a generalized point process model for rainfall this paper further develops a stochastic model for rainfall at a single site in which storms arrive in a poisson process each storm generating a cluster of rain cells with each cell having a random duration and a random intensity the model is generalized by allowing each generated cell to be of n types the duration of each cell is an exponential random variable that has parameter dependent on the cell type the distribution of cell intensity is also dependent on the cell type so that the generalized model provides a correlation between the intensities and durations of the generated rain cells the case for two cell types is considered in some detail the cells are categorized as either heavy or light9 where the heavy cells have a shorter expected lifetime than the light cells which agrees with observational studies on precipitation fields properties are derived and used to fit the model to rainfall data at a single site the adequacy of fit is assessed by considering properties not used in the fitting procedure but which are of interest in applications e g extreme values further properties are derived which enable the model to be fitted to multi site rainfall data

modeling fractal entrainment sets of tracers advected by chaotic temporally irregular fluid flows using random maps abstract we model a two dimensional open fluid flow that has temporally irregular time dependence by a random map n 1 m n n where on each iterate n the map m n is chosen from an ensemble we show that a tracer distribution advected through a chaotic region can be entrained on a set that becomes fractal as time increases theoretical and numerical results on the multifractal dimension spectrum are presented

copula based daily rainfall disaggregation model 1 a daily rainfall disaggregation model which uses a copula to model the dependence structure between total depth total duration of wet periods and the maximum proportional depth of a wet period is presented the wet 1 dry 0 binary sequence is modeled by the nonrandomized bartlett lewis model with diurnal effect incorporated before superimposing the ar 1 depth process submodel unlike previous studies the model is structured such that all wet day data available are considered in the analysis without the need to discard any good quality daily data embedded in a month having some missing data this increased the data size thus improving the modeling process further the daily data are classified according to the total duration of wet periods duration within the day in this way a large proportion of the model parameters become seasonal invariant the overriding factor being the total duration of wet periods the potential of the developed model has been demonstrated by disaggregating both the data set used in developing the model parameters and also a 12 year continuous rainfall data set not used in the model parameterization gross rainfall statistics of several aggregation levels down to 6 min have been very well reproduced by the disaggregation model the copula dependence structure and the variation of the depth process submodel parameters with the total duration of wet periods are also very well captured by the presented model

on the correlation structure of continuous and discrete point rainfall 1 this paper proposes new analytical results based on the theory of stochastic processes that establish a theoretical dependence of point rainfall variance on the scale of observation under somewhat general assumptions it is shown that power law scaling commonly assumed by analyses and models of point rainfall cannot hold over all scales of observation and that an inner regime exists for small sampling intervals where the variance is a quadratic function of interval length a power law scaling regime is shown to exist for larger aggregation intervals whose characteristics depend on the memory of the continuous rainfall process the presence of a transition regime between the inner and the scaling regime is also shown and may explain the deviations from a power law scaling behavior in observed rainfall reported in the literature furthermore the application of the theory to rainfall data from a representative variety of climate types shows that fractal and multifractal analysis techniques and models may indeed fail to capture observed rainfall properties over wide ranges of aggregation scales finally correlation structures with short and long term memory are considered and different theoretical relationships expressing rainfall variance as a function of aggregation interval are derived and tested against rainfall observed at different locations with aggregation timescales ranging from 15 min to 2 days the theoretical and observational analyses show that both finite and infinite memory rainfall processes may be observed in nature

general model of n d system with variable coefficients and its reduction to 1 d system with variable structure a general model of an n d linear discrete system with variable coefficients and its solution are presented a method for reduction of this model to an equivalent 1 d system with a variable structure is given

rational covariance functions for nonstationary random fields this correspondence propose rational type covariance functions for nonstationary gaussian stochastic processes or random fields which are rational functions of given variograms typically have long range dependence and include some stationary covariance functions as special cases

the mathematical structure of rainfall representations 3 some applications of the point process theory to rainfall processes part 3 constitutes the final part of three parts devoted to the mathematical structure of rainfall the objective is to illustrate the scope of the tools developed in part 2 in the mathematical description of rainfall and rainfall driven processes a general overview of the three part series is given in part 1

stochastic modeling of multidimensional precipitation fields considering spectral structure a multidimensional stochastic precipitation model with major emphasis on its spectral structure is proposed as a hyperbolic type of stochastic partial differential equation this model is characterized by a small set of easily estimable parameters these characteristics are similar to those of the noise forced diffusive precipitation model but the representation of the physical and statistical features of the precipitation field is similar to that of the waymire gupta rodriguez iturbe wgr precipitation model the derivation was based on the autoregressive process considering advection and diffusion the dominant statistical and physical characteristics of the precipitation field propagation the model spectrum showed a good match with the global atlantic tropical experiment spectrum this model was also compared with the wgr model and the noise forced diffusive precipitation model both analytically and through applications such as the sampling error estimation from spaceborne sensors and rain gauges the sampling error from spaceborne sensors based on the proposed model was similar to that of the noise forced diffusive precipitation model but much smaller than that of the wgr model similar results were also obtained in the estimation of the sampling error from rain gauges

statistical inference for point process models of rainfall 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 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

fractal tracer distributions in complicated surface flows an application of random maps to fluid dynamics measurements of the local dynamics on the surface of a fluid undergoing complicated motion allow prediction of the measured fractal dimension of an aggregate of passive floating tracers this realization of a strange attractor in physical space is a rare instance where there is a firm quantitative coonection between the dimension of an experimentally observed fractal spatial pattern and the process producing it the results also show that the fractal dimension of a tracer distribution is a potential diagnostic for the dynamics of the underlying flow the analysis of experimental results is performed in terms of random mappings indicating that this mathematical abstraction provides a potentially useful conceptual framework to bring some spatio temporal problems into the domain of low dimensional dynamics

accumulations of t points in a model for solitary pulses in an excitable reaction diffusion medium we consider a family of differential equations that describes traveling waves in a reaction diffusion equation modeling oxidation of carbon oxide on a platinum surface near the onset of spatio temporal chaos the organizing bifurcation for the bifurcation structure with small carbon oxide pressures turns out to be a codimension 3 bifurcation involving a homoclinic orbit to an equilibrium undergoing a transcritical bifurcation we show how infinitely many t point bifurcations of multi loop heteroclinic cycles occur in the unfolding

a spatial temporal point process model with a continuous distribution of storm types 1 a point process rainfall model is further developed that has storm origins occurring in space time according to a poisson process where each storm origin has a random radius so that storms occur as circular regions in two dimensional space where the storm radii are taken to be independent exponential random variables each storm origin is of random type z where z follows a continuous probability distribution cell origins occur in a further spatial poisson process and have arrival times that follow a neyman scott point process each cell origin has a radius so that cells form discs in two dimensional space where the cell radii are independent exponential random variables each cell has a random lifetime and an intensity that remains constant over both the cell lifetime and cell disk area statistical properties up to third order are given for the model using these properties the model is fitted to 10 min series taken from 23 sites across the rome region italy distributional properties of the observed annual maxima are compared to equivalent values sampled from series that are simulated using the fitted model the results indicate that the model will be of use in urban drainage projects for the rome region

conditional distributions of neyman scott models for storm arrivals and their use in irrigation scheduling count for the widely recognized but often neglected statistical dependence in storm arrivals gabriel and newmann 1957 1962 smith and schreiber 1973 kavvas and delleur 1975 recently authors like kavvas and delleur 1975 and gupta and waymire 1979 have argued that this dependence is due to clustering of rainfall in time and space they suggest modeling storm arrivals with cluster point processes like the neyman scott model this work uses the concept of cluster models for storm arrivals the conditional distribution of the occurrence of future events given a history of the timing of past events is derived for the neyman scott model this new result is then used with a reformulated model of the irrigation scheduling problem now the decision of when and how much to irrigate is based on the time since the last observed storm the effective soil moisture content and the amount of available irrigation water a case study is presented to illustrate the solution significantly there is no important difference between the theoretical net benefits derived from assuming a homogeneous independent poisson model or a dependent neyman scott n s model during the growing season a nonhomogeneous poisson model yields higher theoretical benefits as expected the conditional n s model leads to improved irrigation control as compared to the unconditional n s model from this case study however no definite conclusion can be drawn regarding whether the neyman scott model better represents the process of storm arrivals as opposed to the poisson model however the expected opportunity loss of choosing the wrong model can be significant interesting implications of the results are that data manipulation can have a significant effect on the ultimate model

fractal distribution of floaters on a fluid surface and the transition to chaos for random maps abstract the long time spatial distribution of particles floating on the surface of a confined fluid whose flow velocity has complicated time dependence is considered it is shown that this distribution can be either a fractal or else can clump at several or one discrete points the transition from the latter type of distribution to the former occurs when the lyapunov exponent characterizing the particle motion passes through zero from negative values to positive values the characteristic features of this type of transition are investigated using random maps it is shown that near the transition there are extremely intermittent temporal fluctuations in the particle cloud and their scaling with a parameter is elucidated

compatibility of continuous rainfall occurrence models with discrete rainfall observations inferences about the underlying rainfall generating mechanism made through a model fitted to the sampled rainfall series at a particular time scale are meaningful only if they are invariant under the time scale of measurement at which the fitting of the model is made following the work of i rodriguez iturbe et al 1984 this assertion is tested by examining the compatibility of the neyman scott n s model for the continuous underlying rainfall with sampled realizations taken over intervals ranging from one hour to one day the results suggest that the n s model is inconsistent at different time scales and therefore mathematical descriptions of rainfall using this model are inevitably limited to the time scale of measurement at which a description of rainfall is sought these findings are quite general in that they apply to any model with rainfall deposited instantaneously at times described by a n s process regardless of how the amounts associated with the events are distributed and of the dependence structure of the amounts

a point process model for rainfall further developments a stochastic model for rainfall at a single site is studied in which storms arrive in a poisson process each storm consisting of a cluster of a random number of rain cells each cell having random duration and depth a model studied in an earlier paper is extended to provide a better fit to empirical experience the extension being based on the attachment of a single random variable to each storm to achieve in particular some correlation between the durations of the cells within a single storm the properties of the new model are developed its fitting to two sets of empirical data is described and the examination of adequacy of fit is studied in some detail via properties not used in the fitting procedure finally a theoretical study is made of short term prediction from the model

nonseparable stationary covariance functions for space time data geostatistical approaches to spatiotemporal prediction in environmental science climatology meteorology and related fields rely on appropriate covariance models this article proposes general classes of nonseparable stationary covariance functions for spatiotemporal random processes the constructions are directly in the space time domain and do not depend on closed form fourier inversions the model parameters can be associated with the data s spatial and temporal structures respectively and a covariance model with a readily interpretable space time interaction parameter is fitted to wind data from ireland

a hybrid model for point rainfall modeling a hybrid point rainfall model a product of two random processes is presented the model uses a jitter model a general technique for improving the fit of any point process based model we use the nonrandomized bartlett lewis rectangular pulse and an autoregressive model as a jitter first the five parameters of the bartlett lewis model are estimated by the method of moments using the mean of one aggregation level and the dry probabilities of all aggregation levels considered second the parameters of the autoregressive model are derived from the moments of the historical data and those given by the bartlett lewis model without additional cost in terms of parameter calibration using 15 min point rainfall data of capella central queensland australia as a test case the results of the hybrid model were better than the best results given by the randomized bartlett lewis model on its own the nonrandomized bartlett lewis model produced very poor results

two site modeling of rainfall based on the neyman scott process on the basis of the neyman scott process a model is proposed to generate rainfall at two sites simultaneously this model takes into account the correlation which is produced by generating bivariate correlated variates the cross correlation between the sites has been computed in order to estimate the parameters finally the model is validated for three pairs of stations situated in the swiss plateau region

general response formula and minimum energy control for the general singular model of 2 d systems the general response formula for the general singular model of 2 d linear systems is derived the concepts of local reachability and local controllability are extended to the singular model necessary and sufficient conditions for local reachability and local controllability are established the minimum energy control problem for the singular model is formulated and solved

generalized scale invariance in the atmosphere and fractal models of rain in recent years there has been considerable interest in stochastic rain models by developing new ideas about scale invariance and intermittency we argue that the scope of such models can be greatly extended the notions of scale invariance intermittency and the associated idea of fractal dimension have lately gained considerable ground particularly in the context of extremely variable phenomena such as those found in the mesoscale and in hydrodynamic turbulence we review some relevant work and argue that the atmosphere respects a symmetry principle that we call generalized scale invariance in which the statistical properties of the small and large scale are related to each other by a magnification coupled with a differential stratification due to gravity and differential rotation due to the coriolis force we further argue that the extremely erratic intermittent nature of the atmosphere is characterized by scale invariant fractal measures leading to hyperbolic fat tailed probability distributions of the fluctuations the standard statistical methods based on exponential fall offs in both correlations and probability distributions are inappropriate when the variability is of this type we show how both the scaling and intermittency may be exploited to develop extremely variable stochastic models of rain although the models examined here are the simplest of a family of extremely variable processes depending on only two radar determined parameters they lead to realizations possessing many realistic features of the rainfield including complexity of form clustering and bands at all scales as well as differential stratification and rotation finally we point to weaknesses in the models in particular their monodimensional nature and suggest possible improvements

a geomorphologic synthesis of nonlinearity in surface runoff the geomorphic approach leading to a representation of an instantaneous unit hydrograph iuh which we developed earlier is generalized to incorporate nonlinear effects in the rainfall runoff transformation it is demonstrated that the nonlinearity in the transformation enters in part through the dependence of the mean holding time on the rainfall intensity under an assumed first approximation that this dependence is the sole source of nonlinearity an explicit quasi linear representation results for the rainfall runoff transformation the kernel function of this transformation can be termed as the instantaneous response function irf in contradistinction to the notion of an iuh for the case of a linear rainfall runoff transformation the predictions from the quasi linear theory agree very well with predictions from the kinematic wave approach for the one small basin that is analyzed also for two large basins in illinois having areas of about 1100 mi2 the predictions from the quasi linear approach compare very well with the observed flows a measure of nonlinearity naturally arises through the dependence of the mean holding time kb i0 on the rainfall intensity i0via kb i0 i0 computations of for four basins show that approaches as basin size decreases and approaches zero as the basin size increases a semilog plot of versus the square root of the basin area gives a straight line confirmation of this relationship for other basins would be of basic importance in predicting flows from ungaged basins

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

a nonparametric wet dry spell model for resampling daily precipitation a nonparametric wet dry spell model is developed for resampling daily precipitation at a site the model considers alternating sequences of wet and dry days in a given season of the year all marginal joint and conditional probability densities of interest e g dry spell length wet spell length precipitation amount and wet spell length given prior to dry spell length are estimated nonparametrically using at site data and kernel probability density estimators procedures for the disaggregation of wet spell precipitation into daily precipitation and for the generation of synthetic sequences are proffered an application of the model for generating synthetic precipitation traces at a site in utah is presented

an entropic stochastic representation of rainfall intermittency the origin of clustering and persistence 1 the well established physical and mathematical principle of maximum entropy interpreted as maximum uncertainty is used to explain the observed dependence properties of the rainfall occurrence process including the clustering behavior and persistence the conditions used for the maximization of entropy are as simple as possible i e that the rainfall processes is intermittent with dependent occurrences intermittency is quantified by the probability that a time interval is dry and dependence is quantified by the probability that two consecutive intervals are dry these two probabilities are used as constraints in a multiple scale entropy maximization framework which determines any conditional or unconditional probability of any sequence of dry and wet intervals at any timescale thus the rainfall occurrence process including its dependence structure is described by only two parameters this dependence structure appears to be non markovian application of this theoretical framework to the rainfall data set of athens indicates good agreement of theoretical predictions and empirical data at the entire range of scales for which probabilities dry and wet can be estimated from one hour to several months

on binary time series obtained from continuous time point processes describing rainfall recent work on modeling rainfall occurrence has used continuous time point process models in many cases the available data is not sufficiently precise to properly fit such a model for example if only occurrence or nonoccurrence of precipitation is given daily a more appropriate model would be a binary time series in this work we relate some parameters of the time series thus obtained to parameters of the underlying but not observed continuous time process we carry out the details for several models proposed in the literature such as the neyman scott poisson cluster process and the smith karr renewal cox process an application of the theory is illustrated on some washington state rainfall data

fractal snapshot components in chaos induced by strong noise in systems exhibiting transient chaos in coexistence with periodic attractors the inclusion of weak noise might give rise to noise induced chaotic attractors when the noise amplitude exceeds a critical value an extended attractor appears along the fractal unstable manifold of the underlying nonattracting chaotic set a further increase of noise leads to a fuzzy nonfractal pattern by means of the concept of snapshot attractors and random maps we point out that the fuzzy pattern can be decomposed into well defined fractal components the snapshot attractors belonging to a given realization of the noise and generated by following an ensemble of noisy trajectories the pattern of the snapshot attractor and its characteristic numbers such as the finite time lyapunov exponents and numerically evaluated fractal dimensions change continuously in time we find that this temporal fluctuation is a robust property of the system which hardly changes with increasing ensemble size the validity of the kaplan yorke formula is also investigated a superposition of about 100 snapshot attractors provides a good approximant to the fuzzy noise induced attractor at the same noise strength

a spectral theory of rainfall intensity at the meso β scale the available empirical descriptions of extratropical cyclonic storms are employed to formulate a physically realistic stochastic representation of the ground level rainfall intensity field in space and time the stochastic representation is based on three component stochastic point processes which possess the general features of the embedding of rain cells within small mesoscale areas within large mesoscale areas within synoptic storms certain scale idealizations and assumptions on functional forms which qualitatively reflect the physical features lead to a closed form expression for the covariance function i e the real space time spectrum of the rainfall intensity field the theoretical spectrum explains the empirical spectral features observed by zawadzki almost a decade ago of particular interest and importance in this connection is an explanation of the empirical observation that the taylorian propogation of the fine scale structure via a transformation of time to space through the storm velocity holds only for a small time lag and not throughout the results here indicate the extent of this lag in terms of the characteristic scales associated with cell durations cellular birthrates and velocities etc

on space time scaling of cumulated rainfall fields in the study of space time rainfall it is particularly important to establish characteristic properties to guide both theoretical and modeling research efforts in the present paper new observational analyses on the scaling properties of time evolving cumulated rainfall fields are presented and a theoretical framework for their interpretation is introduced it is found that the time evolution of the spatial organization of a cumulated rainfall field produces scaling relationships of spatial variance versus time and characteristic values for the scaling exponent the reproduction of these values constitutes a basic requirement for spatial temporal field generators in order to model important properties of real rainfall fields it is then shown on theoretical grounds what properties the instantaneous rainfall intensity fields must obey in order to reproduce the experimental observations and how the size of the observation domain affects the scaling relationships some current stochastic models of space time precipitation are finally discussed and analyzed in the light of the tools introduced to show under what circumstances the models considered give acceptable results furthermore it is shown that the assumption of an exponential time correlation function used in many current rainfall models is not compatible with observations

continuous time versus discrete time point process models for rainfall occurrence series several authors have had apparent success in applying continuous time point process models to rainfall occurrence sequences in this paper it is shown that if rainfall occurrences are interpreted as the events of a point process and not as a censored sample the continuous time point process methodology and estimation procedures are not directly applicable since they fail to account for the time discreteness of the sample process this is demonstrated analytically by studying the effects of discretization on selected statistical properties of a poisson process a neyman scott process and a renewal cox process with markovian intensity in general the study of rainfall occurrences under the continuous time point process framework may result in misleading inferences regarding clustering dispersion and consequently incorrect interpretations of the underlying rainfall generating mechanisms for example daily rainfall occurrence structures underdispersed relative to the poisson process are usually overdispersed relative to the bernoulli process the discrete time analogue of the poisson these findings are confirmed by the statistical analysis of six daily rainfall records representative of a range of u s climates two of which are described in detail

the singular general model of 2d systems and its solution a general singular model for 2d linear discrete systems with constant coefficients is defined necessary and sufficient conditions for the existence of a solution to the model are established the considerations of the author s previous work 1986 can be extended to singular models of 2d linear systems with variable coefficients

the mathematical structure of rainfall representations 2 a review of the theory of point processes part 2 constitutes the second of three parts devoted to the mathematical structure of rainfall an expository yet current treatment of those features of point process theory which are essential to descriptions of the hydrologic processes is developed a general overview of this three part series is given in part 1

a generalized spatial temporal model of rainfall based on a clustered point process the objective in this paper is to present and fit a relatively simple stochastic spatial temporal model of rainfall in which the arrival times of rain cells occur in a clustered point process in the x y plane rain cells are represented as discs each disc having a random radius the locations of the disc centres being given by a two dimensional poisson process the intensity of each cell is a random variable that remains constant over the area of the disc and throughout the lifetime of the cell the lifetime being an exponential random variable the cells are randomly classified from 1 to n with different parameters for the different cell types so that the random variables of an arbitrary cell e g radius and intensity are correlated multi site second order properties are derived and used to fit the model to hourly rainfall data taken from six sites in the thames basin uk

straight line reachability of roesser model a new definition of straight line reachability for the roesser model is presented necessary and sufficient conditions for straight line teachability of the roesser model are established a relationship between local reachability and straight line reachability of the roesser model is discussed

the mathematical structure of rainfall representations 1 a review of the stochastic rainfall models this is the first of a three part series on the mathematical structure of rainfall models several important attempts at modeling rainfall are reviewed special attention is given to the mathematical structures that arise in the rainfall descriptions a general overview of the three part series is given as preface to this part

the effect of certain restrictions imposed on the interarrival times of flood events on the poisson distribution used for modeling flood counts in the application of partial duration series models of flood analysis it is occasionally observed that successive exceedances are correlated to reduce this correlation some investigators tend to impose certain restrictions on the interarrival times of flood events in order that these events will not occur close together in bunches we show analytically how such restrictions interfere with the underlying hypotheses of the poisson process commonly used to model flood counts and we caution against imposing restrictions that may render this simple and appealing model inapplicable

modelling extremes of the areal rainfall process risk assessment for many hydrological structures requires an estimate of the extremal behaviour of the rainfall regime within a specified catchment region in most cases it is the spatially aggregated rainfall which is the key process though in practice only pointwise rainfall measurements from a network of sites over the region are available in this paper we address the problem of making inferences about the extremal properties of the aggregated process from the pointwise data working within the usual extreme value paradigm a model is derived in which the resulting distribution is determined by the marginal tail behaviour and spatial dependence at extreme levels of the process data collected from a region in the south west of england are used to illustrate the procedure

multiscaling pulse representation of temporal rainfall 1 we develop a pulse based representation of temporal rainfall with multifractal properties in the small scale limit the representation combines a traditional model for the exterior process at the synoptic scale with a novel hierarchical pulse model for the event interiors for validation we apply the model to a temporal rainfall record from florence italy although the model has only six parameters four for the exterior process and two for the event interiors it accurately reproduces a wide range of empirical statistics including the distribution of dry and wet periods the distribution of rainfall intensity up to extreme fractiles the spectral density the moment scaling function k q and the distribution of the partition coefficients for rainfall disaggregation the model also reproduces observed deviations of physical rainfall from perfect scaling multiscaling behavior

pollution release tied to invariant manifolds a case study for the coast of florida high resolution ocean velocity data has become readily available since the introduction of very high frequency vhf radar technology the vast amount of data generated so far however remains largely unused in environmental prediction in this paper we use vhf data of the florida coastline to locate lagrangian coherent structures lcs hidden in ocean surface currents such structures govern the spread of organic contaminants and passive drifters that stay confined to the ocean surface we use the lagrangian structures in a real time pollution release scheme that reduces the effect of industrial contamination on the coastal environment

further developments of the neyman scott clustered point process for modeling rainfall this paper provides some useful results for modeling rainfall it extends work on the neyman scott cluster model for simulating rainfall time series several important properties have previously been found for the model for example the expectation and variance of the amount of rain captured in an arbitrary time interval rodriguez iturbe et al 1987a in this paper additional properties are derived such as the probability of an arbitrary interval of any chosen length being dry in applications this is a desirable property to have and is often used for fitting stochastic rainfall models to field data the model is currently being used in rainfall time series research directed toward improving sewage systems in the united kingdom to illustrate the model s performance an example is given where the model is fitted to 10 years of hourly data taken from blackpool england

analytically derived runoff models based on rainfall point processes this work introduces four stochastic and lumped in space rainfall runoff models they arise combining two stochastic rainfall models with two conceptualizations of a basin s response rainfall is given by rectangular pulses whose arrivals are described by poisson and neyman scott models runoff is obtained routing such pulses via two alternative conceptualizations both based on simple linear reservoirs general expressions for the first and second order moments are given for both the instantaneous and locally averaged processes the mathematical structure of the alternative parameterizations is illustrated via computer simulations

dynamics of a three dimensional incompressible flow with stagnation points abstract the nonlinear dynamics of the streamlines of a two parameter family of incompressible flows is investigated in a range of parameters the flow contains two fixed points or stagnation points which are created and destroyed by saddle node and saddle node hopf bifurcations between the parameter values where these local bifurcations occur there is a global bifurcation that creates an unstable closed orbit further global bifurcations lead to chaotic scattering and horseshoes the detailed nature of the scattering depends upon whether the stagnation points are spiralling i e upon whether their eigenvalues have imaginary parts in the vicinity of a curve in the parameter space a reversed cascade of period doubling bifurcations occurs after which kam kolmogorov arnold and moser tori appear giving rise to nonhyperbolic chaotic scattering as the parameter value for the saddle node hopf bifurcation is approached the region of chaos shrinks rapidly leaving a vortex structure of kam tori of size where is the distance in parameter space to the saddle node hopf bifurcation

some models for discretized series of events a discretized series of events is a binary time series that indicates whether or not events of a point process in the line occur in successive intervals such data are common in environmental applications we describe a class of models for them based on an unobserved continuous time discrete state markov process which determines the rate of a doubly stochastic poisson process from which the binary time series is constructed by discretization we discuss likelihood inference for these processes and their second order properties and extend them to multiple series an application involves modeling the times of exposures to air pollution at a number of receptors in western europe

daily spatiotemporal precipitation simulation using latent and transformed gaussian processes 1 a daily stochastic spatiotemporal precipitation generator that yields spatially consistent gridded quantitative precipitation realizations is described the methodology relies on a latent gaussian process to drive precipitation occurrence and a probability integral transformed gaussian process for intensity at individual locations the model reduces to a markov chain for precipitation occurrence and a gamma distribution for precipitation intensity allowing statistical parameters to be included in a generalized linear model framework statistical parameters are modeled as spatial gaussian processes which allows for interpolation to locations where there are no direct observations via kriging one advantage of such a model for the statistical parameters is that stochastic generator parameters are immediately available at any location with the ability to adapt to spatially varying precipitation characteristics a second advantage is that parameter uncertainty generally unavailable with deterministic interpolators can be immediately quantified at all locations the methodology is illustrated on two data sets the first in iowa and the second over the pampas region of argentina in both examples the method is able to capture the local and domain aggregated precipitation behavior fairly well at a wide range of time scales including daily monthly and annually

evaluation of cluster based rectangular pulses point process models for rainfall this technical note presents a comparison of cluster based point rainfall models using the historical hourly rainfall data observed between 1949 and 1976 at denver colorado the denver data are used to analyze the performance of three classes of models namely the bartlett lewis model the geometric neyman scott model and the poisson neyman scott model the original formulation of the structure of each model as well as the modified description developed in order to improve the zero depth probability is considered in this study rodriguez iturbe et al 1987a concluded that it is unlikely that empirical analysis of rainfall data can be used to choose between the bartlett lewis model and the neyman scott model in a subsequent paper rodriguez iturbe et al 1987b argued that the choice of the distribution of the number of cells per storm for the neyman scott model either geometric or poisson has no general bias effect on the stochastic structure some investigators e g burlando and rosso 1991 however reported results contradictory to those of the previous authors in light of these observations this note investigates the performance of the cluster based models for the denver data the geometric neyman scott model yields better results compared to the poisson neyman scott model moreover the bartlett lewis model is shown to be very sensitive to the sets of moment equations used in the parameter estimation this sensitivity is not observed in the neyman scott scheme and is believed to be a drawback for applying the bartlett lewis model in hydrologic simulation studies

a regionalization method based on a cluster probability model 1 a regionalization method based on a cluster probability model a mixed multivariate gaussian model is proposed for grouping sites into nonoverlapping contiguous homogeneous regions defined by a voronoi tessellation theisson polygons the cluster probability model is applied to second order standardized annual sample properties mean coefficient of variation and autocorrelation evaluated at the daily level of aggregation taken from each of 234 daily rainfall records with positions located in the basque country spain using the bayesian information criterion four clusters of sites are identified which do not fall into contiguous regions the distances between all neighboring pairs of sites connected by edges from the delaunay planar graph are found the probability that a site belongs to each of the four clusters is extracted from the fitted gaussian model and multiplied into the probability that the neighboring site belongs to the same cluster these products are divided by the squared distance between the sites and are summed to give an overall measure of a site belonging to a cluster that takes into account the classification of neighboring sites regions from the voronoi tessellation of the points are classed on the basis of this measure and according to whether they are spatially isolated from other regions of the same class points that have the least influence on the variance of residual errors of the fitted model are found using a criterion based on wilks lambda for multivariate analysis of variance and the classes of the least influential points are adjusted to ensure the overall regions are contiguous

dynamics of leaking hamiltonian systems in order to understand the dynamics in more detail in particular for visualizing the space filling unstable foliation of closed chaotic hamiltonian systems we propose to leak them up the cutting out of a finite region of their phase space the leak through which escape is possible leads to transient chaotic behavior of nearly all the trajectories the never escaping points belong to a chaotic saddle whose fractal unstable manifold can easily be determined numerically it is an approximant of the full hamiltonian foliation the better the smaller the leak is the escape rate depends sensitively on the orientation of the leak even if its area is fixed the applications for chaotic advection for chemical reactions superimposed on hydrodynamical flows and in other branches of physics are discussed

general response formula for two dimensional linear systems with variable coefficients a new definition of the transition matrix for a general 2 d system model with variable coefficients is presented the general response formula for the general 2 d system model with variable coefficients is derived solution to the first and the second fornasini marchesini models with variable coefficients are also given

a comparative copula based bivariate frequency analysis of observed and simulated storm events a case study on bartlett lewis modeled rainfall 1 because of a lack of historical rainfall time series of considerable length one often has to rely on simulated rainfall time series e g in the design of hydraulic structures one way to simulate such time series is by means of stochastic point process rainfall models such as the bartlett lewis type of model for the evaluation of model performance with a focus on the reproduction of extreme rainfall events often a univariate extreme value analysis is performed recently developed concepts in statistical hydrology now offer other means of evaluating the overall performance of such models in this study a copula based frequency analysis of storms is proposed as a tool to evaluate differences between the return periods of several types of observed and modeled storms first this study performs an analysis of several storm variables which indicates a problem with the modeling of the temporal structure of rainfall by the models thereafter the bivariate frequency analysis of storms defined by their duration and volume illustrates the underestimation and overestimation of the return period of the storms simulated by the models which is partially explained by a large difference in the marginal distribution functions of the storm duration and storm volume the difference in the degree of association between the latter and a different mean storm interarrival time the proposed methodology allows for the identification of some problems with the rainfall simulations from which recommendations for possible improvements to rainfall models can be made

a space time neyman scott model of rainfall empirical analysis of extremes 1 a spatial temporal model of rainfall based on a neyman scott stochastic point process is fitted to hourly data taken from nine sites in the arno basin italy the stochastic model is an extension of the temporal neyman scott rectangular pulses model into two dimensional space and introduces a further parameter into the model in the model storms arrive in a poisson process where each storm consists of discs representing rain cells with centers distributed over an area according to a spatial poisson process the cells have a random radius lifetime and intensity with the intensity remaining constant over the area of the disc and cell lifetime a fitting procedure is proposed which couples the results obtained in two preceding papers the second order properties of the spatial temporal model and the third moment function of the single site model cowpertwait 1995 1998 the model is validated by comparing extreme historical hourly data and equivalent data simulated using the fitted spatial temporal model these comparisons are made using a regional frequency analysis based on l moments and log log plots of the upper distribution tail the results indicate that the model is able to preserve regional extremes and support the use of the model in hydrological applications

enhanced speciation in particle swarm optimization for multi modal problems abstract in this paper we present a novel multi modal optimization algorithm for finding multiple local optima in objective function surfaces we build from species based particle swarm optimization spso by using deterministic sampling to generate new particles during the optimization process by implementing proximity based speciation coupled with speciation of isolated particles and by including turbulence regions around already found solutions to prevent unnecessary function evaluations instead of using error threshold values the new algorithm uses the particle s experience geometric mean and exclusion factor to detect local optima and stop the algorithm the performance of each extension is assessed with leave it out tests and the results are discussed we use the new algorithm called isolated speciation based particle swarm optimization ispso and a benchmark algorithm called niche particle swarm optimization nichepso to solve a six dimensional rainfall characterization problem for 192 rain gages across the united states we show why it is important to find multiple local optima for solving this real world complex problem by discussing its high multi modality solutions found by both algorithms are compared and we conclude that ispso is more reliable than nichepso at finding optima with a significantly lower objective function value

non power law scale properties of rainfall in space and time 1 the identification of general relationships linking statistical properties of rainfall aggregated at different temporal and spatial scales possesses clear theoretical and practical relevance among other properties it is important to characterize the scale dependence of rainfall variability which may for many purposes be summarized by its variance this paper presents theoretical results and observational analyses connecting the variance of temporal and spatial rainfall to the scale of observation under an assumption of second order stationarity previous results regarding the theoretical form of the temporal variance as a function of aggregation are refined and extended to the spatial case it is shown that the variance of aggregated rainfall exhibits an overall nonscaling form within the range of scales of usual hydrological interest it is further shown that the theoretically derived form of the variance as a function of scale is incompatible with power law relations usually assumed for the second order statistical moment within multiscaling approaches to rainfall modeling predictions of the theoretical derivations are validated by use of observations representing a wide variety of resolutions from 2 s to 1 hour in time and 4 km in space climates and measurement instruments the validation shows a good agreement between observations and the theoretically predicted forms of the variance as a function of aggregation the proposed theoretical framework thus provides a useful context for rainfall analysis modeling and downscaling in space and time and suggests careful reexaminations of usual multiscaling assumptions

elliptically contoured random fields in space and time elliptically contoured random fields are natural extensions of the gaussian random field and may or may not have second order moments a second order elliptically contoured random field is determined by its mean and covariance functions just like the gaussian one this paper proposes a class of covariance functions for second order elliptically contoured random fields by using the nonnegative mixture method which possess the following properties nonseparable nonstationary but with the stationary case as a special case and allowing for both positive and negative correlation some long range dependent models are also derived

advection in chaotically time dependent open flows the passive advection of tracer panicles is considered in open two dimensional incompressible flows with chaotic time dependence as illustrative examples we investigate flows produced by chaotically moving ideal point vortices the advection problem can be seen as a chaotic scattering process in a chaotically driven hamiltonian system studying the motion of tracer ensembles we present numerical evidence for the existence of a bounded chaotic set containing infinitely many aperiodic trajectories never leaving the mixing region of the flow these ensembles converge to filamental patterns which however do not follow self similar scaling nevertheless they possess a fractal dimension after averaging over several finite time realizations of the flow we propose random maps as simple models of the phenomenon

componentwise stability of 1d and 2d linear discrete time systems this paper deals with a special type of asymptotic exponential stability namely componentwise asymptotic exponential stability for 1 d and 2 d linear discrete time systems the main motivation for these results is the need particularly felt in the evaluation in a more detailed manner of the dynamical behaviour of 1 d and 2 d linear discrete time systems necessary and sufficient conditions for componentwise asymptotic exponential stability are given

on parameter estimation of temporal rainfall models characteristics and moment estimators of temporal rainfall models such as poisson rectangular pulse prp neyman scott white noise nswn and neyman scott rectangular pulse nsrp are investigated it is shown that prp and nswn have a correlation structure like that of an autoregressive moving average arma 1 1 model whereas the nsrp has a dependence structure like that of an arma 2 2 the admissible regions of lag 1 and lag 2 autocorrelations are derived to demonstrate that in general they are more restricted than their arma counterparts an additional property denoted as variance ratio which is intimately related to the scale of fluctuation of a process is defined and used for model comparison the bias and mean square error properties of the moment estimators are investigated with emphasis on the nswn model for which it is suggested that the temporal scale t defined by t 1 provides the most efficient estimators of the parameters parameter defines the arrival of rain bursts relative to the origin of storm systems all three models are fitted to an extensive data set covering hourly precipitation data at 38 stations in northeastern colorado the correlation and variance ratio plots are used to select the appropriate model for each month however it is shown that the diurnal periodicity of storm occurrence is predominant during the summer months which is a characteristic not built into any of the temporal models used here

scaling limits and self similarity in precipitation fields it has been recognized in the literature on rainfall models in hydrology that the exact calculations of probabilities for models which incorporate cluster structure is typically not possible consequently one is led to considerations of limiting approximations however because of the presence of dependencies and or fat tails in several of the current models the nature of the approximations involves some important issues for both theoretical as well as experimental consideration in precipitation research the objective of the present paper is to systematically describe these issues as they pertain to rainfall models some specific large scale approximations are given for models of the type currently in the literature

a markov renewal model for rainfall occurrences a probabilistic model for the temporal description of daily rainfall occurrences at a single location is presented by defining an event as a day with measurable precipitation the model is cast into the discrete time point process framework in the proposed model the sequence of times between events is formed by sampling from two geometric distributions according to transition probabilities specified by a markov chain the model belongs to the class of markov renewal processes and exhibits clustering relative to the independent bernoulli process as a special case it reduces to a renewal model with a mixture distribution for the interarrival times the rainfall occurrence model coupled with a mixed exponential distribution for the nonzero daily rainfall amounts was applied to the daily rainfall series for snoqualmie falls washington and was successful in preserving the short term structure of the occurrence process as well as the distributional properties of the seasonal rainfall amounts

probabilistic representation of the temporal rainfall process by a modified neyman scott rectangular pulses model parameter estimation and validation the capability of the neyman scott clustered stochastic point process model of rainfall to preserve various observed statistics is considered randomization of the cell duration parameter from storm to storm is shown to considerably improve the wet dry period joint distribution and extreme value statistics a simple procedure for parameter estimation is introduced and applied

the relationship between a continuous time renewal model and a discrete markov chain model of precipitation occurrence a relationship between a continuous wet dry renewal model with gamma distributed dry intervals and a markov chain model for daily rainfall is derived the relationship requires numerical evaluation for the general case but closed form solutions are available for the markov process case in which dry intervals are exponentially distributed the markov process equations are equivalent to those derived by j r green 1964 for first order transition probabilities but are presented in a simplified nonrecursive form the markov process relationships are examined for convergence and accuracy with simulation studies and comparison to observed rainfall records the markov process model is shown to provide a good representation in certain parts of the united states in other areas where the markov property is inappropriate due to event clustering or other phenomena the gamma model provides an improved characterization of the relationship between continuous and discrete rainfall occurrence

stochastic perturbations in open chaotic systems random versus noisy maps we investigate the effects of random perturbations on fully chaotic open systems perturbations can be applied to each trajectory independently white noise or simultaneously to all trajectories random map we compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time dependent conditionally map invariant measure for the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map in random maps we show that the escape rate ensuremath kappa and dimensions d of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation we discuss the accuracy bias and precision variance of finite size estimators of ensuremath kappa and d and show that the improvement of the precision of the estimations with the number of trajectories n is extremely slow ensuremath propto 1 mathrm ln n we also argue that the finite size d estimators are typically biased general theoretical results are combined with analytical calculations and numerical simulations in area preserving baker maps

seasonal generalized exponential probability models with application to interstorm and storm durations a simple generalization of the exponential probability model is presented which provides a flexible method for identifying and fitting probability distributions when coupled with likelihood based estimation this approach enables a parsimonious description of seasonal dependence using harmonic functions a case study involving modeling of interstorm and storm durations for several australian state capitals illustrates practical issues in calibration and identification because the rainfall data were stored in a binned format consisting of rainfall depths accumulated over fixed time intervals the start and end times of storms were known only to the resolution of the fixed time interval a likelihood function is developed which properly makes use of such information likelihood ratio statistics along with monthly distribution plots are used to select the number of harmonics necessary to model seasonal dependence it is shown for melbourne that a single harmonic adequately describes seasonal dependence thereby reducing the number of parameters from 48 to 6