Satellite geodesy (ge-2112)

1. Satellite geodesy (ge-2112) Processing of observations E. Schrama

2. Contents • Preprocessing of observations • example 1: dual frequency ionospheric • example 2: tropospheric range delay • example 3: normal point compression • Classification of problems, function models • Parameters in function model • Parameter estimation procedure • Implementation • Precise Orbit Determination • Variational equations • Organisation parameter estimation

3. Preprocessing of observations • Oftentimes raw observations are NOT suitable for direct application in parameter estimation algorithms • Raw observations typically contain non Gaussian errors like outliers greater than 3 sigma • Often there are very good reasons to inspect and clean up the data before you put it into an estimation procedure • This topic is much depending on the observation technique, we will just show some well known examples.

4. Preprocessing example 1 The problem is: how do you eliminate the ionospheric delay from dual frequency range data? Task: compute the factors for GPS L1/L2

5. Preprocessing example 2 • The air pressure is 1000 mbar, the air temperature is 20 degrees centigrade, the relative humidity is 50%, what is the dry+wet tropospheric delay of a radio signal as a function of the elevation angle for a station at MSL and 50 degrees latitude. The answer is: • Use the Hopfield model (see Seeber p 45 - 49) to calculate the refractive index • Use the integral over (n-1) ds to compute the path delay • For the latter integral various mapping functions exist

6. Dry tropospheric delay example This result is entirely depending on the air pressure P, 1% air pressure change (=10 mbar) gives 1% range change. Since air pressure is usually known to within a millibar the dry tropospheric delay error is small. For low elevation angles the delay error increases due to the mapping function uncertainties. Hence elevation cut-off angles are used (typically 10 degrees).

7. Wet tropospheric delay example The wet tropospheric range depends on the relative humidity which varies more rapidly in time and place compared to air pressure. Variations of the order of 50% are possible. As a result the vertical path delay can vary between 5 and 30 cm. The alternative is the use of a multifrequency radiometer system, see Seeber p 49.

8. Normal point compression Case: red crosses is SLR data, there are too many of them and there are clear blunders that we don’t accept in the parameter estimation procedure. Method: Use a compression technique (splines, polynomials, etc) that fit the crosses. Evaluation of the model results in the compression points (the circles). This procedure filters out the noise. Horizontal: time, vertical: range

9. Classification of problems • Terminology: • Here, a problem refers to an interesting case to study. • Problems in satellite geodesy: • Type of problem • does it contain orbit parameters? • does it contain gravity field parameters? • does it contain any other geophysical parameters? • How do you organize parameter estimation? • it is a batch or a sequential least squares problem? • can you solve it from one observation set or are more sets involved? • Is preprocessing of observations involved or is it in the problem?

10. Function model (1) • The function model aims to relate observations and parameters to another • The unknowns are gathered in vector • The observations are in vector • Usually we begin to approximate reality by a priori estimates and

11. Function model (2) Ze S Rij B Rj Ri Ye Xe

12. Function model (3): Examples • The overdetermined GPS navigation solution for one receiver • VLBI observations of phase delay • Two GPS receivers: double difference processing • SLR network: station, orbit parameters, earth rotation parameters • DORIS with orbit and gravity field improvement • Spaceborn GPS receiver on a LEO

13. Parameter estimation(1): Least squares

14. Parameter estimation (2) • The way the A matrix is computed completely depends on the type of observations and parameters in your problem. • We will distinguish between problems that contain orbit parameters and problems that do not. • Our first task will always be to model an orbit in the best possible way given the existing situation • This task is called orbit prediction

15. Implementation • From our function model we conclude that: • it is by definition a non linear problem • it depends on a priori information • it almost always depends on orbit dynamics • orbit predictions are used to correct the raw observations and to set-up the design matrix • the orbit prediction model is not necessarily accurate the first time you apply it

16. Orbit prediction (1) Part 2 of these lectures mentioned that one needs to integrate the equations of motion. Suitable numerical techniques are used to treat differential equations of the following type: There are numerical procedures like the Runge-Kutta single step integrator and Adams-Moulton-Bashforth multi step integrator that allow the state vector y0 to be propagated from y0 till yn. In this case a state vector at index j coincides with the time index t0+(j-1)*h where h is the integrator step size.

17. Orbit prediction (2) • The orbit prediction problem is entirely driven by the choice of the initial state vector y0, the definition of F(y,t) and g(t). • The basic question is of course, where does this information come from? • F(y,t) and g(t) fully depend on the realism of your mathematical model and its ability to describe reality • However, knowledge of the initial state vector should follow from 1) earlier computations or 2) launch insertion parameters • The conclusion is that it is desirable to estimate initial state parameters from observations to the satellite.

18. Initial state vector estimation in precise orbit determination Task: determine the size, orientation and position of the arrow, it determines whether you hit the bull’s eye

19. Summary Precise Orbit Determination (POD) • In reality orbit parameters are estimated from observations like range, Doppler or camera to the satellite • Orbit prediction scheme • Numerically stable schemes are used • Choice initial state vector • Definition satellite acceleration model • Variational scheme • Define parameters that need to be adjusted using least squares • Iterative improvement of these parameters • Use is made of so-called variational equations

20. Parameters in POD • Station coordinates • Station related parameters (clock, biases) • Initial state vector elements of satellite orbits • Parameters in acceleration models satellite • Other satellite related parameters (clock, biases, etc) • Signal delay related parameters • Earth rotation related parameters • Gravity field related parameters

21. Variational equations Example : initial state vector component, terms in force model etc

22. Organisation parameter estimation • For large scale batch problems: • separation of arc -- and common parameters • combination of normal matrices and right hand sides • choice of optimal weight factors for combination • example: development of earth models like EGM96 • Sequential problems • apart from the adjustment procedure there is a state vector transition mechanism • During transition state vector and variance matrix are advanced to the next time step (normally with a Kalman filter) • Example: JPL’s GPS data processingmethod