Alan F. Karr

Bio: Alan F. Karr is an academic researcher from National Oceanic and Atmospheric Administration. The author has contributed to research in topics: Storm & Statistical model. The author has an hindex of 5, co-authored 5 publications receiving 247 citations.

A point process model of summer season rainfall occurrences

Abstract: A point process model of summer season rainfall occurrences is developed. The model, which is termed an RCM process, is a member of the family of Cox processes (Poisson processes for which the rate of occurrence of events varies randomly over time). Model development is based on counts and interarrival time statistics estimated from Potomac River basin rainfall data. The counting parameters used are the conditional intensity function, index of dispersion, and counts spectrum; the interarrival time parameters are the coefficient of variation and the autocorrelation function. Explicit results are presented for the counts and interarrival time parameters of RCM processes. Of particular importance in this paper is the interpretation of clustering suggested by the form of the RCM process. For the RCM process the rate of occurrence alternates between two states, one of which is 0, the other positive. During periods when the intensity is 0, no events can occur. The form of the intensity process suggests that clustering of summer season rainfall occurrences in the Potomac River basin results from the alternation of wet and dry periods. Computational results are presented for two extensions of the RCM process model of rainfall occurrences: a marked RCM process model of rainfall occurrences and associated storm depths and a bivariate RCM process model of rainfall occurrences at two sites.

Parameter Estimation for a Model of Space‐Time Rainfall

Abstract: In this paper, parameter estimation procedures, based on data from a network of rainfall gages, are developed for a class of space-time rainfall models. The models, which are designed to represent the spatial distribution of daily rainfall, have three components, one that governs the temporal occurrence of storms, a second that distributes rain cells spatially for a given storm, and a third that determines the rainfall pattern within a rain cell. Maximum likelihood and method of moments procedures are developed. We illustrate that limitations on model structure are imposed by restricting data sources to rain gage networks. The estimation procedures are applied to a 240-mi2 (621 km2) catchment in the Potomac River basin.

Statistical Inference for Point Process Models of Rainfall

Abstract: In this paper we develop maximum likelihood procedures for parameter estimation and model selection that apply to a large class of point process models that have been used to model rainfall occurrences, including Cox processes, Neyman-Scott processes, and renewal processes. The statistical inference procedures are based on the stochastic intensity λ(t) = lims→0,s>0 (1/s)E[N(t + s) − N(t)|N(u), u < t]. The likelihood function of a point process is shown to have a simple expression in terms of the stochastic intensity. The main result of this paper is a recursive procedure for computing stochastic intensities; the procedure is applicable to a broad class of point process models, including renewal Cox process with Markovian intensity processes and an important class of Neyman-Scott processes. The model selection procedure we propose, which is based on likelihood ratios, allows direct comparison of two classes of point processes to determine which provides a better model for a given data set. The estimation and model selection procedures are applied to two data sets of simulated Cox process arrivals and a data set of daily rainfall occurrences in the Potomac River basin.

Flood Frequency Analysis Using the Cox Regression Model

Abstract: Procedures for incorporating time-varying exogenous information into flood frequency analyses are developed using the Cox regression model for counting processes. In this statistical model the probability of occurrence of a flood peak in a short interval [t, t + dt) depends in an explicit manner on the values at t of k “covariate” processes Z1, …, Zk. Specifically, letting dN(t) be 1 if a flood peak occurs in [t, t + dt) and 0 otherwise, dN(t) = a(t) exp {∑j=1kbjZj(t)} + dM(t) where a, the “baseline intensity,” is an unknown function, b is a vector of unknown “regression” parameters, and the error dM(t) is (conditionally) orthogonal to the past history. Two applications, assessment of relative importance of physical processes such as snow melt or soil moisture storage on flood frequency at a site and derivation of time-varying flood frequency estimates, are considered.

A statistical model of extreme storm rainfall

Abstract: A model of storm rainfall is developed for the central Appalachian region of the United States. The model represents the temporal occurrence of major storms and, for a given storm, the spatial distribution of storm rainfall. Spatial inhomogeneities of storm rainfall and temporal inhomogeneities of the storm occurrence process are explicitly represented. The model is used for estimating recurrence intervals of extreme storms. The parameter estimation procedure developed for the model is based on the substitution principle (method of moments) and requires data from a network of rain gages. The model is applied to a 5000 mi2 (12,950 km2) region in the Valley and Ridge Province of Virginia and West Virginia.

Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes

Abstract: We argue that the basic properties of rain and cloud fields (particularly their scaling and intermittency) are best understood in terms of coupled (anisotropic and scaling) cascade processes. We show how such cascades provide a framework not only for theoretically and empirically investigating these fields, but also for constructing physically based stochastic models. This physical basis is provided by cascade scaling and intermittency, which is of broadly the same sort as that specified by the dynamical (nonlinear, partial differential) equations. Theoretically, we clarify the links between the divergence of high-order statistical moments, the multiple scaling and dimensions of the fields, and the multiplicative and anisotropic nature of the cascade processes themselves. We show how such fields can be modeled by fractional integration of the product of appropriate powers of conserved but highly intermittent fluxes. We also empirically test these ideas by exploiting high-resolution radar rain reflectivities. The divergence of moments is established by direct use of probability distributions, whereas the multiple scaling and dimensions required the development of new empirical techniques. The first of these estimates the "trace moments" of rain reflectivities, which are used to determine a moment-dependent exponent governing the variation of the various statistical moments with scale. This exponent function in turn is used to estimate the dimension function of the moments. A second technique called "functional box counting," is a generalization of a method first developed for investigating strange sets and permits the direct evaluation of another dimension function, this time associated with the increasingly intense regions. We further show how the different intensities are related to singularities of different orders in the field. This technique provides the basis for another new technique, called "elliptical dimensional sampling," which permits the elliptical dimension rain (describing its stratification) to be directly estimated: it yields del =2.22+0.07, which is less than that of an isotropic rain field (del =3), but significantly greater than that of a completely flat (stratified) two-dimensional field (de1-2).

Regional Climate Information—Evaluation and Projections

Abstract: Contributing Authors R. Arritt, B. Bates, R. Benestad, G. Boer, A. Buishand, M. Castro, D. Chen, W. Cramer, R. Crane, J. F. Crossley, M. Dehn, K. Dethloff, J. Dippner, S. Emori, R. Francisco, J. Fyfe, F.W. Gerstengarbe, W. Gutowski, D. Gyalistras, I. Hanssen-Bauer, M. Hantel, D.C. Hassell, D. Heimann, C. Jack, J. Jacobeit, H. Kato, R. Katz, F. Kauker, T. Knutson, M. Lal, C. Landsea, R. Laprise, L.R. Leung, A.H. Lynch, W. May, J.L. McGregor, N.L. Miller, J. Murphy, J. Ribalaygua, A. Rinke, M. Rummukainen, F. Semazzi, K. Walsh, P. Werner, M. Widmann, R. Wilby, M. Wild, Y. Xue

Multiscaling properties of spatial rainfall and river flow distributions

Abstract: Two common properties of empirical moments shared by spatial rainfall, river flows, and turbulent velocities are identified: namely, the log-log linearity of moments with spatial scale and the concavity of corresponding slopes with respect to the order of the moments. A general class of continuous multiplicative processes is constructed to provide a theoretical framework for these observations. Specifically, the class of log-Levy-stable processes, which includes the lognormal as a special case, is analyzed. This analysis builds on some mathematical results for simple scaling processes. The general class of multiplicative processes is shown to be characterized by an invariance property of their probability distributions with respect to rescaling by a positive random function of the scale parameter. It is referred to as (strict sense) multiscaling. This theory provides a foundation for studying spatial variability in a variety of hydrologic processes across a broad range of scales.

A model fitting analysis of daily rainfall data

Abstract: SUMMARY This paper discusses the fitting and use of models for daily rainfall observations. Nonstationary Markov chains are fitted to the occurrence of rain, and gamma distributions, with parameters which vary with-the time of year, are fitted to the rainfall amounts. Numerical methods are used to derive results from these models that are important in agricultural planning. Examples include the distributions of soil water content and lengths of dry spells. The process of fitting and using these models provides a straightforward and flexible analysis for rainfall records.