The GRACE satellites [Tapley et al., 2004] have been providing high quality observations of the Earth's gravity field ever since 2002. 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].
 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. . 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.
 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 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.
 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 changemeans 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-Dsurface 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.
 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 4.