Horizontal Error Research Articles

A lightweight odometry network for GNSS/INS integration during GNSS outages

In challenging environments like urban canyons and tunnels, the Global Navigation Satellite System (GNSS) signal can be interrupted. When this happens, the integrated Global Navigation Satellite System/Inertial Navigation System (GNSS/INS) navigation system relies solely on INS, resulting in rapid dispersion of positioning accuracy over time. Incorporating odometer information into a filter algorithm is one potential solution for correcting INS errors and improving navigation accuracy. However, this approach increases the cost and power consumption of the system. To implement an odometer-assisted integrated navigation system without inflating cost and power consumption, we propose a lightweight odometer network, LONet. This network can emulate a wheeled odometer, determine the carrier’s forward velocity using Inertial Measurement Unit (IMU) output data, and utilize non-holonomic constraint (NHC) and zero velocity update (ZUPT) to maintain the system’s positioning accuracy. To evaluate the performance of our network, we conducted velocity estimate experiments across different velocity intervals. The results demonstrated that our network requires fewer parameters and produces lower errors in estimated velocity compared to state-of-the-art networks. Furthermore, we integrated LONet into the navigation system to mitigate the effects of GNSS signal outages. The results show that the LONet-assisted integrated navigation system achieved horizontal errors comparable to, and sometimes lower than, those obtained using a real wheeled odometer.

Wear characteristics analysis of variable hyperbolic circular-arc-tooth-trace cylindrical gear transmission considering assembly error theory

Wear is the main failure form of the gear transmission interface, runs through the whole life cycle of gear service, and the distributed error of the tooth surface due to the introduced assembly error will make the actual meshing state of the tooth surface different from the ideal meshing state, thus affecting the tooth surface wear trajectory and state. Therefore, considering the coupling relationship between assembly error and tooth surface contact analysis, the variation laws of the wear amount of the gear transmission interface of the variable hyperbolic circular-arc-tooth-trace cylindrical gear were studied under the consideration of assembly error in this article. Based on the tooth surface contact analysis method and Hertz contact theory, the transient linear load, the combined radius of curvature, the length axis and short axis of the contact ellipse, and the instantaneous sliding velocity were calculated. Through revising the calculation method of adhesive wear of variable hyperbolic circular-arc-tooth-trace cylindrical gears, the correlations between the tooth surface wear amount and gear parameters of variable hyperbolic circular-arc-tooth-trace cylindrical gears under different assembly errors were analyzed. The results show that the maximum wear occurs at the root of the tooth, and the wear at the indexing circle approaches zero, whether it is standard installation or non-standard installation. The vertical axial error and horizontal axial error have little effect on the wear, while the center distance error will aggravate the wear. The proposed method comprehensively considers the influence of assembly error and gear parameters on tooth surface wear, and this method can provide a theoretical reference for the design and analysis of anti-wear, wear reduction, and life extension of variable hyperbolic circular-arc-tooth-trace cylindrical gear tooth surface.

Line feature based self-calibration method for dual-axis scanning LiDAR system

LiDAR calibration is an essential step for acquiring accurate three-dimensional point-cloud. To enhance the convenience of calibration of dual-axis scanning LiDAR, a line feature based self-calibration method is proposed, which corrects point-cloud error by establishing observation model and estimating model parameters. The model parameters are estimated by minimizing the square error of line feature points, referring as straight edges extracted from point-cloud. The experimental results verify the effectiveness of the proposed method on improving point-cloud accuracy, the horizontal and vertical angular errors are reduced to 0.10° and 0.08° after calibration with reductions of 64% and 68%, respectively. The analysis on parameter correlation finds that, parameters can be divided into groups according to the introduced point-cloud error: dimension error, trapezoid and parallelogram shape distortion. A de-correlation strategy combining inter-group and stepwise intra-group de-correlation thus can be implemented, which can be applied for point-cloud quality enhancement in in-situ environmental perception scenes.

Estimating the Location of Acoustic Sources with Oceanic Internal Gravity Waves via Frequency-Difference Source Localization

Frequency-difference source localization (FDSL) is a relatively new method for locating sources emitting acoustic or electromagnetic waves from recorded data. Estimates of location are produced by matching a nonlinear function of the received data with the same nonlinear function of modeled data. This so-called matched-field approach depends on the accuracy of the modeled field. FDSL exhibits less sensitivity to model errors in some cases of interest. Fluctuations of the speed of sound in the ocean are affected by a wide variety of phenomena and internal gravity waves is one where their spectrum is usually well known. There is no analytical theory for predicting how these fluctuations of sound speed affect the accuracy of FDSL, leaving a numerical evaluation as the alternative for the focus of this paper. Experimental application of FDSL was previously made for an acoustic source of 100 Hz bandwidth surrounding 250 Hz for a 129 km section in the Philippine Sea with data recorded on a long vertical array [Geroski and Dowling, Long-range frequency-difference source localization in the Philippine Sea, J. Acoust. Soc. Am. 146 (2019) 4727–4739.]. The role of internal waves is quantified with a realistic FDSL simulation using models based on the parabolic and ray approximations of the linear acoustic wave equation. The modeled vertical variation of FDSL error is similar to observed while the modeled horizontal error is too small. The complicated software model is automatic but is computationally intensive, with our version needing about 1.5 mo to compute on a PC with 16 cpu cores.

Performance Analysis of Relative GPS Positioning for Low-Cost Receiver-Equipped Agricultural Rovers.

Global navigation satellite systems (GNSSs) became an integral part of all aspects of our lives, whether for positioning, navigation, or timing services. These systems are central to a range of applications including road, aviation, maritime, and location-based services, agriculture, and surveying. The Global Positioning System (GPS) Standard Position Service (SPS) provides position accuracy up to 10 m. However, some modern-day applications, such as precision agriculture (PA), smart farms, and Agriculture 4.0, have demanded navigation technologies able to provide more accurate positioning at a low cost, especially for vehicle guidance and variable rate technology purposes. The Society of Automotive Engineers (SAE), for instance, through its standard J2945 defines a maximum of 1.5 m of horizontal positioning error at 68% probability (1σ), aiming at terrestrial vehicle-to-vehicle (V2V) applications. GPS position accuracy may be improved by addressing the common-mode errors contained in its observables, and relative GNSS (RGNSS) is a well-known technique for overcoming this issue. This paper builds upon previous research conducted by the authors and investigates the sensitivity of the position estimation accuracy of low-cost receiver-equipped agricultural rovers as a function of two degradation factors that RGNSS is susceptible to: communication failures and baseline distances between GPS receivers. The extended Kalman filter (EKF) approach is used for position estimation, based on which we show that it is possible to achieve 1.5 m horizontal accuracy at 68% probability (1σ) for communication failures up to 3000 s and baseline separation of around 1500 km. Experimental data from the Brazilian Network for Continuous Monitoring of GNSS (RBMC) and a moving agricultural rover equipped with a low-cost GPS receiver are used to validate the analysis.

Guangdong Lightning Mapping Array: Errors Evaluation and Preliminary Results

AbstractThe Guangdong Lightning Mapping Array (GDLMA), as the first LMA in China, was deployed in Guangzhou, Guangdong Province, China, in November 2018 by the Chinese Academy of Meteorological Sciences and New Mexico Institute of Mining and Technology. An evaluation was conducted using Monte Carlo and an aircraft track. The average timing uncertainty of GDLMA is 35 ns based on the distributions of reduced chi‐square values. Based on the aircraft track, the average horizontal error is 13 m and the average vertical error is 41 m at an altitude of 4–5 km over the network, consistent with the Monte Carlo results. Location errors outside the network exhibit noticeable directionality. The ability to characterize lightning channels varies with different location errors. In locations that are far from the network center, only the basic structure of lightning flash can be presented, while closer to the network, the flash channel structure can be mapped well. Compared with Low‐to‐Mid Frequency E‐field Detection Array (MLFEDA), they were generally similar in overall structure, and some lightning flash characteristics such as flash duration and coverage area exhibited consistency. However, GDLMA demonstrated better flash channel structure characterization capability, while MLFEDA performed better in processes such as leader/return strokes. In addition, based on the comparison of spatial positions of one‐on‐one discharge events, we found that very high frequency sources were more located ahead of low frequency sources in the direction of lightning channel development.

A Vision-Based Method for Simultaneous Instance Segmentation and Localization of Indoor Objects

Visual-based positioning technology plays a pivotal role in spatial artificial intelligence, facilitating precise perception and comprehension of the physical world for robotic platforms and augmented reality devices. In this study, we propose a binocular camera-based method for spatial localization of targets using CNN for instance segmentation while simultaneously providing target location information. The method encompasses image acquisition and correction, target recognition and segmentation, and stereo matching, among other components. Building upon this foundation, we introduce a pedestrian recognition segmentation network model with an attention mechanism. To accurately locate the target, we employ a multi-feature fusion feature point extraction and matching algorithm that combines edge information with semantic information. Finally, our proposed method is evaluated for dynamic pedestrian targets in indoor environments, achieving a horizontal positioning error of less than 0.25 m.

Methodology for performing bathymetric measurements of shallow waterbodies using an UAV, and their processing based on the SVR algorithm

State-of-art methods of bathymetric measurements for shallow waterbodies use Global Navigation Satellite System (GNSS) receiver, bathymetric Light Detection and Ranging (LiDAR) sensor or satellite imagery. Currently, photogrammetric methods with the application of Unmanned Aerial Vehicles (UAV) are gathering great importance. This publication aims to present step-by-step methodology for carrying out the bathymetric measurements of shallow waterbodies with the use of UAV-Structure from Motion (SfM) and Support Vector Regression (SVR) algorithms. Proposed model is explained and verified on the basis of measurement campaign on Raduńskie Górne Lake located in Poland. The campaign included GNSS Real Time Kinematic (RTK) bathymetric measurements for robust training dataset and photogrammetric flight pass for SfM point cloud. The presented methodology will allow to perform bathymetric measurements with the accuracy provided for the International Hydrographic Organization (IHO) Special Order (horizontal position error ≤ 2 m (p = 0.95), vertical position error ≤ 0.25 m (p = 0.95)).

Real-time Detection System for Portable Transient Electromagnetic Unexploded Ordnance

Portable transient electromagnetic systems are not limited by terrain and can detect unexploded ordnance very flexibly. However, existing data processing methods are slow and cannot achieve real-time positioning of targets. In this paper, based on a laboratory-made portable transient electromagnetic detection system, the magnetic dipole model and tensor positioning algorithm are used to achieve fast and accurate target positioning. Using the magnetic gradient tensor algorithm for positioning reduces the calculation load, allowing it to run on embedded systems, and improving target positioning speed while ensuring the accuracy of measurements. Experimental results show that the vertical and horizontal positioning errors of the UXO are within 10 cm, and single-point positioning can be completed within 10 ms.

PPP-RTK with Rapid Convergence Based on SSR Corrections and Its Application in Transportation

Real-time Kinematic (RTK) positioning provides centimeter-level positioning accuracy within several seconds, but it has to rely on a nearby base station. Although Precise Point Positioning (PPP) supplies precise positions with one receiver, its convergence time takes several tens of minutes, which makes PPP not well suited for real-time kinematic applications where a rapid convergence is required. PPP-RTK integrates the benefits of PPP and RTK, which actually is PPP augmented by a ground GNSS network. The satellite orbit, clock offsets, signal biases, ionospheric and tropospheric corrections are determined based on this GNSS network, modeled as state space information and transmitted to PPP users. By applying these State Space Representation (SSR) corrections, a real-time kinematic PPP-RTK approach is developed and implemented, which can instantly resolve the ambiguities to integers and realize rapid convergence. In a static scenario, it realized an instant ambiguity resolution and a rapid convergence within 2 s in more than 90% of 120 hourly sessions. The PPP-RTK has been applied and evaluated in a kinematic scenario on the highway. The horizontal positioning errors are almost lower than 0.1 m except for the time of passing through bridges. After passing bridges, the PPP-RTK successfully resolved ambiguities within 2 s in 90.6% of the cases and achieved convergence in horizontal within 5 s in more than 90% of the cases. The PPP-RTK with a precision of 0.1 m and rapid convergence of several seconds benefits the precise navigation of automobile on the highway, which will support the development of autonomous driving in future.

Analysis and compensation for the dominant tool error in ultra-precision diamond ball-end milling

Diamond ball-end milling is an enabling technology to fabricate complex curved surfaces with a nanometer level roughness. However, limited by the errors of tool manufacturing and installation, the cutting edge profile of a single-bladed diamond milling tool cannot form an ideal sphere crown in rotation, which will significantly worsen the machining accuracy. Although a titled milling method and a tool error reduction method based on milling-test have been proposed to improve the machining accuracy, a complete analysis of the effects of multiple tool errors on the machining accuracy have not been performed. Therefore, the present work contributes an analytical model to accurately calculate the tool rotation profile under various position and orientation errors. The results reveal that the horizontal off-center error of tool cutting edge is the dominant error among all the tool errors. According to the results, a novel high precision identification process is developed to evaluate the horizontal off-center error in view of the cutting-mark, and subsequently a tool center shift method is proposed to compensate for such error. Finally, experiments on millimeter scale spherical surface, micro sphere array and micro sinusoidal surface confirm that the proposed tool error compensation method can significantly improve the form accuracy. The average form errors (PV) of three test surfaces are reduced from 2.29 μm, 0.90 μm and 0.88 μm to 0.61 μm, 0.28 μm and 0.18 μm, respectively.

ANALYSIS OF 3D COASTAL TERRAIN BY UAV IMAGE AND BUNDLE ADJUSTMENT

Recently, studies on the construction of 3D terrain modeling and location accuracy evaluation using UAV have been conducted, but the application to coastal areas lacking image matching points has not been made in detail yet. In coastal topography survey using UAV, ortho-images and DSM (Digital Surface Model) data were constructed through imaging and ground control point (GCP) surveying on the ground. In the case of ortho-images, VRS survey was used to analyze horizontal and vertical errors between GCP points, to evaluate accuracy, and to construct terrain data from the ortho-image and DSM data.

Real-time continuous EOG-based gaze angle estimation with baseline drift compensation under stationary head conditions

Objective:This work aims to propose a novel method to estimate the gaze from electrooculography (EOG) signals while compensating for the baseline drift, and which intrinsically detects fixations, saccades and blinks. In contrast to existing baseline drift mitigation techniques, the proposed framework is real-time-implementable, and does not require the average point of gaze to lie at the primary gaze position, nor does it disrupt the overall ocular pose-induced EOG signal DC characteristics. The EOG data used to validate the proposed method is also being made publicly available. Methods:The proposed method is based on the dual Kalman filter, which estimates the gaze angles (GAs) and the baseline concurrently, taking into consideration the EOG signal’s non-stationary and temporally-multimodal characteristics. In fact, it is a multiple-model technique based on a battery model of the eye wherein fixations, saccades and blinks are modelled separately. Results:When applied to short EOG data segments, a horizontal and vertical GA estimation error of 1.64 ± 0.82°and 1.97 ± 0.34°, respectively, was obtained, which compared well with the corresponding results obtained using the state-of-the-art linear regression models. Conversely, for longer data segments, the proposed framework yielded superior GA estimation performance when compared to the state-of-the-art techniques. Eye movement detection and labelling F-scores exceeding 90% were achieved. Conclusion:The proposed method yields reliable gaze estimation performance, and accurately detects fixations, saccades and blinks. Significance:This work proposes an integrated method to simultaneously estimate the GAs and address the baseline drift issue without the limitations of existing baseline drift mitigation techniques.

Ground-to-Air Tropospheric Mapping Function for Very Low Elevation Angles

This paper proposes a real-time capable tropospheric delay model for terrestrial radio navigation systems used for aviation. Its functional scope ranges from elevation angles below the local horizon to zenith. Flight experiments were carried out to quantify the improvement in service coverage achieved for measurements close to and below the local horizon, with aircraft distances in the order of 250 km. Mismodeling tropospheric delays at these elevation angles and aircraft distances can result in slant-range error several order of magnitude greater than the accuracy of terrestrial nav-aids, and adequately accounting them translates into improved accuracy and tighter integrity bounds. The proposed method uses a polynomial representation of the atmospheric refractivity instead of the traditional exponential distribution used in GNSS. The resulting method provides tropospheric corrections accurate to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula> 0.5 cm for elevation angles above 5 degrees, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula> 15 cm at an elevation of 0.2 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{\circ }$</tex-math></inline-formula> . In addition, the proposed model offers a reduction in the horizontal position error in terms of an order of magnitude from 14 meters (for uncompensated tropospheric delay) to below 1.6 cm for all flight segments, that will enable terrestrial systems to support increasingly demanding performance based navigation systems.

Technical Note: Validation of Recon-3D, iPhone LiDAR for bullet trajectory documentation

Bullet trajectory documentation is an important part of shooting reconstructions. Manual methods are quite common, using a protractor and angle gauge, but more advanced methods using laser scanners and photogrammetry are also used for documentation. Past studies have shown that using terrestrial laser scanners (TLS) can reduce the variability in trajectory documentation. In 2020, Apple Inc. released the iPad Pro & iPhone 12 Pro with a Light Detection and Ranging (LiDAR) sensor, which effectively placed a low-cost laser scanner in mobile devices. In May of 2022, an iOS application focused on forensics called Recon-3D was released. Recon-3D uses Apple’s LiDAR sensor and fuses it with photogrammetry to produce 3D point cloud data in the e57 format. The purpose of this research was to compare the accuracy, repeatability, and quality of 3D data, using Recon-3D to the Faro Focus S350 laser scanner in bullet trajectory documentation. As a first test, twelve trajectory rods were installed on a wooden panel in a controlled, indoor environment, in order to optimize the best settings for scanning trajectory rods. Subsequently, a more practical exercise was run outdoors with a vehicle containing five trajectory rods. The mean horizontal angle error for the twelve trajectory rods installed on the wooden panel was 0.14° with a standard deviation of 0.30°. The mean error for the vertical angle was − 0.05° with a standard deviation of 0.25°. For the outdoor vehicle test, the mean error for the horizontal angle was 0.03° with a standard deviation of 0.28°. The mean error for the vertical angle was 0.22° with a standard deviation of 0.36°. These errors are similar to other studies which utilize the terrestrial laser scanner with mean errors well below 1°. Although further work is required to make Recon-3D a more robust application, preliminary results are promising and in line with previous studies where the terrestrial laser scanner has been used. Thus, Recon-3D appears suitable to document trajectory rods when used in shooting reconstructions.

Fundamental Design Concepts of a Modular Pier System Using Ultra-High-Performance Concrete for Solving Construction Errors

Concrete structures in marine environments are prone to deterioration and damage due to chloride ion penetration, freezing and thawing, and chemical erosion. Ultra-high-performance concrete (UHPC) mixed with steel fibers has been proposed as a solution to enhance the durability and mechanical properties of concrete in marine environments. Although several studies have been conducted in this regard, they have yet to focus on addressing errors that may be caused during the construction of offshore piers. Therefore, this study proposes a modular system to control horizontal and vertical errors during construction using a new connecting core type. UHPC with a fiber content of 0.75% was considered the optimum mix proportion because this met the tensile and compressive strength requirements and the chloride attack resistibility requirements of marine structures. The structural performance of a specimen constructed using modular technology was evaluated. The results of the lateral load resistance experiments showed minimal deformation in the girder and pier. Additionally, both the precast and cast-in-place types met the criterion of load resistance. This study contributes to the advancement of construction technology in marine environments by considering both material performance and construction conditions.

Comparison of temperature and wind observations in the Tropics in a perfect‐model, global EnKF data assimilation system

Flow‐dependent errors in tropical analyses and short‐range forecasts are analysed using global observing‐system simulation experiments assimilating only temperature, only winds, and both data types using the ensemble Kalman filter (EnKF) Data Assimilation Research Testbed (DART) and a perfect model framework. The idealised, homogeneous observation network provides profiles of wind and temperature data from the nature run for January 2018 using the National Center for Atmospheric Research (NCAR) Community Earth System Model (CESM) forced by the observed sea‐surface temperature. The results show that the assimilation of abundant wind observations in a perfect model makes the temperature data in the Tropics largely uninformative. Furthermore, the assimilation of wind data reduces the background errors in specific humidity twice as much as the assimilation of temperature observations. In all experiments, the largest analysis uncertainties and the largest short‐term forecast errors are found in regions of strong vertical and longitudinal gradients in the background wind, especially in the upper troposphere and lower stratosphere over the Indian Ocean and Maritime Continent. The horizontal error correlation scales are on average short throughout the troposphere, just several hundred km. The correlation scales of the wind variables in precipitating regions are half of those in nonprecipitating regions. In precipitating regions, the correlations are elongated vertically, especially for the wind variables. Strong positive cross‐correlations between temperature and specific humidity in the precipitating regions are explained using the Clausius–Clapeyron equation.

Investigating two-dimensional deformations of the head of the Central Route of the South–North Water Diversion Project in China with TerraSAR-X datasets

As a landmark in China’s hydraulic engineering history, the Central Route of the South–North Water Diversion Project (SNWD-CR) plays a major role in alleviating severe water shortages in Northern China. Existing InSAR monitoring of the SNWD-CR based on one-dimensional viewing geometry and Sentinel-1 imaging is insufficient for a comprehensive assessment of deformation causes. Moreover, the poor geocoding accuracy hampers the previous interpretation of the estimated deformations. Here we carry out comprehensive two-dimensional deformation analyses over the head of the SNWD-CR by integrating the multitemporal PS/DS InSAR and the stress–strain model, where the well-designed topography and geocoded error correction can help to identify the causes of deformation. We apply our strategy to ascending and descending TerraSAR-X meter-resolution datasets acquired from June 2020 to February 2021. Geocoding errors in both ascending and descending orbits are corrected, and the relationship between topography error and horizontal positioning error is discussed. The multi-data fusion results reveal insignificant deformations in the east–west direction except for a makeshift rigid-frame bridge, whereas the vertical deformation field shows two uplifts and a subsidence area in the narrow canal with a maximum rate of 23 mm/y, which are likely to be related to rainfall and water diversion pressure. Furthermore, the combination of the external DEM and topographic residual offers the chance to reconstruct the high-accuracy DEM.

LiDAR-change-based mapping of sediment movement from an extreme rainfall event

ABSTRACT Mapping landscape change at fine scales (e.g. <1.0 m resolution) using airborne LiDAR data from manned aircraft is a significant challenge. This challenge is magnified in disaster response contexts. A combination of collection and processing factors contributes to horizontal and vertical errors (and resulting uncertainty) in each pre- and post-LiDAR derived digital elevation model (DEM). Subsequently, the errors in the change surface from the two (or more) DEMs are an accumulation of the errors in the individual DEMs. Thus, reliable mapping erosion/deposition changes at sub-meter precision in change detection studies using LiDAR data is largely the domain of terrestrial LiDAR or sUAS with LiDAR scanners rather than manned aircraft. Unfortunately, terrestrial and sUAS LiDAR scanners are not well suited for mapping large areas and sUAS collections are subject to additional airspace constraints compared to manned aircraft. In this study, we probed one of the significant issues in airborne LiDAR change projects – vertical height errors from sequential flight lines. A simplified solution for determining flight line vertical biases in areas of low topographic relief with natural cover types was developed and tested for normalizing point clouds. The approach was tested in a fine-scale erosion/deposition study from an extreme rainfall event that eroded and deposited sand at depths of about 1.0 m. Airborne LiDAR had been collected prior to the rainfall event, and another airborne LiDAR collection was made 1 month after the event. Eleven field campaigns to collect reference data and visit anomalies in the change surface were conducted in a 15-month period after the event, beginning 25 February 2016 and ending 8 May 2017. The validation results indicate accuracies for the pre-event and post-event LiDAR derived DEMs were 7.8 cm and 13.0 cm RMSE, respectively. After modeling vertical errors and corrections applied to the post-event point clouds, the RMSE for the post-event DEM was 8.3 cm. In the depositional use case, 27 locations were sampled with auger boreholes/sand pits and compared with LiDAR-based change. The LiDAR-based change detection analysis resulted in predicted sand depth accuracies of 94% with a mean error of 4.7 cm.

Improved PPP-RTK by Considering the Non-Homogeneous Uncertainty of the ionosphere with a Spatial Three-Direction Model

The ultimate goal of PPP-RTK is to achieve rapid ambiguity resolution, which is influenced by the prior precision of the external ionospheric information. This study proposes a method for determining the precision of ionospheric corrections for each satellite. In this method, an 8 min piece-wise function linearly related to the spatial three-direction distance components (SDC) within the geocentric coordinate system is constructed. By exploiting the SDC model, the user can calculate the precision of the ionospheric corrections satellite by satellite. Using the German and French stations, we validate this method experimentally and compare it to a method with an 8 min piece-wise function constructed by the baseline length (BLL). The SDC model provides an accuracy better than 10 mm in modeling ionospheric correction precision for each GPS satellite, with an average improvement of 43% compared to the BLL model. In addition, the SDC model offers an accuracy of approximately 5 mm in the reference network with an inter-station distance of less than 100 km, which is about 15% better than that of the BLL model during the active ionospheric period. The SDC model exhibits advantages over ionospheric correction precision modeling, with an average improvement of 73.5% for a reference network with station spacing of 125–155 km. By adopting the adaptive ionospheric precision derived from the SDC model, the GPS/GPS + Galileo PPP-RTK achieves a horizontal error of 50 mm and a vertical error of 100 mm within an average of three to four epochs. Notably, the convergence time is significantly enhanced by 30% in reference networks with inter-station distances of 125–155 km, compared to that of the PPP-RTK solution generated with dynamic ionospheric correction precision from the BLL model for all observed satellites.