Abstract: The main objective of this thesis is to develop novel misfit functions for full-waveform inversion such that (a) the estimation of the long-wavelength model will less likely stagnate in spurious local minima and (b) the inversion is immune to wavelet inaccuracy. 

 First, I investigate the pros and cons of misfit functions based on optimal transport theory to indicate the traveltime discrepancy for seismic data. Even though the mathematically well-defined optimal transport theory is robust to highlight the traveltime difference between two probability distributions, it becomes restricted as applied to seismic data mainly because the seismic data are not probability distribution functions. 

 I then develop a misfit function combining the local cross-correlation and dynamic time warping. With this combination, the proposed misfit function can automatically identify arrivals associated with a phase shift. Numerical and field data examples demonstrate its robustness for early arrivals and the limitation for the mix of early and reflection arrivals. That means that a proper pre-processing step is still required. 

 To discover a misfit function that highlights the traveltime discrepancy without non-trivial human intervention, I introduce the concept of differentiable dynamic time warping distance. Compared to the conventional warping distance, the differentiable version retains the property of representing the traveltime difference but results in the numerically accurate adjoint source. That helps full-waveform inversion converge to informative estimates. 

 Finally, I develop a misfit function entitled the deconvolutional double-difference measurement. The new misfit measures the first difference by deconvolution rather than cross-correlation. I present the derivation of the adjoint source with the new misfit function. Numerical examples and mathematical proof demonstrate that this modification makes full-waveform inversion with the deconvolutional double-difference measurement immune to wavelet inaccuracy.