Notice that low echo frequencies are delayed more than high echo frequencies. This common feature of Bragg diffraction gratings is responsible for the downward chirp. Time dispersion is another familiar feature of inclined Bragg diffraction gratings. The most famous feature at Chichen Itza is the light and shadow show that plays on the temple of Kukulkan around the time of the spring and fall equinox. At that time, an undulating shadow identified as the great plumed serpent moves up or down the balustrade of the staircase.
Cultural connections identify the plumed serpent with the quetzal.
The pyramid was once covered with smooth finish plaster. The plaster has been mising for many hundreds of years. You can see how irregular the steps are Despite this irregularity, the quetzal - like chirp can still be heard in the echo. If smooth finish plaster were reapplied, my acoustician's judgement is that the echo would sound slightly less raspy and more birdlike. This was calculated from a Mathcad program from a simple 2 dimensional ray model.