Jones 2012 Decomposition Volcanic Tremor
-
Upload
christiam-morales-alvarado -
Category
Documents
-
view
216 -
download
0
Transcript of Jones 2012 Decomposition Volcanic Tremor
-
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
1/18
Subband decomposition and reconstruction of continuous volcanic tremor
J.P. Jones a,, R. Carniel b, S.D. Malone a
a University of Washington, Department of Earth and Space Sciences, Seattle, WA 98195-1310, USAb Laboratorio di misure e trattamento dei segnali, DICA, Universit di Udine, Via delle Scienze, 206-33100 Udine, Friuli, Italy
a b s t r a c ta r t i c l e i n f o
Article history:
Received 29 January 2011
Accepted 12 July 2011
Available online 3 August 2011
A new method of analyzing volcanic tremor is presented, which uses properties of undecimated wavelet packet
transforms to lter, decompose, and recover signals from continuous multichannel data. The method preserves
manystandardpropertiesthat areused to characterizetremor, suchas waveeldpolarization andseismic energy.
In this way, we can better understand the (potentially many) seismic sources that combine to form continuous
volcanic tremor, and we can specically address the problem of what causes changing tremor spectral content.
Tests on synthetic data suggest that SDR can recover multiple quasi-continuous signals that differ from one
another by an order of magnitude, even in noisy environments. Tests on real data recorded at Erta 'Ale in 2002
suggest thatSDRcan recover signals with geophysically meaningful interpretations, and corroborates existing
seismic and multiparametricworkby Harriset al.(2005),Joneset al. (2006), and Harris (2008). We suggest that
this algorithm could effectively detect subtle changes in the time-frequency content of volcanic tremor, and
recover signals from real seismic sources that appear buried in background noise (and/or partly masked by one
another). Such an algorithm could allow volcanologists much greater insight into the dynamics of volcanic
systems, and could detect subtle signals that might help address the possibility of unrest.
2011 Elsevier B.V. All rights reserved.
1. Background and motivation
Continuous volcanic tremor is a persistent seismic signal observed
near active volcanoes, which generally lacks impulsive phase arrivals,
and can persist on timescales as long as several years (Chouet, 1996;
Konstantinou and Schlindwein, 2002). It has been observed since the
beginning of volcano monitoring: a few examples of volcanoes with
continuous tremor include Kilauea, Hawai'i (e.g. Goldstein and
Chouet, 1994), Etna, Italy (e.g.Di Lieto et al., 2007), Ambrym, Vanuatu
(Carniel et al., 2003), Stromboli, Italy (e.g. Ripepe et al., 2002), and
Erta 'Ale, Ethiopia (Harris et al., 2005; Jones et al., 2006). Recently, it
has been found that the spectral content of continuous volcanic
tremor can undergo relatively abrupt, characteristic changes on
timescales of hours to weeks, suggesting either a changing source
mechanism for the tremor, or additional seismic sources generating
signals that affect the seismic wave eld: this has been seen at e.g.
Stromboli, Italy (Ripepe et al., 2002), Ambrym, Vanuatu (Carniel et al.,
2003), Erta 'Ale, Ethiopia (Harris et al., 2005; Jones et al., 2006), and
Dallol, Ethiopia (Carniel et al., 2010). This is quite different from
gliding spectral lines observed in tremor data recorded at volcanoes
that erupt explosively, e.g. Montserrat (Powell and Neuberg, 2003)
and Lascar (Hellweg, 2000), in which spectral content changed over
relatively short timescales, but in a continuous (rather than abrupt)way.
The changing spectral content of continuous volcanic tremor
presents a challenging problem, but is also a potentially useful
diagnostic tool, particularly if its relationship to eruptive behavior can
be quantied. Clearly, because spectral transitions are seen to some
degree at each station in each seismic network, it follows that the
tremor source must change somehow. However, it is difcult to
determine from modeling alone whether tremor has a single,
changing source mechanism, or is rather a composite of multiple
source processes with differentspectral energies. It is usefulto resolve
this ambiguitybecause continuous tremorhas been shown to relateto
changes in eruptive behavior at some volcanoes (Ripepe et al., 2002;
Harris et al., 2005), and because this sort of changing spectral content
has been associated with changes in hydrothermal systems (Carniel
et al., 2010), hydrocarbon reservoirs (Dangel et al., 2003), and slow
slip in subduction zones (Obara and Hirose, 2006). Identifying the
physical processes that drive these observed spectral changes could
thus help volcanologists identify, understand, or even predict,
changes in volcanic activity at these systems.
The goal of this work is to develop a quantitative means of
determining whether changing spectral content in continuous volcanic
tremor represents a change in the tremor source mechanism, or an
introduction of new, superimposed, seismic sources with different
spectral peaks. More generally, we wish to introduce a quantitative
means of tracking the signal content of continuously recorded geophys-
ical data. Ideally, if the data are to be treated as a composite of several
Journal o f Volcanology and Geothermal Research 213-214 (2012) 98115
Corresponding author. Tel.: +1 206 930 8065.
E-mail address:[email protected](J.P. Jones).
0377-0273/$ see front matter 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.jvolgeores.2011.07.006
Contents lists available at ScienceDirect
Journal of Volcanology and Geothermal Research
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j vo l g e o r e s
http://dx.doi.org/10.1016/j.jvolgeores.2011.07.006http://dx.doi.org/10.1016/j.jvolgeores.2011.07.006http://dx.doi.org/10.1016/j.jvolgeores.2011.07.006mailto:[email protected]://dx.doi.org/10.1016/j.jvolgeores.2011.07.006http://www.sciencedirect.com/science/journal/03770273http://www.sciencedirect.com/science/journal/03770273http://dx.doi.org/10.1016/j.jvolgeores.2011.07.006mailto:[email protected]://dx.doi.org/10.1016/j.jvolgeores.2011.07.006 -
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
2/18
signals, then any such method should recover some of the standard
parameters about each component signal, e.g. polarization (Montalbetti
and Kanasewich, 1970; Vidale, 1986; Jurkevics, 1988), and signal energy,
which can be used to locate a tremor centroid (e.g.Gottschmmer and
Surono, 2000; Patan et al., 2008).
2. Subband decomposition and reconstruction
It canbe challenging to isolate what signals comprisethe tremordatawithout makingad hocassumptions about the nature of the data. The
general concept of blind source separation methods is not new to
volcano seismology, and waspreviously used by e.g. Cabras et al. (2008)
to study low-frequency tremor at Merapi, Indonesia. Volcanic data can
be quite complex, often consisting of transients superimposed on a
continuous background signal. If LP or VLP events are present, classical
methods based on Fourier analysis can perform poorly, because the
notion of statistical stationarity may not apply. We wish to develop an
algorithm based on more robust wavelet methods, beginning with the
assumption that there exists a subband or subbands (f1, f2) of the
frequency spectrum (0, fN) where seismic energy at some stations is
dominated by energy from a single seismic source. From this, we will
introduce a simple, efcient, top down algorithm based on the
undecimated (a.k.a. maximal overlap) discrete wavelet packet
transform (Walden and Cristan, 1998; Percival and Walden, 2000).
Our algorithm makes use of subband selection methods similar to
the top-down best basis approach of Taswell (1996), based on
Coifman and Wickerhauser (1992), which generally allows for basis
selection without computing a full wavelet packet table. It has several
similarities to algorithms proposed for medical data by Oweiss and
Anderson (2007), though the subband selection criteria are dened in
a way more appropriate for geophysical data sets. We follow a simple
3-step procedure:
1.) Determine the wavelet decomposition in which a single
principal component of the multichannel data most complete-
ly dominates each subband comprising some part of the
frequency spectrum 0 ffN.
2.) Hierarchically cluster the wavelet packets using quantitativeinformation about the signal content of each.
3.) Inverse transform the wavelet packets of each cluster, thereby
ltering the input data to frequencies where the same signal
dominates the seismic energy.
3. Wavelet decomposition and representation
For purposes of this manuscript we use the following notation:
Assume that non-boldface variables(e.g.N) refer to scalars, andboldface
variables (e.g. h) refer to vectors. Assume that uppercase boldface
variables (e.g. Xt) refer to N-length vectors of (or derivedfrom) a single
data channel withNdata points, and that uppercase boldface variables
with a bar (e.g. Xt) refer toNKmatrices of (or derived from) multi-
channel data, havingKchannels andNdata points.Let usrstreview some relevant principles of wavelettransforms for
a single time series Xt, i.e. a single data channel at a single seismic
station. The discrete wavelet transform, or DWT, of an input time series
Xtto some levelJis an orthonormal transform that uses a sequence ofltering operations to obtain wavelet coefcients Wj and scaling
coefcients Vj, associated with weighted differences on scales of
j= 2j1. It captures information about both the frequency and
temporal contents of the input data; this is contrasted with Fourier
transforms, which characterizes the amplitude and phase of the
frequency content only, assuming a stationarity in time that is not
always satised in practice. Whereas each coefcient of a Discrete
Fourier Transform (DFT) is associated with a particular frequency, each
DWTcoefcientWj(t) or scaling coefcientVj(t)canbethoughtofasthe
difference in adjacent averages ofXton scales ofj=2j1
, centered
about some time t. We can succinctly write Wj and Vj as circularlylteredconvolutions ofXtwitha waveletlter hl or scalinglter gl using
the recursion relations
Wj t = L1
t= 0hlVj1;2t+1 = l mod Nj1 Vj t =
L1
t=0glVj1;2t+ 1 = l mod Nj1
1
and by denition V0 Xt. Herelis the index of the coefcient in a lter(hj orgj), L is lterwidth,and Nis the length of theinput vectorXt.Fora
more in-depth review of the meaning of wavelet and scaling
coefcients, see e.g. Chapter 4 of Percival and Walden (2000). A
complete description of theDWT is givenin Strang (1993), with detailed
discussion in Daubechies (1992) and Chui (1997). Wassermann (1997)
previously used the DWT to characterize and locate volcanic tremor at
Stromboli.
A common extension of the discrete wavelet transform is the so-
called maximal overlap discrete wavelet transform (MODWT), which
is equivalent to the undecimated, stationary, translation invariant, or
shift invariant DWT (Greenhall, 1991; Shensa, 1992; Percival and
Guttorp, 1994; Coifman and Donoho, 1995; Nason and Silverman,
1995; Liang and Parks, 1996; Percival and Mojfeld, 1997). The
MODWT carries out
ltering steps nearly identical to the DWT, onlywith no decimation (down sampling by 2) at each successive level j.
Several useful mathematical properties of the MODWT are described
inWalden and Cristan (1998) and Percival and Walden (2000). Here,
we list only those relevant to this research:
I. Because the MODWTdoes not down sample, the length Nof an
input time series Xtneed not be a power of 2. (Percival and
Guttorp, 1994)
II. Inverse transforming the MODWT coefcientsWj creates the so-
calleddetail coefcients ordetailsDjand smooths SJthat
form a multi-resolution analysis ofXt; that is,
Xt= SJ+ J
j =1Dj
III. Dj and SJare associated with zero phase lters (Percival and
Mojfeld, 1997). Thus the pass band of the detail coefcientsDjassociated with each wavelet coefcient vector Wj is a well-
dened subband of the frequency range [0,fn]. For data sampled
at Hz, the exact corner frequencies (in Hz) of the subband
whose detail coefcients are Djare given by * 2 (j+1)
|f| *2j (Percival and Walden, 2000).
IV. The MODWT preserves energy. For a MODWT of an input time
series Xtto any level J, VJ2
+ J
j =1Wj
2=Xt
2(Percival
and Walden, 2000). This property is not shared by the detail
coefcients, which follows from II. and the Schwarz inequality.
V. Wavelet transforms can estimate the time-varying cross-correlations of 3-component seismic data in a subband of the
frequency spectrum, and hence its polarization and principal
components. (Lilly and Park, 1995; Anant and Dowla, 1997;
Oweiss and Anderson, 2007).
4. Maximal overlap discrete wavelet packet transform
We now briey review some relevant properties of the maximal
overlap discrete wavelet packet transform (MODWPT) ofWalden and
Cristan (1998). This extension of the discrete wavelet transform,
while common in the signal processing literature, is relatively
unknown to volcano seismology. The MODWPT generalizes the
DWT by recursively ltering an input time seriesXtwith all possible
combinations of the (rescaled) wavelet lterh l and scaling lter gl,
99J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
-
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
3/18
withoutdownsamplingWj,n by 2 at each successive transform levelj. A
sample selection oflters used to create the MODWPT to levelj =2 is
given in Fig. 1, along with their squared gain functions at unit
sampling frequency. Observe that, for data sampled times per
second, each wavelet packetWj,nis now associated with frequencies
in the nominal pass band
n2j + 1
bf n + 1
2j +1 :
2
It is clear fromFig. 1 that the MODWPT forms an overcomplete
representation of the frequency range [0,fn]. The idea behind the
MODWPT is thus to compute a generalized, highly redundant table of
non-decimated wavelet packet vectors Wj;n. For each successive levelj
of the MODWPT, one lters each wavelet packet Wj1;nwith (rescaled
wavelet or scaling) coefcients, un;lhn;l=ffiffiffi
2p
or un;lgn;l=ffiffiffi
2p
, to create
vectors Wj;2n and Wj;2n + 1, respectively. At each successive level j, we
can write the equivalent recursive ltering operation to obtain Wj;nas
Wj;n t = Lj1
l = 0
un;l Wj1;
n
2
j k;t2j1l mod Nj1
3
where we dene W0;0Xtand un;l as above.
Using this more general notation, theMODWTcoefcientsWj (and
corresponding detail coefcients Dj) correspond to wavelet packetsWj;1 for any level j. The scaling coefcients VJand corresponding
smooths SJ correspond to wavelet packet WJ;0 and its inverse
transformed detail coefcients DJ;0. For convenience, as the rest of
this manuscript deals exclusively with the MODWPT, we will write a
generalized wavelet packet Wj;n as Wj,nand its corresponding detail
coefcients Dj;n as Dj,n. Henceforth we assume non-decimation.
Percival and Walden (2000) showedthat any complete partitionof
the frequencies [0, fN] using wavelet packets Wj,nis an orthonormal
transform. Thus, any MODWPT basis shares properties IV of the
MODWT. The collection of all wavelet packets for all levels 1 jbJis
called a wavelet packet (WP) table.
5. Wavelet basis selection for multichannel data using
the MODWPT
We canextract many differentorthonormal transformsfrom a WP table.
Many algorithms and cost functionals have been devised to determinethe best wavelet packet basis for singlechannel data (e.g. Coifman and
Wickerhauser, 1992; Taswell, 1996; Oweiss and Anderson, 2007). Such
algorithms select parts of the wavelet packet tree that evaluate the
characteristics of input data Xt using some cost functional m(Wj,n),
which is associated with each wavelet packet vector Wj,n. The wavelet
basis that satises
minC
j;n C
m Wj;n
4
is thebestrepresentation of the data in the wavelet domain.
Unfortunately, algorithms applicable to a single input time series
are not necessarily generalized to multichannel data in any straight-
forward way. For the example of geophysical data, the channels (i.e.
stations) nearest the source generally have the highest associated
cost, and therefore most signicantly affect the calculation. In the
ideal case of a single seismic source recorded by multiple receivers,
with no glitches, transients, or additive noise, wavelet packet vectors
are weighted for each station by the samepower law as the falloff rate
of the energy in the frequency range of Eq. (2). However, this means
that thebest basiscan be skewed by just one channel of bad data.
Additionally, with multiple sources having different relative energies
at each station, summing costs for each node of the wavelet packet
table over each data channel is notnecessarily an appropriate method.
The problem of determining a best wavelet decomposition for
multichannel data was partially addressed byOweiss and Anderson
(2007), who derived a multichannel cost functional relating the eigen
W(1,1)
0 1/16 1/8 3/16 1/4 5/16 3/8 7/16 1/2
Normalized Frequency [Hz]
W(1,2)
W(2,1)
W(2,2)
W(2,3)
W(2,4)
MODWPT Filter Squared Gain
W(1,1)
0 1/16 1/8 3/16 1/4 5/16 3/8 7/16 1/2
Normalized Frequency [Hz]
W(1,2)
W(2,1)
W(2,2)
W(2,3)
W(2,4)
MODWPT Filter Squared Gain
Fig. 1.Wavelet packet lters and corresponding normalized squared gain functions that create a MODWPT to level j =2. Left: The LA16 wavelet. Right: The Daubechies-2 or Haar
wavelet.
100 J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
-
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
4/18
decomposition of each (multichannel) wavelet packet matrixWj;n to
that of the original multichannel data Xt. Their approach was designed
to guarantee that thebestbasis isalsothebestt of thewavelet packet
transform to the input data, and is thus suitable for problems with
broadband sources whose spectra span [0,fn] and have additive noise.
However, geophysical data in general, and almost all volcano-seismic
data, have seismic sources that generate energy in a relatively narrow
frequency band. Due to the band-limited nature of volcanic tremor
sources (e.g. McNutt, 1996; Sherburn et al., 1998; Konstantinou andSchlindwein, 2002), and attenuation and geometrical spreading, the
meaningful part of the frequency spectrum of the recorded data almost
never spans the frequency range [0,fn]. Thus, for volcano-seismic data
sets, it isnotnecessarily true or desirable that the eigen decomposition
of the covariance matrix ofXtis a suitable match toeachsubband.
Rather than approaching the problem of wavelet basis selection
with the expectation that each Wj;n will match the eigen decompo-
sition Xt, we approach this problem with the more geophysically
appropriate expectation that we can nd a wavelet decomposition of
Xtwhose wavelet packetsWj;nare each nearly dominated by a single
principal component, corresponding to a single physical source,
whose mathematical representation is the observed signal content.
This notion is widely assumed in volcano-seismological literature, as
suggested in many of the references above.
Herewe denea cost functional with this goal inmind, for data from
K seismic stations. Whereas conventional cost functionals seek to
minimize an information cost, our goal is to select a basis using the
expectation that a single principal component dominates (each part of)
our observation matrixXtin some subband of [0,fn]. We can quantify
this expectation by making note of the following properties of the
principal components (cf. Pearson, 1901), i.e. eigenvalues j,n,k and
eigenvectorsvj,n,kof the covariance matrix of each wavelet packet:
A. For Gaussian data, the eigenvectors of the covariance matrix of
eachWj;npoint in the direction (in K-dimensional space) of the
independent components, i.e. each eigenvector represents the
relative strength of each independent component at each
seismic station. Even for non-Gaussian or multi-modal Gaussian
data, principal components analysis (PCA) de-correlates the axesof the independent components. (Hyvrinen et al., 2000)
B. The relative eigenvalues of the covariance matrix of eachWj,ncorrespond to the relative energies of the independent
components. (see e.g.Shaw, 2003).
The use of these properties is best illustrated with two conceptual
examples. First, an eigenvector of one wavelet packetWj,n, which aligns
almost exactly inK-space to a single station, and whose corresponding
eigenvalue is very large, could represent a local transient recorded only
at that station, in the frequency range given by Eq. (2). On the other
hand, if the energies of the eigenvalues of one wavelet packet Wj,nare
nearly equal, and the eigenvectors appearrandomlyoriented in K-space,
then it follows that the subband is dominated by either local transients
at each receiver, or by incoherent noise.Principal components analysis can produce erroneous results when
tracedataare out of phase, as the maxima at onestation mayalign intime
with the minima of another station. For this reason, and following from
propertyV of waveletcoefcients,one canalign themultichannel wavelet
packet coefcients in a least-squares sense following the algorithm of
Vandecar and Crosson (1990), prior to computing their principal
components. This introduces some potential for cycle skipping, however,
which must be controlled by carefully choosing a maximum lag for each
pairof stations. We remark that this slows the algorithm slightly, because
unshifted coefcientsare necessaryto computesuccessive levelsofWj,n in
iterative algorithms. However the added step is a necessary precaution.
From these properties of principal components analysis, we may
now dene a cost functional based on how well a single principal
component dominates a subband whose wavelet packet coefcients
areWj,n. ForKstations, whenj,n,1is large relative toj,n,2,j,n,K, we
expect the ratio
K
k = 2j;n;k
j;n;1
to be small. Recalling from Eq.(2)that eachWj,nis associated with a
normalized bandwidth of approximately /2j+1
, we can dene a costfunctional directly from the above ratio, which accounts explicitly for
this bandwidth:
M Wj;n
K
k =2j;n;k
j;n;1 2j +1
: 5
This cost functional behaves similarly to the entropy-based cost
functional ofCoifman and Wickerhauser (1992), but is bounded by
0 M(Wj,n)b(K 1)/2j+1. However, it must be noted that, like the
cost functional ofOweiss and Anderson (2007), this is not strictly anoptimizationproblem, as, by the very nature of geophysical data, the
principal components of the input data are not necessarily inherited
exactly by each subband.Using this cost functional in traditional best basistype algorithms
determines those wavelet packetsWj,nwhich are most dominated by a
single principal component. However, it is not necessarily true that the
same seismic source (or equivalently the same principal component) will
dominate each Wj,n. In fact, due to the notorious complexity of tremor
sources, it is likely that someWj,n, and their associatedpassband frequen-
cies(2)will be dominated by very different seismic sources than others.
Thus, we constrain our basis selection algorithm in the following
way. The similarity of the most energetic principal component in each
wavelet packet is easily quantied by measuring the distance d()
between eigenvectorsvj,n,k,vj,n+1,k, corresponding to adjacent wavelet
packets Wj,n and Wj,n+1. Note, however, that the distance between
eigenvectors must account for a possible sign change. Thus we compute
distance using the trigonometric formula
d vj;n;k; vj;n + 1;k
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi22vj;n;kvj;n +1 ;k
r
: 6
If this distance falls below some predetermined threshold, we say
that the dominant principal component of each wavelet packet vector is
the same. Thus, we apply thefollowing constraint when selectinga basis.
It is clear that the wavelet packet table forms a tree of nodes, with
each parent node Wj,n having two child nodes constructed via
circular convolution in Eq.(3),Wj+1,2n1andWj+1,2n. Thus, at each
parent nodeWj,nin the wavelet packet tree, for wavelet levels 1 jbJ,
we replace child nodes Wj+1,2n1, Wj+1,2n with their parent
nodeWj,nif the following two criteria are met:
1. M(Wj,n) M(Wj+1,2n1)+ M(Wj+1,2n)
2. d(vj+1,2n1,1,vj+1,2n,1)b.
By proceeding down (or up) the wavelet packet table, we use this
conditional basis selector to determine the wavelet decomposition in
which the subbands that span [0, fn] are most dominated by a single
principal component each.
6. Wavelet packet clustering and signal reconstruction
We now dene a recovered signal, using the properties of wavelet
transforms and principal components analysis. The wavelet packetsWj,nin the best basis can be hierarchically clustered using Eq.(6). Because of
Property III of the MODWT, each Wj,n in the best basis can be inverse
101J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
-
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
5/18
transformedto create zero-phasebandpassltered detail coefcients Dj,n.
Because of Property II, summing the (inverse transformed) detail
coefcients Dj,n in each cluster enables us to reconstruct our original
observation matrixXt, ltered exactly to those frequency bands where
each recovered signal dominates. Thus, we dene a recovered signal as
the sum of the detail coefcient vectorsDj,nfor allWj,nin one particular
group of wavelet packets in the best basis.
This nal step clustering, reconstruction, and summation
essentially treats the wavelet
ltersun,lin Eq.(3)as orthogonal zero-phase lter banks. To see why this is possible, recall fromWalden and
Cristan (1998)that wavelet ltersun,l are orthogonal, and consider
the single-channel case Xt. FromPercival and Walden (2000)we have
a convenient equation to dene the wavelet detail coefcientsDj,nfor
a single channel Xtin termsof the circular cross-correlation of wavelet
lters with their corresponding wavelet packets Wj,n:
Dj;n t Lj1
l = ouj;n;lWj;n;t+ lmod N 7
where we dene
uj;n;l 2j =2
L1
k = 0
un;kuj1;
n
2j k
;l2j
1k 8
i.e. uj,n,l is dened as the wavelet packetlter that directly createsWj,nby circular convolution withXt. It is already noted that each wavelet
detail vector Dj,nis equivalent to Xtconvolved with a zero-phase band
pass lter. The idea that we can create a ltered time series merely by
summing detail coefcients follows from expressingXtas the sum of
those detail coefcients Dj,n whose Wj,n belong to the best basis;
expressed using Eq. (5), and noting from Eq. (6) that the detail
coefcients are created from orthonormal un,l, we have
Xt= 2j =2
J
j =1
2j1
n =0
Lj1
l = oL1
k =0un;ku
j1;n
2
j k;l2j1k
0@
1AWj;n;t+ lmod N
0@
1A:
9
Thedesiredresult followsfromtaking theDFT of bothsides of Eq.(8),
multiplying each by its complex conjugate, and noting that cross terms
vanish due to orthogonality ofun,k. The optimal pass band of each detail
coefcientDj,nis, again, given by Eq.(2).
6.1. Algorithm description
We now describe conceptually the algorithm to compute an
iterative, top-down, wavelet decomposition of a multichannel obser-
vation matrixXt, clusters its wavelet packet vectors Wj;nusing the rst
principalcomponentof each,and returns Xtltered to thosefrequencies
where each recovered signal dominates (we'll call this Xt). We rst
describe its use for single-component data, then discuss how togeneralize to the case of 3-component data.
6.1.1. Step 1. Iterative, top-down wavelet decomposition
Foreach levelj, beginning withj =1 (and recalling from above that
we denedWj;nXt):
1. Compute wavelet packet coefcientsWj,n for each channel at
each station. In practice, this is not computed directly using the
convolution of Eq.(3), but is obtained using more efcient
means, e.g. the pyramid algorithm ofMallat (1999).
2. Compute principal components vj,n,1 and cost M(Wj,n) of the
(possibly circularly shifted) wavelet packet coefcients.
a. For single-component data, compute the principal compo-
nents of the vertical component of each station.
b. For three-component data, polarization lter the data, and
computethe principalcomponentsof themost energetic(i.e.z)
component from each station.
3. Compare the costs of the sum of each pair ofchildrennodes,
M(Wj+1,2n) andM(Wj+1,2n+1), with the cost of their parent
node,M(Wj,n).
a. If M(Wj+1,2n)+M(Wj+1,2n+1)NM(Wj,n), and d(vj+1,2n,1,
vj+1,2n+1,1)b, markWj,n asa memberof the best basis,and
do not compute wavelet packet coef
cients for its childrennodes.
b. Otherwise, replace the cost ofWj,nby the sum ofthecostsof
the children nodes, and continue down the wavelet packet
table.
4. Repeat steps 13 as we move down the WP table, computing
wavelet packet coefcients for those nodes whose parents do
not belong to the best basis.
5. Stop either when an arbitrary levelj =Jhas been reached, or
when there are no further childnodes available.
6.1.2. Step 2. Hierarchical clustering
In this step, we cluster the wavelet packets Wj;n belonging to the
best basis. We use the distanced() given in Eq.(6)to measure the
similarity between eigenvectors vj,n,1 of each Wj;n. To determine
whether the input data Xtis a good match to some subband of the
data, therst eigenvector of the original observation matrix (i.e. v0,0,1)
is included in clustering, though this is for comparison purposes only.
6.1.3. Step 3. Signal reconstruction
Inverse transform eachWj,nin each cluster using Eq. (5)to create
its detail coefcientsDj,n; summingDj,nover allj,ncorresponding to a
cluster yields a recovered signalXtfor a particular data channel.
With regard to volcanoseismic data in particular, the following
specic steps are performed to separate the subbands ofKstations of
3-component data:
A.) Data are loaded and detrended. Instrument response is
deconvolved, then reconvolved with a common lter.
B.) For each subbandn at wavelet level j:1) Wavelet coefcientsWj;n are computed via the pyramid
algorithm ofMallat (1999).
2) For 3-component data, the method of Jurkevics (1988)is
used to calculate the 33 covariance matrix from inner
products of the3 components. A similar methodwas used in
Lilly and Park (1995)and specically inAnant and Dowla
(1997)in conjunction with wavelet transform methods.
3) Eigenvalues and eigenvectors are computed for each 3-
component station. Rectilinearity and planarity are com-
puted as dened inJurkevics (1988).
4) Dene Z at a given station/subband as the (3 channel)
Wj;n rotated into the eigenvector corresponding to the
largest eigenvalue. This eigenvector is multiplied by 1
(if necessary) to force the vertical component to bepositive.
5) Dene R and T following the convention of Jurkevics
(1988) by rotating into theazimuth ofZ. This ensures that
detail coefcients Dj,n formed from R and Tare always
aligned identically relative toZ.
6) AlignZcoefcients in time using a variant of the method
ofVandecar and Crosson (1990)if eachZwavelet packet
vector correlates to at least one other Z wavelet packet
vector at a level ofr(1 p) 0.1. Hereris the maximum
cross-correlation computed over a range of lag times
determined from the estimated phase velocity and inter-
station distance.p is probability of that maximum arising
from random chance. A maximum of one under-con-
strained channel is allowed per subband.
102 J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
-
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
6/18
7) The (possibly aligned)Z are now used to calculate the K K
covariance matrix of the principal components. The total
cost of this subbandnat wavelet levelj is given by Eq.(5).
8) The eigenvectorvj,n,1corresponding to the largest eigen-
valuej,n,1is saved, as is the cost and polarization.
9) Successive child nodes of each Wj,n are computed (if
necessary) from the (unaligned) unrotated wavelet co-
efcients corresponding to the parent node. The (un-
aligned) rotated wavelet coefcients are used to form the
detail coefcients Dj,n.
For a matrix Xtwith K stations and Ndata points, and wavelet
packets grouped into Pclusters, the algorithm returns Pltered sets of
3K NoutputsXt. Since this algorithm works from the top down, it isalmost never necessary to compute a full MODWPT. In an ideal case,
where one seismic source completely dominates a wide subband of
the frequency spectrum, and there are not many stations included,
real time implementation is possible on modern computing equip-
ment. However, the lack of decimation when computing the DWPT
does not allow the O(KN) efciency ofOweiss and Anderson (2007).
We remark that, despite the computational inefciency of computing
relative lags for each set ofchildnodes, formingDj;nby this method
preserves the phase of recovered signals Xt, making it a potential
preprocessing technique for array processing.
Fig. 2. Sample synthetics and Fourier power spectra for three sinusoids at f=1.5 Hz,
f=1.8 Hz, andf=3.6 Hz, randomly phase shifted and masked by Gaussian white noise
of 6 unit amplitude. Signal scaling is described in the text.
Fig. 3.Shaded intensity plot of log10(RMS) of recovered signals as a function of noise
amplitude scaling factor. Input signals are sinusoids whose median amplitude is unity.
Fig. 4.Plot of log10(RMS) of recovered signals as a function of number of channels used
inSDR. Solid line indicates mean RMS of recovered signals in each set of simulations.
Dashed lines indicate 2 errors for each set of simulations. Dotted lines indicate
maxima and minima of each set of simulations. Noise amplitude is held constant at a
scale factor of 4. Input signals are sinusoids whose median amplitude is unity.
Fig. 5. Erta 'Ale, Ethiopia, and stations from the 2002 Erta 'Ale experiment,
superimposed on an aerial photograph. Notable features of summit caldera are labeled.
Short-period sensors (L22 and LE3D) are indicated by small white triangles. Broadband
sensor (CMG-40 T) is indicated by large white triangle. Position of lava lake is
consistent with that of the aerial photograph, left and under indicative text.
103J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B5%80http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B4%80http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B3%80 -
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
7/18
7. Tracking signal invariance using principal components
Implicit in this method is that the invariance of recovered signals Xtcan be tracked over time by examining the principal componentsof their
constituent subbands. If allprincipal eigenvectors vj,n,1of the wavelet
packets Wj;n whose detail coefcients Dj;n form a recovered signal Xt1
cluster toallprincipal eigenvectorsvj,n,1of a recovered signal Xt2from a
later period, and the frequencies of their nominal passbands in Eq. (2)
overlap, then it follows from the de
nition of a recovered signal thatX
t1
andXt2 are mathematically identical. This property is potentially useful
for determining the difference between whether a recovered signal Xtchanges over time, or whether there are merely different secondary
signals superimposed upon a signal whose principal components do not
change.
8. Applicability and limitations
A much more detailed discussion of thelimitationsand capabilities
of the above algorithm is found in Jones (2009). Here, we wish to
discuss some important limitations.
Implicit in the use of PCA to select an optimal data transformation is
the assumption that each pair of stations records somecoherent source
energy in some subband(s). Recall that, following the notation ofAki
and Richards (2002), a measured seismic signal can be written as the
ltering of a source time series S(t) by apath effect lterP(t) and areceiver effect lter R(t), i.e.X(t)= S(t)* P(t)* R(t). This methodworks
best when the energy of one (possibly rotated) channel of each pair of
stations has a cross-correlation that is statistically signicant. However,
even if the receiver functions R(t) canbe neglected or deconvolved from
eachseismogram, path effects P(t) changethe frequency content of each
station. Path effects at real volcanic systems can be very complex (see
e.g. Harrington and Brodsky, 2007). Thus, this method will be most
effective when the seismic sourceS(t) has an isotropic component.
It is also true that this approach becomes less appropriate as intrinsic
attenuation and scattering increasingly affect the frequency content and
waveforms recorded at each station. In a sense this method can be
thought of as thecomplement to coda wave interferometry (Pachecoand
Sneider, 2005; Brenguier et al., 2008), which favors array geometries
where scattering dominates the recorded multichannel data Xt.This feature is inherently useful to detect subbands of the frequency
spectrum dominated by transients and/or by path effects. For example,
follows from Eq.(5) that a subband whose cost is low, but which is
nearly aligned with thedirection(in K-dimensional space) of stationk,
could be dominated by transients at station k. Furthermore, a subband
whosecostis high, and which does not align well with the direction
(in K-dimensional space) of a station k, could be dominated by path
effects. In this way, the method introduced here could also be used to
discriminate between regions of the frequency spectrum dominated by
source effects, and regions dominated by path effects.
Now, recall that a non-volumetric source function S(t) (or even a
volumetricsource function in an inhomogeneous medium) generatesat
least two distinct phase arrivals (Pand S) in the far eld, and even a
volumetric source has a transverse near-eld term (Aki and Richards,
2002). In the most general case, PCA decouples the axes of the
mathematically independent inputs; however, these independent
components areanysignal whose mathematical properties (e.g. arrival
time, frequency content) are different. Now, because the radiation
patterns ofPand Sdiffer e.g.the orientation of the maximum energyof
each phase is rotated 45 their relative amplitudesat each station also
differ. Therefore one feature of this algorithm is that itcouldcreate two
Fig. 6.Representative samples of trace data (top) and corresponding spectrograms (bottom) from the Erta 'Ale experiment. Amplitudes of trace data are normalized to illustrate
detail. All data have been detrended, downsampled to 50 Hz, preprocessed using a 3 s cosine taper, and ltered to the instrument response of a Lennartz MarsLite (f0=0.2 Hz). Data
from station L22 are further highpassltered using a 4 pole Butterworth lter atf=0.4 Hz. Spectrogram scaling is in dB computed from ground velocity. a. (Left) 17 m 35 s of raw
data beginning 15 Feb 2002, 16:28:46 GMT, during thelowconvective regime. Spectrogram corresponds to the vertical component of station L22 (Fig. 5). b. (Right) 10 m 54 s of
raw data beginning 15 Feb. 2002, 19:53:23 GMT, during the high convective regime. Spectrogram corresponds to the vertical component of station L22 (Fig. 5). Longer, more
detailed spectrograms from each convective regime can be seen inHarris et al. (2005) and Jones et al. (2006) .
104 J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B6%80 -
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
8/18
(ormore)recoveredsignals foreach uniqueseismic sourceone for the
region of the frequency spectrum where Pdominates (where possible),
another forS.
8.1. Performance testing with synthetic data
Because this algorithm has been developed with continuous vol-
canic tremor in mind, it is instructive to illustrate its ability to recover
quasi-continuous signals under various circumstances that mimic realvolcanogenic signals. To this end, two sets of Monte-Carlo simulations
are performed using synthetic data buried in Gaussian white noise. A
sample of such data is given in Fig. 2. Both tests use theLA-16 wavelet,
which offers a good balance between linear phase and compact length
(Daubechies, 1992), and whose detail coefcients are nearly perfect
bandpasslters (Fig. 1). MODWPT coefcients are computed to level
J=7. All tests use a clustering threshold of =0.3 to group the
principal eigenvectorsvj,n,1of eachWj,n. This is equivalent to a maxi-
mum average angle in K-space of=17 between members of any
two wavelet packet clusters.
In both sets of Monte-Carlo simulations, the inputs are three
sinusoids of unit amplitude. The rst test evaluates the algorithm's
ability to recover signals as background noise becomes more energetic.
The second test evaluates the algorithm's sensitivity to the number of
availablechannels, or stations. In bothtests, the inputs are3 sinusoids at
f=1.5 Hz,f=1.8 Hz, andf=3.6 Hz, sampled at 50 Hz for 81.92 s. The
rst two sources differ in frequency by only slightly more than the pass
band width of each wavelet packet at level J=7 (from Eq. (2),
~0.195 Hz). Their closely spaced spectral peaks test the algorithm's
sensitivity, while the third sinusoid tests whether the algorithm falsely
detects harmonics instead of two unique sources.
For all tests, input amplitudes at each station(i.e. channel) are
scaled by a random multiplier chosen from a normal distribution with
=1 and =0.5. Sinusoids are phase shifted randomly in each
channel by 2to 2radians. Thus the randomly generated param-
eters of each simulation are Gaussian white noise and amplitude and
phase of each of 3 inputs in each data channel. We restrict our K 3
matrix S of input scale factors so that max(rms(cov(S) I))b0.3. Thus
the amplitude falloff of one input is never proportional to another.For the rst set of simulations, 6 data channels (equivalent to
stations) are generated. Noise in each channel is multiplied by a
scaling factor that increases from 0.1 to 100 in increments of 100.2. 100
Monte-Carlo simulations are performed for each scale factor, making
1600 total simulations. The algorithm recovers each sinusoid indepen-
dently in 1589/1600 tests (i.e. 99.3% success rate), without clustering
them together. However, we remark that control tests of pure Gaussian
white noise, containing no sinusoidal input, also recover each band
independently inN95%of tests. A farmore appropriate measure ofSDR's
signalrecovery ability is theRMS error between theoutputsignalYt that
contains each sinusoid, and each (scaled, shifted) input sinusoid Xt.
With 6 channels and 3 signals, each simulation produces 18 RMSvalues.
A shaded intensity plot of log10(RMS) vs. noise scaling factor is given in
Fig. 3. Because the medianamplitudeof each input sinusoid is unity, thisplot suggests that SDR recovers signals even when noise amplitude is
almost an order of magnitude larger than input signal amplitude.
It is similarly important to investigate how the number of data
channelsaffectsthe ability to resolve recoveredsignals.Thus,a second
set of simulations follows a similar process to the rst for generating
noise, signal scaling, and phase shifts, but varies the number of chan-
nels from 3 up to 10. 100 simulations are performed for each number
of channels. In each simulation, background noise is held constant at a
scale factor of 4, which (fromFig. 3) produces typical RMS values of
0.360.08 with 6 data channels.Fig. 4shows a plot of log10(RMS) vs.
number of channels. The mean RMS of the recovered signals, and
variations in RMS, are virtually unchanged when 5 or more channels
are used. However, the mean RMS of recovered signals is factor of two
greater (0.670.41) when the number of data channels is reduced to
3.Jones (2009)found empirically that choosing a value for in the
range 0.25 0.4 produced nearly identical results, with a slight
increase in all values of RMS.
It must be re-emphasized here that sampling interval and sinusoid
frequencies were carefully chosen so that wavelet coefcients on thenest scale (J=7) had nominal passbands narrower than the dif-
ference between the two closest spectral peaks. SDR cannot be
expected to recover signals whose spectral peaks are closer together
than its nominal pass band width. In a hypothetical worst-casescenario, multiple signals with nearly identicalspectral peaks could be
indistinguishable.
8.2. Performance testing with real data: Erta 'Ale, Ethiopia
To illustrate the use and limitations of this algorithm with a real
volcano-seismic example, we present analysis of two samples of data
recorded at Erta 'Ale, Ethiopia, a basaltic shield volcano in the Danakil
Fig. 7. Decomposition of the data sample in Fig. 6a using the SDR algorithm. Top
Dendrogram of subband clustering. Horizontal dashed line indicates distance threshold
distance =0.3 for clustering of principal components eigenvectors. Labels give the
nominal passband in the interval [0,fn] associated with each wavelet packet. Bottom
Wavelet basis selected by subband clustering. Selected wavelet packets are shaded in
grayscale on a scale-frequency representation of the full wavelet packet table. Wavelet
packets belonging to the same recovered signal are the same shade of gray. Y-axis
indicates wavelet scale j, withj =0 corresponding to the wideband (input) signal by
convention. X-axis shows the position in the wavelet packet table (and nominal
passband) for each wavelet packet in the interval [0,fn].
105J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B7%80 -
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
9/18
Table 1
Recovered signals from a data sample recorded during the lowconvective regime in
2002, beginning 15 Feb 2002, 16:28:46 GMT ( Fig. 6a). Signal names use the convention
L(n), described in the text. Values for eigenvectors vj,n,1 correspond to the largest
eigenvalue j,n,1 ofthe principalcomponentsof each waveletpacketWj,n. Thequantities
vj,n, energy (En), azimuth (Az), incidence angle (In), rectilinearity (Rc), and planarity
(Pl) are arranged in columns by station. Az and In are given in degrees. Wavelet packets
are designatedWj,nand nominal passbands fminfmaxfor eachWj,nare given in Hz. Cost
functionalM(Wj,n) is renormalized (multiplied by normalized bandwidth 2j+1) so that
0 M(W)b3. Spectral leakage is tabulated in the form log 10(Eout/Ein), the base 10
logarithm of spectral energy outside the passband to spectral energy inside thepassband. Relative lags of each subband are given in seconds for subbands whose
wavelet cross-correlations are well constrained.
Signal name Station L22 CMG MAR
Wavelet packet vwavelet
fminfmax(Hz) En (A.U.)
M(Wj,n) Lag (s)
log10(Eout/Ein) Az (deg)
In (deg)
Rc (0 rc 1)
Pl (0 pl 1)
L(1)
W7,7 v7,7,1 0.90 0.44 0.02
fminfmax 1.371.56 En 1.4e0 7 5.9e08 1.7e08
M(Wj,n) 0.313 Lag 0.22 0.12 0.34
log10(Eout/Ein) 0.194 Az 160.31 45.96 56.54In 30.36 74.04 84.95
Rc 0.90 0.57 0.66
Pl 0.97 0.90 0.85
W7,8 v7,8,1 0.76 0.64 0.02
fminfmax 1.561.76 En 1.9e0 7 1.5e07 5.1e08
M(Wj,n) 0.123 Lag 0.10 0.12 0.22
log10(Eout/Ein) 0.081 Az 166.97 53.89 116.45
In 28.01 71.45 87.99
Rc 0.97 0.79 0.86
Pl 0.96 0.83 0.97
W6,5 v6,5,1 0.80 0.53 0.27
fminfmax 1.952.34 En 2.5e0 7 1.2e07 5.4e08
M(Wj,n) 0.303 Lag 0.02 0.02 0.04
log10(Eout/Ein) 0.544 Az 173.09 44.29 73.91
In 24.75 69.19 89.27
Rc 0.98 0.74 0.86
Pl 0.98 0.81 0.97W7,12 v7,12,1 0.82 0.57 0.07
fminfmax 2.342.54 En 4.4e0 8 2.9e08 7e09
M(Wj,n) 0.375 Lag 0.04 0.12 0.16
log10(Eout/Ein) 0.139 Az 160.73 163.14 69.21
In 33.73 87.01 71.88
Rc 0.90 0.78 0.79
Pl 0.97 0.90 0.85
W7,24 v7,24,1 0.86 0.52 0.01
fminfmax 4.694.88 En 3.8e0 9 2.8e09 5.4e10
M(Wj,n) 0.615 Lag 0.18 0.10 0.28
log10(Eout/Ein) 0.040 Az 179.80 8.05 55.94
In 45.68 80.52 77.74
Rc 0.70 0.77 0.48
Pl 0.90 0.89 0.55
L(2)
W6,0 v6,0,1 1.00 0.05 0.00
fminfmax 0.000.39 En 1.1e0 8 4.9e09 2.1e09
M(Wj,n) 0.007 Lag 0.02 0.16 0.14
log10(Eout/Ein) 0.708 Az 158.09 20.01 122.18
In 8.23 86.36 80.80
Rc 0.62 0.65 0.63
Pl 0.59 0.88 0.63
W7,14 v7,14,1 1.00 0.00 0.00
fminfmax 2.732.93 En 1.5e0 9 5.9e10 1.8e10
M(Wj,n) 0.367 Az 149.44 111.71 116.15
log10(Eout/Ein) 0.418 In 28.95 85.83 83.82
Rc 0.54 0.75 0.68
Pl 0.44 0.79 0.78
W4,2 v4,2,1 1.00 0.00 0.00
fminfmax 3.124.69 En 4.6e1 0 8.4e11 3.4e11
M(Wj,n) 0.201 Az 161.15 65.41 103.52
log10(Eout/Ein) 0.355 In 39.93 88.80 82.91
Rc 0.54 0.76 0.60
Table 1 (continued)
Signal name Station L22 CMG MAR
Wavelet packet vwavelet
Pl 0.55 0.70 0.88
W7,31 v7,31,1 1.00 0.00 0.00
fminfmax 6.056.25 En 4.3e1 0 3 .3e1 1 1 .8e11
M(Wj,n) 0.614 Az 166.25 69.87 91.47
log10(Eout/Ein) 0.045 In 34.78 85.75 83.46
Rc 0.66 0.80 0.56
Pl 0.67 0.82 0.90W2,1 v2,1,1 0.99 0.14 0.01
fminfmax 6.2512.50 En 2.6e0 7 3 .8e0 8 1 .9e08
M(Wj,n) 0.641 Lag 0.10 0.18 0.28
log10(Eout/Ein) 0.259 Az 157.88 164.44 68.02
In 31.66 88.38 89.43
Rc 0.93 0.54 0.67
Pl 0.96 0.74 0.75
W2,2 v2,2,1 1.00 0.00 0.00
fminfmax 12.5018.75 En 8.6e0 8 3 .4e1 0 2 .3e10
M(Wj,n) 0.506 Az 95.26 65.96 68.87
log10(Eout/Ein) 0.260 In 87.61 85.68 76.79
Rc 0.31 0.61 0.21
Pl 0.46 0.50 0.32
W3,6 v3,6,1 1.00 0.06 0.00
fminfmax 18.7521.88 En 4.2e0 8 9 .4e0 9 6 .1e09
M(Wj,n) 0.258 Lag 0.04 0.26 0.30
log10(Eout/Ein) 0.494 Az 164.27 40.42 54.76In 37.15 74.83 85.66
Rc 0.87 0.70 0.80
Pl 0.97 0.63 0.89
W3,7 v3,7,1 0.98 0.20 0.03
fminfmax 21.8825.00 En 1.1e0 9 4 .9e1 0 2 .3e10
M(Wj,n) 0.120 Lag 0.06 0.16 0.10
log10(Eout/Ein) 0.909 Az 20.72 14.20 128.37
In 10.72 84.66 75.98
Rc 0.77 0.81 0.73
Pl 0.74 0.89 0.67
L(3)
W7,2 v7,2,1 0.59 0.75 0.29
fminfmax 0.390.59 En 6.7e0 9 8 .5e0 9 2 .2e09
M(Wj,n) 0.323 Az 170.34 79.89 53.33
log10(Eout/Ein) 0.061 In 55.42 87.45 65.77
Rc 0.59 0.95 0.87
Pl 0.47 0.94 0.84
W7,6 v7,6,1 0.47 0.86 0.21
fminfmax 1.171.37 En 6.2e0 8 1 .2e0 7 1 .1e08
M(Wj,n) 0.288 Lag 0.20 0.02 0.18
log10(Eout/Ein) 0.170 Az 167.14 67.01 22.29
In 30.11 77.43 87.98
Rc 0.92 0.88 0.54
Pl 0.94 0.89 0.93
L(4)
W7,3 v7,3,1 0.43 0.71 0.55
fminfmax 0.590.78 En 9.1e0 9 2 .4e0 8 1 .8e08
M(Wj,n) 0.368 Lag 0.12 0.10 0.22
log10(Eout/Ein) 0.170 Az 179.54 97.63 53.46
In 43.90 87.27 53.25
Rc 0.89 0.88 0.94
Pl 0.86 0.97 0.91
L(5)
W7,4 v7,4,1 0.28 0.86 0.42
fminfmax 0.780.98 En 1.3e0 8 3 .1e0 8 1 .7e08
M(Wj,n) 0.568 Lag 0.14 0.18 0.04
log10(Eout/Ein) 0.435 Az 173.40 121.89 52.51
In 29.05 80.35 49.56
Rc 0.89 0.85 0.88
Pl 0.95 0.93 0.96
L(6)
W7,5 v7,5,1 0.54 0.84 0.00
fminfmax 0.981.17 En 4.3e0 8 9 .4e0 8 7 .2e09
M(Wj,n) 0.107 Lag 0.12 0.10 0.00
log10(Eout/Ein) 0.655 Az 176.25 93.68 92.81
In 36.12 87.88 59.06
Rc 0.92 0.83 0.56
106 J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
-
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
10/18
depression of northeast Ethiopia (Fig. 5). The summit caldera features
two pit craters, the southernmost of which held a persistent, active
lava lake from (at least) 1967 through late 2004 (Martini, 1969;
Oppenheimer and Francis, 1998; Bardintzeff and Gaudru, 2004).
During a pilot study in February, 2002, seismic, thermal, and video
data were collected for 5 days, to better understand the dynamics of
the shallow magma system that fed the southern crater's persistent
lava lake (Harris et al., 2005; Jones et al., 2006). The three-station
seismic array geometry of the 2002 experiment is shown in Fig. 5.The 2002 campaign found that the lava lake uctuated between
two convective regimes, characterized by low (0.010.08 m s1) and
high (0.10.4 m s1) velocities of cooled crust on the lava lake
surface, which corresponded to sluggish and vigorous convection,
respectively (Harris et al., 2005). The persistent, continuous tremor
was described by Jones et al. (2006), in which we found distinct
spectral characteristics corresponding to each convective regime
(Fig. 6). Because active lava lakes can be considered the exposed
upper surface of a convecting magma column (Swanson et al., 1979;
Harris et al., 1999), these changes could be explained by cooling and
degassingprocesses in theshallow part of theexposed conduit (Harris
et al., 2005). Themodelingwork ofHarris (2008) and the observations
of Harris et al. (2005) strongly suggest that magma feeds the
convecting lava lake at a relatively steady supply rate and constant
viscosity, and the changing rate of convection was driven by shallow
processes within the lava lake.
Because the model of Erta 'Ale's conduit convection is well
supported by observations, thermal modeling, and existing analysis,
it is a suitable test of theSDRmethod to examine whether or not the
method corroborates these results. If the modeling work ofHarris
(2008) and the observations ofJones et al. (2006) are correct, then we
expect to recover at least one, nearly identical seismic signal, from
both convective regimes, which represents the response of a
subsurface conduit to the steady ow of fresh, hot, gas-rich magma
from a deeper reservoir. We expect further that during a period of
high-velocity convection in the lava lake, we will recover signals
whose measurable properties correspond to shallow sources within
the lava lake itself.
Prior to analysis, all data were downsampled to 50 Hz for computa-tional efciency, and a 3 s cosine taper was applied. As inJones et al.
(2006), the instrument response of all stations was corrected to match
that of an LE3D sensor (f0=0.2 Hz). Finally, to prevent low-frequency
artifacts that might result from this convolution, all data recorded by the
L22 geophone was high-pass ltered using a 4-pole Butterworth lter
with corner frequency 0.4 Hz.
8.2.1. 2002 tremor recorded during slow convection
We begin with the quiescent, lowconvective regime described in
Harris et al. (2005), i.e. those periods characterized by lava lake
convection of 0.010.08 ms1.Fig. 6a shows the raw data sample and a
spectrogram from station L22. Observe that the signal's spectral energy
contains few transients and is concentrated mostly below 5 Hz.Fig. 7
shows a dendrogram of the chosen wavelet decomposition, with thenominalcorner frequencies of eachsubbandassociated witheachwavelet
packetWj,n used to label the appropriate nodes. Fig. 7 shows the wavelet
basis that best represents this decomposition, with wavelet packetsWj,npositioned according to waveletlevelj andbandn. Each wavelet packet is
arranged to ll the associated passband, and shaded so that all wavelet
packets with the same shade belong to the same cluster. FromFig. 7, we
see that our algorithm recovers 9 signals, to which we henceforth refer
using theconvention L(n), n being an arbitrarily assigned number foreach
recovered signal. FromFig. 7, we see that most of these are narrowband
signals whose energy is concentrated below 3.13 Hz, i.e. in the regions of
the spectrum where seismic energy is highest.
We wish now to discuss this decomposition in detail, and focus in
particular on its implications for the tremor sources. The frequency
content, wavelet polarization, subband energy, cost M(Wj,n), spectral
Table 1 (continued)
Signal name Station L22 CMG MAR
Wavelet packet vwavelet
Pl 0.94 0.94 0.85
W7,25 v7,25,1 0.38 0.91 0.13
fminfmax 4.885.08 En 1.2e09 2 .4 e0 9 5 .1 e10
M(Wj,n) 0.564 Lag 0.10 0.20 0.30
log10(Eout/Ein) 0.040 Az 36.49 3.92 58.76
In 30.54 87.48 64.04
Rc 0.53 0.83 0.61Pl 0.68 0.86 0.69
L(7)
W7,9 v7,9,1 0.82 0.50 0.28
fminfmax 1.761.95 En 9.5e08 5 .7 e0 8 2 .9 e08
M(Wj,n) 0.473 Lag 0.14 0.22 0.08
log10(Eout/Ein) 0.228 Az 179.88 97.73 73.19
In 16.43 69.55 88.19
Rc 0.89 0.58 0.85
Pl 0.97 0.85 0.96
W7,29 v7,29,1 0.82 0.57 0.08
fminfmax 5.665.86 En 1.1e09 9 .3 e1 0 2 .1 e10
M(Wj,n) 0.814 Lag 0.02 0.06 0.04
log10(Eout/Ein) 0.155 Az 65.97 162.16 121.90
In 25.66 89.20 87.91
Rc 0.65 0.82 0.62
Pl 0.68 0.95 0.65W7,30 v7,30,1 0.79 0.61 0.06
fminfmax 5.866.05 En 1e09 9.2e10 2e10
M(Wj,n) 0.779 Lag 0.16 0.00 0.16
log10(Eout/Ein) 0.056 Az 64.57 15.21 135.41
In 12.07 87.69 68.38
Rc 0.69 0.87 0.63
Pl 0.63 0.94 0.62
L(8)
W7,13 v7,13,1 0.89 0.44 0.13
fminfmax 2.542.73 En 3e08 1.3e0 8 6 .4 e09
M(Wj,n) 0.356 Lag 0.14 0.14 0.26
log10(Eout/Ein) 0.237 Az 176.35 169.59 70.10
In 60.36 80.20 79.14
Rc 0.87 0.54 0.83
Pl 0.97 0.49 0.81
W7,15 v7,15,1 0.96 0.27 0.00
fminfmax 2.933.12 En 5.8e08 1 .4 e0 8 6 .1 e09
M(Wj,n) 0.253 Lag 0.12 0.14 0.26
log10(Eout/Ein) 0.204 Az 155.27 30.83 140.20
In 31.85 83.79 84.67
Rc 0.94 0.63 0.83
Pl 0.98 0.83 0.84
W7,27 v7,27,1 0.95 0.32 0.01
fminfmax 5.275.47 En 1e09 6.1e10 2e10
M(Wj,n) 0.715 Lag 0.14 0.06 0.08
log10(Eout/Ein) 0.140 Az 30.67 21.86 104.19
In 34.67 84.17 81.52
Rc 0.67 0.77 0.52
Pl 0.77 0.85 0.75
W7,28 v7,28,1 0.97 0.21 0.10
fminfmax 5.475.66 En 1.3e09 8 .7 e1 0 2 .5 e10
M(Wj,n) 0.816 Lag 0.06 0.06 0.12
log10(Eout/Ein) 0.163 Az 50.20 22.70 128.22
In 26.45 87.82 81.33Rc 0.74 0.84 0.60
Pl 0.78 0.92 0.75
L(9)
W7,26 v7,26,1 0.02 0.99 0.16
fminfmax 5.085.27 En 9.6e10 1 .5 e0 9 2 .3 e10
M(Wj,n) 0.769 Lag 0.08 0.20 0.28
log10(Eout/Ein) 0.223 Az 27.81 179.59 92.89
In 51.57 89.04 69.32
Rc 0.57 0.87 0.42
Pl 0.68 0.90 0.56
107J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
-
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
11/18
Fig. 8. Trace data,spectrograms, and azimuths of recovered signalL(1) from thelow convective regime. a. (Top left) Therecovered signal L(1). Amplitudes ofeach traceare normalized
to illustrate detail. b. (Top right) Spectrograms of the inputZdata at station L22 (above) and theZ component of recovered signal L(1) at L22 (below). Intensity scaling is in dB. Color
scaling of input data is computed from true ground velocity. c. (Bottom) Azimuths of subbands Wj,n that form recovered signal L(1) are superimposed on an aerial photograph
showing station locations and relevant physical features in the Erta 'Ale summit caldera. Azimuths are scaled according to relative energy of each Wj,nat each station. The largest
azimuth vectorcorresponds toW7,8 atL22, whose nominal passband is 1.561.76 Hz. Boxeson spectrograms indicate the nominal passbands associatedwith the wavelet coefcients
whose detail coefcients form the recovered signal.
108 J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B8%80 -
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
12/18
leakage, and principal eigenvectors for select recovered signals are
given inTable 1.
We begin with the recovered signal whose spectral energy is
highest. Signal L(1) (Fig. 8a) is formed from 5 clustered wavelet
packets:W7,7(nominal passband 1.361.56 Hz),W7,8(1.561.76 Hz),W6,5(1.952.34 Hz),W7,12(2.342.54 Hz), andW7,24(4.694.88 Hz).
It contains the most energetic spectral peak of the input data at
stations L22 and CMG (e.g.Fig. 6a). Spectrograms of theZcomponent
of the reconstructed signal, and theZcomponent of the input data, are
shown in Fig. 8b. The recovered signal's subbands are highly
rectilinear at station L22, with rectilinearity (cf. Jurkevics, 1988)Rc=0.900.98 for frequenciesbelow 2.53 Hz. Its polarizations at MAR
and CMG are also rectilinear for some subbands. The spread in
azimuths of this recovered signal's subbands, noting the 180
ambiguity, is only 33. Fig. 8c shows a plot of the azimuths of the
subbands that form this recovered signal, in which azimuth vectors
Fig. 9. Spectrograms and azimuths of recovered signalL(3) from thelow convective regime. a. (Upperleft)Spectrograms of theinput Ndata atstation L22(top) andtheZ component
of recovered signalL(4) (bottom). b. (Upper right) Spectrograms of the input Edata at station CMG (top) and the Zcomponent of recovered signalL(4) (bottom). Scaling intensity is
in dB computed from true ground velocity. c. (Bottom) Azimuths ofWj,nthat form recovered signalL(3) are superimposed on an aerial photograph showing station locations and
relevant physical features in the Erta 'Ale summit caldera. Azimuths are scaled according to relative energy of each Wj,nat each station.
109J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B9%80 -
8/11/2019 Jones 2012 Decomposition Volcanic Tremor
13/18
are rescaled to the relative energyof that subband at that station; thus
the largest azimuth vector corresponds to W7,8, whose nominal
passband is 1.561.76 Hz. Notably this subband is also the most
rectilinear.
The computed incidenceangles of these subbands are 2433 from
vertical at L22. If we assume that the source underlies this part of the
crater, and that the azimuth points toward the source, then simple
trigonometry (fromFig. 5) constrains the maximum depth to about
100 masl, or about 420 m below the crater
oor. If we further assumethat the sourceof this tremoris related to fresh, hot magma upwelling
roughly in the center of the lava lake (Harris et al., 2005), then the
sourceis probably 40100 m below the lava lake surface. This range of
depths corresponds to estimates of the lava lake depth in Harris et al.
(2005)andOppenheimer and Francis (1998).
Unfortunately, these source depths contraindicate expanding on or
recreating the location work ofJones et al. (2006). 3 stations cannot
adequately constrain a tremor centroid in 3 dimensions, and neglecting
depths comparable to (or greater than) the nearest epicentral distance
canintroduce grievous errorsto thecalculation.However, we canobtain
some additionalclues about this recoveredsignal's source by examining
its spectrogram. From Fig. 8b, observe that the peaks of this signal
correspond roughly to two sets of harmonics: One set of spectral peaks
(i.e. fundamental and rst overtone) at 1.8 and 3.6 Hz, one at 2.4 and
4.8 Hz.We note thatthe narrowband energycentered at 3.6 Hz is notan
aliasing phenomenon, but a result of wavelet lters not being ideal
bandpass lters (see e.g.Percival and Walden, 2000,Ch. 4). It could be
that these are real harmonics, and that their spectra lack very sharp
peaks merely because many other signals are superimposed on them at
similar frequencies.
It is not the case that all recovered signals appear to originate in
the lava lake. An excellent counter-example of this is shown inFig. 9
for signalL(3), formed from wavelet packetsW7,2(0.390.59 Hz) and
W7,6 (1.171.37 Hz). The spectrogram of this recovered signal
(Fig. 9b) suggests the recovered signal consists of three harmonics
(1st, 3rd, and 5th) with a fundamental overtonef=0.45 Hz. It may be
that these are true harmonics mixed with other signals that comprise
the observed tremor, while the other harmonics are masked by the
intense energy ofL(1). Observe fromTable 1that the energy of thisrecovered signal's subbands is greatest at station CMG, which is also
where the subbands are most rectilinearly polarized, and that the
azimuths of this signal's constituent subbands point toward the old
(north) crater, which was a source of fumarolic degassing in 2002
(Harris et al., 2005). Incidence angles at station CMG (Fig. 8b) range
from 77 to 87, suggesting that its source is shallow. At station L22,
azimuths and incidence angles of the recovered signal Z component
point to the lava lake; this is consistent with a weak source in the
north crater being masked by the intense energy of tremor from the
lava lake. Notably, recovered signals L(4), L(5), and L(6) have similar
features, though obviously lack the apparent harmonics ofL(3).
One additional, important conclusion can be drawn from analysis of
Table 1. During the slow lava lake convection, there is no clear
evidence of multiple signals whose sources are in or near the lava lake.The signalsL(1) andL(9), each of which appear to originate near station
L22, cannotbe unambiguouslyassociated with uniquesources.It maybe
that these signals all originate from a single tremor source, and
following the surface morphology described in e.g.Oppenheimer and
Francis (1998) undergo phase conversions in the complex structure
that underlies the Erta 'Ale summit caldera. There is, however, good
evidencethat four signals (L(3) through L(6)) contain energyat CMGthat
originates elsewhere.
8.2.2. 2002 tremor during rapid convection
We now turn to tremor related to more rapid convection of the
lava lake in 2002. This convective regime was characterized byHarris
et al. (2005)by more rapid surface velocities of the lava lake, ranging
from 0.1 to 0.4 ms1
, corresponding to vigorous overturn of cooled
crust on the lava lake surface, with frequent episodes of very small
lava fountains.Fig. 5b shows sample data and a spectrogram for thehigh convective regime at station L22. Note the increase in t high
frequency energy (fN5 Hz) during the highconvective regime.
Fig. 10shows SDR analysis of the data sample ofFig. 6b.Table 2
tabulates the frequency content, wavelet polarization, subband
energy, costM(Wj,n), spectral leakage, and principal eigenvectors for
selected recovered signals from this convective phase. Note that SDR
recovers 16 signals (designated H
(n)
), suggesting more seismicsources are present in thehighconvective regime. This is consistent
with the modeling work ofHarris et al. (2005) and Harris (2008).
Fig. 10 shows a dendrogram of the chosen wavelet decomposition,
with the nominal corner frequencies of each subband associated with
each wavelet packet Wj,nused to label the appropriate nodes.
We wish to focus on which signals have changed, and which signals
persist, between the convective regimes. As noted above, our denition
of a recovered signal enables us to track signal persistence in a very
straightforward way. Comparing the principal eigenvectors vj,n,1 of
Table 1 withthoseofTable 2 showsseveral examples: L(1), L(2), L(3), L(4),
andL(6) persist (and are renamed in the high regime) as H(1) (and
Frequency [Hz]
WaveletScale
3.12 6.25 9.38 12.50 15.62 18.75 21.88 25.00
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00.4
16.016.4
18.825.0
2.73.1
7.89.4
18.018.8
14.114.3
3.16.2
15.415.6
6.66.8
16.817.0
2.02.3
15.615.8
17.017.2
7.27.4
7.47.6
17.218.0
12.312.5
15.015.2
16.416.8
6.26.4
12.514.1
6.87.0
15.816.0
12.112.3
15.215.4
11.712.1
InputData
2.32.7
1.41.6
1.61.8
6.46.6
7.07.2
7.67.8
9.49.6
9.810.0
9.69.8
1.82.0
14.614.8
0.40.6
0.81.0
1.01.2
1.21.4
0.60.8
10.010.2
10.911.1
10.710.9
11.111.3
10.210.4
10.410.5
11.511.7
10.510.7
14.815.0
11.311.5
14.314.5
14.514.6
FirstEigenvectorDistance
Fig. 10. Decomposition of the data sample in Fig. 6b using the SDR algorithm. Top
Dendrogram of subband clustering. Horizontal dashed line indicates distance threshold
distance =0.3 for clustering of principal components eigenvectors. Labels give the
nominal passband in the interval [0,fn] associated with each wavelet packet. Bottom
Wavelet basis selected by subband clustering. Selected wavelet packets are shaded in
grayscale to showclustering. Y-axis shows waveletscalej, withj=0 corresponding to the
wideband (input) signal. X-axis shows the position in the wavelet packet table (and
nominal passband) for each wavelet packet in the interval [0,fn].
110 J.P. Jones et al. / Journal of Volcanology and Geothermal Research 213-214 (2012) 98115
http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_5/image%20of%20Fig.%E0%B1%B0http://localhost