Single-frequency Receivers Research Articles

Tropospheric delay retrieval using cost-effective single-frequency GNSS receivers with established reference stations

This study explores the feasibility of using single-frequency (SF) receivers to achieve Zenith Total Delay (ZTD) retrievals with existing reference stations via the Satellite-specific Epoch-differenced Ionospheric delay (SEID) model. Over 30 days, data from a low-cost SF receiver was collected under varying ionospheric conditions. Two reference network configurations were tested: scheme 1 with an average inter-station distance of 138 km and scheme 2 with a distance of 431 km. The GPS-only solution exhibited Root Mean Square (RMS) of 4.2 mm and 11.3 mm for schemes 1 and 2, respectively. Incorporating GLONASS data had minimal impact on scheme 1 but increased the mean bias in scheme 2, likely due to residual ionospheric delays over the larger area. ZTD estimates from reanalysis products fell between those of the two schemes. This study demonstrates the viability of cost-effective, single-frequency receivers for precise ZTD estimation, advocating for network expansion through additional low-cost receivers.

Improved Ionospheric Total Electron Content Maps over China Using Spatial Gridding Approach

Improved Ionospheric Total Electron Content Maps over China Using Spatial Gridding Approach

A reduced-condition-number algorithm for single-frequency precise point positioning based on regularized Kalman filter

Single-frequency receivers are low cost and portable, thus being widely applied in engineering; the extended Kalman filter (EKF) is commonly used to perform single-frequency precise point positioning (SF-PPP). However, the positioning performance of SF-PPP is seriously influenced by various errors. Due to the large process noise and initial variance of the estimated parameters, the weight matrix of state parameters will be ill-conditioned, and since the noise of the pseudo-range is much higher than that of the carrier phase, the weight matrix of observations presents as ill-conditioned. Additionally, the condition number of the normal matrix will jump on the conditions of cycle slip, new emerging satellites, and signal outages. To reduce the condition number of the normal matrix, the regularized Kalman filter (RKF) algorithm is proposed, with additional support for the maximum variance matrix and singular value decomposition, thereby improving the accuracy and stability of SF-PPP. Through static and dynamic experiments, it is found that the proposed method can reduce both the ill-conditioning of the weight matrices of the observations and the state parameters. The condition number of the normal matrix is <500 per epoch, and the convergence time is shortened by >40%. Compared with the SF-PPP using EKF, centimeter-level static positioning accuracies of 1.13, 0.73, and 2.92 cm and decimeter-level kinematic positioning accuracies of 12.5, 10.8, and 27.3 cm in the east, north, and vertical components, respectively, using RKF; this yielded 38.3, 29.8, and 45.2% and 39.6, 41.9, and 21.3% improvement in the static and kinematic scenarios, respectively.

Characterizing Ionospheric Effects on GNSS Reflectometry at Grazing Angles from Space

Coherent observations in GNSS reflectometry are prominent in regions with smooth reflecting surfaces and at grazing elevation angles. However, within these lower elevation ranges, GNSS signals traverse a more extensive atmospheric path, and increased ionospheric effects (e.g., delay biases) are expected. These biases can be mitigated by employing dual-frequency receivers or models tailored for single-frequency receivers. In preparation for the single-frequency GNSS-R ESA “PRETTY” mission, this study aims to characterize ionospheric effects under variable parameter conditions: elevation angles in the grazing range (5° to 30°), latitude-dependent regions (north, tropic, south) and diurnal changes (day and nighttime). The investigation employs simulations using orbit data from Spire Global Inc.’s Lemur-2 CubeSat constellation at the solar minimum (F10.7 index at 75) on March, 2021. Changes towards higher solar activity are accounted for with an additional scenario (F10.7 index at 180) on March, 2023. The electron density associated with each reflection event is determined using the Neustrelitz Electron Density Model (NEDM2020) and the NeQuick 2 model. The results from periods of low solar activity reveal fluctuations of up to approximately 300 TECUs in slant total electron content, 19 m in relative ionospheric delay for the GPS L1 frequency, 2 Hz in Doppler shifts, and variations in the peak electron density height ranging from 215 to 330 km. Sea surface height uncertainty associated with ionospheric model-based corrections in group delay altimetric inversion can reach a standard deviation at the meter level.

A novel approach to enhancing the Klobuchar algorithm to mitigate the effect of ionospheric delay errors on static single-frequency receivers

Abstract The demand for real-time high-precision positioning for global navigation satellite system applications is difficult to satisfy. In this regard, a single-frequency receiver is found to play an important role in overcoming this challenge, especially in developing countries where economic factors are a major restriction. Hence, the development of built-in models, such as the Klobuchar model, is an important objective for single-frequency users to mitigate the effect of ionospheric delay errors in real-time applications. Accordingly, this study aims to devise a new approach to enhance the behavior of the Klobuchar model and increase its efficiency in resolving the aforementioned problem. The new approach seeks to enhance the behavior of the Klobuchar model without refining or increasing its coefficients. To eliminate the ionospheric delay disturbance, the proposed methodology applies normalization and filtration processes to the raw ionospheric delay probability distribution estimated by the unified least squares technique. A final assessment of the new method for enhancing the Klobuchar behavior in predicting the precise position of a single-frequency static receiver under different weather conditions around the globe is presented in this paper.

GNSS-based baseline vector determination for widely separated cooperative satellites using L1-only receivers

Real time estimation of the relative position and velocity vectors between two satellites in a formation is an integral part of the formation control loop. Relative positioning based on Global Navigation Satellite Systems (GNSS) has been a dominating technology for formation missions in LEO, where extremely precise estimates could be obtained for formations with small inter-satellite distances (1-10 km). Larger baselines between the satellites (>10 km) are more challenging as they pose the problem of huge differences in the ionospheric delays experienced by the signals received by each receiver. This problem could be mitigated by using precise ionospheric-free combinations that could only be obtained by dual-frequency receivers, which is not a cost-efficient option for the modern low-cost miniature missions. In this paper, the problem of GNSS-based relative navigation between two spacecraft with large inter-satellite distance which are equipped with single-frequency receivers is treated through adopting the space-proven Extended Kalman Filter (EKF). Although using an EKF for relative navigation is a common practice, there are many variants of the filter settings, which vary in terms of the state and measurement vectors to be adopted as well as the techniques to be used to handle the ionospheric delay. In this research, optimal settings of the filter are sought for the problem of relative baseline vector estimation between two spacecraft that have large separation and which are equipped with with single-frequency GNSS receivers.

Cycle slip detection and repair based on the unmodeled-error-constrained geometry-free combining geometry-based models for a single-frequency receiver

Accurate cycle slip detection and repair is the prerequisite of high-precision and high-reliability Global Navigation Satellite System (GNSS) positioning and navigation. The cycle slips are widely existent in a challenging environment or a low-cost receiver, where the unmodeled errors are usually significant. Therefore, it is highly urgent to enhance the performance of cycle slip detection and repair for a single-frequency receiver under complex conditions. In this paper, a new cycle slip processing method based on the unmodeled-error-constrained geometry-free (GF) combining geometry-based (GB) models is proposed. Specifically, first, the short-term GF and GB unmodeled errors are predicted based on their temporal behaviours. Second, the predicted GF and GB unmodeled errors are applied to correct the potentially biased GF and GB models. Finally, the cycle slips are detected and repaired by combing the unmodeled-error-constrained GF and GB models. By using the real dataset with multipath effects and ionospheric delays, the proposed short-term prediction of unmodeled errors is effective both in GF and GB models, where the centimetre-level accuracy can be obtained, respectively. Based on this, three different typical situations, i.e., large, small, and mixed cycle slips, are all analysed. The results show that compared with the traditional methods, the proposed method considering the unmodeled errors is the most effective. Precisely, it can process the cycle slips with a detection percentage of 100% and a mean repair percentage of 93% even under challenging environments.

Accuracy assessment of high-resolution terrain data produced from UAV images georeferenced with on-board PPK positioning

In recent years, unmanned aerial vehicles (UAVs) have increased rapidly and successfully established themselves as a tool for the rapid collection of high-resolution images as baseline data in land cover studies and topographic mapping. In photogrammetry using the SfM-MVS method of processing captured images, indirect georeferencing of the digital data through ground control points (GCPs) is usually applied. But selecting, marking, and coordinating GCPs in hardto-reach terrains is time-consuming and sometimes dangerous or impossible. The main objective of this study is to evaluate the accuracy of high-resolution topographic data (HRTD) products of photogrammetric processing of PPK-directly georeferenced images by SfM-MVS workflow. Direct and indirect methods of georeferencing digital products are compared. The planimetric and vertical root mean square error (RMSE) in the position of the validation points were calculated by the differences between measured coordinates in dense point clouds, orthophoto mosaics, and terrain surfaces (DSM), and precisely measured coordinates of the validation points by GNSSRTK receivers. The analysis is based on a statistical evaluation of experimental data obtained from a TAROT X6-based hexacopter equipped with two different image sensor configurations: 1) Sony RX0 action camera and 2) Sony A6000 mirrorless camera, and 3) DJI Phantom 4 Pro quadcopter with integrated additional L1-GNSS module for direct georeferencing by PPKmethod. HRTD generation was performed with three block control configurations for each UAV: 1) Indirect georeferencing via GCP only, 2) PPK direct georeferencing without GCP, and 3) PPK georeferencing using one GCP. Our research showed that when using L1-GNSS onboard receivers for PPK-georeferencing without any GCPs, the point cloud&amp;#39;s planimetric accuracy (RMSExy) was from 0.125 to 0.231 cm, depending on the UAV/camera configuration. However, two flight missions produced significant vertical offsets, most likely due to ionospheric disturbances affecting the resolution of phase cycle ambiguities in the single-frequency receivers used. When adding one control point in the PPK georeferencing method, the planimetric and vertical accuracy of the data is comparable to the indirect GCP referencing method. Furthermore, our results show that camera properties (i.e., focal length, resolution, sensor quality) affect the quality and accuracy of digital products. The HRTDs were also evaluated according to the ASPRS (American Photogrammetry and Remote Sensing Society) Standards for Accuracy of Digital Geospatial Data. Analyzing the accuracy of the HRTDs obtained with the experimental UAV/camera configurations for the test area, the present study shows that the PPK-SfM-MVS workflow can provide quality data with a centimeters accuracy of the photogrammetric products.

An Ionospheric Total Electron Content Model with a Storm Option over Japan Based on a Multi-Layer Perceptron Neural Network

Ionospheric delay has a severe effect on reducing the accuracy of positioning and navigation of single-frequency receivers. Therefore, it is necessary to construct a precise regional ionospheric model for real-time Global Navigation Satellite System (GNSS) applications. The total electron contents (TECs) of 839 GNSS stations affiliated with the GPS Earth Observation Network were used to build a Japanese ionospheric model (JIM) based on a multi-layer perceptron neural network. During quiet space conditions, the correlation coefficient between the targets and the predictions of the JIM was about 0.98, and the root-mean square error (RMSE) of TEC residuals was ~1.5TECU, while under severe space events, the correlation coefficient increased to 0.99, and the corresponding RMSE dropped to 0.96 TECU. Moreover, the JIM model successfully reconstructed the two-dimensional (time vs latitude) TEC maps, and the TEC maps had evident hourly and seasonal variations. Most of TEC residuals accumulated between universal time 01–06 with an averaged magnitude of 1-2TECU. Furthermore, the JIM model had a perfect prediction performance under various kinds of complex space environments. In the quiet days, the prediction accuracy of the JIM was nearly equal to the global ionosphere map (GIM), and in some moments, the JIM was more competitive than the GIM. In the disturbed days, the RMSEs of TEC residuals were proportional to the solar wind speed and were inversely proportional to the geomagnetic Dst value. The maximum RMSE of the JIM was lower than 2TECU, while the corresponding RMSEs for the IRI and TIE-GCM exceeded 5TECU.

Estimation and impact of GLONASS inter-frequency code bias in single-frequency precise point positioning

The current GLONASS system suffers from the inter-frequency code bias (IFCB) and the inter-frequency phase bias (IFPB) due to its employment of the frequency division multiple access (FDMA) technology. The IFCB affects the code observation’s accuracy and further degrades the positioning accuracy, while the IFPB can be absorbed into the float ambiguity term. With increasing demand on precise point positioning (PPP) in mass market applications using low-cost single-frequency (SF) receivers, the impact of IFCB on SF-PPP with low-cost receivers need to be investigated. In this paper, we propose the use of the University of Calgary (UofC) model to estimate the IFCB in SF-PPP. The feasibility of the proposed approach is first verified using GNSS observations from the IGS stations. Comparing to the traditional SF-PPP model, the UofC model enables more stable IFCB estimations with 80.2% of the maximum changes less than 1.0 ns within one week. Then, the stability of IFCB in the low-cost SF receiver is analyzed with static data collected with u-blox M8T, and the STandard Deviation (STD) of IFCB for each satellite within 5 days is less than 1.5 ns when applying real-time orbit, clock and ionospheric corrections. Further, it’s verified that the power off won’t affect the IFCB’s value significantly in the low-cost receiver. Finally, six different strategies are designed to evaluate the impact of the GLONASS IFCB on SF-PPP. Comparing to those strategies ignoring IFCB, the strategies considering IFCB in SF-PPP can improve the positioning accuracy in both static and kinematic tests.

Evaluation of the weighted least square based Receiver Autonomous Integrity Monitoring (RAIM) for single frequency GNSS receivers

Single frequency receivers for Global Navigation Satellite System (GNSS) are low-cost and easily accessible, deeming them cost-effective for commercial applications requiring localization and tracking. However, the positioning solution these receivers provide must conform to desired reliability standards. A reliability check is essential in the signal-degraded environment, where a position fix might be unacceptably inaccurate. Under such conditions, faulty measurements need to be identified and excluded to ensure the system's integrity and get a reliable solution. In this paper, a Receiver Autonomous Integrity Monitoring (RAIM) scheme is devised that employs different tests. The first test performs a global sweep on all epochs and raises the alarm after detecting the fault. Next, a group of tests are performed to identify the satellite responsible for producing the faulty measurements. Once identified, it is then excluded from the measurements to maintain the reliability of the positioning provided by the receiver. The results indicate that a polling scheme based on multiple tests identifies the faulty satellite correctly and minimizes the false alarm rate if either of these tests is performed individually.

MT-e&amp;R: NMEA Protocol-Assisted High-Accuracy Navigation Algorithm Based on GNSS Error Estimation Using Multitask Learning

Accurate location data of ground vehicles is very important for various intelligent transportation applications. As one of the most commonly-used navigation solutions at present, the GNSS/INS integrated navigation still cannot meet the accuracy and stability command of current applications. This paper presents a National Marine Electronics Association (NMEA) protocol data-assisted high-accuracy navigation algorithm based on GNSS position error estimation using multi-task learning (MT-e&R), which can accurately estimate the GNSS position error and GNSS measurement noise covariance matrix with the assistance of protocol data and a multi-task learning model. Extensive experimental results on practical navigation data collected in various urban environments of Beijing demonstrate that our proposed approach can improve the performance of integrated navigation system. The positioning errors of integrated navigation equipped with single-frequency receiver are reduced by 36.17% and 39.58% for double-frequency receiver, which confirms the reasonable environmental adaptability of our proposed MT-e&R algorithm.

Investigation of Precise Single-Frequency Time and Frequency Transfer with Galileo E1/E5a/E5b/E5/E6 Observations

With the rapid upgrade of global navigation satellite system (GNSS) single-frequency (SF) receivers and the increasing market demand for low-cost hardware, SF precise point positioning (PPP) technology has been widely applied in the time and frequency field. The five-frequency signals provided by the whole constellation of Galileo bring more opportunities for the application of SF PPP in time and frequency transfer. In this contribution, using Galileo’s multi-frequency observations, three SF PPP time and frequency transfer models, i.e., the un-combined (UC) model, the ionosphere-free-half (IFH) model, and the ionosphere-weighted constraints (IWCs) model are established. SF PPP time and frequency transfer performance with Galileo E1, E5a, E5b, E5, and E6 multi-frequency observations is evaluated using four links (947.7 km to 1331.6 km) with five external high-precision atomic clocks stations. The results show that the time and frequency transfer performance of SF-UC and SF-IWC is better than that of SF-IFH, and the timing accuracy of SF-UC and SF-IWC is similar. SF PPP time and transfer performance with E5, E5a, E5b, and E6 signals is improved compared with traditional E1 signal. Among them, the frequency stability of E5 improves the most (about 58%), and that of E6 improves the least (about 14%). In addition, the difference in frequency stability between SF and double-frequency (DF) PPP decreases gradually with an increase in average time, and the frequency stability difference between SF and DF PPP can reach 2 × 10−16 in 120,000 s, indicating that SF PPP has the potential to achieve DF PPP frequency stability. Considering the possible frequency data loss during actual observation, the cost of the GNSS SF receiver, and the advantages of Galileo multi-frequency observations, SF PPP can also meet the long-time time and frequency transfer requirements, and the SF-IWC model based on Galileo E5 observations is more recommended.

A BDGIM-Based Phase-Smoothed Pseudorange Algorithm for BDS-3 High-Precision Time Transfer

Single point positioning (SPP) can meet the requirements of the majority of real-time time transfer applications. Meanwhile, a single-frequency (SF) receiver is cheaper than a dual-frequency receiver. However, SPP performance can be greatly affected by large pseudorange observation noise. Phase smoothing the pseudorange is an effective approach to reduce pseudorange noise. Since the classical phase-smoothed pseudorange algorithm does not account for the effect of ionosphere delay, we propose a BDGIM-based phase-smoothed pseudorange algorithm to eliminate the ionospheric delay and apply it to BeiDou Navigation Satellite System (BDS-3) SPP time transfer. In this paper, we first evaluate the performance of the BeiDou global ionospheric delay correction model (BDGIM) and compare it with that of the BeiDou Klobuchar model to determine if it is practical to incorporate the BDGIM into our suggested method. The performance of the BDGIM is better than that of the Klobuchar model. The mean RMS value of the BDGIM is 2.6 Total Electron Content Unit (TECU). The average ionospheric correction rate of the BDGIM is 75.5%. Then, we investigate the performance of the improved SF SPP time transfer. The performance of the improved SPP time transfer is much better than that of the traditional SPP time transfer. Compared with the traditional time transfer, the average Type A uncertainty of the improved time transfer is 2.08 ns, which is reduced by about 11.1% from the time transfer without it. Regarding frequency stability, the modified Allan deviations of the improved time transfer are 1.43E-12 and 1.68E-13 at 960 s and 61,440 s, with improvements of 51.2% and 59.9%, respectively.

Multiple Cycle Slip Detection Algorithm for a Single Frequency Receiver.

A satellite navigation system makes it simple to find and navigate to a specific position. Although a carrier measurement is required to establish a precise position due to the characteristics of the carrier observation, it is difficult to determine a robust position in a poor signal reception environment such as urban areas. Various studies are being carried out to overcome this problem, with cycle slips being the most important factor. With only a single frequency, it is very challenging to detect cycle slips in multiple satellite channels at the same time. A geometry-based technique is proposed in this study as a technical solution for detecting simultaneous cycle slips for multiple channels utilizing only a single-frequency receiver. The method could detect a half-wavelength size of cycle slip for each channel through the geometry information.

An improved time-domain multipath mitigation method based on the constraint of satellite elevation for low-cost single frequency receiver

Multipath effect seriously affects the accuracy of GNSS high precision positioning since it cannot be corrected by empirical model or eliminated by difference techniques. Sidereal filtering method can effectively mitigate multipath error for static positioning by using the characteristics of satellite time-domain repetition. The key of this method is to extract accurate multipath model on the reference day and mitigate multipath error by searching the multipath model accurately on the subsequent observation day. We analyze the relationship between the multipath error and satellite elevation first by using the data sets which collected from the Multi-GNSS Experiment (MGEX) stations. Based on the result, an improved time-domain sidereal filtering multipath mitigation method based on satellite elevation constraints is proposed. The information of satellite elevation is used to constrain the threshold strategy and decomposition layer of wavelet packets transform, which can effectively improve the denoising effect and extract a more precision multipath model. In addition, the performance of positioning of the proposed method at different mask angles is also evaluated. Real short baseline data sets are collected by low-cost single frequency receivers to evaluate the performance of the positioning of the proposed method, experimental results show that using the proposed method in multipath mitigation can improve approximately 25.00, 21.43 and 13.98% in the east, north and up components compared with the original results, respectively. Even compared with the traditional wavelet packets transform method, the improvement rate can reach approximately 10.78% for 3D positioning. Moreover, the proposed method can also improve the performance of positioning at different mask angles to a certain degree, especially for the low mask angle seriously affected by multipath effect.

Gauss process regression for real-time ionospheric delay estimation from GNSS observations

The number of devices equipped with global satellite positioning has exceeded seven billion recently. There are a wide variety of receivers regarding their accuracy and reliability. Low cost, multi-frequency units have been released on the market latterly; however, the number of single-frequency receivers is still significant. Since their measurements are influenced by ionospheric delay, accurate ionosphere models are of utmost importance to reduce the effect. This paper summarizes how Gauss process regression (GPR) can be applied to derive near real-time regional ionosphere models using raw Global Navigation Satellite System (GNSS) observations of permanent stations. While Gauss process is widely used in machine learning, GPR is a nonparametric, Bayesian approach to regression. GPR has several benefits for ionosphere monitoring since it is quite robust and efficient to derive a grid model from data available in irregular set of ionospheric pierce points. The corresponding instrumental delays are estimated by a parallel Kalman filter. The presented algorithm can be applied near real-time, however the results are offline calculated and are compared to two high quality TEC map products. Based on the analysis, the accuracy of the GPR modell is in 2 TECu range. The developed methods could be efficiently applied in the field of autonomous vehicle navigation with meeting both accuracy and integrity requirements.

The implications of ionospheric disturbances for precise GNSS positioning in Greenland

Ionospheric irregularities impair Global Navigation Satellite System (GNSS) signals and, in turn, affect the performance of GNSS positioning. Such effects are especially evident at low and high latitudes, which are currently gaining the attention of research and industry sectors. This study evaluates the impact of ionospheric irregularities on GNSS positioning in Greenland. We assess the performance of positioning methods that meet the demands of a wide range of users. In particular, we address the needs of the users of mass-market single-frequency receivers and those who require a solution of high precision provided by geodetic dual-frequency receivers. We take advantage of the datasets collected during three ionospheric storms: the St. Patrick’s Day storm of March 17, 2015, the storm on June 22, 2015, and another on August 25–26, 2018. We discover a significant impact of the ionospheric disturbances on the ambiguity resolution performance and the accuracy of the float solution in Real Time Kinematics (RTK) positioning. Next, assessing the single-frequency ionosphere-free Precise Point Positioning (PPP), we demonstrate that the model is generally unaffected by ionospheric disturbances. Hence, the model is predestined for the application by the users of single-frequency receivers in the areas of frequent ionospheric disturbances. Finally, based on the observation analyses, we reveal that phase signals on the L2 frequency band are more prone to cycle slips induced by ionospheric irregularities than those transmitted on the L1. Such signal properties explain a noticeable decline in the dual-frequency RTK performance during the ionospherically disturbed period and merely no effect for the single-frequency ionosphere-free PPP model.

Performance of BDS-2/3, GPS, and Galileo Time Transfer with Real-Time Single-Frequency Precise Point Positioning

Single-frequency (SF) receivers are much cheaper than that of dual-frequency (DF). Even though DF precise point positioning (PPP) is nowadays applied in the time community, the cost of equipment is one of the key considerations for time users. Furthermore, the hardware delay calibration of single-frequency devices is simpler than that of dual-frequency devices. In addition, there is no literature to study real-time SF PPP time transfer. In this contribution, the possibility of time transfer using SF PPP was studied. The Un-combined SF PPP was employed for time transfer with ionospheric-constraint using real-time precise products. In this case, 18 multi-GNSS experiment (MGEX) stations and one time lab station were used to study real-time SF PPP time transfer using GPS, Galileo and BDS-2/3 satellites with 20-day. The results suggested that real-time single-frequency PPP can meet time transfer. the standard deviation (STD) of the clock difference obtained from GPS-only, Galileo-only and BDS-2/3 single-frequency PPP are about (0.51, 0.54, 0.91) ns, respectively. The frequency stability of real-time single-frequency PPP can achieve (1E-12, 1E-13, 1E-13) level at short-term and (1E-13, 1E-13, 1E-14) level at long-term, respectively, for BDS-2/3, Galileo-only and GPS-only based.

Influence of the ionosphere on the accuracy of the satellite navigation system

The presence of large and small-scale irregularities TID of the ionosphere in the path of radio wave propagation from the navigation satellite to the GPS receiver is the main challenge for the application of ionospheric research to improve positioning accuracy using single-frequency receivers. In this paper, in contrast to the traditional methods of compensating for the error associated with the ionospheric delay (the Klobuchar method and the use of TEC values), we propose the calculation of the ionospheric delay by the ray tracing method in a model that is as close as possible to the real conditions. To achieve this proximity, a model is proposed that includes various types of TID irregularities. The corresponding algorithms are presented and the numerical results are given. It is shown that single large-scale TIDs can lead to errors of 10–30 m, errors due to small-scale TIDs do not exceed 2 m. © 2001 Elsevier Science. All rights reserved.