Noise Modelling for GRACE Follow-On Observables in the Celestial Mechanics Approach

A key to understanding the dynamic system Earth in its current state is the continuous observation of its time-variable gravity field. The satellite missions Gravity Recovery And Climate Experiment (GRACE) and its successor GRACE Follow-On take an exceptional position in sensing these time-variable components because of their unique observing concept, which is based on ultra precise measurements of distance changes between a pair of satellites separated by a few hundred kilometres. These observations allow for a modelling of the Earth’s gravity field, typically on a basis of monthly snapshots. One of the key components of any model is the accurate specification of its quality. In temporal gravityfield modelling from GRACE Follow-On data one has to cope with several noise sources contaminating not only the observations but also the observation equations via mis-modellings in the underlying background force models. When employing the Celestial Mechanics Approach (CMA), developed at the Astronomical Institute of the University of Bern (AIUB), for gravity field modelling from satellite data a Least-Squares Adjustment (LSQA) is performed to compute monthly models of the Earth’s gravity field. However, as a consequence of the various contaminations with noise, the jointly estimated formal errors usually do not reflect the error level that could be expected but provides much lower error estimates. One way to deal with such deficiencies in the observations and modelling is to extend the parameter space, i.e., the model, by additional quantities, such as pseudo-stochastic parameters, which are co-estimated in the LSQA. These parameters are meant to absorb any kind of noise while retaining the signal in the gravity field and orbit parameters. In the CMA such pseudo-stochastic parameters are typically set-up as Piece-wise Constant Accelerations (PCAs) in regular intervals of e.g., 15 min. The stochastic behaviour of these parameters is unknown because they reflect an accumulation of a variety of noise sources. In the CMA fictitious artificial zero-observations are appended to the vector of observations together with an empirically determined variance to introduce a stochastic model for the PCAs. In order to also co-estimate a stochastic model for the pseudo-stochastic parameters in the LSQA Variance Component Estimation (VCE) is used in this work as a well established tool to assign variance components to individual groups of observations. In the simplest case the magnitude of the constraints of the pseudo-stochastic parameters can be determined fully automatically. Additionally, VCE is applied as an on-the-fly data reviewing method to account for gross outliers in the observations. Addressing the problem of noise contamination from the point of the GRACE Follow-On satellite mission’s observations, this work presents the incorporation of several noise models into the CMA to not only obtain high-quality time-variable gravity field models but also an accurate description of their stochastic behaviour. The noise models applied stem from pre-launch simulations or the formal covariance propagation of a kinematic point positioning process. Furthermore, the derivation and application of empirical noise models obtained from post-fit residuals between the final GRACE Follow-On orbits, that are co-estimated together with the gravity field, and the observations, expressed in position residuals to the kinematic positions and in the inter-satellite link range-rate residuals, is implemented. Additionally, the current operational processing scheme of GRACE Follow-On data is expounded, including the normal equation handling in the CMA with BLAS and LAPACK routines. All implementations are compared and validated with the operational GRACE Follow-On processing at the AIUB by examining the stochastic behaviour of the respective post-fit residuals and by investigating areas on Earth where a low noise is expected. Finally, the influence and behaviour of the different noise modelling techniques is investigated in a combination of monthly gravity fields computed by various institutions as it is done by the Combination Service for Time-variable Gravity fields (COST-G).Ein wesentlicher Baustein für das Verständnis des Systems Erde ist die kontinuierliche Überwachung des zeit-variablen Anteils des Erdschwerefeldes. Die beiden Satellitenmissionen Gravity Recovery And Climate Experiment (GRACE) und GRACE Follow-On spielen hierbei eine gewichtige Rolle, da sie mit ihrem Beobachtungskonzept, das auf einer hochpräsizen Abstandsmessung zwischen einem Satellitenpaar beruht, diesen zeit-variablen Anteil besonders hoch auflösen können. Diese Messungen ermöglichen es, monatliche Schwerefeldmodelle zu bestimmen. Eine der wichtigsten Komponenten eines jeden Modells ist die akkurate Beschreibung seiner Unsicherheiten. Bei der Modellierung von zeit-variablen Schwerefeldern aus GRACE Follow-On Daten treten Effekte auf, die einerseits die Beobachtungen direkt kontaminieren, und andererseits auch durch Hintergrundmodelle der Kräfte in die Beobachtungsgleichungen einfliessen. Der Ansatz des Celestial Mechanics Approach (CMA), der am Astronomischen Institut der Universität Bern (AIUB) entwickelt wurde und in dieser Arbeit angewandt wird, beruht auf einer Kleinste-Quadrate-Parameterschätzung, um aus entsprechenden Satellitendaten Orbit- und Schwerefeldmodelle abzuleiten. Dabei ist zu beobachten, dass die formale Fehlerabschätzung der Parameter deutlich besser ausfällt als es zu erwarten wäre. Eine Möglichkeit mit Unsicherheiten in den Beobachtungen und der Modellierung umzugehen ist es, den Parameterraum zu erweitern. Das bedeutet, dass zusätzliche Grössen bestimmt werden, wie z.B. pseudo-stochastische Parameter. Diese Grössen sind dazu gedacht, Unsicherheiten zu absorbieren, aber gleichzeitig das Signal in den Schwerefeld- und Orbitparametern zu erhalten. Im CMA werden diese pseudo-stochastischen Parameter als stückweise konstante Beschleunigungen (PCAs) für regelmässige Intervalle (von z.B. 15 min) geschätzt. Das stochastische Verhalten dieser Parameter ist unbekannt, da sie eine Summe an Fehlerquellen ausgleichen sollen. Im CMA wird daher ein empirisch ermitteltes stochastisches Modell für die PCAs eingeführt. Um so ein Modell auch schätzen zu können, wird in dieser Arbeit auf die Methode der Varianzkomponentenschätzung (VCE) zurückgegriffen, die sich dadurch auszeichnet, Varianzkomponenten für unterschiedliche Beobachtungsgruppen zu bestimmen. Im einfachsten Falle zeigt sich, dass die Magnitude des stochastisches Modells der PCAs zusammen mit allen Parametern berechnet werden kann. Zusätzlich werden die Varianzkomponenten als Mass eingeführt, um Ausreisserin den Daten zu glätten. Die Problemstellung des Beobachtungsrauschens wird in dieser Arbeit durch unterschiedliche Rauschmodelle betrachtet. Damit soll sichergestellt werden, dass die geschätzten Schwerefeldmodelle nicht nur das zeit-variable Schwerefeldsignal beschreiben, sondern auch nachvollziehbare Informationen zu den zugehörigen Unsicherheiten bieten. Die Rauschmodelle stammen einerseits aus Simulationen zum Instrumentenverhalten, die vor dem Start durchgeführt wurden, und andererseits im Fall der kinematischen Positionsbeobachtungen aus einer formalen Kovarianzfortpflanzung. Des Weiteren wird auf die Ableitung von empirischen Rauschmodellen aus Post-Fit-Residuen eingegangen, die aus dem berechneten Orbit und den kinematischen Positionen bzw. Inter-satellite Link Range-Rates bestimmt werden. Auch wird die operationelle GRACE Follow-On Prozessierung ausgeführt, verbunden mit einer verbesserten Handhabung der Normalgleichungen mittels BLAS- und LAPACK-Routinen. Alle Neuerungen werden mit den operationellen GRACE Follow-On Lösungen verglichen und validiert. Hierbei werden insbesondere das stochastische Verhalten der Post-Fit-Residuen untersucht, ebenso wie Gebiete der Erde, in denen aufgrund physikalischer Prozesse kaum Rauschen zu erwarten ist. Zuletzt wird im Rahmen des Combination Service for Time-variable Gravity fields (COST-G) noch darauf eingegangen wie sich unterschiedliche Rauschmodellierungen in einer Kombination von Schwerefeldmodellen, die mit verschiedenen Ansätzen und Softwarepaketen bestimmt wurden, verhalten

Comparison of empirical noise models for GRACE Follow-On derived with the Celestial Mechanics Approach

A key component of any model is the accurate specification of its quality. In gravity field modelling from satellite data, as it is done with the observation collected by GRACE Follow-On, usually least-squares adjustments are performed to obtain a monthly solution of the Earth's gravity field. However, the jointly estimated formal errors usually do not reflect the error level that could be expected but provides much lower error estimates. We take the Celestial Mechanics Approach (CMA), developed at the Astronomical Institute, University of Bern (AIUB), and extend it by an empirical modelling of the noise based on the post-fit residuals between the final GRACE Follow-On orbits, that are co-estimated together with the gravity field, and the observations, expressed in position residuals to the kinematic positions and in K-band range-rate residuals. We compare and validate the solutions that employ empirical modelling with solutions that do not contain sophisticated noise modelling by examining the stochastic behaviour of the respective post-fit residuals, by investigating areas where a low noise is expected and by inspecting the mass trend estimates in certain areas of global interest. Finally, we investigate the influence of the empirically weighted solutions in a combination of monthly gravity fields based on other approaches as it is done in the COST-G framework

Variance component estimation for co-estimated noise parameters in GRACE Follow-On gravity field recovery

Temporal gravity field modelling from GRACE Follow-On has to cope with several noise sources contaminating not only the observations but also the observation equations via mis-modellings in the underlying background force models. One way to deal with such deficiencies is to extend the parameter space by additional quantities, such as pseudo-stochastic parameters, which are co-estimated in the Least-Squares Adjustment (LSA). These parameters are meant to absorb any kind of noise while retaining the signal in the gravity field and orbit parameters. In the Celestial Mechanics Approach (CMA) such pseudo-stochastic parameters are typically set-up as Piece-wise Constant Accelerations (PCA) in regular intervals of e.g., 15 min. The stochastic behaviour of these parameters is unknown because they reflect an accumulation of a variety of noise sources. In the CMA fictitious artificial zero-observations are appended to the vector of observations together with an empirically determined variance to introduce a stochastic model for the PCAs. In order to also co-estimate a stochastic model for the pseudo-stochastic parameters in the LSA we use Variance Component Estimation (VCE) as a well established tool to assign variance components to individual groups of observation. In the simplest case the magnitude of the constraints of the pseudo-stochastic parameters can be determined fully automatically. We present results for GRACE Follow-On gravity field recovery when extending the CMA by stochastic models for the piece-wise constant accelerations computed with VCE and provide noise and signal assessment applying the quality control tools routinely used in the frame of the Combination Service for Time-variable gravity fields (COST-G)

Spherical acquisition trajectories for X-ray computed tomography with a robotic sample holder

This work presents methods for the seamless execution of arbitrary spherical trajectories with a seven-degree-of-freedom robotic arm as a sample holder. The sample holder is integrated into an existing X-ray computed tomography setup. We optimized the path planning and robot control algorithms for the seamless execution of spherical trajectories. A precision-manufactured sample holder part is attached to the robotic arm for the calibration procedure. Different designs of this part are tested and compared to each other for optimal coverage of trajectories and reconstruction image quality. We present experimental results with the robotic sample holder where a sample measurement on a spherical trajectory achieves improved reconstruction quality compared to a conventional circular trajectory. Our results demonstrate the superiority of the discussed system as it outperforms single-axis systems by reaching nearly 82\% of all possible rotations. The proposed system is a step towards higher image reconstruction quality in flexible X-ray CT systems. It will enable reduced scan times and radiation dose exposure with task-specific trajectories in the future, as it can capture information from various sample angles

The new COST-G deterministic signal model

The precise orbit determination (POD) of Low Earth Orbiters (LEO), e.g. the Copernicus Sentinel Earth observation satellites, relies on the precise knowledge of the Earth gravity field and its variations with time. The most precise observation of time-variable gravity on a global scale is currently provided by the GRACE-FO satellites. But the monthly gravity field solutions are released with a latency of approx. 2 months, therefore they cannot be used for operational POD. We present a deterministic signal model (DSM) that is fitted to the time-series of COST-G combined monthly gravity fields and describe the differences with respect to the available long-term gravity models including seasonal and secular time-variations. To validate the DSM, dynamic POD of the Sentinel-2B, -3B and -6A satellites is performed based on long-term or monthly gravity field models, and on the COST-G DSM. We evaluate the model quality on the basis of carrier phase residuals, orbit overlap analysis and independent satellite laser ranging observations, and study the limitation on orbit altitude posed by the reduced spherical harmonic resolution of the monthly models and the DSM. The COST-G DSM is updated quarterly with the most recent GRACE-FO combined monthly gravity fields. It is foreseen to apply a sliding window approach with flexible window length to allow for an optimal adjustment in case of singular events like major earthquakes

Variance component estimation for co-estimated noise parameters in GRACE Follow-On gravity field recovery

Temporal gravity field modelling from GRACE Follow-On deals with several noise sources polluting the observations and the system of equations, be it actual measurement noise or mis-modellings in the underlying background models. One way to collect such deficiencies is to co-estimate additional pseudo-stochastic parameters in the least-squares adjustment which are meant to absorb any kind of noise while retaining the signal in the gravity field and orbit parameters. In the Celestial Mechanics Approach (CMA) such pseudo-stochastic parameters are typically piece-wise constant accelerations set up in regular intervals of e.g., 15 min, and an empirically determined constraint is added to confine the impact of the additional quantities. As the stochastic behaviour of these parameters is unknown because they reflect an accumulation of a variety of noise sources, Variance Component Estimation (VCE) is a well established tool to assign a stochastic model to the pseudo-stochastic orbit parameters driven by the observations. In the simplest case the magnitude of the constraints of the pseudo-stochastic orbit parameters can be determined fully automatically. We present results for GRACE Follow-On gravity field recovery when extending the CMA by stochastic models for the piece-wise constant accelerations computed with VCE and provide noise and signal assessment applying the quality control tools routinely used in the frame of the Combination Service for Time-variable gravity fields (COST-G)

Time-variable gravity field determination from GRACE Follow-On data usingthe Celestial Mechanics Approach extended by empirical noise modelling

We study gravity field determination from GRACE Follow-On satellite-to-satellite tracking using the inter-satellite K-band link and kinematic positions of the satellites as observations and pseudo- observations respectively. A key component of any model is the accurate specification of its quality. In the case of gravity field modelling from satellite data with the Celestial Mechanics Approach (CMA) a least-squares adjustment is performed to obtain a monthly solution of the Earthâ?Ts gravity field. However, the jointly estimated formal errors usually do not reflect the error level that could be expected but provides much lower error estimates. We present gravity field solutions computed with the CMA and extend it by an empirical modelling of the noise based on the post-fit residuals between the final GRACE Follow-On orbits, that are co-estimated together with the gravity field, and the observations, expressed in position residuals to the kinematic positions and in K-band range-rate residuals. We compare and validate the solutions that use empirical modelling with solutions from the operational GRACE Follow-On processing at AIUB by examining the stochastic behaviour of the respective post-fit residuals, by investigating areas where a low noise is expected and by inspecting the mass trend estimates in certain areas of global interest. Finally, we investigate the influence of the empirically weighted solutions in a combination of monthly gravity fields based on other approaches as it is done by the Combination Service for Time-variable Gravity fields (COST-G) and make use of noise and signal assessment applying the quality control tools routinely used in the frame of COST-G

On the co-estimation of static and monthly gravity field solutions from GRACE Follow-On data

Temporal gravity field modelling from GRACE Follow-On data is usually performed by computing monthly snapshots of spherical harmonic coefficients representing the state of the Earthâ?Ts gravity field. Associated to this, the spherical harmonic series has to be truncated at a certain point, commonly at degree/order 96. Higher degrees and orders are fixed to the a priori used background gravity field model. We present an investigation on the influence of the high degrees and orders of different a priori background gravity field models on monthly gravity field model computations from GRACE Follow-On data. Furthermore, we extend the temporal gravity field modelling to additionally co-estimate a static gravity field for the GRACE Follow-On satellite mission along with the monthly snapshots to provide for a consistent handling of correlations between temporal and static gravity field coefficients. Moreover, we model the stochastic noise of the data with an empirical description of the noise based on the post-fit residuals between the final GRACE Follow-On orbits, that are co-estimated together with the gravity field, and the observations, expressed in position residuals to the kinematic positions and in K-band range-rate residuals, to further study the influence of the high degrees and orders of the a priori background gravity field model on such noise models. We compare and validate the monthly solutions with the models from the operational GRACE Follow-On processing at AIUB by examining the stochastic behaviour of the respective post-fit residuals, by investigating areas where a low noise is expected and by inspecting the mass trend estimates in certain areas of global interest. Finally, we investigate the influence in a combination of monthly gravity fields based on other approaches as it is done by the Combination Service for Time-variable Gravity fields (COST-G) and make use of noise and signal assessment applying the quality control tools routinely used in the frame of COST-G

Image formation and tomogram reconstruction in optical coherence microscopy

In this work we present a model for image formation in optical coherence microscopy. In the spectral domain detection, each wavenumber has a specific coherent transfer function that samples a different part of the object's spatial frequency spectrum. The reconstruction of the tomogram is usually accurate only in a short depth of field. Using numerical simulations based on the developed model, we identified two distinct mechanisms that influence the signal of out-of-focus sample information. Besides the lateral blurring induced through defocusing, an additional axial envelope contributing equally to the signal degradation was found. (C) 2010 Optical Society of Americ