Abstract: We present the first application of full-waveform ambient noise inversion to observed correlation functions that jointly constrains 3-D Earth structure and heterogeneous noise sources. For this, we model and interpret ambient noise correlations as recordings of correlation wavefields, which completely eliminates the limiting assumptions of Green's function retrieval, such as equipartitioning and homogeneous random noise sources. Our method accounts for seismic wave propagation physics in 3-D heterogeneous and attenuating media and also for the heterogeneous and nonstationary nature of the ambient noise field. Designed as a proof of concept, the study considers long periods from 100 to 300 s, thus focusing on the Earth's hum. Treating correlations as self-consistent observables allows us to make separate measurements on the causal and acausal branches of correlation functions, without any need to choose one of them or form the average. We validate our approach by assessing the quality of the obtained models and by comparing them to previous studies. This work is a step toward the establishment of full-waveform ambient noise inversion as a tomographic technique with the goal to go beyond ambient noise tomography based on Green's function retrieval.