Pinatubohavebeenanalyzedusingbothmethods.

Among a variety of spectrogram methods Short-Time Fourier Transform (STFT) and Continuous Wavelet Transform (CWT) were selected to analyse transients in non-stationary tremor signals. Depending on the properties of the tremor signal a more suitable representation of the signal is gained by CWT. Three selected broadband tremor signals from the volcanos Mt. Stromboli, Mt. Semeru and Mt. Pinatubo were analyzed using both methods. The CWT can also be used to extend the definition of coherency into a time-varying coherency spectrogram. An example is given using array data from the volcano Mt. Stromboli.


Introduction
It is most common to represent the information content of a physical quantity either as a function of time or as a Fourier spectrum.However a function of time does not show the spectral content of the signal while the Fourier spectrum cannot extract temporal variations of the signal.A time-frequency representation provides a trade-o between both representations.In the following two approaches to computing a timefrequency representation are compared.

STFT and CWT
The STFT is an integral transform of a signal f(t) de ned as S !; f(t)] = Z 1 ?1 f(t)w(t ?)e ?j!t dt (1) = e ?j! Z f(t)w(t ?)e ?j!(t? ) dt: (2) Equation ( 1) may be interpreted as a Fourier spectrum of a signal seen through a sliding window w(t ?).The position of the window indicates the approximate time for which the spectrum is valid.Side-lobes, leakage e ects and frequency resolution are in uenced by a proper choice of the window function.The time and frequency resolution can be adjusted by modifying the window length t w .The smallest measurable frequency is determined by the window length f min = 1=t w .
On the other hand equation ( 2) can be identi ed as a convolution.It can therefore be interpreted as an in nite channel lter bank (Vaidyanathan, 1993), (Nawab and Quatieri, 1988), see gure 1.The impulse responses h i (t) in gure 1 di er only in a modulation factor e ?j!t .They can be regarded as modulated versions of a prototype low pass impulse response function which is the window function itself.In the frequency domain it is obvious that their transfer functions Ĥi (!) have equal bandwidth but shifted center frequencies depending on the modulation factor.By multiplication with e ?j!t the ltered signal  is shifted into its base band which is equivalent with the computation of the envelope.
Thus we can see that the STFT represents a timefrequency analysis with xed resolution in the time and frequency dimension.In general it does not seem possible to nd a trade-o between time and frequency resolution which is well-adapted to tremor signals from volcanos in their entire frequency range.The CWT which transforms the signal in a time-scale plane called scalogram o ers a more suitable tradeo .The CWT is de ned as where (t) is called a wavelet function when it ful lls the admissibility condition 0 < Z j ^ (!)j 2 j!j d! < 1: (4) Where ^ (!) = Z 1 ?1 (t)e ?j!t dt (5) is the de nition of the Fourier transform and is the conjugate complex of .For a detailed representation of the CWT see (Daubechies, 1992).The parameter a is called the scale which controls the time duration of the wavelet and is again a translation parameter.As can be seen in (4) a wavelet is an oscillating function with mean value equal to zero.
In gure 3 the functions h i (t) = (?t=a) give an example of a wavelet and shows its dependence on the scale.Since (3) represents a scalar product it is obvious that when a trends to 1 the CWT measures signal features on large time scales, and when a trends to zero it maps signal features on ne time scales.Thus it is able to zoom in on the signal.This property makes the CWT a useful tool for the detection of singularities.beginning and end of the brief harmonic event can be resolved well using the CWT.On the other hand, the frequency of the brief harmonic event is measured more precisely using the STFT.
In analogy to the STFT, the CWT's frequency resolution properties can be investigated through a lter bank view.When (3) is interpreted as a convolution formula then the CWT can be identi ed as a nonuniform in nite channel lter bank, see gure 3. The impulse responses h i (t) are derived by dilatation of a prototype band pass impulse response function (?t).In contrast to the STFT their durations depend on the scale a.Because of the dilatation property of the Fourier transform, their corresponding transfer functions Ĥi have the property f i =f i = constant (const.Q-analysis), where f i is the bandwidth and f i the center frequency of Ĥi , see (Bartosch, 1996).Thus when the scale tends to 1, the frequency resolution becomes ner.The converse is true when the scale tends to zero.
Two common types of wavelets are depicted in -gure 2. The Mexican-hat wavelet (t) = (1 ?2t 2 )e ?t 2 (6) is real valued, while the Morlet wavelet (t) = (e ?t 2 =2 ?p 2 e ?b 2 b =4 e ?t 2 ) e jbbt : ( 7 a suitable choice of the wavelet we can therefore interpret the time-scale relationship in terms of time and frequency.In this case the constant Q property implies a logarithmic frequency axis.A time-frequency representation is generated by mapping the tiles corresponding to the coe cients S !; or W a; .Figure 6 shows the tiling of the plane for the STFT and CWT.The size of the tiles is dened by their duration T and the bandwidth B of the related transformation kernels, h !; = w (t ?)e j!t and h a; = ((t?)=a).The tiling properties of the time-frequency plane can be summarized as follows, see (Bartosch, 1996) For computing the STFT and CWT, (2) and (3) must be discretized.See (Nawab and Quatieri, 1988), (Vaidyanathan, 1993) for discrete time STFT and (Daubechies, 1992), (Chan, 1995) for discrete time WT.Our approach to computing a time-frequency representations is based on a dense lattice ( b << T; f << B) when calculating W a;b .Therefore the coe cient map W is plotted instead of the true tiles.
The result is an interpolated spectrogram.
The STFT and the CWT using a Morlet wavelet with 12 cycles (b b = 12) are calculated of a synthetic signal and are pictured in gure 4.This gure illustrates the di erences between the representation of special signal features of the two transforms in the time-frequency plane.The signal is well adapted to the properties of the CWT, as it consists of a large scale (long duration) low frequency part (at 0.2Hz) and small scale high frequency parts (pulse and switch on/o transients).In case of the STFT, the spike can hardly be recognized but the high frequency event is measured with better frequency resolution.
3 Signal Analysis of selected Tremor Signals Besides analyzing selected tremor signals, our intention is to emphasize di erences between spectrogram analysis using STFT and CWT.For better comparison only one STFT spectrogram is computed with a rather large window length in order to include the lowest of the displayed frequency decades.An ingenious application of STFT analysis would be carried out by calculating the STFT separately with a window length adapted to each frequency decade.

Stromboli Tremor
Mt. Stromboli is located on the island Stromboli in the Tyrrhenian Sea.Our analysis is based on data recorded in 1995 by J. Neuberg during an array measurement with Guralp broadband seismometers.Typically tremor at Mt. Stromboli is continuous tremor in several dominant sub-bands with center frequencies above 1Hz and has superimposed broadband shock events.Figure 5 gives a 250sec long example.The signal consists of two visible shock events, continuous tremor during the whole displayed time and a series of spikes at 70 < t < 100sec.
In order to study transients during the occurrence of shocks, spectrogram analysis based on STFT (left) and CWT (right) is applied to the data.Both transforms can reveal the dominant frequency bands of the continuous tremor but the CWT gives a more clear representation of the approximate bandwidth.The rst shock event contains frequencies spread out over the entire range.The STFT resolves narrow fre-quency bands above 3Hz (arrows 1) while the CWT resolves the temporal behavior far better (arrows 3).The CWT gives a blurred representation in time of the shock at lower frequencies but reveals some more complex structures along the frequency axis (arrows 2).Below 7 10 ?2 Hz the window length of the STFT is not large enough to resolve three spectral lines which are correctly displayed in case of the CWT (lower three arrows (2)).The series of spikes between 70 < t < 100sec is resolved in case of the CWT at f > 10Hz.The STFT blurs their temporal behavior because of the long window length.In the CWT a second spike series can be recognized at 200 < t < 250sec by comparison with the rst patterns.
Comparing the modulus of both spectrograms a  The fundamental and tree overtones can be recognized.The rst three spectral lines have onset at the same time while the 4th line consists of a jump in frequency at 25sec.di erent distribution of their maxima is found.The STFT attains its maximum in the low frequency part of the spectrogram (55sec, 5 10 ?2 Hz) while the CWT displays the shock in the lowest tremor band at (150sec, 1.5Hz) as its maximum.
A closer look at the low frequency part of the signal is displayed in gure 7 using a CWT with a Mexicanhat wavelet.Because the wavelet has fewer cycles the spectrogram has improved time and decreased frequency resolution compared to the CWT in gure 5. Thus, the harmonic sub-bands above 1Hz can not be separated.On the other hand, the low frequency signal content can be resolved with acceptable time resolution in a sub-band (5 10 ?3 < f < 5 10 ?2 Hz).The related (band-pass) time signal can be extracted by inverse transformation of a selected segment of the CWT.The bottom in gure 7 contains the  partially (1 10 ?3 < f < 0:15Hz) reconstructed tremor signal.Both shocks are associated with events on a small scale which have a steep slope at the onset superimposed to the event at large scale ( 100sec period).

Semeru Harmonic Tremor
Mt. Semeru is located in East-Java (Indonesia).In 1992 a German-Indonesian Volcano Expedition investigated the seismicity of the volcano.During the experiment a STS-2 3-component broadband seismometer was xed installed in 8km distance to the crater.Interesting events called harmonic tremor have been recovered, see (Hellweg et al., 1994), (Schlindwein et al., 1994).Based on data of this station we studied the transient phenomena at the onset of such an event.
In gure 8 the rst 80 sec of a 160 sec long harmonic tremor is plotted.The event consists of narrow band harmonic lines at frequencies above 0.5 Hz.Since the signal features all have similar scales, the STFT and CWT resolve approximately the same kind of information.For gure 8 a CWT with a narrow bandwidth Morlet wavelet (large b b ) to enhance the frequency resolution was applied to the event.The fundamental and overtones can be resolved.The fundamental mode is splitted into two nar- row lines.The spectrogram for the higher harmonics is not ne enough to determine if they follow similar behavior.The onsets of the rst three spectral lines at (0.9Hz,1.8Hz,2.7Hz)occur at the same time while the 4th spectral line jumps from f = 3:8Hz to its approximate overtone frequency of 3.6Hz at t = 25sec.By inverse transforming the spectral energy of each harmonic line in a surrounding sub-band and calculating its envelope we get the amplitude of each harmonic line as a function of time ( gure 9).The splitting of the fundamental mode can be recognized as a beating phenomenon in the bottom trace.No relation between the amplitudes of the di erent harmonic lines can be recognized.The rst and second harmonic lines seem to be delayed compared to the fundamental and third harmonic lines.
In 1995 a second expedition organized by J. Neuberg, R. Schick observed the volcanos Mt.Semeru and Mt.Bromo on East-Java and Gunung Batur on Bali (Gottschaemmer, 1998).Our search for harmonic tremor events within a whole day of data from two stations located at Mt. Semeru has been unsuccessful.A source mechanism suggested in (Schlindwein et al., 1994) might have been destroyed during large eruptions.

Pinatubo Tremor
In gure 10 a 22.2h long gravity signal recorded at Black Forest Observatory (Schiltach) with an ultra long period ET-19 seismometer is displayed.Too narrow low frequency spectral lines are attributed to a crisis at Mt. Pinatubo volcano, see (Widmer and Z urn, 1992).
In addition to the signal from the volcano, there are several seismic events on very di erent scales.As can be seen by comparison with the STFT in gure 10 the CWT is a more appropriate tool for spectrogram analysis of this data.
The signal begins with several Rayleigh wave trains (R1-R4) which were generated by remote earthqua-kes.The time resolution is ne enough to resolve the dispersion phenomena typical for such wave trains.
The volcanic signal is represented by the two narrow harmonic lines at 3:7 10 ?3 Hz and 4:4 10 ?3 Hz.The spectrograms are adjusted to have approximately the same time and frequency resolution in this region.Through the zoom property of the CWT, several shock events which are superimposed on the two harmonic tremor lines can be resolved.The two strongest ones also show dispersion e ects.
A large scale event which is related to an approaching local cold front at the 7 10 4 < t < 8 10 4 sec can be well localized by the CWT (f 2 10 ?4 Hz).In the STFT the event is blurred in time.
The rst shock event at 7:1 10 4 sec appears quite similar compared to two shock events around 3:9 10 4 sec and is related to a local earthquake while the second event at 7:25 10 4 sec might be caused by the seismometer itself.Its representation in the CWT spectrogram looks very similar to the CWT of the pulse in gure 4.
A comparison between the STFT and the CWT spectrogram shows that the CWT is well adapted to the signal over the entire time-frequency plane.

Local coherency of Stromboli Tremor
The coherency is de ned as C = Sxy q Sxx Syy (8) where Sx;x ; Sy;y and Sx;y are the smoothed power and cross spectra S x;x ; S y;y ; S x;y of the stationary processes x and y.In case of non-stationary processes this de nition is inapplicable as the power and cross spectra become time-dependent.In (Liu, 1994) the de nition of ( 8) is extended in order to be an estimate for non-stationary time series by introducing the CWT for the de nitions of the power and cross spectra S xx = W a; x(t)] W a; x(t)]; S yy = W a; y(t)] W a; y(t)]; S xy = W a; x(t)] W a; y(t)] as time-depended estimators.Translating the scale into a frequency parameter f and using a logarith- mic mapping for f gives a constant resolution along the frequency axis.Smoothing along the frequency axis can therefore be performed by convolution with a constant-length window function.As we implemented our CWT with a constant grid for the translation parameter b in order to get rectangular maps W a; , we must adapt our window length for smoothing along the time axis which depends on the scale.The window length is then W l = W linit =f.Each row of the map C has to be convolved with a window of length w l .The solid line in gure 11 shows w l (f) in sec as a function of frequency.
In gure 11 recordings of time series of tremor with superimposed shock events are plotted for two stations FOS and OBS located near the craters of Mt.Stromboli along with their local coherency.In the frequency band above 0.3Hz the coherency pattern is similar for all shocks.Also the the coherency around 0.2Hz (sea micro-seismic) brakes down during the occurrence of the shocks.For a more detailed interpretation of local coherency applied on data from Mt. Stromboli see (Wassermann, 1997).

Conclusions
The spectral analysis of 3 di erent tremor signals show that the advantages of the CWT compared to STFT depend highly on the properties of the signal itself.The CWT proved to be useful for analyzing signals which consist of information in very di erent time scales, e.g Pinatubo tremor.The CWT can provide spectrograms presenting the spectral information over several frequency decades resulting in an improved insight into the nature of broadband signals.
Using an array of stations the WT can be applied to the determination of the local spatial coherency as a function of time and scale.

Figure 1 :
Figure 1: Interpretation of the STFT as an in nite channel lter bank.The impulse responses of three channels of the lter bank are pictured in the time and frequency domains.The hi's have equal support length while the Hi' have equal bandwidth.The STFT represents a xed resolution time-frequency analysis.

Figure 2 :
Figure 2: Mexican-hat wavelet and Morlet wavelet in the time and frequency domain.

Figure 3 :
Figure 3: Interpretation of the CWT as a non-uniform in nite channel lter bank.The impulse responses are pictured in the time and frequency domain.They are derived by dilatation of a prototype wavelet function.Therefore their support length and bandwidth depends on the scale parameter.tThe time and frequency resolution is varying.

Figure 4 :
Figure 4: STFT (left) and CWT (right) of a synthetic signal consisting of a Hanning windowed 0.2Hz harmonic signal, a spike at 10sec and a switched (30 < t < 33sec) 10Hz harmonic event.The signal is adapted to the properties of the CWT.The pulse and the

Figure 6 :
Figure 6: Tilling of the time-frequency plane of the STFT (left) and CWT (right).

Figure 7 :
Figure 7: CWT with a Mexican hat wavelet for studying the low frequency part of the signal.As the wavelet is real and has few cycles the transform resolves an oscillating and well separated time pattern.The very last boy shows the reconstructed signal which belongs to the low frequency features of the signal (band pass ltered signal).

Figure 8 :
Figure 8: CWT of a harmonic tremor event from Mt. Semeru.

Figure 9 :
Figure 9: shows the envelope of the reconstructed spectral lines in a sub-band with 0.8Hz bandwidth around the center frequencies.

Figure 11 :
Figure 11: Local coherency of two seismic signals recorded at FOS and OBS station at Mt. Stromboli near the craters.All shock events consist of similar correlation pattern especially at frequencies above 1Hz while the correlation of the sea micro-seismic (0.2Hz) breaks down during the occurrence of a shock.The solid line indicates the duration of the smoothing window.