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The fractal and multifractal dimension of volcanic ash particles contour: a test study on the utility and volcanological relevance
Journal of Volcanology and Geothermal Research 113 (2002) 1^18 www.elsevier.com/locate/jvolgeores
The fractal and multifractal dimension of volcanic ash particles contour: a test study on the utility and volcanological relevance P. Dellino , G. Liotino Dipartimento Geomineralogico, Universita' di Bari, Via E. Orabona 4, 70125 Bari, Italy Received 27 October 2000; received in revised form 14 June 2001; accepted 14 June 2001
Abstract Image processing analysis is used to check the ability of the fractal dimension for quantitatively describing the shape of volcanic ash particles. Digitized scanning electron microscopy images of fine pyroclasts from the eruptions of Monte Pilato^Rocche Rosse (Lipari, Italy) are investigated to test the efficiency of the fractal dimension to discriminate between particles of different eruptive processes. Multivariate analysis of multiple fractal components allows distinction between magmatic particles and phreatomagmatic particles, which however is less significant than the discrimination obtained in previous studies by the use of simple ‘adimensional’ shape parameters. Approximation of the actual particle boundary and the not rotation invariant nature of the fractal data frequently result in a significant scatter of data points in the Mandelbrot^Richardson plot. Such behavior obscures in some cases the actual information of particle shape and renders the discriminating power of fractal analysis less effective than classical shape descriptors. Data less affected by scatter reveal that phreatomagmatic particles of the Monte Pilato^Rocche Rosse eruptions are true (mono) fractals, whereas magmatic particles are multifractals. The textural (small-scale) fractal of magmatic particles is similar to the fractal dimension value of phreatomagmatic particles, and is attributed to the rheological behavior of melt upon brittle fragmentation. The structural (large-scale) fractal of magmatic particles refers to the walls of ruptured vesicles that lay on the particle outline. The high difference between the values of the textural and structural fractals of magmatic particles of the Monte Pilato^Rocche Rosse eruptions suggests two distinct and independent processes in the formation of such pyroclasts. At the scales corresponding to the textural fractal, the fragmentation process is independent of vesicles. Magmatic fragmentation is not simply related to growth, expansion, interference and explosion of vesicles but to a brittle cracking of the highly viscous melt, likely related to rapid decompression. 5 2002 Elsevier Science B.V. All rights reserved. Keywords: shape analysis; fractal dimension; brittle fragmentation
1. Introduction * Corresponding author. Fax: +39-080-5442591. E-mail address: [email protected] (P. Dellino).
It is widely accepted in geology that the shape of particles carries information. The investigation of ash particles represents a well known approach,
0377-0273 / 02 / $ ^ see front matter 5 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 7 3 ( 0 1 ) 0 0 2 4 7 - 5
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in volcanology, for identifying the process responsible for pyroclastic deposit formation. The peculiar morphological features that pyroclasts related to di¡erent eruption dynamics show by scanning electron microscopy (SEM) are in fact systematically described in various atlas and publications (Heiken, 1974; Heiken and Wohletz, 1985; Wohletz, 1987). Classi¢cation and interpretation of data is however frequently subjective and interpretation heavily depends on the experience and training of the operator. A big e¡ort has been made, especially in sedimentology, to obtain quantitative shape parameters e¡ective in the process characterization of clastic particles. Conventional parameters, such as those referring to simple geometric forms, are useful for describing particles with a regular shape, however they are not very e¡ective in describing the shape of the irregular particles that frequently occur in sedimentary deposits (Orford and Whalley, 1983). This problem is ampli¢ed with pyroclastic particles that often show very complex morphologies, which, according to Marshal (1987), is the reason why the analysis of volcanic clast shape has remained largely qualitative. Attempts have been made for objectively identifying particle types and particle populations by means of multivariate statistical classi¢catory schemes (Sheridan and Marshall, 1983; Sheridan and Kortemeier, 1987; De Rita et al., 1991). Dellino and La Volpe (1996) recently proposed the use of simple adimensional shape parameters for describing the ash particles of the explosive eruptions of Monte Pilato^Rocche Rosse (Lipari, Italy). Multivariate elaboration of data allowed a good discrimination between ash clasts related to phreatomagmatic and magmatic processes. These authors suggested that, even if a single parameter is not very e¡ective for uniquely describing particle shape, the combination of multiple parameters allowed a useful and e¡ective characterization to be obtained. The reliability of fractal analysis methods to describe particle morphology has increasingly improved in recent times, and they are nowadays widely used in various branches of research, from material science, to biology and to geology (Be¤rube¤ and Je¤brak, 1999; Gonzato et al., 1998). Such
methods revealed quite successful in describing complex^irregular shapes of sedimentary particles, and the use of the fractal and multifractal dimension has proved e¡ective in describing and interpreting detrital clasts (Orford and Whalley, 1983). In a recent paper, Carey et al. (2000), using a methodology similar to that used in the present work, demonstrated the ability of fractal dimension to discriminate between primary and reworked jokulhlaups deposits in Iceland. In the present paper we constrain the use of the fractal and multifractal dimension for describing the shape of volcanic ash particles and in particular for addressing fragmentation dynamics. To test the signi¢cance of the method, the same particles from the Monte Pilato^Rocche Rosse eruptions used by Dellino and La Volpe (1996) have been investigated, and results are compared against the more conventional shape parameters previously utilized.
2. The fractal analysis method The utility of fractal analysis in the description of particles stems from the idea that a shape, however irregular or complex, can be circumscribed by equilateral polygons. The shorter the sidelength, the greater the number of sides, the longer the polygon perimeter becomes and the better the approximation of such polygons is to the particle outline. The actual particle perimeter is eventually reached using a polygon with an in¢nite number of sides. If we perform such analysis using polygons with an increasing number of sides and if we de¢ne as steplength the side of the polygon, we can plot the log of steplength vs. the log of the resulting perimeter on a diagram. If the data points lay on a straight line, the particle is self-similar, in the sense that it has the same degree of irregularity at all scales. This particle has a fractal behavior, it is called a true fractal (mono fractal), and its fractal dimension is de¢ned by D = 13S, where S is the slope of the line, which is easily calculated using least square methods. The rate of increase of perimeter as the steplength decreases is a measure of particle irregularity.
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The log^log diagram is called the Mandelbrot^ Richardson plot (M^R plot), as it was developed by Mandelbrot (1967), on Richardson’s unpublished data. The fractal dimension of a closed bidimensional form (as are SEM images of particles) belongs to the interval 1^2. It re£ects the plane-¢lling tendency of a particle outline (Kaye, 1978). Smooth curves (Euclidean forms) have a fractal dimension equal to 1 and the value increases as the shape acquires irregularities, up to a line so tortuous and irregular as that de¢ned by a Brownian motion (Mandelbrot, 1977), that completely ¢lls a plane and tends to the value 2. The fractal dimension of particles of geological interest actually ranges in a more restricted interval, and Clark (1986) found values between 1 and 1.36. Particles of geological interest generally are not considered true fractals (mono fractals), since in the M^R plot they de¢ne segmented lines (or curvilinear trends). In such an instance the utility of the fractal dimension is retained, and multiple fractal segments are recognized (Orford and Whalley, 1987). Some authors propose the use of the textural (small-scale) and structural (large-scale) fractals, if two segments are clearly distinguished (Kaye, 1978; Flook, 1979). The textural fractal relates to small-scale irregularities, the structural fractal relates instead to large-scale irregularities. When the separation between segments is not easily obtained, or if more segments are identi¢ed, other authors (Kennedy and Lin, 1992) propose that multiple fractal components can be de¢ned by multivariate statistic analysis. Fractal analysis is nowadays performed by means of image processing analysis software, which allows the reconstruction of the particle outline on digitized images. The coordinates of points that lay along the outline represent the particle boundary. In our case analysis was performed on a Macintosh computer using the public domain NIH Image program (developed at the U.S. National Institutes of Health and available on the Internet at http://rsb.info.nih.gov/nih-image/). High-resolution (300 pixels per inch) black and white SEM images were utilized and the thresholding
3
operation allowed the identi¢cation (segmentation) of particle boundary. The boundary was approximated by at least 500 points, and the macro fraccalc, written by M. Warfel, was used to determine the steplengths and relative perimeter lengths, which were necessary for the calculation of the fractal dimension value. The algorithm used was the hybrid method, as described by Clark (1986). The software used is the same as that utilized by Carey et al. (2000), but the algorithm is a little di¡erent, and is brie£y described in the following. If the steplength is de¢ned by r, the calculation starts by determining the distance dp;i from a starting point (xp ,yp ) to the second coordinate (xi ,yi ) using the equation [(xi 3xiþ1 )2 +(yi 3yiþ1 )2 ]1=2 = dp;i . If the distance is less than r the next point (xiþ1 ,yiþ1 ) is then selected, the distance is computed again and the test against the steplength is made again. The process continues until the distance dp;iþk s r. If Mdp;iþk 3rM9Mdp;iþðk31Þ 3rM then the point (xiþk ,yiþk ) is selected, if not, the point (xiþðk31Þ ,yiþðk31Þ ) is selected. Then the calculation is started again from this new point, until the whole particle boundary is covered. As the endpoint of the curve is approached, the distance between the last point and the end-point will be less than r, and a fraction of the steplength is computed for closing the curve. The perimeter is then calculated by P = gN31 i¼1 di;iþ1 . The calculation is then started again with a longer steplength, until a speci¢ed range of r is covered. In the present study 50 steplengths were used, covering the range between three pixels and 150 pixels, which actually corresponds to the interval between 0.5 Wm and 25 Wm. The range is inside the interval of 0.085^0.25 of the particle diameter as suggested by Orford and Whalley (1983). Data points obtained by this method have a certain scatter in the M^R plot (as it happens also using other implementations of the fractal dimension calculation, Carey et al., 2000; Gonzato et al., 1998). The scatter is attributed to three sources of error. The ¢rst source of error is that the boundary may not be fractal. The second source of error results from the fact that the points representing the particle coordinates are an approximation of the real particle boundary.
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The third source of error occurs because the fractal dimension, as implemented on computers, is not rotation invariant (Kennedy and Lin, 1986), in the sense that by changing the starting point along the boundary, the resulting perimeter estimate changes. This problem is clearly addressed at the footnote of Schwarz and Exner (1980), by an anonymous reviewer, and is attributed to the wavelength of irregularities that characterize the particle boundary. As the steplength approaches this characteristic wavelength and if the starting point is in-phase with the wavelength, the perimeter estimate results shorter than expected. To show this drawback a graphic example, based on the synthetic shapes of Fig. 1, is useful. The ¢rst shape represents the silhouette of a perfect square without irregularities, and in the resulting M^R plot the clear Euclidean nature of the form is revealed, as the slope is zero. The second silhouette represents a square with some indentations along the outline that render the shape irregular. The M^R plot of this silhouette reveals how the occurrence of indentations drastically changes the arrangement of data points, and a signi¢cant slope thus results. The third and fourth silhouettes represent the same indented square as the second one, which are rotated 90‡ clockwise and anticlockwise respectively. The M^ R plots reveal how rotations resulted in a di¡erent scatter of points compared to the second silhouette. Such an error, as discussed by Kennedy and Lin (1986), should be smoothed out by calculating the fractal dimension value of di¡erent particle orientations (changing the actual starting point), then averaging data between orientations.
3. Geologic and volcanologic background of study particles Particles investigated in this paper are from the pyroclastic deposits of the 1.4 ka eruptions of
Monte Pilato^Rocche Rosse at Lipari (Aeolian Islands, Italy). These deposits were chosen because eruption characteristics are well known (Cortese et al., 1986; Dellino and La Volpe, 1995). Furthermore, a quantitative shape analysis of ash particles had already been carried out by means of simple adimensional form descriptors (Dellino and La Volpe, 1996). The ability of the fractal dimension in describing pyroclastic particles can be, in this way, directly compared to that of more conventional shape parameters. Monte Pilato^Rocche Rosse eruptions, which were alimented by rhyolitic magma, were characterized by an alternating magmatic/phreatomagmatic style. Such eruptive activity led to the formation of various types of deposits, among which four main lithofacies were identi¢ed and are fully described by Dellino and La Volpe (1995). Lithofacies 1 deposits are well sorted pumice fallout layers of magmatic origin. Lithofacies 2 deposits are massive layers of mixed magmatic^phreatomagmatic origin. Lithofacies 3 deposits are surge layers of phreatomagmatic origin. Lithofacies 4 deposits are accretionary lapillirich layers of phreatomagmatic origin. Dellino and La Volpe (1996) investigated ash particles by SEM, and used adimensional shape parameters, such as circularity, compactness, elongation and rectangularity, to obtain a quantitative shape characterization. Data were elaborated by means of factor analysis and a signi¢cant discrimination resulted between clasts of magmatic and phreatomagmatic origin. In the present study we have selected 89 SEM images, from the same collection used by Dellino and La Volpe (1996), representing as many particles from the four main lithofacies. In particular, particles are between 64 Wm and 125 Wm in diameter. This size fraction was chosen because it is the most useful for discriminating between di¡erent fragmentation dynamics, and represents an important source of knowledge
Fig. 1. On this plate, synthetic silhouettes and the corresponding M^R plots are shown. The in£uence of indentations and rotations on the alignment and scatter of data points is evident. 1 = perfect square; 2 = square with indentations; 3 = the same as 2, with a 90‡ clockwise rotation; 4 = the same as 2, with a 90‡ anticlockwise rotation.
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Table 1 Data of the fractal dimension value (D) of the four test particles of Fig. 2 Particle reference number (orientation)
D value
GR/79 (original) GR/79 (vertical £ipping) GR/79 (horizontal £ipping) GR/79 (90‡ clockwise) GR/79 (90‡ anticlockwise) GI/7 (original) GI/7 (vertical £ipping) GI/7 (horizontal £ipping) GI/7 (90‡ clockwise) GI/7 (90‡ anticlockwise) LR/10 (original) LR/10 (vertical £ipping) LR/10 (horizontal £ipping) LR/10 (90‡ clockwise) LR/10 (90‡ anticlockwise) LI/26 (original) LI/26 (vertical £ipping) LI/26 (horizontal £ipping) LI/26 (90‡ clockwise) LI/26 (90‡ anticlockwise)
1.030 1.031 1.025 1.027 1.033 1.200 1.181 1.197 1.215 1.191 1.021 1.023 1.023 1.023 1.020 1.112 1.118 1.121 1.118 1.135
Data refer to the ¢ve particle orientations used in the analysis.
about the most energetic processes of explosive dynamics, as discussed by Dellino and La Volpe (1996) and by Bu«ttner et al. (1999).
4. Analysis and results A few images showing distinct particle morphologies were ¢rst selected, to allow an easy assessment of the power of fractal dimension on discriminating between di¡erent shapes. The selected particles are shown on Fig. 2. For each particle, to address the error caused by the rotation not-invariant nature of the fractal dimension, ¢ve di¡erent orientations were used. The ¢rst orientation refers to the original position of the particle as shown by SEM ; the second to a 90‡ clockwise rotation of the particle; the third to a 90‡ counterclockwise rotation ; the fourth to a vertical £ipping of the particle; the ¢fth to a horizontal £ipping. On each orientation the fractal dimension was calculated by using 50 steplengths, which covered
the interval between three pixels and 150 pixels, corresponding to 0.5^25 Wm. Data are shown on Table 1. An analysis of variance was then carried out, to test the di¡erence of the fractal dimension among the di¡erent particles (variation among particles) against the di¡erence between the orientations of individual particles (variation within particles). Results show that the fractal dimension among the various particles is di¡erent (at a 95% level of signi¢cance). It is therefore possible to state that even if the di¡erent orientations result in a certain variability of the fractal dimension value of individual particles, this variability is lower than that among particles. The mean value among the ¢ve orientations was then calculated and it represents the average fractal dimension of each particle. It is 1.029 for GR/79; 1.20 for GI/7; 1.023 for LR/10; 1.121 for LI/26. Data con¢rm the visual impression that particles GI/7 and LI/ 26, on having a more irregular boundary, should result in a higher fractal dimension compared to particles GR/79 and LR/10 that have a more regular outline and a lower fractal dimension. The fractal dimension is therefore a useful parameter for distinguishing between the form of ash particles that have evident morphological di¡erences. The analysis of the M^R plots of Fig. 3 (the original position of Fig. 2 images is reported) shows, anyway, that in some cases data points do not lay on a line, and, because of the scatter of points, it is not easy to identify how many segments occur. In order to obtain a more precise characterization of the fractal dimension, we have used, as suggested by Kennedy and Lin (1992), multiple fractal components in the investigation of the complete data set of particles. Five components were used. The ¢rst component is the fractal dimension value resulting from the steplengths between three and 30 pixels ; the second is the one between 33 and 60 pixels and so on. Three orientations were used, the original one, the 90‡ clockwise and anticlockwise rotations. The horizontal and vertical £ipping orientations, on not resulting in additional variations, were omitted. Data from the three orientations were weighted by the correlation coe⁄cient of the least square
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Fig. 2. Photographs of SEM images of the four test particles. Particle reference number is on the bottom-left of photographs.
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Fig. 3. Diagrams showing the M^R plots of the test particles. The original orientation is shown. D = fractal dimension value. R = correlation coe⁄cient. The least square regression line is also drawn. Particle reference number is on the bottom-left of diagrams.
line, and were consequently averaged out using the formula N X Di R i
Dave ¼
i¼1 N X
Ri
i¼1
Where R is the correlation coe⁄cient and N is the number of orientations. Average data of the ¢ve fractal components (Table 2) were then used as variables in a multivariate system that was elaborated by means of factor analysis of the same type as that used by Dellino and La Volpe (1996). Using the ¢rst two factors, which explain more than 70% of the variance of the system, a diagram
(Fig. 4) was constructed, and data points represent individual particles. Particles from lithofacies 1 magmatic deposits frequently plot on the top-left side of the diagram, whereas particles from lithofacies 3 and 4 phreatomagmatic deposits frequently plot on the bottom-right side of the diagram. Particles from the mixed magmatic^phreatomagmatic deposits of lithofacies 2 plot, as expected, both on the right and on the left sides of the diagram. Volcanic ash clasts of di¡erent fragmentation processes are indeed discriminated in the diagram; a phreatomagmatic and a magmatic ¢eld should thus be identi¢ed. A substantial overlap, in the central part of the diagram, exists between particles of di¡erent origin, rendering such a diagram less signi¢cant than the one obtained by Dellino and La Volpe (1996).
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Table 2 Data of the ¢ve fractal components used in the fractal analysis Ref. (lit)
D1
Av R1
D2
Av R2
D3
Av R3
D4
Av R4
D5
Av R5
Score F1
Score F2
LI/16 (lit 1) GI/11 (lit 1) LI/22 (lit 1) GI/18 (lit 1) GI/17 (lit 1) GI/16 (lit 1) LI/15 (lit 1) LI/14 (lit 1) LI/17 (lit 1) GI/6 (lit 1) LI/2 (lit 1) LI/18 (lit 1) LI/3 (lit 1) GR/2 (lit 1) GR/77 (lit 1) LI/4 (lit 1) LI/13 (lit 1) LI/5 (lit 1) LI/9 (lit 1) LI/8 (lit 1) GR/81 (lit 2) GR/83 (lit 2) GR/82 (lit 2) GI/1 (lit 2) GR/55 (lit 2) LR/15 (lit 2) GR/54 (lit 2) GR/11 (lit 2) LR/14 (lit 2) GR/20 (lit 2) GR/21 (lit 2) GR/19 (lit 2) GR/18 (lit 2) GR/61 (lit 2) GR/62 (lit 2) GR/26 (lit 2) GR/25 (lit 2) LI/19 (lit 2) LI/20 (lit 2) GR/6 (lit 2) GR/5 (lit 2) GR/73 (lit 3) GR/72 (lit 3) GR/63 (lit 3) GR/71 (lit 3) GR/65 (lit 3) GR/64 (lit 3) GR/58 (lit 3) GR/16 (lit 3) GR/59 (lit 3) LI/28 (lit 3) LR/12 (lit 3) GR/8 (lit 3)
1.020 1.029 1.013 1.062 1.086 1.060 1.026 1.043 1.080 1.138 1.072 1.091 1.048 1.026 1.028 1.032 1.039 1.053 1.027 1.030 1.027 1.017 1.027 1.020 1.029 1.033 1.014 1.048 1.100 1.034 1.036 1.015 1.016 1.049 1.014 1.040 1.039 1.131 1.060 1.030 1.014 1.024 1.036 1.039 1.030 1.022 1.036 1.019 1.039 1.010 1.043 1.044 1.048
0.970 0.954 0.963 0.960 0.959 0.975 0.978 0.926 0.977 0.955 0.952 0.990 0.986 0.914 0.839 0.902 0.963 0.960 0.955 0.980 0.933 0.930 0.952 0.927 0.950 0.972 0.848 0.952 0.978 0.970 0.958 0.945 0.981 0.976 0.860 0.974 0.966 0.980 0.979 0.984 0.912 0.968 0.963 0.960 0.974 0.968 0.968 0.940 0.982 0.910 0.988 0.936 0.960
1.044 1.037 1.029 1.202 1.185 1.041 1.028 1.104 1.156 1.271 1.152 1.128 1.038 1.041 1.087 1.053 1.054 1.056 1.066 1.049 0.9100 0.9620 0.5250 1.028 0.9880 1.056 1.014 1.030 1.311 1.917 1.027 1.003 1.034 0.9250 1.047 1.032 1.052 1.180 1.139 1.009 1.003 1.027 1.015 1.022 1.109 1.010 1.017 1.036 1.023 1.010 1.049 1.022 1.055
0.696 0.444 0.709 0.847 0.778 0.515 0.571 0.634 0.844 0.730 0.661 0.902 0.772 0.618 0.631 0.722 0.560 0.473 0.726 0.717 0.245 0.294 0.377 0.715 0.195 0.705 0.162 0.369 0.853 0.639 0.551 0.279 0.973 0.286 0.559 0.607 0.544 0.849 0.725 0.639 0.416 0.475 0.223 0.441 0.767 0.281 0.363 0.446 0.661 0.316 0.619 0.253 0.319
1.156 1.033 1.080 0.9210 1.451 1.136 1.038 1.164 1.185 1.331 1.214 1.092 1.035 1.101 1.125 1.079 1.075 1.019 1.138 1.069 0.9280 1.041 1.123 1.056 1.027 1.084 1.120 1.162 1.105 1.089 0.9750 1.055 1.001 1.020 1.059 1.004 0.9460 1.291 1.284 1.002 1.031 1.054 1.041 1.021 1.054 1.022 0.9960 1.039 1.010 0.9830 1.083 0.9850 0.9930
0.758 0.198 0.554 0.513 0.778 0.520 0.211 0.646 0.518 0.504 0.418 0.457 0.479 0.377 0.455 0.398 0.475 0.337 0.756 0.347 0.348 0.265 0.602 0.277 0.234 0.402 0.473 0.362 0.499 0.695 0.257 0.527 0.264 0.322 0.564 0.207 0.229 0.729 0.595 0.365 0.274 0.552 0.282 0.410 0.356 0.247 0.374 0.360 0.600 0.490 0.509 0.140 0.287
1.069 0.9650 1.133 1.224 1.403 0.8530 1.134 0.9310 1.660 0.6100 1.034 1.130 0.9580 0.8970 0.8830 1.152 1.121 1.179 0.8880 1.089 0.6970 0.9470 1.134 1.097 1.031 1.217 1.051 0.9030 1.088 1.096 1.084 1.033 1.030 1.088 0.8440 1.057 1.025 1.245 1.062 1.065 1.166 1.122 0.9530 1.003 0.9630 0.9520 1.090 0.9080 1.024 1.054 1.071 0.9460 0.9980
0.229 0.298 0.245 0.275 0.341 0.354 0.443 0.106 0.418 0.476 0.198 0.461 0.135 0.306 0.369 0.227 0.441 0.132 0.337 0.281 0.242 0.341 0.500 0.326 0.213 0.277 0.538 0.216 0.421 0.581 0.397 0.115 0.160 0.242 0.341 0.292 0.137 0.343 0.211 0.161 0.594 0.268 0.207 0.175 0.220 0.244 0.228 0.226 0.151 0.146 0.338 0.173 0.288
1.233 1.568 1.150 0.8200 0.7960 0.9800 1.224 1.278 0.7920 1.268 1.350 1.127 1.380 1.120 1.215 1.352 1.347 1.802 1.094 1.487 0.7960 1.158 1.481 0.8100 0.8920 0.6080 1.182 0.3120 1.500 1.327 1.108 1.114 0.9890 1.128 0.9550 1.041 1.276 1.034 1.120 1.019 0.9770 1.101 1.046 1.081 1.149 1.171 0.9610 1.046 1.083 0.9940 1.066 0.9350 1.316
0.576 0.420 0.652 0.289 0.569 0.490 0.504 0.546 0.406 0.555 0.465 0.359 0.541 0.460 0.601 0.666 0.700 0.557 0.559 0.692 0.255 0.250 0.647 0.418 0.216 0.546 0.300 0.304 0.595 0.424 0.322 0.331 0.488 0.342 0.379 0.602 0.329 0.440 0.414 0.431 0.282 0.346 0.364 0.298 0.254 0.433 0.205 0.418 0.121 0.196 0.183 0.306 0.637
30.181 0.187 0.224 30.474 33.15 30.398 0.142 30.605 32.26 32.92 31.59 31.42 30.0710 0.211 30.0620 30.261 30.325 30.589 30.00100 30.141 1.62 0.741 1.11 0.371 0.502 30.126 0.189 30.0920 32.21 32.67 0.330 0.506 0.663 0.204 0.693 0.194 0.341 33.15 31.58 0.434 0.490 0.176 0.301 0.229 0.0380 0.584 0.320 0.599 0.241 0.870 30.243 0.414 0.0160
0.282 1.92 30.246 31.27 32.34 0.483 0.0680 1.13 33.29 2.81 1.07 0.0620 1.40 0.664 1.07 0.439 0.578 1.94 0.618 1.14 0.398 0.517 0.406 31.19 30.630 32.29 0.159 32.01 1.66 1.38 30.0360 0.0420 30.294 30.0610 0.321 30.147 0.814 30.645 0.0910 30.309 30.944 30.309 0.240 0.182 0.597 0.614 30.570 0.382 0.111 30.409 30.144 30.0280 1.06
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Table 2 (continued) Ref. (lit)
D1
Av R1
D2
Av R2
D3
Av R3
D4
Av R4
D5
Av R5
Score F1
Score F2
LR/4 (lit 3) GR/10 (lit 3) GR/9 (lit 3) GR/56 (lit 3) GR/32 (lit 3) GR/40 (lit 3) GR/42 (lit 3) GR/41 (lit 3) GR/24 (lit 3) GR/27 (lit 3) GR/78 (lit 3) GR/33 (lit 3) GR/34 (lit 3) GR/35 (lit 3) GR/48 (lit 3) LR/6 (lit 3) GR/50 (lit 3) GR/49 (lit 3) GR/53 (lit 3) LR/3 (lit 3) GR/84 (lit 3) GR/85 (lit 3) GR/67 (lit 3) GR/75 (lit 4) GR/74 (lit 4) GR/76 (lit 4) GR/69 (lit 4) GR/70 (lit 4) GR/51 (lit 4) LR/5 (lit 4) GR/23 (lit 4) GR/66 (lit 4)
1.021 1.020 1.016 1.010 1.023 1.026 1.017 1.015 1.014 1.012 1.021 1.046 1.050 1.040 1.015 1.019 1.024 1.014 1.014 1.047 1.057 1.020 1.079 1.028 1.022 1.020 1.015 1.024 1.008 1.012 1.019 1.010
0.977 0.975 0.915 0.809 0.958 0.987 0.945 0.960 0.848 0.869 0.977 0.992 0.990 0.968 0.932 0.948 0.933 0.879 0.965 0.984 0.985 0.972 0.999 0.978 0.963 0.952 0.977 0.976 0.894 0.946 0.932 0.937
1.008 1.038 1.014 0.9870 1.025 1.020 1.010 1.035 1.007 1.028 1.031 1.032 1.033 1.006 1.021 1.038 1.019 1.010 1.022 1.029 1.013 1.020 1.067 1.034 0.9870 1.026 1.011 1.038 1.017 1.034 1.017 1.028
0.517 0.590 0.365 0.425 0.686 0.557 0.420 0.645 0.180 0.304 0.609 0.676 0.728 0.137 0.527 0.532 0.288 0.303 0.625 0.480 0.392 0.532 0.750 0.584 0.350 0.505 0.284 0.560 0.340 0.663 0.316 0.796
0.9970 1.033 0.9780 1.004 1.036 1.037 1.010 0.7780 1.016 1.050 1.059 1.043 1.047 1.013 1.073 1.083 0.6620 0.9650 1.022 0.9830 1.019 1.053 1.043 1.009 0.9700 1.056 0.9350 1.036 1.049 1.018 1.094 0.9970
0.392 0.310 0.278 0.284 0.239 0.563 0.343 0.462 0.360 0.394 0.390 0.350 0.403 0.247 0.391 0.435 0.514 0.178 0.171 0.215 0.295 0.422 0.243 0.105 0.186 0.520 0.258 0.187 0.487 0.179 0.441 0.444
0.9080 1.084 1.129 0.8470 1.022 1.074 1.064 1.076 1.019 1.069 1.107 1.002 1.199 1.156 1.029 1.004 1.124 1.058 1.061 1.043 1.010 0.9560 1.087 1.099 1.104 1.066 0.9670 1.136 1.106 0.9600 1.083 1.022
0.178 0.340 0.405 0.523 0.188 0.384 0.291 0.560 0.248 0.194 0.441 0.0890 0.740 0.399 0.143 0.395 0.543 0.215 0.216 0.285 0.261 0.184 0.412 0.254 0.452 0.360 0.168 0.520 0.421 0.306 0.284 0.146
1.100 0.9680 0.9880 1.036 1.082 1.009 1.103 1.117 1.149 1.189 0.9130 1.160 0.7250 1.382 1.168 1.210 0.9990 1.282 1.038 1.065 0.9250 1.076 1.072 1.128 1.062 1.123 1.020 1.045 0.9580 1.076 0.6860 0.8830
0.159 0.260 0.332 0.426 0.496 0.192 0.251 0.370 0.529 0.417 0.163 0.348 0.490 0.503 0.288 0.374 0.353 0.482 0.346 0.200 0.140 0.562 0.505 0.239 0.167 0.163 0.127 0.336 0.229 0.217 0.364 0.377
0.789 0.394 0.678 1.11 0.409 0.325 0.592 1.45 0.672 0.432 0.284 30.0310 30.224 30.0230 0.371 0.234 1.77 0.764 0.590 0.175 0.00200 0.497 30.773 0.286 0.690 0.306 1.08 0.217 0.569 0.727 0.333 0.847
0.569 30.586 30.705 0.541 0.0420 30.406 30.0790 0.0800 0.239 0.158 30.881 0.470 31.76 0.549 0.242 0.508 30.416 0.556 30.290 0.0260 30.304 0.266 30.0150 30.0760 30.356 30.0240 0.0820 30.530 30.782 0.255 31.58 30.638
Ref. = particle reference number. D1, D2, D3, D4, D5 are the ¢ve fractal component values calculated for the 50 steplengths covering the range between three pixels and 150 pixels (0.5^25 Wm). D1 covers the range between 0.5 Wm and 5 Wm. D2 covers the range between 5.5 Wm and 10 Wm. D3 covers the range between 10.5 Wm and 15 Wm. D4 covers the range between 15.5 Wm and 20 Wm. D5 covers the range between 20.5 Wm and 25 Wm. Av R1, Av R2, Av R3, Av R4, Av R5 are the average correlation coe⁄cients of the ¢ve fractal components, calculated between the three particle orientations considered (see text for explanation of the formula used). Score F1 is the factor 1 coordinate of particles in the diagram of Fig. 4. Score F2 is the factor 2 coordinate of particles in the diagram of Fig. 4.
The relatively poor discrimination power of Fig. 4 is caused by two main factors. The fractal dimension is very e¡ective in distinguishing between shapes with a di¡erent amount of irregularities. Such a parameter however fails in discriminating between simple geometric forms (Orford and Whalley, 1983), such as those displayed by elongated or isometric or angular clasts, which in volcanology are very important for assessing the particle features related to di¡erent eruptive processes. Such simple geometric proper-
ties are instead easily distinguished by means of adimensional shape parameters as those used by Dellino and La Volpe (1996), i.e. circularity, elongation, compactness and rectangularity. The second reason is that sometimes fractal data are affected by errors (as discussed in 3. Geologic and volcanologic background of study particles), and the averaging operation needed to smooth them out obscures also part of the shape information, weakening the descriptive power of the fractal dimension. Such a drawback is easily shown by the
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example of Fig. 5, on which the M^R plots of the three orientations of an ash particle are shown. The scatter of data points occurring on the ¢rst two orientations is so high (low correlation coef¢cient) that it is very di⁄cult to identify a clear trend. The averaging operation among data of the three orientations eventually washes out also the information on shape that emerges from the third orientation that, thanks to a lower scatter (higher correlation coe⁄cient), revealed a certain alignment of points. One should conclude therefore that the fractal analysis is not completely successful in distinguishing between particles from magmatic and phreatomagmatic fragmentation processes. A further re¢nement of the statistical elaboration technique should be needed to reduce the scatter of points. For example, the use of carefully selected moving average values should serve to re-establish the power of the fractal dimension. In the present paper we have decided to not further implement the statistic elaboration. We have instead chosen to concentrate on some representative particles that, on showing a clear trend in
11
the M^R plot, are easy to interpret by a volcanological point of view. Some very interesting fractal trend emerges in fact from the investigation of ash particles whose M^R plots are not a¡ected by a relevant scatter of points and have a high correlation coe⁄cient (about 50% of the particle collection). The analysis of the M^R plots of these particles shows a peculiar and distinctive behavior between ash from magmatic and phreatomagmatic processes. As shown on Fig. 6, phreatomagmatic particles reveal a clear linear trend on the M^R plot and the correlation coe⁄cient of the regression line is close to 1. Such particles are self-similar over the whole investigated scale range, they show indeed a fractal behavior and can be described as true fractals (mono fractals). The fractal dimension value is quite low; suggesting therefore that particle boundary is moderately irregular (as by the photographs of Fig. 6). On Fig. 7 a few additional M^R plots of phreatomagmatic particles are shown, to represent the range of variation be-
Fig. 4. Diagram of factor 1 vs. factor 2, showing the discrimination obtained by multivariate statistic analysis of the ¢ve fractal components. Data points represent individual particles. In the inset, the key of symbols of the four lithofacies of deposits is shown. The phreatomagmatic, magmatic and overlap ¢elds are also drawn.
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tween these true fractals. The fractal dimension value typically ranges between 1.02 and 1.07. Magmatic particles show instead, as revealed by Fig. 8, two clear and distinct segments in the M^ R plot, they are indeed multifractals and a textural (small-scale) fractal with a lower value is distinguished from a structural (large-scale) fractal that has a higher value. The textural fractal typically occurs in the scale range between 0.5 Wm and 10 Wm, and its value is similar to that of phreatomagmatic particles, indeed, at small scales, magmatic particles are moderately irregular. The structural fractal, which typically occurs in the scale range between 10.5 Wm and 25 Wm, has a signi¢cantly higher value than the textural
one. It is associated to the walls of ruptured vesicles occurring on particle boundary, as is shown on the photographs of Fig. 8. In fact the walls of ruptured vesicles are on the scale range of the structural fractal (10.5^25 Wm). On Fig. 9, a few additional M^R plots of magmatic particles are shown, to report the range of values of these multifractal particles. The textural fractal ranges in value between 1.02 and 1.07, the structural fractal ranges between 1.16 and 1.57. The fractal and multifractal behaviors of magmatic and phreatomagmatic particles mark, in our opinion, some important aspects of the fragmentation processes that characterized the explosive activity of the Monte Pilato^Rocche Rosse erup-
Fig. 5. On this plate, the SEM photograph of an ash particle is presented together with the M^R plots resulting from three different particle orientations. The not rotation invariance of data points is evident. Particle reference number is on the bottom-left of photograph. D = fractal dimension value. R = correlation coe⁄cient. 1 = M^R plot resulting from the original orientation. 2 = M^R plot resulting from the 90‡ clockwise rotation. 3 = M^R plot resulting from the 90‡ anticlockwise rotation.
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13
Fig. 6. On this plate, two SEM images of phreatomagmatic particles and the corresponding M^R plots are shown. Particle reference number is on the bottom-left of photographs. D = fractal dimension value. R = correlation coe⁄cient. Diagrams, which are not a¡ected by a relevant scatter of data points, show a good alignment of data points (see the least square regression line) that result in the true fractal behavior. The fractal dimension value, D, is quite low, suggesting that the irregularities are self-similar at all scales and that the shape is not very complex. This behavior is con¢rmed by the visual analysis of particle photographs, which evidence clasts outline characterized by evenly distributed irregularities and indentations.
tions, as will be tentatively discussed in 5. Discussion and conclusion.
5. Discussion and conclusion In the case of the Monte Pilato^Rocche Rosse eruptions, we have found that the fractal dimension, compared to more conventional shape descriptors, does not add signi¢cant information on the general shape of ash particles. Simple
geometric properties are in fact more easily de¢ned, as done in a previous paper (Dellino and La Volpe, 1996), by using adimensional parameters such as compactness, elongation and circularity. It is however not our intention to cast doubts on the ability of the fractal dimension in describing complex^irregular shapes. The analysis of the M^R plots not a¡ected by a signi¢cant scatter allowed in fact to obtain valuable information on the outlines of ash particles, that in our opin-
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Fig. 7. On this plate, the M^R plots of four phreatomagmatic particles are presented. Diagrams do not show a relevant scatter of data points (see the least square regression lines) and reveal the true fractal behavior. D = fractal dimension value. R = correlation coe⁄cient. The fractal dimension value, D, which ranges between 1.03 and 1.07, shows that phreatomagmatic particles have a not very complex shape. Particle reference number is on the bottom-left of diagrams.
ion is not of trivial importance for the characterization of the Monte Pilato^Rocche Rosse eruptions. Many independent processes, such as fragmentation, transport and alteration, likely occur in the generation of the boundary irregularities of pyroclastic particles. Each of the processes in£uences the irregularity of particle outline at di¡erent scales, likely resulting in their fractal and multifractal behavior. Carey et al. (2000) de¢ned primary jokulhlaup particles as multiple fractals. They associated the textural (small-scale) fractal to fractures, and the structural (large-scale) fractal to the occurrence of broken bubbles along particles outline. They however did not ¢nd true fractal (mono fractal)
shapes, which instead are found in the phreatomagmatic particles of the present paper. In the case of phreatomagmatic particles of Monte Pilato^Rocche Rosse eruptions, shape should be simply related to the fragmentation process, since no additional mechanisms, i.e. transportation or alteration, have signi¢cantly modi¢ed particle shape, as stated by Dellino and La Volpe (1996). During phreatomagmatic explosions, ¢ne ash particles are the result of a brittle fragmentation process (Zimanowski et al., 1997). E¡ective contact between magma and water results in a sudden expansion of water that deforms very fast the melt, which reacts by fracturing. Water intrudes into cracks leading to a cascading process, which
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results in the ¢ne brittle fragmentation (Bu«ttner and Zimanowski, 1998). Particle shape and boundary irregularities are related to branching cracks propagating into the brittle melt, which are self-similar at all scales (Brown and Wohletz, 1995). Phreatomagmatic ash particles are indeed true fractals and the fractal dimension refers to the rheological behavior of magma, indeed mainly to its relaxation time. Such a result shows, in contrast to what is reported in the literature, that particles of geological interest may show indeed
15
fractal behavior. Probably, in the data set presented by Carey et al. (2000) no phreatomagmatic particles were present, therefore no true fractal particles were identi¢ed. Also the shape and irregularities of the magmatic particles of the Monte Pilato^Rocche Rosse eruptions should be simply related to the fragmentation process since, as for the phreatomagmatic ones, they did not su¡er shape modi¢cation by transportation or alteration (Dellino and La Volpe, 1996). Magmatic particles are multifractals
Fig. 8. On this plate, two SEM images of magmatic particles and the corresponding M^R plots are shown. Particle reference number is on the bottom-left of photographs. D = fractal dimension value. R = correlation coe⁄cient. Diagrams, which are not affected by a relevant scatter of data points, show the multifractal behavior that is marked by two distinct segments, which identify the textural fractal, DT, and the structural fractal, DS. Values of DT and DS are shown together with the least square regression lines of the two segments. R = correlation coe⁄cient. The low value of DT is related to small-scale irregularities along particles outline, the high value of DS is related to the relatively strong irregularities resulting from the walls of ruptured vesicles, which render the shape of particles very complex.
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and show a textural and a structural fractal; shape irregularities of such particles occur at two distinct ranges of scales as shown by the two distinct fractal components. The textural component, which relates to smallscale irregularities, has a low value, which is similar to that of phreatomagmatic particles. The structural component, having a higher value, relates instead to large-scale irregularities that represent the walls of ruptured vesicles laying on the particle outline. We suggest therefore that the particle boundary of magmatic particles is de¢ned by two orders of
irregularities that are not self-similar, likely representing two independent factors : one related to vesicle walls, the other independent from vesicles. It seems that magmatic fragmentation, during the Monte Pilato^Rocche Rosse eruptions, was not simply related to the gas-bubble behavior, i.e. to the process of growth, expansion, interference and consequent explosion of vesicles, as suggested by some authors (Sparks, 1978). Even if bubble expansion is assumed during the exsolution processes that joined magmatic activity, vesicles did not completely expand, up to interference, before fragmentation. If such had been the case, particle
Fig. 9. On this plate, the M^R plots of four magmatic particles are presented. Diagrams do not show a relevant scatter of data points and two segments, which reveal the multifractal behavior, are identi¢ed. The textural fractal value, DT, and the structural fractal value, DS, are reported together with the least square regression lines of the two segments. R = correlation coe⁄cient. The textural fractal, which ranges between 1.03 and 1.07, shows that magmatic particles are not highly irregular at small scales. The structural fractal, which ranges between 1.16 and 1.57, relates to the large-scale irregularities de¢ned by walls of ruptured vesicles, which are not self-similar to the small-scale irregularities. Particle reference number is on the bottom-left of diagrams.
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irregularities should have been in£uenced simply by the vesicle distribution along the outline, resulting in a true fractal, which value should have simply re£ected vesicle walls. The textural fractal seems instead not related to vesicles, as also suggested by Carey et al. (2000). It is characterized by a value similar to that of phreatomagmatic particles, and we suggest it was related to the rheological behavior of the melt, which in fact was of the same composition (rhyolitic), likely of the same viscosity, as the melt that characterized phreatomagmatic explosions. Bubble expansion was in our opinion interrupted by a brittle fracturing of the melt, and occurred as a response to a mechanical shock. The results obtained in this paper seem therefore to support recent models, based on experimental research, that refer magmatic fragmentation dynamics to brittle processes (Alidibirov and Dingwell, 1996; 2000). These models point out the role of rapid decompression as the dominant factor producing magmatic fragmentation, especially for highly viscous rhyolitic melts as those that characterized the products studied in the present work. We tentatively suggest that during Monte Pilato^Rocche Rosse eruptions, magmatic fragmentation was a brittle response of the viscous melt to a rapid decompression process that exceeded the tensile strength of magma. It is likely that such rapid decompression should have been caused by the shock waves produced by a previous phreatomagmatic event. A relevant amount of the thermal energy that is converted into a mechanical form is released just as energetic shock waves during explosive magma/water interaction (Zimanowski et al., 1997). Additional work on pyroclastic products representative of di¡erent magma composition and of di¡erent eruption intensities is needed to assess the validity of such argumentation also for other eruptions characterized by an alternating magmatic/ phreatomagmatic style. Finally, a further implementation of the method proposed in this paper (that allows reduction of the scatter in the M^R plots) would be very useful for the quantitative analysis of particles resulting from laboratory experiments, and could be eventually used for scaling of experimental results to natural processes.
17
Acknowledgements The authors are grateful to C. Hagelberg and J. Taddeucci for the thorough revision of the paper.
References Alidibirov, M., Dingwell, D.B., 1996. Magma fragmentation by rapid decompression. Nature 380, 146^149. Alidibirov, M., Dingwell, D.B., 2000. Three fragmentation mechanisms for highly viscous magma under rapid decompression. J. Volcanol. Geotherm. Res. 100, 413^421. Be¤rube¤, D., Je¤brak, M., 1999. High precision boundary fractal analysis for shape characterization. Comput. Geosci. 25, 1059^1071. Brown, W.K., Wohletz, K.H., 1995. Derivation of the Weibull distribution based on physical principles and its connection to the Rosin-Rammler and lognormal distributions. J. Appl. Phys. 78, 2758^2763. Bu«ttner, R., Zimanowski, B., 1998. Physics of thermo hydraulic explosions. Phys. Rev. E57, 5726^5729. Bu«ttner, R., Dellino, P., Zimanowski, B., 1999. Identifying modes of magma/water interaction from the surface features of ash particles. Nature 401, 688^690. Carey, S., Maria, A., Sigurdsson, H., 2000. Use of fractal analysis for discrimination of particles from primary and reworked jokulhlaup deposits in SE Iceland. J. Volcanol. Geotherm. Res. 104, 65^80. Clark, N.N., 1986. Three techniques for implementing digital fractal analysis of particle shape. Powder Technol. 46, 45^ 52. Cortese, M., Frazzetta, G., La Volpe, L., 1986. Volcanic history of Lipari (Aeolian islands, Italy) during the last 10,000 years. J. Volcanol. Geotherm. Res. 27, 117^133. Dellino, P., La Volpe, L., 1995. Fragmentation versus transportation mechanisms in the pyroclastic sequence of Monte Pilato^Rocche Rosse (Lipari, Italy). J. Volcanol. Geotherm. Res. 64, 211^232. Dellino, P., La Volpe, L., 1996. Image processing analysis in reconstructing fragmentation and transportation mechanisms of pyroclastic deposits. The case of Monte Pilato^ Rocche Rosse eruptions, Lipari (Aeolian islands, Italy). J. Volcanol. Geotherm. Res. 71, 13^29. De Rita, D., Frazzetta, G., Romano, R., 1991. The Biancavilla ^ Montalto Ignimbrite (Etna, Sicily). Bull. Volcanol. 53, 121^131. Flook, A.G., 1979. The characterization of textural and structural pro¢les by automated measurement of their fractal dimension. Proceedings of the 2nd European Symposium on Particle Characterization, pp. 591^599. Gonzato, G., Mulargia, F., Marzocchi, W., 1998. Practical application of fractal analysis: problems and solutions. Geophys. J. Int. 132, 275^282. Heiken, G., 1974. An atlas of volcanic ash. Smithsonian Contrib. Earth Sci. 12, 1^101.
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Heiken, G., Wohletz, K., 1985. Volcanic ash. Sharp and Simmons, University of California Press, 246 pp. Kaye, B.H., 1978. Speci¢cation of the ruggedness and/or texture of a ¢ne particle pro¢le by its fractal dimension. Powder Technol. 21, 1^16. Kennedy, S., Lin, W.H., 1986. Fract-A FORTRAN subroutine to calculate the variables necessary to determine the fractal dimension of closed forms. Comput. Geosci. 12, 705^712. Kennedy, S.K., Lin, W.H., 1992. A comparison of Fourier and fractal techniques in the analysis of closed forms. J. Sedim. Petrol. 62, 842^848. Mandelbrot, B.B., 1967. How long is the coast of Britain? Statistical self-similarity and fractal dimension. Science 156, 636^638. Mandelbrot, B.B., 1977. Form, Chance and Dimension. W.H. Freeman, San Francisco, CA, 316 pp. Marshall, J.R., 1987. Shape analysis. In: Marshall, J.R. (Ed.), Clastic Particles. Van Nostrand Reinhold Company, New York, 255 pp. Orford, J.D., Whalley, W.B., 1983. The use of the fractal dimension to quantify the morphology of irregular-shaped particles. Sedimentology 30, 655^668. Orford, J.D., Whalley, W.B., 1987. The quantitative description of highly irregular sedimentary particles: the use of
fractal dimension. In: Marshall, J.R. (Ed.), Clastic Particles. Van Nostrand Reinhold Company, New York, pp. 267^280. Schwarz, H., Exner, H.E., 1980. The implementation of the concept of fractal dimension on a semi-automatic image analyser. Powder Technol. 27, 207^213. Sheridan, M.F., Kortemeier, C.P., 1987. Quantitative analysis of surface features on pyroclasts: An example from the Monte Guardia Sequence, Lipari. In: Marshall, J.R. (Ed.), Clastic Particles. Van Nostrand Reinhold Company, New York, pp. 122^134. Sheridan, M.F., Marshall, J.R., 1983. Interpretation of pyroclast surface features using SEM images. J. Volcanol. Geotherm. Res. 16, 153^159. Sparks, R.S.J., 1978. The dynamics of bubble formation and growth in magmas: A review and analysis. J. Volcanol. Geotherm. Res. 3, 1^37. Wohletz, K.H., 1987. Chemical and textural surface features of pyroclasts from hydrovolcanic eruption sequences. In: Marshall, J.R. (Ed.), Clastic Particles. Van Nostrand Reinhold Company, New York, pp. 79^97. Zimanowski, B., Bu«ttner, R., Lorenz, V., 1997. Premixing of magma and water in MFCI experiments. Bull. Volcanol. 58, 491^495.
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The fractal and multifractal dimension of volcanic ash particles contour: a test study on the utility and volcanological relevance P. Dellino , G. Liotino Dipartimento Geomineralogico, Universita' di Bari, Via E. Orabona 4, 70125 Bari, Italy Received 27 October 2000; received in revised form 14 June 2001; accepted 14 June 2001
Abstract Image processing analysis is used to check the ability of the fractal dimension for quantitatively describing the shape of volcanic ash particles. Digitized scanning electron microscopy images of fine pyroclasts from the eruptions of Monte Pilato^Rocche Rosse (Lipari, Italy) are investigated to test the efficiency of the fractal dimension to discriminate between particles of different eruptive processes. Multivariate analysis of multiple fractal components allows distinction between magmatic particles and phreatomagmatic particles, which however is less significant than the discrimination obtained in previous studies by the use of simple ‘adimensional’ shape parameters. Approximation of the actual particle boundary and the not rotation invariant nature of the fractal data frequently result in a significant scatter of data points in the Mandelbrot^Richardson plot. Such behavior obscures in some cases the actual information of particle shape and renders the discriminating power of fractal analysis less effective than classical shape descriptors. Data less affected by scatter reveal that phreatomagmatic particles of the Monte Pilato^Rocche Rosse eruptions are true (mono) fractals, whereas magmatic particles are multifractals. The textural (small-scale) fractal of magmatic particles is similar to the fractal dimension value of phreatomagmatic particles, and is attributed to the rheological behavior of melt upon brittle fragmentation. The structural (large-scale) fractal of magmatic particles refers to the walls of ruptured vesicles that lay on the particle outline. The high difference between the values of the textural and structural fractals of magmatic particles of the Monte Pilato^Rocche Rosse eruptions suggests two distinct and independent processes in the formation of such pyroclasts. At the scales corresponding to the textural fractal, the fragmentation process is independent of vesicles. Magmatic fragmentation is not simply related to growth, expansion, interference and explosion of vesicles but to a brittle cracking of the highly viscous melt, likely related to rapid decompression. 5 2002 Elsevier Science B.V. All rights reserved. Keywords: shape analysis; fractal dimension; brittle fragmentation
1. Introduction * Corresponding author. Fax: +39-080-5442591. E-mail address: [email protected] (P. Dellino).
It is widely accepted in geology that the shape of particles carries information. The investigation of ash particles represents a well known approach,
0377-0273 / 02 / $ ^ see front matter 5 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 7 3 ( 0 1 ) 0 0 2 4 7 - 5
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in volcanology, for identifying the process responsible for pyroclastic deposit formation. The peculiar morphological features that pyroclasts related to di¡erent eruption dynamics show by scanning electron microscopy (SEM) are in fact systematically described in various atlas and publications (Heiken, 1974; Heiken and Wohletz, 1985; Wohletz, 1987). Classi¢cation and interpretation of data is however frequently subjective and interpretation heavily depends on the experience and training of the operator. A big e¡ort has been made, especially in sedimentology, to obtain quantitative shape parameters e¡ective in the process characterization of clastic particles. Conventional parameters, such as those referring to simple geometric forms, are useful for describing particles with a regular shape, however they are not very e¡ective in describing the shape of the irregular particles that frequently occur in sedimentary deposits (Orford and Whalley, 1983). This problem is ampli¢ed with pyroclastic particles that often show very complex morphologies, which, according to Marshal (1987), is the reason why the analysis of volcanic clast shape has remained largely qualitative. Attempts have been made for objectively identifying particle types and particle populations by means of multivariate statistical classi¢catory schemes (Sheridan and Marshall, 1983; Sheridan and Kortemeier, 1987; De Rita et al., 1991). Dellino and La Volpe (1996) recently proposed the use of simple adimensional shape parameters for describing the ash particles of the explosive eruptions of Monte Pilato^Rocche Rosse (Lipari, Italy). Multivariate elaboration of data allowed a good discrimination between ash clasts related to phreatomagmatic and magmatic processes. These authors suggested that, even if a single parameter is not very e¡ective for uniquely describing particle shape, the combination of multiple parameters allowed a useful and e¡ective characterization to be obtained. The reliability of fractal analysis methods to describe particle morphology has increasingly improved in recent times, and they are nowadays widely used in various branches of research, from material science, to biology and to geology (Be¤rube¤ and Je¤brak, 1999; Gonzato et al., 1998). Such
methods revealed quite successful in describing complex^irregular shapes of sedimentary particles, and the use of the fractal and multifractal dimension has proved e¡ective in describing and interpreting detrital clasts (Orford and Whalley, 1983). In a recent paper, Carey et al. (2000), using a methodology similar to that used in the present work, demonstrated the ability of fractal dimension to discriminate between primary and reworked jokulhlaups deposits in Iceland. In the present paper we constrain the use of the fractal and multifractal dimension for describing the shape of volcanic ash particles and in particular for addressing fragmentation dynamics. To test the signi¢cance of the method, the same particles from the Monte Pilato^Rocche Rosse eruptions used by Dellino and La Volpe (1996) have been investigated, and results are compared against the more conventional shape parameters previously utilized.
2. The fractal analysis method The utility of fractal analysis in the description of particles stems from the idea that a shape, however irregular or complex, can be circumscribed by equilateral polygons. The shorter the sidelength, the greater the number of sides, the longer the polygon perimeter becomes and the better the approximation of such polygons is to the particle outline. The actual particle perimeter is eventually reached using a polygon with an in¢nite number of sides. If we perform such analysis using polygons with an increasing number of sides and if we de¢ne as steplength the side of the polygon, we can plot the log of steplength vs. the log of the resulting perimeter on a diagram. If the data points lay on a straight line, the particle is self-similar, in the sense that it has the same degree of irregularity at all scales. This particle has a fractal behavior, it is called a true fractal (mono fractal), and its fractal dimension is de¢ned by D = 13S, where S is the slope of the line, which is easily calculated using least square methods. The rate of increase of perimeter as the steplength decreases is a measure of particle irregularity.
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The log^log diagram is called the Mandelbrot^ Richardson plot (M^R plot), as it was developed by Mandelbrot (1967), on Richardson’s unpublished data. The fractal dimension of a closed bidimensional form (as are SEM images of particles) belongs to the interval 1^2. It re£ects the plane-¢lling tendency of a particle outline (Kaye, 1978). Smooth curves (Euclidean forms) have a fractal dimension equal to 1 and the value increases as the shape acquires irregularities, up to a line so tortuous and irregular as that de¢ned by a Brownian motion (Mandelbrot, 1977), that completely ¢lls a plane and tends to the value 2. The fractal dimension of particles of geological interest actually ranges in a more restricted interval, and Clark (1986) found values between 1 and 1.36. Particles of geological interest generally are not considered true fractals (mono fractals), since in the M^R plot they de¢ne segmented lines (or curvilinear trends). In such an instance the utility of the fractal dimension is retained, and multiple fractal segments are recognized (Orford and Whalley, 1987). Some authors propose the use of the textural (small-scale) and structural (large-scale) fractals, if two segments are clearly distinguished (Kaye, 1978; Flook, 1979). The textural fractal relates to small-scale irregularities, the structural fractal relates instead to large-scale irregularities. When the separation between segments is not easily obtained, or if more segments are identi¢ed, other authors (Kennedy and Lin, 1992) propose that multiple fractal components can be de¢ned by multivariate statistic analysis. Fractal analysis is nowadays performed by means of image processing analysis software, which allows the reconstruction of the particle outline on digitized images. The coordinates of points that lay along the outline represent the particle boundary. In our case analysis was performed on a Macintosh computer using the public domain NIH Image program (developed at the U.S. National Institutes of Health and available on the Internet at http://rsb.info.nih.gov/nih-image/). High-resolution (300 pixels per inch) black and white SEM images were utilized and the thresholding
3
operation allowed the identi¢cation (segmentation) of particle boundary. The boundary was approximated by at least 500 points, and the macro fraccalc, written by M. Warfel, was used to determine the steplengths and relative perimeter lengths, which were necessary for the calculation of the fractal dimension value. The algorithm used was the hybrid method, as described by Clark (1986). The software used is the same as that utilized by Carey et al. (2000), but the algorithm is a little di¡erent, and is brie£y described in the following. If the steplength is de¢ned by r, the calculation starts by determining the distance dp;i from a starting point (xp ,yp ) to the second coordinate (xi ,yi ) using the equation [(xi 3xiþ1 )2 +(yi 3yiþ1 )2 ]1=2 = dp;i . If the distance is less than r the next point (xiþ1 ,yiþ1 ) is then selected, the distance is computed again and the test against the steplength is made again. The process continues until the distance dp;iþk s r. If Mdp;iþk 3rM9Mdp;iþðk31Þ 3rM then the point (xiþk ,yiþk ) is selected, if not, the point (xiþðk31Þ ,yiþðk31Þ ) is selected. Then the calculation is started again from this new point, until the whole particle boundary is covered. As the endpoint of the curve is approached, the distance between the last point and the end-point will be less than r, and a fraction of the steplength is computed for closing the curve. The perimeter is then calculated by P = gN31 i¼1 di;iþ1 . The calculation is then started again with a longer steplength, until a speci¢ed range of r is covered. In the present study 50 steplengths were used, covering the range between three pixels and 150 pixels, which actually corresponds to the interval between 0.5 Wm and 25 Wm. The range is inside the interval of 0.085^0.25 of the particle diameter as suggested by Orford and Whalley (1983). Data points obtained by this method have a certain scatter in the M^R plot (as it happens also using other implementations of the fractal dimension calculation, Carey et al., 2000; Gonzato et al., 1998). The scatter is attributed to three sources of error. The ¢rst source of error is that the boundary may not be fractal. The second source of error results from the fact that the points representing the particle coordinates are an approximation of the real particle boundary.
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The third source of error occurs because the fractal dimension, as implemented on computers, is not rotation invariant (Kennedy and Lin, 1986), in the sense that by changing the starting point along the boundary, the resulting perimeter estimate changes. This problem is clearly addressed at the footnote of Schwarz and Exner (1980), by an anonymous reviewer, and is attributed to the wavelength of irregularities that characterize the particle boundary. As the steplength approaches this characteristic wavelength and if the starting point is in-phase with the wavelength, the perimeter estimate results shorter than expected. To show this drawback a graphic example, based on the synthetic shapes of Fig. 1, is useful. The ¢rst shape represents the silhouette of a perfect square without irregularities, and in the resulting M^R plot the clear Euclidean nature of the form is revealed, as the slope is zero. The second silhouette represents a square with some indentations along the outline that render the shape irregular. The M^R plot of this silhouette reveals how the occurrence of indentations drastically changes the arrangement of data points, and a signi¢cant slope thus results. The third and fourth silhouettes represent the same indented square as the second one, which are rotated 90‡ clockwise and anticlockwise respectively. The M^ R plots reveal how rotations resulted in a di¡erent scatter of points compared to the second silhouette. Such an error, as discussed by Kennedy and Lin (1986), should be smoothed out by calculating the fractal dimension value of di¡erent particle orientations (changing the actual starting point), then averaging data between orientations.
3. Geologic and volcanologic background of study particles Particles investigated in this paper are from the pyroclastic deposits of the 1.4 ka eruptions of
Monte Pilato^Rocche Rosse at Lipari (Aeolian Islands, Italy). These deposits were chosen because eruption characteristics are well known (Cortese et al., 1986; Dellino and La Volpe, 1995). Furthermore, a quantitative shape analysis of ash particles had already been carried out by means of simple adimensional form descriptors (Dellino and La Volpe, 1996). The ability of the fractal dimension in describing pyroclastic particles can be, in this way, directly compared to that of more conventional shape parameters. Monte Pilato^Rocche Rosse eruptions, which were alimented by rhyolitic magma, were characterized by an alternating magmatic/phreatomagmatic style. Such eruptive activity led to the formation of various types of deposits, among which four main lithofacies were identi¢ed and are fully described by Dellino and La Volpe (1995). Lithofacies 1 deposits are well sorted pumice fallout layers of magmatic origin. Lithofacies 2 deposits are massive layers of mixed magmatic^phreatomagmatic origin. Lithofacies 3 deposits are surge layers of phreatomagmatic origin. Lithofacies 4 deposits are accretionary lapillirich layers of phreatomagmatic origin. Dellino and La Volpe (1996) investigated ash particles by SEM, and used adimensional shape parameters, such as circularity, compactness, elongation and rectangularity, to obtain a quantitative shape characterization. Data were elaborated by means of factor analysis and a signi¢cant discrimination resulted between clasts of magmatic and phreatomagmatic origin. In the present study we have selected 89 SEM images, from the same collection used by Dellino and La Volpe (1996), representing as many particles from the four main lithofacies. In particular, particles are between 64 Wm and 125 Wm in diameter. This size fraction was chosen because it is the most useful for discriminating between di¡erent fragmentation dynamics, and represents an important source of knowledge
Fig. 1. On this plate, synthetic silhouettes and the corresponding M^R plots are shown. The in£uence of indentations and rotations on the alignment and scatter of data points is evident. 1 = perfect square; 2 = square with indentations; 3 = the same as 2, with a 90‡ clockwise rotation; 4 = the same as 2, with a 90‡ anticlockwise rotation.
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Table 1 Data of the fractal dimension value (D) of the four test particles of Fig. 2 Particle reference number (orientation)
D value
GR/79 (original) GR/79 (vertical £ipping) GR/79 (horizontal £ipping) GR/79 (90‡ clockwise) GR/79 (90‡ anticlockwise) GI/7 (original) GI/7 (vertical £ipping) GI/7 (horizontal £ipping) GI/7 (90‡ clockwise) GI/7 (90‡ anticlockwise) LR/10 (original) LR/10 (vertical £ipping) LR/10 (horizontal £ipping) LR/10 (90‡ clockwise) LR/10 (90‡ anticlockwise) LI/26 (original) LI/26 (vertical £ipping) LI/26 (horizontal £ipping) LI/26 (90‡ clockwise) LI/26 (90‡ anticlockwise)
1.030 1.031 1.025 1.027 1.033 1.200 1.181 1.197 1.215 1.191 1.021 1.023 1.023 1.023 1.020 1.112 1.118 1.121 1.118 1.135
Data refer to the ¢ve particle orientations used in the analysis.
about the most energetic processes of explosive dynamics, as discussed by Dellino and La Volpe (1996) and by Bu«ttner et al. (1999).
4. Analysis and results A few images showing distinct particle morphologies were ¢rst selected, to allow an easy assessment of the power of fractal dimension on discriminating between di¡erent shapes. The selected particles are shown on Fig. 2. For each particle, to address the error caused by the rotation not-invariant nature of the fractal dimension, ¢ve di¡erent orientations were used. The ¢rst orientation refers to the original position of the particle as shown by SEM ; the second to a 90‡ clockwise rotation of the particle; the third to a 90‡ counterclockwise rotation ; the fourth to a vertical £ipping of the particle; the ¢fth to a horizontal £ipping. On each orientation the fractal dimension was calculated by using 50 steplengths, which covered
the interval between three pixels and 150 pixels, corresponding to 0.5^25 Wm. Data are shown on Table 1. An analysis of variance was then carried out, to test the di¡erence of the fractal dimension among the di¡erent particles (variation among particles) against the di¡erence between the orientations of individual particles (variation within particles). Results show that the fractal dimension among the various particles is di¡erent (at a 95% level of signi¢cance). It is therefore possible to state that even if the di¡erent orientations result in a certain variability of the fractal dimension value of individual particles, this variability is lower than that among particles. The mean value among the ¢ve orientations was then calculated and it represents the average fractal dimension of each particle. It is 1.029 for GR/79; 1.20 for GI/7; 1.023 for LR/10; 1.121 for LI/26. Data con¢rm the visual impression that particles GI/7 and LI/ 26, on having a more irregular boundary, should result in a higher fractal dimension compared to particles GR/79 and LR/10 that have a more regular outline and a lower fractal dimension. The fractal dimension is therefore a useful parameter for distinguishing between the form of ash particles that have evident morphological di¡erences. The analysis of the M^R plots of Fig. 3 (the original position of Fig. 2 images is reported) shows, anyway, that in some cases data points do not lay on a line, and, because of the scatter of points, it is not easy to identify how many segments occur. In order to obtain a more precise characterization of the fractal dimension, we have used, as suggested by Kennedy and Lin (1992), multiple fractal components in the investigation of the complete data set of particles. Five components were used. The ¢rst component is the fractal dimension value resulting from the steplengths between three and 30 pixels ; the second is the one between 33 and 60 pixels and so on. Three orientations were used, the original one, the 90‡ clockwise and anticlockwise rotations. The horizontal and vertical £ipping orientations, on not resulting in additional variations, were omitted. Data from the three orientations were weighted by the correlation coe⁄cient of the least square
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Fig. 2. Photographs of SEM images of the four test particles. Particle reference number is on the bottom-left of photographs.
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Fig. 3. Diagrams showing the M^R plots of the test particles. The original orientation is shown. D = fractal dimension value. R = correlation coe⁄cient. The least square regression line is also drawn. Particle reference number is on the bottom-left of diagrams.
line, and were consequently averaged out using the formula N X Di R i
Dave ¼
i¼1 N X
Ri
i¼1
Where R is the correlation coe⁄cient and N is the number of orientations. Average data of the ¢ve fractal components (Table 2) were then used as variables in a multivariate system that was elaborated by means of factor analysis of the same type as that used by Dellino and La Volpe (1996). Using the ¢rst two factors, which explain more than 70% of the variance of the system, a diagram
(Fig. 4) was constructed, and data points represent individual particles. Particles from lithofacies 1 magmatic deposits frequently plot on the top-left side of the diagram, whereas particles from lithofacies 3 and 4 phreatomagmatic deposits frequently plot on the bottom-right side of the diagram. Particles from the mixed magmatic^phreatomagmatic deposits of lithofacies 2 plot, as expected, both on the right and on the left sides of the diagram. Volcanic ash clasts of di¡erent fragmentation processes are indeed discriminated in the diagram; a phreatomagmatic and a magmatic ¢eld should thus be identi¢ed. A substantial overlap, in the central part of the diagram, exists between particles of di¡erent origin, rendering such a diagram less signi¢cant than the one obtained by Dellino and La Volpe (1996).
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9
Table 2 Data of the ¢ve fractal components used in the fractal analysis Ref. (lit)
D1
Av R1
D2
Av R2
D3
Av R3
D4
Av R4
D5
Av R5
Score F1
Score F2
LI/16 (lit 1) GI/11 (lit 1) LI/22 (lit 1) GI/18 (lit 1) GI/17 (lit 1) GI/16 (lit 1) LI/15 (lit 1) LI/14 (lit 1) LI/17 (lit 1) GI/6 (lit 1) LI/2 (lit 1) LI/18 (lit 1) LI/3 (lit 1) GR/2 (lit 1) GR/77 (lit 1) LI/4 (lit 1) LI/13 (lit 1) LI/5 (lit 1) LI/9 (lit 1) LI/8 (lit 1) GR/81 (lit 2) GR/83 (lit 2) GR/82 (lit 2) GI/1 (lit 2) GR/55 (lit 2) LR/15 (lit 2) GR/54 (lit 2) GR/11 (lit 2) LR/14 (lit 2) GR/20 (lit 2) GR/21 (lit 2) GR/19 (lit 2) GR/18 (lit 2) GR/61 (lit 2) GR/62 (lit 2) GR/26 (lit 2) GR/25 (lit 2) LI/19 (lit 2) LI/20 (lit 2) GR/6 (lit 2) GR/5 (lit 2) GR/73 (lit 3) GR/72 (lit 3) GR/63 (lit 3) GR/71 (lit 3) GR/65 (lit 3) GR/64 (lit 3) GR/58 (lit 3) GR/16 (lit 3) GR/59 (lit 3) LI/28 (lit 3) LR/12 (lit 3) GR/8 (lit 3)
1.020 1.029 1.013 1.062 1.086 1.060 1.026 1.043 1.080 1.138 1.072 1.091 1.048 1.026 1.028 1.032 1.039 1.053 1.027 1.030 1.027 1.017 1.027 1.020 1.029 1.033 1.014 1.048 1.100 1.034 1.036 1.015 1.016 1.049 1.014 1.040 1.039 1.131 1.060 1.030 1.014 1.024 1.036 1.039 1.030 1.022 1.036 1.019 1.039 1.010 1.043 1.044 1.048
0.970 0.954 0.963 0.960 0.959 0.975 0.978 0.926 0.977 0.955 0.952 0.990 0.986 0.914 0.839 0.902 0.963 0.960 0.955 0.980 0.933 0.930 0.952 0.927 0.950 0.972 0.848 0.952 0.978 0.970 0.958 0.945 0.981 0.976 0.860 0.974 0.966 0.980 0.979 0.984 0.912 0.968 0.963 0.960 0.974 0.968 0.968 0.940 0.982 0.910 0.988 0.936 0.960
1.044 1.037 1.029 1.202 1.185 1.041 1.028 1.104 1.156 1.271 1.152 1.128 1.038 1.041 1.087 1.053 1.054 1.056 1.066 1.049 0.9100 0.9620 0.5250 1.028 0.9880 1.056 1.014 1.030 1.311 1.917 1.027 1.003 1.034 0.9250 1.047 1.032 1.052 1.180 1.139 1.009 1.003 1.027 1.015 1.022 1.109 1.010 1.017 1.036 1.023 1.010 1.049 1.022 1.055
0.696 0.444 0.709 0.847 0.778 0.515 0.571 0.634 0.844 0.730 0.661 0.902 0.772 0.618 0.631 0.722 0.560 0.473 0.726 0.717 0.245 0.294 0.377 0.715 0.195 0.705 0.162 0.369 0.853 0.639 0.551 0.279 0.973 0.286 0.559 0.607 0.544 0.849 0.725 0.639 0.416 0.475 0.223 0.441 0.767 0.281 0.363 0.446 0.661 0.316 0.619 0.253 0.319
1.156 1.033 1.080 0.9210 1.451 1.136 1.038 1.164 1.185 1.331 1.214 1.092 1.035 1.101 1.125 1.079 1.075 1.019 1.138 1.069 0.9280 1.041 1.123 1.056 1.027 1.084 1.120 1.162 1.105 1.089 0.9750 1.055 1.001 1.020 1.059 1.004 0.9460 1.291 1.284 1.002 1.031 1.054 1.041 1.021 1.054 1.022 0.9960 1.039 1.010 0.9830 1.083 0.9850 0.9930
0.758 0.198 0.554 0.513 0.778 0.520 0.211 0.646 0.518 0.504 0.418 0.457 0.479 0.377 0.455 0.398 0.475 0.337 0.756 0.347 0.348 0.265 0.602 0.277 0.234 0.402 0.473 0.362 0.499 0.695 0.257 0.527 0.264 0.322 0.564 0.207 0.229 0.729 0.595 0.365 0.274 0.552 0.282 0.410 0.356 0.247 0.374 0.360 0.600 0.490 0.509 0.140 0.287
1.069 0.9650 1.133 1.224 1.403 0.8530 1.134 0.9310 1.660 0.6100 1.034 1.130 0.9580 0.8970 0.8830 1.152 1.121 1.179 0.8880 1.089 0.6970 0.9470 1.134 1.097 1.031 1.217 1.051 0.9030 1.088 1.096 1.084 1.033 1.030 1.088 0.8440 1.057 1.025 1.245 1.062 1.065 1.166 1.122 0.9530 1.003 0.9630 0.9520 1.090 0.9080 1.024 1.054 1.071 0.9460 0.9980
0.229 0.298 0.245 0.275 0.341 0.354 0.443 0.106 0.418 0.476 0.198 0.461 0.135 0.306 0.369 0.227 0.441 0.132 0.337 0.281 0.242 0.341 0.500 0.326 0.213 0.277 0.538 0.216 0.421 0.581 0.397 0.115 0.160 0.242 0.341 0.292 0.137 0.343 0.211 0.161 0.594 0.268 0.207 0.175 0.220 0.244 0.228 0.226 0.151 0.146 0.338 0.173 0.288
1.233 1.568 1.150 0.8200 0.7960 0.9800 1.224 1.278 0.7920 1.268 1.350 1.127 1.380 1.120 1.215 1.352 1.347 1.802 1.094 1.487 0.7960 1.158 1.481 0.8100 0.8920 0.6080 1.182 0.3120 1.500 1.327 1.108 1.114 0.9890 1.128 0.9550 1.041 1.276 1.034 1.120 1.019 0.9770 1.101 1.046 1.081 1.149 1.171 0.9610 1.046 1.083 0.9940 1.066 0.9350 1.316
0.576 0.420 0.652 0.289 0.569 0.490 0.504 0.546 0.406 0.555 0.465 0.359 0.541 0.460 0.601 0.666 0.700 0.557 0.559 0.692 0.255 0.250 0.647 0.418 0.216 0.546 0.300 0.304 0.595 0.424 0.322 0.331 0.488 0.342 0.379 0.602 0.329 0.440 0.414 0.431 0.282 0.346 0.364 0.298 0.254 0.433 0.205 0.418 0.121 0.196 0.183 0.306 0.637
30.181 0.187 0.224 30.474 33.15 30.398 0.142 30.605 32.26 32.92 31.59 31.42 30.0710 0.211 30.0620 30.261 30.325 30.589 30.00100 30.141 1.62 0.741 1.11 0.371 0.502 30.126 0.189 30.0920 32.21 32.67 0.330 0.506 0.663 0.204 0.693 0.194 0.341 33.15 31.58 0.434 0.490 0.176 0.301 0.229 0.0380 0.584 0.320 0.599 0.241 0.870 30.243 0.414 0.0160
0.282 1.92 30.246 31.27 32.34 0.483 0.0680 1.13 33.29 2.81 1.07 0.0620 1.40 0.664 1.07 0.439 0.578 1.94 0.618 1.14 0.398 0.517 0.406 31.19 30.630 32.29 0.159 32.01 1.66 1.38 30.0360 0.0420 30.294 30.0610 0.321 30.147 0.814 30.645 0.0910 30.309 30.944 30.309 0.240 0.182 0.597 0.614 30.570 0.382 0.111 30.409 30.144 30.0280 1.06
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Table 2 (continued) Ref. (lit)
D1
Av R1
D2
Av R2
D3
Av R3
D4
Av R4
D5
Av R5
Score F1
Score F2
LR/4 (lit 3) GR/10 (lit 3) GR/9 (lit 3) GR/56 (lit 3) GR/32 (lit 3) GR/40 (lit 3) GR/42 (lit 3) GR/41 (lit 3) GR/24 (lit 3) GR/27 (lit 3) GR/78 (lit 3) GR/33 (lit 3) GR/34 (lit 3) GR/35 (lit 3) GR/48 (lit 3) LR/6 (lit 3) GR/50 (lit 3) GR/49 (lit 3) GR/53 (lit 3) LR/3 (lit 3) GR/84 (lit 3) GR/85 (lit 3) GR/67 (lit 3) GR/75 (lit 4) GR/74 (lit 4) GR/76 (lit 4) GR/69 (lit 4) GR/70 (lit 4) GR/51 (lit 4) LR/5 (lit 4) GR/23 (lit 4) GR/66 (lit 4)
1.021 1.020 1.016 1.010 1.023 1.026 1.017 1.015 1.014 1.012 1.021 1.046 1.050 1.040 1.015 1.019 1.024 1.014 1.014 1.047 1.057 1.020 1.079 1.028 1.022 1.020 1.015 1.024 1.008 1.012 1.019 1.010
0.977 0.975 0.915 0.809 0.958 0.987 0.945 0.960 0.848 0.869 0.977 0.992 0.990 0.968 0.932 0.948 0.933 0.879 0.965 0.984 0.985 0.972 0.999 0.978 0.963 0.952 0.977 0.976 0.894 0.946 0.932 0.937
1.008 1.038 1.014 0.9870 1.025 1.020 1.010 1.035 1.007 1.028 1.031 1.032 1.033 1.006 1.021 1.038 1.019 1.010 1.022 1.029 1.013 1.020 1.067 1.034 0.9870 1.026 1.011 1.038 1.017 1.034 1.017 1.028
0.517 0.590 0.365 0.425 0.686 0.557 0.420 0.645 0.180 0.304 0.609 0.676 0.728 0.137 0.527 0.532 0.288 0.303 0.625 0.480 0.392 0.532 0.750 0.584 0.350 0.505 0.284 0.560 0.340 0.663 0.316 0.796
0.9970 1.033 0.9780 1.004 1.036 1.037 1.010 0.7780 1.016 1.050 1.059 1.043 1.047 1.013 1.073 1.083 0.6620 0.9650 1.022 0.9830 1.019 1.053 1.043 1.009 0.9700 1.056 0.9350 1.036 1.049 1.018 1.094 0.9970
0.392 0.310 0.278 0.284 0.239 0.563 0.343 0.462 0.360 0.394 0.390 0.350 0.403 0.247 0.391 0.435 0.514 0.178 0.171 0.215 0.295 0.422 0.243 0.105 0.186 0.520 0.258 0.187 0.487 0.179 0.441 0.444
0.9080 1.084 1.129 0.8470 1.022 1.074 1.064 1.076 1.019 1.069 1.107 1.002 1.199 1.156 1.029 1.004 1.124 1.058 1.061 1.043 1.010 0.9560 1.087 1.099 1.104 1.066 0.9670 1.136 1.106 0.9600 1.083 1.022
0.178 0.340 0.405 0.523 0.188 0.384 0.291 0.560 0.248 0.194 0.441 0.0890 0.740 0.399 0.143 0.395 0.543 0.215 0.216 0.285 0.261 0.184 0.412 0.254 0.452 0.360 0.168 0.520 0.421 0.306 0.284 0.146
1.100 0.9680 0.9880 1.036 1.082 1.009 1.103 1.117 1.149 1.189 0.9130 1.160 0.7250 1.382 1.168 1.210 0.9990 1.282 1.038 1.065 0.9250 1.076 1.072 1.128 1.062 1.123 1.020 1.045 0.9580 1.076 0.6860 0.8830
0.159 0.260 0.332 0.426 0.496 0.192 0.251 0.370 0.529 0.417 0.163 0.348 0.490 0.503 0.288 0.374 0.353 0.482 0.346 0.200 0.140 0.562 0.505 0.239 0.167 0.163 0.127 0.336 0.229 0.217 0.364 0.377
0.789 0.394 0.678 1.11 0.409 0.325 0.592 1.45 0.672 0.432 0.284 30.0310 30.224 30.0230 0.371 0.234 1.77 0.764 0.590 0.175 0.00200 0.497 30.773 0.286 0.690 0.306 1.08 0.217 0.569 0.727 0.333 0.847
0.569 30.586 30.705 0.541 0.0420 30.406 30.0790 0.0800 0.239 0.158 30.881 0.470 31.76 0.549 0.242 0.508 30.416 0.556 30.290 0.0260 30.304 0.266 30.0150 30.0760 30.356 30.0240 0.0820 30.530 30.782 0.255 31.58 30.638
Ref. = particle reference number. D1, D2, D3, D4, D5 are the ¢ve fractal component values calculated for the 50 steplengths covering the range between three pixels and 150 pixels (0.5^25 Wm). D1 covers the range between 0.5 Wm and 5 Wm. D2 covers the range between 5.5 Wm and 10 Wm. D3 covers the range between 10.5 Wm and 15 Wm. D4 covers the range between 15.5 Wm and 20 Wm. D5 covers the range between 20.5 Wm and 25 Wm. Av R1, Av R2, Av R3, Av R4, Av R5 are the average correlation coe⁄cients of the ¢ve fractal components, calculated between the three particle orientations considered (see text for explanation of the formula used). Score F1 is the factor 1 coordinate of particles in the diagram of Fig. 4. Score F2 is the factor 2 coordinate of particles in the diagram of Fig. 4.
The relatively poor discrimination power of Fig. 4 is caused by two main factors. The fractal dimension is very e¡ective in distinguishing between shapes with a di¡erent amount of irregularities. Such a parameter however fails in discriminating between simple geometric forms (Orford and Whalley, 1983), such as those displayed by elongated or isometric or angular clasts, which in volcanology are very important for assessing the particle features related to di¡erent eruptive processes. Such simple geometric proper-
ties are instead easily distinguished by means of adimensional shape parameters as those used by Dellino and La Volpe (1996), i.e. circularity, elongation, compactness and rectangularity. The second reason is that sometimes fractal data are affected by errors (as discussed in 3. Geologic and volcanologic background of study particles), and the averaging operation needed to smooth them out obscures also part of the shape information, weakening the descriptive power of the fractal dimension. Such a drawback is easily shown by the
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example of Fig. 5, on which the M^R plots of the three orientations of an ash particle are shown. The scatter of data points occurring on the ¢rst two orientations is so high (low correlation coef¢cient) that it is very di⁄cult to identify a clear trend. The averaging operation among data of the three orientations eventually washes out also the information on shape that emerges from the third orientation that, thanks to a lower scatter (higher correlation coe⁄cient), revealed a certain alignment of points. One should conclude therefore that the fractal analysis is not completely successful in distinguishing between particles from magmatic and phreatomagmatic fragmentation processes. A further re¢nement of the statistical elaboration technique should be needed to reduce the scatter of points. For example, the use of carefully selected moving average values should serve to re-establish the power of the fractal dimension. In the present paper we have decided to not further implement the statistic elaboration. We have instead chosen to concentrate on some representative particles that, on showing a clear trend in
11
the M^R plot, are easy to interpret by a volcanological point of view. Some very interesting fractal trend emerges in fact from the investigation of ash particles whose M^R plots are not a¡ected by a relevant scatter of points and have a high correlation coe⁄cient (about 50% of the particle collection). The analysis of the M^R plots of these particles shows a peculiar and distinctive behavior between ash from magmatic and phreatomagmatic processes. As shown on Fig. 6, phreatomagmatic particles reveal a clear linear trend on the M^R plot and the correlation coe⁄cient of the regression line is close to 1. Such particles are self-similar over the whole investigated scale range, they show indeed a fractal behavior and can be described as true fractals (mono fractals). The fractal dimension value is quite low; suggesting therefore that particle boundary is moderately irregular (as by the photographs of Fig. 6). On Fig. 7 a few additional M^R plots of phreatomagmatic particles are shown, to represent the range of variation be-
Fig. 4. Diagram of factor 1 vs. factor 2, showing the discrimination obtained by multivariate statistic analysis of the ¢ve fractal components. Data points represent individual particles. In the inset, the key of symbols of the four lithofacies of deposits is shown. The phreatomagmatic, magmatic and overlap ¢elds are also drawn.
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tween these true fractals. The fractal dimension value typically ranges between 1.02 and 1.07. Magmatic particles show instead, as revealed by Fig. 8, two clear and distinct segments in the M^ R plot, they are indeed multifractals and a textural (small-scale) fractal with a lower value is distinguished from a structural (large-scale) fractal that has a higher value. The textural fractal typically occurs in the scale range between 0.5 Wm and 10 Wm, and its value is similar to that of phreatomagmatic particles, indeed, at small scales, magmatic particles are moderately irregular. The structural fractal, which typically occurs in the scale range between 10.5 Wm and 25 Wm, has a signi¢cantly higher value than the textural
one. It is associated to the walls of ruptured vesicles occurring on particle boundary, as is shown on the photographs of Fig. 8. In fact the walls of ruptured vesicles are on the scale range of the structural fractal (10.5^25 Wm). On Fig. 9, a few additional M^R plots of magmatic particles are shown, to report the range of values of these multifractal particles. The textural fractal ranges in value between 1.02 and 1.07, the structural fractal ranges between 1.16 and 1.57. The fractal and multifractal behaviors of magmatic and phreatomagmatic particles mark, in our opinion, some important aspects of the fragmentation processes that characterized the explosive activity of the Monte Pilato^Rocche Rosse erup-
Fig. 5. On this plate, the SEM photograph of an ash particle is presented together with the M^R plots resulting from three different particle orientations. The not rotation invariance of data points is evident. Particle reference number is on the bottom-left of photograph. D = fractal dimension value. R = correlation coe⁄cient. 1 = M^R plot resulting from the original orientation. 2 = M^R plot resulting from the 90‡ clockwise rotation. 3 = M^R plot resulting from the 90‡ anticlockwise rotation.
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Fig. 6. On this plate, two SEM images of phreatomagmatic particles and the corresponding M^R plots are shown. Particle reference number is on the bottom-left of photographs. D = fractal dimension value. R = correlation coe⁄cient. Diagrams, which are not a¡ected by a relevant scatter of data points, show a good alignment of data points (see the least square regression line) that result in the true fractal behavior. The fractal dimension value, D, is quite low, suggesting that the irregularities are self-similar at all scales and that the shape is not very complex. This behavior is con¢rmed by the visual analysis of particle photographs, which evidence clasts outline characterized by evenly distributed irregularities and indentations.
tions, as will be tentatively discussed in 5. Discussion and conclusion.
5. Discussion and conclusion In the case of the Monte Pilato^Rocche Rosse eruptions, we have found that the fractal dimension, compared to more conventional shape descriptors, does not add signi¢cant information on the general shape of ash particles. Simple
geometric properties are in fact more easily de¢ned, as done in a previous paper (Dellino and La Volpe, 1996), by using adimensional parameters such as compactness, elongation and circularity. It is however not our intention to cast doubts on the ability of the fractal dimension in describing complex^irregular shapes. The analysis of the M^R plots not a¡ected by a signi¢cant scatter allowed in fact to obtain valuable information on the outlines of ash particles, that in our opin-
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Fig. 7. On this plate, the M^R plots of four phreatomagmatic particles are presented. Diagrams do not show a relevant scatter of data points (see the least square regression lines) and reveal the true fractal behavior. D = fractal dimension value. R = correlation coe⁄cient. The fractal dimension value, D, which ranges between 1.03 and 1.07, shows that phreatomagmatic particles have a not very complex shape. Particle reference number is on the bottom-left of diagrams.
ion is not of trivial importance for the characterization of the Monte Pilato^Rocche Rosse eruptions. Many independent processes, such as fragmentation, transport and alteration, likely occur in the generation of the boundary irregularities of pyroclastic particles. Each of the processes in£uences the irregularity of particle outline at di¡erent scales, likely resulting in their fractal and multifractal behavior. Carey et al. (2000) de¢ned primary jokulhlaup particles as multiple fractals. They associated the textural (small-scale) fractal to fractures, and the structural (large-scale) fractal to the occurrence of broken bubbles along particles outline. They however did not ¢nd true fractal (mono fractal)
shapes, which instead are found in the phreatomagmatic particles of the present paper. In the case of phreatomagmatic particles of Monte Pilato^Rocche Rosse eruptions, shape should be simply related to the fragmentation process, since no additional mechanisms, i.e. transportation or alteration, have signi¢cantly modi¢ed particle shape, as stated by Dellino and La Volpe (1996). During phreatomagmatic explosions, ¢ne ash particles are the result of a brittle fragmentation process (Zimanowski et al., 1997). E¡ective contact between magma and water results in a sudden expansion of water that deforms very fast the melt, which reacts by fracturing. Water intrudes into cracks leading to a cascading process, which
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results in the ¢ne brittle fragmentation (Bu«ttner and Zimanowski, 1998). Particle shape and boundary irregularities are related to branching cracks propagating into the brittle melt, which are self-similar at all scales (Brown and Wohletz, 1995). Phreatomagmatic ash particles are indeed true fractals and the fractal dimension refers to the rheological behavior of magma, indeed mainly to its relaxation time. Such a result shows, in contrast to what is reported in the literature, that particles of geological interest may show indeed
15
fractal behavior. Probably, in the data set presented by Carey et al. (2000) no phreatomagmatic particles were present, therefore no true fractal particles were identi¢ed. Also the shape and irregularities of the magmatic particles of the Monte Pilato^Rocche Rosse eruptions should be simply related to the fragmentation process since, as for the phreatomagmatic ones, they did not su¡er shape modi¢cation by transportation or alteration (Dellino and La Volpe, 1996). Magmatic particles are multifractals
Fig. 8. On this plate, two SEM images of magmatic particles and the corresponding M^R plots are shown. Particle reference number is on the bottom-left of photographs. D = fractal dimension value. R = correlation coe⁄cient. Diagrams, which are not affected by a relevant scatter of data points, show the multifractal behavior that is marked by two distinct segments, which identify the textural fractal, DT, and the structural fractal, DS. Values of DT and DS are shown together with the least square regression lines of the two segments. R = correlation coe⁄cient. The low value of DT is related to small-scale irregularities along particles outline, the high value of DS is related to the relatively strong irregularities resulting from the walls of ruptured vesicles, which render the shape of particles very complex.
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and show a textural and a structural fractal; shape irregularities of such particles occur at two distinct ranges of scales as shown by the two distinct fractal components. The textural component, which relates to smallscale irregularities, has a low value, which is similar to that of phreatomagmatic particles. The structural component, having a higher value, relates instead to large-scale irregularities that represent the walls of ruptured vesicles laying on the particle outline. We suggest therefore that the particle boundary of magmatic particles is de¢ned by two orders of
irregularities that are not self-similar, likely representing two independent factors : one related to vesicle walls, the other independent from vesicles. It seems that magmatic fragmentation, during the Monte Pilato^Rocche Rosse eruptions, was not simply related to the gas-bubble behavior, i.e. to the process of growth, expansion, interference and consequent explosion of vesicles, as suggested by some authors (Sparks, 1978). Even if bubble expansion is assumed during the exsolution processes that joined magmatic activity, vesicles did not completely expand, up to interference, before fragmentation. If such had been the case, particle
Fig. 9. On this plate, the M^R plots of four magmatic particles are presented. Diagrams do not show a relevant scatter of data points and two segments, which reveal the multifractal behavior, are identi¢ed. The textural fractal value, DT, and the structural fractal value, DS, are reported together with the least square regression lines of the two segments. R = correlation coe⁄cient. The textural fractal, which ranges between 1.03 and 1.07, shows that magmatic particles are not highly irregular at small scales. The structural fractal, which ranges between 1.16 and 1.57, relates to the large-scale irregularities de¢ned by walls of ruptured vesicles, which are not self-similar to the small-scale irregularities. Particle reference number is on the bottom-left of diagrams.
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irregularities should have been in£uenced simply by the vesicle distribution along the outline, resulting in a true fractal, which value should have simply re£ected vesicle walls. The textural fractal seems instead not related to vesicles, as also suggested by Carey et al. (2000). It is characterized by a value similar to that of phreatomagmatic particles, and we suggest it was related to the rheological behavior of the melt, which in fact was of the same composition (rhyolitic), likely of the same viscosity, as the melt that characterized phreatomagmatic explosions. Bubble expansion was in our opinion interrupted by a brittle fracturing of the melt, and occurred as a response to a mechanical shock. The results obtained in this paper seem therefore to support recent models, based on experimental research, that refer magmatic fragmentation dynamics to brittle processes (Alidibirov and Dingwell, 1996; 2000). These models point out the role of rapid decompression as the dominant factor producing magmatic fragmentation, especially for highly viscous rhyolitic melts as those that characterized the products studied in the present work. We tentatively suggest that during Monte Pilato^Rocche Rosse eruptions, magmatic fragmentation was a brittle response of the viscous melt to a rapid decompression process that exceeded the tensile strength of magma. It is likely that such rapid decompression should have been caused by the shock waves produced by a previous phreatomagmatic event. A relevant amount of the thermal energy that is converted into a mechanical form is released just as energetic shock waves during explosive magma/water interaction (Zimanowski et al., 1997). Additional work on pyroclastic products representative of di¡erent magma composition and of di¡erent eruption intensities is needed to assess the validity of such argumentation also for other eruptions characterized by an alternating magmatic/ phreatomagmatic style. Finally, a further implementation of the method proposed in this paper (that allows reduction of the scatter in the M^R plots) would be very useful for the quantitative analysis of particles resulting from laboratory experiments, and could be eventually used for scaling of experimental results to natural processes.
17
Acknowledgements The authors are grateful to C. Hagelberg and J. Taddeucci for the thorough revision of the paper.
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