Intrinsic Noise Level of Noise Cross-Correlation Functions and its Implication to Source

Population of Ambient noises

Ying-Nien Chen1, Yuancheng Gung

2, Ling-Yun Chiao

1 and Junkee Rhie

3

1 Institute of Oceanography, National Taiwan University, Taipei, Taiwan.

2 Department of Geosciences, National Taiwan University, Taipei, Taiwan.

3 School of Earth and Environmental Sciences, Seoul National University

Abbreviated title: On the INL of NCF

Corresponding author: Ying-Nien Chen

e-mail : d98241004@ntu.edu.tw

phone : +886 233661634#213

Summary

We present a quantitative procedure to evaluate the intrinsic noise level (INL) of the noise

cross-correlation function (NCF). The method is applied to realistic NCFs derived from the

continuous data recorded by the seismic arrays in Taiwan and Korea. The obtained temporal

evolution of NCF noise level follows fairly the prediction of the theoretical formulation,

confirming the feasibility of the method. We then apply the obtained INL to the assessment of

data quality and the source characteristics of ambient noise. We show that the INL-based

signal-to-noise ratio provides an exact measure for the true noise level within the NCF and

better resolving power for the NCF quality, and such measurement can be implemented to any

time windows of the NCFs to evaluate the quality of overtones or coda waves. Moreover,

since NCF amplitudes are influenced by both the population and excitation strengths

of noises, while 𝐼𝑁𝐿 is primarily sensitive to the overall source population, with

information from both measurements, we may better constrain the source

characteristics of seismic ambient noises.

Keywords:

Ambient seismic noise, noise cross-correlation function, noise level

1. Introduction

Over the last two decades, the theoretical bases and practical implementations for

extracting Greens Function from noise have been developed in multiple disciplines of

scientific researches (e.g., Godin, 1997, Godin, 2006, Godin, 2007, Weaver and Lobkis, 2005,

Larose et al., 2008, Tanimoto, 2008). The existence of a diffusive wave field is the important

common prerequisite among these seemingly distinctive formulations. The same principle

was applied to seismology by Shapiro and Campillo (2004), who first showed that the

impulse response of elastic waves between two seismic stations resembles the noise

cross-correlation function (NCF) of their continuous records. The result suggests that the

long-time-averaged wave field of ambient seismic noise is approximately diffusive, as the

seismic noise is mainly composed of microseisms generated by the everlasting and complex

interactions between ocean waves and the solid Earth.

The noise-derived empirical Green's functions (EGF) are dominated by fundamental

mode surface waves [e.g., Shapiro et al., 2005] and they are ideal data for the practices of the

crust and uppermost mantle tomography (e.g., Shapiro et al., 2005, Yao et al., 2006, Lin et al.,

2009, Zhan et al., 2010, Harmon et al., 2013, Poli et al., 2012a). Similar to the earthquake

data, quality evaluation of the earthquake-free EGFs is a vital procedure to guarantee the

reliability of the tomographic results. Typically, the EGF quality is simply estimated by its

signal to noise ratio (SNR), in which only the fundamental mode surface waves are taken as

signals, and the rest of NCF are considered as the contrasting noise (e.g., Gerstoft et al., 2006,

Lin et al., 2008).

In fact, those wiggles tailing the major signal of NCF are not structure-unrelated noise,

they might contain the coda signal of the Green’s function, and one may even reconstruct the

EGF between the stations by cross-correlating the NCF coda (e.g., Froment et al., 2011,

Stehly et al., 2008). Additionally, the robust coda train can be applied to the detection of

temporal perturbations in crustal elastic properties (e.g., Yu and Hung, 2012, Brenguier et al.,

2008). The usual definition of signal in the SNR evaluation of NCF arises mainly from the

practical concern in the tomographic applications.

Apparently, there is true noise within the NCFs. According to the theoretical studies,

acoustic experiments and numerical simulations, the noise level of NCFs generally decreases

with the total correlation time (𝑇𝑎𝑙𝑙) (e.g., Larose et al., 2008, Derode et al., 2003, Larose et

al., 2007, Lobkis and Weaver, 2001, Weaver and Lobkis, 2005, Tsai, 2010, Tsai, 2011, Sabra

et al., 2005). Similarly, in seismological applications, it has been shown that the empirically

defined SNR of NCFs also increases with 𝑇𝑎𝑙𝑙 (Seats et al., 2012, Sabra et al., 2006, Gerstoft

et al., 2006, Bensen et al., 2007). However, such SNR is simply a convenient rough measure

of EGF quality in the context of seismic data, and it has little to do with the exact noise level.

So far, there are few studies focusing on the noise in NCFs based on the acoustic

experiments (e.g., Derode et al., 2003, Larose et al., 2008) whereas details regarding to the

noise within seismic-record-based NCFs have never been explored. In the study, we present a

procedure to measure quantitatively the noise in NCF. We will show how the noise content

evolves with the correlation time, and demonstrate a plausible way of invoking this property

to constrain the source population of ambient noise.

2. Background Theory

We consider a case with N independent noise sources. The displacement 𝑈 recorded by

the station at 𝑥 in a specific time window j for frequency 𝜔 can be expressed as

𝑈𝑗(𝑥, 𝑡, 𝜔) = ∑ 𝐴𝑠(𝜔) cos [𝜔 (𝑡 −𝑟𝑠𝑥

𝑐(𝜔)) + 𝜙𝑠𝑗]

𝑁𝑠=1 , (1)

where 𝐴𝑠 represents the amplitude triggered by the source s, 𝑟𝑠𝑥 the distance between

source and station, and 𝜙𝑠𝑗 the initial phase of the noise source s in the jth time window (e.g.,

Boschi et al., 2013, Tsai, 2011). Thus, the NCF of two stations at 𝑥 and y obtained from the

time window j is

𝑁𝐶𝐹𝑗(𝜔, 𝜏) =∑ 𝐴𝑠2(𝜔)cos[𝜔(𝜏 − Δ𝑡𝑠)]

𝑁

𝑠=1+

∑ ∑ 𝐴𝑘(𝜔)𝐴𝑙(𝜔)𝑁𝑙=1𝑙≠𝑘

cos(𝜔𝜏 + 𝜙𝑘𝑙𝑗)𝑁𝑘=1 , (2)

where Δ𝑡𝑠 =𝑟𝑠𝑥−𝑟𝑠𝑦

𝑐(𝜔) is the time delay between the travel times from the source s to the two

stations at x and y, and 𝜙𝑘𝑙𝑗 = 𝜔(𝑟𝑘𝑥−𝑟𝑙𝑦)

𝑐(𝜔)+ (𝜙𝑘𝑗 −𝜙𝑙𝑗) is the phase shift associated to the

time delay of the two sources (𝑘 and 𝑙) traveling to the two stations (x and y) and their

differential initial phases (see Appendix A.) (Tsai, 2010, Tsai, 2011). Note that the first term in

equation (2) is the correlations of records from N common sources, and is the so-called signal

in the NCF, while the second term is contributed from correlations of unrelated source pairs.

Assuming that the distribution and triggering times of noise sources are random, namely,

both 𝜙𝑘𝑗 and 𝜙𝑙𝑗 are randomly distributed, and knowing that a random walk of N2 unit

steps is equivalent to traveling a total distance of N units (Tsai, 2011, equation (19)), the

summation of the N(N-1) elements in equation (2) can be further simplified

∑ ∑ 𝐴𝑘(𝜔)𝐴𝑙(𝜔)𝑁𝑙=1𝑙≠𝑘

cos(𝜔𝜏 + 𝜙𝑘𝑙𝑗)𝑁𝑘=1 ≅ 𝑁𝐴𝑥

2̅̅̅̅ cos(𝜔𝜏 + 𝜙𝑗𝑎𝑣𝑔), (3)

where 𝐴𝑥2̅̅̅̅ =

1

𝑁(𝑁−1)∑𝐴𝑘𝐴𝑙 and the averaged term 𝜙𝑗

𝑎𝑣𝑔 is a random final phase shift (Tsai,

2011). Thus, in contrast to the structure-related signal part, we may refer the cross-source

contribution in NCF as “noise” or “remnant fluctuations” (e.g., Larose et al., 2008).

Given M independent 𝑁𝐶𝐹𝑠 resulted from the same length of time windows, we define

the ensemble averaged NCF as 𝑁𝐶𝐹̅̅ ̅̅ ̅̅ 𝑀 =1

𝑀∑ 𝑁𝐶𝐹𝑗𝑀𝑗=1 . Likewise, since a random walk of

𝑁2 unit steps in the complex plane results in a total distance of N units, the noise term in the

stacked 𝑁𝐶𝐹̅̅ ̅̅ ̅̅ 𝑀 decreases as 1/√𝑀 (Lobkis and Weaver, 2001, Derode et al., 2003, Larose

et al., 2008, Tsai, 2010, Tsai, 2011). Similar results about the noise evolution in NCF can be

obtained using the normal mode formulation (e.g., Tsai, 2010, Tanimoto, 2008).

Based on the mode derivation, the signal part of NCF is related to the time derivative of

the Green’s function (𝐺′(𝜔, 𝜏)) (e.g., Tanimoto, 2008, Tsai, 2010), and the strength of the

NCF signal is proportional to the average source power along the station pair (𝐴𝑠2̅̅ ̅̅ ̅(𝜔)), as

surface waves are the major sources of ambient noise in nature (e.g., Lobkis and Weaver,

2001, Snieder, 2004, Tsai, 2010). We may thus express equation (2) using the coherency form.

𝑁𝐶𝐹̅̅ ̅̅ ̅̅ 𝑀(𝜔, 𝜏)~𝐴𝑠2̅̅ ̅̅ ̅(𝜔)

𝐴2(𝜔)̅̅ ̅̅ ̅̅ ̅̅ ̅ 𝐺′(𝜔, 𝜏) + (

𝑁(𝜔)

√𝑀) (𝐴𝑥2(𝜔)̅̅ ̅̅ ̅̅ ̅̅ ̅

𝐴2(𝜔)̅̅ ̅̅ ̅̅ ̅̅ ̅) cos(𝜔𝜏 + 𝜙𝑀𝑎𝑣𝑔), (4)

where 𝐴2(𝜔)̅̅ ̅̅ ̅̅ ̅̅ is an azimuthal average of source power.

Note that both 𝐴2(𝜔)̅̅ ̅̅ ̅̅ ̅̅ and 𝐴𝑠2̅̅ ̅̅ ̅(𝜔) in equation (4) are directly related to the source

conditions. Hence, if the noise sources remain constant over time, the first term on the right

hand side (R.H.S.) of equation (4), i.e., the signal part of NCF, also remains stable regardless

of the correlation time. On the other hand, the second term on the R.H.S of equation (4), i.e.,

the noise part, decays with the numbers of stacked NCFS, i.e., the correlation time. In such a

condition, the effect of stacking, a common practice to retrieve reliable EGFs, is to suppress

the noise part rather than to enhance the signal.

Defining the intrinsic noise level (INL) of NCF associated with a given stable source

condition by 𝐼𝑁𝐿(𝜔) ≡ 𝑁(𝜔)(𝐴𝑥2(𝜔)̅̅ ̅̅ ̅̅ ̅̅ ̅

𝐴2(𝜔)̅̅ ̅̅ ̅̅ ̅̅ ̅), we rewrite the ensemble averaged NCF as

𝑁𝐶𝐹̅̅ ̅̅ ̅̅ 𝑇(𝜔, 𝜏)~𝐴𝑠(𝜔)2̅̅ ̅̅ ̅̅ ̅̅ ̅̅

𝐴(𝜔)2̅̅ ̅̅ ̅̅ ̅̅ ̅ 𝐺′(𝜔, 𝜏) +

𝐼𝑁𝐿(𝜔)

√𝑇cos(𝜔𝜏 + 𝜙𝑇

𝑎𝑣𝑔). (5)

We replace M in equation (4) by T in the above equation, because each unit NCF used for

stacking is generated with the same time length of record, M is thus linearly proportional to

the correlation time. Note that the INL defined here is independent of correlation time, and

proportional to the source population N.

Obviously, the “stable source distribution” is not a reasonable assumption, because the

source population/strength of ambient noise is highly influenced by the atmospheric

conditions. For instance, the excitation strengths of short-period secondary microseisms

(SPSM) are strongly influenced by the near coast wind speeds and wave heights (Chen et al.,

2011). It turns out that the strength of the signal part of the stacked NCFs also varies with

time, and we cannot evaluate 𝐼𝑁𝐿 at any given time without knowing the exact source

conditions. Nevertheless, we may measure the expected intrinsic noise level (𝐼𝑁�̂�(𝜔)), i.e.,

the 𝐼𝑁𝐿 averaged over a given time period.

In the following, we provide a procedure to evaluate 𝐼𝑁�̂�(𝜔) and apply it to the NCFs

derived from the vertical component continuous seismic data recorded at 57 broadband

stations in Taiwan and Korea (Figure 1) for the period from 2006 to 2007. For all the data set,

we first compute the daily coherency with a 2000s time window and a 50% overlapping

moving window, using the method presented by Welch (1967).

3. Method

In the stable source condition, the noise decays regularly with the correlation time T as

indicated by equation (5). Taking the final 𝑁𝐶𝐹(𝑇𝑟𝑒𝑓) of correlation time 𝑇𝑟𝑒𝑓 as a

reference NCF, i.e., the NCF with total correlation time, the noise part of 𝑁𝐶𝐹(𝑇𝑟𝑒𝑓) is

𝐼𝑁�̂�(𝜔)

√𝑇𝑟𝑒𝑓; likewise, the noise part of NCF stacked over any given correlation time T is

𝐼𝑁�̂�(𝜔)

√𝑇.

Note that the noise part is also a time series, here we quantify 𝐼𝑁�̂� by the root-mean-square

of the waveform residual.

To homogenize the source conditions over time, we generate the NCF of any target

correlation time by stacking over randomly selected individual NCFs. Furthermore, we

generate 50 such randomly stacked NCFs for each target correlation time, and the

representative residual at that target correlation time is given by the mean residual of the 50

estimates. By doing so, we aim to average out the temporal variations of noise sources, and

make sure the signals in the reference NCF and NCF at any target correlation time are similar,

i.e., 𝐴𝑠(𝑇)̅̅ ̅̅ ̅̅ ̅̅

𝐴(𝑇)̅̅ ̅̅ ̅̅ ̅ ≈𝐴𝑠(𝑇𝑟𝑒𝑓)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅

𝐴(𝑇𝑟𝑒𝑓)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ .

With a properly defined time-dependent term √1

𝑇+

1

𝑇𝑟𝑒𝑓 and the above multiple random

stacking procedure, 𝐼𝑁�̂�(𝜔) can be expressed as (Appendix B)

𝐼𝑁�̂�(𝜔) = |𝑁𝐶𝐹(𝜔, 𝑇𝑟𝑒𝑓) − 𝑁𝐶𝐹(𝜔, 𝑇)| √1

𝑇+

1

𝑇𝑟𝑒𝑓⁄ , (6)

where |∙| is a root-mean-square operator. Examples of residual waveforms evolving with

correlation time are shown in Figure 2. It’s clear that the amplitudes of the residual

waveforms are decreasing with correlation time.

In Figure 3a, we compare the evolution of residuals over time for the mean

random-stacked NCFs (black lines) and the straight-stacked NCFs (gray lines). In straight

stacking, the NCFs are stacked from the first day toward the target days sequentially in time.

Obviously, the potential temporal variations in source may leave its signature to the

straight-stacked NCFs. It clearly shows that the relative noise level obtained from

random-stacked NCFs decays fairly with √𝑇 . On the other hand, the pattern of the

straight-stacked NCFs is seriously perturbed by the source temporal variations.

In Figure 3b, we replace the total correlation time in the x-axis of Figure 3a with

√1

𝑇+

1

𝑇𝑟𝑒𝑓. In this new frame, 𝐼𝑁�̂�(𝜔) is equivalent to the estimated regression slope. The

stable slope in Figure 3b, shown by the black dashed line, implies that the errors caused by

source temporal variations are effectively suppressed in this approach. The case shown in

Figure 3 is not a particular example, similar results are obtained in all the NCFs used in this

study and more examples are presented in Figure 4.

4. The noise level of NCF

We have defined the estimated intrinsic noise level, 𝐼𝑁�̂�, of the NCFs, and confirmed

that 𝐼𝑁�̂� can be robustly evaluated. In the following, we demonstrate the applications of

𝐼𝑁�̂� derived in Taiwan and discuss the implications of the results.

4-1. The Exact SNR of NCFs

NCFs have been widely applied to seismic tomography, in which an empirically defined

SNR is usually assigned to each NCF as a measure of data quality. In general, the SNR of

NCFs is defined as

𝑆𝑁𝑅 ≈𝑊𝑠(𝑡)𝐴𝑠(𝜔)̅̅ ̅̅ ̅̅ ̅̅ ̅𝐺′(𝑡)

𝑊𝑛(𝑡)𝐴𝑠(𝜔)̅̅ ̅̅ ̅̅ ̅̅ ̅𝐺′(𝑡)=𝑊𝑠(𝑡)𝐺

′(𝑡)

𝑊𝑛(𝑡)𝐺′(𝑡) , (7)

where 𝑊𝑠(𝑡) is the box function windowing the target major signals, 𝑊𝑛(𝑡) is the box

function widowing the noise or the entire NCF trace, and t is the lag time. Notice that the

source strength 𝐴𝑠(𝜔)̅̅ ̅̅ ̅̅ ̅̅ disappears in such expression. Moreover, considering the fact that

amplitudes of major signals of NCFs generally decay with inter-station distance due to

intrinsic attenuation and geometrical spreading, the above criteria favors data with shorter

inter-station distance (Appendix C). In other words, the implicit distance-dependent selection

criteria may result in a biased data collection.

Using the above empirically defined SNR to measure NCF quality seems reasonable, as

only the fundamental mode surface waves are used as data in tomography studies.

Nevertheless, it is not an appropriate approach to measure the relative strength between the

signal and noise within the NCFs because signal and noise always coexist in NCFs, namely,

there are noise in the signal window 𝑊𝑠(𝑡) and signal in the noise window 𝑊𝑛(𝑡).

Recently, it has been recognized that the NCF coda is meaningful. Studies have showed

that NCF coda can be used for an iterative NCF reconstruction (e.g., Stehly et al., 2008,

Froment et al., 2011). Furthermore, NCF coda contains multiple-reflected P waves, although

amplitudes of these signals are usually much weaker as compared to the dominant signal of

NCFs (e.g., Poli et al., 2012b, Poli et al., 2012a, Lin et al., 2013). To ensure the data quality

in the applications using the NCF coda, we need an alternative measuring approach other than

the one defined in equation (7).

With the obtained 𝐼𝑁�̂�(ω) , we may quantitatively estimate the true noise level

(𝐼𝑁�̂�(ω) √𝑇𝑟𝑒𝑓⁄ ) of the reference 𝑁𝐶𝐹(𝜔, 𝑇𝑟𝑒𝑓). Here we define the 𝐼𝑁�̂�-based 𝑆𝑁�̂� by

𝑆𝑁�̂�(𝜔, 𝑇𝑟𝑒𝑓) ≡ |𝑁𝐶𝐹(𝜔, 𝑇𝑟𝑒𝑓)| ∙ √𝑇𝑟𝑒𝑓 𝐼𝑁�̂�(𝜔)⁄ . (8)

Prior to the application to realistic data, we first present results of a synthetic experiment

to confirm that the 𝐼𝑁�̂�-based 𝑆𝑁�̂� is reliable. The distribution of sources and stations in this

experiment is shown in Figure 5a. Each source is randomly triggered 50 times in the 2000

seconds time window, the source signal is simulated by three cycles of cosine waves in the

frequency range from 0.2 to 0.05 Hz with interval 0.005 Hz, and the wave speed is 2 km/s.

The sample displacement records at the two stations are shown in Figure 5b. In the same

approach, we generated 500 such synthetic records, and measure the 𝐼𝑁�̂� from the resulting

500 NCFs of stations A and B. Figure 5c shows the evolution of NCF waveforms and the

corresponding waveform residuals. The evolution of the remnant noise with the numbers of

stacking is shown in Figure 5d, where the 𝐼𝑁�̂� thus measured is also presented. Note that the

remnant noise decays in the same fashion as those observed in the realistic data. In Figure 5e,

we compare the waveforms between the noise-based NCF and the theoretical Green’s function,

i.e., the NCF from signals of common sources, and the waveform residual represents the exact

remnant noise (Figure 5f). We find that the difference between the exact noise level and the

𝐼𝑁�̂�-based noise level is only about 7%.

We have confirmed that the we may measure the exact noise level of the NCFs using 𝐼𝑁�̂�,

and it is worth noting that the implementation of the 𝐼𝑁�̂�-based 𝑆𝑁�̂� is not restricted to the

predominant signals in the NCF, i.e., the fundamental mode surface waves. It can be applied to

any portion of the NCF. Using a window function 𝑊(𝑇 ) prior to 𝐼𝑁�̂� measurement, we may

evaluate the noise level of any segment of NCFs (Appendix D).

4.2 Characteristic 𝐼𝑁�̂� and implications of to ambient source populations

It is well known that microseisms are the major sources of seismic ambient noise, and

the signal part of NCFs is mainly contributed from sources along the great-circle lines

connecting the station pairs. Thus, the strong variations and asymmetry of signal strengths in

NCFs are commonly attributed to the excitation characteristics of the nearby or distant ocean

waves (e.g.,Stehly et al., 2006, Chen et al., 2011).

On the other hand, 𝐼𝑁�̂� is primarily sensitive to the overall source population around

the station pairs. Here, we first show that the heterogeneities of source strength have little

influence on the remnant noise and 𝐼𝑁�̂� through a synthetic experiment.

In this experiment, the setup of sources is the same as the previous test shown in Figure

5, except that the excitation strength of sources in the left hand side is four times larger than

those in the right hand side (Figure 6). Note that despite the expected strong amplitude

asymmetry in the signal part of the causal and acausal NCF, the strengths of remnant noise on

both sides are similar (Figure 6b), and the INLs estimated from both sides of NCF are

essentially the same. This confirms that INL is not influenced by the heterogeneities of source

strength.

The above features resulted from synthetic experiment can be observed in the realistic

data in Taiwan, where the amplitude asymmetry of NCFs is rather common in the SPSM

frequency band (Chen et al, 2011), nevertheless, such strong asymmetry is not seen in the

remnant noise and 𝐼𝑁�̂� (Appendix D).

We present the experiment on the relationship between the source population and 𝐼𝑁�̂�

in the Appendix E, and the results show that there is a clear positive correlation between the

source population and estimated 𝐼𝑁�̂�.

From the above results of synthetic experiments and realistic data, we may conclude that

the observed variations in 𝐼𝑁�̂� have little to do with the excitation strength of ambient noise.

Accordingly, it is expected that the fluctuations of 𝐼𝑁�̂� derived in a given area should be

much smaller as compared the spatial variations of NCF amplitudes. This is well

demonstrated in Figure 7, where we present the spectra of NCFs and the corresponding 𝐼𝑁�̂�

derived from 40 and 17 broadband stations in Taiwan and Korea, respectively. As expected,

the variations in 𝐼𝑁�̂� is indeed much smaller than those in NCFs for both regions.

We also noticed that the spectra of NCFs and 𝐼𝑁�̂� are characterized by a peak strength

around 7 seconds, which is exactly the significant ocean wave period observed in the nearby

coast of Taiwan and Korea (Kim et al., 2011, Chen et al., 2011), and is related to the

excitations of the near-coast primary microseism (PM). In general, the larger NCF amplitudes

can be explained by the higher source population and/or stronger source excitation.

With the unique constraint to the source population from 𝐼𝑁�̂� spectra, we thus

conclude that the stronger source excitation is the major factor contributing to the larger NCF

amplitudes in Korea. Exceptions may take place for the periods around the PM band, where

rich source population could be an additional factor responsible for the significant larger NCF

amplitudes in Korea. This is comprehensible, since PM is mainly excited in the near-coast

area, and the coastal line in Korean Peninsula is much longer than the island of Taiwan.

5. Conclusion

In this study, we present a recipe to evaluate the noise level of NCFs. We then apply it to

NCFs derived from realistic data recorded by seismic arrays in Taiwan and Korea. The results

confirm that the true noise level of NCFs can be quantitatively measured using this method.

We also show that, the 𝐼𝑁�̂�-based 𝑆𝑁�̂� may provide a more objective evaluation for NCF

quality, and such measurement can be applied to any time windows of the NCF to assess the

reliability of overtones or coda waves, which are potentially important for broader seismic

applications of NCFs. In addition, since 𝐼𝑁�̂� is closely related to the overall source

population of ambient noise, it helps to better constrain the source characteristics, in

complementary to those provided by NCF amplitudes.

Finally, we should point out that, while the stable source condition is an important

prerequisite in our measurement, 𝐼𝑁�̂� is insensitive to the source distribution. Namely, as

long as the stable source condition is empirically satisfied using this measuring procedure, the

expected temporal evolution of noise level can be equally attained even if the source

distribution is highly non-homogeneous. Thus, to guarantee a confident measurement using

this method, a reliable reference NCF resulting from a quasi-diffused wave field, i.e., an

appropriate long correlation time, is necessary.

Acknowledgments

This work was supported by the Ministry of Science and Technology of Taiwan (MOST

105-2116-M-002-005- and MOST 105-2611-M-002-001-MY3). The continuous broadband

seismic data were essential for this study. The authors would like to thank Prof Junkee Rhie

for providing seismic data of Korea Institute of Geoscience and Mineral Resources (KIGAM).

The data from Broad-Band Array in Taiwan for Seismology (BATS) and Central Weather

Bureau Broad-Band array (CWBBB) were available in the following websites:

CWBBB:http://gdms.cwb.gov.tw/index.php ; BATS:http://bats.earth.sinica.edu.tw .

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Figure 1.

Figure 1. Distribution of broad-band seismic stations used in this study. (a) Stations of

Taiwan from two networks are shown by different symbols as indicated in the figure. NCFs

derived from station pairs ANPB-YHNB and CHKB-TPUB are used to demonstrate

convergence of NCF residuals with correlation time in Figure 2, Figure 3, and Figure A2 in

the Appendix D. (b) Distribution of 17 stations of KIGAM network Korea.

Figure 2.

Figure 2: Evolution of NCF of pair CHKB-TPUB (inter-station distance 79 km). The total

correlation time is shown in the left side of each trace. (a) Comparisons of the normalized

waveforms between the reference NCF (the top trace) and the randomly stacked NCFs for

four representative correlation time. (b) The normalized residual waveforms between the

target NCF and the reference NCF.

Figure 3.

Figure 3: A comparison of remnant noise level derived from waveform residuals for

straightforward (gray) and randomly (black) stacks for the same station pair CHKB-TUPB in

figure 2. (a) Waveform residual (rms) as a function of integration time. The cross is the

average residual measurement of the randomly stacked NCF. (b) Waveform residual as a

function of the properly defined time-dependent term, and the dash lines are the regression

results.

Figure 4.

Figure 4. The evolution of remnant noise level of NCFs derived from six representative

station pairs. The NCFs are band-pass filtered to extract the SPSM signals (2~5 seconds).

(a)~(f) The remnant noise level as a function of correlation time. Results of random stacks

and straight stacks are shown in the black and gray lines, respectively. The crosses are the

average estimates from randomly stacking. (g) Station pair distribution. The black triangles

are stations, and the associated NCF pairs are marked by dash lines.

Figure 5

Figure 5. Comparison of the 𝐼𝑁�̂�-based noise level and the exact noise level from a

synthetic modeling. (a) Distribution of 200 sources (stars) and stations (triangles). (b)

Synthetic waveforms observed at station A and B. (c) Evolution of NCF stacked from M

individual NCF. The stacked number is shown in the left side of each trace. The

normalized residual waveforms between the reference NCF (the top trace) and the stacked

NCFs for four representative correlation time are shown in the right. (d) Waveform

residual (rms) as a function of integration time. (e) The comparison of the NCF waveform

of the final stacked NCF (black) and the exact NCF (red). (f) The normalized residual

waveform, where the noise levels and misfit between the exact and the INL-estimated

noise level are also shown.

Figure6.

Figure 6. Comparison of 𝐼𝑁�̂� derived from model with inhomogeneous source

excitation strength. (a) Distribution of sources (stars) and stations (triangles). Excitation

strength of sources in black is four times larger than those in red. (b) The evolution of

NCF of the station pair A-B. The trace color is related to the color of the corresponding

sources in (a) that contribute to the major signals in NCFs. Note that the amplitudes of

major signals are clearly related to the asymmetric source strength. (c) Normalized

residual waveforms between the target NCF and the reference one. Note that the

amplitudes on both sides are compatible. (d) Waveform residual (rms) as a function of

integration time. The estimated 𝐼𝑁�̂� is 0.0108 and 0.0099 for the causal and the acausal

part of NCF, respectively.

Figure7.

Figure 7. Amplitude spectra for NCFs derived from the vertical component continuous noise

in Taiwan (a) and Korean (b), and their corresponding 𝐼𝑁�̂� spectra in Taiwan (c) and Korean

(d).The red lines and blue lines are mean spectra for Taiwan and Korea, respectively. The

mean spectra of Korean NCFs are also shown in the top panels for a comparison. For the

𝑁𝐶𝐹 spectra, the effects of inter-station distance are removed by taking into account of the

geometrical spreading effects.

Appendix A: Derivations for Noise Cross Correlation

To avoid divergence in cross-correlation between infinite-length time series, we define

the normalized cross correlation as:

(𝑡) (𝑡) = 𝑙 𝑇 1

2𝑇∫ (𝑡) (𝑡 + 𝜏) 𝑡𝑇

−𝑇 (A1)

Similarly, the normalized convolution is defined as:

(𝑡) (𝑡) = 𝑙 𝑇

1

𝑇∫ (𝑡) (𝑡 − 𝜏) 𝑡𝑇

−𝑇

= 𝑙 𝑇 1

2𝑇∫ (𝜏 − 𝑡) (𝑡) 𝑡𝑇

−𝑇 (A2)

Then, we have

𝑗(𝜔, 𝜏) = 𝑈𝑗(𝑥, 𝑡, 𝜔) 𝑈𝑗( , 𝑡, 𝜔)

= 𝑈𝑗(𝑥, −𝑡, 𝜔) 𝑈𝑗( , 𝑡, 𝜔)

= ∑ 𝐴𝑘(𝜔) 𝑠 [𝜔 (−𝑡 −𝑟𝑘𝑥

𝑐(𝜔)) + 𝜙𝑘𝑗]

𝑁𝑘=1 ∑ 𝐴𝑘(𝜔)

𝑁𝑙=1 𝑠 [𝜔 (𝑡 −

𝑟𝑙𝑥

𝑐(𝜔)) + 𝜙𝑙𝑗]

= ∑∑𝐴𝑘(𝜔)

𝑁

𝑙=1

𝐴𝑙(𝜔)

𝑁

𝑘=1

𝑇=

∑ cos [𝜔 (𝜏 + 𝑡 −𝑟𝑘𝑥 (𝜔)

) + 𝜙𝑘𝑗] cos [𝜔 (𝑡 −𝑟𝑙𝑥 (𝜔)

) + 𝜙𝑙𝑗]

𝑇

𝑡=−𝑇

= ∑∑𝐴𝑘(𝜔)

𝑁

𝑙=1

𝐴𝑙(𝜔)

𝑁

𝑘=1

𝑙 𝑇=

1

𝑇∑

1

{ 𝑠 [𝜔 (𝜏 −

𝑟𝑘𝑥 − 𝑟𝑙𝑥 (𝜔)

) + (𝜙𝑘𝑗 − 𝜙𝑙𝑗) ] +

𝑇

𝑡=−𝑇

𝑠 [𝜔 ( 𝑡 + 𝜏 −𝑟𝑘𝑥 − 𝑟𝑙𝑥 (𝜔)

) + (𝜙𝑘𝑗 + 𝜙𝑙𝑗)]}

=1

∑∑𝐴𝑘(𝜔)

𝑁

𝑙=1

𝐴𝑙(𝜔) 𝑠 [𝜔 (𝜏 −𝑟𝑘𝑥 − 𝑟𝑙𝑥 (𝜔)

) + (𝜙𝑘𝑗 − 𝜙𝑙𝑗) ]

𝑁

𝑘=1

=1

∑𝐴𝑠

2(𝜔) 𝑠 [𝜔 (𝜏 −𝑟𝑘𝑥 − 𝑟𝑙𝑥 (𝜔)

) ] +

𝑁

𝑠=1

1

2∑ ∑ 𝐴𝑘(𝜔)

𝑁𝑙≠𝑘 𝐴𝑙(𝜔) 𝑠 [𝜔 (𝜏 −

𝑟𝑘𝑥−𝑟𝑙𝑥

𝑐(𝜔)) + (𝜙𝑘𝑗 − 𝜙𝑙𝑗) ]

𝑁𝑘=1 (A3)

Appendix B: Method

Assuming there is no temporal variations in the source condition, the root-mean-square of

the waveform residual between 𝑁𝐶𝐹̅̅ ̅̅ ̅̅ 𝑇1(𝜔, 𝜏) and 𝑁𝐶𝐹̅̅ ̅̅ ̅̅ 𝑇𝑟𝑒𝑓(𝜔, 𝜏) is

|𝑁𝐶𝐹̅̅ ̅̅ ̅̅ 𝑇1(𝜔, 𝜏) − 𝑁𝐶𝐹̅̅ ̅̅ ̅̅ 𝑇𝑟𝑒𝑓(𝜔, 𝜏)|

= [1

𝑃∫(𝐼𝑁𝐿

√𝑇1∙ cos(ωt − 𝜙𝑇

𝑎𝑣𝑔) −

𝐼𝑁𝐿

√𝑇𝑟𝑒𝑓cos (ωt − 𝜙𝑇𝑟𝑒𝑓

𝑎𝑣𝑔))2

𝑃

0

𝑡]

1/2

=𝐼𝑁𝐿

√𝑃[∫1

𝑇1∙ 𝑠2(ωt − 𝜙𝑇

𝑎𝑣𝑔) +

1

𝑇𝑟𝑒𝑓∙ 𝑠2 (ωt − 𝜙𝑇𝑟𝑒𝑓

𝑎𝑣𝑔)

𝑃

0

− 1

√𝑇1 𝑇𝑟𝑒𝑓cos(ωt − 𝜙𝑇

𝑎𝑣𝑔) cos (ωt − 𝜙𝑇𝑟𝑒𝑓

𝑎𝑣𝑔)) 𝑡]

1/2

= 𝐼𝑁𝐿 [1

𝑇1+

1

𝑇𝑟𝑒𝑓−

1

√𝑇1 𝑇𝑟𝑒𝑓 𝑠 (𝜙𝑇𝑟𝑒𝑓

𝑎𝑣𝑔− 𝜙𝑇

𝑎𝑣𝑔)]1/2

, (B1)

where P is the length of NCF.

In this simple case, we are not able to evaluate INL because the last term of equation (B1),

(1

√𝑇1 𝑇𝑟𝑒𝑓 𝑠 (𝜙𝑇𝑟𝑒𝑓

𝑎𝑣𝑔− 𝜙𝑇

𝑎𝑣𝑔) ), remains uncertain. However, the expected value can be

estimated once we have plenty measurements, and this is exactly the case in our procedure.

Specifically, to homogenize the source conditions over time, we generate the NCF of any

target correlation time by stacking over randomly selected individual NCFs. Furthermore, we

generate 50 such randomly stacked NCFs for each target correlation time, and the 𝐼𝑁�̂�(𝜔) is

given by the mean residual of the 50 estimates. By doing so, we aim to average out the

temporal variations of noise sources, and make sure the signals in the reference NCF and

NCF at any target correlation time are similar. Because the phase term, 𝜙𝑇𝑟𝑒𝑓𝑎𝑣𝑔

− 𝜙𝑇𝑎𝑣𝑔

is

random in each NCF generation, thus, the expected value of 𝑠 (𝜙𝑇𝑟𝑒𝑓𝑎𝑣𝑔

− 𝜙𝑇𝑎𝑣𝑔) is zero in the

averaged NCF. The equation (B1) is then

|𝑁𝐶𝐹(𝜔, 𝑇1) − 𝑁𝐶𝐹(𝜔, 𝑇𝑟𝑒𝑓)| = 𝐼𝑁𝐿 (1

𝑇1+

1

𝑇𝑟𝑒𝑓)

1

2. (B2)

Appendix C: Comparison of SNR estimated from two different approaches

As shown in Figure A1(a), the distance-dependence of SNR defined in equation (7) is much

stronger than the 𝐼𝑁�̂�-based 𝑆𝑁�̂�. In addition, compared to the empirically defined SNR, the

𝐼𝑁�̂�-based 𝑆𝑁�̂� provide a better resolving power for NCF quality. This is shown in their

distribution map (see Figure A1(b)), where the distribution of the 𝐼𝑁�̂�-based ̂ spreads

over a wide range, while the distribution of SNR is focused within a narrow band.

Figure A1.

Figure A1. Comparisons of the empirical SNR measurements (blue) and the 𝐼𝑁�̂�-based 𝑆𝑁�̂�

given in this study (red). The number in the parenthesis represents the corresponding frequency

band. Here, we only present results obtained from NCFs in Taiwan. (a) Data quality estimates

as a function of the interstation distance. (b) In general, empirical SNR and 𝐼𝑁�̂�-based 𝑆𝑁�̂�

have a clear linear relationship. It is noticed that 𝐼𝑁�̂�-based 𝑆𝑁�̂� can provide a higher

resolution for data selection, especially for the longer period cases. In this analysis, the

empirically defined SNR is a ratio of the maxima amplitude within the signal window to the rms

of the whole trace.

.

Appendix D: Evaluation of ̂ at different lag time windows of NCFs

We present NCF convergence within four lag time windows, in which the major signals and the

corresponding tailing coda waves of causal and acausal NCFs are considered separately (Figure

A2). Note that the decay of noise within coda windows agrees with the theoretical expectation

as well. In addition, despite a strong amplitude asymmetry, the noise decays within two signal

windows are rather similar, and there is nearly no direction dependence for the obtained 𝐼𝑁�̂� in

Taiwan (Figure A2). This is expected, as 𝐼𝑁�̂� is associated to the overall source population.

Figure A2. An example of NCF convergence within a day (2006,101). (a) The numbers next

to NCFs are the total-correlation time. Using the NCF averaged over a day as a reference

(marked in red), the corresponding waveform residual is shown in the right. The annual

averaged NCF is also shown in the top for comparison. The blue dash lines denote the signal

windows for the causal and acausal parts of NCF and the coda is defined as the rest of NCF. (b)

Disregarding a miner temporal/spatial variation of source strength within a day, straight

stacking is used in this analysis. Apparently, the noise level varies with total correlation time

nicely.

Appendix E: Relationship between INL and Source population

To examine the relationship between the source population and INL, we estimate INL for

a series of source population ranging from 100 to 1000 sources at interval 50. The sample

distribution of sources is shown in Figure A3 (a). In synthetic test, the related INL can be

estimated simply by the rms of the noise part of the NCF.

Note that while the statement “a random walk of 𝑁2 unit steps in the complex plane

results in a total distance of N units” is one of the key approximations in our formulation, the

rationale is true only in a statistic manner, i.e., for cases with abundant sources such as the

microseisms in the nature. In the synthetic experiment, we can only afford computations with

rather limited source number, and the results may not converge to the statistic expectation. To

cope with the computation resources, we have done 400 synthetic experiments for each

source population, and the mean of the resulting 400 INLs is used as the representative one.

Figure A3 (b) shows the mean INLs and the corresponding standard deviations for each

source population.

Given the limited source numbers used in the experiment, we have noticed that the

resulting noise level is not only influenced by the noise population, it is also altered by the

random entry phases of each source triggering. However, the positive correlation between the

source numbers and the representative INLs remains robust.

Figure A3. (a) Distribution of source (colored dots) and station (triangles) in the

synthetic test. 50 sources are evenly distributed on the half-circle, and source numbers is

increased by adding sources to the outer circles at radius interval 20 km. (b) variations of

INL with respect to the source population. The black dots and the gray bars represent the

mean and standard deviation of INL, respectively, and the red dash line is the regression

result.