The minimization of the Sobolev norm during linearized inversion of given data enables to control the model parameters unresolved by the data being fitted. Even if a reasonably looking model can be obtained without minimizing the Sobolev norm, it may be too rough for some computational methods. We may construct models optimally smooth for given computational methods by minimizing the corresponding Sobolev norm during the inversion.
Probably the smoothest models are required by the ray methods. The efficiency of ray tracing can be evaluated in terms of the ``average Lyapunov exponent'' for the model. The ``average Lyapunov exponent'' may be approximated by the square root of the corresponding Sobolev norm of the model, which allows models most suited for ray tracing to be constructed.
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Bulant, P.: Sobolev scalar products in the construction of velocity models: Application to model Hess and to SEG/EAGE Salt Model. Pure and appl. Geophys., 159 (2002), 1487-1506.