For the first case, consider the distributions of wind speed in m/s recorded at two independent weather stations. The discrete distributions of the rvs, called X₁ and X₂, are shown in Figure 1 . The distribution of X₁ has only two impulses (0.3, 0.7) located at (3, 5). The distribution of X₂ has 3 impulses (0.2, 0.6, 0.2) located at (3, 6, 8). The convolution of both rvs, X₁ and X₂, is defined as

$Z=X_1+X_2$ (1)

and is performed numerically by adding together the x-axis and multiplying the corresponding probabilities, as shown in Figure 1 (bottom). The convolution, in this case, implies shifting the distribution to the right and reducing the size of the impulses. The realisation of events X₁ = 5 m/s and X₂ = 6 m/s occurring concurrently is given by

$P\left[Z=11\right]=P\left[X_1=5|X_2=6\right]=P\left[X_1=5\right]P\left[X_2=6\right]=0.42$ (2)

to stress the point, Figure 1 shows the x-axis of the individual rvs below the main x-axis. Discrete distributions have been used to model landslide hazard [42] .

The independent Normal distributions are shown in Figure 2 . The parameter of Normal 1, N₁ (blue curve), are (2,1), i.e. mean (m₁) = 2 m/s, and Variance (Var) = 1 (m/s)². The parameters of Normal 2, N₂ (red curve), are (3.5, 0.25). The corresponding convolution is a Normal with parameters (5.5, 1.25). Again, observe that the net effect is a shifting of the resulting distribution (black curve) to the right and a reduction of the body of the distribution (because of the combination of probabilities). In this case, showing the combination of the individual x-axis is more challenging because the distributions are continuous. Still, as in Figure 1 , we show a few points of the original distributions below the main x-axis.

The equivalent to Equation (2) in continuous distributions requires the cumulative distribution function (CDF) shown in Figure 3 . In this case, the probability of events N₁ < 3 m/s and N₂ < 4 m/s occurring simultaneously is given by

$P\left[Z<7\right]=P\left[N_1<3|N_2<4\right]=P\left[N_1<3\right]P\left[N_2<4\right]=0.91$ (3)

[41] and [37] present more examples of convolutions.

In risk analysis, we are more interested in the complementary event $P\left[Z>z_1\right]=1.0-P\left[Z<z_1\right]$.

If z₁ is large, we are selecting the extreme values of Z, our compound variable. The value $P\left[Z>z_1\right]$ represents the probability that the concurrent extremes of N₁ and N₂ are exceeded. The hazard and risk studies of natural phenomena aim to calculate the frequency with which extreme values repeat. As mentioned before, the main indicator in these types of studies is the Return Period (RP). A curve of RP shows the recurrence of extreme values, usually in years, at a given region. These extreme values are termed Return Levels. Thus, if the 50-year RP of wind speed at a given region is 30 m/s, what we are saying is that 30 m/s is expected to be exceeded, on average, once every 50 years.

The curve of RP can be built from Figure 3 ( [1] [2] ), using the expression,

$\text{RP}=\left(1.0/\left(1.0-P\left[Z<z_1\right]\right)\right)/\text{nopy}$ (4)

where RP is the Return Period corresponding to the z₁ Return Level. The constant “nopy” sets up the right scale for the curve and depends on the physical characteristics of the problem.

The major limitation of this technique is that it allows us to calculate RPs for the same period of the given data, usually 20 or 30 years. In practical problems we are interested in calculating RP for values well beyond the available data. The Australian/NZ regulations [43] , for instance, prescribe wind loads for design of structures corresponding to return periods of 500 years. In this case we should use extreme value distributions to extrapolate RPs to 50, 100, 500 years and beyond. The technique consists in fitting an extreme value distribution to the tail of the distribution of the compound variable (black curve in Figure 2 ). There are a number of families of extreme value distributions available to do this, in our example we will use the Generalised Pareto distribution (GPD), and we will fit the GPD using the method of moments [44] [45] . Figure 4 shows the fitting of the GPD to the tail of the compound extremes (green curve). From the parameters of the GPD we can calculate the parameters of the RP curve, as will be explained in Section 3.2.

More information has to be provided to calculate the curve of RP. Suppose that the Normal distributions of Figure 2 refer to the maximum daily wind speed (m/s) recorded at two independent stations in Australia. The records cover a period of 20 years for each station; hence the convolution of the two distributions refers to a period of 40 years. Furthermore, suppose that the maximum value found in the records for each Normal distribution is 5.5 and 6.5 m/s, respectively. The maximum of the combined distributions will be 12 m/s. Hence, we can say that the 40-year RP of max daily wind speed at the region is 12 m/s, the correct scale to use in Equation (4) is nopy = 9.44. Figure 5 shows the corresponding curve of RP. From this curve, we can say that the combined events, wind speed at station 1 = 5.5 m/s and wind speed at station 2 = 6.5 m/s occur, on average, once every 50 years, as shown by the scales to the left of the y-axis.

Note that this is just an example; in practice, it is inappropriate to fit Normally-distributed data with extreme value distributions [2] .

In this method, the rvs are replaced by a set of parameters called cumulants. These cumulants are linear combinations of the moments of the distribution. The convolution consists on adding together the cumulants of the components to obtain the cumulants of the resulting distribution. From these cumulants the distribution of the resulting rv can be calculated [37] . Consider the linear equation,

$Z=aX+bY+c$ (5)

where,

a, b, c = constants.

X, Y = Independent random variables.

Z = Resulting random variable.

Let m_x and m_y be the mean of X and Y; and let κ_r(x) and κ_r(y) be the cumulants of X and Y respectively. The subscript “r” indicates the order of the cumulant.

In terms of moments and cumulants Equation (5) can be expressed as,

$m_z=a{m}_x+b{m}_y+c$ (6)

${\kappa }_r\left(z\right)=a^r{\kappa }_r\left(x\right)+b^r{\kappa }_r\left(y\right)$ for cumulants of order r > 1 (7)

Note that the operation does not involve any approximation. Equation (6) and (7) can be collapsed into one [37] .

One of the strengths of the method of cumulants for convolution of rvs is that the process can easily be extended to the case of non-independent rvs, i.e. for the case in which there is some degree of covariance between the rvs. The covariance indicates whether both variables follow the same pattern of change. If X tends to increase or decrease along with Y, the covariance will be large and positive. If, on the other hand, X tends to increase as Y decreases their covariance would be a large negative number.

Let us suppose that the covariance between the rvs X and Y is σ_xy, then the second cumulant or Variance of rv Z has to be modified to take this covariance into account. Equation (7), then, becomes,

${\kappa }_2\left(z\right)=a^2{\kappa }_2\left(x\right)+b^2{\kappa }_2\left(y\right)+2ab{\sigma }_{xy}$ (8)

There are basically two techniques to calculate the RP. One is the so-called “block maxima” in which an extreme value distribution is fitted to the annual maxima. The distribution used in this case is the Generalised Extreme Value Distribution (GEV). If more observations are available, for instance, maximum daily observations, “block maxima” is not recommended because it uses only one observation per year; the other observations are not used, which may result in a poor fitting of the data. The second technique is the “peaks-over-threshold”. This technique utilises all values over a given threshold to fit the distribution. In this case, the Generalised Pareto Distribution (GPD) is used. This methodology has several advantages over the “block maxima” method; firstly, it uses a lot more data to fit the distribution, and secondly, by setting the threshold high enough, the data will be better distributed in time, improving the likelihood that the data samples are independent of each other (one of the conditions of EV applications).

The latter is the method recommended when maximum daily observations are available [1] [46] . One of the problems in fitting the GPD is the selection of the appropriate threshold. Return period calculations using GPD distributions are very sensitive to the threshold selection.

In this study, we use the latter technique and calculate the appropriate threshold u using the automatic procedure developed by Sanabria and Cechet [47] .

The parameters of the GPD distribution, location μ_z, scale σ and shape ξ can be calculated from the moment m_z and the second cumulant of Z (κ₂) using the expressions [44] [45] ,

${\mu }_z=m_z$ (9)

$\sigma =m_z\left(\left[m_z^2/{\kappa }_2\right]+1\right)/2$ (10)

$\xi =\left(\left[m_z^2/{\kappa }_2\right]-1\right)/2$ (11)

From these parameters the curve of RP follows ( [1] ), see Figure 5 ,

${\text{RL}}_j=u-\sigma /\xi \left(\left[{\text{RP}}_j\ast \text{nopy}\ast \text{crate}\right]-\xi \right)-1.0)$ (12)

where,

RL_j is the Return Level corresponding to a given Return Period, RP_j, u is the threshold, nopy is the scale factor and crate is the cross rate or proportion of excesses over the threshold. The y-axis in Figure 5 is wind speed in m/s.

We acquired data from the Australian Bureau of Meteorology [51] for this illustrative example. For wind, we have maximum daily wind gust in m/s. For precipitation, we have cumulative daily values in mm. In both cases, the data covers the period 2006 to 2019 (14 years). For sea level, we have 40 years of maximum monthly sea level in m (1981-2020). The recorded data stations are located in the same region (greater Sydney). Referring to the three phenomena to a common axis for the cumulant-based convolution process is convenient. This can be achieved by expressing the variables per unit (pu), and dividing each one by the maximum value found in the respective period. Table 1 shows the maximum value of the three phenomena. Figure 6 and Figure 7 show the histogram and density function of the wind speed and precipitation distributions, respectively. For the latter, we have selected values > 4 mm. The histogram and density function of sea level is presented in Figure 8 . The distributions are assumed to remain static for the study period.

As explained in Section 3.1, the method of cumulants for convolution of rvs can be extended to non-independent rvs by modifying the calculation of the second cumulant. Table 4 shows the covariance of the three rvs used in this example. Compared with the second cumulant (in pu) shown in Table 2 , we can see that the covariance is relatively large, and hence, it must be considered for a proper modelling of the problem. The table shows that the covariance is strongest between sea level and precipitation. The positive value shows that the sea level tends to rise with precipitation while decreasing with wind speed.

To compare the curves of RP considering independent and non-independent rvs, both curves have been plotted in a single graph shown in Figure 10 . This figure shows that the curve of RP considering non-independence between rvs (red curve) is lower than the one considering independence. For the 100-year RP, the return level for the former is about 4.75% less than that of the latter. For larger RPs the difference tends to increase. Consider again the compound event wind speed = 36.4 m/s, precipitation = 145.25 mm and sea level = 2.71 m. It used to have a 500-year RP; now it has a 1000-year RP. In other words, considering the covariance between rvs, the event’s recurrence happens over a larger interval. This shows the importance of modelling the problem correctly. For mitigation purposes, the overestimation of the RP could result in a substantial increase in the cost.