The study investigates the least squares coseismic slip distribution inversion problem combined with the Bayesian method for determining the regularization factor. In response to the significant time consumption of the Bayesian method in coseismic slip distribution inversion, the neglect of uncertainty in earthquake studies by the least squares method, and the issue that using L-curve, U-curve, and EI-curve to determine the regularization factor only provides numerical point estimates, a method is proposed that combines the Bayesian method with least squares coseismic slip inversion. The task of determining the regularization factor is assigned to the Markov Chain Monte Carlo (MCMC) method under the Bayesian framework, while the final slip distribution inversion is performed using least squares. The proposed method was verified using the February 6, 2016, Meinong earthquake and demonstrated its superiority in correspondence with the fault geometry parameter inversion of the Meinong earthquake. The inversion results show that the maximum slip of the Meinong earthquake was 0.54 m, the average slip angle was 44.94°, the seismic moment was 5.26 × 10¹⁸ N·m and the moment magnitude was M_W6.45. This method takes into account the uncertainties in earthquake studies and helps provide new insights into post-seismic deformation mechanisms. Moreover, in the future, the inherent advantages of the Bayesian method can be expanded by incorporating specific geophysical factors of each earthquake into the prior information constraints for the regularization factor.

This study proposes a novel hybrid model integrating least squares (LS) with a spatial attention mechanism (SAM) for 1–10 days polar motion prediction. The LS method primarily predicts principal components, while residual sequences are transformed into image data via Gramian angular field (GAF) representation before being processed by a neural network to forecast future residuals. Three basic sequences with different lengths of 6 a, 10 a, and 14 a are selected as the basic data of this LS + SAM. A total of 400 issues are predicted, with prediction data update at 1-day intervals per issue. The results showed that the mean absolute error (MAE) of PMX and PMY were 0.346–3.319 mas and 0.333–2.113 mas for three different length base sequences, respectively. Notably, predictions based on the 10-year sequence exhibited superior accuracy compared to the other intervals. Meanwhile, comparative analysis with the conventional LS + AR model revealed comparable performance in absolute error (AE), indicating that GAF-based image transformation effectively preserves residual characteristics while demonstrating the feasibility of two-dimensional image approaches for polar motion prediction.

With the continuous improvement of the accuracy of geodetic deformation data, the inversion of seismic source parameters puts forward a higher demand for nonlinear inversion algorithms. In this research, an improved Sparrow Search Algorithm (SSA) is proposed for the seismic source parameter inversion problem. By replacing the original population generation in the improved algorithm with Latin hypercubic sampling, the Sparrow Search Algorithm reduces the repetition of samples in the population initialization. Subsequently, the algorithm introduces adaptive weights in the discoverer generation phase of the sparrow algorithm and combines the Levy flight strategy to make the algorithm more comprehensive and improve the search accuracy during the whole iteration process. Therefore, the improved Latin hypercube-based sparrow search algorithm (ILHSSA) has better advantages in terms of iterative convergence speed and stability. In order to verify the performance of ILHSSA, the basic genetic algorithm (GA) and sparrow search algorithm (SSA) are examined and compared with ILHSSA by simulated earthquakes of two different earthquake types. The simulation experiments show that the improved algorithm ILHSSA outperforms SSA in accuracy and stability. Compared with the GA algorithm, ILHSSA can achieve the same inversion accuracy as GA, and it even surpasses GA in inversion speed and the inversion results of some parameters, demonstrating better stability. Finally, the improved algorithm is used for the 2017 Bodrum-Cos earthquake and the 2016 Amatrice earthquake in Italy. The inversion results all reflect the practicality and reliability of the improved algorithm.

In the variance component estimation (VCE) of geodetic data, the problem of negative VCE is likely to occur. In the ordinary additive error model, there have been related studies to solve the problem of negative variance components. However, there is still no related research in the mixed additive and multiplicative random error model (MAMREM). Based on the MAMREM, this paper applies the non-negative least squares variance component estimation (NNLS-VCE) algorithm to this model. The correlation formula and iterative algorithm of NNLS-VCE for MAMREM are derived. The problem of negative variance in VCE for MAMREM is solved. This paper uses the digital simulation example and the Digital Terrain Mode (DTM) to prove the proposed algorithm's validity. The experimental results demonstrated that the proposed algorithm can effectively correct the VCE in MAMREM when there is a negative VCE.

A novel artificial bee colony algorithm was introduced for the eruption event of the Sakurajima volcano on August 9, 2020, to invert the magma source characteristics below the volcano based on the point source Mogi model. Considering that the Sakurajima volcano is surrounded by sea, all the deformation data are used to obtain the location and magma eruption volume of the volcano. In response to the weak local search capability of the artificial swarm algorithm, the difference between the global optimal individual and the un-roulette screened individual is introduced as the variance component in the onlooker stage. Detailed simulation experiments verify the improvement of the algorithm in terms of convergence speed. In real experiments, the Sakurajima volcano inversion shows closer fitting results and smaller residuals compared to the existing literature. Meanwhile, the convergence speed of the algorithm echoes with the simulation experiments.

Geodetic functional models, stochastic models, and model parameter estimation theory are fundamental for geodetic data processing. In the past five years, through the unremitting efforts of Chinese scholars in the field of geodetic data processing, according to the application and practice of geodesy, they have made significant contributions in the fields of hypothesis testing theory, un-modeled error, outlier detection, and robust estimation, variance component estimation, complex least squares, and ill-posed problems treatment. Many functional models such as the nonlinear adjustment model, EIV model, and mixed additive and multiplicative random error model are also constructed and improved. Geodetic data inversion is an important part of geodetic data processing, and Chinese scholars have done a lot of work in geodetic data inversion in the past five years, such as seismic slide distribution inversion, intelligent inversion algorithm, multi-source data joint inversion, water reserve change and satellite gravity inversion. This paper introduces the achievements of Chinese scholars in the field of geodetic data processing in the past five years, analyzes the methods used by scholars and the problems solved, and looks forward to the unsolved problems in geodetic data processing and the direction that needs further research in the future.

This paper considers setting different dips for different sub-faults to fit the actual rupture situation based on the fault rupture of the 2013 Lushan M_S7.0 earthquake. Meanwhile, combined with the coseismic GNSS data of the Lushan earthquake, the source parameters and sliding distribution of the Lushan earthquake fault are inversed. Firstly, we use the gradient based optimizer (GBO) in nonlinear inversion to obtain the source parameters of this seismic fault. The inversion results indicate that the strike of the fault is 206.52°, the dip is 44.10°, the length is 21.92 km, and the depth is 12.79 km. To refine the sliding distribution of the seismic fault, the seismic fault is divided into 3 × 3 sub-faults. Then, we fix the central sub-fault dip of 44.10°; the dip of other sub-faults is obtained by iteration. After that, the model is further divided into a fault layer model composed of 23 × 19 sub fault slices, and using the Matlab fitting function is used to fit the dip of the 23 × 19 sub faults. Finally, the Lushan seismic fault plane is established as a shovel structure with steep upper and gentle lower, steep south and gentle north. The slip distribution inversion results indicate that the depth of the slip peak is 13 km, the corresponding maximum slip momentum is 0.67 m, the seismic moment is 1.10 × 10¹⁹ N·m and the corresponding moment magnitude is M_W6.66. The results above are consistent with the research results of seismology.

To estimate the parameters of the mixed additive and multiplicative (MAM) random error model using the weighted least squares iterative algorithm that requires derivation of the complex weight array, we introduce a derivative-free cat swarm optimization for parameter estimation. We embed the Powell method, which uses conjugate direction acceleration and does not need to derive the objective function, into the original cat swarm optimization to accelerate its convergence speed and search accuracy. We use the ordinary least squares, weighted least squares, original cat swarm optimization, particle swarm algorithm and improved cat swarm optimization to estimate the parameters of the straight-line fitting MAM model with lower nonlinearity and the DEM MAM model with higher nonlinearity, respectively. The experimental results show that the improved cat swarm optimization has faster convergence speed, higher search accuracy, and better stability than the original cat swarm optimization and the particle swarm algorithm. At the same time, the improved cat swarm optimization can obtain results consistent with the weighted least squares method based on the objective function only while avoiding multiple complex weight array derivations. The method in this paper provides a new idea for theoretical research on parameter estimation of MAM error models.

The use of geodetic observation data for seismic fault parameters inversion is the research hotspot of geodetic inversion, and it is also the focus of studying the mechanism of earthquake occurrence. Seismic fault parameters inversion has nonlinear characteristics, and the gradient-based optimizer (GBO) has the characteristics of fast convergence speed and falling into local optimum hardly. This paper applies GBO algorithm to simulated earthquakes and real LuShan earthquakes in the nonlinear inversion of the Okada model to obtain the source parameters. The simulated earthquake experiment results show that the algorithm is stable, and the seismic source parameters obtained by GBO are slightly closer to the true value than the multi peak particle swarm optimization (MPSO). In the 2013 LuShan earthquake experiment, the root mean square error between the deformation after forwarding of fault parameters obtained by the introduced GBO algorithm and the surface observation deformation was 3.703 mm, slightly better than 3.708 mm calculated by the MPSO. Moreover, the inversion result of GBO algorithm is better than MPSO algorithm in stability. The above results show that the introduced GBO algorithm has a certain practical application value in seismic fault source parameters inversion.

After the first Earth Orientation Parameters Prediction Comparison Campaign (1st EOP PCC), the traditional method using least-squares extrapolation and autoregressive (LS + AR) models was considered as one of the polar motion prediction methods with higher accuracy. The traditional method predicts individual polar motion series separately, which has a single input data and limited improvement in prediction accuracy. To address this problem, this paper proposes a new method for predicting polar motion by combining the difference between polar motion series. The X, Y, and Y-X series were predicted separately using LS + AR models. Then, the new forecast value of X series is obtained by combining the forecast value of Y series with that of Y-X series; the new forecast value of Y series is obtained by combining the forecast value of X series with that of Y-X series. The hindcast experimental comparison results from January 1, 2011 to April 4, 2021 show that the new method achieves a maximum improvement of 12.95% and 14.96% over the traditional method in the X and Y directions, respectively. The new method has obvious advantages compared with the differential method. This study tests the stability and superiority of the new method and provides a new idea for the research of polar motion prediction.