Digisonde spread F and GPS phase fluctuations in the equatorial ionosphere during solar maximum

Authors


Abstract

[1] The Jicamarca (11.95°S, 76.87°W) digisonde and the Arequipa (16.47°S, 71.49°W) GPS receiver observed the equatorial F region irregularities on the western South America from April 1999 to March 2000. The spread F measured by the digisonde were classified into four types, and the GPS phase fluctuations derived from the temporal variation of total electron content were divided into three levels to represent the irregularity strength. The observation shows that the occurrences of all four types of spread F are higher in the D months (January, February, November, and December) than in the E months (March, April, September, and October). For the GPS phase fluctuations, both seasonal and nighttime variations show that the occurrences of strong level irregularities are higher than moderate level irregularities in the E months, but the situation is reversed in the D months. Moreover, the occurrence sequences of four types of spread F and three levels of GPS phase fluctuations all can be explained by the E × B drift variations and the generalized Rayleigh-Taylor instability. For the comparisons between the GPS phase fluctuations and the digisonde spread F/plasma bubbles, results show that the GPS phase fluctuations can represent the appearances of the digisonde spread F, and the strong level of GPS phase fluctuations are associated with the occurrence of topside plasma bubbles. These results imply that the greater GPS phase fluctuation is related to the larger altitudinal range distribution of irregularities.

1. Introduction

[2] Equatorial spread F (ESF), a kind of ionospheric irregularity, often occurs in the equatorial F region after sunset. The ESF is mainly generated by the generalized Rayleigh-Taylor (GRT) instability, which is the gravitational Rayleigh-Taylor instability in conjunction with other processes [e.g., Dungy, 1956; Kelley, 1989; Adbu, 2001]. The instability forms irregularities in the bottomside F layer. Moreover, nonlinear development of the instability may raise these irregularities to ∼1000 km altitudes [Kelley, 1989]. According to previous studies [Fejer et al., 1999; Whalen, 2002; Lee et al., 2005a], the E × B drift is the major mechanism for ESF development by directly increasing the growth rate or indirectly decreasing the collision effect [Kelley, 1989; Lee, 2006]. Additionally, the transequatorial neutral wind may play an important role in suppressing ESF generation [Maruyama, 1988; Mendillo et al., 2000, 2001; Lee et al., 2005a]. For more than 5 decades, ESF observations have been made by many different instruments including ionosonde [King, 1970; Rastogi, 1980; Sales et al., 1996; Whalen, 2001, 2002], Global Positioning System (GPS) receiver [Aarons et al., 1996; Basu et al., 1999; Valladares et al., 2001], VHF radar [Woodman and LaHoz, 1976; Hysell and Burcham, 1998; Abdu et al., 2006a], satellite [Huang et al., 2001; Su et al., 2001], CCD imager [Weber et al., 1978; Mendillo and Baumardner, 1982], and rocket [Rino et al., 1981]. Nevertheless, little attention was paid to concurrent GPS and digisonde observation. Simultaneous, collocated digisonde and GPS observation can provide a better understanding of the physical processes of ESF.

[3] Since Booker and Wells [1938] studied the ESF occurrence on ionograms, many investigators have used ionosondes to study the equatorial F region irregularities. Rastogi [1980] reported extensive statistical results about seasonal and solar cycle variations of ESF by analyzing ionograms from the Huancayo (12.03°S, 75.33°W) ionosonde. Argo and Kelley [1986] investigated the irregularities drift direction and ionosphere height at ESF onset condition and late-night irregularities formation during the CONDOR campaign. Whalen [2001, 2002] examined the interrelation between the bottomside spread F (BSSF) and bubbles, as well as the maximum E × B drift velocity using an array of ionosonde located on the western South America during solar maximum. Abdu et al. [2006b] used ionosondes to determine the ESF statistics for comparison with the International Reference Ionosphere (IRI) predictions. Recently, Lee et al. [2005a] investigated the seasonal variations in the ESF occurrence using the Jicamarca digisonde (11.95°S, 76.87°W). Regarding the ESF study using GPS, Wanninger [1993] first introduced the rate of total electron content (TEC), ROT, to describe the ionospheric irregularities. Pi et al. [1997] defined the rate of the TEC index (ROTI) based on the standard deviation of ROT to study the occurrence of smaller-scale irregularities. Meanwhile, Aarons et al. [1996] introduced the GPS phase fluctuations to characterize irregularities development in the equatorial region. Further, Mendillo et al. [2000, 2001] defined two indices based on the GPS phase fluctuations to characterize known features of ESF and studied the effect of transequatorial neutral wind suppressing ionospheric irregularities.

[4] In this paper, we employed the Jicamarca digisonde (11.95°S, 76.87°W) and Arequipa GPS receiver (16.47°S, 71.49°W) to observe F region irregularities in the equatorial ionosphere from April 1999 to March 2000. The spread F observed by the digisonde and the GPS phase fluctuations recorded by the GPS receiver were used to reveal the seasonal and nighttime behavior of equatorial F region irregularities. Although the seasonal variations for the digisonde spread F at Jicamarca was published by Lee et al. [2005a], a detailed classification of spread F echoes was applied here to study the generation and evolution of spread F, and to examine the relationship between the digisonde spread F and the GPS phase fluctuations.

2. Experiment Setup

[5] Figure 1 shows the location of the Jicamarca digisonde (11.95°S, 76.87°W, 1°N dip latitude) and the Arequipa GPS receiver (16.47°S, 71.49°W, 3.5°S dip latitude). The small circle represents the illuminating area at 400 km height of the digisonde transmit antenna, while the large circle represents the field of view at 400 km height of the GPS receiver for the elevation angle >15°.

Figure 1.

Map of the Jicamarca digisonde and the Arequipa GPS receiver. The small circle represents the illuminated area at 400 km height of the digisonde transmit antenna, and the large circle symbolizes the field of view at 400 km height of the GPS receiver for elevation >15°.

[6] We used the digisonde to detect the F region irregularities at the magnetic equator from April 1999 to March 2000. The digisonde recorded an ionogram every 30 min. When irregularities develop, the diffused F layer echo traces will appear on the ionogram, called spread F. In this work, for studying spread F in detail, we categorized the spreading echo traces into four types based on the U.R.S.I Handbook of Ionogram Interpretation and Reduction [Piggott and Rawer, 1972]. These four types are (1) range spread F type I (RSF-I), (2) range spread F type II (RSF-II), (3) frequency spread F type I (FSF-I), and (4) frequency spread F type II (FSF-II). Figure 2 shows illustrations and examples of these four types of spread F. It is noted that the distribution of spread F echoes symbolize the distribution of irregularities. For the RSF-I type, irregularities appear in the lower part of bottomside F region. The RSF-II and FSF-I types indicate that irregularities fill with all bottomside of F region, and irregularities exist near the F layer peak, respectively. For the FSF-II type, irregularities distribute from the lower part of bottomside F region to the F layer peak, but the spreading is more severe near the F layer peak than in the lower part of bottomside F region.

Figure 2.

Examples of four types of spread F. The left side is illustrations for spread F classification which were reproduced from U.R.S.I. Handbook of Ionogram Interpretation and Reduction [Piggott and Rawer, 1972]. The right side presents examples of spread F type, which are (a) RSF-I, (b) RSF-II, (c) FSF-I, and (d) FSF-II, respectively.

[7] The index Fp of the GPS phase fluctuations proposed by Mendillo et al. [2000] was applied to describe the F region irregularities in this work. This Fp value was derived from the temporal variation of TEC obtained by the Arequipa GPS station by following two formulas:

equation image

and

equation image

where n is the satellite number; hr is the hour (0000–2400 UT); i is the 15-min time section within an hour (i = 1, 2, 3, 4); nsat is the total number of satellite observed within 1 hour; and k is the number of fp value available within each hour (k = 1, 2, 3, 4). Notice that the time resolution of raw GPS data is 30 s and that of the GPS phase fluctuation indices, fp and Fp, are 15 min and 1 hour, respectively. According to Mendillo et al. [2000], the magnitude of Fp indicates the strength of existing irregularities: Fp ≤ 50 means that the GPS phase fluctuations come from background level of irregularities; 200 < Fp means that the GPS signal is influenced by very strong irregularities; 50 < Fp ≤ 200 indicates the existence of moderate irregularities. In following sections, these three Fp levels will be used to examine detailed relationship between the digisonde spread F and the GPS phase fluctuations.

3. Results and Discussions

3.1. Digisonde Spread F

[8] The results of seasonal variations in digisonde spread F was reported by Lee et al. [2005a] (see their Figure 4) based on the same digisonde data used in this study. Lee et al. [2005a] pointed that the spread F occurrence is higher in the D (January, February, November, and December) and E (March, April, September, and October) months and lower in the J months (May, June, July, and August). They confirmed that the seasonal variation in spread F occurrence is related to the vertical E × B drift velocity. Moreover, larger upward drift velocity in the D and E months favor the generation of spread F [Fejer et al., 1999; Lee et al., 2005a; Whalen, 2002].

[9] For a detailed study of spread F occurrence, we have examined the nighttime (1800–0600 LT) variation in relative occurrence for each spread F type for the E, D, and J months (Figure 3). The relative occurrence is the number of each type of spread F in an hour divided by the number of observed ionogram in this hour for the season. For the E months (Figure 3a), the type of RSF-I (thick gray line) occurs only during 1900–2100 LT with a maximum occurrence value of 19%. The type of RSF-II (line with circles) appears from 1900 to 0500 LT, and its maximum value (43%) is at 2100 LT. Regarding the FSF-II type (line with triangles), it appears from 2100 to 0500 LT with a maximum occurrence of 23% at 0200 LT. The FSF-I type (dotted line) also starts occurring at 2100 LT but remains at a low level of ∼10% for the rest of the night. In the Figure 3b (the D months), the RSF-I type exists between 1900 and 2200 LT and its maximum occurrence (29%) is at 2000 LT. The RSF-II type appears during 1900–0500 LT with a maximum value (51%) of occurrence at 2300 LT. The FSF-II type appears during 2100–0500 LT with a maximum value of 32% at 0200 LT. Regarding the FSF-I type, it starts at 2000 LT and has low occurrence values (∼15%) during 2000–0500 LT. In the J months, the relative occurrences in the seasonal variation are low [see Lee et al., 2005a, Figure 4]. Therefore in Figure 3c, the nighttime variation for the occurrence of each type of spread F are also low (<20%). Owing to the low occurrence, it is difficult to identify the behaviors of nighttime variation of each type of spread F in this season.

Figure 3.

The nighttime variation in the relative occurrence of different type spread F in (a) the E months (March, April, September, and October), (b) the D months (January, February, November, and December), and (c) the J months (May, June, July, and August).

[10] Figures 3a and 3b show that the values of the nighttime occurrence for each type of spread F in the D months are slightly higher than that in the E months. This feature is expected from the seasonal variation of total ESF reported by Lee et al. [2005a, Figure 4]. The E × B drift with large enough upward velocity, later reversal time, and smaller early night downward velocity in the D months provide a favorable condition for the development of spread F [Fejer et al., 1999]. Moreover, it is noted that the nighttime occurrence of RSF-II in the D months is obviously different from that in the E months among these types of spread F. The time of maximum occurrence of RSF-II is later in the D months than in the E months. This time delay can be caused by the later reversal time of the E × B drift in the D months.

[11] However, Figures 3a and 3b also indicate that the sequences of each type of spread F in the E and D months are similar. The peak value of RSF-I first appears in 1900–2000 LT, then the RSF-II occurrence peaks at 2100 and 2300 LT in the E and D months, respectively. Finally, FSF-II peaks at 0200 LT. Such kind of the sequence in the E and D months demonstrates that the F region irregularities initially develop at the lower part of the F region (RSF-I). This initiation of spread F at lower altitudes was addressed by earlier studies with digital ionosonde [Argo and Kelley, 1986], VHF radar [Farley et al., 1970], and rocket observation [Kelley et al., 1986]. After spread F initiation, the irregularities ascend to higher altitudes and then distribute in whole F region (RSF-II). While the irregularities at lower altitudes dissipate, they still can persist at higher altitudes (FSF-I/FSF-II). Moreover, this sequence of each type of spread F is consistent with the evolution of F region irregularities described by the GRT theory [Kelley, 1989].

3.2. GPS Phase Fluctuations

[12] In the case of the GPS phase fluctuations, the seasonal variations in relative occurrence for two thresholds, 50 < Fp and 200 < Fp, are shown in Figure 4. The relative occurrence in the figure is the number of day on which at least one premidnight (1800–2400 LT) GPS phase fluctuation event is observed divided by the number of day in a month. Clearly, the occurrence distribution for 50 < Fp (dotted line with circles) is different from that for 200 < Fp (thick line with solid circles). For 50 < Fp, the highest (96%) and lowest (3%) value of occurrence are in December (D months) and June (J months), respectively. For 200 < Fp, two peaks exist in the E months, one in March (72%), the other in October (67%). Further, to compare the 50 < Fp occurrence with the 200 < Fp occurrence, it can be seen that the relative occurrences in the E months (except September) have ∼15% difference in average between two thresholds. This small difference demonstrates that irregularities in the E months are mostly the strong level. In the D months, the larger difference (>30%) between these two threshold distributions suggests that irregularities are predominantly the moderate level, 50 < Fp ≤ 200. For the J months, irregularities of any strength level are rare.

Figure 4.

Seasonal variations in the relative occurrence of GPS phase fluctuation for 50 < Fp and 200 < Fp.

[13] Figure 5 displays the relative occurrences for 50 < Fp ≤ 200 and 200 < Fp during nighttime (1800–0600 LT) in the E, D, and J months. This relative occurrence is the number of event of each level in an hour divided by the total number of the record in that hour in a season. The two Fp levels, 50 < Fp ≤ 200 and 200 < Fp, represent the moderate and strong irregularities, respectively. In Figure 5a (E months), it is found that there are two peaks (29% and 44%) at 1900 and 0100 LT in the nighttime variation for the moderate level (50 < Fp ≤ 200, line with triangles). For the strong level, 200 < Fp (gray line with solid circles), the maximum value (47%) appears at 2100 LT. Generally, the occurrences of 200 < Fp are higher than that of 50 < Fp ≤ 200 in premidnight but lower in postmidnight. In the D months (Figure 5b), two peak values (56%) of relative occurrence exist at 1900 and 2300 LT for the moderate level, while the maximum occurrence (41%) is at 2000 LT for the strong level. In this season, the relative occurrences of 50 < Fp ≤ 200 are higher than that of 200 < Fp in each hour except 2000 LT. Regarding the J months (Figure 5c), the relative occurrences for both 50 < Fp ≤ 200 and 200 < Fp level irregularities are lower than 10% during entire night. Therefore it is unlikely to characterize the variations in this season.

Figure 5.

The nighttime variations in the relative occurrence for two levels of GPS phase fluctuation in the (a) E, (b) D, and (c) J months.

[14] Comparing the nighttime variation in relative occurrence of the E months with that of the D months, the occurrences of 200 < Fp are higher in the E months than in the D months, but the occurrences of 50 < Fp ≤ 200 are lower in the E months than in the D months. These results are similar with the seasonal variations in relative occurrence (Figure 4). The strong and moderate irregularities occur more frequently in the E and D months, respectively. Considering the E × B drift variations [Lee et al., 2005a, Figure 5], the larger upward drift velocity in the E months corresponds with the strong irregularities formation.

[15] At the same time, there are similar features appearing on both Figures 5a and 5b. A one peak distribution for 200 < Fp and a two peak distribution for 50 < Fp ≤ 200 are found in both the E and D months. The first peak of 50 < Fp ≤ 200 appears initially at 1900 LT, and then the peak of 200 < Fp comes next at 2000–2200 LT. Finally, the second peak of 50 < Fp ≤ 200 in the E (D) months appears at 0100 (2300) LT. This sequence reveals that irregularities form at moderate level, then enhance to strong level, and decay to moderate level at last. This result also agrees with the evolution of irregularities predicted by the GRT instability process [Kelley, 1989].

3.3. Comparisons Between Digisonde Spread F and GPS Phase Fluctuations

[16] Figure 6 shows the comparisons of the seasonal variations in relative occurrence between the GPS phase fluctuations and the digisonde spread F as well as the ROCSAT-1 plasma bubbles [Lee et al., 2005b]. The seasonal variations of digisonde spread F and plasma bubbles are adopted from Lee et al. [2005a] and Lee et al. [2005b], respectively. Notice that the digisonde spread F referred here contains all spread F types. It is found that in Figure 6a the occurrence distribution of digisonde spread F (line with solid triangles) is similar to that of 50 < Fp (dotted line with circles), and the maximum and minimum occurrence are in the D and J months, respectively. This similarity suggests that the thresholds, 50 < Fp, of GPS phase fluctuations can represent the occurrence of spread F on ionogram. In Figure 6b, the occurrences of both 200 < Fp (thick line with solid circles) in the GPS phase fluctuations and the topside plasma bubbles (dashed line with asterisks) observed by the ROCSAT-1 at 600 km altitudes show a similar double peak distribution. This similarity indicates that the occurrence of topside plasma bubbles is associated with the strong level of GPS phase fluctuations. On the basis of Lee et al. [2005b], the larger upward E × B drift velocity would lift the bottomside F region irregularities to higher altitudes (topside plasma bubbles). Therefore the appearance of topside plasma bubbles means that irregularities distribute from bottomside to topside (600 km) in F region. In other words, the larger altitudinal range with irregularities distributed could make the Fp value greater. It is worth to mention that the relation between the thickness of irregularity layer and the scintillation strength was also emphasized by Rodrigues et al. [2004].

Figure 6.

(a) The seasonal variations of relative occurrence of 50 < Fp (dotted line with circles) and spread F (line with solid triangles). (b) The seasonal variations of relative occurrence for 200 < Fp (thick line with solid circles) and bubbles (dashed line with asterisks). Notice that the variations of spread F and bubble were reproduced from Lee et al. [2005a, 2005b].

[17] To study the nighttime variation of the GPS phase fluctuations and the digisonde spread F in more detail, the occurrence distributions of four types of spread F were compared with that of GPS phase fluctuations. In the E months (Figure 7), we have found that the nighttime variations of RSF-II and RSF-I+FSF-I+FSF-II are good fits for the 200 < Fp and 50 < Fp ≤ 200, respectively. In Figure 7a (Figure 7b), the occurrence distribution of RSF-II (RSF-I+FSF-I+FSF-II) agrees with that of 200 < Fp (50 < Fp ≤ 200) during 1800–0100 LT. Again, these agreements demonstrate that the greater (smaller) Fp value is related to the larger (smaller) altitudinal range with irregularities distributed. However, during 0100–0600 LT, the occurrences of digisonde spread F are higher than that of GPS phase fluctuations as shown in Figures 7a and 7b. These discrepancies will be discussed in the following paragraph.

Figure 7.

Comparisons of the nighttime variations between the digisonde spread F and the GPS phase fluctuations in the E months: (a) RSF-II (thick line) and 200 < Fp (dotted line), (b) RSF-I+FSF-I+FSF-II (thick line) and 50 < Fp ≤ 200 (dotted line).

[18] According to Pi et al. [1997], the GPS phase fluctuations is formed by the optical path changes of a radio wave because the irregularity scale size is much larger than first Fresnel zone. The irregularity scale size for GPS phase fluctuation is few kilometers for high elevation angle and tens kilometers for low elevation angle [Aarons et al., 1997]. Regarding the digisonde, the cause of spreading echoes may due to partial reflection (scattering) or total reflection. For partial reflection, the digisonde operates as a HF radar and the scale size of irregularity is in tens meters scale range. On the other hand, the spreading echoes caused by total reflection depend on surfaces of constant electron density, although King [1970] already showed that spreading echoes mainly originate from total reflection. However, it is still unable to separate these two mechanisms from an ionogram. Let us now return to the postmidnight irregularities; it might be associated with the magnetic disturbance [Fejer et al., 1999] or the spread F fossil [Argo and Kelley, 1986] which drift from the west to the east. Among these two mechanisms, the fossil would be the major source of the postmidnight F region irregularities because the magnetic disturbance seldom occurred on postmidnight during April 1999 to March 2000. Moreover, Kelley [1989] mentioned that the irregularity spectrum is quite different between development and equilibrium stage. Thus the fossil should be in equilibrium stage which mainly contains large-scale (>1 km) irregularities. Further, the irregularities larger than this scale size could persist more than 4 hours at least [Argo and Kelley, 1986]. However, this concept lead us the question that irregularities with scale size larger than 1 km can be detected by the digisonde after 0300 LT, but not for the GPS phase fluctuations. This question needs more investigations to explain.

[19] In the D months, there is no clear relation could be found before 2300 LT (Figure 8a), and the better fit is the RSF-II and 50 < Fp ≤ 200 during 2300–0600 LT (Figure 8b). In Figure 8a, the relative occurrence of 200 < Fp is same as RSF-I between 1900 and 2000 LT, and there is almost no RSF-II occurrence during this period. After 2000 LT, the relative occurrence of RSF-II sharp increase immediately, but the 200 < Fp and the RSF-I show decreasing gradually. Moreover, the 200 < Fp and the RSF-I maintain ∼10% differences before 0000 LT. This result is different from the E months in which the larger Fp value is directly related to larger altitudinal range with irregularities distributed. Additionally, this result implies that the influence of the altitudinal range before midnight in the D months is not obvious as that in the E months. Considering the E × B drift variation is different between the E and D months. Thus the composition of various scale size irregularities may show different signatures during irregularities development in these two seasons and then cause different GPS phase fluctuation variations. During 2300–0600 LT (Figure 8b), the agreement between the RSF-II and the 50 < Fp ≤ 200 implies that the altitudinal range with irregularities distributed is reduced. Because the diffuse echoes are still filling in the region below the F layer peak (RSF-II), the reduction of the altitudinal range could exist in the topside ionosphere. Recall that the upward E × B drift velocity in the D months is smaller than in the E months. Thus the irregularities in the D months may not rise into the altitudes as high as E months [Lee et al., 2005b]. Therefore the duration of strong level GPS phase fluctuation will be shorter than that in the E months. Figure 6b which shows low occurrences of topside plasma bubbles (high-altitude irregularities) in the D months also supports this argument indirectly. Finally, Figure 8b shows no discrepancies between two curves after 0200 LT. This result implies that the life time of irregularities, which are sensitive for the GPS phase fluctuations, is longer in the D months than in the E months.

Figure 8.

Comparisons of the nighttime variation between the digisonde spread F and the GPS phase fluctuations in the D months: (a) RSF-I (thin line), RSF-II (thick line), and 200 < Fp (dotted line), (b) RSF-II (thick line) and 50 < Fp ≤ 200 (dotted line).

4. Summary

[20] In this study, the digisonde and the GPS receiver data have been analyzed to describe the equatorial F region irregularities during solar maximum. This is the first attempt to systematically study the digisonde spread F and the GPS phase fluctuations at magnetic equator simultaneously. Moreover, a detailed classification of spread F has been introduced, which characterizes the time evolution and altitudinal range distribution of spread F.

[21] For the digisonde spread F, the nighttime occurrences of four types all are higher in the D months than that in the E months. This is due to the later reversal time and smaller early night downward drift velocity in the E × B drift variations of D months. Moreover, the similar sequences of four types of spread F in the E and D months reveal that the spread F evolves from lower to higher parts of F region, as predicted by the GRT instability theory.

[22] For the GPS phase fluctuations, both seasonal and nighttime variations for 50 < Fp and 200 < Fp demonstrate that the irregularities of strong and moderate level appear frequently in the E and D months, respectively. Furthermore, for the nighttime variations, the peak sequence of the distributions of 50 < Fp ≤ 200 and 200 < Fp in the E and D months can also be explained by the GRT instability theory.

[23] Regarding the relationship between the digisonde spread F and the GPS phase fluctuations, the seasonal distribution comparisons suggest that 50 < Fp can represent the appearances of spread F on the ionogram, and the occurrence of topside plasma bubbles can associate with the strong level GPS phase fluctuations (200 < Fp). Further, by comparing the nighttime variations in detail, the results in the E months indicate that the greater (smaller) Fp value is related to the larger (smaller) altitudinal range with irregularities distributed. For the comparison in the D months, it implies that the altitudinal range with irregularities distributed in the topside ionosphere is smaller in the D months than that in the E months. Meanwhile, the lifetime of irregularities, which are sensitive to the GPS phase fluctuations, is longer in the D months than in the E months.

Acknowledgments

[24] WSC and JYL are supported by the National Science Council grant NSC 95-2111-M-008-018. CCL is supported by NSC 95-2111-M-231-001. BWR is supported by AFRL grants F19628-02-C-0092 and FA8718-06-C-0072. The authors would like to thank the International GPS Service (IGS) for Geodynamics for GPS data and the Jicamarca Radio Observatory (J. Chau) and the UML DIDBase archive for the digisonde data. The authors would also like to thank C. Goodwin, the JGR editor's assistant, who helped to improve the readability of this paper.

[25] Amitava Bhattacharjee thanks Krishna Iyer and another reviewer for their assistance in evaluating this paper.

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