[1] Gravity fields produced by the Gravity Recovery and Climate Experiment (GRACE) satellite mission require smoothing to reduce the effects of errors present in short wavelength components. As the smoothing radius decreases, these errors manifest themselves in maps of surface mass variability as long, linear features generally oriented north to south (i.e., stripes). The presence of stripes implies correlations in the gravity field coefficients. Here we examine the spectral signature of these correlated errors, and present a method to remove them. Finally, we apply the filter to a model of surface-mass variability to show that the filter has relatively little degradation of the underlying geophysical signals we seek to recover.

[3] Each gravity field is comprised of a set of spherical harmonic (Stokes) coefficients, complete to degree and order 120. Spatial averaging, or smoothing, of GRACE data is necessary to reduce the contribution of noisy short wavelength components of the gravity field solutions. A number of methods have been proposed to smooth GRACE data: an isotropic Gaussian filter [Wahr et al., 1998], an anisotropic filter based on the calibrated error spectrum [Han et al., 2005], and optimal filters based on a priori estimates of signal and measurement error variances [Swenson and Wahr, 2002; Seo and Wilson, 2005]. However, none of these methods accounts for correlated errors in the data.

[4] A map of surface mass anomalies computed from unsmoothed GRACE data will be dominated by errors in the short wavelength (high degree) Stokes coefficients. The most obvious characteristic of such a map is the presence of long, linear features, commonly referred to as “stripes”. As the level of smoothing is increased the amplitude of the stripes decreases, until geophysical signals become apparent, typically at averaging scales of a few hundred km or larger [Chen et al., 2005].

[5] The presence of stripes indicates a high degree of spatial correlation in the GRACE errors. Here, we identify the spectral signature of the correlations between spherical harmonic coefficients that is manifested in the striped patterns seen in GRACE maps. We then design a filter to remove correlated errors in the Stokes coefficients, and apply the filter to GRACE data. Finally, we apply the filter to a model of surface-mass variability to estimate the extent to which the filter degrades the geophysical signals of interest.

2. Correlated Errors

2.1. Spatial Domain

[6] Because of the absence of suitable ground-truth observations, preliminary validation of GRACE data has taken the form of comparisons between GRACE-derived and model-derived estimates of surface-mass variability. Studies such as Wahr et al. [2004] and Tapley et al. [2004b] have shown good agreement between estimates of water storage changes from GRACE and those from models. Error analyses based on internal consistency tests provide model-independent upper bounds on the errors in the GRACE fields; these analyses indicate root-mean-square errors of approximately 1 cm for surface-mass estimates smoothed with a 1000 km Gaussian filter [Wahr et al., 2004], and 3 mm for geoid-height estimates smoothed with a 600 km Gaussian filter [Tapley et al., 2004b].

[7] However, inadequately smoothed GRACE surface-mass estimates contain significant striping [Ramillien et al.,2005; Chen et al., 2005]. These stripes are classified as errors based on geophysical arguments. For example, the large zonal ocean bottom pressure gradients implied by these stripes would indicate an extremely large meridional transport of water unseen in nature (V. Zlotnicki, personal communication, 2004).

[8]Figure 1 depicts a monthly surface-mass anomaly derived from GRACE, smoothed with Gaussian filters of varying halfwidths. Figure 1a shows a map when no smoothing is applied, and Figures 1b–1d show the result of smoothing with increasing filter halfwidths. Note that the right side scale is ten times smaller than the left side scale. As the halfwidth of the filter is increased, the amplitude of the stripes decreases. At spatial scales of around a few hundred km, expected geophysical signals can be seen in regions such as South America, Africa, and South Asia. After filtering with a 750 km Gaussian (Figure 1d), most continental areas appear to have a signal-to-noise ratio greater than 1, while a few places, such as in the Pacific Ocean, still appear to have unphysical mass anomalies.

2.2. Spectral Domain

[9] The spatial correlations seen in these maps indicate corresponding correlations in the spectral domain. The problem is to determine which coefficients are correlated. An examination of Stokes coefficients for a particular order gives insight into the problem. The first column of Figure 2 shows Stokes coefficients, C_{lm}, as a function of degree (l) for orders m = 0 through 3. The second column of Figure 2 shows the same Stokes coefficients plotted separately as functions of even and odd degrees. A visual inspection reveals no obvious correlations between Stokes coefficients as a function of degree. However, at greater orders, it is apparent that certain coefficients are correlated. In Figure 2c, orders m = 15 through 18 are plotted. It can be seen in Figure 2d that Stokes coefficients of even or odd parity vary fairly smoothly. Even and odd coefficients do not appear correlated with one another. An examination of Stokes coefficients of all orders shows that this behavior begins at approximately m = 8, and can be seen in all higher orders. An examination of coefficients for a particular degree reveals no obvious correlations as a function of order.

3. Correlated-Error Filter Derivation

[10] To determine the connection, if any, between these apparent correlations in the spectral domain and the stripes seen in the spatial domain, we now derive a filter designed to isolate and remove smoothly varying coefficients of like parity. To do this, we smooth the Stokes coefficients for a particular order (m) with a quadratic polynomial in a moving window of width w centered about degree l,

where C_{lm}^{ce} is the smoothed Stokes coefficient, Q_{lm}^{i} is the degree i coefficient of the polynomial fit and p is the order of the polynomial, in this case, p = 2. A similar expression is used for S_{lm}^{ce}.

[11] The polynomial coefficients are obtained by least-squares according to

where

[12] Note that the summation over degree (n) includes only those terms of the same parity as l; that is, if l is odd, then only odd degrees n are summed, and similarly when l is even.

[13] Combining equations (1)–(3), one can express each new coefficient, C_{lm}^{ce}, as a combination of the original coefficients,

where the filter, Λ_{lnm}, is defined by

[14] The spatial representation of the filter can be obtained via

where

[15] Λ(θ, ϕ, θ′, ϕ′) describes the relative contribution of each point (θ′, ϕ′) to the weighted average at a point (θ, ϕ).

4. Filtering GRACE Data

[16] Having constructed the filter, we can now examine its effect on a typical GRACE monthly anomaly. Figure 3 shows the filter applied to a GRACE monthly anomaly, converted to mass. Figure 3a is the filtered anomaly field. Figure 3b shows the residual field produced when the filtered field is removed from the original field. The filtered field in Figure 3a is dominated by stripes, and the residual field in Figure 3b shows a significant decrease in stripes, although the filter appears less effective near the equator. Figure 3c shows the filtered field shown in Figure 3a after a Gaussian filter of 500 km has been applied. This figure is also dominated by stripes. Figure 3d shows the residual field Figure 3b, also filtered by a 500 km Gaussian. In this case, little striping is evident, while the expected geophysical signals are apparent.

[17]Figure 4 shows the filter used to create the residual field shown in Figure 3d; that is, Λ′ = G(I − Λ), where I is the identity matrix, and G is the Gaussian smoother. Figure 4a shows that for a particular point (θ, ϕ), the filter is localized about (θ′, ϕ′), much like a Gaussian. A closer look reveals the spatial characteristics that allow the filter to preferentially remove striping. Figure 4b shows the filter with a scale ten times smaller. White contours represent offscale values. For reference, a black circle corresponding to the half-maximum of a 500 km Gaussian filter is plotted. To the immediate north and south of the filter's center, negative lobes exist, while to the east and west, positive lobes exist. These side lobes will act to offset the large positive central lobe of the filter by reducing the contribution of a central stripe (via the negative lobes) while increasing the contribution of neighboring stripes which are out of phase with the central stripe. Thus, a filter designed to remove correlations between spectral degrees of like parity acts in the spatial domain by differentiating between signals which are correlated north-to-south (noise) and those which are more isotropically correlated (geophysical signals).

5. Testing Filter on a Model

[18] The correlated-error filter clearly reduces the presence of stripes in the GRACE gravity fields, but it may remove real signals as well. An ideal filter would remove all striping, while leaving geophysical signals untouched. We cannot directly determine the level of signal reduction that occurs when the filter is applied to GRACE data, but we can obtain an indirect estimate by applying the filter to models of surface mass variability.

[19]Figure 5 is similar to Figure 3 except that the filter has been applied to model output. The model is comprised of total water storage from the Noah land surface model, forced by output from the Global Land Data Assimiltion System (GLDAS) [Rodell et al., 2004] and ocean bottom pressure from a JPL version of the Estimating the Circulation and Climate of the Ocean (ECCO) model [Lee et al., 2002]. The Noah model [Ek et al., 2003] was forced with observed precipitation and solar radiation, and produced estimates of soil moisture (2 m column depth), snow, and vegetation canopy surface water. The ECCO model was forced with NCEP reanalysis winds and thermal and salinity fluxes. To mimic the GRACE de-aliasing process, the output of a barotropic ocean model [Ali and Zlotnicki, 2003] was removed from the ECCO output.

[20]Figure 5a shows that the correlated-error filter does effect geophysical signals. Some short-wavelength features are removed from the model data, mainly at high latitudes. Fortunately, the filtered field represents a small fraction of the residual field shown in Figure 5b; the spatial rms is reduced from 36.4 mm to 33.2 mm. Because actual GRACE data will still require smoothing to diminish the influence of errors which are not correlated in this way (i.e., random or other systematic errors), Figure 5c is a more pertinent example of the filter's effect. After a Gaussian filter of 500 km is applied, little signal is present in the filtered field relative to the residual field shown in Figure 5d; the spatial rms is reduced from 26.9 mm to 24.9 mm. This implies that the filter is quite orthogonal to the geophysical signals at these spatial scales.

6. Discussion

[21] We have described a filter that preferentially selects and substantially removes the spatially correlated errors (stripes) present in GRACE data. To quantify the filter's effects, we follow Wahr et al. [2004], who used the spatial average of the rms about the annual cycle as a conservative estimate of the GRACE errors as a function of Gaussian halfwidth. Using this metric, the average error in a 1000 km Gaussian average is reduced from 18.6 mm (unfiltered) to 14.5 mm (filtered), and from 41.9 mm to 22.0 mm when a 500 km Gaussian is used. These estimates are conservative because there are contributions from non-annual signals. Removing the portion of the variance predicted by the modeled signal gives error estimates of 15.1 mm (unfiltered) and 9.6 mm (filtered) for a 1000 km Gaussian and 39.5 mm and 17.0 mm for a 500 km Gaussian. Thus, approximately 3/4 the variance in the GRACE errors at 500 km resolution can be attributed to spatially correlated errors, and more significantly, these errors can be removed from the data. Furthermore, the application of a Gaussian post-filter is best-suited to an isotropic error spectrum, which is not the case for the filtered fields. For example, the residual signal near the equator in Figure 3c is mainly due to the near-sectorals (where l ∼ m), which are not entirely removed by this filter. A post-filter designed to de-weight their contribution may provide a more accurate estimate of the residual errors.

[22] Ideally, the source of the correlated errors will be determined and incorporated into the gravity field determination process. In the interim, the correlated-error filter provides a tool to extract much more information from the GRACE gravity fields.

Acknowledgments

[23] We wish to thank Srinivas Bettadpur and Richard Eanes for their insight into the gravity field determination process. This work was partially supported by NASA grant NNG04GF02G and NSF grant OPP-0324721 to the University of Colorado.