Hello everyone. In previous sequences, you have studied methods for both seismic hazard and vulnerability assessment. In this sequence, I will present all these methods can be combined into a so-called cat model. A cat model is a stochastic model which allows to calculate for a given set of buildings the probability of distribution of loss caused by an earthquake. Cat models are widely used in insurance industry to assess the probability for the insured losses caused by natural catastrophes, especially earthquakes, to exceed a given threshold. Let us start by describing the main steps of a cat model. The first step is a stochastic event set, which is a catalog of all earthquakes considered in the model. The second step is hazard, which associate to each earthquake listed before a ground motion footprint. The third step is called the exposure. It consists in assessing the ground motion of each earthquake at the location of each building study. The fourth step called the vulnerability is used to assess the damage state of each building after each earthquake. The fifth step called loss ratio convert the damage state into a financial loss. Last, the sixth step is about applying the financial conditions to each loss amount calculated at the previous step. Let us now dig into the detail of each step. The stochastic event set is a list of all possible damaging earthquakes that are expected to occur over the considered area. It contains both historical earthquakes and earthquakes that have never been observed but still expected to occur. Usually, a stochastic event set contains millions of events. The magnitude range from the lowest magnitude for which the damage is expected up to the maximum magnitude. In this example, from 5 to 8.2. Both shallow and deep earthquakes are considered here from 1 kilometer to 70 kilometers. Regarding the epicenter location, it is localized on a spatial grid, usually at one kilometer square resolution. For the same epicenter location, several earthquakes can be categorized but with different parameters. For example, a different depth or magnitude. Last, additional parameters can be used to characterize each earthquake, like the fault parameters. Next, for each earthquake in the stochastic event set, the ground motion over the entire affected area is calculated using one or several ground motion prediction equation, GMPEs. In most models, the ground motion is quantified in peak ground acceleration or spectral acceleration and represented on a spatial grid at one kilometer square resolution. Such a footprint is very similar to the ShakeMap output released by the USGS when a severe earthquake occurs, as shown on this map. Depending on the earthquake characteristics and the GMPEs used, a footprint can have very different shapes. At the end of the two first steps, the ground motion caused by each earthquake in the stochastic event set has been calculated for any location. At this stage, the set of building study called the exposure needs to be determined. Many building parameters are required to calculate the damage caused by each earthquake to each building. The first is the accurate location, usually given by the spatial coordinates. It is used to extract from each footprint, the ground motion experienced by the building. Next, several parameters like the structure type, the year built, or the number of storeys are also necessary to determine the vulnerability profiles of the building. Last, financial values, including the sum insured of the building and the insurance policies condition are mandatory. Cat models differ from other risk mitigation models, mainly by the scale of the analysis. Indeed, stochastic loss models are used to assess the risk for a limited set of buildings spread over a large area. Only few buildings are considered, plotted in red in this example. It can be either far one from the other at the left-hand side of the map, or accumulated in one neighborhood at the center top of the map. Conversely, the other models are either at a large scale and based on global indicators or at a small scale and based on buildings profile. Furthermore, in cat models, all the buildings do not have the same importance. The larger the sum insured, the higher the materiality of the building in the study. In this map, the large building on the right-hand side is more important than the others, since the sum insured is higher. For each building studied, the fragility curve is retrieved in the vulnerability step, according to the building characteristics. A fragility curve is a function which associate a ground motion. For instance, a peak ground acceleration to a damage state with a given probability. A damage state is a level on a damage scale which captures the level of destruction of a building from no damage up to complete collapse. Such fragility curves can be assessed by a pushover analysis and capacity design. For example, at a peak ground acceleration of 0.2 g, the plotted fragility function shows that the considered building has a 38 percent chance to experience damage state extensive or higher. Thus, using fragility curves of buildings in the exposure and ground motion footprints calculated previously, we calculate the damage state of each building, analyze after each earthquake in the stochastic event set. The next step of the stochastic loss model is to translate the damage state into a loss. For that, the loss ratio is introduced and defined as the repair cost divided by the sum insured. The plot shows the variability of loss ratios between different damage cost relationship in the literature. For the damage grade, slight, variability is limited. For damage grade, moderate, it is significant. For damage grade, extensive, it is even larger, and for damage grade, complete, it is still significant. From an insurance perspective, financial conditions are often attached to the policy. Such conditions must be taken into account to calculate the share of the loss incurred by the insurance company. Let us consider a building ensured by two insurance companies called, A, in orange, and B, in yellow. The graph shows the breakdown of the loss between the policyholder and the two insurance companies. The loss share below the deductible amount and above the limit is that the charge of the policyholder, the remaining A, in orange and yellow colors, is split between the two insurance companies at the share of 30 percent for A and 70 percent for B. Although financial conditions can also be considered like the reinsurance conditions. Finally, a cat model can be used to estimate the loss distribution caused by a natural catastrophe as earthquakes on a given set of buildings. The graph shows the most common output of stochastic loss models called the occurrence exceedance probability curve. For a given loss L on the x-axis, the value on the y-axis gives the probability p, that an earthquake cause a loss equal or higher than L during the year. To conclude, I would rephrase Professor Aaron Levenstein, "Cat models are like bikinis. What they reveal is suggestive, but what they conceal is vital."
