JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, B07406, doi:10.1029/2006JB004917, 2008
Multiobjective genetic algorithm inversion of ground deformation and gravity changes spanning the 1981 eruption of Etna volcano Daniele Carbone,1 Gilda Currenti,1 and Ciro Del Negro1 Received 22 December 2006; revised 11 January 2008; accepted 12 March 2008; published 23 July 2008.
[1] During the last few decades, joint investigations of microgravity and surface
deformation measurements have played an increasingly important role in studying the internal dynamics of active volcanoes. Deformation and microgravity observations have been accomplished at Mt Etna since the eighties. Past data sets collected during important paroxysmal events can be utilized as case-studies to both (1) test the possibilities of nowadays more powerful inversion tools and improved analytical formulations to model the source-mechanisms of volcano-related deformation and gravity changes and (2) in turn obtain new insights into the functioning of the plumbing system of the volcano. Here we analyze a data set spanning the March 1981 eruption of Mt. Etna. Large horizontal displacements were evidenced on the NE and SW flanks of the volcano through electrooptical distance measurements (EDM) during two 20-month periods, both encompassing the March 1981 eruption. Elevation changes, evidenced through leveling measurements, during a 12-month period spanning the eruption, were in general smaller than horizontal displacements with important amplitudes only close to the eruptive fissure. Gravity measurements, carried out together with leveling measurements, evidenced positive changes, spatially well correlated with elevation changes, but having a larger wavelength. The joint inversion of the multimethod geophysical data is regarded as a multiobjective optimization problem and solved through a Genetic Algorithm technique of the nondominated type. We conclude that a composite intrusive mechanism with two tensile cracks, each associated to a zone where preexisting microfractures were filled with new magma, leaded to the 1981 eruption. The results of the present study highlight the advantages of multiobjective evolutionary algorithms, as a powerful tool to jointly invert multimethod geophysical data, and pose important issues on the subject of volcano-monitoring. Citation: Carbone, D., G. Currenti, and C. Del Negro (2008), Multiobjective genetic algorithm inversion of ground deformation and gravity changes spanning the 1981 eruption of Etna volcano, J. Geophys. Res., 113, B07406, doi:10.1029/2006JB004917.
1. Introduction [2] Mt. Etna is nowadays one of the best monitored volcanoes in the world with many arrays of stations, each devoted to measure, in a discrete or continuous fashion, a given geophysical or geochemical parameter. Apart from seismic observation, which can only be carried out continuously, until the Eighties most measurements were accomplished through repeated surveys, while, since the Nineties, with technological improvements allowing most logistic difficulties to be overcame, more and more continuously recording stations have been installed even on the summit zone of the volcano (see Bonaccorso et al. [2004] for a comprehensive review). During the last decades, many flank eruptions have occurred at Etna (1981, 1983, 1985, 1986 – 1987, 1989, 1991 – 1993, 1999, 2001, 2002 –2003, 1 Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Catania, Catania, Italy.
Copyright 2008 by the American Geophysical Union. 0148-0227/08/2006JB004917$09.00
2004 [e.g., see Branca and Del Carlo, 2005; Behncke et al., 2005]). Following what stated before, the bulk of the information about the older eruptions come from discrete data. [3] The 1981 flank eruption of Etna (Figure 1a) [Romano, 1981; Scott, 1983] took place in March (between the 17th and the 23rd), before and after pairs of microgravity and leveling surveys of measurement were performed (in August/September 1980 and July/August 1981 [Sanderson et al., 1983]). Electroptical distance measurements (EDM) were also taken before and after the eruption (campaigns made in October 1979, September 1980, June 1981, and May 1982 [Bonaccorso, 1999]). [4] Sanderson et al. [1983] performed a joint analysis of the microgravity and leveling data covering the August/ September 1980 to July/August 1981 period. The uplift of about 17 cm and the gravity increase up to 63 mGal, both observed close to the new eruptive fissure, were interpreted as due to (1) the shallow magma intrusion which led to the 1981 eruption and (2) a recharge of a deeper part of the plumbing system of the volcano, respectively. Furthermore,
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intrusion of magma. However, in spite of the overall good quality of the data they used, the results obtained by these authors are severely compromised by the fact that gravity and elevation changes were separately inverted, using oversimplified analysis tool. A 2D model with the source-body having fixed ratios among its dimensions was used to calculate the elevation changes, while a 2D model accounting only for the mass added to or subtracted from the source volume was utilized to calculate the gravity changes. [5] Subsequently, Bonaccorso [1999] cross analyzed the leveling data set used by Sanderson et al. [1983] together with the EDM data acquired between October 1979 and May 1982. He did not take into account the microgravity data. Bonaccorso [1999] modeled the observed deformation pattern using a more advanced analytical formulation [Yang and Davis, 1986] than that utilized by Sanderson et al. [1983]. He came to the conclusion that two tensile cracks activated, the first one representing the initial deeper part of the intrusion, the latter reflecting the final shallower stage of the intrusive process, more closely related to the eruptive fissures. Bonaccorso [1999] rejected the view of Sanderson et al. [1983] about the western tensile structure being oriented E-W on the ground of the horizontal displacements (EDM data), which point to a N-S trending structure rather than to an E-W trending one. [6] Starting from the conclusions of Bonaccorso [1999], we aim to gain a more comprehensive picture of the intrusive mechanism related to the 1981 flank eruption of Etna through a joint inversion of the available data (microgravity, leveling and EDM). The joint inversion is regarded as a multiobjective optimization problem and solved through a Genetic Algorithm technique. By taking into account the gravity measurements we can constrain the mass change due to the intrusion. The availability of advanced analytic formulations, to model the observed data, and powerful inversion tools, to search for the best possible model-source, enables to gain new insights into the magmatic processes linked to the 1981 Etna eruption.
2. The 1981 Flank Eruption of Mt. Etna Figure 1. (a) sketch map showing the location of (1) the leveling stations (filled circles), (2) the gravity stations (open circles), and (3) the EDM networks and stations (shaded areas with triangles). The position of the eruptive fissures and lava flow of the 1981 eruption are also reported. (b) August/September 1980 to July/August 1981 elevation changes observed through leveling measurements along the ABC profile in Figure 1a. (c) August/September 1980 to July/August 1981 gravity changes observed along the ABC profile in Figure 1a. The dashed line in Figures 1b and 1c indicates the position of the eruptive fissures.
Sanderson et al. [1983] postulated that another tensile structure activated on the western flank of the volcano and provoked the uplift observed in that sector. This structure was thought to be radial and oriented approximately east – west. The gravity changes observed in the same zone were also assumed to be due to a deeper
[7] The 1981 eruption of Etna took place between 17 and 23 March on the north-northwestern flank of the volcano (Figure 1a). This event was almost unique among Etna’s eruptions in (1) developing an unusually long fracture system, (2) producing a high effusion rate and (3) occurring on a sector of the volcano where no eruptive vents had opened for many centuries. During the six months preceding the eruption, a few episodes of paroxysmal activity occurred at the Northeast Crater, with minor lava flows and lava fountains. The level of seismicity started rising since the 14th of March [Kieffer, 1982] and culminated in a swarm with epicenters migrating toward the northern flank, where a NW-SE-trending fissure system opened on 17 March at the elevation of 2550 m a.s.l. This event was accompanied by lava fountains and minor effusion. The fissure system quickly propagated downslope and, in a few hours, it reached the elevation of 1800 m a.s.l., where the main flow of the eruption was emitted. The lava flow moved at a high velocity (up to 1 km/h) and approached the village of Randazzo (Figure 1a), seriously threatening it.
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[8] On 18 March, the flow had already slowed considerably. The eruptive fissures further propagated down to 1150 m elevation and emitted minor lava flows. Mild explosive activity at the lowermost vents, together with slow advancements of the lava fronts, continued until 23 March, when the eruption came to an end. [9] The estimated total volume of the lava erupted during the 1981 eruption ranges between 18 106 m3 [Murray, 1982] and 35 106 m3 [Romano, 1981]. During the first day of the eruption the peak effusion rate was 300 m3/s, one of the highest values ever observed at Etna.
3. Data Presentation [10] The deformation pattern during the 1981 Etna eruption, along the horizontal and vertical directions was assessed through EDM and leveling measurements, respectively [Bonaccorso, 1999]. EDM measurements on the SW network (Figure 1a) were made in September 1980 and May 1982, while on the NE network (Figure 1a) they were made in October 1979 and June 1981. Leveling measurements were accomplished in August/September 1980 and July/ August 1981, along the array of 51 benchmarks shown in Figure 1a. [11] Along the horizontal direction, a general movement of the benchmarks, away from the fracture line and orthogonally to its direction, was observed (Figure 1a). The magnitude of the displacements ranges between a few and a few tens of centimeters and, according to Bonaccorso [1999], is affected by an error smaller than 1 cm. [12] The strongest elevation changes were observed in the proximity of the eruptive fracture (up to 17.5 cm; Figure 1b). A wider but weaker (up to 3 – 4 cm) positive anomaly occurred on the western flank of the volcano (Figure 1b). Furthermore, a slight (approximately 2 cm) subsidence was observed between 3 and 8 km to the east of the eruptive fissures (Figure 1b). According to Bonaccorso [1999], the volume which would correspond to the subsidence along the BC Profile (Figure 1a) is much smaller than the volume of magma erupted. The error on the elevation changes is within ±4 mm [Sanderson et al., 1983; Bonaccorso, 1999]. [13] Positive gravity changes, observed between August/ September 1980 and July/August 1981 along a network of 24 benchmarks, most of which coincide with leveling benchmarks (Figure 1a), share some features with elevation changes: (1) the maximum variation (63 mGal) occurs in the proximity of the eruptive fracture and (2) another significant change (up to 55 mGal) takes place on the western flank (Figure 1c). However, with respect to the corresponding elevation changes, the gravity changes across the eruptive fracture are broader, while on the west flank they appear to be shifted toward south (Figure 1c). Sanderson et al. [1983] estimated an error of determination of the gravity changes between 7 and 13 mGal (it increases up to 21 mGal for stations measured only once). [14] The overall surface deformation pattern, with horizontal movements away from the eruptive fissures and positive elevation changes, points toward an intrusive mechanism as the source of the observed changes. Furthermore, the observed variation of the gravity field, with amplitude up to 63 mGal, suggests the occurrence of a
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significant mass increase within the part of the plumbing system where the inferred intrusive process took place.
4. Inverse Modeling Technique [15] Inverse methods combine forward models with appropriate optimization algorithms to automatically find out the best model parameters [Beauducel and Cornet, 1999; Murray et al., 1996; Owen et al., 2000; Williams and Wadge, 2000; Tiampo et al., 2004; Jousset et al., 2003; Carbone et al., 2006]. As a rule, geophysical inverse problems are illposed, suffering from the ambiguity and instability of inverse solutions. Because of the inherent nonuniqueness of the inverse problems, there are often a large number of inverse models that could explain the observed geophysical data within the error limits. Moreover, the geophysical inverse problems are usually unstable. Small variations in the observed data may result in dramatic changes of the model parameters. Integrated inversions of multiparametric data has the potential to yield more robust estimates of source parameters and reduce the ambiguity or the range of likely solutions [Currenti et al., 2007a]. The aim of the multiparametric inversion is to find the vector of model parameters m = {m1, . . ., mp} 2 M that minimizes the misfit between the observed and calculated data (M is the model space). The joint inversion of different geophysical parameters implies that the misfits for each i-th data set are simultaneously minimized: i fi ðmÞ ¼k gi ðmÞ dobs k i ¼ 1; . . . ; k
ð1Þ
where fi is an objective function and denotes the difference between the value calculated through gi(m) (forward model) and the observed value diobs for each i-th geophysical parameter in a l2 norm. Therefore the joint inversion of a multiparametric geophysical data set can be regarded as a multiobjective optimization problem (MOP). To solve this problem means to find the set of model parameters m* which satisfies some constraints and optimizes the objective function vector, whose elements are the objective functions: m* ¼min F ðmÞ with mmin mj mmax j ¼ 1; . . . ; p j j m2M
where FðmÞ ¼ ½ f1 ðmÞ; f2 ðmÞ; . . . ; fk ðmÞ
ð2Þ
Since the gi(m) forward models are nonlinear operators, it calls for using robust nonlinear inversion methods. For the solution of the geophysical nonlinear inverse problems there are many examples of successful implementation of gradient-type methods [Tarantola, 1987; Zhdanov, 2002]. However, nonlinear models can be difficult to optimize due to the presence of inherent discontinuities or local optima, where techniques based on gradient methods are likely to get stuck [Beauducel and Cornet, 1999; Cervelli et al., 2001; Currenti et al., 2005, 2007b; Tiampo et al., 2004]. Among several techniques, Genetic Algorithms (GAs) have been proven to be a robust global search procedure, suitable for nonlinear and multimodal inverse problems, which are typically encountered in geophysics [Boschetti et al., 1996; Tiampo et al., 2000, 2004; Currenti et al., 2005; Moorkamp et al., 2007]. When the parameter space is very large, multimodal, and poorly understood, GA is able to perform a
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broad search over the model space, with a greater likelihood of finding the global optimal solution [Holland, 1975; Goldberg, 1989]. In the framework of a multi-objective Genetic Algorithm, the estimate obtained through minimization of equation (2) represents a Pareto optimal solution [Fonseca and Fleming, 1993; Deb and Gupta, 2005; Sobol, 1985]. Although the joint inversion of multiparametric geophysical data is not a problem with highly contradictory objectives, it is unlikely that the same set of model parameters is the best for all the objectives. That is caused by the natural uncertainty (noise) that affect the data and the limited number of available measurements. Hence any Pareto optimal solution is the result of a trade-off made between the objectives. [16] Classical optimization methods convert the multiobjective optimization problem of minimizing the vector F(m) into a single-objective optimization problem. Usually, that operation is accomplished by constructing a weighted sum of all the objectives (weighted sum strategy). [17] The minimization of the weighted sum provides an estimate for the model parameters vector m: m* ¼ min m2M
k X
wi fi ðmÞ
ð3Þ
i¼1
However, the noncommensurability of the different objective functions means that the contribution to equation (3) of each data set needs to be normalized on the basis of the number of data points, the accuracy of the data and the physical unit of the geophysical parameter. If the weight coefficients are not properly chosen, the minimization of the functional in equation (3) may yield solutions that are dominated by one data set. [18] To avoid this drawback, GAs based on the Pareto approach [Deb et al., 2000] were developed, which differ from other similar inversion schemes mainly in the population sorting procedure. The latter is based on the nondominance concept: a solution m1 is said not to be dominated by m2 if the following two conditions are satisfied: 1 fi ðm1 Þ fi ðm2 Þ for i ¼ 1; . . . ; k 2 fi ðm1 Þ < fi ðm2 Þ for at least one i
ð4Þ
A solution m* is said to belong to the Pareto optimal set P if and only if no other m 2 M exists such that fi (m) fi (m*), for all i = 1,. . ., k and fi(m) 6¼ fi(m*) for at least one i. In other words, m* is a Pareto optimal solution if there exists no feasible solution m which would decrease some criterion without causing a simultaneous increase in at least one other objective function. The set of the objective function vectors whose solutions are in the Pareto optimal set forms the Pareto front. [19] The NSGA-II algorithm (Nondominated Sorting Genetic Algorithm), devised by Deb et al. [2000], classifies the population of possible solutions into several sets of nondomination levels [Deb et al., 2002]. A rank is assigned to each solution depending on the nondomination level to which it belongs [Deb et al., 2000]. Obviously, the firstlevel of nondominated solutions are candidates for Paretooptimal solutions. Besides the ranking procedure, the
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population is evolved using well-known GA operations (selection, crossover and mutation operators on real numbers). The use of NSGA-II algorithm has two main advantages. First, no weights need to be defined a priori since the scalarization of the objective function is avoided [Schwarzbach et al., 2005]. Secondly, in a single run of inversion, a set of solutions is achieved (instead of a single one as in traditional approaches) underlining the inherent ambiguity of geophysical inversions and giving a representation of it through the Pareto front. It makes possible to understand whether the objectives ‘‘cooperate’’ (have similar optimal solutions) or ‘‘conflict’’ (have different optimal solutions) with each other. Conflicting objectives imply a Pareto front spread over a large region. Conversely, if all the objectives tend to converge toward similar solutions, the Pareto front is well confined in the vector space of the objectives [Deb et al., 2000]. The size and shape of the Pareto front provides insights into the interactions among the objective functions. By investigating the distribution of the objectives in the vector space, it is possible to understand whether multiple inverse solutions, each able to satisfactorily match a family of geophysical observations, are likely to occur. That provides a depiction of the trade off between competing objectives and thus allows to better quantify the ambiguity of the inverse solutions. Among all the Pareto optimal solutions, the models that provide an adequate fit to the data are selected by examining all the models whose chi-square value c2 is less than a particular 2 for the value c2a obtained by the F ratio test. Given the copt 2 2 optimal models, all the models with c < ca are consistent with the optimal models at the 100 a% confidence level, where: c2a ¼ c2opt 1 þ
p F ð p; n p; 1 aÞ np
ð5Þ
and p is the number of model parameters, n is the number of data and F is the F distribution with p and n-p degrees of freedom [Draper and Smith, 1981]. The models having c2 below these thresholds for all the data set are valuable inverse solutions. In this selected set, the c2tot is computed as a sum of the c2 of all the data set, and the model with the minimum c2tot is taken as optimal solution. This procedure warrants that the optimal solution is valuable for each data set. Moreover, a confidence region for each parameter is determined from the range of values given by all the models consistent with the optimal model at the specified confidence level [Murray et al., 1996]. If a parameter is well resolved, it will have a narrow range of parameter values for which the model misfit is near the minimum.
5. Choice of the Forward Models [20] A key point in any inversion procedure is the choice of the appropriate forward model to be used to invert the observations. Both the gravity and elevation changes encompassing the 1981 eruption show two main anomalies placed respectively on the Northwestern sector of the volcano, in the zone where the eruptive fractures opened, and 4 km to the west of the summit of the volcano (see Figure 1a and section 2). EDM data point to strong
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horizontal displacements on the NE and SW flanks of the volcano (see Figure 1a and section 2). [21] As reported above, we perform our analysis starting from the conclusions of Bonaccorso [1999] and formulate the inversion problem accordingly, under the following a priori assumptions. [22] 1. Two tensile structures activated during the period encompassing the 1981 eruption; [23] 2. The northernmost tensile structure strikes along the eruptive fissures of the 1981 eruption; [24] 3. The second tensile crack is placed in more central position and works as a joint between the central conduit and the eruptive fissures. [25] A series of analytical solutions have been derived and widely used to model tensile source mechanisms leading to ground deformation and gravity changes [e.g., Bonafede and Mazzanti, 1998]. In the present study we utilize the sets of analytical expressions by Okada [1985, 1992] and Okubo [1992]. Both analytical formulations account for the effects arising from tensile faults buried in a homogeneous half-space and, while the former allows to calculate the displacement, shear and tilt due to the faulting processes, the latter allows to calculate all the contributions to the corresponding gravity change. Namely (1) the density changes due to the compressibility of the medium, (2) the surface mass redistribution accompanying the uplift of the ground surface, and (3) the input of new mass into the cavity generated by the tensile fracturing. Thus by coupling these two sets of analytical expressions we are able to model the surface deformation (horizontal and vertical) and gravity changes due to a given tensile source simultaneously.
6. Inversion Procedure [26] As stated before, the aim of the present study is to invert the multimethod data set encompassing the 1981 Etna eruption using the analytical formulations by Okada [1985, 1992] and Okubo [1992], as forward models, and the NSGA-II [Deb et al., 2000] as the inversion tool. We utilize two tensile dislocation sources (see previous section), each implying nine different parameters: m = {d, a, D, W, L, U, Xc, Yc, Dr}. The description and assumed range of variability for each source parameter are reported in Table 1. The ranges of variability for the position on the horizontal plane and length of the northernmost tensile dislocation model are set under the assumption that the tensile source follows the trend of the eruptive fissures (see previous section). Furthermore, since it is supposed to be closely related to the eruptive fissures, the top depth of this tensile source is expected to be very shallow and is kept fixed to a value of 20 m. We also assume the density contrast between the intruding material and the medium to be the same for both tensile structures. The ranges of variability of the second tensile source parameters (Table 1) are set according to the available geophysical and volcanological evidences [Sanderson et al., 1983; Bonaccorso, 1999]. As for the values of the elastic parameters, we adopted the Lame’s constants to be equal to each other (l = m), which leads to a value of the Poisson’s ratio of 0.25 [Bonaccorso et al., 2005]. The NSGA-II was run for 800 generations, using a population of 500 individuals. The number of iterations was
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sufficient to assure a good convergence of the algorithm. Additional tests showed that increase the number of interactions does not lead to improvements in the objective function vector. 6.1. Model 1: Two Tensile Sources [27] The simultaneous inversion of the leveling, EDM and gravity data spanning the 1981 eruption can be regarded as a MOP with one objective function for each of the three data sets (see section 4). Through the NSGA-II strategy, a family of solutions is achieved. From Figure 2a it appears that the calculated values of the objective function vectors form a Pareto front spread over the parameter space, an evidence of the conflicting behavior of the objective functions. Furthermore, no solution has c2 values below the thresholds c2a at 95% confidence level contemporaneously for EDM, leveling and gravity data. The root mean square errors (RMSE) between observed and computed values (Table 2) shows that (1) while the Pareto front crosses the zones below 2s for leveling and gravity data, it does not reach the zone with low misfit for the EDM data; (2) the objectives relative to both horizontal and vertical deformation are minimized within a region of the Pareto front where the RMSE relative to gravity are unsuitably high (Figures 3a – 3c). The first above point suggests that it is not possible to assess a solution able to well fit the observed horizontal deformation. That could be due to the shift in the timing of the EDM measurement campaigns with respect to leveling and gravity surveys (see section Data presentation). Since EDM was measured over longer time intervals than leveling and gravity, the horizontal deformation could be affected by sources which do not affect the other two geophysical data set. As shown in Figure 2, to move along the Pareto front toward the region where the gravity objective is minimized, means to increase the value of the objective of both horizontal and vertical deformation. Accordingly, a model (black square in Figure 2c, Model 1a in Table 1), chosen in the region where the objective of the deformation parameters are minimized (with a RMSE between observed and calculated data of 2.8 cm and 6 cm, for vertical and horizontal deformation, respectively) allows to explain only a low percent of the observed gravity changes (Figure 2b; the RMSE between observed and calculated data is 23.5 mGal). Conversely, a higher gravity effect can be obtained using tensile sources which overestimate the observed deformation (black triangle in Figure 2c and Model 1b in Table 1; RMSE are of 12.6 cm, 11.5 cm, 15.0 mGal for EDM, leveling and gravity, respectively). These results indicate that the chosen forward model is not able to provide a satisfactory solution to the proposed multiobjective problem (Figure 2c). 6.2. Model 2: Two Tensile Sources With Zones of Increasing Mass [28] From the results presented in the previous section, it follows that the model parameters that optimizes the fit between observed and calculated anomalies for the deformation parameters, through the analytical formulation by Okada [1985, 1992], allow to explain a low percent of the observed gravity variation through the model of Okubo [1992]. One way to account for this apparent discrepancy is to assume that a mass increase, independent from the ensuing deformation, took place in conjunction with the
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Table 1. Summary of the Parameters Range Used in the Inversiona Parameter Z1, depth of the top, m b.s.l. L, length, m H, height, m W, tensile opening, m f , azimuth (from the north) X, northing of top center, m Y, easting of top center, m d , dip (from the east) Dr , density contrast, kg/m3 D, depth, m H, height, m U r , thickness density, kg/m2
Minimum
Maximum
Model 1a
Model 1b
20 4,000 200 0.5 35 4,181,250 496,750 45 100
North Source 20 8,000 500 2 15 4,186,250 499,250 145 500
20 4,530 566 1.87 29 4,184,892 497,708 69.72 461
20 4,620 600 1.97 33 4,186,250 497,961 110.38 496
905 2,291 1,999 3.36 29.9 4,180,540 499,497 93 461
970 4,940 1,710 5.22 9 4,182,500 499,395 106 496
500 100 0
Z1, depth of the top, m b.s.l. L, length, m H, height, m W, tensile opening, m f , azimuth (from the north) X, northing of top center, m Y, easting of top center (m) d , dip (from the east) Dr, density contrast, kg/m3
100 1,000 500 2 30 4,180,000 497,500 45
D, depth, m H, height, m U r, thickness density, kg/m2
500 100 0
North Infiltration Zone 2000 2000 50,000 South Source 1,000 5,000 2,000 6 10 4,182,500 501,000 145
South Infiltration Zone 2,000 2,000 50,000
Model 2 20 6,703 ± 231.7 ± 0.93 ± 16 ± 4,184,924 ± 497,970.7 ± 88.1 ± 116.8 ±
425 20 0.2 1.29 226 109 7.26 78
1,325 ± 180 576.5 ± 200 13,146.76 ± 7873 404 3,589 1,140 5.2 30 4,181,004 499,988.3 131.1 116.8
± ± ± ± ± ± ± ± ±
140 413 119 0.6 3.5 194 117 6 78
1,585 ± 293 1,409 ± 213 34,485.43 ± 6,234
a The source parameter for Models 1 and 2 are reported. The confidence ranges at 95% confidence level of the parameters for the Model 2 are also computed.
intrusive event which leaded to the 1981 eruption. The good spatial correlation between observed gravity and elevation changes would suggest that the inferred additional mass increase was triggered by the same overall process leading to the observed surface deformation. Accordingly, we assume that, besides activating the tensile structures, the intrusive process leaded to an almost deformation-free increase in the whole mass of the studied system. The assumption of a single overall process triggering the intrusive event, together with the need to restrict the number of model parameters to be found by the inversion scheme, leads to the assumption that the above mass increase occurs within two volumes (1) having a rectangular parallelepiped shape and (2) sharing some geometrical parameters with the corresponding tensile structure (namely: horizontal position, length, and dip). The other geometrical parameters are independent from those of the corresponding tensile cracks and are to be found by the inversion scheme (top depth, width, thickness, and density change; Figure 4). [29] Since the gravity change caused by a rectangular prism linearly depends on the product U * r, with U and r being the thickness and the density of the prismatic body, respectively, the inverse problem becomes undetermined if U and r are searched separately. Therefore we searched for the product U * r which represents the mass change for unit of area. [30] The new inversion step, which takes into account the additional mass, is also accomplished through the NSGA-II algorithm. As for the parameters of the tensile cracks, the same ranges of variability used in the previous inversion are utilized. The ranges of variability of the independent parameters of the volumes containing the inferred new mass
are reported in Table 1. The Pareto front resulting from the values of the objective function vectors is restricted within a smaller region with respect to the previous inversion step (compare Figures 2c and 5c), thus confirming the more cooperative behavior of the objectives. A set of inverse solutions were found to have c2 values below the thresholds c2a at 95% confidence level for EDM, leveling and gravity data contemporaneously. The model parameters of the optimal model together with their confidence ranges are reported in Table 1. Neither in this case the Pareto front intersects the region below 2s for all the data set (Figures 3d– 3f). However, there exists a subset of Pareto optimal solutions having RMSE values for leveling and gravity below 2s (Figure 3f). The optimal solution falls into this subset with RMSE between observed and calculated data of 13.51 mGal and 9.2 mm for gravity and leveling data, respectively (Table 2). Conversely, the RMSE for EDM data is 69 mm, thus larger than 2s. However, it is important to stress that Bonaccorso [1999], using only EDM and leveling data and a grid search inversion method obtained residuals between observed and calculated data (both EDM and leveling) outside the measurement errors.
7. Discussion and Conclusions [31] The cross-analysis of deformation and microgravity data allows constraints to be set on both the volume and mass involved in the source mechanism, and thus furnishes a more complete picture of the volcanic processes than attainable through the use of only one technique [Battaglia and Segall, 2004; Carbone et al., 2006].
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[32] We jointly invert a multimethod geophysical data set encompassing the 1981 eruption of Mt Etna and comprising leveling, EDM and microgravity measurements. We aim at completing, through the information contributed by the gravity data, the partial picture drawn by Bonaccorso [1999], who only considered the ground deformation data. As a forward model, we use two analytical formulations [Okada, 1992; Okubo, 1992] which allow to calculate, at the surface of a half-space, both the ground deformation and the changes in the gravity field. Therefore we significantly improve the results of Sanderson et al. [1983] who (1) inverted separately leveling and gravity data, (2) used 2D forward models, (3) assumed that the observed gravity changes were due exclusively to the addition of mass in the source-volume and (4) did not take into account the EDM measurements at the flanks of the volcano. [33] We show that, when only two tensile cracks are used as a model-source, both vertical and horizontal ground deformations can be explained satisfactorily, but the observed gravity changes are underestimated. On the other hand, solutions which better suit the gravity data, overestimate both leveling and EDM data (Figure 2c). This finding leads to the hypothesis that a mechanism allowing mass redistributions to occur without deformation acted during the period encompassing the 1981 eruption. The use of a composite forward model, with two volumes where additional mass increases occur, each associated to a tensile crack, allows a satisfactory fit to be achieved between observed and calculated gravity and ground deformation data. Accordingly, the objective functions in the Pareto front interact cooperatively with each-other (Figure 5c). [34] A simple way to account for bulk mass increases without ground deformation is to assume that the ascending magma fills a network of preexisting interconnected microfractures [Sanderson et al., 1983; Rymer and Brown, 1986; Budetta and Carbone, 1998]. The exceptionally high effusion rates during the 1981 eruption [Romano, 1981; Murray, 1982] would suggest that it was fed by magma with low viscosity, which could have easily filled open spaces during its ascent from the deep storage to the surface. Under the assumption that the inferred mass increase is due to the arrival of new magma with density of 2500 kg/m3, the best parameters found by the inversion procedure (Table 1) lead to a minimum volume of magma of about 20 106 and 60 106 m3, for the southern and northern infiltration zones, respectively. Compared with the estimated volume of the lava erupted during the 1981 eruption (section 2), these findings imply an infiltrated-to-erupted-magma ratio between 2 and 4. [35] The implications of our results are relevant from the geophysical/volcanological point of view and raise some important issues on the subject of volcano-monitoring. Most of the observed surface deformation associated to the 1981
Figure 2. Objective function values of the Pareto front obtained from Model 1a (green lines) and Model 1b (red lines). Observed (blue) and modeled (red and green) EDM (a), leveling and gravity data (b). The arrows point to the selected models (c).
Table 2. Root Mean Square Errors Between Observed and Calculated Data for the Selected Solutions in Models 1a and 1b and the Optimal Solution in Model 2 RMSE Value
Model 1a
Model 1b
Model 2
RMSEgravity RMSEelevation RMSEEDM
23.5 mGal 28 mm 60 mm
15.0 mGal 115 mm 126 mm
13.5 mGal 9.2 mm 69 mm
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Figure 3. RMSE values of all the solutions in the Pareto optimal set for the Model 1 (a –c) and the Model 2 (d –f). The gray regions show the values bellow 2s. The solutions for the Model 1a (black square) and Model 1b (black triangle) are evidenced (Figures 3a – 3c). The black square is the optimal solution for the Model 2 (Figures 3d– 3f). eruption occurred on the flanks of the volcano, along the horizontal direction (movements of the benchmarks between 4 and 47 cm). Our calculation shows that this deformation is most controlled by the deeper and central
part of the intrusion. Vertical deformations along the profile crossing the eruptive fissures (BC Profile in Figure 1a) are in general very small (within a few cm) and reach relatively high values (up to 17 cm) only within a restricted zone in
Figure 4. Parameters of (a) the Okada model and independent parameters of (b) the volumes where an additional mass increase occurs. 8 of 10
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it follows that, while the horizontal deformation on the flanks of the volcano could have occurred over a relatively long period before the eruption, the vertical deformation along the line crossing the eruptive fissures is likely to be an almost syn-eruptive effect. Hence even if more measurements had been carried out, allowing a better time resolution to be achieved (or even if continuously running stations were available), only generic inferences about an impending intrusive event could have been made, while it is unlike that inferences about the sector where the new eruption would have occurred could have been drawn, significantly in advance of the eruption start, on the basis of the observed ground deformation only. [36] Strong gravity changes (up to 63 mGal) were also observed in correspondence of the eruptive fissures (Table 1). The associated mass increase is found to have occurred within a deeper volume than the tensile crack. Obviously, speculations on the timing of the filling of the inferred network of interconnected microfractures below the eruptive fissures are difficult to advance. Anyway, it cannot be ruled out that, if gravity measurement had been performed more frequently before the start of the 1981 eruption, insights about the sector of the volcano which was being charged with new magma could have been obtained. In any case, our results show that the cross-analysis and joint inversion of different geophysical parameters is crucial to ensure information useful from the volcano monitoring standpoint, especially in complex environment such as Mt Etna, where composite active (magma pushing its way upward)/passive (magma filling open spaces) intrusive mechanisms can occur. Naturally, suitable tools are needed to effectively handle multiparameter data sets. [37] We show that the use of multi-objective evolutionary algorithms, and in particular the use of the NSGA-II [Deb et al., 2000], is especially suitable for the joint inversion of multimethod geophysical data. Besides offering the common advantages of GAs, such as their ability to globally optimize highly nonlinear problems with limited amount of a priori information, the NSGA-II finds multiple Pareto optimal solutions in one single run (instead of converting the multi-objective into a single-objective optimization problem), a feature which is especially suitable when dealing with geophysical inverse problems, often characterized by ambiguous solutions. We thus encourage the development of a near-real-time system based on this algorithm to be used at volcanic areas where different geophysical parameters are routinely monitored.
Figure 5. Objective function values of the Pareto front obtained from Model 2. Observed (blue) and modeled (red) EDM (a), leveling and gravity data (b). The arrow points to the optimal solution (c). correspondence of the eruptive fracture. Accordingly, we found them to be the effect of the shallower part of the intrusive body, i.e., the very last part of the path followed by the magma to reach the surface. From the above statements
[38] Acknowledgments. This work was developed in the frame of the TecnoLab, the Laboratory for the Technological Advance in Volcano Geophysics organized by DIEES-UNICT and INGV-CT. This study was performed with the financial support from the ETNA project (DPC – INGV 2004 – 2006 contract) and the European Commission, 6th Framework Project – ‘‘VOLUME’’, contract 08471. We are grateful to Alessandro Bonaccorso for discussions about the ground deformation modeling. We thank the editor Patrick Taylor, the associate editor Antonio Rapolla and three anonymous reviewers for their useful comments on this paper.
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D. Carbone, G. Currenti, and C. Del Negro, Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Catania, Piazza Roma 2, 95123 Catania, Italy. (
[email protected])
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