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Objectives: Co-seismic deformation monitoring is of great significance for the interpretation of co-seismic deformation characteristics and intuitive understanding of fault geometric characteristics. For large earthquakes with surface rupture, global navigation satellite system(GNSS) technique has a low spatial resolution, and interferometric synthetic aperture radar(InSAR) technology will cause phase decorrelation due to large deformation gradients, it is impossible to obtain specific deformation around the fault. Optical image correlation(OIC) and pixel offset tracking(POT) based on sub-pixel cross-correlation can solve these problems well. Methods: The main idea of the sub-pixel cross-correlation method is to use the pre-event image in the two images as a reference, and then compare the post-event image with it to evaluate the similarity between the two images, and then retrieve the displacement between the homonymous points. OIC can obtain east-west and north-south deformations, and POT can obtain range and azimuth deformations. In this paper, the deformation in each direction of the Kaikoura earthquake obtained from Sentinel-1 data and Sentinel-2 data is used to form different combinations, and the least square method is used to calculate the three-dimensional deformation. Results: The combination of OIC+POT_As_Des has the best constraint. The accuracy of the north-south deformation obtained by OIC+POT_Range is not as good as that of OIC+POT_As_Des. Compared with POT_As_Des, the north-south deformation precision obtained by it is higher, and the deformation performance on the right side of the Kekerengu fault and the left side of the Papatea fault is also better. In the northeast of the Kaikoura earthquake epicenter, very complex and huge surface deformation and multiple ruptures were detected in two close areas, and the vertical defoamation was mainly uplift. Conclusions: For Sentinel-1 and Sentinel-2, the combination of OIC+ POT_As_Des is most suitable for obtaining the three-dimensional deformation of the Kaikoura earthquake. This earthquake is a dextral strike-slip earthquake with reverse. [ABSTRACT FROM AUTHOR]

Objectives: When using variance component estimation to determine the relative weight ratio, the least square solution is generally used as the initial value of the iteration. In geodetic joint inversion, the least squares method will cause ill ⁃ posed problems, so the regularized solution is used instead of the least square solution. Regularization introduces bias to reduce variance, but when using variance component estimation to determine the relative weight ratio, the influence of bias is not considered, and the introduction of bias will cause inaccurate variance component estimation. Methods: This paper adopts the bias - corrected variance component estimation method to eliminate the influence of bias introduced by regularization. The residual - based bias - corrected variance component estimation and the variance component estimation method are used for simulation experiments, the Visso earthquake and Norcia earthquake are used for verification. Results and Conclusions: Simulation experimental result shows that the variance component estimation method after bias correction can better reverse the slip distribution. The bias-corrected variance component estimation method takes into account the bias introduced by the iterative initial value, and the theory is more rigorous. Two real earthquakes results show that the bias correction is reasonable and advantageous. [ABSTRACT FROM AUTHOR]

Objectives: As nonlinear models become more complex and data sources become more abun⁃ dant,people have higher and higher requirements for nonlinear methods and precision.However, the existing research mainly considered the precision evaluation of the algorithm, and ignored how to comprehensively evaluate the applicability of the algorithm based on precision information obtained by calculation. Methods: Considering the above problems, we propose a method to evaluate the advantages of algorithms based on precision information—non-linear comprehensive evaluation method. The method firstly evaluates the accuracy by the traditional Monte Carlo (MC) method, and uses the calculated deviation and the median error as the evaluation index, and calculates the comprehensive evaluation value according to the proposed comprehensive evaluation formula. We use the proposed method to evaluate the precision information of simulated annealing algorithm (SA), genetic algorithm (GA) and neural network algorithm (NNA) in volcano com⁃ pound dislocation model (CDM) and seismic Okada model inversion. Results: The experimental results show that whether it is based on the CDM model or the Okada model, the parameter estimation gap between the above three methods is small, but the precision of the different methods varies greatly. The MC is used to calculate the precision of SA, GA and NNA. It is found that in the volcano CDM, mean square errors in some parameters calculated by SA and GA are relatively large, and the comprehensive evaluation values 11.398 8 and 3.982 0 of SA and GA are larger than the comprehensive evaluation value 3.613 1 of NNA. In the Okada model, the SA has a higher mean square error than GA and NNA, and in the Lushan earthquake, the comprehensive evaluation value of the SA is 11.656 2, which is much larger than the comprehensive evaluation values 4.060 4 and 3.625 4 of the GA and NNA . Conclusions: The precision information of the NNA is higher when the proposed method is used for evaluation, and the result is more convincing, followed by GA, and SA has lower precision. Objectives: As nonlinear models become more complex and data sources become more abun⁃ dant,people have higher and higher requirements for nonlinear methods and precision.However, the existing research mainly considered the precision evaluation of the algorithm, and ignored how to comprehensively evaluate the applicability of the algorithm based on precision information obtained by calculation. Methods: Considering the above problems, we propose a method to evaluate the advantages of algorithms based on precision information—non-linear comprehensive evaluation method. The method firstly evaluates the accuracy by the traditional Monte Carlo (MC) method, and uses the calculated deviation and the median error as the evaluation index, and calculates the comprehensive evaluation value according to the proposed comprehensive evaluation formula. We use the proposed method to evaluate the precision information of simulated annealing algorithm (SA), genetic algorithm (GA) and neural network algorithm (NNA) in volcano com⁃ pound dislocation model (CDM) and seismic Okada model inversion. Results: The experimental results show that whether it is based on the CDM model or the Okada model, the parameter estimation gap between the above three methods is small, but the precision of the different methods varies greatly. The MC is used to calculate the precision of SA, GA and NNA. It is found that in the volcano CDM, mean square errors in some parameters calculated by SA and GA are relatively large, and the comprehensive evaluation values 11.398 8 and 3.982 0 of SA and GA are larger than the comprehensive evaluation value 3.613 1 of NNA. In the Okada model, the SA has a higher mean square error than GA and NNA, and in the Lushan earthquake, the comprehensive evaluation value of the SA is 11.656 2, which is much larger than the comprehensive evaluation values 4.060 4 and 3.625 4 of the GA and NNA . Conclusions: The precision information of the NNA is higher when the proposed method is used for evaluation, and the result is more convincing, followed by GA, and SA has lower precision. [ABSTRACT FROM AUTHOR]

Objectives: For multiplicative error model, the parameters estimated by weighted least squares (WLS) are nonlinear functions of observations, and the weights of observations are nonlinear functions of the estimated parameters. The existing parameter estimation methods of multiplicative error model can theoretically achieve second‑order unbiased, but the precision of uncertainty can only achieve first‑order unbiased. Therefore, a new method is used to improve the accuracy of uncertainty. Methods: The effect of nonlinear iterative processes on WLS parameters is considered, and the relationship between the estimated parameters and the observations in iterative WLS process is regarded as a nonlinear nested function. The derivative‑free Sterling interpolation method with symmetric sampling is used to calculate the expectations of estimated parameters and the standard deviation. Results: From the analysis of two synthetic experiments, the following results are obtained: (1) Considering the impact of the randomness of each iteration on the parameter estimation, the proposed Sterling interpolation method can get better estimated parameters than the existing methods. (2) When the nonlinearity of model is high, the effectiveness of the Sterling interpolation method is more significant. (3) The uncertainty estimation method in this paper can achieve secondorder precision. Conclusions: The feasibility and effectiveness of the Sterling interpolation method for parameter estimation and precision estimation of multiplicative error model are verified. [ABSTRACT FROM AUTHOR]

Objectives: Partial errors⁃in⁃variables (PEIV) model is a more general form of errors in variables (EIV) model. It has been widely used in the data processing of global navigation satellite system (GNSS), coordinate conversion, deformation monitoring and data fitting in the past ten years.Methods: First, the development history of PEIV model is reviewed. Then, five core problems of PEIV model are summarized and analyzed, including parameter estimation algorithm, precision estimation, random model estimation, expansion algorithm and data processing application. Finally, the applications of PEIV model are outlined. Results: We point out three questions for further research, including how to process big data by PEIV parameter estimation method, how to correct the bias of weighted total least squares(WTLS) adjustment by sampling method, and how to derive the formula of higher order parameter estimation covariance matrix are.Conclusions: This review paper aims to further promote the development of surveying and mapping da⁃ ta processing and provide readers with suggestions and references. [ABSTRACT FROM AUTHOR]

Objectives: Inversion of seismic fault parameters using geodetic observation data is a hotspot in geodetic inversion, and it is also the focus of studying the mechanism of earthquake occurrence. Aiming at the low accuracy of particle swarm optimization (PSO) currently used in fault parameter inversion, this paper analyzes the nonlinear characteristics of seismic fault parameter inversion and the characteristics of basic PSO. Basic PSO is easy to fall into a local optimal solution in highly nonlinear problems, and the local optimal solution and the global optimal solution may affect each other during the PSO solution process. This paper proposed a new particle swarm algorithm for inversion of fault parameters to solve the local optimization. Methods: In this paper, we adopted the strategy of segmentally and dynamically adjusting the parameters, including the inertia factor that affects the particle velocity and the acceleration factors that affect the local and global optimal solutions.Results: The proposed algorithm was applied to inverse the fault parameters for the simulation earthquake and the L ' Aquila earthquake. The results of the simulation earthquake show that the proposed algorithm is stable, and the inclination and sliding angle obtained by the proposed algorithm are closer to the true value. The results of the L ' Aquila earthquake show that the root mean squared error (RMSE) of surface observations and deformation variables obtained by the proposed algorithm is 5.2 mm, which is better than 6.7 mm obtained by multi‐peak particle swarm optimization (MPSO). Conclusions: The experiment results show that the fault model obtained by the proposed algorithm is more consistent with the true fracture condition, and the proposed algorithm has practical application value. [ABSTRACT FROM AUTHOR]

As a further extension of traditional regression model, the regression prediction model not only involves the fixed parameter estimation of regression model, but also incorporates the model prediction into part of adjustment, which is more in line with the solutions of actual requirements. Focusing on the issues of predicted non‐common points (independent variables) polluted with errors and inaccurate stochastic model, this paper proposes a new complete solution with a sufficient consideration to all errors of each variables based on errors‐in‐variables (EIV) model. Meanwhile, performed with the methodology of variance‐cova‐ riance component estimation, stochastic model and prior cofactor matrix of the predicted non ‐ common points have been corrected. The corresponding formulas are derived and the iterative algorithm is also pre‐ sented. Experimental design shows that the presented approach can effectively achieve the estimation of variance components for various types of observations. It provides a feasible means for retrieving more rea‐ sonable parameter results and achieving higher prediction accuracy. In addition, the prediction effect of our presented approach is better over other control schemes, especially for the situation where there is a certain correlation between the observed data and the random elements in coefficient matrix. [ABSTRACT FROM AUTHOR]

As for the ill-posed coefficient matrix in the process of co-seismic slip distribution inversion, a two-step solution method is proposed, and the Laplace second-order smoothing matrix is used for smooth- ing constraint in this paper. This method not only solves the ill-posed problem of the coefficient matrix, but also suppresses the large gradient variation between adjacent fault patches. During the inversion process of two-step solution method, the L curve method is used to determine the regularization parameters. In this paper, systematic simulation experiments are carried out and the Lushan earthquake is used as an actual earthquake case. The maximum slip inversion results of simulation experiments show that the inversion result of two-step solution method has been improved between 3.34% and 19% than that of one-step solu-tion method in accuracy. The root mean square error inversion result of two -step method is less between 3.3% and 13.3% than that of one - step solution method. The inversion results of the Lushan earthquake show that it is feasible to use the two-step solution for slip distribution inversion. [ABSTRACT FROM AUTHOR]

In regard to the joint adjustment problem with different types of dataset, the functional model of each type of dataset is affected by random errors, which indicates the observation vector and coefficient ma⁃ trix are not error-free. In this paper, the weight total least squares (WTLS) method is applied to joint adjustment model. An iterative WTLS method for joint adjustment model is derived, which uses the weight scaling factor to adjust the contribution of each type of dataset. In view of the determination of the weight scaling factor, more schemes are designed, which includes the minimum discrimination function method. The results show that the prior unit weight variance method and the total least squares variance component estimation (TLS-VCE) method have their limitations. When the prior information is inaccurate or the variance components are not estimable while using the TLS-VCE method, the minimum discriminate function method withΣi = 1 n1 eˉ̂1i as its discriminate function can achieve the relative effective results. [ABSTRACT FROM AUTHOR]

On April 20th 2013, Ms 7.0 earthquake occurred on the southern section of the Longmen Shan fault, in Sichuan Province. In this research, the three-dimensional coseismic displacement field that GPS data from 33 continuous stations are used to invert for fault geometry parameters and the slip distribution. The MPSO (multiple peak particle swarm optimization) algorithm has been successfully applied to invert for the fault model parameters.Compared with the commonly used algorithm, such as genetic algorithm, simulated annealing and the overall minimum norm optimization algorithm, MPSO algorithm has the advantages of less control parameters, high global convergence efficiency, and is very suitable for nonlinear inversion of fault parameters. The result indicates that the seismogenic fault is a thrust with a strike of 206.47°, a dip of 44.11°, and length of 21.94 km. The minimum and maximum depth is 7.66 km and 17.84 km, respectively. The fracture surface was determined as listric surface by analysis. In order to obtain the detailed slip distribution, the listric surface model is set with the default dip angle varying from 50° on the top boundary of the surface to 21° on the bottom boundary. Our inversion result shows that the rupture surface has two slip peaks at different depths. The maximum slip is 0.68 m at the depth of 13 km. The inverted geodetic moment is 1.47×1019 N·m and the moment magnitude Mw is 6.74, which is consistent with the result of seismology. [ABSTRACT FROM AUTHOR]

Among existing theories on precision estimation of nonlinear adjustment, the simulation number of Monte Carlo method generally is chosen subjectively and its result also cannot be controlled directly. Besides those, the biases of parameter estimates, corrections of observations and the estimate of variance of unit weight are not taken into consideration simultaneously. The adaptive Monte Carlo method is combined with precision estimation of nonlinear adjustment for solving problems given above in this paper. By calculating biases of estimates and covariances matrix of parameter estimates, the complete process of precision estimation based on adaptive Monte Carlo is given. With the help of the term of antithetic variates, the antithetic and adaptive Monte Carlo algorithm is proposed for biases of parameter estimates. Results from two examples of straight line fitting model and ellipse fitting model show that the adaptive Monte Carlo method in this paper can obtain the stable and reasonable effects for precision estimation of nonlinear adjustment with extensive applicability, the antithetic and adaptive Monte Carlo in this paper is better at convergence and computational efficiency for calculating biases of parameter estimates. [ABSTRACT FROM AUTHOR]

In this paper, a total least squares joint (TLS-J) adjustment method is proposed to the inversion of Mogi model with vertical and horizontal observational data. The proposed method considers the errors in both observation vector and coefficient matrix of the functional model of joint adjustment problem. Three forms of the minimum discriminate function methods are adopted to determine the weight scaling factor which are used to weigh the vertical and horizontal observation data. In view of the existing ill-posed problems in the joint adjustment, the L-curve method is adopted to determine the ridge parameter. Through practical examples, the total least squares joint method is systematically applied to the inversion of the Mogi model of Changbaishan Tianchi volcano. The results show that the discriminant function as the minimum can obtain the reasonable value of pressure source parameters and the relative weight ratio, which has a certain referential value to practical applications. [ABSTRACT FROM AUTHOR]

The errors-in-variables (EIV)model with inequality constraints is transformed into a standard nonlinear optimization program, which can be solved by existing optimization methods such as the active set method or sequential quadratic programming(SQP). Since weighted total least squares with inequality constraints (ICWTLS) is limited by the complexity of a Hessian matrix, which is the second partial derivative of objective function. In this paper, the Hessian matrix in SQP is replaced by an approximation based on Quasi-Newtonian method.The algorithm we propose can deal with the ICWTLS problem with a general weight matrix, and has the ability to handle large-scale problems. Eexamples illustrate that this new algorithm is efficient. [ABSTRACT FROM AUTHOR]

The weighted total least squares method based on partial errors-in-variables (PEIV for short) model is used to solve the inversion parameters of crustal strain model. It not only considers the error of observation (displacement or velocity field). but also the error effects from the coefficient matrix, generally composed of monitoring points coordinates. When taking the special structure of the coefficient matrix in the geodetic inversion model into account, we insure that the repeated coordinates have the same residual and that the constants are not allocated any correction. The method usedin this paper can meet these requirements as it separates the random elements from the constant elements tak­ing advantage of the partial errors-in-variables model. All calculation formulae for crust strain (rate) parameters inversion based on partial errors-in-variables using monitoring point displacement or veloc­ity fields are deduced. In addition, the derivate correction of weighted least squares (WLS) is used to analyze the effect of the random coefficient. The discrepancy between the weighted least squares solu­tion and WTLS solution was also investigated. Because of the complexity of the WTLS solution, we propose a formulation to relate the WLS\and WTLS solutions based on Xu (J Geod 86:661-675, 2012). A simulation using data from the Sichuan-Y unnan region permits a comparison and analysis of the effect of the random design matrix. The experimental results reveal that the effect of the random coefficient matrix on adjustment of the inversion of crustal strain (rate) parameters model is mainly depend on the order of value of the GPS coordinates and the crustal strain parameters themselves. [ABSTRACT FROM AUTHOR]

Objectives: The coseismic surface three-dimensional (3D) deformation field is the most intuitive description of the surface deformation caused by earthquakes in real space. In the study of seismic deformation, the recovery of coseismic surface 3D deformation field is of great significance. In view of the shortcomings and defects of the current coseismic 3D deformation calculation method, a coseismic surface 3D deformation field joint calculation method considering deformation gradient is proposed. Methods: First, we use the deformation gradient information to decompose the deformation region obtained by interferometric synthetic aperture radar (InSAR) into disjoint subregions with approximately equal deformation gradient. Then, according to the number of observations in the sub-region, the weighted least square (WLS) method or InSAR 3D deformation calculation method based on stress-strain model (SM) and variance component estimation (VCE) are selected to obtain 3D deformation field of the coseismic surface. Results: Compared with the traditional WLS and SM-VCE methods, the results of simulation experiments and actual earth⁃ quake examples show that the deformation gradient information can effectively restore the surface 3D deformation of the earthquake rupture zone and obtain more complete and reliable 3D deformation field of the coseismic surface. Conclusions: This proposed method can avoid the defect that the traditional SM-VCE method cannot identify the differential deformation information on both sides of the fault in the cross-fault area, and make up for the lack of adjacent point information in the traditional WLS solution method. [ABSTRACT FROM AUTHOR]

As a prior stochastic model contains inaccurate information, the weight matrices of observation and coefficient matrix are unreasonable. To address this problem, we investigate the total least squares adjustment of partial errors-in-variables(PEIV)model with a weight scaling factor that adaptively adjusts the contribution of the observation and coefficient matrix to parameter estimation. A prior unite weight variance and minimum discriminate function method are deduced, so the proposed method is valid for a structured coefficient matrix. Some conclusions are drawn from simulations of straight line fitting and coordinate transformation. When the prior unit weight variances of observation and coefficient matrix are known and accurate, the prior unit weight variance method is very effective; the minimum discriminate function method with the ??? as its discriminate function to determine weight scaling factor yielded the best performace. [ABSTRACT FROM AUTHOR]

The Total Leasl Squares (TLS) method is applied 10 joint adjustment. An algorithm for total least squares joint adjustment with a weight scaling factor is derived. The weight scaling factor is the key to deal with joint adjustment, and methods for determining the weight scaling factor are discussed. The difference norm between the estimated and true values is used to evaluate the TLS joint adjustment simulation. The influence of different noises on the weight scaling factor is also analyzed for two simulated examples. The results show that the estimated values is related to the weight scaling factor. When priori information is accurate, the prior unit weight variance method performs the best, and when priori information is inaccurate, the minimum discriminate function method with n1Σ i=1 ǀ + n2Σ j=1ǀ V2j and ǀ n1Σ i=1 ǀ V1iǀ + n2Σ j=1ǀ V2j ǀ (1 - XTX) as its discriminate function can achieve effective results. [ABSTRACT FROM AUTHOR]

In the current research on the total least squares method in the conversion of GPS height, the calculation of the conversion parameter and elevation abnormities of the check points are generally performed in two steps, and only consider the error in the coefficient matrix used to calculate the parameters; errors in the coordinate of the check point are ignored. In view of this gap, we put forward a total least squares fitting estimation model of GPS height transformation, that combines the calculation of fitting parameters with the calculation of elevation abnormities at inspection points, and considers the position error of all points. Collocation calculation experiemental results verify the feasibility of this method. These test results show that the method can effectively improve the accuracy of elevation conversion. [ABSTRACT FROM AUTHOR]

Considering the situation that the weight matrix of observation vector and coefficient matrix may be inaccurate, an available algorithm is introduced in this paper, which is derived on the basis of combining the Helmert variance component estimation with a kind of fast weighted total least squares algorithm in the errors-in-variables models. And the derivative process of the fast weighted total least squares is described in detail and the comparison with three other algorithms is implemented in this paper. Using the fast weighted total least squares algorithm combining Helmert variance component estimation derived in this paper, the stochastic model and the unknown parameters of the functional model can be solved simultaneously. Three empirical examples, two straight line fitting and one linear parameter estimation, are also used to investigate the application of posteriori estimation of stochastic model on weighted total least squares problem. Results show that the algorithm is very effective. [ABSTRACT FROM AUTHOR]