Time-variable gravity determination from the GRACE gravity solutions filtered by Tikhonov regularization in Sobolev subspace

The GRACE mission has provided scientific community Time-variable gravity field solutions with high precision and on a global scale. The GRACE mission was launched on March 2003. This mission consists of two satellites that pursue each other in their orbit. Distance between two satellites in orbit is measured continuously to an accuracy of better than 1 micron using KBR system placed in satellites. As the satellite fly in the gravity field, this distance changes and by monitoring those changes the gravity field can be determined. To reduce non-gravitational accelerations, each satellite has an on-board accelerometer to measure these accelerations (Wahr and Schubert, 2007). Providing profiles of the atmosphere using GPS measurements for gaining knowledge about the atmosphere is another goal of this mission.
One of the products of this mission is GRACE LEVEL-2. This product consists of monthly spherical harmonic coefficients up to degree 120. One application of these coefficients is to determine time-variable gravity field. The time-variable gravity field is then used to solve for the time-variable-mass field (Wahr and schubert, 2007).
A mathematical model for determining the surface density (mass) variations using spherical harmonic coefficients is presented by Wahr et al. (1998). This mathematical model is as follows:
                                                          
Where , , , , ,  ,   ,   are surface density variations, mean earth density, mean earth radius, Love number of degree , GRACE potential changes, fully normalized Legendre functions, degree and order respectively.
Spherical harmonic coefficients from the GRACE are noisy which increase rapidly with increasing degree of geopotential coefficients. In addition, monthly surface mass variations map shows the presence of long, linear features, commonly referred as stripes (Swenson and Wahr, 2006).
Hence, in different methods of filtering it is tried to solve both problems. Filtering of the GRACE gravity solutions has been studied extensively. For some of the recent contributions we refer to Wahr et al. (1998), Chen et al.(2005), Swenson and Wahr (2006), Kusche (2007), Sasgen et al. (2006), , Swensonand Wahr (2011), Save et.al. (2012).
In this paper, for filtering the GRACE gravity solutions, we propose a new way of determining the surface mass change formula under the assumptions considered in Wahr  et al. (1998) by means of Singular Value Expansion of the Newton’s Integral equation as an Inverse Problem.
Let be the potential change caused by just Earth's surface mass change, then:
                                                                        
or in operator form:
                                                                                                  
Where is an integral operator with kernel . Series expansion of the kernel  based on Associated Legendre functions is as follows:
                                              
Now, by means of singular value expansion, singular system for this operator is as follows:
                                                                                                   
where and , ,  are singular values, right singular vectors, left singular vectors, respectively. In terms of singular value expansion, the surface density variation can be written as follows:
                                                                                            
where s are filter coefficients that are determined by regularization methods. In this paper, filter coefficients are determined from regularization methods, such as Truncated SVE, Damped SVE and the Standard and Generalized Tikhonov methods in Sobolev subspace.
The numerical results show a good performance of the method “Generalized Tikhonov in Sobolev subspace”, which effectively reduces the noise and the stripes.