Determination of volumes of tephra deposits Erupted volume and erupted mass are amongst the most important model input parameters and also the most difficult to derive from field data. In fact, volume (and mass) estimation strongly depends on the technique used, on the deposit exposure and on data distribution and density. Most erupted volumes derived from field data should be considered to be minimum values unless the data sets extend hundreds of kilometers away from the vent (the higher the plume the larger the deposit to investigate). A review of several used methods can be found in Froggatt (1982), whereas the most recent techniques is summarized below. In addition, recent applications of inversion techniques applied to analytical models have shown very promising results (Connor and Connor 2006).
Determination of erupted volume based on the exponential thinning of tephra deposits This approach was firstly introduced by Pyle (1989) adopting the preliminary observation of Thorarinsson (1954) that both thickness and grainsize of tephra deposits mostly follow an exponential decay with distance from the vent. As a result, the logarithm of tephra thickness can be easily described by straight lines (i.e. exponential segments) when plotted against distance from vent or square root of isopach area :
(1)
where T_{o} is the maximum thickness of the deposit and k defines the rate of thinning of the deposit (i.e. slope of the associated exponential segment). Assuming that isopachs have an elliptical shape, the erupted volume is:
(2)
where . Fierstein and Nathenson (1992) and Bonadonna and Houghton (2005) developed this method further, mainly to account for abrupt changes in the rate of thinning of some tephra deposits:
(3)
where Tn0, kn and Sn are the intercept, slope and breakinslope of the line segment n. Their approach to estimating volume by defining several exponential segments (i.e. different values of k from proximal to distal portions of the deposit) is consistent with the observations of wellpreserved tephra deposits (e.g. Hildreth and Drake 1992; Scasso et al. 1994) and with the results of some analytical models (Bonadonna et al. 1998; Bursik et al. 1992b; Sparks et al. 1992). The approach of Pyle (1989) was also modified to estimate erupted volumes in cases when only one proximal isopach line can be defined based on the available data (Legros 2000) and when most distal data are missing (Sulpizio 2005). The first technique is derived from the empirical investigation of 74 tephra deposits and gives estimated minimum volumes of the same order of magnitude as only the first two segments on semilog plots of thickness vs square root of the area were available: (4)
where A_{x} (m^{2}) is the area enclosed within the isopach line with thickness T_{x} (m). In contrast, Sulpizio (2005) presents three different techniques for the determination of distal volume based on the extrapolation of the distribution of proximal deposits to distal areas (up to thickness >1cm). In particular, he suggests (i) the compilation of distal isopachs in case of sparse distal data assuming same elliptical shape (same eccentricity) and same dispersal axis of proximal isopachs, (ii) the empirical determination of a breakinslope between proximal and distal data in case only one isopach line is available, and (iii) the empirical calculation of distal thinning when only proximal data area available. These three techniques give good agreement with field data for the 2030 deposits used in his case study. Finally, Mannen (2006) suggests an analytical method to derive the total erupted mass of relatively small eruptions by adopting the model of Bursik et al. (1992b) and integrating two exponential segments determined from isopleth maps (as cumulation of erupted mass of individual particle sizes).
Determination of erupted volume based on the powerlaw thinning of tephra deposits Based on the results of analytical investigations (Bonadonna et al. 1998) and on the observations of wellpreserved deposits that show how tephradeposit thinning can be either described by 4 exponential segments or by a powerlaw fit on a semilog plot of thickness vs. square root of isopach areas, Bonadonna and Houghton (2005) suggest deriving the total erupted volume by integrating the powerlaw best fit of field data. In particular, the powerlaw best fit can be described as:
(5)
where C_{pl} and m are the powerlaw coefficient and exponent respectively. The associated volume can be calculated as: (6)
which in a dimensionless form becomes as: (7)
where and are two arbitrary integration limits. In particular, should be taken as the downwind extent of the whole deposit, whereas can be taken as the distance of the maximum deposit thickness and calculated from eq. (5) (where is the maximum thickness that can be derived from eq. (1)).
Caveats of volume determination using the exponential and powerlaw techniques Sensitivity analysis of volume calculations have shown that integration of less than 3 exponential segments can underestimate the volume when distal data are missing (Bonadonna and Houghton 2005). As an example, the integration of only two exponential segments described by data within 10km from the vent of the 1996 Ruapehu eruption resulted in an underestimation of half of the actual deposit. Such an underestimation does not affect the classification of the eruption in terms of VEI, but is significant when simulating tephra dispersal and compiling hazard assessments. In contrast, the powerlaw fit is a good approximation to wellpreserved deposits and is consistent with theoretical models, but it is also problematic because integration limits have to be chosen. In particular, the volume of tephra deposits characterized by limited dispersal (m>2) is very sensitive to the choice of but not to the choice of . In contrast, the volume of tephra deposits characterized by wide dispersal (m<2) is very sensitive to the choice of but not to the choice of . Given that can be fixed by eq. (5) but is difficult to constrain, the powerlaw method should be used for small deposits (m>2) but is not recommended for widely dispersed deposits (m<2). Essentially, empirical fitting of poor data sets can be problematic and unsafe especially for large eruptions even with a powerlaw fitting because there is no theoretical simple relation between proximal and distal thinning. In fact, proximal deposition is controlled by high Reynolds number particles, whereas distal deposition is controlled by low Reynolds number particles.


email your questions or comments to Costanza Bonadonna or Simona Scollo last modified: 15 July 2013 