Table 2 Equations used in VoLREst

Equation 1

P(NO_tw) = 1 − P(E_tw1+)

3

Equation 2

\( P\left({NO}_h\right)=P{\left({NO}_{tw}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$h$}\right.} \)

5

Assumes binomial distribution, stationary probability, and independent events

Equation 3

P(E_h) = 1 − P(NO_h)

6

Equation 4

P(E_hL) = P(E_h)/100

7

See Table 3

Equation 5

P(E_hM) = (P(E_h)/10) − P(E_hL)

7

See Table 3

Equation 6

P(E_hS) = P(E_h) − P(E_hL) − P(E_hM)

7

See Table 3

Equation 7

P(F_x| E) = P(F_x| X) × P(X| E)

13

Equation 8

P(F_hx) = P(E_h) × P(F_x| E)

14, 24

Equation 8 illustrates principle with E_h, but in spreadsheet this in fact is done for all three eruption sizes (E_hL, E_hM, E_hS)

Equation 9

\( P\left({b}_{hit}\right)=\pi {\left(\frac{W+b}{2L}\right)}^2 \)

20a, 21

See Fig. 6. This equation ignored edge effects, and assumes that one person and one ballistic are within the reference area. The equation is the probability that the ballistic partially or completely overlaps with the person: it is the probability of the center of the ballistic landing in the area covered by the person, in other words, in zone with diameter W + b. The probability of a hit is the area of a circle with diameter W + b divided by the reference area L². We note this is a slightly different approach than taken in Jolly et al. (2014), and Massey et al. (2014a), but consistent with the approach in Massey et al. (2014b).

Equation 10

\( P\left({b}_{hit}\right)=\frac{W+b}{L} \)

20b, 21

See Fig. 6. This equation ignored edge effects, and assumes that one person is on the reference length, and one ballistic perpendicularly crosses the reference length. The equation is the probability that the center of the ballistic partially or completely crosses the person, corresponding to a length of W + b, divided by the reference length L. We note this is a slightly different approach than taken in Jolly et al. (2014), and Massey et al. (2014a), but consistent with the approach in Massey et al. (2014b).

Equation 11

P(S_nb) = (1 − P(b_hit))^nb

22

Equation 12

P(F_b| E) = 1 − P(S_nb)

23

Equation 13

P(S_{x. size}) = (1 − P(F_{hx1. size})) × (1 − P(F_{hx2. size})) × . … × (1 − P(F_{hx. size}))

25

Equation 14

P(F_{hx. size}) = 1 ‐ P(S_{x. size})

26

Equation 15

P(F_hsite) = P(F_{h. site. small}) + P(F_{h. site.  mod erate}) + P(F_{h. site. l arg e})

27