K-band Range Rate Research Articles

An Improved Average Acceleration Approach of Modelling Earth Gravity Field Based on K-Band Range-Rate Observations

The conventional average acceleration approach relies on K-band range observation, containing an unknown bias, which leads to possible degradation of the precision of Earth’s gravity field modelling. It also suffers from correlated errors caused by three-point numerical differentiation. In this study, an improved approach is proposed that makes use of K-band range-rate observations instead and overcoming the influence of correlated errors by introducing a whitening filter. GRACE-Follow On data spanning the period from January 2019 to December 2022 were processed by the proposed approach and a series of time-varying gravity field models was derived, referred to as SSM-AAA-GFO in this paper. This model series is compared comprehensively with three official model series. Results demonstrate that all model series are highly coincident below degree 30 and reflect similar time-varying gravity field signals in both large and small basins. After filtering, SSM-AAA-GFO shows uncertainty, in the form of equivalent water height below 2.5 cm, which is comparable with three official model series. The comparison results confirm the effectiveness of the proposed approach for precisely modelling a time-varying gravity field based on K-band range-rate observations.

An Improved Acceleration Approach by Utilizing K-Band Range Rate Observations

During gravity field modeling, the conventional acceleration approach rarely incorporates KBR inter-satellite range rate data from the GRACE mission. To propose an improved acceleration method, this study introduces initial orbital position and velocity vectors to be estimated along with a combination of Cowell, KSG, and Adams integrators. In addition to achieving a full-rank design matrix regarding orbit corrections when constructing observation equations, the proposed method is capable of utilizing range rate observations for gravity field estimation. To verify the reliability of this approach, GRACE data from April 2002 to December 2016 was used to calculate a time series of monthly gravity solutions up to a degree and order of 96, referred to as Tongji-Acc RL06 in this paper. The computed time series are compared with the official models (i.e., CSR RL06, GFZ RL06, and JPL RL06) in terms of geoid degree variances, signal contents over distinct areas, and noise levels in desert regions. The investigations lead to the following conclusions: (a) the geoid degree variances indicate that Tongji-Acc RL06 exhibits comparable signal levels (approximately below 20 degrees) to the other three models while demonstrating lower noise at higher degrees (above 40 degrees); (b) the analysis over the globe, typical river basins, and land–ice regions illustrates that the solutions derived using the proposed acceleration method agree well with the official models based on the dynamic approach; (c) especially over the two large-scale river basins (i.e., Amazon and Zambezi) and another two small-scale river basins (i.e., Tennessee and Irrawaddy), Tongji-Acc RL06 significantly improves the SNR values; and (d) in the cases of the Sahara and Karakum deserts, Tongji-Acc RL06 achieves noise reductions of over 55.8% and 61.5% relative to CSR RL06, respectively. In general, the signal and noise analyses demonstrate that the proposed acceleration-based approach can effectively extract gravity field signals from KBR inter-satellite range rate observations with improved SNR, while significantly reducing the high-frequency noise.

Recovery of Rapid Water Mass Changes (RWMC) by Kalman Filtering of GRACE Observations

We demonstrate a new approach to recover water mass changes from GRACE satellite data at a daily temporal resolution. Such a product can be beneficial in monitoring extreme weather events that last a few days and are missing by conventional monthly GRACE data. The determination of the distribution of these water mass sources over networks of juxtaposed triangular tiles was made using Kalman Filtering (KF) of daily GRACE geopotential difference observations that were reduced for isolating the continental hydrology contribution of the measured gravity field. Geopotential differences were obtained from the along-track K-Band Range Rate (KBRR) measurements according to the method of energy integral. The recovery approach was validated by inverting synthetic GRACE geopotential differences simulated using GLDAS/WGHM global hydrology model outputs. Series of daily regional and global KF solutions were estimated from real GRACE KBRR data for the period 2003–2012. They provide a realistic description of hydrological fluxes at monthly time scales, which are consistent with classical spherical harmonics and mascons solutions provided by the GRACE official centers but also give an intra-month/daily continuity of these variations.

Evaluating non-tidal atmospheric products by measuring GRACE K-band range rate residuals

In order to reduce high frequency non-tidal mass changes, while inverting for the Earth’s time-variable gravity fields from the Gravity Recovery And Climate Experiment (GRACE) measurements, it is usual to apply the Atmospheric and Oceanic De-aliasing (AOD1B) products. However, limitations in these products count as a potential threat to the accuracy of time-variable gravity fields derived from GRACE, as well as its follow-on mission(s). Therefore, in this study, we show to what extent the GRACE-type gravity recovery procedure is sensitive to different non-tidal atmospheric background models. For this, we evaluate the atmospheric parts of the GeoForschungsZentrum (GFZ)’s AOD1B RL05 and RL06, as well as those computed as a part of the European Space Agency Earth System Model ESA-ESM, and the ITG3D model. These data products employ different atmosphere fields (operational and reanalysis data or their combination) from the European Centre for Medium-Range Weather Forecasts (ECMWF) as inputs, and they are also computed by implementing different 2-dimensional or 3-dimensional (2-D or 3-D) integration methods. The accuracy of these products is assessed by comparing the resulting GRACE K-Band Range-Rate (KBRR) residuals computed for time-variable gravity field inversions using each of them separately as a background model. Our investigations during 2006 indicate that: (i) applying ESA-ESM and ITG3D decreases averaged KBRR residuals by 2.8 nm/s and 3.4 nm/s compared to those reduced by the official RL05 products. (ii) Projecting these residuals onto the spatial domain indicates that the improvement covers 78.4% and 78.9% of the globe, respectively. (iii) We find that, compared to ESA-ESM, ITG3D can further reduce the KBRR residuals by 1.8 nm/s at regions of high latitudes, which likely improve the uncertainty of ice mass estimations. Our investigation of the AOD1B RL06 products covers 2006-2010, which indicates the advantage of using the higher temporal sampling, i.e. 3-hourly reanalysis data. Applying the RL06 reduces the averaged KBRR residuals by 44.2 nm/s with respect to the use of the RL05 for gravity field inversion. We, therefore, conclude that the integration method of ITG3D and utilizing reanalysis data with higher (than 6-hourly) temporal sampling rate are beneficial for GRACE-like gravity inversion such as the GRACE Follow-On mission with laser interferometric ranging system.

A Terrestrial Reference Frame realised on the observation level using a GPS-LEO satellite constellation

Applying a one-step integrated process, i.e. by simultaneously processing all data and determining all satellite orbits involved, a Terrestrial Reference Frame (TRF) consisting of a geometric as well as a dynamic part has been determined at the observation level using the EPOS-OC software of Deutsches GeoForschungsZentrum. The satellite systems involved comprise the Global Positioning System (GPS) as well as the twin GRACE spacecrafts. Applying a novel approach, the inherent datum defect has been overcome empirically. In order not to rely on theoretical assumptions this is done by carrying out the TRF estimation based on simulated observations and using the associated satellite orbits as background truth. The datum defect is identified here as the total of all three translations as well as the rotation about the z-axis of the ground station network leading to a rank-deficient estimation problem. To rectify this singularity, datum constraints comprising no-net translation (NNT) conditions in x, y, and z as well as a no-net rotation (NNR) condition about the z-axis are imposed. Thus minimally constrained, the TRF solution covers a time span of roughly a year with daily resolution. For the geometric part the focus is put on Helmert transformations between the a priori and the estimated sets of ground station positions, and the dynamic part is represented by gravity field coefficients of degree one and two. The results of a reference solution reveal the TRF parameters to be estimated reliably with high precision. Moreover, carrying out a comparable two-step approach using the same data and models leads to parameters and observational residuals of worse quality. A validation w.r.t. external sources shows the dynamic origin to coincide at a level of 5 mm or better in x and y, and mostly better than 15 mm in z. Comparing the derived GPS orbits to IGS final orbits as well as analysing the SLR residuals for the GRACE satellites reveals an orbit quality on the few cm level. Additional TRF test solutions demonstrate that K-Band Range-Rate observations between both GRACE spacecrafts are crucial for accurately estimating the dynamic frame’s orientation, and reveal the importance of the NNT- and NNR-conditions imposed for estimating the components of the dynamic geocenter.

Along-Track Geopotential Difference and Deflection of the Vertical from GRACE Range Rate: Use of GEOGRACE

We present a theory and numerical algorithm to directly determine the time-varying along-track geopotential difference and deflection of the vertical at the Gravity Recovery and Climate Experiment (GRACE) satellite altitude. The determination was implemented using the GEOGRACE computer program using the K-band range rate (KBRR) of GRACE from the Level-1B (L1B) product. The method treated KBRR, GPS-derived orbit of GRACE and an initial geopotential difference as measurements used in the least-squares estimation of the geopotential difference and its formal error constrained by the energy conservation principle. The computational procedure consisted of three steps: data reading and interpolation, data calibration and estimations of the geopotential difference and its error. The formal error allowed removal of KBRR outliers that contaminated the gravity solutions. We used the most recent models to account for the gravity changes from multiple sources. A case study was carried out over India to estimate surface mass anomalies from GEOGRACE-derived geopotential differences. The 10-day mass changes were consistent with those from the MASCON solutions of NASA (correlation coefficient up to 0.88). Using the geopotential difference at satellite altitude avoids the errors caused by downward continuation, enabling the detection of small-scale mass changes.

WHU-Grace01s: A new temporal gravity field model recovered from GRACE KBRR data alone

A new temporal gravity field model called WHU-Grace01s solely recovered from Gravity Recovery and Climate Experiment (GRACE) K-Band Range Rate (KBRR) data based on dynamic integral approach is presented in this paper. After meticulously preprocessing of the GRACE KBRR data, the root mean square of its post residuals is about 0.2 micrometers per second, and seventy-two monthly temporal solutions truncated to degree and order 60 are computed for the period from January 2003 to December 2008. After applying the combination filter in WHU-Grace01s, the global temporal signals show obvious periodical change rules in the large-scale river basins. In terms of the degree variance, our solution is smaller at high degrees, and shows a good consistency at the rest of degrees with the Release 05 models from Center for Space Research (CSR), GeoForschungsZentrum Potsdam (GFZ) and Jet Propulsion Laboratory (JPL). Compared with other published models in terms of equivalent water height distribution, our solution is consistent with those published by CSR, GFZ, JPL, Delft institute of Earth Observation and Space system (DEOS), Tongji University (Tongji), Institute of Theoretical Geodesy (ITG), Astronomical Institute in University of Bern (AIUB) and Groupe de Recherche de Geodesie Spatiale (GRGS), which indicates that the accuracy of WHU-Grace01s has a good consistency with the previously published GRACE solutions.

Tongji-GRACE01: A GRACE-only static gravity field model recovered from GRACE Level-1B data using modified short arc approach

The modified short arc approach, where the position vector in force model are regarded as pseudo observation, is implemented in the SAtellite Gravimetry Analysis Software (SAGAS) developed by Tongji university. Based on the SAGAS platform, a static gravity field model (namely Tongji-GRACE01) complete to degree and order 160 is computed from 49months of real GRACE Level-1B data spanning the period 2003–2007 (including the observations of K-band range-rate, reduced dynamic orbits, non-conservative accelerations and altitudes). The Tongji-GRACE01 model is compared with the recent GRACE-only models (such as GGM05S, AIUB-GRACE03S, ITG-GRACE03, ITG-GRACE2010S, and ITSG-GRACE2014S) and validated with GPS-leveling data sets in different countries. The results show that the Tongji-GRACE01 model has a considered quality as GGM05S, AIUB-GRACE03S and ITG-GRACE03. The Tongji-GRACE01 model is available at the International Centre for Global Earth Models (ICGEM) web page (http://icgem.gfz-potsdam.de/ICGEM/).

Sequential estimation of surface water mass changes from daily satellite gravimetry data

We propose a recursive Kalman filtering approach to map regional spatio-temporal variations of terrestrial water mass over large continental areas, such as South America. Instead of correcting hydrology model outputs by the GRACE observations using a Kalman filter estimation strategy, regional 2-by-2 degree water mass solutions are constructed by integration of daily potential differences deduced from GRACE K-band range rate (KBRR) measurements. Recovery of regional water mass anomaly averages obtained by accumulation of information of daily noise-free simulated GRACE data shows that convergence is relatively fast and yields accurate solutions. In the case of cumulating real GRACE KBRR data contaminated by observational noise, the sequential method of step-by-step integration provides estimates of water mass variation for the period 2004–2011 by considering a set of suitable a priori error uncertainty parameters to stabilize the inversion. Spatial and temporal averages of the Kalman filter solutions over river basin surfaces are consistent with the ones computed using global monthly/10-day GRACE solutions from official providers CSR, GFZ and JPL. They are also highly correlated to in situ records of river discharges (70–95 %), especially for the Obidos station where the total outflow of the Amazon River is measured. The sparse daily coverage of the GRACE satellite tracks limits the time resolution of the regional Kalman filter solutions, and thus the detection of short-term hydrological events.

Regional GRACE-based estimates of water mass variations over Australia: validation and interpretation

Abstract. Time series of regional 2° × 2° Gravity Recovery and Climate Experiment (GRACE) solutions have been computed from 2003 to 2011 with a 10-day resolution by using an energy integral method over Australia (112° E–156° E; 44° S–10° S). This approach uses the dynamical orbit analysis of GRACE Level 1 measurements, and specially accurate along-track K-band range rate (KBRR) residuals with a 1 μm s−1 level of errors, to estimate the total water mass over continental regions. The advantages of regional solutions are a significant reduction of GRACE aliasing errors (i.e. north–south stripes) providing a more accurate estimation of water mass balance for hydrological applications. In this paper, the validation of these regional solutions over Australia is presented, as well as their ability to describe water mass change as a response of climate forcings such as El Niño. Principal component analysis of GRACE-derived total water storage (TWS) maps shows spatial and temporal patterns that are consistent with independent data sets (e.g. rainfall, climate index and in situ observations). Regional TWS maps show higher spatial correlations with in situ water table measurements over Murray–Darling drainage basin (80–90%), and they offer a better localization of hydrological structures than classical GRACE global solutions (i.e. Level 2 Groupe de Recherche en Géodésie Spatiale (GRGS)) products and 400 km independent component analysis solutions as a linear combination of GRACE solutions provided by different centers.

Antarctica, Greenland and Gulf of Alaska land-ice evolution from an iterated GRACE global mascon solution

AbstractWe have determined the ice mass evolution of the Antarctic and Greenland ice sheets (AIS and GIS) and Gulf of Alaska (GOA) glaciers from a new GRACE global solution of equal-area surface mass concentration parcels (mascons) in equivalent height of water. The mascons were estimated directly from the reduction of the inter-satellite K-band range-rate (KBRR) observations, taking into account the full noise covariance, and formally iterating the solution. The new solution increases signal recovery while reducing the GRACE KBRR observation residuals. The mascons were estimated with 10 day and 1 arcdeg equal-area sampling, applying anisotropic constraints. An ensemble empirical mode decomposition adaptive filter was applied to the mascon time series to compute annual mass balances. The details and causes of the spatial and temporal variability of the land-ice regions studied are discussed. The estimated mass trend over the total GIS, AIS and GOA glaciers for the time period 1 December 2003 to 1 December 2010 is −380 ± 31 Gt a−1, equivalent to −1.05 ± 0.09 mm a−1 sea-level rise. Over the same time period we estimate the mass acceleration to be −41 ± 27 Gt a−2 , equivalent to a −0.11 ± 0.08 mm a−2 sea-level acceleration. The trends and accelerations are dependent on significant seasonal and annual balance anomalies.

Constrained Regional Recovery of Continental Water Mass Time-variations from GRACE-based Geopotential Anomalies over South America

We propose a “constrained” least-squares approach to estimate regional maps of equivalent-water heights by inverting GRACE-based potential anomalies at satellite altitude. According to the energy integral method, the anomalies of difference of geopotential between the two GRACE vehicles are derived from along-track K-Band Range-Rate (KBRR) residuals that correspond mainly to the continental water storage changes, once a priori known accelerations (i.e. static field, polar movements, atmosphere and ocean masses including tides) are removed during the orbit adjustment process. Newton's first law merely enables the Difference of Potential Anomalies from accurate KBRR data and the equivalent-water heights to be recovered. Spatial constraints versus spherical distance between elementary surface tiles are introduced to stabilize the linear system to cancel the effects of the north-south striping. Unlike the “mascons” approach, no basis of orthogonal functions (e.g., spherical harmonics) is used, so that the proposed regional method does not suffer from drawbacks related to any spectrum truncation. Time series of 10-day regional maps over South America for 2006–2009 also prove to be consistent with independent data sets, namely the outputs of hydrological models, “mascons” and global GRACE solutions.

Improving global mass flux solutions from Gravity Recovery and Climate Experiment (GRACE) through forward modeling and continuous time correlation

We describe Earth's mass flux from April 2003 through November 2008 by deriving a time series of mascons on a global 2° × 2° equal‐area grid at 10 day intervals. We estimate the mass flux directly from K band range rate (KBRR) data provided by the Gravity Recovery and Climate Experiment (GRACE) mission. Using regularized least squares, we take into account the underlying process dynamics through continuous space and time‐correlated constraints. In addition, we place the mascon approach in the context of other filtering techniques, showing its equivalence to anisotropic, nonsymmetric filtering, least squares collocation, and Kalman smoothing. We produce mascon time series from KBRR data that have and have not been corrected (forward modeled) for hydrological processes and find that the former produce superior results in oceanic areas by minimizing signal leakage from strong sources on land. By exploiting the structure of the spatiotemporal constraints, we are able to use a much more efficient (in storage and computation) inversion algorithm based upon the conjugate gradient method. This allows us to apply continuous rather than piecewise continuous time‐correlated constraints, which we show via global maps and comparisons with ocean‐bottom pressure gauges, to produce time series with reduced random variance and full systematic signal. Finally, we present a preferred global model, a hybrid whose oceanic portions are derived using forward modeling of hydrology but whose land portions are not, and thus represent a pure GRACE‐derived signal.

Evaluating terrestrial water storage variations from regionally constrained GRACE mascon data and hydrological models over Southern Africa – preliminary results

A concentration of surface mass has a distinct, localized signature in Gravity Recovery and Climate Experiment (GRACE) K-band range rate (KBRR) data. This fact is exploited in the regional solutions for mass concentration parameters (mascons) made at the Goddard Space Flight Center (GSFC). In this paper we explore an experimental set of regionally constrained mascon blocks over Southern Africa where a system of 1.25° × 1.5° and 1.5° × 1.5° blocks has been designed. The blocks are divided into hydrological regions based on drainage patterns of the largest river basins, and are constrained in different ways. We show that the use of regional constraints, when solving mascon parameters of different hydrological regions independently, yields more detail and variation than comparable spherical harmonic solutions and mascon solutions using isotropic constraints. We validate our results over Lake Malawi with water level from altimetry. Results show that weak constraints across regions in addition to intra-regional constraints are necessary, to reach reasonable mass variations.

A simulation study of the errors of omission and commission for GRACE RL01 gravity fields

This paper examines the influence that certain omission and commission errors can have on the gravity field models estimated from the initial release of data (RL01) from the Gravity Recovery And Recovery Experiment (GRACE) satellite mission. The effects of omission errors were analyzed by limiting the degree and order to which the GPS and K-band range-rate (KBR) measurement partials were extended in the solution process. The commission error studies focused on the impact of an imperfect mean reference gravity field model on the solution. Combinations of both of these error sources were also explored. The nature of these errors makes them difficult to distinguish from the true gravity signal, so the exploration of these error sources was performed using simulations; however, comparisons to real-data solutions are provided. The results show how each of the specific error sources investigated influences the gravity field solution. The simulations also show how all of the errors examined can be sufficiently mitigated through the appropriate choice of processing parameters.