Bull Volcanol (1994) 56:193-206Bulletin of

Springer-Verlag 1994

Quantifying the effect of rheology on lava-flow marginsusing fractal geometryB. C. Bruno1, G. J. Taylor1, S. K. Rowland1, S. M. Baloga2*

1 Hawaii Institute of Geophysics and Planetology, University of Hawaii at Manoa, 2525 Correa Road, Honolulu,Hawaii 96822, USA

2 Jet Propulsion Laboratory, NASA, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109, USA

Received: June 6, 1993/Accepted: March 18, 1994

Abstract. This study aims at quantifying the effect ofrheology on plan-view shapes of lava flows using frac-tal geometry. Plan-view shapes of lava flows are impor-tant because they reflect the processes governing flowemplacement and may provide insight into lava-flowrheology and dynamics. In our earlier investigation(Bruno et al. 1992), we reported that flow margins ofbasalts are fractal, having a characteristic shape regard-less of scale. We also found we could use fractal di-mension (D, a parameter which quantifies flow-marginconvolution) to distinguish between the two endmem-ber types of basalts: a'a (D: 1.05-1.09) and pahoehoe(D: 1.13-1.23). In this work, we confirm those earlierresults for basalts based on a larger database and overa wider range of scale (0.125 m-2.4 km). Additionally,we analyze ten silicic flows (SiO2: 52-74%) over a sim-ilar scale range (10 m-4.5 km). We note that silicicflows tend to exhibit scale-dependent, or non-fractal,behavior. We attribute this breakdown of fractal be-havior at increased silica contents to the suppression ofsmall-scale features in the flow margin, due to thehigher viscosities and yield strengths of silicic flows.These results suggest we can use the fractal propertiesof flow margins as a remote-sensing tool to distinguishflow types. Our evaluation of the nonlinear aspects offlow dynamics indicates a tendency toward fractal be-havior for basaltic lavas whose flow is controlled by in-ternal fluid dynamic processes. For silicic flows, or bas-altic flows whose flow is controlled by steep slopes, ourevaluation indicates non-fractal behavior, consistentwith our observations.

Key words: fractals - lava - rheology - remote sens-ing

* Present address: Proxemy Research Inc., 20528 Farcroft Lane,Laytonsville, Maryland 20882, USA

Correspondence to: B. C. Bruno

Introduction

Plan-view shapes of lava flows reflect the processesgoverning flow emplacement; they are frozen snap-shots of the final moments of flow. As such, they pro-vide insight into the final stages of lava-flow dynamicsand rheological state. Plan-view shapes and other mor-phological characteristics have been studied extensive-ly and important quantitative parameters have beendeveloped to extract rheological properties and erup-tion and emplacement processes of lava flows. Usefulparameters include flow length and width as indicatorsof eruption rate and duration (Walker 1973; Hulmeand Fielder 1977); widths and thicknesses of flows toestimate yield strengths (Hulme 1974); widths of distallobes to deduce rheological properties and SiOz con-tent (Wadge and Lopes 1991); channel depth andwidth and surface speed to estimate viscosity (Shaw etal. 1968); total area and volume to estimate maximumflow rates and minimum emplacement times (Shawand Swanson 1970); flow length and width coupledwith levee and channel width to yield effusion rate(Crisp and Baloga 1990); average thickness and the ra-tio of maximum width to maximum length to calculateeruption duration (Lopes and Kilburn 1990); and ridgeheights and spacings to estimate viscosity of flow inter-iors (Fink and Fletcher 1978; Fink 1980). Use of thesemeasurements has led to improved insight into lava-flow dynamics and planetary volcanism, but manyquestions about their quantitative use remain.

We have been using a new approach to quantita-tively characterize lava-flow morphology: the fractalproperties of flow margins. In our preliminary report(Bruno et al. 1992), we showed that the perimeters ofbasaltic flows are fractal, and have characteristic fractaldimensions. Fractals are objects (real or mathematical)that look the same at all scales (Mandelbrot 1967,1983). Many geologic features exhibit such 'self-simi-lar' behavior (e.g. rocky coastlines, topography, rivernetworks). A qualitative example of self-similar behav-ior of a lava-flow margin appears in Fig. 1. We believethat measurement of fractal properties of lava flows

194

1972 Mauna UluPahoehoe Flow

1kmInside flow

margin —'

Inside flow margin

20m

Fig. 1. Margin of a typical pahoehoe flow from the 1972 eruptionof Mauna Ulu, Kilauea volcano with small section enlarged toshow self-similarity. The similar shapes of the entire flow marginand the enlarged section at different scales suggests fractal behav-ior

Fig. 2 Fractal dimensions (D) of selected curves: a D = 1.00; bD = 1.01; c D = 1.10. The complex curves (b, c) are longer and aremore plane-filling than (a) and thus have D>1. Since thesecurves are contained in a plane (D = 2), they have D between 1and 2 (following Garcia 1991)

will shed light on flow dynamics, eruption rates, andrheology, and will prove to be a useful method forquantifying the morphology of lava flows in inaccessi-ble areas of the Earth as well as on other planets bymeans of remote sensing.

The key parameter we derive is fractal dimension.Fractal dimension (D) is based on a similar concept astopological dimension (DT). For example, a line can becontained in a plane; thus a line (DT = 1) has a lowertopological dimension than a plane (DT = 2). Similarly,a plane can be contained in a volume; thus a volumehas a greater topological dimension (DT = 3) than aplane. Fractal dimensions are also measures of theamount of space occupied, but they do not have integ-er values. The following example illustrates the differ-ence between D and DT. Any curve, such as thoseshown in Fig. 2, can be contained in a plane; thusDT = 1. However, the complex curves (Fig. 2b, c) havea much greater length than do simple curves (Fig. 2a);therefore, these convoluted and involuted curves haveD>1. As curves becomes increasingly complex (i.e.plane-filling) in a self-similar fashion, D continues toincrease, approaching an upper limit at the topologicaldimension of a plane (since no curve can take up morespace than a plane). Thus, the fractal dimensions of all

plan-view shapes of self-similar objects are in therange: 1<D<2. The method by which fractal dimen-sion is calculated is described below.

Bruno et al. (1992) showed that the flow margins ofboth endmember types of basaltic lavas (a'a and pa-hoehoe) are fractal, with the scale of self-similarity ex-tending from about 0.5 m to over 2 km. This suggeststhat the processes that control the shapes of basalticflows at a small (say, 1 m) scale are dynamically similarto the processes that control flow shapes at a 10 m or100 m scale. For pahoehoe flows, this implies that thesame factors that control the outbreak of a small toecontrol the outbreak of a larger eruptive unit. For a'aflows, which have crenulation-like features superim-posed upon larger flow lobes, self-similarity impliesthat the same factors that cause these crenulations toform (presumably related to differential shear stress)are also responsible for forming the lobes themselves;i.e. the lobes are large-scale crenulations. Kilburn(1990) made a similar point in describing the fractalproperties of the surfaces of a'a flows. Also, Bruno etal. (1992) discovered that the margins of a'a and pa-hoehoe flows have different fractal dimensions. Pahoe-hoe margins have higher D (typically >1.15) than doa'a flows (usually <1.09). This is consistent with ourobservation that outlines of pahoehoe margins arequalitatively different from a'a margins (Fig. 3); pahoe-hoe margins tend to have many more embayments andprotrusions than the more 'linear' a'a margins.

These differences in geometry do not reflect differ-ences in composition, but rather differences in rheolo-gy and emplacement mechanisms. Whether an erupt-ing basalt becomes a'a or pahoehoe depends on a crit-ical relationship between volumetric flow rate (largelycontrolled by effusion rate and ground slope), effectiveviscosity and shear strength (Shaw et al. 1968; Shaw1969; Peterson and Tilling 1980; Kilburn 1981). Pahoe-hoe flows are associated with low terminal volumetric

Fig. 3. Digitized outlines of typical a'a and pahoehoe flows fromthe 1935 eruption of Mauna Loa volcano. The pahoehoe marginis more convoluted than the a'a margin, and would be expectedto have a higher D (following Bruno et al. 1992)

195

flow rates (typically <10m3/s for Hawaiian eruptions)and/or fluid lavas (Rowland and Walker 1990). Theytend to be thin (< 2 m) and advance with a smoothrolling motion (Cas and Wright 1987). Pahoehoe flowsare formed in compound flow fields composed of nu-merous thin overlapping units. In contrast, a'a flowsform at higher terminal flow rates. They are generallyassociated with higher effusion rates (typically>10 m3/s for Hawaiian eruptions) and/or viscous lavas(Rowland and Walker 1990). A'a flows are generallythicker (typically a few meters), and have massive in-teriors and clinkery exteriors. Unlike most pahoehoeflows, they are erupted as a single unit. A'a and pahoe-hoe lavas also differ in mode of transport. Lava tubescan play crucial roles in transport of pahoehoe lavas,enabling flow over long distances with small radiativeheat losses; a'a lavas typically flow in open channels.All of these differences in terminal flow rates, flowstyles and emplacement mechanisms lead to differentfractal dimensions for a'a and pahoehoe flows.

One of the objectives of investigations of flow mor-phology is to determine rheological properties and per-haps lava-flow composition, particularly SiO2 and vola-tile content. So, in addition to basalts, we have studied

more silicic flows with SiO2 contents ranging from 52to 74 wt %. Silicic flows can erupt as single-flow unitscharacterized by a blocky morphology. They are alsooften associated with channel formation. Thus, interms of both morphology and emplacement mecha-nism, some high-silica flows are similar to a'a flows anddifferent from pahoehoe flows. We have found thathigher silica contents and the accompanying increasein viscosity and presumable yield strength lead to qual-itative as well as quantitative differences in plan-viewshapes. Figure 4a shows a basaltic a'a flow, character-ized by fairly linear margins, superimposed upon whichare small-scale features that resemble crenulations.Figure 4b (basaltic andesite) has finger-like lobes,hundreds of meters in diameter, and appears less 'lin-ear'. Like basaltic a'a, this basaltic andesite has a cre-nulated appearance. Figure 4c (andesite) also has mul-tiple lobes. Here, the lobes appear shorter, stubbierand wider (approaching 1 km), and the crenulationsappear to be absent. Figure 4d (dacite) is characterizedby the highest silica content. Here the lobes are stillwider (> 1 km) and protrude less from the main massof the lava flow, causing the flow to assume a morebulbous appearance. We note that silica content is just

Kg. 4a-d. Plan-view shapes of lava flows of various compositions US); d dacite (Chao, Chile). As silica content increases, flow lobes(in order of increasing silica content): a basalt (Galapagos Islands); tend to widen, thicken and protrude less from the main mass ofb basaltic andesite (Hekla, Iceland); c andesite (Mount Shasta, the lava flow, and the smaller-scale features become suppressed

196

one controlling factor on plan-view shape; there aremany other controlling factors (e.g. overall volume, vo-latile content, eruption rate). Nevertheless each rangeof silica content (basalts, basaltic andesites, andesites,and dacites/rhyolites) appears to show qualitative dif-ferences in plan-view shape. In this paper, we quantifythe effect of rheology on perimeters of lava flows usingfractal analysis. Our objective is to define quantitativeparameters that vary with rheology, which in combina-tion, can be used to remotely distinguish flow types.

Methodology

The fractal analysis employed in this study uses threequantitative parameters: correlation coefficient (R2),fractal dimension (D), and quadratic coefficient (a),These parameters are all calculated in accordance withthe 'structured-walk' method (Richardson 1961). Al-ternative methods include 'equipaced polygon', 'hybridwalk' and 'cell-count' methods; these are discussed indetail in Longley and Batty (1989). We selected thestructured-walk method because it can be readily ap-plied, both in field measurements and from remote-sensing images. According to the structured-walkmethod, the apparent length of a lava-flow margin ismeasured by walking rods of different lengths alongthe margin. For each rod length (r), flow margin length(L) is determined according to the number of rodlengths (N) needed to approximate the margin; that is,L = Nr. By plotting log L vs log r (called a 'Richardsonplot', after Richardson 1961). fractal behavior can bedetermined.

Calculating correlation coefficient (R)

A linear trend on a Richardson plot indicates the dataform a fractal set, indicating self-similarity over therange of rod lengths used. Our criterion for linearity(i.e. fractal behavior) is an R2 value exceeding 0.95,where R is the correlation coefficient of the linear leastsquares fit. This criterion is chosen somewhat arbitrari-ly, but follows that used by Mueller (1987). Care wastaken to ensure that the data array did not artificiallyflatten out at long rod lengths as a result of choosingrod lengths that are so large such that they approachthe length of the object. One can avoid this problemaltogether by letting r approach the length of the ob-ject (that is, letting N approach 0) and plotting all thedata on a Richardson plot. One can then visually selectthe linear portion of the curve and fit a least squaresline to the selected segment. Although we have foundthis technique suitable in measurements of lava flowstaken from aerial photographs, it is quite impractical inthe field, as it would involve a large number of time-consuming measurements. We have found that choos-ing our longest rod length such that it can be placed atleast five times along a flow margin (i.e. N = 5 is a min-imum value) is sufficient to prevent this artifact fromcompromising our results.

Calculating fractal dimension (D)

The fractal dimension of a curve (such as a lava-flowmargin) is a measure of the curve's convolution, or de-viation from a straight line. The fractal dimension (D)can be calculated as:

where m is the slope of the linear least squares fit tothe data on the Richardson plot (see Turcotte 1991 forderivation and more detailed discussion). Becauselava-flow margins are characterized by embaymentsand protrusions and smaller rods traverse more ofthese features, L increases as r decreases. Thus, theRichardson plot has a negative slope (m<0) and

Calculating quadratic coefficient (a)

In the above discussions of calculating fractal dimen-sions and correlation coefficients, the data on the Rich-ardson plot are fit with a least squares line. Alternate-ly, the data can be approximated by a second-orderleast squares fit and the quadratic coefficient (a) canprovide insight into fractal tendency. An ideal fractalwould be expected to have a — Q. (We tested this meth-odology on an ideal, computer-generated fractal andfound a = 0.002). A negative value of a on a Richard-son plot (concave-downward) translates to an increasein slope with increasing rod length, indicating a relativelack of small-scale features. A positive value of a (con-cave-upward) correlates with a decrease in slope withincreasing rod length, or a relative lack of large-scalefeatures.

Field measurement technique

We applied our methodology to lava-flow marginsboth in the field and on aerial photographs and otherimages. The field technique requires two people, a tapemeasure, and measuring rods of various lengths. Weuse wooden dowels to define the smaller rod lengths(1/8, 1/4, 1/2 and 1 m) and lightweight chains to definethe longer rod lengths (2, 4, 8 and 16 m). First, we iso-late a section of flow margin to be measured and,somewhat arbitrarily, choose a point along the marginas the starting point. When the selected section of flowmargin is sufficiently long to permit, the measurementbegins with one person holding one end of the 16 mchain at the starting point (a). A second person walksalong the flow margin until the other end of the tautchain exactly intersects the outline. This new point (b)becomes the next starting point. Now, as the secondperson holds the end of the 16 m chain fixed over pointb, the first person walks along the boundary until thenext intersection point (c) is found. This process con-tinues until a given number of lengths (N) are mea-sured, and the ending point is marked. To maximizeaccuracy, the measurement is replicated using the same'

197

Table 1. Directional analysis. N values obtained by replicatingfield measurement in opposite direction

Table 2. Error analysis (field data of 1972 Mauna Ulu pahoehoe,segment 1)

r (meters) N,

16842

1

5.0011.5226.7363.61

140.10

5.0210.8125.5563.63

140.19

chain length, but this time the persons walk in the op-posite direction (from the ending point to the startingpoint). We have found that the N values from both di-rections match well (Table 1). The results (N) are aver-aged and L (in meters) is calculated as L = Nr. Ideally,this first length calculation (L:) will be based on fivelengths of a 16 m chain, so L, =80 m.

We then recalculate the length of the same segment(L2), using a chain half of the original length (r = 8 m).Since the 8 m chain will, in all probability, record someundulations in the flow margin that were not encoun-tered by the 16m chain, L2>Li, implying N2>10.Note that it is possible (and likely) that N2 will be afraction. We continue dividing the chain length by twoand repeating the procedure until at least five measur-ements of L have been made using five different rodlengths, i.e. the Richardson plots have a minimum offive data points.

For sufficiently long flow-margin segments, thesedata points generally correspond to chain lengths of 1,2, 4, 8, and 16 m. In some cases, we included an addi-tional rod length of 0.5 m. For shorter flow-margin seg-ments that cannot accommodate five lengths of a 16mchain, the first (longest) chain length we chose is thelongest chain length that can be walked along the flowmargin at least five times. In these cases, rod lengthssmaller than 1 m are necessarily used to meet the min-imum requirement of five measuring rods/chains, sepa-rated by a factor of two in length. The smallest rodlengths used were 0.25 m for a'a flows and 0.125 m forpahoehoe flows.

Error and variation analyses of field measurementtechnique

We conducted analyses, based on field measurements,to confirm both the field measurement technique'sprecision ('error analysis') as well as its applicability tothe entire flow margin ('variation analysis'). To assessthe precision, we conducted five replicate measure-ments of a typical Hawaiian pahoehoe margin: a por-tion of the 1972 Mauna Ulu pahoehoe flow (KilaueaVolcano). We began each measurement at the samestarting point, and measured off five lengths of a 16mchain. Therefore, the ending points of each measure-ment did not necessarily coincide, but instead werechosen such that Lx = 80 in each case. Each measure-ment consisted of five data points, corresponding tochain lengths of 1, 2, 4, 8 and 16 m. The results of this

Trial number D

12345

Mean D value:Standard deviation:

1.1631.1731.1771.1821.182

1.1750.008

0.9800.9770.9880.9800.990

Table 3. Variation analysis (field data of 1972 Mauna Ulu, pahoe-hoe)

Segment number D

1 (avg.)234567

Mean D value:Standard deviation:

1.1751.2071.3151.1861.1831.1611.185

1.2020.052

0.9870.9580.9600.9970.9840.9800.956 -

error analysis are summarized in Table 2. Note the ne-gligible variance of D: cr=0.008. Although this erroranalysis implies that the technique is precise, it doesnot suggest that the calculated D of a given flow-mar-gin segment is representative of the entire flow. Differ-ent segments of a flow margin may have different frac-tal dimensions, and this error analysis does not meas-ure this segment-to-segment variation. Therefore, weperformed an additional analysis on the 1972 MaunaUlu pahoehoe flow to rigorously study variation alonga flow margin. We measured D of seven adjacent seg-ments of a flow margin in the field, with each segmentdefined as five lengths of a 16 m chain (Lt =80). Theseresults, summarized in Table 3, show a significantlylarger variation, with <r=0.05.

Photographic measurement technique

A form of the same 'structured-walk method' was uti-lized to determine fractal dimensions of lava flowsfrom aerial photographs and radar images, at scalesranging from 1:6000 to 1:70 000. We tried to use flowmargins in the centers of the images to avoid distor-tion.

The margins were digitized and the fractal dimen-sions calculated using the EXACT algorithm (Hay-ward et al. 1989). Computerization facilitates changingthe rod lengths in small increments, improving the pre-cision of the calculated D. We used. 30 rod lengths,equally spaced on a log scale. (Using more than 30 rodlengths did not significantly improve the calculated D.)

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Consistent with the field methodology, the minimumflow-margin segment included in the aerial photogra-phic data set corresponds to N = 5 for the longest rodlength, and fractional N-values were permitted for sub-sequent rod lengths. The actual length of this longestrod depends on the scale of the image, and ranges upto 2.4 km. The minimum rod length was chosen to besufficiently large as to exceed both the noise inherentin the digitization process as well as the spatial resolu-tion of photographic images.

Error and variation analyses of photographicmeasurement technique

Analogous with our analyses of the field technique, weconduct error and variation analyses to confirm thephotographic measurement technique. Since this tech-nique is computerized, it is perfectly reproducible; ev-ery measurement taken from a given starting pointwill, after a certain amount of rod lengths are mea-sured, result in the exact same ending point. Thus, anyerror analysis of fractal dimension would necessarilyyield a—G. In order to assess variation of fractal di-mension among different segments of flow margin, weselect the longest flow margin in the photographic da-tabase (Hell's Half Acre, pahoehoe). We divide thismargin, which contains over 8000 data points, into sev-en overlapping flow-margin segments. Each of thesesegments contains 2000 points and overlaps adjacentsegments by 1000 points. Thus segments 1, 3, 5, and 7are non-overlapping, as are segments 2, 4 and 6. To beconsistent with our field variation analysis, we wouldideally like to have seven non-overlapping flow seg-ments. However, data limitations prevent this. The re-sults of this analysis, summarized in Table 4, show acomparable variation, with <r=0.04.

Data

The database consists of 55 lava flow margins (or seg-ments thereof). The selected margin may be of an indi-vidual eruptive unit or a compound flow field. Inchoosing suitable candidates for measurement, we

used the following 'simple-case' criteria: (1) The mar-gin is continuous, well-preserved and unambiguous(e.g. not obscured by forest or younger flows); (2) It isunaffected by external controls, such as a steep groundslope or preexisting topography; (3) The segment isrepresentative of the entire margin. We categorize theanalyzed flows based on composition, separating thebasalts from the more silicic flows. We further dividethe more silicic flows based on silica content. This da-tabase is an extension of that considered by Bruno etal. (1992), which included 28 basaltic lava flows.

Basaltic lava flows

This analysis of basaltic lava flows is based on twotypes of data: (1) field studies of 27 lava flows on Ki-lauea, Mauna Loa and Hualalai volcanoes on Hawaii.These flows have different morphologies, and includeseven a'a, 16 pahoehoe and four 'transitional' flows,i.e. flows with morphologies intermediate between a'aand pahoehoe; (2) aerial photographs of 18 lava flowsin Hawaii, the western US, Iceland, and the GalapagosIslands. These flows include eight pahoehoe and tena'a. No transitional flows are included in the photo-graphic database. Scales of photographs range from1:6000 to 1:60000, which determine the rod lengthswhich range from 12 m to 2.4 km. Including the fielddata, the scale extends down to 0.125 m for pahoehoeflows and 0.25 m for a'a flows. The database for basal-tic flows is summarized in Table 5a.

Silicic lava flows

This analysis of silicic lava flows is based exclusively ondata obtained from aerial photographs and radarimages; no field data have been taken to date. The da-tabase, summarized in Table 5b, consists of ten flowswith silica contents ranging from 52 to 74%. We dividethese flows into two categories based on silica content:basaltic andesites (SiO2: 52-58%) and more silicicflows (SiO2: 61-74%), the latter being primarily dacitesand rhyolites. These images have scales ranging from1:8250 to 1:70000, which determine the lengths ofrods used (10 m-4.5 km).

Table 4. Variation analysis (photographic data of Hell's HalfAcre, pahoehoe)

Segment number D R2

1234567

Mean D value:Standard deviation:

1.2041.2631.2431.1881.1771.2181.270

1.2230.036

0.9700.9530.9360.9540.9690.9600.953

Results and discussion: basaltic lava flows

Basaltic lava flow margins are fractals

Our preliminary results (Bruno et al. 1992) indicatedthat both a'a and pahoehoe flow margins are fractalswithin the range of scale studied (r: 0.5 m-2.4 km).Richardson plots are linear (Fig. 5), demonstratingself-similarity. The present analysis confirms that con-clusion based on a larger database (45 flows) and overa wider range of scale (r: 0.125 m-2.4 km). Further-more, transitional flows have also been shown to befractal. The only cases where the margins of basaltic

199

Table 5a. Database of basaltic flows

Flow description

Kilauea Volcano, Hawaii1971 Mauna Ulu1972 Mauna Ulu1972 Mauna Ulu1972 Mauna Ulu1972 Mauna Ulu1972 Mauna Ulu1972 Mauna Ulu1974 Mauna Ulu1974 Mauna Ulu1974 Mauna Ulu1974 Mauna Ulu1974 Mauna Ulu1974 Mauna Ulu1977 Pu'u O'o1982 Kilauea1990 Pu'u O'o

Mauna Loa Volcano, Hawaiiprehistoric, nr Saddle Rdprehistoric, nr Pu'u Kiprehistoric, nr Pu'u Ki1843 Mauna Loa1843 Mauna Loa1852 Mauna Loa1855 Mauna Loa1855 Mauna Loa1855 Mauna Loa1855 Mauna Loa1859 Mauna Loa1859 Mauna Loa1881 Mauna Loa1899 Mauna Loa1935 Mauna Loa1935 Mauna Loa1935 Mauna Loa1942 Mauna Loa

Hualalai Volcano, Hawaii1800 Hualalai1800 Hualalai1800 Hualalai1800 Hualalai

Non-Hawaiian VolcanoesHell's Half Acre, IdahoVolcano Peak, CaliforniaFernandina, GalapagosFernandina, GalapagosFernandina, GalapagosKrafla, Iceland

Flow type

pahoehoepahoehoepahoehoepahoehoepahoehoea'aa'apahoehoetransitionaltransitionala'aa'aa'aa'apahoehoepahoehoe

pahoehoepahoehoepahoehoea'apahoehoepahoehoepahoehoepahoehoepahoehoetransitionala'apahoehoepahoehoea'aa'apahoehoepahoehoea'a

a'atransitionala'aa'a

pahoehoepahoehoea'aa'aa'apahoehoe

D

1.191.20 (avg)1.181.211.201.051.061.151.101.121.071.091.081.051.211.18

1.231.231.121.111.151.131.191.171.191.091.071.141.171.131.081.201.151.07

1.061.151.091.08

1.211.231.071.091.051.16

R2

0.9620.9940.9730.9870.9820.9900.9880.9630.9750.9770.9870.9630.9650.9670.9890.995

0.9880.9970.9540.9720.9690.9920.9600.9860.9790.9610.9650.9700.9700.9810.9730.9560.9880.973

0.9680.9920.9670.995

0.9810.9630.9720.9520.9850.971

Data type

fieldfieldfieldfieldfieldfieldfieldfieldfieldfieldfieldfieldfieldphotofieldfield

fieldfieldfieldphotofieldphotophotofieldfieldfieldphotofieldphotophotophotophotofieldphoto

photofieldfieldfield

photophotophotophotophotophoto

Substrate(field data only)

ashpahoehoepahoehoepahoehoepahoehoepahoehoepahoehoepahoehoepahoehoepahoehoepahoehoepahoehoepahoehoe

ashpahoehoe

a'aa'apahoehoe

pahoehoe

pahoehoea'aa'a

a'a

a'a

pahoehoepahoehoepahoehoepahoehoe

flows are not fractal are on steep slopes. In these caseswhere the margin is externally controlled by a steepground slope, the margin becomes more linear, withfewer convolutions.

The fractal behavior of pahoehoe and a'a flowsmight be predicted by their basaltic composition. Lowviscosities of the order of 1000 Pa-s for typical eruptiontemperatures of 1150°C, coupled with a negligibleyield strength for most basalts, offers no obstacle toprevent self-similar features from being formed on awide range of scales. We note that at some small scale

below the detection limit of this study, fractal behaviorwill eventually break down due to material proper-ties.

Pahoehoe and a'a have different D

We find that over a wide range of geographic locations(Hawaii, Iceland, western US, Galapagos Islands), bas-altic lavas divide into two populations with regard totheir fractal dimensions. A'a flows generally have D

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Table 5b. Database of silicic flows

Flow description SiO2(%) Flow type Scale of image Reference

1 Andes Mountains 522 Andes Mountains 523 1980 Hekla, Iceland 554 1991 Hekla, Iceland 555 SP Flow, Arizona 516 Lava Park Flow, California 617 Ludent, Iceland 658 1104 Hekla. Iceland 659 Chao, Chile 66

10 Glass Mountain, California 74

Bas. AndesiteBas. AndesiteBas. AndesiteBas. AndesiteBas. AndesiteAndesiteDaciteDaciteDaciteRhyolite

1:270001:135001:82501:21500

:36000:30000:8250:8250: 70 000

1:12000

Thorpe et al. (1984)P Francis (personal communication)Gudmundsson et al. (1991)Gudmundsson et al. (1991)Ulrich and Bailey (1987)Smith and Carmichael (1968)Nicholson (personal communication)Sigmarsson (personal communication)Guest and Sanchez (1969)Eichelberger (1975)

2 ABasalts

2.2

2.0

SK = a'aD =1.07R2=0.99

0=phhD =1.21R2= 0.99

-0.5 0.5Logr (m)

1.0 1.5

Fig. 5. Richardson plots of typical a'a and pahoehoe flows, in me-ters, based on field data. High R2 values (>0.95) indicate fractalbehavior. The more convoluted margins of pahoehoe flows trans-late to higher D

S. 3

-Q

Basaltic flows • A'aD Pahoehoe

I1.00 1.05 1.10 1.15 1.20

Fractal dimension (D)1.25

Fig. 6. Histogram of D values of a'a and pahoehoe flows based onfield and photographic data. Both field and photographic meas-urements show pahoehoe flows have higher D than a'a flows.Transitional flows (not shown) tend to have intermediate D

ranging between 1.05 and 1.09 whereas pahoehoe flowstypically have D ranging between 1.15 and 1.23. Figure6 summarizes our results for basaltic flows. Most (12 of14) of the Hawaiian a'a flows have D between 1.05 and1.09; all have D between 1.05 and 1.13. Most Hawaiianpahoehoe flows (18 of 21) have D between 1.15 and1.23; all have D between 1.12 and 1.23. The two pahoe-hoe flows in the western US yield measurements of1.21 and 1.22, consistent with the range of Hawaiianpahoehoe flows. Similarly, the Krafla, Iceland basalt(pahoehoe) falls into the Hawaiian pahoehoe range,with a fractal dimension of 1.16. The three Galapagosflows measured, all a'a, yield D values of 1.05, 1.07 and1.09, in agreement with the range of Hawaiian a'aflows. This is good evidence that regardless of the ex-act nature of the eruption, the pahoehoe flows consis-tently have higher D than a'a flows.

By definition, fractals should have constant rangesof fractal dimensions, regardless of the rod lengthsused to measure D. Thus, if lava flows are fractals overthe range of scale studied, the fractal dimensions ob-tained at the field scale (0.125-16 m) should be similarto the range of fractal dimensions obtained at the aer-ial photographic scale (12 m-2.4 km) for a'a as well aspahoehoe. This is confirmed by our results. All sevena'a flows measured in the field have D between 1.05and 1.09 (Fig. 6), the same range we find for photo-graphic data of a'a flows (Fig. 6). All 16 pahoehoe fieldmeasurements have D between 1.12 and 1.23, com-pared with a range of 1.13-1.23 for photographic dataof pahoehoe flows.

For three flows (all pahoehoe), we measured mar-gins of the same flow in the field and from aerial pho-tographs. The fractal dimensions as measured fromaerial photographs are 1.19 (1855 Mauna Loa), 1.14(1859 Mauna Loa) and 1.20 (1935 Mauna Loa). Fieldmeasurements yielded corresponding D of 1.17, 1.16and 1.15, respectively. These variations in D are withinthe variation of Table 3, and indicate fractal behav-ior.

Flows that we have determined to be transitionalbetween a'a and pahoehoe based on field observationstend to have intermediate fractal dimensions, as mightbe expected. Of the four field measurements of transi-tional flows, three have D between 1.09 and 1.12; thefourth has a slightly higher D of 1.15.

201

Substrate typeA'aAshPahoehoe

Table 6. Slope analysis. Effect of slope on fractal properties of1972 Mauna Ulu a'a flow (field data)

1.12 1.14 1.16 1.18 1.20

Fractal dimension (D)1.22

Fig. 7. Histogram of D values of pahoehoe flows on three differ-ent substrates - preexisting a'a or pahoehoe flow, or ash - basedon field data. There is no apparent correlation between D andsubstrate type

One might expect that the fractal dimensions offlow margins would be affected by the nature of thesubstrate over which they flowed. A pahoehoe marginmight be different on a preexisting a'a flow comparedto a preexisting pahoehoe flow. However, a detailedanalysis shows that D values are unaffected by differ-ences in substrate. We took 16 field measurements ofHawaiian pahoehoe flows. Some of these lavas flowedupon preexisting a'a lava flows (5), others upon preex-isting pahoehoe flows (9), still others atop ash deposits(2). Figure 7 shows the lack of correlation between Dand substrate for these 16 flows. In one case (1855Mauna Loa pahoehoe), we performed a controlled ex-periment on the effect of substrate on D. We measuredD in one location where this pahoehoe flowed over apreexisting pahoehoe, and again nearby (within100 m), where the same flow covered an a'a substrate.The D values obtained for this flow overlying pahoe-hoe and a'a substrates (1.17 and 1.19, respectively) arewell within the observed variation of D along a flowmargin with a constant substrate (see Table 3).

Clearly, a pattern emerges for the fractal dimen-sions of terrestrial basaltic lava flows. Regardless ofgeographic location, lengths of rods used, or substratematerial, pahoehoe flow margins consistently havehigher D than a'a flow margins within the range ofscale studied. This is consistent both with the prelimi-nary results of Bruno et al. (1992) and also the obser-vation that the outlines of pahoehoe and a'a flows arequalitatively different.

A note about topographically controlled flows

Topographically controlled flows have been excludedfrom this analysis because these external controls canhave a significant effect on D. Positive topography (e.g.hills) may deflect or bifurcate flows, increasing the de-gree of flow-margin convolution and therefore increas-ing D. Negative topography (e.g. channels) serves to

Flow Description Flow type D Slope

1972 Mauna Ulu1972 Mauna. Ulu1972 Mauna Ulu

a'aa'aa'a

1.0461.0551.023

0.9900.9880.778

11.6°14.7°27.8°

confine or channelize flows, causing the margin to be-come more linear and thus decreasing D. In manycases, these external controls interfere with the devel-opment of self-similar features, and prevent fractal be-havior. Similarly, we have found fractal behavior tobreak down, with an accompanying decrease in D, onsteep (> 15-28°) slopes (see Table 6). This tendencytoward nonfractal behavior as the gravity-driven forceon the flow increases is consistent with the results pre-sented in Baloga et al. (1992).

Implications for flow dynamics

The fractal properties of lava flows may offer insightinto the dynamics of flow emplacement because frac-tals reflect nonlinear processes (e.g. Campbell 1987).We have made a preliminary evaluation of the nonli-near aspects of flow dynamics to obtain a qualitativeindication of the tendency toward fractal behavior.Following earlier fluid dynamic models (e.g. Balogaand Fieri 1986; Baloga 1987), we depict variations inthe free surface of a lava flow as due to a balance be-tween a gravitational transport term and the fluid dy-namic ('magmastatic') pressure gradient. Baloga et al.(1992) define two dimensionless parameters (p and q)to describe the relative importance of these two in-fluences. The parameter p is the ratio of the pressuregradient to gravitational terms; the parameter q is anabsolute measure of the gravitational force on theflow. Baloga et al. (1992) developed a governing equa-tion for the three-dimensional surface of a lava flowduring emplacement, based on simplifying assump-tions:

ah/at + q h2 (dh/dx) = p q d/dy [h3 (dh/dy)]wherep = cot#h0L/(3w2)q = gsin#hz,T/(j'L)

and where x and y are the downstream and cross-stream directions respectively, h = flow thickness,t = time; h0, L, w and T are scales for thickness, length,width and time, respectively; 0—slope, v—kinematicviscosity and g = gravitational acceleration.

By assuming dh/dt is on the order of 1, Baloga et al.(1992) evaluated this equation for selected values of pand q (Fig. 8). High p values (right column of the ma-trix) indicate the magmastatic pressure gradient is im-portant relative to gravity. Low q values (top row ofmatrix) indicate a weak gravitational term. Thus, incase Ic (large p, small q), the gravitational term is theleast important, both relatively (to the pressure gra-dient) and absolutely, and the magmastatic pressure

202

p « 1 p » 1

q « 1 Case la

3h/3t = 0

Case Ib

3h/3t= 0

Case Ic

Assume pq = 0(1)

3h/3t = 3/3y[h3(3h/3y)]

q = 1 Case 2a

3h/3t + h2(3h/3x) = 0

Case 2b

3h/3t + h2(3h/3x) = 3/3y[h3(3h/3y)]

Case 2c

0 = 3/3y[h3(3h/3y)]

q»l Case 3a

Assume pq = 0(1)

h2(3h/3x) = 0

Case 3b

h2(3h/3x) = 3/3y[h3(3b/3y)]

Case 3c

0 = 3/3y[h3(3h/3y)]

Fig. 8. Matrix of special cases of thegoverning equation for selected val-ues of p and q, obtained by assumingdh/dt is on the order of 1. Some ofthe equations in the matrix are linear;others are nonlinear. The linearequations would not be expected toproduce fractals, whereas the non-linear equations could be expected toproduce fractals. See text for details

gradient dominates. Thus, since the lava flow is beinglargely driven by internal fluid dynamic forces in caseIc, we predict that this combination of p and q is likelyto produce fractal behavior. As expected, the resultingdiffusion equation is explicitly nonlinear.

For the same q (q<^l), consider the cases corre-sponding to p values that are low (case la) and moder-ate (case Ib). Both of these equations are linear, andwould therefore not be expected to produce fractals.Since p is proportional to the ratio of magmastaticpressure gradient to gravitational driving force, this hasimportant implications for the effect of gravity on frac-tal behavior. When gravity plays a non-negligible role(small or moderate p), the matrix predicts that thelava-flow margin would not be fractal. This is consis-tent with our field observations on Hawaii that flowoutlines are not fractals when slopes are steep.

Case 2b is nonlinear diffusion with a kinematictransport term. Case 3b is the steady-state nonlineardiffusion equation. These are also likely candidates forproducing fractals. Cases 2c and 3c are both nonlinearand are dominated by the pressure gradient term(p> 1). These cases may be expected to produce fractalbehavior, but are difficult to interpret physically.

This analysis suggests that nonlinear processes arecommon in lava flows, particularly in those caseswhere the magmastatic pressure gradient influence issignificant relative to the influence of gravitationaltransport. These nonlinear equations are candidatesfor producing fractals, provided they are physicallyplausible. Further studies are underway to (1) test thisphysical plausibility by continued comparison of theor-etical prediction and field measurements and (2) ex-tend the underlying physics to include more complexrheologic properties for lava flows of different compo-sitions.

Results and discussion: silicic lava flows

Silicic lava flows are generally not fractals

Silicic lava flows are generally not fractals within therange of scale studied (r: 10m-4.5 km). Typical Rich-ardson plots for basalt, basaltic andesite, and dacite areshown in Fig. 9. Unlike the basaltic case, the Richard-son plots for basaltic andesite and dacite are nonlinear,characterized by relatively low R2 values. Instead offractal behavior, these Richardson plots exhibit scale-dependent behavior: longer rod lengths have steeperslopes, most notably for the dacite. Thus, D tends toincrease as r increases, contradicting the definition ofD as a scale-independent parameter. This breakdownof fractal behavior at increased silica content is pre-sumably related to the higher viscosities and yieldstrengths, which suppress smaller-scale features andthus prevent self-similarity over a wide range ofscales.

Quantifying the effect of silica content on D

We seek to develop parameters that can be used re-motely to quantify the effect of increasing silica con-tent on fractal properties by comparing basalts, basalticandesites, and dacites/rhyolites for two main purposes:(1) to gain insights into yield strength and rheologicalprocesses, and (2) to develop a remote-sensing toolthat can differentiate flow type based on plan-viewshape. Our approach is to use the study of basalticflows as a benchmark for comparison with the moresilicic flows. However, we restrict our basaltic 'bench-mark' to a'a flows, which are similar to silicic flows interms of both morphology and emplacement mecha-nism.

203

3.75

3.70

E-J

g>

3.65

3.604.10

400

-§ 3.90_j

O)

° 3.80

3.70

3.604.60

4.55

4.50

4.45

4.40

a Basalt

b Basaltic andesite

c Dacite

1 2 3 4

Log r(m)

Fig. 9a-c. Richardson plots of representative a a'a basalt (Gala-pagos Islands); b basaltic andesite (Hekla, Iceland); c dacite(Chao, Chile), based on image data. Note that the data in (a) areclosely approximated by a straight line, whereas the data for thehigher silica flows (b, c) are not linear

Ideally, we would like to compare D of silicic flowsto basaltic flows, to see how D changes with silica con-tent. However, this approach is tricky because, asnoted above, silicic flows are generally not fractals; in-stead D tends to increase with r for the majority of thesilicic flows. Hence, the concept of a scale-independentfractal dimension for silicic flows is not valid. However,small regions of logr can be locally fit with a line. Herewe introduce the concept of a 'local fractal dimension'.This does not imply the data set is fractal, nor that thelocal fractal dimension is scale-independent. It simplyexploits our observation that select regions of the datacan be fit by a line and we can estimate locally the de-gree of convolution for a selected range of rod lengths.Here we describe two methods used to compare silicicand basaltic flows. Both of these methods are sensitive

to - and based on - our observation that silicic flowsdo not have scale-independent fractal dimensions.

Method 1: disjoint subsets of logr

This method dissects the abscissa of the Richardsonplot into disjoint subsets of logr. The specific choice ofsubsets (summarized in Table 7a) is constrained by thedata. Each of these subsets is fit locally by a leastsquares line; that is, the Richardson plot is fit by a se-ries of lines. For each line, the slope (m) is calculated,and local fractal dimension D is calculated as 1 - m,consistent with our methodology for basaltic flows.Since this method can be used to describe fractals aswell as non-fractals, it can be employed to comparebasaltic and silicic lava flows.

Figure 10 shows sample Richardson plots of basalt,basaltic andesite and dacite, with the abscissa dissectedaccording to the methodology described above. Thedata on these plots are the same as shown in Fig. 9; theonly difference is the number of lines used to fit thedata. Note that for the basalt, the three segments haveessentially the same slope. This is consistent with ourconclusion that basalts are fractals. Unlike the basalts,the basaltic andesite and dacite show noticeable differ-ences in slope among the various subsets.

By plotting D of these segments vs. logr for the en-tire database of silicic flows, patterns begin to emergeamong the basaltic andesites and the more silicic flows(primarily dacites and rhyolites). The basaltic andesiteshave roughly the same D values for the first two sub-sets (Fig. 11). At rod lengths of about 100m(logr = 2m), D starts to increase, and the values alsohave a greater scatter. For the first three subsets oflogr, the dacites/rhyolites have D plotting in a rathercompact area, showing only negligible differencesamong the various ranges. At logr-2.5m, D appar-ently begins to increase. We can use this technique todistinguish basaltic andesites from the more silicicflows. Both have a general increase in D with longer r,but the basaltic andesites tend to have greater D foreach of these categories. Furthermore, the fact that da-cites/rhyolites show negligible changes within the firstthree subsets (logr < 2.5 m), whereas the basaltic ande-sites only remain relatively constant for the first two

Table 7. Ranges of logr (meters) for a Method 1 and b Meth-od 2

METHOD 1Log r (meters)

b

Range 1:Range 2:Range 3:Range 4:

METHOD 2

Range 1:Range 2:Range 3:

<1.51.5-2.02.0-2.52.5-3.3

Log r (meters)1.7-2.81.7-2.51.3-2.0

204

3.75

3.70

3.65

3.504.10

4.00

3.90

3.80

3.70

3.604.60

4.55

4.50

4.45

4.40

a Basalt

c Dacite

Log r(m)

Fig. 10. Dissected Richardson plots of representative samples of aa'a basalt (Galapagos Islands); b basaltic andesite (Hekla, Ice-land); c dacite (Chao, Chile), based on photographic data (samedata as Fig. 9). Here, the abscissa is dissected into ranges of logr(Table 7a), and each range is locally fit with a straight line. Localfractal dimension is calculated as D = 1 — m, where m is the localslope as calculated according to Method 1. From left to right ofthese Richardson plots (increasing r), these local fractal dimen-sions are: a D = 1.07 for each segment; b D = 1.17,1.19,1.19,1.46.(c) 1.02, 1.03, 1.05, 1.20. Note that D values are constant for (a),but not for (b) and (c). See text regarding Method 1

subsets (log r< 2.0m), is apparent. Figure 11 also em-phasizes that D is not a constant function of logr forboth basaltic andesites and dacites/rhyolites, indicatingscale-dependent (non-fractal) behavior. Fractals suchas basalts have relatively constant fractal dimensionsacross the various subsets. However, a sufficientlylarge range of logr is needed to discern fractal andnon-fractal behavior. Note that a'a basalts and daciteshave a similar range of fractal dimensions for the firstthree categories. Since data limitations often preventobtaining such a large range of logr, we invoke a sec-

1.0

1.1

1.2

1.3

1.4

1.5

Silicic flowsI

AA6ooo

-

I

A

iAO

O8

iA

O

OO

O

i

A :

i ;

AO

o ;o

o

<1.5 1.5-2.0 2.0-2.5 2.5-3.3Ranges of log r (m)

Fig. 11. Summary of 'local fractal dimension' (D) based on photo-graphic data for entire database of silicic flows. Basaltic andesitesare shown as open circles, whereas dacites and rhyolites areshown as solid triangles. Note that D is not a constant function oflogr indicating non-fractal behavior

ond method to differentiate a'a basalts from dacites,described below.

Method 2: overlapping subsets of logr

Like Method 1, this method dissects the abscissa of theRichardson plot into distinct regions of logr. However,it is different from the previous method in two re-spects. First, the selected ranges (as summarized in Ta-ble 7b) of logr are overlapping. Although the exactchoice of ranges is again constrained by the data, theywere intentionally chosen to overlap. This is to expli-citly show the effect of a restricted range of rod lengthson local fractal dimension. For example, by comparingRegion 1 (logr: 1.7-2.8 m) and Region 2 (logr: 1.7-2.5 m), we can explicitly see the effect of a restrictedrange of logr (2.5-2.8 m) on D and R2. Second, asthese regions span a greater range of logr than those inMethod 1, we have sufficient data points to fit a sec-ond-order least squares curve to the data, in additionto the standard first-order least squares line. In thismethod, we fit a curve of the form y = ax2 + bx + c, andnote the value of the leading (or quadratic) coefficienta. In summary, this method compares three quantita-tive parameters (D, R2, a) for three overlapping rangesof logr.

Applying Method 2 to dacites and rhyolites, wenote systematic variation in D, R2 and a with range ofrod length. In four out of five cases, the longest rodlengths (Range 1) have the highest D and the lowest

corresponding R2 values, whereas the shortest rodlengths (Range 3) have the lowest D and the highestR2 values. Fitting a quadratic curve to Range 1 in eachcase yields a negative a. For Range 3, a can be eitherpositive or negative. The D, R2 and a for the dacites/rhyolites provide remarkably consistent results: all sug-gest scale-dependent (or non-fractal) behavior, charac-terized by an increase in D with increasing r. We attri-bute this to the suppression of small-scale features, dueto the higher viscosities and yield strengths of silicicflow. We suggest the margin appears 'linear' to a cer-tain range of small rod lengths because the scale of fea-tures they would otherwise detect are suppressed. Thisexplains a fractal dimension close to 1 for the shortestrange of rod lengths and, as expected, the correspond-ing R2 values are quite high. We interpret these resultsto suggest that the shortest range of rod lengths (logr:1.3-2.0 m) detects features in the flow margin that arebelow the limit of self-similarity.

We now present the results of Method 2 applied tobasalts (i.e. those basalts which we measured using theranges of r shown in Table 7b). Fractal dimension andthe corresponding R2 values show no systematic varia-tion with range of rod length. The R2 values are high,generally exceeding 0.95. The parameter a can be posi-tive or negative, is generally close to zero, and againshows no systematic pattern among the various ranges.These results for D, R2 and a for basalts all suggestfractal behavior.

We can use these fractal parameters to remotely dif-ferentiate flow types. Basaltic a'a and basaltic ande-sites can be distinguished primarily by their D values;basaltic andesites generally have higher D (>1.15)than basaltic a'a (D: 1.05-1.09) and are less likely toexhibit fractal behavior. Although dacites/rhyolitesand basalts have similar fractal dimensions (1.05-1.10)for extensive ranges of logr, dacites and rhyolites dis-tinctly show non-fractal behavior. Systematic evalua-tion of D, R2, and a at different range of rod lengths(as done in Method 2) can be used to distinguish da-cites and rhyolites from basalts remotely.

There may be a critical value of r, related to silicacontent, which serves as a boundary for self-similar be-havior (i.e. a value of r above which the flow appearsfractal). This critical value may be related to lobe di-mensions and/or the degree of suppression of smaller-scale features. Note that Fig. 11 shows a marked in-crease in D for dacites after about logr of 2.5m(r = 300m). This may be related to the lobe width ofdacites, typically hundreds of meters. If so, we wouldexpect the apparent D of basaltic andesites to increaseat shorter rod lengths. This may be suggested by Fig.11 but our database is too small to be conclusive. Webelieve that a larger database of silicic flows would re-veal a critical value of breakdown of fractal behaviorrelated to silica content. The fact that basaltic andesitesappear to have relatively constant fractal dimensionsup to log r = 2 m while dacites/rhyolites appear to haverelatively constant fractal dimensions up tolog r = 2.5m suggests an effect of yield strength whichis related to silica content. Our field observations

205

shows that fractal behavior for basalts also breaksdown, but at r < 10 cm.

The suppression of smaller-scale features in silicicflows implies that nonlinear instabilities are also sup-pressed inside the flows. Either the sluggish rheologyprevents their formation, or it prevents their growth byrapidly damping out feedback mechanisms. The gener-ally non-fractal nature of the margins of silicic flows isconsistent with our simplified flow model (Fig. 8). Vis-cosities of silicic flows are very large, > 106 for basalticandesites and >108 for dacites and rhyolites, so q iscertainly <f 1. Thus, unless the flows have very largeinitial pressures, it is likely that their behavior wouldtend to be linear.

Conclusions

1. Basaltic lava flows are fractals

Bruno et al. (1992) suggested that basaltic lava flowsare fractals, with pahoehoe flow margins having higherfractal dimension (1.13-1.23) than a'a flow margins(1.05-1.09). This study, based on a larger database (45flows) and over a wider range of scale (0.125 m-2.4 km), confirms that earlier conclusion. Richardsonplots are consistently linear, characterized by high R2

values. Furthermore, we have shown that basaltic lavaflows having transitional morphologies also exhibitfractal behavior, and tend to have dimensions interme-diate between a'a and pahoehoe. This indicates thatbasaltic lavas, regardless of the emplacement mecha-nism, exhibit self-similar behavior. We interpret this tosuggest that basalts are sufficiently fluid and lack asizeable yield strength, offering no obstacle to deterthe formation of small-scale self-similar features.

2. Silicic flows are generally not fractals

Unlike basalts, silicic lava flows tend to exhibit scale-dependent (non-fractal) behavior within the range ofscale studied (r: 10 m-4.5 km). Typical Richardsonplots for basaltic andesites and (especially) the moresilicic dacites and rhyolites are non-linear. This break-down of fractal behavior at increased silica content ispresumably related to the higher viscosities and yieldstrengths, which suppress smaller-scale features.

3. Flow dynamics are nonlinear

Our observations that basaltic lava flows have fractaloutlines when they are internally controlled yet havenon-fractal outlines when they are controlled by gravi-tational forces are consistent with our theoretical mod-el. An assessment of flow dynamics suggests that nonli-near processes operate for lava-flow emplacement onrelatively flat slopes. These nonlinear mechanisms aredamped out in silicic flows, leading to non-fractal mar-gins, especially at small rod lengths.

206

4. Quantifying the effect of rheology

One of the primary objectives of this study is to re-motely distinguish flow types. We suggest that fractaldimension (or local fractal dimension), correlationcoefficient, and quadratic coefficient can be used, incombination, to attain this objective. We define 'localfractal dimensions' for select ranges of logr, and findthat D tends to increase with increasing r after certaincritical rod lengths are exceeded. We can use localfractal dimension to differentiate basaltic andesitesfrom dacites and rhyolites. Although basaltic a'a anddacites have similar fractal dimensions over a widerange of r, the parameters R2 and a can be used to re-motely differentiate between these flow types.

Acknowledgements. This research was funded by the followingNational Aeronautics and Space Administration (NASA) grants:NOT 50930 (NASA Graduate Student Researchers Program),NAGW 3684 and NAGW 1162 (NASA Planetary Geology andGeophysics Program). Additional support was provided by theUniversity of Hawaii Project Development Fund and Harold TStearns Foundation. This manuscript benefited from reviews byH Shaw and G Valentine. Images were kindly provided by Dun-can Munro (Galapagos Islands mosaic), Rosaly Lopes-Gautierand Peter Mouginis-Mark (Western US), Peter Francis (Andes)and Thorvaldur Thordarson (Iceland). Paul Lucey and HaroldGarbeil provided software support. Many thanks go to MichelleTatsumura for field assistance. This is IGP Publication No. 764and SOEST Contribution No. 3578. This work forms part of thePhD dissertation of B Bruno.

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