PUBLICATIONS Journal of Geophysical Research: Solid Earth RESEARCH ARTICLE 10.1002/2014JB011470 Key Points: • Modified short-arc approach • Monthly gravity field models • Temporal mass change signals
Correspondence to: Y. Shen,
[email protected]
Citation: Chen, Q., Y. Shen, X. Zhang, H. Hsu, W. Chen, X. Ju, and L. Lou (2015), Monthly gravity field models derived from GRACE Level 1B data using a modified short-arc approach, J. Geophys. Res. Solid Earth, 120, 1804–1819, doi:10.1002/2014JB011470. Received 22 JUL 2014 Accepted 26 JAN 2015 Accepted article online 31 JAN 2015 Published online 9 MAR 2015
Monthly gravity field models derived from GRACE Level 1B data using a modified short-arc approach Qiujie Chen1,2,3, Yunzhong Shen1,2, Xingfu Zhang4, Houze Hsu5, Wu Chen3, Xiaolei Ju1, and Lizhi Lou1 1
College of Surveying and Geo-informatics, Tongji University, Shanghai, China, 2Center for Spatial Information Science and Sustainable Development, Shanghai, China, 3Department of Land Surveying and Geo-informatics, Hong Kong Polytechnic University, Hong Kong, 4Department of Surveying and Mapping, Guangdong University of Technology, Guangzhou, China, 5 State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and Geophysics, CAS, Wuhan, China
Abstract In this study, a new time series of Gravity Recovery and Climate Experiment (GRACE) monthly solutions, complete to degree and order 60 spanning from January 2003 to August 2011, has been derived based on a modified short-arc approach. Our models entitled Tongji-GRACE01 are available on the website of International Centre for Global Earth Models (http://icgem.gfz-potsdam.de/ICGEM/). The traditional short-arc approach, with no more than 1 h arcs, requires the gradient corrections of satellite orbits in order to reduce the impact of orbit errors on the final solution. Here the modified short-arc approach has been proposed, which has three major differences compared to the traditional one: (1) All the corrections of orbits and range rate measurements are solved together with the geopotential coefficients and the accelerometer biases using a weighted least squares adjustment; (2) the boundary position parameters are not required; and (3) the arc length can be extended to 2 h. The comparisons of geoid degree powers and the mass change signals in the Amazon basin, the Antarctic, and Antarctic Peninsula demonstrate that our model is comparable with the other existing models, i.e., the Centre for Space Research RL05, Jet Propulsion Laboratory RL05, and GeoForschungsZentrum RL05a models. The correlation coefficients of the mass change time series between our model and the other models are better than 0.9 in the Antarctic and Antarctic Peninsula. The mass change rates in the Antarctic and Antarctic Peninsula derived from our model are 92.7 ± 38.0 Gt/yr and 23.9 ± 12.4 Gt/yr, respectively, which are very close to those from other three models and with similar spatial patterns of signals.
1. Introduction The twin satellites of Gravity Recovery and Climate Experiment (GRACE) launched on 17 March 2002 have achieved a breakthrough in terms of both accuracy and resolution of the gravity field determination [Tapley et al., 2004]. The most significant part of the GRACE mission is the precise K band ranging (KBR) system to measure the ranges between the twin satellites. The precise orbits of the twin satellites are determined using onboard GPS receivers. Additionally, the nonconservative forces acting on the satellites can be effectively detected by the high-precision accelerometers. The attitudes of the twin satellites are determined by the star cameras that are set up on each of the satellites. With the GRACE observations, a number of monthly gravity field models have been developed and published by the Centre for Space Research (CSR), the GeoForschungsZentrum (GFZ), the Jet Propulsion Laboratory (JPL), and the Bonn University, such as the models of CSR RL05 [Bettadpur, 2012], GFZ RL05a [Dahle et al., 2012], JPL RL05 [Watkins and Yuan, 2012], and ITG-GRACE2010 [Mayer-Gürr, 2006]. These time variable gravity field models have been successfully used to detect the mass changes of the Earth [Tapley et al., 2004; Chambers et al., 2004; Wu et al., 2009], including the Antarctic and Greenland ice mass melting [Velicogna and Wahr, 2005; Chen et al., 2006, 2009; Riva et al., 2009], decreases in groundwater [Scanlon et al., 2007; Feng et al., 2013], and coseismic deformation [Chen et al., 2007]. Although these gravity field models agree well with each other in the spatial pattern and accuracy, their theoretical backgrounds are quite different. The CSR RL05, GFZ RL05a, and JPL RL05 models are computed using a dynamic approach [Bettadpur, 2012; Dahle et al., 2012; Watkins and Yuan, 2012]. This dynamic approach is also used to develop the time variable gravity field models of 10 days temporal resolution by the Centre National d’Etudes Spatiales [Bruinsma et al., 2010]. The ITG-GRACE2010 monthly models are generated using a short-arc approach, the DMT-1 monthly models are recovered using a mean acceleration approach [Ditmar and van Eck van der Sluijs, 2004; Ditmar et al., 2006; Liu, 2008] by the Delft Institute of Earth Observation and Space
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Systems, and the AIUB models are computed based on a celestial mechanics approach [Beutler et al., 2010a, 2010b; Jäggi et al., 2012; Meyer et al., 2012]. In the dynamic approach, the satellite positions are integrated from six initial states and a priori force model [Reigber et al., 2005; Tapley et al., 2005], and the design matrix of the observation equation is integrated via a variation equation; therefore, the computational burden is very high. The dynamic approach can be modified to deal with long-arc integration [Xu, 2008]. The mean acceleration approach builds up the relationship between the mean acceleration and three-points range combination, which enables the observation equation to directly connect the potential gradients with KBR ranges [Liu, 2008]. Chen et al. [2013] further improved the mean acceleration approach by taking the position errors in the force model into consideration. The celestial mechanics approach is similar to the dynamic method to some extent, which requires a priori gravity field model, the initial state parameters, and the numerical integration of variation equation. Owing to the distinct parametric method of the celestial mechanics method, such as adopting the Kepler orbit parameters, the empirical acceleration, and other dynamical parameters, this celestial mechanics method can achieve a high computational efficiency [Beutler et al., 2010a, 2010b]. The short-arc approach was first presented by Schneider [1968], one of whose basic characteristics compared to the classical dynamic method is that the length of integral arc is significantly shorter. The approach is also based on Newton’s equation of motion formulated as a boundary value problem in the form of an integral equation [Mayer-Gürr et al., 2005]. Any point in each arc could be described as the numerical function of the satellite positions at two boundaries of the integral arc. The gravity field solution based on the short-arc approach is very stable and highly accurate due to the condition number of normal equations being much smaller. However, the coefficient matrix was calculated using the kinematic orbit directly when Mayer-Gürr et al. [2005] solved for ITG-CHAMP01 model, which is a linearization solution that ignores the impact of kinematic orbit error on the coefficient matrix. To consider the effects of kinematic orbit errors, the gradient corrections are applied in GRACE gravity field recovery [Mayer-Gürr, 2006]. However, correcting the satellite orbit errors and resolving the gravity field model in two steps are not theoretically rigorous. Shen et al. [2013] have proposed a modified short-arc approach, in which both the corrections of satellite orbits and range rate observations are solved together with the unknowns of gravity field model, as well as the accelerometer biases, by using a weighted least squares adjustment. In this study, we will elaborate this modified short-arc approach in detail and use it to develop a new time series of monthly solution (namely, Tongji-GRACE01) covering the period January 2003 to August 2011 up to degree and order 60 from the GRACE Level 1B RL02 observations officially released by JPL. Comparing our monthly model with the RL05 models of CSR and JPL as well as the GFZ RL05a model, our model agrees with all of them in terms of geoid degree powers and mass change signals. The rest of this paper is organized as follows. Section 2 will give the theoretical background of the modified short-arc approach. Section 3 presents data processing strategy for solving the Tongji-GRACE01 gravity field model. Section 4 will evaluate the Tongji-GRACE01 model and compare it with the monthly models of CSR RL05, JPL RL05, and GFZ RL05a. Section 5 will estimate the global mass variation signals and the mass change signals in Amazon, Antarctic, and Antarctic Peninsula, to demonstrate the accuracy of the derived gravity models. The concluding remarks will be presented in section 6.
2. Modified Short-Arc Approach The mathematical-physical model of the short-arc approach is given as [Mayer-Gürr et al., 2005; Mayer-Gürr, 2006],
∫
1
rðτ Þ ¼ r0 ð1 τ Þ þ rN ðτ Þ T 2 0 K ðτ; τ ′ Þaðrðτ ′ Þ; u; pÞdτ ′
(1)
and r˙ðτ Þ ¼ ðrN r0 Þ=T þ T
∫0 1
∂K ðτ; τ ′ Þ aðrðτ ′ Þ; u; pÞdτ ′ ∂τ
(2)
where r(τ) and r˙ðτ Þ are the satellite’s position and velocity vectors at normalize time τ, r0 and rN are the satellite position vectors at two boundaries of the integral arc, T is the time interval of the arc, and K is the integral kernel, described as, K ðτ; τ ′ Þ
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¼
τ ð1 τ ′ Þ; τ ≤ τ ′ τ ′ ð1 τ Þ; τ ≥ τ ′
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(3)
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The vector a stands for the force acting on unit mass satellite, including the conservative force ag and the nonconservative force ang, aðrðτ ′ Þ; u; pÞ ¼ ag ðrðτ ′ Þ; uÞ þ ang ðrðτ ′ Þ; pÞ
(4)
and u and p denote the unknowns of gravity field model and the biases of nonconservative acceleration, respectively. The conservative force ag is a function of r(τ), and the nonconservative force is measured by accelerometer and can be expressed as ang ðrðτ ′ Þ; pÞ ¼ CðqÞT ðSf acc þ pÞ
(5)
where C is the transformation matrix from satellite-fixed system to inertial system, q is the attitude measurement, and facc is the nonconservative acceleration measurement in satellite-fixed system. As in developing the EIGEN-6C2 models, the scale of nonconservative force S is not estimated [Dahle et al., 2012], so we calibrate the scale with the value derived by GFZ. The relationship between the range rate measurement ρ˙ ðτ Þ and the velocity vectors of the twin satellite A and B is ρ˙ ðτ Þ ¼ eTAB ðτ Þ ðr˙B ðτ Þ r˙ A ðτ ÞÞ ¼ f ðrA ðτ Þ; rB ðτ Þ; u; pÞ
(6)
where eAB(τ) = (rB(τ) rA(τ))/ρ(τ) and ρ(τ) is the range between satellite A and B; the symbol “” denotes the inner product of two vectors. We introduce the correction vector vr(τ ’) to the satellite’s kinematic (or reduced-dynamic) orbit rk(τ’) and the correction term v˙ρðτ Þ to the range rate as rðτ ′ Þ ¼ rk ðτ ′ Þ þ vk ðτ ′ Þ
(7)
ρ˙ ðτ ′ Þ ¼ ρ˙ k ðτ ′ Þ þ vρ˙ðτ ′ Þ
(8)
By substituting equation (7) into equations (1) and (2), carrying out linearization with respect to the kinematic orbit rk(τ ’) and discretization via numerical integration, we can derive the linear observation equations for satellite’s position and velocity vectors at epoch i as N X rðτ i Þ þ vr ðτ i Þ ¼ ðr0 þ vr0 Þð1 τ i Þ þ ðrN þ vrN Þðτ i Þ T 2 αk K ðτ i ; τ k Þ k¼0 ∂aðrk ; u0 ; p0 Þ ∂aðrk ; u0 ; p0 Þ ∂aðrk ; u0 ; p0 Þ aðrk ; u0 ; p0 Þ þ δu þ δp þ v rk ∂u ∂p ∂rk
rð˙τ i Þ þ vr˙ðτ i Þ ¼
N X ðrN þ vrN Þ ðr0 þ vr0 Þ ∂K ðτ i ; τ k Þ þT βk T ∂τ k¼0 ∂aðrk ; u0 ; p0 Þ ∂aðrk ; u0 ; p0 Þ ∂aðrk ; u0 ; p0 Þ aðrk ; u0 ; p0 Þ þ δu þ δp þ vrk ∂u ∂p ∂rk
(9)
(10)
where u0 and p0 are the priori values of u and p, δu and δp are the unknowns to be estimated, αk and βk are the integration coefficients, and N is the maximum index of epochs in each arc. One can refer to Mayer-Gürr [2006] for the details of the partial derivatives of vector a with respect to u, p and r. Analogously by substituting equations (8)–(10) into equation (6), the observation equation of range rate measurement at epoch i is derived as ∂f δu ∂u N N X X ∂f ∂f ∂f ∂f δpA þ δpB þ v rAi þ v rBi þ ∂pA ∂pB ∂r ∂r i¼0 Ai i¼0 Bi
ρ˙ ðτ i Þ þ vρ˙ðτ i Þ ¼ f ðrA ðτ i Þ; rB ðτ i Þ; u0 ; pA0 ; pB0 Þ þ
(11)
in which the subscripts A and B stand for the GRACE A and GRACE B, respectively. The partial derivatives of vector f with respect to u, p and r can also be found in Mayer-Gürr [2006]. Thus, the observation equations (9) and (11) can be set up at every epoch for the kinematic orbit and range rate observations, respectively. All the observation equations of one arc, i.e., arc j, are briefly expressed as follows: Aj xj þ Bj vj ¼ yj
(12)
where xj is the vector of unknowns to be estimated, including the geopotential coefficients δu and the arc-specific T bias parameters δw j ¼ δpTAj ; δpTBj , vj is the vector of corrections of range rate observations and pseudo observations of both GRACE satellite’s orbits, Aj and Bj stand for the coefficient matrices of xj and vj, and yj is the CHEN ET AL.
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constant vector. Using a weighted least squares adjustment, we can readily derive the normal equation for solving the unknowns as 1 1 T T Aj Bj Qj Bj Aj xj ¼ ATj Bj Qj BTj yj (13) and the corrections
1 yj Aj xj vj ¼ Qj BTj Bj Qj BTj
(14)
where Qj is the variance-covariance matrix of range rate and orbit observations. Since the geopotential coefficients are recovered with multiple arcs of observations, such as 1 month, the arc-specific parameters δwj must be first eliminated. Thus, we briefly rewrite the normal equation (13) as follows: " #" # " # Lu j Nuj uj Nuj wj δu ¼ (15) Nwj uj Nwj wj Lwj δwj 1 where Nuj uj , Nwj wj , and Nuj wj are the block matrices of ATj Bj Qj BTj Aj corresponding to thegeopotential 1 parameters δu and arc-specific parameters δwj, Luj , and Lwj are the subvectors of ATj Bj Qj BTj yj . After eliminating the arc-specific parameters, we obtain the normal equation for solving geopotential unknowns 1 1 Nuj uj Nuj wj N (16) wj wj Nwj uj δu ¼ Luj Nuj wj Nwj wj Lwj Then we merge the normal equations of all arcs together, NJ δu ¼ LJ
(17)
J X here J is the number of arcs for deriving one gravity field model, NJ ¼ Nuj uj Nuj wj N1 wj wj Nwj uj and LJ ¼ j¼1 J X Luj Nuj wj N1 wj wj Lwj . After solving the geopotential unknowns with (17), the arc-specific parameters are j¼1
available with
1 δwj ¼ N wj wj Lwj Nwj uj δu
(18)
Then corrections of observations are computed with equation (14), and the variance of unit weight is calculated by J X
σ 20 ¼
j¼1
vTj Q1 j vj nt
(19)
where n denotes the number of observation equations and t is the number of unknowns. The formal errors σc,s of geopotential coefficients are estimated with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (20) σc;s ¼ σ 0 diag N1 J where diag denotes the operator for taking the diagonal elements from a matrix. Unlike the traditional short-arc approach [Mayer-Gürr, 2006], the modified short-arc approach need not to compute the gradient corrections for the GRACE orbits beforehand, and the boundary position parameters are also not to be estimated.
3. Data Processing Methods 3.1. GRACE Data and Force Models The GRACE Level 1B observations of RL02 version released by JPL, spanning the period January 2003 to August 2011, have been used for estimating our monthly gravity field models. They are the reduced-dynamic orbits with a sampling rate of 5 s, the K band range rate measurements with a sampling rate of 5 s, and the acceleration and attitude data with the sampling rate of 1 s and 5 s, respectively. The nonconservative accelerations are measured by the accelerometers, while the conservative accelerations are computed using force models. Our monthly gravity field solutions use the following force models: CHEN ET AL.
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1. Background gravity model. The ITG-GRACE2010 static gravity field model is used as a background model for correcting the impacts of the coefficients from degree 61 to 160, since our monthly gravity field solution is complete to degree and order 60. 2. N-body perturbations. The perturbation caused by the Sun and the Moon are directly computed. Both the direct and indirect J2 effects for the satellite are taken into account according to the GFZ GRACE Level 2 document [Dahle et al., 2012]. Additionally, the DE421 planetary ephemerides [Folkner et al., 2008] are applied for calculating the position and velocity of both the Sun and the Moon. The third-body gravitational impacts from other planets are excluded, since their effects appear relatively small. 3. Solid Earth tides. The solid Earth tides refer to the International Earth Rotation Service (IERS) 2010 conventions [Petit and Luzum, 2010]. Both the frequency-independent terms in the geopotential coefficients from degree 2 to 4 and the frequency-dependent terms in the coefficients of degree 2 are needed. In addition, the permanent tide in C20 is taken to be 4.1736 × 10 9 [Dahle et al., 2012]. 4. Ocean tides. EOT11a model up to degree and order 80 is taken into account, which contains 18 waves (eight long periodic, four diurnal, five semidiurnal, and one nonlinear waves). The 238 secondary waves are interpolated with admittance theory [Rieser et al., 2012]. 5. Solid Earth pole tides. The corrections for the term C21 and S21 due to the solid Earth pole tides are calculated based on the anelastic Earth model according to the IERS 2010 conventions. 6. Ocean pole tides. The self-consistent equilibrium model [Desai, 2002] up to degree and order 30 is adopted to correct the geopotential coefficients caused by the ocean pole tides. 7. Variations of the atmosphere and ocean are removed by applying the AOD1B RL05 dealiasing products [Flechtner and Dobslaw, 2013] up to degree and order 100, available through the Integrated System and Data Center. Note that the geopotential coefficients of AOD1B RL05 are available as 6-hourly time series up to degree and order 100. Thus, linear interpolation is used for the values of the harmonics at intermediate epochs. 8. General relativistic perturbations are computed according to the IERS 2010 conventions. 3.2. Data Processing Scenario In our data processing scenario, the estimated monthly gravity field model is complete to degree and order 60 with the spatial resolution of about 400 km, the arc length is chosen as 2 h with a sampling rate of 5 s, and the calibration parameters of accelerometer observations are chosen as three biases (along, cross, and radial) per hour (six biases per arc for one satellite). The gravity field coefficients beyond degree and order 60 are fixed at the ITG-GRACE2010 static gravity field model values. The weights of the pseudoorbit observations and range rate observations are taken as constant, which are determined based on their priori accuracies. The priori accuracies for the orbit and range rate measurements are regarded as 2 cm and 0.2 um/s, respectively [Kang et al., 2009; Beutler et al., 2010b]. The reasons for the above data processing scenario are as follows. The spatial resolution of the present GRACE mission is only about 400 km [Tapley et al., 2004] which is very close to degree and order 60 and the CSR RL05 model is also complete to degree and order 60. We also notice that the GFZ RL05a and JPL RL05 models are up to degree and order 90, and the ITG-GRACE2010 monthly model from Bonn University is even complete to degree and order 120. Therefore, in the future version we will work on monthly models up to degree and order 90 or higher. Most of the GRACE observations, except for the accelerometer data, are released with the sampling rate of 5 s, and the time variable gravity models from GFZ, CSR, and JPL are also computed with this sampling rate data [Dahle et al., 2012; Bettadpur, 2012; Watkins and Yuan, 2012]; hence, the accelerometer data should be converted to 5 s beforehand. Unlike the ordinary short-arc approach using 1 h arcs [Mayer-Gürr et al., 2010], we use the arc length of 2 h for our modified short-arc approach, since a longer arc can reduce the impact of orbit errors due to the velocity with the scale factor 1/T shown in equation (2). As proposed by Bettadpur [2009] and used by Dahle et al. [2012] and Mayer-Gürr et al. [2010] in developing the static gravity field model EIGEN-6C2 and ITG-GRACE210S, we only estimate three bias parameters per hour for each accelerometer and do not solve the scale parameters, since the scale parameters are more correlated with secular geopotential coefficients than the bias parameters [Helleputte et al., 2009]. We use the reduced-dynamic orbits in computation, although these orbits certainly contain priori gravity field information [Ditmar et al., 2006; Jäggi et al., 2007]. However, the main contribution to the gravity field solution is the range rate measurements. Figure 1 shows the postfit residuals of the GRACE orbits CHEN ET AL.
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Figure 1. Postfit residuals of reduced-dynamic orbits of (left) GRACE-A and (right) GRACE-B on 1 January 2008.
on 1 January 2008. Since the reduced-dynamic orbits are very smooth and our linearization of observation equations of orbits and range rates are based on this kind of orbits, the residuals shown in Figure 1 are very small. The estimated root-mean-square errors (RMS) for the coordinates (X, Y, Z) in inertial Cartesian system are only (4.73 mm, 2.62 mm, 2.57 mm) and (4.53 mm, 2.67 mm, 2.56 mm) for the GRACE-A and GRACE-B, respectively, which are much smaller than the real orbit accuracy of about 2 cm [Kang et al., 2009]. The RMS of range rate observations derived from the postfit residuals in Figure 2 is 0.215 μm/s, consistent with the results from other groups [Beutler et al., 2010b; Meyer et al., 2012]. For this reason, we determine the weight based on the priori accuracy information of the observations mentioned above. 3.3. Arc Length Determination Longer arcs, such as 1 day arcs, are often used for gravity filed solution in the dynamic approach [Bettadpur, 2012], for instance, the RL05 model from CSR is computed using 1 day arcs. On the contrary, the time variable gravity field model ITG-GRACE2010 from Bonn University is obtained using the short-arc approach and 1 h arcs. Although the CSR RL05 and ITG-GRACE2010 models are developed using different approaches and arc lengths, their geoid degree powers agree well as shown in Figure 3 for the solutions in January 2008, February 2008, January 2009, and February 2009. In order to determine the proper arc length for our modified short-arc approach, we compute the monthly models in January 2008 with 1, 2, and 3 h arcs, respectively, and the results are shown in Figure 4 and the corresponding time costs are given in Table 1. As for Dahle et al. [2012] and Mayer-Gürr et al. [2010], three bias parameters per hour for each accelerometer are estimated in the three solutions. Figure 4 demonstrates that the solutions with 2 and 3 h arcs are not only consistent with each other but also closer to CSR RL05 model than that of 1 h arcs in terms of geoid degree power. However, as shown in Table 1, the computation time using 3 h arcs is almost twice as long as that using 1 h arcs. Therefore, we decide to use 2 h arcs in developing our monthly gravity field models.
4. Tongji-GRACE01 Monthly Gravity Field Models
Figure 2. Postfit residuals of range rate observations on 1 January 2008.
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A new time series of monthly gravity field models from January 2003 to August 2011, called Tongji-GRACE01, available on the International Centre for Global Earth
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Figure 3. Geoid degree powers in January 2008, February 2008, January 2009, and February 2009 between CSR RL05 and ITG-GRACE2010 monthly models (EIGEN-6C2 as reference field).
Model (ICGEM) website (http://icgem.gfz-potsdam.de/ICGEM/), have been computed according to the data processing scenario. Since the variance factor is computed from postfit residuals, the variance-covariance matrices are overly optimistic. Unfortunately, the gravity field solutions are unavailable for 4 months, i.e., June 2003, January 2004, January 2011, and June 2011, due to the poor or insufficient measurements in these months. In order to demonstrate the quality of Tongji-GRACE01 monthly model, we present the root-mean-square errors (RMS) of 2 months (e.g., January 2008 and January 2009) in Figure 5. The results show that the coefficients for the degrees less than 40 and orders less 20 are estimated well with the RMS less than 2 × 10 12. However, the higher degree and order coefficients are relatively worse.
Figure 4. Solutions with different arcs in January 2008 (EIGEN-6C2 as reference field).
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Since the EIGEN-6C2 model developed by GFZ with various gravity observations has been used as a reference field by ICGEM for assessing the quality of gravity field model submitted to the ICGEM website, we also use the EIGEN-6C2 model as a reference field for comparisons. In Figure 6, we show the geoid degree powers (dashed lines) of Tongji-GRACE01 and the RL05 models of CSR, JPL, and the GFZ RL05a model as well as the ITG-GRACE2010 monthly model relative to the reference model EIGEN-6C2, and the
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Table 1. Time Cost of Three Solutions With Different Arcs
formal errors (solid lines) of GFZ Rl05a, JPL RL05, ITG-GRACE2010, and Tongji-GRACE01 models (the formal Arc with 1 h 6.3 error of CSR RL05 is unavailable). Note Arc with 2 h 11.0 Arc with 3 h 20.1 that the ITG-GRACE2010 monthly model is also derived using a short-arc approach by Bonn University, but it is only available for the period August 2002 to August 2009. The geoid degree powers of Tongji-GRACE01 in Figure 6 are very consistent with those of GFZ RL05a, CSR RL05, and ITG-GRACE2010 monthly models, especially with those of CSR RL05 model. The geoid degree powers of the JPL RL05 model indicate relatively larger difference in the degrees from 30 to 60 compared to other four models. However, in the next section we will demonstrate that the time variable signals of JPL RL05 model agree well with other models after decorrelation and Gaussian filtering are employed. Figure 6 also suggests that our model shows the smallest formal error compared to other models, but it just reflects how well our model fits the data we use. To compare the quality of the whole time series of models, we calculate the yearly mean models spanning from 2003 to 2011 (ITG-GRACE2010 yearly models are only available from 2003 to 2008) and show the geoid degree powers in Figure 7. Our yearly mean models in Figure 7 are consistent with those from GFZ, CSR, JPL, and Bonn University, especially with those from GFZ and CSR as well as Bonn University. However, the JPL yearly mean solutions from 2007 to 2010 tend to have more powers at higher degrees from 40 to 60 with respect to the other four models. Arc
Time Cost/h
We plot the C20 time series (effects of atmospheric and oceanic have been removed from GRACE) for the Tongji-GRACE01 and the RL05 models of CSR and JPL, the GFZ Rl05a model and the ITG-GRACE2010 monthly model in Figure 8, as well as that derived from satellite laser ranging (SLR) data [Cheng and Tapley, 2004]. The C20 time series of SLR is also produced by CSR; its force models are consistent with that of GRACE RL05 solutions, including the use of the same atmosphere-ocean dealiasing product. In Figure 8, the C20 time series of Tongji-GRACE01 model is the closest to that of SLR among the C20 time series of all solutions. The mean values and standard deviations as well as the correlation coefficients derived from the differences of the C20 time series of five GRACE solutions and SLR spanning from January 2003 to August 2009 are presented in Table 2, which demonstrates that the C20 from our solution has the least mean and standard deviation and the higher correlation coefficient. The C20 time series of Tongji-GRACE01 model becomes larger after August 2009. However, it should be mentioned here that all the C20 values of the GRACE-based solution are typically replaced with those of SLR in time variable signals analysis, because the determination of C20 from GRACE is problematic, in general [Meyer et al., 2012].
5. Time Variable Mass Change Signals 5.1. Processing Method To demonstrate the quality of our monthly gravity field models, we compare the time variable signals of our models with that of the RL05 models of CSR and JPL, and GFZ RL05a models, respectively. Degree 1 coefficients
Figure 5. Root-mean-square errors of the coefficients of Tongji-GRACE01. (left) January 2008 and (right) January 2009.
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Figure 6. Geoid degree powers (dashed line) and formal errors (solid line) of Tongji-GRACE01 and the RL05 models of CSR and JPL, GFZ RL05a, and ITG-GRACE2010 monthly models. (left) January 2008 and (right) January 2009.
of each monthly solution are taken from Swenson et al. [2008]. The C20 coefficients of all solutions are replaced by that of SLR [Cheng and Tapley, 2004]. Usually, the time variable mass change signals are expressed in the forms of equivalent water height (EWH). The expressions for computing EWH are according to Wahr et al. [1998] and Wahr [2007]. Some filtering techniques, such as the Gaussian smoothing [Jekeli, 1981], the Wiener filter [Sasgen et al., 2006], and the Fan filter [Zhang et al., 2009], can be used to decrease the high-frequency noises of
Figure 7. Geoid degree powers of yearly mean models of Tongji-GRACE01 and RL05 models of CSR and JPL, GFZ RL05a models, and ITG-GRACE2010 monthly models relative to EIGEN-6C2.
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monthly gravity field model. We adopt the Gaussian smoothing filtering with 300 km radius except where otherwise noted, because it is convenient and the result after filtering with the Gaussian smoothing will not change significantly compared with other more complicated filters [Meyer et al., 2012]. Moreover, we use the P4M6 decorrelation filtering (4 degree polynomial fit treating with odd and even orders of the same degrees from 6 to 60) to remove the north-south stripes as Chen et al. [2009]. For the analysis of the time variable Figure 8. The C20 time series of Tongji-GRACE01 and the RL05 models of signals in the Amazon basin, the CSR and JPL, GFZ RL05a, and ITG-GRACE2010 monthly models as well as Antarctic, and the Antarctic Peninsula, satellite laser ranging (SLR). the GIA effect and leakage effect are taken into account. The leakage effects are corrected by two parts, one is the Global and Land Data Assimilation System (GLDAS) terrestrial water storage change [Rodell et al., 2004] and the other is the truncation of GRACE geopotential coefficients and spatial filtering applied to GRACE data which is corrected by the leakage bias estimated with the GLDAS data [Klees et al., 2007; Velicogna and Wahr, 2013]. The mass change of each grid point of 1 × 1° is estimated in terms of EWH spanning from January 2003 to August 2011. The uncertainty includes three parts: the first is the least squares slope (95.5% confidence, correspond to 2 sigma values) by fitting trend, annual, semiannual, and S2 alias sinusoids; the second is the uncertainty of the IJ05_R2 model given by Velicogna and Wahr [2013]; and the last one is the leakage error estimated by using the GLDAS model [Ivins et al., 2013] and leakage biases. 5.2. Global Mass Change Signals After applying the P4M6 decorrelation filtering and 300 km Gaussian filtering, we extract the global mass change signals from Tongji-GRACE01 model and the RL05 models of CSR and JPL, and GFZ RL05a model with respect to the mean model for the period January 2003 to August 2011, and present the results of January 2009 in Figure 9. We can see from Figure 9 that the global mass change signals of the four models are consistent with each other. The annual amplitude of the mass change of total water storage is presented in Figure 10 for the period January 2003 to August 2011. We can find from Figure 10 that the spatial patterns of the four monthly gravity field models show a great agreement with each other. The areas which have the larger annual amplitude of total water storage variation mainly focus on the Amazon basin in South America, the Niger River in West Africa, the Ganges and Yangtze Rivers in Southeast Asia, and the Zambezi basin in Africa. 5.3. Mass Change Signals in Amazon Basin, Antarctic, and Antarctic Peninsula The monthly gravity models have been used to analyze the mass changes in the Amazon basin [Tapley et al., 2004] and the Antarctic [Chen et al., 2006, 2009]. We also choose these two regions to compare the results of Tongji-GRACE01 model with the RL05 models of CSR and JPL, and GFZ RL05a model. The mask of the Amazon basin we select is a rectangular region ([21° S, 5° N], [45° W, 80° W]) except for the ocean area. After truncating at degree and order 60 and applying P4M6 decorrelation filtering and 300 km Gaussian filtering and the leakage effect corrections, the mass change time series derived from the CSR RL05, JPL RL05, GFZ RL05a, Table 2. Statistics of C20 Time Series Between Different GRACE Solutions With Respect to SLR Models CSR RL05 GFZ RL05a JPL RL05 ITG-GRACE2010 Tongji-GRACE01
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Mean
Standard Deviation
Correlation Coefficient
2.27e10 2.57e10 2.14e10 2.82e10 1.23e10
1.17e10 3.87e10 1.30e10 1.48e10 9.63e11
0.4 0.2 0.4 0.3 0.4
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Figure 9. Global mass change signals in January 2009 derived from the RL05 models of CSR and JPL, GFZ RL05a model, and Tongji-GRACE01 model after applying the P4M6 decorrelation filtering and 300 km Gaussian filtering (in EWH).
Figure 10. Annual amplitudes of total water storage mass change from CSR RL05, GFZ RL05a, JPL RL05, and Tongji-GRACE01 models after applying the P4M6 decorrelation filtering and 300 km Gaussian filtering.
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and Tongji-GRACE01 models in the Amazon basin are presented in Figure 11. We can see from Figure 11 that the four mass change time series are very consistent and the annual mass change is the dominant signal in the Amazon basin. The corresponding annual amplitudes of mass variation in the Amazon basin are 27.6 ± 2.5 cm, 27.3 ± 2.5 cm, 27.8 ± 2.4 cm, and 26.0 ± 2.4 cm, respectively. The spatial distributions of annual amplitudes of mass change derived from the four models are shown in Figure 12, in which the patterns are almost the same. Figure 11. Mass changes in terms of equivalent water height in the Amazon River for the period January 2003 to August 2011 from CSR RL05, GFZ RL05a, JPL RL05, and Tongji-GRACE01 models.
The GIA model IJ05_R2 [Ivins et al., 2013] and the leakage bias are considered for analyzing the Antarctic mass changes. The mass change time series of the
Figure 12. Annual amplitudes of mass changes in the Amazon River during the period January 2003 until August 2011 from CSR RL05, GFZ RL05a, JPL RL05, and Tongji-GRACE01 models. The accurate numbers from CSR RL05, GFZ RL05a, JPL RL05, and Tongji-GRACE01 models are 27.6 ± 2.5 cm, 27.8 ± 2.4 cm, 27.3 ± 2.5 cm, and 26.0 ± 2.4 cm, respectively.
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Figure 13. The mass change time series in terms of equivalent water height in the whole of Antarctic, East Antarctic, and West Antarctic for the period January 2003 to August 2011 from CSR RL05, GFZ RL05a, JPL RL05, and Tongji-GRACE01 models.
whole of Antarctica, East, and West Antarctica are presented in Figure 13 for the CSR RL05, GFZ RL05a, and JPL RL05 and Tongji-GRACE01 models. The correlation coefficients of mass changes from the other three models with respect to that from Tongji-GRACE01 model are shown in Table 3, which demonstrates that the four series are also consistent well with each other, especially between the results of Tongji-GRACE01 and CSR RL05 models with the correlation coefficient as large as 0.98. We can see from Figure 13 that the mass loss is very significant in Antarctic, especially in west Antarctic, and the whole Antarctic is experiencing a significant mass loss after 2006. After the GIA effects and the leakage effects as well as their uncertainties are taken into account, the mass loss rates derived from the RL05 models of CSR and JPL, the GFZ Rl05a model and Tongji-GRACE01 model are 0.66 ± 0.28 cm/yr, 0.80 ± 0.28 cm/yr, 0.77 ± 0.29 cm/yr, and 0.68 ± 0.28 cm/yr, respectively, in other words, 90.0 ± 38.2 Gt/yr, 109.3 ± 38.4 Gt/yr, 104.7 ± 39.0 Gt/yr, and 92.7 ± 38.0 Gt/yr, respectively. The four mass loss rates are very consistent, especially between Tongji-GRACE01 and CSR RL05 models. The spatial distributions of the mass loss rates of the four models are shown in Figure 14, and the patterns are also very similar. After removing the annual, semiannual, trend, and the S2 alias, the RMS values of CSR RL05, JPL RL05, GFZ RL05a, and Tongji-GRACE01 models are 1.92 cm, 2.64 cm, 2.09 cm, and 1.86 cm, respectively, which actually reflects the noise level of GRACE solutions, indicating the noise level of our solution is closer to that of CSR RL05. In Figure 15, we present the mass change time series in the Antarctic Peninsula from the RL05 models of CSR and JPL, the GFZ RL05a model, and Tongji-GRACE01 model with 75 km Gaussian smoothing and P4M6 decorrelation filtering. The correlation coefficients between the time series of our model and the other three models are shown in Table 4, which are all over 0.92. After taking into account the GIA and leakage effects as well as their uncertainties, the mass loss rates in the Antarctic Peninsula, derived from the RL05 models of CSR and JPL, the GFZ Rl05a model, and the Tongji-GRACE01 model, are Table 3. Correlation Coefficients of Mass Changes Time Series From 2.35 ± 1.29 cm/yr, 1.95 ± 1.29 cm/ Different Solutions With Respect to Tongji-GRACE01 yr, 2.26 ± 1.31 cm/yr, and 2.53 Solutions Entire Antarctic ± 1.31 cm/yr, respectively, in other CSR 0.98 words, 22.2 ± 12.2 Gt/yr, 18.4 GFZ 0.90 ± 12.3 Gt/yr, 21.4 ± 12.4 Gt/yr, and JPL 0.95 23.9 ± 12.4 Gt/yr, respectively. After CHEN ET AL.
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Figure 14. Spatial distribution of Antarctic mass change rate (a) CSR, (b) GFZ, (c) JPL, and (d) Tongji-GRACE01.
removing the annual, semiannual, trend, and the S2 alias, the RMS values for CSR RL05, JPL RL05, GFZ RL05a, and Tongji-GRACE01 models are 1.78 cm, 1.86 cm, 2.05 cm, and 1.96 cm, respectively. Therefore, the performance of our monthly model is also as good as the CSR RL05, JPL RL05, and GFZ RL05a models in a limited area such as the Antarctic Peninsula.
Figure 15. The mass changes time series in terms of equivalent water height in the Antarctic Peninsula for the period January 2003 to August 2011 from CSR RL05, GFZ RL05a, JPL RL05, and Tongji-GRACE01 models.
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From Figures 13 and 15, we can see that the mass change time series from our models are consistently smoother than those from other models, which may be caused by using modified short-arc approach, since the other models are all
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Antarctic Peninsula
CSR GFZ JPL
0.95 0.97 0.92
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developed from the dynamic approach. Further study is required to understand the reasons for that.
6. Concluding Remarks
We have presented in detail the modified short-arc approach for recovering gravity field models with GRACE data, in which the kinematic or reduced-dynamic orbits are regarded as the pseudo observations. Using a weighted least squares adjustment, the corrections of orbits and range rate observations are estimated together with the geopotential coefficients and the accelerometer biases. Compared to the traditional short-arc approach, the main advantages of our approach are that the gradient corrections of orbits need not be computed beforehand and 2 h arcs are suitable for the gravity field solution. Finally, a new time series of monthly Earth’s gravity field models complete to degree and order 60, named Tongji-GRACE01, have been developed using our modified short-arc approach with the GRACE Level 1B observations from January 2003 to August 2011. The Tongji-GRACE01 monthly models are now available on the ICGEM website (http://icgem.gfz-potsdam.de/ICGEM/).
Acknowledgments The GRACE Level 1B data in this paper are provided by JPL, which can be accessed from their ftp website (podaac. jpl.nasa.gov) through the internet of the first author’s university, the Hong Kong Polytechnic University. And the GRACE monthly models are available at the ICGEM (http://icgem.gfz-potsdam.de/ ICGEM/). This work is mainly sponsored by National key Basic Research Program of China (973 Program; 2012CB957703) and National Natural Science Foundation of China (41474017, 41104002, and 41274035). It is also sponsored by State Key Laboratory of Geodesy and Earth’s Dynamics (SKLGED2013-3-2-Z and SKLGED2014-1-3-E) and State Key Laboratory of Geo-information Engineering (SKLGIE2014-M-1-2). We also would like to thank John Ries and Jianli Chen from the Center for Space Research for their help. We also are grateful to the Editors’ and two anonymous reviewers’ comments for improvement of our original manuscript. Walid Darwisl from Hong Kong Polytechnic University also should be acknowledged for checking and correcting the English grammar in this paper.
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According to the geoid degree powers of the mean monthly models from 2003 to 2011, we can conclude that the Tongji-GRACE01 model is consistent with the RL05 models of CSR and JPL and the GFZ RL05a model in term of geoid degree power. Its C20 coefficients are closer to that of SLR than that of CSR, JPL, and GFZ, but we also recommend to use the C20 from SLR. The power of the coefficients of our model from 40 to 60 is lower than that of JPL. After carrying out Gaussian filtering and P4M6 decorrelation filtering, the global mass change signals derived from the RL05 models of CSR, JPL, and the GFZ RL05a model are also very consistent with those of the Tongji-GRACE01 model. The annual amplitudes of mass changes in the Amazon basin computed from the RL05 models of CSR and JPL, the GFZ RL05a, and our models are 27.6 ± 2.5 cm, 27.8 ± 2.4 cm, 27.3 ± 2.5 cm, and 26.0 ± 2.4 cm, respectively. The mass loss rates in the Antarctic derived from RL05 models of CSR and JPL, the GFZ Rl05a model, and Tongji-GRACE01 model are 90.0 ± 38.2 Gt/yr, 109.3 ± 38.4 Gt/yr, 104.7 ± 39.0 Gt/yr, and 92.7 ± 38.0 Gt/yr, respectively. These four spatial distributions of the Antarctic mass change rate also agree well. After removing the annual, semiannual, trend, and the S2 alias, the RMS of our solution is close to CSR RL05, JPL RL05, and GFZ RL05a models. The correlation coefficients of mass changes in the Antarctic with all correlations larger than 0.90 show that our models are close to CSR RL05, JPL RL05, and GFZ RL05a models, especially to CSR RL05 models. The performance of our solution in the Antarctic Peninsula also supports that our model is close to these models.
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