**3.1 Spherical harmonics solutions**

The Level-2 products are obtained using a dynamic approach, which relies on the Newtonian formulation of the satellite motion equation evaluated in an inertial reference frame having the origin at the Earth's center. The formulated solution is combined with a dedicated modeling of the gravitational and non-conservative forces, which act on the spacecrafts [6]. During the process of data reduction, the known gravitational contributions are removed from observations using a priori information from meteorological and global ocean circulation models as well as the non-gravitational forces that were measured by the on-board accelerometers [10, 11]. The residual values represent mainly the contribution of the continental hydrology and errors of the correcting models in the measured gravity field. More details about the pre-treatment of the GRACE data reduction can be found in [12]. These solutions are provided as monthly or weekly lists of Stokes coefficients, i.e. dimensionless Spherical Harmonic (SH) coefficients of the geopotential [13], up to degree and order 96 or less that correspond to a spatial resolution of 200–300 km [14–17]. The range

**63**

*Continental Water Storage Changes Sensed by GRACE Satellite Gravimetry*

of an ideal resolution for GRACE products for hydrology is discussed in [18]. While the correcting models represent a reasonable dealiasing of high-frequency changes, the errors due to tide modeling remain in the GRACE solutions, especially for diurnal S2 tides [18–22]. These SH solutions are affected by north–south striping, especially dominant in the tropical band where the coverage of the satellite is insufficient mainly because of three reasons including the sparsity of GRACE track sampling in the longitudinal direction due to the polar orbit plane; propagation of systematic errors from the correcting model acceleration [19–21]; and the numerical correlations generated by solving the underdetermined systems of normal equations for the high-degree Stokes coefficients [23]. Average of each Stokes coefficient Cnm and Snm versus time is computed and removed to estimate the time anomalies for each monthly or weekly period. It is necessary to eliminate this noisy effect on the GRACE solutions, which are impacted by short North–South wavelength components. The monthly Stokes coefficients ΔCnm and ΔSnm that have been destriped by filtering have been used in very numerous studies on the evaluation of water storage on lands and the oceans [4, 24–32]. A simple strategy to attenuate the short-wavelength striping of the GRACE solutions consists of computing the weighted average of the solutions taken from the official centers for each period of time, and by considering the a posteriori uncertainty levels of the fitted Stokes coefficients to define these

Maps of water storage anomalies over lands directly based on [34] are:

θ

*n m n*

derived from GRACE data were explicated as total water mass change.

<sup>+</sup> ∆ = <sup>∆</sup> + ∆

2 1 ( ,,) sin ( )cos ( )sin 1

where Δσ is the change in surface density (mass/area), ρw is the density of the water and kn is the Love numbers [35]. The change of surface mass is usually expressed in meters of equivalent water thickness (EWH). All the mass anomalies

Other GRACE solutions can also be obtained by other research centers where different numerical approaches are used to reach temporal resolution of one day to one month in the form of SH coefficients (global approaches) or spatial grids (local or regional approaches). As alternative to the SH approach, which is based on frequency representation instead of pure spatial localization [36], other types of base functions are used to represent surface water mass densities, mass concentration elements or mascons. In this case, water mass anomalies are estimated in specified concentrated surfaces on the Earth's locations. The GRACE mascons have been proposed by several research groups such as Goddard Space Flight Center (GSFC) [37–41], Jet Propulsion Laboratory (JPL) [42, 43] and Center of Space Research (CSR) [44] at the University of Texas, Austin, where they are processed differently. As an instance, the 1° equatorial equivalent sampled mascons developed at CSR are computed by no temporal smoothing and regularization, as they are only based on GRACE information, whereas more recent mascons solutions are derived by using partial derivatives to relate KBRR observations to EWH to be determined [44]. In a second version, the mascons are related to the range rate or the range acceleration using SH that remain truncated at certain degrees and orders, as proposed by [38]. Mascons can also be estimated by post-processing of Level-2 GRACE SH solutions without a direct use of range rate observations (see examples in [44–47]). Global

*e w nm nm nm*

*<sup>n</sup> ta P C m t t S m k*

0 0

= =

 ρ

**3.2 Mascons and regional solutions**

*n*

∞

( ) ( )

<sup>+</sup> ∑∑ (2)

λ λ

*DOI: http://dx.doi.org/10.5772/intechopen.96109*

weights for combination [33].

σ λθ *Continental Water Storage Changes Sensed by GRACE Satellite Gravimetry DOI: http://dx.doi.org/10.5772/intechopen.96109*

*Geodetic Sciences - Theory, Applications and Recent Developments*

of the GRACE mission are listed below [3]:

bands at frequencies of 26 and 32 GHz.

acting on each satellite along three axis.

**3. GRACE data products**

**3.1 Spherical harmonics solutions**

K-band for the KBR system at the desired frequency.

satellite relatively to the positions of fixed stars.

each GRACE satellite in the geocentric reference frame.

The payload was composed of five instruments on-board, the satellite components

• The K-band ranging system (KBR) for inter-satellite distance an accuracy of 10 μm. It uses the phases of carrier electromagnetic waves in the K and Ka

• The Ultra-Stable Oscillator (USO) for generating electromagnetic waves in the

• The SuperSTAR accelerometers (ACC) for accurate measurement of the forces

• The Stellar Camera ASSEMBLY (SCA) for determining the orientation of the

• The Black-Jack GPS receivers and Instrument Processing Unit for providing three coordinate components of the position and the ones of the velocity of

The three official processing centers forming the GRACE Science Data Center (GSDC), i.e. the Center for Space Research (CSR) in Austin, Texas, United States; the GeoFoschungsZentrum (GFZ) in Potsdam, Germany; the Jet Propulsion Laboratory (JPL) in Pasadena, California, United States, produce the Level-1B parameter products and the Level-2 solutions derived from measurements of the GRACE mission. Level-1B products are constituted by the processed positions and velocities, which were measured by the on-board GPS receivers, accelerometers and the accurate K-band measurements of the variations in distance between the two vehicles. Using these measurements, the monthly gravity field models or Level-2 products for continental hydrology are computed. These products are expressed by means of geoid heights and Equivalent-Water Heights (EWH). The latter products are distributed by the GFZ's Integrated System Data Center (ISDC, [8]), and the JPL's Physical Oceanography Distributive Active Data Center (PODAAC, [9]).

The Level-2 products are obtained using a dynamic approach, which relies on the Newtonian formulation of the satellite motion equation evaluated in an inertial reference frame having the origin at the Earth's center. The formulated solution is combined with a dedicated modeling of the gravitational and non-conservative forces, which act on the spacecrafts [6]. During the process of data reduction, the known gravitational contributions are removed from observations using a priori information from meteorological and global ocean circulation models as well as the non-gravitational forces that were measured by the on-board accelerometers [10, 11]. The residual values represent mainly the contribution of the continental hydrology and errors of the correcting models in the measured gravity field. More details about the pre-treatment of the GRACE data reduction can be found in [12]. These solutions are provided as monthly or weekly lists of Stokes coefficients, i.e. dimensionless Spherical Harmonic (SH) coefficients of the geopotential [13], up to degree and order 96 or less that correspond to a spatial resolution of 200–300 km [14–17]. The range

**62**

of an ideal resolution for GRACE products for hydrology is discussed in [18]. While the correcting models represent a reasonable dealiasing of high-frequency changes, the errors due to tide modeling remain in the GRACE solutions, especially for diurnal S2 tides [18–22]. These SH solutions are affected by north–south striping, especially dominant in the tropical band where the coverage of the satellite is insufficient mainly because of three reasons including the sparsity of GRACE track sampling in the longitudinal direction due to the polar orbit plane; propagation of systematic errors from the correcting model acceleration [19–21]; and the numerical correlations generated by solving the underdetermined systems of normal equations for the high-degree Stokes coefficients [23]. Average of each Stokes coefficient Cnm and Snm versus time is computed and removed to estimate the time anomalies for each monthly or weekly period. It is necessary to eliminate this noisy effect on the GRACE solutions, which are impacted by short North–South wavelength components. The monthly Stokes coefficients ΔCnm and ΔSnm that have been destriped by filtering have been used in very numerous studies on the evaluation of water storage on lands and the oceans [4, 24–32]. A simple strategy to attenuate the short-wavelength striping of the GRACE solutions consists of computing the weighted average of the solutions taken from the official centers for each period of time, and by considering the a posteriori uncertainty levels of the fitted Stokes coefficients to define these weights for combination [33].

Maps of water storage anomalies over lands directly based on [34] are:

$$\Delta\sigma(\lambda,\theta,t) = a\_{\epsilon}\rho\_w \sum\_{n=0}^{\overline{\alpha}} \sum\_{m=0}^{\overline{\alpha}} P\_{nm}(\sin\theta) \frac{2n+1}{1+k\_{\mu}} (\Delta C\_{nm}(t)\cos m\lambda + \Delta S\_{nm}(t)\sin m\lambda) \tag{2}$$

where Δσ is the change in surface density (mass/area), ρw is the density of the water and kn is the Love numbers [35]. The change of surface mass is usually expressed in meters of equivalent water thickness (EWH). All the mass anomalies derived from GRACE data were explicated as total water mass change.

### **3.2 Mascons and regional solutions**

Other GRACE solutions can also be obtained by other research centers where different numerical approaches are used to reach temporal resolution of one day to one month in the form of SH coefficients (global approaches) or spatial grids (local or regional approaches). As alternative to the SH approach, which is based on frequency representation instead of pure spatial localization [36], other types of base functions are used to represent surface water mass densities, mass concentration elements or mascons. In this case, water mass anomalies are estimated in specified concentrated surfaces on the Earth's locations. The GRACE mascons have been proposed by several research groups such as Goddard Space Flight Center (GSFC) [37–41], Jet Propulsion Laboratory (JPL) [42, 43] and Center of Space Research (CSR) [44] at the University of Texas, Austin, where they are processed differently. As an instance, the 1° equatorial equivalent sampled mascons developed at CSR are computed by no temporal smoothing and regularization, as they are only based on GRACE information, whereas more recent mascons solutions are derived by using partial derivatives to relate KBRR observations to EWH to be determined [44]. In a second version, the mascons are related to the range rate or the range acceleration using SH that remain truncated at certain degrees and orders, as proposed by [38]. Mascons can also be estimated by post-processing of Level-2 GRACE SH solutions without a direct use of range rate observations (see examples in [44–47]). Global

grids of mascons solutions can be easily downloaded from [48, 49] for CSR and JPL Releases 06, respectively. Note that these latter solutions need to be scaled by a gain factor that varies geographically. A sequential Kalman Filtering (KF) approach for estimating regional maps of water mass changes by progressive integration of daily along-track GRACE geopotential anomalies has been recently proposed by [1, 50]. This iterative Kalman filter procedure has been successfully applied to determine 2° x 2° surface water mass density solutions over continental regions instead of using the SH or mascons representation [51–53].

## **3.3 Spatial resolution and accuracy of the GRACE products on lands**

The GRACE products contribute in continental hydrology research witn a novel information: the terrestrial water storage or integrated water content, i.e. the sum of the water contained in the column from the different hydrological reservoirs: surface water, soil water, groundwater and snow cover. An early study showed an expected measurement accuracy of a few millimeters of EWH in terms of surface density for a reference water density of 1000 kg/m3 , over areas of 400 km by 400 km, this work was based on Land Surface Models (LSM) outputs as soil moisture, evapotranspiration and run-off. It is expected that the presence of noise in the shorter wavelengths affects the TWS retrieval [34]. In addition, errors due to the spectra truncation increase as the area of the studied basin decrease. Based on LSM outputs and the expected accuracy of the GRACE land water solutions, it was proved that the changes in TWS could be detected by the GRACE system if they exceed 1.5 cm of EWH over an area of 200 000 km2 [54]. The accuracy of the GRACE land water solutions was expected to be about 0.7 cm of EWH for a drainage area of 400 000 km2 and 0.3 cm for a drainage basin of 4 million of km2 [55].

Current GRACE Products have a spatial resolution of a few hundred kilometers (around 200 km for the mascons and for the regional solutions, and 330 km for the releases 03 and 05 for a typical degree of truncation of n = 60–90). Errors were estimated to be around 4 cm at the Equator, and decreasing to 1.5 cm in Polar regions due to denser GRACE satellite tracks coverage [56].
