GRACE反演南极冰盖质量变化的高斯与Wiener滤波比较

本文利用UTCSR 2003年1月到2008年8月间的GRACE Level-2 RL04重力场模型估计了南极冰盖质量变化。计算过程中分别采用高斯和Wiener滤波两种平滑方法,分别采用22、43和65个月重力场模型计算Wiener滤波信号与噪声函数,得出以下结论:在实际的计算过程中需要具体计算Wiener滤波平滑因子值,65个月GRACE重力场模型计算得到的Wiener滤波权值非常接近于平滑半径为540km高斯滤波权值;采用两种不同的滤波方法在相同区域质量变化率基本相同。     

Reducing errors in the GRACE gravity solutions using regularization

The nature of the gravity field inverse problem amplifies the noise in the GRACE data, which creeps into the mid and high degree and order harmonic coefficients of the Earth’s monthly gravity fields provided by GRACE. Due to the use of imperfect background models and data noise, these errors are manifested as north-south striping in the monthly global maps of equivalent water heights. In order to reduce these errors, this study investigates the use of the L-curve method with Tikhonov regularization. L-curve is a popular aid for determining a suitable value of the regularization parameter when solving linear discrete ill-posed problems using Tikhonov regularization. However, the computational effort required to determine the L-curve is prohibitively high for a large-scale problem like GRACE. This study implements a parameter-choice method, using Lanczos bidiagonalization which is a computationally inexpensive approximation to L-curve. Lanczos bidiagonalization is implemented with orthogonal transformation in a parallel computing environment and projects a large estimation problem on a problem of the size of about 2 orders of magnitude smaller for computing the regularization parameter. Errors in the GRACE solution time series have certain characteristics that vary depending on the ground track coverage of the solutions. These errors increase with increasing degree and order. In addition, certain resonant and near-resonant harmonic coefficients have higher errors as compared with the other coefficients. Using the knowledge of these characteristics, this study designs a regularization matrix that provides a constraint on the geopotential coefficients as a function of its degree and order. This regularization matrix is then used to compute the appropriate regularization parameter for each monthly solution. A 7-year time-series of the candidate regularized solutions (Mar 2003–Feb 2010) show markedly reduced error stripes compared with the unconstrained GRACE release 4 solutions (RL04) from the Center for Space Research (CSR). Post-fit residual analysis shows that the regularized solutions fit the data to within the noise level of GRACE. A time series of filtered hydrological model is used to confirm that signal attenuation for basins in the Total Runoff Integrating Pathways (TRIP) database over 320 km radii is less than 1 cm equivalent water height RMS, which is within the noise level of GRACE.     

Decorrelated GRACE time-variable gravity solutions by GFZ,and their validation using a hydrological model

We have analyzed recent gravity recovery and climate experiment (GRACE) RL04 monthly gravity solutions, using a new decorrelating post-processing approach. We find very good agreement with mass anomalies derived from a global hydrological model. The post-processed GRACE solutions exhibit only little amplitude damping and an almost negligible phase shift and period distortion for relevant hydrological basins. Furthermore, these post-processed GRACE solutions have been inspected in terms of data fit with respect to the original inter-satellite ranging and to SLR and GPS observations. This kind of comparison is new. We find variations of the data fit due to solution post-processing only within very narrow limits. This confirms our suspicion that GRACE data do not firmly ‘pinpoint’ the standard unconstrained solutions. Regarding the original Kusche (J Geod 81:733–749, 2007) decorrelation and smoothing method, a simplified (order-convolution) approach has been developed. This simplified approach allows to realize a higher resolution—as necessary, e.g., for generating computed GRACE observations—and needs far less coefficients to be stored.     

Attenuation effect on seasonal basin-scale water storage changes from GRACE time-variable gravity

In order to effectively recover surface mass or geoid height changes from the gravity recovery and climate experiment (GRACE) time-variable gravity models, spatial smoothing is required to minimize errors from noise. Spatial smoothing, such as Gaussian smoothing, not only reduces the noise but also attenuates the real signals. Here we investigate possible amplitude attenuations and phase changes of seasonal water storage variations in four drainage basins (Amazon, Mississippi, Ganges and Zambezi) using an advanced global land data assimilation system. It appears that Gaussian smoothing significantly affects GRACE-estimated basin-scale seasonal water storage changes, e.g., in the case of 800 km smoothing, annual amplitudes are reduced by about 25–40%, while annual phases are shifted by up to 10°. With these effects restored, GRACE-estimated water storage changes are consistently larger than model estimates, indicating that the land surface model appears to underestimate terrestrial water storage change. Our analysis based on simulation suggests that normalized attenuation effects (from Gaussian smoothing) on seasonal water storage change are relatively insensitive to the magnitude of the true signal. This study provides a numerical approach that can be used to restore seasonal water storage change in the basins from spatially smoothed GRACE data.     

Accuracy assessment of the monthly GRACE geoids based upon a simulation

The purpose of this paper is to demonstrate the effect of geophysical background model errors that affects temporal gravity solutions provided by the Gravity Recovery And Climate Experiment (GRACE). Initial performance estimates by Dickey et al. (1997) suggested a formal geoid RMS error better than 0.1 mm up to spherical harmonic degree 5. Now that the GRACE gravity models and data are available, it is evident that these original expectations were too optimistic. Our hypothesis is that this is partially explained by errors in geophysical background models that need to be applied in the GRACE data reduction, and that this effect was not considered by Dickey et al. (1997). We discuss the results of a closed-loop simulation, where satellite trajectory prediction software is used for the generation of GRACE range-rate data and GRACE orbit solutions with the help of the Global Positioning System (GPS). During the recovery step in our closed-loop simulation, we show that simulated nuisance signals (based on tide and air pressure model differences) map to a 0.7 mm geoid effect for periods longer than 3 months and to less than 0.4 mm for periods shorter than 3 months. The long-period geoid hydrology signal is at a level of 4.5 mm, while the short-period hydrology is at 0.25 mm. The long-period ocean bottom pressure (OBP) signal maps at 0.8 mm and for short periods it is 0.4 mm. We conclude that short-period effects are difficult to observe by GRACE and that long-period effects, like hydrology, are easier to recover than OBP variations.     

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 un-modeled high-frequency variations in the gravity signal. Recent work by Ball Aerospace & 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 ~0.6 nm/s as compared to ~0.2 μm/s 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 (~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 ~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.     

High-frequency signal and noise estimates of CSR GRACE RL04

A sliding window technique is used to create daily-sampled Gravity Recovery and Climate Experiment (GRACE) solutions with the same background processing as the official CSR RL04 monthly series. By estimating over shorter time spans, more frequent solutions are made using uncorrelated data, allowing for higher frequency resolution in addition to daily sampling. Using these data sets, high-frequency GRACE errors are computed using two different techniques: assuming the GRACE high-frequency signal in a quiet area of the ocean is the true error, and computing the variance of differences between multiple high-frequency GRACE series from different centers. While the signal-to-noise ratios prove to be sufficiently high for confidence at annual and lower frequencies, at frequencies above 3 cycles/year the signal-to-noise ratios in the large hydrological basins looked at here are near 1.0. Comparisons with the GLDAS hydrological model and high frequency GRACE series developed at other centers confirm CSR GRACE RL04’s poor ability to accurately and reliably measure hydrological signal above 3–9 cycles/year, due to the low power of the large-scale hydrological signal typical at those frequencies compared to the GRACE errors.     

根据人卫激光测距、GRACE和地球物理模型求解地球低阶重力场季节变化

本文利用2002年4月至2010年10月的Lageos1和Lageos2两颗激光卫星观测数据、GRACE以及地球物理模型三种独立的方法计算地球低阶重力场系数J 的变化,根据大气压强数据计算 J 时分别按反变气压计（IB）和非反变气压计(NIB)两种假设进行计算。通过分析 J 的季节特性表明,大气在NIB假设下得到的周年振幅比在IB假设下得到的振幅大3倍左右,相位相差47°;大气和陆地水的质量变化对 J 周年变化的贡献占主导地位,海洋的影响最小;大气、海洋和陆地水得到 J 半年振幅和相位值与SLR得到的振幅和相位值吻合较差,尤其是在IB假设下大气得到的结果与SLR结果相差最大; SLR、GRACE和地球物理模型三种独立方法得到的 J 周年项之间吻合相对较好,GRACE得到的周年振幅比SLR得到的周年振幅大50%左右, SLR观测得到的 J 周年振幅介于在NIB和IB两种假设下地球物理模型得到的结果之间;GRACE与SLR得到的 J 半年项的振幅相同,在IB假设下AOW得到的 J 半年振幅和相位与SLR结果差异最大。     

Combination of GRACE monthly gravity field solutions from different processing strategies

We combine the publicly available GRACE monthly gravity field time series to produce gravity fields with reduced systematic errors. We first compare the monthly gravity fields in the spatial domain in terms of signal and noise. Then, we combine the individual gravity fields with comparable signal content, but diverse noise characteristics. We test five different weighting schemes: equal weights, non-iterative coefficient-wise, order-wise, or field-wise weights, and iterative field-wise weights applying variance component estimation (VCE). The combined solutions are evaluated in terms of signal and noise in the spectral and spatial domains. Compared to the individual contributions, they in general show lower noise. In case the noise characteristics of the individual solutions differ significantly, the weighted means are less noisy, compared to the arithmetic mean: The non-seasonal variability over the oceans is reduced by up to 7.7% and the root mean square (RMS) of the residuals of mass change estimates within Antarctic drainage basins is reduced by 18.1% on average. The field-wise weighting schemes in general show better performance, compared to the order- or coefficient-wise weighting schemes. The combination of the full set of considered time series results in lower noise levels, compared to the combination of a subset consisting of the official GRACE Science Data System gravity fields only: The RMS of coefficient-wise anomalies is smaller by up to 22.4% and the non-seasonal variability over the oceans by 25.4%. This study was performed in the frame of the European Gravity Service for Improved Emergency Management (EGSIEM; http://www.egsiem.eu) project. The gravity fields provided by the EGSIEM scientific combination service (ftp://ftp.aiub.unibe.ch/EGSIEM/) are combined, based on the weights derived by VCE as described in this article.     

Statistically optimal estimation of Greenland Ice Sheet mass variations from GRACE monthly solutions using an improved mascon approach

We present an improved mascon approach to transform monthly spherical harmonic solutions based on GRACE satellite data into mass anomaly estimates in Greenland. The GRACE-based spherical harmonic coefficients are used to synthesize gravity anomalies at satellite altitude, which are then inverted into mass anomalies per mascon. The limited spectral content of the gravity anomalies is properly accounted for by applying a low-pass filter as part of the inversion procedure to make the functional model spectrally consistent with the data. The full error covariance matrices of the monthly GRACE solutions are properly propagated using the law of covariance propagation. Using numerical experiments, we demonstrate the importance of a proper data weighting and of the spectral consistency between functional model and data. The developed methodology is applied to process real GRACE level-2 data (CSR RL05). The obtained mass anomaly estimates are integrated over five drainage systems, as well as over entire Greenland. We find that the statistically optimal data weighting reduces random noise by 35–69%, depending on the drainage system. The obtained mass anomaly time-series are de-trended to eliminate the contribution of ice discharge and are compared with de-trended surface mass balance (SMB) time-series computed with the Regional Atmospheric Climate Model (RACMO 2.3). We show that when using a statistically optimal data weighting in GRACE data processing, the discrepancies between GRACE-based estimates of SMB and modelled SMB are reduced by 24–47%.     

On the feasibility of using satellite gravity observations for detecting large-scale solid mass transfer events

The main focus of this paper is to assess the feasibility of utilizing dedicated satellite gravity missions in order to detect large-scale solid mass transfer events (e.g. landslides). Specifically, a sensitivity analysis of Gravity Recovery and Climate Experiment (GRACE) gravity field solutions in conjunction with simulated case studies is employed to predict gravity changes due to past subaerial and submarine mass transfer events, namely the Agulhas slump in southeastern Africa and the Heart Mountain Landslide in northwestern Wyoming. The detectability of these events is evaluated by taking into account the expected noise level in the GRACE gravity field solutions and simulating their impact on the gravity field through forward modelling of the mass transfer. The spectral content of the estimated gravity changes induced by a simulated large-scale landslide event is estimated for the known spatial resolution of the GRACE observations using wavelet multiresolution analysis. The results indicate that both the Agulhas slump and the Heart Mountain Landslide could have been detected by GRACE, resulting in \({\vert }0.4{\vert }\) and \({\vert }0.18{\vert }\) mGal change on GRACE solutions, respectively. The suggested methodology is further extended to the case studies of the submarine landslide in Tohoku, Japan, and the Grand Banks landslide in Newfoundland, Canada. The detectability of these events using GRACE solutions is assessed through their impact on the gravity field.     

Optimal orbits for temporal gravity recovery regarding temporal aliasing

The temporal changes of the Earth’s gravity field can be observed on a global scale with low–low satellite-to-satellite tracking (SST) missions. One of the largest restrictions of the quality of low–low SST gravity fields is temporal aliasing. This study investigates the design of optimal satellite orbits for temporal gravity retrieval regarding temporal aliasing. We present a method with which optimal altitudes for the orbit of a gravity satellite mission with the goal of temporal gravity retrieval can be identified. The two basic orbit frequencies, the rates of the argument of the latitude and the ascending node, determine the mapping of the signal measured along the orbit onto the spherical harmonic (SH) spectrum. The main spectral characteristics of temporal aliasing are maxima at specific SH orders. The magnitude of the effects depends on the basic frequencies. This is analyzed with numerical low–low SST closed-loop simulations including both tidal and non-tidal background models and GRACE-like observation noise. Analyses of actual monthly GRACE solutions show that these characteristics do not depend on the low–low SST processing method. Optimal orbits are found in specific altitude bands. The best altitude bands regarding temporal aliasing for polar low Earth orbiters (LEOs) are around 301, 365, 421 and 487 km. In these bands, major aliasing effects do not occur for SH degrees and orders below 70. This study gives unique and in-depth insights into the mechanism of temporal aliasing. As it provides an important orbit design approach, it is independent of any (post-) processing method to reduce temporal aliasing.     

S<Subscript>2</Subscript> tide aliasing in GRACE time-variable gravity solutions

Errors in high-frequency ocean tide models alias to low frequencies in time-variable gravity solutions from the Gravity Recovery and Climate Experiment (GRACE). We conduct an observational study of apparent gravity changes at a period of 161 days, the alias period of errors in the S₂ semidiurnal solar tide. We examine this S₂ alias in the release 4 (RL04) reprocessed GRACE monthly gravity solutions for the period April 2002 to February 2008, and compare with that in release 1 (RL01) GRACE solutions. One of the major differences between RL04 and RL01 is the ocean tide model. In RL01, the alias is evident at high latitudes, near the Filchner-Ronne and Ross ice shelves in Antarctica, and regions surrounding Greenland and Hudson Bay. RL04 shows significantly lower alias amplitudes in many of these locations, reflecting improvements in the ocean tide model. However, RL04 shows continued alias contamination between the Ronne and Larson ice shelves, somewhat larger than in RL01, indicating a need for further tide model improvement in that region. For unknown reasons, the degree-2 zonal spherical harmonics (C₂₀) of the RL04 solutions show significantly larger S₂ aliasing errors than those from RL01.     

各向异性组合滤波法反演陆地水储量变化

地球时变重力场模型反演陆地水储量变化已为全球气候变化研究作出巨大贡献,考虑到时变重力场模型球谐系数中存在相关性,其高阶次项具有较大的误差,需采用最优的滤波方法进行空间平滑。本文提出一种新的各向异性组合滤波方法,其基本思想是将改进的高斯滤波法与均方根(root mean square,RMS)滤波法组合,即对球谐系数的低阶次采用改进的高斯滤波法,而高阶次采用RMS滤波法。首先分析了最新的GRACE RL05系列时变重力场模型系数误差特性,基于全球水储量变化反演结果,分析比较了高斯滤波、改进的高斯滤波、RMS滤波和DDK滤波与本文提出的组合滤波法的有效性及精度,并利用模型结果进行了验证,计算结果表明,组合滤波法的中误差最小。研究结果表明,本文提出的组合法相比于先前的滤波方法,可有效地过滤高阶次的噪声,消除南北条带误差,同时减少信号泄漏,提高信噪比,保留更多有效的地球物理信号,进而提高反演精度。     

Quantifying FES2004 S<Subscript>2</Subscript> tidal model from multiple space-geodesy techniques,GPS and GRACE,over North West Australia

Unmodeled sub-daily ocean S₂ tide signals that alias into lower frequencies have been detected in the analysis of gravity recovery and climate experiment (GRACE) space gravity fields of GRGS. The most significant global S₂ aliased signal occurs off the northwest coast of Australia in a shallow continental shelf zone, a region with high tidal amplitudes at a period of 161 days. The GRACE S₂ aliased equivalent water height grids are convolved with Green’s functions to produce GRACE aliased tidal loading (GATL) vertical displacements. The analysis of hourly global positioning system (GPS) vertical coordinate estimates at permanent sites in the region confirms the presence of spectral power at the S₂ frequency when the same ocean tide model (FES2004) was used. Thus, deficiencies in the FES2004 ocean tide model are detected both directly and indirectly by the two independent space geodetic techniques. Through simulation, the admittance (ratio of amplitude of spurious long-wavelength output signal in the GRACE time-series to amplitude of unmodeled periodic signals) of the GRACE unmodeled S₂ tidal signals, aliased to a 161-day period, is found to have a global average close to 100%, although with substantial spatial variation. Comparing GATL with unmodeled S₂ tidal sub-daily signals in the vertical GPS time-series in the region of Broome in NW Australia suggests an admittance of 110–130%.     

Understanding data noise in gravity field recovery on the basis of inter-satellite ranging measurements acquired by the satellite gravimetry mission GRACE

Spectral analysis of data noise is performed in the context of gravity field recovery from inter-satellite ranging measurements acquired by the satellite gravimetry mission GRACE. The motivation of the study is two-fold: (i) to promote a further improvement of GRACE data processing techniques and (ii) to assist designing GRACE follow-on missions. The analyzed noise realizations are produced as the difference between the actual GRACE inter-satellite range measurements and the predictions based on state-of-the-art force models. The exploited functional model is based on the so-called “range combinations,” which can be understood as a finite-difference analog of inter-satellite accelerations projected onto the line-of-sight connecting the satellites. It is shown that low-frequency noise is caused by limited accuracy of the computed GRACE orbits. In the first instance, it leads to an inaccurate estimation of the radial component of the inter-satellite velocities. A large impact of this component stems from the fact that it is directly related to centrifugal accelerations, which have to be taken into account when the measured range-accelerations are linked with inter-satellite accelerations. Another effect of orbit inaccuracies is a miscalculation of forces acting on the satellites (particularly, the one described by the zero-degree term of the Earth’s gravitational field). The major contributors to the noise budget at high frequencies (above 9?mHz) are (i) ranging sensor errors and (ii) limited knowledge of the Earth’s static gravity field at high degrees. Importantly, we show that updating the model of the static field on the basis of the available data must be performed with a caution as the result may not be physical due to a non-unique recovery of high-degree coefficients. The source of noise in the range of intermediate frequencies (1–9?mHz), which is particularly critical for an accurate gravity field recovery, is not fully understood yet. We show, however, that it cannot be explained by inaccuracies in background models of time-varying gravity field. It is stressed that most of the obtained results can be treated as sufficiently general (i.e., applicable in the context of a statistically optimal estimation based on any functional model).     

日本Mw9.0地震前GRACE卫星重力变化

利用GRACE重力卫星月重力场数据,通过去相关与高斯滤波等方法获取日本大地震震前震源区周缘年度、季度和年度差分重力变化,以及若干点位时间序列重力变化;分析结果表明:日本大地震前5年内在震源区周边出现了比较明显的卫星重力异常正负交替和迁移现象,至震前1－2年,震区周边形成了明显正负异常区,正重力异常区重力增加现象明显;点位重力时间序列分布指出了日本MW9.0级地震前存在与1976年唐山地震类似的重力变化现象。这些结果反映了日本大地震震前震源区周边地下物质运动、质量迁移和能量积累等问题,为研究该地震的孕育过程提供了证据。     

Wavelet Modeling of Regional and Temporal Variations of the Earth’s Gravitational Potential Observed by GRACE

This work is dedicated to the wavelet modeling of regional and temporal variations of the Earth’s gravitational potential observed by the GRACE (gravity recovery and climate experiment) satellite mission. In the first part, all required mathematical tools and methods involving spherical wavelets are provided. Then, we apply our method to monthly GRACE gravity fields. A strong seasonal signal can be identified which is restricted to areas where large-scale redistributions of continental water mass are expected. This assumption is analyzed and verified by comparing the time-series of regionally obtained wavelet coefficients of the gravitational signal originating from hydrology models and the gravitational potential observed by GRACE. The results are in good agreement with previous studies and illustrate that wavelets are an appropriate tool to investigate regional effects in the Earth’s gravitational field. Electronic Supplementary Material Supplementary material is available for this article at     

重力场恢复与气候实验的石羊河流域水储量变化

为了更好地从空间尺度显示石羊河流域水储量的变化情况,该文利用2003年1月至2014年12月GRACE RL06重力场数据计算其陆地水储量变化,对数据进行预处理以及去相关滤波和300km高斯平滑处理,并利用流域内4个气象站的降水平均数据与GRACE、GLDAS的结果进行对比分析。结果表明,GRACE反演结果与GLDAS水文模型变化趋势基本一致,时间上呈现明显季节性变化,研究区水储量整体呈现0.93mm/a的上升趋势,期间水储量变化波动较大主要与季节性降水有关。