Comparison of Mount Etna, Kilauea, and Piton de la Fournaise by a quantitative modeling of their eruption histories

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. B3, PAGES , MARCH 10, 2001 Comparison of Mount Etna, Kilauea, and Piton de la Fournaise by a quantitative modeling of their eruption histories Keiiti Aki Department of Earth Sciences, University of Southern California, Los Angeles Valdrie Ferrazzini Observatoire Volcanologique du Piton de la Foumaise, Institut de Physique du Globe de Paris, La Plaine des Cafres, La Reunion, France Abstract. From published data we found characteristic relations between the amount V of erupted lava and the duration d of eruption for Mount Etna, Kilauea, and Piton de la Fournaise. The relation is similar between Mount Etna and Kilauea, where the increase of V with increasing d is slow, showing a trend of a lower flow rate for a larger eruption. For Piton de la Fournaise, however, the trend is distinctly different, showing a higher flow rate for a larger eruption. We constructed quantitative models of a magma system with reservoirs at various levels and tested hypotheses about the existence of large reservoirs under these volcanoes using the observed V-d relations. We found that the observed V-d relation is consistent with the presence of a large reservoir at a shallow depth under Kilauea, and with the presence of a large reservoir near the bottom of the volcanic edifice under Piton de la Fournaise. The above models for Kilauea and Piton de la Fournaise are characterized by a simple path from the mantle reservoir which leads to the shallowest reservoir connected to a hierarchy of channels with varying resistance to eruption sites. We obtained less satisfactory agreement between the observed V-d relation and the predicted using our model for Mount Etna. Time histories of pressures in the reservoirs at various levels obtained by the modeling explain why inflation-deflation cycles observed at Kilauea have not been reported for Piton de la Fournaise. The absence of volcano-tectonic earthquakes with magnitude greater than -2.5 under Piton de la Fournaise is attributed to the simplicity of the magma path from the mantle to the shallowest reservoir and the underdeveloped rift zone, which result in a stress concentration localized only beneath the summit area. 1. Introduction Mount Etna, Sicily, Kilauea, Hawaii, and Piton de la Fournaise, Reunion Island, are all basaltic volcanoes and grossly similar in the chemical composition and physical properties of produced lava [Williams and McBirney, 1979; Bach lery, 1999]. They are, however, different in the style of eruption and associated phenomena. It is widely accepted that a shallow magma reservoir exists at depths 3-7 km beneath the summit of Kilauea, which slowly inflates over periods of months to years during the intervals between eruptions and then rapidly deflates during flank eruptions or lateral intrusions into the rift zones [Decker, 1987]. On the other hand, such a regular cycle of inflation and deflation has not been reported at Piton de la Fournaise, although the period of deformation monitoring there is much shorter than at Kilauea. Recent results on the pre eruption deformation from spaceborne radar interferometry appear to indicate also the current absence of large shallow 1Now at Observatoire Volcanologique du Piton de la Fournaise, Institut de Physique du Gloge de Paris, La Plaine des Cafres, La R union, France. Copyright 2001 by the American Geophysical Union. Paper number 2000JB /01/2000JB reservoirs under Mount Etna [Massonnet et al., 1995] as well as under Piton de la Fournaise [Sigmundsson et al., 1999]. The issue of the presence or absence of large magma reservoirs within the volcanic edifices of Mount Etna and Piton de la Fournaise, however, is still an open question. For Mount Etna, for example, Bonaccorso [1996] refuted the conclusion of Massonnet et al. [1995] that the deflation center during the eruption was as deep as 16 km and proposed its depth at-3 km below sea level (bsl) on the basis of the electronic distance measurement (EDM), GPS, tilt, and leveling data. The result of Massonnet et al. [1995] was also questioned by Beauducel et al. [2000]. Lanari et al. [1998] reported deflation during the end of 1993 attributed to a source at a depth of 9 km followed by inflation for attributed to a source at depths km. Murru et al. [1999] also claims the presence of shallow magma chambers under Mount Etna from a 3-D mapping of the b value. For Piton de la Fournaise, there are three pieces of recent seismological observations in support of a large reservoir near the bottom of the volcanic edifice. The first observation is the upward migration of the focal depths of volcano-tectonic events from-5 km bsl to -2 km asl within 1.5 days preceding the major eruption of March 1998 [Staudacher e! al., 1998]. The second is the temporal-spatial pattern of the amplitude of coda waves from the volcano-tectonic earthquakes during the 2 -year period before the 1998 eruption interpreted by Aki and Ferrazzini [2000] as due to filling of two large magma 4091

2 4092 AKI AND FERRAZZINI ß MODELING OF THREE ACTIVE VOLCANOES amount of lave (Mm e) looo Etn summit o flank loo lo 1 lo loo looo loooo looo Kilauea * summit rift zone loo r lo Piton de Io Fournoise * summit ½ rift zone duration in days Figure 1. The relation between the amount of erupted lava and the duration of eruption observed for Mount Etna since 1700, for Kilauea since 1952, and for Piton de la Fournaise since Solid diamonds are used for the eruptions from the summit, and shade diamonds represent those from the rift zone or on the flank. reservoirs within the volcanic edifice. The third is the occurrence of long-period events concurrent with the coda localization and the increased rate of the volcano-tectonic earthquakes as described by Aki and Ferrazzini [2000]. The purpose of this paper is to pursue the above issue by comparing the eruption history of the three volcanoes through modeling by a magma system composed of reservoirs and channels. This is not a simple problem because the style of eruption changes with time at each of these volcanoes. At Mount Etna, there must have been once a large shallow reservoir as evidenced by the caldera collapse at the summit during the 1669 flank eruption [Wadge et al., 1975], and petrological and geochemical evidence supports magma fractionating in a quite regularly replenished magma reservoir in early seventeenth century [Corsaro, et al., 1996]. The 1669 eruption with the output of nearly 1 km 3 apparently marks the change in the style of eruption. At Kilauea the activity alternates with that at nearby Mauna Loa, and the current active period started in 1952 [Klein, 1982]. For Piton de la Foumaise, Ldnat and Bachblery [1988] suggest two different periods, type A and B, alternating rather rapidly. The type A periods mark the arrival of new batches of magma in the shallow reservoir and high rate of output, while eruptions during the type B period (including the period between 1977 and 1998) come from small shallow reservoirs replenisheduring the preceding type A period. Taking the above temporal variation into account, we selected eruptions which may be considered belonging to a period of temporal uniformness and compiled the time of eruption, the duration of eruption and the amount of erupted lava for each volcano. We found a characteristic relation between the amount of lava and the duration of eruption for each volcano as shown in Figure 1. The relation is very similar between Mount Etna and Kilauea, but Piton de la Fournaise shows a different behavior. If the average flow rate is the same for all eruptions, we expect a linear relation between the amount of magma and duration. At Mount Etna and Kilauea the increase in the amount of lava with the duration is slower than in the proportional case,

3 AKI AND FERRAZZINI: MODELING OF THREE ACTIVE VOLCANOES 4093 amount of lava (Mm 3) 1000 Piton de Io Fournoise ß summit o rift zone loo lo t duration in days Figure 2. The relation between the amount of erupted lava and the duration of eruption for Piton de la Fournaise for the type A period, defined by Lgnat and Bachdlery [1988] as the period during which new magma may be supplied from the mantle. Solid diamonds are used for eruptions from the summit, and shadediamonds represent those from the rift zone. showing a horizontally elongated trend of plots in the diagram of the erupted amount versus duration. On the other hand, the increase is more rapid at Piton de la Fournaise than the proportional case, showing a vertically elongated trend of plots (Figures 1 and 2). In other words, the average flow rate is lower for larger eruptions at Mount Etna and Kilauea but higher for larger eruptions at Piton de la Fournaise. We shall construct a model of magma system to explain these observations. Our model is composed of "reservoirs" at various levels connected to the source of magma at depth, to the eruption site at the surface, and to each other through "channels". A similar idea of characterizing the magma system by an analog electric circuit consisting of capacity and resistance components was suggested by Decker [1968, 1987] and Shimozuru [1981]. As far as we know, however, they did not pursue the idea for simulating actual eruptions. 2. Data We relied on published data for the time of eruption, the amount of erupted lava, and the duration of eruption. For Mount Etna we used the compilation by Tanguy and Patan [1996]. It was checked by comparing with the listings by Wadge [1977], and Mulargia et al. [1987]. All eruptions except four are on the flank of the volcano. For Kilauea we used the listing by Klein [1982], who distinguished a summit eruption from a rift-zone eruption. For Piton de la Fournaise the data sources are Benard and Kraft [1986] and Stieltjes and Moutou [1989], and the distinction between the summit eruption and the rift-zone eruption was made according to the criterion developed by Aki and Ferrazzini [2000]. They are also classified into the type A and B periods following Ldnat and Bach lery [ 1988]. Figure 1 shows the observed relation between the amount of erupted lava and the duration of eruption for Mount Etna (since 1700), Kilauea (since 1952), and Piton de la Foumaise (since 1931). Solid diamonds are used for the summit eruption, and shaded diamonds are used for the rift-zone (flank) eruptions. We plot the relation for the Piton de la Fournaise separately for the whole period (Figure 1) and the type A period (Figure 2). For all three volcanoes the trend of the relation does not appear to depend on whether the eruption is from the summit or the rift-zone (flank), although the amount of magma for a given duration, i.e., the average flow rate, is significantly greater during the rift-zone eruption than during the summit eruption for Mount Etna and Piton de la Fournaise, while there is no obvious difference for Kilauea. The selection of the type A period for Piton de la Fournaise makes the relation slightly more systematic than the whole period. It is clear from these figures that the relation is similar between Kilauea and Mount Etna, while the relation observed at Piton de la Fournaise is distinctly different from them as mentioned in the introduction. We shall now describe our model that will be used to explain the observed relation between the amount V of erupted lava and the duration d of eruption. Hereafter we shall refer to this relation as "the V-d relation". 3. Model Let us first explain the elements of our model for the simplest case of a single reservoir that receives magma from below at a constant flow rate Q0 and connected to an eruption site by a channel. When the overpressure (pressure in excess of the hydrostatic pressure) in the reservoireaches a critical value Pc, the channel to the surface opens and an eruption starts. The overpressure P in the reservoir and the flow rate Q through the channel is related by P = RQ, where R is the resistance of the channel. On the other hand, the rate of pressure increase in the reservoir is related to the net inflow by the relation dp/dt = (Qo- Q)/C, where C is the capacity of the reservoir. For an episodic eruption we need another parameter for stopping the flow. We introduced the critical flow rate Qc, below which the channel closes or freezes. Qc must be greater than Q0 for an eruption to be episodic. Otherwise, the eruption becomes stationary at a flow rate of Q0. This model is essentially static, neglecting the dynamic effects of magma flow and dike propagation. We also neglect the pressure change in the reservoir due to phase transition, chemical reaction, etc. In spite of these simplifications, we shall find complex interactions among reservoirs and channels with nonlinear characteristics. In a single reservoir system the flow rate starts with the value Pc/R and decreases nearly exponentially with the decay time constant RC. Such decays were observed repeatedly in the tremor amplitude history during the 1998 eruption of Piton de la Fournaise. The time interval of consecutiveruptions, on the

4 4094 AKI AND FERRAZZINI ß MODELING OF THREE ACTIVE VOLCANOES Figure 3. A model of magma system consisting of two reservoirs in the mantle and the other two within the volcanic edifice. Reservoirs are connected by single channels between them, but the shallowest one is connected to eruption sites by a hierarchy of channels with varying resistance. Equations governing Pi and Qi are given in the text. other hand, is given by CPc/Qo, and the amount of magma in each eruption is CPc. For example, if we assume a critical overpressure of 100 bars, the capacity of 0.1 million cubic meters per bar corresponds to a magma storage of 10 million cubic meters. In the present paper, we shall limit our purpose to highlight the observed difference in the V-d relation among the three volcanoes rather than trying to explain all available observations accumulated about these volcanoes over many years. Here we shall use a model of four reservoirs; two in the volcanic edifice and the other two in the mantle as shown in Figure 3. According to Aki and Ferrazzini [2000], the depth of the shallowest reservoir under the Piton de la Fournaise is about sea level, and that of the intermediate reservoir is -5 km bsl. In addition to three channels connecting between the reservoirs, we introduce four more channels connecting the shallowest reservoir to eruption sites at the surface in order to representhe range of resistance needed to simulate the observed V-d relation. The choice of two reservoirs in the mantle is for the model versatility, especially for generating an active period and a quiet period, which are seen in the actual eruption activities for all three volcanoes. The V-d relation, however, is affected very little whether we consider one or two reservoirs in the mantle. Each reservoir is characterized by its capacity, and each channel is characterized by its resistance, critical overpressure to open it, and critical flow to close it. Thus our model requires 4 parameters to define the reservoirs and 21 parameters to define the channels. In addition, we need two parameters to assign the input flow rate to the two mantle reservoirs, making the number of model parameters described so far 27 (see Table 1 for examples of model parameters). A large number of channels from the shallowest reservoir to the eruption site is needed to representhe diversity of eruption paths of actual volcanoes, but we must keep it manageable for the simulation purpose. Thus we restricted the number of the eruption channels to four, but made each channel renewable in the following manner. After each eruption, the critical overpressure is increased by a certain amount and afterward goes back to the assigned value approaching exponentially (the difference between the actual critical overpressure and the assigned one decreases exponentially with time) with a given time constant. We selected a common value of the critical overpressure increase for all the eruption channels. Thus we have an additional five parameters to define the eruption channels. Finally, we allowed the initial values of the actual critical overpressure for eruption channels to be free parameters, making the total number of model parameters 36. Two basic assumptions of our model are (1) the stationary supply of magma into the mantle reservoirs and (2) the movement of magma only in the upward direction. The initial state of the system is given by P/(t: 0)- 0 i = 0, 1, 2, 3 Qi(t = 0) = 0 i = 0, 1, 2, 3, 4, 5, 6. The continuity equation gives the pressures in the reservoirs as function of the fluid flow and the reservoir capacity: dp (t ) : Qoi - Qi(t) i= O, 1, dt C i dpa (t) Q o (t) + Q,(t)- Q 2 (t) dt C 2 alp3(')_ dt C 3 Also the flow rate in the channel i, as soon as the pressure reaches P (t) in the reservoir at the lower end of the channel and as long as the flow in the channel has not decreased to Qa, obeys the equations: eo(,)- 7o(,)- Ro e,(,): 2, Oi(t)- P3(t) i:3,4,5,6. Ri Otherwise there is no flow in the channel. When a surface channel is closed P (t) is set to Pa+DP, where DP is the post eruption increase of critical overpressure. Once the channel is closed, P 8) is a function of time tc measured from the closure time: P i(t)= Pci + DP e -t /vr' i=3,4,5,6, and the initial values of P (t) at (t = 0) are set as P&y(t : O)= P i j:3,4,5,6. It turned out that the V-d relation simulated by our model is affected very little by the choice of the input flow into the mantle reservoirs as described later. Assumption 2 requires the increase of the overpressure with the depth. We have chosen rather arbitrarily (but probably not unreasonably from common sense about the tensile strength of crustal rocks) the range of the overpressure from around 100 bars at the shallowest reservoir to 300 bars at the deepest reservoir. We shall explain how we assigned the remaining parameters in section 4. Our model is deterministic because the output of the simulation is completely determined by the assigned model parameters and initial conditions. We allow, however, some stochastic elements by including a hierarchy of eruption channels connecting the shallowest reservoir with the eruption

5 AKI AND FERRAZZINI: MODELING OF THREE ACTIVE VOLCANOES 4095 Table 1. Parameters of Models Used for Generating Eruption Histories Shown in Figures 5-7. Parameters Etna-type Kilauea-type Fournaise-type Input flow rate into mantle reservoirs, m 3 h - Reservoir 0 Qo ,200 Reservoir 1 Qoo 1, ,200 Critical overpressure, bars Channel 0 Pco Channel 1 P Channel 2 Pc Channel 3 Pc Channel 4 P Channel 5 P s Channel 6 P Channel resistance, bar m -3 h - Channel 0 Ro Channel 1 R Channel 2 R Channel 3 R Channel 4 R Channel 5 Rs Channel 6 R Critical freezing flow, m 3 h - Channel 0 Q o 500 1, Channel 1 Q 666 1, Channel 2 Qc2 22,000 30,000 22,000 Channel 3 Q 3 30,000 16,000 30,000 Channel 4 Q 4 10,000 8,000 10,000 Channel 5 Q s 3,000 4,000 3,000 Channel 6 Qc6 1,000 2,000 1,000 Reservoir capacity, 106 m 3 bar - Reservmr 0 Co Reservoir 1 C Reservoir 2 C Reservoir 3 C Time constant for renewal of eruption, 103 hour Channel 3 TR Channel 4 TR Channel 5 TRs Channel 6 TR Post-eruption increase of critical overpressure, bars Channels 3, 4, 5, 6 DP Initial value of the critical overpressure, bars Channel 3 P, Channel 4 P, Channel 5 P,s Channel 6 Pt sites as explained in section 4. In this way, we try to integrate deterministic and stochastic elements rather than giving up the deterministic element entirely as in the cellular automaton approach of Lahaie and Grasso [1998], which would not offer any explanation for the observed//-d relation. 4. Assigning Model Parameters In order to focus on the issue of the presence or absence of large magma reservoirs in the volcanic edifice we shall adopt the following working hypothesis accepting the observation by the radar interferometry mentioned in the introduction, although we are aware of controversie surrounding the issue as mentioned there. We shall assign the capacity of magma reservoirs under Piton de la Fournaise roughly in agreement with the upper limit allowed within the error of observations by Sigmundsson et al. [1999], namely, 2 million cubic meters at a depth of 2.5 km (sea level) and 50 million cubic meters at 7 km bsl. We shall follow the generally accepted idea about the magma reservoir under Kilauea [e.g. Decker, 1987] and assume the existence of a large reservoir at a shallow depth. Klein [1982] gave an estimate of the volume of the reservoir under Kilauea as 80 million cubic meters from the mean repose time and the supply rate. We do not have a consensus view about the presence of a large shallow reservoir under Etna and we shall tentatively assume that the capacities of reservoirs within the volcanic edifice of Etna are very small (C2 and C3 in Table 1). Since the maximum volume of magma stored in the reservoir in our model is equal to the product of the critical overpressure and the capacity of the reservoir, we can assign the capacity of each reservoir using the assumption made earlier on the critical overpressure. For simplifying the comparison of the three volcanoes we shall consider the following three cases of reservoir capacities (in the unit of million cubic meters per bar): (1) C2=0.05 and C3=0.05, (2) C2:0.05 and C3:0.35, and (3) C2:0.35 and C3=0.05, representing the Etna type, Kilauea type, and Fournaise type respectively. We shall find that these three cases generate distinctly different V-d relations and behaviors of the temporal variation in the reservoir pressure associated with the eruption. We now consider a set of parameters to characterize channels connecting the shallowest reservoir to the eruption sites. We found from the simulation exercise that the amount V of magma and the duration d of eruption from each individual channel are related approximately linearly and that the ratio V/d (the mean flow rate) is close to the ratio Pc/R of the critical overpressure to

6 4096 AKI AND FERRAZZINI ß MODELING OF THREE ACTIVE VOLCANOES amount of lava (Mm 3) looo Etna * simulation loo lo looo Kilauea simulation loo lo loo Piton de la Fournoise * simulation lo duration in days Figure 4. The simulated relation between the amount of erupted lava and the duration of eruption for the Etnatype, Kilauea-type, and Fournaise-type model. The main difference among the models is the location of the largest reservoir. It is shallow under Kilauea, intermediate under Piton de la Fournaise, and deep under Mount Etna. the resistance (the maximum flow rate) of the channel. The observed V-d relation can thus determine the range of resistance to be assigned to the surface channel since we already made assumptions the critical overpressures. In order to assign the critical freezing flow we introduce a nondimensional parameter Pc/(RQc), which is the critical overpressure divided by the product of the channel resistance and the critical freezing flow. This parameter is the ratio of the maximum flow rate to the minimum flow rate and can be considered as the linearity range of flow for each channel. We found from the simulation exercise that the flow becomes regular when this parameter is greater than -3 and becomes irregular if it is below -3. We tried the following three cases: (1) this parameter is -3 for all the eruption channels, (2) it increases up to -10 with increasing channel resistance, and (3) it increases up to -10 with decreasing channel resistance. The above three cases have some different effects on the V-d relation, but the differences are not so significant within the scatter of simulated plots shown later in Figure 4. We found that case 1 gives a more satisfactory result for Kilauea, and case 3 gives a more satisfactory result for Piton de la Fournaise. The values of the critical overpressure (P½3, P½4, P½5 and P½6), resistance (R3, R4, R5 and R6) and critical freezing flow (Q½, Qc4, Qc5 and Qc6) for eruption channels were obtained by trial and error to fit the observed V-d relation, and the best ones in simulating the eruption histories as shown in Figures 5-7 for the Etna, Kilauea, and Fournaise type, respectively, are listed in Table 1. The critical overpressures chosen for these eruption channels are close to 100 bars, but a small variation was introduced to minimize the simultaneous occurrence of eruptions from different channels. Such a coupling between two different channels has an artifact of making the duration of eruptions from these channels close to each other. The time constant for the renewal of each channel as well as the amount of increment for the critical overpressure after the eruption affect the frequency of eruptions but have little influence on the V-d relation. Likewise, the effect of the rate of

7 AKI AND FERRAZZINI ß MODELING OF THREE ACTIVE VOLCANOES e+08-4e+08 - Etna 3e+08-2e+08 - le ] ii!il -, 0 I, [ ] [ I! [iii [ ] ] [ ] I! I [ 4 I [ I" I I [! I [ I' [ [ [ /' [ [ [ I! [ ii I [ [ i,! I ] I i [ I 0 [ [ t [ [ [ ['[ [ I [ "] [ ]"[ [ [ ] i i [ I [ I I [ [ [ ] [ i' [ [ [ [ [' [ [ [ ] I'[ [ [ [, '[' [ [ 0 [, ] [, [, ] [ [ i [ [ i I I I I I I i I I I I I I I I I I I I I I I I I I' I soooo -I V Q2 iii :j i i :}l' it![ i } ' i 0 I I I I I 1 I I I I I I I I I I I I I [ I [ I I I [ I I I I [ I I I I [ I I [ I I I I I oo- o o years Figure 5. Time history of pressures in reservoirs and flow rates through channels for a 50 -year period for the Etna-type model. Note irregular behaviors of both P3 and P2. input flow to the mantle reservoirs on the V-d relation is also insignificant. In the case of the Fournaise type, the increase in input flow only increases the frequency of eruptions. In the case of the Kilauea and Etna type, the increase will generate few additional eruptions with large volumes and long durations, but it does not cause a significant change in the general trend of the V-d relation. The total flow into the mantle reservoir listed in Table 1 for the Foumaise type agrees with the long term average rate of erupted lava at Piton de la Foumaise, that for the Etna type is -1.5 times the long-term rate at Mount Etna, and that for the Kilauea type is about two thirds of the long term rate at Kilauea. They were chosen to allow the same computer graphic program for the three cases. The differences of these magnitudes in absolute values of the total mantle flow do not affect the simulated V-d relation. On the other hand, the resistances Ro, R, and R 2 of the channels connecting the mantle reservoirs and the lower and upper reservoirs within the volcanic edifice (Figure 3) have

8 ..:.... ß 4098 AKI AND FERRAZZINI ß MODELING OF THREE ACTIVE VOLCANOES 5e+08-4e+08 - Kilauea -= 3e+08 - C7 2e+08 - le I...,...i'22Z7L; T... ;'... :..'::.:.: 2 i '".'.' '."?' 7'77"7'."':'"'" ":".' "=====================......'::5... :?..:.:..:?, x.:½::::::::.-.:: :=. ::.:.m :: :: : ::.:...: :::::;.;i....:: ::.; I I I*" ' I I I'; I I 1' 'l I I "1; I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I oooo Q6 0 I I I I I I I! I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I i I I I '1 I i I I I I I I i I '1' I I I I I I I I I I '1 T:! I I I I I 0 I I I I, I 300 /.'.' :...i.. : ' 0 ' ; ' I ' '! ' ' '' ' I... I... I... I... I... I... I... I Figure 6. The time history of pressures in reservoirs and flow rates through channels for a 50 -year period for the Kilauea-type model. Note regular inflation-deflation cycles of P3 associated with eruptions. years important effects on the V-d relation. In general, a lower for the three volcanoes in Figure 4. In this plot we selected channel resistance generates a smaller number of large batches several cases which give essentially the same V-d relation as of magma transported through these channels. those corresponding to the model parametershown in Table 1. This was done in order to increase the number of plots for the V- d diagram by varying the input flow to the mantle reservoir and 5. Best Fitting Models some of the parameters of the eruption channels that have little We made nearly 400 trials of simulating eruption history, each for 50 -year period. We plotted the simulated V-d relation effect on the V-d relation. The observed characteristics of the V- d relation are well reproduced by the simulation for Kilauea and Piton de la Fournaise. For Mount Etna the model produces the

9 ß _ AKI AND FERRAZZINI ß MODELING OF THREE ACTIVE VOLCANOES e+08-4e+08 - Piton de la Fournaise z= 3e+08 - CY 2e+08 - le I'"'"'i... i i... i... i... i... i... i... i i... i... i"" 'l I i... I' I I I I I I I I i I i i i I I Q I I I I I I I I I I I I I I I I I I :':1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 10ooo Q5 0 i i i i i i i i I I I I I I I I I' I i I I I I I ii I I I I' I I I' I I I I I I I! I I Q4 0 I I I I I I! I I I I I I! I I! I I I i I I I I I I I I I I I I I I I I I I I I I I I I I I I Q3 0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ' I I I I I I I 0 i i i i i i i i i i i i i i i i i I I I i I I I I I I I I I I I I I I ' ' 200- P2 lo0-0 0 P3 /... I 5 Figure 7. The time history of pressures in reservoirs and flow rates through channels for a 50 -year period for the Fournaise-type model. Note irregular behaviors of P3 but regular ones for P2 associated with eruptions. years observed range of V and d, but there is a significant difference in amount of magma transported is also shown for each channel at the trend of the relation between the observed and simulated. the top of Figures 5-7. Although the results shown in Figure 5-7 The simulated relation shows a strong trend of concave upward, are for a particular choice of the initial values of the critical while the observed shows a slight trend of concave downward. overpressures of the surface channels shown in Table 1, their We shall discuss the implication of these results in section 6. features discussed below apply to all cases of initial values Figures 5-7 show the 50 year time histories of the pressure in studied. The pressure time histories for the mantle reservoirs each reservoir and the flow rate in each channel for the model (Po and P:) show more or less regular saw-tooth shapes, and are parameters as given in Table 1 for the Etna, Kilauea, and not shown in Figures 5-7 to avoid clutter. Fournaise types, respectively. The cumulative flow rate or the The pressure (P3) in the shallowest reservoir for the Kilauea

10 4100 AKI AND FERRAZZINI: MODELING OF THREE ACTIVE VOLCANOES type shows a gradual increase during the interval between eruptions followed by a sudden decrease after the commencement of the eruption in agreement with observations at Ki auea mentioned earlier. On the other hand, there is no such gradual increase for the Etna and Fournaise type. P3 for the Fournaise type tends to be constant between eruptions and suddenly increases just before the eruption, and that for the Etna type shows very erratic behavior. The pressure (P2) in the lower reservoir in the volcanic edifice, however, shows a gradual increase between eruptions for the Fournaise type. We believe that this increase was detected during the 2 -year period preceding the 1998 eruption of Piton de la Fournaise by monitoring the coda localization and long-period events as described by Aki and Ferrazzini [2000]. P2 for the Etna type is as erratic as P3, indicating the difficulty in both long-term and short-term prediction of eruption for this type of volcano model. Another interesting comparison among the three models is regarding the flow (Q2) through the volcanic edifice from the lower reservoir to the shallowest upper reservoir. Because of the small capacity of the lower reservoir for the Etna and Kilauea types, the batches through this channel are small and numerous. For the Foumaise type model the large capacity of the lower reservoir allows occasional large batches through this channel. This difference in Q2 affects the mode of eruption as shown in Figures 5-7. The many small batches can produce eruptions of long duration at highe resistance channels for the Etna and Kilauea types, while the small number of large batches are responsible for large eruptions through lower resistance channels for the Fournaise type. Since lower resistance means higher average flow rate, we find that the average flow tends to be lower for larger eruptions for the Etna and Kilauea types, and it tends to be higher for larger eruptions for the Fournaise type. The three cases shown in Table 1 and Figures 5-7 were chosen primarily for the fit to the observed V-d relation, but it turned out that all the parameters of the Kilauea type model are identical to those of the Etna type, except for the capacity of the shallowest reservoir and the flow rate into one of the mantle reservoirs. Since the effect of the latter factor is relatively minor, we may say that the drastic difference seen between the Etna type and the Kilauea type in both the V-d relation (Figure 4) and the time histories of reservoir pressure (Figures 5 and 6) is primarily due to the presence or absence of the large shallow reservoirø The parameters of the model best fitting to the V-d relation observed at Piton de la Fournaise deviates from the case for Kilauea and Etna types in the following way. As mentioned above, the presence of a large reservoir near the bottom of the volcanic edifice is the main factor causing this difference. In addition, we recognize that the channel resistances under Piton de la Fournaise are, in general, lower than under Kilauea and Etna. This is required to explain the observed relatively short duration of eruptions from Piton de la Fournaise. Another difference is that the greater linearity range for the lower resistance eruption channel helped to explain better the observed higher flow rate for larger eruptions at Piton de la Fournaise. In contrast, the best models for Kilauea and Etna implied selfsimilarity with regard to the linearity range of flow through eruption channels. 6. Discussion The agreement between the observed and simulated V-d relation for Kilauea and Piton de la Fournaise supports our hypothesis about the presence of a large reservoir at a shallow depth in the volcanic edifice for the former and that near the bottom of the volcanic edifice for the latter. Let us now give a simple explanation for why these hypotheses are consistent with the observed V-d relation. When a large-capacity reservoir is directly connected to a hierarchy of eruption channels with varying resistance, the amount of magma to be erupted in an eruption is largely determined by the total amount of magma stored in the reservoir. Since the flow rate depends on the channel resistance, the duration of eruption will vary, depending on the channel resistance. The net result is the tendency to have a common amount of erupted lava and a diverse range of the duration, as observed at Kilauea. In the case of the Fournaise type, magma is transported from the large lower reservoir to the small upper reservoir in the volcanic edifice. Because of this, the duration of a batch through this channel tends to be constrained. When a batch from the lower reservoir arrives at the upper reservoir, an eruption can occur through different eruption channels with different resistances. The amount of erupted lava will be larger for the lower-resistance channel, and it will vary greatly if the range of resistance variation is large among different channels. The net result in this case is the tendency to have a common duration of eruption and a diverse range of the amount of erupted lava, as observed at Piton de la Fournaise. The above reasoning throws some light on the difficulty in matching the observed and simulated V-d relation for Mount Etna. The simulated V-d relation for the Etna type shown in Figure 4 consists roughly of two branches; one nearly horizontal at the amount of erupted magma 3-10 million cubic meters and the other nearly vertical at the duration of days. The former branch is associated with the supply of magma from the mantle reservoir with a small capacity connected to a high resistance channel, and the latter branch is associated with the supply of magma from the mantle reservoir with a large capacity connected to a very low resistance channel. The former branch is produced by the diversity of the eruption channel as in the Kilauea case but with much less amount of erupted lava because of the small capacity of contributing reservoirs. The latter branch is produced by the characteristic duration of the batch from the mantle as in the Fournaise case. Thus we needed a combination of the Kilauea type relation and the Fournaise type relation to cover the observed range of V and d for the Etna type within the framework of our model. Since the observed V-d relation for Mount Etna is remarkably similar to that observed for Kilauea, we are tempted to imagine a model of diversity of eruption channels connected to a large reservoir under Mount Etna also. Such a model may be possible if the large mantle reservoir is connected to the eruption site not through one or two channels, as in our current model, but through a hierarchy of channels with a broad range of resistance. In other words, observed V-d relations may be explained by assuming one or two channels from the mantle to the upper reservoir for Kilauea and Piton de la Fournaise, but for Mount Etna we need a diversity of channels from the mantle reservoir. This is an acceptable proposition considering the well-known complexity of tectonics and crustal structure of the region where Mount Etna is situated (plate boundary), as compared to Kilauea and Piton de la Fournaise (mid-plate). It must be noted, however, that the above discussion is based on the assumption of the absence of a large shallow reservoir under Mount Etna. Why has a large shallow reservoir developed under Kilauea and not under Piton de la Fournaise? The answer to this question may be found in the much more extensive rift zone developed under Kilauea than under Piton de la Fournaise as discussed by Aki and Ferrazzini [2000]. Nakarnura [ 1980] was the first to ask why long rift zones develop in Hawaiian

11 AKI AND FERRAZZINI: MODELING OF THREE ACTIVE VOLCANOES 4101 volcanoes. Comparing with the Galapagos islands, another intraplate basaltic volcano lacking a rift zone and located on a young lithosphere, he suggested that the intrusion into rift zones in Hawaii is sustained by repeated Kalapana type earthquakes caused by a slip along the top of the oceanic crust which may have anomalously low friction due to a thick sediment and a high pore pressure [Hubbert and Rubey, 1959]. This idea has been supported by more recent works [e.g., Dieterich, 1988; Thurber and Gripp, 1988]. Piton de la Fournaise is located on an old oceanic lithosphere with a flexural rigidity [Walcott, 1970; Bonnevill et al., 1988] comparable to the Hawaiian lithosphere, and the oceani crust upon which the volcano was built must have had a thick sediment cover. There is, however, a major difference in the speed of lithosphere relative to the mantle under these two volcanoes. In Hawaii the old oceanic crust is always present in front of the youngest volcanowing to the high plate speed, and the Kalapana type earthquake occurs along the weak plane formed by the deep-sea sediment, sustaining the intrusion into the rift zone. On the other hand, the African plate, on which Piton de la Fournaise is located, moves so slowly that a fluctuation in the path of magma ascent in the crust can move the volcano back and forth, allowing the remains of proto- Fournaise volcano, as evidenced from the old magma chamber encountered by deep drilling [Rancon et al., 1989] as well as the gravity highs deviating from the current summit area [Rousset al., 1989], to preventhe full development of a rift zone. Thus it is more difficult for magma to intrude into the rift zone in Reunion Island than in Hawaii, and that explains why Piton de la Fournaise has a remarkable central cone, as high as 500 m from the floor of the Enclos caldera, due to repeated summit eruptions, which is absent in the Kilauea volcano. We suggest that the highly developed rift zone under Kilauea made the magma transport in the horizontal direction from the summit area relatively easy, and repeated lateral movements of magma helped to create a permanent shallow reservoir under the summit. Aki and Ferrazzini [2000] found a source of long period events under the summit of Piton de la Fournaise, which was attributed to lateral movement of magma during the period from 1985 to This source generated nearly identical wave forms and arrival times at three summit stations apart-1 km from each other, suggesting a laterally extended region containing magma beneath the summit. None of the longperiod events from 1995 to 2000, however, showed the above coherency among the three summit stations. This suggests the temporary nature of the shallow reservoir under Piton de a Fournaise, if it exists. Another characteristic of Piton de la Fournaise different from other major basaltic volcanoes is the absence of large nearby earthquakes. The magnitude of the largest volcano-tectonic earthquake observed so far is -2.5, while it is well known that earthquakes with magnitude >7 occur near Kilauea, Mount Etna, and volcanoes in Iceland. As Gudrnundsson and Hornberg [1999] propose using the example of southern Iceland, an earthquake is large only if its rupture area is large, and a large earthquake is possible only when a large region is homogeneously stressed. Under Piton de la Fournaise the stress is concentrated to a small volume under the summit area as indicated by the strong clustering of hypocenters of volcanotectonic earthquakes beneath the summit. The small volume is also reflected in the extremely short duration of the swarm of volcano-tectonic events precursory to eruptions as compared to other volcanoes (<1 hour for summit eruptions). The absence of large earthquakes under Piton de la Fournaise implies that the source of stress concentration is localized only under the summit, preventing homogenization of the stress field in a large volume, necessary for the occurrence of a large earthquake. The ability of our model having a simple channel from the mantle to the shallowest reservoir in explaining the observed V-d relation and the absence of well-developed rift zones as discussed above support a localized stress concentration solely under the summit. In the present paper we focused on simulating the V-d relation because it is more easily quantified than the spatial and temporal characteristics of the eruption history. Our original intention in this study was the use of a quantitative modeling for the prediction of eruption at Piton de la Fournaise. The present study offers a starting point toward this goal, and the relative simplicity of Piton de la Fournaise as compared to other volcanoes revealed by the present study encourages us to proceed. 7. Conclusion 1. From published data, we found characteristic relations between the amount V of erupted lava and the duration d of eruption for Mount Etna, Kilauea, and Piton de la Fournaise. The relation is similar between Mount Etna and Kilauea, where the increase of V with the increasing d is slow, showing a trend of a lower mean flow rate for a larger eruption. For Piton de la Fournaise, however, the trend is distinctly different, showing a higher mean flow rate for a larger eruption. 2. The presence or absence of a large magma reservoir in the volcanic edifices of Mount Etna and Piton de la Fournaise is still an open question, although the presence of a large shallow reservoir is widely accepted under Kilauea. We constructed quantitative models of magma systems with reservoirs at various levels in order to test hypotheses about the existence of large reservoirs under these volcanoes using the observed V-d relations. 3. We found that the observed V-d relation is consistent with the presence of a large reservoir at a shallow depth under Kilauea and with the presence of a large reservoir near the bottom of the volcanic edifice under Piton de la Fournaise. 4. The above models for Kilauea and?iton de la Fournaise are characterized by a simple path from the mantle reservoir to the shallowest reservoir, which is connected to a hierarchy of eruption channels with varying resistance. We could not find, however, a satisfactory agreement between the observed and simulated V-d relation for Mount Etna within the framework of our model, suggesting that there must exist a hierarchy of channels with varying resistance from the mantle reservoir to the eruption sites under Mount Etna if there is no large reservoir at a shallow depth. 5. Time histories of pressures in the reservoirs at various levels obtained for the best fitting models explain the inflationdeflation cycle observed at Kilauea and its absence at Piton de la Fournaise. 6. The absence of volcano-tectonic earthquakes with magnitude greater than -2.5 near Piton de la Fournaise is attributed to the simplicity of the magma path from the mantle to the shallowest reservoir and the underdeveloped rift zone, which result in a stress concentration localized solely beneath the summit area. Acknowledgments. We thank Jean-Francois L nat for sending us a copy of the table of eruptions at Mount Etna compiled by Tanguy and Patan. We also thank Alan Linde, Alessandro Bonaccorso, and an anonymous reviewer for valuable comments, upon which the original manuscript was revised. This work was supported in part by the W. M. Keck Foundation and in part by the INSU under the PNRN 1995 project, CDP

12 4102 AKI AND FERRAZZINI: MODELING OF THREE ACTIVE VOLCANOES References Aki, K., and V. Ferrazzini, Seismic monitoring and modeling of an active volcano for prediction, d. Geophys. Res., 10.5, 16,617-16,640, Bach lery, P., Le fonctionnement des Volcans-Boucliers, m moire, Fac. des Sci. de la Terre, Univ. de la R union, Saint-Denis, La R union, France, Beauducel, F., P. Briole, and J. L. Froger, Volcano-wide fringes in ERS synthetic aperture radar interferograms of Etna ( ): Deformation or tropospheric effect?, d. Geophys. Res., 10.5, 16,391-16,402, Benard, R., and M. Kraft, Au Coeur de la Fournaise, 220 pp., Nourault- Benard, Saint-Denis, La R union, France, Bonaccorso, A., Dynamic inversion of ground deformation data for modeling volcanic sources (Etna ), Geophys. Res. Lett., 23, , Bonneville, A., J.P. Barriot, and R. Bayer, Evidence from geoid data of a hotspot origin for the southern Mascarene plateau and Mascarene islands (Indian Ocean), d. Geophys. Res., 93, , Corsaro, R.A., R. Cristofolini, and L. Patan, The 1669 eruption at Mount Etna: Chronology, petrology and geochemistry, with inferences on the magma sources and ascent mechanisms, Bull. Volcanol., 58, , Decker, R. W., Kilauea volcanic activity: An electrical analog model (abstract), Eos Trans. AGU, 49, , Decker, R. W., Dynamics of Hawaiian volcanoes: An overview, in Volcanism in Hawaii, edited by R.W. Decker, T.L. Wright, and P.H. Stauffer, U.S. Geol. Surv. Prof Pap., 13.50, , Dieterich, J.H., Growth and persistence of Hawaiian rift zones, d. Geophys. Res., 93, , Gudmundsson, A. and C. Homberg, Evolution of stress fields and faulting in seismic zones, Pure Appl. Geophys., 154, , Hubbert, M.K., and W.W. Rubey, Mechanics of fluid filled porous solids and its application to overthrust faulting, Geol. Soc. Am. Bull., 70, , Klein, F.W., Patterns of historical eruptions at Hawaiian volcanoes, d. Volcanol. Geotherm. Res., 12, 1-35, Lahaie, F., and J.R. Grasso, A fluid-rock interaction cellular automaton of volcano mechanics: Application to the Piton de la Fournaise, d. Geophys. Res., 103, , Lanari, R., P. Lundgren, and E. Sansosti, Dynamic deformation of Etna volcano by satellite radar interferometry, Geophys. Res. Lett., 2.5, , L nat, J.-F., and P. Bach lery, Dynamics of magma transfers at Piton de la Fournaise volcano (R union Island, Indian Ocean), in Modeling of Volcanic Processes edited by C.-Y. King and R. Scarpa, pp , Fried. Vieweg, und Sohn, Brunswick, Germany, Massonnet, D., P. Briole, and A. Arnaud, Deflation of Mount Etna monitored by spaceborne radar interferometry, Nature, 37.5, , Mulgargia, F., P. Gasparini, and S. Tinti, Identifying different regimes in eruptive activity: An application to Etna volcano, d. Volcanol. Geotherm. Res., 34, , Murru, M., C. Montuori, M. Wyss and E. Privitera, The locations of magma chambers at Mt. Etna, Italy, mapped by b-values, Geophys. Res. Lett., 26, , Nakamura, K., Why do long rift zones develop in Hawaiian volcanoes: A possible role of thick oceanic sediments, Bull. Volcanol. Soc. dpn., Ser. 2, 25, , Rancon, J.P., P. Lerebour, and T. Auge, The Grand Brule exploration drilling: New data on the deep framework of the Piton de la Fournaise volcano, part 1, d. Volcanol. Geotherm. Res., 36, , Rousset, D., A. kesquer, A. Bonneville, and J.F. L nat, Complete gravity study of Piton de la Fournaise volcano, Reunion Island, d. Volcanol. Geotherm. Res., 36, 37-52, Shimozuru, D., Magma reservoir systems infered from tilt patterns, Bull. Volcanol., 44, ,1981. Sigmundsson, F., P. Durand, and D. Massonnet, Opening of an eruptive fissure and seaward displacement at Piton de la Fournaise volcano measured by RADARSAT satellite radar interferometry, Geophys. Res. Lett., 26, , Staudacher, T., P. Bachelery, M.P. Semet, and J.L. Cheminee, Piton de la Fournaise, Bull. Global Volcanism Network, 23, (3), 2-4, Sfieltjes, L., and P. Moutou, A statistical and probabilistic study of the historic activity of Piton de la Fournaise, Reunion island, Indian Ocean, d. Volcanol. Geotherm. Res., 36, 67-86, Tanguy, J.C., and G. Patan, L 'Etna et le Monde des Volcans, 278 pp., Diderot, Arts et Sciences, Paris, Thurber, C.H., and A.E. Gripp, Flexure and seismicity beneath the south flank of Kilauea volcano and tectonic implications, d. Geophys. Res., 93, , Wadge, G., The storage and release of magma on Mount Etna, d. Volcanol. Geotherm. Res., 2, , Wadge, G., G.P.L. Walker, and J.E. Guest, The output of the Etna volcano, Nature, 255, , Walcott, R.I., Flexure of the lithosphere at Hawaii, Tectonophysics, 9, , Williams, H., and A.R. l cbirney, Volcanology, W.H. Freeman, New York, K. Aki and V. Ferrazzini, Observatoire Volcanologique du Piton de la Fournaise, 14 Route Nationale 3, La Plaine des Cafres, La R union, France. (aki@iremia.univ-reunion.fr) (Received December 2, 1999; revised April 25, 2000; accepted August 18, 2000.)