Digital Filtering Algorithms for Decorrelation within Large Least Squares Problems

The GOCE (Gravity Field and steady-state Ocean Circulation Explorer) mission is dedicated to the determination of the Earth's gravity field. During the mission period of at least one year the GOCE satellite will collect approximately 100 million highly correlated observations. The gravity field will be described in terms of approximately 70,000 spherical harmonic coefficients. This leads to a least squares adjustment, in which the design matrix occupies 51 terabytes while the covariance matrix of the observations requires 72,760 terabytes of memory. The very large design matrix is typically computed in parallel using supercomputers like the JUMP (Juelich Multi Processor) supercomputer in Jülich, Germany. However, such a brute force approach does not work for the covariance matrix. Here, we have to exploit certain features of the observations, e.g. that the observations can be interpreted as a stationary time series. This allows for a very sparse representation of the covariance matrix by digital filters. This thesis is concerned with the use of digital filters for decorrelation within large least squares problems. First, it is analyzed, which conditions the observations must meet, such that digital filters can be used to represent their covariance matrix. After that, different filter implementations are introduced and compared with each other, especially with respect to the calculation time of filtering. This is of special concern, as for many applications the very large design matrix has to be filtered at least once. One special problem arising by the use of digital filters is the so-called warm-up effect. For the first time, methods are developed in this thesis for determining the length of the effect and for avoiding this effect. Next, a new algorithm is developed to deal with the problem of short data gaps within the observation time series. Finally, it is investigated which filter methods are best adopted for the application scenario GOCE, and several numerical simulations are performed.Digitale Filteralgorithmen zur Dekorrelation in großen kleinste-Quadrate Problemen Die GOCE (Gravity Field and steady-state Ocean Circulation Explorer) Mission ist der Bestimmung des Erdschwerefeldes gewidmet. Während der Missionsdauer von mindestens einem Jahr wird der GOCE Satellit circa 100 Millionen hoch korrelierte Beobachtungen sammeln. Das Erdschwerefeld wird durch circa 70.000 sphärisch harmonische Koeffizienten beschrieben. Dies führt zu einem kleinste-Quadrate Ausgleich, wobei die Designmatrix 51 Terabytes benötigt während die Kovarianzmatrix der Beobachtungen 72.760 Terabytes erfordert. Die sehr große Designmatrix wird typischerweise parallel berechnet, wobei Supercomputer wie JUMP (Juelich Multi Processor) in Jülich (Deutschland) zum Einsatz kommen. Ein solcher Ansatz, bei dem das Problem durch geballte Rechenleistung gelöst wird, funktioniert bei der Kovarianzmatrix der Beobachtungen nicht mehr. Hier müssen bestimmte Eigenschaften der Beobachtungen ausgenutzt werden, z.B. dass die Beobachtungen als stationäre Zeitreihe aufgefasst werden können. Dies ermöglicht es die Kovarianzmatrix durch digitale Filter zu repräsentieren. Diese Arbeit beschäftigt sich mit der Nutzung von digitalen Filtern zur Dekorrelation in großen kleinste-Quadrate Problemen. Zuerst wird analysiert, welche Bedingungen die Beobachtungen erfüllen müssen, damit digitale Filter zur Repräsentation ihrer Kovarianzmatrix benutzt werden können. Danach werden verschiedene Filterimplementierungen vorgestellt und miteinander verglichen, wobei spezielles Augenmerk auf die Rechenzeit für das Filtern gelegt wird. Dies ist von besonderer Bedeutung, da in vielen Anwendungen die sehr große Designmatrix mindestens einmal gefiltert werden muss. Ein spezielles Problem, welches beim Benutzen der Filter entsteht, ist der sogenannte Warmlaufzeiteffekt. Zum ersten Mal werden in dieser Arbeit Methoden entwickelt, um die Länge des Effekts zu bestimmen und um den Effekt zu vermeiden. Als Nächstes wird ein neuer Algorithmus zur Lösung des Problems von kurzen Datenlücken in der Beobachtungszeitreihe entwickelt. Schließlich wird untersucht, welche Filtermethoden man am besten für das Anwendungsszenario GOCE verwendet und es werden verschiedene numerische Simulationen durchgeführt

TOLEOS: Thermosphere Observations from Low-Earth Orbiting Satellites

The objective of the TOLEOS project is to process the CHAMP, GRACE, and GRACE-FO accelerometer measurements with improved processing standards to obtain thermosphere density and crosswind data products. These new data products will cover the entirety of the accelerometer missions and complement the existing ESA databases for Swarm and GOCE. The improvements in the processing focus on the radiation pressure modelling, which is expected to have a significant effect on the density and crosswind data, in particular at altitudes above 450 km during solar minimum conditions. Substantial validation activities are performed since the project’s start in June 2021 and will continue until the end of the project in July 2022

Lower-thermosphere–ionosphere (LTI) quantities: current status of measuring techniques and models

The lower-thermosphere-ionosphere (LTI) system consists of the upper atmosphere and the lower part of the ionosphere and as such comprises a complex system coupled to both the atmosphere below and space above. The atmospheric part of the LTI is dominated by laws of continuum fluid dynamics and chemistry, while the ionosphere is a plasma system controlled by electromagnetic forces driven by the magnetosphere, the solar wind, as well as the wind dynamo. The LTI is hence a domain controlled by many different physical processes. However, systematic in situ measurements within this region are severely lacking, although the LTI is located only 80 to 200 km above the surface of our planet. This paper reviews the current state of the art in measuring the LTI, either in situ or by several different remote-sensing methods. We begin by outlining the open questions within the LTI requiring high-quality in situ measurements, before reviewing directly observable parameters and their most important derivatives. The motivation for this review has arisen from the recent retention of the Daedalus mission as one among three competing mission candidates within the European Space Agency (ESA) Earth Explorer 10 Programme. However, this paper intends to cover the LTI parameters such that it can be used as a background scientific reference for any mission targeting in situ observations of the LTI.Peer reviewe

Model data comparison with an improved data set of thermosphere density and crosswind observation

Within the TOLEOS project improved processing standards are used to process GRACE, GRACE-FO and CHAMP accelerometer data resulting in a new unique thermosphere density and crosswind observations data set. The improvements in the processing focus on the radiation pressure modelling, which is expected to have a significant effect on the density observations at altitudes above 450 km during solar minimum conditions and crosswind observations in general. Because these observations will complement ESA’s Swarm and GOCE data bases and achieve a full coverage of all accelerometer missions, they are subject to substantial validation. The validation of the densities is performed by comparing models with observations from multiple missions as well as via models. This validation study will involve density observations of CHAMP, GOCE, GRACE, GRACE/GFO and Swarm. The involved models will be the empirical model DTM2020 and the physics-based models CTIPe and TIE-GCM. The comparisons with model predictions aims at determining a scale difference between the model and observations. In addition, we are comparing with other thermosphere density data, too, e.g. comparing CHAMP-GRACE and GRACE-Swarm, as well as with HASDM density data for example. For the validation of the wind observations, the available wind data and wind models are much less and less accurate. Thus, the level of accuracy and confidence of the wind product validation is lower than for the density products. This presentation will summarize the validation results of the improved thermosphere density and crosswind observations generated in the TOLEOS project