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 Quasi-periodic scintillations are characterized as primary deep fadeout in field strength, associated with regular ringing patterns before and after it. In this paper, observations of quasi-periodic scintillations using geostationary satellite (FLEETSAT) transmissions operating at frequency 250 MHz at low-latitude ground station, Varanasi (geomagnetic latitude 14°55′N, longitude 153°59′E), are reported. The results indicate that the quasi-periodic scintillations are most likely produced by plasma blobs/bubbles present in the E and F regions of the ionosphere which are helpful in identifying the generation mechanism of the associated irregularities. The various characteristic features of the different types of quasi-periodic scintillations observed at low latitude are discussed for the first time in detail based on a highly comprehensive analysis of longer data sets using autocorrelation, power spectrum, and scintillation index analysis. The computed horizontal scale size of the quasi-periodic scintillations producing irregularity varies from 100 to 1300 m which shows that the irregularities are of intermediate-scale sizes. The spectral index obtained from the slopes of power spectrum varies from −2 to −8. All of these observed results are important for identifying the generation mechanism of ionospheric irregularities associated with quasi-periodic scintillations. The observed fading patterns, especially the modulation of the diffraction patterns (fading envelopes), can be explained by considering an obstacle called radio lens in the ionosphere elongated in one direction. For the first time, we have successfully simulated the amplitude versus time plots of almost all types of quasi-periodic scintillation patches and found that our modeled and observed characteristics of quasi-periodic scintillation patches compare well with each other.
 Scintillations are irregular changes in the amplitude, phase and frequency of the radio wave propagating through the ionosphere [Aarons et al., 1971]. The radio signals received from the radio beacon transmitters give us information for the ionospheric investigation. Small-scale irregularities cause changes in the refractive index and produce scintillations in the strength of the received signals. These irregularities generally occur in large numbers forming a random diffracting screen.
 In general, ionospheric scintillation is of random type in which no regular variation in the amplitude and phase are recognized [Maruyama, 1991]. However, in some cases peculiar periodic fluctuations in the field strength with a systematic change in amplitude and phase are observed which persist for one to several minutes when a geostationary satellite is used [Davies and Whitehead, 1977; Karasawa et al., 1985; Maruyama, 1991] and for some 10 s when an orbiting satellite is used [Hajkowicz and Dearden, 1988; Bowman, 1989]. These phenomena are called quasi-periodic scintillations, distinguished from random scintillations. These are characterized as primary deep fadeouts in signal strength, associated with regular ringing patterns before and after it [Kelleher and Martin, 1975]. An example is shown in Figure 1c. Generation of quasi-periodic scintillation (QPS) is believed to be due to an obstacle in the radio wave propagation path, which is a small-scale density enhancement/depletion or blob/bubble of ionization, which acts as convex/concave radio lens [Titheridge, 1991; Davies and Whitehead, 1977]. Davies and Whitehead  and Karasawa et al.  reproduced quasi-periodic Fresnel diffraction patterns due to columnar shaped obstacle using thin phase screen model that advances the phase for the radio wave passing through it. Bowman  assumed horizontal wavefronts in the sporadic E layers as the density irregularities for the reproduction of certain quasi-periodic patterns. He further explained that if the irregularity is assumed to be at F layer height, the column like structure elongated along the magnetic field may be a pertinent model whereas when at E layer height, field-aligned large structures at midlatitude are unrealistic. Comparative results of QPS with random scintillation records and ionograms given by Maruyama  support the fact that quasi-periodic scintillations are related to sporadic E. Similar results are given by Hajkowicz et al. , who explained that some properties of sporadic E helps in creating QPS. The existence of density enhancements within horizontal structure in the sporadic E layer is directly shown by incoherent scatter radar [Miller and Smith, 1975]. Franke et al.  explained that QPS patterns occur either at the beginning or end of scintillation patch. They further explained that these match up with those that have east-west (E-W) scale size of few hundred meters which are associated with the walls of equatorial plasma bubbles (EPBs). Chen et al.  introduced a new time-frequency analysis method of Hilbert-Huang Transform (HHT) to analyze the quasi-periodic scintillation data taken at Ascension Island to understand the characteristics of corresponding ionospheric irregularities. They concluded that the quasi-periodic scintillation diffraction patterns were produced by the steep irregularity structure located very near the edges or walls of the plasma bubble/blob. After cross-correlation analysis they computed the density irregularity pattern having an eastward drift velocity of ∼130 m/s. They further estimated the height location of the irregularities that cause diffraction to be between 310 and 330 km height (around F peak). Recently, Beach and Baragona  have studied the spectrograms of high-rate Global Positioning Systems power data at L1 frequency (1.57542 GHz) that commonly show the fluctuations of signal strength with time resembling the QPS. They further demonstrated that the source of these fluctuations is generally not edge-type diffraction from steep ionospheric gradients, as is often believed for quasi-periodic scintillations, despite some features being consistent with ionospheric generation rather it is intersatellite interference pattern. Haerendal  proposed the theory that the plasma instability creates the irregularities of 3–4 orders of magnitude in scale size simultaneously. He gave the concept of alternate horizontal and vertical growth of irregularity with each coming instability (collisional Rayleigh-Taylor instability) giving small structures. He further showed that these structures of horizontal wave vectors and scale sizes of 50–500 m are responsible for the QPS [Franke et al., 1984]. In general, the horizontal scale lengths of the QPS producing structures ranges from several meters to 1000 m [Whitehead, 1989; Ogawa et al., 1989].
Kelleher and Martin  have given various features of QPS in their results from traveling satellite and geostationary satellite. Their traveling satellite results showed that annual variation of occurrence of QPS was high in June and July and with zero occurrence in December–January. Considering geostationary satellite, Kelleher and Martin  observed quite extensive ringing patterns in the time interval 0500–0800 LT and 1800–2200 LT whereas, they were single and last longer in that of 0800–1800 LT. They concluded that no common diurnal variation was obtained. Kelleher and Kasensally  also observed maximum occurrence of QPS in June month. Hajkowicz et al.  also found, the occurrence of QPS patches to peak in June from the scintillation data observed and recorded at Slough (geographic latitude and longitude 51.5°N, 0.6°W) using signals at frequency of 140 and 360 MHz from the geostationary satellite ATS-6 from November 1975 to August 1976. They reported that only one event of QPS was obtained in December. They further showed that out of total records of QPS, 57% were asymmetrical, 7% had an extended initial part of the ringing pattern, 13% were symmetrical, and 23% had only the trace of ringing pattern. Ireland and Preddey  quoted the height range of QPS producing irregularities in the range of 90–140 km. Kelleher and Sinclair  found the height of QPS producing irregularities corresponding to Es region and lower F region. At 40 MHz signal frequency size varies from 2 to 25 km and at 136 MHz, from 0.5 to 13 km [Kelleher and Martin, 1975]. Ireland and Preddey  attributed their observations of QPS to E region whereas Elkins and Slack  attributed their records of QPS to F region irregularities.
 Many theoretical models have been developed to explain the QPS observations [Singleton, 1964; Titheridge, 1991; Heron, 1979; Davies and Whitehead, 1977; Franke et al., 1984]. Heron  modeled the random amplitude scintillations from small number of discrete and possibly regular, ionospheric irregularities. The use of ionospheric lenses on the propagation of transionospheric radio signals have been studied successfully by a number of workers [Titheridge, 1991; Heron, 1979; Davies and Whitehead, 1977]. Davies and Whitehead  modeled the irregularities considering the cylindrical lens of about 100 m across in the ionosphere. They modeled theoretically the observed fading patterns of QPS on 140 MHz and 360 MHz from geostationary satellite ATS-6, showing quasi-periodic fading before and after a deep central minimum. Singleton  had investigated the lens focusing of radio-star signals, while Ireland and Preddey , Elkins and Slack , and Slack  observed quasi-periodic fading of radio signals from both orbiting and geostationary satellites. These authors explained the fading patterns by considering the interference of two signals. Some considered the lens in E region of ionosphere and some in the F region. Titheridge  produced the Gaussian distribution of phase across the wavefront, using Kirchoff integral process to get the diffraction pattern due to the radio lens in the ionosphere. He gave procedures for the rapid calculations of refraction and diffraction pattern produced by isolated irregularities. Giving the theory of large, small and dense irregularities, he further explained that for large irregularities compared to first Fresnel zone, the ground pattern can be obtained by a simple ray theory. Calculations should be made through diffraction theory for small irregularities and different patterns. Hajkowicz  has pointed out that the QPS may be closely linked with an interference effect of waves simultaneously transmitted by two satellites moving within the bandwidth of the receiving antenna. Apart from lots of study of QPS, it was found that still the generation mechanism of QPS producing irregularities is not well understood as well as most of the studies are of short period, and long-term observation and analysis of QPS are lacking, especially for the low-latitude region. A long-term data analysis of QPS patches will be helpful for identifying the generation mechanism of ionospheric irregularities associated with QPS.
 This paper reports the observations of QPS using transmission from geostationary satellite FLEETSAT operating at frequency 250 MHz at low-latitude station Varanasi (geomagnetic latitude 14°55′N, longitude 153°59′E). The various characteristic features of different types of QPS at this location are discussed for first time in detail; based on autocorrelation function, power spectrum and scintillation index analysis. We have analyzed VHF amplitude scintillation recorded at Varanasi for the period 1991–1999 and tried to study the generation mechanism of QPS producing irregularities. Data selection and analysis of the QPS patches are explained in section 2. The experimental results of QPS patches and their observed characteristic are given in section 3. The QPS patches were theoretically modeled and reproduced. The theoretical modeling is shown in section 4. Section 5 describes the various modeled parameters for computation. The theoretical results obtained and simulated QPS patches are shown in section 6. Section 7 discusses the experimental and theoretical results and their comparison. Finally, section 8 comprises the summary of the results and conclusions.
2. Experimental Setup and Data Analysis
 The fluctuations in the amplitude of 250 MHz signals transmitted from geostationary satellite, FLEETSAT situated at 73°E longitude were recorded continuously at Varanasi (geomagnetic latitude is 14°55′N, longitude is 153°59′E, dip angle is 37.3°, subionospheric dip is 34°) using a fixed frequency VHF receiver and a strip chart recorder [Singh, 1993]. The receiver was calibrated on the basis of the method described by Basu and Basu  by disconnecting the antenna from the receiver and connecting with a calibrating signal source using RF signal generator at 250 MHz. The calibration was imparted every three months. The dynamic range of the receiver was about 20 dB and our recording was mainly done on strip chart recorder that was calibrated as 1 cm = 2.54 dB. On a few nights, data were recorded digitally, at the sampling rate of 1 Hz. At Varanasi, amplitude scintillations observed mostly in the nighttime, predominantly in the premidnight periods, in small patches with duration <30 min [Singh and Singh, 1997; Singh et al., 2004]. For the present study, VHF amplitude scintillation data recorded from January 1991 to December 1993, the declining phase of the solar cycle and April 1998 to December 1999, the ascending phase of the next solar cycle have been analyzed to study the characteristic of QPS.
 The scintillations having peak to peak amplitude greater than 3 dB were considered and examined. The patches of observed QPS were selected both during day and night periods, and were analyzed to study symmetric as well as asymmetric patterns of the QPS patches. We have obtained total of 188 numbers of QPS patches from the complete scintillation records for the period January 1991 to December 1993 and April 1998 to December 1999. Since the scintillations were recorded mostly during nighttime at Varanasi, we have obtained about 75% patches during nighttime and 25% during daytime. The obtained quasi-patches are classified into seven different types as shown by Maruyama . Typical examples of different types of QPS patches are shown in Figure 1. Figure 1 clearly shows QPS patches of type 1: primary fadeout followed by a ringing pattern and not ringing beforehand (Figure 1a); type 2: primary fadeout accompanied by short ringing before and long extended ringing afterward (Figure 1b); type 3: primary fadeout accompanied by extended ringing before and short ringing after (Figure 1c); type 4: burst-like scintillations (Figure 1d); type 5: valley-like scintillations (V-type) (Figure 1e); type 6: spike-type scintillations with periodicity (Figure 1g); and type 7: long-duration ringing pattern (Figure 1f). Apart from these seven types of QPS patches, we have also obtained some QPS patches which we have categorized under unidentified ones as their category is not clearly defined. From the classification of total obtained patches at Varanasi, we have obtained only 2.7% of type 1, 4.8% of type 2, 4.3% of type 3, and 33% of type 4. In addition to it, we have obtained about 25% valley-type, 7% spike-type, 10% long extended form and 12% were unidentified. Figure 2 shows the percentage occurrence of types of QPS patches showing that the maximum occurrence is for type 4 (bursts-like scintillations) and type 5 (valley-like scintillations). We have studied the various characteristic features of the QPS patches obtained at Varanasi like their shape, characteristic length, strength, occurrence rate on the basis of analysis of autocorrelation function, power spectrum and scintillation index.
3. Experimental Results
 The diurnal variation of all types of QPS patches (percentage occurrence of day and nighttime patches) is shown in Figure 3. It shows that the occurrence peak during nighttime is maximum (20–25%) in the premidnight hours, from 2000–2400 h IST, as well as during 0200–0400 h IST in the postmidnight hours. During daytime, maximal peak occurs in between 1600 and 1800 h IST and minimal from 0600 to 0800 h IST interval. The reason for the nighttime maximum is that VHF scintillation recording was mainly done in the nighttime than in the daytime because of larger occurrence in nighttime scintillations [Singh et al., 2004, 2006]. As for example in initial period of recording in the first four months of the year 1991 the percentage occurrence of nighttime scintillations was about 17.2% whereas in daytime it was only 4.1%. These patterns of nighttime observations of scintillations are attributed to the spread F phenomena and the daytime scintillation phenomenon is attributed to sporadic E phenomena. [Das Gupta and Kersley, 1976; Maruyama, 1991; Singh et al., 2006; Patel et al., 2009]. The daytime maximum occurrence observations of QPS from 1600 to 1800 h IST are induced by sporadic E irregularities over low latitude [Patel et al., 2009]. The monthly variation of the mean percentage occurrence of QPS observed at Varanasi during January 1991 to December1993 and April 1998 to December 1999 is shown in Figure 4. Figure 4 shows clearly that the occurrence rate is maximum in the year 1992 and minimum in 1999. The seasonal variation of occurrence of QPS patches shows that the QPS producing irregularities are found to be maximum in summer months and minimum in winter months. To study the effect of solar activity on the occurrence of QPS, we have plotted the variation of percentage occurrence of QPS with the mean sunspot number (Rz), which is shown in the Figure 5. The scattered plot does not show any clear trend except a slightly increasing trend (correlation coefficient is 0.24). This “plot” does not indicate any trend for the effect of solar activity. Many workers have found both positive and negative correlation between the QPS occurrence and solar activity [Briggs, 1964; Tyagi, 1967; Walker and Chen, 1970; Koster, 1972].
 The characteristic features of QPS phenomena produced by the plasma bubbles/blobs have been derived from the digital scintillation data. The digitized data give the autocorrelation function, power spectrum and scintillation index. We have computed the characteristic length of the irregularities by the analysis of autocorrelation function. The characteristic length of the irregularities is the distance at which the autocorrelation falls to 0.5 [Khastgir and Singh, 1960]. The time observed at 0.5 value of autocorrelation function is called half decorrelation time (τ) which when multiplied by the horizontal drift velocity gives the characteristic length of the irregularity [Khastgir and Singh, 1960; Singh et al., 2006]. Singh et al.  computed the maximum and minimum drift velocity of irregularities as 200 m/s and 75 m/s, respectively, after the analysis of the VHF scintillation data recorded at Varanasi during the same period of analysis of the present study of QPS patches. Since the analyzed QPS patches are from the same period of analysis (i.e., January 1991 to December 1999), we have assumed the average value of horizontal drift velocity as 100 m/s for the computation of characteristic lengths of QPS producing irregularities. We have also computed the drift speed for the E region QPS patches as 63.6 m/s and for F region QPS patches as 116.6 m/s, which is in the range of our assumed drift speed of 100 m/s. Franke et al.  has also computed the drift speed of QPS in the range of 100–150 m/s. Figures 6a–6f show the six typical examples of the autocorrelation function of QPS data recorded on 20 August 1992 at 0202–0207 h IST (Figure 6a), 26 March 1993 at 0948–0956 h IST (Figure 6b), 25 April 1998 at 1955–1957 h IST (Figure 6c), 19 May 1998 at 2221–2225 h IST (Figure 6d), 26 March 1993 at 0934–0937 h IST (Figure 6e), and 3 March 1993 at 1515–1522 h IST (Figure 6f), with their respective half decorrelation time as 3.0 s, 2.1 s, 4.8 s, 2.8 s, 3.7 s, and 6.2 s, respectively. The percentage occurrence of the characteristic lengths of irregularities obtained for the QPS patches recorded during January 1991 to December 1993 and April 1998 to December 1999 is shown in a bar diagram in Figure 7. The computed characteristic length ranges from 100 to 1300 m which shows that the QPS producing irregularities lie in the intermediate scale range with peak occurrence rate between 300 and 400 m.
Figures 8a–8f show the six typical examples of the power spectrum plotted from the digitized QPS data recorded on 23 July 1992 at 0224–0227 h IST (Figure 8a), 9 July 1992 at 2214–2216 h IST (Figure 8b), 25 April 1998 at 1955–1957 h IST (Figure 8c), 3 March 1993 at 1509–1515 h IST (Figure 8d), 19 August 1992 at 0319–0321 h IST (Figure 8e), and 10 June 1991 at 2324–2327 h IST (Figure 8f) with their corresponding spectral indices calculated as −3.7, −2.4, −7.3, −4.5, −7.4, and −2.5, respectively. The spectral indices of all QPS patches recorded at Varanasi lie within the range of −1 to −8 with mean value of −4.5. The scintillation index S4 is the standard deviation of the normalized intensity which is used to measure strength of the amplitude of scintillation on a particular signal [Whitney et al., 1969; Bhattacharya, 2003]. At our low-latitude station Varanasi, the obtained scintillation index ranges between 0.2 and 0.8, which show that the irregularities are of weak or moderate type. Figure 9 shows the variation of half decorrelation time with scintillation index S4 which is a scattered plot. From Figure 9, it is clear that the half decorrelation time is almost independent of scintillation index. This may be due to the fact that half decorrelation time is controlled by scale size and not by the perturbation level [Franke and Liu, 1983]. All the above characteristics of QPS patches observed at Varanasi are tabulated in Table 1.
Table 1. All Observed and Computed Parameters of Quasi-Periodic Scintillations at Low-Latitude Station, Varanasia
Characteristic Parameters of QPS
QPS, quasi-periodic scintillations.
Half-decorrelation time (τ) (s)
Characteristic scale (m)
Spectral index (p)
−1 to −8
−1.3 to −8.7
Scintillation index (S4)
4. Theoretical Formulation
 For theoretical modeling of the observed events of QPS patches recorded at Varanasi we have used the radio lens theory model as given by Davies and Whitehead  which is the extension of classical Cornu Spiral [Jenkins and White, 1957, chapter 18]. Jenkins and White  have introduced a new variable ω to plot the theoretical curves for the diffraction patterns produced by small wires and have assumed Δω = 0.5, 2.5, 3.0. They further showed that the opaque strips produce irregular fading whereas the regular fading is produced by the interference of two signals. The ionospheric irregularities can be explained as a “cylindrical lens” in the ionosphere. The geometry of the cylindrical lens is shown in Figure 10, which produces a Gaussian distribution of phase (θ) across the emergent wavefront as [Davies and Whitehead, 1977]
where θ is the phase across the emergent wavefront, θ0 is the phase advance for the wave passing through the center of the lens at S0 from the perpendicular from the observing point P to the incident wavefront, S is the distance along the emerging wavefront, S0 is the center of the lens, and ℓ is the scale size of the lens. S, S0, and ℓ are measured in terms of the radius of the first Fresnel zone. Here,
where v is the velocity in m/s, t is the time in seconds, L is the distance along the wavefront, h is the distance between the irregularity and the observation point, i.e., for E region and F region in meters, and λ is the wavelength of the radio signal.
 From the theory of Fresnel Integrals given by Jenkins and White , we get that the “x” and “y” coordinates of the Cornu Spiral can be expressed by two integrals given as
which represents the phase lag in the wave from any element of the wavefront.
 The integrals given by equations (4) and (5) are called as Fresnel Integrals. They cannot be integrated in closed form but gives the never-ending sequence that can be solved in several ways. In case of cylindrical lens, the in-phase and quadrature amplitude components A1 and A2 are obtained from the modified Fresnel Integrals [Davies and Whitehead, 1977]:
where L is the distance from the center of the diffraction pattern in meters, h is the distance between the irregularity and the observation point, i.e., for E region and F region in meters, and f is the frequency of the wave in (Hz),
 The first Fresnel minima have been obtained from power spectrum of day and nighttime QPS patches obtained and computed the velocity of irregularity which is further used to calculate the distance along the wavefront (L). The computed velocity of irregularities is 63.6 m/s for E region QPS and 116.6 m/s for F region QPS.
 For our theoretical modeling of QPS patches obtained at Varanasi, we have used equations (9), (10), (11), (12′), (13), and (14) and tried to simulate the recorded QPS patches using appropriate modeled parameters for low-latitude region.
5. Modeled Parameters
 For the theoretical modeling of the observed QPS patches recorded at Varanasi, we have computed the amplitude of the signals using equations (9), (10), (11), (12′), (13), and (14) and have tried to plot the amplitude against the time choosing some appropriate parameters for low-latitude region. We have made an effort to model up the QPS observed at Varanasi on the basis of radio lens model in the ionosphere [Davies and Whitehead, 1977]. We have chosen the model input parameters: scale size of the lens (ℓ), phase shift (θ0) for the E and F regions at certain frequencies and drift velocity for the irregularity patches in the ionosphere [Franke et al., 1984].
 Depending on the obliquity of the raypath (∼45°), Davies and Whitehead  considered the distance (h) from the lens to the receiver as 140 km and 420 km for the lens height of 100 km and 300 km in the E and F regions, respectively. They computed amplitude versus time pattern for cylindrical lenses for signal frequencies at 40, 140, and 360 MHz. Corresponding to the scale size of the lens they considered the width of the lens equal to 97 m, if the lens is in the E region, and 168 m if the lens is in the F region whereas the distance along the ground from the center of the diffraction pattern to the first maximum was 1100 m for E region and 1900 m for F region. At 140 MHz signal frequency, for h = 140 km and 420 km they have considered velocity of irregularities as 50 m/s and 86 m/s, respectively, whereas at signal frequency of 360 MHz, for h = 140 km and 420 km, the drift velocity was considered as 51 m/s and 89 m/s, respectively [Davies and Whitehead, 1977]. Franke et al.  assumed the distance between the irregularity and observation point (h) as 350 km, and the drift velocity as 135 m/s. Similar values of drift velocity (eastward) at Ascension Island were obtained by Weber et al.  and Basu et al. . Franke et al.  further obtained the extent of the diffraction pattern at VHF (from the effective center) in the range of 10–12 km and thus obtained the θ0/ℓ corresponding to a value in the range of 0.15–0.18. For θ0/ℓ = 0.15, they obtained the required phase shift θ0 = 54 radians for the scale size of lens as 350 m.
 For an attempt to model up the QPS patches, we have also assumed the irregularities to be at E region (daytime) and F region (nighttime), for which we have taken the irregularity height as 150 km (E region) and 350 km (F region). We have considered the frequency of signals as 250 MHz for our present computation of QPS patches.
 Using equation (14) we have computed the scale size of the lens which is 0.16 km at 250 MHz for the E region whereas for F region, the scale size of the lens is obtained as 0.26 km at 250 MHz [Franke et al., 1984]. By taking product of scale size of the lens (ℓ) and the ratio θ0/ℓ, the phase shift (θ0) for E region is obtained in the range of 4–7 radians and for F region it is from 30 to 45 radians. Using equation (12′), we have obtained the θ0/ℓ values as 0.03–0.04 for E layer and 0.14–0.17 for F layer at signal frequency 250 MHz. We have computed the distance from the center of diffraction pattern (L) as 1–1.2 km for E layer and for F layer as 10–12 km. The drift velocity for E layer is 63.6 m/s and for F layer is 116.6 m/s [Franke et al., 1984].
 Using equations (9), (10), (11), and other computed modeled parameters for low-latitude region we have computed the amplitude of the signal at 250 MHz for E and F regions of the ionosphere and plotted against time. In all, six different types of simulated QPS patches are shown in Figures 11a–11f. From our computed diffraction patterns we observed that our model represents a very narrow and intense cylindrical lens in the ionosphere [Davies and Whitehead, 1977]. We have compared our computed amplitudes (as shown in Figure 11) at 250 MHz frequencies with that of observed amplitudes (as shown in Figure 1) of QPS patches at our low-latitude station Varanasi radiated from FLEETSAT satellite and found that they are in good agreement with the modulation envelope being reproduced. Comparing both the Figures 1 and 11 we found that we have simulated successfully almost all types of observed QPS patches except type 6 (i.e., spike-type QPS patch) which is hard to be simulated using this cylindrical lens theory. Davies and Whitehead  found a good agreement between the theoretical and experimental amplitude versus time patterns at 140 and 360 MHz whereas a poor agreement at 40 MHz.
 The characteristics of the simulated QPS producing irregularities at E and F regions of the ionosphere are also studied theoretically using the computed amplitudes at different frequencies and then further computing the power spectrum, autocorrelation function and scintillation index. Figures 12a–12f show six typical examples of the autocorrelation functions of different types of the simulated diffraction pattern of QPS patches at 250 MHz. The half decorrelation time, τ and characteristic length computed from Figures 12a–12f are tabulated in Table 2. Figures 13a–13f show six typical example of power spectrum of the different types of simulated diffraction QPS patterns at 250 MHz. The spectral indices of the corresponding power spectra are tabulated in Table 2. After comparing case-by-case basis of six different simulated and observed characteristics of QPS patches from Table 2 and Figures 6 and 12 as well as Figures 8 and 13 we found, in general, a good agreement between computed and observed parameters of QPS patches, which validate our modeling results. The spectral indices of all the simulated QPS patches are computed which lie in the range −2 to −8. The characteristic lengths of all the simulated QPS producing irregularities are computed considering the drift velocity to be 100 m/s, which lies in the range of 200–650 m for the E and F regions irregularities. This shows that the QPS producing irregularities are of intermediate scale sizes. The computed scintillation index S4, ranges from 0.2 to 0.9 which shows that the irregularities are weak and moderate in strength.
Table 2. Six Typical Observed and Computed Parameters of Quasi-Periodic Scintillations From Figures 6, 8, 12, and 13
 All the above theoretically computed results are tabulated in Tables 1 and 2 and found to be in very much close agreement with that of experimental results obtained for QPS patches recorded at our low-latitude station Varanasi.
 From the diurnal variation of QPS events, we observed that the increase of sporadic E activity increases the QPS events [Maruyama, 1991]. The QPS event occurrence is maximum from 1600 to 1800 h IST which is also in agreement with that reported by Hajkowicz et al. . The month-to-month variation of the mean percentage occurrence of the QPS patches observed for the period January 1991 to December 1993 and April 1998 to December 1999 at Varanasi shows the peak occurrence in the summer months and minimum in the winter months. This seasonal variation of QPS patches is analogous to the maximum occurrence of sporadic E irregularities in summer months and minimum in winter months [Hajkowicz et al., 1981; Karasawa et al., 1985; Maruyama, 1991]. Franke et al.  have shown that the scale size of small structure irregularities responsible for quasi-periodic patches were of 50–500 m. Chen et al.  reported the east-west scale of the QPS producing irregularity about 24 km. Several workers found the range of the horizontal scale length of the QPS pattern from few meters to 1000 km [Whitehead, 1989; Ogawa et al., 1989]. Our results show that the scale size of the QPS producing irregularities varies from 100 to 1300 m which is of intermediate scale size.
 Strength of the irregularities specified by scintillation indices varies from 0.2 to 0.8 which shows the weak and moderate character of the QPS phenomena. From the in situ measurements of the electron density, Jahn and LaBelle  reported the spectral indices as −1.7 and −5. At Varanasi, daytime scintillation spectrum was found to be steeper than the nighttime scintillations [Fujita et al., 1982; Singh et al., 2006]. Our results show the variation of the spectral indices from −2 to −8 which are very much similar to the other reported results. All of these observed results are important for identifying the generation mechanism of ionospheric irregularities associated with QPS.
 Observed amplitude and phase variations of the QPS are due to the diffraction and refraction on an isolated electron density irregularity. The previous works were based on the simple ray optics [Slack, 1972] and on a phase changing screen model where the diffraction on the ground is computed from the Kirchoff diffraction formula [Franke et al., 1984]. Comparing the geostationary data with the theoretical one made the workers unable to find the height of obstacle as observed by Davies and Whitehead  and Karasawa , whereas Bowman  when compared the orbiting satellite data with the theoretical one found the obstacle in the E region of the ionosphere. The presence of the barrier in the E region was supported by the work of Hajkowicz et al.  which produces quasi-periodic scintillations [Maruyama, 1991]. With the use of incoherent scatter radar, Miller and Smith  showed that the irregularities are in the E region giving plasma blobs within a reserved height. The presence of irregularity in the F region gives QPS due to columnar shaped irregularity and when present in the E region then, due to the layers of sporadic E [Maruyama, 1991]. Chen et al.  have shown that the height of density irregularities that caused the QPS diffraction pattern were found in the F region between 310 and 330 km.
 We have taken the cylindrical Gaussian model due to the requirement that maximum phase shift should be large and the scale size of the lens should be small [Davies and Whitehead, 1977]. Our obtained amplitude diffraction patterns show the presence of very narrow and intense cylindrical lens as shown by Davies and Whitehead . The asymmetrical (modulated) diffraction patterns are due to the narrow opaque strip and symmetrical (unmodulated) are due to the interference of two signals [Jenkins and White, 1957, chapter 18]. Our calculated and observed amplitude of the QPS patches on 250 MHz are found in good agreement and the modulation envelope in general being reproduced as shown in Figures 1 and 11. To test our modeled results further, we have done the autocorrelation analysis and power spectrum analysis of our calculated amplitudes of QPS patches and computed the spectral index and characteristic length of irregularities which is shown in Table 1. From Table 1 we found good agreement between the observed and the modeled characteristics of the QPS producing irregularities, which supports the theory of cylindrical lens in the ionosphere. To validate this modeling approach we have made a specific comparison between simulated autocorrelation function (Figures 12a–12f) with that of observed autocorrelation function (Figures 6a–6f) as well as between simulated power spectrum (Figures 13a–13f) with that of observed power spectrum (Figures 8a–8f). Different characteristics of the six specific observed and simulated QPS patches are shown in Table 2. After comparing case-by-case basis we found, in general, a good agreement between computed and observed parameters of QPS patches, which validate our modeling results. By the autocorrelation analysis, most of the low-latitude QPS events are caused by relatively few, isolated irregularity structures [Franke et al., 1984]. Thus, our modeling approach has simulated successfully almost all types of observed QPS patches observed at low latitude as well as it also addressed some new features of QPS other than those attempted earlier.
8. Summary and Conclusions
 The characteristic features of the QPS at a low-latitude station Varanasi are discussed. The observations predict that QPS are the outcome of the plasma blobs/bubbles present in the E and F regions of the ionosphere. Most of the QPS are nonsymmetrical in nature with the maximum occurrence of burst-like scintillations, produced by the plasma blobs which are nonsymmetrical in nature with the steep density gradient at the backside.
 The seasonal variation of QPS patches shows the peak occurrence in the summer months and minimum in winter months which is analogous to the occurrence statistics of sporadic E irregularities. The diurnal variation of QPS events shows that the increase of sporadic E activity increases the QPS events. The horizontal scale sizes of the QPS producing irregularities vary from 100 m to 1300 m which shows that the irregularities are of intermediate-scale size. The spectral indices obtained from the slopes of power spectrum vary from −2 to −8, and the obtained scintillation index shows that the QPS are of weak and moderate strength. Our above obtained characteristics of QPS are found in good agreement with that of other workers.
 The observed fading patterns, especially the modulation of the diffraction pattern (fading envelope) can be explained by considering an obstacle called radio lens in the ionosphere elongated in one direction. We have simulated successfully the amplitude versus time plot of almost all types of QPS patches observed at low latitude. To validate this modeling approach the simulated QPS patches were again used to work out the autocorrelation function, power spectrum to find out theoretically the characteristic length, spectral index, and scintillation index that matches case-by-case very well with the experimentally observed values.
 Thus, we conclude that our simulated and observed characteristic of QPS patches compare well with each other and also with those of the earlier published works. Our modeling approach also addressed some new features of QPS other than those attempted earlier. This validates the radio lens in the ionosphere theory to explain the observed QPS patches at low latitudes.
 The work is partly supported by ISRO, Bangalore, and partly by UGC, New Delhi, under MRP. We are thankful to both the reviewers for their valuable comments and suggestions.