5.1. Simulation models for ground-motion generation using urn games
To start out, we will look at two connected random processes in the form of a simple game. At first glance it seems to have little to do with seismic hazard. Looking closer, however, it will turn out that it contains features which make it quite useful as a very simple simulation model for seismically generated ground motion. It consists of five urns which are shown in Figs. 5.1.1 and 5.1.2. Each urn contains 100 balls.
Figure 5.1.1 Urn with 2 red, 8 green, 22 blue and 68 gray balls. Make it interactive by allowing a ball to be drawn randomly.
The urn in Fig. 5.1.1 contains 2 red, 8 green, 22 blue and 68 gray balls. In addition, there are four urns with numerical labels shown in Fig. 5.1.2, one urn for each of the colors present in the urn in Fig. 5.1.1.
Figure 5.1.2 Four urns with 100 balls, each carrying a numerical label. The colors of the balls correspond to the colors of the balls in Fig. 5.1.1. Make it interactive by allowing balls to be drawn randomly.
Each of the balls in the four urns in Fig. 5.1.2 is labeled with a numerical value according to the scheme shown in Fig. 5.1. 3 and listed in Table 5.1.
Figure 5.1.3 Histograms showing the numbers of particular numerical labels of the balls in the four urns in Fig. 5.1.2. The colors of the histograms correspond to the colors of the balls in Fig. 5.1.1.
Table 5.1.1. Numbers of particular numerical labels of the balls in the four lower urns in Fig. 1.1.
Let us assume that we have some automatic mechanism to randomly draw balls from any of the urns. We can now combine random draws from the urn in Fig. 5.1.1 and the urns in Fig. 5.1.2 in a single experiment. We first have a ball drawn from the urn in Fig. 5.1.1, followed by a draw from that urn in Fig. 5.1.2 which contains the balls in the color which we obtained from the draw from the urn in Fig. 5.1.1. So if the first draw results in a blue ball, the second draw will be done from that urn in Fig. 5.1.2 which contains only blue balls. The ball obtained in the second draw will contain a numerical label, which is the result of our experiment. If we repeat this experiment a thousand times, we call this a game with thousand draws. Fig. 5.1.4 shows the resulting histograms of the values of the numerical labels for four games with thousand draws. You will note that the resulting histogram changes slightly but not by much.
Potential extra figure???: Stripped down version of what is now Fig. 5.6 to ilustrate the generation of a single histogram.
Figure 5.1.4 Histogram of numerical labels obtained from playing the urn game with thousand draws from the urns shown in Fig. 5.1.1 and Fig. 5.1.2 for four times.
In other words, the number of times which a particular numerical label will be obtained in a game of thousand draws seems to be a more or less stable feature of the experimental setup. The histograms indicate that that the majority of the values of the numerical labels which were obtained show values slightly below 1.0 with their number decreasing with increasing value of the numerical label.
One might wonder how this can be related to seismic hazard. Well, if we think of the colors in the top urn to represent earthquake magnitudes as illustrated in Fig. 5.1.5 (the magnitude concept is discussed in more detail in chapter 3) and the numerical labels of the colored balls in the lower urns to represent ground acceleration in , the connection becomes rather obvious. Each single experiment consists of two random processes, one from which a magnitude value is generated which is followed by a second, dependent one from which a ground motion value is generated. As in nature, the distributions of ground motion values shift to larger values with increasing magnitudes (see Fig. 5.1.3 and Table 5.1). For this example to hold, the distance of the earthquakes to the site of interest is assumed to be constant. For the sake of the following arguments, however, this is acceptable.
Figure 5.1.5 The same urn as in Fig. 5.1.1, but with numerical labels. Mathematically this defines a random variable. Make it interactive by allowing a ball to be drawn randomly.
We can make the connection to earthquake generated ground shaking even stronger if we add a temporal element to the simulation, by considering that the experiments are conducted sequentially in time. In this case, we can define a quantity which will later be refered to as occurrence or activity rate (in units of number of earthquakes per time) which is simply the total number of values observed divided by the total duration over which we assume the experiments to take place.
There are several ways in use to describe the results of this experiment in different types of histograms. e. g.
by binning the particular numerical values and count the number in each bin as in Fig. 5.1.4 (Counts).
by summing the counts from the bin with the smallest numerical values to the largest one (CumulativeCount).
by counting the number of values in each bin above the selected value (SurvivalCount).
by normalizing the counts over all bins so that they sum up to 1 (Probability). For large numbers of draws in a random experiment this can be related to the so-called frequentist definition of probability which is discussed in chapter 2.
by normalizing the product of the counts times the bin width in which they occur over all bins so that they sum up to 1 (PDF). This way, the area under the histogram is normalized to one. This can be related to the so-called probability density function (PDF) of a continuous distribution which is also discussed in chapter 2.
by cumulating the product of the counts times the bin width in which they occur over all bins from the bin with the smallest to the bin with the largest value (CDF). This can be related to the so-called cumulative distribution function (CDF) which is also discussed in chapter 2.
by taking 1 – CDF. This function is sometimes called survival function (SF).
All of the histogram types above are related to each other through simple operations and/or a single normalization constant as can be seen in detail in the following interactive figure (Fig. 5.1.6). It allows to display the outcome of the urn game for a varying number of experiments and for the mentioned histogram types. In addition it allows the read to choose between a autoscaled plot or a plot with a given maximum value.
Figure 5.1.6 Displaying the outcome of the urn game for a varying number of single experiments. Note: Make yourself familiar with the properties of the different histogram types and their mutual relations. Make restart button. Replace label “number of games” by “number of draws” .
.............. 5.1. Simulation models for ground-motion generation using urn games
.............. 5.1.1. How safe do you want to be?