Jones 2012 Decomposition Volcanic Tremor

8/11/2019 Jones 2012 Decomposition Volcanic Tremor

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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


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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


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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


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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



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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.



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.

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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) .

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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].



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)



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



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)



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



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.



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.



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].

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Jones 2012 Decomposition Volcanic Tremor

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