The relation between star formation rates (SFRs) and stellar masses, i.e., the galaxy main sequence, is a useful diagnostic of galaxy evolution. We present the distributions relative to the main sequence of 55 optically selected PG and 12 near-IR-selected Two Micron All Sky Survey (2MASS) quasars at z0.5. We estimate the quasar host stellar masses from Hubble Space Telescope or ground-based AO photometry, and the SFRs through the midinfrared aromatic features and far-IR photometry. We find that PG quasar hosts more or less follow the main sequence defined by normal star-forming galaxies while 2MASS quasar hosts lie systematically above the main sequence. PG and 2MASS quasars with higher nuclear luminosities seem to have higher specific SFRs (sSFRs), although there is a large scatter. No trends are seen between sSFRs and SMBH masses, Eddington ratios, or even morphology types (ellipticals, spirals, and mergers). Our results could be placed in an evolutionary scenario with quasars emerging during the transition from ULIRGs/mergers to ellipticals. However, combined with results at higher redshift, they suggest that quasars can be widely triggered in normal galaxies as long as they contain abundant gas and have ongoing star formation.

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Full-waveform inversion (FWI) promises a high-resolution model of the earth. It is, however, a highly nonlinear inverse problem; thus, we iteratively update the subsurface model by minimizing a misfit function that measures the difference between the measured and the predicted data. The conventional [Formula: see text]-norm misfit function is widely used because it provides a simple, high-resolution misfit function with a sample-by-sample comparison. However, it is susceptible to local minima if the low-wavenumber components of the initial model are not accurate. Deconvolution of the predicted and measured data offers an extended space comparison, which is more global. The matching filter calculated from the deconvolution has energy focused at zero lag, like an approximated Dirac delta function, when the predicted data matches the measured one. We have introduced a framework for designing misfit functions by measuring the distance between the matching filter and a representation of the Dirac delta function using optimal transport theory. We have used the Wasserstein [Formula: see text] distance, which provides us with the optimal transport between two probability distribution functions. Unlike data, the matching filter can be easily transformed to a probability distribution satisfying the requirement of the optimal transport theory. Though in one form, it admits the conventional normalized penalty applied to the nonzero-lag energy in the matching filter, the proposed misfit function is metric and extracts its form from solid mathematical foundations based on optimal transport theory. Explicitly, we can derive the adaptive waveform inversion (AWI) misfit function based on our framework, and the critical “normalization” for AWI occurs naturally per the requirement of a probability distribution. We use a modified Marmousi model and the BP salt model to verify the features of the proposed method in avoiding cycle skipping. We use the Chevron 2014 FWI benchmark data set to further highlight the effectiveness of the proposed approach.

Full Waveform Inversion updates the subsurface model iteratively by minimizing a misfit function, which measures the difference between observed and predicted data. The conventional l 2 norm misfit function is widely used as it provides a simple, sample by sample, high resolution misfit function. However it is susceptible to local minima if the low wavenumber components of the initial model are not accurate. A deconvolution of the predicted and observed data offers an extend space comparison, which is more global. The matching filter calculated from the deconvolution has energy focussed at zero lag, like a Dirac Delta function, when the predicted data matches the observed ones. We use the Wasserstein distance to measure the difference between the matching filter and a Dirac Delta function. Unlike data, the matching filter can be easily transformed to a distribution satisfying the requirement of optimal transport theory. Compared with the conventional normalized penalty applied to non-zero lag energy in the matching filter, the new misfit function is a metric and has solid mathematical foundation based on optimal transport theory. Both synthetic and real data examples verified the effectiveness of the proposed misfit function.