Frequency structure of deep low-frequency tremors occurring ...

Geophys. J. Int. (2010) 181, 406–416 doi: 10.1111/j.1365-246X.2010.04500.xG

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Frequency structure of deep low-frequency tremors occurringin western Shikoku region, Japan

Sho Nakamula∗ and Minoru TakeoEarthquake Research Institute, University of Tokyo, Tokyo, Japan. E-mail: [email protected]

Accepted 2009 December 31. Received 2009 December 24; in original form 2008 June 2

S U M M A R YSince the discovery of deep low-frequency tremors in southwestern Japan, various types oflow-frequency oscillation phenomena have been reported and analysed. Among these phe-nomena, non-volcanic, deep, low-frequency tremors in southwestern Japan are outstandinglow-frequency oscillation phenomena occurring over a wide region along the Philippine SeaPlate. However, the weakness of these signals makes it difficult to apply usual analyses to derivesource characteristics. In this paper, we apply a new algorithm to obtain a frequency structurefrom highly noisy data and reveal the characteristic frequency structure of deep low-frequencytremors with peaks lined up from 1 to 5 Hz at intervals of 0.5 Hz. In addition, we present onepiece of evidence that the deep low-frequency tremors and low-frequency earthquakes have acommon source mechanism.

Key words: Time series analysis; Earthquake source observations; Seismicity and tectonics.

1 I N T RO D U C T I O N

Recently, various types of low-frequency oscillation phenomenahave been discovered in various regions (e.g. Obara 2002; Rogers& Dragert 2003; Ohmi et al. 2004; Nadeau & Dolenc 2005). Non-volcanic, deep, low-frequency tremors along the Philippine SeaPlate (Obara 2002) are outstanding phenomena occurring over awide region of 600 km in length. Most active and strong genera-tions of the non-volcanic, deep, low-frequency tremors in south-western Japan are observed in the Shikoku region (Fig. 1). Depthsof these tremors were estimated to be around 30 km where the nor-mal lithosphere was usually ductile and regarded as aseismic region(e.g. Scholz 1988). Dominant frequencies of these tremors werelower than those of tectonic earthquakes with similar amplitudes.Non-volcanic, deep, low-frequency tremors were also observed indeep regions of inland crust such as beneath the San Andreas Fault(Nadeau & Dolenc 2005) and beneath the seismogenic fault of the2000 Western Tottori earthquake, Japan, (Ohmi et al. 2004). There-fore, these tremors are considered to be different from tectonic earth-quakes and to represent phenomena that are specific to the subduc-tion zone and/or the deep part of crust in the active tectonic regions.In the subduction zone of the Philippine Sea Plate, the Japan Meteo-rological Agency (JMA) has distinguished a class of events that hadclear isolated S-wave phases with dominant frequencies lower thanthose of tectonic earthquakes with similar amplitudes: these eventsare referred to as low-frequency earthquakes in their seismicity cat-alogue. Several characteristics of the epicentre distributions, such asclustering (Obara et al. 2004), slow migration of the non-volcanic,

∗Now at: Ministry of the Environment, Tokyo, Japan

deep, low-frequency tremors along the strike of the Philippine SeaPlate with a velocity of about 10 km day−1 (Obara 2002), and amigration of the low-frequency earthquakes perpendicular to thestrike (Shelly et al. 2007) are also known in this region. However,the amplitude of non-volcanic, deep, low-frequency tremors is verysmall, and it can hardly be distinguished from the background noiseif it is observed at a single station. The weakness of the signal makesit difficult to apply usual methods to derive source characteristics ofthe non-volcanic, deep, low-frequency tremors, such as hypocen-tres, frequency structures and so on. In this paper, we focus on thefrequency structure of non-volcanic, deep, low-frequency tremorsand apply a new algorithm to reveal the characteristic frequencystructure of non-volcanic, deep, low-frequency tremors from highlynoisy data, intending to get knowledge about its source mechanism.

2 AV E R A G E D I S S I PAT I O N S P E C T RU MB A S E D O N T H E T H E O RY O FK M 2O - L A N G E V I N E Q UAT I O N S

Average dissipation spectrum (ADS) proposed by Takeo et al.(2006) is an efficient algorithm to obtain the characteristic frequencystructure of the coherent signals. This method is on the basis of thetheory of KM2O-Langevin equations (Okabe 1999, 2000; Matsuura& Okabe 2001), which is popular in the fields of economics and en-gineering but is not widely used in the field of geophysics. Here webriefly explain the basic concept of the theory of KM2O-Langevinequations, and then describe the algorithm of ADS.

Let X = (X (n); 0 ≤ n ≤ N ) be any Rd-valued stochastic processon a probability space (�,B, P) such that all random variables X (n)are square integrable. We introduce the forward KM2O-Langevin

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Freq. structure of deep low-frequency tremors 407

Figure 1. Distributions of the epicentres of non-volcanic, deep, low-frequency tremors in western Shikoku region from 5 to 30 August 2002 estimated bythe envelope correlation method (Obara 2002). The lower panel shows an example of non-volcanic, deep, low-frequency tremors observed at 21:00 on 2003September 1.

fluctuation process v+ = (v+(n)0 ≤ n ≤ N ) associated with X byprojecting X (n) on the vector space Mn−1

0 (X) which is spanned by{X j (k); 0 ≤ k ≤ n − 1, 1 ≤ j ≤ d}:v+ (0) ≡ X (0) (1)

v+ (n) ≡ X (n) − PMn−10 (X) X (n) (1 ≤ n ≤ N ) (2)

where PMn−10 (X ) denotes the projection operator to Mn−1

0 (X).In the same way, we introduce the backward KM2O-Langevin

fluctuation process

v− = (v− (l) ; −N ≤ l ≤ 0)

by

v− (0) ≡ X (N ) (3)

v− (1) ≡ X (N + l) − PMNN+l+1(X) X (N + l) (−N ≤ l ≤ −1) (4)

To simplify, we mention the KM2O-Langevin equation associ-ated with non-degenerate stochastic process, which means that Xsatisfies the following independent conditions:

{X j (n); 0 ≤ n ≤ N , 1 ≤ j ≤ d} is linearly independent in MN0 (X).

The Following results are extended naturally for degenerate stochas-tic processes, and the theoretical study of degenerate stochastic

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408 S. Nakamula and M. Takeo

processes is detailed in Matsuura & Okabe (2001). Then, we canderive the stochastic difference equation that describes the timeevolution of non-degenerate stochastic process X in the followingway:

X (n) = −n−1∑k=0

γ+ (n, k) X (k) + v+ (n) (5)

X (N − n) = −n−1∑k=0

γ− (n, k) X (N − k) + v− (−n) (6)

where γ±(n, k) is a d × d matrix that is determined uniquely fromthe non-degenerate stochastic process X (Okabe 1999; Matsuura &Okabe 2001). We call this equation the KM2O-Langevinequation. Further, the matrix functions γ+(n, k) andγ−(n, k) (0 ≤ k < n ≤ N ) are referred to as the forwarddissipation matrix and the backward dissipation matrix, re-spectively. In particular, for each n (1 ≤ n ≤ N ), we putδ±(n) ≡ γ± (n, 0), and denote by V±(n) the inner productsof v±(±n) : V±(n) ≡ (v±(±n),t v±(±n)). It is to be noted that(v±(±n), tv±(±m)) = δnm V±(n) (0 ≤ m, n ≤ N ). Now, we define asystem LM(X) by LM(X) ≡ {γ±(n, k), δ±(n), V±(m); 0 ≤k < n ≤ N , 0 ≤ m ≤ N }, and call this system the KM2O-Langevin matrix associated with the non-degenerate d-dimensionalstochastic process X. Physically, the dissipation term represents thepart that can be explained by the previous data and the fluctuationterm represents the information newly added at the time n.

We shall define the weakly stationary property for X as follows:

(a) The mean value of X is zero (E(X (n)) = 0).(b) The correlation matrix (R) of X (n) and X (m) depends only

on the difference between times n and m, such that (X (n), t X (m)) =R(n − m) (0 ≤ m, n ≤ N ) .

If X is a local and weakly stationary process, there exist specialrelationships among the elements ofLM(X) [Characterization The-orem for Stationary Property (Okabe 1999)]; that is, the elements ofLM(X) satisfies the Dissipation–Dissipation Theorem (DDT) andthe Fluctuation-Dissipation Theorem (FDT). On the other hand, wecan uniquely construct a KM2O-Langevin matrix systemLM(R) ={γ R

± (n, k) , δR± (n) , V R

± (m) ; 0 ≤ k < n ≤ N , 0 ≤ m ≤ N}

that sat-isfies the (DDT) and (FDT) from any non-negative definite d × dmatrix R = (R(n); |n| ≤ N ) with the Toeplitz condition (Okabe2000). This theorem is called the Construction Theorem. Thesetheorems demonstrate that the KM2O-Langevin matrix is obtainedwith no priori assumptions when stationarity of the data is assured.Therefore, the important characteristics of this theory are that itrequires no priori information about the data whose stationarity isassured and that we do not have to define any parametric modelsbefore data analysis is performed. An appropriate model can beextracted directly from the analysed data in the form of differenceequations.

Now X = (X (n); 0 ≤ n ≤ Na) is a stochastic timeseries. When we extract a local time series Xr =(X (n); r ≤ n ≤ N + r ) (0 ≤ r ≤ Na − N ) with a length N + 1from X, we obtain the KM2O-Langevin matrix LM(Xr ) uniquelyfor each interval Xr when it satisfies local stationarity. Although weobtain a different KM2O-Langevin matrix for each Xr , the value ofeach matrix is determined by the physical mechanism behind thedata. If the specific physical process continues for some length oftime and is found in the data, the dissipation terms obtained musthave similar values. Conversely, we can obtain the fundamental

property of the background physical process by averaging LM(Xr )for intervals that satisfy stationarity. From this viewpoint, Takeoet al. (2006) propose an algorithm to obtain an average KM2O-Langevin matrix for plural local stationary time series, satisfyingthe Characterization Theorem for Stationary Property. The detailsare given in Takeo et al. (2006).

Using the obtained average KM2O-Langevin matrix, we can re-construct the time series X̄ in the form of KM2O-Langevin equationsX̄ (n) = − ∑n−1

k=0 γ (n, k)X̄ (k) + v(n). The fluctuation term is whitenoise, and we assume that this term includes no significant infor-mation about the physical process we want to elucidate. The afore-mentioned equation then becomes X̄ (n) + ∑n−1

k=0 γ (n, k)X̄ (k) = 0,being an equivalent to an nth-order linear difference equation. Ap-plying a Z-transformation to this equation, we can solve it in ap-propriate conditions. The solution can be described by the super-position of nth characteristic frequencies (complex in general) asX (n) = ∑n

s=1 αs exp (�s t). Characteristic frequencies �s can beexpressed by �s = 2π (−gs + i fs). The terms αs, fs and gs , all ofthem are real constants, represent the intensity of relevant element,the characteristic real frequency and the transient amplitude withwhich the relevant element decays exponentially with time, respec-tively. In this paper, we refer to gs as the ‘decay rate’. The ADS isdefined as the set of elements (αs, gs, fs) representing the discretespectrum of the dissipation part of the average KM2O-Langevinmatrix (Takeo et al. 2006). These elements represent the character-istic frequency structure of the analysed data. Detailed descriptionsof how a set of complex frequencies like the ADS is obtained fromthe linear homogeneous difference equation are given in Takeo et al.(2006).

3 S I M U L AT I O N T E S T O F A D SF O R H I G H LY N O I S Y DATA

There is another method, called Sompi method, to obtain charac-teristic frequencies and decay rates of a linear dynamic system(Kumazawa et al. 1990). The basic concept of the Sompi method isestimating a governing difference equation of a hypothetical lineardynamic system that has yielded the given time series data by fittingthe autoregressive (AR) model. Differences between the ADS andthe Sompi method mainly come from the way to reveal the coeffi-cients of the difference equation from the data. The Sompi methoduses a least square fitting to obtain the AR model coefficients, whilethe ADS uses the Construction Theorem, which is closely relatedto an approach of minimizing a prediction error, and the averagingalgorithm to obtain an average KM2O-Langevin matrix.

To check the performance of the ADS for highly noisy data, wecompared these two methods using synthetic time series data withvarious noise levels. The synthetic data consisted of three signalswith (frquency, decay rate) = (1.0 Hz, 1.0/2π) , (2.0 Hz, 1.5/2π),(3.0 Hz, 2.0/2π) and a certain level of random noise as follows:

e−2π ·t sin (2π · t) + 1.5e−1.5·(2π ·t) sin (2π · 2t) + 1.2e−2·(2π ·t)

× sin (2π · 3t) + σ (t) ,

σ (t) =⎧⎨⎩

0.3 · N (t)0.6 · N (t) ,

1.5 · N (t) (7)

where N (t) was a uniform random variable between −1.0 and 1.0.The synthetic data with the noise level of 0.3 is shown in Fig. 2a(a).The sampling interval of data were 0.01 s, and the signal was excitedfor every 12 s. The length of the particular local time window wastaken to be 500 points (5 s), and we checked stationarity of the local

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Figure 2. (a) An example of synthetic data composed for the comparison between the ADS and the Sompi method. (b) Frequency versus decay rate plot forthe spectrum obtained by the ADS and the Sompi method. The noise level of the synthetic data is 0.3. Average values of the spectrum from each methodare shown by the symbol and the maximum and minimum values are shown by the lines. Spectrum obtained by the ADS, the Sompi method applied to thestationary time-series, and the Sompi method applied to the fixed length data are indicated by the blue circles, green triangles and purple triangles, respectively.(c) Frequency versus decay rate plot for the spectrum obtained by the ADS and the Sompi method. The noise level of the synthetic data is 0.6. (d) Frequencyversus decay rate plot for the spectrum obtained by the ADS and the Sompi method. The noise level of the synthetic data is 1.5.

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time-series. If it satisfied stationarity, the end point of the time-series was plotted in green; if it did not, the end point was plottedin red. We applied the ADS and the Sompi method to the time-series that satisfied stationarity. We also applied the Sompi methodto the fixed length of data (1000 points) regardless of whether itsatisfied stationarity or not. In the case of the low noise level (σ (t) =0.3 · N (t)), the frequencies and the decay rated were determinedcorrectly by the Sompi method using the long fixed length data,because the least square fitting for the longer data could draw outthe information more efficiently (Fig. 2a(b)). In the case of the highnoise levels (σ (t) = 0.6·N (t)and σ (t) = 1.5·N (t)), the frequenciesand the decay rates estimated from the Sompi method became lessaccurate [Figs 2a(c) and 2a(d)], because the fitting algorithm of theSompi method misread the random noise as the signal, extractingwrong coefficients of the AR model. On the other hand, the ADS wasnot so influenced by the growth of noise level as the Sompi methodbecause the ADS did not work for the direction on minimizing thenoise component. The averaging algorithm went more effectivelydue to increment of local time-series satisfying stationarity eventhough the high noise level. The accuracy of the estimation wasevidently better for the ADS than the Sompi method in the cases ofboth high noise level simulations. Moreover, it is difficult to knowabout times of signal excitations for the case with low signal-to-noise ratio such as the non-volcanic, deep, low-frequency tremors.We cannot take fixed length of data to apply the Sompi method at anappropriate timing. For these reasons, we conclude that the ADS isvery useful for the purpose of obtaining the averaged characteristicfrequency structure from the non-volcanic, deep, low-frequencytremors.

In the actual data analyses in this paper, we used data of 1-hr length: therefore, we performed another simulation test undersimilar situation of the actual data analyses. We simulated a longsimulation data with a length of 1-hr under the highest noise levelcondition (r = 1.5), dividing them every 12 s. Then, we had 300 sim-ulate time-series data. In order to apply the ADS method, we took thelength of the particular local time window to be 500 points, check-ing stationarity of the local time-series on each divided time-seriesdata. If it satisfied stationarity, we applied the ADS method to getthe frequencies and the decay rates from each divided time-series.The average values and the standard deviations of the frequenciesand the decay rates obtained by ADS were shown by blue circlesand bars in Fig. 2b. The Sompi method was applied to each dividedtime-series using the fixed time window (500 points) regardless ofwhether it satisfied stationarity or not. Shifting the starting point ofthe time window on each time-series, we got 500 sets of coefficientsof the AR model, and averaged them. We applied the Sompi methodto the 300 simulate time-series, obtaining the average values and thestandard deviations of the frequencies and the decay rates, whichwere shown by purple triangles and bars in Fig. 2b. Because thefitting algorithm of the Sompi method misread the random noise assignal, the averaging procedure worked negatively; it was clear thatthe performance of Sompi method was quite poor for highly noisydata.

4 DATA A N D I N S T RU M E N T S

The data set that we used for the analysis consists of the ve-locity waveform of non-volcanic, deep, low-frequency tremorsrecorded by the densely distributed high-sensitivity borehole seis-mograph network (Hi-net) operated by the National Research In-stitute for Earth Science and Disaster Prevention (NIED) located

in the Shikoku region, Japan. Our analysis covered the time peri-ods between 5 and 30 August 2002 and between 27 August and5 September 2003 because the activity of non-volcanic, deep, low-frequency tremors were very high in these two periods in 2002 and2003, and the Hi-net data were available in these periods. To dimin-ish the influence of the ground structure, we analysed the tremor datarecorded at the stations that are thought to be directly above theseclusters. Fig. 1 shows the epicentre distributions of non-volcanic,deep, low-frequency tremors in western Shikoku region obtained bythe envelope correlation method (Obara 2002) for these two periodsand an example of waveforms.

5 F R E Q U E N C Y C H A R A C T E R I S T I C SO F N O N - V O L C A N I C , D E E P,L OW- F R E Q U E N C Y T R E M O R S B Y A D S

By using the ADS, we revealed the characteristic frequencies anddecay rates of non-volcanic, deep, low-frequency tremors that oc-curred in 2002 and 2003. Five periods were selected from eachyear with a length of 1 hr in which the tremors occurred actively.Four stations (KWBH, HIYH, OOZH, YNDH), as the stations lo-cated directly above the sources of the tremors occurring in theseperiods, were used for the analysis. Fig. 3(a) shows the frequencyversus decay rate plots of the tremors acquired by the ADS. Thelength N + 1 of the each interval Xr to be analysed was taken to be10 s (500 points). This window length was determined based onlocal stationarity of the data. When we took a long window lengthsuch as longer than 50 s, a lot of local time series did not satisfystationarity, causing a barrier to apply the ADS method. We adoptedthe window length ranging several seconds to several tens seconds,and obtained the similar frequency characteristics discussing in thefollowing sentence. We extracted a local time series that satisfiedlocal stationarity sequentially for longer than 20 s, and obtained thesets of the KM2O-Langevin matrix by changing r from 0 to Na − N .The ADS spectrum was calculated for these sets and all pairs of thefrequency and the decay rate derived from the sets were plotted ascircles in the Fig. 3(a). The intensity of the spectrum was normal-ized by the maximum intensity of each set, and the colour bar of theintensity is shown in logarithmic-scale. The spectra whose relativeintensities were less than 0.005 were eliminated. The instrumentalresponse of seismograph was not removed because the deconvolu-tion for the Hi-net seismograph mainly increased the intensity ofthe frequency range under 1 Hz, which is masked by microseismsor other background noise. Beyond controversy, part of the tremor’ssignal higher than 1 Hz was also affected by the short-period seis-mograph’s response with corner frequency of 1 Hz, reducing thespectrum amplitude around 1–2 Hz slightly. In these figures, we canrecognize that several peaks commonly exist among all stations inthe frequency range between 0 and 5 Hz. A frequency domain withhigh-intensity also exists in the range over 5 Hz. However, the fre-quency domain varied with every station, suggesting that the maincause of this domain of high-intensity over 5 Hz was the scatteringfrom the local velocity structure. Therefore, this domain contains lit-tle useful information about the non-volcanic, deep, low-frequencytremors source.

At all stations, two strong peaks exist less than 1 Hz. One is near0.3 Hz and the other is near 0.7 Hz. The ADS spectrum at the fourstations derived from the periods with no non-volcanic, deep, low-frequency tremor are shown in Fig. 3(b). It seems to be acceptableto consider these two peaks as the microseisms occurring constantlydue to the ocean waves or some other causes because these peaks

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Figure 3. (a) The frequency versus decay rate plots of the non-volcanic, deep, low-frequency tremors occurring in the period spanning from 27 Augustto 5 September 2003 acquired by the ADS at the stations named HIYH, KWBH, OOZH and YNDH, which are labeled as ‘2003 HIYH’, ‘2003 KWBH’,‘2003 OOZH’ and ‘2003 YNDH’, respectively. Arrows indicate the spectral peaks identified in the ADS spectrum, which are listed in Table 1. Longitudinalbroken lines in each panel represent 2 Hz and 4 Hz, respectively. The frequency versus decay rate plot of the non-volcanic, deep, low-frequency tremors atKWBH, which occurred in 2002, is also shown in this figure labeled as ‘2002 KWBH’. (b) The frequency versus decay rate plots of the period not includingnon-volcanic, deep, low-frequency tremors at HIYH, KWBH, OOZH and YNDH in 2003. Blue arrows in each panel indicate first-, second- and third-peaksof dominant frequencies. Longitudinal broken lines in each panel represent 2 and 4 Hz, respectively.

also exist in the ADS spectrum obtained from the waveforms of theperiods with no tremors. The two peaks are obviously recognizedat other stations. In Fig. 3(b), all stations have another peak near1.1 Hz, and the dominant frequencies of the microseisms were

considered to be these three peaks shown by blue arrows in Fig. 3(b).The small decay rates of these peaks compared to the other peaksobserved in non-volcanic, deep, low-frequency tremors also indicatethat these frequencies existed at all times in the analysed periods.

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Figure 3. (Continued.)

Since the frequencies and decay rates of these peaks were similarat all stations, we took the average of the frequencies and decayrates for all sets and all stations. The averaged frequency and decayrate of the first two peaks were as follows: (frequency, decay rate) =(0.43 Hz, 0.09–0.12) and (0.73–0.80 Hz, 0.14–0.16). The deviationsof the decay rate were larger than the deviation of the frequencybecause the decay rate depends on the length or the trend of theselected interval. In comparison with the frequencies and decay

rates of the period including the tremors, the first two peaks ofthe periods without the tremors were almost the same. However, inthe third peak, i.e. 1.12 Hz, the frequency and the decay rate werelower than those of the period including the tremors, i.e. 1.3 Hz.From this finding it was assumed that the non-volcanic, deep, low-frequency tremors also had a peak at a frequency a bit higher thanthe microseism peak and that these two peaks were too close todistinguish. The lower decay rate of the peak from the period without

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the non-volcanic, deep, low-frequency tremor also suggests thisassumption.

We can recognize other peaks in the range from 1 to 5 Hz inthe frequency versus decay rate plots of non-volcanic, deep, low-frequency tremors (see Fig. 3a). The peaks, which commonly existedamong all stations and were not the same as the microseisms peaks,were considered to result from the source process of non-volcanic,deep, low-frequency tremor.

If the frequency structure derived by the ADS method is dueto the effects of the crustal structure beneath the stations, tectonicearthquakes whose hypocentres are near the non-volcanic, deep,low-frequency tremor sources must also excite these frequencies.If this excitation occurs, the particular frequency peaks recognizedin the non-volcanic, deep, low-frequency tremors should be ob-served in the spectra of the tectonic earthquakes recorded at theplural stations. Therefore, we selected three earthquakes with mag-nitudes of 3.6–4.7 lying near the tremor sources, and calculatedthe spectra at the plural stations. Fig. 4(a) shows the spectra ofground displacements at the four stations (KWBH, HIYH, YNDH& OOZH) excited by these three earthquakes. The amplitudes ofthese earthquakes were several thousands larger than those of thenon-volcanic, deep, low-frequency tremors. Because the amplitudesof non-volcanic, deep, low-frequency tremors were slightly higherthan those of microseisms, we could not use small earthquakes hav-ing similar amplitudes with the tremors in the spectral analyses dueto low signal-to-noise ratio. There remained a little ambiguity inthe evaluation of effects of crustal structure due to the differenceof source sizes and/or structure of the source spectrum. However,the spectral peaks common among the all four stations were notobserved from these three earthquakes especially in the low fre-quency range from 1 to 5 Hz. Because seismograms of tectonicearthquakes do not satisfy stationarity, we can’t apply the ADSmethod to these records. Therefore, to check the crustal structureresponse beneath the stations, we calculated a stacking spectrum ofseismograms using 40 tectonic earthquakes with magnitude rangingfrom 2.5 to 4.2, which occurred in the western Shikoku region. Thestacking velocity spectra of vertical component records at KWBHand HIYH are shown in Fig. 4(b) with the seven frequency peaks inthe non-volcanic, deep, low-frequency tremors revealed by the ADSmethod. These peaks are not consonant with the spectral peaks ofthe stacking spectra of both stations. The similar features were con-firmed at other stations. These facts strongly suggest that the peakscommonly exited among all stations were considered to result fromthe non-volcanic, deep, low-frequency tremors.

Table 1 lists the average values of the recognizable peaks between1 and 5 Hz derived from the 2003 data for each station. The sevenpeaks obtained in 2003 were plotted at about 0.5 Hz intervals startingfrom 1.3 Hz and the decay rates were found to be around 0.20for all stations. On the other hand, the peaks obtained in 2002started from a relatively low frequency (about 1.2 Hz) with thesame interval of 0.5 Hz and the decay rates were distributed fromabout 0.15 to 0.18 (Fig. 3a). Since the ADS provides the discretefrequencies, there is a possibility that these peaks were not real peaksbut apparent peaks produced by the discretization. The number ofdiscrete frequencies obtained by the ADS is the same as the ordernumber of the difference equation; however, the intervals of thesefrequencies do not need to be equally spaced frequencies like as theDiscrete Fourier Transform. For example, the third-lowest frequencycomponents of non-volcanic, deep, low-frequency tremors derivedfrom the ADS did not always have a value near 1.30 Hz (thirdpeak). Moreover, the derived frequencies and decay rates variedslightly among stations and years, indicating that the derived peaks

100

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frequency (Hz)

10-1

100

101

2 4 6 8 2 4 6 8 2 4

M = 4.7

M = 3.7

M = 3.6

KWBH

HIYH

YNDH

OOZH

(a)

(b)

1

10

100

1000

KWBH

HIYH

1 2 4 04 01 86 20

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Figure 4. (a) Spectra of tectonic earthquakes whose hypocentres are near thenon-volcanic, deep, low-frequency tremor sources. The spectra are obtainedfrom the horizontal component of the displacement waveforms at the fourstations. The black, red, green and purple lines indicate the spectra at KWBH,HIYH, YNDH and OOZH, respectively. For each earthquake, the spectraare moved along the longitudinal axis to overlap these spectra in the lowfrequency region (from 0.1 to 1 Hz) for the purpose of comparing the spectralpeaks among the four stations. (b) Stacking spectra of velocity seismogramsat KWBH and HIYH. Forty tectonic earthquakes with magnitude rangingfrom 2.5 to 4.2, which occurred in the western Shikoku region, were used inthe calculation of the stacking spectra. The seven frequency peaks obtainedin the deep low-frequency tremors are shown by grey broken lines. The sharppeaks in the frequency region higher than 10 Hz were caused by electronicnoises in the recording system.

truly exist. A peak with maximum spectral intensity of the tremorswas determined around 1.8 Hz. In view of the intensity, the obtainedfrequencies and decay rates seem to bear the main part of the energyfrom the tremors, as these peaks are considered to represent the

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Table 1. Average values of the frequencies, decay rates and relative intensity for seven peaks derived from the non-volcanic,deep, low-frequency tremor data from 2003. The number of sets used for the average is shown in the rightmost field. Thespectra whose relative intensities were less than 0.005 were eliminated.

Frequency (Hz) Decay rate Relative intensity Deviation (Hz) Deviation (decay) Number of sets

HIYH 1.26 0.20 0.14 0.14 0.06 3751.75 0.21 0.10 0.17 0.08 3092.20 0.19 0.07 0.16 0.06 2882.76 0.20 0.04 0.14 0.10 2423.26 0.20 0.03 0.20 0.06 1753.84 0.20 0.03 0.17 0.08 1684.35 0.19 0.05 0.17 0.06 216

KWBH 1.28 0.19 0.24 0.13 0.06 2661.74 0.19 0.16 0.14 0.07 2412.30 0.21 0.05 0.20 0.07 1752.81 0.17 0.12 0.15 0.06 2373.28 0.17 0.14 0.14 0.07 2433.77 0.19 0.05 0.20 0.06 1734.32 0.19 0.07 0.19 0.06 162

OOZH 1.30 0.19 0.20 0.12 0.06 4271.87 0.18 0.31 0.14 0.07 3992.35 0.19 0.39 0.15 0.07 3942.79 0.19 0.34 0.15 0.08 3933.28 0.20 0.18 0.18 0.07 3733.78 0.20 0.14 0.17 0.07 3484.25 0.19 0.12 0.17 0.07 368

YNDH 1.35 0.21 0.25 0.14 0.08 3891.79 0.19 0.42 0.13 0.07 4142.27 0.20 0.35 0.15 0.07 4222.70 0.19 0.27 0.14 0.07 4063.18 0.19 0.08 0.15 0.07 3393.56 0.19 0.03 0.15 0.05 1414.35 0.18 0.01 0.23 0.04 46

fundamental frequency and the decay rate structure of the non-volcanic, deep, low-frequency tremors which occurred in 2002 and2003.

6 D I S C U S S I O N

Since 1999, the JMA has distinguished a class of events denoted aslow-frequency earthquakes in their seismicity catalogue. The visi-ble arrivals from these events are primarily S waves, and if arrivalsfrom the same source are distinguished at enough stations, the eventis included in the JMA seismicity catalogue. Shelly et al. (2006)made a precise hypocentres relocation using the double-differencetomography method (Zhang & Thurber 2003) and found that thelow-frequency earthquakes were occurring at the plate boundary.On the basis of the sum total of cross-correlation values betweenthe non-volcanic, deep, low-frequency tremor’s records and sev-eral low-frequency earthquake’s records over many stations, Shellyet al. (2007) concluded that the non-volcanic, deep, low-frequencytremors beneath the Shikoku, Japan, can be explained as a swarmof low-frequency earthquakes. However, it is difficult to concludethat both phenomena had the same mechanism, because the cross-correlation values calculated for a single station is not very higheven though both occurred in the same region. To confirm whetherthe non-volcanic, deep, low-frequency tremors and low-frequencyearthquakes represent the same phenomenon, we applied the ADSto the low-frequency earthquake’s data. The obtained frequency ver-sus decay rate plots are shown in Fig. 5. Note that the low-frequency

earthquakes are represented as short-term isolated phases, and thelengths of the successive stationary time series become shorter. Al-though the result shown in Fig. 5 becomes less accurate, the obtainedfrequency structure is almost the same as that of the non-volcanic,deep, low-frequency tremors except that the intensity of the peaksnear 2.2 Hz is slightly stronger. This coincidence suggests that thesame physical mechanism is behind both phenomena. The spectralpeaks lined up from 1 to 5 Hz at intervals of 0.5 Hz seems to suggesta harmonic oscillation in the source, but it is not clear at the momentwhat kind of physical process can excite such oscillation in the deepsubduction zone.

In 2006, we deployed a seismic array observation at Aichi Pre-fecture with Nagoya University to help us avoid the problem ofa low signal-to-noise ratio of non-volcanic, deep, low-frequencytremors (Nakamichi et al. 2006). The array observation allows aslant-stacking process that greatly improves the signal-to-noise ra-tio. Using the slant-stacking process for a fixed slowness vector canremove not only the incoherent noises but also the noises that arecoherent but have other slowness vectors. This is effective for reduc-ing the microseism caused by the ocean. Fig. 6 shows a comparisonof the non-volcanic, deep, low-frequency tremors, earthquake andbackground noise spectra. The spectrum of the slant-stacking traceof the non-volcanic, deep, low-frequency tremors shows a clear re-duction of the noise higher than around 1 Hz. This reduction couldmake a clear distinction between the spectrum of the tremors andthe spectrum of background noises. An inflection point at 1.5 Hzclearly distinguishes the two spectra. We believe that the right-sidepart of the tremor spectrum from the inflection point, which has a

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Figure 5. The frequency versus decay rate plot of low-frequency earth-quake’s records at KWBH. The periods were selected from 2002 and 2004.Arrows indicate the spectral peaks identified in the ADS spectrum. Longi-tudinal broken lines in each panel represent 2 and 4 Hz, respectively.

flat nature until about 5 Hz and then decreases mildly, was causedby the non-volcanic, deep, low-frequency tremor signals. This noisereduction also allows us to find characteristic peaks in the frequencyrange. We note that there is a spectral peak at 1.8 Hz, which wasobserved in the non-volcanic, deep, low-frequency tremor spectrabut not in the spectra of the earthquake and background noise. Thispeak was also found in the ADS shown in the lower panel of Fig. 6,where relative high intensity ADS spectrum concentrated in a nar-row region. We recognize two other spectral peaks in the tremorspectra pointed by blue arrows. The corresponding frequencies inthe ADS spectrum are also represented by blue arrows in the lowerpanel of Fig. 6. There were several ADS spectral peaks with strongintensity, shown by black circles, but the concentration of the ADSspectral peaks was ambiguous due to lack of available data. Thealignment of the peaks at 0.5 Hz intervals in the Shikoku regionwere not observed clearly in the ADS spectrum calculated fromthe non-volcanic, deep, low-frequency tremor records in Aichi Pre-fecture. The peak of the maximum spectral intensity observed inthe Shikoku region around 1.8 Hz is consistent with that observedin Aichi Prefecture. The spectral peak around 2.1 Hz for both theordinary earthquakes and the tremors seems to be caused by thecrustal structure beneath the array observation network. We cannotnegate this possibility because only one array observation was avail-able. However, the spectral peak at 1.8 Hz observed in the tremorswas not found for both the earthquakes and the background noises.This indicates that this particular spectral peak was considered toresult from the non-volcanic, deep, low-frequency tremors in AichiPrefecture.

Shelly et al. (2007) showed a spectrum stacked for long stretchesand over the Hi-net stations and concluded that there are no char-acteristic peaks in the non-volcanic, deep, low-frequency tremorsignals. However, they investigated only the whole shapes of the

0.5

0.0

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Figure 6. Comparison of non-volcanic, deep, low-frequency tremor, earth-quake, and background noise spectra. The red solid line represents thespectrum of the non-volcanic, deep, low-frequency tremor signals obtainedby the slant-stacking trace of HOU array. The grey sold line shows the av-eraged spectrum of 33 normal intraplate earthquakes observed at the array.The black dotted line indicates the spectrum of background noise calculatedfrom 19:00 to 20:00 on 2006 July 15. All of the spectra in this figure are thedisplacement spectra. A green longitudinal broken line shows an inflectionpoint at 1.5 Hz. A red and two blue arrows represent the spectral peaks foundin the non-volcanic, deep, low-frequency tremor spectrum. A grey brokenline indicates the overall trend of the non-volcanic, deep, low-frequencytremor spectrum higher than 1.5 Hz. The lower panel shows the frequencyversus decay rate plot of the non-volcanic, deep, low-frequency tremor sig-nals. A spectral peak exists at around 1.8 Hz, indicated by an red arrow. Twoblue arrows correspond to the two higher spectral peaks in the non-volcanic,deep, low-frequency tremor spectrum shown in the upper panel.

spectra and neglected the small peaks as artefacts. As describedearlier, the overall trend of the spectrum were also observed in ouranalysis of the array-stacked spectrum. The characteristic frequencystructure and peaks that we discovered in this study all overlappedin this overall shape of the spectrum. Thus, the characteristic peakswere not highly visible as characteristics of the overall shape. Thischaracteristic frequency structure may be difficult to distinguish

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from the normal spectrum. However, a more detailed investigationcould confirm the characteristic peaks from the array-stacked spec-trum and the ADS. Such an investigation would be a useful tool forthe purpose of obtaining the characteristic frequency structure.

7 C O N C LU S I O N S

Through the application of the ADS method to the continuousrecords observed at the Hi-net station in western Shikoku region, thecharacteristic frequency structure of the non-volcanic, deep, low-frequency tremor with peaks lined up from 1 to 5 Hz at intervals of0.5 Hz was obtained. The peak of maximum spectral intensity wasdetermined around 1.8 Hz, which is consistent with that derived atAichi Prefecture. The ADS of the low-frequency earthquakes in theShikoku regions showed that the non-volcanic, deep, low-frequencytremors and the low-frequency earthquakes in western Shikoku re-gion have the same frequency characteristics. This is one piece ofevidence that the non-volcanic, deep, low-frequency tremor andlow-frequency earthquake have a common source mechanism.

A C K N OW L E D G M E N T

This research was supported by Grant-in-Aid for JSPS Fellow (No.17·11814) and Grant-in Aid for International Scientific Research(B) (No. 14340127) from the Ministry of Education, Culture, Sports,Science and Technology, Japan.

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Nakamichi, H. et al., 2006. Array observations of deep low-frequency non-volcanic tremor beneath Tokai area, central Japan, Eos Trans. Am. geo-phys. Un., 87(52), Fall Meet. Suppl., Abstract V41A-1698.

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