5.3. The hazard integral and its visual interpretation
In the last section we have seen how one can construct the hazard curve for seismically active region in which earthquake epicenters are assumed to be spatially uniformly distributed within a circular area and for which the magnitude-frequency distribution can be described by a doubly-truncated exponental distribution (Gutenberg-Richter distribution). This involved first spatially discretizing the circular area into circular bands of finite width and secondly discretizing the magnitude-frequency distribution into magnitude bins of width (cf. Fig. 5.2.16).
We are now going to extend our view and look at the determination of hazard curves from a very general, theoretical perspective. This will lead to an expression in integral form, the so-called hazard integral. In the context of its derivation, we will make use of some relationships of probability theory which will facilitate the rearrangement of some of the expressions which we will need. These are the relationships between conditional and unconditional probability distributions, and between joint and marginal distributions. We will also use the principle of marginalisation of joint distributions and will make use of the properties of statistical independence all of which can be found discussed in detail in chapter 1.
Finally we will take a very pragmatic look (literally) at the ingredients of the hazard integral which will result in a very intuitive visual interpretation of what the hazard integral is all about.
.............. 5.3. The hazard integral and its visual interpretation
.............. 5.3.1. Infinitesimally small magnitude/distance bins
.............. 5.3.2. From probabilities to probability densities
.............. 5.3.3. Marginalization
.............. 5.3.4. Several areal sources
.............. 5.3.5. The anatomy of the hazard integral illustrated for a simple source geometry