Average Convergence Research Articles

Comprehensive analysis of zenith tropospheric dealy and precipitable water vapor retrieved from BDS-3 B1C and B2a signals

This study presents a comprehensive analysis of ZTD and PWV estimated from BDS-3 new B1C/B2a signal combination. ZTD are firstly estimated using B1C and B2a signals and evaluated with the IGS final products. The average convergence time of ZTD was 1980 s, which was 12.0 % shorter than that from the legacy B1I/B3I signals. This indicated their superiority for real-time operational meteorology. The average RMS value of ZTD were 5.9 mm for all stations, corresponding to an ∼ 7.8 % improvement with respect to the legacy B1I/B3I signal combinations. ZTD error characteristics analysis revealed relationships with station latitudes and seasons, mainly due to the more active water vapor variation in equator region and summer. Afterwards, the converted PWV was assessed using multiple inter-technique datasets. For radiosonde-derived PWV, the RMS for GPS L1W/L2W, BDS-3 legacy B1I/B3I, and BDS-3 new B1C/B2a signals were 2.5, 2.4, and 2.5 mm, respectively, whereas those for the remotely-sensed PWV from MODIS were 4.9, 5.0, and 4.9 mm, respectively. Besides, evaluation with ERA5 showed the average RMS were all 1.7 mm for different GNSS signal combinations. These results prove the feasibility of PWV retrieval using BDS-3 new B1C/B2a signals, which are expected to provide scientific references for further applications.

An optimization method for deformable mirror configuration in multi-conjugate adaptive optics systems

Multi-conjugate adaptive optics (MCAO) is a crucial technology for achieving high-resolution imaging over a wide field of view with modern ground-based optical telescopes. The configuration of deformable mirrors (DMs) is a key component in the analysis and optimization of MCAO performance. Currently, the search for the optimal DM configuration often relies on iterative and time-consuming Monte Carlo simulations. This issue arises from the lack of an appropriate optimization method for DM configurations. The primary objective of this paper is to establish an optimization method for DM configurations in MCAO systems. We established a quantitative criterion for evaluating DM configurations by analyzing their correction capabilities for turbulence aberrations at different altitudes. Then, we optimized the DM configurations based on this criterion. This method provides a new theoretical foundation and practical tool for the design and performance optimization of MCAO systems. Based on the pupil phase structure function, we established a DM configuration evaluation criterion, namely the non-conjugate correction index (NCCI). Using NCCI as the optimal criterion, combined with the particle swarm optimization algorithm, we searched for the optimal solution across different DM configuration spaces. We conducted simulations based on the turbulence profiles of typical telescope sites. We validated our proposed theoretical model against Monte Carlo simulation models and find that the NCCI error ranges from 0.05 to 0.1. For optimizing DM conjugate heights, the results of our optimization algorithm differ by less than 1 km from those obtained via Monte Carlo simulations. Regarding the performance of the DM optimization algorithm, the average convergence accuracy error is less than 0.1 km, and the average convergence speed is approximately ten iterations. Additionally, our optimization method runs in just a few minutes; Monte Carlo simulations, in comparison, require several dozen hours.

Practical Probabilistic Model-Based Reinforcement Learning by Integrating Dropout Uncertainty and Trajectory Sampling.

This article addresses the prediction stability, prediction accuracy, and control capability of the current probabilistic model-based reinforcement learning (MBRL) built on neural networks. A novel approach to dropout-based probabilistic ensembles with trajectory sampling (DPETS) is proposed, where the system uncertainty is stably predicted by combining the Monte Carlo dropout (MC Dropout) and trajectory sampling in one framework. Its loss function is designed to correct the fitting error of neural networks for more accurate prediction of probabilistic models. The state propagation in its policy is extended to filter the aleatoric uncertainty for superior control capability. Evaluated by several Mujoco benchmark control tasks under additional disturbances and one practical robot arm manipulation task, DPETS outperforms related MBRL approaches in both average return and convergence velocity while achieving superior performance than well-known model-free baselines with significant sample efficiency. The open-source code of DPETS is available at https://github.com/mrjun123/DPETS.

Estimation of GNSS satellite bias corrections using non-common frequency model with application to PPP-AR

Abstract Corrections of satellite phase biases (SPBs) enable precise point positioning with ambiguity resolution (PPP-AR), which shortens the convergence time and improves positioning precision compared with precise point positioning (PPP). Extending PPP-AR from dual-frequency to multi-frequency can further enhance the positioning performance by correcting satellite code biases (SCBs) at the third frequency and above. In formulating a multi-frequency model to estimate satellite bias corrections (SBCs), including SPBs and SCBs, existing studies use only satellite observations collected at common frequencies (CF). This study proposes a non-common frequency (NCF) model that utilizes dual- and multi-frequency satellites to estimate SBCs simultaneously. To validate the NCF model, we collect data from an Australian network to estimate the SBCs and apply these corrections to the PPP-AR user positioning. The results show that the NCF model can provide SBCs for more satellites and frequencies than the CF model. After correction of SBCs, the proportion of fractional parts of float PPP ambiguities, which are less than 0.1 cycles, has a significant increase from 20% to
70%, indicating the successful restoration of the integer properties of PPP ambiguities by the
SBCs. Moreover, a comparable experiment between the NCF and CF PPP-AR models is conducted. The NCF PPP-AR model demonstrates superior performance over the CF model using triple-frequency satellites in convergence speed and positioning precision, achieving improvements of 41.16% and 39.52%, respectively. Meanwhile, compared to the CF model using dual-frequency satellites, the NCF PPP-AR model shows improvements of 17.16% and 7.46% in these aspects. The positioning performance of the NCF PPP-AR model is evaluated across ten user stations sampled over three days. The results indicate the root mean square
(RMS) positioning errors for the east, north and up components are 0.41, 0.51 and 2.46 cm, respectively. Additionally, the average convergence time for achieving 3D positioning errors within 1 dm is 9.75 min.

PERBANDINGAN RETURN PENGGUNAAN MOVING AVERAGE CONVERGENCE DIVERGENCE DAN RELATIVE STRANGE INDEX PADA INDEKS SAHAM SYARIAH INDONESIA

This study compares the return levels from the use of the technical indicators Moving Average Convergence Divergence (MACD) and Relative Strength Index (RSI) applied to the simulation of the movement of the Indonesia Sharia Stock Index (ISSI). This research is a population study using panel data in the form of daily closing values over 12 years of the Indonesia Sharia Stock Index movement. The data was processed using a technical analysis approach with two momentum indicators, MACD and RSI, to obtain buy and sell signals based on the principles of each indicator, applied to the ISSI. The use of the MACD and RSI technical indicators on the ISSI shows that MACD yielded a profit of 207% with a success rate (win rate) of 72%, while RSI provided a profit of 125% with a win rate of 55%.

Market resilience in turbulent times: a proactive approach to predicting stock market responses during geopolitical tensions

PurposeThe Indian stock market can be tricky when there's trouble in the world, like wars or big conflicts. It's like trying to read a secret message. We want to figure out what makes investors nervous or happy, because their feelings often affect how they buy and sell stocks. We're building a tool to make prediction that uses both numbers and people's opinions.Design/methodology/approachHybrid approach leverages Twitter sentiment, market data, volatility index (VIX) and momentum indicators like moving average convergence divergence (MACD) and relative strength index (RSI) to deliver accurate market insights for informed investment decisions during uncertainty.FindingsOur study reveals that geopolitical tensions' impact on stock markets is fleeting and confined to the short term. Capitalizing on this insight, we built a ground-breaking predictive model with an impressive 98.47% accuracy in forecasting stock market values during such events.Originality/valueTo the best of the authors' knowledge, this model's originality lies in its focus on short-term impact, novel data fusion and high accuracy. Focus on short-term impact: Our model uniquely identifies and quantifies the fleeting effects of geopolitical tensions on market behavior, a previously under-researched area. Novel data fusion: Combining sentiment analysis with established market indicators like VIX and momentum offers a comprehensive and dynamic approach to predicting market movements during volatile periods. Advanced predictive accuracy: Achieving the prediction accuracy (98.47%) sets this model apart from existing solutions, making it a valuable tool for informed decision-making.

Nonlinear FOPID controller design for pressure regulation of steam condenser via improved metaheuristic algorithm

Shell and tube heat exchangers are pivotal for efficient heat transfer in various industrial processes. Effective control of these structures is essential for optimizing energy usage and ensuring industrial system reliability. In this regard, this study focuses on adopting a fractional-order proportional-integral-derivative (FOPID) controller for efficient control of shell and tube heat exchanger. The novelty of this work lies in the utilization of an enhanced version of cooperation search algorithm (CSA) for FOPID controller tuning, offering a novel approach to optimization. The enhanced optimizer (en-CSA) integrates a control randomization operator, linear transfer function, and adaptive p-best mutation integrated with original CSA. Through rigorous testing on CEC2020 benchmark functions, en-CSA demonstrates robust performance, surpassing other optimization algorithms. Specifically, en-CSA achieves an average convergence rate improvement of 23% and an enhancement in solution accuracy by 17% compared to standard CSAs. Subsequently, en-CSA is applied to optimize the FOPID controller for steam condenser pressure regulation, a crucial aspect of heat exchanger operation. Nonlinear comparative analysis with contemporary optimization algorithms confirms en-CSA’s superiority, achieving up to 11% faster settling time and up to 55% reduced overshooting. Additionally, en-CSA improves the steady-state error by 8% and enhances the overall stability margin by 12%.

Approximation by Subsequences of Matrix Transform Means of Some Two-Dimensional Rectangle Walsh–Fourier Series

In the present paper we discuss the rate of the approximation by the matrix transform of special partial sums of some two-dimensional rectangle (decreasing diagonal) Walsh-Fourier series in Lp(G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p(G^{2})$$\\end{document} space (1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p <\\infty $$\\end{document}) and in C(G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C(G^{2})$$\\end{document}. It implies in some special case norm convergence. We also show an application of our results for Lipschitz functions. At the end of the paper we show the most important result, the almost everywhere convergence theorem. We note that T summation is a common generalization of the following known summation methods Cesàro, Weierstrass, Riesz and Picar and Bessel methods.

Stock Market Prediction in Nepali Stock Market using Machine Learning Model

The stock market, a key driver of the global economy, presents a formidable challenge for accurate prediction due to its intricate, chaotic, and dynamic nature. This study explores machine learning approaches, specifically comparing three prediction models—Back Propagation Neural Network (BPNN), Long Short-Term Memory (LSTM), and Gated Recurrent Unit (GRU). The analysis incorporates historical stock trading data, including open, high, low, and close prices, and technical analysis indicators such as moving average, Relative Strength Index (RSI), Moving Average Convergence Divergence (MACD), and Commodity Channel Index (CCI) so on. The evaluation focuses on the performance of these models in predicting future trends in stock prices, utilizing data from the Nepal Stock market spanning from 2013 to 2023. The selected stocks for analysis include two stock NIC Asia Bank Limited and, NMB Bank Limited, two index NEPSE and Hydropower index. Through rigorous assessment, it is determined that LSTM demonstrates superior overall performance compared to GRU and BPNN. This research contributes valuable insights into the effectiveness of machine learning models in forecasting stock market trends, with implications for investors and financial practitioners.

An ultraweak-local discontinuous Galerkin method for nonlinear biharmonic Schrödinger equations

This paper proposes and analyzes a fully discrete scheme for nonlinear biharmonic Schrödinger equations. We first write the single equation into a system of problems with second-order spatial derivatives and then discretize the space variable with an ultraweak discontinuous Galerkin scheme and the time variable with the Crank–Nicolson method. The proposed scheme proves to be computationally more efficient compared to the local discontinuous Galerkin method in terms of the number of equations needed to be solved at each single time step, and it is unconditionally stable without imposing any penalty terms. It also achieves optimal error convergence in L2 norm both in the solution and in the auxiliary variable with general nonlinear terms. We also prove several physically relevant properties of the discrete schemes, such as the conservation of mass and the Hamiltonian for the nonlinear biharmonic Schrödinger equations. Several numerical studies demonstrate and support our theoretical results.

An optimally convergent Fictitious Domain method for interface problems

We introduce a novel Fictitious Domain (FD) unfitted method for interface problems associated with a second-order elliptic linear differential operator, that achieves optimal convergence without the need for adaptive mesh refinements nor enrichments of the Finite Element spaces. The key aspect of the proposed method is that it extends the solution into the fictitious domain in a way that ensures high global regularity. Continuity of the solution across the interface is enforced through a boundary Lagrange multiplier. The subdomains coupling, however, is not achieved by means of the duality pairing with the Lagrange multiplier, but through an L2 product with the H1 Riesz representative of the latter, thus avoiding gradient jumps across the interface. Thanks to the enhanced regularity, the proposed method attains an increase, with respect to standard FD methods, of up to one order of convergence in energy norm. The Finite Element formulation of the method is presented, followed by its analysis. Numerical tests on a model problem demonstrate its effectiveness and its superior accuracy compared to standard unfitted methods.

A hybrid sampling-based RRT* path planning algorithm for autonomous mobile robot navigation

The path-planning algorithms for autonomous mobile robot navigation are crucial, often relying on sampling-based methods. RRT* is a robust, sampling-based path planning algorithm. The sampling process in RRT* plays a pivotal role, where uniform sampling can lead to slow convergence, while non-uniform sampling offers faster convergence but may struggle in complex environments due to its limited exploration. Thus, achieving a balance between exploitation and exploration is essential when selecting the sampling method for the RRT* path-planning algorithm. To address this issue, this research paper introduces Hybrid-RRT*, a path planning method that utilizes hybrid sampling. This unique approach generates samples using both non-uniform and uniform samplers. Hybrid-RRT* is evaluated against three baseline path planning algorithms—RRT*-N, Informed RRT*, and RRT*—across three different 384x384 2D simulation environments. Compared to these baseline methods, Hybrid-RRT* achieves superior results across all five performance metrics: convergence rate, success rate, number of nodes visited, path length, and planning time. According to the numerical results, the proposed algorithm achieves a faster average convergence rate that is 76.14% higher than RRT*, 24% higher than Informed RRT*, and 3.33% higher than RRT*-N. Moreover, it reduces node exploration by an average of 48.53% compared to RRT* and 40.83% compared to Informed RRT*. The simulation results demonstrate that the proposed Hybrid-RRT* algorithm effectively addresses the issue of slow convergence with uniform sampling and the challenge of limited exploration with non-uniform sampling methods.

Calibration of Receiver-Dependent Pseudorange Bias and Its Impact on BDS Augmentation Positioning Accuracy

Pseudorange bias refers to the receiver-dependent and satellite-dependent constant bias in the pseudorange resulting from the nonideal characteristics of a signal. The impact of pseudorange bias on high-precision satellite navigation services has long been ignored. This paper proposes a pseudorange bias calibration method for two collocated receivers. Then, we calibrate pseudorange biases for two types of collocated receivers at a monitoring station within China and evaluate their impact on two high-precision services: BeiDou Navigation Satellite System 3 (BDS-3) dual-frequency pseudorange augmentation and precise point precision (PPP). Theoretical analysis reveals that the calibrated pseudorange biases contribute 17.2% and 7.7% to the user equivalent ranging error (UERE) of BDS-3 and Global Positioning System (GPS) dual-frequency pseudorange augmentation, respectively, and that the convergence time of the GPS static and kinematic PPP increases from 6 min and 26 min to 19 min and 58 min, respectively. The experimental results indicate that the calibrated pseudorange biases are consistent as the receiver location and time vary. The spatial distribution consistency is generally better than 0.1 m, and the temporal consistency is better than 0.15 m. The pseudorange biases for BDS-3 B1C and B2a are approximately 0.7 m and 0.1 m, respectively, whereas those for GPS L1C/A and L2P are both approximately 0.25 m. Furthermore, The results show that after correction of the pseudorange biases, the average convergence time for BDS-3/GPS static PPP decreases from 48.83/24.03 min to 38.54/21.12 min, respectively, a decrease of approximately 21%/12%. For BDS-3/GPS/BDS-3 + GPS kinematic PPP, the average convergence time decreases from 109.53/45.10/39.15 min to 62.99/40.83/22.94 min, respectively, a decrease of approximately 42%/41%/9%. Similarly, the three-dimensional positioning accuracy for BDS-3/GPS/BDS-3 + GPS dual-frequency pseudorange augmentation improves from 3.25/3.94/2.49 m to 2.65/3.69/2.16 m, respectively, increasing by approximately 6.3%, 18.5%, and 13.3%, respectively. The above analysis and experiments demonstrate that pseudorange bias is an important error source affecting both dual-frequency pseudorange augmentation and PPP services.

Unconditionally energy stable high-order BDF schemes for the molecular beam epitaxial model without slope selection

In this paper, we consider a class of k-order (3≤k≤5) backward differentiation formulas (BDF-k) for the molecular beam epitaxial (MBE) model without slope selection. Convex splitting technique along with k-th order Douglas-Dupont regularization term τnk(−Δ)kD_kϕn (D_k represents a truncated BDF-k formula) is added to the numerical schemes to ensure unconditional energy stability. The stabilized convex splitting BDF-k (3≤k≤5) methods are unique solvable unconditionally. Then the modified discrete energy dissipation laws are established by using the discrete gradient structures of BDF-k (3≤k≤5) formulas and processing k-th order explicit extrapolations of the concave term. In addition, based on the discrete energy technique, the L2 norm stability and convergence of the stabilized BDF-k (3≤k≤5) schemes are obtained by means of the discrete orthogonal convolution kernels and the convolution type Young inequalities. Numerical results are carried out to verify our theory and illustrate the validity of the proposed schemes.

Combine Study of Technical Indicator Bollinger Bands and Moving Average Convergence Divergence in Nepal Stock Exchange Limited

Stock price forecasting involves predicting a company's future condition based on time series analysis. Investors need precise forecasts of future indices or stock price changes to address the Buy-Sell-Hold problem effectively. Data were collected for Nepal Stock Exchange Limited, Nepse Alpha, and Chukul. For this study, data were collected over six months (December 2023 to May 2024), utilizing technical trading tools like Bollinger Bands (BB) and Moving Average Convergence Divergence (MACD). Combining BB and MACD on a single chart yields better results than analyzing each indicator separately. This integrated approach provides a more thorough understanding of market dynamics, improves decision-making accuracy, and leads to more profitable outcomes. Integrating the analysis of BB and the MACD on one chart can provide more benefits than using each indicator independently. This method offers market behavior and improves decision-making precision, resulting in more profitable outcomes.

Norm convergence rate for multivariate quadratic polynomials of Wigner matrices

We study Hermitian non-commutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root growth at its edges and prove an optimal local law around these edges. Combining these two results, we establish that, as the dimension N of the matrices grows to infinity, the operator norm of such polynomials q converges to a deterministic limit with a rate of convergence of N−2/3+o(1). Here, the exponent in the rate of convergence is optimal. For the specific reducible cases, we also provide a classification of all possible edge behaviors.

Compact RTK for expanded area (COREA): a new method for carrier-phase-based satellite augmentation system

This study proposes a new concept of carrier-phase-based satellite augmentation system named “Compact Real-time Kinematic for Expanded Area (COREA),” which provides centimeter-level positioning services across nationwide coverage. The proposed system’s architecture is very similar to that of the satellite-based augmentation system (SBAS), a meter-level aviation safety service. While network real-time kinematic and precise-point-positioning-RTK (PPP-RTK) rely on several densely positioned reference stations, COREA provides carrier-phase-based corrections using a few reference stations with a distance of 400–1000 km. Furthermore, the COREA corrections can be transmitted by satellite signals with extremely low-speed data links of 250 bps, similar to SBAS. This study focused on the generation method for satellite code/phase clock (CPC) corrections, which is the most significant part of the system. We analyzed the user performance of the COREA system constructed in the Midwest and South of the United States with six reference stations. Consequently, satellite CPC corrections are resilient to communication failures and highly accurate in identifying user integer ambiguity. The 95% position accuracy is approximately 1.8 cm horizontally and 7.1 cm vertically, with an average convergence time of 1–3 min using only GPS triple-frequency signals. In summary, the COREA system preserves the hardware architecture of the legacy SBAS while providing centimeter-level services with fast convergence times by utilizing extremely low-speed satellite data links across the country.

The stabilized nonconforming virtual element method for the Darcy–Stokes problem

A stabilized nonconforming virtual element method is designed in order to solve the Darcy–Stokes problem, which preserves a divergence-free approximation to the velocity. The same degrees of freedom as the usual Crouzeix–Raviart-type virtual element is used, but a different virtual element space is obtained by modifying the conforming Stokes virtual element with the H1-projection operator. The proposed stabilized scheme contains two jump penalty terms over edges. One is the penalty for jumps of velocity approximation and the other one is the penalty for jumps of its normal component. We analyze this method’s well-posedness and prove its uniform convergence in a discrete energy norm. Finally, we verify the validity of this stabilized scheme by some numerical experiments.

Improvement of BDS-3 inter-frequency clock bias estimation and its application in multi-frequency precise point positioning

BDS-3 broadcasts five civilian signals, establishing a fundamental basis for multi-frequency real-time precise point positioning (RT PPP). Nevertheless, the existence of inter-frequency clock bias (IFCB) renders the current satellite clock error product unsuitable for multi-frequency PPP processing. Consequently, there is a critical need to conduct a thorough analysis of BDS-3 IFCBs and propose real-time service methods for RT PPP users. In this study, 8-day BDS-3 multi-frequency data obtained from 130 International GNSS Service (IGS) stations are employed to scrutinize the characteristics of BDS-3 IFCBs and validate the proposed real-time service method. The results disclose a noticeable disparity between the constant part of IFCBs computed using the differential code bias (DCB) product and those estimated from observations. The time-varying parts of different IFCBs, estimated using various combinations, are examined using the Fast Fourier Transform (FFT). The outcomes indicate that these IFCBs exhibit identical periods of 12, 6, and 4 h, although their fourth-period terms differ. To model the sequences of IFCBs, a polynomial with a fourth-order harmonic model is utilized, and the coefficients of the fitted model are subsequently transmitted for RT PPP processing. Experimental results show that the predicted accuracies of IFCBs for most satellites exceed 90 %, albeit with a noticeable decline as the prediction time increases. When applied in RT PPP for 20 IGS stations, the method significantly shortens the convergence time in dual-frequency and triple-frequency RT PPP when predicted IFCBs are provided. In the B1C/B2a, B1C/B3I, and B1I/B2a dual-frequency combinations, the average convergence times for RT PPP are reduced by 58 %, 46 %, and 59 %, respectively. In the B1I/B2a/B3I triple-frequency combinations, it is reduced by 48 %.

BDS-3 full-frequency precise point positioning: models and performance comparison

With the BDS-3 six-frequency integration, there are more choices to implement the precise point positioning (PPP) technology. To take full advantage of the multi-frequency combination, six typical BDS-3 six-frequency PPP models using uncombined observation (UC), five dual-frequency ionospheric-free (IF) combinations (IF2), four triple-frequency IF combinations (IF3), three four-frequency IF combinations (IF4), two five-frequency IF combinations (IF5), and a single six-frequency IF combination (IF6) were constructed and compared. The results indicate that the positioning accuracies of the six models are similar. The static positioning accuracies reach 4, 3–5, and 12–14 mm in the east, north, and up directions, respectively, while the kinematic positioning accuracies are 24–26, 16–17, and 42–43 mm, respectively. Regarding the convergence times, the UC model is slightly worse than the five IF combined models, except for some cases in the up direction. In the static mode, benefiting from the smallest noise amplification factor, the average convergence times of the IF6 model are the shortest, reaching 12.1, 5.5, and 13.4 min in the three directions, respectively, and are 7%, 15%, and 6% shorter than those of the UC model. In the kinematic mode, with the combination of more signals into a single IF combination, the noise level of the IF combined observables gradually decreases, but the convergence times gradually increase. The kinematic average convergence times of the IF2 model are 15.3, 3.6, and 18.0 min in the three directions, which are 6%, 22%, and 10% and 1%, 16%, and 20% shorter than those of the UC and IF6 models, respectively. In addition, the observation residuals, and the estimates of inter-frequency bias, tropospheric zenith wet delay, and receiver clock offsets from different models were compared and analyzed. The specific selection of the model should be based on actual situations.