#### 3.1. The Closed-Loop Simulation Environment

In order to perform the closed-loop simulations, we employ the closed-loop mission simulator described by [

18] to simulate GRACE-like observations and to calculate monthly GRACE-like time variable gravity field solutions. Acceleration differences along the line of sight between the two satellites serve as observations. GRACE-like white noise with standard deviation of

$2\times {10}^{-9}\mathrm{m}/{\mathrm{s}}^{2}$ is added to the observations as stochastic error on the level of accelerations. A discussion of different noise models, such as white noise and colored noise, can be found in [

14]. This GRACE-like white noise can be accounted for being a good first approximation for the stochastic error budget. This is confirmed by the matching magnitude of the formal error degree root mean square (RMS) and the degree RMS of the error derived from residual ITSG-Grace2014 solutions from spherical harmonic (SH) degree 20 onwards. The input data for the observations are almost 12 years of 6 hour hydrological, ice, and solid earth signal (HIS) data from the ESA Earth system model [

19]. The spherical harmonic degree of expansion used within the simulations is 120. To simulate non-tidal atmospheric and ocean signal (AO) de-aliasing errors, AO observations are added with a weighting of 10% and 20% to the HIS components at the level of normal equations. The choice of 10% and 20% is derived from the investigations presented by [

20]. Another option, not pursued within this study, would be the use of AO errors described in [

21]. Ocean tide (OT) de-aliasing errors are accounted for by ocean tide model differences between FES2004 [

22] and EOT08a [

23] in the same way as the AO de-aliasing errors. The AO and OT error budget are considered as coarse upper bounds of error budget contributions. Other key inputs for the simulations are the orbits. For the presented study we use GRACE orbits from [

24] for the period between January 2003 and November 2014. Obviously the timeframe of the orbits and the Earth model do not match. For the investigations regarding filter design and performance this does not cause any disadvantage. The simulated time series just contains geophysical signal from another decade but the conclusions drawn are nevertheless meaningful and valid as if the timeframe was matching because the orbit positions are not used as observations, but only the gravitational acceleration differences computed are at the orbit positions from the underlying time variable gravity field.

Main outputs from the simulations are the monthly sets (timeframe is defined by the respective GRACE orbits) of SH coefficients by solving the corresponding full normal equation systems (NEQs). The input N in Equation (1) is derived from the NEQs of the simulation runs.

The closed-loop simulations are run for each of the 143 months of the timeframe investigated. The different post-processing strategies are compared on the basis of the residuals with respect to the true reference, which is the mean HIS signal of each individual month.

#### 3.2. Determination of a Favorable Filter Design

With the closed-loop simulator the effect of employing different populations of the error covariance matrices and different designs of the signal variance matrices, building the filter matrix ${W}_{\alpha}$, are tested. They are for $N$ (cf. Equation (1)) in particular full, order-wise block diagonal, and diagonal population. For $M$ the effect of using signal variance composed by the true reference for each month, the variance of the annual mean (calendar year), a calendar month mean, and an overall mean (in time) are evaluated. Monthly mean HIS signal of the Earth model is the basis for deriving the signal variance used within the simulation runs. The true reference is rejected for further application since it is not known in real world environments and since applying the same signal in a circular way is not advisable. Accordingly, more generalized representations of the signal variance are applied for further processing. In addition, the respective signal variance is not applied directly, but as a degree dependent approximation, leading to a population of the signal variance matrix only along its diagonal.

The SH degree

$l$ dependent approximation of the signal variance

${\sigma}_{M}$ reads

Since the approximate signal variance is now only degree dependent, it is equivalent to a Kaula rule [

25].

Figure 2 illustrates the variability of the

$a$ and

$b$ coefficients in Equation (2), derived from the 143 month time series of the HIS dataset.

$a$ and

$b$ coefficients are estimated from each month, and the median for each calendar month is indicated in the figure. For comparison, not only the a and b coefficients derived from the Earth model are shown, but also the ones derive from 143 months of real GRACE data, filtered with the DDK4 filter. The pre-filtered datasets are downloaded from the International Centre for Global Earth Models (ICGEM) web portal (

http://icgem.gfz-potsdam.de/ICGEM/). A similar analysis regarding the variability of signal variance can be found in [

15]. The amplitude of the

$\mathrm{a}$ coefficients from the DDK4 filtered ITSG-Grace-2014 solution is higher than the one derived from the model (about factor 2; illustrated by dashed red line). One reason is that the signal contained in the filtered GRACE solutions also contains—contrary to the coefficients derived from the Earth model—signal from residual AO and OT de-aliasing errors. A second effect is the difference in signal strength itself; analysis showed that the Earth model tends to show smaller signal amplitudes than the real world situation [

26]. The difference of a factor 2 for the

$\mathrm{a}$ coefficient can be accounted for when selecting a suitable

$\alpha $ value. The dashed red line reveals that the characteristics of the

$a$ coefficients derived from the Earth model data and the DDK4 filtered ITSG-Grace2014 data is, apart from the scaling issue, very similar. The

$b$ coefficients, representing the exponent in Equation (2), agree very well (apart from the difference of the local maxima in June and July) with the ones derived from the DDK4 filtered solution. But one should keep in mind that the underlying datasets do not cover the same time period. Accordingly some minor deviations must be expected although both results represent decadal average values. Due to the fact that decadal average values are used for building the signal variance for the VADER filter, a bias towards the Earth models characteristics is avoided. The impact of using static, cyclo-stationary, and non-stationary signal variance is evaluated in detail later in this section.

Another way for determining the signal variance is the one embedded in the DMT-1 series [

27], which estimates the signal variance from the solution itself, and is therefore on the one hand independent of external model data. On the other hand, separation of different geophysical signal constituents is not possible with such a technique. Using the spectral characteristics of specific target signals like HIS for building the filter allows tailoring of the regularization more precisely for the respective filter scenario.

The impact of different populations of the error covariance matrices is analyzed in the following.

Figure 3 shows cumulative geoid errors in mm for different VADER filters. The median from all 143 monthly solutions is shown for the respective filter strength, steered by the choice of the

$\alpha $ value, which is expressed by the Gaussian smoothing radius according to

Table 1. The impact of using different setups of the inverse of the normal equation matrix

$N$ in equation (1) is demonstrated. The smallest residual is found using the full information from the covariance matrix (red curve). Approximating the matrix and keeping only blocks containing the correlations among coefficients of the same order (order blocks) causes inferior recoverability of the original signal (black curve). Degrading the covariance matrix even more to a diagonal structure (blue curve), which implies not considering any correlation among coefficients, delivers the largest residual geoid error of the three options. For the diagonal setup, not only is the minimum residual achievable significantly higher than for the other two options, but also stronger filtering is necessary (larger average filter radius) to reach this minimum. However, still all VADER filter setups are able to achieve better signal recoverability than the Gaussian filter (cyan dashed line).

The average smoothing radius should not be misinterpreted as the exact number of the spatial resolution of the final result. The average smoothing radius can, due to the way it is derived, be seen as an approximation of the spatial resolution. If the convolution kernel was Gaussian type shaped (meaning isotropic), the half weight radius would correspond exactly to the spatial resolution of the filtered solution [

28]. The filter in general acts as a kind of low-pass filter omitting high frequency signals. Since the VADER filter is highly anisotropic and the indicated average smoothing radius is a global approximation, the real resulting spatial resolution cannot be described by a single number.

The results of the closed-loop simulations so far indicate receiving best performance by using fully populated error covariance matrices. The analysis regarding the approximated signal variance indicates a varying behavior of the signal variance in a cyclo-stationary manner. A stationary signal variance is expected to deliver inferior results as this state does not represent the real variability. Using true month-to-month signal variance is on the other hand most promising from a theoretical perspective, but not possible to realize in real world applications, because the true signal is simply not known.

In the following an in-depth analysis of the impact of stationary or non-stationary error covariance and signal variance is presented.

Figure 3 shows the median values of cumulative geoid error from residuals of the filtered set of solutions. Analyzing the scattering of the respective clusters of filtered solutions gives valuable insight into the properties and characteristics of the filter design.

Figure 4 shows the individual monthly cumulative geoid errors in mm derived from the residuals (filtered solution minus true reference). The colored clusters indicate different filter strengths. The black x-marks indicate the median of the respective cluster in terms of cumulative geoid error (y-axis) and average smoothing radius (x-axis). Accordingly the x-marks correspond to the median curves like the ones shown in

Figure 3.

Figure 4a shows the results, building the VADER filter with static (stationary) error covariance and variable (cyclo-stationary) signal variance. Panel (b) shows results with both ingredients, signal and error covariance, with non-stationary properties (non-stationary N and cyclo-stationary M). Especially for the clusters around the local minimum (green to blue colored clusters) the scattering is significantly reduced. One can conclude that using non-stationary error covariance significantly improves the quality of filtered time series. Panel (c) indicates in black the performance of a stationary design of the VADER filter, meaning stationary M and N as used also by the DDK filter. For this scenario the average smoothing radius is the same for each month since the filter matrix

${W}_{\alpha}$ is identical for each month. For comparison, the performance of a 350 km Gaussian filter in magenta and an empirical decorrelation filter [

9] in green are shown (label S&W).

Table 2 summarizes the results presented in

Figure 4 by giving some performance metrics for the different post-processing methods. The average smoothing radii are indicated to allow a rough comparison of the respective filter strength. For the empirical S&W decorrelation filter a window size of 5 with a second degree polynomial was chosen. The minimum SH order for the first part of the filter cascade is 10 and for the second part of the filter cascade a 250 km Gaussian filter is applied. For details regarding the assessment of the empirical decorrelation refer to [

9,

14].

The lowest residuals are derived when using the VADER filter in non-stationary design. All results originate from the respective local minima of the different post-processing techniques. The impact of switching the properties of the error covariance matrix are significantly larger than for the signal variance (as long as it is chosen within reasonable bounds). The scattering is drastically reduced using time variable decorrelation, comparing the max values of the outliers scaling down from 6.84 and 7.14 to 0.44 and 0.43. The empirical decorrelation (S&W) does show better performance than the standard Gaussian filter, but is clearly outperformed by the VADER filter. Months showing large deviations from the median mostly show problems in data coverage and orbital configuration (short repeat cycles). The improvement of the median of cumulative geoid errors from residuals is about 15% switching from stationary to non-stationary filter design. For individual months this improvement is significantly higher as one can see from the single scattered results (

Figure 4). The maxima are reduced by more than one order of magnitude.

Figure 5 shows the performance of post-processing (VADER compared with Gaussian) within the closed-loop experiments for one individual month. Panel (a) shows SH degree RMS in mm geoid height of the residuals of the raw and the filtered solutions. Residual means in this case the difference between the true HIS signal, represented by the mean HIS input signal, as an approximation, for the simulation run of the respective month (black curve). To analyze the effect of deterministic error sources like ocean tide (OT) signal, an OT difference signal is added to the observations, leading to the residual SH degree RMS drawn in the dashed lines. As described in [

18], characteristic RMS peaks, induced by incomplete OT de-aliasing affecting specific SH order bands, can be seen in the dashed gray line around SH degrees 46, 61, and 107. As one can conclude from the dashed red line, the OT signal is almost completely removed from the observations and the residuals are almost as small as for the case without deterministic errors. For the Gaussian filter, this is not the case. Here one can still see significantly increased residuals after filtering (dashed blue line). Another notable effect is the intersection of the solid blue and solid red line with the solid black line, the latter representing the average HIS signal for the specific month. The VADER filter allows signal recovery (positive signal to noise ratio) up to SH degree 55 instead 30, using the Gaussian filter.

Figure 5b shows cumulative geoid errors in mm for different filter radii computed from the residuals of all 143 monthly solutions (the median is shown for each filter strength) with respect to the corresponding HIS signal filtered with VADER (solid lines) and Gaussian (dashed lines). Complete de-aliasing (blue) is compared with effects of ocean tide model differences (red) and of 20% of the non-tidal atmospheric and oceanic (AO) signals (black). The minima of the solid curves are always lower (for same colors) than the dashed lines and correspond to a shorter average smoothing radius. The increase of the residuals is stronger in the case where deterministic OT error contributions are added instead of the 20% AO error. This circumstance is of course related to the choice of error contributions and will change using different assumptions regarding the error budgets. The minimal residuals for both filter techniques are shifted to higher average smoothing radii, implying the need for stronger filtering, by inducing additional errors. This effect is stronger for the Gaussian filter than for the VADER filter which shows hardly any noticeable shifts in the smoothing radius. The ability to better separate between target signals and noise is one of the key advantages of the VADER filter compared to the Gaussian filter.