**
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).