Probability models for estimation of number and costs of landslides

The objective of this report is to describe the development of probability models for estimation of the number and costs of landslides during a specified time. Important philosophical ideas about natural processes and probability models are presented first. Then two probability models for the number of landslides that occur during a specified time are investigated: a continuous-time model (Poisson model) and a discrete-time model (binomial model). Estimation theory is developed for the estimation of the parameters of both of the models. The exceedance probability of one or more landslides during a specified time is formulated for both models. The estimation theory and probability formulation of the Poisson model are applied to the future occurrence of landslides in Seattle, Washington, using historical data from 1909 to 1997. Theoretical and numerical comparisons between the Poisson and binomial models are conducted that show the binomial model is an approximation to the Poisson model. An economic probability model is developed as an addition to the Poisson model for the estimation of the total damage from future landslides in terms of economic loss as costs in dollars. For illustrative purposes the economic probability model is applied to damaging landslides caused by El Nino rainstorms within the winter season 1997-98 in the San Francisco Bay region, California. Philosophy of Probability Models Natural Processes Important philosophical ideas about natural processes: Determinism or the law of cause and effect is the doctrine that all events in the universe are deterministic: every event has a cause. At the scale of geologic and atmospheric hazards (e.g., landslides, earthquakes, floods, tsunamis, volcanoes, and storms), nature is deterministic: every hazardous event has a cause. A hazard process is a physical process involving the occurrence of point (hazardous) events in time. Beginning at some point in time, after a certain amount of time, the first hazardous event occurs. Then, after a certain amount of time, the second hazardous event occurs. And so forth. The time between hazardous events is certainly irregular. We cannot predict exactly when a hazard event will occur because of the limitations to our knowledge of nature. The limitations to our knowledge of nature are explained by the following: Heisenberg Uncertainty Principle and Godel's Theorem; chaos theory and fractal geometry; algorithmic and computational complexity; physical and financial constraints. Chaos is the apparent randomness from extremely complex behavior occurring in a deterministic process due to excessive sensitivity of an event to small changes in initial conditions. Probability Models Important philosophical ideas about probability models: Probability is a numerical measure of our uncertainty regarding nature. A probability model is a mathematical model that incorporates our uncertainty. Probability models are an approach to deal with the limitations to our knowledge of natural processes. Probability models are used for purposes of description and prediction of physical processes in nature. Randomness is an assumption of probability models, not natural processes. Hazards do not occur at random in nature, but they do occur at random in the models. It is not correct to say that a natural process follows a particular probability model. (This would be putting the cart before the horse.) We will always be uncertain of nature because of our limitations in understanding. In summary, hazard processes are deterministic, but because of our limitations when studying hazards, we resort to probability models that incorporate our uncertainty. Probability Models for Landslides Consider the occurrence of landslides during a specified time in a particular area. Denote N(t): Number of landslides that occur during time tin a particular area We are interested in deriving a formula for calculating the probability of one or more landslides during a specified time t. That is, Two probability models for N(t) will be .investigated: first, a continuous-time model and second, a discrete-time model. Poisson Model for Number of Landslides The Poisson model is a continuous-time model consisting of the occurrence of random point-events (landslides) in ordinary time which is naturally continuous. The Poisson model is the most commonly used model for the occurrence of random point-events in time and has been used in modeling the occurrence of earthquakes. Assumptions of the Poisson model: The numbers of events (landslides) which occur in disjoint time intervals are independent. The probability of an event occurring in a very short time interval is proportional to the length of the time interval. The probability of more than one event in such a short time interval is negligible. The probability distribution of the number of events remains the same for all time intervals of a fixed length. It is important to acknowledge that these assumptions may not completely hold for the occurrence of landslides, especially the independence assumption. However, given a certain lack of understanding of the physical processes that control landslides, the Poisson model represents the best first-approximation model in attempting to model their occurrence. A first-approximation model is often applied in mathematical modeling when the assumptions are not completely satisfied by the physical process. Usually the firstapproximation model is relatively easy to work with and is mathematically tractable. A more accurate model might be extremely complex and not mathematically tractable. Poisson Distribution Probability of n landslides during time t: P{N(t) = n} = e~^ n = 0,1,2,... n\ where \: Rate of occurrence of landslides Note that time t is specified, whereas rate A, is estimated. Definition of recurrence intervals {7), i 1, 2, ..., n}: T\: Time until the first landslide Ti\ Time between the (i l)st and the /th landslide for i > 1 Note that n landslides will have n recurrence intervals. Theorem Recurrence intervals { T,-, / = 1 , 2, . . . , n } are independent identically distributed exponential random variables having mean recurrence interval (ji) equal to the reciprocal of the rate of occurrence, i.e., \JL = 1 A. For landslides, the mean recurrence interval (ji) is the average time interval between landslides.