GIPSIE ® (6.0.0)

# Computational models

# Satellite Orbits

Satellite state vectors are computed within the simulation for each simulation epoch and used to determine the pseudorange measurements as well as satellite visibility and depending parameters. Two options exist for state vector computation:

  • orbit integration
  • ephemeris-based computation

Using ephemeris-based computation the given perturbed Keplerian elements from the scenario are directly used to compute the satellite state. In the other case these Keplerian elements are used only for initialization of the orbit integration module which is a numerical integration of the satellite state based on gravitational forces.

The simulation of satellite orbits using orbit integration is useful to simulate satellite orbit errors as they would appear in reality due to the non-perfect representation of the perturbed orbit by using Keplerian elements as used within the navigation message.

Using orbit integration the satellite state vectors are propagated based on the gravitational forces of earth, sun and moon. While for the earth a gravity field model up to a certain (configurable) degree is used, sun and moon are modelled as point masses only, which is sufficient due to large distance to these bodies. The propagation is computed using numerical integration based on a Runge-Kutta scheme that is taken from [1].

# Satellite Clock Noise - Allan Deviation

Based on the satellite clock model, the Allan Variance parameter differ. The appropriate values are given in the table below.

Allan Variance:

where

  • WFM: white frequency modulation
  • FFM: flicker frequency modulation
  • RWFM: random walk frequency modulation
Clock Type Description
Rubidium
  • h0: 1.0e-23
  • h1: 3.0e-27
  • h2: 4.0e-31
Cesium
  • h0: 1.0e-22
  • h1: 1.0e-27
  • h2: 2.0e-32
Hydrogen-maser
  • h0: 2.0e-24
  • h1: 7.2e-29
  • h2: 1.76e-34

# Receiver Path

Two models (Interpolation or Propagation) for the receiver path computation are provided. Depending on the settings, a certain model can be chosen.

In general, the following characteristics apply for both models:

  • The earth-fixed receiver state (position, velocity, acceleration) is either set or computed for each simulation epoch.
  • User defined input specifies the receiver states at certain relative timestamps (in order to be independent of changing simulation start and end times).
  • If only one receiver state position (no velocity or acceleration) is set, the receiver is treated as static.

# State Interpolation

  • When state interpolation is chosen, only the defined positions (nodes) are utilized (velocitiy and acceleration is neglected).
  • The movement is simulated as geodetic lines (on the WGS84 ellipsoid) between these nodes. The geodetic line maintains its velocity after the last defined node.
  • Interpolation:
    • The travelled distance is interpolated as a cubic spline interpolated function of time.
    • Additionally the cubic spline of travelled distance has to be monotone (rising) to ensure that the absolute velocity is never below zero.
    • Absolute velocity and acceleration are computed as derivatives of the travelled distance and applied to the unit vector of receiver direction at each point.
    • The receiver position is computed as first geodetic principle on the ellipsoid.

Note

The monotone cubic spline interpolation follows the algorithm published in [2] to ensure monotonicity (further information on the implementation can be found on Monotone Cubic Interpolation (opens new window))

# First and Second Geodetic Principal

The computation of the receiver position is done as mentioned above using an interpolation on the ellipsoid for which solutions of the first and second geodetic principal on the ellipsoid are necessary. The first geodetic principal denotes the computation of the coordinates of a target point given the start point coordinates as well as direction and distance to the target point and the second geodetic principal is the reverse operation of computing direction and distance based on the two coordinate pairs. In Cartesian space these operations are quite trivial, however on the surface of an ellipsoid this is not true anymore. Since especially for longer distances there is a difference between the Cartesian and ellipsoidal solution the ellipsoidal formulae are implemented within the simulation. The principals on the ellipsoids are solved using the algorithm presented by Carl Friedrich Gauß[3] with the extension that also the height above the ellipsoid can be different for start point and target point. The formulae for implementation are taken from [4].

# State Propagation

  • When state propagation is chosen, the set position, velocity and acceleration are utilized.
  • The position has the be provided as absolute WGS84 coordinates, whereas the velocity and acceleration are defined in a LLF (north, east, up).
  • Every state is propagated via the previous state for every simulation epoch (an initial state is needed).
  • Only one component (position, velocity, acceleration) has to be provided (the other two are optional).
  • If a defined state is provided, it is directly applied for that epoch.

# Propagation Model

Provided Component Description

# Atmospheric Effects

Atmospheric effects are simulated in two ways: ionospheric delay and tropospheric delay including respective noise components (noise simulation can be activated and deactivated independently). Delays are computed based on the model selected within the scenario and follow the algorithms presented in:

# Troposphere models

  • Troposphere mapping functions for GPS and VLBI [5]
  • Estimation of tropospheric delay for microwaves from surface weather data [6]
  • GPT2w [7]
  • Galileo Reference Model [8]

# Ionosphere models

  • Klobuchar [9]
  • ESA Nequick G model [10]

# Multipath

Multipath simulation within GIPSIE® can be activated/deactivated and is modelled using a custom obstruction mask for each simulated receiver. This mask defines direction from which satellite signals arrive as line-of-sight (LOS) signal only, echoes only, LOS and echoes or no signal at all. In case echoes arrive from a satellite signal these echoes will have a certain delay as well as lower power compared to the original LOS signal, which is based on a statistical model using a Rayleigh distribution. The algorithm to compute the multipath simulation is taken from [11].

# Interference

An arbitrary number of interference signals can be added to the simulation for the simulation of unintentional RF interference or jamming signals using various customizeable waveforms. Each signal has its own transmission location and transmit power, which is used to compute the received signal power at each receiver based on a Friis free-space path-loss equation.

# Spoofing

Multiple spoofers can be added to the simulation, which represent replicas of the respective satellite's authentic signal with a certain delay that is based on the location of spoofer and receiver as well as the simulated receiver position of the spoofer. Furthermore it is possible to add an additional code offset (in terms of pseudorange offset) to the spoofing signal to simulate imperfections of the spoofing signal alignment with the authentic ones.

  1. O. Montenbruck and E. Gill, Satellite Orbits. Springer, Wien, NewYork, 2001. ↩︎

  2. F. N. Fritsch and R. E. Carlson, "Monotone Piecewise Cubic Interpolation" SIAM J. Numer. Anal., vol. 17, no. 2, pp. 238–246, Apr. 1980. ↩︎

  3. C. F. Gauß, "Untersuchungen über Gegenstände der Höheren Geodäsie" in Ostwald’s Klassiker der exakten Wissenschaften, 177th ed., J. Frischauf, Ed.Leipzig: Wilhelm Engelmann, 1910. ↩︎

  4. B. Hofmann-Wellenhof and G. Kienast, "Bezugssysteme", Graz University of Technology, Graz, 2006. ↩︎

  5. J. Boehm, B. Werl, and H. Schuh, "Troposphere mapping functions for GPS and very long baseline interferometry from European Centre for Medium-Range Weather Forecasts operational analysis data" J. Geophys. Res. Solid Earth, vol. 111, no. B2, p. n/a-n/a, Feb. 2006. ↩︎

  6. J. Askne and H. Nordius, "Estimation of tropospheric delay for microwaves from surface weather data" Radio Sci., vol. 22, no. 3, pp. 379–386, May 1987. ↩︎

  7. K. Lagler, M. Schindelegger, J. Böhm, H. Krásná, and T. Nilsson, "GPT2: Empirical slant delay model for radio space geodetic techniques" Geophys. Res. Lett., vol. 40, no. 6, pp. 1069–1073, Mar. 2013. ↩︎

  8. A. Martellucci, “Galileo reference troposphere model for the user receiver” ESA-APPNG-ReF/00621-AM v2, vol. 7, 2012. ↩︎

  9. Klobuchar, J., 1987. Ionospheric Time-Delay Algorithms for Single-Frequency GPS Users. IEEE Transactions on Aerospace and Electronic Systems (3), pp. 325-331. ↩︎

  10. Ionospheric Correction Algorithm for Galileo Single Frequency Users, Issue 1.1, June 2015 [ESA Nequick G Model] ↩︎

  11. C. Abart, "Development of a GNSS IF Signal Generator", Graz University of Technology, 2009. ↩︎