1. A method of enhancing well logs produced by a well logging tool, comprising: receiving raw logging data from the well logging tool; calculating a nonlinearly enhanced logging data from the raw logging data, comprising: (a) selecting a resolution; (b) calculating an estimate of a formation profile; (c) calculating a nonlinear forward modeling operator corresponding to the well logging tool; (d) calculating an enhancement factor from the raw logging data, the estimate of the formation profile, and the nonlinear forward modeling operator; (e) calculating a resolution matched stability factor; (f) updating a previous nonlinearly enhanced logging data with the enhancement factor and the resolution matched stability factor to produce the nonlinearly enhanced logging data; and (g) repeating actions (a), (b), (c), (d), (e), and (f) until a predetermined criterion is satisfied; and outputting the nonlinearly enhanced logging data.