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The Square Root Normal Field (SRNF) framework is a method in the area ofshape analysis that defines a (pseudo) distance between unparametrizedsurfaces. For piecewise linear (PL) surfaces it was recently proved that theSRNF distance between unparametrized surfaces is equivalent to the WassersteinFisher Rao (WFR) metric on the space of finitely supported measures on $S^2$.In the present article we extend this point of view to a much larger set ofsurfaces; we show that the SRNF distance on the space of Lipschitz surfaces isequivalent to the WFR distance between Borel measures on $S^2$. For the spaceof spherical surfaces this result directly allows us to characterize thenon-injectivity and the (closure of the) image of the SRNF transform. In thelast part of the paper we further generalize this result by showing that theWFR metric for general measure spaces can be interpreted as an optimizationproblem over the diffeomorphism group of an independent background space.

Square Root Normal Fields for Lipschitz surfaces and the Wasserstein Fisher Rao metric