multiscale fracturing as a key to forecasting volcanic

$: Multiscale fracturing as a key to forecasting volcanic$
Multiscale fracturingas a key to forecasting volcanic eruptions

Christopher R.J. Kilburn �

Ben¢eld Hazard Research Centre, Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, UK

Received 10 October 2002; accepted 7 April 2003

Abstract

Among volcanoes reawakening after long repose intervals, the final approach to eruption (V1^10 days) isusually characterised by accelerating rates of seismicity. The observed patterns are consistent with the slow extensionof faults, which continue to grow until they connect a pre-existing array of subvertical fractures and so open a newpathway for magma to reach the surface. Rates of slow extension are here investigated assuming that gravitationalloading and magma overpressure create a fluctuating stress field in the country rock. Fluctuations are due to theintermittent growth of small cracks that cannot be detected by monitoring instruments. The rate of detected events isthen determined by the frequency with which the concentration of strain energy around the tips of macroscopic faultsbecomes large enough to permit fault extension. The model anticipates oscillations in seismic event rate about a meantrend that accelerates with time, and identifies the rate of increase in peak event rate as a key indicator of theapproach to eruption. The results are consistent with earlier empirical analyses of seismic precursors and, applied todata before the andesitic eruptions of Mt Pinatubo, Philippines, in 1991, and of Soufriere Hills, Montserrat, in 1995,they suggest that, for such volcanoes, reliable forecasts of magmatic activity might eventually be feasible on the orderof days before eruption.3 2003 Elsevier Science B.V. All rights reserved.

Keywords: eruption forecasting; fracture mechanics; slow cracking; Boltzmann distribution

1. Introduction

Volcanic eruptions are normally preceded bymeasurable signs of growing unrest. Although arange of geochemical and geophysical precursorshave been identi¢ed (McGuire et al., 1995; Scarpaand Tilling, 1996; Sigurdsson, 2000), the mostevident are increases in seismicity and ground de-

formation, caused by magma distorting the crustas it pushes its way to the surface.

Although magma will naturally follow existingfractures through a volcano, only under the spe-cial conditions of persistent activity is it likely to¢nd an open pathway to the surface. More fre-quently, a new pathway must be formed by frac-turing. For a magma body to migrate, therefore,it must normally exert a stress large enough topropagate a new fracture and to open the fracturesu⁄ciently wide to permit £ow against its ownrheological resistance. Rates of magma ascent

0377-0273 / 03 / $ ^ see front matter 3 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0377-0273(03)00117-3

* Fax: +44 (20) 7387 1612.E-mail address: [email protected] (C.R.J. Kilburn).

VOLGEO 2623 23-6-03

Journal of Volcanology and Geothermal Research 125 (2003) 271^289

R

Available online at www.sciencedirect.com

www.elsevier.com/locate/jvolgeores

mailto:[email protected]

http://www.elsevier.com/locate/jvolgeores

will thus be limited either by the rate of fracturegrowth, or by the rate of magma £ow through thefracture.

Standard models of dyke propagation (Shaw,1980; Spence and Turcotte, 1985; Lister, 1991)assume that fracturing is too fast to restrict mag-ma ascent, so that the rate-limiting process is theability of buoyant magma to overcome its viscos-ity. However, although rapid, dynamic failure isperhaps the most familiar form of fracturing, suchbehaviour is often preceded or replaced by a slow-er mechanism (Fig. 1), in which large fracturesextend by the growth and coalescence of smallercracks around their tips (Anderson and Grew,1977; Heimpel and Olson, 1994; Main, 1999,2000). Slow edi¢ce fracturing, therefore, cannotbe excluded as a control on rates of magma as-cent. Indeed, given that standard models have yet

to provide a basis for forecasting eruptions, it ispertinent to consider the case when slow edi¢cefailure is the rate-limiting mechanism. This paperfocusses on how populations of cracks interact topropagate a new pathway for magma to reach thesurface. It will argue that similar styles of crack-ing can be expected over a large range of lengthscales and use this characteristic to propose a newprocedure for forecasting eruptions from precur-sory accelerations in seismic event rate.

2. Empirical analyses of self-acceleratingprecursors

A slow-fracturing control is consistent with thecharacteristic of pre-eruptive signals that theyshow a restricted range of acceleration before aneruption, independent of the speci¢c precursor inquestion. Voight (1988, 1989) quanti¢ed thisrange through an empirical power-law relationbetween the rate of change of a precursor (d6/dt) and its acceleration (d26/dt2) :

d26 =dt2 ¼ Aðd6 =dtÞK ð1Þ

where A is a constant and K lies between 1 and 2.Since an in¢nite d6/dt implies an uncontrolledrate of change, this condition is associated withthe collapse of resistance to magma ascent and, asa result, with the start of an eruption. The fore-casting potential of Eq. 1 is in describing the rateat which d6/dt approaches uncontrolled condi-tions.

Comparing precursory trends with the acceler-ations to failure recorded for engineering materi-als, Voight (1989) implicitly identi¢ed cracking asa probable control on pre-eruptive signals. Thisassumption was made explicitly in analyses ofmovements before dome explosions at Mount StHelens (McGuire and Kilburn, 1997) and of pre-cursory seismicity before the 1995 eruption ofSoufriere Hills volcano on Montserrat (Kilburnand Voight, 1998). These studies argued in partic-ular that: (a) for a population of cracks, the ex-ponent K in Eq. 1 increases from 1 to 2 as thedominant crack mechanism changes from cracknucleation to crack growth, and (b) since growthdominates the ¢nal stages of deformation, the ap-

Fig. 1. Regimes in rates of crack growth. As a crack extends,it releases strain energy by relaxing surrounding rock, butconsumes energy in creating and separating new crack surfa-ces. For large cracks, strain energy release dominates lossesand the crack propagates dynamically. For cracks below athreshold size (determined by the fracture mechanical proper-ties of rock and imposed stress ¢eld), the energy released isjust greater than that consumed, so that the crack extendsslowly under conditions of quasi-equilibrium (shaded). Con-ventional studies of dyke propagation assume that crackingis in the dynamic regime. This analysis focusses on crackingunder quasi-equilibrium conditions. Note that, for step-likegrowth, crack velocity is proportional to event rate (see alsoEq. 8; simpli¢ed after Main, 1991).

VOLGEO 2623 23-6-03

C.R.J. Kilburn / Journal of Volcanology and Geothermal Research 125 (2003) 271^289272

proach to uncontrolled propagation is describedby the condition KW2.

When K=2, Eq. 1 can be re-expressed in theform:

ðd6 =dtÞ31 ¼ ðd6 =dtÞ310 3Aðt3t0Þ ð2Þ

where t is time, and (d6/dt)0 is the value of (d6/

dt) when t= t0. Thus, a plot of inverse rate againsttime follows a negative linear trend, so that thetime at which the inverse rate is zero, correspond-ing to the uncontrolled condition when d6/dttends to in¢nity, can be obtained by simple linearextrapolation of the measured trend to the timeaxis (Fig. 2). The linear form is important, be-

Fig. 2. Pre-eruptive conditions expected from Voight’s empirical relation for K=2 (Eqs. 1 and 2. Although the change in eventrate with time (panel a) may qualitatively appear to be exponential, it is in fact hyperbolic, so that the inverse rate shows a clearlinear decrease with time (panel b).

VOLGEO 2623 23-6-03

C.R.J. Kilburn / Journal of Volcanology and Geothermal Research 125 (2003) 271^289 273

cause it is simpler to extrapolate than non-lineartrends and so is less prone to observer error dur-ing an emergency.

The fracturing interpretation, however, wasbased on a comparison between Eq. 1 and anew relation for 6 derived from the growth ofmicroscopic cracks (McGuire and Kilburn, 1997;Kilburn and Voight, 1998). The assumption thatlarge-scale fracturing, which triggers measuredseismic events, shows behaviour similar to micro-scopic cracking was justi¢ed retrospectively by thegood agreement between expected and observedrates of seismicity. The assumption, however,has not been formally justi¢ed.

Eq. 1 has been applied to changes in precursoryrates of seismic energy release, ground deforma-tion and seismic event rate (Voight, 1988, 1989;Voight and Cornelius, 1991; Cornelius andVoight, 1995, 1996; McGuire and Kilburn,1997; Kilburn and Voight, 1998; De la Cruz-Rey-na and Reyes-Da¤vila, 2001; Reyes-Da¤vila and Dela Cruz-Reyna, 2002). Of these, seismic event rateis the most useful during an emergency because itis simple to measure, even with a temporary mon-itoring network, and requires minimal data pro-cessing. Accordingly, this paper will focus on howseismic event rate can be related to the macro-scopic development of crack populations and thepropagation of a pathway that allows magma tobreach the surface. By concentrating on the exten-sion of a fracture system, the analysis is implicitlydirected towards changes in the rate of volcano-tectonic (VT) events, rather than the long-period(LP) events that are commonly associated withpressure changes induced by £uids (e.g. magma,released magmatic volatiles or meteoric £uids) mi-grating through fractures (Chouet, 1996; McNutt,2000).

3. Features of pre-eruptive seismicity

Short-term pre-eruptive seismicity normallyevolves at depths of less than 5^7 km (McNutt,1996). Events tend to cluster (a) beaneath the vol-cano’s axis, within an approximately cylindricalregion some 1^3 km across (Fig. 3) and, especiallyduring the initial stages of unrest, also (b) around

one or more locations with epicentres o¡set fromthe axis by several kilometres (Aspinall et al.,1998; De Natale et al., 1991; McNutt, 1996;Harlow et al., 1996). Although events of magni-tude 5^6 have been recorded (Smith and Ryall,1982; Savage and Cockerham, 1984), most havemagnitudes less than 2 (Hill et al., 1990; McNutt,1996). For magnitudes of 0^2, the events are as-sociated with movements of about 1^10 mm overcorresponding distances of 10^100 m (McNutt,2000).

During the ¢nal acceleration to eruption, whichevolves over intervals of V1^10 days and is oftendominated by events within the ‘seismic cylinder’beneath the volcano’s axis, the total number ofevents with magnitudes of about 1 or more isfrequently V102^103 (Figs. 6 and 7). The impliedcumulative displacement is thus V10 m or less. Incomparison, magma bodies triggering the precur-sors have notional lengths of V1 km and mustforce their way at least through V1 km of a vol-canic edi¢ce. Apparently, the detected events can-not be explained by the simple upward displace-ment of a dyke. Indeed, events often occurthroughout the whole axial seismic cylinder dur-ing the entire precursory sequence without astrongly preferred upward migration (Aspinall etal., 1998; Harlow et al., 1996). The simplest inter-pretation is that cracks grow throughout thewhole axial seismic cylinder before they eventuallyjoin together an array of previously isolated frac-tures. Once connected, these fractures form thenew pathway for magma ascent. Because thekey objective of this paper is to investigate theuse of accelerating seismicity for forecasting erup-tions, the analysis concentrates on events trig-gered within the axial seismic cylinder (Fig. 3).The role of non-axial seismic clusters is left forevaluation elsewhere.

Precursory events are commonly dominated byreverse and strike-slip faulting. Such behaviour isconsistent with slip movements under compres-sional stress (due to gravitational loading and tomagmatic overpressure) in rock surrounding anarray of subvertical fractures (Fig. 3). Thus,even in a compressional ¢eld, subvertical fracturescan be preferentially reactivated when compres-sion is smaller horizontally than vertically, while

VOLGEO 2623 23-6-03


slip movements are favoured by the net uplift ofcountry rock between a magma body and the sur-face. Seismic events are triggered as the countryrock slips within restricted volumes, each with acharacteristic dimension L (notionally 10^100 m)between the subvertical fractures (Fig. 3). The re-stricted volumes may contain either (a) a collec-tion of smaller fractures that eventually coalesceinto a new fault, or (b) a continuous fault thatslips after small fractures have grown and coa-lesced around its ends. In either case, displace-ment along the length L, which triggers a detect-able seismic event, is determined by the growth ofthe small cracks. With repeated cycles of fractureextension and slip, faults within the restricted vol-umes eventually link with the subvertical fracturesto create a major new pathway to the surface.

4. Time-dependent rock fracture

Repeated failure can occur under constant ap-plied stress. Also called static fatigue, such a styleof failure is a well-known property of compositematerials, as well as rock (Anderson and Grew,1977). It is encouraged by high temperatures,which elevate the rate of chemical interactionswith a circulating £uid, especially water, andalso increase pore pressures in the host rock,

thereby further reducing frictional resistancealong existing faults.

Fractures remain stable until subjected to astress large enough to overcome sliding frictionand to provide the energy for opening new cracksurfaces (Scholz, 1990). Although not importantat very small scales, friction is the dominant fac-tor stabilising macroscopic fractures. Expressedenergetically, when failure occurs the strain en-ergy Em supplied by intruding magma is absorbedby deforming the country rock (Scholz, 1990):

Em ¼ Ed þ Ef þ Enc þ Es ð3Þ

where Ed is the energy used for ground uplift, Ef

is the frictional energy lost in sliding along frac-tures, Enc is the energy required for creating newcrack surfaces, and Es is the radiated seismic en-ergy.

Typically, Enc is much smaller than the otherterms on the right-hand side of Eq. 3 and so isusually neglected in deformation and seismic stud-ies. However, whereas Ed, Ef and Es are associ-ated with instantaneous responses to magmaintrusion, Enc is inherently time-dependent. More-over, since new crack surfaces must be formedbefore an existing fault can be displaced (therebya¡ecting all the other energy terms), Enc emergesas a small but essential control on rates of crustaldeformation and seismicity. The present analysis

Fig. 3. (Left) Pre-eruptive seismicity is commonly concentrated within an approximately cylindrical region around a volcano’saxis between a magma body and the surface (especially during the ¢nal approach to eruption; see text). Events occur throughoutthe ‘seismic cylinder’ for the entire pre-eruptive crisis ; an upward migration of events is not usually obvious. Such behaviour isinterpreted (centre and right) in terms of the extension of, and movements along, faults connecting pre-existing subvertical frac-tures above the magma body (black). The characteristic length L of the connection between vertical fractures (e.g. small box,right) is V10^100 m. See Fig. 4 for details of fault extension. As discussed in the text, additional seismicity may occur in clusterswith epicentres o¡set from the volcano’s axis, but this is not incorporated in the present analysis. (Not to scale.)

VOLGEO 2623 23-6-03


therefore focusses on how energy is supplied forcreating new crack surfaces and for controllingobserved rates of seismicity.

5. Seismic event rates and stress £uctuations

Detected seismic events are produced as faultsgrow between the ends of widening, subverticalfractures (Fig. 3). Fault extension itself is con-trolled by processes operating within the zone ofstress concentration, or process zone, that devel-ops around the fault’s tip when it is put understress (Atkinson, 1984; Main, 1991; Main andMeredith, 1991; Lockner, 1993; Cowie and Ship-ton, 1998). Within this zone, arrays of smaller,tensile cracks and slip surfaces grow and coalesceuntil the whole zone is broken, so extending theparent fault (Fig. 4). However, just as a large faultgrows by the extension and coalescence of smallcracks around its tips, so each of the small cracksalso grows by the extension and coalescence ofthe next smaller population of cracks in its ownprocess zone. Fracturing is thus viewed as a multi-scale process, major fractures being the result ofcracking at successively smaller scales, down tothe submicroscopic dimensions at which cracks

nucleate. Excluding nucleation events, therefore,it should be possible to resolve a recorded eventinto a sequence of growth steps among smallercracks whatever the scale of observation (Fig. 4).

As a crack of any size extends, it releases en-ergy by relaxing the strain in the rock immediatelyaround its new extension, but consumes energy increating and separating the new crack surfaces(e.g. Lawn, 1993). Extension continues as longas the energy released exceeds the amount con-sumed. Among a population of cracks, however,the overall pattern of growth is complicated byinterference between neighbouring cracks. Thus,the zone of stress concentration around a cracktip is enclosed by regions of reduced stress (Cow-ie, 1998). Accordingly, the growth of one crackcan be stalled if it extends into the zone of re-duced stress (or stress shadow) from a neighbour-ing crack. The ¢rst crack may then be reactivatedas the local stresses change to a more favourabledistribution, in response to the extension of an-other neighbour.

Slow cracking is thus characterised also by £uc-tuating stresses in the process zones of cracks ofall sizes. Events due to cracks smaller than athreshold value (determined by observing condi-tions, such as instrument sensitivity, the size and

Fig. 4. Faults extend by activating existing cracks in the process zones around their tips. Even if in absolute compression, thehorizontal component is smaller than the vertical component, yielding a tensile di¡erential horizontal stress. Hence tensile cracksopen, extend and coalesce around the tips of the fault. Upon coalescence, the fault can move along its new extension (left andcentre). (Right) The separation between new crack surfaces is controlled by relaxation of stress in previously unbroken rock(shaded), which is on the order of the extension rather than the length of the existing cracks (white) in the process zone. Notethat the ¢gures are scale-independent.

VOLGEO 2623 23-6-03


geometry of the monitoring array, the depth offracturing, and the attenuating properties of thefracturing medium) cannot be distinguished indi-vidually but, instead, they control the frequencyof the background variation in stress. The numberof events detected seismically then depends on therate at which the background stress is locally highenough to extend fractures that are larger thanthe threshold value.

5.1. Quantifying rates of cracking

Assuming that stress £uctuations occur ran-domly in a volume under stress, the probabilityof crack propagation is given by the proportion ofcracks that have the required additional strainenergy around their tips at any particular mo-ment. If the volume consists of P potential con-trolling cracks (where P is large), then the numberof ways 3 that random energy £uctuations willyield mi active cracks, with the strain energy Oi

necessary for propagation, is :

3 ¼ r!=Q

imi! ð4Þ

Importantly, it is not necessary to specify whichparticular members of the crack population areactive at any given moment, but only the numbermi that are active. The problem thus becomesanalogous to the classical Boltzmann analysis instatistical physics of describing the energy distri-bution among a large population of randomlymoving gas molecules (and, hence, of derivingthe associated gas pressure) for which it is impor-tant to know the total number of molecules in aparticular energy state, rather than to follow thechanges in energy state of individual molecules(Ruhla, 1992). Following the Boltzmann analysis(Ruhla, 1992; Landau and Lifshitz, 1999), there-fore, the probability P(Oi) that a crack tip is sup-plied with the extra energy Oi is given by determin-ing the maximum value of 3, which correspondsto the state of statistical equilibrium:

PðO iÞ ¼½Number of ways of containing energy O i�½Total number of energy states available�

¼ ð1=ZÞe3O i=nRT ð5Þ

where the total internal energy nRT consists of

the ideal gas constant, R (8.314 J mol31 K31),absolute temperature, T, and the number of molesn of rock involved in cracking a process zone;although originally determined from gas thermo-dynamics, the use of R is valid also for the inter-nal energy of solids. The partition function Z=gie3O i=nRT measures the total number of energystates in which a crack tip could exist.

The probability of cracking in Eq. 5 can betransformed into expected rates of cracking, byintroducing the mean frequency f of stress £uctu-ations, so that dN(t)/dt, the expected crack rate(or rate of successful stress £uctuations for crackextension), becomes fP(Oi). Given that f and Z areconstant for a given system, it follows that:

dNðtÞ=dt ¼ f �e3O i=nRT ð6Þ

where f* = f/Z.Eq. 6 combines cracking at di¡erent scales

through the exponent and pre-exponential termon its right-hand side. Thus, while f* describesthe rate of stress £uctuations (involving thosedue to cracking at sizes too small to be resolved),the term e3O i=nRT describes the proportion ofthose £uctuations that provide the energy Oi foractivating the cracks that can be detected. Impor-tantly, no constraints have been imposed on ab-solute crack size, so that the form of Eq. 6 shouldbe valid for rates of cracking at all scales(although some limits will be introduced later).

Eq. 6 provides a second link between fracturingat di¡erent scales, by noting that each event Ncorresponds both (a) to a growth step for onemember of a population of fractures of a partic-ular size (at, say, Scale 1), and (b) to a cycle ofgrowth and coalescence among the smaller cracks(Scale 2) in the process zone of the Scale-1 frac-ture (Fig. 4). Coalescence terminates the acceler-ation in Scale-2 event rate and so coincides withthe peak rate achieved by the Scale-2 cracks (Fig.5). Accordingly, a Scale-1 growth step can be ex-pressed in terms of the peak rate of Scale-2 crack-ing by:

dN1=dt ¼ 8 dN2P=dt ð7Þ

where, to simplify notation, N(t) is replaced by Nand understood to be a function of time, the sub-scripts 1 and 2 denote the scale of cracking, sub-

VOLGEO 2623 23-6-03


script P denotes peak value, and 8 is the scalingfactor (86 1). This relation will be used later torelate detected seismicity to fracturing at di¡erentscales.

By utilising the Boltzmann distribution in Eq.5, the derivation of Eq. 6 assumes an equilibriumsize^frequency distribution for the values of back-ground energy Oi, smaller values having a greaterfrequency of occurrence. For the event rate toaccelerate, therefore, the preferred value of Oi forfurther cracking must decrease with time (suchthat the associated frequency of occurrence in-creases). The next section describes how such acondition occurs naturally as propagating frac-tures release the strain energy already stored inthe host rock.

5.2. Event rate and rate of crack growth

Persistent slow cracking is maintained while therate of energy released by crack growth just ex-ceeds that required for new cracks to form. Fromstandard theory, this quasi-equilibrium conditioncan be related to the total change in energy Oi inEq. 6 by (e.g. Gri⁄th, 1920; Jaeger, 1969; Scholz,1990; Lawn, 1993):

O i ¼ ONS3OG ð8Þ

where ONS is the energy required to create newfracture surfaces, and OG is the excess strain en-ergy released by the surrounding rock as a crackgrows (equivalent to the total strain energy minusthe energy required to move new crack surfacesapart). In terms of di¡erential applied stress S andmaterial properties (Jaeger, 1969; Scholz, 1990;Lawn, 1993):

O i ¼ 2Kwvc3BðS2=Y 0Þcwvc ð9Þ

where K is the energy per unit area to create newsurfaces, w and c are the width and length of theextending crack, and vc is the increase in lengthper crack step. The constant B describes the ge-ometry of loading, and for straight-edged frac-tures is VZ, while the e¡ective Young’s modulusYP is a function of the actual modulus Y andPoisson’s ratio Xp (about 0.3 for crustal rock),the form of which depends on deformation con-ditions (e.g. for plane stress, YP=Y, whereas for

plane strain, YP=Y/(13X2p) (Lawn, 1993); to a

¢rst approximation, YPVY). Crack width (i.e.the length of its propagating edge), K and YPare assumed to be constant.

The prevailing stress ¢eld appears in the secondterm on the right-hand side of Eq. 9. Importantly,this term incorporates only the magnitude of Sand so has the same form for compressional andtensile ¢elds, as well as for di¡erent modes of rockfailure (Scholz, 1990; Lawn, 1993).

For a single crack:

c ¼ c0 þNvc ð10Þ

where c0 is the initial crack length, vc is the in-crease in length per crack step (assumed con-stant), and N is the number of steps already takenin extending from c0.

Eqs. 8^10 can be combined to relate the energyfor crack extension to initial crack size and to theprevious amount of cracking:

O i=nRT ¼ R3Q ðc0=vcÞ3QN ð11Þ

where R=2Kwvc/nRT (the ratio [Surface energyconsumed in extending crack by vc]/[Total inter-nal energy]) and Q=B(S2/YP)wvc2/nRT (the ratio[Strain energy released in extending crack by vc]/[Total internal energy]). As expected from the pre-vious section, as N increases with continued frac-turing, so the extra energy Oi for crack extensiondecreases. Moreover, the fact that the change in Oi

depends on the number of incremental increasesin length also demonstrates how the total energyin the fracture system can be viewed in terms ofthe net release of packets of energy, each packetassociated with a crack extension of vc.

Eq. 11 has been developed for the growth of asingle crack. A similar type of relation is antici-pated for a population of cracks, replacing valuesof w and c0 for a single crack with those averagedover the whole population, especially if the cracklengths follow a fractal frequency distribution(Main, 1991). Accordingly, it is assumed thatEq. 11 is valid for a collection of cracks usingaverage values for initial crack dimensions. Eq.11 can then be substitued for Oi in Eq. 6 to yieldfor the event rate:

dN=dt ¼ f �e3R eQ c0=vceQN ð12Þ

VOLGEO 2623 23-6-03


recalling that N is a function of time (N(t)).According to Eq. 12, event rates during crack

growth are expected to increase exponentiallywith the total number of growth steps N alreadycompleted. Such an exponential dependence is im-portant because relations of the form dg/dtOeKg

(where K is a constant and g is the relevant eventnumber) yield d2g/dt2OeKgdg/dtO(dg/dt)2, iden-tical to Voight’s empirical relation (Eq. 1) withK=2. Indeed, after manipulation (Appendix),Eq. 12 can be re-expressed in the same form asEq. 2:

ðdN=dtÞ31 ¼ ðdN=dtÞ310 3Q ðt3t0Þ ð13Þ

(dN/dt)0 is the value of dN/dt when t= t0. Eqs. 12and 13 thus provide a direct link between macro-scopic rock fracture and the Voight model.

5.3. Detected seismicity and the opening ofmagmatic pathways

Applied to volcanic precursors, the detectedevent rate immediately before an eruption is at-tributed to the frequency of growth steps alongfaults in the axial seismic cylinder under stress(Fig. 3). This rate accelerates with time until atleast one fault has coalesced with a subverticalfracture. At this stage, some of the excess strainenergy for new cracking is dissipated, because thefault has propagated into the void of an openfracture, rather than into country rock. Theopen fracture thus extends by the length of thenewly coalesced fault. Under the maintained mag-matic stress, however, cracking continues as otherfaults and fractures grow in the seismic cylinder.Against an accelerating overall trend, therefore,the variation of event rate with time (Fig. 5) isexpected to show episodes of acceleration (faultextension and coalescence) separated by intervalsof reduced event rate (energy dissipation at coa-lescence).

While detected events correspond to growthsteps along faults, the local peak values corre-spond to growth steps among the subvertical frac-tures. The two cases are related to cracking atdi¡erent scales : faults extend by the coalescenceof cracks in their process zones, whereas subver-tical fractures extend by opening newly coalesced

faults. Accordingly, each sequence of acceleratingevent rate and the change in peak value from onesequence to the next are both expected to increaseaccording to trends of the form of Eq. 12. Thus,setting the detected seismicity (fault steps) asScale-2 events and their peak values (main-frac-ture steps) as Scale-1 events, Eqs. 7 and 12 leadto:

dN2=dt ¼ f �2e3R 2 eQ 2c02=vc2 eQ 2N2 ð14aÞ

ðdN2=dtÞ31 ¼ ðdN2=dtÞ310 3Q 2ðt3t0Þ ð14bÞ

and

dN2P=dt ¼ ð1=8 ÞdN1=dt ¼

ðf �1=8 Þe3R 1 eQ 1c01=vc1 eQ 18N2P ð15aÞ

ðdN2P=dtÞ31 ¼ ðdN2P=dtÞ310 38 Q 1ðt3t0Þ ð15bÞ

noting, from Eq. 7, that N1 =8N2P.Given that the variation with time in peak

event rate describes the extension of the primarysubvertical fractures, it follows that Eq. 15 is themore appropriate for monitoring the growth of anew magmatic pathway. Hence, in contrast tochanges in actual event rate, as used in previousstudies (e.g. Cornelius and Voight, 1996; Kilburnand Voight, 1998), the change in peak event ratemay be the more reliable trend for forecasting thetime of an eruption.

5.4. Submicroscopic limits

Although no macroscopic constraints have beenplaced on Eq. 12, a submicroscopic limit can beperceived when growing cracks are the equivalentof crack nuclei. By de¢nition, such nuclei are thesmallest cracks available, so that the extension oftheir process zones must depend on mechanismsother than the growth of even smaller cracks. Inthis limit, rates of crack growth are controlled bymolecular processes, of which the most favouredis chemically enhanced stress corrosion (Charlesand Hillig, 1962; Wiederhorn and Bolz, 1970;Lockner, 1993). Such corrosion assumes that £u-ids chemically weaken molecular bonds in a ma-

VOLGEO 2623 23-6-03


terial, so promoting the growth of crack nucleiwhen an external stress is applied. The chemicalreactions require repeated attempts by excitedmolecules to break the bonds around a cracktip. The rate of crack growth thus depends on

the mean frequency X of molecular attempts tobreak bonds and on the probability P(Ei) thatan attempt will have su⁄cient energy Ei forbreakage to occur. Replacing the backgroundrate of stress £uctuation f with X and the proba-

Fig. 5. Schematic change in peak event rate as precursory trend before an eruption. (a) The recorded event rate (all triangles) in-creases with time about an oscillating trend (small dashes). Plotted on an inverse-rate diagram (panel b), however, di¡erent linearinverse trends become apparent between sequences of cracks coalescing aound the tips of faults (open triangles with small dashedlines) and in the extension of the faults themselves, given by the peaks in monitored event rate (¢lled triangles with large dashedline).

VOLGEO 2623 23-6-03


bility P(Oi) with P(Ei), analogy with the develop-ment of Eq. 6 yields for the frequency of crackevents:

dN=dt ¼ ðX=ZÞe3Ei=nRT ð16Þ

where n is now the number of moles involved inthe chemical reactions that cause a crack to ex-tend, and Z is the appropriate partition coe⁄-cient.

As with Oi in Eq. 6, Ei decreases linearly withthe excess energy released by previous crackevents, such that E=E*3EG, where E* is the ac-tivation energy for a chemical reaction to occurand EGO(c0+Nvc) is the energy available fromthe N previous crack events. Accordingly, usingthe same arguments as for Eqs. 6 and 12, Eq. 16again leads to a result of the form dg/dtOeKg ,identical to that of Eqs. 14 and 15. Althoughthe controlling factors di¡er, therefore, the crackgrowth law at the smallest scale is expected tohave the same form as that at larger scales, thusaccounting for the previous success in extrapolat-ing submicroscopic crack models to macroscopicconditions (McGuire and Kilburn, 1997; Kilburnand Voight, 1998).

6. Application to eruption precursors

For initial veri¢cation of the slow-crackingmodel, Eqs. 14 and 15 have been tested againstthe seismic precursors recorded before the erup-tions of Soufriere Hills on Montserrat in 1995 andof Pinatubo, in the Philippines, in 1991. In bothcases, the eruptions began with the extrusion ofandesitic lava domes. However, whereas Pinatu-bo’s activity evolved within days into a plinianeruption (Newhall and Punongbayan, 1996), ac-tivity at Soufriere Hills has continued to 2003 (thetime of writing) to be dominated by dome growthwith minor explosivity (Young et al., 1998). Ac-cordingly, precursory data from the two eruptionsmay cover the spectrum of behaviour expected atleast from andesitic volcanoes.

6.1. Soufriere Hills, Montserrat, 1995

During the 2 weeks before magma breached the

surface on 15 November 1995, after a repose in-terval of some 350 years, precursory activity atSoufriere Hills was characterised by a 10-fold in-crease in the rate of earthquake occurrence (Kil-burn and Voight, 1998). Recorded by an array ofshort-period, vertical-component seismometers(Aspinall et al., 1998), most events appear tohave had magnitudes of less than 2.5, althoughsome achieved magnitude 4 (Power et al., 1998).

The recorded seismic event rates and their in-verse are shown in Fig. 6. Kilburn and Voight(1998) analysed these trends using a slow-fracturemodel extrapolated from conditions for micro-scopic cracking. They identi¢ed a mean lineartrend among the inverse-rate data, but, owing tothe scatter, speculated that the event rates mighthave been controlled by the growth of more thanone fracture system. Their analysis, however, didnot account for the possibility ^ highlighted bythe present analysis ^ that accelerating fracturepropagation may be recorded as a sequence ofincreasing peaks in seismic event rate. Indeed,when the variations in peak rate (dNP/dt) are an-alysed using the sequence of minimum values onthe inverse-rate plot, a clear negative linear trendemerges with time (Fig. 6), described by:

ðdNP=dtÞ31 ¼ 0:03730:0024t ð17Þ

with an r2 regression coe⁄cient of 0.99 (for dNP/dt in events per day and t in days; see Kilburnand Voight (1998) for an error analysis). At thesame time, event-rate sequences between peaksshow negative linear inverse rates with gradientsof about 0.01 (estimated by eye from Fig. 6).

The inverse-rate trend in earthquake occurrenceis thus consistent with the present model, forwhich peak seismic event rate is the key factorfor monitoring the rate of fracture growth. Thehigh-value data on the inverse-rate trend can thenbe attributed not to scatter, but to lower eventrates between sequences of accelerating seismicity.Importantly, had this interpretation been avail-able at the time, a linear negative trend amonginverse-rate minima could have been identi¢edat least by 09 November. The four minima be-tween 02 and 09 November yield the regressionline (dN/dt)31 = 0.03130.0019t (r2 = 0.99) and anexpected eruption date of 16 November, 1 day

VOLGEO 2623 23-6-03


Fig. 6. (a) Change in recorded seismicity before the 1995 eruption of Soufriere Hills on Montserrat. Filled and open trianglescorrepond to symbols used in panel b. Arrow E marks onset of magmatic activity. On an inverse-rate diagram (panel b), the¢lled triangles, which correspond to inverse-rate minima, lie along a linear trend (large dashed line). Such a trend might havebeen recognised by Day 8^9 and so have provided a maximum 6-day warning of eruption. Note that peak values are more read-ily identi¢ed from the inverse-rate minima than the actual event-rate maxima, and that steeper linear trends occur among individ-ual sequences (small dashed lines).

VOLGEO 2623 23-6-03


Fig. 7. (a) Change in recorded seismicity beneath the summit before the 1991 eruption of Pinatubo on Luzon in the Philippines.Filled and open triangles correspond to symbols used in panel b. Arrow E marks onset of magmatic activity. On an inverse-ratediagram (panel b), the ¢lled triangles, which correspond to inverse-rate minima, lie along a linear trend (large dashed line). Sucha trend might have been recognised by Hour 112 and so have provided a maximum 48-h warning of eruption. Note that peakvalues are more readily identi¢ed from the inverse-rate minima than the actual event-rate maxima, and that steeper linear trendsoccur among individual sequences (small dashed lines).

VOLGEO 2623 23-6-03


later than observed. By 12 November, the esti-mate would have been revised to 15 November.Potentially, therefore, a reliable forecast couldhave been made 3^6 days ahead of the eruption.

6.2. Mount Pinatubo, Luzon, Philippines, 1991

Seismic monitoring provided the primary datafor evaluating the possibility of renewed activityat Mt Pinatubo in 1991, following a repose inter-val of some 500 years (Cornelius and Voight,1996; Harlow et al., 1996). After magma brokethe surface as a lava dome on 07 June, activityevolved to a climactic plinian eruption on 15June. The present analysis focusses only on seis-micity before the appearance of the lava dome.

Seismicity during the week before dome em-placement was characterised by an accelerationin event rate from about 10 to 60 events every4 h (Fig. 7; Harlow et al., 1996). Between 01and 04 June, most of the recorded earthquakeshad magnitudes greater than or equal to 0.4 andwere located at depths of 1^5 km about 4^5 kmnorthwest of the volcano’s summit; after 04 June,in contrast, seismicity was dominated by eventswith magnitudes of 1.2 or greater and located atless than about 3 km below the future summitvent (Harlow et al., 1996). The seismicity shownin Fig. 7 refers only to events recorded within theaxial seismic cylinder (Harlow et al., 1996).

Harlow et al. (1996) postulated that the accel-erating rate of seismicity was produced by intensebrittle fracturing as magma pushed its way to thesurface. Indeed, from 03 to 04 June, the 4-hourlyevent rate for sub-summit earthquakes shows aclear oscillation about an increasing mean trend(Fig. 7). When plotted on an inverse-rate diagram(Fig. 7), the sequence of minimum values followsa negative linear trend with time, described by:

ðdNP=dtÞ31 ¼ 0:251530:0064t ð18Þ

with an r2 regression coe⁄cient of 0.98 (for dNP/dt in events per 4 h and t in units of 4 h; for tmeasured in hours, as in Fig. 7, Eq. 18 becomes(dNP/dt)31 = 0.251530.0016t). In contrast, event-rate sequences between peaks show negative linearinverse rates with gradients of about 0.2^0.6 (esti-mated by eye from Fig. 7; for t in units of 4 h).

Once again, the inverse-rate trend among mini-mum values is consistent with that expected fromthe fracture-growth model. Moreover, consideringonly the trend that would have been apparentbetween 03 and 05 June (the four minima fromhours 48 to 112 in Fig. 7), extrapolation of theassociated regression line of (dNP/dt)31 =0.246330.006t with r2 = 0.97 (almost identicalwith Eq. 18) would have forecast an eruption dur-ing the evening of 07 June (hour 164), as wasobserved. In this case, therefore, the analysiscould have provided a 48-h warning of eruption.

6.3. Implications for short-term fracture growthbefore eruptions

From Eqs. 14b and 15b the inverse-rate gra-dients for individual sequences of events and forpeak event rate are, respectively, Q2 and 8Q1,where:

Q i ¼ BðS2=Y 0Þwivc2i =nRT ð19Þ

and the subscript i denotes the scale of crackingand all other terms are scale-independent.

For rock in compression within a volcanic edi-¢ce, S2/YPV105 J m33 (Rocchi, 2002). Temper-atures may range from about 300 K at the surfaceto about 1200 K at the tip of the dyke; a repre-sentative value is here taken to be TW700 K. Thenumber n of moles in the stressed volumeVi( = Niwivci) is bNiwivci/Wm, where b and Wm

are the rock’s mean density and molar mass,and Ni is the mean thickness of rock involved inreleasing energy while creating the Scale-i cracks.Accordingly, Eq. 19 becomes:

Q i ¼ ðvci=N iÞ½BðS2=Y 0ÞWm=bRT � ð20Þ

which shows that, in addition to the scale-inde-pendent terms in square brackets, the gradient forindividual sequences depends on the geometric ra-tio vci/Ni between the step length of crack exten-sion and the adjacent thickness of rock acrosswhich strain energy is released. It also followsfrom Eq. 20 that the inverse-rate gradient forpeak event rates 8Qi =8(vci/Ni)B(S2/YP)Wm/bRT,showing that this gradient should be smaller by afactor 8 ( =Ni/Nðiþ1ÞP) than the gradient for theindividual sequences leading to each peak value.

VOLGEO 2623 23-6-03


For a fractured volcanic edi¢ce, bW2200 kgm33 and WmW0.2 kg mol31. Hence, assumingvariations about the representative values of10% on Wm and b, 30% on T, and 20% onS2/YP, and setting BWZ, Eq. 20 yields values of:(1) Q2 in the range (4.5 U 3.2)U1033(vc2/N2), and(2) the gradient among inverse peak rates in therange (4.5 U 3.2)U10338(vc1/N1).

To a ¢rst approximation, the thickness of rockrelaxing molecular bonds (and so releasing en-ergy) along a new crack surface is similar to themean separation between the new surfaces of thecrack (Marder and Fineberg, 1996), such that vc1/N1 approximately describes the geometry of the c1crack tip via the ratio [Step length]/[Separationthickness]. From geometry, the mean separationof a Scale-i crack is also on the order of thegrowth step of the coalescing Scale-(i+1) cracks,such that NiVvciþ1 (Fig. 4). Accordingly, vc1/N1Vvc1/vc2.

Geometric arguments also link vc1/vc2 to8=N1/N2P. Since the same growth step is beingconsidered, the extension due to N1 steps oflength vc1 must be similar to N2P steps of lengthvc2. Thus, N1vc1WN2Pvc2, from which:

8 ¼ N1=N2PWvc2=vc1WN 1=vc1 ð21Þ

Eq. 21 shows that, because episodes of coales-cence among c2 cracks also de¢ne step advancesin the larger c1 fractures, the scaling factor 8 isdirectly linked to the shape of the c1 fracture tips,through the ratio [Separation thickness]/[Steplength]. As a result, the product 8(vc1/N1)W1,so removing any explicit dependence on crackshape for the gradient among inverse peak rates,which simpli¢es from 8Q1 to Q*=B(S2/YP)Wm/bRT= (4.5U 3.2)U1033, embracing the observedvalues of 2.4U1033 for Soufriere Hills and6.4U1033 for Pinatubo.

A key point here is that the condition 8(vc1/N1)W1 occurs because the growth of the mainfracture system (Scale 1) is being monitored indi-rectly through the peak event rates of cracks atthe next smaller scale (Scale 2). The same condi-tion does not apply to individual sequences ofaccelerating event rate (between peak values),since the cracks being monitored are those di-rectly responsible for the detected events (in this

case, all at Scale 2). Indeed, for each precursorysequence, the ratio of the gradient for an inverse-rate sequence to the gradient among inverse peakrates is Q2/8Q1 = (1/8)(vc2/vc3)(vc2/vc1)W1/8, as-suming that the di¡erence in mean crack dimen-sions remains the same among any pair of succes-sive length scales. From Figs. 6 and 7, 1/8=0.01/0.0024W4 for Soufriere Hills and about (0.2 to0.6)/0.0064W30 to 90 (notionally 60) for Pinatu-bo. Compared with conditions at Soufriere Hills,therefore, each growth step of the main Pinatubofracture apparently required the coalescence of agreater number, by a factor of about 15, of small-er process-zone cracks. In other words, the tips offractures extending beneath Pinatubo may havebeen about 15 times more elongate with respectto those extending beneath Soufriere Hills.

7. Discussion

A remarkable feature of the inverse-rate trendsfor seismic precursors at Soufriere Hills and Pina-tubo is that, although the gradients for individualsequences di¡er by more than an order of magni-tude, the gradients for inverse peak values arewithin a factor of three of each other (Eqs. 17and 18). Such a coincidence suggests a commoncontrolling mechanism. Thus, combining Eqs. 20and 21 with Eq. 15a,b yields for the trends amongpeak rates dNP/dt :

dNP=dt ¼ L eQ�NP ð22aÞ

ðdNP=dtÞ31 ¼ ðdNP=dtÞ310 3Q

�ðt3t0Þ ð22bÞ

where Q*=B(S2/YP)Wm/bRT and L is a constantcontaining the remaining energy terms; Q* is theratio [Strain energy per unit volume]/[Total inter-nal energy per unit volume], but can also beviewed as 1/N* where N* is the e¡ective numberof seismic events required to release the total in-ternal energy as strain energy (if such a completeconversion were possible). The linear dependenceof inverse peak rate with time (Eq. 22b) is espe-cially favourable to forecasting since it requiresonly the extrapolation of a straight line to thetime axis in order to identify a possible time of

VOLGEO 2623 23-6-03


eruption. Moreover, the inverse peak values (min-ima on the inverse-rate plot) may be easier toidentify by eye than the actual peak values (onthe event-rate plot) under emergency conditions(compare, for example, Fig. 6a,b).

Applied to the observed precursors (Figs. 6 and7), the measured values of Q* are within the rangeof (4.5 U 3.2)U1033 expected for conditions with-in a subvolcanic system. Such agreement is excel-lent and not only supports the validity of theslow-fracture model, but also suggests that reli-able forecasts of eruption might eventually be fea-sible at least 2^3 days ahead of time. Warningintervals of days are relevant in practice becausethey may permit successful evacuation of threat-ened districts.

The restricted range for Q* is another featurewith possible forecasting implications. It is a nat-ural consequence of using peak rates of process-zone cracking as a proxy measure for growthsteps among the parent fractures, since this sub-stitution cancels terms that depend on crack-tipgeometry. Seismic monitoring systems, however,are deployed to detect as many events as possible.It is thus by happy circumstance, rather than bydesign, that the majority of detected events ap-pear to be those relevant to cracking in the pro-cess zones of the dominant existing fractures. De-tected events commonly have magnitudes ofabout 0^2, consistent with movements of 1^10mm over 10^100 m (McNutt, 2000). The factthat a negative gradient of Q* (rather than asteeper gradient) is found for their inverse-peak-rate trend therefore suggests that the detectedevents relate either (a) to movements in the pro-cess zones of primary fractures that are typicallyseveral tens of metres long, such that each eventre£ects slip along a primary fracture, or (b) tomovements of fractures tens of metres long thatform part of the process zones of primary frac-tures (perhaps V102 m long), in which case slipalong the parent fractures is detected as a swarmof small events, rather than a small number ofevents of larger magnitude. Whatever the inter-pretation, if such a condition exists among volca-noes in general, it follows that the opening of anew magmatic pathway will be characterised byinverse-peak-rate trends with gradients of Q*, as

opposed to the steeper gradients that characterisethe inverse-rate trends of individual process-zonesequences (Fig. 5). An inverse-rate gradient of Q*might thus also be indicative of the ¢nal approachto eruption.

The present model has been tested against onlytwo examples, both of them related to andesiticvolcanoes reawakening after centuries of quies-cence. However, because it focusses on the frac-ture behaviour of host rock (in response to stress-es applied due to increased magma pressure),there is no immediately obvious reason why itshould not apply generally to conditions in whichthe opening of a new magmatic pathway is essen-tial for an eruption to occur. It may thus be founduseful, for example, also for describing the lateralpropagation of fractures from the central feedingsystem of persistently active basaltic volcanoes,such as Etna or Stromboli. On the other hand,it would be less useful for describing seismic pre-cursors to eruptions from vents that are alreadyopen, corresponding to conditions that may ap-ply, for example, (a) to summit e¡usions at per-sistently active volcanoes, and (b) to subsequenteruptive episodes through a pathway that hadbeen created only shortly beforehand during thereawakening of a long-quiescent volcano.

The present model further introduces the no-tion of describing an opening magmatic pathwayin terms of the progressive linkage of a popula-tion of existing fractures. This notion accommo-dates the observation that the locations of seismicprecursors do not always migrate towards the sur-face with time. It also provides the basis for treat-ing macroscopic seismic event rates statisticallythrough random £uctuations in the release ofstress-related energy, an approach which naturallyexplains the similar types of acceleration in eventrate with time observed for both microscopic andmacroscopic fracturing.

However, although the model describes thegrowth of a major fracture system, it cannot guar-antee that the new fracture will intersect both amagma body and the surface. Thus, if only themagma body is intersected, the result may be anupward intrusion, rather than eruption. Similarly,if the magma body is not reached, displacementalong the new fracture may instead trigger an

VOLGEO 2623 23-6-03


earthquake or an intense seismic swarm. Thus,although eruptions may be preceded by seismicsequences similar to those analysed here, it isnot evident that all such sequences inevitablylead to eruption. They may instead be followedby a permanent decay in activity or by a tempo-rary lull in seismicity (McNutt, 1996). To guaran-tee an eruption, therefore, additional criteria mustalso be satis¢ed. Such criteria may be determinis-tic (e.g. the volcanic edi¢ce may ¢rst have to bestretched by a critical amount) or stochastic (e.g.whether or not a fracture intersects a magmabody may contain an element of chance, perhapsdue to edi¢ce heterogeneity). In either case, thepresent analysis provides a basis on which newstudies can build for identifying the additionalpre-eruption requirements.

The present model also assumes that, in the¢nal approach to eruption, detected seismicity isdominated by events driven by magmatic pressureapplied axially below the future eruptive vent. Itdoes not account (a) for seismicity induced byfactors other than magma pressure, such as ther-mal cracking, or (b) for detected events due tofracturing at locations away from the axial seis-mic cylinder. Future analyses may be able to ¢lterout signals produced by non-magmatic stresses,while the eventual real-time location of eventswould distinguish fracturing within and outsidethe axial seismic cylinder.

The slow-fracture model is a natural extensionof the pioneering studies by Voight (1988, 1989)and their development by Voight and Cornelius(1991) and Cornelius and Voight (1994, 1995,1996). Unlike the present model, which focusseson changes in peak seismic event rate, the pre-vious studies concentrated on changes in meanevent rate. The Voight and Cornelius studiesalso suggested that, in addition to linear decreaseswith time, inverse rates may on occasion decayexponentially with time (corresponding to thecondition K=1 in Eq. 1). Such decay would ap-proach a zero inverse rate (time of eruption) afteran unspeci¢ed interval and so cannot be used foreruption forecasting, unless independent argu-ments can also identify a non-zero thresholdrate of seismicity at which an eruption is expected.This behaviour has been attributed to the growth

of cracks that do not eventually coalesce and sodo not lead to the opening of a new magmaticpathway (McGuire and Kilburn, 1997; Kilburnand Voight, 1998); as discussed above, contribu-tors to such cracking may include non-magmaticstresses and events occurring outside the axialseismic cylinder. However, such behaviour mightalso represent a condition for which a rate-depen-dent resistance, perhaps along moving faults (Die-terich, 1992), becomes large enough to inhibitfurther slow fracture. Thus, a more completeform of Eq. 22a may be:

dNP=dt ¼ L eQ�NP3f ðRÞ ð23Þ

where the resisting term f(R) is small for condi-tions that ¢nally result in eruption. The signi¢-cance of such a resisting term awaits future study.

8. Conclusions

The present model argues that the ¢nal stagesof magma ascent are controlled by the slow frac-turing of host rock between a magma body andthe surface. Fracturing occurs by the growth andcoalescence of pre-existing cracks. Each crack it-self has a process zone containing even smallercracks. Failure is thus perceived as a multiscaleprocess for which microscopic cracking can ulti-mately lead to macroscopic fracturing. Since sim-ilar processes are assumed to operate at all scales,the form of the governing crack-growth equationremains the same, independent of the size andduration of cracking being investigated. Similarseismic patterns are therefore anticipated amongeruption precursors, irrespective of the time scaleused to measure event rate. Indeed, the examplesfrom Soufriere Hills and Pinatubo (Figs. 6 and 7)illustrate how the same essential pattern emergesfor event rates measured, respectively, every 4 hand per day, and suggest that reliable forecasts oferuption might be feasible at least 2^3 days aheadof time. Further studies, however, are required tocon¢rm the present results and, also, to identifythe broader range of conditions that distinguishaccelerations in seismic event rate that may decaywithout volcanic activity from those that will in-evitably lead to eruption.

VOLGEO 2623 23-6-03


Acknowledgements

This paper bene¢tted from discussions with Pe-ter Sammonds and Valentina Rocchi (both atUCL) and from helpful reviews by SebastienChastin (University of Edinburgh) and SteveMcNutt (Alaska Volcano Observatory). The re-search was funded by the European Commission(DG Research, Environment and Climate pro-gramme, Natural Risks, A. Ghazi, Head ofUnit, Maria Yeroyanni, Scienti¢c O⁄cer) forProject Volcalert (CEC Contract No. EVG-2001-00040) and by the Gruppo Nazionale perla Vulcanologia, Italy.

Appendix. Manipulation of the event-rate equa-tion

From Eq. 12 in the main text:

dN=dt ¼ CeQN ðA1Þ

where the constant C= f*e3R eQ c0=vc.Rearrangement of Eq. A1 leads to:

RNN0expð3QNÞdN ¼ C

Rtt0dt ðA2Þ

which yields:

ðe3QN0=CÞ3ðe3QN=CÞ ¼ Q ðt3t0Þ ðA3Þ

Noting from Eq. A1 that (dN/dt)31 = e3QN /C,Eq. A3 becomes:

ðdN=dtÞ31 ¼ ðdN=dtÞ310 3Q ðt3t0Þ ðA4Þ

where (dN/dt)0 is the value of dN/dt at t= t0, iden-tical with Eq. 13 in the main text.

The values of t0 and, hence, of (dN/dt)0 arenormally determined empirically from graphs ofaccelerating event rate. However, (dN/dt)0 can inprinciple also be calculated from Eq. A1 asf*e3R eQ ci=vceQN0 or, using Eq. 11, as f*e3O i0=nRT

where Oi0 is the excess energy required to startthe acceleration in rate of rock fracturing.

References

Anderson, O.L., Grew, P.C., 1977. Stress corrosion theory of

crack propagation with applications to geophysics. Rev.Geophys. Space Sci. 15, 77^104.

Aspinall, W.P., Miller, A.D., Lynch, L.L., Latchman, J.L.,Stewart, R.C., White, R.A., Power, J.A., 1998. Soufrie'reHills eruption, Montserrat, 1995-1997: Volcanic earthquakelocations and fault plane solutions. J. Geophys. Res. 25,3397^3400.

Atkinson, B.K., 1984. Subcritical crack growth in geologicalmaterials. J. Geophys. Res. 89, 4077^4114.

Charles, R.J., Hillig, W.B., 1962. The kinetics of glass failureby stress corrosion. In: Symposium sur la resistance mecha-nique du verre et les moyens de l’ameliorer. Union SciencesContinentale du Verre, Charleroi, 511 pp.

Chouet, B.A., 1996. New methods and future trends in seis-mological volcano monitoring. In: Scarpa, R., Tilling, R.I.(Eds.), Monitoring and Mitigation of Volcano Hazards.Springer, Berlin, pp. 23^97.

Cornelius, R.R., Voight, B., 1994. Seismological aspects of the1989-1990 eruption at Redoubt volcano, Alaska: the mate-rials failure forecast method (FFM) with RSAM and SSAMseismic data. J. Volcanol. Geotherm. Res. 62, 469^498.

Cornelius, R.R., Voight, B., 1995. Graphical and PC-softwareanalysis of volcano eruption precursors according to theMaterials Failure Forecast Method (FFM). J. Volcanol.Geotherm. Res. 64, 295^320.

Cornelius, R.R., Voight, B., 1996. Real-time seismic amplitudemeasurement (RSAM) and seismic spectral amplitude mea-surement (SSAM) analyses with the Materials Failure Fore-cast Method (FFM), June 1991 explosive eruption at MountPinatubo. In: Newhall, C.G., Punongbayan, R.S. (Eds.),Fire and Mud: Eruptions and Lahars of Mount Pinatubo,Philippines. PHIVOLCS, Quezon City, and University ofWashington Press, Seattle, WA, pp. 249^267.

Cowie, P.A., 1998. A healing-reloading feedback control onthe growth rate of seismogenic faults. J. Struct. Geol. 20,1075^1087.

Cowie, P.A., Shipton, Z.K., 1998. Fault tip displacement gra-dients and process zone dimensions. J. Struct. Geol. 20, 983^997.

De la Cruz-Reyna, S., Reyes-Da¤vila, G.A., 2001. A model todescribe precursory material-failure phenomena: applica-tions to short-term forecasting at Colima volcano, Mexico.Bull. Volcanol. 63, 297^308.

De Natale, G., Pingue, F., Allard, P., Zollo, A., 1991. Geo-physical and geochemical modeling of the 1982-1984 unrestphenomena at Campi Flegrei caldera (Southern Italy).J. Volcanol. Geotherm. Res. 48, 199^222.

Dieterich, J.H., 1992. Earthquake nucleation on faults withrate- and state-dependent strength. Tectonophysics 211,115^134.

Gri⁄th, A.A., 1920. The phenomenon of rupture and £ow insolids. Philos. Trans. R. Soc. London A 221, 163^197.

Harlow, D.H., Power, J.A., Laguerta, E.P., Ambubuyong, G.,White, R.A., Hoblitt, R.P., 1996. Precursory seismicity andforecasting of the June 15, 1991, eruption of Mount Pinatu-bo. In: Newhall, C.G., Punongbayan, R.S. (Eds.), Fire andMud: Eruptions and Lahars of Mount Pinatubo, Philip-

VOLGEO 2623 23-6-03


pines. PHIVOLCS, Quezon City, and University of Wash-ington Press, Seattle, WA, pp. 285^305.

Heimpel, M., Olson, P., 1994. Buoyancy-driven fracture andmagma transport through the lithosphere: models and ex-periments. In: Ryan, M.P. (Ed.), Magmatic Systems. Aca-demic Press, San Diego, CA, pp. 223^240.

Hill, D.P., Ellsworth, W.L., Johnston, M.J.S., Langbein, J.O.,Oppenheimer, D.H., Pitt, A.M., Reasenburg, P.A., Sorey,M.L., McNutt, S.R., 1990. The 1989 earthquake swarm be-neath Mammoth Mountain, California: an initial look atthe 4 May through 30 September activity. Bull. Seism.Soc. Am. 80, 325^339.

Jaeger, J.C., 1969. Elasticity, Fracture and Flow, 3rd edn.Chapman and Hall, London, 268 pp.

Kilburn, C.R.J., Voight, B., 1998. Slow rock fracture as erup-tion precursor at Soufriere Hills volcano, Montserrat. Geo-phys. Res. Lett. 25, 3665^3668.

Landau, L.D., Lifshitz, E.M., 1999. Statistical Physics, Part 1,3rd edn. Butter¢eld Heinemann, 546 pp.

Lawn, B., 1993. Fracture of Brittle Solids, 2nd edn. CambridgeUniversity Press, Cambridge, 378 pp.

Lister, J.R., 1991. Steady solutions for feeder dykes in a den-sity-strati¢ed lithosphere. Earth Planet. Sci. Lett. 107, 233^242.

Lockner, D., 1993. Room temperature creep in saturated gran-ite. J. Geophys. Res. 98, 475^487.

Main, I.G., 1991. A modi¢ed Gri⁄th criterion for the evolu-tion of damage with a fractal distribution of crack lengths:application to seismic event rates and b-values. Geophys. J.Int. 107, 353^362.

Main, I.G., Meredith, P.G., 1991. Stress corrosion constitutivelaws as a possible mechanism of intermediate-term andshort-term seismic event rates and b-values. Geophys. J.Int. 107, 363^372.

Main, I.G., 1999. Applicability of time-to-failure analysis toaccelerated strain before earthquakes and volcanic erup-tions. Geophys. J. Int. 139, F1^F6.

Main, I.G., 2000. A damage mechanics model for power-lawcreep and earthquake aftershock and foreshock sequences.Geophys. J. Int. 142, 151^161.

Marder, M. Fineberg, J., 1996. How things break. PhysicsToday (September) 24^29.

McGuire, W.J., Kilburn, C.R.J., 1997. Forecasting volcanicevents: some contemporary issues. Geol. Rundsch. 86,439^445.

McGuire, W.J., Kilburn, C.R.J., Murray, J.B. (Eds.), 1995.Monitoring Active Volcanoes: Strategies, Procedures andTechniques. UCL Press, London, 421 pp.

McNutt, S.R., 1996. Seismic monitoring and eruption forecast-ing of volcanoes: a review of the state-of-the-art and casehistories. In: Scarpa, R., Tilling, R.I. (Eds.), Monitoringand Mitigation of Volcano Hazards. Springer, Berlin, pp.99^146.

McNutt, S.R., 2000. Volcano seismicity. In: Sigurdsson, H.(Ed.), Encyclopedia of Volcanoes. Academic Press, San Die-go, CA, pp. 1015^1033.

Newhall, C.G., Punongbayan, R.S. (Eds.), 1996. Fire andMud: Eruptions and Lahars of Mount Pinatubo, Philip-pines. PHIVOLCS, Quezon City, and University of Wash-ington Press, Seattle, WA.

Power, J.A., Wyss, M., Latchman, J.L., 1998. Spatial varia-tions in the frequency-magnitude distribution of earthquakesat Soufriere Hills Volcano, Montserrat, West Indies. Geo-phys. Res. Lett. 25, 3653^3656.

Reyes-Da¤vila, G.A., De la Cruz-Reyna, S., 2002. Experience inthe short-term eruption forecasting at Volca¤n de Colima,Me¤xico, and public response to forecasts. J. Volcanol. Geo-therm. Res. 117, 121^127.

Rocchi, V., 2002. Fracture of basalts under simulated volcanicconditions. Ph.D. thesis, University of London, 390 pp.

Ruhla, C., 1992. The Physics of Chance. Oxford UniversityPress, Oxford, 222 pp.

Savage, J.C., Cockerham, R.S., 1984. Earthquake swarm inLong Valley caldera, California, January 1983: evidencefor dike injection. J. Geophys. Res. 89, 8315^8324.

Scarpa, R., Tilling, R.I. (Eds.), 1996. Monitoring and Mitiga-tion of Volcano Hazards. Springer, Berlin, 841 pp.

Scholz, C.H., 1990. The Mechanics of Earthquakes and Fault-ing. Cambridge University Press, Cambridge, 439 pp.

Shaw, H.R., 1980. The fracture mechanisms of magma trans-port from the mantle to the surface. In: Hargraves, R.B.(Ed.), Physics of Magmatic Processes. Princeton UniversityPress, Princeton, NJ, pp. 201^264.

Sigurdsson, H. (Ed.), 2000. Encyclopedia of Volcanoes. Aca-demic Press, San Diego, CA, 1417 pp.

Smith, G.M., Ryall, F.D., 1982. Bulletin of the SeismologicalLaboratory for the period January 1 to December 13, 1980.Mackay School of Mines, University of Nevada, Reno, NV,50 pp.

Spence, D.A., Turcotte, D.L., 1985. Magma-driven propaga-tion of cracks. J. Geophys. Res. 90, 575^580.

Voight, B., 1988. A method for prediction of volcanic erup-tions. Nature 332, 125^130.

Voight, B., 1989. A relation to describe rate-dependent materi-al failure. Science 243, 200^203.

Voight, B., Cornelius, R.R., 1991. Prospects for eruption pre-diction in near real-time. Nature 350, 695^698.

Wiederhorn, S.M., Bolz, L.H., 1970. Stress corrosion and stat-ic fatigue of glass. J. Am. Ceram. Soc. 53, 543^548.

Young, S.R., Sparks, R.S.J., Aspinall, W.P., Lynch, L.L.,Miller, A.D., Robertson, R.E.A., Shepherd, J.B., 1998.Overview of the eruption of Soufriere Hills volcano, Mont-serrat, 18 July 1995 to December 1997. Geophys. Res. Lett.25, 3389^3392.

VOLGEO 2623 23-6-03


multiscale fracturing as a key to forecasting volcanic

Documents