We formulate a full wave field inversion for ambient seismic noise recorded by large and dense seismograph arrays. Full wave field inversion exploits the constraints on the gradients of the wave field that array data inherently possess. We pose full wave field inversion as a partial differential equation (PDE) constrained inverse problem resulting in a joint estimation of a reconstructed wave field and the medium parameters. The inverse problem is solved by variable projection. We explicitly allow for non-unique solutions to the PDE system that is imposed as a constraint. The boundary conditions of the wave field do not need to be specified, and can remain unknown. This makes the algorithm suitable for inverting observations of ambient seismic noise by dense arrays. The result is that the inverse problem for subsurface properties becomes insensitive to the character and distribution of the noise sources that excited the seismic wave field. In principle the formulation holds equally for ambient noise wave fields and for wave fields excited by controlled sources. The theory is supported with examples in one dimension in the time domain, and in two dimensions in the frequency domain. The latter are of interest in the inversion of surface wave ambient noise for phase velocity maps.