The LeafGeometry() operation returns the spatial geometry (GM_Point) that this Moving Feature possesses at the time (TM_Coordinate) specified by the input parameter is the shape of the Moving Feature expressed in the Local CRS. Since Trajectories only convey location, only GM_Point geometries are supported.

5.2.5.7.  Orientation

5.2.5.7.1.  MF_PrismGeometry

If an application focuses on only the linear movement (i.e., the spatiotemporal line string) of moving points based on World Geodetic System 1984, with longitude and latitude units of decimal degrees, and the ISO 8601 standard for representation of dates and times using the Gregorian calendar, the application can share the trajectory data by using only IETF GeoJSON, called MF-JSON Trajectory. For other cases, MF-JSON Prism can be used for expressing more complex movements of moving features. MF-JSON Prism is a GeoJSON-like format reserving new members of JSON objects (“temporalGeometry,” “temporalProperties,” “crs,” “trs,” “time,” and others) as “foreign members” to represent spatiotemporal geometries, variations of measure, coordinate reference systems, and the particular period of moving features in a JSON document.

A trajectory provides the location of a Moving Feature as a function of time. Prism Geometry represents the full geometry (location, orientation, and shape) of the Feature as a function of time.

Figure 16 — Foliation

The key concepts in the Prism model are as follows.

Leaf: A leaf is the geometry of the Moving Feature at time (tn).

Foliation: A collection of leaves where there is a complete and separate representation of the geometry of the Feature for each specific time (tn).

Trajectory: A curve that represents the path of a point in the geometry of the Moving Feature as it moves with respect to time (t).

Prism: The union of the geometries (or the union of the trajectories) in a foliation.

Like a Temporal Trajectory, a Prism is a subclass of MF_TemporalGeometry.

Figure 17 — Prism Context

A MF_PrismGeometry class has the following characteristics.

The association role “originTrajectory” associates a Temporal Trajectory with a Prism geometry. For any TM_Position the following hold true.

1. The associated Temporal Trajectory provides the location of the Moving Feature in the Global CRS.

2. This location serves as the origin of the Local CRS.

3. The prism geometry is defined in that specific Local CRS.

The localCoordinateSystem() operation returns a SC_CRS for the design coordinate reference system in which the moving feature’s shape is defined. This is usually the same as the local coordinate system.

The rotationAtTime() operation accepts a time in the domain of the prism geometry and returns the rotation matrix that embeds the local geometry into geographic space at a given time (TM_Coordinate). The vectors of the rotation matrix allow the feature to be aligned and scaled as appropriate to the vectors of the global coordinate reference system.

This one association and two operations provide the location, orientation, axis definition, and units of measure needed to define the local CRS and to transform geometries between the Local and Global CRSs.

Finally, the geometryAtTime() operation accepts a time in the domain of the prism geometry and returns the geometry of the moving feature, as it is at a given time in the global coordinate reference system. The return type is a GM_Object so this operation is not limited to points. It is fully capable of representing a 3D surface and volume.

In short, a MF_PrismGeometry provides the shape, location, and orientation of a Moving Feature as a function of time (tn).

5.4.1.  Coordinate System Definitions and Specifications

According to the CCSDS, in order to define a coordinate system unambiguously the following elements are needed.

• A coordinate frame which is defined as an associated set of mutually orthogonal Cartesian axes.

• A frame origin which is the common origin of the Cartesian axes (e.g., center of mass of Earth, satellite etc.).

• A reference plane which is the xy-plane in a coordinate frame and therefore defines the direction of the z-axis such as the Earth equator or the elliptical plane.

• A reference direction which defines the direction of the x-axis (e.g., vernal equinox).

The definition of a reference plane and direction can be done in different ways. Some realizations would be as follows.

• Pointing to a fixed direction in inertial space (e.g., toward a quasar)

• Parallel to the distance vector between one object and another

• Parallel to an object’s velocity vector

• Pointing from the origin through the intersection of two defined planes

• Parallel to an object’s spin axis

• Along a body axes

• Normal to an object’s orbit

In current practice, most navigation messages between agencies use only a small subset of reference systems and frames. The most used are the International Celestial Reference System (ICRS) and the International Terrestrial Reference System (ITRS) defined by the IERS. A detailed discussion follows later in this report. In short, the ICRS and ITRS could be summarized as follows.

ICRS

Inertial barycentric reference system whose axes are defined by the measured positions of extragalactic sources (mainly quasars).

ITRS

Terrestrial Earth-fixed reference system for measurements on locations relatively near the Earth’s surface.

Since many coordinate systems use some sort of equator as the reference plane, motions of this plane must be compensated for. The effects can be separated into long-term (precession )and short-term effects (nutation). Due to those effects, the CCSDS distinguishes between True of Date (TOD) and Mean of Date (MOD). The mean equator and vernal equinox of a given date define a MOD coordinate system, which includes long-term, but not short-term effects. A given date’s true equator and equinox, which can be obtained by applying short-term effects to the mean values, define a TOD coordinate system. A detailed description of long- and short-term effects is done in Clause 6. The Greenwich True of Date Coordinate System (GTOD) is the last important coordinate reference system on the list. In contrast to different realizations of the ITRS (the so-called frames, e.g., ITRF2000), the GTOD corresponds to the respective true point in time. GTOD is defined as follows.

• Frame origin: Center of the Earth

• Reference plane: The Earth’s true-of-date Equator (therefore the z-axis is directed along the Earth’s true-of-date rotational axis and is positive north)

• Reference direction: The positive x-axis is directed toward the prime meridian

• The y-axis completes a right-handed system

Important reference frames defined using the orbital position and velocity at a given time are used in aerospace for attitude estimation, attitude control, and orbital relative motion. All the local orbital frames described in this convention are rotating coordinate systems unless specified otherwise in the context of some specific data exchange between participants. These systems can be used to study the relative motion between spacecraft.

5.4.2.  Time

Since the CCSDS sees the world from the aerospace point of view,in contrast to the ISO Standard, a main chapter deals with coordinate systems in reference to time and time systems. In detail, the exact definition and understanding of time systems is essential for the following.

• The modeling of satellite orbits and attitude

• Processing of navigation data

• Satellite ground operations

Furthermore, different time systems are represented in the convention. Since those are specified in detail in Clause 8, here a detailed representation is renounced. The time systems can be summarized as follows.

Terrestrial Time (TT)

Conceptually uniform time scale that would be measured by an ideal clock on the surface of a geoid.

Geocentric Time (GT)

Differs from Terrestrial Time by removing the effects of the Earth’s motion.

International Atomic Time (TAI)

Practical realization of a uniform time scale based on atomic clocks and agrees with TT (except for offset of 32.184s).

GPS Time

Differs from TAI in the chosen offset and the choice of atomic clocks used in its realization.

Greenwich Mean Sidereal Time (GMST)

Defined as the Greenwich hour angle of the mean vernal equinox of date.

Universal Time (UT)

Defined as solar time,

Universal Time Coordinated (UTC)

Atomic time, which is adapted to the universal time.

Barycentric Dynamical Time (TDB)

Independent variable of current barycentric solar system ephemerids.

Barycentric Coordinate Time (TCB)

Relativistic time coordinate of the 4-dimensional barycentric coordinate system.

5.5.1.  Time Systems

Every event that can be calculated in SPICE is associated to an epoch. An epoch is an instant in time. Clocks are used for the realization of this. Clocks count epochs specified by events such as: “regular” oscillations of a pendulum, quartz crystal, or electromagnetic radiation from a specified source, measured from an agreed upon reference epoch. A time system is an agreed upon standard for “naming” epochs, measuring time, and synchronizing clocks. SPICE divides all time systems into three categories.

• Ephemeris Time (ET) or (the newer name) Barycentric Dynamical Time (TDB)

• Universal Time Coordinated (UTC) which corresponds to the standard time when using calendars

• Spacecraft Clocks

The transformations between those time systems are implemented in the SPICE-toolkit.

5.5.2.  Reference Frames and Coordinate Systems

The next important calculation step is the definition of the reference frame in which the calculations should take place. In SPICE, a reference frame is an ordered set of three mutually orthogonal (possibly time dependent) unit-length direction vectors, coupled with a location called the reference frame’s center or origin. All reference frames in SPICE are right-handed and a reference frame’s center must be a SPICE ephemeris object whose location is coincident with the origin of the frame. Reference frames can be divided into two main columns.

6.1.1.  Hierarchy within the space-fixed systems

Within the space-fixed reference system there are different approximations, in which the term “inertial” is defined more or less strictly.

6.2.1.  Hierarchy within the Earth-fixed system

Further, there are different gradations within the Earth-fixed reference systems. A true and a conventional terrestrial system is necessary to describe the position of the rotation axis, analogous to the inertial systems. These systems use cartesian coordinates, but in practice latitude, longitude, and height are preferred. Therefore, a third coordinate system is introduced with the use of a reference ellipsoid.

6.3.1.  Hierarchy within the local systems

The local reference systems can also be subdivided. The differences arise from the choice of the north and zenith directions.

6.4.1.  International Celestial Reference Frame

This is the realization of the ICRS by the IERS. The International Celestial Reference Frame (ICRF) is defined by the coordinates of over 600 extraterrestrial points that have been observed by Very Long Baseline Interferometry (VLBI) in J2000. The position of the quasars, which are extragalactic radio sources, is determined by their right ascension $\alpha$ and declination $\delta$. Thus, the ICRF is a realization in the radio frequency band.
Classically, star coordinates have been measured in the optical waveband. This has resulted in a series of fundamental catalogues, such as FK5. Due to atmospheric refraction, these coordinates cannot compete with VLBI-derived coordinates. However, in the early 1990s, the astrometry satellite HIPPARCOS collected the coordinates of over 100 000 stars with a precision better than 1 milliarcsecond. The HIPPARCOS catalogue constitutes the primary realization of an inertial frame at optical wavelengths. Multiple versions of the ICRF exist, with the latest version being ICRF 3 from the year 2020.

6.4.2.  International Terrestrial Reference Frame

In addition to the conventional inertial reference system (ICRS), the IERS also defines a conventional terrestrial reference system, the International Terrestrial Reference System (ITRS). The realization of this is the International Terrestrial Reference Frame (ITRF). The ITRF is updated in irregular intervals, with ITRF2020 being the most current version.
The transformation between different frames of the ITRF is stated as a 7-parameter transformation. Of these, three are parameters for translation, three for rotation, and one is a scale factor. Since the rotation angles between different frames are very small, one approximates $\sin \left(x\right)=x$.

The latest frames additionally contain ranges of change, i.e., coordinate velocities, resulting in a total of 14 parameters. All parameters, therefore, are a function of time.

6.4.3.  World Geodetic System 1984

The World Geodetic System (WGS84) is perhaps the best known reference frame, especially since it is used in Global Navigation Satellite Systems such as GPS. WGS84 itself defines a reference system, with multiple realizations, done with Doppler, SLR, VLBI, and GNSS. The transformation between ITRF and WGS84 is performed in a similar fashion to the transformation between two ITRF versions. The parameters can be found in literature (Handbook of GNSS) or online.

7.1.1.  Precession

This transformation from the conventional inertial reference system at epoch $T_0$ to the mean inertial one at epoch $T$ describes the phenomenon of precession. Simon Newcomb (1835–1909), formulated the transition of the coordinates $r$ as follows.

First, a rotation around the north celestial pole at epoch $T_0$ (NCP₀) shifts the mean equinox at epoch $T_0$ over the mean equator at $T_0$. This is $R_3(-{\zeta }_0)$. Next, the NCP₀ is shifted along the cone towards the mean pole at epoch $T$ (NCP). This is a rotation $R_2(\theta )$, which also brings the mean equator at epoch $T_0$ is brought to the mean equator at epoch $T$. Finally, a last rotation around the new pole, $R_3(-z)$ brings the mean equinox at epoch $T$ (${\overline{\text{♈︎}}}_T$) back to the ecliptic.
The required precession angles are given with a precision of 1” using the following.

[SOURCE: Handbook of GNSS]

The time $T$ is counted in Julian centuries (of 36,525 days) since J2000.0, i.e., January 1, 2000, 12^h UT1. The time in Julian centuries is calculated from calendar date and universal time (UT1) by first converting to the so-called Julian day number (JD), which is a continuous count of the number of days. A detailed discussion of time systems will follows later in this ER, for more information on UT1 see Clause 8.

7.1.2.  Nutation

The following transformation describes the transition from the mean inertial reference system (MOD) to the true inertial (TOD) one. This transformation deals with the phenomenon of nutation.
The mathematical expression for this is:

First, the mean equator at epoch $T$ is rotated into the ecliptic around ${\overline{\text{♈︎}}}_T$. This rotation, $R_1(\epsilon )$, brings the mean north pole towards the NEP. Next, a rotation $R_3(-\Delta \psi )$ brings the mean equinox over the ecliptic towards the true inertial epoch. Finally, the rotation $R_1(-\epsilon -\Delta \epsilon )$ transforms back to an equatorial system.
The nutation angles are known as nutation in obliquity ($\Delta \epsilon$) and nutation in (ecliptical) longitude ($\Delta \psi$). Together with the obliquity $\epsilon$ itself, which is minimally time-dependent.

The obliquity $\epsilon$ is given in arcseconds. Converted into degrees it equals $\epsilon \approx {23.4}^{\circ }$. It also changes by some 47” per Julian century. The nutation angles are not exact. They are often realized using a Fourier series:

which represents a linear combination of fundamental arguments of solar and lunar orbits.

$l$

Mean anomaly of the Moon

$l\prime$

Mean anomaly of the Sun

$F$

Mean longitude of the Moon minus the mean longitude of the Moon’s ascending node

$D$

Mean elongation of the Moon from the Sun

$\Omega$

Mean longitude of the ascending node of the Moon

The coefficients can be taken from the Observatoire de Paris or the IERS. Only regarding the two main frequencies, while still calculating the angles with precision of 1”, the formula can be simplified to:

with $f_1={125.0}^{\circ }-{0.05295}^{\circ }d$ and $f_2={200.9}^{\circ }+{1.97129}^{\circ }d$. The coefficients to the variable $d$ are frequencies in units of degree/day so $f_1$ corresponds to a period of 18.6 years and $f_2$ to a semiannual period.

7.1.3.  New convention

The first realization of the transformation regarding a conventional inertial and a true inertial system would be to simply concatenate the two transformations of precession and nutation:

Another approach is a convention defined by the IERS in 2010. In this approach both the precession and nutation are taken into account, reducing the calculation to three instead of six rotations:

For this transformation the location of the true or instantaneous pole is described via the co-declination, $d$, and the right ascension, $E$, with respect to the reference pole. The following is a conversion into cartesian coordinates:

Figure 24 — Coordinates of the instantaneous pole in the celestial reference system

The matrix $Q$ which represents the transformation can be rewritten to:

with $a=\frac{1}{1+\cos \left(d\right)}$, which is, with an accuracy of 1 μas (micro-arcsecond), equal to $a=\frac{1}{2}+\frac{1}{8}\left(X^2+Y^2\right)$.

The coordinates $X,Y$ and the last rotation $s$ are calculated with the following formula:

[SOURCE: Handbook of GNSS]

8.2.1.  LAST

A description of sidereal time begins with the fundamental astronomical triangle on the celestial sphere, cf. Figure 27. This figure shows LAST as an angle between local meridian (the $x$-axis of the hour angle system) and true equinox (the $x$-axis of the instantaneous inertial system). Moreover, by including the hour circle of a given star,this figure relates LAST to the hour angle $h$ and the right ascension $\alpha$:

Figure 27 — Projection of the fundamental astronomical triangle on the equator plane

8.2.2.  GAST

In Figure 28, the Greenwich meridian is taken instead of the local meridian. The difference between them is the astronomical longitude $\Lambda$. This results in the simple but fundamental relation:

which says that time and longitude are intimately connected. It is possible to obtain astronomical longitude from the combined measurement of time (through GAST) and observation to a given star (through $h$), using the star’s right ascension $\alpha$ from a catalogue. This observation is even simpler for stars passing the local meridian, in which case $h=0$ applies. Such transits are called upper culmination if the star passes between zenith and north pole. A transit below the pole star is known as lower culmination.

Figure 28 — Comparison of LAST and GAST

8.2.3.  Comparison of LMST and GMST

One of the criteria for a useful time system is the stability of the time scale. This stability is not given in case of actual sidereal time. Because of nutation, the equinox will move back and forth over the ecliptic by the angle $\Delta \psi$, the nutation in longitude. The angles LAST and GAST are therefore relative to a time dependent reference point.

The circles drawn so far represent the equator. To correct for nutation, $\Delta \psi$ is projected on the equator. With the obliquity $ϵ$ between ecliptic and equator, this projection becomes $\Delta \psi \cos ϵ$. This is called the Equation of the Equinox or Eq.E:

The variability in the Equation of the Equinox is only of the order of magnitude of roughly $1s$. However, for a stable definition of a sidereal time system this is relevant.

One sidereal day is given as the time interval between two successive crossings of the mean equinox ${\overline{\text{♈︎}}}_{\text{T}}$ through the local meridian. Such a crossing, during which $\text{LMST}=0$ is the sidereal noon. One might subdivide the sidereal day in 24 sidereal hours of 60 sidereal minutes of 60 sidereal seconds. It is important to emphasize the word sidereal here, since these time units are slightly different from the solar equivalents discussed in the next section ny a factor $F$.

All sidereal time angles are presented in Figure 29, from which the following formulas are derived:

Figure 29 — The four basic types of sidereal time.

8.3.1.  LT, UT

Figure 31 graphically explains the basic definition of (LT) where LT is the hour angle of the Sun ☉, i.e., the angle between local and solar meridian. However, since noon is associated with a zero-hour angle, 12^h are added:

Similarly, by taking the Greenwich meridian, results are in Greenwich solar time. This is known as true universal time (UT). It is called true because it refers to the real Sun:

Local and Greenwich solar time are simply related by adding or subtracting the longitude:

Figure 31 — The basic local (left) and Greenwich (right) solar time

8.3.2.  Refinement 1: UT0

Similar to the true equinox, which was not suitable to define a stable sidereal time system, the motion of the true Sun is not homogeneous enough to define a stable solar time system. The reason for the non-uniform motion is twofold:

1. Ellipticity of the Earth’s orbit around the Sun (according to Kepler’s area law, the Earth moves faster near the perihelion (the point closest to the Sun)); and

2. Obliquity between equator and ecliptic plane.

Therefore, a fictitious mean Sun ($\overline{\odot }$) is introduced that moves along the equator with uniform speed. The difference between the true and fictitious Sun, projected onto the equator, is called Equation of Time:

Figure 32 — Equation of Time (left) and definition of UT0 (right)

The Equation of Time can reach values of up to 15 minutes.

Employing the mean Sun, the first refinement to UT consists of correcting for the inhomogeneous solar motion. The result is called UT0:

8.3.3.  Refinement 2: UT1

Due to polar motion, the instantaneous longitude of any meridian and therefore for this variability, the conventional terrestrial pole, is referred to the corresponding solar time for the conventional Greenwich meridian, which is called Greenwich mean time or UT1:

$\text{UT1}=\text{UT0}+\Delta {\Lambda }_{\text{p}}$
$=\text{UT0}-\left(x_{\text{p}}\sin \Lambda +y_{\text{p}}\cos \Lambda \right)\tan \Phi$
$=h_{\overline{\odot }}^{\overline{Gr.}}\pm {12}^h$

The result is a relatively stable time system for civilian time-keeping purposes, based on a physical and observable phenomenon: the orientation of the Earth with respect to the mean Sun.

Figure 33 — Mean Greenwich meridian and UT1 (left) and conversion between solar and sidereal time (right)

8.3.4.  Length Of Day

Still after these two refinement steps, UT1 will still not be completely stable. Due to mass redistributions on and in the Earth, the length of a mean solar day will not be constant. These fluctuations will become apparent after comparing every length of a solar day (LOD) to 86400 atomic seconds, cf. next section. The difference, or excess LOD is at the millisecond level.

[SOURCE: IERS]

8.3.5.  Conversion between solar and sidereal times

The conversion between solar and sidereal time is straightforward. Figure 33 (right) shows the mean equinox, the mean solar meridian and the mean Greenwich meridian:

The right ascension of the mean Sun is given by the internationally adopted formula (IAU 1976):

in which $T$ is the time since the reference epoch J2000.0, counted in Julian centuries of 36525 days. Thus, $T=d/\mathrm{36,525}$ with $d=\text{JD}-\mathrm{2,451,545.0}$. The right ascension of the mean Sun is counted in seconds. Thus, the factor in the second term at the right-hand side has units of second per Julian century. Indeed, the number 8640184.812866 corresponds to one full circle ($={24}^h$) per year or $3^m{56}^s.33$ per day. Therefore the conversion is appropriate:

Again, this formula is in seconds. Moreover, an integer times 24^h must be added in order to keep $\text{GMST}\in 0;{24}^h$.

8.4.1.  UTC

The highly stable atomic TAI and the universal time UT1 will diverge over the years, as the daily LoD values accumulate. The difference UT1-TAI will increase in a non-homogeneous manner. This effect is caused by the slowing of the Earth’s rotation and by irregularities in the spin rate.

Figure 35 — Atomic time scales and UT1.

As a compromise — to keep atomic and solar time close to each other — Universal Time Coordinated (UTC) is introduced. The stable time scale of UTC is inherited from TAI. To stay close to UT1, leap seconds have been introduced such that the difference $\Delta \text{UT}=\text{UTC}-\text{UT1}\le 0.9s$. The resulting time system meets all criteria, mentioned in the introduction:

1. stability: stable definition of the second,

2. accessibility: through atomic clocks and radio broadcast, and

3. meaningfulness: very close to mean solar time, i.e., the rotation phase of the Earth.

The corresponding formulas:

Leap seconds are issued by the IERS, either on July 1 or January 1. The IERS makes a decision whenever $\Delta \text{UT}$ comes close to the condition Formula (34). This system started in 1972 with $n=10s$. The last leap second was issued January 1, 2017, bringing the total to $n=37s$.

The CGPM published a statement in the Resolution 4 of the 27^th November CGPM (2022) that until the year 2035 the maximum difference of UTC and UT1 of 0.9 seconds will be increased. This is the result of a long discussion since the introduction of leap seconds creates discontinuities that risk causing serious malfunctions in critical digital infrastructures including the Global Navigation Satellite Systems (GNSSs), telecommunications, and energy transmission systems.

At the start of GPS system time at midnight January 6, 1980, the difference between $T_{GPS}$ and TAI was $19s$. Leap seconds are neither applied to $T_{GPS}$ nor to TAI. Consequently, UT1 and UTC are also drifting away from GPS time. Thus, $T_{GPS}$ is now $13s$ apart from UTC. In general:

[SOURCE: IERS]

8.5.1.  Geocentric and barycentric coordinate time system

Since those two systems mainly vary in their reference point they can be discussed together. The zero point of both scales is set so that the event January 1, 1977, $0^h{0}^m{0.000}^s$ TAI at the geocenter corresponds to the time January 1, 1977, $0^h{0}^m{32.184}^s$ for both TCB and TCG.
The measurement of a clock is called proper time $\tau$ and can be converted to the coordinates time systems $t$ via:

with $U$ being the gravitational potential. A stationary clock on the Earth has a fixed position $r$ in a geocentric coordinate system, rotating at the angular velocity $\Omega$ (of the Earth’s rotation) rotating in the inertial frame. The velocity of the clock in a non-rotating geocentric reference system is expressed as:

$v=\Omega \times r$.

Therefore, the proper time results in:

The geopotential $U_{\text{geo}}$ additionally includes the potential created by centrifugal force. Therefore, clocks at locations with identical geopotential number (the numerical difference between two different equipotential surfaces) are equally fast in respect to TCG. Non-moving clocks on the geoid have equal speeds.

Transformation between TCB and TCG
For this transformation, the entire trajectory since $T_0$ (January 1977 $0^h{0}^m{32.184}^s$) must be known. Therefore, the calculation is very complex with high accuracy. A closed expression for TCG is:

Here $U_{\text{ext}}$ is the Newtonian gravitational potential generated by all bodies in the solar system except the Earth.

8.5.2.  Terrestrial Time

The TCG is faster than the proper time measured by a clock. The Terrestrial Time TT is defined as a variation of the TCG, in which this higher rate is compensated:

The discrepancy from a relative progress rate of 1 is $L_G=6.969290134\cdot {10}^{-10}\approx 22\frac{\text{ms}}{\text{a}}$. The value of $L_G$ corresponds to the gravitational time dilation $\frac{U_{\text{geo},0}}{c^2}$ for the geopotential $U_{\text{geo},0}=\mathrm{62,636,856.0}\frac{{\text{m}}^2}{{\text{s}}^2}$ which is the best value for the geopotential of the geoid known when the IAU-resolution was passed in 2000. Thus, while the definition of TT becomes independent of the intricacy and temporal changes inherent to the definition and realization of the geoid, the TT-second on the rotating geoid is, with very high accuracy, the SI-second and TT is further approximated very well by $\text{TAI}+32.184s$.

8.5.3.  Barycentric time

The barycentric time was originally intended to serve as an independent time argument of barycentric ephemerides and equations of motion. The transformation between TCB and BT is similar to the transformation between TT and TCG:

with the same $T_0$ as in TT and $L_B=1.550519768\cdot {10}^{-8}\approx 0.49\frac{\text{s}}{\text{a}}$.
The value of $L_B$ was chosen so that TDB and TT have the same average progress rate at the center of mass of the Earth, so the difference remains limited.

8.6.1.  Julian Days

Civilian time is counted in the Gregorian Calendar in days ($D$), months ($M$), and years ($Y$). Since the length of months is variable and since some years (leap years) have an additional day, it is difficult to calculate time intervals in terms of days. For geodetic, astronomic, and chronological purposes a chronological counting of days would be more practical, which is what Julian Day Numbers are.

The Julian Day system begins at noon on January 1, -4712. A new Julian Day starts at noon. This makes sense for astronomical purposes (hour angle equals zero). Many algorithms to compute Julian Day numbers from calendar dates exist. For instance:

The latter is the Modified Julian Day, which starts at midnight November 16-17, 1858. It guarantees that at most 5 digits are required in the period from 1859 to about 2130. The 0.5 at the end means that a new MJD starts at midnight.

[SOURCE: Astronomical Algorithms]

9.1.1.  Size and shape of ellipse

First, the ellipse itself must be described. Figure 37 shows the basic elements of an ellipse. In the case of a satellite orbiting the Earth, the focus would be the center of mass of the Earth and the satellite follows the outline of the drawn ellipse. When orbiting around the Earth, the terms perigee and apogee are used for the peri- and apofocuses. The apofocus is the point on an elliptic orbit at the greatest distance from the principal focus. The (first) eccentricity $e$ is defined by the semi-major axis $a$ and the semi-minor axis $b$:

9.1.2.  Orientation of orbital plane in space

The coordinate axes are as defined in the conventional inertial reference system. The orbital plane is tilted with respect to the equator as seen in Figure 38. The corresponding angle $I$ is called Inclination. The intersection line between orbital and equator plane is the nodal line. The node, in which the satellite crosses the equator from south to north, is the ascending node. The angle in the equatorial plane between the vernal equinox (♈︎) and ascending node is the right ascension of the ascending node $\Omega$. The angle within the plane between the equator plane and the perigee is called argument of perigee $\omega$.

9.1.3.  Position within orbital plane

A simple way to define the position within the orbital plane is to refer to the time that passes during one full revolution, which is called orbit period $T$. The commonly used alternative is the mean motion $n$ that describes the mean angular velocity within the orbital plane.