A dynamic balance between magma supply and eruption rate

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B8, PAGES 18,091-18,100, AUGUST 10, 1997 A dynamic balance between magma supply and eruption rate at Kilauea volcano, Hawaii Roger P. Denlinger Cascades Volcano Observatory, U.S. Geological Survey, Vancouver, Washington Abstract. The dynamic balance between magma supply and vent output at Kilauea volcano is used to estimate both the volume of magma stored within Kilauea volcano and its magma supply rate. Throughout most of 1991 a linear decline in volume flux from the Kupaianaha vent on Kilauea's east rift zone was associated with a parabolic variation in the elevation of Kilauea's summit as vent output initially exceeded then lagged behind the magma supply to the volcano. The correspondence between summit elevation and tilt established with over 30 years of data provided daily estimates of summit elevation in terms of summitilt. The minimum in the parabolic variation in summitilt and elevation (or zero elevation change) occurs when the magma supply to the reservoir from below the volcano equals the magma output from the reservoir to the surface, so that the magma supply rate is given by vent flux on that day. The measurements of vent flux and tilt establish that the magma supply rate to Kilauea volcano on June 19, 1991, was 217, ,000 m3/d (or km3/yr). This is close to the averageruptive rate of 0.08 km3/yr between 1958 and In addition, the predictable response of summit elevation and tilt to each east rift zone eruption near Puu Oo since 1983 shows that summit deformation is also a measure of magma reservoir pressure. Given this, the correlation between the elevation of the Puu Oo lava lake (4 km uprift of Kupaianahand 18 km from the summit) and summit tilt provides an estimate for magma pressure changes corresponding to summit tilt changes. The ratio of the change in volume to the change in reservoir pressure (dv/dp) during vent activity may be determined by dividing the ratio of volume erupted to change in summit tilt (dv/dtilt) by the ratio of pressure change to change in summit tilt (dp/dtilt). This measure of dv/dp, when combined with laboratory measurements of the bulk modulus of tholeitic melt, provides an estimate of 240 +_ 50 km 3 for the volume of Kilauea's magma reservoir. This estimate is much larger than t3aditional estimates but consistent with seismic tomographic imaging and geophysical modeling of Kilauea's magma system. Introduction When basaltic volcanoes have active lava lakes in addition to other active vents, the lakes can be used as manometers for magma pressure [Tilling, 1987]. Such a vent configuration existed during the current eruption along Kilauea volcano's east rift zone (ERZ) (Figure 1) between 1989 and 1996 [Mangan et al., 1995; U.S. Geological Survey, (USGS), unpublished data, 1996]. A linear decline in lava flux from the Kupaianaha vent was observed beginning in 1991 and lasting until the vent died in February 1992 [Kauahikaua et al., 1996]. When Kupaianaha was the only outlet, this linear decline in flux was associated with a parabolic variation in tilt measured near the summit of magma supply rate to the volcano and the volume of melt stored within its magma reservoir in a way that is independerlt of the geometry of the magma system. Previous estimates of magma reservoir volume and pressure at Kilauea volcano have been based on surface deformation data [Tilling and Dvorak, 1993] using models that assume a local summit source with a spherical magma chamber geome- try. With a Mogi (spherical chamber) model, changes in sur- fac elevation, strain, and slope are used to calculate the depth and change in volume of the spherical reservoir [Delaney and McTigue, 1994], and there is a trade-off between chamber size and pressure. Such models for Kilauea give a 3-4 km depth to the center of the reservoir, a total volume not exceeding 15 the volcano. Previous studies have established that eruptions at km 3, high values of reservoir pressure, and locations for the Puu Oo (Figure 1) are faithfully and predictably represented reservoir that may vary by 2-3 km. For small summit eruptions both by changes in summit tilt and elevation [Wolfe et al., 1988; the change in reservoir volume given by the Mogi model is Okamura et al., 1988], with summit inflation prior to the erupcomparable (within a factor of 2) to the volume of surface tion and summit deflation beginning with lava extrusion. This subsidence and the volume of magma erupted that caused the correspondence is strengthened in this paper by establishing a subsidence [Dzurisin et al., 1984; Delaney and McTigue, 1994]. strong correlation between the level of an active lava lake The latter comparison is used to estimate both the magma within Puu Oo, summit elevation, and summit tilt. The tilt, lava supply rate to the volcano and the amount of magma supplied lake, and vent flux data are then analyzed to estimate both the to the volcano that remains underground. The magma supply This paper is not subject to U.S. copyright. Published in 1997 rates are in the range of km3/yr [Dvorak and Dzuriby the American Geophysical Union. sin, 1993] and are particularly robust when establisheduring Paper number 97JB long-lived eruptions that produce little or no changes in sum- 18,091

2 18,092 DENLINGER: A DYNAMIC BALANCE AT KILAUEA VOLCANO ISLAND OF MAUNA LOA vo..o,, rd Park Uwe Vault'* 5KM 1 Kupaianaha.. Kflauea Caldera EPE o' Figure 1. (a) A map of Kilauea volcano showing the locations of Bird Park, Hawaiian Volcano Observatory (HVO) 35, Puu Oo, Kupaianaha, and the Uwekahuna vault. An active lava pond has been present in Puu Oo since 1989 in addition to active vents along the east rift zone. (b) Map of the summit region showing locations of laser ranging lines 1 and 2 relative to HVO 35, the Uwekahuna vault (Uwe vault), Bird Park, and the faults forming Kilauea caldera. mit subsidence [Swanson, 1972]. However, neither the summit reservoir volume nor its pressure are so well determined as the magma supply rate. Instead of providing better definition of a summit reservoir centered in the caldera, many previous studies have hinted at the existence of a large magma system mainly to the south of Kilauea caldera. More than a decade ago, gravity and seismicity data were used to infer the existence of two magma reservoirs: a small summit reservoir near the center of Kilauea caldera and a large deeper reservoir that underlies the Koae fault system and extends along the rift zones [Ryan et al., 1981; Ryan, 1988]. Recently, this view has received additional support from detailed studies of the gravity and magnetic anomalies overlying the Koae and rift zones (T. Hildebrand and J. Kauahikaua, written communication, 1996), which produced magmatic sources that are much deeper and more extensive than most studies of the summit have recognized. The best definition comes from detailed seismic tomography with a cell size of a few hundred meters (H. Benz, written communication, 1997) that is then averaged to cells 2 km by 2 km by 1 km deep (Figure 2). This provides a three-dimensional (3-D) image that shows a large high-velocity anomaly south of Kilauea at a depth of 5-7 km that extends along the rift zones. This seismic tomographic anomaly occupies the same map area as the combined gravity and magnetic anomaly over Kilauea and along its rift zones, and modeling of these potential fields also produces a dense source centered at a depth of between 5 and 7 km (J. Kauahikaua, USGS, written communication, 1996). The volume obtained for the magma system from each of these seismic and potential field modeling studies is about 500 km 3 and, presumably, is composed of dense, hot, high-velocity material. Plots of earthquake hypocenters observed from 1970 to 1990 show that this volume is aseismic at depths of between 4 and 7 km beneath the Koae and along the rift zones [Denlinger and Okubo, 1995] despite high rates of flank extension in these areas [Owen et al., 1995]. The existence of a large magmatic system for Kilauea volcano is evident from the petrology of the erupted lava as well. The erupted melt is missing large quantities of olivine, yet the presence and chemistry of deformed dunite xenoliths and clusters of deformed crystals in erupted lavas show that the olivine not only precipitated within Kilauea's magma system but was later remobilized during eruption [Clague and Denlinger, 1994]. The volume of missing olivine in erupted lavas is about!5% of the volume of the volcano, and this discrepancy supports the model of a hot, dense core of picritic mush for Kilauea that gives rise to the gravity, magnetic, and seismic anomalies. In this paper I assume that the summit deformation data is a measure of the inflation and deflation of the volcano's magma reservoir, as is justified by a long history of observation of summit deformation associated with volcanic activity [Tilling and Dvorak, 1993]. I compare the tilt data from the Uwekahuna vault (Figures 1 and 2) with the leveling of benchmarks surrounding the vault and in the summit region to show the way the vault tilt data represent inflation and deflation of the summit of the volcano. I test the correspondence between three different measurements of summit deformation using nonparametric statistics and use the results to relate summit tilt to summit elevation. Then I show that there is a dynamic balance between summit elevation, magma supply, and vent output and that these data support the existence of a large volume of melt within Kilauea volcano. The volume estimates I obtain are 20 times larger than traditional geodetic estimates of summit reservoir volume and more in accord with the petrologic, geophysical, and structural evidence for a large magma system. Summit Tilt, Summit Elevation, and Horizontal Summit Strain The history of water tube tilt at the Uwekahuna vault, surface elevation at Hawaiian Volcano Observatory (HVO) 35, and frequently measured changes in the length of line 1 (all plotted together in Figure lb) are shown in Figure 3. The "elevation" in Figure 3 is the elevation difference between benchmark HVO 35 and Bird Park. These differences were determined by spirit leveling benchmarks to National Geodetic Survey (NGS) second-order standards from Bird Park and around the caldera, then evenly distributing the misclosure error (due to the single loop closure around the caldera) to all the benchmarks in the loop. Typically, these errors are 2 orders

3 ... DENLINGER: A DYNAMIC BALANCE AT KILAUEA VOLCANO 18, :-:,'.5;i!!':..5:. '":': '...5.:.Z'.." :" '... ' '"' :: : : ::;: ::.:: :: :. --:. %? ;, ':..::..:.:....: :.,?......,' ,.. ':.... ;;'.:-, :. ;:.;..::.;. : ::,; : :';,..:?.:..:'**:'...,:?:: '"...;,.:?:.: '-; Z :*:-*½/ ' **'.**, :..' :%; :::,..., ;; :--.-½:, ::... : :... ; iii::::::i:: ;::i::i.:.:. ':""':* X:... ;... *½*?*:'::':"':::"*:::* ; :: :*:'?:*:*:*:*:*:*:*:*:*:*: '".:::***::::;;;;i;:,::},... :'*;:";;:',:;-,:,'," ""': ½ ß - :.:,:,.... :; :::;;:? ;;.;.;:...".' :,... ;;): ; :( ½:4. ß...;: ;;: ;;;;:'"'": :.: :.½. : ½;, ::::::... ;}... ;; ; : ß.:.: ::.;:.,:... :..:.... ;:,,;. : :: :: :.:::.:...::::...: Figure 2. Map view of seismic anomaly at a depth of 3-5 km determined using tomographic methods (H. Benz, written communication, 1997). The high-velocity anomaly coincides with a gravity high and magnetic low anomaly over the caldera and along rift zones. These anomalies also are centered over the areas of inflation and deflation at the summit and along the east rift zone and are indicative of a large magma system for Kilauea volcano. of magnitude less than the elevation changes due to volcanic and related tectonic activity within the caldera. The time series for the three different measurements of surface deformation in Figure 3 are remarkably coherent with each other in time, reflecting the volcanic events and large earthquake that have marked the history of the caldera since 1968 [Klein et al., 1987]. The last three decades of summit deformation, as shown in Figure 3, can be broadly characterized as summit uplift prior to the M earthquake and summit subsidence thereafter, interrupted only by the inflation prior to the eruption of Puu Oo in The patterns of uplift and subsidence are shown by contouring height changes (labeled as Z in the legends) in Figure 4 for four periods. The contouring was done without gridding by triangulating the benchmark arrays, a method that does not create spurious elevation changes by extrapolation. The four periods were chosen to coincide with events on the volcano: is the inflationary period prior to the 1974 southwest rift zone intrusion; includes both the southwest rift zone intrusion and the 1975 earthquake; is the period between the earthquake and the eruption of Puu Oo; and begins with the eruption of Puu Oo and shows the steady subsidence of the summit since that time. Except for the epoch all periods exhibit a remarkably circular pattern of uplift or subsidence that is closely matched by the seismic tomography anomaly in Figure 2. Despite the circular pattern of uplift and subsidence the Uwekahuna vault tilts primarily on a north-south axis in response to inflation and deflation of the summit, as is evident in Figure 5. In Figure 5 the location of the vault is superimposed on the surface determined by contouring height changes (labeled Z in the legends) on benchmarks surrounding the vault. The contours are obtained by a Kriging interpolation over the rectangular area shown in Figure 5 and fit the height changes in a least squares sense. A comparison of Figures 3, 4, and 5 shows that except for the period (which contains the 1975 earthquake and the 1974 opening of the southwest rift zone), the vault tilts north as the summit inflates and south as it deflates. In contrast, the period is dominated by height changes whose contours parallel the primary extensional faults near the vault and along the southwest rift zone, so that the tilt of the vault was in a N60øW-S30øE direction when the rifting of the caldera and southwest rift zone occurred.

4 18,094 DENLINGER: A DYNAMIC BALANCE AT KILAUEA VOLCANO NS Uwekahuna Tilt F, ' ' - EW Uwekahuna Tilt : ß Length Line 1,, o_ Elevation HVO o 12oo -._(2 E :. -. '.;.. ' ooo- '".., I I I Time (year) Figure 3. History of surfacelevation at HVO 35, water tube tilt at Uwekahuna vault, and frequently measured changes in the length of line 1 (Figure la) since Each data set is independent, yet the time series are remarkably coherent. The divergence of the EW tilt reflects the opening of the southwest rift zone after The N-S component of water tube tilt faithfully represents (0.079 km3/yr) from linear interpolation of actual flux meathe change in elevation of the summit, as is evident from surements. These flux values have been corrected for the dif- Figures 3, 4, and 5, and the relationship between summit ele- ference between reservoir density (2650 kg/m 3) and the density vation and N-S vault tilt is linear (Figure 6). To test this, a of melt erupted at the surface (1960 kg/m 3) by multiplying by nonparametric Spearman test of correlation [Gibbons, 1976] the ratio of these two densities. was made between the leveling and tilt data near the vault, and A parabolic variation in summit tilt with systematic changes the results are shown in Table 1. The linearity between the N-S in vent flux occurred again between April 1992 and August summitilt and the elevation of HVO 35 for the epochshown 1992 (see Figure 8a), during which time the parabolic variation in Figures 4 and 5 is highly correlated and reflects the predom- in tilt is inverted from that shown in Figure 7 and the vent flux inate rotation of the surface of the vault on a nearly E-W axis was gradually increasing time. However, there are only two as the summit inflates and deflates over periods of months and flux measurements at the endpoints and no measurements years. For this reason the N-S component of tilt will be re- near the apex of the parabolic variation in tilt. Therefore, ferred to as summit tilt and used as a measure of summit though the pattern was repeated, the measurements needed to elevation in the remainder of this paper. check for repeatability were not obtained. Variation of Summit Tilt During Linear Changes Variation of Summit Tilt With Volume of Erupted Melt in Vent Output The relationship between a change in reservoir volume A linear decline in output flux from the Kupaianaha vent (since flux data are corrected to reservoir density) and a [Kauahikaua et al., 1996] was associated with systematic change in tilt required to flatten the parabola in Figure 7 with changes in lake level and summitilt between February 1991 the linear vent flux data alone is 0.8 x 10 ' m 3 of lava produced and September The measurements of vent flux are ro- per microradian of N-S Uwekahuna water tube tilt. Alternabust: the observed linear decline in flux was used to predict, 6 tively, the same relationship may be determined from a linear months in advance, when the Kupaianaha vent flow would fit to a plot of the volume of eruptions with rapid extrusion stop, and this prediction was accurate to within a few days rates along the east rift zone versusummitilt for the period [Kauahikaua et al., 1996]. While the flux was declining, the [Epp, 1983]. Using all east rift zone eruptions with summitilt was parabolic in time, as shown in Figure 7, until extrusion rates at least 2 x 10 ' m3/d (10 times the magma rising lava lake levels in Puu Oo ruptured the side of that supply rate), a value of x 106 of volum erupted edifice [Mangan et al., 1995]. Using a Marquardt regression, per microradian of tilt is obtained. the summitilt data in Figure 7 are found to closely fit a parabolic polynomial with a correlation coefficient of 0.91 and Variation of Summit Inflation and Deflation With the Elevation of a Lava Lake in Puu Oo the minimum at a time of _ (June 19, 1991). The measurements of vent flux show that the flux rate on June 19 During the early eruptive episodes at Puu Oo, there was a was 205,000 m3/d (0.075 km3/yr) from the straight line fit to the consistent pattern of inflation of Kilauea' summit prior to vent flux data in Figure 7, or 217,000 m3/d _+ 10,000 m3/day each eruption followed by rapid deflation when the eruption at

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':.::::::::::::: -"'<-{ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: -o.81 ' :::::::-'.-'. -:..:.:'::::""' '-'-:..-:.::. :. ::::::::::::::::::::::::::::::..:::.::. ::::::::::::::: :_..,.' j:',_::::...:..:...:::...:> ::: ,.....:-:<-.-..< :... <.. ::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::: :..: ::.:..::::::... :::.:,:::::..:... ::,:..-.:...:.-...:.-. ;..:.::? e:--'-:... ::::::::::::::::::::::::::::::.....,...-:-:..:..::.,1_, _ / -'-... Q?...., --.-,,,., : :.::::..-: _ Figure 4. The elevation changes on each benchmark (shown as a dot) relative to Bird Park (last dot at NW corner of array) are contoured here by firstriangulating the array and then contouring the triangles. This methodoes not produce spurious elevations by extrapolation and reveals a circular pattern for the uplift and subsidence of the summit region that is south of the caldera. Compare this pattern to Figure 2. Puu Oo began. The mean time delay between the impulsive After 1994 the deepening lake levels, increasing fume, and onset of each fountaining episode and summit deflation was changing of observers and methods degrade the measureabout 2 hours [Wolfe et al., 1988]. As the elevation of the ments. The lake elevation dropped by 15 m between April 23 summit subsided between 1983 and 1986, the maximum height and June 15 (a 52 day period) and rose by 36 m between June achieved by lava fountains also systematically declined, and in 15 and August 29 (a 75 day period). As shown in Figure 7, the 1988 a lava lake formed within the vent at Puu Oo. As the summit also deflated (elevation and tilt declined) between Kupaianaha vent flux continued its decline the latter half of April 23 and June 19 and inflated between June 19 and August 1991, the summitilt and summit elevation, which had indi- 29. However, the lake level observations are episodic and are cated steady deflation since 1983 [Delaney et al., 1993] (Figure 3), easily affected by local pressure fluctuations near Puu Oo. reversed and indicated summit inflation as shown in Figure 7. These pressure disturbances may be caused by changes in lava The systematic changes in the lava lake elevation Puu Oo flow out of vents near Puu Oo, by changes in storage in the between 1991 and 1994 correlate to changes in summit eleva- conduits feeding Puu Oo and Kupaianaha, or by gas piston tion and hence to inflation and deflation of the volcano (Figure activity in nearby vents [Greenland et al., 1988]. Since the last 8). The lake level was determined by measuring angles and two causes are unconstrained by data, a 4 year time history was distances from fixed points on the rim of Puu Oo to the rim of constructed, as shown in Figure 8, and nonparametric tests the lake (C. C. Heliker et al., Episodes of the Puu were done for any long period correlation between lake level Oo-Kupaianaha eruption of Kilauea volcano, submitted to fluctuations and summit tilt. The results, shown in Table 2, Bulletin of Volcanology, 1996). Before 1994 the variance of the indicate that both the lake level elevation and summit elevalake level measurements is 5 m, or about one tenth of the tion are well correlated to summit tilt, as is expected from the observed range of lake level variation between 1991 and field observations.

6 18,096 DENLINGER: A DYNAMIC BALANCE AT KILAUEA VOLCANO to Z 1974 to Bird Park---- -* Uwekahuna Vault ?.::.: O 22 I, i,, I,, to 1983 Z to 1993 z -:.:-:.:.:.:.x :..-::-: a-0.12 : ':: :-:- :: 67 :' :'"'" i :""'"": :.. :-:i '*' :... "" """ ::::::::::::::::::::::::::::::::::::::::::::::::: ,, I,,, I,, I,,,, I I Figure 5. The surface defined by the elevation changes on benchmark surrounding the Uwekahuna vault, where summit tilt is measured, is determined here in a least square sense. The contoured change show that the vault tilts north when the summit inflates and south when the summit deflates over periods of months and years. The period , which includes the 1974 southwest rift zone eruption and the M earthquake, is the only exception as it is dominated by movement of extensional faults near the vault. Method of Analysis The observation of a parabolic change in tilt with linear decline in vent flux is expected from continuity of mass and is independent of reservoir geometry or fluid rheology. The combination of a constant magma supply a to Kilauea's magma reservoir with a steady decline in vent flux b t at the Kupaianaha vent in 1991 will produce a change in reservoir volume V with time t that may be written dv d- = a - bt (1) where the magma supply rate a is given by the vent flux b t when dv/dt is zero. At time to the magma supply to the reservoir, a, equals the outflow from the vent, bto, so the equation is dv d- -= b(to- t) (2) where a has been set to bt o. Integration of this equation over time yields an equation that is parabolic in time and therefore gives a change in reservoir volume (and pressure) that is parabolic in time that is consistent with the observed parabolic variation in summit tilt. This equation is b V= V0+ (t-t0) 2 (3) where Vo is the volume of the reservoir at time to (this time corresponds to the minimum in the parabolic curve shown in Figure 7). This concept may be used to interpretilt data during an eruption and is supported by concurrent observations of tilt and flux at Kilauea (Figure 7). A parabolic change in tilt throughout an eruption means that the vent flux is either linearly increasing with time or linearly decreasing with time and at some time during the eruption equaled the magma supply rate. If the tilt be-

7 DENLINGER: A DYNAMIC BALANCE AT KILAUEA VOLCANO 18, [] 1965-Nov 1975 /x Nov 1975-Jan 1983 O o 11o8 - ß Tilt atuwe ß Measured vent flux Linear fit to vent flux ,; ra Parabo fit to tilt ,i i' ' 11oo I I I i i Elevation of HVO 35, meters Figure 6. The N-S summit tilt at Uwekahuna is well correlated to the elevation of HVO 35 (see Table 1). Both quantities are measured as elevation differences: the tilt from the difference along a 10 m long water tube and the elevation from the elevation difference between HVO 35 and Bird Park. The rise and fall of the N-S tilt in 1991 corresponds to a rise and fall of summit elevation, and the correlation with the lake level data shows that these changes resulted from changes in magma pressure within Kilauea volcano. The time periods are the same as for Figures 4 and 5, with the 1974-!975 and periods lumped together to form the period here. Time, year Figure 7. The parabolic variation in summit tilt (solid symbols) during a steady decline in vent flux (open circles) at Kupaianaha is expected for a simple overpressurized magma system with a nearly constant input flux and steadily declining output flux. An increase in summit tilt corresponds to an increase in summit elevation and inflation of a summit magma chamber, whereas a decrease corresponds to deflation. At the parabolic minimum in tilt, input equals output, so that a vent flux measurement provides the magma supply rate. For Kilauea volcano the magma supply rate on June 19, 1991, was 217,000 m3/d _+ 10,000 m3/d, (or km3/yr) and is close to the averag eruptive rate of 0.08 km3/yr between 1958 and comes linear with time, then the vent flux has stabilized at a constant value either less than (for increasing tilt) or greater than (for decreasing tilt) the magma supply to the reservoir. The vent flux, lake level, and summit tilt may also be used to estimate the total reservoir volume by estimating the change in reservoir pressure that corresponds to a change in reservoir volume. The volume erupted between 1990 and 1995 is small (0.1 km3/yr) relative to previous estimates (10-15 km 3) of the volume of magma stored within Kilauea [Tilling and Dvorak, 1993]; thus a linear relationship is assumed between changes in volume and changes in pressure. For linear compressibility of the magma in the reservoir and rigid host rock dv/dp = V/K (4) where K is the bulk modulus of the reservoir fluid and P is the pressure. The assumption of a rigid host rock produces a negligible error of 0.2%, as shown in finite element tests discussed later. In a laboratory setting with a known volume of liquid and a rigid container this equation may be used to measure the bulk modulus of the liquid. Here, with a measured bulk modulus for Kilauea's tholeitic melt at reservoir conditions [Fujii and Kushiro, 1977] this equation may be used to calculate reservoir volume. The ratio of change in volume to change in pressure may be estimated by relating both quantities to summit tilt. Table 1. Correlation of Leveling and Tilt Near Uwekahuna Vault A B Spearman Correlation of A and B If.4 and B Correlate, Correlation Coefficient of A = xb Rate of Change of.4 With Time , ppm/yr Rate of Change of B With Time N-S water tube tilt at Uwe vault E-W WT tilt at Uwe vault N-S WT tilt at Uwe vault E-W WT tilt at Uwe vault N-S WT tilt at Uwe vault N-S dry tilt at Uwe vault E-W dry tilt at Uwe vault N-S tilt component of elevation differences near Uwe vault E-W tilt component of elevation differences near Uwe vault Elevation difference between HVO 35 and Bird Park a ppm/yr +0.4 ppm/yr ppm/yr +.3 ppm/yr m/yr (-19. ppm/yr is N-S vector between HVO 35 and Bird Park) x = 145. _+ 4. ppm/m of elevation difference.

8 18,098 DENLINGER: A DYNAMIC BALANCE AT KILAUEA VOLCANO ; ,', 850 t ' Time, year -,:: - Lake level after 10 day.,.'::"-' pause in eruption,&,_.if,,7 '& &ik, z. '., b, Time, year E 840,, "- '... I Confidence 830 : ;;,!...[; _ 820- =o 8 0 -'--'--"" =':"'-i i... > 800,:, I 790 T_..... "-,', :: 780 -I ---,,... l l-,- '.,".w 'T.L 1' r'--- > 770- ' T"Z... J... _L ' 760- c ' I I I I i Summit Tilt, microradians Figure 8. Figures 8a and 8b show all data for days when both the summit tilt and the variation of elevation of an active lava lake in Puu Oo were measured during the epoch ] Figure 8c plots the lake elevation versus tilt. The systematic long-term trends Jn the data are statisticall), significant (Table 2) and are used to corrcjat½ pressure with summit tilt using a reservoir magma densit7 of 2650 k /m. The linear regression gives ].42 m/y ad of summit tilt, which corresponds to MPa/y ad of NS summit tilt. Interpretation of Reservoir Volume Changes From Changes in Summit Tilt Estimation of the rela. tionship between changes in reservoir volume and tilt either may be derived from the vent flux versus the tilt data in Figure 7 or from the total change in tilt with the 97 total erupted volume for historic ERZ eruptions [Epp, 1983]. If the latter type of data are used, then the parabola in Figure 7 is overcorrected, and the Kupaianaha vent produced more melt than can be accounted for by changes in summit tilt. This discrepancy between the two estimates of change in reservoir volume with change in tilt cannot be reconciled within the limits of the data, and one estimate must be discarded. Using the Kupaianaha data alone, the ratio between volume and tilt is 0.8 x 106 m3//zrad, and the flux a into the magma reservoir from the mantle is constant in time. Alternatively, discrete eruptions could be used to give the ratio between volume and tilt; in which case the Kupaianaha data suggest that the supply of melt to Kilauea's magma reservoir is inversely related to reservoir pressure. If the supply rate increases as pressure drops and decreases as pressure rises, then there is a parabolic modulation of the magma supply that is the inverse of the parabolic variation in tilt in Figure 7. These separate results cannot be proven or discounted, so both will be used to bracket the estimate of magma reservoir volume. Interpretation of Reservoir Pressure Changes From Lake Level Changes and Summit Tilt limit I... Regression Prediction Jimit fit Estimation of the relationship between changes in reservoir pressure with tilt may be made from the correlation between lake level, summit tilt, and summit elevation. As shown in Table 2, a Spearman nonparametric correlation test [Gibbons, 1976] of the data plotted in Figure 8 gave a rank correlation of 62% and a probability of no correlation much less than 1%, indicating that summit tilt and lake level variations are well correlated. This correlation is compared in Table 2 to nonparametric correlations for Mauna Ulu and Alae lava lakes between 1971 and 1974 [Tilling, 1987; Tilling et al., 1986] as well as to nonparametric correlations between summit elevation and summit tilt (Figure 3b). Mauna l 11u and Alae lava lake elevations were measured during a period with negligible longterm variations in N-S summit tilt [Dvorak and Dzurisin, 1993, Figure 3), and these lake levels are apparently dominated by short-term variations in magma pressures in response to processesuch as flow out of nearby vents or gas piston activity. In contrast, long period variations both in Puu Oo lake elevations and in summit tilt are recovered in the nonparametric analysis. On the basis of the strong nonparametric correlation the lake level at Puu Oo is plotted against summit tilt in Figure 8, and the data are fitted to a straight line using a linear regression. The result is a mean slope of 1.42 m//zrad (the straight middle line), and the slope has 95% confidence limits of 1.05 and The outer lines that bracket the data are the upper and lower confidence limits for thls data set based upon the regression analysis, where the data a e weighted by their errors. The mean slope equates to a pressure change of _ MPa//zrad of tilt for a magma column with a reservoir Table 2. Statistical Tests of Lava Lake Data Summit Tilt and Crater Floor Elevation If A Correlated to B, Spearman Correlation of Correlation of A andb, % A = xb, % NS Uwe tilt Puu Oo lake elevation 61 NS Uwe tilt Mauna Ulu lake elevation 14 EW Uwe tilt Mauna Ulu lake elevation 13 NS Uwe tilt Alae lake level elevation 20 Mauna Ulu lake Alae lake level elevation 16 elevation 58

9 DENLINGER: A DYNAMIC BALANCE AT KILAUEA VOLCANO 18,099 density of 2650 kg/m 3 (densities from Mangan et al. [1995]). This is one fourth or less of the pressure changes per microradian of summit tilt estimated by Decker et al. [1983] or Johnson [1992] on the basis of geodetic analyses of summit tilt and local subsidence with small spherical magma chamber models. The correlation between lava lake level and summit tilt was tested by two 10 day pauses in lava extrusion, one during February 6-16, 1992, and the other during February 3-14, These data, shown as gray diamonds in Figure 8b, provide an additional measurement of the relationship between tilt and lake level. Fountaining events at Puu Oo in 1983 and 1984 show that it takes on average about 2 hours for pressure transients generated at Puu Oo to be felt at the summit [Wolfe et al., 1988]. Ten days is sufficiently long for pressure transients due to flow to have dissipated in the magma system so that these 10 day pause measurements may be regarded as a measure of static pressure in the magma system. The difference between the two measurements gives a value of 1.8 m of lake level change per microradian of tilt, which is comparable to the estimate determined using all of the data within the error in the measurements. Volume of Kilauea's Magma System The volume of interconnected liquid in Kilauea's magma system may be estimated using (4), where dv/dp is obtained using summit tilt. No assumptions are necessary regarding reservoir shape, and the change in volume due to elastic straining of the crust is neglected. The bulk modulus of Hawaiian tholeiite at reservoir pressures and temperatures is 11.4 GPa [Fujii and Kushiro, 1977], whereas seismic velocities (H. Benz, written communication, 1997) show that the rock bulk modulus is about 49 GPa. Thus the bulk modulus of the host rock is 4-5 times greater than the magma. Assuming a negligible response of the host rock, dv/dp is estimated by dividing the ratio of erupted volume to tilt by the ratio of pressure change to tilt obtained from lake level data. Dividing the value of MPa//xrad obtained from lake level changes into either value for the change in reservoir volume per microradian provides the ratio dv/dp. The bulk modulus and dv/dp values are used in (3) to estimate the total volume of interconnected liquid that makes up the reservoir affecting summit tilt and feeding the active vent(s) along the east rift zone of Kilauea volcano. The result of using the value of 0.68 X 106 m3//xrad obtained from historic ERZ eruptions is a mean value for reservoir volume of 204 km 3, with' 95% confidence limits between 162 and 277 km 3. Alternatively, the value of 0.8 x 106 m3//xrad (obtained by reconciling the vent flux with the parabolic summit response in Figure 7) yields a reservoir volume of 240 km 3, with 95% confidence limits of 190 Kilauea volcano over time, the total volume of magma supplied to Kilauea since 1950 is about 4 km. This is about 1.5% of my estimate for the total volume stored and implies long residence times for the magma reservoir. At a magma supply rate in the neighborhood of 0.08 km /yr a reservoir volume of 240 km would take 3000 years to fill with no eruptions at all and tens of thousands of years to develop when surface eruptions subtract some of the supply. Such long residence times allow for the cooling of primary magmas from mantle temperatures (1300øC) to erupted temperatures (1150øC) and are in accord with estimates of magma residence times for midocean ridges [Goldstein et al., 1994]. and 326 km 3. These volume estimates are times larger than previous estimates for the summit magma reservoir Implications for Determining Magma Supply Rates [Dieterich and Decker, 1975] and more in accord with previous at Remote Volcanoes [Ryan, 1988] and current (H. Benz, written communication, 1997) estimates for the total volume of melt stored within This same method may provide an estimate of magma supply rate on remote basaltic volcanoes such as midocean ridges. Kilauea volcano. If the tilt recorded during an eruption on the ocean floor is A three-dimensional finite element model of Kilauea was parabolic in time (or even approximately so), then the eruptive used to test the assumption of a rigid host rock. I converted a digital elevation base of the topography and bathymetry into a three-dimensional finite element grid of 10 node tetrahedra. flux was changing linearly in time if magma supply was constant. In this case, given eruption duration from the tilt records or nearby hydrophones and total volume of the eruption from This mesh and images created with seismic tomography (H. Benz, written communication, 1997) were put into the same geographic database. The tetrahedra inside the high velocity seismic anomaly were assigned the fluid properties shown in Table 3. Properties Used to Test Coupling of Magma Reservoir to Basaltic Host Rock Bulk Modulus, Poissons Density, GPa Ratio kg/m 3 Basaltic host rock Magma Table 3. A I MPa drop in fluid pressure caused the reservoir volume to contract, and the volume of contraction was calculated. The change in volume caused by the pressure drop is a coupled response between the fluid and the rock when the body forces due to gravity are included. The total volume change is combined with the pressure change to calculate an apparent bulk modulus. The apparent bulk modulus is GPa (as compared to 11.4 GPa), giving a negligible 0.2% correction to the calculation that assumed a rigid host rock. Implications for Reservoir Volume and Magma Supply Rate to Kilauea Volcano This analysis gives a magma supply rate considerably less, and a stored magma volume significantly greater, than previous estimates. Both the total volume and the supply rate have important implications for the evolution of the magma system within Kilauea volcano. The magma supply rate implied by the flux measurements is the actual flux on June 19, 1991, or km3/yr, and is comparable to the averag eruptive rate of 0.08 km3/yr between 1958 and 1993 [Tilling and Dvorak, 1993; Dvorak and Dzurisin, 1993]. However, this value is less than that estimated during voluminous eruptions in 1952 and 1960, when negligible summit tilt change occurred during significant lava extrusion (as originally used by Swanson [1972] to estimate the magma supply rate). Integrating these published values of magma supply rate to some method of mapping, the magma supply rate can be estimated. The interpolated value of magma flux at the time when the parabolic tilt record is at a minimum is the magma supply rate.

10 18,100 DENLINGER: A DYNAMIC BALANCE AT KILAUEA VOLCANO Conclusions The volume of melt stored within Kilauea volcano is about 20 times larger than can be attributed to a summit magma reservoir, and the current magma supply rate is 20% less than that determined during the voluminous Mauna Ulu eruptions 25 years ago. Using vent flux and lake level measurements, the interconnected volume of magma stored within Kilauea volcano in 1991 was 240 km, with 95% confidence limits of 190 and 326 km. The dynamic balance between summit inflation from melt input and summit deflation from vent output established that the magma supply rate to Kilauea volcano on June 19, 1991, was 217,000 _ 10,000 m /d (or km /yr) and is well constrained because the vent at Kupaianaha was the only outlet at that time. Measurements of the lava produced by Kilauea volcano in the last 13 years show that if this magma supply rate is representative of this period, then the magma reservoir has been draining since the onset of the Puu Oo eruption in This conclusion is consistent with the nearly steady rate of summit subsidence since that time. Simultaneous observations of vent flux and summit tilt dem- onstrate a delicate balance between summit inflation and eruptive output at Kilauea volcano. If the summit tilt is parabolic in time, then observations at Kilauea volcano show that the vent flux is either linearly increasing or linearly decreasing with time, and its range includes the magma supply rate. If summit tilt is linear in time, then the vent flux has become constant at a value either greater than (for summit deflation), equal to (no change), or less than (summit inflation) the magma supply rate to the volcano. In remote environments, such as midocean ridges, then an ocean floor tilt record combined with the duration and total volume of a single eruption may be used to estimate magma supply rate to that part of the ridge during the eruption period. Acknowledgments. I would like to thank the staff of the Hawaiian Volcano Observatory for their help while I was stationed there. I am also indebted to D. Clague of MBARI and to C. Heliker, J. Kauahikaua, M. Mangan, A. Okamura, and C. Thornber of the Hawaiian Volcano Observatory for numerous discussions about volcanoes and for their often astute insights into the behavior of Kilauea volcano. References Clague, D. A., and R. P. Denlinger, The role of cumulate dunite in destabilizing basaltic volcanoes, Bull. Volcanol., 56, , Decker, R. W., A. T. Okamura, and J. J. Dvorak, Pressure changes in the magma reservoir beneath Kilauea volcano, Hawaii (abstract) Eos Trans. AGU, 64(45), 901, Delaney, P. T., and D. McTigue, Volume of magma accumulation or withdrawal estimated from surface uplift or subsidence, with application to the 1960 collapse of Kilauea volcano, Bull. Volcanol., 56, , Delaney, P. T., A. Miklius, T. Arnadottir, A. T. Okamura, and M. K. Sako, Motion of Kilauea volcano during sustained eruption from the Puu Oo and Kupianaha vents, , J. Geophys. Res., 98, 17,801-17,820, Denlinger, R. P., and P. Okuba, Structure of the mobile south flank of Kilauea volcano, Hawaii, J. Geophys. Res., 100, 24,499-24,507, Dieterich, J. H., and R. W. Decker, Finite element modeling of surface deformation associated with volcanism, J. Geophys. Res., 80, , Dvorak, J. J., and D. Dzurisin, Variations in magma-supply rate at Kilauea volcano, Hawaii, J. Geophys. Res., 98, 22,255-22,268, Dzurisin, D., R. Koyanagi, and T. English, Magma supply and storage at Kilauea volcano, Hawaii, , J. Volcanol. Geotherm. Res., 21, , Epp, D., Relation of summit deformation to east rift zone eruptions on Kilauea volcano, Hawaii, Geophys. Res. Lett., 10, , Fujii, T., and I. Kushiro, Density viscosity and compressibility of basaltic liquid at high pressures, in Annual Report of the Director , pp , Geophys. Lab., Carnegie Inst., Washington, D.C., Gibbons, J. D., Nonparametric Methods for Quantitative Analysis, 463 pp., Holt, Rinehart, and Winston, Austin, Tex., Goldstein, S. J., M. R. Perfit, R. Batiza, D. J. Fornari, and M. T. Murrell, Temporal variations in East Pacific Rise magmatic activity based on Uranium-series dating of basalts, Nature, 367, , Greenland, L. P., A. T. Okamura, and J. B. Stokes, Constraints on the mechanics of the eruption, in The Puu 0o Eruption of Kilauea volcano, Hawaii: Episodes 1 Through 20, January 3, 1983 Through June 8, 1984, U.S. Geol. Surv. Prof. Pap. 1463, , Johnson, D., Dynamics of magma storage in the summit reservoir of Kilauea volcano, Hawaii, J. Geophys. Res., 97, , Kauahikaua, J.P., M. T. Mangan, C. C. Heliker, and T. N. Mattox, A quantitative look at the demise of a basaltic vent--death of Kupianaha Kilauea volcano, Hawaii, Bull. Volcanol., 57, , Klein, F. W., R. Y. Koyanagi, J. S. Nakata, W. R. Tanigawa, The seismicity of Kilauea's magma system, in Volcanism in Hawaii, U.S. Geol. Surv. Prof. Pap., 1350, , Mangan, M. T., C. C. Heliker, T. N. Mattox, J.P. Kauahikaua, and R. T. Helz, Episode 49 of the Puu Oo-Kupianaha eruption of Kilauea volcano--breakdown of a steady state eruptive era, Bull. Volcanol., 57, , Okamura, A. T., J. J. Dvorak, R. Y. Koyanagi, and W. R. Tanigawa, Surface deformation during dike propagation, in The Puu 0o Eruption of Kilauea Volcano Hawaii: Episodes I Through 20, January 3, 1983 Through June 8, 1984, U.S. Geol. Surv. Prof. Pap., 1463, , Owen, S., P. Segall, J. Freymueller, A. Miklius, R. Denlinger, T. Arnadottir, M. Sako, and R. Burgmann, Rapid deformation of the south flank of Kilauea volcano, Hawaii, Science, 267, , Ryan, M.P., The mechanics and three-dimensional internal structure of active magmatic systems: Kilauea volcano, Hawaii, J. Geophys. Res., 93, , Ryan, M.P., R. Y. Koyanagi, and R. S. Fiske, Modeling the threedimensional structure of macroscopic magma transport systems: Application to Kilauea volcano, Hawaii, J. Geophys. Res., 86, , Swanson, D. A., Magma supply rate of Kilauea volcano, , Science, 175, , Tilling, R. I., Fluctuations in surface height of active lava lakes during Mauna Ulu eruption, Kilauea volcano, Hawaii, J. Geophys. Res., 92, 13,721-13,730, Tilling, R. I., and J. J. Dvorak, Anatomy of a basaltic volcano, Nature, 363, , Tilling, R. I., R. L. Christiansen, W. A. Duffield, R. T. Holcomb, and D. W. Peterson, Determinations of the depth to the surfaces of active lava lakes at Mauna Ulu and Alae Kilauea volcano, Hawaii, U.S. Geol. Surv. Open File Rep., , Wolfe, E. W., C. A. Neal, N. G. Banks, and T. J. Duggan, Geologic observations and chronology of eruptive events, in The Puu 0o Eruption of Kilauea Volcano Hawaii: Episodes 1 Through 20, January 3, 1983 Through June 8, 1984, USGS Prof. Paper, 1463, 1-98, R. P. Denlinger, Cascades Volcano Observatory, U.S. Geological Survey, 5400 MacArthur Boulevard, Vancouver, WA ( røger@mailvan'wr'usgs'gøv) (Received May 8, 1996; revised February 27, 1997; accepted April 11, 1997.)