Introduction to Space Geodesy

• earth rotation

• gravity field

• geometry

They all come together in reference frames.

Coordinate systems (definition) define the mathematical base functions in an abstract metric space; the set of coordinates enable a reference to the unique position in a spatial domain (i.e. earth, space).

Reference systems (convention) are a coordinate system which is defined in a physical model of the real world; origin and axis directions are associated with elements that exist in the real world. The reference system includes a discussion about time as well. Reference Systems are, for example, the ICRS.

Reference frames (realization) are a realization of the reference systems; the outcome is a set of coordinates that have been measured and are associated with reference points of observing stations.

So from Coordinate System -> Reference System -> Reference frame it gets more concrete. Coordinate systems are purely mathematical while in Reference Systems, the mathematics are applied to real objects. For Reference Frames, examples would be the earths surface or the earth rotation.

For Coordinate Systems, the set of coordinates have to be chosen, i.e. either spherical, cartesian or ellipsoidal. The change from one set of coordinates to another is called conversion.

For Referene Systems, the “physical oberver”, so the origin of the Reference System has to be chosen. It can be either topocentric (centered in a specific location on Earth’s surface, uses local viewpoint, i.e. local horizon; good for tracking celesial objects from earth), barycentric (center is the center of mass in a system, for solar system near sun’s center) or geocentric (center is the earth, good for satellite trajectories and earth-based measurements)

Cartesian: c = c(x,y,z)

Spherical: u = u(φ, λ, r)

Ellipsoidal e = e(B,L,h)

u1 = x, u2 = y, u3 = z

The coordinate system of the ICRS is an orientational coordinate system (distance is projected onto the unit sphere, so r = 1 or unspecified) which uses spherical coordinates. So, the ICRS uses a spherical coordinate system without r, but just with the two angles.

Therefore, the ICRS is based on distant sources (fixed relative to distant, non-moving objects such as distant galaxies or quasars) and not affected by the motion of the Earth or the Sun.

The metric tensor defines the distance between to points.

For cartesian coordinates, it is:

For spherical coordinates, it is:

For ellipsoidal/geodetic coordinates, it is:

ICRS: barycentric, Origin is Solar system Barycenter SSB, Orientation is celestial; time: Barycentric coordinate time TBC; quasi-inertial

GCRS: geocentric, orientation is celestial, time: Geocentric coordinate time TCG

ITRS: geocentric, orientation is global terrestrial, time: Geocentric coordinate Time TCG

Local horizon system: topocentric, orientation is local terrestrial, time: TT

In this context, a celestial orientation means the Reference System is aligned with distant astronomical objects such as quasars outside the Solar system. A global terrestrial orientation means it’s aligned with Earth’s surface and rotation, so it’s fixed relative to Earth’s features. Local terrestrial means the orientation is specific to the observers location on Earth’s surface.

ITRS: global terrestrial

ECEF: Earth-centered Earth-fixed, used for satellites

WGS84: geodetic reference system, used for GPS and satellite coordinates

Earth gravity field

The connecting element here is the geocenter.

• translation

• rotation: either reference frame or vectors are rotated using rotation matrices, i.e. with euler angles or cardan angles

• scaling: vectors get “longer” or “shorter”

• reflection: special case of scaling where only one axis is scaled; switching from right-handed to left-handed system

Inertial is a reference system that is, dynamically speaking, not accelerating, so non-rotating. In an inertial system a mass particle experiences no apparent forces.

ICRS is sometimes labeled as quasi-inertial, which means it is non-rotating in a kinematic sense, but is rotating in a dynamic sense.

In a non-inertial system (i.e. ITRS which is fixed to earth and therefore also to earths rotations) a free mass particle experiences accelerations due to inertial forces, which are:

• centrifugal force

• coriolis force

• euler force

1.: two “different vectors” and keep reference system

2.: two “different reference systems” while keeping vector -> what we do!

Polar motion is the motion of earth’s rotaional axis relative to its surface, meaning it’s a shift of geographic poles. It’s the difference between the convential pole to the actual pole and caused by changes in the distribution of Earth’s mass resulting from factors such as tectonic activity, seasonal shifts, melting of glaciers. Movement is small and irregular.

The rotation is just the rotation of Earth around the Earth’s axis.

Precession and nutation have the same physical cause (gravitational influence of objects in space), only differ mathematically.

Precession is the slow (cycle of 26.000 years), conical motion of earths rotational axis, caused by gravitational forces by the Sun and the Moon. Affects orientation of GCRS

Nutation is the periodic oscillation of Earth’s rotational axis which causes the axis to “wobble” slightly, mainly caused by the gravitational influence of the Moon’s orbit.

Polar motion is a movement of earth’s axis relative to its own surface while precession changes the orientation of Earth’s axis in space.

EOP: Earth orientation parameters, which are

• celestial pole offsets (X, Y) given through the precession-nutation model

• ERA: Earth Rotation Angle

• x, y polar motion components

Have to be observed by VLBI, are only predictable for about 7 years into the future

ERP: Earth rotation parameters, which are

• LOD or UT1: length of day, which has by far the biggest effect

• x, y polar motion components

Are observable by VLBI and satellite-based geodetic techniques

• classical equinox based approach

• new (since 2000) conventional “non-rotating origin” approach

• External causes

  • gravitational forcing by Sun and Moon, explains precession & nutation

  • tidal friction (tides deform Earth, deformations cause friction, and friction has rotational energy -> slows rate of rotation (3 ms/100 yrs, therefore the days were much shorter hundreds of million years ago, influences LOD) while keeping angular momentum of Earth-Moon-System which means the distance between Earth and Moon increases; the gravitational interaction between Moon and Earth slows them so that the Moon is tidally locked

• intrinsic causes

  • surface processes (mass variations, i.e glaciers and ice sheets as well as ocean mass, water storage; long-term trends in polar motion are caused by mass variations!)

  • interior processes (long-term trend in polar motion not fully explained by surface processes -> plate tectonics, anomalies in the mantle and changing masses in the mantle)

EOP (Earth orientation parameters) are:

• polar motion components

• celestial pole offsets (X, Y) -> precession-nutation

• ETR (Earth rotation angle)

which describe differences between GCRS and ITRS.

All systems that have acces to both ITRS and GCRS can measure/observe the EOP, which are the classical ground based space geodetic techniques like GNSS, LLR, SLR, VLBI, DORIS.

All techniques are classical space geodetic techniques.

You can measure the earth rotation with inertial sensors such as ring laser gyroscopes (i.e. ROMY ring laser in Munich) with which it’s possible to monitor the instantaneous eath rotation in real time with respect to the local inertial system. But (!) space geodetic techniques and inertial sensors refer to different intermediate reference systems, they have their own local inertial frame. Which is why they are good for monitoring earthquakes, but for the general earth rotation space geodetic techniques are probably better.

• planning and monitoring of earth-orbiting, interplanetary and deep space spacecraft launches and trajectories

• transformation between celestial and terrestrial reference systems

• application of remote sensing, altimetry, other space geodetic techniques and VLBI which is useful for navigation

• establishment of time scales: now solar time UT1, UTC

useful for

• free core nutation

• UT1 and LOD variations, i.e. tidal friction spinning, atmospheric oceanic hydrologic angular momentum exchange (for example: strong El Nino -> angular momentum of solid earth decreases -> earth rotation slows down)

• polar motion: climate-change driven (i.e. glaciers metling) and earth interior causes of polar motion

El Nino results in a decrease in earths rotation rate -> increase in LOD -> decrease in strength of the Coriolis force because if the Earth rotation slows, the force is not as strong.

During an El Nino event, warm water moves from the western to the eastern (southern) pacific, so there is a mass distribution towards the equator where the rotational speed of earth is faster compared to the poles. This mass distribution leads to a higher LOD (like a skater spinning with arms extended vs. with arms pulled in).

La Nina tends to have the opposite effect.

1. Navigation is defined as determining or planning the position and course of a vessel or aircraft using various methods like geometry, astronomy, and radio signals.

2. Fundamental concepts include Position, Velocity, and Time (PVT).

1. To find a position using known points, range measurements (distances) from a receiver to transmitters with known coordinates are used. These measurements form a "line of position."

2. With only one measurement, there is ambiguity about the receiver's exact position. At least three to four measurements are needed to resolve the coordinates.

GNSS uses (artificial) satellites to determine positions on earth.

Measurements between the receiver and satellite allow for calculating the receiver's position.

GNSS requires at least four satellites to be visible for accurate positioning.

GNSS comprises a space segment (satellites), a control segment (tracking stations and ground infrastructure), and a user segment (receivers and processing software).

The range is the true distance between a receiver (on the ground) and a satellite (in space), based on the time it takes from a signal to travel from the satellite to the receiver (time of Flight ToF).

r = c * ( Tr - Ts) with

c = speed of light

Tr = time of signal arrival (Time of Arrival)

Ts = signal transmission time (Time of fly)

Because i.e. the clocks of the satellite and the receiver are not perfectly synchronized, the measurements are not ranges but pseudoranges, which is an estimate of the distance. Therefore the pseudorange includes errors.

Thus, it is defined as

1. Code-phase measurements: provide the initial range estimate but are influenced by instrumental noise which makes them less precise.

2. carrier-phase measurements are more precise but come with the added complexity of phase ambiguity (carrier phase ambiguity bias). When resolved, they are very precise.

GNSS satellites transmit continuous electromagnetic waves; the phase of the carrier wave which has a high frequency is tracked by the receiver. The receiver then measures the phase difference between the transmitted and the received signal - this difference allows for the calculation of the distance between the satellite and the receiver.

With Carrier-phase measurements, there is an ambiguity in the number of complete wavelengths between satellite & receiver which needs to be resolved to determine the true distance accurately. Once that is resolved, carrier-phase measurements can achieve millimeter-level precision.

Carrier waves can also carry data by using a Modulator which alters a carrier wave to encode data onto it. It can modify either the amplitude, frequency or phase. Phase modulation is particularily relevant for GNSS because phase modulation helps to encode the satellite’s signal (position, timing data) onto the carrier.

The Demodulator is then responsible for extracting the original information.

Code-phased measurements use a known, repeating sequence of signals (pseudorandom noise PRN, series of binary numbers 0 or 1) which is transmitted by the satellite at a very high speed. For GPS, this is often the C/A (Coarse/Acquisition) code. The receiver generated an identical PRN code and shifts it until it matches the one it receives from the satellite.

The time delay between when the receiver's code matches the satellite's incoming code indicates the signal’s time of arrival (ToA).

The receiver calculates how long it took for the signal to travel from the satellite to the receiver by determining the shift required to synchronize the codes. This time is called Time of Flight.

Code-phase measurements have an accuracy of 3 -10 meters without augmentation.

They are used for the initial estimation of the receivers position.

The satellite clock (usually an atomic clock) is usually a little bit faster than the receiver clock (usually a quartz clock) due to the smaller gravitation at the altitude of the satellite.

But then, a moving clock runs slower moves slower than a clock at rest.

Both effects together cause a relativistic Doppler shift.

This Doppler shift can be used to estimate the velocity of the receiver relative to the satellite.

GNSS has 4 main segments, which are:

• Space segment: constellation of satellites (GPS: min. 24, currently: 31 GPS satellites) that broadcast signals to earth; 4 satellite should be visible from any point on Earth’s surface to calculate an accurate position

• control segment: consist of Operational Control Segment (OCS) which includes tracking/monitoring stations (typically 11 globally which track satellites and send data to Master Control station), ground antennas and master control stations (processes data they get i.e. from tracking stations). They are responsible for monitoring, maintaining and managing the health of the satellites and their orbits. They i.e. also correct their clocks.

• User segment: all GNSS receivers, which can be used for i.e. navigation, surveying and mapping

• Ground segment: networks and services like the International GNSS service (IGS) that delivers data products to its users; it consists of a network of globally distributed tracking stations that observe GNSS signals. Products include satellite orbits, clock corrections and Earth rotation parameters.

GPS: Global Positioning System

• managed by the U.S. Space Force

• fully operational since 1995

• Medium Earth Orbit at ca. 20000 km

• currently 31 active satellites

• widely used!

GLONASS: Global Navigation Satellite System

• managed by the Russian Space Forces

• fully operational since 1995

• Medium Earth Orbit at ca. 19000 km

• currently 23-26 operational satellites

Galileo

• managed by the ESA

• fully operational since the 2020s

• Medium Earth Orbit at around 23000 km

• targets 30 satellites

• designed to be very precise (more precise than GPS), like 1 meter

BeiDou (BDS):

• managed by the China Satellite Navigation Office

• regional service since 2000, global service since 2020

• combination of orbits, MEO for global coverage and GEO (Geostationary Earth Orbit) and IGSO (Inclined Geosynchronous Satellite Orbit) for a more regional coverage

• over 30 satellites for global coverage

• similar to GPS but offers unique features such as SMS -> data communication through satellites

• Doppler observables, used for velocity determination

• Code pseudo-ranges, derived from PRN code

• Carrier-phase measurements

• Signal-to-noise-ratio (SNR), indicated signal quality

A GPS signal transmitted by satellites is a combination of carrier waves, PRN codes and navigation data.

GPS satellites transmit signal using radio frequencies which are

L1 (civilian C/A & encrypted P(Y) code), L2 (encrypted code) and L5 (newer civilian signal for enhanced accuracy & safety.

THe L-Band ranges from 1,1 to 1,6 GHz.

1. GEO (Geostationary Earth Orbit): Satellites remain fixed relative to a point on Earth, useful for communication and augmentation services but not typically for global GNSS coverage.

2. MEO (Medium Earth Orbit): This is where most GNSS satellites (e.g., GPS, GLONASS, Galileo) operate, at altitudes of 10,000–20,000 km. MEO provides a good balance of coverage and orbital period.

3. LEO (Low Earth Orbit): Satellites here operate between 750 and 2,500 km, like those used for communication (e.g., Iridium) and specialized positioning systems. LEO has a small coverage footprint and shorter orbit times.

• Total Electron content (number of electrons in a vertical column of the ionosphere), which is closely related to the propagation of weather fronts; also can be used to monitor space weather such as solar flares and their impact

• Integrated water vapor (high in summer, low in winter) which is rising and related to the rising of global temperatures and therefore is good data for climate change modelling

• Zenith/Slant total delay, which represents the total delay experienced by a GNSS signal as it travels through the Earth's atmosphere. It consist of the dry delay (90% of delay, due to dry gases like nitrogen and oxygen in the atmosphere) and the wet delay (caused by water vapor content in the atmosphere) and can be used to improve short-term weather forecasts

• Atmospheric sounding (measures the vertical structure of the atmosphere, providing data on temperature, pressure, humidity, and other variables at different altitudes)

  • ground-based (measurements are made using instruments located on the Earth's surface and receiving GNSS satellite signals, i.e. radiosondes, GNSS ground stations, and weather radars; GNSS ground stations measure i.e. the Zenith Total Delay ZTD and the Integrated Water Vapor IWV; limited to lower atmosphere, high resolution but less coverage)

  • satellite-based (measurements are made from satellites orbiting Earth, providing global coverage and data across different altitudes, including remote areas like oceans and polar regions; lower resolution but global coverage)

    • GNSS Radio Occultation (Low-Earth-Orbit satellite receives signal from a GNSS satellite like GPS/BeidOu/whatever; during the travel from the GNSS satellite to the LEO-satellite the signal bends due to changes in air density, temperature, and water vapor and then the bend & delay can be analzyed which gives info about: Temperature, pressure, humidity, and electron density); examples for LEO-satellites are: CHAMP, Grace, Metop - also an advantage: radio signals can penetrate clouds!

• Reflectometry: relatively new method, uses GNSS signal that are reflected from the ground and picked up by specialized receivers (on the ground or in the atmosphere); can be used to monitor the ocean. i.e. the rise of sea levels or the roughness of the ocean, land surface monitoring as i.e. the soil moisture and climate studies as for example studying the ice sheet dynamics

• Scatterometry: works similar to Reflectometry but uses signals that are scattered; this scattering happens when signals interact with rough or complex surfaces like ocean waves or forest canopies; considers diffuse scattering, where signals are scattered in multiple directions due to surface irregularities; can be used i.e. to analyze ocean wind speed & direction (ocean surfaces appear “rougher” with higher wind speeds, which scatters the signals)

1. GPS-Radio Occulation can be used to improve hurricane forecasts; radio signals can penetrate clouds and precipitation so it works in every weather condition, it has a high vertical resolution and are not affected by instrument biases which makes them highly accurate!

2. GNSS Reflectometry

   • GNSS signals reflected from ocean surfaces are analyzed to estimate wind speeds and directions.

   • The roughness of the ocean surface (caused by winds) modifies the reflected signals.

   Application in Hurricanes:

   • Helps in tracking wind speeds over oceans, a critical factor in hurricane intensity prediction.

   • Missions like CYGNSS (Cyclone GNSS) specifically monitor tropical cyclones, providing frequent updates on ocean winds and surface conditions.

3. GNSS is also highly effective for tsunami detection, monitoring, and forecasting, both through direct measurements and by supporting other observational techniques. Here’s an overview of GNSS-based methods for dealing with tsunamis:

   1. GNSS-Based Sea Level Monitoring

   How it Works:

   • Coastal GNSS stations detect changes in sea surface height by measuring vertical land motion and comparing it with sea level changes.

   • GNSS-equipped buoys in the ocean measure sea level changes directly.

   Application in Tsunamis:

   • Tsunamis are preceded by abrupt changes in sea level due to underwater earthquakes or landslides.

   • GNSS-based observations can detect these changes in real time, providing critical early warning signals.

   2. GNSS Reflectometry (GNSS-R) for Ocean Monitoring

   How it Works:

   • GNSS-R uses reflected GNSS signals from the ocean surface to measure sea level and wave characteristics.

   • Anomalies in sea surface elevation or wave patterns can indicate a tsunami.

   Application in Tsunamis:

   • Monitors ocean surface changes far from land, detecting tsunamis soon after they are triggered.

   • GNSS-R can complement traditional tsunami buoys (like DART) by providing additional spatial coverage.

MOSAIC: The ship Polarstern which is drifting through the arctic ice and has GNSS Reflectometry equipment installed on the ship which analyzes the reflections from the ice.

CYGNSS: A constellation of satellites used for ocean speed determination (using AI and deep learning methods) and can i.e. be used to monitor tropical cyclones or derive global soil moisture; uses GNSS Reflectometry

COSMIC: Missions COSMIC-1 (FORMOSAT-3) and COSMIC-2, which are using GNSS Radio Occultation to do atmospheric research, analyzing Total Electron Content for monitoring Space Weather and can also improve hurricane forecasts, like Ernesto in 2006

MeTop: a series of satellites which use GNSS Radio Occultation for weather forecasts, climate monitoring and disaster response

Smartphones and other devices such as smartwatches are a huge data source! For example, ZTD can also be derived from smartphone data and although it’s not completely accurate, this can change over the following years!

For example with the CAMALIOT Android app, GNSS data can be crowdsourced which collected 5 TB of raw data.

1. GNSS Atmospheric Sounding (GNSS-RO): Ideal for studying vertical atmospheric structures, weather prediction, and ionospheric research. However, it is complex and resource-intensive.

2. GNSS Reflectometry (GNSS-R): Excellent for monitoring Earth’s surface (e.g., ocean, soil, snow) with lower cost and broad applicability. It complements GNSS-RO but lacks detailed atmospheric profiling.

On a calm ocean, the GNSS signal is reflected directly towards the receiver from a single, precise point (Specular reflection point).

On a rough ocean surface (due to winds or waves), the reflection no longer comes from a single point, but the GNSS signal scatters over a wider area (the glistening zone) due to the varying angles of the waves. The size of the glistening zone depends on the roughness of the ocean surface. Due to the scattering, the signal is weaker and requires more advanced processing. But the size and characteristics of the glistening zone help with estimation of the surface roughness which is linked to wind speed and wave height; GNSS-R methods like CYGNSS use this principle to estimate ocean wind speeds i.e. for hurricane forecasts!

1. Satellite issues: Orbital inaccuracies, clock errors.

2. Signal propagation: Ionospheric/tropospheric delays, multipath effects.

3. Receiver-related: Instrumental noise, antenna variations.

4. Environmental factors: Radio interference.

Absolute Positioning: Determines receiver’s position directly with respect to a global reference frame such as WGS84, using only the signals received from satellites.

• Single Point Positioning (SPP) using pseudo-ranges from code-phase measurements. Requires no external correction data - easy to implement but less accurate due to uncorrected biases & errors, typically an accuracy between 10-30m. Used in Navigation where high accuracy isn’t critical (recreational/handheld GPS devices)

• Precise Point Positioning (PPP), requiring accurate satellite clock and orbit data; uses code- and carrier-phase measurements combined with correction data for satellite clocks and orbits; removes/models atmospheric effects, accuracy: sub-meter to centimeter-level, depending on processing. Applications: Surveying, geodesy, precise navigation

Relative Positioning: compares position of “rover” receiver (which position needs to be determined) with a “reference” receiver at a known location

• Requires a known reference station and a rover station.

• Techniques:

  • Static: Both receivers are stationary, collect data over a long time period. Highest precision (mm to cm accuracy), long observation times. Used for geodetic surveys and tectonic plate monitoring

  • Rapid Static: Like static, but shorter observation periods (minutes)

  • Kinematic: Continuous tracking for moving receivers. Accuracy typically 10 cm, used for vehicle tracking and dynamic surveys.

  • "Stop-and-Go": Combines static and kinematic methods, receiver stops at each point to collect static data, then goes to the next point. 3-5cm accuracy

  • Real-Time Kinematic (RTK): High-accuracy carrier-phase-based trelative positioning technique, requires a reference station transmitting corrections to the rover; both receivers must track the same satellites.

Conclusion

• Use Absolute Positioning when cost, simplicity, or global coverage is more critical, and moderate accuracy suffices.

• Use Relative Positioning when high accuracy is essential, particularly for scientific, engineering, or precision navigation tasks.

Augmentation systems improve GNSS performance by correcting errors in satellite signals, such as:

• Satellite orbit and clock inaccuracies

• Atmospheric delays (ionospheric and tropospheric effects)

• Multipath errors (signal reflections)

• Instrumental biases

They are particularly useful in applications where high accuracy and reliability are critical, such as aviation, precision farming, surveying, and autonomous navigation.

There are space-based (SBAS) and ground-based (GBAS) augmentation systems.

1. SBAS: Broad coverage over a wide area, useful for applications like aviation where users are distributed across large regions. Examples: WAAS (US), EGNOS (Europe), MSAS (Japan), GAGAN (India). Uses geostationary (orbital speed matches earth rotation -> appears to remain fixed in the same position) satellites to broadcast correction signals. Therefore SBAS is limited to areas within the footprint of geostationary satellites. Also requires SBAS-compatible receivers

2. GBAS: Focused on specific areas, ideal for applications requiring very high accuracy, like landing planes at airports. Use local ground reference stations typically for a limited area. Example: LAAS (primarily for aviation, approach & lansding). Is limited to local coverage and infrastructure-dependent.

   1. Differential GNSS (DGNSS); involves ground reference stations that compute corrections and transmit them to users in real-time, can use code-phase & carrier-phase corrections. Is widely used, i.e. in marine navigation. Limited coverage based on reference station network & Communication link required between reference station and rover

1. First Law: The orbit of a planet is an ellipse with the Sun at one focus. This implies that orbits are not perfect circles but can vary in shape.

2. Second Law: A line joining the planet and the Sun sweeps equal areas in equal times, meaning that planets travel faster when closer to the Sun and slower when farther away.

3. Third Law: The square of a planet's orbital period (T2T^2T2) is proportional to the cube of its semi-major axis (a3a^3a3). This relationship helps predict orbital periods and distances for celestial bodies.

It defines the boundary between Earth's atmosphere and space, located at 100 km above sea level. It’s defined like this because Aircraft equipped with wings can generate lift and maintain flight at altitudes < 80 - 90 km. Anything beyond is flying in inertial space, requiring the high speed of the Kepler orbits.

A Hohmann transfer orbit is a two-step, energy-efficient method to move a spacecraft between two circular orbits around the same celestial body.

1. First Impulse: Accelerates the spacecraft to enter an elliptical transfer orbit.

2. Second Impulse: At the transfer orbit's apogee, the spacecraft accelerates again to enter the desired circular orbit. This method minimizes fuel consumption, making it ideal for interplanetary missions or satellite repositioning.

1. Circular Orbit: The orbital radius and velocity remain constant. Gravitational and centrifugal forces are perfectly balanced.

2. Elliptical Orbit: The distance and velocity vary. Objects move faster near the pericenter (closest point to the central body) and slower near the apocenter (farthest point). Governed by Kepler’s laws.

Lagrangian points are five positions in space where a small object can remain stationary relative to two larger celestial bodies (e.g., Sun and Earth).

• L1, L2, and L3: Unstable points along the line connecting the two bodies.

• L4 and L5: Stable points forming equilateral triangles with the two bodies. Applications include placing telescopes (e.g., James Webb Telescope) and observing phenomena like solar winds.

1. LEO (Low Earth Orbit): 200–2,000 km altitude, used for remote sensing and manned space missions.

   1. Polar Orbit, used for global mapping. high inclination (90°) wrt the Earth Equator; when Earth rotates, it can map the entire earth (see picture).

   2. ELEO (Equatiorial LEO), low inclination (0°) wrt the Earth equator

2. MEO (Medium Earth Orbit): 5,000–20,000 km altitude, used for navigation systems like GPS, GLONASS, Galileo.

3. GEO (Geostationary Orbit): 35,790 km altitude, where satellites appear fixed relative to Earth's surface, useful for communication and broadcasting. Is a special type of Geosynchronous Orbits, can see almost 1/3 of Earths suerface from high above -> Only few satellites required for global coverage, but no coverage near the poles right now. Orbit is very stable, but high launch costs.

• Euler method: single-step method, Simple, not for practical use - large errors for longer integration times

• Runge-Kutta Methods: single-step method, easy to use and widely applied

• Multistep methods: high effiency, but need storage of past data points

• Encke Method: higher accuracy & less integration steps

The goal of Orbit Integration is to improve observations by minimizing the difference between the calculated satellite orbit and the actual observed orbit.

The calculation of the orbit is what orbit integration does (solution to the Euler-Newtonian equation of motion), and the more accurate it is, the lower is the difference.

The observations come from space geodetic techniques like GNSS, SLR or DORIS.

We need it to improve our data; if we want to measure the ocean level but calculate the wrong position for the satellite the measurement is useless.

Due to gravity effects the satellite orbit isn’t a clear ellipse but instead it “bops up and down” additionally.

The Euler-Newtonian equation of motion describes the position of a satellite.

Numerical integration of the Euler-Newtonian equation of motion provides a vector with position and velocity at any time.

There are 2:

• Euler method

• Runge-Kutta methods

The advantages of single-step methods are:

• easy to use

• In every step a new step size can be used: well suited for functions with rapid changes (which is usually not the case for satellite orbits)

Euler method: Start with known values (t0, y0) and proceed with a time step of size h along the tangent to the graph of y.

y(t0 + h) = y0 + h * f(t0, y0) where h is the step size and f(t,y) the function describing the changing rate of y.

The method is only first-order accurate and errors grow with step size h, which is why it needs to be small.However, even with very small step sizes the errors are large if we follow the graph over several steps, and the small step sizes increase the compuational effort.

Therefore, the Euler Method is not of practical use.

Runge-Kutta Methods: These methods consider additional intermediate point within each step. The classical method is the Runge-Kutta 4th Order method or short “4th order method”. It evaluates only the function f and avoids the calculation of derivatives (e.g. in contrast to 4th order Taylor polynomial). Instead, it uses a weighted mean of 4 slopes/tangents. 2 of the slopes are slopes at the start/end of the interval and 2 are slopes from midpoints.

This method is widely used for practical problems and easy to use.

The stepsize h depends on the satellite altitude due to the influence of the gravity field, so LEO would use sth like 5 seconds (CHAMP/GRACE) and MEO would use sth like 30s (GPS).

The runge kutta method is a single step method because all integration steps are independent and no use is made of function values calculated in earlier steps.

Multistep methods store values from previous steps. They are most efficient & for differential equations defined by complicated functions with a lot of arithmetic operations

An example are Adams-Bashforth-methods. They calculate a polynomial of 3rd order from 4 known values which leads to the Adams-Bashford coefficients. In practice multistep methods of order 10-12 are used.

The Encke Method is a high accuracy method used for determining the satellites position at any time.

It integrates the pertubation of an orbit (DE: Unruhe, Störung) relative to a known reference orbit, for example the Kepler orbit -> this is useful for near-Keplerian orbits.

The motion of the satellite is split to the reference orbit and the pertubation vector, which accounts for the deviations from the reference orbit due to e.g. gravitational forces.

Note that in Encke’s method from time to time it is necessary to select a new osculating Keplerian reference orbit to prevent the perturbations from growing too large. This process is called “rectification” of the orbit

Spherical Harmonics are mathematical functions used to represent functions defined on the surface of a sphere; they are, for example, a tool to represent gravity fields for bodies with inhomogeneous density distribution (such as Earth!).

They are functions which fulfill the Laplace equation deltaV = 0.

Currently, some Missions measure the Gravity field of Earth and and its changes. to do so, some use HL SST (High-Low Satellite-toSatellite-Tracking). They measure the Gravity field by tracking the motion of a LEO satellite (itself) relative to a high-altitude satellite like GPS.

For example, GRACE measures temporal changes in the Gravity field of Earth for lower scale features such as ice mass loss and groundwater depletion.

CHAMP provides static coefficients for large-scale features like Earth's flattening and has a relatively low resolution with n = 100.

GOCE studies ocean circulation patterns and refines global gravity field models with a high resolution n > 100, providing static models.

There are already formulas for calculating the gravitational potential of simple bodies like solid spheres:

V = GM/r

but Earth does have inhomogenities and therefore needs a better approximation.

To approximate the gravitational potential, polynomials (like ax + ax² etc.) are used.

Then, the equation is organzied by degree. For example, x² and xy have the degree of 2, y³x has the degree of 4.

But we don’t use just any polynomial for approximating, we use homogenous harmonic polynomials.

Harmonic in this case means that the function satisfies the Laplace equation deltaV = 0 (deltaV is the Laplace operator).

For a function to be harmonic, its value must "balance out." For example, if it's high in one spot, it must drop elsewhere to compensate. This property makes harmonic functions excellent for describing smooth things like gravity or temperature fields.

Homogenous means that all terms in a function have the same degree. Example: f = ax² + xy + y² + xz (all have degree 2). TO see the Himalean Plateau for example, we need a degree of 30.

The gravitational potential is then an infinite sum of these harmonic polynomials, so that each polynomial captures more and more detail about the gravitational field. The higher the degree n is, the more detailed but also more complex the model is.

Each of these harmonic polynomials is represented as a linear combination of m linear independent base functions Hnm.

While these are great for 3D variations, we want the equations to be restricted to the surface of a sphere, and as such only be dependant on the latitude (theta) and longitude (lambda) - r is constant.

Therefore, the general harmonic polynomials H needs to be transformed into the base function Y. This is done by switching to spherical coordinates and neglecting the radial component (this is for variations with distance, so not only on the surface).

Further splitting the equations up, Y can be expressed as being a combination the base functions C and S, which again are expressed by the Legendre function P multiplied by sin(l) or cos(l).

Spherical Harmonics then also vary with longitude. The order m of spherical harmonics represents how the function varies in the longitude direction around the sphere. There are different types of Harmonics:

• m = 0: Zonal Harmonics, independent on longitude and only dependant of latitude; describes large-scale features such as earth’s flattening at the poles.

• |m| = n: Sectorial Harmonics: These have maximum variation in longitude, creating patterns like "slices" through the sphere; captures longitudinal asymmetry.

• 0 < |m| < n: Tesseral Harmonics, these have variation in both longitude and latitude, creating a checkerboard-like pattern, model localized variations like mountains and trenches.

When taking the base functions S and C into account:

Both functions consist of Legendre functions multiplied by coefficients.

C has coefficients os the cos(ml) term, which is why Cn,0 can represent zonal Harmonics while S with coefficients sin(ml) can not! (sin(0) = 0.)

If you imagine the Earth:

• n=2, m = 0: Describes the Earth's flattened shape (oblate spheroid).

• n=2,m=2: Captures variations like the equatorial bulge and additional longitudinal variations.

1. Degree n: Controls the latitude and overall detail of the pattern.

2. Order m: Controls the variation along longitude.

n = degree, m = order.

For each degree nnn, we can systematically construct 2n+1 independent harmonic functions.

Why?

• all terms must have the same degree n.

• all functions must satisfy the Laplace equation - here, this means: -> this is the Laplace Operator!

Further, for knowing which equation relates to which order, the equations need to be expressed in spherical coordinates.

theta = t (latitude), lambda = l (longitude)

x = rsin(t)cos(l)

y = rsin(t)sin(l)

z = rcos(t)

The order m describes how the harmonic function varies with longitude lambda.

m = 0: No variation in longitude -> no lambda in the equation! This is true for x² + y² - 2z², which is why this is H20.

The other ms require trigonomic identities and such, but in the end cos(2l) means m = 2, sin(l) means m = 1 and so on.

The Laplace operator deltaV relates to the Laplace equation

deltaV = 0.

Functions who satisfy the Laplace equation are called harmonic.

In rectangular/cartesian coordinates this means:

So the sum of the 2nd partial derivatives must be 0.

In spherical coordinates, this gets a bit more complicated but in the end the Operator is dependent on r (radius of the sphere).

If we neglect the dependency on r, the Laplace operator is reduced to a spherical shell and is now called the Bertrami operator (sometimes represented by deltaY). This is important because often, we are interested in how the field varies on the surface.

(Associated) Legendre Functions P are functions derived by Legendre polynomials which describe variation with latitude on a sphere, which is why they are used in Spherical Harmonics for approximating the gravity field of the Earth’s surface.

The base functions of spherical harmonics C and S consist partly of (associated) Legendre Functions.

VBLI or Very long Baseline Interferometry is a very precise technique in space geodesy which allows

• observing quasars which realize the ICRF (Intrnational Celestial reference frame) -> good for maintenance & realization of global reference frames!

• providing the complete set of Earth Orientation parameters, is unique for the determination of DUT1 (difference to UT1) and long-term nutation

• providing precise length of very long intercontinental baselines (distance between two radio telescopes used in observation) which support realization & stability of the ITRF (International Terrestrial Reference Frame)

The products of VLBI are mainly:

• Celestial Reference Frame CRF

• Terrestrial Reference Frame TRF and baseline lengths

• Station velocities which can be used for plate-kinematic models in case of e.g. earthquakes

• complete set of EOP

The VLBI2010 Global Observing System (VGOS) aims to enhance the technique's capacity and accuracy for future geodetic and astrometric needs

IVS is the International VLBI service for Geodesy and Astrometry and is a service of:

• IAG (International Association of Geodesy)

• IAU (International Astronomical Union)

• WDS (International Science Councel World Data System)

The goals of the organization is to provide a service to support research & developemnt, interact with the comminuty of users of VLBI products and to integrate VLBI into a global earth observing system.

The main tasks are to coordinate VLBI components, guarantee provision of products.

Each telescope in the VLBI network observes the same distant radio source (e.g., a quasar, a radio galaxy which emit strong radio signals) simultaneously.

The radio waves travel vast distances, arriving at the telescopes with a slight time delay due to differences in their positions.

Each telescope is equipped with a very precise atomic clock, which timestamps the incoming signals and allowing for synchronization. Together with the time stamps, the signals are recorded in two main frequency bands which are the X-band (8 channels) for high resolution & precision and the S-Band (6 channels) to help filter out ionospheric effects.

The recorded signals are sent to a central processing facility called a correlator, which aligns the signals and analyszes the interference patterns of the signals. Out of these, the position & movement of the source can be reconstructed.