In Farihi et al. (2013) (it's a Science paper, unfortunately I'm not sure its content is freely accessible), they actually measured metal excess in the white dwarf GD 61 (for an astronomer, everything that is nor hydrogen neither helium is a metal). Due to high surface gravity in white dwarfs, any heavy element should sink rapidely in its atmosphere; therefore, they infer that, if you find traces of metals, it means that it comes from accreted material, "polluting" its atmosphere.
So if you find oxygen in a white dwarf spectrum, you know that it has to come from somewhere else, and if the white dwarf has a circumstellar disk, it has to come from planetesimals that are accreted from the disk.
Now, what they did was to measure abundances (or upper limit for their abundance) for a bunch of chemical elements (O, Mg, Al, Si, Ca, Fe) and carbon. Then, they could infer different things to determine the total oxygen budget:
- There is a carbon deficiency, that means that carbon abundance has no impact on the total oxygen budget;
- Elements Mg, Al, Si, Ca, are assumed to be carried as MgO, Al2O3, SiO2, and CaO at their maximum abundance, so once they had these abundances, they could substract it to the total oxygen budget;
- The remaining oxygen excess, found in FeO and debris, is interpreted as water-rich (if you want to form FeO, you need water somehow).
That's how they were able to determine this water abundance in debris around the white dwarf GD 61.
I'm not sure the oxygen abundance would be very meaningful here, but since you asked for it, it is of -5.95+/-0.13 dex.