  Knowing a quantum system's environment is critical for its practical use as a
quantum device. Qubit sensors can reconstruct the noise spectral density of a
classical bath, provided long enough coherence time. Here we present a protocol
that can unravel the characteristics of a more complex environment, comprising
both unknown coherently coupled quantum systems, and a larger quantum bath that
can be modeled as a classical stochastic field. We exploit the rich environment
of a Nitrogen-Vacancy center in diamond, tuning the environment behavior with a
bias magnetic field, to experimentally demonstrate our method. We show how to
reconstruct the noise spectral density even when limited by relatively short
coherence times, and identify the local spin environment. Importantly, we
demonstrate that the reconstructed model can have predictive power, describing
the spin qubit dynamics under control sequences not used for noise
spectroscopy, a feature critical for building robust quantum devices. At lower
bias fields, where the effects of the quantum nature of the bath are more
pronounced, we find that more than a single classical noise model are needed to
properly describe the spin coherence under different controls, due to the back
action of the qubit onto the bath.
