Temporal Aliasing Errors Research Articles

Constellation design and performance of future quantum satellite gravity missions

Temporal aliasing is currently the largest error contributor to time-variable satellite gravity field models. Therefore, the evolution of sensor technologies has to be complemented by strategies to reduce temporal aliasing errors. The most straightforward way to improve temporal aliasing is through extended satellite constellations because they improve the observation geometry and increase the achievable temporal resolution. Therefore, strategies to optimize the design of larger satellite constellations are investigated in this contribution. A complete constellation modeling procedure is presented, starting from primary design variables (such as the required targeted resolutions) and concluding with concrete orbital elements for the individual satellites. In parallel, it is evaluated if improved instrument sensitivities based on quantum technologies (cold atom interferometry) can be fully exploited in the case of larger constellations. For this, future quantum satellite gravity missions adopting the gradiometry concept (similar to the GOCE mission) and the low-low satellite-to-satellite tracking concept (similar to GRACE/-FO) are simulated on optimized constellations with up to 6 satellites/pairs. The retrieval performance of a 6-pair mission in terms of the global equivalent water height RMS can be improved by a factor of roughly 3 compared to an inclined double-pair mission. 3D-gradiometry intrinsically has a better de-aliasing behavior but has extremely high accuracy requirements for the gradiometer (about 10 µEotvos) and the attitude reconstruction to be of any benefit. All simulations show that when incorporating improved sensor technologies, such as future quantum sensing instruments in extended constellations, temporal aliasing will remain the dominant error source by far, up to five orders of magnitude larger than the instrument errors. Therefore, improving sensor technologies has to go hand in hand with larger satellite constellations and improved space–time parameterization strategies to further reduce temporal aliasing effects.Graphical

Satellite Gravity Field Recovery Using Variance‐Covariance Information From Ocean Tide Models

AbstractMonthly gravity field recovery using data from the GRACE and GRACE Follow‐On missions includes errors limiting the spatial and temporal resolution of the estimated gravity fields. The major error contributions, besides the noise of the accelerometer instruments, arise from temporal aliasing errors due to imperfections in the non‐tidal atmospheric and oceanic de‐aliasing models and ocean tide models. We derive uncertainty information for the eight major tidal constituents from five different ocean tide models and introduce it into the gravity field recovery process in terms of a constrained normal equation system while expanding the parameter space by additional tidal parameters to be adjusted. We prove the effectiveness of the ocean tide variance‐covariance information through realistic simulations and we assess its potential based on microwave and laser interferometer observations from the GRACE Follow‐On mission. We show that errors are reduced by more than 20% ocean wRMS for a Gaussian filter radius of 300 km if uncertainty information for ocean tides is considered and stochastic modeling of instrument errors is applied, compared to the latest GFZ release 6.1. Our results also show the limited visibility of the effectiveness of the ocean tide variance‐covariance information due to the dominating error contribution of non‐tidal atmospheric and oceanic mass variations. Additionally, we investigate the option of estimating ocean tide parameters over a 1‐year period while including ocean tide uncertainty information in order to improve ocean tide background modeling.

Using a Multiobjective Genetic Algorithm to Design Satellite Constellations for Recovering Earth System Mass Change

The Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) provided twenty years of data on Earth’s time-varying gravity field. Due to their design, GRACE and GRACE-FO are inherently limited in their spatiotemporal coverage, limiting their resolution to a few hundred kilometers and temporally to roughly monthly solutions. To increase the global spatiotemporal resolution and allow for the determination of submonthly time-varying gravity field signals, a constellation of GRACE-type satellite pairs is a possible path forward. Advances in small form factor instrumentation for small satellites have become progressively inexpensive, reliable, and of higher quality. This leads us to consider that a constellation of GRACE-type small satellites could be part of future gravimetric satellite missions. In this work, we investigate the viability and limitations of a genetic-algorithm-based optimization and its objective function to generate satellite constellations to recover daily Earth system mass changes. The developed approach is used to create satellite constellations that are optimally designed to recover gravity variations of sufficient resolution at a range of temporal frequencies (i.e., daily to monthly). We analyze a constellation’s performance using a combination of observability in space, accounting for directionality, and homogeneity in time. This allows us to navigate through a vast search space in a relatively short period of time and estimate the relative performance of constellations to each other. Using evolutionary theory, we converge towards a set of optimally selected orbits. The characteristics of the designed constellations have been validated using high-fidelity numerical simulations. We summarize these results and discuss their implications for possible future constellations of small GRACE-like satellite pairs. The resulting constellations have an inherently improved spatiotemporal performance, which reduces temporal aliasing errors and allows the characterization of daily mass-change effects.

Mitigating temporal aliasing effects of high-frequency geophysical fluid dynamics in satellite gravimetry

SUMMARY Temporal aliasing errors induced by high-frequency tidal and non-tidal mass variability in the Earth system are among the three most important error sources that limit the accuracy of present-day surface mass estimates from satellite gravimetry. By means of end-to-end simulations, we demonstrate that the Kalman Smoother approach developed by Kurtenbach et al. effectively captures non-tidal submonthly variability, and thereby reduces temporal aliasing errors way beyond the level of simply subtracting the standard dealiasing model AOD1B. Validation against in situ ocean bottom pressure observations confirms that the Kalman Smoother solutions published together with the ITSG-Grace2016 monthly gravity fields contain high-frequency signal over the oceans not predicted by AOD1B. The daily gravity fields therefore reduce aliasing artefacts in the monthly gravity fields, and at the same time provide observational evidence on submonthly bottom pressure variability presently not reflected in state-of-the-art numerical ocean circulation models. It is thus recommended to include a Kalman Smoother approach into any standard GRACE processing scheme. For a hypothetical double-pair configuration currently under consideration as a future mass change mission, we find that the benefit of the Kalman Smoother is much smaller thanks to the increased number of observations taken at different inclinations, which lead to generally reduced aliasing errors and much more isotropic spatial error correlations. We also reassess the idea of pre-eliminating low-resolution daily gravity fields and find large distortions in the monthly mean gravity solution at spatial wavelengths around the cut-off-degree of the daily fields. We thus recommend further study for any satellite gravity mission concept that critically relies on such pre-elimination schemes for reaching its science objectives.

Gravity Field Recovery Using High-Precision, High–Low Inter-Satellite Links

Past temporal gravity field solutions from the Gravity Recovery and Climate Experiment (GRACE), as well as current solutions from GRACE Follow-On, suffer from temporal aliasing errors due to undersampling of the signal to be recovered (e.g., hydrology), which arise in terms of stripes caused by the north–south observation direction. In this paper, we investigate the potential of the proposed mass variation observing system by high–low inter-satellite links (MOBILE) mission. We quantify the impact of instrument errors of the main sensors (inter-satellite link and accelerometer) and high-frequency tidal and non-tidal gravity signals on achievable performance of the temporal gravity field retrieval. The multi-directional observation geometry of the MOBILE concept with a strong dominance of the radial component result in a close-to-isotropic error behavior, and the retrieved gravity field solutions show reduced temporal aliasing errors of at least 30% for non-tidal, as well as tidal, mass variation signals compared to a low–low satellite pair configuration. The quality of the MOBILE range observations enables the application of extended alternative processing methods leading to further reduction of temporal aliasing errors. The results demonstrate that such a mission can help to get an improved understanding of different components of the Earth system.

Earth's surface mass transport derived from GRACE, evaluated by GPS, ICESat, hydrological modeling and altimetry satellite orbits

Abstract. The Gravity Recovery and Climate Experiment (GRACE) delivered the most accurate quantification of global mass variations with monthly temporal resolution on large spatial scales. Future gravity missions will take advantage of improved measurement technologies, such as enhanced orbit configurations and tracking systems, as well as reduced temporal aliasing errors. In order to achieve the latter, sub-monthly to daily innovative models are computed. In addition, non-conventional methods based on radial basis functions (RBFs) and mascons will give the ability to compute models in regional and global representations as well. We show that the RBF modeling technique can be used for processing GRACE data yielding global gravity field models which fit independent reference values at the same level as commonly accepted global geopotential models based on spherical harmonics. The present study compares for the first time a complete global series of solutions in order to quantify recent ice mass changes. We further compare the ice-induced crustal deformations due to the dynamic loading of the crustal layer with the Global Positioning System (GPS) uplift measurements along Greenland's coastline. Available mass change estimates based on Ice, Cloud, and land Elevation Satellite (ICESat) laser altimetry measurements both in Greenland and Antarctica are used to assess the GRACE results. A comparison of GRACE time series with hydrological modeling for various basin extensions reveals overall high correlation to surface and groundwater storage compartments. The forward computation of satellite orbits for altimetry satellites such as Envisat, Jason-1 and Jason-2 compares the performance of GRACE time-variable gravity fields with models including time variability, such as EIGEN-6S4.

Finding the suitable drag-free acceleration noise level for future low-low satellite-to-satellite tracking geodesy missions

This paper evaluates the impact of residual acceleration noise on the estimation of the Earth’s time-varying gravity field for future low-low satellite-to-satellite tracking missions. The goal is to determine the maximum level of residual acceleration noise that does not adversely affect the estimation error. The Gravity Recovery And Climate Experiment (GRACE) has provided monthly average gravity field solutions in spherical harmonic coefficients for more than a decade. It provides information about land and ocean mass variations with a spatial resolution of ∼350 km and with an accuracy within 2 cm throughout the entire Earth. GRACE Follow-on was launched in May 2018 to advance the work of GRACE and to test a new laser ranging interferometer, which measures the range between the two satellites with higher precision than the K-Band ranging system used in GRACE. Moreover, there have been simulation studies that show, an additional pair of satellites in an inclined orbit increases the sampling frequency and reduces temporal aliasing errors. Given the fact that future missions will likely continue to use the low-low satellite-to-satellite tracking formation with laser ranging interferometry, it is expected that the residual acceleration noise will become one of the largest error contributor for the time-variable gravity field solution. We evaluate three different levels of residual acceleration noise based on demonstrated drag-free systems to find a suitable drag-free performance target for upcoming geodesy missions. We analyze both a single collinear polar pair and the optimal double collinear pair of drag-free satellites and assume the use of a laser ranging interferometer. A partitioned best linear unbiased estimator that was developed, incorporating several novel features from the ground up is used to compute the solutions in terms of spherical harmonics. It was found that the suitable residual acceleration noise level is around 2 × 10−12 ms−2 Hz−1/2. Decreasing the acceleration noise below this level did not result in more accurate gravity field solutions for the chosen mission architecture.

Treatment of ocean tide aliasing in the context of a next generation gravity field mission

Current temporal gravity field solutions from Gravity Recovery and Climate Experiment (GRACE) suffer from temporal aliasing errors due to undersampling of signal to be recovered (e.g. hydrology), uncertainties in the de-aliasing models (usually atmosphere and ocean) and imperfect ocean tide models. Especially the latter will be one of the most limiting factors in determining high-resolution temporal gravity fields from future gravity missions such as GRACE Follow-On and Next-Generation Gravity Missions (NGGM). In this paper a method to co-parametrize ocean tide parameters of the eight main tidal constituents over time spans of several years is analysed and assessed. Numerical closed-loop simulations of low-low satellite-to-satellite-tracking missions for a single polar pair and a double pair Bender-type formation are performed, using time variable geophysical background models and noise assumptions for new generation instrument technology. Compared to the single pair mission, results show a reduction of tide model errors up to 70 per cent for dedicated tidal constituents due to an enhanced spatial and temporal sampling and error isotropy for the double pair constellation. Extending the observation period from 1 to 3 yr leads to a further reduction of tidal errors up to 60 per cent for certain constituents, and considering non-tidal mass changes during the estimation process leads to reductions of tidal errors between 20 and 80 per cent. As part of a two-step approach, the estimated tide model is used for de-aliasing during gravity field retrieval in a second iteration, resulting in more than 50 per cent reduction of ocean tide aliasing errors for a NGGM Bender-type formation.

Treatment of temporal aliasing effects in the context of next generation satellite gravimetry missions

AbstractTemporal aliasing effects have a large impact on the gravity field accuracy of current gravimetry missions and are also expected to dominate the error budget of Next Generation Gravimetry Missions (NGGMs). This paper focuses on aspects concerning their treatment in the context of Low‐Low Satellite‐to‐Satellite Tracking NGGMs. Closed‐loop full‐scale simulations are performed for a two‐pair Bender‐type Satellite Formation Flight (SFF), by taking into account error models of new generation instrument technology. The enhanced spatial sampling and error isotropy enable a further reduction of temporal aliasing errors from the processing perspective. A parameterization technique is adopted where the functional model is augmented by low‐resolution gravity field solutions coestimated at short time intervals, while the remaining higher‐resolution gravity field solution is estimated at a longer time interval. Fine‐tuning the parameterization choices leads to significant reduction of the temporal aliasing effects. The investigations reveal that the parameterization technique in case of a Bender‐type SFF can successfully mitigate aliasing effects caused by undersampling of high‐frequency atmospheric and oceanic signals, since their most significant variations can be captured by daily coestimated solutions. This amounts to a “self‐dealiasing” method that differs significantly from the classical dealiasing approach used nowadays for Gravity Recovery and Climate Experiment processing, enabling NGGMs to retrieve the complete spectrum of Earth's nontidal geophysical processes, including, for the first time, high‐frequency atmospheric and oceanic variations.

Gravity field recovery in the framework of a Geodesy and Time Reference in Space (GETRIS)

The study “Geodesy and Time Reference in Space” (GETRIS), funded by European Space Agency (ESA), evaluates the potential and opportunities coming along with a global space-borne infrastructure for data transfer, clock synchronization and ranging. Gravity field recovery could be one of the first beneficiary applications of such an infrastructure. This paper analyzes and evaluates the two-way high-low satellite-to-satellite-tracking as a novel method and as a long-term perspective for the determination of the Earth’s gravitational field, using it as a synergy of one-way high-low combined with low-low satellite-to-satellite-tracking, in order to generate adequate de-aliasing products. First planned as a constellation of geostationary satellites, it turned out, that an integration of European Union Global Navigation Satellite System (Galileo) satellites (equipped with inter-Galileo links) into a Geostationary Earth Orbit (GEO) constellation would extend the capability of such a mission constellation remarkably. We report about simulations of different Galileo and Low Earth Orbiter (LEO) satellite constellations, computed using time variable geophysical background models, to determine temporal changes in the Earth’s gravitational field. Our work aims at an error analysis of this new satellite/instrument scenario by investigating the impact of different error sources. Compared to a low-low satellite-to-satellite-tracking mission, results show reduced temporal aliasing errors due to a more isotropic error behavior caused by an improved observation geometry, predominantly in near-radial direction within the inter-satellite-links, as well as the potential of an improved gravity recovery with higher spatial and temporal resolution. The major error contributors of temporal gravity retrieval are aliasing errors due to undersampling of high frequency signals (mainly atmosphere, ocean and ocean tides). In this context, we investigate adequate methods to reduce these errors. We vary the number of Galileo and LEO satellites and show reduced errors in the temporal gravity field solutions for this enhanced inter-satellite-links. Based on the GETRIS infrastructure, the multiplicity of satellites enables co-estimating short-period long-wavelength gravity field signals, indicating it as powerful method for non-tidal aliasing reduction.

Mitigation of Oceanic Tidal Aliasing Errors in Space and Time Simultaneously Using Different Repeat Sub-Satellite Tracks from Pendulum-Type Gravimetric Mission Candidate

This contribution investigates two different ways for mitigating the aliasing errors in ocean tides. This is done, on the one hand, by sampling the satellite observations in another direction using the pendulum satellite mission configuration. On the other hand, a mitigation of the temporal aliasing errors in the ocean tides can be achieved by using a suitable repeat period of the sub-satellite tracks.The findings show, firstly, that it is very beneficial for minimizing the aliasing errors in ocean tides to use pendulum configuration; secondly, optimizing the orbital parameter to get shorter repeat orbit mode can be effective in minimizing the aliasing errors. This paper recommends the pendulum as a candidate for future gravity mission to be launched in longer repeating orbit mode with shorter “sub-cycle” repeat periods to improve the temporal resolution of the satellite mission.

Sub-monthly gravity field recovery from simulated multi-GRACE mission type

Monthly solutions of the current GRACE mission are affected by the aliasing problem. In fact, sub-monthly temporal sampling may reduce the temporal aliasing errors but this will be done at the cost of reduced spatial sampling. Reducing the effects of temporal aliasing can be achieved by setting two pairs of satellites in different orbital planes. In this paper, we investigate the so-called Multi-GRACE constellation to improve temporal and spatial resolution for the GRACE-type mission without deteriorating accuracy. We investigate two scenarios: the Multi-GRACE ΔM that improves the temporal sampling only and the Multi-GRACE ΔΩ that improves the spatial sampling besides the temporal one in time span of only 12 days for the hydrological signal as a time-varying gravity field component. Our findings indicate that the hydrological signal can be submonthly recovered and the aliasing errors can be reduced as well by increasing temporal resolution (sub-month) via the Multi-GRACE ΔΩ constellations.

Expected improvements in determining continental hydrology, ice mass variations, ocean bottom pressure signals, and earthquakes using two pairs of dedicated satellites for temporal gravity recovery

[1] The Gravity Recovery and Climate Experiment mission has demonstrated the ability to quantify global mass variations at large spatial scales with monthly to sub-monthly temporal resolution. Future missions of this type taking advantage of improved measurement technologies will be limited by temporal aliasing errors. We suggest the addition of a second pair of satellites to reduce these errors. Using an optimized mission architecture consisting of a polar pair of satellites coupled with a lower inclined pair of satellites (72°), both in 13-day repeating orbits, we quantify the expected scientific improvements that having two pairs of satellites will provide over one pair. Numerical simulations to spherical harmonic degree 100 are run over one full year. Analysis using empirical orthogonal functions reveals that two satellite pairs determine annual mass variations in small basins which are undetected using one pair of satellites. Averaging kernels are used to show that two satellite pairs offer an 80% reduction in the level of error in determining mass variations in 53 hydrological basins and 12 Greenland basins over the year. After standard GRACE post-processing techniques have been applied to the one-pair solutions, it is seen that two satellite pairs (with no post-processing) still offer a 25%–75% improvement in determining the mass variations. Spatiospectral localization analysis is used to show increased spatial resolution and higher signal-to-noise ratios in recovering hydrology in the Amazon River basin, ocean bottom pressure signals in the Southeast Pacific basin, and a simulated earthquake signal representative of the 2010 Maule, Chile earthquake.

Simulation study of a follow-on gravity mission to GRACE

The gravity recovery and climate experiment (GRACE) has been providing monthly estimates of the Earth's time-variable gravity field since its launch in March 2002. The GRACE gravity estimates are used to study temporal mass variations on global and regional scales, which are largely caused by a redistribution of water mass in the Earth system. The accuracy of the GRACE gravity fields are primarily limited by the satellite-to-satellite range-rate measurement noise, accelerometer errors, attitude errors, orbit errors, and temporal aliasing caused by unmodeled high-frequency variations in the gravity signal. Recent work by Ball Aerospace and Technologies Corp., Boulder, CO has resulted in the successful development of an interferometric laser ranging system to specifically address the limitations of the K-band microwave ranging system that provides the satellite-to-satellite measurements for the GRACE mission. Full numerical simulations are performed for several possible configurations of a GRACE Follow-On (GFO) mission to determine if a future satellite gravity recovery mission equipped with a laser ranging system will provide better estimates of time-variable gravity, thus benefiting many areas of Earth systems research. The laser ranging system improves the range-rate measurement precision to approximately 0.6 nm/s as compared to approx. 0.2 micro-seconds for the GRACE K-band microwave ranging instrument. Four different mission scenarios are simulated to investigate the effect of the better instrument at two different altitudes. The first pair of simulated missions is flown at GRACE altitude (approx. 480 km) assuming on-board accelerometers with the same noise characteristics as those currently used for GRACE. The second pair of missions is flown at an altitude of approx. 250 km which requires a drag-free system to prevent satellite re-entry. In addition to allowing a lower satellite altitude, the drag-free system also reduces the errors associated with the accelerometer. All simulated mission scenarios assume a two satellite co-orbiting pair similar to GRACE in a near-polar, near-circular orbit. A method for local time variable gravity recovery through mass concentration blocks (mascons) is used to form simulated gravity estimates for Greenland and the Amazon region for three GFO configurations and GRACE. Simulation results show that the increased precision of the laser does not improve gravity estimation when flown with on-board accelerometers at the same altitude and spacecraft separation as GRACE, even when time-varying background models are not included. This study also shows that only modest improvement is realized for the best-case scenario (laser, low-altitude, drag-free) as compared to GRACE due to temporal aliasing errors. These errors are caused by high-frequency variations in the hydrology signal and imperfections in the atmospheric, oceanographic, and tidal models which are used to remove unwanted signal. This work concludes that applying the updated technologies alone will not immediately advance the accuracy of the gravity estimates. If the scientific objectives of a GFO mission require more accurate gravity estimates, then future work should focus on improvements in the geophysical models, and ways in which the mission design or data processing could reduce the effects of temporal aliasing.

Design considerations for a dedicated gravity recovery satellite mission consisting of two pairs of satellites

Future satellite missions dedicated to measuring time-variable gravity will need to address the concern of temporal aliasing errors; i.e., errors due to high-frequency mass variations. These errors have been shown to be a limiting error source for future missions with improved sensors. One method of reducing them is to fly multiple satellite pairs, thus increasing the sampling frequency of the mission. While one could imagine a system architecture consisting of dozens of satellite pairs, this paper explores the more economically feasible option of optimizing the orbits of two pairs of satellites. While the search space for this problem is infinite by nature, steps have been made to reduce it via proper assumptions regarding some parameters and a large number of numerical simulations exploring appropriate ranges for other parameters. A search space originally consisting of 15 variables is reduced to two variables with the utmost impact on mission performance: the repeat period of both pairs of satellites (shown to be near-optimal when they are equal to each other), as well as the inclination of one of the satellite pairs (the other pair is assumed to be in a polar orbit). To arrive at this conclusion, we assume circular orbits, repeat groundtracks for both pairs of satellites, a 100-km inter-satellite separation distance, and a minimum allowable operational satellite altitude of 290 km based on a projected 10-year mission lifetime. Given the scientific objectives of determining time-variable hydrology, ice mass variations, and ocean bottom pressure signals with higher spatial resolution, we find that an optimal architecture consists of a polar pair of satellites coupled with a pair inclined at 72°, both in 13-day repeating orbits. This architecture provides a 67% reduction in error over one pair of satellites, in addition to reducing the longitudinal striping to such a level that minimal post-processing is required, permitting a substantial increase in the spatial resolution of the gravity field products. It should be emphasized that given different sets of scientific objectives for the mission, or a different minimum allowable satellite altitude, different architectures might be selected.

Estimating low resolution gravity fields at short time intervals to reduce temporal aliasing errors

The Gravity Recovery and Climate Experiment (GRACE) satellite mission has been estimating temporal changes in the Earth’s gravitational field since its launch in 2002. While it is not yet fully resolved what the limiting source of error is for GRACE, studies on future missions have shown that temporal aliasing errors due to undersampling signals of interest (such as hydrological variations) and errors in atmospheric, ocean, and tide models will be a limiting source of error for missions taking advantage of improved technologies (flying drag-free with a laser interferometer). This paper explores the option of reducing the effects of temporal aliasing errors by directly estimating low degree and order gravity fields at short time intervals, ultimately resulting in data products with improved spatial resolution. Three potential architectures are considered: a single pair of polar orbiting satellites, two pairs of polar orbiting satellites, and a polar orbiting pair of satellites coupled with a lower inclined pair of satellites. Results show that improvements in spatial resolution are obtained when one estimates a low resolution gravity field every two days for the case of a single pair of satellites, and every day for the case of two polar pairs of satellites. However, the spatial resolution for these cases is still lower than that provided by simply destriping and smoothing the solutions via standard GRACE post-processing techniques. Alternately, estimating daily gravity fields for the case of a polar pair of satellites coupled with a lower inclined pair results in solutions with superior spatial resolution than that offered by simply destriping and smoothing the solutions.