This dissertation presents a detailed analysis of recorded seismic waves in terms of their source and their propagation through the Earth in multiple scenarios. First, I investigate the source mechanisms of some highly unusual seismic events associated with the formation of a large sinkhole at Napoleonville salt dome, Assumption Parish, Louisiana in August 2012. I implemented a grid-search approach for automatic detection, location and moment tensor inversion of these events. First, the effectiveness of this technique is demonstrated using low frequency (0.1-0.2 Hz) displacement waveforms and two simple 1D velocity models for the salt dome and the surrounding sedimentary strata for computation of Green’s functions in the preliminary analysis. In the revised, and more detailed analysis, I use Green’s functions computed using a finite-difference wave propagation method and a 3D velocity model that incorporates the currently known approximate geometry of the salt dome and the overlying anhydrite-gypsum cap rock, and features a large velocity contrast between the high velocity salt dome and low velocity sediments overlying and surrounding it. I developed a method for source-type-specific inversion of moment tensors utilizing long-period complete waveforms and first-motion polarities, which is useful for assessing confidence and uncertainties in the source-type characterization of seismic events. I also established an empirical method to rigorously assess uncertainties in the centroid location, MW and the source type of the events at the Napoleonville salt dome through changing network geometry, using the results of synthetic tests with real seismic noise. During 24-31 July 2012, the events with the best waveform fits are primarily located at the western edge of the salt dome at most probable depths of ~0.3-0.85 km, close to the horizontal positions of the cavern and the future sinkhole. The data are fit nearly equally well by opening crack moment tensors in the high velocity salt medium or by isotropic volume-increase moment tensors in the low velocity sediment layers. The addition of more stations further constrains the events to slightly shallower depths and to the lower velocity media just outside the salt dome with preferred isotropic volume-increase moment tensor solutions. I find that Green’s functions computed with the 3D velocity model generally result in better fit to the data than Green’s functions computed with the 1D velocity models, especially for the smaller amplitude tangential and vertical components, and result in better resolution of event locations and event source type. The dominant seismicity during 24- 31 July 2012 is characterized by the steady occurrence of seismic events with similar locations and moment tensor solutions at a near-characteristic inter-event time. The steady activity is sometimes interrupted by tremor-like sequences of multiple events in rapid succession, followed by quiet periods of little of no seismic activity, in turn followed by the resumption of seismicity with a reduced seismic moment-release rate. The dominant volume- increase moment tensor solutions and the steady features of the seismicity indicate a crack- valve-type source mechanism possibly driven by pressurized natural gas. Accurate and properly calibrated velocity models are essential for the recovery of correct seismic source mechanisms. I retrieved empirical Green’s functions in the frequency range ~ 0.2–0.9 Hz for interstation distances ranging from ~1 to ~30 km (~0.22 to ~6.5 times the wavelength) at The Geysers geothermal field, northern California, from cross-correlation of ambient seismic noise recorded by a wide variety of sensors. I directly compared noise- derived Green’s functions with normalized displacement waveforms of complete single-force synthetic Green’s functions computed with various 1D and 3D velocity models using the frequency-wavenumber integration method, and a 3D finite-difference wave propagation method, respectively. These comparisons provide an effective means of evaluating the suitability of different velocity models to different regions of The Geysers, and assessing the quality of the sensors and the noise cross-correlations. In the T-Tangential, R-Radial, Z- Vertical reference frame, the TT, RR, RZ, ZR and ZZ components (first component: force direction, second component: response direction) of noise-derived Green’s functions show clear surface-waves and even body-wave phases for many station pairs. They are also broadly consistent in phase and relative inter-component amplitudes with the synthetic Green’s functions for the known local seismic velocity structure that was derived primarily from body wave travel-time tomography, even at interstation distances less than one wavelength. I also found anomalous large amplitudes in TR, TZ, RT and ZT components of noise-derived Green’s functions at small interstation distances (≲4 km) that can be attributed to ~10°-30° sensor misalignments at many stations inferred from analysis of longer period teleseismic waveforms. After correcting for sensor misalignments, significant residual amplitudes in these components for some longer interstation distance (≳ 8 km) paths are better reproduced by the 3D velocity model than by the 1D models incorporating known values and fast axis directions of crack-induced shear-wave anisotropy in the geothermal field. I also analyzed the decay of Fourier spectral amplitudes of the TT component of the noise-derived Green’s functions at 0.72 Hz with distance in terms of geometrical spreading and attenuation. While there is considerable scatter in the amplitudes of noise-derived Green’s functions, the average decay is consistent with the decay expected from the amplitudes of synthetic Green’s functions and with the decay of tangential component local-earthquake ground-motion amplitudes with distance at the same frequency.