Volcanic Earthquake Foreshocks During the 2018 Collapse of K

Geophys. J. Int. (2020) 220, 71–78 doi: 10.1093/gji/ggz425 Advance Access publication 2019 September 25 GJI Heat flow and volcanology

Volcanic earthquake foreshocks during the 2018 collapse of K¯ılauea Caldera Downloaded from https://academic.oup.com/gji/article-abstract/220/1/71/5573810 by University of Hawaii at Manoa Library user on 01 December 2019 Rhett Butler Hawaii Institute of Geophysics and Planetology, University of Hawaii at Manoa, Honolulu, HI 96822, USA. E-mail: [email protected]

Accepted 2019 September 23. Received 2019 September 12; in original form 2019 May 23

SUMMARY The summit collapse of the K¯ılauea Caldera—due to magma chamber drainage being directed to the Volcano’s lower east rift zone—was accompanied by 50 large, nearly identical magnitude Mw 5 earthquakes between 29 May and 2 August 2 2018. I have examined the seismicity associated with these 50 primary earthquakes, and find that the typical pattern of earthquake aftershocks decaying in number and magnitude is not evident. Rather, immediately after the primary shock there is a hiatus of one-to-several hours before the associated earthquakes grow in number and magnitude up to the next primary shock. In essence, the associated seismicity consists of thousands of foreshocks. The magnitude of completeness is estimated at ML = 2.5. The trend of foreshocks does not fit an Omori power-law model. Rather, the pattern of foreshocks (number per hour) is fit well by a semi-Gaussian curve, which initially grows rapidly and slows hours prior to the primary earthquakes. The Gaussian fits (r 2 > 0.98) for three different magnitude thresholds are self-similar with a common half-width, σ ∼ 13 hr. The pattern of foreshock seismic moments aligned with and stacked for the 50 primary events is fit by an exponential trend, growing at the mean (stacked intervals) rate of 17 per cent per hour. The power of foreshocks measured within each interval also grows with time—the total foreshock power per interval (J hr–1) increases by a factor of 18 through the first half of the sequence (May 29 through June 26), and then declines by half through to the end. Key words: Time-series analysis; Earthquake dynamics; Earthquake interaction, forecasting, and prediction; Volcano seismology; Magma chamber processes.

The Omori relation (Omori 1894; Utsu & Ogata 1995)and 1 INTRODUCTION its variants as t approaches/follows the origin time t0 for expo- In May of 2018 an unprecedented swarm of Mw ∼5.3 earthquakes nent p, is the traditional power law formulation for foreshocks, −p −p began at K¯ılauea Caldera when 50 events occurred between 29 May ∼ (t − t0) and aftershocks, ∼ (t0 − t) (Papazachos 1973;Ka- and 2 August 2018 immediately beneath the summit (Fig. 1), as- gan & Knopoff 1978; Jones & Molnar 1979; Helmstetter et al. sociated with >20 000 events (ML ≥ 2.2) contemporaneous with 2003). Nevertheless, the Omori relations do not fit this K¯ılauea the K¯ılauea Caldera collapse. These events occurred in close con- data set, whether as aftershocks or foreshocks. Instead, the inter- junction with new K¯ılauea eruptive activity and summit collapse val frequency-count data are best fitted—as foreshocks—to a semi- (see Neal et al. 2018, for chronology and extensive analysis). Butler Gaussian curve, which increases rapidly over tens of hours, and then (2019) analysed the focal mechanisms for the 50 primary shocks, slows several hours before the subsequent Mw 5 earthquake. Plau- and constrained the vertically downward propagating energy from sible ramifications for the K¯ılauea Caldera collapse sequence are the earthquake sources with near-antipodal data. considered. This analysis herein focuses on documenting the properties of the earthquakes associated with the intervals between successive pri- mary shocks, which are binned into the hourly data segments within 2DATA each interval. In particular, I review whether these associated shocks are precursory foreshocks or aftershocks (or both). Finding char- The data source for this study comes from the USGS Hawai- acteristics of foreshocks, the analysis focuses on the sequencing, ian Volcano Observatory (HVO) earthquake catalogue (earth- number, and seismic moments of the associated earthquakes. These quake.usgs.gov). Over 20 000 earthquakes ML ≥ 2.2 were down- K¯ılauea analyses are then compared with prior tectonic and volcano loaded from the origin time of the first, Mw 5.3 swarm event May foreshock series—those which occur within tens of hours prior to 29 through the last event on August 2, supplemented by associated an event, such as seen at K¯ılauea summit during this swarm. earthquakes immediately prior to and following the Mw 5events.In

C The Author(s) 2019. Published by Oxford University Press on behalf of The Royal Astronomical Society. 71 72 R. Butler Downloaded from https://academic.oup.com/gji/article-abstract/220/1/71/5573810 by University of Hawaii at Manoa Library user on 01 December 2019

Figure 1. Large map shows HalemaAuma` Au` crater within K¯ılauea Caldera. Upper-right inset map locates the K¯ılauea events within the Island of HawaiAi,` where the red star shows the location of the POHA seismic station. The Mw 5.3 ± 0.1 earthquake swarm (29 May–2 August 2018) associated with the collapsing crater is plotted within the light blue circle (3 km radius) outlining the study area, where earthquake Mw magnitude is annotated by colour (yellow 5.2; orange 5.3; red 5.4). The density of epicentres (>12 000) within the light blue circle are plotted both reduced in size and offset by the dotted blue lines into the grey circular inset (3 km radius) at the lower-right. addition, the pre-May 29 and post-August 2 swarm data, are com- that the uncertainty (standard deviation) σ =∼0.14 in each of pared and contrasted with the primary swarm results. All events se- three regions tested. lected were located within a 3 km radius centred on Halema’uma’u Therefore, within the uncertainty in measuring Mc, the estimate crater, where all of the Mw 5 events occurred in a tight cluster of the Mc is ∼2.5 for the aggregated data set of 49 intervals. An (Fig. 1). For more detailed analysis and discussion of the Mw 5 Mc estimate was also derived for each interval between primary primary earthquake sources and the listing of their locations, please events. Of these 49 intervals between primary earthquakes, only refer to Butler (2019). three intervals exhibited Mc in the range 2.5–2.6, whereas all others showed Mc ≤ 2.5. Finally, since the numbers of events increase prior to the subsequent primary earthquake, Mc was determined for hourly bins prior to primary shocks. For the ten 1-hr bins occurring 2.1 Magnitude completeness 1–10 hr before the subsequent primary event, Mc = 2.5 in each ‘Magnitude completeness’ (M ) can vary prior, during, and follow- instance. Hence, the aggregated, interval, and hourly-binned data c ∼ ing swarm activity (Hainzl 2016). The high level of seismic activity all exhibit magnitude completeness Mc 2.5. To extend the completeness to lower magnitudes will require us- associated with the 2018 swarm advises reviewing Mc. Preliminary b-value analysis by HVO of the 2018 eruption sequence indicates ing all micro-earthquakes recorded during the entire sequence as templates, and scanning through the continuous waveforms (i.e. so- an approximate Mc in the range 2.2–2.4, which exceeds by a magni- tude unit the estimates before and after the sequence. (B. Shiro & J. called template matching method) to detect any potential missing Chang, HVO, personal communications 2019). Following Wiemer foreshocks that were not listed in the standard catalogue but were recorded by the network (see Peng et al. 2007;Katoet al. 2012; &Wyss(2000), Mc is defined as the magnitude at which 90 per Walter et al. 2015). This is beyond the scope of this initial effort cent of the data can be modeled by a power law fit; for which Mc ∼2.3–2.4 for this K¯ılauea data set. Woessner & Wiemer (2005) to classify the unique characteristics of the K¯ılauea foreshock se- quence. Nonetheless, HVO has plans to relocate swarm events (B. compared Mc among five alternate methods (e.g. Wiemer & Wyss 2000; Cao & Gao 2002; and Woessner & Wiemer 2005) and found Shiro, HVO, personal communication 2019) K¯ılauea earthquake foreshocks 73

2.2 Moment, hypocentral depth, b-values With the final primary shock on August 2 concluding the primary shock events (Fig. 2), a ‘finale’ sequence immediately followed. In The summary data for the swarm are plotted versus time in Fig. 2, the first hour, there is one M 2.2 shock, increasing in number where magnitude is converted to approximate moment (Hanks & L to about 15 per hour at 24 hr. The peak magnitude (M 4.0) oc- Kanamori 1979). The intervals between the M 5 events are not L w curs at 26 hr. Unlike other sequences, there is no finale primary constant, but rather decreased from about 50 hr initially to 24 hr shock, and the frequency-count trend may be fitted by a complete by the 7th event, and then increased linearly by ∼0.5 hr per event, Gaussian curve. The aftershock rate decreased to nearly zero events concluding at an interval of about 50 hr (Butler 2019). The median (M ≥ 2.2) after 53 hr—for the next two weeks, only five earth- Downloaded from https://academic.oup.com/gji/article-abstract/220/1/71/5573810 by University of Hawaii at Manoa Library user on 01 December 2019 depths in each interval are shallow, near 0.5 km below sea level, L quakes (2.2 ≤ M ≤ 2.5) occurred in the caldera. which is 1.6 km below the summit. Further detail on the depth L distribution will require event relocation. The Gutenburg–Richter b-value power law relating the frequency of earthquakes with their magnitude is calculated for each interevent 3.3 Stacking interval. For the K¯ılauea associated earthquakes the mean b-value Data for each time interval between primary earthquakes were ana- b¯ = 1.4 ± 0.2. The b-values beneath K¯ılauea’s South Flank are lyzed in hourly bins. The data were stacked together with a common high, b ∼ 1.3–1.7 at depths above 8 km and considered to be origin at the time of the subsequent primary temblor as an aggre- due to the presence of active magma bodies beneath the East Rift gated seismic synthesis of the K¯ılauea Caldera collapse. Fig. 3 shows zone (Wyss et al. 2001). Observed for other volcanoes, the vol- the summary synthesis of cumulative earthquake frequency count umes surrounding active magma chambers and conduits exhibit and seismic moment by hourly bin over the 49 intervals for three ML anomalously high b-values—for example Mt Pinatubo, Philippines magnitude ranges: ≥ 2.5, ≥ 2.7 and ≥ 3.0. The frequency data plot (Sanchez´ et al. 2004), Off-Ito Volcano, Japan (Wyss et al. 1997), on relatively smooth curves. Initially growing rapidly in number, Long Valley, California (Wiemer et al. 1998) and Mt Etna (Murru the rate decreases from about 7 hr prior to the primary shock. The et al. 1999). cumulative seismic moment plot shows a linear increase in log10 moment, corresponding to an exponential increase in moment. As the growth rate of frequency-counts slows, so do the cumulative 3 ANALYSIS AND RESULTS moments. 3.1 Foreshock or aftershock swarm A first glance may not see the unique relationship between the 3.4 K¯ılauea trend analysis ‘primary’ temblors (M ∼ 1017 Nm), and the associated shocks 0 To fit the K¯ılauea foreshock frequency-count versus hourly bin, (1012 < M < 1016 Nm). Earthquake seismology since Omori (1894) 0 I tested several functional forms, including the Omori power-law has recognized the sequence of a main shock followed by after- relations. However, these Omori relations did not match the initially shocks. However, for these K¯ılauea events there is a hiatus (several rapid growth rate followed by a decreasing rate. The functional form hours of relative quiescence) of seismicity immediately following of the frequency-count curve is best fitted by Gaussian functions, each primary shock. In contrast, viewing the associated earthquakes ⎛ ⎞ as foreshocks, the activity (number and moment) builds over >24 t − b 2 ⎝ ⎠ hr from relative quiescence immediately following a primary shock, − = c culminating in the next primary shock. This is clearly shown in an f (t) ae (1) animation of summit earthquakes (see Data and Resources) com- as plotted in Fig. 3.ForML ≥ Mc = 2.5 the Gaussian fit parameters piled by Nathan Becker (personal communication 2018). a, b,andc are 724, 0.9 and 18.6, respectively (b is the mean, and the Therefore, two unique observations are evident in each interval standard deviation σ = √c ). The fit is ‘semi-Gaussian’ in the sense 2 between primary shocks. First, the primary shocks do not exhibit that the primary shock occurs as the seismicity rate decreases within a traditional aftershock sequence decaying with time (e.g. Omori’s several hours prior. The coefficient of determination, r 2 = 0.99. Law of aftershocks; Omori 1894; Utsu & Ogata 1995). Rather, the Fig. 3 also shows the Gaussian fits to an abridged date sequence primary shocks are preceded by extensive foreshocks which build which excludes the 10 hr immediately prior to the subsequent pri- over tens of hours to a subsequent primary shock. The mean time mary event. The self-similarity of the Gaussian curves is striking. of the largest foreshock for the intervals occurs about 6 hr prior to The width of the Gaussian, characterized by its standard deviation, the primary (the maximum span is about 21 hr). Furthermore, given is 13 ± 0.5 hr for each of the magnitude levels plotted in Fig. 3. the repetition of the primary shocks, it is likely that the earthquakes These Gaussian trends are manifest not only in the stacked data, in the interval are comprised by a combination of aftershocks from but also for the 49 individual intervals. Fig. S1 compares the fits of prior primary shocks, as well as foreshocks. both the inverse Omori power law and the semi-Gaussian curves for each of the 49 intervals. In each instance, as measured quantitatively by the coefficients of determination r2, the semi-Gaussian gives a 3.2 Pre-swarm and post-swarm earthquake patterns superior fit to the data. In addition to the 50-fold repeating sequence of the K¯ılauea 29– The foreshock cumulative moments in each hourly bin (not in- August 2 swarm, I reviewed the earthquakes immediately prior to cluding the Mw 5 events) are also plotted in Fig. 3 on a log10 and following the swarm of Mw 5 events. Prior to the first Mw 5.3 scale. Where the increasing seismicity rate slows, the cumulative on May 29, a ML 4.7 earthquake occurred on May 26 showing moment rate also declines. A linear model was fitted to the log10, the basic features characteristic of the primary swarm intervals. cumulative moment observations (green triangles, ML ≥ 2.5). Data Between May 17 and May 26, eleven earthquakes 4.8 ≤ ML ≤ 5.1 within about 7 hr of the primary shock show a slowing rate of in- occurred at the summit, where only the ML 4.9 event on May 22 crease. For events ML ≥ Mc the linear intercept is 24.2 and the slope displayed a foreshock pattern similar to the 50 primary events. b = 0.068, hence the foreshock cumulative moments increase by 74 R. Butler

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Figure 2. Seismicity within K¯ılauea Caldera from May 26 through August 5 plotted versus apparent seismic moment M0 and Mw, as derived from ML by Hanks & Kanamori (1979). The largest events are Mw 5.2–5.4, and the smallest are ML = 2.2. The associated earthquakes between the 50 Mw ∼5.3 events are plotted as black dots. Note the quiescent hiatus (white space) following each primary event, followed by seismicity increasing to the next primary. Grey dots are associated earthquakes beginning and concluding the event sequence, prior to the first and following the final Mw ∼5.3 event. The foreshock interval ending on June 26 is the proximal peak, foreshock power during the swarm sequence (Fig. 4). The grey BEGINNING sequence is initiated by a ML = 4.7 earthquake. The FINALE concludes with a 2-week hiatus where only five ML ≤ 2.5 earthquakes occur—this hiatus continues through 2018. Time is in hours from the origin time of the ML 4.7 grey earthquake which precedes the first primary event.

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r2>0.92 300 10 14 .0 Number of earthquakes per bin ML≥ 3 200

Cumulative seismic moment (Nm) per bin r2>0.98 100

10 13 0 60 50 40 30 20 10 0 Time, Hourly bins prior to primary event

Figure 3. The aggregated counts (blue) of ∼12 000 K¯ılauea summit earthquakes in each hourly bin prior to primary shocks are stacked in three groups corresponding to minimum ML ≥ 2.5, 2.7 and 3.0. Time proceeds left-to-right, measured in hourly bins prior to the primary events, and the counts correspond with the right axis. Each group is fitted with a Gaussian curve (only semi-Gaussian fit shown). Estimates of magnitude completeness indicate Mc∼2.5. Each of the Gaussian curves (black) well fit the temporal progression, are self-similar, and possess a significant coefficient of correlation r 2 > 0.98. To examine the sensitivity to the Gaussian fits to the 10 hr immediately preceding the primary events, the data were fit (red) excluding these last 10 hr. The quality of these abridged fits essentially overlay the full data set. This suggests that the Gaussian fit is governed by the initial rapid increase in the data, which also matches the slowing trend in the final 10 hr. Individual intervals are plotted in Figs S1 and S2. The cumulative seismic moments (green triangles) of all the earthquakes (ML ≥ 2.5) in each bin—which are dominated by the larger events—refer to the logarithmic axis at the left. The binned, log10 cumulative moments (open 2 triangles) are fitted with a linear model, and the 95 per cent confidence limits are plotted (r > 0.92). The linear trend in log10, cumulative seismic moment corresponds to an exponential fit wherein the total moment increases at 17 per cent per hour. However, within about 7 hr of subsequent primary events, the growth of foreshock moment slows. Gaussian fit coefficients are shown in Table S1. K¯ılauea earthquake foreshocks 75 afactorof10b = 1.17 per hour. This exponential increase in mo- the primary event. However, there are no instances in the literature ment, ∼17 per cent per hour, is starkly different from the primary of the frequency-count of foreshocks following a Gaussian trajec- shocks which differ by only ±0.1 magnitude units. For each interval tory, or that of the cumulative moment growing exponentially— between primary earthquakes, Fig. S2 plots the cumulative moment indicating that the foreshock swarm associated with the K¯ılauea increase as the log10, seismic energy increase, such as derived for Caldera collapse is unique, unprecedented. Without belaboring the Fig. 4 in the following paragraphs. The range of the exponential details, I note prior foreshock references for both tectonic and vol- increases in seismic energy over the 49 intervals is 4 per cent to canic earthquakes. A pattern of foreshock activity for tectonic earth-

32 per cent per hour, with a geometric mean at ∼13 per cent per quakes following the ‘inverse Omori’ power law trend (i.e. linear Downloaded from https://academic.oup.com/gji/article-abstract/220/1/71/5573810 by University of Hawaii at Manoa Library user on 01 December 2019 hour. The slowest rate (4.0 per cent per hour) occurred in the last slope on a log–log plot) is reported by: Papazachos (1973); Kagan & interval–49. Knopoff (1978); Jones & Molnar (1979) and Jones 1985; Yamashita Butler (2019) noted that the intervals between primary shocks & Knopoff (1989); Davis & Frohlich (1991); Shaw (1993); Dodge initially decreased in duration from May 29. However, from 8 June et al. (1995); Ogata et al. (1995); Maeda (1999); Yamaoka et al. 2018 (event 7) through the final Mw 5.3 earthquake on 2 August 2018 (1999); Helmstetter et al. (2003) and Peng et al. (2007). Helmstetter (event 50), the swarm is characterized by interevent times increasing & Sornette (2003) find ‘In contrast with the direct Omori law, which at a rate of about a half hour per event. These moment values may be is clearly observed after all large earthquakes, the inverse Omori law recast (e.g. Kanamori 1977) as the minimum elastic strain energy is observed only when stacking many foreshock sequences.’ drop in joules before and after the earthquake by the relationship Three significant earthquakes exhibited foreshock sequences W = (σ/2μ)M0,whereσ is stress drop and μ is the shear within tens of hours of the main shock (i.e. the same time scale modulus. Prior estimates of σ for Hawaiian earthquakes range shown in Fig. 2), but do not follow the inverse Omori relationship; from: 0.12 to 9 MPa (Ando 1979; Butler 1982; Li & Thurber 1988; instead, their numbers of foreshocks abate rather than increase while Dvorak 1994;Yamadaet al. 2010; Butler 2018) with a geometric approaching the main shock. Two of these examples are the among mean of 1.6 MPa. the largest earthquakes ever recorded: the 1960 Chilean earthquake The Mw moments were derived assuming a source depth of main shock (Mw 9.5) was preceded 33 hr by a Mw 8.1 earthquake 11.5 km and a PREM crustal shear modulus (e.g. Butler 2019). (Cifuentes 1989); and the 2011 Tohoku, Japan earthquake (Mw 9.1) Correcting the shear modulus to shallow K¯ılauea Caldera values, which experienced a Mw 7.3 foreshock nearly 51 hr prior (e.g. Kato ∼9.4 GPa (Dawson et al. 1999), the apparent moments and Mw et al. 2012; Marsan & Enescu 2012). The 1975 Haicheng, China are shifted lower, as shown by the dotted blue line in Fig. 4,and earthquake is famous for its large foreshock sequence, which re- brought into correspondence with moments calculated locally via sulted in one of the few successful earthquake predictions (Adams corner frequency methods. 1976; Raleigh et al. 1977; Jones et al. 1982;Zhu&Wu1982;Chen Using data from Global Seismographic Network (Butler et al. et al. 1999;Wanget al. 2006). 2004) station POHA, located 47 km NNW of the K¯ılauea summit Studies of foreshocks prior to volcanic eruptions have observed (Fig. 1), I measured the minimum stress drops of 12 primary events power law dependence on the seismicity rate (e.g. Voight 1988; by using four ω−2 source models (Kaneko & Shearer 2014), the Chastin & Main 2003; Lemarchand & Grasso 2007; Bell & Kilburn P and S corner frequencies, low-frequency amplitude spectra, and 2012; Schmid & Grasso 2012;Bellet al. 2013). Using stacked shear modulus for K¯ılauea (Dawson et al. (1999). Geometric mean data prior to K¯ılauea eruptions, Bell & Kilburn (2013) found two minimum stress drops are estimated at 1.4 and 2.5 MPa, for P and different pre-cursory styles at K¯ılauea: (1) a small proportion of S waves, respectively. For these values of stress drop and shear eruptions and intrusions that are preceded by accelerating rates of −4 modulus, minimum elastic strain energy drop is ∼(10 )M0 joules, volcano-tectonic earthquakes over intervals of weeks to months and where M0 is measured in N-m. (2) a much larger number of eruptions that show no consistent The trends in Fig. 4 may be compared with other measures of increase until a few hours beforehand. The foreshocks discussed energy in the volcanic system, i.e., mechanical, thermal, and gravi- herein are intermediate to these extremes, starting tens of hours prior tational work. For example, a caldera plug of radius 1 km, thickness to a primary shock. Comparable to the observation by Helmstetter & 500 m and density ∼2700 kg m–3, dropping 1 meter in Earth’s Sornette (2003) for tectonic earthquakes, Schmid & Grasso (2012) gravity yields ∼8 × 1013 J, exceeding the minimum elastic energy note that data averaging is necessary to retrieve the power law released, 0.5–1.1 × 1013 J per primary. As the intervals between trends. primary earthquakes are not of constant duration, Fig. 4 also plots Two references of seismic foreshocks to volcanic caldera collapse the foreshock energy per hour for each interval. Unlike the linear have similarities with the K¯ılauea events: Miyakejima in 2000 and growth of the interval duration from June 8 forward (Butler 2019), Piton de la Fournaise in 2007. The latter event occurred at the the power shows a systematic increase by a factor of eighteen from Dolomieu summit crater which collapsed on April 5 following a May 29 through June 26 (interval 24). Then, the hourly power de- large eruption at a vent 7 km distant on the Piton de la Fournaise clines by a factor of two from June 26 through August 2. The shield volcano on La Reunion´ Island, Indian Ocean (Staudacher stability of the Mw of the primary shocks (±0.1) is in contrast to the et al. 2009). The caldera collapse was accompanied by a single foreshock energy, which varies by more than an order of magnitude. mb = 5.3 earthquake (USGS earthquake catalogue), preceded by ∼8 hr of local seismic activity and followed by several hours of relative quiescence (i.e. Fig. 15 in Staudacher et al. 2009). The seismic activity associated with the collapse of the Mt Oyama 4 DISCUSSION volcano caldera on Miyakejima in 2000 is perhaps the closest ana- logtotheK¯ılauea sequence. Forty-one earthquakes ranging from 4.1 Foreshocks: tectonic and volcanic Mw 5.1 to 5.6 were observed from 9 July through 18 August 2000, I review the foreshock literature for both earthquakes and volcan- discussed as Very Long Period (VLP) pulses by Ukawa et al. (2000) ism, limiting the focus to foreshock occurrence during the same and Kobayashi et al. (2003). Four of these 41 events showed pre- time intervals as the K¯ılauea data, that is within tens of hours of cursory seismic activity starting one to two hours before a VLP 76 R. Butler

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June 26 10 11 x 2 10 11 x 18 Seismic energy rate, Joules/hour Seismic strain energy drop, Joules 10 10 Foreshocks, J/hr 10 10 primary earthquake energy, J primary energy corrected for μ foreshock energy, J foreshock energy rate, J/hour 10 9 10 9 050510 15 202530354045 Interval number

Figure 4. Earthquake moments are converted to minimum elastic strain energy drop in joules following Kanamori (1977). Primary events ending each interval are plotted as blue triangles using the the USGS–calculated Mw moments. As these were calculated for a standard minimum depth of 11.5 km, the dotted blue line below corrects their energy levels, as appropriate for a shallow shear modulus (9.4 GPa) beneath the caldera (Dawson et al. 1999). For each of the 49 intervals between the primary earthquakes, the cumulative foreshock energy is plotted in red. The cumulative foreshock energy rate (black, right axis) is scaled by interval width in hours. Intervals are listed in time order—interval 1 begins May 29 and interval 49 starts July 31. During the first month, the foreshock power increases by a factor of 18 through to its peak at June 26 (Fig. 2), and then declines through the end of the sequence to about half of its peak rate. Note that part of the decline is due to the increasing interval times between primary earthquakes observed by Butler (2019). pulse and stopping with its occurrence (Ukawa et al. 2000;Fu- destabilizing the foundational support of the caldera, causing asso- jita et al. 2001; Kobayashi et al. 2003). This quiescence after the ciated fracture earthquakes building in time and magnitude, con- event is similar to K¯ılauea. Kobayashi et al. (2003) continue, ‘As cluding with a major collapse earthquake, after which the caldera approaching to the VLP pulse, the time intervals of the pre-swarm floor is buttressed by and in equilibrium with magma chamber, and earthquakes gradually decrease. The maximum amplitudes of the then repeated 50 times until the drainage of magma chamber ceased, earthquakes are initially nearly constant, and then suddenly start and the eruption in the lower East Rift Zone halted. A major chal- decaying rapidly few minutes before the occurrence of the VLP lenge for evoking the ‘piston model’ is physically connecting the pulse.’ seismic activity beneath the summit caldera floor (∼1.6 km above The 2018 K¯ılauea data do not follow these latter trends observed sea level) with the volume and rate of lava erupting 43 km down for Miyakejima. Reviewing 1-hr windows of vertical data prior the lower East Rift Zone (primarily fissure 8, Ahu’aila’au) at 216 m to each primary shock for the GSN (Butler et al. 2004) station elevation. POHA, located 47 km NNW of the K¯ılauea summit (Fig. 1), the Over the ∼2-month sequence, the caldera floor (top of the ‘pis- K¯ılauea windows are substantially more variable in character and ton’) subsides ∼500 m (Neal et al. 2019) from –1.1 to –0.6 km do not exhibit the simple, geometrically decreasing time span in (relative to sea level), whereas the median locus of seismic energy the foreshocks as observed for the four Miyakejima earthquakes by remains centred near +0.5 km depth throughout the swarm. Based Kobayashi et al. (2003). Further, substantial decreases at K¯ılauea upon Vp/Vs ratios, Dawson et al. (1999) estimated that the depth in foreshock amplitudes (i.e. moment) minutes prior were also not of the magma chamber east and south of Halema’uma’u extends evident. from –0.5 to 3 km depth—amidst the median depths of foreshock earthquakes. I have fit these foreshock sequences with a Gaussian process, wherein the number of precursory earthquakes follows a semi- 4.2 Qualitative observations Gaussian curve, increasing rapidly initially and then slowing prior to the primary shock. This relationship exists not only for the stacked Armed with physical seismic measurements of the caldera collapse, aggregated sequences (Fig. 3), but also for the individual sequences I qualitatively review the seismic data in the context of the broader between the primary shocks (Fig. S1). events at K¯ılauea. The Miyakejima volcano collapse in 2000 is In addition to modelling the frequency count of foreshocks as a perhaps the most analogous to K¯ılauea, featuring similar repeating semi-Gaussian, the cumulative seismic moments for each foreshock earthquake source mechanisms and foreshock activity tens of hours sequence are found to follow an exponential trend before a primary prior. For Miyakejima, a ‘piston-model’ is evoked to explain the earthquake (Fig. 3). Since the minimum strain energy drop in an seismic activity (Kumagai et al. 2001). earthquake is proportional to seismic moment (Kanamori 1977), the The K¯ılauea sequence of volcano-earthquake events follows a increasing cumulative moment translates into seismic strain energy pattern of magma withdrawal from the summit magma chamber, increasing ∼17 per cent each hour. Within ∼7 hr of the primary, the K¯ılauea earthquake foreshocks 77 hourly energy increase slows. Individual intervals show the same REFERENCES hourly energy trends (Fig. S2). This seismic energy curve may serve Adams, R.D., 1976. The Haicheng, China, earthquake of 4 February 1975: as a new constraint on the collapse process. the first successfully predicted major earthquake, Earthq. Eng. Struct. There is no obvious connection between a piston model per se Dyn., 4, 423–437. and either the semi-Gaussian foreshock process and its exponential Ando, M., 1979. The Hawaii earthquake of November 29, 1975: low dip angle faulting due to forceful injection of magma, J. geophys. Res.: Solid accumulation of seismic moment. Furthermore, this K¯ılauea fore- Earth, 84(B13), 7616–7626. shock process appears to be unique in the seismological literature. Bell, A.F.& Kilburn, C.R., 2012. Precursors to dyke-fed eruptions at basaltic Gaussian processes are generally in the realm of probability distri- volcanoes: insights from patterns of volcano-tectonic seismicity at Ki- Downloaded from https://academic.oup.com/gji/article-abstract/220/1/71/5573810 by University of Hawaii at Manoa Library user on 01 December 2019 butions. Here, we observe a repeating, semi-Gaussian growth rate lauea volcano, Hawaii, Bull. 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Peng, Z., Vidale, J.E., Ishii, M. & Helmstetter, A., 2007. Seismicity rate immediately before and after main shock rupture from high-frequency Figure S1. Detail for Fig. 3 plotting the frequency-counts versus waveforms in Japan, J. geophys. Res.: Solid Earth, 112(B3), B03306. hourly bin data for each event interval. Same format as Fig. 3,but Raleigh, C.B. et al., 1977. Prediction of the Haicheng earthquake, EOS, showing the fits for each individual interval by both the Gaussian Trans. Am. Geophys. Un., 72, 236–272. trend (green) and the Omori relation (red). The 49 intervals occur Sanchez,´ J.J., McNutt, S.R., Power, J.A. & Wyss, M., 2004. Spatial variations in the frequency-magnitude distribution of earthquakes at Mount Pinatubo between each of the 50 primary earthquakes. In each instance, the Volcano, Bull. seism. Soc. Am., 94(2), 430–438. Gaussian fit is superior to the Omori fit, as measured quantitatively Schmid, A. & Grasso, J.R., 2012. Omori law for eruption foreshocks and by r2. aftershocks. J. geophys. Res.: Solid Earth, 117(B7). Figure S2. Detail for Figs 3 and 4. The linear trend in Fig. 3 (green) Shaw, B.E., 1993. Generalized Omori law for aftershocks and foreshocks of cumulative seismic moment per hourly bin—converted to log10 from a simple dynamics. Geophys. Res. Lett., 20(10), 907–910. energy as in Fig. 4—are shown for each interval. The rate of energy Staudacher, T., Ferrazzini, V., Peltier, A., Kowalski, P., Boissier, P., Cather- increase ( per cent per hour) corresponding to the linear trends are ine, P., Lauret, F. & Massin, F., 2009. The April 2007 eruption and the indicated for each interval window. Dolomieu crater collapse, two major events at Piton de la Fournaise (La Table S1. Gaussian coefficients (with 95 per cent confidence Reunion´ Island, Indian Ocean), J. Volc. Geotherm. Res., 184(1-2), 126– bounds) for three magnitude thresholds fitted to the Gaussian curve 137. = ∧ − t−b1 2 Ukawa, M., Fujita, E., Yamamoto, E., Okada, Y. & Kikuchi, M., 2000. The f (t) a1 e [ ( c1 ) ]. 2000 Miyakejima eruption: crustal deformation and earthquakes observed Please note: Oxford University Press is not responsible for the con- by the NIED Miyakejima observation network, Earth, Planets Space, 52(8), xix–xxvi. tent or functionality of any supporting materials supplied by the Utsu, T. & Ogata, Y., 1995. The centenary of the Omori formula for a decay authors. Any queries (other than missing material) should be di- law of aftershock activity, J. Phys. Earth, 43(1), 1–33. rected to the corresponding author for the paper.