Static and time-variable gravity fields of the Earth, computed from GRACE data by the Celestial Mechanics Approach at AIUB.

U. Meyer, A. Jaeggi, G. Beutler

The static field AIUB-GRACE03S

The multi-annual mean gravity field AIUB-GRACE03S was computed from kinematic positions based on high-low GPS and low-low K-band inter-satellite range-rate observations collected on board the two GRACE satellites between August 2003 and December 2009. The static coefficients were estimated up to degree and order 160 in the course of a generalized orbit adjustment procedure. Secular and periodic variations with annual and biannual frequencies were solved in parallel up to degree and order 30. EGM96 served as a priori gravity field model and no regularizations have been applied. Periods of orbit resonances in autumn 2004 and in 2009 were excluded from the mean field.

AIUB-GRACE03S is available on the website of the ICGEM (ICGEM).

Figure 1: Gravity anomalies calculated on the basis of AIUB-GRACE03S (mgal).

Monthly gravity field solutions AIUB_YYMM_6045

Based on the static part of AIUB-GRACE03S (up to degree and order 160) monthly gravity field solutions up to degree and order 60 were estimated for the timespan from July 2003 to December 2009 (also available during resonance periods with no regularizations been applied). Some gaps remain (November 2003, January 2004 and November 2004) due to data gaps, observations of reduced quality, or processing artefacts.

The monthly gravity field solutions reveal secular variations due to ice mass change over Greenland, Alaska, and Antarctica, as well as Global Isostatic Adjustment (GIA), e.g., in Fennoscandia and northern Canada. Periodic variations are mainly caused by the hydrological cycle and are prominent in river basins like the Amazon. A parametric model of these changes was derived a posteriori from the monthly solutions by representing each gravity field coefficient with a static term, a linear trend, and periodic variations on annual and biannual frequencies. The trends from this model are visualized in Fig. 2, the periodic variations up to degree and order 30 in Fig. 3. These variations estimated a posteriori agree very well with those estimated directly in AIUB-GRACE03S (to a reduced resolution of degree and order 30).

Figure 2: Secular variations in geoid heights (m) from 2003 to 2009. Shown are statistically significant terms up to degree and order 60. At northern latitudes the effects of ice mass loss and isostatic adjustment are well visible. The instantaneous change in gravity due to the Sumatra earthquake 2004 shows up in the trends as an artefact. Over the oceans the noisy longitudinal stripes typical for GRACE are dominant.

Figure 3: Periodic gravity variations in equivalent water heights (-20 cm ... +20 cm). One cycle corresponds to one year. Only statistically significant terms up to degree and order 30 are shown.

The secular and periodic variations estimated a posteriori from the monthly fields were statistically checked for their significance. Only few biannual variations were tested positive (not shown). In contrast annual periodic variations as well as trends can be determined significantly at least up to degree 60 and order 16 (Figs. 4 and 5).

Figure 4: Triangle plot of significance of periodic annual variations per coefficient (in the left half of the triangle are the S-coefficients, in the right half are the C-coefficients). Blue corresponds to high significance, red to insignificance.

Figure 5: Triangle plot of significance of secular variations per coefficient (in the left half of the triangle are the S-coefficients, in the right half are the C-coefficients). Blue corresponds to high significance, red to insignificance.

To get a measure for the accuracy of the single gravity coefficients all variations that could not be modeled by linear or annual/biannual periodic parameters, were interpreted as noise. Therefore the modeled time variations were subtracted from the monthly solutions and the mean errors of the corrected coefficients against the static field determined (Fig. 6).

Figure 6: Triangle plot of the standard deviations per coefficient of the monthly gravity field solutions (secular and annual/biannual variations removed). Resonant orders (around 16, 32 and 46) are well visible. Beyond order 46 the noise dominates the time-variable signal.

As a consequence of the significance tests and the error calibration the monthly solutions were recomputed with a reduced maximum order of 45. The monthly fields were completed to order 60 by the static coefficients of the a priori gravity field AIUB-GRACE03S. The difference degree amplitudes of the monthly solutions with respect to the static field (Fig. 7) show the continuously high quality of the coefficients of moderate order (up to 45) beyond degree 45.

Figure 7: Difference degree amplitudes of the new monthly solutions (to degree 60 and order 45) with respect to the static part of AIUB-GRACE03S.

It is common practice to smooth (e.g., by a Gaussian filter with halfwidth radius between 300 and 700 km) the differences between the monthly solutions of the gravity field and a static field to suppress noise. This leads to a down-weighting of time variations in high degrees (independent of their order). If non-isotropic filters (e.g., Han, DDK1-3, etc.) are used, in particular the variations of coefficients with high orders are damped. In these cases coefficients beyond order 45 are practically confined to their static part and the reduction of the solution space does not pay out. Whenever small scale variations are of interest and only small smoothing radii are used, then the new fields show significantly less stripes (Fig. 8).

Figure 8: Differential signal (geoid heights (m)) between a sample month and the static field. Only orders 46 to 60 were taken into account, smoothed by a Gaussian filter with halfwidth radius 250 km. The stripes have to be interpreted mainly as noise. This noise is removed in the monthly gravity fields AIUB_YYMM_6045.

In Fig. 9 the effect of different filters on a sample month (to full degree and order 60) is visualized. The static part of AIUB-GRACE03S was subtracted and the remaining time variations were scaled to equivalent water height. This transformation leads to a roughening of the spectrum, since high degrees are weighted stronger. The resulting field is dominated by noise (top left), but the interesting signal can be restored by adequate smoothing technics.

Figure 9: Time variable signal in march 2007 (in equivalent water heights (-20 cm ... +20 cm)). If no filter is applied, the signal is obscured by noise (top left). Pellinen smoothing (top right) corresponds to a uniformly weighted mean on the globe. Gauss smoothing (middle left) is a weighted mean, resulting in a isotropic (only degree depending) weighting in the spectral domain. Han (middle right) refined this method by introducing degree dependent smoothing radii, which leads to a non isotropic filter (down-weighting high orders). Kusche finally proposed a regularization towards physically modeled signal, using only a block-diagonal (left) or a full (right) weighting matrix.

A very demonstrative view of the gravity-variations within the time span observed by GRACE is gained, when point values for selected areas are compared. Again the signal has to be smoothed, especially when equivalent water heights are shown. Figure 10 shows results for the Amazon basin and inner Greenland in water height, smoothed by a 500 km Gaussian filter. In the Amazon basin the hydrological cycle of dry and wet seasons is dominant, in Greenland the ice mass loss caused by global warming becomes visible (superimposed by a small seasonal signal due to accumulation of snow).

Figure 10: Point values (blue) of gravity variations (in equivalent water heights (m)) in the Amazon basin and in inner Greenland. The variations were smoothed by a Gaussian filter with 500 km halfwidth radius. A model including trend and annual/biannual variations was fitted to the point values (red).

The monthly gravity field solutions AIUB_YYMM_6045 are available on the website of the ICGEM (ICGEM).