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The results obtained through GRACE/GRACE Follow-On spherical harmonic coefficient model exhibit noticeable north–south strip noise. While applying low-pass filtering along the latitude circle direction can effectively eliminate the north–south strip error, it unavoidably introduces east–west strip noise. This paper introduces a method that utilizes the cosine function to determine the cutoff frequency for the low-pass filtering in the latitude direction. Additionally, a low-pass filtering step is added in the meridian circle direction. This combined approach successfully mitigates the north–south strip error and simultaneously suppresses the east–west strip noise. The analysis results highlight the improvement in signal to noise ratio of global surface quality anomaly estimation achieved by utilizing the cosine function to determine the cutoff frequency. The use of data from five institutions reveals that the dual low-pass filtering (DLP) improves the signal to noise ratio by more than 2% compared to the traditional efficient low-pass-filtering (ELP). When analyzing the uncertainty in the Yangtze River Basin using the three-cornered hat method, it is observed that the uncertainties of the DLP solution, traditional ELP solution, and mascon solution are comparable. Furthermore, the DLP solution exhibits the smallest uncertainty, measured at 4.89 cm. In terms of root mean square error, the DLP almost consistently yields the lowest values across various regions, with the difference from the smallest root mean square error value remaining within 2 mm in certain regions.

Layer precipitable water (LPW), a water vapor product similar to precipitable water vapor (PWV), reports partial moisture content within a specified vertical range. Compared with PWV data, the latest LPW products can describe more refined distributions and variations in water vapor in the troposphere. Global Navigation Satellite Systems (GNSSs), as a powerful water vapor sensing tool, only provide the opportunity to retrieve all-weather PWV, not LPW products. To this end, we develop the first deep-learning-assisted, global water vapor stratification (GWVS) model to estimate the GNSS LPW within any given vertical range. The proposed model is trained and tested using the global radiosonde data, with the training and testing root mean square error (RMSE) of 0.94 and 1.10 mm for radiosonde LPW, indicating the excellent generalization of the GWVS model. Furthermore, the model is comprehensively validated using the data from the two regional GNSS networks and one global network. The RMSEs of the predicted GNSS LPW from the three GNSS networks compared with the co-located radiosonde LPW are 1.52, 1.80, and 1.54 mm, respectively. To study potential applications, we use the model-derived GNSS LPW products to calibrate Geostationary Operational Environmental Satellite-16 (GOES-16) LPW products and improve the GNSS water vapor tomography technique. Results show that the accuracy of three GOES-16 LPW products is improved by 31.3%, 23.3%, and 17.9%, respectively, and the RMSE of the tomography results is reduced from 2.28 to $1.67~\text {g}/\text {m}^{3}$ . Both validation and application results highlight that the GWVS model retrieves the required GNSS LPW products and provides additional value for water-vapor-related studies.

The global navigation satellite systems (GNSSs) water vapor tomography technique has been successfully used as a promising tool for sensing atmospheric water vapor and applied to weather forecasting in recent years. In most tomography models, the single grid resolution, i.e., the same horizontal resolution, is widely adopted to divide the 3-D tomographic domain into many small voxels. However, the single-resolution GNSS tomography (SRGT) method implements the grid-based parametrization of the physical domain and does not follow the vertical spatial heterogeneity of atmospheric water vapor. To this end, we develop a new multi-resolution GNSS tomography (MRGT) method that incorporates the vertical decline tendency of water vapor. The MRGT method generates different-resolution tomographic water vapor products in the lower, middle, and upper domains of the troposphere. Besides, a new indicator, known as the integrated water vapor (IWV) lapse rate, is introduced to determine the appropriate nonuniform stratification strategy. Eight tomography schemes were analyzed to compare the tomography results obtained from different tomography models based on the GNSS data in the Hong Kong region during June and July 2015. The results show that with respect to radiosonde data, the MGRT method reconstructs a more accurate water vapor distribution than the SRGT approach, with the average root mean square error of tomographic results reduced by 12%. Moreover, in rainfall conditions, the tomographic water vapor profiles from the MGRT model agree well with the radiosonde profiles, which highlights the potential of multi-resolution tomographic water vapor products for rainfall-related studies.

Slepian functions are commonly used to analyze band-limited signals over arbitrary spatially limited regions on the sphere. However, the classical algorithm struggles to compute Slepian functions with large band-limits due to the heavy computational burden and large storage requirements; the efficient algorithm reduces the computational burden and storage requirements but also reduces the computational accuracy of Slepian functions. In this letter, we propose an improved efficient algorithm for computing Slepian functions over arbitrary regions on the sphere. Due to the utilization of the fast calculation method of Slepian functions over the polar cap region on the sphere, the proposed algorithm exhibits computational efficiency comparable to the efficient algorithm; while using more Slepian functions over the spherical cap region for numerical integration leads to computational accuracy comparable to the classical algorithm. The proposed algorithm supports fast analysis of modern data with large band-limits over arbitrary spatially limited regions on the sphere. In addition, we provide a program package for the proposed algorithm.

The downward continuation (DWC) of airborne gravity data usually adopts the Poisson integral equation, which is the first kind Fredholm integral equation. To improve the stability and accuracy of the DWC, based on the traditional regularization methods and Landweber iteration method, we introduce two other iterative algorithms, namely Cimmino and component averaging (CAV). In order to cooperate with the use of the iterative algorithms, in response to the discrepancy principle (DP) stopping rule, we further investigate the monotone error (ME) rule and normalized cumulative periodogram (NCP) in this paper. The numerically simulated experiments are conducted by using the EGM2008 to simulate airborne gravity data in the western United States, which is a mountainous area. The statistical results validate that Cimmino and CAV are comparable to Landweber, and iterative methods are better than generalized cross validation and least squares in the test examples. Furthermore, the results also show that three stopping rules succeed in stopping the iterative process, when the resolution of gravity anomalies grid is 5′. When the resolution is 2′, DP fails to stop the iteration, and ME is unstable, and only NCP performs well.

The Global Navigation Satellite System (GNSS) precise point positioning (PPP) technology is currently used to process GNSS water vapor observations in real time or near real time. Further developments are required to improve the accuracy and real-time performance of processing tropospheric delays from which the water vapor observations are extracted. In real-time BDS/GPS precise clock correction estimation with square-root information filtering and PPP solution with Kalman filtering, a fixed variance is often assigned for the process noise of the troposphere dynamics model. However, this fixed value may deviate from reality as the weather conditions change, especially for extreme weather. In this paper, a new adaptive Kalman filter is proposed for tropospheric delay processing, in which the variance of the process noise for the zenith tropospheric delay (ZTD) dynamics model is tuned in real time using the least-squares variance component estimation technique. MGEX/IGS data of 15 consecutive days were processed in a stepwise manner. Namely: Real-time BDS/GPS precise clock corrections were estimated firstly, followed by comparison among ZTD solutions of four schemes based on these real-time clocks of first step and GFZ multi-GNSS precise clock (GBM) final clocks: (1) real-time solution of ZTD (G(GPS), GC(GPS and BDS)) without VCE; (2) real-time solution of ZTD (G, GC) with VCE; (3) final solution of ZTD (G, GC, GR (GPS and GLONASS), GE (GPS and GALILEO), GRC (GPS, GLONASS and BDS), GREC (GPS, GLONASS, GALILEO and BDS) without VCE; and (4) final solution of ZTD (G, GC, GR, GE, GRC, GREC) with VCE. The performance of the ZTD and positioning solution was analyzed. Results showed that the accuracy of estimated real-time satellite clock correction was 0.27, 1.31, 0.29, and 0.21 ns for GPS, BDS/GEO, BDS/IGSO, and BDS/MEO, respectively. For the ZTD solutions, the results of schemes 1 and 2 were 12.8 mm (GC) and 10.1 mm (GC) in terms of mean root-mean-square (RMS) values and 2.1 mm (GC) and 1.6 mm (GC) in terms of minimum RMS values, respectively, thereby showing an improvement for scheme 2 of 1.8–81.4% over scheme 1 with average increasing rates of 20.7% (GC) and 20.2% (G). The results for schemes 3 and 4 were 7.6 mm (G) and 6.3 mm (GRC) in terms of mean RMS values and 2.1 mm (G) and 1.9 mm (GRCE) in terms of minimum RMS values, respectively, thereby showing an improvement in scheme 4 over scheme 3 by 1.3–85.6% with average increasing rates of 22.1% (GRCE), 21.9% (GRC), 18.4% (GR), 15.9% (GC), 15.2% (GE), and 12.1% (G). Similar results can be observed for the positioning solution, especially in the height component. These findings clearly show the advantage of the proposed method, which is consistent with the theoretical analysis. Notably, the advantage of the adaptive VCE becomes significant with the inclusion of additional satellite systems.

Single receiver positioning has been widely used as a standard and standalone positioning technique for about 25 years. To detect the slowly growing faults caused by satellite and receiver clocks in single receiver positioning, the Autonomous Integrity Monitoring with an Extrapolation method (AIME) was proposed based on the Kalman filter measurement domain. However, AIME was designed with the assumption of there is the same number of visible satellites at each epoch, which limits its application. To address this issue, this paper proposes a state-domain Robust Autonomous Integrity Monitoring with the Extrapolation Method (SRAIME). The slowly growing fault detection statistics is established based on the difference between the estimates of the state propagator and the posterior state estimation in Kalman filtering. Meanwhile, singular value decomposition is adopted to factor the covariance matrix of the difference to increase computational robustness. Besides, the relevant formulas of the proposed method are theoretically derived, and it is proven that the proposed method is suitable for any positioning model based on the Kalman filter. Additionally, the results of two experiments indicate that SRAIME can detect slowly growing faults in single receiver positioning earlier than AIME.

ABSTRACTDifferential synthetic aperture radar interferometry (D-InSAR) is limited when exploited in high-intensity mining areas, because large deformation gradients lie beyond the maximum measurable value of the D-InSAR technique which breaks the prerequisite for successfully employing of the method. The SAR amplitude-based pixel-tracking method provides an alternative way to efficiently and robustly extract the large deformation distribution particularly when the D-InSAR technique is limited by loss of coherence. In addition, the deformation in the line-of-sight direction and the deformation along the azimuth direction are also presented in this paper with 24-day interval repeat-pass high-resolution Rardarsat-2 imagery. Combining both of these techniques can help to better understand the deformation mechanisms associated with underground mining activities. The accuracies of 0.12 m in slant-range direction and 0.19 m in the azimuth direction were achieved, respectively. Besides, the profiles across the maximum deformation region have verified that the deformation occurred during two acquisition periods is far beyond the theoretical maximum deformation gradient corresponding to high-resolution C-band SAR data. The obtained surface motion infers to the mining activities and assessed damage caused by the large deformation.

Wide baseline stereo matching is a challenging task because of the presence of significant geometric deformations and illumination changes within the images. Based on the scale invariant feature transformation (SIFT) algorithm, this study proposes a new hybrid matching scheme that uses both the feature-based and the area-based methods to find reliable matches from sparse to dense under different geometric constraints. Firstly, the authors propose a SIFT-based robust weighted least squares matching (LSM) method modelled by a two-dimensional (2D) projective transformation to establish the initial correspondences and their local homographies. In this method, a normalised cross correlation metric modified with an adaptive scale and an orientation of the SIFT features (SIFT-NCC) is proposed to find a good initial alignment for the SIFT-LSM. Secondly, a robust matching propagation using the SIFT-NCC starts from the initial matches under an epipolar geometry and the local homography constraints; geometrical consistency checking is used simultaneously to identify the false matches. Thirdly, they use an improved, feature-based SIFT matching method to find the correspondences from the points that are not coplanar in the 3D space under an epipolar constraint only. A bidirectional selection strategy is used to remove the error matches.