The red arcs are determined just by working out the accurate distance of the ping source from one single satellite (IOR).
The ping messages are very accurately timestamped using very accurate clocks at both ends. Even though this communication system is not for GPS purposes it uses similarly accurate clocks. So we know that the plane was, say 38000km, from the satellite at 0:11Z.
That gives a circle on the surface of the earth.(The so-called 40 degree circle).
No other satellites could be used as there are only four satellites altogether in space listening for ACARS pings. They do not cover the relevant longitudes: they are below the horizon from most of the Indian Ocean and middle Eurasia.
The fact that that the other satellites (POR and AOR-E) did not hear the 0:11Z ping rules out some chunks of the red circle.
This leaves the two red arcs.