Summary Temporal changes in subsurface properties, such as seismic wavespeeds, can be monitored by measuring phase shifts in the coda of two seismic waveforms that share a similar source-receiver path but that are recorded at different times. These nearly identical seismic waveforms are usually obtained either from repeated earthquake waveforms or from repeated ambient noise cross-correlations. The five algorithms that are the most popular to measure phase shifts in the coda waves are the Windowed Cross Correlation (WCC), Trace Stretching (TS), Dynamic Time Warping (DTW), Moving Window Cross Spectrum (MWCS), and Wavelet Cross Spectrum (WCS). The seismic wavespeed perturbation is then obtained from the linear regression of phase shifts with their respective lag times under the assumption that the velocity perturbation is homogeneous between (virtual or active) source and receiver. We categorize these methods into the time domain (WCC, TS, DTW), frequency domain (MWCS), and wavelet domain (WCS). This study complements this suite of algorithms with two additional wavelet-domain methods, which we call Wavelet Transform Stretching (WTS) and Wavelet Transform Dynamic Time Warping (WTDTW), wherein we apply traditional stretching and dynamic time warping techniques to the wavelet transform. This work aims to verify, validate, and test the accuracy and performance of all methods by performing numerical experiments, in which the elastic wavefields are solved for in various 2D heterogeneous halfspace geometries. Through this work, we validate the assumption of a linear increase in phase shifts with respect to phase lags as a valid argument for fully homogeneous and laterally homogeneous velocity changes. Additionally, we investigate the sensitivity of coda waves at various seismic frequencies to the depth of the velocity perturbation. Overall, we conclude that seismic wavefields generated and recorded at the surface lose sensitivity rapidly with increasing depth of the velocity change for all source-receiver offsets. However, measurements made over a spectrum of seismic frequencies exhibit a pattern such that wavelet methods, and especially WTS, provide useful information to infer the depth of the velocity changes.

Summary Cross-correlations of ambient seismic noise are widely used for seismic velocity imaging, monitoring, and ground motion analyses. A typical step in analyzing Noise Cross-correlation Functions (NCFs) is stacking short-term NCFs over longer time periods to increase the signal quality. Spurious NCFs could contaminate the stack, degrade its quality, and limit its use. Many methods have been developed to improve the stacking of coherent waveforms, including earthquake waveforms, receiver functions, and NCFs. This study systematically evaluates and compares the performance of eight stacking methods, including arithmetic mean or linear stacking, robust stacking, selective stacking, cluster stacking, phase-weighted stacking, time-frequency phase-weighted stacking, Nth-root stacking, and averaging after applying an adaptive covariance filter. Our results demonstrate that, in most cases, all methods can retrieve clear ballistic or first arrivals. However, they yield significant differences in preserving the phase and amplitude information. This study provides a practical guide for choosing the optimal stacking method for specific research applications in ambient noise seismology. We evaluate the performance using multiple onshore and offshore seismic arrays in the Pacific Northwest region. We compare these stacking methods for NCFs calculated from raw ambient noise (referred to as Raw NCFs) and from ambient noise normalized using a one-bit clipping time normalization method (referred to as One-bit NCFs). We evaluate six metrics, including signal-to-noise ratios, phase dispersion images, convergence rate, temporal changes in the ballistic and coda waves, relative amplitude decays with distance, and computational time. We show that robust stacking is the best choice for all applications (velocity tomography, monitoring, and attenuation studies) using Raw NCFs. For applications using One-bit NCFs, all methods but phase-weighted and Nth-root stacking are good choices for seismic velocity tomography. Linear, robust, and selective stacking methods are all equally appropriate choices when using One-bit NCFs for monitoring applications. For applications relying on accurate relative amplitudes, the linear, robust, selective, and cluster stacking methods all perform well with One-bit NCFs. The evaluations in this study can be generalized to a broad range of time-series analysis that utilizes data coherence to perform ensemble stacking. Another contribution of this study is the accompanying open-source software, which can be used for general purposes in time-series stacking.

This a preprint and has not been peer reviewed. Data may be preliminary.

Summary Temporal changes in seismic velocities are an important tool for tracking structural changes within the crust during transient deformation. While many geophysical processes span the crust, including volcanic unrest and large-magnitude earthquakes, existing methods for seismic monitoring are limited to the shallow subsurface. We present an approach for deep seismic monitoring based on teleseismic receiver functions, which illuminate the crustal velocity structure from the bottom-up. Using synthetic waveform modeling, we show that receiver functions are uniformly sensitive to velocity changes throughout the crust and can locate the depth of the perturbation. We introduce a novel method based on optimal transport for measuring the nonlinear time-amplitude signal variations characteristic of receiver function monitoring. We show that optimal transport enables comparison of full waveform distributions rather than relying on representative stacked waveforms. We further study a linearized version of optimal transport that renders time-warping signal variations into simple Euclidean perturbations, and use this capability to perform blind source separation in the space of waveform variations. This disentangles the effects of changes in the source-receiver path from changes in subsurface velocities. Collectively, these methods extend the reach of seismic monitoring to deep geophysical processes, and provide a tool that can be used to study heterogeneous velocity changes with different spatial extents and temporal dynamics.

This a preprint and has not been peer reviewed. Data may be preliminary.