We operated ionosondes at four stations in a rapid-run mode to obtain ionograms every minute for 102 hours during the period of the Leonid meteor shower in November 2001. Plenty of radio echoes that looked like echo traces produced by sporadic E reflection were seen in the ionograms, and this allowed statistical analysis in a single meteor shower event, including discrimination of the backscattering by meteor trails from reflection by a horizontally stratified sporadic E layer. The radio echoes seen in the ionograms were categorized into three types. The first were spontaneous echoes, which were distributed across a wide range of virtual heights; at times during the period of maximum meteor activity, there was a statistically good correlation among the echoes seen at the four stations. The echoes of this type appear to be produced by Fresnel backscattering from meteor trails. The second were also spontaneous echoes observed during the shower period but persisted for several tens of minutes at the same virtual height as the typical sporadic E layer, and the top frequency of these echoes decayed with time. Echoes of this type are thus attributed to the reflection from a meteor-induced sporadic E patch. Echoes of the last type appeared outside the period of maximum activity of the meteor shower in the same range of virtual heights as the conventional sporadic E layer, and there was no time correlation between the events observed at the four stations. These events are attributed to a periodical increase in fOEs, which is modulated by planetary-wave activity, and have no relation with the meteor shower.