A Unified Approach To Probabilistic Risk Assessments for Earthquakes, Floods, Landslides, and Volcanoes

River flow extremes constitute a complex random process in both space and time. In this paper, some basic characteristics of flood flows considered as a spatial process are reviewed and discussed. Of particular interest are both marginal and joint distributional properties of annual peak flows at a collection of points in some large region. Two widely used regionalization methods for modeling the dependence of the marginal distribution of annual flood peaks on drainage area -the index-flood method and regional quantile regression -are compared. In addition, problems associated with characterizing spatial correlation of flows are discussed, and generalized least squares as used by USGS to account for spatial correlation in quantile regression is briefly introduced. Finally the problem of estimating regional flood probabilities, with the goal of assessing frequency of occurrence of flow extremes somewhere in a large region, is discussed. An example is presented showing how this frequency may be estimated using annual peak flow data from the state of Washington. Introduction A large part of flood-frequency analysis is concerned with the space-time process of annual peak flows in a region. In this paper we shall consider in particular properties of peak flows as a spatial process. Let x stand for a point in a region R of interest, and denote by Q(x) the peak flow at x in a given year. There are two factors that are responsible for much of the interesting spatial behavior that we see in Q(x). The first is channelization of flow by the drainage network. This channelization results in a high degree of dependence of the distribution of Q(x) on the drainage area, A(x), associated with the point x. The second factor is the spatial coherence of precipitation patterns. Both of these factors result in a high degree of spatial correlation in the Q(x) process. We shall look at a sampling of some approaches to modeling these two factors and at some implications for regionalization and determination of regional flood potential. Properties of Annual Peaks Let us first consider the marginal distribution of the annual peak random variable Q(x) at a point x. Hydrologists often choose to express properties of this distribution in terms of the quantile function , defined by , (1) where is an exceedance probability. If (taking ), is the so-called T-year flood. Considered as a spatial process, the most prominent feature of Q(x) is that it is highly nonstationary, and the main reason for this arises from the dependence of the distribution of Q(x) on drainage area, A(x). Much of the work in regionalization has been, in general terms, directed toward understanding this dependence on area and trying to find a way that data from basins with different areas may be compared or combined for purposes of staqα x ( ) α Prob Q x ( ) qα x ( ) > [ ] = 0 α 1 ≤ ≤ T 1 α ⁄ = 0 α < 1 2 ⁄ ≤ qα x ( ) Troutman and Karlinger: Flood risk assessment Probabilistic Risk Assessments 29 tistical analysis. We shall illustrate the problems here by considering two approaches that have been used to look at how depends on A(x): (1) the index-flood approach, and (2) the quantile regression approach. In all that follows we shall refrain from showing the dependence on x when there is no chance of confusion. Rather than considering flood dependence only on area, we can be more general and look as well at dependence on other physiographic and climatic characteristics, such as main channel slope or mean annual precipitation, that are believed to be related to flood magnitude; let us denote these variables by , where it is understood that the depend on x. The major assumption for the index-flood approach is that there exists a scale factor, say , that is a function of the ,