低纬电离层TEC起伏与相位闪烁的统计特征

穿过电离层不规则体传播后的无线电波,其振幅和相位出现快速随机起伏,即电离层闪烁.为了量化电离层不规则体和相位闪烁的强度,本文提出用TEC起伏δTEC作为特征参量,并用δTEC的标准差构建一种新指数σtec.文中证明指数σtec与相位闪烁指数完全等效.在电离层强闪烁期间,经常出现信号短暂失锁和周跳,导致TEC值突跳和不连续.为此,本文设计了一种周跳检测与校正的批处理算法,用于消除TEC值突跳.在此基础上,利用位于我国中南部电离层闪烁监测台网2012—2015年的观测数据,考察了GPS信号相位闪烁和不规则体的统计特征.结果表明,我国低纬电离层不规则体和相位闪烁与振幅闪烁随地方时和月份变化的特征类似,一天之中主要出现在日落后至黎明前,一年之中,春季不规则体出现最频繁、秋季次之,呈现春秋不对称性,冬夏季出现很少.此外,我们还比较分析了指数S4与σtec的联系,两者之间显著正相关表明,小于第一菲涅尔带尺度的小尺度不规则体和大于第一菲涅尔带尺度的大尺度不规则体一般同时存在.     

2004年11月强磁暴期间武汉电离层TEC的响应和振幅闪烁特征的GPS观测

本文利用设在武汉(11436°E,3053°N,磁纬194°)的GPS电离层TEC和电波闪烁监测仪的测量数据,分析了2004年11月强磁暴期间TEC的响应以及电波闪烁和TEC起伏的特征.结果表明,在这次强磁暴期间,武汉及其邻近地区电离层TEC的响应以正暴相为主,正暴相分别出现在两次主相期间,最大正偏离达到50 TECU.这次磁暴另一个重要影响是主相期间L波段振幅闪烁的活动性及其强度显著增强.S4指数最大接近10.伴随增强的闪烁活动,多次观测到深度耗尽的等离子体泡与TEC起伏,TEC变化率的标准差ROTI指数也显著增强.分析揭示, ROTI指数与S4指数呈正相关,相关系数达到097.线性回归计算得到,ROTI和S4的比率为964.     

Spatial structure of auroral day-time ionospheric electron density irregularities generated by a powerful HF-wave

We describe an experiment in satellite radio-wave probing of the ionosphere, modified by powerful waves from the HF heating facility at Tromsø (Norway) in May 1995. Amplitude scintillations and variations of the phase of VHF signals from Russian navigational satellites passing over the heated region were observed. We show that both large-scale electron density irregularities (several tens of kilometers in size) and small-scale ones (from hundreds of meters to kilometers) can be generated by the HF radiation. Maximum effects caused by small-scale irregularities detected in the satellite signals are observed in the directions sector approximately parallel to the geomagnetic field lines although large-scale structures can be detected within a much larger area. The properties of small-scale irregularities (electron density fluctuations) are investigated by applying a statistical analysis and by studying experimental and model mean values of the logarithm of the relative amplitude of the signal. The results indicate that satellite radio probing can be a supporting diagnostic technique for ionospheric heating and add valuable information to studies of effects produced by HF modification.     

Plasma inhomogeneities and radiowave scattering in experiments with electron pulses in the ionosphere

The absorption of telemetry radiosignals at frequencies of 250 and 75 MHz, transmitted from rockets, was observed in the ARAKS and Zarnitza 2 rocket experiments, respectively, with electron pulses in the ionosphere. The signals were registered with ground receivers. Four cases of complete signal absorption on the propagation path were observed in the ARAKS experiment. The radio absorption at frequencies substantially higher than the plasma and upper hybrid frequencies can be related to wave scattering by plasma inhomogeneities. It has been indicated that plasma inhomogeneities were generated when electrostatic oscillations damped in the region with decreased plasma density at a decrease in the natural oscillation phase volume in the frequency-wave vector space with decreasing plasma density. The observed radio absorption could be related to reflectionless wave scattering in an inhomogeneous plasma structure.     

Using GPS–GLONASS–GALILEO data and IRI modeling for ionospheric calibration of radio telescopes and radio interferometers

VHF and HF radio signals are widely used to observe the Sun and pulsars. Nowadays, large low-frequency radio astronomical arrays (LOFAR, 30–240 MHz; MIRA, 80–300 MHz) are being constructed to record radiation of pulsars at the maximum distance. registration of the solar radio emission intensity at fixed frequencies and in the spectral VHF band is very important along with other methods of monitoring of coronal mass ejections. Interpreting radio astronomical data is known to be necessary to take into account possible distortions of these signals in the Earth ionosphere. However, in contrast to modern navigation systems (Global Position System (GPS), GLObal NAvigation Satellite System (GLONASS), GALILEO), in which a very accurate reconstruction of ionosphere parameters is a built-in function, in present-day radio astronomy a retrieve of ionosphere transfer characteristics has not been appropriately worked out. This collides with increasing requirements to accuracy of the analysis of a radio emission amplitude profile and to the angular and polarizing resolution of radio telescopes of new generation (LOFAR, SKA, etc.). We have developed a method and software to calculate the ionosphere rotation measure (RM) and dispersion measure (DM). We used the ionosphere model IRI-2001, magnetic field model IGRF-10, and the ionosphere total electron content values obtained from GPS measurements. The obtained values of DM and RM were recalculated into characteristics of the phase delay, Faraday amplitude modulation, and polarization changes. We calculated ones for different levels of geomagnetic activity as well as different angular positions of radio sources. Our main idea is to use a signal of navigation satellites (GPS, GLONASS, GALILEO) as a testing signal from a “reference” source located at minimal angle distance from a source studied. Our project allows development of methods and systems of ADAPTIVE RADIO ASTRONOMY, adaptive to the non-uniform and non-stationary ionosphere, by analogy with known systems of adaptive optics intended to adapt optical telescopes to varying conditions of the optically non-uniform and non-stationary troposphere.     

Analysis of the Characteristics of Low-Latitude GPS Amplitude Scintillation Measured During Solar Maximum Conditions and Implications for Receiver Performance

Ionospheric scintillations are fluctuations in the phase and/or amplitude of trans-ionospheric radio signals caused by electron density irregularities in the ionosphere. A better understanding of the scintillation pattern is important to make a better assessment of GPS receiver performance, for instance. Additionally, scintillation can be used as a tool for remote sensing of ionospheric irregularities. Therefore, the study of ionospheric scintillation has both scientific as well as technological implications. In the past few years, there has been a significant advance in the methods for analysis of scintillation and in our understanding of the impact of scintillation on GPS receiver performance. In this work, we revisit some of the existing methods of analysis of scintillation, propose an alternative approach, and apply these techniques in a comprehensive study of the characteristics of amplitude scintillation. This comprehensive study made use of 32?days of high-rate (50?Hz) measurements made by a GPS-based scintillation monitor located in S?o José dos Campos, Brazil (23.2°S, 45.9°W, ?17.5° dip latitude) near the Equatorial Anomaly during high solar flux conditions. The variability of the decorrelation time (τ₀) of scintillation patterns is presented as a function of scintillation severity index (S ₄). We found that the values of τ₀ tend to decrease with the increase of S ₄, confirming the results of previous studies. In addition, we found that, at least for the measurements made during this campaign, averaged values of τ₀ (for fixed S ₄ index values) did not vary much as a function of local time. Our results also indicate a significant impact of τ₀ in the GPS carrier loop performance for S ₄?≥?0.7. An alternative way to compute the probability of cycle slip that takes into account the fading duration time is also presented. The results of this approach show a 38% probability of cycle slips during strong scintillation scenarios (S ₄ close to 1 and τ₀ near 0.2?s). Finally, we present results of an analysis of the individual amplitude fades observed in our set of measurements. This analysis suggests that users operating GPS receivers with C/N ₀ thresholds around 30?dB-Hz and above can be affected significantly by low-latitude scintillation.     

Effects of substorms during HF propagation in the auroral oval

The specific features of radio propagation from the viewpoint of physics of processes in the polar ionosphere have been studied in the present work based on the oblique-incidence sounding of the ionosphere (OISI) on the St. Petersburg-Belyi Nos (Amderma) polar radio path during substorm activity in the summer months of 1997. The OISI data were used to find the following parameters: maximum observable frequency during signal reflection from the E _s layer (EsMOF), maximum observable frequency during signal reflection from the F ₂ layer (F2MOF), and lowest observable frequencies for the E _s and F ₂ layers (EsLOF and F2LOF, respectively). Absolute MOF and LOF values were also found out. The total number of received rays was determined in addition to the above parameters. Isolated substorms against a quiet background were selected for the studies. These substorms resulted in substantial changes in the ionospheric radio channel and propagation conditions along the path. The results of the studies are as follows. (1) The following distinct regularities in the HF propagation along the path have been determined: (i) the range of operational frequencies Δ = MOF-LOF becomes substantially narrower during substorms; (ii) the radio propagation mechanism changes during a substorm; (iii) during substorms, the auroral absorption substantially and partially increases in the course of the expansion and recovery phases, respectively; (iv) multiray effect sharply increases at the beginning of the substorm active phase (T ₀). (2) The indications of changes in the radio propagation parameters, which can possibly be used to predict the beginning of substorm development, have been formulated. (3) All revealed regularities in the HF propagation in the auroral zone have been explained from the geophysical viewpoint. It is important to use these regularities to organize radio communication and to solve the problems within the scope of the Space Weather Program.     

VLF phase and amplitude: daytime ionospheric parameters

Experimental observations of the daytime variations of VLF phase and amplitude over a variety of long subionospheric paths have been found to be satisfactorily modelled with a D-region ionosphere, described by the two traditional parameters, H′ and β (being measures of the ionospheric height and the rate of increase of electron density with height, respectively). This VLF radio modelling uses the NOSC Earth–ionosphere waveguide programs but with an experimentally deduced dependence of these two ionospheric parameters on solar zenith angle. Phase and amplitude measurements from several VLF Omega and MSK stations were compared with calculations from the programs LWPC and Modefinder using values of H′ and β determined previously from amplitude only data. This led to refined curves for the diurnal variations of H′ and β which, when used in these programs, give not only calculated amplitudes but also, for the first time, calculated phase variations that agree well with a series of observations at Dunedin, New Zealand, of VLF signals from Omega Japan, Omega Hawaii, NPM (Hawaii) and NLK (Seattle) covering a frequency range of 10–25 kHz.     

Waveguide radio-wave propagation in the equatorial ionosphere according to the Interkosmos-19 data

Additional strongly remote (up to 2000 km) radio-signal reflection traces on Intercosmos-19 ionograms obtained in the equatorial ionosphere have been considered. These traces, as a rule, begin at frequencies slightly lower than the main trace cutoff frequencies, which indicates that an irregularity with a decreased plasma density exists here. The waveguide stretched along the magnetic-field line is such an inhomogeneity in the equatorial ionosphere. The ray tracing confirm that radio waves propagate in a waveguide and make it possible to determine the typical waveguide parameters: ?δN _e ≥ 10%, with a diameter of 15–20 km. Since the waveguide walls are smooth, an additional trace is always recorded distinctly even in the case in which main traces were completely eroded by strong diffusivity. Only one additional trace (of the radio signal X mode) is usually observed one more multiple trace is rarely recorded. Waveguides can be observed at all altitudes of the equatorial ionosphere at geomagnetic latitudes of ±40°. The formation of waveguides is usually related to the formation of different-scale irregularities in the nighttime equatorial ionosphere, which result in the appearance of other additional traces and spread F.     

Earth’s lower ionosphere during partial solar eclipses according to observations near Nizhny Novgorod

The results of observations in the Vasil’sursk Laboratory (56.1° N, 46.1° E) of partial solar eclipses of August 11, 1999, August 1, 2008, and March 20, 2015 are discussed. Ionospheric observations in the eclipse periods and on control days were conducted by the method of resonant scatter of radio waves at artificial periodic irregularities of the ionospheric plasma and the partial reflection method based on radio wave scatter by natural irregularities in the D region. The lower ionosphere reaction to solar eclipses, including variations in the electron concentration and characteristics of the signals scattered by APIs, was studied. An intensification of the lower ionosphere turbulization, an increase in the signal amplitudes backscattered by APIs in the E region, stratification of the D region, and the arrival of scattered signals from mesopause heights were observed during the eclipses. A decrease in the electron concentration of the D region up to a factor of 3–5 was found by the partial reflection method. Above 88 km, the ionospheric response was delayed by 20–25 min relative to the moment of the eclipse maximum phase, whereas this delay in the lower part of the D region was 2–4 min.     

A study of GPS ionospheric scintillations observed at Guilin

The occurrence of strong ionospheric scintillations with S4≥0.2 was studied using global positioning system (GPS) measurements at Guilin (25.29°N, 110.33°E; geomagnetic: 15.04°N, 181.98°E), a station located near the northern crest of equatorial anomaly in China. The results are presented for data collected from January 2007 to December 2008. The results show that amplitude scintillations occurred only during the first five months of the considered years. Nighttime amplitude scintillations, observed mainly in the south of Guilin, always occurred with phase scintillations, total electron content (TEC) depletions, and Rate Of change of TEC (ROT) fluctuations. However, TEC depletions and ROT fluctuations were weak during daytime amplitude scintillations, and daytime amplitude scintillations usually occurred in most of the azimuth directions. GPS scintillation/TEC observations recorded at Guilin and signal-to-noise-ratio measurements obtained from GPS-COSMIC radio occultation indicate that nighttime and daytime scintillations are very likely caused by ionospheric F region irregularities and sporadic E, respectively.     

Variability of parameters of the F2-layer maximum in the quiet midlatitude ionosphere under low solar activity: 2. Strong fluctuations of critical frequency

This paper presents a qualitative analysis of the properties and particular examples of strong (10% < |δfoF2| < 30%) and very strong (|δfoF2| > 30%) fluctuations in the critical frequency of the F2 layer (foF2) of the quiet ionosphere at midlatitudes under low solar activity according to the Irkutsk station data for 2007–2008. It is found that strong day-to-day fluctuations in foF2 are mainly related to changes in thermospheric parameters, which have a nature of planetary waves and tides. Evidently, very strong day-to-day fluctuations in foF2 are caused by superposition of the effects in the ionosphere caused by changes in the thermospheric parameters and those related to a complex of processes of solar wind interaction with the magnetosphere, including the effects caused by the reversal of the vertical component of the solar wind magnetic field southwards. The increase in foF2 during nighttime hours in winter up to values typical for the daytime maximum in foF2 is the brightest example of very strong changes in foF2 in the quiet ionosphere.     

Dynamics of large-scale ionospheric inhomogeneities caused by a powerful radio emission of the Sura facility from the data collected onto ground-based GNSS network

The measurements of variations in the total electron content of the Earth’s ionosphere along the GPS satellite signal propagation path are described. The signal parameters were measured at a network of receivers at three distant sites: Sura (Vasilsursk), Zelenodolsk, and Kazan. They are arranged along the geomagnetic latitude of the Sura Facility under short-wave radio irradiation of the ionosphere. One feature of the experiment is the crossing of a disturbed region by the radio path between a GPS satellite and Vasilsursk. This resulted from the angular sizes of the Sura array pattern; the radio paths between a GPS satellite and Zelenodolsk and a GPS satellite and Kazan did not cross. Variations in the total electron content of up to 0.15?0.3 TECU were revealed at all three sites during four experimental campaigns (March 2010, March 2013, May 2013, and November 2013). The lateral scale of an ionospheric disturbance stimulated by a high-power radio wave and the velocity of its west-to-east propagation along the geomagnetic latitude were 30–60 km and 270–350 m/s, respectively. A decrease in the total electron content (down to 0.55 TECU) was recorded along the Kazan–Zelenodolsk–Vasilsurks line, which is connected with the solar terminator transit; the lateral scale of the related ionospheric inhomogeneities was ~65–80 km.     

Response of the topside ionosphere over Arecibo to a moderate geomagnetic storm

We analyze the data obtained using the Arecibo incoherent scatter radar to examine the response of the topside ionosphere to a moderate geomagnetic storm that occurred during the period March 7–11, 2008. During this time period a magnetic storm with a non-monotonic main phase decrease in the Dst index occurred. The recovery phase also exhibited a secondary Dst decrease. During the initial phase of the storm, Te and Ti increased coincident with the arrival of the solar wind. The main phase registered an increase in proton concentration proportional to Ne while temperatures reached the lowest values. Variations in O⁺ concentration were not significant but a reduction in helium fraction was observed. Soon after the peak of the storm, the transition height between the topside ionosphere and the protonosphere, where H⁺ ions dominate composition, was lower than would be expected during quiet conditions and this behavior lasted for approximately 12 h.     

GPS L1-Frequency Observations of Equatorial Scintillations and Irregularity Zonal Velocities

In this work, the climatology of ionospheric scintillations at global positioning system (GPS) L-band frequency and the zonal drift velocities of scintillation-producing irregularities were depicted for the equatorial observatory of São Luis (2.33°S; 44.21°W; dip latitude 1.3°S), Brazil. This is the first time that the hourly, monthly, and seasonal variations of scintillations and irregularity zonal drifts at São Luis were characterized during periods of different solar activity levels (from December 1998 to February 2007). The percentage occurrence of scintillations at different sectors of the sky was also investigated, and the results revealed that the scintillations are more probable to be observed in the west sector of the sky above São Luis, whereas the north–south asymmetries are possibly related to asymmetries in the plasma density distribution at off-equatorial latitudes. The scintillations on GPS signals occurred more frequently around solar maximum years, but it is also clear from the results of a strong variability in the scintillation activity in the years with moderate solar flux during the descending phase of the solar cycle. The equatorial scintillations occur predominantly during pre-midnight hours with a broad maximum near the December solstice months. In general, weak level of scintillations (S ₄ index between 0.2 and 0.4) dominated at all seasons; however, during the winter months around solar maximum years (although the scintillation occurrence is extremely low), stronger levels of scintillations (S ₄ > 0.6) may occur at comparable rate with the weak scintillations. The irregularity zonal velocities, as estimated from the GPS spaced-receiver technique, presented a different scenario for the two seasons analyzed; during the equinoxes, the magnitude of the zonal velocities appeared not to change with the solar activity, whereas during the December solstice months, the larger magnitudes were observed around solar maximum years. Other relevant aspects of the observations are highlighted and discussed.     

SPEAR-induced field-aligned irregularities observed from bi-static HF radio scattering in the polar ionosphere

Experimental results from SPEAR HF heating experiments in the polar ionosphere are examined. Bi-static scatter measurements of HF diagnostic signals were carried out on the Pori (Finland)–SPEAR–St. Petersburg path at operational frequencies of 11,755 and 15,400 kHz and the London–SPEAR–St. Petersburg path at frequencies of 12,095 and 17,700 kHz, using a Doppler spectral method. The SPEAR HF heating facility generates heater-induced artificial field-aligned small-scale irregularities (AFAIs), which can be detected by HF diagnostic bi-static radio scatter techniques at St. Petersburg at a distance of about 2000 km. In accordance with the Bragg condition, HF bi-static backscatters were sensitive to small-scale irregularities having spatial sizes of the order of 9–13 m across the geomagnetic field line. The properties and behaviour of AFAIs have been considered in the winter and summer seasons under quiet magnetic conditions and under various status of the polar ionosphere (the presence of “thick” and “thin” sporadic Es layers, different structures of the F2 layer). The experimental results obtained have shown that AFAIs can be excited in the F as well as in the E regions of the polar ionosphere. The excitation of a very intense wide-band spectral component with an abrupt increase in the spectral width up to 16–20 Hz has been found in the signals scattered from striations. Along with a wide-band component, a narrow-band spectral component can be also seen in the Doppler sonograms and in the average spectra of the signals scattered from the SPEAR-induced striations. AFAIs were excited even when the HF heater frequency was up to 0.5 MHz larger than the critical frequency. A simulation of the ray geometry for the diagnostic HF radio waves scattered from AFAIs in the polar ionosphere has been made for the geophysical conditions prevailing during experiments carried out in both the winter and summer seasons.     

Earthquake responses in the high-latitude ionosphere

The data, obtained using the methods of partial reflections and ionosphere vertical sounding on the Kola Peninsula and in Scandinavia, at Tumannyi (69.0° N, 35.7° E) and Sodankyla (67.37°N, 26.63°E) observatories, have been analyzed in order to detect earthquake responses. The strong earthquakes have been considered: one earthquake with a magnitude of 7.7 occurred at 0819:25 UT on July 17, 2006, on the western coast of Indonesia (9.33° S, 107.26° E), and another earthquake with a magnitude of 6.2 occurred 2253:59 UT on May 26, 2006, on Yava (7.94° S, 110.32° E). These earthquakes, the epicenters of which were located in the same region and at identical depths (10 km), were observed under quiet conditions in the geomagnetic field (ΣK _p = 5.7 and 6.3) and during small solar flares. The response of the ionosphere to these flares was mainly observed in the parameters of the lower ionosphere in the D and E regions. It has been found out that the period of variations in the ordinary component of the partially reflected signal at altitudes of the E region increased before the earthquake that occurred on July 17, 2006. The f _min variations at Sodankyla observatory started 20 h before the earthquake. The periods of these variations were 3–6 h. The same periods were found in the variations in other ionospheric parameters (foEs and h’Es). The variations in the ordinary component of partially reflected signals with periods of 2–5 hours were observed on the day of another earthquake (May 26, 2006). Internal gravity waves with periods of several hours, which can be related to the earthquakes, were detected in the amplitude spectra of the ordinary component of partially reflected signals and in other parameters in the lower ionosphere.     

Variations in the amplitude and phase of VLF radiowaves in the ionosphere during the August 1, 2008, solar eclipse

Time variations in the amplitude and phase of signals of the Russian telecommunication station (the frequency is 25 kHz) on the Arkhangelsk—Kharkov path with a length of about 1600 km on the day of the August 1, 2008 solar eclipse (SE) and on the adjacent days are analyzed. Two types of effects are detected. An increase of the signal amplitude by approximately 32% in comparison with the background days and the 2.1 μs time shift of the signal during 2—2.5 h is referred to the first type. Changes in the spectral composition of the quasiperiodic disturbances in the ionosphere presented the second type of the effects. For spectral analysis of the quasiperiodic variations in the amplitude and phase of the radio signal, the window Fourier transform, adaptive Fourier transform, and wavelet transformation were applied simultaneously. In the period of SE and after it, oscillations with periods of 10—15 min (according to the amplitude data) and also about 10 and 18 min (according to the phase data) were intensified. Based on radio signal characteristics, the parameters of ionospheric disturbances are estimated.     

Interplanetary medium condition effects in the South Atlantic Magnetic Anomaly: A case study

One way to investigate the magnetosphere–ionosphere coupling is through the simultaneous observation of different parameters measured at different locations of the geospace environment and try to determine some relationships among them. The main objective of this work is to examine how the solar energetic particles and the interplanetary medium conditions may affect the space and time configuration of the ring current at low-latitudes and also to get a better understanding on how these particles interfere with the lower ionosphere in the South Atlantic Magnetic Anomaly region (SAMA). To accomplish this, the cosmic noise absorption (CNA) and the horizontal component of the Earth's magnetic field data measured from sites located in the SAMA region were compared with the proton and electron fluxes, interplanetary medium conditions (solar wind and the north–south component of the interplanetary magnetic field measured on board satellites), the SYM-H index and magnetometer data from Kakioka (KAK-Japan), located significantly outside the SAMA region. The time series analyzed correspond to the geomagnetic disturbance that occurred on August 25–30, 1998. The analysis was performed by implementing wavelet techniques, with particular attention to singularities detection, which highlights the presence of transient signals. The results are discussed in terms of the first three wavelet decomposition levels of the parameters. The magnitude of wavelet coefficients of the solar wind and proton flux at the two energy ranges analyzed is timely well correlated, indicating that these two signals are energetically linked. The larger wavelet coefficient amplitude of KAK and VSS magnetograms shows time delays that are compatible with an asymmetric configuration of the ring current, considering that at the storm time, VSS was at the dawn sector of the magnetosphere and KAK at the dusk side. The wavelet analysis of CNA signals reveals that the signal may be sensitive to the ionization produced by energetic electrons and protons as well. The time delays observed in wavelet coefficients may give an indication of the different accelerating process to which the particles are submitted when traveling along the magnetic field lines, from higher to lower latitudes, and the likely contribution of these particles to the ionization measured as an absorption of the cosmic noise in the lower ionosphere.     

Statistical modeling of relative variations in the critical frequencies of the ionospheric F region

Relative variations in the critical frequency of the ionospheric F, region are considered as one of the main sources of information about the characteristics of plasma inhomogeneities in the region of the main ionization maximum and as a substantial factor responsible for the statistics of the decameter signal parameters on radio paths with different lengths. The functions defined on finite intervals are also used in statistical modeling in addition to the available methods. Such a consideration made it possible to remove a restriction, peculiar to previous models, in the form of the requirement that the fourth statistical invariant—excess— should be positive. This makes the generalized statistical model more universal, which is of special importance for high4atitude radio paths. The specified mo del more adequately corresponds to the experimental data, which are characterized by finite intervals where the quantities are measured.