Flow banding in volcanic rocks: a record of

multiplicative magma deformation

Helge M. Gonnermann a Michael Manga b

aCorresponding author. Earth and Planetary Science, University of California,307 McCone Hall, Berkeley, CA 94720-4767, USA, email:

hmg@seismo.berkeley.edu, phone: (510) 643-8328, fax: (510) 643-9980bEarth and Planetary Science, University of California, 307 McCone Hall,

Berkeley, CA 94720-4767, USA, email: manga@seismo.berkeley.edu, phone: (510)643-8532, fax: (510) 643-9980

Abstract

Banding in obsidian from Big Glass Mountain (BGM), Medicine Lake volcano, Cal-ifornia, and Mayor Island (MI), New Zealand, provides a record with a 1/wavenum-ber power-spectral density and multifractal characteristics. The samples are compo-sitionally homogeneous, with banding defined by variable microlite content (BGM)or vesicularity (MI). In both samples, banding formation is well explained by acontinuous deformational reworking of magma and by a concurrent change in crys-tallinity (vesicularity) that is a small, random multiple of the total amount alreadypresent. Banding formation, therefore, represents a multiplicative process. We com-plement our spectral and multifractal analysis with several null-hypothesis tests andpropose repeated brittle deformation, concurrent development of textural hetero-geneity (microlite content or vesicularity), reannealing, and viscous deformation asa viable process for the formation of flow banding in these samples. A brittle defor-mational component in a simple flow geometry, for example during magma ascentin the volcanic conduit, provides a suitable mechanism for the spatial redistributionof textural heterogeneity over a broad range of length scales, enhanced open-systemmagma degassing via a temporary network of highly permeable cracks and fractures,and the development of spatially variable microlite content (vesicularity) throughvariable degassing of magma located at different proximities to cracks and fractures.

Key words: volcanology, mixing, flow banding, obsidian, fragmentation, fractalPACS: 91.40.-k, 91.50.Hw, 91.60.Ba, 83.50.-v, 83.70.Hq, 83.80.Nb, 47.20.Ft,47.50+d, 47.52.+j, 47.53.+n, 47.54.+r, 47.55.-t, 47.70-n

Text: 4802 wordsAbstract: 201 wordsEstimated print length: 15 pages.

Preprint submitted to Elsevier Science 21 October 2004

1 Introduction

Banding is common in silicic volcanic rocks and preserves a record of deforma-tion associated with magma ascent, eruption, and emplacement. Alternatinglight and dark bands ranging in width from tens of microns to decimeters areusually caused by differences in composition, crystallinity, or bubble content.The processes that lead to banding formation can be viscous magma mix-ing (e.g., [1–4]), repeated autobrecciation and reannealing during flow at thesurface (e.g., [5,6]), welding and rheomorphism of pyroclastic fragments (e.g.,[7–10]), vapor-precipitated crystallization in stretched vesicles [11], or repeatedepisodes of brittle deformation and reannealing during magma ascent in thevolcanic conduit [12–15].

1.1 Textural heterogeneity

As predicted from theoretical considerations for chaotic fluid mixing flows(e.g., [16,17]), Perugini et al. [4] have found that mingling of compositionallydifferent parent magmas during ascent in the volcanic conduit can producea compositionally heterogeneous hybrid magma of alternating light and darkregions. The spatial distribution of compositional heterogeneity within thishybrid magma is scale invariant (fractal), but with the added complexity ofdifferent scaling properties over different regions of a given sample. In otherwords, it can be characterized as multifractal. Although textural heterogeneitycaused by mixing of preexisting parent magmas has received considerable at-tention, the origin of compositionally homogeneous lavas with banding causedby spatially varying crystallinity or vesicularity still remains relatively unex-plored. This is an important area of study, because crystal or bubble nucleationand growth are intimately linked to magma degassing [18–20], which in turnis responsible for the buildup of overpressure required for the transition fromeffusive to explosive eruptive behavior [14,21–26].

1.2 Brittle deformation and degassing

It has been suggested that banding formation may, at least in some cases, beassociated with, and perhaps be integral to, brittle deformation and enhanceddegassing during magma ascent [12–15]. Brittle deformation of magma occurswhen strain rates locally exceed the ability of the melt to deform viscously(discussed in more detail in section 4), causing it to break like a brittle solidinto fragments. If brittle deformation is predominantly caused by shear, itmay be similar to brecciation or comminution of rocks in fault zones. Brittledeformation of vesicular magma will break bubble walls and allow volatiles

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to be released into adjacent cracks and fractures created during the fragmen-tation process [12,15,27]. Volatiles may thereby escape from the magma atan enhanced rate, reducing the buildup of overpressure required for explosiveeruptive behavior [14,21–26]. Banding has been observed within dissected con-duits [12,15] and can also be found in close proximity to the volcanic edifice,indicating that banding may indeed form during magma ascent in the conduit.

1.3 Viscous vs. brittle deformation

Microlites (crystals < 30 µm) may form in response to magmatic degassing(e.g., [18]) on eruptive timescales [19,20]. Accordingly, Castro and Mercer [28]find that differences in the degree of volatile loss could account for somevariability in microlite content in obsidian. However, they ultimately con-clude that varying microlite content is more likely to record different ascentrates and residence histories in the volcanic conduit, rather than variationsin magma degassing. They suggest that slower magma ascent rates near theconduit walls permit the development of higher crystallinity compared to thatof faster ascending magma further in from the conduit walls. Similarly, it hasbeen suggested that varying vesicularity may be caused by variable ascentrates [29], or by a spatially variable temperature distribution [30]. Regardless,the formation of banding, on scales as small as tens of microns, requires anefficient and extensive spatial redistribution through some process of magmadeformation. If deformation is purely viscous, then repeated stretching andfolding of magma is required. For a highly viscous silicic magma, this requiresa considerable complexity of the flow geometry, in which nearby fluid ele-ments can diverge strongly from each other (e.g., [31] and references therein).If, however, viscoelastic rheology of the magma is taken into consideration,then the effectiveness of mixing in a relatively simple flow may be consider-ably increased [32]. For magma, viscoelasitc behavior occurs near the glasstransition (discussed in more detail in Section 4) and involves both viscousand brittle deformation. Potentially analogous behavior has been observed ex-perimentally in shear flows of polymers (e.g., [33] and references therein). Thepossibility of brittle deformation is attractive, because obsidians and othereffusive silicic lavas are thought to have lost most of their volatiles throughopen-system degassing during ascent (e.g., [12,21,24–26,34]), and as discussedpreviously, brittle deformation may potentially provide an effective mecha-nism for enhanced degassing. Regardless of brittle or viscous deformation, itis important to understand whether the deformational process that resultsin banding is subsequent to or concurrent (and integral) with the actual de-velopment of variable crystallinity and vesicularity. At this point, we wish toemphasize that we distinguish the mechanical deformation process that resultsin banding formation from the development of textural heterogeneity, such asthe apparent spatial differences in nucleation and growth of mircolites or vesi-

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cles. The former will hereafter be referred to as the “deformational process”of banding formation, and we will refer to the latter as the “development oftextural heterogeneity.”

1.4 Hypothesis and outline

We test whether banding in obsidian bears evidence of a multiplicative process(discussed in more detail below) of banding formation, whereby deformationand the development of textural heterogeneity are concurrent and integral toone another; and whether a statistical analysis of banding permits discrimina-tion between such a multiplicative process, as opposed to the mixing of pre-existing heterogeneity. We focus on obsidian because the optical transparencyallows measurements of structures to be made over a broad range of scales.We report results from a quantitative petrographic analysis of obsidian withevidence of a genetic relation between brittle magma deformation and band-ing formation. This obsidian was collected from Big Glass Mountain (BGM),a large obsidian flow at Medicine Lake Volcano, California. We compare re-sults with obsidian from Mayor Island (MI), New Zealand. The MI obsidiansample analyzed here is thought to have formed by rheomorphism of fountain-fed spatter [7,35,36] and therefore provides a control sample of known origininvolving magma breakup, reannealing, and subsequent viscous deformation[37]. A petrographic description of both samples is provided in Section 2.

Our working hypothesis is that of repeated brittle deformation, concurrentdevelopment of textural heterogeneity, reannealing, and viscous deformationinto flow-aligned bands (Figure 1); in other words, a multiplicative process.We conduct a powerspectral and a multifractal analysis of both samples (Sec-tion 3). Powerspectral analysis permits the identification of a scale-invariantrecord, whereas multifractal analysis potentially allows the identification of asignal that preserves a record of a multiplicative process. If magma is subjectedto a multiplicative process, it will undergo a change in some attribute duringeach step of this process. In the context of brittle deformation and the twosamples analyzed here, this change corresponds to either an increase in crys-tallinity possibly associated with magma brecciation or enhanced degassingof fragments (BGM), or a change in vesicularity resulting from different de-grees of degassing associated with magma fragmentation (MI). Throughoutthe entire process (i.e., during all steps), the probability distribution of thefractional change in crystallinity or vesicularity is assumed to remain constant,as it represents a characteristic scaling of the underlying physics. However, thespatial distribution will be nonuniform and vary randomly from step to step.Because of the repeated reworking of the magma, the change in crystallinity orvesicularity from one step to the next is a small random multiple of the totalamount of crystallinity or vesicularity already present. This is what is meant

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by a multiplicative process, and it will be elaborated on further in Section 3.2.It differs from a nonmultiplicative process, in which development of texturalheterogeneity takes place before (and decoupled from) the actual formation ofbanding. Our analysis confirms that banding in the BGM and MI samples ismultifractal and likely preserves a record of a multiplicative process. To ver-ify the reliability of our result, we use a series of null hypothesis tests, whichconfirm that our hypothesis of a multiplicative formational processes is sound(Section 3.3). Because we suggest brittle deformation of magma as a likelyphysical mechanism for this multiplicative process, we provide a discussion ofthe underlying physics in Section 4 before concluding our report in Section 5.

2 Samples

Banding in silicic volcanic rocks is generally defined by alternating “layers” ofvarying color or texture. In their shortest dimension (i.e., cross section), theselayers have the appearance of parallel bands that range in thickness frommicrons to decimeters, with lengths of up to several meters in their longestdimensions. In the third dimension, these layers can be somewhat distortedor folded [37].

2.1 BGM obsidian

BGM is a 900-year-old rhyolite-to-dacite obsidian flow [38–41]. The BGMeruptive center, the youngest part of the Medicine Lake composite shieldvolcano [38], provides an example of the effusive emplacement of a cubickilometer-sized volume of silicic melt (Figure 2). The effusive eruptive phaseof BGM was preceded by an explosive phase that resulted in the depositionof tephra over an area of approximately 320 km2 [42]. Thus, BGM providesan excellent example of a sizable silicic eruption with a transition from explo-sive to effusive behavior. The youngest eruptive sequence of the BGM eruptivecenter includes obsidian that is ubiquitously banded (Figure 3), and some sam-ples bear evidence for a genetic relation between fragmentation and banding(Figure 4). Banded BGM obsidian ,therefore, is well suited for testing our hy-pothesis of banding formation through a multiplicative process that includesbrittle magma deformation, and hence, the possiblilty of enhanced magmadegassing associated with the transition from explosive to effusive eruptivebehavior.

Banding is predominantly caused by varying concentrations of pyroxene mi-crolites (Figure 5), with a higher microlite content resulting in a darker color.At times, dark bands also contain increased concentrations of oxides (e.g., the

5

darkest band in Figure 5). However, overall there appears to be no composi-tional difference or difference in water content (J. Castro, personal communi-cation) between individual bands of BGM obsidian. Some BGM samples showa continuous gradation from angular, reannealed fragments (right third of Fig-ure 4), to deformed fragments (middle of Figure 4), to well-developed banding(left third of Figure 4), indicating that brittle deformation, reannealing, andsubsequent deformation can result in banding.

2.2 MI obsidian

MI is the visible portion of a 700-m-high and 15-km-wide Quaternary shieldvolcano. Despite its uniform peralkaline magma composition, MIs history in-cludes virtually the full range of known eruption styles over a wide rangeof eruption sizes [35,36], including Hawaiian fire-fountaining and spatter-fedlava flows. Hawaiian eruptions involve rapid ascent of relatively low viscositymagma disrupted by gas bubbles. The immediate product is usually a pumicecone formed of magma fragments. These fragments may reanneal to form aspatter-fed lava flow, such as the 8-ka obsidian flow from which our samplewas collected [7,36]. Banding in the MI obsidian sample under considerationis predominantly a result of varying bubble content (Figures 6 & 7), withhigher vesicularity corresponding to lighter bands. Rust et al. [37] infer frombubble orientation that flow deformation was predominantly two-dimensionalpure shear. Given a presumably vigorous nature of magma ascent, it is rea-sonable to assume that variations in vesicularity may represent a record ofvarying degrees of repeated magma breakup and degassing during ascent. TheMI sample provides a sample of banded obsidian in which banding formationis known to have involved magma breakup and reannealing. It is thereforeideally suited for comparison with the BGM sample.

2.3 Banding data

Our subsequent analysis is based on digital photomicrographs of polished ob-sidian samples. Because of the degree of magnification, each image actuallyrepresents a composite of a number of high-resolution digital photographs.Our goal is to test banding from each sample for evidence of a multiplicativeformational process. Our analysis, however, should be viewed as exploratory,because the ability to make detailed quantitative estimates of governing para-meters is limited by the nonunique nature of multifractal analysis [43]. Never-theless, through the use of null-hypothesis tests, we can establish the generalnature of underlying processes that are most likely to result in the formationof banding in the obsidian samples under consideration.

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The digitized sample of BGM obsidian has a length of 200 mm and a res-olution of 2 µm/pixel (Figure 8a). Equivalently, we obtain a digital recordwith 5 µm/pixel resolution from a 55-mm-long sample of MI obsidian, col-lected by K. V. Cashman, University of Oregon (Figure 8b). The cumulativefrequency distribution of resulting measures (amplitudes) for each sample isshown in Figure 9. Each digitized sample will hereafter also be referred to asthe “record.”

We normalize gray-scale color index, Ci, for each pixel, i, as

φi = [Ci −min (C)]

[N∑

i=1

Ci −N min (C]

)−1

whereN∑

i=1

φi = 1. (1)

Here 0 ≤ Ci ≤ 255, and pi is the normalized gray-scale color index, hereafteralso referred to as the “measure.” We do not obtain a precise calibration ofgray-scale color index to textural heterogeneity in our samples because, unlikePerugini et al. [4], our bands are not defined by compositional variations, andit is not our goal to draw detailed quantitative estimates of process parametersfrom our analysis.

3 Analysis of banding

The main goal of our analysis is to assess whether banding in the BGM and MIobsidian samples preserves a record from which we can gain insight into thegeneral nature of the underlying formational process, and hence about magmaascent processes inaccessible to direct observation. The wave-number powerspectrum, S(k), is one of the most often used tools for studying complex spatialpatterns, because of its compact quantitative description of the presence ofmany spatial scales [44]. Multifractal analysis of signals, in contrast is provento be successful in discerning formational processes of a multiplicative nature[45].

3.1 Power spectrum

As spectral estimator we use the multitaper method (e.g., [46] and referencestherein), which is well suited for detailed spectra with a large dynamic range.Both BGM and MI spectra (Figure 10) are characterized by a 1/wavenum-ber scaling of spectral power, S(k) ∼ k−1, indicating the presence of scaleinvariance.

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3.2 Multifractal analysis

Monofractal records are often characterized by k−1 power spectra. They havethe same scaling properties throughout the entire signal and can be char-acterized by a single fractal dimension. Records produced by multiplicativeprocesses, in contrast are usually found to be multifractal [47,48]. Instead ofbeing characterized by a single fractal dimension, they have locally varyingscaling properties that contain valuable information regarding the formationalprocess (e.g., [48]). For example, the energy-cascading process embodied byfluid turbulence generates a multifractal timeseries of velocity fluctuationsthat is mathematically well represented by a multiplicative process [48]. Mul-tifractal analysis has also been successful in discriminating between healthyand heart failure patients on the basis of a time series of human heartbeatintervals [45]. In the latter case, other methods such as spectral analysis areinsufficient in discriminating between the two “processes” (healthy vs. heartfailure). We therefore use a multifractal analysis to test whether banding inBGM and MI obsidian is multifractal and, hence, possibly preserves a recordof a multiplicative formational process.

3.2.1 Method

Multifractal records can be characterized by the singularity spectrum, f(α), ofsome measure, φ(x), which in our case represents the normalized color indexas a proxy for microcrystallinity (BGM) or vesicularity (MI). It is possible toevaluate the presence of different scaling laws throughout the record via theintegrated measure

µi =

i+L/2∫i−L/2

φ dx, whereN∑

i=1

µi = 1. (2)

Here L is the length of a record segment centered at position i, with the entirerecord of length X subdivided into N = X/L segments. The f(α) singularityspectrum provides a mathematical description of the spatial distribution of φover the entire record, with the singularity strength, α, defined by

µi(L) ∼ Lαi . (3)

Counting the number of segments, N(α), where µi(L) has a singularity strength,α, allows the loose definition of f(α) as the fractal (Hausdorff) dimension ofthe part of the record with singularity strength of α [47,49]

N(α) ∼ L−f(α). (4)

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For a given segmentation length, or an equivalently box-counting dimension,L, the value of α will vary with position i. As a consequence, f(α) charac-terizes the spatial complexity of the actual record. If the record is uniformlyfractal (monofractal) or not fractal at all, then there will be only one valueof α, reducing f(α) to a single point. If the record is complexly (though notrandomly) structured so that its fractal dimension changes with position, thenthere exists a range of α with different Hausdorff dimensions, f(α). The morecomplexly structured the record is, the broader the spectrum will be.

The spectrum f(α) can be obtained by considering different moments, µqi , of

the integrated measure. Here the parameter φ provides a “microscope” forexploring different regions of the banding structure. As φ is varied, differentsubsets of the record, which are associated with different scaling exponents, α,become dominant. The singularity spectrum can then be obtained from [49]

f (q) = limL→0

∑i µ

qi (L) log [µq

i (q, L)]

log L(5)

and

α (q) = limL→0

∑i µ

qi (L) log [µi (L)]

log L(6)

In practice, the slopes of∑

i µqi (L) log [µq

i (q, L)] and∑

i µqi (L) log [µi(L)] versus

log L are estimated to obtain f(α) and α, respectively. Care must be taken toonly consider the range of log L over which these graphs do actually representlinear trends. If they do not represent linear trends, then the record is not trulymultifractal, but could still result in a singularity spectrum with multifractalcharacter. More details are provided in the caption to Figure 11; meanwhile,let it suffice to say that in our subsequent analysis, we were careful to avoidthe potential pitfalls associated with multifractal analysis ([43] and referencestherein).

3.2.2 Results

The singularity spectrum (Figure 11) indicates that banding structure of BGMand MI obsidian are a multifractal, indicating that banding may have formedthrough a multiplicative processes. To illustrate that this is indeed feasible, weshow f(α) for a multifractal record produced by a simple mathematical modelfor multiplicative processes (Figure 11). This record is based on the two-scaleCantor set (MC) [47]. To construct this record, a unit interval is randomlysubdivided into three segments: two of given length l2, and one of length l1(Figure 12). The two former intervals each receive a proportion, p2, of the totalmeasure, and the latter interval receives a proportion given by p1. This process

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is recursively repeated for each segment from the previous iteration (Figure12). In an idealized sense, this is equivalent to a fragmentation process witha given scaling for fragment size (l1 and 2l2) and an embedded enrichmentor depletion (p1 and p2) of some measure (e.g., volatile content, crystal orvesicle content, chemical or isotopic component). This idealized multiplicativeprocess with l1 = p1 = 1/5 and l1 + 2l2 = p1 + 2p2 = 1 results in the bandingshown in Figure 8c and provides an excellent match to the BGM singularityspectrum (Figure 11). The important point here is that microlite growth ordevelopment of nonuniform vesicularity may be integral to the formation ofbanding. Banding in BGM and MI obsidian samples, indeed, seems to preservea record of such a multiplicative process.

3.3 Null hypotheses

To assess the limitations of our results, we perform several ‘null-hypothesis’tests, summarized in Table 1.

3.3.1 Black and white banding

Figure 8d shows the banding structure generated by the same multiplicativeprocess as shown in Figure 12, but with banding defined by only two alter-nating values of the measure φ (black or white). In contrast to the BGM andMI samples, as well as the MC model, this binary banding is not multifractal,and its singularity spectrum consists of a single point, shown in Figure 13.As already discussed by Turcotte [50], this result indicates the importance ofa binomial or log-normal measure to produce a multifractal record, and thisexample illustrates that the formation of binomial measures is well explainedby a multiplicative enrichment or fractionation process (e.g., [51–53]).

3.3.2 Randomized BGM

Figure 8e shows the measure of the BGM sample (Figure 8a), sorted fromdarkest to lightest normalized color index, φ. For our second test, we randomlyredistributed this measure. The resultant record (Figure 8f) is not multifractal(Figure 13). Not surprisingly, we can conclude that a binomial measure alonedoes not suffice for a multifractal record, implying either that a random forma-tional process resulting in a log-normal measure, or that a random “mixing”process of a preexisting binomial measure, is not a viable alternative for theformation of banding in BGM or MI samples.

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3.3.3 Chaotic mixing via Baker’s map

In our third test, we examine the possibility that a preexisting measure (e.g.,microlite content) of identical probability distribution as the BGM sample(Figure 8e) could be redistributed through a chaotic mixing process to resultin banding with similar characteristics as the BGM sample. This hypothesis ismotivated by the possibility that microlite content or vesicularity may vary, forexample, radially or vertically before the actual mixing process, as suggestedby Castro and Mercer [28].

A simple mathematical model that has been used to to represent the char-acteristics of chaotic mixing processes is the generalized Baker’s map (e.g.,[16,17,44]). We performed calculations using several different parameteriza-tions of the generalized Baker’s map. Although details varied between differ-ent versions, the overall results were the same. We therefore only show resultsfor one version of this map. This version has a ratio of segmentation lengthsidentical to that of a multiplicative Cantor process. The Baker’s map spa-tially redistributes a given measure on a unit square and is equivalent to theprocess of stretching and folding in chaotic fluid mixing, illustrated in Figure14. Mathematically, the process is given by the mapping

xn+1

yn+1

=

l∗2xn, yn/l∗2 if yn < l∗2

l∗2 + l∗1xn, (yn − l∗2) /l∗1 if l∗2 < yn ≤ l∗2 + l∗1

l∗2 + l∗1 + l∗1xn, (yn − l∗2 − l∗1) /l∗1 if l∗2 + l∗1 < yn

(7)

Here l∗1 = l1 + θ, l∗2 = (1− l∗1)/2, and θ is a number chosen randomly at eachiteration, with θ ≤ l1/2 ≤ θ. The Baker’s map is analogous to closed-systemchaotic mixing, in which the original measure is conserved throughout (i.e.,for all iterations

∑Ni=1 φi = 1). With respect to BGM, the Baker’s map implies

that microlite growth and banding formation are decoupled. The singularityspectrum (Figure 11) obtained from this map is multifractal and is very similarto the BGM sample (Figure 13). However, the power spectrum allows a cleardistinction. Whereas power spectra of BGM, MI (Figure 10), and the MCmodel are characterized by S ∼ k−1, the power spectrum for the Baker’smap with initial measure derived from the BGM sample has S ∼ k−β, withβ ≈ 1.5− 2. This result indicates, consistent with results from mixing studies(e.g., [4,16,17]), that a magma with preexisting nonuniform microlite content,or vesicularity, can be mixed, resulting in banding with multifractal properties.However, the power spectrum is not likely to have the S ∼ k−1 scaling observedin BGM and MI samples, and the mixing process itself is not multiplicative, so

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the generation of variable microlite content or vesicularity still requires someprior process that yields a binomial distribution.

4 Brittle deformation of magma

In this section, we provide a brief review of the physics of brittle magmadeformation. If strain rates during viscous deformation of silicic melt exceedthe viscous relaxation rate, then the melt will no longer deform in a viscousmanner but, rather, in a brittle manner. This transition is referred to as theglass transition (e.g., [54] and references therein). In general, it is thoughtthat magma fragmentation occurs when strain rates, for example, resultingfrom flow deformation or bubble expansion [23,55] exceed the glass transi-tion of the melt. Recent work indicates that shear strain rates during theascent of silicic magmas exceed the glass transition [14,15,56], even for effu-sive eruptions that produce lava flows and domes. The glass transition forsilicate melts is γc ∼ 0.01 G/µ, where G is the elastic modulus of order 10GPa (e.g., [54]), µ is viscosity, and γ is the local shear strain rate. Joseph [57]suggests that a liquid undergoing simple shear, such as magma ascending ina conduit or dike, can cavitate if the maximum tensile stress, 2µγ, exceedsambient pressure and the liquid’s tensile strength (i.e., the glass transition).At the glass transition, γ = γc, implying a tensile stress of the order of 100MPa. As a consequence, shear-induced fragmentation should be accompaniedby cavity formation at depths of up to several kilometers. This is importantbecause it indicates that a shear-induced fragmentation event can create ashort-lived network of fractures at significantly lower pressures than the sur-rounding magma. Moreover, these fractures should be highly permeable to gasflow, and the adjacent magma would be subjected to significant undercooling.The latter, in turn, may result in highly variable crystallization kinetics [18–20,58] over relatively short distances, thereby providing a possible mechanismfor the generation of textural heterogeneity in direct association with the de-formation/fragmentation process.

Because of the displacement and rotational component associated with simpleshear, fragments from different regions in a conduit or a flow can easily be-come juxtaposed. Once strain rates have relaxed to a value smaller than theviscous relaxation rate, the fragmented magma can reanneal and once againdeform in a viscous manner. If the juxtaposed fragments are texturally (orcompositionally) heterogeneous, either initially or as a consequence of the de-formation process itself, viscous stretching of reannealed fragments will resultin banding. The degree of textural heterogeneity recorded in bands is oftensmall. For example, clinopyroxene microlite content, which defines bandingin BGM and similar obsidian samples, varies by less than a few volume per-cent [59]. It appears reasonable that this degree of heterogeneity can exist or

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form over relatively small distances within the conduit. Fragmentation can berepeated during continued ascent and is, except for reannealing and viscousdeformation, essentially equivalent to the comminution of rocks in fault orshear zones banding [60].

5 Conclusion

The essential feature of a multiplicative process, within the context of bandingformation in silicic volcanic rocks, is the repeated reworking of magma andconcurrent development of textural heterogeneity. In the case of the analyzedbanded obsidian from BGM, textural heterogeneity is predominantly a resultof variable microlite content, whereas in the spatter-fed obsidian sample fromMI it is caused by varying vesicularity. A multiplicative process capable ofbanding formation requires a deformational process that repeatedly results in aspatial redistribution of preexisting banding structure and subsequent viscousdeformation into thin, flow-aligned bands. This creates increasingly finer scalesof banding or in other words, a cascade to shorter and shorter wavelengthsuntil some cutoff value, given by bubble or crystal size, is reached [44]. In theBGM obsidian sample, for example, the boundaries between individual bandsare often very sharp, even at the smallest length scales. In contrast to typicalmixing processes (e.g., [4]), this indicates that microlite dispersion or moleculardiffusion were relatively insignificant during banding formation. We thereforeconclude that the observed binomial distribution of textural heterogeneity isbest explained if there is a change in crystallinity or vesicularity that is a smallrandom multiple of the amount already present and that this change occursconcurrently with, or is integral to, the mechanical deformation process.

Although banding formation in the samples under consideration probablyinvolved some deformation during surface emplacement, considerations ofcrystal nucleation and growth kinetics [18–20,58] are perhaps indicative ofa process occuring, at least in part, within the conduit during magma ascent.There is observational evidence for a genetic relationship between brittle de-formation and banding (Figure 4), and furthermore, brittle deformation ofmagma in the conduit seems plausible on the basis of theoretical considera-tions (Section 4). One may therefore speculate that breakup of magma andconcurrent enhanced degassing of individual fragments may result in spatiallyvariable crystallinity or vesicularity. Nonetheless, it should be noted that it ispossible to produce multifractal banding through viscous mixing of a magmawith an “inherited” textural heterogeneity. However, such a scenario wouldimply some prior formational process to result in the required binomial dis-tribution of heterogeneity. Probably the most plausible scenario for a viscousmixing origin of banding would be the development of textural heterogeneityconcurrent with the mixing process. One aspect of possible importance in this

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context might be viscous dissipation [61]. However, pervasive banding forma-tion over the observed spatial scales requires repeated stretching and folding,in which nearby parcels of magma can diverge strongly from each other. Suchlarge degree of divergent stretching is not easily achieved in a highly viscousmagma in the volcanic conduit, where flow is presumably dominated by simplesimple shear. Because of these considerations, and based on the powerspec-tral and multifractal analysis presented here, we conclude that banding in theBGM and MI obsidian samples analyzed here is best explained as the recordof a multiplicative process, represented by repeated cycles of magma breaking,reannealing, and viscous deformation. On the basis of theoretical considera-tions, it is feasible that such a process can occur during ascent in the volcanicconduit, as well as during surface emplacement.

6 Acknowledgments

We thank A. Rust, A.M. Jellinek, and A. Namiki for comments on an ear-lier version of the manuscript; K.V. Cashman for collecting the Mayor Islandsample; and T. Teague and M. Calaleta for sample preparation. This workwas supported by a National Science Foundation grant to M.M. and by theTurner Fellowship to H.M.G.

7 Figure Captions

Figure 1: Schematic illustration of our working hypothesis for banding for-mation: brittle deformation of magma (see Section 4 for a detailed discussion).(a) Brittle deformation during simple shear. The single fragment shown hereundergoes a change in texture (e.g., microlite content or vesicularity), indi-cated by the darker color. Note the presence of fractures and the sigmoidalgeometry, which is similar to tension gashes. Tension gashes often form in enechelon arrays. They may be precursors to shear localization (e.g., [62] andreferences therein), and they can be rotated by deformation either during orafter formation. (b) The magma has reannealed and is deforming viscouslyby simple shear into a band. (c) Repeated brittle deformation and multiplica-tive change in microlite content or vesicularitiy indicated by the change incolor. Note that the band that formed during the previous cycle is broken intoseveral pieces and becomes “reworked”. Repetition of this process results ina broad range of textural heterogeneity and in an increasingly finer bandingstructure.

14

Figure 2: Map of Big Glass Mountain, Medicine Lake volcano and vicinity.

Figure 3: Photograph of banded obsidian sample from Big Glass Mountain,Medicine Lake Volcano, California.

Figure 4: Photograph of obsidian from Big Glass Mountain (cut at mutuallyperpendicular directions). This sample shows a continuous gradation from an-gular, reannealed fragments (right third), to deformed fragments (middle), towell-developed banding (left third), indicating that brittle deformation, rean-nealing, and subsequent deformation can result in banding.

Figure 5: Photomicrograph of obsidian from Big Glass Mountain. Banding isgenerally defined by varying mircolite content in a glassy matrix. Some darkbands also contain oxides.

Figure 6: Polished sample of Mayor Island banded obsidian (collected by K.V. Cashman, University of Oregon).

Figure 7: Photomicrograph of obsidian from Mayor Island. Banding is de-fined by varying vesicularity.

Figure 8: Banding for (a) Big Glass Mountain (BGM) (200 mm length), (b)Mayor Island (55 mm length), (c) Multiplicative cascade (MC) based on thetwo-scale cantor set, (d) MC with alternating black and white bands only,(e) BGM pixels sorted by color index, (f) BGM with random redistribution ofpixels, (g) Baker’s map of the sorted BGM from panel (e) after four iterations,(h) Baker’s map of sorted BGM from panel (e) after six iterations.

Figure 9: Cumulative frequency distribution of the measure (normalized gray-scale color index) for Big Glass Mountain and Mayor Island obsidian samples.

Figure 10: Power spectra for Big Glass Mountain and Mayor Island obtainedfrom averaging the spectra from 200 rasters of the digital photomicrograph ofeach obsidian sample. As discussed by Namenson et al. [44], spectral averagingis a robust and recommended procedure in the search for power-law spectralcharacteristics. Spectral estimation is made using the multitaper method (e.g.,[46] and references therein), where data windows are given by discrete pro-late spheroidal sequences [63] that effectively minimize spectral leakage. We

15

tested different taper sequences to minimize leakage effects or other artifactsand find that an estimator with NW = 4, the time bandwidth for the Slepiansequences, was well suited for our analysis.

Figure 11: Singularity spectra obtain by the Chhabra-Jensen method [49].f(α) for Big Glass Mountain (BGM) and Mayor Island (MI) are broad, in-dicating a multifractal record. The spectrum of the multiplicative cascadeprovides an excellent match to BGM. f(q) and α(q) are determined fromthe slope of the graphs

∑i µ

qi (L) log [µq

i (q, L)] and∑

i µqi (L) log [µi (L)] ver-

sus log L. The error of estimation is evaluated from the variance in the es-timation of the slope. To evaluate the slope, we use only the linear segmentof the aforementioned graphs. Therefore, f(q) and α(q) were estimated formax(δBGM , δMI) ≤ L ≤ min(XBGM , XMI)/10, where δBGM represents thewidth of the smallest resolved band of the BGM sample, δMI is the averagediameter of the largest bubbles of the MI sample, XBGM is the length of theBGM sample, and XMI is the length of the MI sample. These ranges corre-spond to linear segments of the aforementioned graphs. They are also identicalfor both samples, so there is no bias in the analysis. Furthermore, it should benoted that we compared our results for the entire BGM sample (200 mm inlength) with analyses of segments of the BGM sample with lengths identicalto that of the MI sample (55 mm). In all cases, the BGM singularity spectrumessentially has the multifractal characteristics shown in this figure. Finally,we compared results from the Chhabra-Jensen method with the Legendre-transform method [47] and found the results shown here to be robust.

Figure 12: Multifractal record generated by a multiplicative cascade (afterfour iterations) based on the two-scale Cantor set [47]. The segment of unitlength and uniform measure (denoted by gray-scale color) shown at step 0 iscut into three segments, one of length l1 and two of length l2. The measure ofthe first segment is multiplied by a factor of p1, and that of the two other seg-ments is multiplied by a factor of p2, where p1 > p2. Each of the resulting threesegments shown at step 1 are, in turn, cut into three segments, one of lengthl1lseg and two of length l2lseg. Here lseg denotes the length of the given seg-ment shown at step 1. The resultant segment lengths are either l1 l2, l21, or l22.Concurrently, the measure of each segment from step 1 is again multiplied byeither p1 if its length is l1lseg, or by p2 otherwise. The resultant segments andmeasures, denoted by different shades of gray (p1 p2, p2

1, or p22), are shown at

step 2. The process is then repeated for steps 3 and 4. It should be noted thatthe spatial sequence into which each segment at any given step is subdivided(i.e., whether l1lseg is at the first, second, or third position) is chosen randomly.

16

Figure 13: Singularity spectra for the different null-hypothesis tests. BigGlass Mountain with randomized pixels (Figure 8f) and the multiplicativecascade with only black-and-white bands (Figure 8d) are monofractal andplot as single points. The spectrum for the generalized Baker’s map aftersix iterations (Figure 8g) is multifractal and similar to that of the Big GlassMountain sample (Figure 8a), shown as the dashed line.

Figure 14: Schematic representation of the modified Baker’s map (e.g., [16,17]),where the initial measure is based on the sorted Big Glass Mountain of Figure8e. This map represents a chaotic redistribution of a preexisting measure.

bFracturea c

Fig. 1. (b&w)

17

Fig. 2. (b&w)

18

Fig. 3. (b&w)

19

Fig. 4. (b&w)

20

Fig. 5. (b&w)

21

20 mm

Fig. 6. (b&w)

22

Fig. 7. (b&w)

23

a

c

d

e

f

g

h

b

1 pixel

1 pixel

Fig. 8. (b&w)

24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Amplitude

Cum

ulat

ive

Freq

uenc

yBGM MI

Fig. 9. (b&w)

25

100

10-1

10-2

10-3

10-4N

orm

aliz

ed sp

ectra

l pow

er, S

10-1 100 101

Wavenumber, k (m-1)

[width]-1 of smallest resolved bands

S ~ k-1

[diameter]-1 oflargest bubbles

BGM

MI

Fig. 10. (b&w)

26

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

α

f(α)

BGMMIMC

Fig. 11. (b&w)

27

p1, l1 p2, l2 p2, l2

p22, l22p2

2, l22 p1 p2 l1 l2

p2 p1 l2 l1

p2 p1 l2 l1

p12

l12 p22, l22

p1 p2 l1 l2

p22, l22

1

0

2

3

4

Fig. 12. (b&w)

28

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

α

f(α)

Baker (6 iterations)

BGM randomMC binary

BGM

{

Fig. 13. (b&w)

29

1. Break/fold 2. Stretch 3. Anneal/foldx

y

l*1

l*1

l*2

Fig. 14. (b&w)

30

Table 1: Comparison of obsidian samples Cantor map with null hypothesis tests

Record MF a (S ∼ k−1) b MP c Implications

Big Glass Moun-tain (BGM)

Y Y Y Concurrent microlite growth andbanding formation

Mayor Island (MI) Y Y Y Concurrent formation of variablevesicularity and banding

Cantor (MC) Y Y Y Concurrent development of het-erogeneous measure and banding

Cantor binary N N N No binomial measure

BGM randomized N N N Decoupled microlite growth andbanding formation

Baker’s map Y N N Decoupled microlite growth andbanding formation

a Multifractal.b S is spectral power and k is wavenumber.c Multiplicative process.

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