We formulate a full wavefield inversion (FWFI) for ambient seismic noise recorded by large and dense seismograph arrays. FWFI exploits the constraints on the gradients of the wavefield that array data inherently possess. We pose FWFI as a partial differential equation (PDE) constrained inverse problem resulting in a joint estimation of a reconstructed wavefield 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 wavefield 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 wavefield. In principle the formulation holds equally for ambient noise wavefields and for wavefields 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.