IAVCEI Commission on Tephra Hazard Modelling

Commonly used techniques for the calculation of the total grainsize distribution of tephra-fall deposits

Unweighted average of all available grainsize analyses

Example: Rotongaio ash (Walker 1981a)

Average weighted by deposit thickness or mass

Examples: Walker (1980; 1981a; 1981b) prepared isomass maps for each grainsize class for four deposits at Taupo and integrated these data to get the total mass for each size class, which could then be summed to derive a total deposit grainsize. Murrow et al. (1980) calculated average grainsize distribution for the regions enclosed by each grainsize isopach and then weighted that data with respect to the enclosed volume to arrive at the total grainsize distribution. Parfitt (1998) calculated average grainsize distributions for the regions between each pair of isopachs for the Kilauea Iki 1959 tephra-fall deposit and then combined these data with volume estimates and clast density data to derive the total mass of material in each size class for each zone.

Sectorization of the deposit

Examples: Sparks et al. (1981) divided the isopach map of the 1875 Askja Plinian fall into segments and integrated grainsize data weighted by enclosed volume. Carey and Sigurdsson (1982) integrated data for the 18 May 1980, Mount St. Helens tephra-fall deposit by dividing the dispersal area into a series of thirteen polygons and calculating the total mass within each polygon and the average grainsize of all samples from within the polygon. They then weighted the latter by the former to arrive at a total grainsize distribution.

Voronoi Tessellation

Method of spatial analysis that can be defined as the partitioning of the plane such that, for any set of distinct data points, the cell associated with a particular data point contains all spatial locations closer to that point than to any other (e.g. Okabe et al. 1992).

Description: an edge (say between sample point SP1 and sample point SP2) of a Voronoi cell is a line segment that is a subset of the perpendicular bisector of the line segment connecting SP1 and SP2. The mass/unit area value and the grainsize distribution of each sample point SP1, SP2, SP3, etc..., are assigned to the enclosing Voronoi cells VC1, VC2, VC3, etc....As a conclusion, the tephra-fall deposit is divided into Voronoi cells whose interior consists of all grid points which are closer to a particular sample point than to any other (Fig. 1). Then the total grainsize distribution is obtained as the area-weighted average of all the Voronoi cells over the whole deposit.

Algorithm: there are hundreds of different algorithms for constructing various types of Voronoi diagrams (e.g. Brown 1979; Gowda et al. 1983; Klein 1989). Here you can download a MatLab function for the calculation of the total grainsize distribution of tephra-fall deposits based on the Delaunay Triangulation (VORONOI_TOTGS).

Æ click here to download VORONOI_TOTGS

Example: Ruapehu, 17 June 1996

(Bonadonna and Houghton 2004)

Fig. 1 Map showing the Voronoi Tessellation applied to the tephra-fall deposit produced by the 17 June 1996 eruption of Ruapehu (New Zealand). Each polygon represents a Voronoi cell built for each sample point (black circles), and is assigned with the same mass/unit area values and grainsize distributions as the corresponding sample points. The most external line represents the isoline of zero mass. The thin line indicates the NE cost of New Zealand. All polygons outside the zero line and in the ocean are given mass zero (corresponding to the blue crosses).