Introduction

Dike swarms are present in all basaltic volcanic systems, but in only a few cases erosion allows us to view them directly. Much of what is known about those swarms, comes from the monitoring of ground deformation and seismicity in active rift zones. Accordingly, many basic questions about the emplacement of these intrusions have not been resolved, like the intrusion geometry and distribution of dikes swarms, the variations in the dike population and the swarm intensities at varying distances from the volcanic focus. How magma is partitioned between dikes and sills is still unclear. Our long-term general objective is to make a comparative study of the dike characteristics, from quantitative field measurements, anisotropy of magnetic susceptibility (AMS) and rock magnetism data of the intrusive complexes in Hawaii. Today we know that, during the development of Hawaiian volcanoes, a relative rapid growth occurs as well as a volcanic flank collapse of the central volcano due to endogenic and exogenic processes (Voight and Elsworth 1997; Keating and McGuire 2000; Reid et al. 2001; Rubin and Pollard 1988; Elsworth and Day 1999; Silva et al. 2012, 2018). These large mass wasting phenomena have been identified and studied in great detail on a global scale (e.g., Lipman et al. 1988; Moore et al. 1989; Presley et al. 1998; Day et al. 1999; Moore and Clague 2002; Yokose 2002; Coombs et al. 2004; Hildenbrand et al. 2006; Battaglia et al. 2011; Costa et al. 2015).

In the last three decades, it has been clearly demonstrated that paleomagnetism and rock magnetism of magmatic rocks have mainly advanced the knowledge in geophysics, geology and volcanology. They provide a correlation or indirect dating tool to document the tilting and rotation of the rocks, and a way of assessing the thermal history of rocks. Flow direction in dikes is difficult to determine, particularly in fine-grained rocks such as basalts. Those determinations are generally based on the analysis of the preferred orientation of the contained minerals. Various approaches, based on texture analysis (Leiss et al. 2000), have been attempted for basaltic lava flows, but the interpretation remains difficult and disputable because of the nearly isotropic distribution of small grains in volcanic rocks. This is why the AMS technique is now frequently used to study the flow of lavas and dikes in general (e.g., Hargraves et al. 1991; Ellwood 1978; Fanjat et al. 2012; Féménias et al. 2004; Staudigel et al. 1992; Raposo 1997, 2011; Raposo and Ernesto 1995; Moreira et al. 1999, 2014; Rochette et al. 1999; Aifa et al. 1999; Callot et al. 2001; Herrero-Bervera et al. 2001, 2002a, b; Archanjo et al. 2002; Callot and Geoffroy 2004; Cañón-Tapia 1996, 2004; Aubourg et al. 2008; Silva et al. 2004, 2008, 2010, 2014, 2018; Chadima et al. 2009; Delcamp et al. 2014; Hrouda et al. 2015).

Due to the collapse of half of the Wai’anae volcano in Hawaii (Fig. 1), the internal part of this volcano is now exposed, revealing dike swarms. For the inner part of this volcano, studies concerning the emplacement of these dikes are aimed at providing fundamental new insights into the origin of these dikes and the flow direction of the magmas.

Fig. 1
figure 1

a Simplified view of Oahu with Digital Elevation Model (NOAA—Digital Elevation Models: https://www.ngdc.noaa.gov/mgg/coastal/coastal.html) and the two main volcanic structures, the Koolau and Wai’anae volcanic ranges. The sampling sites area over the Wai’anae volcano is indicated, b elevation contour map of the Wai’anae volcano, dike orientations (adapted from Zbinden and Sinton 1988) and site locations of this study

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An AMS study of seven pilot dikes sampled inside and outside the Wai’anae caldera was then performed. It was complemented with a detailed analysis of the magnetic properties of the samples, to look for possible magnetic mineral alterations that would prevent a scientifically sound interpretation of the AMS. Knowledge of the dike flow emplacement of the seven pilot dikes in the Wai’anae volcano helps to understand the inflation and flank collapse processes that affected the volcano.

Previous geologic works

The petrology of the volcano was studied, determined and published with the original geological map by Stearns (1940) and McDonald (1940). The geology of the Wai’anae volcano has been described in detail by Sinton (1986) and briefly summarized by Herrero-Bervera and Coe (1999). The Wai’anae volcano has also been mapped by Sinton (1979, 1986) by means of structural, geochemical and lithologic tools to determine the map units. The first paleomagnetic study of the Wai’anae volcano was published by Doell and Dalrymple (1973) who provided reliable K–Ar age determinations, indicating that the subaerial volcanic activity took place before 3.8 Ma to approximately 2.4 Ma. The magnetostratigraphy and the dating of the Gauss–Gilbert, lower and upper Mammoth geomagnetic reversals were studied by Herrero-Bervera and Coe (1999) and Herrero-Bervera and Valet (1999, 2005). These radiometric and paleomagnetic studies corroborate the fact that Hawaiian volcanoes were built very rapidly, with an eruption rate averaging 1000 years as with the flows of Mauna Loa and Kilauea volcanoes. Today, we know the Mauna Loa volcano on the Big Island of Hawaii has erupted 33 times since 1843, which yields an average of one eruption every 5 years (Trusdell 2012). The first study of the dike swarms emplaced on the Wai’anae volcano was published originally by McDonald (1940) and subsequently by Zbinden and Sinton (1988). Figure 1 depicts the distribution and orientation, as well as the location of the sampling sites of the dike swarm both inside and outside the caldera.

Sampling

Seven thin pilot dikes were selected in three areas, called hereafter sites 1, 2 and 3 following a NNW–SSE transect, west of the deep internal part of the Wai’anae volcano and outside the caldera (Fig. 1). Area 1 is located close to Kaena Point at the NW limit of the Kamai’leunu Member, which is the oldest member of the Wai’anae volcanic stratigraphy within the Gauss Chron (e.g., Herrero-Bervera and Valet 2005) of the Wai’anae volcanics. The dikes in this area are usually slightly weathered and show steady directions from NW–SE to NNW–SSE. A single dike (DK6) presenting non-weathered chilled margins was sampled. Area 2, nearly 8 km southeast of Area 1 and also belonging the Kamai’leunu Member, is located in the southern border of Makua Valley. Most observed dikes show a NW–SE to WNW–ESE trend. Three dikes from this area were selected (DK2, DK3, DK4). Dikes DK2 and DK3, located close to Kaneana Cave (Fig. 1b) are parallel, E–W trending and only separated by a few meters. Field relationships suggest that several magma pulses were emplaced in these dikes. Dike DK4 is trending roughly N–S with moderate dip (i.e., 46°; Table 1). Area 3 is located in the central part of the Wai’anae volcano close to Maili Point, roughly 16 km southeast of Area 2. The dikes here show a predominant NNE–SSW trend. Three dikes (DK1, DK5 and DK7) were sampled. We note that dike DK5 has a complex shape with a local inclusion of host rocks. The average thickness of the dikes in the three areas is about 0.4 m up to 2.5 m (see Table 1).

Table 1 Geometry of the dikes and mean AMS data
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For magnetic fabric studies, it is very important to study imbrication angles that are observable on the dike margins. Most of the samples have to be drilled on these borders (e.g., Cañón-Tapia and Herrero-Bervera 2009). The samples for this study were drilled following a geometry that resembles an “H” letter. The vertical strokes of the H, on the dikes borders, are parallel to the dike’s margins with a small spacing between the samples (for analysis of the imbrication angles). The horizontal stroke of the H cuts across the dike, with a larger spacing of the drilled specimens for study of the magma flow within the dike. Letters W, M and E, associated with the name of the dike, will correspond to samples from the Western margin, Middle part and Eastern margin, respectively (DK2M for example). The drilled cores were oriented both by means of magnetic and sun compasses. Only the sun compass orientation of the samples was used for the AMS study.

Rock magnetism

In order to characterize the magnetic mineralogy and associated grain size of the studied dikes, several rock magnetic experiments have been performed. These data are key information for the magnetic fabric interpretation. The eventual presence of several magnetic minerals and/or magnetic grains with different sizes is factors that can significantly affect the AMS.

Magnetic susceptibility experiments

The mean susceptibility [Km = (Κ1 + Κ2 + Κ3)/3] has been calculated from the AMS data that will be presented in “Anisotropy of magnetic susceptibility (AMS) determinations” section. The average value of the mean susceptibility for all dikes is 15.5 ± 13.8 10−3 SI units, corresponding to a large scatter in susceptibility values for the sample scale with a bimodal distribution (Fig. 2d), indicating possibly heterogenous magma sources in two populations of samples. Dikes DK4, DK5 and DK7 show low mean susceptibility values (Km < 10 × 10−3 SI). Dikes DK2 and DK6 display the highest susceptibilities (Km > 30 × 10−3 SI) and dikes DK1 and DK3 have intermediate susceptibilities (Fig. 2a). The susceptibility variation across the dikes is diverse according to the studied dikes. In dikes DK1, DK2, DK3 and DK5, the mean minimum Km occurs in the middle of the dike. Moreover, on the western margin of DK2, the six samples, close to the margin, show susceptibility values about half that of the other samples of this dike. No particular indication of stronger weathering (see Krasa and Herrero-Bervera 2005) appears in these samples. They could therefore correspond to another intrusive phase or to local contamination by melted xenoliths. It is also possible that the Km is lower due to an heterogeneity of the magma, i.e., a heterogeneous distribution of the magnetic minerals. These six samples will be considered separately in this paper and referred to as site DK2WL. For the two non-vertical dikes (i.e., DK4 and DK7), the minimum mean susceptibility corresponds to the upper part of the dikes. The susceptibility variation across each dike is weak, except in DK3 where samples from the center of the dike show values of susceptibility much lower than those close to the margins (Fig. 2a). Such a strong variation indicates a very heterogeneous body, possibly reflecting multiphase intrusions. For dikes yielding weak susceptibility variation between the core and margin (e.g., DK4, DK5 and DK7), the variation could correspond to slightly different grain sizes related to faster cooling on the dike border and/or to slightly different weathering degrees in the different parts of the dike.

Fig. 2
figure 2

AMS scalar parameters. a–c Mean values of Km (in 10–3 SI): P′, and T in the western (W), middle (M) and eastern (E) parts of the studied dikes, d distribution of Km for all the specimens, e P′T diagram (Jelinek 1981) for mean values in different parts of the dikes (full squares and open symbols correspond to the marginal and central parts of the dikes, respectively; see b, c for details)

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Thermomagnetic K(T) curves

Continuous and partial thermomagnetic cycles of low-field magnetic susceptibility versus temperature K(T) curves were determined for different samples, at least one for each studied dike (curves are shown on Fig. 3 for three selected samples). The experiments were performed in a CS4 furnace coupled to a MFK1-FA Multi-Function Kappabridge, operating at a low field of 200 A/m and a frequency of 976 Hz. The susceptibility was measured from room temperature up to a maximum of 620 °C, with a heating and cooling rate of 9 °C/min, and under an Argon atmosphere using the instrument at the Petrofabrics and Paleomagnetics Laboratory of HIGP-SOEST of the University of Hawaii as well as an identical instrument at the Instituto Dom Luís (Lisboa), Portugal.

Fig. 3
figure 3

Selected plots of magnetic susceptibility versus temperature (T) for three samples DK3 (above), DK7 (in the middle) and DK2 (below). First, heating–cooling cycles (left plots, a, d, g) and second heating–cooling cycles (central plots, b, e, h). The susceptibility values are normalized relative to the initial susceptibility at room temperature (ki, given on the left diagram). Right plots (c, f, i) show the partial heating–cooling cycles

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The Curie temperature (Tc) was estimated according to the Petrovský and Kapicka (2006) approach, which proposes that, if a sharp Hopkinson peak is present, the temperature corresponding to the peak is a good estimate for the Curie temperature. However, the K(T) curve shows a wide maximum with a smooth increase and decrease in the susceptibility over a large temperature range. The best estimate for the Curie temperature should then be then the temperature of the observed inflection point in the decreasing branch of susceptibility.

All samples show irreversible K(T) curves (Fig. 3a, d, g), which indicate the occurrence of mineral alterations during the heating phase (hereafter called “first” cycle). A “second” heating–cooling cycle was performed in all samples with the powder already used in the previous cycle. The heating–cooling curves of these second cycles show a practically reversible process as observed in the sample from DK2 (Fig. 3h), or very similar paths as observed in the samples from DK7 and DK3 (Fig. 3b, e).

“Partial” thermomagnetic cycles of K(T) were also performed for all samples. For each partial cycle, the powder that had been processed in the previous lower temperature cycle was used. These experiments revealed reversible behavior of the susceptibility heating and cooling curves for temperatures lower than 250–300 °C (Fig. 3c, f). The low-temperature hump (T < 300 °C), observed during the heating process in all samples except sample from DK2, is then related to a stable magnetic component that is not altered during heating up to 300 °C. With increasing temperature, another hump is observed between 450 and 500 °C in “first” (Fig. 3a, d, g), as in partial cycles (Fig. 3c, f, i). All K(T) curves up to 610 °C (first, second, and partial cycles) yield a Curie temperature Tc = 560 °C, but that could correspond to a primary mineral as well as a magnetic phase resulting from mineral alteration during the heating. For temperatures above 575–610 °C, all samples show virtually zero magnetic susceptibility. The cooling curve is complex, showing multiple phases, and the final magnetic susceptibility of the first cycles, after the cooling process, is always slightly higher than the initial magnetic susceptibility before the heating (Fig. 3a, d, g). A sample from DK2 (Fig. 3g) shows a different behavior with a continuous (hyperbolic type) rise of the K(T) heating curve up to a sharp peak that occurs at 480 °C, followed by an abrupt decrease in susceptibility. The cooling curve for the partial loop up to 510 °C suggests that this decrease is related to mineralogical alteration and that the sharp peak at 480 °C during heating is not related to the Hopkinson effect (e.g. Herrero-Bervera and Valet 2009) . All other samples (Fig. 3a–d) show a Hopkinson peak at temperatures between 90 and 180 °C. With increasing temperature, mineralogical alteration occurred after 300 °C heating.

An interesting observation can be made from partial K(T) curves (Fig. 3c, f). For cycles with a maximum temperature between 300 and 510 °C, the susceptibility strongly increases before the cooling beginning, showing an unexpected appearance of mineralogical alteration due to cooling. This is actually partially related to the relatively fast heating rate (Jordanova and Jordanova 2016), but also because the minerals formed during heating have Curie temperatures lower than the temperature during their formation. Such behavior clearly appears on the subsequent cycles with a higher maximum temperature, where the heating curve is similar to the cooling curve of the previous cycle. This highlights the major interest of the partial K(T) experiments (e.g., Henry 2007).

The analysis of the thermomagnetic curves suggests that most samples display a mixture of magnetic phases.

  • One phase is revealed by the low-temperature Hopkinson peak observed in the K(T) curves and should be a Ti-rich titanomagnetite. Indeed, titanomagnetites (Fe3−x Tix O4) of the TM60 composition (stoichiometric for x = 0.6) have Curie temperatures of ≈ 150–200 °C. On samples from DK1, DK3 (Fig. 3a), and DK4, the estimated values of Tc around 225–250 °C point to a titanomagnetite with x ≈ 0.4, while for DK5 the lowest value of Tc ≈ 160 °C suggests a value of x ≈ 0.6. These minerals are clearly primary, as evidenced by the reversibility of the K(T) curves for low temperatures.

  • After heating at temperatures higher than 250 °C, the cooling curves (first, second and partial heating–cooling cycles) show a susceptibility increase, which indicates the formation of a mineral of high susceptibility. The partial K(T) experiments show that the newly formed minerals correspond to a wide spectrum of Curie temperatures. That suggests conversion of titanomaghemite into titanomagnetite possibly associated with an exsolution phenomena and grain size variation. A second magnetic phase in our samples should therefore be titanomaghemite.

  • A third phase is associated with the high-temperature reversible variations that results in Curie temperatures of 525–550 °C, and points to the contribution of Ti-poor titanomagnetite, possibly produced from the conversion of Ti-poor titanomaghemite (Dunlop and Özdemir 1997).

IRM/SIRM acquisition curves

Uniaxial isothermal remanent magnetization (IRM) was induced in four specimens (one sample per dike for DK2, DK5, DK6 and DK7). The specimens were first demagnetized by an alternating magnetic field at 100 mT. Magnetization measurements were performed with an AGICO JR-6 Spinner Magnetometer at regular intervals when samples were further submitted to a progressive uniaxial increasing DC magnetic field up to 1.2 T (saturation of isothermal remanent magnetization—SIRM—acquisition), and later when they were submitted to a progressive uniaxial increasing DC magnetic field in the opposite direction of the previous SIRM (back-field “demagnetization”). Fields were applied with an ASC Scientific IM-10-30 Pulse Magnetizer. The obtained IRM acquisition curves and associated backfield demagnetization curves provide information on the magnetic mineralogy. Low-coercivity phases such as magnetite, maghemite, greigite or ferromagnetic pyrrhotite grains are characterized by saturation for low applied fields (< 0.3 T). Higher coercivity phases such as hematite, goethite, and ferrimagnetic pyrrhotite (which reach saturation field of ~ 500 mT and not higher than 1.0 T) do not reach saturation until fields higher than 1.0 T. Moreover, the SIRM intensity gives an indication about the relative concentration of these magnetic minerals in a given volume. According to Robertson and France (1994) and Kruiver et al. (2001), the IRM acquisition curves theoretically follow a log-normal distribution and are cumulative in intensity. The obtained magnetization curves can therefore be described by three main parameters: (1) SIRM intensity; (2) field at which half of the SIRM is reached (B1/2); and (3) a dispersion parameter (DP), which reflects the distribution of the coercivities of the magnetic mineral phases and characterizes the homogeneity of the population in terms of grain size and composition. The IRM acquisition process can be visualized in a diagram with the log applied field in the abscissae scale and the magnetization acquisition or the gradient magnetization acquisition in a linear ordinate scale, referred to as LAP (for the linear acquisition plot) or GAP (for the gradient acquisition plot) diagram. These diagrams reveal the occurrence and contribution of magnetic components with different coercivities (i.e., different “stages” in the IRM acquisition curve) and the heterogeneity of the magnetic assemblage of phases (width and overlapping of the peaks) in the total IRM. In our samples, results show small differences including saturation being reached in fields lower than 0.4 T, and even lower than 0.2 T for one sample (Fig. 4a). The remanent coercive force (HCR), obtained from the backfield demagnetization curves, has low values, varying from 19 to 50 mT (Fig. 4a, inset). The SIRM intensities show also small variations, spanning between 300 and 750 A/m.

Fig. 4
figure 4

IRM/SIRM acquisition curves. a Plot of IRM acquisition and backfield demagnetization curves (inset) for four representative samples, b, c linear acquisition plot (LAP) and gradient acquisition plot (GAP) for two representative samples

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The LAP and GAP (Fig. 4b, c) reveal two components: one of low coercivity (25 mT < B1/2 < 56 mT) that usually represents 95% of the total coercivity, and the other component of higher coercivity (100–300 mT) representing only 5% of the total coercivity (only the sample from DK7 shows a slightly different high coercivity contribution reaching 19%). That could correspond to titanomagnetite and titanomaghemite, respectively.

Hysteresis properties and magnetic grain size determination

To determine hysteresis parameters (Mrs, Ms, Hcr and Hc) and magnetic grain sizes, we prepared powder samples of approximately 200 mg of seven specimens representative of the seven studied dikes. The hysteresis loop measurements were performed on a variable field translation balance (VFTB) with a measuring range of 10−8–10−2 A/m of the Petrofabrics and Paleomagnetics Laboratory of the HIGP-SOEST, University of Hawaii.

A wasp-waisted shape appears at least for part of the obtained curves, confirming the presence of several magnetic phases. As depicted by the magnetic granulometry diagram (Day et al. 1977) modified by Dunlop (2002) presented in Fig. 5. Only specimen number 5 should be of SD size (the sample shows a high coercivity with respect to the other wasp-waisted specimens, see Additional file 1). Most grain sizes are scattered within the PSD range, close to the mixing curves between SD and Multi-Domain (MD) grains. The presence of both SD and larger grains suggests the presence of both normal and inverse fabrics (e.g., Potter and Stephenson 1988; Rochette 1988; Hrouda and Ježek 2017; see also Table 1), possibly resulting in composite “abnormal” fabrics. It is, however, very important to point out that domain limits in this plot apply to pure magnetite only. In our case, magnetic mineralogy corresponds to at least two phases, including titanomaghemite and Ti-rich titanomagnetite. The analysis of the thermomagnetic curves suggests that most samples display a mixture of magnetic phases resulting in a variety of different magnetic fabrics.

Fig. 5
figure 5

Results of the hysteresis experiments performed on seven representative samples displayed in a Mrs/Ms versus Hcr/Hc, Day plot with limits between single domain (SD), pseudo-single domain (PSD) and multi-domain (MD) grains after Dunlop (2002). The figure also shows mixing curves calculated for magnetite using the equations developed by Dunlop (2002). Numbers along the curves are volume fractions of the soft component (SP or MD) in mixtures with SD grains. The dikes analyzed are DK1; DK2; DK3; DK4; DK5; DK6 and DK7. The plot shows green numbers 1, 2, 3, 4, 5, 6 and 7 corresponding to the seven dikes studied

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Anisotropy of magnetic susceptibility (AMS) determinations

The AMS measurement yields a susceptibility ellipsoid of principal susceptibility axes Kmax, Kint and Kmin (Kmax ≥ Kint ≥ Kmin) and mean susceptibility Km = (Kmax + Kint + Kmin)/3. Therefore, the intensity and shape of this ellipsoid are characterized Jelinek’s (1981) parameters P′ and T. The parameter P′ (corrected degree of anisotropy) is, at the grain scale, controlled by the geometry and orientation of the crystallographic system. P′ expresses the departure from a spherical AMS ellipsoid (P′ = 1). The degree of anisotropy measures the strength of the preferred orientation of the magnetic minerals in a rock. The parameter T characterizes the shape of the AMS ellipsoid. If 0 < T < + 1 the AMS ellipsoid is oblate (the magnetic fabric is planar); T = +1 means that the AMS ellipsoid is rotationally symmetric (uniaxially oblate). If − 1 < T < 0 the AMS ellipsoid is prolate (i.e., the magnetic fabric is linear); T = −1 means that the AMS ellipsoid is uniaxially prolate (e.g., Hrouda et al. 2015).

Measurements of AMS were made on KLY2 Kappabridge (AGICO, Brno) at the Petrofabrics and Paleomagnetics Laboratory of the Hawaii Institute of Geophysics and Planetology (HIGP), School of Ocean and Earth Science and Technology (SOEST), University of Hawaii at Manoa. The interpretation of the magnetic fabric at the site scale was done using the bivariate statistics with weighting by the measurement of the k precision parameter (Fisher 1953) values, therefore integrating the measurement uncertainty (Henry and Le Goff 1995), and the classical normalized tensor variability statistics (Hext 1963; Jelinek 1978).

Magnetic fabric in dikes

Today, it is well known that it is possible to identify in volcanic rocks, the predominant magnetic mineral phases (i.e., very small percentages of ferromagnetic minerals) that often control the measured magnetic fabric of mafic rocks (e.g., Stacey 1960; Khan 1962).

A good correlation between the orientation of the magnetic fabric and shape-preferred orientation of minerals, as observed in thin sections has been found in various rocks (e.g., Khan 1962; Grégoire et al. 1998). In addition, Knight and Walker (1988), who studied mafic dikes from the collapsed Koolau volcano (Oahu, Hawaii), confirmed the agreement between the orientation of the principal axes of AMS and macroscopic flow related structures (such as surface lineations consisting of parallel, millimeter-high wrinkles projecting out from the weathered chilled margins and cross sections that show imbricated sheet joints along dike margins (see for example Knight and Walker 1988, figure 3). Mafic dikes on the Isle of Skye, from the highly dissected Cuillin Hills magmatic center, also corroborate the agreement between orientation of the AMS principal axes and structures related to magma flow (Herrero-Bervera et al. 2001).

Rochette et al. (1991, 1999) and Tauxe et al. (1998) found that it is possible to distinguish between “normal” and “abnormal” magnetic fabrics of dikes. They defined the normal magnetic fabric as characterized by a mean minimum axis Kmin subperpendicular to the dike wall (Fig. 6a). An abnormal fabric could have a primary origin, related to mixtures of the different grain types (Soriano et al. 2016), to interaction of particles during dike emplacement (e.g., Aifa et al. 1999; Dragoni et al. 1997; Cañón-Tapia 2004; Cañón Tapia and Chávez Álvarez 2004; Schöbel and de Wall 2014), or be due to the presence of SD magnetite, giving, in this case, an inverse fabric (e.g., Potter and Stephenson 1988; Ferré 2002; Silva et al. 2008; Schöbel and de Wall 2014; Hrouda and Ježek 2017). Inverse fabric is characterized by mean maximum axis Kmax subperpendicular to the dike wall (Fig. 6b). The presence of both MD and SD grains of magnetite can give composite or intermediate fabric, which is between normal and inverse fabrics (Fig. 6c) (Rochette et al. 1999; Ferré 2002). A secondary origin could be related to post-emplacement modification of the fabric during cooling or under the effect of tectonic strain (Henry 1974; Ellwood 1978; Park et al. 1988). Knight and Walker (1988) founded that, along the dike margins, there is commonly an imbrication angle between magnetic foliation and the dike walls, reflecting the tilting of mineral particles (Fig. 6). The Isle of Skye dikes also showed such an imbrication (Herrero-Bervera et al. 2001). Since the work of Knight and Walker (1988), the Kmax in dikes was often used to infer the magma flow direction. However, like in lava flows (Cañón-Tapia et al. 1995, 1996; Henry et al. 2003), Kmax is not always parallel to the assumed flow direction, as shown by Geoffroy et al. (2002) who proposed to use the imbrication of the magnetic foliation to infer the flow vector in dikes (Figs. 6, 7). The flow direction is in the dike plane, perpendicular to the intersection between magnetic foliation and the dike margin; the sense of magma motion is given by the orientation of the imbrication angle that points in the same direction as the magma flow (Fig. 7). Callot and Guichet (2003) applied this methodology in dikes from the Greenland volcanic margin and confirmed that the Kmax is not always parallel to flow direction. The magnetic zone axis (Henry 1997) represents the best intersection of the magnetic foliation measured in different samples from the same site. As for the imbrication angle, it is a good indicator of the direction perpendicular to the flow direction [see Fig. 7(1), (2), (3)].

Fig. 6
figure 6

Determination of the flow direction in dikes using magnetic imbrication angles. Assuming a Newtonian laminar flow without along-plane displacement of the borders (mode I dike of Knight and Walker 1988), near a dike margin, the flow induces a strain regime characterized by simple shear. So, even with low particle concentrations the phenocrysts would interact and align their long axis predominantly along with the flow direction, in a plane parallel or making a small imbrication angle with the dike margin. Magnetic particles that form and develop between the phenocrysts will replicate the orientations and alignments of the phenocrysts. Considering a normal magnetic fabric, the magnetic foliation planes near the borders will be coincident with the particle’s imbrication planes. Therefore, the flow sense can be specified for cases with clearly visible imbrication on both dike borders

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Fig. 7
figure 7

Conceptual block diagrams and Schmidt (equal area) plots (maximum Kmax, intermediate Kint and minimum Kmin axes represented by squares, triangles and circles, respectively; projection on the lower hemisphere showing different relationships between crystal orientation, magma flow direction, and principal susceptibilities). The blocks represent a section of a north–south vertical dike with vertical upward magma flow direction. The equal area plots are from the eastern dike margins. For realism, it is considered that the crystal orientations show some, but limited, scattering around a mean direction. a “Normal” fabric Kmax on the Schmidt plot corresponds to the magnetic zone axis. b “Inverse” fabric related to the presence of SD grains. c Rolling effect of large grains with Kmax oriented normal to flow direction (within the plane of the dike, Kint and Kmin are parallel to flow direction according to the Fig. 6). Original diagrams are from Cañón-Tapia (2004)

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Application to Hawaiian dikes: the Wai’anae volcano case

In the data set obtained in the present pilot study, there is a relatively good agreement between the results of the bivariate (Figs. 8, 9, 10, 11; Henry and Le Goff 1995) and normalized tensor variability (Table 1; Hext 1963; Jelinek 1978) statistics, indicating homogeneous measurement uncertainty in most samples. To obtain information about the imbrication, the results from each margin and the middle part of the dikes are considered as three independent sites: western (W), middle (M), and eastern (E). Within each of these sites, we retained only data calculated with the neighboring mean susceptibility values. Therefore, few data with very large measurement uncertainty were rejected. The anisotropy is weak, with mean P′ values mostly between 1.01 and 1.02, reaching a maximum value of 1.05 in very few samples (Table 1; Fig. 2b). The susceptibility ellipsoid is mostly slightly oblate (Table 1; Fig. 2e) and the fabric in several dikes is more prolate in the middle part than close to margins (Fig. 2c). Uncertainty on the mean orientation of the axes is relatively large (Figs. 8, 9, 10), as is the case in these lava flows (e.g., Cañón-Tapia et al. 1995, 1996; Hrouda et al. 2015).

Fig. 8
figure 8

AMS stereoplots for dike DK6 (Area 1). Maximum (blue squares) and minimum (red circles) susceptibility axes with confidence zone at 95% (Henry and Le Goff 1995) at the western (W) and eastern (E) margins of the dike. Orientation of the dike plane is indicated by a great circle (Schmidt diagrams, lower hemisphere

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Fig. 9
figure 9

AMS stereoplots for dikes DK2, DK2L, DK3 and DK4 (Area 2). Same symbols as indicated in Fig. 8. M middle part of the dikes

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Fig. 10
figure 10

AMS stereoplots for dikes DK1, DK5 and DK7 (Area 3). Same symbols as indicated in Fig. 8. M middle part of the dikes

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Fig. 11
figure 11

Above: from dikes yielding reliable data (after rejection of abnormal fabric; see text), minimum confidence zone at 95% (Henry and Le Goff 1995). For Kmin axes(the great circle represents orientation of the dike plane, from field measurements). Below: magnetic zone axis (MZA), except for DK6 where it cannot be determined (too few normal fabrics), with its confidence zones at 95% and 63% (Henry 1997). Inferred flow direction is indicated by triangles (Schmidt diagrams, lower hemisphere). The orientation of the dike plane, suggested as materialized by the plane including the lengthening of the MZA confidence zone (red line), is slightly different from that obtained by the field measurement. The later is often only a rough estimate, particularly for the Hawaiian dikes, and the data from the MZA probably allow a more precise determination of this plane

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A major constraint could be given by the possible presence of SD magnetite grains (Fig. 5), which opens the possibility that inverse fabric plays a significant role in the measured AMS (Fig. 6b). Figures 8, 9 and 10 show that a normal fabric was obtained in most samples of all sites, except for dikes DK2W, DK2WL, DK3W, DK3E, DK6W and DK6E (all corresponding to the dike margins). These abnormal fabrics correspond to an inverse fabric clearly only at site DK3E and to intermediate fabrics in the other cases. The superposition of normal and inverse fabrics probably explain the low anisotropy in most samples (Hrouda and Ježek 2017). Surprisingly, on the Day plot (Fig. 5), the normal fabrics correspond to samples (except from DK1) with highest amount of SD grains and the abnormal fabrics to those with more MD grains, i.e., a relationship opposite to what is expected. This clearly indicates that a factor, different from the only presence of both SD and MD grains according to the Day et al. (1977) diagram, has here a role for the type of fabric. In our dikes, magnetic mineralogy corresponds to at least two phases, including titanomaghemite and titanomagnetite. Moreover, the presence of superparamagnetic (SP) grains is also possible. Some samples, in particular those with dominant SD size on the Day plot, display wasp-waisted hysteresis loops, confirming that contain several magnetic phases. The classical interpretation of the Day et al. (1977) plot (see Dunlop 2002) about grain size could be inappropriate for our samples. However, grain sizes could be not homogeneous within a dike (on the Day diagram (Fig. 5), the two samples taken far away from the dike border, such as from dikes DK1 and DK2, have clearly PSD grain size, suggesting a cooling rate effect on grain size). Similar cases have been already encountered (see discussion in Soriano et al. 2016). That will be the subject of further studies, in particular with anisotropy of anhysteretic magnetization (AARM) determination.

Note that the orientation of the principal minimum axes is very close to the dike plane in sites DK2W and DK2WL That suggests emplacement at the same time, and then that Km difference between these two sites could be due to contamination of DK2WL magma by fully melted local xenoliths. In DK6, a relative symmetry pertaining to a horizontal direction appears for the Kmin axes, suggesting a possible permutation of Kmin–Kint axes. However, Kint axes are too scattered to be used instead of Kmin to highlight an imbrication.

Discussion

AMS and implications for flow direction in the Wai’anae dike swarms

In the studied dikes, the difference in orientation of the magnetic foliation from one side of the dike to the other is mostly small, and the uncertainty on the orientation of the principal axes is large, which makes the determination of the flow direction from the simple comparison of the imbrication angle difficult. Therefore, a statistical approach is important to analyze the AMS data. For that reason, we determined the magnetic zone axis, with confidence zones at 95 and 63% (from 10,000 bootstrap samplings; Henry 1997), for each dike. To take into account the probable superposition of normal and inverse fabrics, a criterion was first applied to reject data with a clearly abnormal fabric: Sample data corresponding to angular differences between the minimum axis and the dike plane lower than 45° were not retained for determination of the zone axis. Following this criterion, for DK6, very few data represent a normal fabric, and the magnetic zone axis could not be determined.

Figure 11 presents the magnetic zone axis obtained for each dike. Confidence zones are relatively small in DK1, DK3, and DK4, but have an elongate shape in DK2 and DK5. Magnetic zone axes have a weak plunge in DK2, DK3, and DK7 (< 25°) and a steep plunge in DK1 (74°; Table 1). The magnetic zone axes have a similar orientation in neighboring DK2 and DK3. In the four dikes where Kmax axes have a relatively coherent orientation, the magnetic zone axis is roughly parallel to the Kmax axis (DK1) or perpendicular to this axis (DK2, DK4 and DK5).

Following the same selection criterion, confidence zones for minimum axes were determined (Fig. 11). In dikes DK2 and DK3, one of the margins did not give reliable results and the magnetic zone axis is actually mainly related to data from one margin and the middle dike area. In dikes DK1 and DK4, the difference between the confidence zones for the minimum axes concerns the declination, indicating a weak plunge in the flow direction. Observed imbrication allows to specify that the latter is slightly downward toward the north in DK1 and upward toward the same direction in DK4. In dikes DK5 and DK7, the difference between the confidence zones is related to inclination, which highlights a steeply plunging flow direction. In both cases, the imbrication shows that the flow is downward. All of these data suggest inflation (locally favoring downward backflow of the magma, e.g., Silva et al. 2010) of this inner part due to magma accumulation The flow sense is unclear in the other sites due to the large scattering and the presence of abnormal fabrics.

Formation of dikes within a volcano implies opening of new spaces that are filled by magma, i.e., an accumulation of new magma (e.g., Yang and Davis 1992). Such magma accumulation at different levels of central part of a volcano is favored by steep flow direction (feeding from below) within dikes (dikes DK2, DK3, DK5 and DK7), but here also by subhorizontal flow in dikes DK1 and DK4 toward the inner part of the volcano (Figs. 1, 11). All of these data suggest inflation of this inner part due to accumulation of new magma, from below but also from a lateral feeder, causing subsequent destabilization, and flank collapse of the Wai’anae volcano.

Conclusion

Magnetic fabric is a very strong tool for the analysis of the emplacement conditions of dikes within a volcano. Our sampling strategy (i.e., large number of samples and location of most of the samples at dike margins) is a key to the understanding of the Wai’anae volcano dike petrofabrics and magmatic flow directions. Thanks to this strategy, and despite the small number of studied dikes, our analysis yielded important and meaningful results that improve our volcanological understanding of the destabilization and flank collapse of the entire Wai’anae volcano. Alteration may deeply modify the magnetic fabric of dikes (see Krasa and Herrero-Bervera 2005). In the present study, magnetic experiments show that most samples have retained part of their original magneto-mineralogical composition, allowing reliable AMS data. Part of the latter are, however, abnormal fabrics, likely related to the presence of several magnetic phases. We used a simple criterion to eliminate most of the abnormal fabrics, in order to keep only reliable data. That allowed a better way to interpret the data.

The results indicated dominantly vertical magma movements around the central parts of the Wai’anae volcano. This caused inflation in this portion of the volcano. This process of dike emplacement in the Wai’anae volcano is one of the endogenetic factors that has greatly contributed to the collapse of the edifice. Other factors that could have contributed to this collapse of the volcano are (Keating and McGuire 2000): rapid accumulation of lavas, high angle of repose of the flanks of the volcano causing a lack of stability in the pile of lavas, unstable foundation, thermal alteration, edifice pore pressure, un-buttressed structure; and buried faults.

Our results are in contrast with lateral magma flows, deduced from the AMS study of dikes in other highly dissected volcanic edifices, such as the Koolau volcano (Knight and Walker 1988; Herrero-Bervera et al. 2002a, b; Walker 1986, 1987, 1990; Walker et al. 1995) and the volcanic system of the Isle of Skye, Scotland (Herrero-Bervera et al. 2001).

A better evaluation of the flank collapses of the Wai’anae edifice requires further studies, especially of AARM, but also of more AMS investigations dikes from the north, east, and west parts of the caldera (Fig. 1).