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Ninety years of historical landslide records were used as input to the Poisson and binomial probability models. Results from these models show that, for precipitation-triggered landslides, approximately 9 percent of the area of Seattle has annual exceedance probabilities of 1 percent or greater. Application of the Poisson model for estimating the future occurrence of individual landslides results in a worst-case scenario map, with a maximum annual exceedance probability of 25 percent on a hillslope near Duwamish Head in West Seattle. Application of the binomial model for estimating the future occurrence of a year with one or more landslides results in a map with a maximum annual exceedance probability of 17 percent (also near Duwamish Head). Slope and geology both play a role in localizing the occurrence of landslides in Seattle. A positive correlation exists between slope and mean exceedance probability, with probability tending to increase as slope increases. Sixty-four percent of all historical landslide locations are within 150 m (500 ft, horizontal distance) of the Esperance Sand/Lawton Clay contact, but within this zone, no positive or negative correlation exists between exceedance probability and distance to the contact.

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.

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Fi gur e 1.-Discovered and undiscovered recoverab l e quantities of crude oi l and na t u r al ga s in the United States. Undiscovered values are mean estima t es. Crude oil in bi ll ion barrels; total natura l gas in t ri ll ion cubic fee t.

A volumetric approach was used to estimate in-place gas in the Upper Cretaceous through lower Tertiary age basin-centered, low-permeability gas accumulation in the Wind River Basin of central Wyoming. The accumulation was subdivided into a total of 22 plays. Eight different stratigraphic units were considered. These units were then subdivided into as many as three plays based on variations in thermal maturity, presentday formation temperatures, and in the case of the lower member of the Paleocene Fort Union Formation, the presence or absence of an overlying shale seal. Highly overpressured plays occur where present-day formation temperatures exceed 300° F. Pressure gradients in these plays average about 0.73 pound per square inch per foot of depth (psi per ft). Moderately overpressured plays (0.52 psi/ft average pressure gradient) occur where present-day thermal maturity using vitrinite reflectance is 1.1% or greater. Transition plays occur where present-day vitrinite reflectance values are between 1.1% and 0.73%. These plays include a combination of tight gas reservoirs, gas reservoirs with conventional permeabilities, and water-bearing reservoirs,The lower member of the Paleocene Fort Union Formation, the stratigraphically youngest unit included in the basin-centered gas accumulation, is unique in that gas occurs where thermal maturates are as low as a vitrinite reflectance of 0.5 to 0.6%. The overlying lacustrine Waltman Shale Member of the Fort Union Formation appears to be acting as a seal inhibiting the vertical migration of gas out of the basin-centered accumulation. The lower member was divided into two plays, the first where the overlying Waltman Shale is present, and the second where the lacustrine shale has been replaced marginward by deltaic and fluvial deposits. The seal is assumed to be absent in the second play.A combination of overburden maps and isopach maps of total sandstone in beds 10 ft thick or greater is used to subdivide each play into as many as 120 subplay areas, each with a unique sandstone thickness and average depth. Point estimates were made of each attribute for each subplay using the maps, and a point estimate or mean estimate of in-place gas for each subplay was then calculated. The gas in-place in each of the subplays in a play was summed to derive a mean estimate of total gas in-place for a play. Estimates were then made of the ranges for each play attribute at the 95th and 5th percentile levels. The attributes were treated as independent continuous random variables, and ranges of gas in-place for each subplay were then calculated using probability theory. The subplays were aggregated assuming perfect positive correlation to calculate the range of gas in-place for each play. Finally, all of the plays were aggregated assuming perfect positive correlation to assess the total in-place gas in all of the plays in the Wind River Basin.Estimates of mean gas in-place for the twenty two plays are: 1) Upper Cretaceous Frontier Formation highly overpressured-118. trillion cubic feet (tcf);...

LIST OF FIGURES Figure 1. Structure contour map of the top of the Rollins or Trout Creek Sandsone. Contour interval: 500 ft. 2. Generalized Stratigraphic charts for the Piceance basin: a) northern part of study area, b) central part of study area, 3) southern part of study area. From Johnson and Finn (1986). 3. Isopach of the lies Formation (excluding the Castlegate Sandstone). Contour interval: 100 ft. 4. Isopach map of the Williams Fork Formation. Contour interval: 250 ft, 5. Generalized east-west cross section showing the lies and Williams Fork Formations, and the gas, and gas-water transition zones. 6. Cross section showing lithologies, environments of deposition, present-day formation temperatures, vitrinite reflectance, gas shows, and perforation recoveries. From Chancellor and Johnson (1986). Location of cross section on figure 1. 7. Chart showing relationship between Ro and hydrocarbon generation for type I, type II and type III organic matter. From Dow (1977). 8. Isopach map showing total thickness of sandstones ten feet or greater in the Williams Fork Formation. Contour interval: 100 ft. 9. Isopach map showing total thickness of sandstones ten feet thick or greater in the ILes Formation. Contour interval: 50 ft. 10. Approximate distance above or below the top of the Rollins or Trout Creek Sandstone to the Ro 1.1 thermal maturity level. Contour interval: 1000 ft. 11. Approximate distance above or below the top of the Rollins or Trout Creek Sandstone to the Ro 0.73 thermal maturity level. Contour interval: 1000 ft.

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