GNSS global real-time augmentation positioning: Real-time precise satellite clock estimation,prototype system construction and performance analysis

Lots of ambiguities in un-differenced (UD) model lead to lower calculation efficiency, which isn’t appropriate for the high-frequency real-time GNSS clock estimation, like 1 Hz. Mixed differenced model fusing UD pseudo-range and epoch-differenced (ED) phase observations has been introduced into real-time clock estimation. In this contribution, we extend the mixed differenced model for realizing multi-GNSS real-time clock high-frequency updating and a rigorous comparison and analysis on same conditions are performed to achieve the best real-time clock estimation performance taking the efficiency, accuracy, consistency and reliability into consideration. Based on the multi-GNSS real-time data streams provided by multi-GNSS Experiment (MGEX) and Wuhan University, GPS + BeiDou + Galileo global real-time augmentation positioning prototype system is designed and constructed, including real-time precise orbit determination, real-time precise clock estimation, real-time Precise Point Positioning (RT-PPP) and real-time Standard Point Positioning (RT-SPP). The statistical analysis of the 6 h-predicted real-time orbits shows that the root mean square (RMS) in radial direction is about 1–5 cm for GPS, Beidou MEO and Galileo satellites and about 10 cm for Beidou GEO and IGSO satellites. Using the mixed differenced estimation model, the prototype system can realize high-efficient real-time satellite absolute clock estimation with no constant clock-bias and can be used for high-frequency augmentation message updating (such as 1 Hz). The real-time augmentation message signal-in-space ranging error (SISRE), a comprehensive accuracy of orbit and clock and effecting the users’ actual positioning performance, is introduced to evaluate and analyze the performance of GPS + BeiDou + Galileo global real-time augmentation positioning system. The statistical analysis of real-time augmentation message SISRE is about 4–7 cm for GPS, whlile 10 cm for Beidou IGSO/MEO, Galileo and about 30 cm for BeiDou GEO satellites. The real-time positioning results prove that the GPS + BeiDou + Galileo RT-PPP comparing to GPS-only can effectively accelerate convergence time by about 60%, improve the positioning accuracy by about 30% and obtain averaged RMS 4 cm in horizontal and 6 cm in vertical; additionally RT-SPP accuracy in the prototype system can realize positioning accuracy with about averaged RMS 1 m in horizontal and 1.5–2 m in vertical, which are improved by 60% and 70% to SPP based on broadcast ephemeris, respectively.     

联合北斗导航与星间链路的大椭圆卫星定轨方法

传统的地面测控和GNSS均无法实现HEO卫星全弧段的跟踪观测.在分析北斗导航信号及其星间链路信号对典型HEO的观测几何及覆盖特性的基础上,利用北斗导航及其星间链路对HEO测控支持形成互补的特点,提出了一种卫星导航与星间链路相结合的自主导航方法.对HEO定轨进行分段划分并基于EKF设计了卫星导航与星间链路数据融合定轨的自主导航算法.分析结果表明,本文提出的方法能够从全弧段上改善HEO的观测几何,定轨精度比仅使用卫星导航提高了2个数量级,并且仅需较少的星间链路资源.      

BeiDou-3 orbit and clock quality of the IGS Multi-GNSS Pilot Project

Within the Multi-GNSS Pilot Project (MGEX) of the International GNSS Service (IGS), precise orbit and clock products for the BeiDou-3 global navigation satellite system (BDS-3) are routinely generated by a total of five analysis centers. The processing standards and specific properties of the individual products are reviewed and the BDS-3 orbit and clock product performance is assessed through direct inter-comparison, satellite laser ranging (SLR) residuals, clock stability analysis, and precise point positioning solutions. The orbit consistency evaluated by the signal-in-space range error is on the level of 4–8 cm for the medium Earth orbit satellites whereas SLR residuals have RMS values between 3 and 9 cm. The clock analysis reveals sytematic effects related to the elevation of the Sun above the orbital plane for all ACs pointing to deficiencies in solar radiation pressure modeling. Nevertheless, precise point positioning with the BDS-3 MGEX orbit and clock products results in 3D RMS values between 7 and 8 mm.     

Precise coseismic displacements from the GPS variometric approach using different precise products: Application to the 2008 MW 7.9 Wenchuan earthquake

The Global Positioning System (GPS) variometric approach has emerged as an attractive alternative to traditional well-developed positioning techniques including relative positioning and precise point positioning. Previous studies have demonstrated the capability of the variometric approach to retrieve coseismic displacements at centimeter-level precision, in a real-time manner using only readily available broadcast ephemeris. This study presents the first results comparing the performance of the variometric approach by using a variety of precise satellite orbit and clock products. Totally six kinds of products are included in our evaluation, namely the broadcast, IGS (International GNSS Service) ultra-rapid (predicted), ultra-rapid (observed), rapid, final (30-s clock) and CODE (Center for Orbit Determination in Europe) final (5-s clock) products. Static and dynamic experiments are conducted using 1-Hz GPS data covering a relatively large area in China during the 2008 Wenchuan M_W 7.9 earthquake. After removing the linear trend, the displacements using broadcast, ultra-rapid (predicted), ultra-rapid (observed) and rapid products reach nearly equivalent precisions at centimeter level. By using final and CODE final products, the precision of displacements can be significantly improved from 1.9–2.0 cm to 0.4–0.7 cm horizontally, and from 6.0–6.2 cm to 1.0–1.7 cm vertically for the dynamic experiments. The displacements using the CODE final products achieve the best precision, improved by more than 40% compared to those using the IGS final products. With the availability of IGS high-rate real-time precise products, this approach is promising to capture coseismic displacements more precisely in real time, which is crucial for earthquake and tsunami early warning.     

Performance assessment of real-time orbit determination for the Haiyang-2D using onboard BDS-3/GPS observations

The Global Navigation Satellite System (GNSS) receivers equipped on the Haiyang-2D (HY-2D) satellite is capable of tracking the signals of both the third generation of BeiDou satellite navigation System (BDS-3) and the Global Positioning System (GPS), which make it feasible to assess the performance of real-time orbit determination (RTOD) for the HY-2D using onboard GNSS observations. In this study, the achievable accuracy and convergence time of RTOD for the HY-2D using onboard BDS-3 and GPS observations are analyzed. Benefiting from the binary-offset-carrier (BOC) modulation, the BDS-3 C1X signal includes less noise than the GPS C1C signal, which has the same signal frequency and chipping rate. The root mean squares (RMS) of the noises of C1X and C1C code measurements are 0.579 m and 1.636 m, respectively. Thanks to a ten-times higher chipping rate, the code measurements of BDS-3 C5P, GPS C1W and C2W are less noisy. The RMS of code noises of BDS-3 C5P, GPS C1W, and C2W are 0.044 m, 0.386 m, and 0.272 m, respectively. For the HY-2D orbit, the three-dimensional (3D) and radial accuracies can reach 31.8 cm and 7.5 cm with only BDS-3 observations, around 50 % better than the corresponding accuracies with GPS. Better performance of the BDS-3 in RTOD for the HY-2D is attributed to the high quality of its broadcast ephemeris. When random parameters are used to absorb ephemeris errors, substantial improvement is seen in the accuracy of HY-2D orbit with either BDS-3 or GPS. The 3D RMS of HY-2D orbit errors with BDS-3 and GPS are enhanced to 23.1 cm and 33.6 cm, and the RMS of the radial components are improved to 6.1 cm and 13.3 cm, respectively. The convergence time is 41.6 and 75.5 min for the RTOD with BDS-3 and GPS, while it is reduced to 39.2 and 27.4 min after the broadcast ephemeris errors are absorbed by random parameters. Overall, the achievable accuracy of RTOD with BDS-3 reaches decimeter level, which is even better than that with GPS, making real-time navigation using onboard BDS-3 observations a feasible choice for future remote sensing missions.     

卫星编队飞行的相对轨道自主确定研究

为了研究卫星编队飞行相对轨道的自主确定,基于相对轨道根数建立编队卫星间的相对运动方程,利用测量所得到的星间距离和方位信息作为观测量。不同于目前广泛采用的扩展卡尔曼滤波算法,设计Unscented Kalman Filter(UKF)算法实现卫星编队飞行的相对轨道自主确定。仿真结果表明这种相对轨道自主确定方案能获得较高的定轨精度。     

Comprehensive outage compensation of real-time orbit and clock corrections with broadcast ephemeris for ambiguity-fixed precise point positioning

The main challenge in real-time precise point positioning (PPP) is that the data outages or large time lags in receiving precise orbit and clock corrections greatly degrade the continuity and real-time performance of PPP positioning. To solve this problem, instead of directly predicting orbit and clock corrections in previous researches, this paper presents an alternative approach of generating combined corrections including orbit error, satellite clock and receiver-related error with broadcast ephemeris. Using ambiguities and satellite fractional-cycle biases (FCBs) of previous epoch and the short-term predicted tropospheric delay through linear extrapolation model (LEM), combined corrections at current epoch are retrieved and weighted with multiple reference stations, and further broadcast to user for continuous enhanced positioning during outages of orbit and clock corrections. To validate the proposed method, two reference station network with different inter-station distance from National Geodetic Survey (NGS) network are used for experiments with six different time lags (i.e., 5 s, 10 s, 15 s, 30 s, 45 s and 60 s), and one set of data collected by unmanned aerial vehicle (UAV) is also used. The performance of LEM is investigated, and the troposphere prediction accuracy of low elevation (e.g., 10–20degrees) satellites has been improved by 44.1% to 79.0%. The average accuracy of combined corrections before and after LEM is used is improved by 12.5% to 77.3%. Without LEM, an accuracy of 2–3 cm can be maintained only in case of small time lags, while the accuracies with LEM are all better than 2 cm in case of different time lags. The performance of simulated kinematic PPP at user end is assessed in terms of positioning accuracy and epoch fix rate. In case of different time lags, after LEM is used, the average accuracy in horizontal direction is better than 3 cm, and the accuracy in up direction is better than 5 cm. At the same time, the epoch fix rate has also increased to varying degrees. The results of the UAV data show that in real kinematic environment, the proposed method can still maintain a positioning accuracy of several centimeters in case of 20 s time lag.     

基于实时SSR数据的LEO卫星精密单点定位研究

对目前低轨卫星实时定位的方法进行了研究,现在通常采用GPS定位,使用广播星历和普通晶振,实时定位精度一般在10m以内,不能满足高精度实时定位的需求。IGS组织在全球范围内对GPS跟踪分析,生成精密星历、精密钟差产品、按SSR格式的广播星历和钟差修正产品并在网上发布。对这些IGS产品进行了调查,提出在现有测控支持情况下,可以通过高密度上注SSR信息流实现在轨高精度定位。以某型号低轨微小卫星在轨导航增强载荷为应用背景,用IGS03产品中的1057和1058数据对双频GPS接收机的星历和钟差进行修正,采用递推最小二乘估计和LAMDA模糊度固定过对载波相位和伪距信息进行处理,在短时间内获得亚米级定位结果。     

北斗导航星座星间通信速率控制方法

北斗导航星座可以通过星间测距和传输链路实现自主定轨和性能增强。导航卫星间进行数据传输时,卫星相对位置时变,传输信道特性也随之不断发生变化。针对导航卫星间传输链路时变特性,提出了一种基于星历的星间通信速率控制方法。在满足传输服务质量的需求下,根据导航卫星自有的高精度星历资源定量计算星间最优通信速率,通过速率的动态调整提高星座的传输效能。仿真结果表明,采用所提方法,北斗导航星座星间传输效能可以提高1.92倍,验证了方法的有效性。      

Near real-time BDS GEO satellite orbit determination and maneuver analysis with reversed point positioning

The Geostationary Earth Orbit (GEO) satellite is a crucial part of the BeiDou Navigation Satellite System (BDS) constellation. However, due to various perturbation forces acting on the GEO satellite, it drifts gradually over time. Thus, frequent orbit maneuvers are required to maintain the satellite at its designed position. During the orbit maneuver and recovery periods, the orbit quality of the maneuvered satellite computed with broadcast navigation ephemeris will be significantly degraded. Furthermore, the conventional dynamic Precise Orbit Determination (POD) approach may not work well, because of a lack of publicly available satellite information for modeling the thrust forces. In this paper, a near real-time approach free of thrust forces modeling is proposed for BDS GEO satellite orbit determination and maneuver analysis based on the Reversed Point Positioning (RPP). First, the station coordinates and receiver clock offsets are estimated by GPS/BDS combined Single Point Positioning (SPP) with single-frequency phase-smoothed pseudorange observations. Then, with the fixed station coordinates and receiver clock offsets, the RPP method can be conducted to determine the GEO satellite orbits. When no orbit maneuvers occur, the proposed method can obtain orbit accuracies of 0.92, 2.74, and 8.30?m in the radial, along-track, and cross-track directions, respectively. The average orbit-only Signal-In-Space Range Error (SISRE) is 1.23?m, which is slightly poorer than that of the broadcast navigation ephemeris. Using four days of GEO maneuvered datasets, it is further demonstrated that the derived orbits can be employed to characterize the behaviors of GEO satellite maneuvers, such as the time span of the maneuver as well as the satellite thrusting accelerations. These results prove the efficiency of the proposed method for near real-time GEO satellite orbit determination during maneuvers.     

Satellite monitoring of anomalous effects in the ionosphere probably related to strong earthquakes

The paper presents simultaneously a satellite monitoring of plasma density disturbances and quasi-static electric fields on the one hand, and a ground earthquake monitoring, on the other. The INTERCOSMOS-BULGARIA-1300 satellite was launched on 7th August 1981. It had a perigee of 825 km, an apogee of 906 km and orbit inclination of 81.2°. For satellite’s orbits over sources of earthquakes M ⩾ 5.5 (seismic data of the World Data Center – Denver, Colorado), in the time period 12.08.1981–30.12.1981, some ionospheric effects are investigated. These effects can be considered eventually as possible earthquake precursors. The paper discusses specific anomalous effects observed in the ionosphere, which cannot be explained by factors of solar-magnetospheric origin. They could be associated with the earthquake growing processes in the lithosphere and troposphere zones above the source. In conclusion the results of ionosphere anomalous effects monitoring are proposed.     

OPTIS – A Satellite test of Special and General Relativity

OPTIS has been proposed as a small satellite platform in a high elliptical orbit (apogee 40,000 km, perigee 10,000 km) and is designed for high precision tests of foundations of Special and General Relativity. The experimental set-up consists of two ultrastable Nd:YAG lasers, three crossed optical resonators (monolithic cavities), an ensemble of atomic clocks, an optical comb generator, laser tracking devices and a drag-free control system. OPTIS enables improved tests of (1) the isotropy and (2) constancy of the speed of light, (3) special relativistic time dilation, (4) the universality of the gravitational redshift by comparison of various clocks, can measure (5) the absolute value of the gravitational redshift, (6) the Lense–Thirring effect and (7) the perigee advance and (8) can make a test of a hypothetical Yukawa part in the gravitational potential. To avoid any influence from atmospheric drag, solar radiation, or Earth albedo, the satellite needs drag-free control to depress the residual acceleration down to 10⁻¹⁴ m/s² in the frequency range between 10⁻² and 10⁻³ Hz. Precise thermal control must be used to stabilize the cavity temperature to within one part in 10⁷ at time scales of 100 s and to one part in 10⁵ on the orbit time scale.     

VLBI observations to the APOD satellite

The APOD (Atmospheric density detection and Precise Orbit Determination) is the first LEO (Low Earth Orbit) satellite in orbit co-located with a dual-frequency GNSS (GPS/BD) receiver, an SLR reflector, and a VLBI X/S dual band beacon. From the overlap statistics between consecutive solution arcs and the independent validation by SLR measurements, the orbit position deviation was below 10?cm before the on-board GNSS receiver got partially operational. In this paper, the focus is on the VLBI observations to the LEO satellite from multiple geodetic VLBI radio telescopes, since this is the first implementation of a dedicated VLBI transmitter in low Earth orbit. The practical problems of tracking a fast moving spacecraft with current VLBI ground infrastructure were solved and strong interferometric fringes were obtained by cross-correlation of APOD carrier and DOR (Differential One-way Ranging) signals. The precision in X-band time delay derived from 0.1?s integration time of the correlator output is on the level of 0.1?ns. The APOD observations demonstrate encouraging prospects of co-location of multiple space geodetic techniques in space, as a first prototype.     

The impact of orbital errors on the estimation of satellite clock errors and PPP

Precise satellite orbit and clocks are essential for providing high accuracy real-time PPP (Precise Point Positioning) service. However, by treating the predicted orbits as fixed, the orbital errors may be partially assimilated by the estimated satellite clock and hence impact the positioning solutions. This paper presents the impact analysis of errors in radial and tangential orbital components on the estimation of satellite clocks and PPP through theoretical study and experimental evaluation. The relationship between the compensation of the orbital errors by the satellite clocks and the satellite-station geometry is discussed in details. Based on the satellite clocks estimated with regional station networks of different sizes (∼100, ∼300, ∼500 and ∼700 km in radius), results indicated that the orbital errors compensated by the satellite clock estimates reduce as the size of the network increases. An interesting regional PPP mode based on the broadcast ephemeris and the corresponding estimated satellite clocks is proposed and evaluated through the numerical study. The impact of orbital errors in the broadcast ephemeris has shown to be negligible for PPP users in a regional network of a radius of ∼300 km, with positioning RMS of about 1.4, 1.4 and 3.7 cm for east, north and up component in the post-mission kinematic mode, comparable with 1.3, 1.3 and 3.6 cm using the precise orbits and the corresponding estimated clocks. Compared with the DGPS and RTK positioning, only the estimated satellite clocks are needed to be disseminated to PPP users for this approach. It can significantly alleviate the communication burdens and therefore can be beneficial to the real time applications.     

Towards accurate atmospheric mass density determination Using precise positional information of space objects

This paper presents a new method of deriving atmospheric mass densities with a high temporal resolution from precise orbit data of low earth orbiting (LEO) space objects. This method is based on the drag perturbation equation of the semi-major axis of the orbit of LEO space objects which relates the change rate of the semi-major axis to the atmospheric mass density. The effectiveness of the new method is evaluated using the GFZ-ISDC GPS rapid science orbit (RSO) products of the CHAMP satellite over a time period of 3 months. The densities derived using this new method and obtained from accelerometer data are compared and good agreements are achieved. An example of using the derived density to generate good orbit prediction for CHAMP is presented.     

Inter-satellite ranging and inter-satellite communication links for enhancing GNSS satellite broadcast navigation data

Recently the European Space Agency (ESA) has initiated a number of exploratory Projects, within the General Studies Programme (GSP), to analyze what potential improvements on a GNSS system navigation determination and dissemination performance could be brought by introducing inter-satellite ranging & inter-satellite communication-links. The key improvements targeted by these Projects are the enhancement of the orbit and clock prediction accuracy and the reduction of the dependency from ground infrastructure. Both projects adopted the Galileo system architecture as the initial working point.     

Spectral filter for signal identification in the kHz SLR measurements of the fast spinning satellite Ajisai

The high repetition rate satellite laser ranging (SLR) measurements to the fast spinning satellites contain a frequency signal caused by the rotational motion of the corner cube reflector (CCR) array. The spectral filter, developed here, is based on the Lomb algorithm, and is tested with the simulated and the observed high repetition rate SLR data of the geodetic satellite Ajisai (spin period ∼2 s). The filter allows for the noise elimination from the SLR data, and for identification of the returns from the single CCRs of the array – even for the low return rates. Applying the spectral filter to the simulated SLR data increases the S/N ratio by a factor 40–45% for all return rates. Filtering out the noise from the observed data strengthens the frequency signal by factor of ∼25 for the low return rates, which significantly helps to determine the spin phase of the satellite. The spectral filter is applied to the Graz SLR data and the spin rates of Ajisai are determined by two different methods: the frequency analysis and the phase determination of the spinning retroreflector array.The analysis of more than 8 years of the Graz SLR measurements indicates an exponential spin rate trend: f = 0.67034 exp(−0.0148542 Y) [Hz], RMS = 0.085 mHz, where Y is the year since launch. The highly accurate spin rate information demonstrates periodic changes related to the precession of the orbital plane of Ajisai, as it determines the amount of energy received by the satellite from the Sun. The rate of deceleration of Ajisai is not constant: the half life period of the satellite’s spin oscillates around 46.7 years with an amplitude of about 5 years.     

Performance assessment of RTPPP positioning with SSR corrections and PPP-AR positioning with FCB for multi-GNSS from MADOCA products

The state-space representation (SSR) product of satellite orbit and clock is one of the most essential corrections for real-time precise point positioning (RTPPP). When it comes to PPP ambiguity resolution (PPP-AR), the fractional cycle bias (FCB) matters. The Japan Aerospace Exploration Agency (JAXA) has developed a multi-GNSS (i.e., global navigation satellite system) advanced demonstration tool for orbit and clock analysis (MADOCA), providing free and precise orbit and clock products. Because of the shortage of relevant studies on performance evaluation, this paper focuses on the performance assessment of RTPPP and PPP-AR by real-time and offline MADOCA products. To begin with, the real-time MADOCA products are evaluated by comparing orbit and clock with JAXA final products, which gives an objective impression of the correction. Second, PPP tests in static and simulated kinematic mode are conducted to further verify the quality of real-time MADOCA products. Finally, the offline MADOCA products are assessed by PPP and PPP-AR comparisons. The results are as follows: (1) Orbit comparisons produced an average error of about 0.04–0.13 m for the global positioning system (GPS), 0.14–0.16 m for the global navigation satellite system (GLONASS), and 0.07–0.08 m for the quasi-zenith satellite system (QZSS). The G15 satellite had the most accurate orbit, with a difference of 0.04 m between the JAXA orbit products and MADOCA’s counterpart, while the R07 satellite had the least accurate orbit with a difference of 0.16 m. Clock products had an accuracy of 0.4–1.3 ns for GPS, 1.4–1.6 ns for GLONASS, and 0.7–0.8 ns for QZSS in general. The G15 satellite had the most accurate clock with a difference of only 0.40 ns between the JAXA clock products and MADOCA products, and the R07 satellite had the least accurate clock with a difference of 1.55 ns. The orbit and clock products for GLONASS performed worse than those of GPS and QZSS. (2) After convergence, the positioning accuracy was 3.0–8.1 cm for static PPP and 8.1–13.7 cm for kinematic PPP when using multi-GNSS observations and precise orbit and clock products. The PFRR station performed the good performance both in static and kinematic mode with an accuracy of 2.99 cm and 8.08 cm, respectively, whereas the CPNM station produced the worst static performance with an error of 8.09 cm, and the ANMG station produced the worst kinematic performance with a counterpart of 13.69 cm. (3) The PPP-AR solution was superior to the PPP solution, given that, with respect to PPP, post-processing PPP-AR improved the positioning accuracy and convergence time by 13–32 % (3–89 %) in GPS-only mode by 2–15 % (5–60 %) in GPS/QZSS mode. Thus, we conclude that the current MADOCA products can provide SSR corrections and FCB products with positioning accuracy at the decimeter or even centimeter level, which could meet the demands of the RTPPP and PPP-AR solutions.     

Analysis of electromagnetic and electrostatic effects of particle impacts on spacecraft

Particle impacts on spacecraft can cause considerable damage, even leading to complete failure. A theory for the resulting vehicle potential changes and the electromagnetic radiation from impact-induced plasma has been published by Close et al. (2010). Here we compare this theory to impacts registered by the Radio and Plasma Wave Science instrumentation on the Cassini spacecraft. We study both low-velocity (16 km/s) large particles (2.6 μm radius) detected in Saturn’s rings and high-velocity (450 km/s) small particles (1 nm radius) in the solar wind. The agreement with the theory is quite good. We also apply these results to earth orbit and conclude that both Electrostatic Discharge and Electromagnetic Pulse radiation are likely and could lead to spacecraft failure.     

The role of drift and convection–diffusion mechanisms in small energy global cosmic ray modulation: Application to the hysteresis phenomenon of proton and alpha particle satellite data

The hysteresis effect for small energies of galactic cosmic rays is due to two effects. The first is the same as for neutron monitor energies – the delay of the interplanetary processes responsible for cosmic ray modulation with respect to the initiating solar processes, according to the effective velocity of solar wind and shock waves propagation. Then, the observed cosmic ray intensity is connected to the solar activity variations during many months before the time of cosmic ray measurement. The second is caused by the time delay of small energy cosmic ray diffusion from the boundary of modulation region to the Earth’s orbit. The model describing the connection between solar activity variation and cosmic ray convection–diffusion global modulation for neutron monitor energies is here developed by taking into account also the time-lag of the small energy particle diffusion in the Heliosphere. We use theoretical results on drifts and analytically approximate the dependences of drifts from tilt angle, and take into account the dependence from the sign of primary particles, and from the sign of polar magnetic field (A > 0 or A < 0). The obtained results are applied on proton and alpha-particle satellite data. We analyze satellite 5-min data of proton fluxes with energies >1 MeV, >2 MeV, >5 MeV, >10 MeV, >30 MeV, >50 MeV, >60 MeV, >100 MeV, and in intervals 10–30 MeV, 30–60 MeV, and 60–100 MeV during January 1986–December 1999. We exclude periods with great cosmic ray increases caused by particle acceleration in solar flare events. Then, we determine monthly averaged fluxes, as well as 5-month and 11-month smoothed data. We analyze also satellite 5-min data on alpha-particle fluxes in the energy intervals 60-160 MeV, 160–260 MeV and 330–500 MeV during January 1986–May 2000. We correct observation data for drifts and then compare with what is expected according to the convection–diffusion mechanism. We assume different dimensions of the modulation region (by the time propagation X₀ of solar wind from the Sun to the boundary of modulation region), for X₀ values from 1 to 60 average months, by one-month steps. For each value of X₀ we determine the correlation coefficient between variations of expected and observed cosmic ray intensities (the estimation of cosmic ray intensities values is given in Section 3 by Eq. (9), and the determination of correlation and regression coefficients in Section 3 by Eq. (8)). The dimension of modulation region is determined by the value of X_0 max, for which the correlation coefficient reaches the maximum value. Then the effective radial diffusion coefficient and residual modulation in small energy region can be estimated.