Signal Stacking: PhaseWeighted Stack (PWS)
Contents
1 Introduction
In realistic world, seismic signals are contaminated by random noise, but we need signals with high signaltonoise ratios (SNRs) to conduct the following work of, such as probing the interiors of our planet. Therefore, signal processing experts or seismologist came up with some signal stacking strategies to enhance the SNRs. Linear stacking is usually used to suppress the noise, and generally, we can obtain clear signals using linear stacking. However, we want to accelerate the convergence of signals in ambient noise crosscorrelation applications (e.g., Shapiro & Campillo, 2004; Dias et al., 2015 ) if we only have small data sets. Linear stacking may not work well in such situation. Fortunately, seismologists proposed some novel signal stacking rules to accelerate the convergence of coherent signals. Here, we introduce the phaseweighted stack rule ( Schimmel & Paulssen, 1997 ) and give two examples to present this signal stacking rule.
2 Basic principles
Closely following the study of Schimmel & Paulssen (1997), analytical signal can be obtained from the real signal and its corresponding HIlbert transfom (the imaginary signal). That is
$$
\begin{aligned}
s(t) &= x(t)+iy(t) \\
&= A(t)e^{i\phi (t)}
\end{aligned} \tag{1},
$$
where, $x(t)$ is the real signal, $y(t)$ is the imaginary signal, $A(t)$ is the envelope of the analytical and $e^{i\phi (t)}$ is the phase term.
No amplitude information is involved, the coherence can be defined by $$ c(t)=\frac{1}{N}\sum_{k=1}^Ne^{i\phi_k(t)} \tag{2}. $$ We can obtain phaseweight stacked signal by the multiplication of the real signal and coherence terms, $$ s(t)=\frac{1}{N}\sum_{j=1}^Nx(t)_jc(t)^\nu \tag{3}, $$ specially, this formula means linear stacking when $\nu=0$.
3. A numerical example
Here, we give an example of measuring the coherence of several signals. First, we add different noise levels to the original signal. Using eq. $(2)$, the coherence of these signals can be obtained.


In the time window of $510 \ s$, the coherence $c(t)$ is approximately $1$ , because this window contains the lowerfrequency coherent signals.
4. A realistic application
Some researchers utilize the PWS to accelerate the covergence of noise crosscorrelation time functions (e.g., Dias et al., 2015; Dantas et al., 2018), and we use the PWS strategy to suppress the noise contaminating the noise crosscorrelation functions from hurricane events. The PWS is given by the following Python code.


Here, we compare the stacked signals from PWS with different power numbers $\nu$ and linear stacking.
The power number $\nu$ of coherence $c(t)$ are indicated by the numbers on the right panel. Note that the SNRs become higher with the increase of the power number $\nu$. However, one must understand that the waveforms from PWS may change, especially for the larger power number $\nu$.
5 An Extension of PWS: tfPWS
The stacked waveform using PWS changes much due to the nonlinear influence. Here, Schimmel et al. (2011) have developed an extension of the PWS: tfPWS in timefrequency domain. The steps are
(1) Transform the trace into timefrequency domain with Stockwell transform or S transform; $$ S(\tau, f)=\int_{\infty}^{\infty}u(t)w(\taut, f)e^{i2\pi ft} \tag{4}, $$ where $w(\taut, f)=\frac{f}{k\sqrt{2\pi}}e^{f^2(\taut)^2/(2k^2)}$ with $k>0$ is an Gaussian window function. The new coherence function is defined by $$ c(\tau, f)=\frac{1}{N}\sum_{n=1}^N \frac{S_n(\tau, f)e^{i2\pi ft}}{S_n(\tau, f)}^\nu \tag{5}. $$ The stacked waveform with tfPWS in timefrequency domain is $$ S_{tfPWS}(\tau, f)=c(\tau, f)S_{linear}(\tau, f) \tag{6}, $$ where, $S_{linear}(\tau, f)$ is the linear stacking of all traces in timefrequency domain using S transform.
We apply an inverse S transform to $S_{tfPWS}(\tau, f)$ and obtain the tfPWS waveform $s(\tau)$.
6 An Application of tfPWS
We use the same noise crosscorrelations to check the tfPWS. You need to install the Python library stockwell first if you want to try the following example.


We can find that the stacked waveforms with tfPWS do not change much but noise is suppressed. We compare the stacked waveforms with the PWS and tfPWS, and we can see that the difference is more obvious when the power number $\nu$ increases.
References
Dias, R.C., Julià, J. & Schimmel, M. RayleighWave, GroupVelocity Tomography of the Borborema Province, NE Brazil, from Ambient Seismic Noise. Pure Appl. Geophys. 172, 1429–1449.
Dantas, Odmaksuel Anísio Bezerra, Do Nascimento, A. F. , & Schimmel, M. . (2018). Retrieval of bodywave reflections using ambient noise interferometry using a smallscale experiment. Pure & Applied Geophysics.
Schimmel, M., and H. Paulssen (1997), Noise reduction and detection of weak, coherent signals through phase weighted stacks, Geophys. J. Int., 130, 497 – 505.
M. Schimmel, E. Stutzmann2and J. Gallart. Using instantaneous phase coherence for signal extraction from ambient noise data at a local to a global scale. Geophysical Journal International(1), 494506.
Shapiro, N. M., and M. Campillo (2004). Emergence of broadband Rayleigh waves from correlations of the ambient seismic noise, Geophys. Res. Lett. 31, no. 7, doi: 10.1029/2004GL019491.
Author Geophydog
LastMod 20210116