Properties of the near-field term and its effect on ... · for long- (LP) and very-long-period (VLP) volcanic seismicity, including: a single force, compensated linear vector dipole

Journal of Volcanology and Geothermal Research 192 (2010) 35–47

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Journal of Volcanology and Geothermal Research

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Properties of the near-field term and its effect on polarisation analysis and sourcelocations of long-period (LP) and very-long-period (VLP) seismic events at volcanoes

Ivan Lokmer ⁎, Christopher J. BeanSeismology and Computational Rock Physics Laboratory, School of Geological Sciences, University College Dublin, Belfield, Dublin 4, IrelandComplex and Adaptive Systems Lab (CASL), University College Dublin, Dublin 4, Ireland

⁎ Corresponding author. School of Geological SciencBelfield, Dublin 4, Ireland. Tel.: +353 1 7162079.

E-mail address: [email protected] (I. Lokmer).

0377-0273/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.jvolgeores.2010.02.008

a b s t r a c t
a r t i c l e i n f o
Article history:Received 29 September 2009Accepted 12 February 2010Available online 20 February 2010

Keywords:volcanoseismicitylong-period events (LP)very-long-period events (VLP)volcanic sourceslocationpolarisationnear-field effect

Seismicity that originates within volcanic magmatic and hydrothermal plumbing systems is characterised bywavelengths that are often comparable to or longer than the source–receiver distance. The effect of such anear-field configuration must be explored when analysing these signals. Herein, we summarise properties ofnear-field observations for both a single force and moment-tensor seismic sources. We show radiationpatterns of the near-, intermediate- and far-field terms for the source types that are most likely candidatesfor long- (LP) and very-long-period (VLP) volcanic seismicity, including: a single force, compensated linearvector dipole (CLVD), a tensile crack and a pipe-like source. We find that the deviation of the first motionpolarisation from the radial direction is significant in all planes except one whose normal is parallel to thesymmetry axis (if there is one) of the source mechanism. However, this deviation is less pronounced (oreven negligible), when there is a considerable volumetric component in the source (as in the case of a tensilecrack or pipe). Our location test shows that the accuracy of locations obtained using the semblance or cross-correlation techniques is very significantly affected by the near-field geometry. This effect is especiallypronounced for shallow sources, such as often encountered on volcanoes, and decreases with increasingsource depth. Hence, in practical applications on volcanoes, 3D full waveform numerical simulation(including topography and structural heterogeneities) should be used in order to both validate locationtechniques and as an interpretational aid to reduce misinterpretations of location results.

es, University College Dublin,

l rights reserved.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Magmatic and hydrothermal activities beneath a volcano lead todifferent types of seismic activity observable at the surface: volcano-tectonic (VT) events, very-long-period (VLP) events, long-period (LP)events and volcanic tremor (a detailed review and explanation ofthese types of events can be found in Chouet (2003) and McNutt(2005)). These events span a broad frequency range from less than0.01 Hz to more than 20 Hz, radiating seismic waves with wave-lengths ranging between tens of metres to hundreds of kilometres,that often propagate small distances (a few hundredmetres to severalkilometres) before they are recorded by a volcano's surveillancenetwork. Hence, many observations of longer period seismicity onvolcanoes are in the “near-field” (i.e., the source–receiver distance issmaller than or comparable to the radiated wavelength). The near-field effect has already been generally discussed in the literature (e.g.,Aki and Richards, 2002), but its properties have not been systemat-ically outlined in connection with volcano seismology.

The near-field nature of volcanic seismic observations (VLP, LP andtremor) has a two-fold effect: (i) due to long periods and smalldistances/travel-times, different wavetypes cannot be clearly sepa-rated on recorded seismograms and (ii) in addition to P- and S-wavescommonly observed in the far-field, an additional near-field wavebecomes important at close distances. In practice, this near-field wavemust be studied by numerical simulations due to its inseparableinteraction with the medium (heterogeneity and pronounced volcanotopography). However, if we neglect the heterogeneity of themediumand free-surface effects (for simplicity and clarity), the near-fieldproblem can be studied analytically, thus providing us withknowledge about the properties of the near-field radiation. In thefollowing, we will present the simple cases of a seismic point-sourcedescribed by a vertical force, and a suite of moment-tensor sourcesrelevant to volcanic environments. We will analyse properties of thenear-field term, such as (i) the polarisation and radiation pattern ofthe near-field waves, (ii) the shape of the near-field waveform and(iii) the relative importance of near- vs. far-field waves, depending onthe source–receiver distance. There follows a section with severalexamples of synthetic tests in which we illustrate the near-field effecton the wavefield polarisation and source location determined bysemblance techniques. In the concluding section, we summarise the

36 I. Lokmer, C.J. Bean / Journal of Volcanology and Geothermal Research 192 (2010) 35–47

general implications that near-field observations could have foranalysing volcanic seismic signals.

2. Near-field properties of the wavefield generated by a verticalsingle force

A single forcepoint-source is the simplest analytical case for studyingthe full seismic wavefield. Thewidely accepted formalism of Backus andMulcahy (1976), where a seismic source is represented by equivalentbody forces, implies a symmetrical moment-tensor and absence of a netsingle force in the seismic source. However, Takei andKumazawa (1994)argue that the absence of the total force and total torque component inthe source volume is not required by the conservation of the total linearandangularmomenta of the Earth. Accordingly, a single force and torqueseismic sources can exist if there is mass transport in the medium, aprocess likely to be very important in volcanic areas. Thus, the simplecase presented here (single force source) is not only a theoretical step toderive more realistic cases, but instead it may have realistic physicalimplications in volcanic environments.

The solution of the inhomogeneous (in the source sense) elasticwave equation for a homogeneous infinite medium, with a vertical

Fig. 1. a) Reference coordinate system, b) Radiation pattern of near-field, P and S waves for apolarisation of the near-field term, the radial and transverse components of the near-field racomponents travel with the P-wave velocity. The absolute value of the near-field radiation

body force with a time history F(t) acting at the source can be,according to Aki and Richards (2002, eq. 4.23), written as follows:

u r; tð Þ = RN

4πρ r3∫r =β

r =α

τF t−τð Þdτ +R

P

4πρα2rF t− r

α

� �

+R

S

4πρβ2rF t−r

β

� �;

ð1Þ

where u(r,t) is the displacement measured at position r and time t, αand β are P- and S-wave velocities, respectively, ρ is the density of themedium, r is the source–receiver distance, and RN, RP and RS are thenear-field, P- and S-wave radiation patterns, respectively. For thecoordinate system in Fig. 1a, the latter three terms are defined as:

RN = 2cos ðθÞur + sinðθÞuθ;

RP = cosðθÞur;

RS = − sinðθÞuθ:

ð2Þ

Note that all threewavetypes are linearly polarised, but thedirectionof polarisation of the near-field component does not coincide witheither of the far-field wave types. A graphical representation of all three

vertical single force applied at the source (see Eqs. (1) and (2)). In order to illustrate thediation pattern are shown separately (first two panels in the first row). Note that bothpattern is given in the third panel of the first row.

Fig. 2. a) Source-time function of duration T. Insert: Normalised Fourier amplitudespectrum of the source-time function. b) Near-field term (the first term in Eq. (1)). Notethat near-field disturbance arrives at the receiver at time r/α and lasts until time r/β+T,that is, on observed seismogram it is intertwined with both P and S waves.

Fig. 3. Dependence of the ratio between the near-field and far-field P-wave for a singleforce source as a function of distance measured in number of P-wave wavelengths. Notethat for distances smaller than a half of the P-wave wavelength (marked by the dashedline), both the near-field term and the P-wave have the same dependence withdistance. Radiation patterns RN and RP were not included in the calculation of the ratio.

37I. Lokmer, C.J. Bean / Journal of Volcanology and Geothermal Research 192 (2010) 35–47

radiation patterns is given in Fig. 1b. Note that the displacement in the θdirection, uθ, is positive from the positive to the negative z-axis (asopposed to the “curl” convention, when displacement is positive whenin the anticlockwise direction). The sameconvention is used throughoutthe paper. It can be seen that, as opposed to the radiation pattern of far-field waves, the near-field component radiated by the vertical forcepoint-source does not have nodal planes or axes. Since observed near-field seismograms are a combination of all three wavetypes, it impliesthat additional difficulties arise in the near-field, related to separatingdifferent wavetypes based on their polarisation.

Returningback to Eq. (1), it canbe seen that the far-field termsdependon distance as 1/r and their waveforms are identical to that of the source-time function. In contrast, the near-field term can be viewed as aconvolution between the source-time history, F(t), and a ramp functionwhich is zero everywhere except between the times r/α and r/β, withheight proportional to time. An example of the near-field waveform isgiven in Fig. 2. Note that near-field disturbance arrives at the receivertogether with the P-wave and its duration is r/β−r/α+T, where T isduration of the source-time function. Hence the near-field term willalways intertwine with both P and S arrivals, thus “contaminating” theirobserved polarisations. In this example, the duration of the source-timefunction is comparable to the timedifference r/β−r/α. However, if r/β−r/α is much longer than the duration of F(t), then F(t) is practically a deltafunction and the first term in Eq. (1) will be zero everywhere exceptbetween r/α and r/β,with height proportional to time. Consequently, themaximum amplitude of the near-field term in such a casewill decaywithdistance as 1/r2. On the contrary, if r/β−r/α is much smaller than theduration of F(t), the near-field waveform will be practically identical tothat of the source-time function. This is the case when the wavefield isobserved very close to the source, where r/β−r/α is small compared tothe dominant radiated wavelengths. This condition is usually satisfied forrecorded LP events on volcanoes. It is more convenient to study thedependence of the near-field amplitude on small distances in thefrequency domain. If we denote the near-field, P- and S-wave terms byNF, FP and FS, respectively, all three terms of Eq. (1) can be re-written inthe frequency domain as follows:

NF = RN FðωÞ4πρα2r

i2πnλ

− 12πnλ

� �2� �⋅ei2πnλ− α

βi

2πnλ− 1

2πnλ

� �2� �⋅e

αβi2πnλ

� ;

FP = RP FðωÞ4πρα2r

⋅ei2πnλ ;

FS = RS FðωÞ4πρβ2r

⋅eαβi2πnλ :

ð3Þ

F(ω) in the above equations is the spectrum of the source-timefunction F(t), ω is angular frequency, and nλ= rω/2πα is the distancefrom the source measured in number of the P-wave wavelengths. Inorder to obtain the behaviour of the near-field term at small distancesfrom the source, we search for the limit of real and imaginary parts ofthe NF, respectively, as nλ tends to zero. Repeatedly applyingL'Hospital's rule, it can be shown that:

limnλ→0

NF = RN FðωÞ8πρα2r

αβ

� �2−1

� �: ð4Þ

It can be seen that for distances of the order of fraction of the P-wavewavelength, the near-field termdecayswith distance as 1/r, the same asthe far-field terms. This implies that the waves with wavelengths muchlonger than the source–receiver distance will decay as 1/r, while thosewith shorter wavelengths as 1/r2. Thus, apart from a spatially varyinggeometrical spreading factor, the near-field term is characterised by a

distance-dependent spectrum. Such complicated properties prevent usfrom finding a unique formula for estimating the degree of “contam-ination” of the P-wave polarisation by the near-field term in the timedomain. The only way to address this issue is to calculate full wavefieldsynthetic seismograms for a suit of different source-time functions andsource–receiver geometries of interest.

So far, distance-dependent and polarisation properties of the near-field term have been presented. However, it is of practical interest toestimate distances for which the amplitudes of the near- and far-fieldterms are comparable. For this purpose, we estimate the absolute value ofthe spectral ratio:

NFFP

=R

N

RP

i2πnλ

− 12πnλ

� �2� �− α

βi

2πnλ− 1

2πnλ

� �2� �⋅e

αβ−1ð Þi2πnλ

� ;

ð5Þ

assuming a Poisson's ratio of σ=0.25 (α/β=31/2) and neglecting theradiationpattern terms. The result is given inFig. 3. It canbeseen that fordistances smaller than a half of the P-wave wavelength, the near-fieldterm has the same dependence with distance as the P-wave, and bothterms have equal spectral amplitudes. In order to obtain near- field vs.


far-field ratio as a function of receiver position, the function given inFig. 3 must be multiplied by the radiation pattern terms RN/RP given byEq. (2) and in Fig. 1. Thus, for example, at a distance of one P-wavewavelength, in direction θ=0, the ratio between the near-field termand P-wave will be equal to 1, that is, the near-field amplitude will beequal to the P-wave amplitude (here, the distance is measured innumber of dominant P-wavelengths). For 2 s long signal like in Fig. 2a,the cut-off frequency is about 0.5 Hz (see insert in Fig. 2a). For a P-wavevelocity of 2000 m/s, this gives awavelength of 4 km. Thus, at a distanceof up to 4 km in direction θ=0, the amplitude of the near-field termwillbe comparable to that of the P-wave (in volcano seismologyseismograms are often recorded only a few hundred metres from thesource). This is illustrated in Fig. 4. The figure shows the verticalcomponent of displacement calculated for three directions (θ=0, π/9andπ/2, respectively), at a distanceof 4 km fromthevertical forcepoint-sourcewith the source-time function as in Fig. 2a. It can be seen that thenear-field term has a detrimental effect on the shape of seismograms.

As previously mentioned, a near-field term polarisation is differentto either P- or S-wave polarisations, thus compromising the use ofpolarisation analysis to facilitate source locations. Especially interest-ing is the seismogram recorded at the horizontal direction (θ=π/2,right column in Fig. 4): the combination of the near-field and S-waveforms a waveform arriving at the receiver at the time of P-wavearrival, and has linear vertical S-wave polarisation.

Based on the analysis above, the properties of the wavefield observedin the near-field of a single force point-source can be summarised asfollows:

1. the wavefield radiated from a single force source consists of a near-field term and far-field P and S waves

2. the near-field term arrives at the receiver together with the P-wavearrival and lasts until the S-wave passes

3. for distances smaller than half of the dominant P-wave wave-length, the dependence of the amplitude of the near-field termwith distance is r−1, while for greater distances it decays as r−2

4. the near-field term is linearly polarised in a direction whichgenerally does not coincide with either P- or S-wave polarisation

5. as opposed to P and S waves, the radiation pattern of the near-fieldterm does not have any nodal planes; consequently, in the nodal

Fig. 4. Vertical component of the near-field (top), far-field (middle row) and totaldisplacement (bottom row) recorded at 4 km (~1 P-wave wavelength) from thevertical force source (F0=1010N), for three different orientations: θ=0 (left column),θ=π/9 (middle column) and θ=π/2 (right column). The source-time function is thesame as in Fig. 2, with T=2 s, that is, a cut-off frequency of 0.5 Hz. P- and S-wavevelocities were 2000 and 1150 m/s, respectively, and the density 2500 kg/m3. Note thatat θ=π/2 first motions are dominated by the near-field term and have oppositepolarities to far-field P-waves for θbπ/2.

directions of P-wave, only the near-field wave is observed at the P-wave arrival time, and it is emergent and polarised like an S-wave

6. for small distances, the waveform of the near-field term is identicalto those of P and S waves, while its shape changes as the distancegrows; in other words, the near-field term is characterised by adistance-dependent spectrum.

3. Near-field properties of the wavefield radiated by a generalmoment-tensor source

The seismicwavefield excited by amoment-tensor source—such asdouble-couple source, tensile crack, pipe, explosive source—hassomewhat different properties than that excited by a single force.Thus, such a source will produce a displacement u(r,t), which, whenrecorded inside an unbounded homogeneous medium at position rand time t, has a form (Aki and Richards, 2002, eq. 4.32):

u r; tð Þ = RN

4πρ r4∫r =β

r =α

τM t−τð Þdτ +R

IP

4πρα2r2M t− r

α

� �

+R

IS

4πρβ2r2M t−r

β

� �+

RFP

4πρα3rM t− r

α

� �

+R

FS

4πρβ3rM t−r

β

� �;

ð6Þ

whereM(t) is the source-time history of the force couples and dipoles,and RN, RIP, RIS, RFP and RFS are near-field, intermediate-field P,intermediate-field S, far-field P and far-field S radiation patterns,respectively. It should be noted that alongside near-field and far-fieldterms like in the case of a single force presented above, the wavefieldalso comprises so-called intermediate-field P- and S-waves. Theseintermediate P- and S-waves have waveforms identical to those of thesource-time function, and they depend on distance as 1/r2, while far-field P- and S-waves depend on distance as 1/r and their waveformsare equal to the time derivatives of the source-time function. If themoment, M(t), never returns permanently to zero (as for standarddouble-couple earthquakes where slip on the fault has a final non-zero value), then the near-field and intermediate-field terms persistindefinitely, thus yielding the final static deformation of the medium.

Like in Section 2, it is convenient to transform the Eq. (6) to thefrequency domain in order to study the distance dependence of thenear-field term:

NF = RN MðωÞ4πρα2r2

i2πnλ

− 12πnλ

� �2� �⋅ei2πnλ− α

βi

2πnλ− 1

2πnλ

� �2� �⋅e

αβi2πnλ

� ;

IP = RIP MðωÞ

4πρα2r2⋅ei2πnλ ;

IS = RIS MðωÞ4πρβ2r2

⋅eαβi2πnλ ;

FP = RFP MðωÞ

4πρα2r2⋅i2πnλ⋅e

i2πnλ ;

FS = RFS MðωÞ

4πρβ2r2⋅iαβ2πnλ⋅e

αβi2πnλ :

ð7Þ

In the above equation, NF, IP, IS, FP and FS denote near-field term, Pand S intermediate-field terms and far-field P and Swaves, respectively,M(ω) is Fourier spectrumof the source-time functionM(t),α andβ are Pand S wave velocities, ρ is the density of the medium, and nλ is thenumber of the P-wave wavelengths.

Fig. 5. Dependence of the ratios between the near-field and far-field P-wave (blue line),and the intermediate- and far-field P-wave (red line), respectively, for a general moment-tensor source as a function of distancemeasured in number of P-wave wavelengths. Notethat both intermediate P-wave and near-field term decay faster than the far-field P-wavefor all distances. All 3 terms are comparable at distances of about 1/2π of the P-wavewavelength (marked by the dashed line). When considering a particular sourcemechanism, the radiation patterns RN, RIP and RFP of that mechanism have to be includedin the calculation of these ratios.


Comparing Eq. (7) to Eq. (3), and neglecting the radiation patterns, itcan be seen that the relationship between the near- and intermediate-field terms for a general moment-tensor source is identical to thatbetween the near-field and far-field terms for a single force source. Fig. 5shows relationships between all three terms for P-waves. It can be seenthat both the near- and the intermediate-field terms decays as 1/r2 toabout a half of the wavelength, while for greater distances the near-fieldterm decays as 1/r3. All three terms are comparable (denoted by thedashed line) at a distance of about 1/2π of the P-wave wavelength.However, this value will exhibit great variations when the radiationpattern of a particular source mechanism is considered. Hence in thefollowing, the radiation patterns of a purely isotropic source, a verticalCompensated Linear Vector Dipole (CLVD) (the symmetry axis is parallelto z-axis from Fig. 1a), a horizontal tensile crack (lying in the x–y planefrom Fig. 1a, with the symmetry axis parallel to z-axis) and a vertical pipe(the symmetry axis is parallel to z-axis from Fig. 1a) are examined indetail. The latter two sourcemechanisms are thought to be themost likelycandidates to generate long-period seismicity on volcanoes (e.g., Chouet,1985, 1988; Neuberg et al., 2000). Our decision to include both purelyisotropic and CLVD sources in the analysis stems from the fact that thesourcemechanisms of both a tensile crack andpipe can be represented byaweighted sumof these two typesof sources. In an idealised case,where atensile crack can be exactly represented by the theoretical point-sourcemoment-tensor, the percentages of the isotropic and CLVD sources in thecrack and/or pipe source mechanisms depend on the elastic properties ofthe source region (to calculate thesepercentages, seee.g. Vavryčuk, 2001).In a realistic case, the coupling between the source and surroundingmedium (boundary conditions) and the dynamics of the source can alsoplay an important role in determining volumetric change in the source, i.e.its relative isotropic/CLVD content.

Using the general expressions for radiation patterns given in Eq.(4.29) in Aki and Richards (2002), the radiation pattern of an isotropicsource can be expressed as:

RN = 0;

RIP = ur;

RIS = 0;

RFP = ur;

RFS = 0;

ð8Þ

of a vertical CLVD source:

RN = 9

41 + 3cos 2θð Þ½ �u

r+ 9

2sin 2θð Þu

θ;

RIP = 1 + 3cos 2θð Þ½ �ur +

32sin 2θð Þu

θ;

RIS = −3

41 + 3cos 2θð Þ½ �u

r−9

4sin 2θð Þu

θ;

RFP = 1−3

2sin

2θð Þ

� �ur

;

RFS = −3

4sin 2θð Þu

θ;

ð9Þ

of a horizontal tensile crack (for a Poisson's ratio of 0.25):

RN = ½3 + 9cos ð2θÞ�ur + 6sinð2θÞuθ;

RIP = ½3 + 4cosð2θÞ�ur + 2sinð2θÞuθ;

RIS = −½1 + 3cosð2θÞ�ur−3 sinð2θÞuθ;

RFP = ½2 + cosð2θÞ�ur;

RFS = − sinð2θÞuθ ;

ð10Þ

and of a vertical pipe (for a Poisson's ratio of 0.25):

RN = 3−9 cos2 θð Þ

h iu

r−3 sin 2θð Þuθ;

RIP = 1−2 cos 2θð Þ½ �ur− sin 2θð Þuθ;

RIS = 2−3 sin2 θð Þ

h iur+ 3

2sin 2θð Þu

θ;

RFP = 2− cos2 θð Þ

h iur;

RFS = 1

2sin 2θð Þu

θ:

ð11Þ

It can be seen from the above sets of equations that, except for anisotropic source, the polarisations of the intermediate P- and S-wavesare both linear and different to each other and to the polarisations ofthe near-field term and far-field P- and S-waves. This leads to anextremely complicated polarisation of the complete seismogram atsmall distances from the source for which all these terms are ofcomparable magnitudes. Graphical representations of the radiationpatterns given by Eqs. (9)–(11) are shown in Figs. 6–8, respectively.Since near- and intermediate-field terms comprise both the radial andtransverse components of polarisation, the absolute values of theirradiation patterns are also shown in the figures. As can be seen, onlythe S-wave has a nodal plane at θ=π/2 for all three mechanisms,while all the other wavetypes do not have such planes. Thus, the near-and intermediate-field terms will always “contaminate” the far-fieldobservations, regardless of the source–receiver direction. As seenfrom Eq. (8), the radiation pattern of a purely isotropic source, anotherpossible candidate for the generation of long-period volcanicseismicity, comprises only intermediate- and far-field terms, bothradially polarised. Consequently, the direction of polarisation for sucha source is always radial, regardless of the distance at which thewavefield is observed. However, the relative magnitude of theintermediate vs. far-field term will change with distance, producinga distance-dependent shape of the complete waveform.

Fig. 6. Radiation pattern of the near-field term, intermediate P and S waves, and far-field P and S waves for the vertically oriented CLVD source (Mxx=−1/2, Myy=−1/2, Mzz=1).The first column represents the radiation pattern of the radial component of motion, ur, the second of the transverse component, uθ, and the radiation pattern of the absolutedisplacement is shown in the third column (see Eq. (9)). Note that the radial components of all the waves have nodal cones (rather than nodal planes) at θ≅55° and θ≅125°, wherethe transverse components have about 94% of their peak value; since the near-field term and intermediate P-wave travel at the P-wave velocity, thus at these cones, the first arrivalwill be polarised as an SV wave (analogous situation to the vertical force at θ=π/2, as shown in Figs. 1b and 4).


As mentioned before, the radiation pattern must be considered whenestimating the ratio between the near- and far-field terms. For example,for the observation in direction θ=0, the near-field term and far-field P-wave radiated from a horizontal tensile crack will still be comparable at adistanceof about 2/3of theP-wavewavelength (thevalue1/2π fromFig. 5must be multiplied by 4 (the ratio of the near-field and far-field P-waveradiation patterns for θ=0 in Fig. 7)). In a volcanic environment, wherekilometre long wavelengths are often radiated by several hundred metredeep sources, this effect cannot be neglected. This is illustrated in Fig. 9.The figure shows radial and transverse components of displacementrecorded 4 km (note that real observation are often made at much closer

distances, where the effect is more pronounced) from a horizontal tensilecrack source in direction θ=π/9. Although less pronounced than for asingle force source (see Fig. 4), it can be seen that the near-field effect stillhas a significant influence on the shape of the waveform (last row in thefigure). It is even more pronounced for source mechanisms where boththe far-field wavetypes have nodal planes. Thus, for example, usingnumerical simulationsVavryčuk(1992) found that “contamination”of thefar-field polarisation by near-field terms can be found even at distances aslarge as 10 ormorewavelengths. Despite this and other points above, thenear-field term is largely ignored and there are not many publishedstudies on this matter. While this may be justified for earthquake

Fig. 7. Radiation pattern of the near-field term, intermediate P and S waves, and far-field P and S waves for a horizontal tensile crack (see Eq. (10)). Notice that the far-field P-wavedoes not have nodal planes for this source mechanism; hence the polarisation of the first arrival will be generally less contaminated by the near- and intermediate-field terms than inthe CLVD case (see Fig. 6). However, for the angle θ=π/2, the near-field term has opposite sign to the far-field P-wave, so it will decrease the amplitude of the first arrival, making itdifficult to detect in the presence of noise.


seismology where most observations are performed in the far-field, thisis not the case for volcano seismology where observations are mainlyundertaken in the near- and intermediate-field.

4. Examples of the near-field effect in seismic observationsand analyses

So far, we have analysed the properties of the near- andintermediate-field terms and showed some examples of how theyaffect recordedwaveforms (see Figs. 4 and 9). For a better illustration ofthe near-field effect in real applications, in this section we will usesynthetic seismograms to show examples of the near- and intermedi-

ate-field terms influence on the observed polarisation of groundmotionand source location techniques for LP events. The synthetic seismo-grams were calculated at the locations of 25 seismic stations deployedon Etna during an experiment in June 2008 and 7 stations belonging tothe permanent seismic Etna network (Fig. 10). In order to isolate thenear-field from the topography and heterogeneity effects, the seismo-grams in an infinite homogeneous medium (vP=2000 m/s,vS=1150m/s, ρ=2500 kg/m3) were calculated, using the Eqs. (1)and (6) (that is, the topographymerely defines station elevations in ourcalculation; wave scattering and waveform distortion from topographyare not included). The source-time function used in the calculationswasthe Gaussian pulse with a dominant period of 2 s (see Fig. 2a).

Fig. 8. Radiation pattern of the near-field term, intermediate P and S waves, and far-field P and S waves for a vertical pipe (see Eq. (11)). As for a horizontal tensile crack, there are nonodal planes for the far-field P-wave. For the angle θ=0, far-field P-wave is of opposite sign to the near-field term and intermediate-field P-wave, so the amplitude of the first arrivalwill be decreased for this angle.


4.1. First motions polarisation

In order to illustrate the near-field effect on the first-motionpolarisations, we calculated synthetic seismograms for a realisticsource position located by Saccorotti et al. (2007), 800 m beneath thesummit of Mt. Etna. The particle motions were calculated for the timewindows starting at the beginning of the signals and ending at the S-wave arrivals. We used a suite of the source mechanisms, namely:vertical single force, vertical CLVD, horizontal CLVD, a double-couple(DC) strike–slip fault, vertical tensile crack and a vertical pipe.According to the literature, the latter two source mechanisms are themost likely candidates for LP sources. However, since every sourcemechanism can be decomposed into a purely isotropic, CLVD and DCpart (e.g., Vavryčuk, 2001), it is of interest to analyse the polarisationproduced by these source mechanisms as well (apart from a purelyisotropic source that, as stated before, will always produce radialparticle motions). The results of the polarisation analysis for alloutlined sources are shown in Fig. 11.

It can be seen that signals radiated by the source mechanisms thatpossess a single axis of symmetry and produce no net volume change,such as the vertical force (Fig. 11a), vertical CLVD (Fig. 11b) andhorizontal CLVD (Fig. 11c), do not exhibit deviations from the radialdirection in the plane whose normal is parallel to the axis ofsymmetry. However, they do exhibit deviations in all other directions.For the vertical force and vertical CLVD, the plane whose normal isparallel to the symmetry axis is horizontal, while for the horizontalCLVD with the axis of symmetry in x-direction, it is the y–z plane. Amore complicated case can be observed for a strike–slip DC source(Fig. 11d), where deviations from the radial direction can be observedin all planes. Although it is thought that the latter is not a likelycandidate for LP sources, it is included here for completeness.

A much weaker near-field effect on the wave polarisation can beobserved for a tensile crack source (Fig. 11e) and almost none for apipe source (Fig. 11f). The reason for this is that both of these sourcescomprise a strong isotropic component, namely 56% for a tensile crackand 83% for a pipe (see Eq. (8a) in Vavryčuk (2001)). The isotropic

Fig. 9. Near-field, intermediate-field, far-field and total displacement recorded at 4 km(~1 P-wave wavelength) from a horizontal tensile crack source (M0=1013Nm) for thedirection angle θ=π/9. Displacement is projected into radial, ûr, and transverse, ûθ,directions (left and right columns, respectively). It can be seen that the far-field P and Swaves are strongly affected by the near- and intermediate-field terms, thus making theseparation of P and S waves at near-field stations a difficult (if not impossible) task. Thesource-time function is the same as in Fig. 2a, with T=2 s, that is, a cut-off frequency of0.5 Hz. P- and S-wave velocities were 2000 and 1150 m/s, respectively, and the density2500 kg/m3.


component, according to our Eq. (8), tends to diminish the deviationof the ground motion from the radial direction. This is clearlyillustrated by an example shown in Fig. 12. The figure shows particlemotion analysis of the first arrivals (everything up to the S-wavearrival time) for an oblique CLVD source (the orientation of thesymmetry axis is θ=72° and φ=35°), an isotropic source, and anoblique tensile crack which is a weighted sum of the CLVD andisotropic sources. For each of these three sources, particle motions offirst arrivals were calculated at 250 m spaced grid points along three

Fig. 10. Receiver positions and Mt. Etna topography. The receiver positions coincidewith the 32 seismic stations deployed during the experiment on Etna in June 2008(open circles) and with 7 stations belonging to the permanent seismic network (blackdiamonds). The star denotes the epicentre of the source used for synthetic calculations.Epicentral distances from recording stations range from 50 m to 3.8 km. The elevationstep between adjacent contours is 200 m, starting at the centre from 3200 m above sealevel.

perpendicular planes passing through the source location. It can beseen that in the case of the CLVD source, the first motion polarisationsare strongly contaminated by the near-field effect. However, due tothe isotropic component, the tensile crack wavefield exhibits mostlyradial motion, where in certain regions particle motions just slightlydeviate from linear trajectories. This result leads to an optimisticperspective as most of the shallow LP and VLP sources on volcanoesare thought to have a strong isotropic component. A purely isotropicsource, as a special case, produces radial motion for the whole signalduration, so it could theoretically be distinguished based only onparticle motion analysis. However, in practice, the topography andheterogeneity strongly complicate the wavefield, so full waveform 3Dnumerical simulations (e.g., using the numerical scheme described inO'Brien and Bean (2004)) are needed to strip out these effects (seeexamples in Lokmer (2008)).

4.2. Locating LP signals using a dense near-field seismic network

Locating LP events is a challenging task due to the usuallyemergent onset of LP signals. Since classical techniques based onarrival times cannot be applied, a suite of different methods wasdeveloped for locating LP seismicity, such as the semblance method(e.g., Neidell and Taner, 1971; Kawakatsu et al., 2000; Almendros andChouet, 2003), cross-correlation methods (e.g., De Barros et al., 2009),array techniques with frequency-slowness analysis (e.g., Almendroset al., 2001), amplitude decay techniques (Battaglia et al., 2003),coupled inversion for location and moment tensor (e.g., Kumagaiet al., 2002) or travel time inversion with improved onset timeidentification achieved through stacking similar events (Rowe et al.,2004; Saccorotti et al., 2007). The choice of method depends on thenetwork configuration (a sparse network or arrays, azimuthalcoverage, near-field vs. far-field density of deployment etc.) and thetype of recorded seismicity (individual events or swarms, coherencyof the waveforms across the network, strong isotropic component inthe signals, etc.). A strong limiting factor for the location accuracy,especially depth, is usually a poorly resolved superficial velocitymodel (hence in most studies a homogeneous model has been used)and strong scattering of the wavefield on topography. According toLokmer et al. (2007) and Bean et al. (2008), the distortion ofwaveforms is minimised when they are recorded at close distancesfrom the source. In such a case, where there is a large number of near-field records and where a source is purely or nearly isotropic, thesemblance and cross-correlation locating techniques appear to be agood choice for locating the source.

The cross-correlation locating technique is, in fact, a reversedversion of the semblance method. Both of them assume a purelyisotropic source and a high degree of the waveforms coherency acrossthe network. Almendros and Chouet (2003) showed that theirdefinition of semblance is equivalent to an averaging of the cross-correlation coefficients of all possible channel pairs. Their locatingprocedure is based on the time-shifting of the recorded waveformsbased on the travel-times calculated for a number of sources in thesource region, until the semblance reaches its maximum. In the cross-correlation approach, the lag times, τij, of the maximum cross-correlation between all station pairs and corresponding theoreticaltravel-time differences, tij, are calculated; the source location is foundby minimising the sum of differences ||τij–tij||.

In the following, we apply the cross-correlation locating techniqueto a suite of synthetic seismograms generated by a purely isotropicsource at 11 different depths beneath the summit of Mt. Etna, rangingfrom 50 to 3250 m. The stations configuration is as in Fig. 10, and thesource-time function, structural model and synthetic seismograms areas described at the beginning of Section 4. A grid-search for sourcelocation was performed in spatial steps of 25 m. Again, in order toisolate the near-field effect, the seismograms were calculated for ahomogeneous infinite medium, using Eq. (6). In order to test our

Fig. 11. Polarisation analysis of the first arrivals for a suite of different source mechanisms: a) a vertical force, b) a vertical CLVD source (Mxx=Myy=−1/2,Mzz=1), c) a horizontalCLVD source (Mxx=1,Myy=Mzz=−1/2), d) a horizontal double-couple strike–slip source (Mxy=Myx=1), e) a vertical tensile crack (Mxx=3,Myy=Mzz=1), and f) a vertical pipe(Mxx=Myy=2, Mzz=1).


Fig. 12. Polarisation analysis of the first arrivals (before S-wave) for an oblique CLVD source (the orientation of the symmetry axis is θ=72° and φ=35°), an isotropic source, and anoblique tensile crack (the orientation of the symmetry axis is the same as for the CLVD source). The particle motions were calculated at the 250 m spaced grid points along threeperpendicular planes passing through the source location. The amplitudes of the particle motions are normalised at each grid point. Note that although a strong near-field effect canbe observed for the CLVD source, due to the influence of the isotropic component, this effect diminishes for the tensile crack.


implementation of the location method, we first use the seismogramscalculated by using far-field terms from Eq. (6) only. As expected, allrecovered locations were correct. Then we repeat the procedure usingcomplete seismograms (far-, intermediate- and near-field terms). Theresults are listed in Table 1. It can be seen that the epicentres arelocated correctly, while there are quite large errors in recovereddepths, especially for shallow depths (e.g., 425 m error for 50 m

Table 1Location errors for different source depths (the location errors in the East, North and verticacross-correlation technique (see text). The column printed in bold corresponds to the locat

Source depth, h(m)

50 150 250 550 750

Δh(m) 425 375 300 175 150Δx(m) 25 25 0 0 0Δy(m) −25 −25 0 0 0

source depth). These errors decrease as depth increases for thefollowing reasons: (i) the intermediate-term becomes smaller thanthe far-field term, and/or (ii) the relative importance of theintermediate-term becomes uniformly distributed across the net-work, and hence it ceases to cause the fluctuations in waveformcoherency. For visual illustration, the time residuals (the average timedifference between a pair of stations) for the depth of 250 m (the third

l directions are denoted by Δx, Δy and Δh, respectively). The events were located usingion result shown in Fig. 13.

950 1150 1350 1550 1750 3250

125 100 75 75 75 250 0 0 0 0 00 0 0 0 0 0

Fig. 13. 3Dmap of the location residuals for a source depth of 250 m below the summit of Mt. Etna. The residuals, denoted by the colour bars, represent the mean time difference (inseconds) between themeasured and theoretical cross-correlation lag times for all station pairs; a) grid-search residuals for the case where only far-field waves are taken into account(the source location is correctly recovered), and b) grid-search residuals where complete seismograms (including near- and intermediate-field waves are analysed (the recoveredsource location is 300 m deeper than the true source location).


column in Table 1, printed in bold) are shown in Fig. 13. Fig. 13a showsthe correctly recovered location using just far-field terms, whileFig. 13b shows residuals obtained using complete seismograms,suggesting an overestimated source depth.

Our results show that, even in the noise-free environment andwith a perfectly known non-scattering medium, the near- andintermediate-field terms can significantly influence the accuracy ofthe shallow source locations. On the other hand, as stated before, thewaveforms recorded at distant stations will be strongly influenced bythe joint heterogeneity-topography effect. Moreover, in the realworld, P-to-S waves conversions may occur on the structuralheterogeneities and/or sources may differ significantly from purelyisotropic, accentuating near-field effects. All these factors willinevitably introduce errors to the retrieved source locations. Althoughwe cannot avoid these errors, we can minimise them by using 3Dnumerical simulations in the heterogeneous medium with topogra-phy, where the near-field effect is inherent to the computations. Forexample, based on such simulations, De Barros et al. (2009) found anoptimal way of weighting the dataset that minimised the locationerror in their cross-correlation location procedure for real LPseismicity at Mt. Etna. Further improvements in the accuracy of thesource locations and inversions will depend on improved accuracy innear-surface velocity determination.

5. Conclusions

The aim of this study was to highlight the important contributionthe near-field term makes to long-period wavefields on volcanoes. Asis shown, it can have significant implication for polarisation analysisof LP and VLP events. An optimistic aspect of the analysis presentedherein is that this effect is not strongly pronounced for sources with astrong isotropic component, the most likely candidates for volcanic LPand VLP sources.We also showed that the near-field term can stronglyinfluence the accuracy of retrieved source locations. Apart from thecases that are illustrated in this paper, there are many more exampleswhere the near-field term can compromise the reliability of the

obtained results. For example, since the near-field term has variabledecaying properties for small distances, it can also introduce errors tothe amplitude decay location techniques. Furthermore, a distance-dependent spectrum of the near-field termmay distort the coherencyof the wavefield, thus compromising array locating techniques inwhich a sparse network is used as a large-aperture array (small-aperture arrays will not be affected by this effect). The outlined issuesstem from the fact that the near-field term has different polarisation,radiation pattern and spectrum to the far field terms. However, abeneficial aspect of near-field stations is that they lead to lesspronounced path effects and hence they are favourable for performingmore reliable source inversions on volcanoes. Also, due to differentproperties of the near-, intermediate- and far-field terms, moreinformation about the seismic source is contained in a signal recordedat close distances. Hence such a signal will impose more constraintswhen performing source inversions. An important aspect of this studyis to further encourage the use of 3D full waveform numericalsimulations in volcano seismology, as the near-field (as well as thetopography distortion) effect is inherent in these simulations.

Acknowledgements

Financial Support from Science Foundation Ireland (SFI) isacknowledged.

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Documents

Properties of the near-field term and its effect on ... · for long- (LP) and very-long-period (VLP) volcanic seismicity, including: a single force, compensated linear vector dipole