An effective filtering approach is essential for accurately interpreting GRACE (Gravity Recovery and Climate Experiment) monthly gravity field models. The Improved Parameter Filtering (IPF; Zhang et al. [1]) simultaneously estimates signal components with deterministic harmonic parameters and time-variable irregular signals through Kalman Filtering (KF), followed by signal denoising based on their signal and noise covariance matrices. However, it has two critical limitations: (a) excessive computational burden due to redundant dynamic calculations of deterministic parameters in KF, and (b) signal attenuation resulting from a suboptimal two-step estimation framework. For this reason, this letter proposes an Enhanced Parameter Filtering (EPF) based on a more rigorous parameter estimation criterion, which independently resolves deterministic parameters and irregular signals, thereby avoiding dynamic estimations of deterministic parameters while effectively integrating the two-step procedures of IPF. Here, we employ EPF to denoise the ITSG-Grace2018 model at degree 96 from April 2002 to December 2023, comparing its performance with IPF. Results demonstrate that the computational efficiency of EPF is improved by 92.3%, with fitting errors reduced by 62.4% and signal-to-noise ratios enhanced by 4.4%. Spatial analysis of filtered global Terrestrial Water Storage Anomalies (TWSAs) shows EPF better matches CSR Mascon (CSRM) RL06, JPL Mascon (JPLM) RL06, and NOAH products, with average Nash-Sutcliffe Coefficients increased by 1.7%, 2.9%, and 8.3%, respectively. Further comparisons of TWSAs across 30 global basins, mass changes in Greenland and Antarctica, and co-seismic gravity signals of the 2004 Sumatra-Andaman and 2010 Chile earthquakes, reveal the superior performance of EPF over IPF and four recently proposed filters.

Linear ill-posed models are widely encountered in various problems in geophysics and remote sensing. Regularization technique can significantly improve the accuracy of the estimates since the biases introduced by the regularization are much smaller than the errors reduced by regularization. However, from the spectral point of view, certain low-frequency terms with large singular values might become over-regularized, whereas other high-frequency terms with small singular values might be insufficiently regularized for a given regularization parameter. For this reason, we developed a recursive regularization approach to further improve the regularized solution via additional regularization of some high-frequency terms and restricted regularization of some low-frequency terms. The analytical conditions to determine the terms to be further regularized are derived based on the criterion that the introduced biases should be smaller than the reduced errors; in other words, the mean squared error (MSE) should be reduced. Furthermore, the universal form of the recursive regularized solution is derived. Two examples from remote sensing are designed to demonstrate the performance of the developed approach. The first example involves solving the Fredholm integral equation of the first kind, a fundamental mathematical model used in many inverse problems in remote sensing. The results indicate that the proposed method outperforms the ordinary Tikhonov regularization, partial regularization and adaptive regularization, with roots of MSE reduced by 25.8%, 14.5%, and 8.1%, respectively. Subsequently, we apply the proposed method to estimate regional mass anomalies based on the mascon modeling using the GRACE (Gravity Recovery and Climate Experiment) time-variable gravity field models. The results demonstrate that the proposed method preserves more signal than conventional regularization methods.

Geophysical time series derived from remote sensing and ground-based observations contain rich signals reflecting diverse Earth processes. Wavelet filtering is widely employed to extract these signals from noisy datasets. However, due to many factors, geophysical time series inevitably contain missing data, complicating the direct use of wavelet filtering without prior interpolation. The extended wavelet filtering (EWF) method (Ji et al. [1]) addresses this by linking residuals (signals) to missing data and estimating signals by minimizing the quadratic norm of residuals across both observed epochs and missing epochs. However, EWF is suboptimal for filtering observed data, as residual estimates for missing epochs are less reliable, particularly in time series with consecutive data gaps. This study proposes a new approach, minimum norm least squares wavelet filtering (MWF), for processing gappy time series. MWF offers two main advantages over EWF: (a) it filters incomplete time series more effectively through optimal temporal filtering by minimizing the quadratic norm of residuals only for observed epochs. This can avoid disturbances in filtering caused by distorted residuals at missing epochs in EWF. To address the rank deficiency in the parametric model, an additional constraint is applied to minimize the L2 norm of unknown parameters (missing data), ensuring a unique solution. (b) it provides a more concise and effective way to consider prior precision information of the time series by incorporating it into the cost function, without the need for iterative normalization required by the EWF method. We apply the proposed method to extract deformation signals from daily position time series of 27 Global Navigation Satellite System (GNSS) permanent stations in the Chinese mainland and compare the results with the EWF method. Results show that MWF outperforms EWF in signal extraction, as indicated by smaller fitting errors and higher signal-to-noise ratios. Additional simulations confirm that MWF yields signals closer to the true signals regardless of gap sizes, noise characteristics (pure white or colored noise), and gap distributions (uniform or consecutive).

Wavelet multiresolution analysis (MRA) is a widely used and effective method for filtering noisy time series and investigating the detailed characteristics of time series at different resolutions. However, geophysical time series often exhibits unavoidable data gaps stemming from diverse factors, impeding the straightforward implementation of ordinary MRA (OMRA). Moreover, geophysical time series is typically heterogeneous, as evidenced by the dynamic fluctuations in their precision. Regrettably, the prevailing approach of OMRA neglects to incorporate this aspect of heterogeneity. This study develops an extended MRA (EMRA) approach that solves for the missing values based on the best approximation in the temporal domain. The proposed method can directly analyze incomplete time series without prior interpolation and accounts for the formal errors of time series to improve the filtering performance. To validate the proposed approach, we use the EMRA method to extract crustal deformation signals from the daily position time series of 27 permanent global navigation satellite system (GNSS) stations in the Chinese mainland from 1999 to 2019. We compare the results with those obtained through OMRA using interpolation methods. The results reveal that EMRA can extract more signals than OMRA, particularly when considering formal errors. Repeated simulations further depict that the signals extracted by EMRA are closer to the simulated true signals than those by OMRA using interpolation methods.

When ordinary principal component analysis (PCA) is employed to analyze the position time series of a regional GNSS station network, the GNSS time series are assumed to be homogeneous, and the missing data in the time series must be restored beforehand. To directly process incomplete and heterogeneous GNSS position time series, we develop the extended PCA (EPCA) and weighted EPCA approaches to solving for the missing values based on the best low-rank approximation in the spatiotemporal domain. The proposed approaches are used to process the real GNSS position time series of 24 stations in North China spanning 2011 to 2019 and successfully extract the common mode errors (CMEs). The proposed approaches are compared with modified PCA (MPCA) and weighted MPCA, in which an additional optimization criterion needs to be introduced in the frequency domain. The results show that EPCA can extract more CMEs than MPCA for both the unweighted and weighted cases. Consequently, EPCA outperforms MPCA in reducing noise and improving the accuracy of site velocity estimates. Repeated simulation experiments show that the CMEs extracted by EPCA are closer to the simulated true values than those extracted by MPCA. When the formal errors of the time series are considered, both weighted EPCA and weighted MPCA outperform their unweighted counterparts, and the former outperforms the latter. In addition, EPCA is computationally more efficient than MPCA since fewer unknowns need to be estimated.

The ill-posed models are widely encountered in various inversions of geodesy and remote sensing. The regularization approaches can significantly stabilize the solution to ill-posed models since the high-frequency noise is effectively suppressed. Although the famous Tikhonov regularization and truncated singular value decomposition (TSVD) regularization have been widely applied in various geodetic applications, there still remain theoretical drawbacks for either single regularization. For Tikhonov regularization, given a regularization parameter, the low-frequency terms are over regularized, and high-frequency terms are under regularized. For TSVD regularization, some medium-frequency terms will be mistaken for high-frequency terms to be truncated and the hidden signals will be lost. For this reason, we propose an adaptive regularized solution in spectral form, which adaptively divides the terms of different frequencies into three kinds: (i) the low-frequency terms are not regularized; (ii) the medium-frequency terms are regularized by the Tikhonov method; (iii) the high-frequency terms are regularized by TSVD method. The analytical conditions for determining the term sets are derived based on the criteria that the introduced biases should be smaller than the reduced errors, in other words, the mean squared error (MSE) should be reduced. The two examples are presented to demonstrate the performance of our adaptive regularization. The first numerical example is solving the Fredholm integral equation of the first kind, which is widely encountered in remote sensing inversions. The simulations clearly demonstrate that the adaptive regularized solution can improve the MSE of ordinary Tikhonov and TSVD regularized functions by 25.00% and 9.09%, respectively; In the second example, we apply the new method to investigate the mass variation of the Yangtze River Basin based on the Gravity Recovery and Climate Experiment (GRACE) time-variable gravity field model. The Tongji-Grace 2018 monthly gravity field solutions from April 2002 to December 2016 are used to construct the mascon observation equation. The results show that our method also outperforms the ordinary Tikhonov and TSVD regularized solutions, with mean MSE reductions of about 13.40% and 11.69%, respectively. Furthermore, the spatial resolution of secular trend derived by our method are improved and the signal-to-noise ratio (SNR) of mass variation series is higher than the other two regularizations.