To explain how gradient-moment nulling works, it is necessary to make use of a fact developed in the Advanced Discussion section of the prior Q&A on Even-Echo Rephasing. Here we showed that for constant velocity flow (v) through a constant gradient (G), total phase shift (Φ) increases quadratically with time (T).  In other words

Φ =  KT²

where K = ½ γ G v is a constant when velocity and readout gradient strength are also constant.

This quadratic dependence of total phase gain upon time is shown in the figure below.  Note that after 1, 2, 3, and 4 gradient intervals each of duration T, the phase has increased quadratically to KT², 4KT², 9KT², and 16KT² respectively.  The incremental phase contributions of the 1st, 2nd, 3rd, and 4th gradient intervals are therefore  KT²3KT²5KT², and 7KT², respectively.

Let us now consider in a more quantitative way the first-order gradient-moment nulling pulse sequence previously diagrammed..

To analyze what happens to a spin moving at constant velocity, we will do an incremental phase calculation similar to that performed for the even-echo rephasing case.  After the first block, the phase should again be  KT².  The incremental phase contribution from the two blocks that compose the second lobe is 2 x (−3KT² = −6KT².  The net phase of the moving spins immediately after the double block is therefore KT²  −5KT² = .−5KT². Finally, the first readout block contributes its incremental +5KT², yielding a net phase of zero at t = TE. By adding these extra gradient blocks, we have compensated or corrected for phase dispersions attributable to constant velocity.

The above example describes first-order (velocity) gradient-moment nulling. The method can be easily extended by to compensate for phase dispersions due to second-order (acceleration), third-order ("jerk"), and even higher degrees of motion. These higher order corrections require more gradient lobes (specifically, n+1 additional lobes, where n = motion order to be compensated). These extra lobes take time to play out, so using higher order GMN even further lengthens minimum TE and reduces number of slices for a given TR. Also, the effectiveness of the method decreases as the order increases. In practice, therefore, velocity-compensated GMN is nearly exclusively used, with acceleration-compensated GMN performed only occasionally.  