Echo signals are acquired from operation of a nuclear magnetic resonance logging (NMR) tool in a borehole. A multi-inversion process is performed on the echo signals. The multi-inversion process includes a downhole processor performing a downhole inversion. The multi-inversion process includes reconstructing echoes from coefficients in the downhole inversion. Reconstructing echoes is performed by one of the downhole processor or an uphole processor. The multi-inversion process includes transmitting one of: (a) the coefficients in the downhole inversion or (b) the reconstructed echoes to the uphole processor. The multi-inversion process includes the uphole processor performing an uphole inversion on the reconstructed echoes to produce final coefficients used in a spectrum.