Combination of research tools beginning with the units I have been in or know from personal experience (most recently the 157th and 160th FA), my Distinctive/Regimental Unit Insignia collection (about 100 - mostly US Army from different branches), books on this subject that I own and the great world wide web (US Army Institute of Heraldry and unit webpages).
Oddly enough, some regiments are harder than others to track down as the DUI has no motto on it and they are not units that are currently on the active rolls. Without asking a unit member or seeing the Battalion, Regimental or Brigade Colors, it is almost impossible to find the info out.
But it is a challenge I enjoy and you are most welcome.