Using nonlinear programming to correct leakage and estimate mass change from GRACE observation and its application to Antarctica

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2012jb009480, 2012 Using nonlinear programming to correct leakage and estimate mass change from GRACE observation and its application to Antarctica Jingshi Tang, 1,2 Haowen Cheng, 3 and Lin Liu 1,2 Received 27 May 2012; revised 3 October 2012; accepted 8 October 2012; published 27 November [1] The Gravity Recovery And Climate Experiment (GRACE) mission has been providing high quality observations since its launch in Over the years, fruitful achievements have been obtained and the temporal gravity field has revealed the ongoing geophysical, hydrological and other processes. These discoveries help the scientists better understand various aspects of the Earth. However, errors exist in high degree and order spherical harmonics, which need to be processed before use. Filtering is one of the most commonly used techniques to smooth errors, yet it attenuates signals and also causes leakage of gravity signal into surrounding areas. This paper reports a new method to estimate the true mass change on the grid (expressed in equivalent water height or surface density). The mass change over the grid can be integrated to estimate regional or global mass change. This method assumes the GRACE-observed apparent mass change is only caused by the mass change on land. By comparing the computed and observed apparent mass change, the true mass change can be iteratively adjusted and estimated. The problem is solved with nonlinear programming (NLP) and yields solutions which are in good agreement with other GRACE-based estimates. Citation: Tang, J., H. Cheng, and L. Liu (2012), Using nonlinear programming to correct leakage and estimate mass change from GRACE observation and its application to Antarctica, J. Geophys. Res., 117,, doi: /2012jb Introduction [2] The GRACE satellites [Tapley et al., 2004] have been providing high quality observations of the Earth s gravity field ever since The temporal field has shown scientists numerous geophysical, hydrological and many other processes which are currently taking place on the surface and in the interior of the Earth [e.g., Han et al., 2006; Rodell et al., 2009; Jacob et al., 2012]. Although the temporal gravity field has unprecedented accuracy, it is limited to about 300 kilometers in spatial resolution and therefore the GRACE Level-2 monthly solutions are truncated to degree 60. Besides, significant errors exist in high degree gravity field, which require smoothing before use [e.g., Wahr et al., 1998; Swenson and Wahr, 2006]. 1 School of Astronomy and Space Science, Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Institute of Space Environment and Astrodynamics, Nanjing University, Nanjing, China. 2 Key Laboratory of Science and Technology on Aerospace Flight Dynamics, Beijing Aerospace Control Center, Beijing, China. 3 National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China. Corresponding author: J. Tang, School of Astronomy and Space Science, Nanjing University, 22 Hankou Rd., Nanjing , China. (jstang@nju.edu.cn) American Geophysical Union. All Rights Reserved /12/2012JB [3] Because of the truncation and smoothing in processing temporal field, the gravity signal at one point is leaked into surrounding areas. The smoothing process also attenuates the gravity signal at every point, therefore it is impossible to correctly estimate regional mass change without restoring the gravity signals. Effective methods have been proposed to eliminate leakage. One typical way is to separate gravity signals in ocean and on land in spatial or spectral domain [e.g., Wahr et al., 1998; Shum et al., 2011]. Another method to estimate the mass change with leakage corrected, which inspires the new method proposed in this paper, is the forward modeling technique used in Chen et al. [2007]. The forward modeling method assumes that the apparent mass change observed by GRACE is caused by the mass changes in a set of limited areas. These mass changes are iteratively adjusted so that after being forward modeled with the same preprocesses (decorrelation and smoothing, see section 2.1) the computed apparent mass changes match the GRACE observation. This method has been tested in other cases [e.g., Chen et al., 2009, 2011] and is proved to be successful in estimating regional mass changes. [4] In this paper, the new method assumes that long-term gravity signals observed by GRACE entirely come from mass change on land. The mass change is estimated on a given grid, unlike forward modeling which concentrates on a few selected areas. Land mass change is also forward modeled using the same filters as for GRACE observations and is adjusted to match GRACE-observed apparent mass change over the region. The land mass change can be iteratively 1of13

2 corrected from the observed and forward modeled apparent mass change, but it is difficult to obtain reasonable mass distribution without proper external constraints due to the large number of variables. The additional constraint turns an unconstrained least-squares problem into a constrained one and can be solved with nonlinear (quadratic) programming. With this method, the mass change for any region of interest can be simplified as some routine procedures and can be flexibly applied. Only the GRACE observations and certain post-glacial rebound (PGR) model are required to obtain the true mass change. [5] To better clarify the problem, some terms used in this paper are explained first. The land mass change means the mass change solved from the NLP problem, before PGR is removed. Removing PGR from land mass change yields the true mass change. Both land mass change and true mass change are considered only on land grid points. The apparent mass change means the GRACE-observed field after being preprocessed. It can also be mathematically obtained after smoothing procedures are applied to land mass change. The difference between the observed and computed apparent mass changes is used as feedback to solve for the land mass change. It should be noted that GRACE has no vertical resolution and it is common to assume that mass changes take place in a thin layer on Earth surface [Wahr et al., 1998]. Therefore the true mass change is exactly 2-D surface mass [Chao, 2005] and all regional mass changes or change rates in the results also refer to surface mass, although the word surface may be omitted in some occasions for simplicity. [6] This paper is organized as follows. In section 2, the method will be introduced with details including the data preprocess and the mathematical modeling of the method. A case study is given in section 3, where the method will be applied to Antarctica and verified with different parameters or procedures. The stability of the method will be analyzed based on these tests. Some problems and future works are discussed in section Data and Method 2.1. Data [7] To obtain the long-term surface mass change, the GRACE Level-2 monthly solutions are used. The errors in high degree and order spherical harmonics (SH) are eliminated with a common two-step process. A cubic polynomial is fitted and removed from SH in the first step to decorrelate SH starting from degree 6 [Swenson and Wahr, 2006], which is named P3M6 after Chen et al. [2009]. In the second step, the SH coefficients are filtered and smoothed with an anisotropic Fan filter [Zhang et al., 2009], with averaging radiuses of 300 km for both degree and order. The temporal C 2,0 series from GRACE are replaced by solutions from Satellite Laser Ranging (SLR) [Cheng and Tapley, 2004]. Assuming that the mass change takes place in a thin layer on Earth surface [Wahr et al., 1998], it can be quantified with the temporal variations of equivalent water height (EWH) or surface density, which is then fitted with a bias, a linear trend and 2 periodic terms (annual and semi-annual). The apparent linear rate is the input to subsequent NLP problem. [8] ICE-5G model [Peltier, 2004] is used to remove PGR effect. Since the mass change rate is to be recovered at each grid point, contributions from PGR will be removed from the land mass change point by point after it is obtained from NLP and thus we have the final true mass change Method Mathematical Modeling of the Problem [9] Assuming the land surface mass change (Ds 0 ) causes the apparent mass change observed by GRACE, the EWH can be expanded and the corresponding SH coefficients are computed as follows [Wahr et al., 1998]: ( ) DC ð0þ lm 3 1 þ k l ¼ DS ð0þ 4par ave 2l þ 1 lm P lm ðcos qþ Z 2p 0 cos mf sin mf Z p df 0 sin qdqds 0 ðq; fþ ; ð1þ where r ave is the Earth s average density, k l is the loading Love numbers and Ds 0 (q, f) is the land mass change at longitude f and colatitude q. The expansion starts from degree 2 assuming constant total mass and unchanged geocenter and is truncated to degree and order 60 as for GRACE observations. [10] Applying the same Fan filter to the SH from equation (1), the filtered SH DC lm, DS lm read DC lm DS lm ¼ ð2pw l Þð2pW m Þ ( ) DC ð0þ lm DS ð0þ ; ð2þ where W l and W m are Gaussian smoothing scales [Wahr et al., 1998] for degree l and order m. The decorrelation is not applied because the correlation in high degree and order SH is considered as an artificial error in GRACE observations caused by the orbit geometry of the twin satellites [Wahr et al., 2004] and is not supposed to exist in real gravity field. [11] With the filtered SH, the apparent mass change (Ds ) at any point (q j, f j ) in a selected region can be computed as Ds q j ; f j X 60 ar ¼ ave 2l þ 1 X l P lm cos q j 3 1 þ k l¼2 l m¼0 DC lm cos mf j þ DS lm sin mf j : ð3þ [12] The integral equation (1) over the sphere can be computed as summation over the grid points. For a specific region of interest, the mass change outside the region does not contribute (for the leakage effect, see the explanation of buffer area in section 2.2.2) and is excluded. Therefore the summation reads as follows: ( ) DC ð0þ lm 3 1 þ k l X ¼ DS ð0þ 4par ave 2l þ 1 lm i cos mf i P lm ðcosq i Þ sin mf i lm sin q i Dq i Df i Ds 0 ðq i ; f i Þ ; ð4þ where the subscript i denotes the ith grid point on land and the total grid point number is I. 2of13

3 Figure 1. (a) Grid used in the application. Red points denote land points, while blue and green points respectively denote the included (b =3 ) and excluded ocean points when comparing the observed and computed mass change in equation (10). (b) Selected areas in Antarctica. Surface mass changes in these areas are estimated and listed in Table 2. 3of13

4 (a) (b) Figure 2. (a) GRACE-observed apparent mass change rate (PGR not removed) over the entire region (EWH in mm/yr). Obtained from steps in section 2.1. (b) True mass change rate recovered from NLP (PGR removed), only on land (surface mass loss in Gt/yr). 4of13

5 Table 1. Comparison of Total Antarctic Surface Mass Loss From Various Sources Source Time Span Mass Loss Velicogna [2009] Gt/yr Gt/yr Chen et al. [2009] Gt/yr Gt/yr Jacob et al. [2012] Gt/yr This paper Gt/yr [13] Taking equation (4) back into equations (2) and (3), the apparent mass change at any point j in the region with totally J grid points (land and ocean) is computed as below. X 60 Ds q j ; f j ¼ p X l X I sin q i Dq i Df i P lm ðcos q i Þ l¼2 m¼0 i¼1 P lm cos q j Wl W m cos m f i f j Ds0 ðq i ; f i Þ: ð5þ [14] It should be pointed out that not every grid point in the region of interest needs to be included. Unless some significant geophysical process takes place offshore (like the 2004 Sumatra-Andaman Earthquake as in Han et al. [2006]), signals at ocean grid points far from land are likely to result from various errors instead of leakage effect. In actual computation only land grid points and nearshore ocean grid points (within angular radius b offshore) are included to be compared with GRACE observations. [15] Denoting Ds 0 and Ds as I-dimensional and J-dimensional vectors respectively, F as the J I-dimensional coefficient matrix and F j;i ¼ p X60 X l X I l¼2 m¼0 i¼1 W l W m cos m f i f j ; sin q i Dq i Df i P lm ðcos q i the previous equation is simplified as below Ds ¼ FDs 0 : ÞP lm cosq j [16] For such a multivariable problem, simple leastsquares (LSQ) methods do not always lead to a reasonable solution. Tests show that the LSQ solution to this equation set is likely to fall into some unreasonable local minimum and the recovered land mass change may appear unreasonably drastic. Table 2. Surface Mass Change Rates in West Antarctica, East Antarctica, and Other Selected Areas Area Rate (Gt/yr) West East A B C D E ð6þ Table 3. Estimates of Surface Mass Change Rates (Gt/yr) Using Different Empirical Parameters in NLP a Case ID Parameters Total A B A1 a =3,e = A2 a =4,e = A3 a =4,e = A4 a =3,e = A5 a =4,e = A6 a = 0.1, e = A7 a =8,e = a Units are degree for a and mm/yr for e. a =3,e = 200 is given in Tables 1 and 2 and is used in section 3. The last two lines correspond to loose and tight constraints respectively. [17] To make sure the true mass change is smooth over the land, external constraint needs to be considered. One reasonable constraint is to require the difference between the true mass changes at one grid point and the average of the neighboring points not to exceed a given tolerance. The angular radius a (in degree) of the vicinity and the tolerance e are two empirical parameters to quantify the constraint. Mathematically, if the kth land grid point has N neighboring points numbered k 1, k 2,, k N within the vicinity, the following inequality must be satisfied ðds 0 Þ k 1 h i ð N Ds 0Þ k1 þ ðds 0 Þ k2 þ þ ðds 0 Þ kn e: ð7þ [18] For all the land grid points, the following inequality constraints are obtained jads 0 j e; ð8þ where A is an I I square matrix. [19] In mathematics, the least-squares problem constrained in this way is preferably solved with nonlinear programming (NLP) and it is always convenient to convert inequality constraints into equality constraints with slack variables. Introducing an I-dimensional vector y (0 y 2e), equation (8) is converted into the following equalities ADs 0 þ y ¼ e: [20] Therefore if the GRACE-observed apparent mass change Ds G is computed from monthly solutions, the problem is equivalent to the following nonlinear (quadratic) programming T min f ¼ Ds G FDs 0 Ds G FDs 0 Ds s:t: ðaeþ 0 ð10þ ¼ e; 0 y 2e; y where E is an I I identity matrix. Both the true mass change Ds 0 and the slack variables y are to be estimated. [21] Two points should be noted: i). Recovered surface mass change at single grid point may not be accurate due to limited spatial resolution of GRACE observation, but regional true mass change can be properly estimated by integrating surface masses over the grid points in the corresponding area; ii). The original solutions from equations (1) to (10) are temporal mass changes. They can then be fitted to obtain the linear rate of mass change over the area. Since the linear relation and the integral in equation (1) (or linear ð9þ 5of13

6 (a) (b) Figure 3. True mass change rate (in Gt/yr) using (a) loose (a = 0.1, Case A6) and (b) tight (a =8, Case A7) constraints. 6of13

7 Table 4. Total Surface Mass Change in Antarctica With Different Averaging Radiuses Case ID R ave (km) b( ) Total (Gt/yr) B B B B equation set in equation (6)) are independent and therefore the order can be swapped, this method can be applied to individual temporal mass changes, as well as to the long-term mass change rate Brief Outline [22] To summarize how one can obtain the true mass change with this method, a brief outline is listed below. [23] 1. Choose the region of interest. Note that i). The boundary area is subject to leakage anyway from areas outside your region. If the boundary is in ocean, the leakage from outside can be mild or even negligible. But if the boundary is on land, it is better to start from a larger region including a buffer area, so that the leakage into the boundary buffer area does not contaminate the region of interest; ii). Common rectangular grid is highly inhomogeneous at high latitude, which could pose a potential problem for mass inverse. For high latitude areas, especially polar region, it is recommended to use relatively homogeneous grid for mass recovery. [24] 2. Properly process and fit the GRACE monthly solutions. If the long-term mass change rate is to be recovered, fit the monthly solutions and obtain the apparent rate on the grid (section 2.1). [25] 3. Use the land mass change to compute apparent mass change. The initial guess can be arbitrary. Simply using GRACE-observed apparent mass change from step 2 is fine. [26] 4. Compare the computed and observed apparent mass change, the difference is used to adjust the land mass change. External constraints are needed to make sure that the land mass change is smooth. [27] 5. If the difference between the computed and observed mass change is smaller than a given threshold, or if the difference can no longer be improved, stop the iteration. Otherwise repeat from step 3. This problem is solved with NLP. [28] 6. After the iteration (Steps 3 5) converges, remove the PGR effect from the land mass change at each land grid point to obtain the true mass change. The true mass change can be integrated in the region of interest or over the globe to get the regional or global surface mass change. 3. Application and Results 3.1. Mass Loss Estimate for Antarctica [29] The method is applied to Antarctica to estimate the continental surface mass change. The region of interest is a 30-degree cap covering the South Pole. The GRACE Level-2 monthly solutions are produced by the Center for Space Research, University of Texas at Austin. The coastline is taken from Global Self-consistent Hierarchical Highresolution Shorelines (GSHHS) [Wessel and Smith, 1996]. Sixty-four RL04 GRACE Level-2 monthly solutions from January 2006 to May 2011 (January 2011 is unavailable) are used to compute the apparent mass rate over the region (Figure 2a), following the steps in section 2.1. According to the method, all estimates of land mass change from NLP are subject to Figure 2a and forward modeled apparent mass distributions need to match it as close as possible. The NLP problem equation (10) is solved using the open-source OPTPACK based on a dual algorithm [Hager, 1993]. [30] The grid is constructed in the following way. The 30-degree cap is divided with 30 circles of latitude starting from 89.5 Sto 60.5 S with an interval of 1 degree. Starting from the 89.5 S circle and counting the circles northward, on the ith circle evenly distribute 6i grid points starting from the prime meridian (Figure 1a). The surface area or solid angle that one grid point covers is different on different circles of latitude but equal among the points with the same latitude. [31] The true mass change rate on land grid is plotted in Figure 2b, after PGR effect is removed using ICE-5G model. The rates can be integrated over any region of interest for regional mass change, as long as the size of the region is larger than the limit of GRACE spatial resolution. The total Antarctic surface mass loss can be computed by integrating over the entire cap, which is estimated to be Gt/yr. The error includes the formal errors of monthly SH solutions and the fitting error after NLP converges. The uncertainty in PGR model is not considered. This estimate is a little larger than other GRACE-based Antarctica mass loss, but still agrees well (see Table 1). [32] Regional surface mass change rates can also be estimated. For Area A (Graham Land), the estimated mass change rate is Gt/yr, which generally agrees with Chen et al. [2009] ( 28.6 Gt/yr) and Ivins et al. [2011]. Ivins et al. [2011] combines GRACE and Global Positioning System bedrock uplift data between 2003 and 2009 and estimates a 26 6 Gt/yr mass change rate on Graham Land. Area B (Amundsen Sea Embayment) is losing mass quickly, which is estimated to be Gt/yr. It is larger than the previous estimates using GRACE, such as Chen et al. [2009] ( Gt/yr, with CSR data from 2006 to 2009) and Sasgen et al. [2010] ( Gt/yr, with GFZ data from 2002 to 2008). Sasgen et al. [2010] also used independent Interferometric Synthetic Aperture Radar (InSAR) data and a combination of GRACE and InSAR, the estimates from which are Gt/yr and Gt/yr respectively. [33] Large differences exist between other previous estimates and this paper when the regional mass changes in East and West Antarctica are compared. The estimated change rate in West Antarctica is considerably larger than others. The rate is Gt/yr, double of the previous estimates (about 132 Gt/yr) (e.g., Chen et al. [2009] and Rignot et al. [2008] with InSAR). In East Antarctica it was generally considered in balance [Rignot et al., 2008] or losing mass [Chen et al., 2009], but here it shows that East Antarctica is likely to be gaining mass recently, with a surface mass change rate of Gt/yr. Different PGR models and different area encompassment can contribute to the difference in change rate estimates, but a major portion is likely to result from drastic change of mass balance trend in Antarctica after The differences (including Area B) 7of13

8 Figure 4. True mass change rate (in Gt/yr) using (a) 1000 km-smoothed apparent rate (b =8, Case B4) and (b) 100 km-smoothed apparent rate (b =2, Case B2). 8of13

9 Table 5. Surface Mass Change Rates From Recover-and-Fit Steps Area Rate (Gt/yr) Area Rate (Gt/yr) Total B West C East D A E will be further discussed in section where the time series are recovered Stability of the Method Under Different Conditions [34] The previous section shows that this method can give reasonable results with certain parameters and procedures. It is also necessary to know whether this method is stable when the parameters or procedures change. In this section, some parameters are changed and this method is applied to Antarctica again to see how the results respond accordingly. The case above in section 3.1 will be referred to as the original case hereinafter Different Parameters in NLP Constraint [35] To assure smoothness in the map of surface mass change rate, two parameters are introduced in the constraint. One is the size of the vicinity a in which the constraint for a point is applied. The other is the tolerated difference e (from equations (7) to (10)) to describe the smoothness. In the original case, the vicinity is defined as a 3-degree area and the tolerance e is 200 (in the unit of mm/yr for EWH rate). The first five lines in Table 3 (Cases A1 A5) show different estimates of Antarctic surface mass changes with different a and e. The results indicate that different parameters cause little differences between different NLP solutions. [36] To further test this method, another two extreme cases (Cases A6 A7) are computed. One uses a = 0.1 and the other uses a =8. These two cases represent very loose (constraint is only effective in an area smaller than the grid resolution) and very tight constraints respectively. The results are listed in the last two lines in Table 3, where we can see some differences from previous cases. One difference is that, with loose constraint, the estimated rate in Area B is a little smaller while the total surface mass change in Antarctica does not change much. This is mainly because the large gravity signals become more isolated when recovered with loose constraint, thus the regional mass change is more sensitive to the shape of the area. If we extend the latitude range of Area B from 73 S 77.5 S to 73 S 80.0 S, a larger region to contain more large signals in this area, the loosely constrained result becomes Gt/yr. Now the surface mass change in extended Area B in the original case is Gt/yr, still fairly close to Case A6. The other difference is the large error in total mass change with loose constraint. This is because the errors are currently estimated by scaling the apparent formal errors land masschange with a factor of apparent mass change. The solution under loose constraint is much less smooth, which in turn would lead to large scaling factors. The figures for Cases A6 and A7 are plotted in Figure 3. The different color scale used under loose constraint (A6) indicates that large signals are much isolated in this case, but from the results in Table 3 we find that it does not degrade the regional mass estimates and they agree quite well with other cases. [37] The extension of Area B also shows how this method works flexibly when the area of interest changes. In fact, as long as the grid is not changed, the NLP does not need to be computed again. We can directly use the available surface mass distribution over the grid and include the grid points as desired when integrating the surface mass changes Different Averaging Radiuses [38] The original case uses 300 km-fan-filtered field as input. Now two more examples are shown to test the method under other extreme conditions. The temporal fields are filtered with 100 km and 1000 km averaging radiuses respectively. Although both cases are rarely used, they are simply introduced to test the stability of the method. The final results are listed in Table 4. [39] Since the areas affected by leakage are subject to averaging radius, they need to be adjusted when the averaging radius is changed. The parameter b is used to indicate the range of the buffer area (see section and Figure 1a), which is 3 degrees in the original case. Table 4 shows the results where b remains 3 degrees for both 100 km and 1000 km, as well as the results where b is correspondingly decreased or increased for 100 km and 1000 km respectively. With the tests, following conclusions can be drawn: [40] 1. The size of the buffer area b needs to be adjusted according to the averaging radius. NLP cannot reach a proper estimate with an inappropriate b. [41] km is too large even for isotropic Gaussian filter without decorrelation [Chen et al., 2005]. The temporal field is highly smoothed at the cost of details. Although the statistics of Case B4 shows the recovered mass change well matches the GRACE observation (fitting RMS = 0.47 mm/yr), the lack of details still prevents the recovered mass change from correctly revealing the true mass distribution (Figure 4a). [42] 3. Using 100 km averaging radius cannot sufficiently remove errors in temporal fields. But thanks to the details preserved in the temporal field, adjusted buffer area can help to dramatically improve the solution. Recovered surface mass change rates in Areas A and B are Gt/yr and Gt/yr respectively, which are consistent with other previous cases. It is clear that using 100 km averaging radius is more favorable than 1000 km (Figure 4b) Recover and Fit [43] All previous test cases are based on a fit-and-recover procedure given in section 2.1, where the apparent rate is obtained from temporal solutions and serve as the input into NLP. In this section, the test is carried out in different recover-and-fit steps, where the temporal EWH series are first recovered from apparent temporal EWH and then fitted to get the land mass change rate. Considering that the temporal mass over the region could change more severely than its linear rate, the tolerated difference in the constraint is allowed larger in this test. In this case, e is set to be 600 mm/yr while the other parameters remain the same as the original case (section 3.1) and the results are listed in Table 5. The results are in good consistency with the previous values. Figure 5a shows the distribution of surface mass change rate over Antarctica. As a comparison, the original e = 200 is also tested. Consistent with previous analysis, the estimates are almost the same (e.g. total surface 9of13

10 Figure 5. True mass change rate from recover-and-fit steps with (a) e = 600 mm/yr and (b) e = 200 mm/yr. 10 of 13

11 Figure 6. Time series of recovered monthly mass changes for (a) Area A (red square), Area B (blue triangle), total Antarctica (black circle) and (b) East (circle) and West Antarctica (triangle). PGR is not removed from temporal solutions. Linear fit to either East or West time series is in two segments: and of 13

12 mass loss is Gt/yr) only except the exact mass distribution (Figure 5b). [44] Figure 6 shows the time series of land mass changes (NLP solutions) before PGR is removed (Figure 6a for Areas A, B and overall Antarctic continent; Figure 6b for East and West Antarctica), where we find that mass balance trends in both East and West Antarctica Ice Sheets drastically changed after In East the trend changes from losing mass to gaining mass while in West the mass loss becomes faster. The mass loss acceleration is also observable in Area B in Figure 6a. Due to the nature of PGR, the uplift rate can be considered consist over just a few years. So although PGR is not removed in the apparent mass time series, the change in the apparent rate can also be regarded as the change of actual mass balance rate. For the two ice sheets, the mass accumulation in the East and mass loss acceleration in the West since 2009 cause the differences between the previous estimates and the results in this paper. 4. Discussion [45] This paper introduces a new method for gravity recovery using nonlinear programming. The method is an improvement over present forward modeling technique. It is able to recover surface mass change on a given grid and flexibly estimate regional surface mass change. This method is applied to Antarctica as a case study and is verified with several tests. The results are in good agreement with other studies using different techniques and the method is proved stable under different conditions. [46] This method requires input from the apparent temporal mass changes or their linear rate over the grid and solves for the recovered temporal mass or the change rate. Two empirical parameters are needed in NLP to constrain the solution. Although different constraints affect the exact surface mass distribution over the grid, tests show they affect much less on estimated regional surface mass changes. Extreme small and large averaging radiuses are also tested. Although they are rarely applied and are not meant for real cases, it is still possible to recover surface mass change with small averaging radius. Using large averaging radius should be avoided since too much details have been compromised. Changing the order of fit and recovery is shown to have negligible effect on estimating regional surface mass change. [47] The constraint proposed in this paper is intended to assure smoothness of mass distribution. There can be other strategies for this purpose. However, complicated constraints tend to consume more CPU time when solving the NLP problem. The current strategy equation (9) requires modest computing efforts and the iteration quickly converges with the open-source OPTPACK in a minute or two on a general laptop. [48] Grid distribution is another problem. Although currently no further tests are given in the paper, one may naturally ask whether less grid points in mass recovery require weaker constraints. Generally we can expect a positive answer based on the mathematical mechanism. However, in current method GRACE-observed apparent mass change and recovered true mass change share the same grid and the difference over the grid between the observed and the computed is the critical feedback to correct land mass change through iterations. Reducing the grid points will weaken the effect of the performance index f in equation (10). Preliminary tests show that simply reducing the grid points will degrade the results although the fitting RMS from NLP seemingly becomes smaller. However, if the grids for holding GRACE observation and NLP solutions are separated, it is possible to reduce the constraints by reducing the grid points only for mass recovery. The original forward modeling method can be considered as a similar example that uses very few grid points (a limited number of areas) for mass recovery but a 1 1 rectangular grid for GRACE apparent field. Less grid points may reduce the resolution and are not favorable for visually observing how the surface mass changes over the area, nevertheless it works well to estimate regional surface mass change. [49] The last topic is about the application of this method to the global surface mass change recovery. A major difference between regional and global mass recovery is the number of variables in NLP. With a dense grid over the globe, there would be numerous variables in NLP, which considerably slow down the computation. This problem can be settled in two ways: i) Using high performance machines or clusters to directly solve this CPU-intensive NLP problem. Some parallel computing techniques can also be involved; ii) Adopting the method in the previous paragraph and using a relatively sparse grid for surface mass recovery. If a high resolution map of surface mass distribution is not a concern, the grid points only need to cover certain areas with major mass changes. Another problem in global recovery is that in the global case true geophysical signals could exist offshore and they cannot just be ignored. Coseismic and/or postseismic deformation from major offshore earthquakes have been reported observable for GRACE [e.g., Han et al., 2006; Matsuo and Heki, 2011]. A simple approach is that we examine the gravity signals over the globe and wherever a true geophysical signal is identified we mark this point/area as land in the mask and recover the surface mass change accordingly. [50] Acknowledgments. The GRACE Level-2 monthly solutions (RL04) and the C 2,0 series from SLR are produced by CSR, University of Texas at Austin. The Antarctica coastline (GSHHS v2.2.0) is downloaded from NOAA database and the PGR model ICE-5G is provided by W. R. Peltier. The authors also would like to thank W. W. Hager from University of Florida for making OPTPACK publicly available and offering help with it. 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