This is a nice bit of work, thanks for sharing it.
It is similar in nature to what some of us do in geophysics, using seismic records to infer the directionality and rupture pattern of an earthquake over a fault zone. In that problem the measurements are not always ideally placed, as we only have a certain number and often restricted locations of seismographs. Additionally, unlike clean air, the propagation medium is strongly heterogeneous and the waves bounce and refract and scatter along the way from source to receiver. Finally, there are both sound waves and elastic shear waves, which offers more information but a more complex recording (3-axes of movement, rather than a scalar pressure field).
The basic problem of determining the magnitude and fault orientation for an earthquake, called the "moment tensor solution" (which has 6 independent components for a "double-couple" moment in 3D, 3 unknowns for location, and 1 unknown for event time) is carried out in a way that is very similar to the approach you describe here, involving an analytical expansion of the solution in terms of coupled spherical harmonics and orthogonal basis functions that account for radial variations in seismic wave velocities inside the Earth. Combining data from hundreds of seismograph stations results in a non-square sensitivity matrix...typically in seismology we make up for non-ideal coverage of seismic stations by adding as many recordings as possible. The inversion is performed using singular value decomposition (SVD), which returns the least squares solution in terms of generalized eigenvalues and eigenvectors, from which one can infer the dominant modes of the solution as well as the covariances. It is a beautiful technique and I highly recommend SVD. The general theory is described in a book titled "Theoretical Global Seismology" by Dahlen and Tromp.
For inferring rupture patterns and heterogeneous energy radiation along a fault zone (asperities, etc.), other techniques such as "beam-forming methods" are used. This is another approach you could look into, if you want to fall further down the rabbit hole.
In any case, I'd be happy to give you feedback or point you to more detailed references if you're interested.