I was wondering how researchers are able to establish that a geographic song variation in terrestrial birds is actually a song dialect.
What is the threshold of acoustic indices which could identify that a song vocalization is actually a song dialect?
I'm pretty confident that the answer is "No" - there is no threshold that can reliably be used to make this decision!
It is common for researchers to argue for the presence of dialect by identifying within-species differences such as:
(a) song units that are used in one population and not another, and/or
(b) differences in measured features ("acoustic indices" you've called them), such as pitch, duration. When looking at differences in measured features, you can imagine plotting the features on a graph and then measuring whether two populations show up as different clusters on that graph. The most basic way to turn this into a "threshold" is to look at how well these clusters are separated, e.g. a measure of intra-group variation as a fraction of overall variation.
For a specific technical example, you can read: O'Reilly et al (2018) "Measuring vocal difference in bird population pairs". A more general but older review is given in Chapter 4 of the textbook "Nature's Music: the Science of Birdsong" by Marler et al (eds.).
However, animal behaviour is also important here, such as studying/knowing the extent of vocal learning in the species, and sometimes investigating behavioural responses to song from other populations by playback experiments. Birdsong dialect is a complex phenomenon, and for wild populations it's rare to have enough data that the conclusions are obvious. So the conclusion usually involves some combination of the acoustic features, the geographic separation, and the known attributes of the species concerned.
In earli(er) papers, different dialects were defined if songs of different individuals of one species differed "by more than three element substitutions / deletions / additions" (van Dongen & Mulder 2006 J Avian Biology). Of course this only works if the song repertoire is known and fairly small...
Maybe this gives you a starting point for your own analysis?
Just to expand and what Dan said about acoustic spaces: if we project songs from several populations into a low dimensional space (let's say 2 dimensions) representing variation in acoustic structure and in which each point is an individual (so points close to each other have similar songs), then, if dialects are present, we can expect to find clusters, and the individuals within a cluster should belong to the same population. This can be used as a decision rule if we add a statistical test (like a Mantel test of acoustic vs spatial distances). This approach can be pretty flexible given that the acoustic space can be estimated using pretty much anything: measured features, pairwise similarity measures, repertoire composition, syntax, etc.
