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Quantifying eruption source parameters from tephra fallout deposits

Physical volcanology course

Sébastien Biass, Riccardo Simionato, Corin Jorgenson, Tom Sheldrake

April 24 2023

Introduction

The aim of this lab is to apply techniques to quantify eruption source parameters (ESP) from tephra fall deposits. This includes:

  • From field measurements at the outcrop level, producing isopach and isopleth maps;
  • Calculating critical ESP such as plume height, tephra volume and mass and mass eruption rate;
  • Estimating the magnitude and the intensity of an eruption;

Compute all parameters and discuss all points that are highlighted in the text. The lab uses the tephra fall deposit Layer 5 of Cotopaxi volcano in Ecuador, which is a black scoriaceous lapilli fallout with an age of 1,180±80 years B.P. and a whole-rock silica content of 58 wt.% 12. Thickness and maximum clast measurements at each outcrop are provided. It is time consuming to estimate the areas of isopach and isopleth, so use the data provided separately.

You are provided with the following files:

Volume of the tephra fallout deposit

The volume of tephra deposits is estimated from isopach maps by integrating the area below a curve plotting the logarithm of the thickness of isopach contours (y-axis) against the square-root of the isopach area (x-axis). On such plots, the exponential segment method of Fierstein & Nathenson (1992)3 states that a thickness \(T\) at any \(x\) value can be expressed as:

Equation 1
\[ T(x) = T_{0}e^{-k\sqrt{A}} \]

with \(T_0\) being the maximum deposit thickness, \(k\) the slope of the exponential segment and \(\sqrt{A}\) the square root of the isopach area. Based on the assumption of ellipsoidal shapes of isopachs Fierstein & Nathenson (1992)3 estimate the volume as:

Equation 2
\[ V = \frac{2T_0}{k^2} \]

Exercise

Estimate the volume of Layer 5 using the 1-exponential segment method of Fierstein & Nathenson (1992)3 using the isopach map shown in Figure 1:

  • In Excel, import the isopach data provided in Table 1 and plot the thickness (\(cm\)) versus the square-root of the area (\(km\)) as a scatter plot. Change the y-axis to a logarithmic scale

  • Fit an exponential trendline and display its equation to estimate the intersect (\(T_0\)) and the thinning rate (\(k\)) as in Equation 1

Note on units

\(T_0\) as expressed from the equation is now in the same unit as the y-axis. You need to convert it to a unit consistent with the x-axis in order to calculate the volume, which will have the same unit to the cube.

For instance, if both \(sqrt(A)\) and \(T_0\) are in \(km\), then the volume will be in \(km^3\).

  • Calculate the volume of the tephra fallout deposit using Equation 2

  • Estimate the corresponding VEI using the diagram in Fig. 2 from Newhall and Self (1982)4

  • Convert the volume to a mass using a bulk density of 1000 \(kg/m^3\) and calculate the associated magnitude following Pyle (2000)5:

Equation 3
\[ M = log_{10}(mass\ [kg]) - 7 \]

Figure 1: Isopach map of Layer 5.

Figure 2: VEI scale according to Newhall and Self (1982).

Table 1: Isopach data for Layer 5 of Cotopaxi volcano

Isopach area (km\(^2\)) Thickness (cm)
46 100
85 50
130 30
330 20
387 10
685 5

Plume height

The method of Carey & Sparks (1986)6 relies on the construction of theoretical envelopes within which the vertical velocity of the column and the terminal velocity of a clast of specified size and density are equal. Based on this, the crosswind and downwind ranges of isopleth maps can be used to estimate plume height and wind speed. This method method was further updated by Rossi et al. (2019)7 to account for a better parametrisation of physical processes in the plume (e.g., plume rise, settling velocity of particles).

Exercise

Calculate the plume height (\(km\) above vent) with the method of Rossi et al. (2019)7 (Fig. 3). The provided map contains measurements of maximum lithics (density of \(2500\ kg/m^3\)). This method works with isopleth contour values of 1.6 and 3.2 \(cm\), so make sure to contour isopleth accordingly. Assume a mean sampling elevation of 2500 \(m\ asl\). Cotopaxi has an elevation of 5700 \(m\ asl\).

  • Contour isopleths on the provided map. Make sure you contour values that are presented in Figure 3.
  • Measure the downwind and crosswind ranges of the deposit and report them on Figure 3 to estimate a plume height above mean sampling elevation.
  • Calculate the plume height and the wind speed as an average value of the results obtained from the different plots considered. Please also indicate the associated variation (i.e. ±(max-min)/2). Make sure to subtract the average height of sampling from the height obtained with the nomograms in order to derive the height above the vent.

Figure 3a: Graphical method to estimate the plume height (km above the mean sampling elevation) and the wind speed for an isopleth of maximum lithics with a diameter of 1.6 cm.

Figure 3b: Graphical method to estimate the plume height (km above the mean sampling elevation) and the wind speed for an isopleth of maximum lithics with a diameter of 3.2 cm.

Mass eruption rate

Based on early theoretical studies of plume dynamics, Wilson & Walker (1987)8 relate the height of a plume to the MER, with the height \(H\) being proportional to the fourth root of the MER (\(kg\ s^{−1}\)). More recently, Degruyter and Bonadonna (2012)9 have proposed a new analytical expressions relating height and MER that accounts for the variability of the plume parameters and atmospheric conditions:

Equation 4
\[ MER = \pi\frac{\rho_{a0}}{g'}\left(\frac{\alpha^2\bar{N}^3}{10.9}H^4 + \frac{\beta^2\bar{N}^2\bar{v}}{6}H^3\right) \]

  • Use the following values:

    • \(\rho_{a0}\): reference density of the surrounding atmosphere (use 1.2259 kg m\(^{-3}\))
    • \(g'\): reduced gravity at the source (use 45.6525 m s\(^{-2}\))
    • \(\alpha\): radial entrainment coefficient (use 0.1)
    • \(\bar{N}\): average buoyancy frequency across the plume height (use 0.0156 s\(^{-1}\))
    • \(H\): plume height (m above the vent)
    • \(\beta\): wind entrainment coefficient (use 0.5)
    • \(\bar{v}\) average wind velocity across the plume height (use 8.7 m s\(^{-1}\)). Note that the average wind speed along the plume should be smaller than the wind speed that you have derived with the method of Rossi et al. (2019), which is the maximum wind speed at tropopause

Exercise

  • Estimate the mass eruption rate (MER; \(kg\ s^{-1}\)) using the method of Degruyter and Bonadonna (2012)9 and the height obtained from isopleth maps. Note that this technique provides the peak MER of the eruption.

  • Use the MER to calculate the associated intensity following Pyle (2000)5

Equation 5
\[ I = log_{10}(MER [kg\ s^{-1}]) + 3 \]

Summary

This exercise provided an introduction on the characterisation of eruption source parameters from tephra fallout deposits, which is a critical process to infer the eruptive histories of volcanic systems from their stratigraphy. Namely, we learned:

  • How to calculate the volume of tephra deposits from isopach maps;
  • How to estimate the maximum plume height and wind speed from isopleth maps;
  • How to compute the peak mass eruption rate from plume height and wind speed;
  • How to estimate VEI, magnitude and intensity of eruptions.

Further reading

This list contains some references for the characterisation of tephra deposits.

Characterization of tephra-fall deposits

  • Thorarinsson (1954)10
  • Wilson (1972)11
  • Walker (1973)12
  • Wright et al (1980)13
  • Walker (1980)14
  • Carey and Sparks (1986)6
  • Sparks (1986)15
  • Wilson and Walker (1987)8
  • Cas and Wright (1988)16
  • Houghton and Carey (2015)17

Volume calculation

  • Pyle (1989)18
  • Fierstein and Nathenson (1992)3
  • Legros (2000)19
  • Sulpizio (2005)20
  • Bonadonna and Houghton (2005)21
  • Bonadonna and Costa (2013)22
  • Burden et al (2013)23
  • Daggitt et al (2014)24
  • Engwell et al (2015)25
  • Yang and Bursik (2016)26
  • Nathenson (2017)27

Mass eruption rate

  • Wilson and Walker (1987)8
  • Degruyter and Bonadonna (2012)9
  • Woodhouse et al (2013)28
  • Mastin et al (2009)29

Uncertainty assessment

  • Biass and Bonadonna (2011)2
  • Cioni et al (2011)30
  • Engwell et al (2013)31
  • Biass et al (2014)32
  • Klawonn et al (2014a)33
  • Klawonn et al (2014b)34
  • Bonadonna et al (2015)35

  1. Barberi F, Coltelli M, Frullani A, Rosi M, Almeida E. Chronology and dispersal characteristics of recently (last 5000 years) erupted tephra of Cotopaxi (Ecuador): implications for long-term eruptive forecasting. Journal of Volcanology and Geothermal Research 1995;69:217–39. ↩

  2. Biass S, Bonadonna C. A quantitative uncertainty assessment of eruptive parameters derived from tephra deposits: the example of two large eruptions of Cotopaxi volcano, Ecuador. Bulletin of Volcanology 2011;73:73–90. ↩↩

  3. Fierstein J, Nathenson M. Another look at the calculation of fallout tephra volumes. Bull Volcanol 1992;54:156–67. ↩↩↩↩

  4. Newhall CG, Self S. The volcanic explosivity index (VEI)- An estimate of explosive magnitude for historical volcanism. Journal of Geophysical Research 1982;87:1231–8. ↩

  5. Pyle DM. Sizes of volcanic eruptions. In: Sigurdsson H, Houghton BF, Ballard RD, editors. Encyclopedia of volcanoes, San Diego: Academic Press; 2000, p. 263–9. ↩↩

  6. Carey S, Sparks R. Quantitative models of the fallout and dispersal of tephra from volcanic eruption columns. Bull Volcanol 1986;48:109–25. ↩↩

  7. Rossi E, Bonadonna C, Degruyter W. A new strategy for the estimation of plume height from clast dispersal in various atmospheric and eruptive conditions. Journal of Volcanology and Geothermal Research 2019;505:1–2. https://doi.org/10.1016/j.epsl.2018.10.007. ↩↩

  8. Wilson L, Walker G. Explosive volcanic eruptions - VI. Ejecta dispersal in plinian eruptions: the control of eruption conditions and atmospheric properties. Geophys J R Astr Soc 1987;89:657–79. ↩↩↩

  9. Degruyter W, Bonadonna C. Improving on mass flow rate estimates of volcanic eruptions. Geophys Res Lett 2012;39. https://doi.org/10.1029/2012GL052566. ↩↩↩

  10. Thorarinsson S. The eruption of Hekla, 1947-48, 3, The tephra-fall from Hekla, March 29th, 1947. Visindafélag Ĺslendinga 1954:1:3. ↩

  11. Wilson L. Explosive Volcanic Eruptions-II The Atmospheric Trajectories of Pyroclasts. Geophysical Journal International 1972;30:381–92. https://doi.org/10.1111/j.1365-246X.1972.tb05822.x. ↩

  12. Walker GPL. Explosive volcanic eruptions — a new classification scheme. Geologische Rundschau 1973;62:431–46. https://doi.org/10.1007/BF01840108. ↩

  13. Wright JV, Smith AL, Self S. A working terminology of pyroclastic deposits. Journal of Volcanology and Geothermal Research 1980;8:315–36. https://doi.org/10.1016/0377-0273(80)90111-0. ↩

  14. Walker GPL. The Taupo pumice: product of the most powerful known (ultraplinian) eruption. J Volcanol Geotherm Res 1980;8:69–94. ↩

  15. Sparks R. The dimensions and dynamics of volcanic eruption columns. Bull Volcanol 1986;48:3–15. ↩

  16. Cas RAF, Wright J. Volcanic successions, modern and ancient: a geological approach to processes, products, and successions. London: Allen & Unwin; 1988. ↩

  17. Houghton BF, Carey RJ. Pyroclastic fall deposits. In: Sigurdsson H, Houghton BF, McNutt S, Rymer H, Stix J, editors. Encyclopedia of volcanoes, 2nd edition, London: Academic Press; 2015, p. 599–615. ↩

  18. Pyle D. The thickness, volume and grainsize of tephra fall deposits. Bull Volcanol 1989;51:1–5. ↩

  19. Legros F. Minimum volume of a tephra fallout deposit estimated from a single isopach. J Volcanol Geotherm Res 2000;96:25–32. ↩

  20. Sulpizio R. Three empirical methods for the calculation of distal volume of tephra-fall deposits. J Volcanol Geotherm Res 2005;145:315–36. ↩

  21. Bonadonna C, Houghton B. Total grain-size distribution and volume of tephra-fall deposits. Bull Volcanol 2005;67:441–56. ↩

  22. Bonadonna C, Costa A. Plume height, volume, and classification of explosive volcanic eruptions based on the Weibull function. Bulletin of Volcanology 2013;75:1–9. ↩

  23. Burden RE, Chen L, Phillips JC. A statistical method for determining the volume of volcanic fall deposits. Bulletin of Volcanology 2013;75:1–0. https://doi.org/10.1007/s00445-013-0707-4. ↩

  24. Daggitt ML, Mather TA, Pyle DM, Page S. AshCalc–a new tool for the comparison of the exponential, power-law and Weibull models of tephra deposition. Journal of Applied Volcanology 2014;3:7. https://doi.org/10.1186/2191-5040-3-7. ↩

  25. Engwell SL, Aspinall WP, Sparks RSJ. An objective method for the production of isopach maps and implications for the estimation of tephra deposit volumes and their uncertainties. Bulletin of Volcanology 2015;77:1–8. https://doi.org/10.1007/s00445-015-0942-y. ↩

  26. Yang Q, Bursik M. A new interpolation method to model thickness, isopachs, extent, and volume of tephra fall deposits. Bulletin of Volcanology 2016;78:68. https://doi.org/10.1007/s00445-016-1061-0. ↩

  27. Nathenson M. Revised tephra volumes for Cascade Range volcanoes. Journal of Volcanology and Geothermal Research 2017;341:42–52. https://doi.org/https://doi.org/10.1016/j.jvolgeores.2017.04.021. ↩

  28. Woodhouse MJ, Hogg AJ, Phillips JC, Sparks RSJ. Interaction between volcanic plumes and wind during the 2010 Eyjafjallajökull eruption, Iceland. Journal of Geophysical Research: Solid Earth 2013;118:92–109. https://doi.org/10.1029/2012JB009592. ↩

  29. Mastin L, Guffanti M, Servranckx R, Webley P, Barsotti S, Dean K, et al. A multidisciplinary effort to assign realistic source parameters to models of volcanic ash-cloud transport and dispersion during eruptions. Journal of Volcanology and Geothermal Research 2009;186:10–21. ↩

  30. Cioni R, Bertagnini A, Andronico D, Cole PD, Mundula F. The 512 AD eruption of Vesuvius: complex dynamics of a small scale subplinian event. Bulletin of Volcanology 2011;73:789–810. https://doi.org/10.1007/s00445-011-0454-3. ↩

  31. Engwell SL, Sparks RSJ, Aspinall WP. Quantifying uncertainties in the measurement of tephra fall thickness. Journal of Applied Volcanology 2013;2:1–2. https://doi.org/10.1186/2191-5040-2-5. ↩

  32. Biass S, Bagheri G, Aeberhard W, Bonadonna C. TError: towards a better quantification of the uncertainty propagated during the characterization of tephra deposits. Statistics in Volcanology 2014;1:1–27. https://doi.org/10.5038/2163-338X.1.2. ↩

  33. Klawonn M, Houghton BruceF, Swanson DonaldA, Fagents SarahA, Wessel P, Wolfe CecilyJ. Constraining explosive volcanism: subjective choices during estimates of eruption magnitude. Bulletin of Volcanology 2014;76:1–6. https://doi.org/10.1007/s00445-013-0793-3. ↩

  34. Klawonn M, Houghton BruceF, Swanson DonaldA, Fagents SarahA, Wessel P, Wolfe CecilyJ. From field data to volumes: constraining uncertainties in pyroclastic eruption parameters. Bulletin of Volcanology 2014;76:1–6. https://doi.org/10.1007/s00445-014-0839-1. ↩

  35. Bonadonna C, Biass S, Costa A. Physical characterization of explosive volcanic eruptions based on tephra deposits: Propagation of uncertainties and sensitivity analysis. Journal of Volcanology and Geothermal Research 2015;296:80–100. https://doi.org/10.1016/j.jvolgeores.2015.03.009. ↩