A SCALING CRITERION TO ESTIMATE AND COMPARE THE VOLCANIC HAZARD AMONG DIFFERENT VOLCANOES

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1 A SCALING CRITERION TO ESTIMATE AND COMPARE THE VOLCANIC HAZARD AMONG DIFFERENT VOLCANOES Servando De la Cruz-Reyna 1, A.T Mendoza-Rosas 2 1. Instituto de Geofísica,, Universidad Nacional Autónoma de México, Ciudad Universitaria D. F. México 2. Posgrado en Ciencias de la Tierra, Universidad Nacional Autónoma de México, Ciudad Universitaria D. F. México

2 Volcanic hazard is defined as the probabilities of occurrence of eruptions and their potentially destructive manifestations. These probabilities may be estimated analyzing the sequence of past eruptions in a volcano, characterizing the eruptions by size, and assuming that the impact and effects of an eruption are proportional to both, the energy (magnitude) and the rate of energy release (intensity). A quantity that characterizes eruptions based on those parameters is the Volcanic Explosivity Index (VEI, Newhall and Self, 1982). VEI Description Small Moderate Nonexplosive Moderatelarge Large Very large Volume of ejecta (m 3 ) (magnitude) < 10, >10 12 Column Height (km) (intensity related) Continuous blast duration (hr) (intensity related) Tropospheric injection (intensity - related) Stratospheric injection (intensity - related) <0,1 0, > <1 < > negligible minor moderate substantial large none none none possible definite significant large -- --

3 First, we analyze the energy released by explosive eruptions through different mechanisms: thermal, kinetic, deformation, material failure, seismic, acoustic, etc. Estimates of the total energy release of several eruptions have been made by different authors (e.g., Yokoyama, 1957, 1988). VEI values for those eruptions have been independently assigned. A correlation between the energy released and the VEI values may be described by: log (Em) = a M + c (1) where Em is the energy released by an explosive eruption in the VEI magnitude class M. The best fit to 21 well-documented historical eruptions yields a 0.79 and c = 14, log energy (Joules) when the energy of the eruptions is measured in joules (De la Cruz- Reyna, S. (1991) Poissondistributed patterns of explosive activity. Bulletin of Volcanology, 54: 57-67) VEI

4 Similarly, the analysis of the world-wide occurrence rates of volcanic eruptions during the last 500 years shows that the logarithm of the occurrence rate Km of eruptions in the VEI class M, is linearly related to M through the scaling relationship: log (Km) = b M + d (2) The best fit between the reported rates and VEI magnitudes yields b = d = 2.5, when Km is measured in eruptions per year. In the VEI range 3-6, the goodness of fit is over 99.9% (De la Cruz-Reyna, S. (1991) Poisson-distributed patterns of explosive activity. Bulletin of Volcanology, 54: 57-67) K m (er/yr) VEI

5 Combining equations (1) and (2), we obtain that the global annual energy release rate of explosive eruptions in each VEI magnitude class (for VEI 2) is approximately constant EmKm constant joule/yr (3) (De la Cruz-Reyna, S. (1991) Poisson-distributed patterns of explosive activity. Bulletin of Volcanology, 54: 57-67) Individual volcanoes do not necessarily release energy in that fashion. Some prefer to release their energy in many minor to moderate eruptions. Others may behave otherwise, releasing most of their energy as infrequent large eruptions. log K m Em (Joules/yr) A large set of volcanoes would average both behaviors tending to release energy as stated by (3) VEI

6 For an individual volcano, we can combine equations (1): log (Em) = a M + c and (2): log (Km) = b M + d obtaining after adding them log (Km Em) = (a+b) M + (c+d) The left term is the log of the energy release rate by eruptions in the VEI cateogry M. If the slope (a+b) is zero or near zero, as in the global case, the volcano behaves as the average of the world s volcanoes. If the slope is positive, [(a+b)>0] the volcano prefers to release more energy by means of a few large eruptions. If the slope is negative [(a+b)<0], the volcano tends to produce more energy through a larger number of small eruptions. The constant (c+d) is a parameter measuring the energy potential of the volcano.

7 First we obtain a robust eruption time-series combining historical and geological records, and using the relationship between the energy released by eruptions and the rate of eruption occurrences in the corresponding energy category given by equation 2: log (Km( Km) ) = b M + d Second, the distribution of repose periods is fitted to a Weibull distribution with survival function where α is a scale parameter, and k is the shape parameter. In the Third stage, a non-homogenous Poisson-Pareto Process (NHPPP) with a generalized Pareto distribution N u β( x u) λ ( A) = λ 1 Gβε, ( x u) = 1 t ε with shape parameter β and scale parameter ε as intensity function is proposed to estimate the effective eruption rates and thus assess the volcanic hazard. (These methods are described in detail in Mendoza-Rosas and De la Cruz-Reyna, Journal of Volcanology and Geothermal Research 176 (2008): / β

8 We present here, as an example, the assessment of four Mexican volcanoes: Colima, Citlaltépetl (aka Pico de Orizaba), Popocatépetl, and El Chichón.

9 Example: Colima Volcano 1. Independence test, serial correlation analysis. Null or very low correlation between succesive repose intervals is required for independence 5. Assemblage of historical and geological eruption data using equation 2 2. Stationarity test. Stability of the eruption rate. The rate varies about a constant mean value 6. Testing of the magnitude assignations assuming a nonhomogeneous Poisson-Pareto Process (NHPPP) and using Extreme-values theory 3. Running means analysis on consecutive repose intervals for stationarity testing. Reveals the amplitude and duration of rate fluctuations 4. Survival Weibull distribution fitting to determine eruption rate stability and the nature of the time dependence of the rate through the repose times distribution

10 Colima (Col1) Historical: (last 446 years) VEI Occurrences Occurrence rate er/yr KmEm = Energy release rate J/yr /446 = x /446 = x10 14 Geological: (Holocenic, last 7040 yr) VEI Occurrences Occurrence rate er/yr KmEm = Energy release rate J/yr 5* 2* 2/7040 = x x * 1* 1/7040 = x x10 14 * Col1 NHPP model, Mendoza-Rosas and De la Cruz-Reyna, Journal of Volcanology and Geothermal Research 176 (2008):

11 COLIMA VOLCANO log (Km Em) = (a+b) M + (c+d) Directly form dating Statistical estimate Col1 model a = b = 0.79 c = d = 14 Slope a+b = Energy potential c+d = Energy release rate (joule/yr) 1E+16 1E+15 1E+14 1E VEI

12 Citlaltépetl Historical: (last 473 years) VEI Occurrences Occurrence rate er/yr KmEm = Energy release rate J/yr 2 6 6/473 = x10 13 Geological: (last 13,000 yr) VEI Occurrences Occurrence rate er/yr KmEm = Energy release rate J/yr 4* 2 2/8500 = x x * 1 1/13000 = x x10 13 *Citla2 NHPP model, Mendoza-Rosas and De la Cruz-Reyna, Journal of Volcanology and Geothermal Research 176 (2008):

13 CITLALTÉPETL (PICO DE ORIZABA) VOLCANO log (Km Em) = (a+b) M + (c+d) Directly form dating Statistical estimate CITLA2 model a = b = 0.79 c = d = 14 Slope a+b = Energy potential c+d = Energy release rate (joule/yr) 1E+16 1E+15 1E+14 1E VEI

14 Popocatépetl Historical: (last 494 years) VEI Occurrences Occurrence rate er/yr KmEm = Energy release rate J/yr /494 = x /494 = x10 14 Geological: (last 23,000 yr) VEI Occurrences Occurrence rate er/yr KmEm = Energy release rate J/yr 4* 7 7/10700 = x x * 2 2/23000 = x x10 13 *Pop2 NHPP model, Mendoza-Rosas and De la Cruz-Reyna, Journal of Volcanology and Geothermal Research 176 (2008):

15 POPOCATÉPETL VOLCANO log (Km Em) = (a+b) M + (c+d) Directly form dating Statistical estimate POP2 model a = b = 0.79 c = d = 14 Slope a+b = Energy potential c+d = Energy release rate (joule/yr) 1E+16 1E+15 1E+14 1E VEI

16 El Chichón Historical+Geological VEI Occurrences Occurrence rate er/yr KmEm = Energy release rate J/yr 3 7 7/7772 = 9.007x x /7772 = x x /7772 = x x10 14 *Chichón B model, Mendoza-Rosas and De la Cruz-Reyna. Hazard estimates for El Chichón volcano, Chiapas, México: a statistical approach for complex eruptive histories. Nat. Hazards Earth Syst. Sci., 10, , (2010)

17 EL CHICHON VOLCANO log (Km Em) = (a+b) M + (c+d) Directly form dating Statistical estimate Chichon B model a = b = 0.79 c = d = 14 Slope a+b = Energy potential c+d = Energy release rate (joule/yr) 1.E+16 1.E+15 1.E+14 1.E VEI

18 El Chichón Colima Popocatépetl Citlaltépetl Energy release rate (joule/yr) 1.E+13 1.E+14 1.E+15 1.E+16 Colima directly form dating Statistical estimate COL1 model Citlaltepetl directly from dating Statistical estimate from CITLA2 model Popocatepetl directly form dating Statistical estimate form POP2 NHPP model El Chichon directly from datings Statistical estimate Chichon B model VEI

19 Energy release rate (joule/yr) 1.E+13 1.E+14 1.E+15 1.E+16 Colima directly form dating Citlaltepetl directly from dating Popocatepetl directly form dating El Chichon directly from datings Specific preventive measures may thus be specifically designed for each of those conditions VEI Statistical estimate COL1 model Statistical estimate from CITLA2 model Statistical estimate form POP2 NHPP model Statistical estimate Chichon B model Citlaltépetl volcano has an even lower overall energy release rate, but its slightly positive slope indicates that it may have a moderate preference to release energy by means of large, rather than by small eruptions We may conclude that Colima volcano has a large energy potential as well as the higher energy release rate of the four volcanoes. However,, Colima also has a moderate preference to release energy by means of smaller eruptions, as indicated by the negative slope El Chichón n has a relatively low energy potential, but a very high preference preference to release energy by very large eruptions revealed by the steep positive slope. Popocatépetl petl volcano has a lower energy potential than Colima, and a similar preference to release energy by smaller eruptions

20 what else can we say about the global eruption rates and magnitudes? Let s go back to the global scaling relationship log (Km) = b M + d (2) The best fit between the reported global rates (last 500 years) and VEI magnitudes yields b = d = 2.5, when Km is measured in eruptions per year. In the VEI range 3-6, the goodness of fit is over 99.9% K m (er/yr) VEI

21 Assuming that the relation for 500 years can be extrapolated, we may estimate the mean global recurrence times of large eruptions (De la Cruz-Reyna, S. (1991) Poisson-distributed patterns of explosive activity, Bulletin of Volcanology. 54: 57-67) Mean recurrence time (years) La Primavera Caldera 40 km VEI Aira caldera 110 km 3 Yellowstone Lava Creek 1000 km 3

22 Gracias THANK YOU