I believe that it has something to do with the emissions pump pulley or whatever that thing is but I'm not 100% sure. Let me know any ideas or if you've run into this problem.
@CamKenyon Welcome to fordfullsize.com From the video it sounded worse on the driver's side. Those smog pumps are priced near a Benjamin after core charge. Def a bearing on one of the drive accessories. A mechanics stethoscope could help narrow it down by placing the end on each accessory and blipping the throttle. The auto parts store that has "Free Loan-a-Tool" program may have one to use. Call first for availability. Harbor Freight has them for $5 NAPA has them for $10 Depending on which accessory it is, a bearing or rebuild kit for that specific accessory would be cost effective. Then again it depends on your budget and the mileage of the part in question.