Frequency domain analysis of errors in cross-correlations of ambient seismic noise

Abstract

We analyse random errors (variances) in cross-correlations of ambient seismic noise in the frequency domain, which differ from previous time domain methods. Extending previous theoretical results on ensemble averaged cross-spectrum, we estimate confidence interval of stacked cross-spectrum of finite amount of data at each frequency using non-overlapping windows with fixed length. The extended theory also connects amplitude and phase variances with the variance of each complex spectrum value. Analysis of synthetic stationary ambient noise is used to estimate the confidence interval of stacked cross-spectrum obtained with different length of noise data corresponding to different number of evenly spaced windows of the same duration. This method allows estimating Signal/Noise Ratio (SNR) of noise cross-correlation in the frequency domain, without specifying filter bandwidth or signal/noise windows that are needed for time domain SNR estimations. Based on synthetic ambient noise data, we also compare the probability distributions, causal part amplitude and SNR of stacked cross-spectrum function using one-bit normalization or pre-whitening with those obtained without these pre-processing steps. Natural continuous noise records contain both ambient noise and small earthquakes that are inseparable from the noise with the existing pre-processing steps. Using probability distributions of random cross-spectrum values based on the theoretical results provides an effective way to exclude such small earthquakes, and additional data segments (outliers) contaminated by signals of different statistics (e.g. rain, cultural noise), from continuous noise waveforms. This technique is applied to constrain values and uncertainties of amplitude and phase velocity of stacked noise cross-spectrum at different frequencies, using data from southern California at both regional scale (∼35 km) and dense linear array (∼20 m) across the plate-boundary faults. A block bootstrap resampling method is used to account for temporal correlation of noise cross-spectrum at low frequencies (0.05-0.2 Hz) near the ocean microseismic peaks.

Seismic Signal Denoising and Decomposition Using Deep Neural Networks

“…This important feature renders DeepDenoiser with the ability to preserve noise signals with or without the presence of earthquakes at reasonable computational cost. It could have significant applications in data preprocessing for ambient seismic noise studies where the contamination of noise data with earthquake signals can heavily affect the cross-correlation results [58] and the current way of dealing with earthquake contamination is to simply discard the associated windows [59,60].…”

Improving the quality of empirical Green's functions, obtained by cross-correlation of high-frequency ambient seismic noise

“…As discussed in the "Introduction" section, there are several methods based on weighted stacking of cross-correlation functions in the time, frequency, and time-frequency domains, which allow us to increase the quality of extracted EGFs (Schimmel et al, 2011;Cheng et al, 2015;Liu et al, 2016;Li et al, 2018, etc.). We showed in previous sections that our time-domain algorithm based on the global optimization of the signal-to-noise ratio makes it possible to exclude incoherent cross-correlation functions from stacking and generally allows obtaining EGFs of even better quality.…”

Quantifying the Effects of Nondiffuse Noise on Ballistic and Coda Wave Amplitude From Variances of Seismic Noise Interferometry in Southern California

“…For each frequency, we compute the histogram of the real part of the cross‐spectral (Fourier transform of the cross‐correlation) values from all available time windows. The mean and variance of the average cross‐spectrum are derived from the histogram (Liu et al, ; Liu & Ben‐Zion, ), and we derive the time domain equivalent of these statistical estimators below.…”

“…In the synthetic test, we compute the stacked cross‐spectrum from 3,600 time windows following equation along with error bars showing one standard error (Figure c). The standard errors for the real and the imaginary parts of the cross‐spectrum are approximately equal (Liu et al, ), and their squared sum equals the variance of the cross‐spectrum Var[ R k,ab ].…”