Austrian geodesists and Viennese Waltzes
After speaking of future gravity missions and before going back to eigenvalues and eigenvectors, let’s make a short trip to Austria. One very talented group in gravity field determination is the one from the Technical University of Graz. They are also, along with Stellar Space Studies, CNES and others, members of the COST-G combination initiative. TUGRAZ produces high-quality gravity-field models. They made a fantastic job with their orbitography software (GROOPS), and it is open source on GitHub. To help put the story on a map, Austria is a country in Europe with a rich history, majestic architectures, musical symphonies, and a mighty spirit. It is the home of Viennese Waltzes, Wolfgang Amadeus Mozart, and Arnold Schwarzenegger. It is also the birth place of an impressive list of scientific figures, such as Boltzmann, Doppler, Schrödinger, Pauli, Ehrenfest and, if we consider the larger borders of the then Austrian Empire, Mach, Gödel, Freud, and Tesla. Below is a picture of the Salzburg city.
Austrian mighty spirit (Salzburg)
We also would like to state that, contrary to some widespread allegations, there are no kangaroos in Austria.
Collaboration with TUGRAZ
TUGRAZ software is a powerhouse for geodesy. Download GROOPS and you will soon be able to compete at gravity field restitution with NASA, CNES, and other space agencies who are using proprietary softwares based on the work of tens and tens of developers. So, how do we collaborate with TUGRAZ? They are good at making normal equations, and we are good at inverting them. You guessed it. Maybe we could invert their normal equations. As you know, our inversion process is like a magic potion that can reveal hidden qualities in solutions, in a way that traditional filtering can’t. So we had high expectations, and we weren’t disappointed.
Noise and music
The difficulty in the SVD process is the truncation of eigenvalues. Should we declare a fixed threshold (which one?), or should we craft a more complicated truncation process, by bringing the eigenvalues down to zero gradually between two thresholds? This is hard to decide. It takes the intuition of a master chef to cook the recipe to perfection. The thing we know for sure, however, is that the higher the threshold, the more noise remains. Let’s keep it really simple for a start and show you the differences between a CNES and a TUGRAZ normal equation, with an identical truncation level of 3500 eigenvalues.
CNES normal equation truncated at 3500 eigenvalues
TUGRAZ normal equation truncated at 3500 eigenvalues
The quality of their normal equation is amazing. It almost seems that there is no noise in Austrian solutions. Only music.
The Johann Strauss statue in Vienna.
This must have been be the inspiration for TUGRAZ normal equations (zero noise, only music).
Throwing magic dust at the TUGRAZ normal equations
So after showing you the potential of TUGRAZ normal equations, we would like to show you how our technique can improve their product. As all groups, TUGRAZ only deliver unconstrained solutions. It is up to the user to filter them as they wish. Since you cannot calculate a SVD solution after the full Cholesky inversion has been performed, your usual choices are the DDK filters, from DDK1 to DDK8 according to the level of smoothness or detail you want (both vary in the opposite direction). Here is the TUGRAZ solution filtered with DDK5. Very smooth indeed, but a lot of signal has been lost in the filtering (watch the triangle on the top-right).
TUGRAZ filtered with DDK5
And here is the TUGRAZ solution filtered with DDK8. The spatial resolution is much better. But those annoying stripes are back again.
TUGRAZ filtered with DDK8
And now, let the drums roll… Here is the inversion with a truncation of 3500 eigenvalues.
TUGRAZ inverted by SVD (simple truncation of 3500 eigenvalues)
Watch the cleanness of the solution. There is no vertical stripe whatsoever (this is not surprising, we left them out with then truncation here). Watch the spatial resolution. The level of detail. The signal in the triangle. Isn’t it the best of both worlds?
No Kangaroos in Iceland either
After travelling to Austria, let’s make a quick roundtrip to Iceland to finish off the article. We would like to show you another argument in favor of the SVD technique over the traditional DDK filters. The picture below shows two time series of equivalent water heights over Iceland. They were both obtained from the same CNES normal equations. The green one was inverted by Cholesky and then filtered with DDK5. The red on was inverted by SVD (our special recipe). We also plotted the linear trend in order to quantify the rate of the ice melt (cm/year).
Trend over Iceland (CNES normal equations, SVD VS DDK5)
What do you notice? The trend is clearly sharper with the SVD inversion. -5.62 cm/year instead of -2.72 cm/year. This is a hell of a difference. But both series initially contain the same information, they come from the same normal equations. Since the process can only mitigate signal and not add more signal, the series that contains the more signal has to be the more realistic one. This is another argument in favor of our SVD technique.
Where are the kangaroos?
So after all this demonstration, you might still wonder… Where are the kangaroos?
They are in Australia!
A happy kangaroo jumping at the SVD results.
