Parameter Estimation for a Model of Space‐Time Rainfall
TL;DR: In this article, the spatial distribution of daily rainfall in a 240-mi2 (621 km2) catchment in the Potomac River basin is modeled using a network of rainfall gages.
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.
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TL;DR: In this paper, 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.
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).
1,064 citations
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TL;DR: In this article, the chain-dependent process stochastic model of daily precipitation is extended to simultaneous simulation at multiple locations by driving a collection of individual models with serially independent but spatially correlated random numbers.
510 citations
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TL;DR: In this paper, a general class of continuous multiplicative processes is constructed to provide a theoretical framework for these observations, namely, the log-log linearity of moments with spatial scale and the concavity of corresponding slopes.
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.
499 citations
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TL;DR: This work shows how to characterize this variability of cloud and rain fields using scale invariant codimension functions which are exponents characterizing the probability distributions and forms a three parameter universality class for infrared and visible satellite cloud images.
Abstract: The extreme variability of cloud and rain fields poses serious problems in quantitative use of remotely sensed satellite and radar data. We show how to characterize this variability using scale invariant (sensor resolution independent) codimension functions which are exponents characterizing the probability distributions. These codimension functions in turn form a three parameter universality class. We review the properties of these multifractal measures and empirically evaluate the codimension functions as well as the universality classes for infrared and visible satellite cloud images using the new probability distribution/multiple scaling technique, refining previously published results and relating these to the established lognormal rain and cloud phenomenologies. We then show how to solve the radar observers' problem for multifractal radar reflectivity factors and to estimate the codimension function of rain from the radar. Finally, we reexamine some earlier (monofractal) analysis techniques in the light of our findings.
263 citations
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TL;DR: In this paper, the sensitivity of distributed hydrological models to different patterns that account for the spatial distribution of rainfall: spatially averaged rainfall or rainfall field was analyzed and three models were tested using different runoff production models: storm-runoff coefficient, complete or partial interception.
231 citations
References
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TL;DR: In this paper, a general framework is developed to estimate the variance of the sample long-term mean areal precipitation and mean wereal rainfall of a storm event, expressed as a function of correlation in time, correlation in space, length of operation of the network, and geometry of the gaging array.
Abstract: A methodology for the design of precipitation networks is formulated. The network problem is discussed in its general conception, and then focus is made on networks to provide background information for the design of more specific gaging systems. The rainfall process is described in terms of its correlation structure in time and space. A general framework is developed to estimate the variance of the sample long-term mean areal precipitation and mean areal rainfall of a storm event. The variance is expressed as a function of correlation in time, correlation in space, length of operation of the network, and geometry of the gaging array. The trade of time versus space is quantitatively developed, and realistic examples are worked out showing the influence of the network design scheme on the variance of the estimated values.
403 citations
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TL;DR: In this paper, a physically realistic stochastic representation of the ground level rainfall intensity field in space and time is presented, which is based on three-component point processes which possess the general features of the embedding of rain cells within small mesoscale areas within large mesoscales within synoptic storms.
Abstract: 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.
263 citations
01 Jan 1961
TL;DR: In the early part of the war years Pierre Masse organized a group for the purpose of studying optimal procedures of development and management of the French hydroelectric and steam power system.
Abstract: : During the early part of the war years Pierre Masse organized a group for the purpose of studying optimal procedures of development and management of the French hydroelectric and steam power system. The problems encountered by this group included the evaluation of probabilities of excessive discharges, the evaluation of probabilities of excessive droughts, as well as the development of optimal management procedures for the big and small hydroelectric reservoirs.
169 citations
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TL;DR: In this article, a non-stationary multidimensional rainfall generator is proposed to simulate historical storm exteriors and interiors, assuming the validity of Taylor's hypothesis of turbulence within the storm interior.
Abstract: Most existing rainfall models concentrate on storm exterior characteristics. Those that generate storm interiors generally do so at only one point. Very few models attempt to generate exterior and interior rainfall characteristics everywhere in space, and those that do have limiting assumptions of stationary behavior at all levels of storm activity. This work suggests a nonstationary multidimensional rainfall generator. The model, capable of simulating historical storm exteriors and interiors, assumes the validity of Taylor's hypothesis of turbulence within the storm interior. First-order statistics of storm interiors as well as the correlation in time and space of the storm interiors are preserved.
96 citations