Mutual conversion between GCRS and ITRS, time system and conversion

1 Introduction

The conversion of the coordinate system and the time system has always stayed in the general overview of the textbooks. The actual engineering application is more detailed and involves more content than the textbooks. The goal to be achieved here is the conversion of the coordinate system, and some records will be made of the content involved. Among them, it is inevitable that there are inappropriate descriptions or inappropriate understandings. Welcome to point out and discuss.

GCRS is a coordinate system used to observe celestial bodies on the earth. It is mainly used for earth astronomical observations, such as controlling the attitude of the telescope and the alignment of observation equipment. It is important for astronomical applications on Earth, but not for the study of the solar system as a whole.

The International Terrestrial Reference System ITRS is our commonly used coordinate system, and the coordinates involved in general measurement are all in this coordinate system. In order to study satellite problems more conveniently, it is often necessary to convert between these two coordinate systems. IAU's standard basic astronomical program library SOFA provides the library functions of basic astronomical calculations. Using these library functions, the transformation between the two coordinate systems can be realized conveniently.

This paper mainly introduces the development of ITRS and GCRS and the realization of different conversion methods between ITRS and GCRS based on SOFA. The Standard Library for Fundamental Astronomy (SOFA) is a set of programs for the Earth's attitude, time scale, and calendar provided by the International Earth Rotation Services (IERS) protocol.

Tip: The currently widely used protocol celestial coordinate system is the International Celestial Coordinate System ICRS stipulated by IAU. According to the different origins of coordinates, ICRS can be divided into two types: the barycentric celestial reference system of the solar system, BCRS, and the geocentric celestial reference system, GCRS. The origin of the coordinates of the former is located at the barycenter of the solar system, which is used to calculate the orbits of planets and compile star catalogs; the origin of coordinates of the latter is located at the center of the earth, and is used to calculate satellite orbits and compile satellite ephemeris. In the blog post, some content wants to express the conversion between GCRS and ITRS, and it is written as the conversion between ICRS and ITRS (the concept of ICRS and GCRS is blurred in some documents, but it should be written clearly), which is not rigorous enough, but The meaning of the expression is the conversion between GCRS and ITRS!

2. GCRS

2.1 GCRS

GCRS (Geocentric Celestial Reference System) is an astronomical coordinate system with the earth as its origin, which is relative to the earth. In GCRS, the positions of celestial bodies are relative to the Earth, not the center of the solar system.

GCRS has a coordinate system structure similar to ICRS, where the origin of one coordinate axis is aligned with the center of the earth, the other coordinate axis is due south on the earth, and the third coordinate axis is due east on the earth. The difference is that GCRS is a dynamic coordinate system that changes with the movement of the Earth and is affected by factors such as Earth perturbations.

GCRS is a coordinate system used to observe celestial bodies on the earth. It is mainly used for earth astronomical observations, such as controlling the attitude of the telescope and the alignment of observation equipment. It is important for astronomical applications on Earth, but not for the study of the solar system as a whole.

2.2 Precession, nutation, pole shift

Understanding the coordinate system requires a certain understanding of Earth's precession (precession), nutation, and pole shift , which is to have an understanding of the object you are referring to.

Precession: Precession is caused by the gravitational pull of the Sun, Moon, and planets on Earth.
(1) Precession of the sun and the moon: the attraction of the sun and the moon to the equatorial bulge of the earth produces an irreversible couple effect, which makes the earth's rotation axis precess around the ecliptic axis in inertial space. The north celestial pole precesses around the ecliptic axis. The precession period is about 25,700 years, and it moves westward by 50″.37 every year. At the same time, there is also nutation. (2) Planetary precession: due to the gravitational effect of other planets in the solar system, the annual motion of the earth does
not Kepler's law is not strictly followed, the position of the ecliptic plane is constantly changing, and the change of the ecliptic plane will also lead to the change of the vernal equinox, which makes the vernal equinox move eastward along the celestial equator by about 0″.13 every year.

Nutation: The celestial pole makes a circular motion around the ecliptic axis without considering the nutation. At this time, the celestial pole is called the flat celestial pole. If the influence of nutation is considered, it is true.
Pole shift: The earth's rotation axis moves relative to the earth's body, and the resulting position of the earth's poles on the earth's surface changes with time. This phenomenon is called earth's pole shift, or pole shift for short.

Pole shift and precession nutation are two different geophysical phenomena. Precession and nutation are the movement of the earth's rotation axis in inertial space, and its relative position inside the earth does not change, so it only causes changes in the coordinates of celestial bodies, and does not cause changes in the longitude and latitude of the earth's surface. On the contrary, the pole shift is the change of the earth's rotation axis in the earth itself, and its direction in the inertial space does not change, so it will cause changes in the latitude and longitude of various places on the earth's surface.

To sum up:
(1) The precession of the sun and the moon will cause the change of the celestial pole and the vernal equinox;
(2) The planetary precession will cause the change of the vernal equinox;
(3) The nutation will cause the change of the celestial pole and
the vernal equinox; Changes in latitude and longitude on the surface;

Content source: Understanding of J2000.0 coordinate system and WGS84 coordinate system .

2.3 What is the equatorial coordinate system

To understand ICRS, you must first know what is the equatorial coordinate system, because it is established based on the concept of the equatorial coordinate system. Reference: International Celestial Reference System - Introduction to ICRS .

The pole of the equatorial coordinate system is the intersection of the earth's rotation axis and the celestial sphere, and its equator is the earth's equator. The diagram below shows how the equatorial coordinate system is constructed from the Earth's rotation.

2.4 Introduction to ICRS

The International Celestial Reference System (ICRS) is implemented by the International Celestial Reference Frame (ICRF). At the 23rd General Assembly of IAU in 1997, it was approved and decided to adopt ICRS in the fields of astronomical research, space exploration, geodesy and geodynamics from January 1, 1998. The coordinate system in ICRS is basically consistent with the equatorial coordinate system, and the key point is how to select the reference object : the average pole of ICRS J2000.0 is 17.3±0.2 mas in the direction of 12 h, and 5.1±0.2 mas in the direction of 18 h; The point moves 78 ± 10 mas from the origin of ICRS right ascension in a direction perpendicular to the polar axis.
Reference: International Celestial Reference System - Introduction to ICRS , Wikipedia , Baidu Baike , www.iers.org

The figure below depicts the celestial coordinate system .

2.5 Implementation of ICRS

We all know that to determine the motion of an object (motion is actually position) requires a reference. According to a specific reference object, choose a coordinate system, we will get a physical reference system.

In the usual equatorial coordinate system, we stipulate that the position of the vernal equinox of the sun is the zero point of right ascension, but in fact, the vernal equinox is not constant. The equinoxes are in motion all the time due to the precession and nutation of the Earth. Precession is when the Earth's spin axis rotates around another axis in space, and nutation is the smaller movement during precession. P and N are precession and nutation, respectively.

Therefore, ICRS chooses to use almost "permanent" extragalactic celestial bodies as reference objects, which is ICRF (International Celestial Reference Frame) - the International Celestial Reference Frame.

The ICRF is the precise location of a series of extragalactic sources, usually distant quasars, far enough away to guarantee the "inertial" stability and stability of this frame of reference, among which is the famous and first Discovery of the quasar 3C 273. With VLBI - Very Long Baseline Radio Interferometry, we can pinpoint the position of these radio sources to within a few milliarcseconds to ensure the accuracy of ICRS.

The origin of ICRS is the center of mass of the earth , the direction of the center of mass of the earth pointing to the north pole of the celestial sphere is the direction of the Z axis, the direction of the center of mass of the earth pointing to the vernal equinox is the direction of the X axis, and the direction of the Y axis is determined by the right-hand rule. This coordinate system describes the motion state of the satellite when it is in orbit . When using this coordinate system, the agreement celestial coordinate system J2000 launched by international organizations in 1984 is generally adopted. The figure below is a schematic diagram of a space-fixed inertial reference system.

2.6 J2000

J2000 (Julian Year 2000) is an astronomical coordinate system and a reference point of ICRS (International Celestial Reference System). Its origin is at the center of gravity of the solar system, and its three coordinate axes point to the three directions of the equatorial plane of the instantaneous celestial sphere.

J2000 takes January 1, 2000 at 12 o'clock midnight as the reference point, which is used as the coordinate system of the solar system object. It is a commonly used reference system in astronomical research, mainly used to calculate the position and motion of celestial bodies, and to classify and group celestial bodies.

The J2000 coordinate system is very precise and important in the study of astronomy, astrophysics, and astrodynamics. However, as time goes by, the position and motion of celestial bodies will change, so the J2000 coordinate system needs to be corrected and updated continuously in the long run. J2000 can be understood as a kind of International Celestial Inertial Reference Coordinate System ICRS.

This coordinate system is often used as the inertial coordinate system of earth satellites, and satellite motion integrals are calculated in this coordinate system.

3. ITRS

3.1 Introduction to ITRS

In 1988, the International Union of Geodesy and Geophysics (IUGG) and the International Astronomical Union (IAU) jointly established the International Earth Rotation Service (IERS). Take ITRF's annual solution, these solutions are published by IERS annual report and technical memorandum, so far ITRF has published 12 versions, namely ITRF88, ITRF89, ITRF90, ITRF91, ITRF92, ITRF93, ITRF94, ITRF96, ITRF97, ITRF2000, ITRF2005, ITRF2008 . Among them, ITRF2008 is the latest version.

ITRS (International Terrestrial Reference System) is what we often call a ground-fixed coordinate system. Its origin is at the center of gravity of the earth (including the mass of the atmosphere and oceans), and the xy plane of the coordinate system is the equatorial plane of the earth. , the x-axis points to the intersection of the Greenwich meridian and the equatorial plane. This coordinate system is fixed on the earth, and the measurement and control of the ground station, as well as the coefficient of the earth's gravitational field are all defined under this coordinate system. This coordinate system describes the position and flight speed of the satellite. When using this coordinate system, the (World Geodetic System 84, WGS84) coordinate system is often used instead. Since the two are very close, their differences are ignored during use. The use of the WGS84 coordinate system can effectively eliminate the influence of factors such as the curvature of the earth and the rotation of the satellite during satellite operation. The figure below is a schematic diagram of the ground-fixed coordinate system, which is obviously different from the ICRS in that the X-axis points. In space, the ITRS coordinate system rotates with the earth (relative to the earth’s invariance), and the ICRS does not change with the rotation of the earth, because Its X-axis points to the vernal equinox (fixed direction in the celestial sphere).

It is a basic coordinate system for geodesy and geodynamics research. If the earth's tides and crustal movement are ignored, the earth's gravitational field and the position of the ground point are fixed in this coordinate system. That is to say, this coordinate system only rotates with the rotation of the earth and is fixed on the earth, so it is also called a ground-fixed coordinate system. The establishment of the earth coordinate system has a history of more than 100 years. Before 1980, optical observation was mainly used. With the development of space geodesy, the observation of man-made or natural celestial bodies has broken the unique observation tradition of groups or countries, and it is urgent to establish and use a common earth coordinate system. But there cannot be absolutely fixed things in the universe, so the establishment of this coordinate system can only be reflected through an agreement result, so this coordinate system is also called the Agreement Terrestrial Reference System (CTRS), which is an international agreement Uniformly adopted earth reference system. The International Terrestrial Reference Frame, which is often used by countries around the world, is the realization of this coordinate system.

The main task of the International Earth Rotation Service (IERS) is to provide accurate and timely rotation parameters. One of its purposes is to establish and maintain this earth reference frame. The framework was established by the International Union of Geodesy and Geophysics (IUGG), the International Association of Geodesy (IAG), and the International Astronomical Union (IAU). The relevant work is carried out by the Department of Earth Reference Frames under IERS. Specifically, it is hosted by the Geodetic Laboratory (LAREG) of the National Geographic Institute (IGN) in Paris, France. The space geodetic techniques used are: Laser Lunar (LLR); Laser Satellite (SLR); Very Long Baseline Interferometry (VLBI); Global Positioning System (GPS); Doppler Satellite Tracking and Radio Positioning System (DORIS), etc. .

The International Terrestrial Reference Frame (ITRF, International Terrestrial Reference Frame) is the implementation of ITRS, and new ITRF solutions are produced every few years, using the latest mathematical and measurement techniques to achieve ITRS as accurately as possible. Due to experimental error, any given ITRF will be slightly different from existing ITRFs. ITRS and ITRF solutions are maintained by the International Earth Rotation and Reference Systems Service (IERS).

4. Time system

The time system meets two conditions: (1) It can perform continuous and uniform periodic motion, and the motion cycle is very stable; (2) The motion cycle has good reproducibility, that is, the periodicity in different periods and places All motions can be reproduced through observation and experimentation.

For details, please refer to the introduction in the SOFA software package . SOFA provides the time conversion codes of Fortran77 and C language , and the Matlab forum provides the corresponding Matlab version .

Timekeeping is the measurement of time according to an agreed method, and for a long time in human history, the rotation of the earth has been the best timer available. The Gregorian calendar date we are familiar with is composed of year, month and day, with an average year length of 365.25 days. Astronomers use 365.25 days and 36525 days as the Julian year and
Julian century, respectively. For larger intervals, other astronomical phenomena also play a role, notably the orbital periods of the Earth and Moon.

Computations in any science may involve precise time, but various time systems currently exist. There are several reasons for this: astronomers must continue to deal with the phenomena behind obsolete time scales, especially Earth's rotation and planetary motion; as new time scales are introduced, the continuity with the past continues; in astronomical applications, the "clock "The physical context of an object is important, whether it's on Earth, moving or stationary, or on a spacecraft.

Commonly used time systems are as follows:

project	Value
TAI (International Atomic Time)	official timekeeping standard.
UTC（Coordinated Universal Time）	The basis of civil time.
UT1（Universal Time）	Based on the Earth's rotation.
TT（Terrestrial Time）	Used for solar system ephemeris lookups.
TCG（Geocentric Coordinate Time）	For geocentric spatial computing.
TCB（Barycentric Coordinate Time）	For calculations beyond Earth orbit.
TDB（Barycentric Dynamical Time）	A form of scaling for TCB that is averaged to keep with TT.

Of the seven time scales described here, one is atomic time (TAI), one is solar time (UT1), one is atomic/solar hybrid time (UTC), and four are kinetic time (TT, TCG, TCB , TDB). Each has a different role, some with tens of seconds of offset between them: when planning astronomical calculations, it is crucial to choose the correct one. A particularly common mistake is assuming there is only one precise time, UTC, which is compatible with everything from pointing telescopes (which actually requires UT1) to finding planetary positions (which requires TDB, which can be approximated with TT). In fact, UTC itself is almost never the time scale used for astronomical calculations, except perhaps for records. The figure below is a "road map" showing the relationship between the seven time scales and how time on one scale can be transformed into the same time on another scale. Timescales are represented by rectangular boxes, and their relationships are shown in circular boxes.

4.1 Time system conversion relationship

∆UT1 = UT1−UTC
∆AT = TAI−UTC
∆T = TT − UT1 = 32.184s + ∆AT − ∆UT1
∆TT = TT−UTC

TAI 1997 Jan 1 00:00:00
TCG 1997 Jan 1 00:00:00 + 32.184 s
TCB 1997 Jan 1 00:00:00 + 32.184 s
TDB 1997 Jan 1 00:00:00 + 32.184 s − 65.5 µs

TT = TAI + 32.184 s.

TCG = TT + LG × (JDTT − TT0)

Among them, TT0 = 2443144.5003725, JDTT is TT expressed as a Julian day, LG = 6.969290134 × $10^{-10}$。

TDB = TCB − LB × (JDTCB − T0) × 86400 + TDB0

Among them, JDTCB is the Julian day of TCB, T0 = 2443144.5003725, LB = 1.550519768 × $10^{-8}$ and TDB0 = −6.55 × $10^{-5}$s.

4.2 Ephemeris time (ET－ephemeris time)

Due to the unevenness of the earth's rotation, the time system based on the earth's rotation cannot meet the increasingly high precision requirements. Therefore, in 1958, the International Astronomical Union (iau) decided that since 1960, countries began to use the almanac based on the earth's revolution period to compile the astronomical calendar.

Almanac time is counted from the moment of 279°41′48.04″ in mean ecliptic longitude (mean ecliptic longitude), which is defined as 12:00 on January 1, 1900 in ephemeris time. The basic time unit of ephemeris time is ephemeris second (ephemeris second) is equal to 1/31556925.9747 of the length of the tropical year at 12 o'clock on January 1, 1900.

The observation accuracy of almanac time has been gradually enhanced with the improvement of observation equipment and technology, from the early et al. , et1 developed to later et2, its precision can reach $10^{-11}$~$10^{-9}$ magnitudes. This accuracy is higher than the time system based on the earth's rotation, such as sidereal time and universal time, but because it cannot give accurate ephemeral time in real time, and its accuracy is limited, it has certain limitations in use. In 1976, the International Astronomical Union decided to replace the almanac time with the geodynamic time and the solar system center-of-mass mechanical time since 1984.

4.3 Sidereal time, true solar time, mean solar time, zone time

(1) Sidereal time: The sidereal time is based on the vernal equinox as the reference point. Due to the rotation of the earth, the time interval between two consecutive vernal equinoxes passing through the local meridian is one sidereal day.

(2) True solar time: The true solar time is based on the center of the sun as the reference point, and the time interval between the center of the sun passing through the meridian circle of a certain place twice is called a true solar day. Elliptical orbits and different angular velocities lead to different lengths of true solar time.

(3) Mean solar time: A fake sun is used to replace the time system constructed by the real sun. It departs from the vernal equinox at the same time as the real sun every year, runs from west to east on the celestial equator at a constant speed, which is equivalent to the average speed of the real sun on the ecliptic, and finally returns to the vernal equinox at the same time as the real sun.

(4) Zone time: The world is divided into 24 standard time zones. In the same time zone, the mean solar time on the central meridian of the time zone is uniformly adopted, which is called zone time.

4.4 Universal Time (UT—universal time)

The universal time is the Greenwich mean solar time calculated from midnight on midnight. Universal time and mean solar time have the same unit scale, but the starting point is different. The former is 12 hours later than the latter. Universal time is a time system based on the periodic rotation of the earth, and its accuracy is mainly affected by the uneven speed of the earth's rotation and pole shift.

Universal time problem:
the speed of the earth's rotation is not uniform, it not only has a long-term slowing down of the total potential, but also seasonal changes and short-term changes, the situation is more complicated; the position of the earth's poles on the earth is not
fixed , but is constantly moving, that is, there is a phenomenon of pole shift. This means that universal time no longer strictly meets the basic conditions of being a time system, because it is actually not a completely uniform time system.

4.5 Atomic time, international atomic time (TAI-international atomic time)

Due to the high stability and reproducibility of the electromagnetic wave frequency radiated and absorbed during the atomic transition in the material, the atomic time established thus is the most ideal time system at present. The international definition of the atomic time second is: the two hyperfine energy levels of the ground state of the cesium atom at sea level, and the duration of the transition radiation oscillation of 9162631770 cycles in a zero magnetic field, which is an atomic second (atomic second). The origin of atomic time is then defined by the following formula;

In order to unify the atomic time of all countries in the world, more than 100 atomic clocks are compared with each other in the world, and a unified atomic time system is calculated through data processing, which is called international atomic time (international atomic time). The stability of TAI is about $10^{-\; Around13}$ . In GPS positioning, atomic time is used as a high-precision time reference to measure the propagation time of satellite signals.

4.6 Coordinated Universal Time (UTC-universal coordinated time)

Coordinated Universal Time is a time measurement system that is based on the length of atomic time and second, and is as close as possible to universal time in terms of time. It is a compromise time system between atomic time and universal time.

International Atomic Time is accurate to nanoseconds per day, while Universal Time is accurate to milliseconds per day. Many application sectors require a time system close to UT, for which case a compromise time scale known as Coordinated Universal Time was introduced in 1972. To ensure that UTC does not deviate from UTC by more than 0.9 seconds, positive or negative leap seconds are added to UTC where necessary. Therefore, there will be a difference of several integer seconds between the Coordinated Universal Time and the International Atomic Time. 1s. It not only maintains the uniformity of the time scale, but also approximately reflects the change of the earth's rotation. According to the UTC amendment adopted by the International Radio Consultative Committee (CCIR), the difference between UTC and UT1 (obtained by adding pole shift correction to UT) from January 1, 1972 can reach a maximum of ±0.9s. The International Central Bureau for Earth Rotation Affairs in Paris is responsible for deciding when to add the leap second. Generally, adjustments will be made at the last second of June 30 and December 31 each year. The specific date is arranged and announced by the International Earth Rotation and Reference System Service (IERS).

At present, the time broadcasted by countries in the world is based on UTC.

4.7 GPS Time System (GPST - gps time)

For precise navigation and measurement needs, the Global Positioning System has established a dedicated time system, which is controlled by the atomic clock of the GPS master control station. GPS time belongs to the atomic time system, and the second is the same as the international atomic time, but the origin of the international atomic time is different, that is, there is a constant deviation between GPST and TAI at any instant.

TAI-GPST = 19s

The time of GPST and UTC is stipulated to be the same at 0:00 on January 6, 1980. With the accumulation of time, the difference between the two will be expressed as an integer multiple of seconds. The relationship between GPST and UTC

GPST=UTC+1S×n−19sGPST=UTC+1S×n−19s

By 1987, the adjustment parameter n was 23, and the difference between the two systems was 4 seconds. By 1992, the adjustment parameter was 26, and the difference between the two systems had reached 7 seconds.

4.8 Julian Date (JD)/Modified Julian Date (MJD) (Julian Date /Modified Julian Date)

Calendar dates are inconvenient for many purposes. For this, the Julian day numbering system can be used. JD zero is located 7000 years ago, well before the historical era, defined as Greenwich Mean Noon; for example, JD 2449444 begins at noon (12 o'clock) on April 1, 1994. JD 2449443.5 is midnight (0:00) starting April 1, 1994. Due to the large value of the Julian day and the clumsiness of the half-day offset, it is a recognized practice to delete the leading "24" and "tail 0.5", resulting in the so-called corrected Julian day:

MJD = JD − 2400000.5

Test in Matlab:

% 2018年8月7日0时0分0秒
mjd = mjuliandate(2018,8,7,0,0,0)
jd = juliandate(2018,8,7,0,0,0)

result

mjd = 58337                 jd = 2458337.5

MJD is usually used in computer applications, rather than JD itself, to reduce the risk of rounding errors.

For some astronomical purposes it is convenient to use fractional years, for example a given date and time near the end of 2009 could be written as "2009.93".

5. Implementation of ITRS to GCRS theory

The process in the blog post is the process of converting from ICRS to ITRS. The conversion from ITRS to ICRS can be realized by matrix inversion on the basis of the following process. The SOFA software package adopted for the conversion, the Basic Astronomical Standard Library (SOFA), is a project sponsored by the IAU, which aims to provide authoritative and effective algorithm programs and constant values ​​for astronomical calculations. In 1997, the SOFA review committee was formally established, and a SOFA center for releasing codes was set up. In this paper, the basic code used for conversion between coordinate reference systems comes from this center. A SOFA link is provided at the end of the blog post.

5.1 Coordinate system conversion

At present, the ICRF and ITRF conversion methods mainly include the classical conversion method based on the vernal equinox, and the new method based on the celestial intermediate origin (CIO).

The classic method of ICRF and ITRF conversion based on the vernal equinox
In the conversion method based on the vernal equinox, the conversion from ICRF to ITRF has to go through the conversion of ICRF into the instantaneous flat celestial coordinate system, the conversion of the instantaneous flat celestial coordinate system into the instantaneous celestial coordinate system, and the conversion of the instantaneous celestial coordinate system There are four processes of transforming the instantaneous earth coordinate system and the instantaneous earth coordinate system into ITRF. The conversion from ITRF to ICRF undergoes four opposite conversion processes, and the inverse matrix from ICRF to ITRF is mainly used in the actual conversion. The main process of the classical method based on vernal equinox ICRF conversion to ITRF conversion is as follows:

CIO-based ICRF and ITRF conversion method
In the CIO-based system, because it does not depend on the vernal equinox, adopts a new precession and nutation model, and improves the pole shift correction model, compared with the traditional vernal equinox-based system, the ICRF and ITRF The methods and processes of mutual conversion are different. The new conversion method mainly includes three processes: converting ICRF to celestial intermediate reference system, converting celestial intermediate reference system to instantaneous earth coordinate system, and converting instantaneous earth coordinate system to ITRF. The main flow of the classic method based on the transformation from ICRF to ITRF based on the rotation-free origin is shown in the figure below.

5.2 Realization of coordinate system conversion

At present, the latest version of SOFA includes the precession model announced by IAU in 2006. It mainly consists of two parts: astronomy library and vector matrix operation library. Among them, the former has 131 subroutines, which can perform astronomical calendar calculation, time calculation, ephemeris calculation, precession and nutation calculation, stellar space motion calculation, and conversion of main star catalog systems, etc. The latter has 52 subroutines, and its main function is vector and matrix operations.

5.3 Transformation Required Parameters

[GCRS] = Q(t)R(t)W(t)[ITRS]

Among them, [ITRS] and [GCRS] correspond to the coordinates of the same position vector in the ITRS and GCRS coordinate systems, respectively. W(t), R(t) and Q(t) respectively correspond to the pole shift matrix, the earth rotation matrix, precession and nutation matrix, refer to Chapter 5 of IERS2010 . The three transformation matrices are all 3X3 in size.

There are 5 parameters required to calculate the transformation matrix:

t       - modified julian day referenced to UTC  修正儒略日MJD
dt      - UTC-TAI offset (leapsecs)    [seconds] 通过SOFA软件包可以计算
dut     - UT1-UTC offset               [seconds] 通过SOFA软件包可以计算
xp,yp   - coordinates of the pole      [arcsec ] 根据IERS发布的极移参数线性插值（预测）得到MJD对应的xp和yp

The time t is used to calculate the four parameters of dt, dut, xp and yp.

5.4 Matlab program for converting GCRS to ITRS

Calculate the conversion matrix from GCRS to ITRS. The process of calculating W(t), R(t) and Q(t) matrices is reflected in the code.

%% IERS4m 
% These classes facilitates the CIO based celestial to terrestrial transformation. 
% The main function is GCRS2ITRS which provides the 3x3 celestial to terrestrial transformation matrix.  
% In order to obtain the necessary EOP information use helper object USNO.m.  
%
% There is a detailed write up in the docs/latex folder which explains this transformation in detail as well as explains the matrix formulation implemented in MATLAB.  
% All relevant references are listed in the References section.  Select papers are avaliable in the docs/refs folder. 
%
% Default 2010 IERS convention.
%
%% Example Usage
% This example is taken from David Vallado's paper listed in the References section.  

% date time UTC: 2004/04/06 07:51:28.386 mjd = mjuliandate(2004,4,6,7,51,28.386)
fMJD_UTC = 53101.3274118751;

% init EOP object
eopobj = USNO(); 

% pull latest EOP data from USNO servers
% eopobj = eopobj.initWithFinalsHttp(); 

% interpolate EOP information for date and time
[xp,yp,du,dt] = eopobj.getEOP(fMJD_UTC);

% Vallado et al. 2006, AIAA                        [meters]
X_itrs = [-1033.4793830, 7901.2952754, 6380.3565958]';

% dx,dy = 0                                        [meters]
X_gcrs = [5102.5089592, 6123.0114033, 6378.1369247]';

% compute the 3x3 transformation matrix
GC2IT = IERS.GCRS2ITRS(fMJD_UTC,dt,du,xp,yp);

% perform the coordinate/position conversion: C -> T
X = GC2IT*X_gcrs;

% compute the error in meters
err = sum(sqrt((X-X_itrs).^2))

% X = inv(GC2IT)*X_itrs;
% X = GC2IT\X_itrs;
% err = sum(sqrt((X-X_gcrs).^2))

The conversion matrix of ITRS to GCRS
, ITRS to GCRS and GCRS to ITRS can be reversed. For example, the conversion matrix from GCRS to ITRS obtained earlier is GC2IT. From GCRS to ITRS can be expressed as: X_itrs = GC2IT *X_gcrs, on the contrary from ITRS to GCRS can be expressed as: X_gcrs = inv(GC2IT )*X_itrs.

The complete calculation is available for download at the Matlab forum ( IERS4m ). The technical methods involved here can be understood in IERS4m with the documents in the reference below, so this blog post does not have too detailed records. Here you are welcome to leave a message to consult and learn from each other.

6. Reference

IERS Chapter 5

IRISH 2010，www.iers.org/IERS/EN/Publications/TechnicalNotes/tn36.html
IRISH 2003，www.iers.org/IERS/EN/Publications/TechnicalNotes/tn32.html

IERS FTP Chapter 5 Fortran Code

Open and download via FileZilla
ftp://tai.bipm.org/iers/conv2010/chapter5/
ftp://maia.usno.navy.mil/conv2010/chapter5/

SOFA Library

Main: http://www.iausofa.org
Time Reference Cookbook: http://www.iausofa.org/2012_0301_C/sofa/sofa_ts_c.pdf

C language, Fortran, Matlab SOFA program Validation Routines:
C: http://www.iausofa.org/2012_0301_C/sofa/t_sofa_c.c

FORTRAN:
http://www.iausofa.org/2012_0301_F/sofa/t_sofa_f.for

Tutorial: www.iausofa.org/publications/sofa_pn.pdf

Matlab version:
https://www.mathworks.com/matlabcentral/fileexchange/74523-standards-of-fundamental-astronomy

NEW Comparison: www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA543243