Mechanics of Inflationary Deformation During CalderaCollapse: Evidence From the 2018 Kı̄lauea Eruption

Paul Segall1 , Kyle R. Anderson2 , Ingrid Johanson3 , and Asta Miklius3

1Geophysics, Stanford University, Stanford, CA, USA, 2California Volcano Observatory, U.S. Geological Survey,Menlo Park, CA, USA, 3Hawaiian Volcano Observatory, U.S. Geological Survey, Volcano, HI, USA

Abstract During the 2018 Kı̄lauea eruption the caldera floor dropped 500 m in 62 nearly periodicevents of up to 8 m. Caldera collapse maintains pressure in the magma reservoir necessary to sustainhigh-rate eruptions. The 2018 collapses were accompanied by inflationary tilts and displacements, similarto observations at other basaltic calderas. Collapse is modeled in 2-D by uniform slip dislocationsintersecting a magma chamber. Collapses occurred rapidly, such that mass within the chamber wasconstant. For vertical faults surface deformation outside the collapse results only from chamberpressurization with displacements antiparallel to precollapse deflation, similar to observations. For inwarddipping faults the predicted ratio of horizontal to vertical displacements during collapse is greater than forthe precollapse deflation, as observed. For outward dipping faults the horizontal displacements are inward,contrary to data. The average deformation trends during the eruption depend on fault dip and whetherstress required to trigger collapses was constant or changed with time.

Plain Language Summary When large volumes of magma are erupted from a volcano the roofof the evacuated magma chamber can founder leading to caldera formation. The weight of the roof blockmaintains pressure on the magma chamber providing a driving force to sustain the eruption. During the2018 Kı̄lauea Volcano Hawaii eruption the caldera collapse occurred in discrete, nearly periodic events.Understanding the mechanics of these events is important for predicting how caldera collapse initiates andevolves with time. We use simple elastic models of collapse into a magma chamber, together withmeasurements of surface deformation during collapse events, to elucidate the geometry of the chamber-roofsystem at Kı̄lauea.

1. IntroductionThe 2018 Lower East Rift Zone eruption of Kı̄lauea Volcano resulted in the best monitored collapse of abasaltic caldera in history (Neal et al., 2019). Between May and early August 2018 the floor of Halema'uma'ucrater dropped up to 500 m and the volume of Kı̄lauea caldera increased by ∼0.8 km3. The collapse wasnot steady but occurred in over 60 events in which the caldera floor dropped from several to ∼8 m andwas accompanied by MW 5.2 to 5.4 very long period earthquakes. The rate of volcano-tectonic earthquakesincreased leading up to collapse events and then subsided dramatically immediately afterward.

Collapse events were accompanied by remarkable, nearly periodic inflationary deformation transients,specifically radially outward tilts and GPS displacements as well as uplift. These episodic GPS displacementsreached as much as 0.18 m (horizontal) at station UWEV, with tilts exceeding 89 μrad at UWD (Figure 1).The inflationary steps were followed by decelerating deflations (Figure 2). Events from middle to late Maywere smaller in amplitude than subsequent events but accompanied by ash-charged explosions. Similartransient deformations have been recorded during other basaltic caldera collapses, notably at Miyakejimain Japan and Piton de la Fournaise in Reunion Island (Kumagai et al., 2001; Michon et al., 2009).

Inflation accompanying eruptions is unexpected, as ejection of mass should lead to a pressure reduction(deflation) in magma reservoirs. The apparent mystery was clarified by measurement of rapid subsidenceat a GPS station adjacent to Halema'uma'u (NPIT, in Figure 1) accompanying the explosions. This revealedcollapse as the driving mechanism, with ash emission as a relatively secondary phenomenon. Two ideashave been proposed in the literature to explain inflationary deformation associated with collapse events:One is that collapsing block(s) decrease the magma chamber volume, transiently increasing the pressure of

RESEARCH LETTER10.1029/2019GL084689

Key Points:• Two-dimensional elastic model

predicts deformation due to collapseon caldera faults that bottom in amagma chamber

• For vertical faults deformationexternal to caldera is solely due tocollapse-induced pressurization ofchamber

• Outward dipping faults predicthorizontal displacements opposite insign to observations

Correspondence to:P. Segall,segall@stanford.edu

Citation:Segall, P., Anderson, K. R.,Johanson, I., & Miklius, A. (2019)Mechanics of inflationary deformationduring caldera collapse: Evidencefrom the 2018 Kı̄lauea eruption.Geophysical Research Letters, 46.https://doi.org/10.1029/2019GL084689

Received 25 JUL 2019Accepted 25 SEP 2019Accepted article online 17 OCT 2019

©2019. American Geophysical Union.All Rights Reserved.

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Figure 1. Average horizontal GPS displacements (red vectors) and tilts (blue vectors) for collapse events between 21June 2018 and the end of the eruption. Tiltmeter UWD is colocated with GPS station UWEV. GPS station KOSM isreference station for the high-rate GPS analysis. Shaded region indicates collapse area.

the resident magma (Kumagai et al., 2001); subsequent deflation reflects accelerated outflow in response tothe pressure increase. The second hypothesis is that elastic rebound associated with slip on block-boundingfaults leads to uplift and outward tilt of the caldera flanks (Michon et al., 2009).

The expected signals due to elastic rebound do not appear to have been previously quantified and are com-plicated by the fact that the block-bounding faults bottom in (or near) a magma chamber, as opposed to solid

Figure 2. (a) Radial component of the UWD tiltmeter showing inflationaccompanying explosion/collapse events with intervening deflation. Up isinflation (tilting away from the caldera). (b and c) Radial time series at GPSstations UWEV and CRIM, uncorrected for motion of reference stationKOSM. Inflation is up. Station locations are shown in Figure 1.

crust. Here we explore simple models that predict surface deformation inresponse to collapse into a magma chamber and compare to observationsfrom the 2018 Kı̄lauea collapse.

Large volume rate eruptions likely require caldera collapse to main-tain sufficient pressure on the magma reservoir to sustain outflow (e.g.,Gudmundsson, 2016). Elucidating the mechanics of collapse, includ-ing the geometry of the bounding faults and how load is transferredto the magma chamber, is thus important for understanding high-rateeruptions.

2. Boundary Element CalculationsConsider an elliptical reservoir in an elastic medium with faults inter-secting the edge of the chamber (Figure 3a). We idealize the problem bytaking the slip on the faults to be specified, spatially uniform, and thesame on both faults. The problem is further simplified by restricting toa 2-D plane-strain formulation. This is sufficient to build understandingof the expected deformations, but insufficient for quantitative compari-son to data. For simplicity, we focus solely on the shallow Halema'uma'ureservoir and do not model deeper components of the summit magma sys-tem (e.g., Cervelli & Miklius, 2003), which may have played a subordinaterole in the collapse-induced deformation. The magma chamber geometryis illustrative and not meant to mimic the actual 3-D geometry at Kı̄lauea.

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Figure 3. (a) Definition sketch for magma chamber with vertical faults. (b) Precollapse deflation driven by a pressuredecrease prior to fault slip. (c) Vertical displacement during fault slip at constant mass (red) for 𝛽m∕𝛽c = 5, and atconstant pressure relative to precollapse state (blue). Dashed red curve shows the contribution to the displacement atconstant mass due to chamber pressurization. (d) Vector displacements in the x-z plane at the surface, plotted onlyexternal to the collapse. Co-col: co-collapse; Pre-col: precollapse. Distance scales in kilometer; displacements in meters.

The boundary conditions on the magma reservoir are spatially uniform normal traction, equal to the(negative of the) magma pressure and vanishing shear traction. Employing the displacement discontinuityboundary element method, we relate the opening 𝛿n and shear dislocations 𝛿s to the boundary conditionson the magma chamber walls,

G[𝛿n𝛿s

]

⏟⏟⏟Tractions due to chamber

+ G𝑓 s⏟⏟⏟

Tractions due to faults

=[−Δp10

]

⏟⏞⏞⏟⏞⏞⏟Boundary conditions

(1)

where G are the Green's functions relating the normal and shear displacement discontinuities to thetractions acting on the magma chamber and Δp is the magma pressure change. Similarly, Gf relates thescalar fault slip s to the chamber wall tractions (1 and 0 are appropriate length vectors of ones and zeros).The normal and shear displacement discontinuities are then determined by

[𝛿n𝛿s

]= G−1

[−Δp10

]− G−1G𝑓 s. (2)

Given 𝛿n and 𝛿s, the surface displacements are computed through standard relations for elastic dislocations(e.g., Segall, 2010, Chapter 3 ).

ui(x𝑗) = Fi(x𝑗)[𝛿n𝛿s

]+ F̃i(x𝑗)s, (3)

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where ui(xj) gives the ith component of displacement at observation point xj. The first term representsdisplacement due to the boundary tractions through Fi(xj), the second displacement due to fault slip throughF̃i(x𝑗).

A decrease in pressure within the reservoir without fault slip (s = 0) leads to subsidence as illustrated inFigure 3b. When a critical stress is reached on the faults they slip and the central block above the chambermoves downward. The fault slip occurs rapidly; according to high-rate GPS data at station NPIT, this takesroughly 5–10 s. During this short interval negligible magma leaves the chamber so the reservoir is effectivelymaintained at constant mass. The mass erupted in the early, ash-charged explosions was minor and isneglected.

To compute the pressure change accompanying fault slip requires two relations. The first is the volumechange ΔV (per unit depth in 2-D) of the reservoir, equal to the integral of the outward displacements actingon the magma chamber wall. From (2) and (3) the displacements are functions of the (unknown) pressurechange Δp and fault slip; thus, the volume change can be represented compactly by

ΔV = ΨΔp + Φs, (4)

where Ψ ≡ 𝜕V∕𝜕p and Φ ≡ 𝜕V∕𝜕s. Equation (4) gives the volume change in terms of pressure change andslip; Δp here refers to the pressure change during fault slip. The second relation comes from a linearizedequation of state for the magma, and the condition that there is no mass change during collapses,

Δm∕𝜌 = V𝛽mΔp + ΔV = 0. (5)

where 𝜌 and 𝛽m are the magma density and compressibility. Combining equations (4) and (5)

Δp = −ΦsΨ + V𝛽m

= −ΦsV𝛽c

(𝛽m∕𝛽c + 1

) (6)

where the chamber compressibility 𝛽c is defined by 𝛽c ≡ (1∕V)𝜕V∕𝜕p = Ψ∕V. Note that for a perfectlycompressible magma 𝛽m → ∞ so that Δp → 0, whereas in the incompressible limit (from (6) and (4))ΔV = 0. The pressure change from (6) is then used in (2) to compute [𝛿n, 𝛿s]T , and from (3) the surfacedeformations.

Figure 3c shows the vertical displacements due to fault slip at constant mass for 𝛽m∕𝛽c = 5 for verticalfaults (in red). (The effect of changing 𝛽m∕𝛽c is described in section 3.) The block between the faults hasdowndropped 5 m, while there is uplift on the flanks. The deformation external to the collapse block resultsfrom magma pressurization, not elastic rebound, as can be seen from the dashed red curve which showsthe contribution solely from pressure change in the chamber. The blue curve in Figure 3c shows the dis-placements at constant pressure—that is, no change relative to the prefaulting state, for reference. Uniformslip on vertical faults with no change in magma chamber pressure is a rigid body motion, with no defor-mation on the caldera flanks. The constant mass solution is thus the sum of a rigid body motion andinflation. The time-dependent displacements following collapse will decay from the Δm = 0 curve towardthe Δp = 0 curve, as magma drains from the reservoir and the pressurization resulting from collapsedissipates. Figure 3d shows vector displacements in a vertical section and highlights that the collapsedisplacements are antiparallel to the precollapse deflation.

The situation is different if the faults are not vertical. Figure 4a shows the geometry with inward dippingfaults (5◦ from vertical). In this case the instantaneous (constant mass) response exhibits more uplift onthe flanks relative to the case with vertical faults (Figure 4b, red), and some of this uplift remains after thepressure transient decays to zero (blue curve in 4b). With this geometry and magma compressibility mostof the immediate “inflation” is due to pressure increase in the magma chamber (dashed red curve), theremainder results from elastic distortion due to faulting (elastic rebound). For this geometry the cumulativedisplacement through the cycle in which the collapse-induced pressure increase decays to zero would be anet uplift (and outward tilt) of the caldera flanks. This would accumulate over time with additional collapses,as the wedge-shaped block is forced downward. Figure 4c shows that, relative to the precollapse deflation,the co-collapse deformation has a larger ratio of horizontal to vertical displacement.

For the outward dipping geometry (5◦ from vertical) the instantaneous collapse generates uplift, for thespecified ratio 𝛽m∕𝛽c (Figure 4e); however, the horizontal displacements are inward toward the caldera

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Figure 4. (a) Definition sketch for inward dipping faults. (b) Vertical displacement during fault slip at constant mass (red) for 𝛽m∕𝛽c = 5 and at constantpressure (blue) for inward dipping faults. Dashed red curve shows the contribution to the displacement at constant mass due solely to chamber pressurization.(c) Vector displacements at the surface, plotted only external to the collapse. Co: co-collapse; Pre: precollapse. (d–f) Same as above for outward dipping faults.Distance scales in kilometers; displacements in meters.

(Figure 4f). This would thus not be identified as inflationary deformation. For more compressible magmaseven the instantaneous (constant mass) response is subsidence. Regardless of compressibility, as the inducedpressure increase decays to zero, the caldera flanks subside (the constant pressure curve).

3. DiscussionNumerical results are presented here for 𝛽m∕𝛽c = 5, which is toward the upper end of estimates forKı̄lauea (Anderson & Poland, 2016). As 𝛽m∕𝛽c increases the instantaneous collapse response approaches theconstant pressure limit. As 𝛽m∕𝛽c decreases the co-collapse pressure change increases (6) and the chamberpressurization becomes more dominant relative to faulting in the surface deformation. Thus, the contribu-tion of elastic rebound relative to chamber pressurization depends on both 𝛽m∕𝛽c and fault dip. It should benoted that in a fully dynamical model the amount of fault slip will depend on the magma compressibility.

The theoretical predictions can be compared to deformations observed at Kı̄lauea. Displacement and tiltvectors during collapse events in 2018 are illustrated in Figure 1, while Figure 2 shows radial time series fortilt at UWD and displacement at two GPS stations close to the caldera rim, UWEV and CRIM.

There is an extensive literature on the development of ring fault structures due to magma chamber depres-surization from both analog laboratory (Acocella, 2007; Ruch et al., 2012) and particle based numerical(Geyer & Martí, 2014; Holohan et al., 2011) experiments. Broadly, these studies show initial development ofan inner reverse ring fault with subsequent growth of a peripheral fault that may have a normal geometry.

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Figure 5. (a) Comparison of precollapse deflation rate (blue) and co-collapse displacements (red). Shaded region indicates collapse area. (b) Similar comparisonin vertical cross section (view is from the west). The two data sets have different units; they are scaled to have comparable vector size.

In addition to mechanical properties, the orientation of the outer fault depends on the roof aspect ratio andthe chamber shape, with shallow sill-like chambers favoring normal faulting. Regional extension also favorsnormal faulting (Acocella, 2007), which may be applicable to Kı̄lauea given seaward motion of the volcano'ssouth flank (Denlinger & Morgan, 2014; Owen et al., 2000). It should be noted, however, that these studiesmodel the formation of collapse structures in an initially intact crust, whereas Kı̄lauea has been subjectedto countless past collapses and preexisting structures at least influenced the 2018 deformation.

As noted above, many simulations and interpretive cross sections show shallow normal faulting with moreextensive reverse faults at depth. However, if the faults were outward dipping the instantaneous, co-collapsedisplacements would be directed inward toward the caldera (Figure 4f), which is inconsistent with the GPSdisplacements at all stations (Figure 1). This observation would appear to rule out reverse faulting as thedominant collapse mechanism.

To first order the data seem roughly consistent with nearly vertical faulting. A prediction for this geometryis that sites outside the caldera experience deformation during collapse events that is opposite in sign to thatduring precollapse deflation (Figure 3d). (Note that this must also be true for collapse in three dimensions,because the deformation is a sum of rigid body translation of the roof block and pressurization of thechamber.) This is largely consistent with observations as shown in Figure 5, where the precollapse velocityand co-collapse displacement vectors are essentially antiparallel in map view. However, the ratio of horizon-tal to vertical displacement is larger during collapse events (Figure 5b) so the displacements are not strictlyantiparallel. Furthermore, the ratio of co-collapse displacement to precollapse velocity is roughly a factor of2 larger at CRIM and UWEV in comparison to the other stations. These stations are both close to the calderarim, suggesting some role for faulting in the observed deformation.

Although calculations here are for a homogeneous half-space, the observed deformation will depend onspatially variable elastic structure (Browning & Gudmundsson, 2015; Masterlark et al., 2010) and topography(Johnson et al., 2019; Williams & Wadge, 2000). These effects will modify the kernels F and F̃ in equation (3)as well as G and Gf . It should be noted that any effects due to heterogeneous elastic structure in F arecommon to both precollapse and co-collapse deformation and therefore do not bias the comparison betweenthese fields as in Figure 5.

As the magma pressure decreased due to outflow from the magma chamber, the shear stress acting on thecollapse bounding faults increased. If the shear stress threshold to trigger successive collapses did not changewith time, the deflation between events would equal the inflation generated during collapses. This is mostconsistent with the time series in late June and July, when the fault system was fully developed; during thistime the interevent deflation nearly canceled the inflation during the collapse events, as recorded by theGPS displacements (Figure 2). The more pronounced long-term deflationary tilt at UWD during this time

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period is not well understood. The tilt time series record offsets due to numerous earthquakes, which doescomplicate interpretation.

The collapse bounding structures are normal faults at the surface, although as noted above the geometry maychange with depth. Slip on inward dipping faults predicts a larger ratio of horizontal to vertical displacementcompared to precollapse deflation (Figure 4c), which is qualitatively consistent with observations at Kı̄lauea(Figure 5b). This observation is thus suggestive of some component of steep normal faulting. Furthermore,it is noteworthy that the longer-term trends (averaged over many collapse cycles) in the GPS time series atUWEV and CRIM (Figure 2) changed at times roughly coincident with changes in the evolution of faultstructures at the surface, suggesting that faulting played a significant role in the surface deformation.

Does the long-term deflationary trend in the data (Figure 2) present an inconsistency with a normal fault-ing geometry? If the pressure decrease between collapse events equaled the pressure increase generated bycollapse, which would be the case if the stress level needed to trigger successive collapses did not changeover time, then the normal faulting geometry implies a long-term increase in outward tilt and displacement(Figure 4b, constant pressure curve), contrary to observations (Figure 2). However, if the stress requiredto initiate successive collapse events increased with time, then the chamber pressure decrease must haveexceeded the pressure increase accompanying collapse. This could generate a long-term deflationary trendconsisting of episodic inflations and somewhat larger deflations. It is worth recalling that in the calculationsof surface deformation here fault slip is imposed kinematically. With the inward dipping geometry the com-pressive stress acting on the block-bounding faults must increase with slip. This coupling to shear strengthsuggests that increased shear stress, and thus, greater deflation would be required to trigger subsequent slipevents. This could explain the longer-term deflationary trends in the data, although other processes includ-ing deflation of a deeper component of the summit magma system could potentially explain these long-termtrends as well.

3.1. Conclusions• Instantaneous collapse without outflow of magma from the chamber produces uplift of the caldera flanks,

except when there are outward dipping faults and the chamber is filled with highly compressible magma.• Radially outward displacements observed during collapse events at Kı̄lauea appears to rule out an outward

dipping fault geometry, which predicts inward directed displacements.• Much of the data are consistent with the inflationary signals accompanying collapse events being driven

primarily by rapid pressurization of the shallow magma reservoir.• Inward dipping faults imply that the inflationary deformation should accumulate with time, assuming the

stress required to trigger successive collapse events is stationary in time. Yet the GPS time series duringthe period when the surface collapse structure was fully developed (post middle to late June) exhibit adeflationary trend. This could be understood if the intercollapse deflations slightly exceed the co-collapseinflations. This would be expected if the shear stress threshold for collapse increased with time as wouldbe expected if, as for inward dipping faults, the normal stress acting on the faults increased with slip.

• The relatively large co-collapse displacements at GPS stations near the caldera rim relative to deflation inthe precollapse period, as well as the increase in the ratio of horizontal/vertical displacement relative tothe precollapse deflation suggest some contribution of normal faulting in the observed deformation.

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