One or more steeply sloped traces have been found in evening ionograms taken from the Kwajalein Atoll (4.3°N dip latitude) during July 1979. Their resemblance to the normal F trace suggests that they are echoes that have undergone a large number of reflections from the F layer. These multi-reflected echoes (MREs) are interpreted in terms of focusing produced by curved isodensity contours in the bottomside F layer, which appear to be associated with large-scale wave structure (LSWS) that develops in the bottomside F layer. MREs appear to be another signature for LSWS, together with satellite traces that appear later in time, closer to the onset of plasma structure referred to as equatorial spread F. MREs are interesting because they display, for the data set examined, a strong preference to occur during the post-sunset rise of the F layer, which includes E-region sunset. How this finding affects our understanding of LSWS is discussed.
 There is mounting evidence that large-scale wave structure (LSWS) is intimately associated with plasma structure that develops in the nighttime equatorial F ionosphere, often referred to generically as equatorial spread F (ESF). Results obtained with ALTAIR, a steerable incoherent-scatter (IS) radar, were the first to show that smaller-scale irregularities, those responsible for radio scintillations and radar backscatter, tend to occur in the bottomside F layer, where isodensity contours are locally displaced upward in altitude [Tsunoda, 1981, 1983; Tsunoda and White, 1981]. These regions, called upwellings or crests, turn out to be organized and distributed in longitude by an LSWS [Tsunoda and White, 1981]. The finding that an LSWS precedes development of smaller-scale irregularities is evidence that ESF generation process is likely initiated by the development of LSWS.
 Given that ESF occurs in regions characterized by downward-concave isodensity contours, it is not surprising to find that the appearance of “satellite” traces in ionograms is a direct precursor of ESF. Careful examination of ionograms, taken with high temporal resolution [Lyon et al., 1961] and over an extended period of time [Abdu et al., 1981], has shown that there is virtually a one-to-one occurrence of satellite traces followed by ESF. We have recently shown in a case study, using simultaneous measurements by ALTAIR and a collocated ionosonde, that concave isodensity contours were indeed present overhead, when doubling of the second-hop F (2F) trace was detected [Tsunoda, 2008]. A low-altitude satellite in a low-inclination orbit has also provided evidence that LSWS precedes occurrence of ESF [Singh et al., 1997]. With this set of results, we have concluded that satellite traces are signatures of LSWS, and that LSWS must participate directly in the initiation of ESF [Tsunoda, 2005, 2008].
 Although we now have good reason to seek a better understanding of LSWS, an obstacle to doing so is the paucity of sensors that can detect, let alone describe, an LSWS. The reason is that LSWS does not appear to display much zonal drift, at least not during its early growth phase [Tsunoda, 2005]. This means that overhead measurements made with a sensor at a fixed location will not contain temporal oscillations, which would be produced if an LSWS drifted. We, therefore, must find ways to detect stationary spatial structure, for example, from backscatter patterns obtained with steerable or multi-beam radars [e.g., Tsunoda and Ecklund, 2007], or from total electron content variations obtained by using a low altitude satellite in a low-inclination orbit [Tsunoda and Towle, 1979]. There is also useful, though less direct, information about LSWS that can be extracted from ionogram signatures, such as satellite traces [Tsunoda, 2008].
 In this paper, we describe what appears to be another ionogram signature of LSWS, that is, echoes that have undergone an unusually large number of reflections. Interpretation of these multi-reflected echoes (MREs) is that signal strength is enhanced, when reflected from concave isodensity contours. Motivation for this investigation comes from a report on MREs by Rastogi , who noted their occurrences around local sunset, and suggested that focusing must occur from concave reflecting surfaces that likely form, when a rapid post-sunset rise (PSSR) of the F layer is followed by an equally rapid descent, especially during years of high solar activity. Instead, we show that MREs occur quite frequently during the PSSR, even prior to E-region sunset; we interpret this finding in terms of concave isodensity contours that are associated with an LSWS, rather than the end of the PSSR.
2.1. Examples of MREs
 Our evidence for MREs were obtained with a Vertichirp sounder, which was operated in the Kwajalein Atoll, Marshall Islands (9.4°N, 167.5°E, geographic; 4.3°N dip latitude) during July 1979. Local time (LT) there leads universal time (UT) by 11 hr 10 min. Examples of MREs, from 23 and 28 July 1979 (Days 204 and 209), are presented in Figure 1; both were taken at 0751 UT (1901 LT). Ionograms were taken every 5 to 10 min during this period on Day 204, but only every 20 min on Day 209. The normal F-region traces from both days are similar. Both contain one hop (1F) and two hop (2F) reflection traces with the virtual height of the 2F trace precisely twice that of the 1F trace at all frequencies, which makes the slope of the 2F trace twice that of the 1F trace. Usual interpretation for such features is in terms of overhead reflections from a horizontally stratified F layer.
 The MREs appear in both ionograms as several discrete traces, whose intensities are comparable to, but whose slopes are steeper than, those of the 1F and 2F traces. The steepness of the MRE traces suggests that these echoes were received after undergoing a large number of reflections [Rastogi, 1977]. For example, the slope of the 1F trace near 9 MHz on Day 204 is 25°, whereas that of the MRE is 70°, which implies that the MRE could be the 6F trace, which ended up near the 1F trace because of range aliasing. The MRE that appeared around 5 MHz on Day 209 could be the 8F trace. These estimates are similar to those by Rastogi , who found MREs over Kodaikanal and Huancayo to be 8F to 10F traces.
 Interestingly, the separation in virtual height between MRE traces that appear in an ionogram is not always the same as that between the 1F and 2F traces. For example, the separation at 5 MHz on Day 209 is the same, but the separation at 8 MHz on Day 204 is not. Also, the additional traces on Day 204 occurred at frequencies that did not seem to produce a 2F trace. According to Rastogi , this behavior can be explained in terms of multiple reflections from a concave surface. The unequal separation distances suggest that more complex paths, including oblique reflections, could have been involved.
 On Day 204, a weak MRE appeared first around 0721 UT, was slightly stronger at 0731 UT, then disappeared at 0741 UT before reappearing strongly at 0751 UT. MREs continued to appear until 0816 UT, which means MREs persisted on this day for nearly an hour. Doubling of the 1F trace occurred at 0816 UT, which led to range spreading by 0841 UT. On Day 209, whether MREs occurred at 0731 UT is not obvious, but strong MREs were observed at 0751, 0811, and 0831 UT, or for about 40 min. A 2F doublet occurred at 0851 UT, which was followed by rapid onset of ESF around 1011 UT. On Day 192, weak MRE appeared at 0731 UT, followed by the strong MRE at 0811 UT, and disappearance by 0831 UT. Both 1F and 2F doublets appeared at 1001 UT. Range spreading, which commenced around 1011 UT, occurred only at the lowest frequencies (<4 MHz). A second MRE occurred later, and that occurrence seemed to be followed by ESF. Hence, MREs seem to persist for 30 min to an hour, and are often followed soon thereafter by doublets and ESF. MREs, however, are not always followed by ESF. For example, out of 18 nights in July, MREs were detected on four nights (22 percent), when ESF did not appear to develop, while MREs were detected on 14 nights (78 percent), when ESF did develop.
 To place these MRE signatures in context, we have plotted the virtual height of the bottomside F layer, at 3 MHz, h′(3), as a function of UT for the three nights in Figure 2. Two are for nights corresponding to the examples in Figure 1. Results from a third night, Day 192, are included to illustrate additional behavioral traits. (Ionograms on Day 204 were obtained every 10 min, up to 0811 UT, then every 5 min until 1121 UT; those on Day 209 were obtained every 20 min; and those on Day 192 were obtained every 30 min.) All three curves are virtually identical up to about 0730 UT (1840 LT). One reason could be that the apparent rise of the F layer, when it was below 300 km, is controlled more by recombination-loss chemistry than by transport [e.g., Tsunoda and White, 1981]. The PSSR is evident from 0730 to around 0900 UT, at least for the two active nights. (The altitude of the geomagnetic field line over the dip equator is about 35 km higher than over Kwajalein.) The strongest MREs all occurred with 10 min of each other, between 0751 and 0801 UT, as indicated by the enlarged black symbols (circle, square, and triangle).
2.2. Occurrence Times
 The most important finding, as seen in Figure 2, is a tendency for MREs to occur during the PSSR, even prior to the time of E-region sunset (vertical gray line). Using ionograms from 18 days (July 10–14, 17–29), we have plotted the occurrences of MREs, during hourly intervals, as a function of UT, which are presented in Figure 3. The LT scale is shown along the top of the plot. We see that occurrences of MREs were tightly clustered between 1800 and 2000 LT, which appears to includes all of the PSSR and E-region sunset (the latter occurs around 1900 LT in July). This finding differs from the interpretation of MREs by Rastogi , that focusing is produced by the concave isodensity contours that are associated with the end of the PSSR. If his interpretation were correct, MRE occurrences should maximize at the end of the PSSR, which typically occurs around the time of F-region sunset, or about 1930 LT. In fact, the hourly interval of maximum occurrences was from 0800 to 0900 UT, which does bracket 1930 LT, but almost an equal number of MREs occurred between 0700 and 0800 UT. Moreover, indications are that MREs seem to occur during the PSSR, when the F layer is still rising, as seen from the curves in Figure 2.
2.3. A New Interpretation of MREs
 Given the apparent preference of MREs to occur during the PSSR, we suggest that the concave isodensity contours associated with an LSWS are more likely to be responsible for MREs than those associated with the end of the PSSR. We begin by showing that focusing by an LSWS is likely to be stronger than that by a sunset effect. Relative echo strength associated with total reflection from a flat surface can be estimated from the size of the first Fresnel zone. For example, for an ionosonde frequency of 7 MHz and reflection at an altitude of 300 km, the diameter of the first Fresnel zone is about 7 km. Echo strength is enhanced if reflection occurs from a concave surface; it maximizes, when the radius of curvature of the isodensity contours is equal to the reflection altitude.
 First, we consider the reflection geometry that is thought to exist at the end of the PSSR. During a period of high solar activity (e.g., 1979), the maximum PSSR rate can be 50 to 60 m/s around E-region sunset, which should produce a mean tilt in isodensity contours of about 6°. This tilt should decrease as the end of the PSSR is approached. If the descent is as rapid as the ascent, the isodensity contours should become concave and symmetric, as envisioned by Rastogi . The PSSR and descent are represented by the gray line segments in Figure 4a. Here, we have selected an idealized geometry, in which the ionosonde is located beneath the end of the PSSR. (Although shown here as fixed tilts, we assume there would be a gradual variation, which could resemble a circular arc segment with a radius equal to the reflection altitude.) However, because the tilt is no more than 6°, the angular width over which reflections would return to the ionosonde would be no more than 12°, as shown. The idealized, circular-arc length, therefore, would not exceed 32 km for a radius of 310 km. This means that the echo strength associated with the end of the PSSR would be no more than about 6 dB above that, which would be associated with the Fresnel zone of a reflection from a horizontal surface.
 Next, we consider the enhancement that could occur from an LSWS. For comparison, we have sketched a black wavy curve for an LSWS with a horizontal wavelength of 400 km and an amplitude of 10 km. If a half wavelength of the LSWS can be approximated by a circular arc with a radius of curvature of 310 km, the angle over which we can expect in-phase reflection is ±19° or an arc length of 205 km. In this case, the enhancement associated with an LSWS would be as much as 15 dB above a reflection from a horizontal surface. As the amplitude of the LSWS increases, the enhancement could become proportionally larger. Another difficulty with an MRE produced by the end of the PSSR is that it would have to be associated with the location of the end of the PSSR, which moves westward at a speed of 15°/hr in longitude, or about 500 m/s. On the other hand, a crest in an LSWS could occur anywhere, and would grow in amplitude without significant zonal transport [Tsunoda, 2005]. The finding that MREs appear during the PSSR, and not always at the end of the PSSR, favors the conclusion that MREs are produced by LSWS. The tendency for MREs to occur during the PSSR, and up to the end of the PSSR, is likely also associated with the fact that the F-region plasma is still unstructured during that period, but becomes structured after the PSSR. Once structure develops, the reflection coefficient would decrease, and MREs should become diffuse and weaker.
 We must also consider what may occur during the PSSR. This geometry is sketched in Figure 4b. Around this time, especially near E-region sunset, isodensity contours associated with the PSSR would not have any curvature; hence, MREs are not likely to occur. On the other hand, we can see that the isodensity contour associated with an LSWS would have a section that is concave, which could support MREs. We further note that soon after E-region sunset, we can expect an eastward neutral wind (U) to drive the gradient-drift instability along the west wall of the concave structure [Tsunoda, 1983], as shown by closely spaced tick marks, labeled ΔN. This possibility may be relevant because we can imagine that the background tilt would reverse during the descent of the F layer. If symmetry held, we might expect MREs to occur sometime after the PSSR; however, the reflection coefficient may be reduced by the development of ΔN, which would decrease the likelihood of MREs. The scenario just described is consistent with ALTAIR IS measurements of LSWS during the PSSR [Tsunoda and White, 1981].
 We have shown that MREs can be explained by the focusing of radio energy reflected by concave isodensity contours associated with an LSWS. The most intense focusing is expected to occur when the radius of curvature of isodensity contours matches the altitude of reflection, and this condition is approached by an LSWS with a zonal wavelength of comparable scale (i.e., 300 to 400 km). This high degree of focusing cannot be attained with curvature that is associated with the end of the PSSR. This finding is consistent with the occurrence of a satellite trace associated with the 2F but not the 1F trace, which can be explained with concave isodensity contours with similar radii of curvature [Tsunoda, 2008]. We, thus, have reason to believe that MREs are manifestations of LSWS. The existence of a preferred zonal wavelength seems to be supported by both observations and theory. Observations include: (1) LSWS with this wavelength [Tsunoda and White, 1981], and (2) ESF patches with a distribution of separation distances that has a median value of 380 km [Röttger, 1973]. There is also a collisional-shear instability, proposed as a seed source for ESF, which favors amplification of perturbations with a comparable zonal wavelength [Hysell and Kudeki, 2004].
 To better understand the nature of MREs, we examined the ionograms containing MREs published by Rastogi . Two are from Kodaikanal (3°N dip), one taken at 1830 LT on 21 November 1958, and the other at 1930 LT on 24 February 1958, when E-region sunset occurred around 1816 and 1848 LT, respectively. The MREs, therefore, occurred 14 and 42 min after E-region sunset, respectively. The ionogram from Huancayo (2°N dip) was taken at 1930 LT on 29 August 1958, when E-region sunset occurred around 1834 LT. Hence, MREs occurred 56 min after E-region sunset. These occurrences would fall in the hour interval after E-region sunset, which is the hour for maximum number of MRE occurrences. It is also interesting to note that in the first example, a 1F doublet is evident in the ionogram. In the second example, a thin layer of ESF was present beneath the base of the clean 1F trace. And, in the third example, ESF with a fair amount of range spreading occurred at overlapping frequencies with those for the MREs.
 These results differ from those obtained with ionograms taken at Kwajalein (9°N dip) in that doublets and ESF were detected simultaneously with MREs. A reason for this difference could be that MREs may be confined to earlier times in locations where the magnetic dip is steeper, whereas, MREs may be detectable to later times over the dip equator, where isodensity contours that become aligned with geomagnetic field lines remain horizontally aligned over the dip equator, but not at higher dip latitudes. The only differentiation between MREs and satellite traces may be that MREs may be more sensitive to the radii of curvature of isodensity contours than are satellite echoes.
 The most interesting finding is that MREs and, hence, LSWS, are observed during the PSSR. The implication is that occurrences of LSWS around E-region sunset are common, and not rare, as might have been inferred from earlier, isolated reports of LSWS [Tsunoda and White, 1981; Singh et al., 1997]. This finding raises the question of whether LSWS could be excited by the collisional-shear instability at a time when velocity shear is virtually non-existent or very weak. If the source is an atmospheric gravity wave [e.g., Singh et al., 1997], we must still determine the source of its day-to-day variability.
 This research was supported by the National Science Foundation under grant ATM-0318674.