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5.5 The world is not a disc

The focus of  the last section was on the key concepts of PSHA. The whole discussion was done in the framework of a very simple model world, because our primary goal was to develop a conceptual understanding for the principles on which PSHA is based.  The main purpose of the present section, however, is to extend our view and look across the boundaries of our simple model world. We will take a look at  some of the changes which take place when we go from the extremely simple model of a disc like areal source to more complex models of the types which are commonly used in practical applications. This will not be a comprehensive treatment, but a glimpse at the kind of changes one  has to deal with in PSHAs in the real world. This will gradually lead us to into discusions of actual “real world problems”. The good news is that none of the fundamental concepts needs to be rethought. Parts of the model, or in more technical terms, some parts of the contributions to the hazard integral will change, but the basic framework remains. Exercises in the present section will be used to check your conceptual understanding as we leave the model world.

This is also a good time to link our previous views on the individual elements of PSHA with the sequential view which you may find in most other text books on PSHA (e. g. Reiter, 1990).  Within this view, I refer to it as procedural or sequential view (Fig. 5.5.1), the calculation of a hazard curve is usually explained as a sequence of procedural steps in which one starts out with the geometrical source characterization (which for our model world so far consisted of what one would call a circular  areal source).  Subsequently, the activity for the source is specified (which in our cases so far consisted of assuming a doubly truncated exponential, or in other words a so-called Gutenberg-Richter distribution). Once this is done, the ground motion model has to be specified (which in our cases consisted of assuming the validity of an empirical ground motion prediction equation (GMPE), mostly for PGA). Finally, the hazard curve is calculated by one of the approaches which were discussed in the last section.

Figure 5.5.1       Sketch of the sequence of steps usually performed when a PSHA is conducted. This procedural or sequential view is the persective in which PSHA is usually explained in textbooks (e.g. Reiter, 1990). The upper left panel sketches the charactarization of the source geometry, which in our case was simply a circular areal source, uniformly filled with seimicity). The central upper panel refers to the characterization of the source in terms of magnitude-frequency relation, which in our case was assumed to follow a doubly truncated exponential distribution (Gutenberg-Richter relation). The upper right panel refers to the caracterization of the ground motion in terms of a (conditional on magnitude, distances, etc.) probability density function for the distribution of ground motion values (e. g. PGA, or spectral acceleration). Shown here are the PDFs as function of distance for three different magnitudes. FInally the central lower plot sketches the corresponding hazard curve.

Exercise  5.5.1 Which parts of the hazard integral are affected by the source geometry, which ones by the source activity, and which ones by the ground motion part of the hazard model?

The reason why I prefered to look at seismic hazard from the ground motion side was that the sequential view is accompanied by the danger to overemphasize the role of the earthquake occurrence in comparison to the ground motion part. I trust that it has become clear by now that it is the ground motion which poses the hazard.  Within the sequential perspective of looking at PSHA, however,  the  different contributions to the hazard integral (5.26) can be nicely related to the individual steps in the sequence. The characterization of the source geometry correspond to the determination of the PDF for the distances . The determination of the activity model for a source corresponds to the determination of and ,  and finally the ground motion model is nothing else than the conditional exceedance probability  P(Sa>a|M,R). Therefore, when we now leave the simple model world we can monitor easily which parts of the hazard integral are affected.

We will first drop the assumption that our model world is a disc, which is assumed to be uniformly seismically active  and look at the effects of accepting  arbitrarily shaped polygonal boundaries. This will be followed by a brief illustration of the effects of non-uniformly distributed seismicity. Subsequently, we will discuss the influence of distribution of earthquake hypocenters in depth (so far we have assumed that epicentral distance is sufficient to characterize the source-site distance). We will also look at some of the consequences of the fact that earthquakes have a spatial extension. This leads to a whole suite of phenomena which can affect the ground motion generation from extended sources. The discussion of the spatial extension of earthquakes will then bring us naturally to the fact that (at least in the real world) earthquakes are related to faults. The assumption of areal sources with spatially uniformly distributed seismicity, is commonly only reflecting the lack of knowledge regarding the locations and properties of faults in that region. In regions where the major faults are assumed to be known, source models can accomodate that both geometrically as well as in terms of the models of earthquake activity. In this context we will look at magnitude frequency relations which have been suggested as alternatives to the Gutenberg-Richter  relation for faults and we will also consider the geometrical effect of having faults instead of an areal source. Before I wrap up with a brief look at Poissonian model for the temporal behaviour of earthquake activity and its characterisitcs, I will take a glimpse at  the effects of differing site conditions, which I have silently ignored up to now by assuming that the site of interest is a rock site with a surface velocity of 760 m/s. A  discussion on how site effects for specifically well studied sites can be accomodated into PSHA in a probabilistic way will follow in the second part of this book.

Frank Scherbaum (2015), Fundamental concepts of Probabilistic Seismic Hazard Analysis, Hazard Classroom Contribution No. 001