A logging while drilling (LWD) and measuring while drilling (MWD) method and device are disclosed for reducing the sensitivity of NMR measurements to tool motions. The invention is based on NMR relaxation measurements determining longitudinal relaxation times T1 instead of the standard T2 measurements, and involves saturating a relatively wide sensitive region of the formation and processing NMR echo signals which originate approximately from the center of the sensitive region. In another aspect, the invention uses novel pulse sequences and processing algorithms to reduce the time for the relaxation measurement and to provide real-time transfer of data uphole.