Regularization of the Gravity Field Inversion Process with High-Dimensional Vector Autoregressive Models

Earth’s gravitational field provides invaluable insights into the changing nature of our planet. It reflects mass change caused by geophysical processes like continental hydrology, changes in the cryosphere or mass flux in the ocean. Satellite missions such as the NASA/DLR operated Gravity Recovery and Climate Experiment (GRACE), and its successor GRACE Follow-On (GRACE-FO) continuously monitor these temporal variations of the gravitational attraction. In contrast to other satellite remote sensing datasets, gravity field recovery is based on geophysical inversion which requires a global, homogeneous data coverage. GRACE and GRACE-FO typically reach this global coverage after about 30 days, so short-lived events such as floods, which occur on time frames from hours to weeks, require additional information to be properly resolved. In this contribution we treat Earth’s gravitational field as a stationary random process and model its spatio-temporal correlations in the form of a vector autoregressive (VAR) model. The satellite measurements are combined with this prior information in a Kalman smoother framework to regularize the inversion process, which allows us to estimate daily, global gravity field snapshots. To derive the prior, we analyze geophysical model output which reflects the expected signal content and temporal evolution of the estimated gravity field solutions. The main challenges here are the high dimensionality of the process, with a state vector size in the order of 103 to 104, and the limited amount of model output from which to estimate such a high-dimensional VAR model. We introduce geophysically motivated constraints in the VAR model estimation process to ensure a positive-definite covariance function.

Accuracy Evaluation of Altimeter-Derived Gravity Field Models in Offshore and Coastal Regions of China

Satellite radar altimetry has made unique contributions to global and coastal gravity field recovery. This paper starts with a general introduction followed by the progress of satellite radar altimetry technology. Then, the methods of marine gravity field recovery and dominating gravity models are described briefly. Finally, typical gravity models are compared with shipboard gravity measurements to evaluate their accuracies in offshore and coastal regions of China. The root mean squares of deviations between gravity models and shipboard gravity are all more than 7 mGal in offshore regions and within the range of 9.5–10.2 mGal in coastal regions. Further analysis in coastal regions indicates that the new gravity models with new satellite missions including Jason-2, SARAL/Altika, and Envisat data have relatively higher accuracy, especially SARAL/Altika data, significantly improving the coastal gravity field. Accuracies are low in areas with strong currents, showing that tide correction is very important for altimetry-derived marine gravity recovery as well as shipboard measurements in coastal gravity field determination. Moreover, as an external check, shipboard gravity data need more operations to improve their precision, such as higher instrument accuracy and finer data processing.

Gravity Field Recovery and Error Analysis for the MOCASS Mission Proposal Based on Cold Atom Interferometry

Satellite missions providing data for a continuous monitoring of the Earth gravity field and its changes are fundamental to study climate changes, hydrology, sea level changes, and solid Earth phenomena. GRACE-FO (Gravity Recovery and Climate Experiment Follow-On) mission was launched in 2018 and NGGM (Next Generation Gravity Mission) studies are ongoing for the long-term monitoring of the time-variable gravity field. In recent years, an innovative mission concept for gravity measurements has also emerged, exploiting a spaceborne gravity gradio-meter based on cold atom interferometers. In particular, a team of researchers from Italian universities and research institutions has proposed a mission concept called MOCASS (Mass Observation with Cold Atom Sensors in Space) and conducted the study to investigate the performance of a cold atom gradiometer on board a low Earth orbiter and its impact on the modeling of different geophysical phenomena. This paper presents the analysis of the gravity gradient data attainable by such a mission. Firstly, the mathematical model for the MOCASS data processing will be described. Then numerical simulations will be presented, considering different satellite orbital altitudes, pointing modes and instrument configurations (single-arm and double-arm); overall, data were simulated for twenty different observation scenarios. Finally, the simulation results will be illustrated, showing the applicability of the proposed concept and the improvement in modeling the static gravity field with respect to GOCE (Gravity Field and Steady-State Ocean Circulation Explorer).

Earth’s Time-Variable Gravity from GRACE Follow-On K-Band Range-Rates and Pseudo-Observed Orbits

During its science phase from 2002–2017, the low-low satellite-to-satellite tracking mission Gravity Field Recovery And Climate Experiment (GRACE) provided an insight into Earth’s time-variable gravity (TVG). The unprecedented quality of gravity field solutions from GRACE sensor data improved the understanding of mass changes in Earth’s system considerably. Monthly gravity field solutions as the main products of the GRACE mission, published by several analysis centers (ACs) from Europe, USA and China, became indispensable products for quantifying terrestrial water storage, ice sheet mass balance and sea level change. The successor mission GRACE Follow-On (GRACE-FO) was launched in May 2018 and proceeds observing Earth’s TVG. The Institute of Geodesy (IfE) at Leibniz University Hannover (LUH) is one of the most recent ACs. The purpose of this article is to give a detailed insight into the gravity field recovery processing strategy applied at LUH; to compare the obtained gravity field results to the gravity field solutions of other established ACs; and to compare the GRACE-FO performance to that of the preceding GRACE mission in terms of post-fit residuals. We use the in-house-developed MATLAB-based GRACE-SIGMA software to compute unconstrained solutions based on the generalized orbit determination of 3 h arcs. K-band range-rates (KBRR) and kinematic orbits are used as (pseudo)-observations. A comparison of the obtained solutions to the results of the GRACE-FO Science Data System (SDS) and Combination Service for Time-variable Gravity Fields (COST-G) ACs, reveals a competitive quality of our solutions. While the spectral and spatial noise levels slightly differ, the signal content of the solutions is similar among all ACs. The carried out comparison of GRACE and GRACE-FO KBRR post-fit residuals highlights an improvement of the GRACE-FO K-band ranging system performance. The overall amplitude of GRACE-FO post-fit residuals is about three times smaller, compared to GRACE. GRACE-FO post-fit residuals show less systematics, compared to GRACE. Nevertheless, the power spectral density of GRACE-FO and GRACE post-fit residuals is dominated by similar spikes located at multiples of the orbital and daily frequencies. To our knowledge, the detailed origin of these spikes and their influence on the gravity field recovery quality were not addressed in any study so far and therefore deserve further attention in the future. Presented results are based on 29 monthly gravity field solutions from June 2018 until December 2020. The regularly updated LUH-GRACE-FO-2020 time series of monthly gravity field solutions can be found on the website of the International Centre for Global Earth Models (ICGEM) and in LUH’s research data repository. These operationally published products complement the time series of the already established ACs and allow for a continuous and independent assessment of mass changes in Earth’s system.

Revisiting the light time correction in gravimetric missions like GRACE and GRACE follow-on

The gravity field maps of the satellite gravimetry missions Gravity Recovery and Climate Experiment (GRACE ) and GRACE Follow-On are derived by means of precise orbit determination. The key observation is the biased inter-satellite range, which is measured primarily by a K-Band Ranging system (KBR) in GRACE and GRACE Follow-On. The GRACE Follow-On satellites are additionally equipped with a Laser Ranging Interferometer (LRI), which provides measurements with lower noise compared to the KBR. The biased range of KBR and LRI needs to be converted for gravity field recovery into an instantaneous range, i.e. the biased Euclidean distance between the satellites’ center-of-mass at the same time. One contributor to the difference between measured and instantaneous range arises due to the nonzero travel time of electro-magnetic waves between the spacecraft. We revisit the calculation of the light time correction (LTC) from first principles considering general relativistic effects and state-of-the-art models of Earth’s potential field. The novel analytical expressions for the LTC of KBR and LRI can circumvent numerical limitations of the classical approach. The dependency of the LTC on geopotential models and on the parameterization is studied, and afterwards the results are compared against the LTC provided in the official datasets of GRACE and GRACE Follow-On. It is shown that the new approach has a significantly lower noise, well below the instrument noise of current instruments, especially relevant for the LRI, and even if used with kinematic orbit products. This allows calculating the LTC accurate enough even for the next generation of gravimetric missions.

GRACE Follow-On Gravity Field Recovery from Combined Laser Ranging Interferometer and Microwave Ranging System Measurements

<p>Besides a K-Band Ranging System (KBR), GRACE-FO carries a Laser Ranging Interferometer (LRI) as a technology demonstration to provide measurements of inter-satellite range changes. This additional measurement technology provides supplementary observations, which allow for cross-instrument diagnostics with the KBR system and, to some extent, the separation of ranging noise from other sources such as noise in the on-board accelerometer (ACC) measurements.</p><p>The aim of this study is to incorporate the two ranging systems (LRI and KBR) observations in ITSG-Grace2018 gravity field recovery. The two observation groups are combined in an iterative least-squares adjustment with variance component estimation used to determine the unknown noise covariance functions for KBR, LRI, and ACC measurements. We further compare the gravity field solutions obtained from the combined solutions to KBR-only results and discuss the differences with a focus on the global gravity field and LRI calibration parameters.</p>

Time-variable gravity field recovery from kinematic positions of Low Earth Orbiting satellites

<p>For the monitoring of mass transport and mass distribution in the Earth’s system, the gravity field and its temporal variations provide an important source of information. Dedicated satellite missions like GRACE and GRACE-FO allow to resolve the Earth’s time-variable gravity field based on ultra-precise inter-satellite ranging. In addition, any (non-dedicated) Low Earth Orbiting (LEO) satellite equipped with an on-board GNSS receiver may also serve as a gravity field sensor. For this purpose, the collected GNSS data is used to derive kinematic LEO orbit positions that can subsequently be utilized as pseudo-observations for gravity field recovery. Although this technique is less sensitive and restricted to the long wavelength part of the gravity field, it provides valuable information, particularly for those months where no inter-satellite ranging measurements are available from GRACE or GRACE-FO. Furthermore, the increasing number of operational LEO satellites makes it attractive to produce combined Multi-LEO gravity field solutions that will take advantage of the variety of complementary orbital configurations and can offer additional sensitivities to selected coefficients of solutions based on inter-satellite ranging.</p><p>At the Astronomical Institute of the University of Bern (AIUB) GPS-based kinematic orbits are routinely processed for various LEO satellites like missions dedicated to gravity (GOCE, GRACE/-FO), altimetry (Jason, Sentinel), or further constellations of Earth-observing satellites like SWARM. Beside conventional ambiguity-float orbits, also ambiguity-fixed orbits are recently being computed based on new phase bias and clock products of the Center for Orbit Determination in Europe (CODE). The kinematic orbit positions offer the opportunity to derive time series of monthly gravity field solutions for the different LEO satellites that are eventually combined on the level of normal equations.</p><p>In this contribution, we will present first results of our effort to generate a combined time series of monthly gravity field solutions based on the kinematic orbits of multiple LEO satellites. In a first step, the focus is laid on the GRACE/-FO missions that provide the longest time series in terms of collected GNSS data and that will therefore serve as a backbone for future combinations. We analyze the impact of accelerometer data on the recovery of time-variable mass variations. This will be particularly important for the handling of non-dedicated gravity missions, for which accelerometer measurements are usually not available. Furthermore, we study and compare the performance of gravity field recoveries based on ambiguity-float and ambiguity-fixed kinematic orbit solutions. We assess our results with respect to superior gravity field models based on inter-satellite ranging for selected areas with strong mass change signals like in Greenland, West-Antarctica or the Amazon river basin.</p>

Assessment of COSMIC-2 reduced-dynamic and kinematic orbit determination

<p>The Constellation Observing System for Meteorology, Ionosphere, and Climate 2 (COSMIC-2) mission was launched on June 25, 2019 into six evenly spaced circular orbital planes of 24° inclination with initial altitudes of 725 km. By February 2021 the COSMIC-2 satellites will be lowered to an operational altitude of about 520 km. The satellites carry an advanced Tri‐GNSS (Global Navigation Satellite System) Radio-occultation System (TGRS) instrument to provide high vertical resolution profiles of atmospheric bending angle and refractivity, as well as measurements of ionospheric total electron content, electron density, and scintillation. The TGRS payload tracks GPS and GLONASS signals on two upward looking antennas used for precise orbit determination (POD). We compute one- and two-antenna GPS and GPS+GLONASS POD solutions at both orbit altitudes and assess the orbit quality and systematic orbit errors using different metrics. In particular, we also use different POD setups to compute kinematic solutions employing single-receiver ambiguity fixing and test their contribution to selected months of gravity field recovery based on Swarm GPS data.</p>

GROOPS: An open-source software package for GNSS processing and gravity field recovery

<p>The Gravity Recovery Object Oriented Programming System (GROOPS) is a software package written in C++ that enables the user to perform core geodetic tasks. The software features gravity field recovery from satellite and terrestrial data, the determination of low-earth-orbiting satellite orbits from global navigation satellite system (GNSS) measurements, and the computation of GNSS constellations and ground station networks. For an easy and intuitive setup of complex workflows, GROOPS contains a graphical user interface to create and edit configuration files. The source code of GROOPS is released under the GPL v3 license and is available on GitHub (https://github.com/groops-devs/groops) together with documentation, a cookbook with guided examples, and installation instructions for different platforms. In this contribution we give a software overview and present results of different applications and data sets computed with GROOPS.</p>