Wave field inversion of ambient seismic noise

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 (the sum is greater than its parts). Consequently, we can relax the spatial and temporal constraints on the wave field source functions in the seismological inverse problem. The result is that we become insensitive to the noise sources in, and changing character of, the ambient seismic field. In principle the formulation holds equally for ambient noise wave fields and for wave fields excited by controlled sources. We support the theory with examples in one dimension in the time domain, and in two dimensions in the frequency domain. The latter being of interest to invert surface wave ambient noise for phase velocity maps. We include checker board tests for geometries mimicking USArray and Ocean Bottom Cables.