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Newtonian machine learning (NML) is a wave-equation inversion method thatinverts single-dimensional latent space (LS) features of the seismic data forretrieving the subsurface background velocity model. The single-dimensional LSfeatures mainly contain the kinematic information of the seismic data, whichare automatically extracted from the seismic signal by using an autoencodernetwork. Because its LS feature dimension is too small to preserve the dynamicinformation, such as the waveform variations, of the seismic data. Thereforethe NML inversion is not able to recover the high-wavenumber velocity details.To mitigate this problem, we propose to invert multi-dimensional LS features,which can fully represent the entire characters of the seismic data. We denotethis method as multi-dimensional Newtonian machine learning (MNML). In MNML, wedefine a new multi-variable connective function that works together with themulti-variable implicit function theorem to connect the velocity perturbationsto the multi-dimensional LS feature perturbations. Numerical tests show that(1) the multi-dimensional LS features can preserve more data information thanthe single-dimensional LS features; (2) a higher resolution velocity model canbe recovered by inverting the multi-dimensional LS features, and the inversionquality is comparable to that of FWI; (3) the MNML method requires a muchsmaller storage space than conventional FWI because only the low-dimensionalrepresentations of the high-dimensional seismic data are needed to be stored.The disadvantage of MNML is that it can more easily get stuck in local minimacompared to the NML method. So we suggest a multiscale inversion approach thatinverts for higher dimensional LS features as the iteration count increase.

Seismic Inversion by Multi-dimensional Newtonian Machine