PUBLICATIONS. Journal of Geophysical Research: Solid Earth

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1 PUBLICATIONS Journal of Geophysical Research: Solid Earth RESEARCH ARTICLE Key Points: Lithosphere is broken below Hawaii Seismicity forms a donut where plate is flexed P axes point to a stress center Lithospheric flexure under the Hawaiian volcanic load: Internal stresses and a broken plate revealed by earthquakes Fred W. Klein 1 1 U. S. Geological Survey, Menlo Park, California, USA Supporting Information: Supporting Information S1 Table S1 Table S2 Table S3 Correspondence to: F. W. Klein, klein@usgs.gov Citation: Klein, F. W. (2016), Lithospheric flexure under the Hawaiian volcanic load: Internal stresses and a broken plate revealed by earthquakes, J. Geophys. Res. Solid Earth, 121, , doi: / 2015JB Received 16 DEC 2015 Accepted 5 MAR 2016 Accepted article online 10 MAR 2016 Published online 8 APR 2016 Published This article is a U.S. Government work and is in the public domain in the USA. Abstract Several lines of earthquake evidence indicate that the lithospheric plate is broken under the load of the island of Hawai`i, where the geometry of the lithosphere is circular with a central depression. The plate bends concave downward surrounding a stress-free hole, rather than bending concave upward as with past assumptions. Earthquake focal mechanisms show that the center of load stress and the weak hole is between the summits of Mauna Loa and Mauna Kea where the load is greatest. The earthquake gap at 21 km depth coincides with the predicted neutral plane of flexure where horizontal stress changes sign. Focal mechanism P axes below the neutral plane display a striking radial pattern pointing to the stress center. Earthquakes above the neutral plane in the north part of the island have opposite stress patterns; T axes tend to be radial. The M6.2 Honomu and M6.7 Kiholo main shocks (both at 39 km depth) are below the neutral plane and show radial compression, and the M6.0 Kiholo aftershock above the neutral plane has tangential compression. Earthquakes deeper than 20 km define a donut of seismicity around the stress center where flexural bending is a maximum. The hole is interpreted as the soft center where the lithospheric plate is broken. Kilauea s deep conduit is seismically active because it is in the ring of maximum bending. A simplified two-dimensional stress model for a bending slab with a load at one end yields stress orientations that agree with earthquake stress axes and radial P axes below the neutral plane. A previous inversion of deep Hawaiian focal mechanisms found a circular solution around the stress center that agrees with the model. For horizontal faults, the shear stress within the bending slab matches the slip in the deep Kilauea seismic zone and enhances outward slip of active flanks. 1. Introduction 1.1. Hawaiian Earthquakes Hawaiian earthquakes can loosely be thought of in three classes according to their proximity to active magma. The first class is associated with the propagation of magma and dikes, and these earthquakes are located at 2 4 km depth under active calderas and rift zones [e.g., Klein et al., 1987]. The second class of earthquakes are those within and at the base of the active volcano flanks between the active rift zone and the coastline. Hawaii is among the most seismically hazardous locations in the United States and has produced earthquakes of magnitude 7.9 [Wyss, 1988; Klein et al., 2001]. The largest historic earthquakes have been of this flank type, and were the M w 7.9 great Kau earthquake in 1868 on the south flank of Mauna Loa, and the M s 7.2 (M w 7.7) Kalapana earthquake on 29 November 1975 on the south flank of Kilauea [e.g., Ando, 1979]. These flank earthquakes can rupture some or even most of the length of the flank and are between 5 and 13 km depth. The southern flanks of Kilauea and Mauna Loa volcanoes are mobile and move laterally on a 9 km deep décollement surface [e.g., Tilling and Dvorak, 1993]. The stress causing flank earthquakes is largely derived from growth of the adjacent rift zone and from gravitational spreading [e.g., Denlinger and Morgan, 2014]. The third class of earthquakes is more broadly distributed around the whole island and has depths down to 60 km. These deeper earthquakes in the upper mantle have reached M6.7, as they did for the Kiholo Bay earthquake at a depth of 39 km on 15 October These earthquakes also occur under the older, inactive volcanoes north of Mauna Loa. We broadly think of these earthquakes as related to flexure and subsidence under the volcano load, as have previous authors [e.g., Walcott, 1976; Moore, 1970; Klein and Koyanagi, 1989; Wolfe et al., 2004; Pritchard et al., 2007]. Earthquakes of this class are the subject of this paper, and the stress driving them is due to broad deformation and only partly to direct magmatic stress. By this we mean that flexure influences the stress orientation and accumulated stress of earthquakes, but magma and volcano growth may push faults to failure. While only a few percent of the probabilistic ground motion hazard of Hawaii results from earthquakes deeper than 20 km [Klein et al., 2001], it is important to understand their cause. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2400

2 Earthquakes are associated with the vertical magma conduits feeding both Kilauea and Mauna Loa [e.g., Klein et al., 1987; Wright and Klein, 2006; Okubo and Wolfe, 2008]. The Kilauea conduit earthquakes can be thought of as a special class of flexural earthquakes localized by a thermally weakened magma conduit and influenced by tractive stresses of the overlying flanks as well as flexural stresses [Klein et al., 1987]. Klein et al. [1987] proposed the effect of magma on making the Kilauea conduit earthquake zone more seismically active than the surroundings is partly from weakening the conduit, permitting regional stresses to concentrate there. The Mauna Loa magma conduit is revealed by a swarm of long-period microearthquakes [Okubo and Wolfe, 2008] Lithospheric Flexure The initial studies of lithospheric plate flexure in Hawaii have concentrated on the big picture of depression of an elastic plate under the load of the island chain and the arch of the plate peaking approximately km from the axial central swell of the island chain [Watts and ten Brink, 1989; Watts and Cochran, 1974; Wessel,1993; Watts, 2001]. The models have assumed a thin plate (thickness small compared to the wavelength of the arch), often an unbroken plate, and had little data about the internal structure of the plate. These studies determined an effective elastic thickness of the lithosphere of km and a flexural rigidity of 1 to N m. Tomographic shear velocities under Hawai`i Island peak at a depth around 40 km and have a minimum around 120 km [Cheng et al., 2015], and the lithosphere-asthenosphere transition between these depths is a different measure of lithospheric thickness. Kunze [1980] determined that the lithosphere must be viscoelastic with a viscosity of about poise ( Pa s) to experience the additional subsidence between the young island of Hawai`i and the older, more subsided Oahu. Zhong and Watts [2013] use flexure profiles to derive a three-dimensional rheological model with low temperature plasticity that yields very high MPa stresses to produce flexure earthquakes to the observed depths near 50 km. Watts [2001, chapter 3] and Hetenyi [1979] provide analytical solutions of localized loads on infinite and semi-infinite (broken) beams, but their solutions only provide plate deflections but not internal stresses. Refraction studies [e.g., Zucca and Hill, 1980;Hill and Zucca, 1987] indicate the Moho is at a depth of about 13 km at the Hawaiian coastline. These authors show the Moho dips about 3 under the submarine flanks and 6 10 under the subaerial flanks toward the island under the volcanic pile, consistent with downward and downward curving flexure under the island load. Seismic profiles near Oahu and spanning the arch and trough show crustal reflectors with downward curvature between the arch and the Oahu coastline but suggest upward curvature crossing the axis of the island chain [Watts and ten Brink, 1989]. A question addressed by some authors is whether the lithospheric plate is broken under the Hawaiian load or is continuous. The submarine topography and gravity profiles of the moat and arch on the distant volcano flanks are similar with either broken and unbroken plates [Walcott, 1970, 1976; Watts and Cochran, 1974]. Walcott [1970] found that the amplitudes of displacement and gravity give a better fit if the plate is broken along the ridge axis. Wessel [1993] found that the high downward curvature of flexure requires a thinner elastic thickness of 33 km under the island of Hawai`i and 38 km along the axis of the islands, versus 44 km thickness on the flanks, implying a weakened volcanic axis. Wessel [1993] interpreted his flexural profile between Oahu and Molokai as intermediate between a continuous and broken plate, though he modeled the break along the Molokai fracture zone instead of the ridge axis. Crustal reflectors in the seismic profiles of ten Brink and Brocher [1987] and Watts and ten Brink [1989] do not provide direct evidence for a broken plate under Oahu, though reflectors show a vertical offset of about 1 km under the island axis, with the NE profile curving downward as it would if the plate were broken. Thus, the studies suggest weakening along the axis but do not provide conclusive evidence for a broken plate near Oahu. If the plate is greatly thinned or weakened under the center of the load, the radius of upward curvature and the distance between downward bending flanks may be so small as to make a distinction between plate breakage and thinning meaningless. A calculation of stresses within a bending, horizontal cantilever with a vertical load at the free end is a twodimensional problem simpler than the three-dimensional problem of a depressed, broken plate under a central load. The 2-D bending cantilever problem was solved by Timoshenko and Goodier [1934, p. 41]. The stress solution within the plate has maximum horizontal extension at the top surface, no extension at the neutral plane in the center of the plate, and maximum horizontal compression at the base of the downward curving plate. A two-dimensional elastic-plastic bending model with application to the Hawaiian swell using oceanic rock properties [Liu and Kosloff, 1978] calculates internal bending stresses within a realistic 60 km thick plate and allows for a broken plate at the volcanic axis. For positive curvature (convex upward), their model predicts a neutral depth (lateral tension above and lateral compression below) of 24 km. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2401

3 An association of the stresses producing Hawaiian mantle earthquakes with lithospheric loading may have begun in Although Hawaiian earthquakes were not a constraint to Liu and Kosloff s [1978] elastic model, Liu was visiting the University of Washington in 1977 and had contact with two students who were studying Hawaiian earthquake focal mechanisms. Rogers and Endo [1977] and Endo et al. [1979] found four deep earthquake clusters whose principal stresses formed a radial pattern they compared to a split elastic beam under a Hawaiian load. This University of Washington visit may have been the first qualitative association of deep Hawaiian earthquake stress and lithosphere loading. Prior studies of deep Hawaiian earthquakes [e.g., Eaton and Murata, 1960; Eaton et al., 1971] recognized volcanic conduits but not lithospheric loading as a cause of mantle earthquakes. Calculations for a mascon, treated as a distributed load of Gaussian profile, have been done for the Moon and Mars, and it is not unlike an oceanic island load. Melosh [1978] derived internal stresses for a thick plate loaded with a mascon and which floats on a fluid asthenosphere. McGovern and Solomon [1993] applied the load calculations to an evolving load of Martian volcanoes on a lithosphere. Both papers assumed an unbroken plate; thus, their models are concave upward directly under the load. This unbroken plate assumption means the model s distribution of stresses does not apply if the plate is broken. The models predict large differential stresses uncompensated by buoyancy from the underlying asthenosphere under the center of the load, which may not be valid once the plate is broken. This means there are geometrical reasons for an unbroken plate to fracture and break under the load center, a factor which may contribute to the eventual breaking of the plate in a location like Hawaii. More refined flexure models for Hawaii using topography and numerical integration of load Green s functions on a lithospheric plate over a viscous asthenosphere were calculated by Pritchard et al. [2007] and McGovern [2007]. Both studies assume a continuous, unbroken plate. Pritchard et al. s [2007] discussion focuses mainly on whether flexural stresses cause the seismicity in the mantle in the primary earthquake zone 30 km below Kilauea. They conclude that the load favorably enhances stress in that zone, but there are other factors such as the presence of fluids that are needed to concentrate the seismicity at that zone within a larger region of flexural stress. Features of their stress model, such as a neutral depth near km, are very helpful in explaining aspects of earthquake patterns explored in this paper. McGovern s [2007] stress calculations, on the other hand, are focused on the north island region surrounding the 2006 Kiholo Bay earthquakes just north of Hualalai volcano. He finds the stress orientations of his flexure model match that of the M6.7 main shock at 39 km depth and the large M6.0 aftershock at 19 km depth, and there are peaks in calculated differential stress at those depths as well. McGovern assumes a continuous, unbroken plate and also assumes a linear volcanic load to the NW parallel to the Hawaiian Ridge. See the section 6 for reasoning that a simple broken plate model fits more earthquake data than the features focused on in Pritchard et al. s [2007] and McGovern s [2007] numerical studies. 2. Seismic Data Earthquake data analyzed in this paper are from the Hawaiian Volcano Observatory earthquake catalog [e.g., Hawaiian Volcano Observatory, 2002; Nakata, 2007]. Locations are from the hypoinverse location program [Klein, 2002] using a layered linear gradient crustal model [Klein, 1981]. The catalog from 1970 to 2007, as used in this study, is based on picks from a Develocorder viewer and an expanded network (starting in 1970) or from digital displays on a computer screen (starting in 1980). First-motion readings are those from the catalog, were carefully made by the seismic analysts at the time, and have been the basis of numerous focal mechanism studies [e.g., Crosson and Endo, 1981]. The use of a linear gradient model, with rays calculated as curved lines, aids dispersion of stations on the focal sphere because fewer arrivals are refracted from model layers with a constant source takeoff angle. Studies using relocated earthquake catalogs [e.g., Got and Okubo, 2003; Wolfe et al., 2004; Matoza et al., 2013; Lin et al., 2014] have greatly sharpened the structures visible within earthquake clouds. Relocation studies improve locations of half or less of Hawaiian earthquakes [Wolfe et al., 2004; Lin et al., 2014], but uncorrelated, small-magnitude earthquakes often do not relocate. Because the goal of this study is to analyze stress directions revealed by earthquake focal mechanisms and general earthquake patterns island wide, it is not necessary to refine relative locations or attempt to reveal spatial fine structure by producing a double-difference earthquake catalog. In our catalog, small (M < 2.5) and poor locations (root-mean-square (RMS) residual >0.15 s) have been left out, and we do not rely on every station polarity being correct. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2402

4 Figure 1. Map of historical M6.0 and larger earthquakes with Hawaii s five volcanoes (submarine Loihi to the south is not shown). The four active rift zones are to the north of the flanks and stress the mobile and seismically active south flanks of Mauna Loa and Kilauea. The larger M > 6.0 crustal earthquakes (black dots) are confined to the active mobile volcano flanks of Kilauea, Mauna Loa, and Hualalai, but the deeper, upper mantle earthquakes (red dots) are broadly distributed. Lower hemisphere focal mechanisms are shown for three deep earthquakes of interest in this paper. 3. Hawaiian Seismicity Hawaii has broadly distributed earthquakes with depths down to 50 or 60 km, which potentially can be triggered by loading stresses. These flexure earthquakes are in addition to large earthquakes on the décollement surfaces at 9 15 km depth under active flanks. Figure 1 maps the magnitude 6.0 and larger earthquakes from Hawaii s historical catalog using both isoseismal maps and instrumental records [Wyss and Koyanagi, 1992a; Klein and Wright, 2000; Klein et al., 2001]. The largest earthquakes are under the south flanks of Mauna Loa and Kilauea, the west flank of Mauna Loa, and the south flank of weakly active Hualalai (black dots, Figure 1). Crustal earthquakes larger than M6 have not occurred historically under the inactive Mauna Kea and Kohala volcanoes. By contrast, the deeper, upper mantle earthquakes (red dots, Figure 1) are more broadly distributed. Patterns of earthquakes also show the locations of active flanks and broader areas of high stress. These wellrecorded earthquakes (M 2.5) are plotted in map and NW-SE cross-section views in Figure 2. Shallow earthquakes less than 13 km deep are dominated by the active, mobile south flanks of Kilauea and Mauna Loa. The west flank of Mauna Loa is also mobile toward the west and has a band of earthquakes following the coastline between 8 and 14 km depth [Gillard et al., 1992, Figures 2a and 2c; Wolfe et al., 2004]. The band continues to the northwest merging into Hualalai s west rift zone and flank. These two flank zones are capable of magnitude 6 and larger earthquakes [Wyss and Koyanagi, 1992b; Klein et al., 2001]. Kohala and the north flank of Mauna Kea also produce shallow earthquakes, some of which may represent northward motion of Mauna Kea s north flank [Wolfe et al., 2004]. The band of earthquakes in Figure 2c between 8 and 14 km depth NW of Mauna Loa can be interpreted as motion at the base of a mobile flank and are not necessarily a result of lithospheric flexure, though we will consider bending stresses as a possible contributor. Loihi submarine volcano to the south (Figure 2a) experiences earthquake swarms that can also be interpreted as caldera, rift, and mobile flank earthquakes [Klein, 1982; Caplan-Auerbach and Dunnebier, 2001]. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2403

5 Journal of Geophysical Research: Solid Earth Figure 2. Earthquakes from the 1970 to 2006 HVO catalog with magnitudes greater than 2.5 and RMS residual less than 0.15 s. (a) Map view of 0 22 km depth. (b) Map view of km depth. (c) NW-SE cross section through the center of the island including earthquakes in the box in Figure 2b. The pronounced gap in earthquakes near km depth separates the two map views and is interpreted as the neutral plane of flexure at which stress is a minimum. Above 13 km depth, active mobile flanks dominate. The most active earthquake zone below the neutral plane is centered 30 km below Kilauea caldera and surrounds and is part of Kilauea s vertical magma conduit. Mauna Loa s deep conduit is defined by a short-lived poorly located cluster of small, long-period earthquakes. Except for this swarm, Mauna Loa s magma column is mostly aseismic. Deeper earthquakes below 22 km depth (Figures 2b and 2c) are particularly relevant to revealing flexure stresses. The most prominent mantle earthquake zone, which also has produced magnitude 6 earthquakes, is centered about 30 km depth below and to the south of Kilauea caldera. Wolfe et al. [2003] interpret this zone as primarily tectonic faulting, and Klein et al. [1987], Wright and Klein [2006], and Wech and Thelen [2015] note that it is part of an earthquake-defined magma conduit from depth, with long-period earthquakes above the north end and below the south end. Wech and Thelen [2015] use double-difference relocation to resolve this mantle fault zone into a subhorizontal north-south earthquake band about 35 km long. Pritchard et al. [2007] note that flexural stresses contribute to faulting in this 30 km deep zone, but their model does not produce peak stresses that coincide with the seismic zone. Each of these interpretations likely provides part of the story of this complex 30 km deep source. Mauna Loa s vertical magma conduit is revealed by a cluster of small, short-lived, long-period earthquakes (Figure 2c) that collapse to a source less than 2 km diameter when relocated using double difference [Okubo and Wolfe, 2008]. The cross section through the island (Figure 2c) shows a prominent gap in the depth of earthquakes centered at about 21 km depth. This gap is visible under Kilauea where depths are well determined and also island KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2404

6 wide. There are no layer boundaries in the crustal model used to locate earthquakes nor are there other known artificial means to explain this gap. A large halo of mantle earthquakes about 160 km in diameter around Hawaii is a major feature of the earthquake pattern (Figure 2b). The halo includes the 30 km deep Kilauea earthquake zone and the prominent magnitude 6 and larger earthquakes. Earthquakes in this deep halo are mostly dispersed; there is some clustering into aftershock zones and clusters associated with magma conduits, but there is no clumping into dense zones that is characteristic of flank earthquakes. Some of this dispersal is a result of poorer location quality under a more widely spaced seismometer network, but even double-difference relocation [Wolfe et al., 2004; Matoza et al., 2013; Lin et al., 2014] does not pull earthquakes in the halo into small clusters. This dispersal of locations might be an indication of a broad source of stress such as regional bending, rather than concentrated sources of stress such as magma conduits or long faults. The center of the halo (near the star symbol in Figure 2b) is nearly aseismic, except for the cluster of long-period events in Mauna Loa s magma conduit. The largest earthquakes within the halo of deep earthquakes north of Kilauea are the M6.7 Kiholo Bay earthquake of 15 October 2006 and the M6.2 Honomu earthquake of 26 April Aftershocks of these earthquakes (Figure 3) illustrate the largest recent ruptures within the mantle under Hawaii. The Kiholo aftershock cluster of original catalog locations can be partly reduced with double-difference locations [Yamada et al., 2010], but the trend is still NW-SE (Figure 3) and is parallel with the NW-SE nodal plane (Figure 1). The large M6.0 aftershock that occurred 7 min after the main shock (Figure 3, blue symbol) is detached from the main shock zone, is 21 km shallower, has a different focal mechanism (Figure 1), and as we will demonstrate, is in a different stress orientation regime within the bending plate. McGovern [2007] interprets the M6.7 and M6.0 earthquakes to be within different differential stress maxima for each earthquake depth. The HVO catalog also places the Kiholo main shock and large aftershock on opposite sides of the lithosphere s neutral plane, and there is radial compression below and tangential compression above. The aftershock zone of the slightly smaller M6.2 Honomu earthquake is correspondingly smaller than the Kiholo zone. 4. Focal Mechanisms and P Axis Azimuth Getting good focal mechanisms in Hawaii faces the challenge of getting good seismic station coverage of the focal sphere. Seismic stations are concentrated on the active volcanoes in the south of the island with only a few stations in the north, and there are no stations offshore. Therefore, parts of the focal sphere have limited coverage, and depending on the position of the nodal planes, it is often difficult to unambiguously constrain both nodal planes. Therefore, we seek to get a sense of the tectonic orientation of the stress field revealed by simpler parameters such as the P axis and T axis azimuths, rather than a precisely determined set of fault planes in every case. This approach is better suited to regimes where the stress field is more consistent than the orientation of the individual nodal planes. The focal mechanism technique we use is to plot P first motions on the focal sphere and to fit planes using the FPFIT computer program [Reasenberg and Oppenheimer, 1985]. FPFIT assumes that the nodal planes are orthogonal with no net expansion or contraction of the moment tensor. Often, there are only first motions available, but our use of a multilayer linear gradient crust model helps disperse stations on the focal sphere instead of constant-angle refractions from the same model layer boundary. We examined earthquakes of magnitude 2.5 and larger, and found smaller magnitudes than M2.5 typically did not provide useful focal mechanisms in north Hawaii. All focal sphere plots were visually inspected. Full two-nodal-plane mechanisms were retained in unambiguous cases and rejected if an unambiguous mechanism relied on a single polarity. All P axis directions that we retained were visually unambiguous to an error of about 20. FPFIT provides a plot of the possible scatter of the P and T axis distribution, and we used this to exclude P axis azimuth errors that were larger than 20. There were not enough broadband seismometers in Hawaii to unambiguously determine a moment tensor. Teleseismic centroid moment tensor (CMT) solutions often require magnitudes of 5 and larger; Wolfe et al. [2004] were only able to obtain 21 mechanisms from the Harvard CMT catalog (and their extensions to it) for 1979 to 2004 that were deeper than 13 km, and for those the local P polarity and the CMT mechanisms were fairly similar. Therefore, we rely primarily on traditional P first-motion focal mechanisms to get enough mechanisms to get good spatial coverage and to constrain a simpler measure like the P axis azimuth. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2405

7 Figure 3. Hawaiian earthquakes within 31 days after the M6.7 Kiholo Bay earthquake of 15 October 2006 at 1707 UTC and the M6.2 Honomu earthquake of 26 April 1973 at 1726 UTC. All earthquakes with their HVO catalog locations in the two time periods are plotted. Symbol color and shape depend on depth, and symbol size depends on magnitude. The main shocks are large black triangles. Because the depths of the aftershocks of the two earthquakes are mostly greater than 30 km, most red symbols in the cross section are aftershocks. The M6.0 earthquake just north of and shallower than the Kiholo Bay earthquake is the major aftershock (blue color) that occurred 7 min after the M6.7 main shock and is plotted with other secondary aftershocks detached from the main aftershock zone P Axis Azimuth We can obtain unambiguous focal mechanisms for some events for which both nodal planes are constrained on the focal sphere (Figure 4a, for example). We can nevertheless determine very useful tectonic information from even an incomplete mechanism, such as the azimuth of the P axis. The P axis azimuth is well constrained, for example, if there is a large cloud of dilatational first motions (Figure 4b). We consider the P axis azimuth to be bidirectional because it is at opposite azimuths on the upper and lower hemispheres and because we want to determine the general azimuth of tectonic compression. We use the azimuth of the P axis when it is determined to within 20 and exclude azimuths when the P axis is within 45 of the vertical because the azimuth is less well determined. Thus, if the focal mechanism is unambiguous, it may still not yield a usable P axis azimuth if the P axis is approximately vertical. If the FPFIT program yielded two possible but distinct mechanisms that both had the same P axis azimuth, only the P axis was considered determined and not the entire mechanism. We are not attempting to invert for the orientation of maximum compressive stress; though the P axis approximates maximum compression, the difference in azimuth between the two is relatively small and is not important for our purposes. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2406

8 Figure 4. Representative P first-motion focal mechanisms of the types identified in this study. These are typical, not the best or worst. (a) Unambiguous orientation with both nodal planes constrained. (b) Nodal planes not constrained, but azimuth of P axis is determined to better than 20. (c) A horizontal fault mechanism with one nodal plane dipping 15 or less. Horizontal fault mechanisms are common at the décollement at the base of volcano flanks where flank blocks slip seaward and are sometimes also found at deeper locations. The P axis direction is visually easier to interpret than a focal sphere symbol, which requires 3 numbers to specify (strike, dip, and rake). The two representations provide a picture of the stress field in different ways. Full mechanisms allow an interpretation of the nodal planes, although when no single plane is apparent in an earthquake cluster, the P axis direction as an indicator of tectonic stress can be nearly as useful as complete focal mechanisms. P axis azimuths can be a useful way to summarize a tectonic direction in shallow Hawaiian earthquakes even when full mechanisms are available [Johnson et al., 2015] Variations of Focal Mechanisms All candidate focal mechanisms were categorized into one or more of five categories: (1) unambiguous with both nodal planes and P and T axes determined; (2) a subclass of category 1 termed horizontal fault mechanisms, in which one nodal plane dips 15 or less; (3) mechanisms where the azimuth of the P axis found by FPFIT is determined to within 20 and is not vertical; (4) mechanisms where the azimuth of the T axis is determined to within 20 and is not vertical; and (5) rejected mechanisms too poor to qualify for any of the above. Table S1 in the supporting information lists the events in classes 1 and 2 for which complete mechanisms were determined. Supporting information Tables S2 and S3 list the events in classes 3 and 4, respectively, for which nonvertical P and T axis azimuths were determined. Horizontal fault mechanisms are common at the décollement at the base of volcano flanks and sometimes are at deeper locations (Figure 4c, for example) [Wolfe et al., 2003]. Horizontal fault mechanisms are a category of interest because they identify potential flank block movement near 9 13 km depth and possibly sympathetic slip on deeper horizontal planes. Horizontal faults are typical of Kilauea s south flank, and the horizontal plane is almost always the preferred slip plane because of agreement with hypocenter distributions and geodetic inversions [e.g., Crosson and Endo, 1981; Bryan and Johnson, 1991; Owen et al., 2000]. Note that the azimuths of the P and T axes are nearly the same for horizontal fault mechanisms, and thus, these mechanisms do not distinguish different compression and extension directions when the two are being contrasted. Some solutions yield apparent horizontal planes (like Figure 4b), but they are rejected from this category if the plane dips steeper than 15 to keep them as a distinct horizontal fault class. Faulting between 9 and 13 km depth is dominated by flank processes. The depth range 9 13 km contains the décollement surface on which flank blocks slip seaward. The horizontal fault mechanism of the M7.2 (M w 7.7) 29 November 1975 earthquake is typical of hundreds of Kilauea south flank earthquakes that we do not plot (Figure 5a)[e.g., Crosson and Endo, 1981]. Horizontal fault mechanisms with seaward directed slip vectors away from the expanding rift zones are also visible on Mauna Loa s southeast flank in the Kaoiki and Hilea seismic zones [e.g., Wyss et al., 1992; Jackson et al., 1992] and on Mauna Loa s west flank [Gillard et al., 1992; Wolfe et al., 2004]. There is also a horizontal fault mechanism on Hualalai s south flank, suggesting block motion in the rupture zone of the M6.5 earthquake of 6 October 1929, interpreted as flank rupture similar to Kilauea s south flank [Wyss and Koyanagi, 1992b; Klein et al., 2001]. Lower crustal (9 13 km) faulting in north Hawaii under Mauna Kea volcano (Figure 5a) does not show consistent focal mechanisms with either similar stress orientation or fault planes. Mauna Kea does have three horizontal fault mechanisms of differing orientation, suggesting consistent flank block motion has ceased at KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2407

9 Journal of Geophysical Research: Solid Earth Figure 5. (continued) KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2408

10 Figure 5. Maps of lower hemisphere focal mechanisms and P axis directions in 4 km thick depth slices, Solid quadrants are compressional first P wave motions. Inward pointing arrows are the P (pressure) axis, which is in the white dilatational quadrant. Kilauea and Mauna Loa volcanoes are represented by their summit calderas, and the summit of the dormant Mauna Kea volcano is shown as a triangle. (a) The shallowest layer are Mauna Loa s southwest and northeast rift zones, and Kilauea s rift southwest and east zones together with major south flank normal faults. The P axis arrows are shown for events where the azimuths could be identified within 20, and we exclude azimuths when the P axis is within 45 of the vertical. Thus, the earthquake sets with mechanisms and with P axis azimuths may be different. Magnitude 3 and larger earthquakes are shown. Magnitude 6 and larger earthquakes (and the deep M5.2 1 February 1994 earthquake in Figure 5f) are plotted with larger focal symbols for emphasis. Depth slices begin at 9 km to include some of the décollement earthquakes at the base of the volcanic pile. Earthquakes shallower than 9 km were purposely omitted because they result more from rift and flank processes than from lithospheric flexure. Similarly, hundreds of Kilauea south flank earthquakes in the 9 13 km layer were omitted for clarity, and the M s 7.2 (M w 7.7) 29 November 1975 earthquake is typical of the horizontal fault mechanisms of the south flank. To avoid overcrowding, the magnitude cutoff in the Kaoiki seismic zone (Figures 2 and 5a) and Kilauea s deep seismic zone (Figures 5e and 5f) is higher, about M Typical mechanisms for these two crowded seismic zones, including the Hilea seismic zone to the southwest (Figure 5a), are shown. The star in the deeper layers represents the location of the stress center identified from the orientation of P axes. Note the dominance of radially oriented P axis in the deeper layers indicating radial compression. this time in the volcano s history. Figure 5a provides a depiction of the dominance of flank block motion away from the rift zones toward the sea on subhorizontal, basal décollement faults on the active volcanoes. To avoid overcrowding, Figure 5 plots both focal spheres and P axis direction for only magnitude 3.0 and above. Figure 6 shows magnitude 2.5 and above but shows only the P axis. For clarity, the hundreds of nearly identical horizontal fault mechanisms on Kilauea s south flank have been omitted from the figures. The event labeled M7.2 for the Kalapana earthquake is typical of the south flank, which is not the subject of this study. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2409

11 Figure 6. (continued) KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2410

12 Figure 6. Maps of P axis directions in 4 km thick depth slices using magnitude 2.5 and larger earthquakes. The depth slices are the same as Figure 5. Kilauea and Mauna Loa volcanoes are represented by their summit calderas, and the dormant Mauna Kea volcano is shown as a triangle. The P axis arrows are shown for events where the azimuths could be identified within 20, and we exclude azimuths when the P axis is within 45 of the vertical. Note that nearly all P axes below about 27 km depth radiate about the stress center, represented by a star. The focal sphere symbols for the deeper zones (below 13 km) in Figures 5 and supporting information reveal a variety of different mechanism types, though earthquake clusters or small regions often have similar mechanisms. Except for horizontal faulting, it is difficult to see a tectonic uniformity of mechanisms, common fault planes, or a structural grain for large areas of the island Radial P Axes The most prominent feature of earthquake mechanisms below 21 km depth is the increasing tendency of P axes to become radial, i.e., pointing outward from the center of the island. The radial pattern of P axes is prominent for M > 3 earthquakes in Figures 5 and supporting information and is reinforced by the M > 2.5 events of Figure 6. The radial pattern is striking especially near 35 43depth(Figures5gand 5h and 6g and 6h) where the P axes align like compass needles around a magnetic pole. This depth range of strongly radial P axes is also where the two large M > 6 earthquakes (Honomu, 26 April 1973 and Kiholo, 15 October 2006) are located. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2411

13 Figure 7. The focal mechanisms (black), P axis arrows (red) and T axis arrows (blue) from 35 to 48 km depth (from Figures 5g to 5i) plotted on a single map. The earthquakes are magnitude 3 and larger. P and T axis azimuths are known to about 20 uncertainty and excluded if within 45 the vertical, as in Figure 5. This figure makes clear the donut of earthquakes under the edges of the island and the hole of low seismicity under the center of the island. The radial direction of P axes and the tangential direction of T axes are also shown. The radial pattern of P axes is emphasized by grouping the km depth layers together (Figure 7, red arrows). T axes tend to be tangential in the same depth range (Figure 7, blue arrows). Figure 7 also emphasizes the donut of seismicity around the edge of the island and the lack of seismicity in the center of the island. These patterns are strong evidence for a circular stress pattern and what we interpret as a downward bending lithosphere surrounding a weak center. The stress center shown by a star (Figures 5 7) is visually determined by the convergence of P axes. The stress center is close to the center of the island volcanic load; Mauna Loa is the most massive volcano and Mauna Kea is the second most massive (see topography of Figure 1). The stress center is near the summit of Mauna Loa but is slightly offset in the direction of Mauna Kea, perhaps where the combined loads reach a maximum. The coincidence of the stress center derived from earthquake mechanisms and the visual center of the topographic load emphasizes the current topography as the cause of the flexural bending. Earthquake P axes in the north of Hawaii Island illustrate a shifting of stress above and below the neutral plane. P axes NE of Mauna Kea and under Hualalai are tangential from 13 to 17 km depth, mixed between 17 and 22 km, but clearly show radial compression under the neutral plane between 22 and 27 km (Figures 5 and 6). In contrast, Kohala at the north tip of the island, and between 13 and 17 km depth and above the neutral plane, P axes are directed NW away from the stress center. Under Kohala, north-south compression thus occurs at all examined depth ranges P Axis Azimuths Relative to a Stress Center of a Flexing Lithosphere The radial distribution of the P axes below about 22 km depth is profiled with a histogram of the angle between the P axes and the stress center (Figure 8a). The P axes below 22 km depth are more highly radial KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2412

14 Figure 8. Distribution of (a) P axis azimuths and (b) T axis azimuths relative to the island stress center (star symbol in Figures 5 7) for (Figure 8a) The shallow (9 22 km depth) earthquake P axes (blue bars) are randomly distributed, but the deeper km P axes (red bars) are highly radial. (Figure 8b) The T axes of shallow events (blue bars) are slightly radial, but deeper T axes (red bars) tend to be tangential, but less strongly than thep axes are radial. Radial P axes below the neutral plane and radial T axes above it indicate a lithosphere that is flexed concave downward about a central stress point. with respect to the island center, but those above 22 km are variously oriented relative to the stress center. The half width of the deep P axis distribution is about 20 30, in part because we estimated the individual errors of azimuths to be about 20 from the P axis distributions from the FPFIT program. The T axis azimuth, for the fewer events where it can be determined, shows a tendency to be tangential below 22 km and a weak tendency to be radial above 22 km (Figure 8b). If we characterize earthquakes by the direction of their P axis relative to the stress center, we might see where different types of faulting are dominant. Figure 9 plots intermediate and deep map views and a NW-SE cross KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2413

15 Figure 9. Maps and cross section of earthquakes, symbolized according to the azimuth of their P axes relative to the island s stress center, shown as a star. The figure is a way of showing where earthquakes conform to the simplest pattern of radial P axes below the neutral plane and tangential P axes above the neutral plane and where they deviate. Violet plus symbols show horizontal fault earthquakes (a fault plane within 5 of horizontal) for all azimuths of the slip vector. (a) Map of events 9 22 km depth (above the neutral plane), (b) The km depth (below the neutral plane), and (c) NW-SE cross section corresponding to the mapped rectangle. Location of the Kilauea, Mauna Loa, and Mauna Kea summits and magnitude 6 and larger earthquakes are shown. The section shows the depth of the neutral plane inferred from a gap in seismicity (Figure 2) and the décollement plane of seaward motion of volcano flanks relative to the underlying oceanic crust. The P axes of earthquakes below the neutral plane are overwhelmingly radial but locally of mixed direction above the plane. Horizontal fault mechanisms are common on Kilauea s south and Muana Loa s west flanks, at Kilauea s 30 km deep seismic zone, and can be found at the décollement and sometimes near the neutral plane. Note that only a few of the horizontal fault symbols are plotted of the hundreds on Kilauea s southflank. section of earthquake P axis symbols through Hawaii. Earthquake symbols are distinguished by the orientation of their P axis relative to the stress center: either radial (within 30 ), tangential (within 30 ), or intermediate (the remaining 30 ). The principal feature of P axis orientation is the radial orientation of mechanisms below the neutral plane (Figures 9b and 9c). An exception is the cluster under Kilauea s southcoast (green squares, Figures 9b and 9c) where P axes are perpendicular to the coastline (Figure 7). The P axis orientation perpendicular to the coast matches the slip direction of the vast majority of south flank mechanisms, a point we will return to in the discussion. The places where the P axis orientation differs from the pattern of its neighbors is likely where there is influence of local flank deformation. Thehorizontal fault mechanisms (Figure 9, plus symbols) are largely confined to two regions: (1) thebase of the volcanic edifice interpreted as the décollement on which mobile flanks slide seaward and (2) the 30 km deep seismic zone under Kilauea. The west flank of Mauna Loa and north flank of Mauna Kea between 9 and 13 km depth also exhibit seaward motion on horizontal faults. We will see that a flexing lithosphere under Hawaii exerts shear stresses on horizontal planes that reinforce the seaward motion of the upper block away from the stress center. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2414

16 5. Mantle Stress Under Hawaii 5.1. Stress Inversion A stress inversion of the set of M > 3 focal mechanisms of Figure 5, supporting information Table S1, and Figure S1 (Ross et al. [2007] and S. Ross, personal communication, 2007) reveals stress that forms a radial pattern around the island s stress center. The stress does not align with the linear axis of the island chain of the Hawaiian ridge as hypothesized by Pritchard et al. [2007]. Thus, a 3-D stress model will ultimately be required and not a 2-D model with a linear axis. The stress inversion assumes that a stress orientation can account for the faulting even though individual slip planes can vary widely. Ross et al. [2007] inverted focal mechanisms from six different depth ranges and found significant stress alignment in two depth zones. Between 9 and 12 km depth, the least compressive stress axis σ 3 (approximately the T axis) significantly dipped radially toward the stress center at about 50. The σ 3 axis in the layer between 12 and 20 km also dipped about 50, but with a higher misfit that the authors indicate is too high for a meaningful result. In the crucial flexural zone between 25 and 48 km depth, the most compressional stress axis σ 1 (approximately the P axis) significantly dipped radially away from the stress center at about 30. The σ 1 axis in the layer between 20 and 25 km also dipped about 30, but with a higher misfit. Between the shallow and deep zones (12 25 km depth) the stress inversion of mechanisms had a high misfit anddidnotreveala reliable set of the three principal stress axes. This zone of weak stress alignment coincides with the neutral plane of flexure. Thus, the stress field from inversions of focal mechanisms, when rotated into radial coordinates, shows a strongly radial pattern between 9 and 12 km (T axis) and between 25 and 48 km depth (P axis). We will later compare the stress orientation with a simple Timoshenko flexure model of a downward flexing plate and find good general agreement Stress Model It is beyond the scope of this paper to produce a theoretical or numerical 3-D elastic model of a lithosphere with a broken center and a gravitational load based on volcano topography. But a simple 2-D model of the free edge of a plate with a vertical load at its end is informative and fits the data surprisingly well. We seek a theory of stresses within a bending plate to compare with the stresses and P and T axes of earthquake focal mechanisms. The flexural bending of the lithosphere under Hawaii has conventionally been treated with a thin plate theory [e.g., Watts and ten Brink, 1989; Watts, 2001]. The thin plate theory provides a good determination of the effective elastic thickness and a fittothewidthofthehawaiiantroughandarchbutdoesnotenumerate internal stresses. A point force acting on an elastic half space is the traditional Boussinesq problem [Timoshenko and Goodier, 1934] that has been treated as the Green s function for loads applied to the Earth s surface[e.g., Farrell, 1972]. An analytical solution of 3-D stresses within a broken plate of finite thickness under a concentrated vertical load may not exist, but a simpler 2-D geometry may qualitatively match the load under the Big Island and illustrates key features of the stress field. The geometry of a downward bending plate which is broken under the center of the load but is fixed at a distance far from the center of the island is what we consider. Timoshenko and Goodier [1934, p. 42] provide a solution for the bending of a horizontal cantilever loaded with a vertical load at one end and fixed at the other end. Timoshenko and Goodier s solution is two dimensional (y for depth and x for horizontal distance from the edge of the plate) and does not have the circular 3-D geometry of Hawaii. But the stresses in a vertical and radial plane of this model exhibit features like a neutral plane with extension above and compression below that are useful for Hawaii. The load P is directed downward in the direction of +y at the x = 0 free end of a beam. The beam is of thickness 2c with the y =0 horizontal line at its center and free surfaces at y = +c and y = c. The beam is held fixed at a distance x = L from the load, where L>>c. No other forces are applied to the surface of the beam, though a uniform layer of rock at the surface y = c would create a compressional stress σ y within the beam, and body (gravity) forces within the beam would increase σ y with depth. The Timoshenko and Goodier result is simpler to express in terms of the moment of inertia of the cross section of the cantilever I = (2/3)c 3. Here the stress σ is positive in compression and negative in extension and follows the geologic convention rather than Timoshenko s engineering convention. The shear stress τ is positive for left lateral and negative for right lateral. Then the 2-D solution, which is satisfactory at considerable distances from the ends of the beam, is σ x ¼ Pxy=I σ y ¼ 0 τ xy ¼ ðp=2iþðc 2 -y 2 Þ KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2415

17 Figure 10. The inset shows a cantilever beam with the left end fixed and the right end free that is a 2-D simplified analog along a radius through the Hawaiian lithosphere. The main figure is a schematic cross section of stresses in the interior of this slab of half width c, undergoing a downward point force P at the free x = 0 end. The left end at some large x is held fixed. The neutral plane separating extensional from compressional regimes is shown as a dashed line. The slab or cantilever bends concave downward, and the normal σ and shear τ stresses within it are calculated by the equations of Timoshenko and Goodier [1934, p. 42]. Double arrows show the sense of shear stress and slip on horizontal faults. Shown are the various stress functions that vary with depth within the slab. These stress functions (without a body force σ y ) are for some fixed point x within the slab. The plotted internal stress functions are to the left, and the value of the dip θ of a principal stress at five depth points is in the center. The stress orientation symbol is shown at five depths within the slab; note that it rotates 90 from top to bottom. The data stress orientations were derived by Ross et al. [2007] from focal mechanisms for two depth ranges above and below the neutral plane. The correspondence between the two model and observed dips is very good. The focal mechanism tectonics under Hawai`i generally have radial T axes above the neutral plane and radial P axes below the neutral plane. Horizontal faulting (with the upper block moving away from the island center) is seen at several depths including the Kilauea deep seismic zone near 30 km depth and at the bases of the mobile volcano flanks at about 9 12 km depth. Introducing a body (gravitational) force σ y (light dashed line, not to scale) would either move the neutral plane up or increase the slab thickness. The approximate depths under Hawai`i are 21 km for the neutral plane and about 40 km for the slab thickness ignoring body (gravity) forces or perhaps about 60 km for the slab thickness considering body forces. The 60 km is the maximum earthquake depth where the lithosphere becomes ductile. The horizontal σ x stress is zero at the neutral plane y = 0, extensional above the neutral plane where y < 0, and compressional below where y > 0. The shear stress τ xy is a maximum at the neutral y = 0 plane and 0 at the top and bottom free surfaces. If the load is to the right (south) end (at x = 0) of a N-S south Hawaii cross section, then the shear stress is left lateral (when viewed from the west) with the upper part of the beam being sheared northward in the positive x direction. This geometry is analogous to the north side of Hawai`i Island with a load at the islands center where the beam is broken. If the load is to the left (north) end (at x = 0) of a N-S south Hawaii cross section, then the shear stress is right lateral with the upper part of the beam being sheared southward in the positive x direction. This would represent the south side of Hawai`i Island. We can calculate the orientation of the principal stresses σ 1 and σ 2 in two dimensions in the Timoshenko model, to compare with the orientation of stresses determined from focal mechanisms. σ 1 is the most compressive principal stress. The dip of a principal stress θ (from Mohr s circle arguments like those in Jaeger and Cook [1969], for example) is tanð2θþ ¼ 2τ xy = σ x σ y We are only dealing with the orientations of stresses and not their magnitudes and are only looking at a cross-sectional plane where the least compressive stress σ 3 of the focal mechanism inversion becomes σ 2 in two dimensions. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2416

18 Figure 10 shows a cross section of a slab with a point load P. The inset shows a cantilever beam and the main figure amplifies stress functions and the orientations of σ 1 and σ 2 within it. The left end extends to an unloaded region far from the load. This geometry is analogous to a cross section of north Hawaii seen from the west looking east, much like the left half of Figure 2c or Figure 9c. Superimposed on Figure 10 are the various 2-D stress functions that vary with depth within the slab. These schematic functions are for some point x within the slab and are not meant to change along its length as it may appear in the figure. Then in simplified form, σ x ~ y and τ xy ~(c 2 y 2 ) as plotted on the left side of Figure 10. The dip θ of a principal stress at five depth points is in the center of the figure. The stress orientation symbol rotates 90 from top to bottom. Next to the model stress orientations are the two stress orientations that Ross et al. [2007] derived from focal mechanisms above and below the neutral plane. The stress orientations in the Hawaiian lithosphere matches the 2-D bending slab stress model very well, but the top and bottom of the slab are not free surfaces, have other influences on stress, and are more complicated. We believe that the simple bending slab stress model explains the first-order stresses driving deep Hawaiian earthquakes. Qualitatively, we can see the effect of body (gravity) forces and surface load forces on the bending Timoshenko model. The internal weight of the slab increases the vertical compressive stress σ y from 0 at the free surface to approximately the weight of rock above a given depth. The component of the horizontal compressive stress σ x also increases with depth due to body forces, but only at a fraction of the σ y increase. Linear elasticity implies that σ x (body) = σ y (body)/3 for a typical Poisson s ratio of For a fixedslabthickness,thedepthoftheneutral plane decreases because σ x and σ y become equal at a shallower depth with the body force added. Alternatively and more realistically, if we fix the neutral plane at 20 km depth, the slab thickness must increase to larger than 40 km, perhaps to 60 km or so. We see radially compressive earthquakes to about 60 km depth (Figure 2b), and elastic flexure probably extends to about that depth. Including a body (gravity) force indicates the slab thickness below the neutral plane is likely greater than the thickness above it, as observed. 6. Discussion 6.1. The Flexing Lithosphere Is a Broken Plate A flexing lithosphere that is concave downward because it is broken in the center is the simplest explanation for the seismicity and earthquake stress observations. The stress distribution within a two-dimensional cantilever plate that is vertically loaded at one end, even though it is a simple 2-D model, generally explains the earthquake stress distribution if the plate is broken. The generalized models of a load on the lithosphere (Figure 11) (taken from Watts [2001, p. 134]) all have concave upward curvature near the load unless the plate is broken or curves so sharply in the center that it acts like a break. The center of the island below the volcanic pile is nearly aseismic below 15 km depth and within km distance of the stress center near Mauna Loa, except for Mauna Loa s vertical magma conduit (Figure 2). In this aseismic region it is impossible to identify a rigid plate and determine stress within it from earthquakes. The active and hot conduit of Mauna Loa, combined with the gravitational load, have broken the lithosphere and made a relatively warm area that is free from stress. The P wave velocity structure from tomography reveals a low velocity structure [Tilmann et al., 2001, their Figures 4a and 4b] that coincides very well with the aseismic zone and stress center. The lowest velocity in their 35 km depth layer is under the summit region of Mauna Loa, and the slowness is about 5% compared to the NE and SW edges of the island. The 35 and 50 km depth layers show a velocity low extending NW in the direction of the island chain. The lowest velocities are in the center of the island and thought to be thermal in origin, which will reduce the effective viscosity and lower the strength. Farther to the northwest, the island chain moves away by plate motion from the intensely active volcanoes. An elastic plate with an elastic thickness of about km explains the lithospheric structure, the Hawaiian moat, and flanking bulge between Oahu and Molokai where the studies have been made [e.g., Watts and ten Brink, 1989]. There are 2 Myr of plate motion and cooling between Mauna Loa and Oahu. It is not clear whether the cooled plate behaves as broken or continuous, but modeling of geophysical data suggests an intermediate case between the two [Walcott, 1970; Watts and Cochran, 1974; Watts and ten Brink, 1989; Wessel, 1993]. Seismicity diminishes markedly moving northwest along the island chain [Klein et al., 2001] KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2417

19 Figure 11. Schematic drawings of a load on the oceanic lithosphere (from Watts [2001, p. 134]). All have concave upward curvature near the load unless the plate is broken (case c with arrow) which produces downward concave curvature and opposite internal stresses. Te is the effective elastic thickness, and case d is for underplating of magma under the island center. indicating that new load stresses are not being added, and stress relief is taking place. Erosion and subsidence of the islands also redistributes and broadens, as the lithospheric load moves to the northwest. We hypothesize the stress pattern within the plate also changes as it cools, but a search for earthquakes large enough to have known moment tensors revealed no earthquake stress data from deep earthquakes northwest of Hawaii. It seems likely that the downward curvature formed during shield building remains frozen into the lithosphere, and the curvature of seismic reflectors near Oahu may reflect this curvature [Watts and ten Brink, 1989]. The lithosphere would not have to change composition as the plate moves to the northwest, but the internal ductility and stress would change to that of a more rigid lithosphere that may be intermediate between broken and unbroken. A greatly thinned plate may act as if it is broken. Ten Brink [1991] derived the relation between lithosphere thickness and distance to the inflection radius (with relative tension above the neutral plane outside the inflection radius, and compression inside) from a suite of worldwide volcanoes. The observed inflection radius in the case of the island of Hawai`i sislimited by the 30 km radius of the aseismic circle (Figure 13) and where the radial compression in the lower lithosphere is dominant (Figures 6f and 6g) or about 40 km. This 40 km maximum inflection radius predicts that the elastic plate thickness directly under Hawaii is very small, about 16 km. Thus, the elastic lithosphere under the center of the island may be so thin as to be effectively broken. In addition to a broken lithosphere, there are some indications that the lithosphere under Hawaii is less rigid or thinner in the region surrounding the lithospheric stress-free hole. Though the stress model developed here of a two-dimensional elastic plate loaded near its free end matches the stress patterns revealed from earthquakes and is an acceptable first-order model, it does not include the three-dimensional effects of a hole in the plate or the finite width of the load. One can visualize the edge of the lithospheric hole deriving some strength from being surrounded on all sides by brittle lithosphere in 3-D, which would not be a factor in two dimensions where the unsupported edge extends to infinity. If the lithosphere around the hole under Hawaii were as rigid and thick as the rest of the Pacific plate, the plate as a whole would probably subside under the load but with upward curvature as if the hole were not there as in Figure 11a. Our hypothesis based on P and T axis orientations is a downward curving plate near the lithospheric hole. Timoshenko and Woinowsky- Krieger [1959, chapter 3, section 17] consider several theoretical elastic cases of a hole in a loaded circular plate which show the same upward curvature as if the hole were not there. Those theoretical cases assume the plate has the same thickness and elastic rigidity right up to the edge of the hole. This comparison suggests that the KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2418

20 Figure 12. Schematic NW-SE cross section through Hawai`i Island showing lithosphere bending and stress directions inferred from earthquakes. Depths of features are approximate, but the vertical exaggeration is about 2. The volcanic edifice overlies the brittle lithosphere (shaded gray), separated by a layer of horizontal faulting (thick yellow line) which in the south is referred to as the décollement. The volcanic edifice shown with satellite imagery on the topography is seismically very active but is not shown with any faults or seismicity in this diagram. Earthquakes in the brittle lithosphere approximately extend as deep as the fine dashed line, which is about 60 km deep in the south. The gray areas of brittle lithosphere show the interpreted places of maximum bending corresponding to a ring of seismicity surrounding the island center. The areas of brownish shading in the gray lithosphere represent areas of more intense seismicity from Figure 2c. The center of the island is mostly aseismic below about 15 km depth (except for a unique swarm in Mauna Loa s magma conduit), and this aseismic zone is interpreted as the weak or broken center of the plate. The island center and the underlying asthenosphere are shown dark orange which indicates ductility and lack of earthquakes but is not meant to show a large magma body. The orange tubes schematically show generalized magma conduits feeding Mauna Loa and Kilauea. The magma conduits are probably narrow, and their location under Kilauea is indicated by short- and long-period earthquakes and passage through the 30 km deep earthquake zone. The approximate elastic thickness of the bending plate is shown with a black dashed line and is inferred from the shape of the Hawaiian arch and moat outside this section. The elastic thickness is not a structure revealed by Hawaiian earthquakes. The elastic thickness is just an approximation for a plate that becomes gradually softer with depth at is base. The neutral plane (black solid line) is aseismic under most of Hawai`i and marks a change from radial extension above to well-defined radial compression below. The compressional stress directions defined from P axes of earthquake concentrations are shown with converging red arrows, and extensional stress from T axes is indicated with diverging blue arrows. Horizontal faulting, with movement of the overlying volcanoes away from the island center, is characteristic of the décollement and the 30 km deep Kilauea seismic zone, where weak horizontal planes slip laterally. The location of the Kiholo (M6.7, and M6.0) and Kalapana earthquakes (M s 7.2 and M w 7.7) are shown. The dominant stress feature is radial P axes below the neutral plane interpreted as downward bending under the island load. The seismic data on which this schematic interpretation is based are taken from Figures 2c and 9c. The Hawai`i graphic is from Google Earth using Landsat imagery. lithosphere surrounding the hole under the island of Hawai`i is weak and can only bend downward as we propose because it is hotter, thinner, and more depressed than the surrounding lithosphere. These arguments are qualitative and a numerical stress model of a loaded lithosphere with a hole and variable thickness or rigidity will be required to completely model lithospheric stress under the island of Hawai`i. KLEIN LITHOSPHERIC FLEXURE UNDER HAWAII 2419