Hello, I am Maria Lancieri. I am a researcher at the Institute for Radioprotection and Nuclear Safety here in France. With my team the BERSSIN, we work on Probabilistic Seismic Hazard Assessment, the PSHA, the topic I will discuss with you during this sequence. The PSHA expresses the annual probability of exceedance of the intensity of the seismic ground motion shaking at a given site. Where the ground motion is usually expressed in terms of Peak Ground Acceleration, PGA, or Spectral Amplitude, SA. The information is summarized in a hazard curve expressing such a probability of a range of intensity levels. During this sequence, we will see together how such a probability is assessed in its basic ingredients. Let's start with a simple case. The probability of exceeding the x intensity of ground motion from a single seismic source. The first term appearing is the probability density function of earthquake magnitude. It expresses the probability of occurrence of an earthquake of given magnitude on the seismic source. The second term is the probability density function of distance from the seismic rupture, occurring on the source, to the site of interest. The third term is the probability of exceeding the x intensity of ground motion for a given magnitude distance scenario. Finally, for a single source, the PSHA is expressed as the integral over all magnitude and distance, or the product of these three terms. It's not that complicated. So, where is the fit for? In input data and model used to retrieve each of these term. The PSHA ingredients are basically three. The first is the source models. Earthquakes occur on faults, but it's not always possible to precisely identify the faults generating all the earthquakes occurred in this region. For this reason, a seismic hazard assessment, in addition to faults, seismotectonic zoning is introduced. A seismotectonic zone is a volume having a homogeneous seismic potential. In other words, all the points in the volume have the same probability of generating an earthquake. The second is the seismic catalog, delivering information on magnitude and location of earthquakes occurred in a given area. The third ingredient is the ground motion prediction equation, predicting the ground motion as a function of magnitude, distance, among other parameters. Now, let's see how seismic zoning in catalog are used to compute the probability density function of magnitude. It's well-known that geological sources can produce earthquakes having different sizes. In 1944, Benno Gutenberg and Charles Richter observed that in a given region the distribution of an earthquake, having magnitude greater than a given size, follows this kind of distribution where lambda is the rate of earthquake having magnitude greater than M, a expresses the overall rate of earthquakes in the region, and b the ratio between small and large earthquakes. In its original form, the Gutenberg-Richter is an unbounded distribution. This is not physical model because the faults in a given region can generate earthquakes until a given maximum magnitude. For this reason, the Gutenberg-Richter is truncated and the maximum magnitude is defined. How? It's often an expert's opinion based on geology and the seismicity of the region. Indeed, small earthquakes are not considered in PSHA because they do not induce any damage. The hazard assessment is thus computed in a minimum and maximum magnitude range. The probability density function of earthquake magnitude is obtained at deriving the truncated Gutenberg-Richter. It's form is the following; and it basically depends on b value of the Gutenberg-Richter relation, the minimum magnitude, and the maximum magnitude. The question arising now is how b value is computed using source model and seismic catalog. It's a matter of counting but in a smart way. First, from the seismic catalog, we extract the earthquakes that occurred over the seismic source we are focusing on. Then we must count how many of these earthquakes have magnitude greater than a given value ranging in the minimum and maximum magnitude bounds. But before counting, a further step has to be accomplished to estimate the catalog completeness period for each magnitude. It's easy to figure out that to estimate b, we need a rich catalog describing the seismicity of the zone. Indeed, instrumental seismicity delivers a little information on the seismic activity in a given area, barely covering 50 years. For this reason, it's necessary to go back in time integrating information on historical seismicity. The question of completeness is, starting from which date all the earthquakes of a given magnitude are recorded in the catalog? This step is of capital importance because in counting the earthquakes, any bias related to the lack of records must be avoided. To better understand, let's see this figure where the magnitude of earthquakes is plotted as a function of year. We can see that the witness of large historical earthquakes are available since 14th century. The earthquakes with a magnitude greater than five seems to be complete from the second half of 18th century. Finally, small earthquakes are systematically recorded only after the development of seismic networks during the second half of 20th century. How the completeness period are estimated, usually by expert opinion based on this kind of information or applying the statistical methods to the catalog. Once the completeness period has been estimated for each magnitude value, it is finally possible to count the earthquakes with magnitude greater than a series of values and to estimate the p-value. Well, we defined the probability density function of magnitude and we described how to infer from data the p-value controlling this probability. Now, what about the probability density function of distance? This term is computed making the assumption that the earthquakes will occur with equal probability at any location of the seismic source. The form of this distribution depends on the shape of the source and the site location. Example of simple distribution are given in Baker 2018. There is the case of circular source. The site is located in the center. The third term is the probability of exceeding the x intensity of the ground motion for a given magnitude distance scenario. It is given by the ground motion prediction equation. Let's see how. In the figure, the ground motion is described by the peak ground acceleration plotted as a function of magnitude for a fixed distance from the source. X is the value for which the probabilistic hazard is assessed. For a given magnitude value, the ground motion prediction equation expresses the log-normal distribution of peak ground acceleration. The probability of exceedance is thus given by the integral of the log-normal distribution between the value x and a given number of standard deviation, usually three or four. Now, we have all the elements to assess a probabilistic seismic hazard at a given site by multiplying the probability density function of magnitude of distance and the probability of exceedance of the x value for the ground motion for this scenario. By repeating the separation of a multiple distance magnitude scenario, we can finally compute the PSHA for a given intensity of the seismic ground motion by summing up all the probability products or the procedure illustrated during this sequence can be repeated over multiple seismic sources and by introducing the rate of occurrence of earthquakes greater than the minimum magnitude over each source, we obtain the complete formulation of probabilistic seismic hazard assessment that is now expressed in terms of rate of exceedance instead of the probability. Beyond this formulation, there is the assumption of Poissonian occurrence of earthquakes. Under this assumption, the probability of observing at least an earthquake in a given time interval is given by this equation, where Lambda is the rate of occurrence of earthquakes. For small Lambda, the probability can be expressed as Lambda time t. Finally, after such a big effort, what we have computed, just one point of the hazard curve, the one corresponding to the x intensity of ground motion we test so far. By repeating the computation over a range of x value, the PSHA curve is built in a given site. Repeating and assembling the computation for a set of sites, we obtain seismic hazard maps. It's worth to notice that the hazard maps are indeed expressed in terms of probability of exceedance over a given return period. Often, it is equal to 50 years.
