Most of what is routinely labeled ‘stochastic hydrology’ does not contain any hydrology at all and could be more properly identified as the fitting of stochastic models to samples of data of hydrologic origin. T0 engage in this enterprise, no hydrologic knowledge is necessary, nor do the results contribute to hydrologic knowledge. Moreover, the bulk of the current stochastic hydrology does not appreciably enhance the quality of water management decisions-an aim which provided the original impetus for its development. It seems that the mainstream of stochastic hydrology follows in the steps of ‘dam theory’, the only difference being that while the latter has become a self-contained branch of pure probability theory, the former is on the way to becoming a branch of pure mathematical statistics.

Statistical methods for analysing the extreme values of a time series may be based on the observed exceedances of the series above a high threshold level. Todorovic (1979) has developed this approach in detail; other relevant references are North (1980) and the English Flood Studies Report (1975). One way of motivating these models is by reference to the theory of extremes in stationary sequences, due to Leadbetter and others.

The strong impetus for the research in the theory of point processes comes from applications, real or potential, to a multitude of engineering, industrial and biological problems. Here we discuss some particular topics in this area and their applications in hydrology. Specifically, we consider stochastic models of a variety of geophysical phenomena, such as the rainfall, floods, sediment transport, dispersion in porous media and some others. The first part of this presentation is concerned with point processses on R+. Our discussion includes the stochastic intensity, some remarks on the martingale approach and a brief expose of the elements of renewal theory. In addition, we discuss in some detail the case when the counting random function represents a Markov process. In the second part we give an introduction to the theory of point processes on an abstract topological space. Elements of marked point processes, which are of particular interest in hydrological investigations, are also included. The rest of the paper is concerned with some hydrological applications.

Stochastic modeling of vector hydrologic time series exhibiting spatial as well as temporal correlations is examined with the general class of STARIMA, space-time autoregressive integrated moving-average models.

Both temporal and spatial information on rainfall distributions are important in various types of hydrologic studies concerning the determination of runoff characteristics. In reality, rainfall is measured at a point. Hence, for practical purposes, it is necessary to establish some relationships for the transformation of point rainfall at a fixed location to average rainfall over a catchment area in order to provide a more reliable input for rainfall-runoff models. The least equivocal method for deriving values of areal correction factor seems to be directly from frequency curves of areal and point rainfalls. Knowledge of the probability distribution of mean areal rainfall, therefore, is of paramount importance in solving this complex practical problem in hydrology.

The sedimentation of a monodisperse suspension of solid particles in a viscous fluid is a complex process. Mathematically, a slurry may be viewed as a collection of interacting particle paths. Interest focuses primarily on various global aspects of the system, but the behaviour of individual particles during sedimentation is also important. Kynch's flux theory and continuum mechanics models explicitly avoid the treatment of individual particles. At the other extreme, Brenner's fluid dynamic approach characterizes individual motion in terms of the exact (and detailed) configuration of the entire slurry. We replace this chaos of complex interaction with the formal chaos of stochastic processes.

In this paper we develop maximum likelihood procedures for parameter estimation and hypothesis testing for three classes of point processes that have been used to model rainfall occurrences; renewal processes, Neyman-Scott processes, and RCM processes (which are members of the family of Cox processes). The statistical inference procedures developed in this paper are based on the intensity process

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, …}, having state space J ⊆ {1, 2, …}. If Xn, = j, then there is a random. ‘input’ Vn(j) (a negative input implying a demand) of ‘type’ j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn(k)}, for k ≠ = j. Here, the random variables Vn(j) represent instantaneous ‘inputs’ of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (zn, Xn) and (Zn, OnLn), where Zn, is the level of storage at time n, Qn is the cumulative overflow at time n, and Ln is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

The growth rate of a cancerous tumour as a function of its age is a subject of intellectual and practical importance, as it influences both the effectiveness of proposed screening programs and strategy of treatment. Obtaining direct evidence on the growth rate is quite difficult owing to the ethical necessity to intervene when cancer is confirmed.

The linear compartmental model arises when ‘particles’ move independently between (or out of) a system of compartments in a stochastically similar way. With a given ‘initial’ particle count, the subsequent compartmental particle counts follow multinomial probability distributions (Faddy (1976)) for Markov or semi-Markov movement processes. One immediate consequence of this is that the variance of the compartmental particle count is always less than the mean, with the result that the coefficient of variation is very small for large mean counts. In applications (e.g, Faddy, Jones and Edwards (1976)) this underestimation of the variation can be a shortcoming of a compartmental analysis. Clustering, introduced by Matis and Wehrly (1981), where ‘clusters’ of particles may move together, is a way in which increased variability may be attained. Increased variability in general will result from relaxing the main assumption that gives rise to a linear model: independent behaviour of the particles. Such nonlinear compartmental models can be generally difficult to handle; in this paper some particular cases are discussed and illustrated with reference to one- and twocompartment systems.

The interpretation of mark-recapture data depends on a probabilistic model for the biological system. The assumptions of the general model must be verified against what is known about the particular system. This talk considers the application of markrecapture methods to the estimation of salmon smolt migrations from Babine Lake, British Columbia.

One of the more important parameters in the study of plant growth is the length of the root system.

The determination of the kinetics of early cancer growth has received considerable attention from cancer researchers interested in reducing the time required to assess carcinogenic risk. The three main difficulties in this study are: (1) tumors can only be identified after the animal has been sacrificed, (2) each animal exhibits considerable variability in tumor sizes and (3) tumors can only be identified once they have grown above a threshold size. Thus, the only data available to the researcher summarizes the sizes of observable tumors at the time of sacrifice. In this paper, we give techniques to help the researcher decide the type of tumor growth (e.g., exponential, logistic, Gompertz), the time at which the cancerous growth began, as well as the principal source of variability in tumor sizes (either the time of the initial growth, or the rate of growth). Additional techniques can be used to decide if the rate of growth increases with time (a consequence of the increased malignancy of the tumor) or decreases with time (due to the lack of oxygenation at the core of a large tumor). These techniques are applied to data on microscopic liver tumors (often called foci) from the lab of Dr Stanley Goldfarb (Department of Pathology, University of Wisconsin-Madison).

Offers become available from time to time. The offers constitute a sequence of i.i.d. positive random variables. Upon arrival of an offer a decision is taken whether to accept it or not. When an offer is accepted the process terminates. Otherwise, the offer is lost and the process continues until the next arrival or until a moment where it terminates by itself. The probability of self-termination depends on the age of the underlying process.

From the point of view of practical usefulness, the results of modelling exercises should not be too sensitive to the detailed assumptions built into the model. Consequently, it is a great advantage if structural assumptions are kept to a minimum.

Venereal disease is among one of the more active serious epidemics in the world today. Despite its prevalence and the ever-increasing literature on the mathematical treatment of epidemics very little progress has been made toward modelling the spread of this disease. Key features of the disease that help to account for this lack of progress are: (i) VD involves the cross-infection between two groups of individuals and (ii) VD is a recurrent disease. The heterosexual spread .of VD ties in the modelling of this disease with the yet unsolved problem in mathematical biology concerning the growth of two interacting species. In this paper we review the current state of the art in the stochastic modelling of this disease.

Polymeric materials are encountered everywhere: rubber, plastics, paint, soft contact lenses; the list is endless. There are two basic types of polymer, linear and non-linear. The linear polymers can be envisioned as a mass of long spaghetti-like strands. The molecules of non-linear polymers have a branched tree-like structure and can form ‘infinite’ networks. The basic building blocks of polymers are monomers. Monomers have reactive sites: chemical bonds can form between sites on different monomers leading to the formation of many monomers into a large molecule (polymer). If all monomers have one or two reactive sites, linear polymers result. If some of the monomers have three or more reactive sites, a non-linear (branched) molecule results. For many materials the reaction between monomers can be modelled as a random process; thus probability theory is the natural tool for studying polymers.

This paper reviews a variety of results for genealogical (or line-of-descent) processes that arise in connection with the theory of some classical selectively neutral haploid population genetics models. While some new results and derivations are included, the principal aim of the paper is to demonstrate the central importance and simplicity of genealogical Markov chains in this theory. Considerable attention is given to ‘diffusion time scale’ approximations of such genealogical processes. A wide variety of results pertinent to (diffusion approximations of) the classical multi-allele single-locus WrightFisher model and its relatives are unified by this approach. Other examples where the genealogical process plays an explicit role (for example, the infinite-sites models) are discussed.

Plasmids are subcellular particles somewhat similar to, but sirnpler than, viruses. A small part of the host cell's genome, they multiply independently within host cells and are apparently inherited in a random manner. They code important genetic traits not normally encoded and are responsible for several plant and animal diseases; they can also confer the ability to resist a wide variety of toxic agents.

The M/M/∞ queue can be considered as a compartmental system which may be modified by allowing its arrival and service rates to depend on an exogenous ‘environment process’. The states of the environment process are known as environments. We present some examples of this, chosen for their analytic or algorithmic tractability. We discuss (i) models with arrivals in only one environment, (ii) models with service in only one environment, (iii) reversible models and (iv) a model in which the environment process is also a queue.