1. Introduction
After more than 15 productive years in orbit, the U.S./German Gravity Recovery and Climate Experiment (GRACE) satellite mission has provided unprecedented insights into how our planet is changing (mainly the surface mass variations) by tracking the continuous movement of liquid water, ice, and the solid Earth [
1]. Generally, there are two categories of processing methods to obtain the surface mass variations from the GRACE observations. The first one is the mascon approach, directly using the intersatellite range-rate and acceleration observation to estimate the mass changes of mass concentration blocks (or “mascons”) on the Earth’s surface [
2,
3,
4]. These mascon solutions are computed in presence of regularization constraint and no additional smoothing or empirical de-striping or filtering is applied. The second approach uses the monthly gravity field solutions (GRACE Level-2 data) as spherical harmonic (SH) Stokes coefficients provided by GRACE analysis centers [
5]. Due to the errors of the satellite instruments and the imperfections of the background models [
6], the SH coefficients are seriously polluted by errors. Therefore, the direct use of GRACE Level-2 data to measure the surface mass changes of the Earth is affected by different kinds of noises, and if they are not filtered in the post processing, it is difficult to extract useful geophysical signals from the Level-2 data.
The noises among the Level-2 data are mainly manifested in two types. One is the high-frequency (short-wavelength) noise, namely mainly existing in the high degree part of the gravity field. And the other is the north–south stripe error which is shown as north–south stripe distribution features. In order to obtain the pure geophysical signals, it is necessary to isolate the signals from the SH coefficients accurately. The principle of the separation is to obtain the signals as completely as possible (without any attenuation or damage) from the global observations of the GRACE mission. To solve this problem, many algorithms have been proposed.
One type of the algorithms is spatial smoothing filtering method mainly to weaken the impact of high-frequency noise. The first such filter is the degree-dependent isotropic Gaussian filter [
5,
7]. The higher degree of the SH coefficients, the more serious the effect caused by high-frequency noise is. Therefore, by reduction of the weight of high degree of the SH coefficients, this filter can obviously reduce the effects of the shortwave-length noise. Unlike the isotropic Gaussian filter, the non-isotropic filter [
8], such as Fan filter [
9], is constructed depending on both the degree and the order of the SH coefficients, and it has been proved to be a better method. The nature of this type of methods is to achieve a smooth effect by sacrificing spatial resolution of signals. Therefore, the selection of the smoothing radius of the filter is a problem. Only if the smoothing radius is large enough can the noise be eliminated significantly and can the signal be clearly displayed. However, the short-wavelength features of the original signals will be removed, and simultaneously the original spatial resolution will also be destroyed. Moreover, the smoothing radius will change with time and latitude, since the satellite orbit height changes with time, and the data on different latitudes on the Earth will have different spatial resolution. For example, a smoothing radius of 350 km may be appropriate for areas near the equator, but it is too large for denser ground track coverage near the poles. Another problem of this spatial smoothing filtering method is that it will yield signal leakage. Even though researchers have put forward the forward-modeling method [
10] and the scaling technique [
11,
12,
13,
14,
15] to restore the signal, and the effect is obvious, the signals cannot be completely restored.
Focusing on the north–south stripe errors, Swenson and Wahr [
16] developed another type of algorithm called the de-striping filter, which aims to reduce the empirical errors owing to certain even–odd degree correlations found in the GRACE SH coefficients. Duan et al. [
17] empirically designed a moving window filter to reduce these correlations. The core idea of this kind of filtering method is to use the polynomial to fit the correlation errors between certain SH coefficients. Next, by subtracting the fitted values from the original SH coefficients, the residual values are treated as the signals. The de-striping filter for the Stokes’ coefficients with order m or higher using a polynomial of degree n is also known as the PnMm filter, such as P7M7 and P11M11 [
18], P3M10 [
19], P3M6 [
20,
21], P4M6 [
22],and P5M8 [
23]. Different values of the parameter n and m in the de-striping filter can suppress the stripe features to a different extent in different research contexts. However, there are still many stripe features around the equator area after they are by de-striping filter method. After that, a spatial smoothing filtering method such as Gaussian or Fan smoothing is usually required. The so-called correlation and the parameters n and m are all empirical, so they cannot be accurately obtained. Because of a certain width of the moving window, the high order parts of the SH coefficients which have much more serious north–south stripe errors cannot be filtered [
9]. Moreover, some shortwave-length features of the geophysical signals are removed after this filter method, mainly at high latitudes [
16].
In addition to the algorithms above, some other algorithms have also been proposed to filter the observations to obtain useful signals. Chen et al. [
24] constructed an optimized smoother based on a criterion of maximizing the ratio of variance over land relative to that over the oceans to suppress high-degree noise in the monthly gravity field solutions. Empirical orthogonal function (EOF) analysis [
25,
26] was developed to extract signals from the GRACE Level-2 data. Frappart et al. [
27] denoised satellite gravity signals by independent component analysis (ICA).
Even though the filtering methods described above can successfully suppress the high-frequency noise and the north–south stripe errors, they do not take advantage of the prior information. Therefore, it is impossible to obtain the rigorous estimates of north–south stripe errors of SH coefficients and the signals are also suppressed at the same time. However, using some prior information that reflects both the spatial spectral features and temporal correlations among the data, Kurtenbach et al. [
28] obtained daily gravity field solutions based on GRACE Level 1B data. Wang et al. [
29] developed a new stochastic filter technique for the statistically rigorous separation of gravity signals and correlated “stripe” noises in the GRACE Level-2 data. This stochastic filter is able to remove the correlated stripe noise even without a spatial smoothing procedure. However, the stochastic model designed to reflect monthly GRACE correlated errors for all months is not time-varying, so this is not in line with the actual situation.
The study in this article is inspired by Kurtenbach et al. [
28] and Wang et al. [
29], who focused on the construction of stochastic models and used the Kalman filter to improve GRACE gravity field solutions. The correction of the truncation error is also the subject of our research. The main contents of this article are as follows: In
Section 2, the Kalman filter approach is explained, and the prior covariance information used in the Kalman filter and multiple gridded-gain factors to correct the truncation error are given.
Section 3 shows the results, and
Section 4 is the discussion. Conclusions are presented in
Section 5.
3. Results
In order to evaluate the performance of the methods as described above, we used the Kalman filter and multiple gridded-gain factors to process the monthly GRACE Level-2 data. The data sets provided by GFZ, JPL, and CSR start in April 2002 and end in June 2017, but the data sets provided by ITSG-Grace2014 lack a substantial amount of the month, so we selected the monthly data start in January 2003 and end in December 2014 (a total of 133 months) from the four GRACE analysis centers above.
According to our convention, the results are expressed as equivalent water height (EWH) [
5]. Firstly, the mean gravity field of these months was subtracted from the GRACE monthly solutions so that the EWH can reflect the effects of variations in the gravity field. We then replaced the C
2,0 terms with results observed by satellite laser ranging [
37], and the Degree-1 terms obtained by the method of Swenson et al. [
38] in the form of Stokes coefficients. The updated time series for these terms can be downloaded from PODAAC. Lastly, the results of the Kalman filter expressed as EWH Were compared with the traditional filtering results (P3M6+Fan filter, smoothing radius of 300 km) and the mascon products released by CSR. After separating geophysical signals from GRACE Level-2 data, we needed to correct the truncation error. Therefore, we first used simulation data to test the performance of multiple gridded-gain factors proposed in this article.
3.1. The Performance of Multiple Gridded-Gain Factors
We first expanded the synthetic model data to the degree of 60 in the form of harmonic coefficients, and then converted the harmonic coefficients into EWH as the truncated result. Next, the single gridded-gain factors and the multiple gridded-gain factors were calculated respectively using the synthetic model and the truncated result. At last, the truncated result was restored by using these two types of gridded-gain factors. As shown in
Figure 3, we chose two typical grid points in the Amazon basin of time series to show the performance of these two restored methods. Moreover, we also tested this method for the averaged surface mass variations in the Mississippi Basin, the Yangtze River basin, Greenland, and Antarctica.
Table 1 shows the RMS value of the surface mass variation differences between the original synthetic model and the data derived from different approaches in different regions.
Although the signals restored by these two methods were somewhat different from the original synthetic model data in
Figure 3, we could also see that the signals restored by multiple gridded-gain factors were more consistent with the original synthetic model data than those restored by single gridded-gain factors intuitively. This is confirmed from statistical results in
Table 1. Therefore, the multiple gridded-gain factors had better performance in the correction of truncated errors when compared with the single factor method. Therefore, the method of multiple gridded-gain factors was applied to restore signals.
3.2. Monthly Variations of the Separated Signal and Stripe Noise
For the sake of convenience, we show results for six consecutive months.
Figure 4 shows the geophysical signal (surface mass) separated from the GRACE Level-2 data by the Kalman filter and recovered by multiple gridded-gain factors. It can be seen from the results in
Figure 4 that the Kalman filter could effectively extract geophysical signals from the monthly gravity field model. High-frequency noise and stripe noise were successfully suppressed. Simultaneously, the short-wavelength features of the signals were well preserved since no spatial smoothing (such as Gaussian smoothing and Fan filtering) was applied. Seasonal signals in the mid-low latitudes (such as the Amazon basin, Northern India, Central Africa, and Northern Australia) could be intuitively represented. Long-term changes in high latitudes (mainly Greenland and Antarctica) can be clearly seen if the results of all months are observed. With the application of multiple gridded-gain factors, the attenuation and leakage of the geophysical signals due to the truncation error were effectively corrected.
The monthly variations in the separated stripe noise are shown in
Figure 5, in which we can see the temporal and spatial variation characteristics of this noise. From the spatial perspective, the stripe errors are manifested by the linear characteristics of the north–south elongation. Furthermore, the stripe errors in the mid-low latitudes are much more serious than those in the high latitudes. This may be partly explained by the system’s polar orbit. It is important to emphasize that the stripe error varies significantly over time. This also proves that it is not reasonable to adopt the invariant stochastic models of north–south stripe errors and that the time-varying month-by-month stochastic models based on GRACE data quality information in this article are feasible.
3.3. Comparison with a Commonly Used Filter
Figure 6 compares the commonly used filtering method (the P3M6 and smoothed with the 300 km Fan filter) with the method used in this article. The left column is filtered with the common method, and the right column applies the Kalman filter. In order to better show the performance of the post-processing method, we selected three months (February 2003, December 2005, and January 2014) with relatively poor data quality. In addition, multiple gridded-gain factors were not used to eliminate the effect of signal correction for the truncation error. Hence, the signal leakage due to the truncation error using these two methods can be clearly seen in
Figure 6. In the left column of
Figure 6, some stripe noises have not been removed by the common method, primarily in the middle and low latitudes. However, with the Kalman filter, this phenomenon becomes inconspicuous.
It is important to note that signal distortions obtained by the common method are severe and produce many false signals, mainly in high latitudes, such as in the central and northeastern parts of Greenland [
16,
39]. Since the Kalman filter was used without any spatial smoothing, the original spatial resolution was preserved, and it seems that the resolution is higher because this method retains a substantial amount of detail signals. Moreover, the effect of the common filtering on signal attenuation is obvious, especially where the surface mass changes greatly. The signal amplitude is significantly smaller than the signal amplitude separated by the Kalman filter. This is verified by GPS results.
As we know, temporal mass variations at the Earth’s surface cause surface displacements that can be measured by GPS receivers. Therefore, monthly vertical station displacements of the global network of permanent GPS stations can act as independent observations to be compared with the monthly height series derived from GRACE data. The vertical station displacements of the time series used in this article are available at
ftp://sopac-ftp.ucsd.edu/pub/timeseries/measures/ats/. All the GPS time series data used here span more than 5 years and we tried to ensure that these stations (a total of 175) were evenly distributed globally.
For each station’s daily height coordinate time series, a trend was removed and monthly means were computed, in order to correspond to monthly vertical displacements obtained by GRACE data series. The reduction ratios of the WRMS (weighted root-mean-square) values of the GPS observations when the GRACE solutions were subtracted were calculated [
40]. The WRMS reduction ratio reflects the agreement of the GPS and GRACE time series in both amplitude and phase. A value of 1.0 indicates perfect agreement between GPS-observed and GRACE-observed annual plus semiannual seasonal displacements.
Figure 7 shows the WRMS reduction ratio calculated by the GRACE monthly height series derived from the results of the Kalman filter and the commonly used filtering method (the P3M6 and smoothed with 300 km Fan filter), respectively.
Table 2 lists the statistical results. Overall, the height time series with the Kalman filter are more consistent with GPS data than that with the commonly used filter. In addition, 69.7% (122 of 175) of the stations with the Kalman-filtered results have a larger WRMS reduction than those with the commonly used filtered results. Since GPS data reflect changes in a local point, we can say that the Kalman-filtered results reflect more details.
The secular trends and annual amplitudes of the global surface mass variations derived from the P3M6 and Fan (300 km) filter, the Kalman filter, and the mascon method are compared in
Figure 8. Because the GIA effects are removed in the mascon products provided by CSR, the GIA models have to be added back. Note that the mascon solutions have restored the GAD products, representing the atmospheric and oceanic effects over the ocean. So here the effects of GAD are deducted from the mascon solution. In addition to the global results, we also compare the average mass changes of basin scales estimated by the Kalman filter with the commonly used filtered solution and the mascon solution.
Figure 9 compares the average mass estimates at various spatial scales ranging from ∼950,000 km
2 (the Orinoco River Basin) to ∼2,160,000 km
2 (Greenland).
Figure 9a–f are ordered by increasing area.
Moreover, the indexes of the SNR as mentioned in
Section 2.2 are calculated with the results derived from these three methods, and they are 2.45 (P3M6 and Fan filtered), 3.35 (Kalman-filtered), 4.94 (Kalman-filtered and with truncated errors corrected), and 4.50 (mascon), respectively. When compared with the Kalman filter, it can also be seen that the common method has removed a substantial amount of detail as mentioned above. The signals obtained by the common method are attenuated in many places, while the other two methods perform well. When comparing the results with mascon solutions that do not need processing of de-striping or smoothing, the results obtained by the Kalman filter show good consistency in both secular trend and annual amplitude.
4. Discussion
When different temporal modes of mass variations are taken into consideration, the multiple gridded-gain factors method improves the signal recovery performance compared with the single gridded-gain factors method [
14]. However, in some areas, where the temporal modes of mass changes vary simply, such as desert areas, the effect of this improvement will be weakened. Conversely, the more complex mass variations are at time scale, the better performance of the multiple gridded-gain factors method will be. The two grid points in the Amazon basin in
Figure 3 are representative of this. Therefore, the reduced value of RMS is substantial at some grid points. However, in this basin, the reduction in average surface mass variations is smaller than the reduction at these grid points.
It should be noted that there is no signal attenuation and leakage caused by spatial filtering, because the Kalman filter is used to obtain the signal, and the signal recovery here is mainly aimed at truncation error. As we know, the more serious the signal is damaged, the harder it is to recover. Therefore, it is relatively easy to recover the signal here. In order to achieve better effects in a specific application of a certain area, we can divide the time series of mass changes into several periods to restore the signal separately.
In the separation of geophysical signals from GRACE gravity data based on the Kalman filter, the determination of the prior covariance information that reflects the spatial and temporal features of the geophysical signals and the correlated errors is crucial. Obviously, the more accurate the prior information, the more accurate the signal separated by the Kalman filtering. Therefore, we carefully used a variety of geophysical models to synthesize a priori models in order to achieve better results. Since there is no time-varying model to reflect the information of the stripe noise, we designed an approximate method to describe the temporal-spatial characteristics of the stripe error.
After obtaining the signals by Kalman filtering, we compared them with signals derived from the commonly used filter and the mascon solution. The data of a total of 175 GPS stations, which are evenly distributed globally, were used to evaluate the result of GRACE. The WRMS reduction ratio, which reflects the agreement of the GPS and GRACE time series, shows that the Kalman filter can obtain more details regarding geophysical signals than can the P3M6 and Fan filter. The approximate estimate of the SNR is based on the fact that GRACE measurement errors are at approximately the same level over both land and ocean, but the surface mass variability is stronger in the continents than that in the oceans [
24]. We used this estimate to evaluate the signals derived from different methods. Furthermore, we used the average mass changes of different basins to compare the secular trend and annual amplitude of the signal.
5. Conclusions
We used the Kalman filter to separate the geophysical signals from GRACE gravity data with prior information derived from a variety of models and observations. In order to correct the truncation error, a multiple gridded-gain factors method considering different temporal frequencies of the geophysical signal was developed to recover the signal. The results show that this method can considerably reduce the high-frequency noises and correlated errors. Without spatial filtering, the estimated signal preserves its original resolution and retains more detailed information. This is confirmed by the GPS data and the SNR value. Good consistency is shown when comparing the results with mascon solutions in both secular trend and annual amplitude.
The covariance matrix of process noise based on time-varying prior information was designed, and the results of the separated stripe noise prove that this error does vary with time. By increasing the number of observations with an iterative process, the accuracy of the results is improved. Therefore, the Kalman filter can fuse the information of the solutions provided by each GRACE analysis center. Because no spatial smoothing is needed, more details on the separated signal are retained. Without the signal distortion due to spatial smoothing, after using the multiple gridded-gain factors method to correct the truncation error, the method performs satisfactorily. In addition, full covariance matrices in the prior information are not urgently needed, so we can use the diagonal components of the formal errors when the full formal error covariance matrices are not accessible.
These methods can be used as a complete post-processing method for monthly GRACE Level-2 data, and the results can then be used in continental water, ice, ocean mass, and atmospheric mass research. We believe that, as these models become more refined in the future, the results of this algorithm will greatly improve. In application, more prior information will lead to better results.