generalized hydraulic geometry derivation based on a multiscaling formalism 1 relationships between channel characteristics e g mean depth water surface width mean velocity and discharge known as hydraulic geometry hg have been extensively used by hydrologists and geomorphologists since the seminal work of leopold and maddock 1953 on the basis of recent empirical evidence that the parameters of at site hg depend systematically on the contributing area scale and that the parameters of downstream hg depend on the frequency of discharge we propose a multiscaling formalism within which to model and interpret both at site and downstream hg in a homogeneous region in particular we postulate and test multiscaling models for cross sectional area and discharge and derive generalized hg relationships that explicitly account for scale frequency dependence the multiscaling formalism is tested in several basins in oklahoma and kansas for drainage areas ranging from 2 to 20 000 km2 and shows good agreement with the data to quantify the effects that scale dependence in hg has on the hydrologic response of a basin a geomorphologic nonlinear cascade of reservoirs model has been used to compute attributes of a representative hydrologic response function for various levels of catchment averaged effective rainfall and different basin orders the numerical experiment shows substantial differences in hydrologic response when using classical versus generalized hg finally a preliminary effort is reported to generalize even further the hg relationships such that they can account for deviations from a single power law e g consideration of two different power laws in low and high flow regimes through the introduction of a bivariate mixed multiscaling framework

on mass response functions field transport of reactive solute species is investigated through a class of stochastic models here termed mass response functions mrfs which incorporate simplified concepts of chemical physical nonequilibrium kinetics in the formulation of transport by travel time distributions mrfs are probability density functions pdfs associated with solute particles travel time within transport volumes the theory hinges on recent advances in modeling transport of solutes in groundwater and in basin scale transport volumes and links the approaches of surface hydrologists with recent subsurface transport models the relationship between mrfs and the theory of solute transport by continuous motions is investigated it is found that mrfs extend the basic formulation of transport of inert solutes to a particular case of sorption process the relationship between mrfs and the basic differential convection dispersion equation incorporating linear sorption is also investigated it is found here that not only are transfer functions of solutes consistent with any mechanistic three dimensional 3 d model of convection dispersion but also that they are under limit conditions the product of the travel time distribution of the carrier flow with a bounded continuous function the latter is the solution to an initial value problem which results from solving the general 3 d differential equations of convection dispersion with sorption under some simplifying assumptions and formally coincides with the resident concentration included as an assumption in the original mrf formulation travel time distributions and mrfs underlain by statistical constraints rather than by dynamical models are proposed non gaussian distributions are studied by statistical mechanical tools and are found to represent the norm rather than the exception in this formulation of transport of reactive solutes the concepts are applied to a field study and are shown to yield reliable models of solute migration in nonpoint pollution problems

on the main channel length magnitude formula for random networks a solution to moon s conjecture this is a technical note which solves moon s conjecture for the exact asymptotic formula for the expected length of the longest path in shreve s random network for large magnitude source number

stochastic model of the width function a new class of probabilistic models of the width function based on so called iterated random pulse irp processes is proposed irp processes reproduce the main characteristics of empirical width functions nonnegativity nonstationarity and power law decay of the spectrum and require few and easily accessible parameters irp models are based on a simple conceptualization of the geometrical structure of river basins and exploit in a natural way the self similarity of natural channel networks a result that is derived from the irp representation is that the exponent of hack s law l a and the exponent of the power spectral density of the width function s are related as 1 empirical values of are typically in the range 1 8 2 0 and are consistent with this theoretical result and the usual range of

distributed low overhead schemes for multi stream mimo interference channels our aim in this paper is to propose fully distributed schemes for transmit and receive filter optimization the novelty of the proposed schemes is that they only require a few forward backward iterations thus causing minimal communication overhead for that purpose we relax the well known leakage minimization problem and then propose two different filter update structures to solve the resulting nonconvex problem though one leads to conventional full rank filters the other results in rank deficient filters that we exploit to gradually reduce the transmit and receive filter rank and greatly speed up the convergence furthermore inspired from the decoding of turbo codes we propose a turbo like structure to the algorithms where a separate inner optimization loop is run at each receiver in addition to the main forward backward iteration in that sense the introduction of this turbo like structure converts the communication overhead required by conventional methods to computational overhead at each receiver a cheap resource allowing us to achieve the desired performance under a minimal overhead constraint finally we show through comprehensive simulations that both proposed schemes hugely outperform the relevant benchmarks especially for large system dimensions

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

approximations on the peano river network application of the horton strahler hierarchy to the case of low connections a network analysis is used to investigate the low connections of natural river channels at the basin scale the river networks are analyzed according to the horton strahler hierarchy we propose a quantitative criterion for the average junction degree as a function of a fixed hierarchical order of the network and independent of the usual scaling laws the numerical results of this analysis are compared with exact results of the peano river network showing differences of the order of 10 ensuremath 3 this aspect is especially relevant for the characterization of transport and diffusion processes at the basin scale

on the main channel length area relationship for channel networks motivated by the empirical relationship between the main channel length and the basin area for river networks a similar equation is derived analytically using the random model postulates and the additional assumption that link lengths have a common exponential probability distribution specifically it is shown that in networks with large magnitude m the main channel length l m and the magnitude are related by l m 2 2 where is mean link length this result is utilized to explain another empirical relationship between the distance to the center of gravity lc m and the main channel length lc m l m 2 the observed deviations of the random model prediction regarding the main channel length from empirical observations is used to discuss some important open problems

rotation invariant texture classification using a complete space frequency model a method of rotation invariant texture classification based on a complete space frequency model is introduced a polar analytic form of a two dimensional 2 d gabor wavelet is developed and a multiresolution family of these wavelets is used to compute information conserving microfeatures from these microfeatures a micromodel which characterizes spatially localized amplitude frequency and directional behavior of the texture is formed the essential characteristics of a texture sample its macrofeatures are derived from the estimated selected parameters of the micromodel classification of texture samples is based on the macromodel derived from a rotation invariant subset of macrofeatures in experiments comparatively high correct classification rates were obtained using large sample sets

nash model relation to horton order ratios the nash model of the instantaneous unit hydrograph iuh is parameterized in terms of horton order ratios of a catchment on the basis of a geomorphologic model of catchment response the shape parameter of the nash model is found to depend on horton s numbers ra rb and rl of a catchment thus catchment geomorphology can provide a synthesis of the shape of the hydrologic response the scale parameter of the nash model results to be a time varying character depending on both geomorphology and average streamflow velocity along the stream network thus the time scale of the iuh could account for the variability of the hydrologic response for different storms and throughout a storm

statistics of lattice animals the authors investigate the large n behaviour of the number of lattice animals with n vertices having alpha cycles per vertex they prove concavity and continuity properties for the corresponding growth constant and in particular show that lattice trees are exponentially scarce in the set of lattice animals they also consider the corresponding generating function and prove a number of theorems which bound it and set other limits on its possible behaviour

interference alignment algorithms for the k user constant mimo interference channel this paper considers the degrees of freedom dof for a k user multiple input multiple output mimo m n interference channel using interference alignment ia a new performance metric for evaluating the efficacy of ia algorithms is proposed which measures the extent to which the desired signal dimensionality is preserved after zero forcing the interference at the receiver inspired by the metric two algorithms are proposed for designing the linear precoders and receive filters for ia in the constant mimo interference channel with a finite number of symbol extensions the first algorithm uses an eigenbeamforming method to align sub streams of the interference to reduce the dimensionality of the interference at all the receivers the second algorithm is iterative and is based on minimizing the interference leakage power while preserving the dimensionality of the desired signal space at the intended receivers the improved performance of the algorithms is illustrated by comparing them with existing algorithms for ia using monte carlo simulations

reaction diffusion wave fronts on comblike structures from the hamilton jacobi formalism an explicit expression for the speed of wave front propagation along the backbone of comblike structures is obtained this expression through the waiting time distribution function takes into account the number of sites and their distribution in the secondary branches our theoretical results are supported by numerical simulations of the reaction random walk process on the structure finally a more complex situation such as the peano basin structure is also considered both theoretically and numerically exhibiting a good agreement too

assessment of the instantaneous unit hydrograph derived from the theory of topologically random networks an instantaneous unit hydrograph iuh based on the theory of topologically random networks topological iuh is evaluated in terms of sets of basin characteristics and hydraulic parameters hydrographs were computed using two linear routing methods for each of two drainage basins in the southeastern united states and are the basis of comparison for the topological iuh s elements in the sets of basin characteristics for the topological iuh s are the number of first order streams only n or the nuber of sources together with the number of channel links in the topological diameter n d the hydraulic parameters are values of the celerity and diffusivity constant sensitivity analyses indicate that the mean celerity of the internal links in the network is the critical hydraulic parameter for determining the shape of the topological iuh while the diffusivity constant has minimal effect on the topological iuh asymptotic results source only indicate the number of sources need not be large to approximate the topological iuh with the weibull probability density function

communication over mimo x channels interference alignment decomposition and performance analysis in a multiple antenna system with two transmitters and two receivers a scenario of data communication known as the x channel is studied in which each receiver receives data from both transmitters in this scenario it is assumed that each transmitter is unaware of the other transmitter s data noncooperative scenario this system can be considered as a combination of two broadcast channels from the transmitters points of view and two multiple access channels from the receivers points of view taking advantage of both perspectives two signaling schemes for such a scenario are developed in these schemes some linear filters are employed at the transmitters and at the receivers which decompose the system into either two noninterfering multiple antenna broadcast subchannels or two noninterfering multiple antenna multiple access subchannels the main objective in the design of the filters is to exploit the structure of the channel matrices to achieve the highest multiplexing gain mg it is shown that the proposed noncooperative signaling schemes outperform other known noncooperative schemes in terms of the achievable mg in particular it is shown that in some specific cases the achieved mg is the same as the mg of the system if full cooperation is provided either between the transmitters or between the receivers in the second part of the paper it is shown that by using mixed design schemes rather than decomposition schemes and taking the statistical properties of the interference terms into account the power offset of the system can be improved the power offset represents the horizontal shift in the curve of the sum rate versus the total power in decibels

on the fractal interpretation of the mainstream length drainage area relationship the exponent of the mainstream length drainage area relationship is interpreted as a fractal quantity superseding the previous allometric interpretation the fractal interpretation is based on the assumption that cartographic generalization is applied evenly to all map scales twenty three drainage basins of the eaton river quebec canada were delineated from topographic maps at three different scales 1 20 000 1 50 000 and 1 125 000 the exponent of the length area relation is much lower 0 546 at the largest scale than at the smallest scales 0 65 and its value corresponds to that obtained from a richardson analysis of 10 interior stream segments at the 1 20 000 map scale the exponent is entirely fractal the larger exponent values obtained at the smallest scales exceed the fractal value and incorporate an allometric component this component is not functional however and it merely reflects the generalization process of cartographic abstraction of stream heads as scale is reduced the fractal dimension of streams should not be inferred from the exponent of the length area relation because its value is likely to be scale dependent

application of directed lattice animal theory to river networks it is shown that the theory of directed lattice animals is a good candidate for determining the exponents which describe river networks

best linear unbiased design hyetograph under the conditions of system linearity and stationarity of the precipitation process we derive the hyetograph associated with any given flood discharge q using best linear unbiased estimation blue theory the blue hyetograph depends explicitly on the correlation characteristics of the rainfall process and the instantaneous unit hydrograph iuh of the basin when the correlation time of the rainfall process is short relative to the dispersion of the iuh the shape of the blue hyetograph is the mirror image of the iuh itself in the opposite case the design hyetograph is the mirror image of the rainfall correlation function the procedure is extended from lumped to semidistributed and distributed basins in the latter case we obtain simultaneous design hyetographs for each subbasin or each site in the basin the analytical findings and the possibility of regionalization are verified using synthetic and recorded rainfall time series with excellent agreement in both cases because of its simplicity robustness and theoretical basis the proposed procedure should be appealing for practical use particularly when not only the peak but the entire flood hydrograph is required

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

on feasibility of interference alignment in mimo interference networks we explore the feasibility of interference alignment in signal vector space based only on beamforming for k user mimo interference channels our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry it is well known e g from bezout s theorem that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables following this intuition we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables rigorous connections between feasible and proper systems are made through bernshtein s theorem for the case where each transmitter uses only one beamforming vector the multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems in this case we show that the connection between feasible and proper systems can be further strengthened since the equivalency between feasible and proper systems does not always hold by including standard information theoretic outer bounds in the feasibility analysis

trees networks and hydrology this paper reviews theoretical and observational material on form and function of natural networks appeared in somewhat disparate contexts from physics to biology whose study is related to hydrologic research moving from the exact result that drainage network configurations minimizing total energy dissipation are stationary solutions of the general equation describing landscape evolution we discuss the properties and the dynamic origin of the scale invariant structure of river patterns and its relation to optimal selection we argue that at least in the fluvial landscape nature works through imperfect searches for dynamically accessible optimal configurations and that purely random or deterministic constructs are clearly unsuitable to properly describe natural network forms we also show that optimal networks are spanning loopless configurations only under precise physical requirements that arise under the constraints imposed by continuity in the case of rivers every spanning tree proves a local minimum of total energy dissipation this is stated in a theorem form applicable to generic networks suggesting that other branching structures occurring in nature e g scale free and looping may possibly arise through optimality to different selective pressures we thus conclude that one recurrent self organized mechanism for the dynamic origin of fractal forms is the robust strive for imperfect optimality that we see embedded in many natural patterns chief and foremost hydrologic ones copyright 2006 by the american geophysical union

gibbs distribution on drainage networks we propose a spatial stochastic model for drainage networks defined on a square lattice of points the probability of a particular spanning tree s draining a given basin represented by a set of lattice points is proportional to exp h s where is a parameter to be estimated and h s is defined to be the difference between total flow distance to the outlet through the drainage tree and total distance along shortest paths to the outlet thus h s is a global measure of sinuosity of the channels constituting the drainage tree this probability distribution on trees is known as gibbs distribution and is well known in statistical mechanical contexts the distribution with 0 a uniform distribution has been studied in a previous paper by the authors and was found to yield networks that are too sinuous in this paper is allowed to be greater than zero which produces more realistic networks that are not as sinuous the question of spatial variability of is addressed from an equilibrium point of view using ideas from statistical mechanics to examine the question of spatial variability the parameter is defined in a bayesian sense and allowed to vary randomly from basin to basin in a case study we express in terms of a log linear model with three independent variables pixel size grid spacing drainage area and average channel slope we then estimate the parameters of this model with a set of 50 drainage trees obtained using digital elevation data for willow creek in montana parameter estimates indicate that tends to vary directly with slope and inversely with drainage area which suggests heterogeneities within larger basins that are not adequately accounted for by gibbs distribution with a constant parameter value

hortonian scaling and effect of cutoffs on statistics of self affine river networks 1 the planar structure of river networks exhibits fractal properties in particular available data indicate a tendency for drainage areas a to be distributed according to a power law their boundaries and main channels to form self affine curves the characteristic lengths of drainage areas to be independently self affine in the one dimensional space z of total channel length rendering the network elongated and anisotropic the inverse network density a z to be either constant or self affine in z depending on whether or not the network fills the two dimensional space of a and as found in a recent study areas of given size vegetative cover and mean steady state soil moisture weighted by their flow distance from the basin outlet to be self affine in the one dimensional space of this distance various theoretical and semiempirical relationships have been proposed among exponents defining some of these and other scale dependencies expressions have been proposed for ensemble moments of a and some length measures associated with finite size river basins we present a new model that views any self affine basin property y x as belonging to an infinite hierarchy of mutually uncorrelated statistically homogeneous random functions defined on elementary subbasins each of which is characterized by a unique integral spatial correlation scale we cite a mathematically rigorous hydrologic rationale for our model and use it in conjunction with published scaling relations to obtain the probability density function of to derive analytical expressions for ensemble analogs of horton s scaling laws to deduce from them that streams of any horton strahler order are associated with integral scales 1 where the ratio 1 is a constant independent of to develop analytical expressions relating statistical moments of y x to arbitrary lower and upper cutoff scales that may but need not be taken to represent data support and maximum watershed size to describe ways of estimating the corresponding parameters and to provide a theoretical basis for the heretofore unexplained observation that self affine amplitude fluctuations of basin boundaries and main channels as measured by their variance having a common hurst exponent are larger in the former than in the latter

on the use of digital elevation model data for hortonian and fractal analyses of channel networks a common method of channel network extraction from digital elevation model dem data is based on specification of a threshold area ath that is the minimum support area required to drain to a point for a channel to form current efforts to predict ath from dem data are inconclusive and usually an arbitrary constant ath value is chosen for channel network extraction in this paper we study the effects of threshold area selection on the morphometric properties such as drainage density length of drainage paths and external and internal links and scaling properties such as horton s laws and fractal dimension of a channel network we also study the related problem of dem data resolution and its effect on estimation of scaling properties the results indicate that morphometric properties vary considerably with ath and thus values reported without their associated ath are meaningless and should be used in hydrologic analysis with caution also the completeness of a channel network in terms of having the outlet stream flowing directly into a higher order stream is found to depend on ath in a random unpredictable way even if only the complete channel networks are used in the analysis the statistical variability of scaling properties estimates due to ath selection is significant and can be of comparable size to the variability due to dem resolution and variability between estimates of different river networks our analysis highlights the need to carefully study the problem of network source representation or channel initiation scale from dems which will point to an appropriate ath for channel network extraction and estimation of morphometric properties

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

relation between grid channel and peano networks in high resolution digital elevation models the topological interconnection between grid channel and peano networks is investigated by extracting grid and channel networks from high resolution digital elevation models of real drainage basins and by using a perturbed form of the equation describing how the average junction degree varies with horton strahler order in peano networks the perturbed equation is used to fit the data observed over the hortonian substructures of real networks the perturbation parameter denoted as uniformity factor is shown to indicate the degree of topological similarity between hortonian and peano networks the sensitivities of computed uniformity factors and drainage densities to grid cell size and selected threshold for channel initiation are evaluated while the topological relation between real and peano networks may not vary significantly with grid cell size these networks are found to exhibit the same drainage density only for specific grid cell sizes which may depend on the selected threshold for channel initiation

on self similarity and self affinity of drainage basins an approach is proposed which allows consideration from common positions the self similar and self affine properties of river drainage basins and which introduces respective characteristics of their fractal structure horton s 1945 model has been assumed as the basis of this study it is shown that self similarity of river drainage basins is characterized by the self similarity dimension ds ln rb ln rl in the case of self affinity two scaling indices dl ln rb ln rl and dw ln rb ln ra rl are required the combinations of which determine the hurst h dl dw index characterizing the degree of drainage basins self affinity and the lacunary dimension dg 2dldw dl dw characterizing the degree of lacunarity or noncompactness of drainage basins on the basis of analysis of published data from rivers in the united states italy russia and romania the conclusion is drawn that river drainage basins are generally noncompact dg 2 and self affine h 1 fractal objects besides for a quantitative description of river drainage basins a new parameter is introduced which characterizes the degree to which a large catchment is filled with smaller ones

fractal relation of mainstream length to catchment area in river networks institute of hydraulics university of genoa italymandelbrot s 1982 hypothesis that river length is fractal has been recently substantiated byhjelmfelt 1988 using eight rivers in missouri the fractal dimension of river length d is derived herefrom the horton s laws of network composition this results in a simple function of stream length andstream area ratios that is d max 1 2 log rl og ra three case studies are reported showing thisestimate to be coherent with measurements of d obtained from map analysis the scaling properties ofthe network as a whole are also investigated showing the fractal dimension of river network d todepend upon bifurcation and stream area ratios according to d min 2 2 log rb log ra theseresults provide a linkage between quantitative analysis of drainage network composition and scalingproperties of river networks

wavelet based texture retrieval using generalized gaussian density and kullback leibler distance we present a statistical view of the texture retrieval problem by combining the two related tasks namely feature extraction fe and similarity measurement sm into a joint modeling and classification scheme we show that using a consistent estimator of texture model parameters for the fe step followed by computing the kullback leibler distance kld between estimated models for the sm step is asymptotically optimal in term of retrieval error probability the statistical scheme leads to a new wavelet based texture retrieval method that is based on the accurate modeling of the marginal distribution of wavelet coefficients using generalized gaussian density ggd and on the existence a closed form for the kld between ggds the proposed method provides greater accuracy and flexibility in capturing texture information while its simplified form has a close resemblance with the existing methods which uses energy distribution in the frequency domain to identify textures experimental results on a database of 640 texture images indicate that the new method significantly improves retrieval rates e g from 65 to 77 compared with traditional approaches while it retains comparable levels of computational complexity

directed recursion models for fractal growth fractals constructed by recursive processes are introduced to model growth phenomena these fractals are simultaneously directed and self similar in analogy with patterns growing under diffusion limited conditions the multifractal nature of the harmonic measure associated with laplacian interfaces is qualitatively interpreted using the models calculation of the largest singularity exponent allows one to make conclusions about the behaviour of diffusion limited aggregates

regional flood probabilities 1 the t year annual maximum flood at a site is defined to be that streamflow that has probability 1 t of being exceeded in any given year and for a group of sites the corresponding regional flood probability rfp is the probability that at least one site will experience a t year flood in any given year the rfp depends on the number of sites of interest and on the spatial correlation of flows among the sites we present a monte carlo method for obtaining the rfp and demonstrate that spatial correlation estimates used in this method may be obtained with rank transformed data and therefore that knowledge of the at site peak flow distribution is not necessary we examine the extent to which the estimates depend on specification of a parametric form for the spatial correlation function which is known to be nonstationary for peak flows it is shown in a simulation study that use of a stationary correlation function to compute rfps yields satisfactory estimates for certain nonstationary processes application of asymptotic extreme value theory is examined and a methodology for separating channel network and rainfall effects on rfps is suggested a case study is presented using peak flow data from the state of washington for 193 sites in the puget sound region it is estimated that a 100 year flood will occur on the average every 4 5 years

exploiting image indexing techniques in dct domain abstract this paper is concerned with the indexing and retrieval of images based on features extracted directly from the jpeg discrete cosine transform dct domain we examine possible ways of manipulating dct coefficients by standard image analysis approaches to describe image shape texture and color through the mandala transformation our approach groups a subset of dct coefficients to form ten blocks each block represents a particular frequency content of the original image two blocks are used to model rough object shape nine blocks to describe subband properties and one block to compute color distribution as a result the amount of data used for processing and analysis is significantly reduced this can lead to simple yet efficient ways of indexing and retrieval in a large scale image database experimental results show that our proposed approach offers superior indexing speed without significantly sacrificing the retrieval accuracy

degrees of freedom of the k user m times n mimo interference channel we provide inner bound and outer bound for the total number of degrees of freedom of the k user multiple input multiple output mimo gaussian interference channel with m antennas at each transmitter and n antennas at each receiver if the channel coefficients are time varying and drawn from a continuous distribution the bounds are tight when the ratio max m n min m n r is equal to an integer for this case we show that the total number of degrees of freedom is equal to min m n k if k r and min m n r r 1 k if k r achievability is based on interference alignment we also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some mimo interference channels with constant channel coefficients

unified feature analysis in jpeg and jpeg 2000 compressed domains retrieving images compressed by different algorithms typically involves a pre processing operation to decompress them onto the spatial domain from which features are extracted for further analysis our objective is to investigate common features that can be found in jpeg compressed and jpeg 2000 compressed images so that image indexing can be done directly in their respective compressed domains a fundamental difference between jpeg and jpeg 2000 is their transforms the former uses a block based discrete cosine transform bdct while the latter uses a wavelet transform wt direct comparison on bdct blocks and wt subbands cannot reveal their relationship by employing our proposed subband filtering model the bdct coefficients can be concatenated to form structures similar to wt subbands our theoretical studies show that the concatenated bdct and wt filters share common characteristics in terms of passband regions magnitude and energy spectra in particular their low pass filters are identical for haar wavelets and highly similar for other wavelet kernels despite the fact that compression can affect features that can be extracted our experimental results confirm that common features can always be extracted from jpeg and jpeg 2000 compressed domains irrespective of the values of the compression ratio and the types of wt kernels used as a result similar jpeg compressed and jpeg 2000 compressed images can be retrieved from one another without requiring a full decompression

statistical self similarity in river networks parameterized by elevation a stochastic theory is developed to describe spatial variability in link heights in topologically random river networks both the systematic and random spatial variability in link heights are reflected in a scaling invariance property of their probability distributions with drainage area serving as a scale parameter tests of theoretical predictions against empirical observations show that link height distributions exhibit scaling invariance under magnification or reduction of the scale parameter this invariance property is referred to as statistical self similarity it provides a fundamental theoretical basis for some existing empirical relationships on gradients and other river geometries in channel networks and points to important research directions in river basin hydrology

degrees of freedom for the mimo interference channel in this correspondence we show that the exact number of spatial degrees of freedom dof for a two user nondegenerate full rank channel matrices multiple input multiple output mimo gaussian interference channel with m1 m2 antennas at transmitters 1 2 and n1 n2 antennas at the corresponding receivers and perfect channel knowledge at all transmitters and receivers is min m1 m2 n1 m2 max m1 n2 max m2 n1 a constructive achievability proof shows that zero forcing is sufficient to achieve all the available dof on the two user mimo interference channel we also show through an example of a share and transmit scheme how the gains of transmitter cooperation may be entirely offset by the cost of enabling that cooperation so that the available dof are not increased

fat fractal scaling of drainage networks from a random spatial network model an alternative quantification of the scaling properties of river channel networks is explored using a spatial network model whereas scaling descriptions of drainage networks previously have been presented using a fractal analysis primarily of the channel lengths we illustrate the scaling of the surface area of the channels defining the network pattern with an exponent which is independent of the fractal dimension but not of the fractal nature of the network the methodology presented is a fat fractal analysis in which the drainage basin minus the channel area is considered the fat fractal random channel networks within a fixed basin area are generated on grids of different scales the sample channel networks generated by the model have a common outlet of fixed width and a rule of upstream channel narrowing specified by a diameter branching exponent using hydraulic and geomorphologic principles scaling exponents are computed for each sample network on a given grid size and are regressed against network magnitude results indicate that the size of the exponents are related to magnitude of the networks and generally decrease as network magnitude increases cases showing differences in scaling exponents with like magnitudes suggest a direction of future work regarding other topologic basin characteristics as potential explanatory variables

on hack s law hack s law is reviewed emphasizing its implications for the elongation of river basins as well as its connections with their fractal characteristics the relation between hack s law and the internal structure of river basins is investigated experimentally through digital elevation models it is found that hack s exponent elongation and some relevant fractal characters are closely related the self affine character of basin boundaries is shown to be connected to the power law decay of the probability of total contributing areas at any link and to hack s law an explanation for hack s law is derived from scaling arguments from the results we suggest that a statistical framework referring to the scaling invariance of the entire basin structure should be used in the interpretation of hack s law

what controls the width function shape and can it be used for channel network comparison and regionalization 1 the width function captures the essential features of the catchment s giuh s response this paper aims to identify the morphometric properties which control the shape of the width function and assess whether these properties can be used as similarity indices for catchment comparison a new deterministic iterative model of the width function is proposed on the basis of a conceptualization of the topology of the channel network and exploiting the morphometric characteristics of internal and external nodes tests are carried out on eleven french catchments and compared to the reference peano catchment results show that the morphometric properties of three main internal and external nodes such as the drained area the distance to the outlet and the position on the channel network are useful descriptors for modeling the width function and for representing the scaling properties of a channel network while the giuhs based on horton strahler ratios are strongly related to the method used to extract the channel network from the dem the new indices are independent of the method used they are sufficient descriptors to reproduce the main shape of the width function the peak the time to peak and the main properties such as non negativity non stationarity and power law decay of the spectrum they may be used to establish catchment typology to compare catchments and to classify the width function peaks for catchment regionalization

joint space decomposition and synthesis approach and achievable dof regions for k user mimo interference channels the paper proposes a new approach of joint space decomposition and synthesis jsds for deriving achievable degree of freedom dof region by interference alignment without symbol extension for k user mimo interference channel solving the interference alignment problem is to find the precoders that align desired signal and interference with disjoint subspaces at receivers a related property of these precoders is that the solution range for each one of them is a union of linear spaces termed transmit spaces based on this property the jsds approach is developed to derive transmit spaces that fulfill alignment conditions from this approach achievable dof regions for 2 and 3 user interference channels are obtained the region for the 2 user channel is shown to coincide with the existing result of the dof capacity of the channel for the k user k 3 case inner as well as outer bounds are given

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

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

pattern recognition experiments in the mandala cosine domain the problem of recognition of objects in images is investigated from the simultaneous viewpoints of image bandwidth compression and automatic target recognition a scenario is suggested in which recognition is implemented on features in the block cosine transform domain which is useful for data compression as well while most image frames would be processed by the automatic recognition algorithms in the compressed domain without need for image reconstruction this still allows for visual image classification of targets with poor recognition rates by human viewing at the receiving terminal it has been found that the mandala sorting of the block cosine domain results in a more effective domain for selecting target identification parameters useful features from this mandala cosine domain are developed based upon correlation parameters and homogeneity measures which appear to successfully discriminate between natural and man made objects the bhattacharyya feature discriminator is used to provide a 10 1 compression of the feature space for implementation of simple statistical decision surfaces gaussian and minimum distance classification imagery sensed in the visible spectra with a resolution of approximately 5 10 ft is used to illustrate the success of the technique on targets such as ships to be separated from clouds a data set of 38 images is used for experimental verification with typical classification results ranging from the high 80 s to low 90 percentile regions depending on the options choosen

new results for self similar trees with applications to river networks in a little known series of papers beginning in 1966 tokunaga introduced an infinite class of tree graphs based on the strahler ordering scheme as recognized by tokunaga 1984 these trees are characterized by a self similarity property so we will refer to them as self similar trees or ssts ssts are defined in terms of a generator matrix which acts as a blueprint for constructing different trees many familiar tree constructions are absorbed as special cases however in tokunaga s work an additional assumption is imposed which restricts from ssts to a much smaller class we will refer to this subclass as tokunaga s trees this paper presents several new and unifying results for ssts in particular the conditions under which ssts have well defined horton strahler stream ratios are given as well as a general method for computing these ratios it is also shown that the diameters of ssts grow like m where m is the number of leaves in contrast to many other tree constructions here need not equal 1 2 thus ssts offer an explanation for hack s law finally it is demonstrated that large discrepancies exist between the predictions of shreve s well known model and detailed measurements for large river networks while other ssts fit the data quite well other potential applications of the sst framework include diffusion limited aggregation dla lightning bronchial passages neural networks and botanical trees

fast algorithms for the discrete cosine transform several fast algorithms for computing discrete cosine transforms dcts and their inverses on multidimensional inputs of sizes which are powers of 2 are introduced because the 1 d 8 point dct and the 2 d 8 8 point dct are so widely used they are discussed in detail algorithms for computing scaled dcts and their inverses are also presented these have applications in compression of continuous tone image data where the dct is generally followed by scaling and quantization

unit hydrograph approximations assuming linear flow through topologically random channel networks the instantaneous unit hydrograph iuh of a drainage basin is derived in terms of fundamental basin characteristics z where parameterizes the link channel segment length distribution and is a vector of hydraulic parameters z is one of three basin topological properties n n d or n m where n is magnitude number of first order streams d is diameter mainstream length and m is order the iuh is derived based on assumptions that the links are independent and identically distributed random variables and that the network is a member of a topologically random population linear routing schemes including translation diffusion and general linear routing are used and constant drainage density is assumed by using n as the fundamental basin characteristics asymptotic for large n considerations lead to a weibull probability density function for the iuh with time to peak given by tp 2n where is mean link length and is a scalar hydraulic parameter usually average celerity this asymptotic iuh is identical for all linear routing schemes

a random spatial network model based on elementary postulates a model for generating random spatial networks that is based on elementary postulates comparable to those of the random topology model is proposed in contrast to the random topology model this model ascribes a unique spatial specification to generated drainage networks a distinguishing property of some network growth models the simplicity of the postulates creates an opportunity for potential analytic investigations of the probabilistic structure of the drainage networks while the spatial specification enables analyses of spatially dependent network properties in the random topology model all drainage networks conditioned on magnitude number of first order streams are equally likely whereas in this model all spanning trees of a grid conditioned on area and drainage density are equally likely as a result link lengths in the generated networks are not independent as usually assumed in the random topology model for a preliminary model evaluation scale dependent network characteristics such as geometric diameter and link length properties and topologic characteristics such as bifurcation ratio are computed for sets of drainage networks generated on square and rectangular grids statistics of the bifurcation and length ratios fall within the range of values reported for natural drainage networks but geometric diameters tend to be relatively longer than those for natural networks

can one gauge the shape of a basin this paper investigates the effects of geometrical factors characterizing the shape of a river basin on the features of its hydrologic response in particular we wonder if by measuring the hydrologic response i e gauging the salient geomorphic features of the basin can be recovered we argue that the basic structure of the channel network tends in ideal conditions to yield some universal characters of the width function w x defining the relative proportion of a contributing area at a distance x from the outlet w x exhibits low frequency features which are geometry dominated and high frequency features determined by recurrent aggregation patterns it is suggested that given the shape of the basin one can indeed forecast in a rational manner the main characters of the hydrologic response which are imprinted in reproducible width functions however the inverse problem i e the determination of the shape from the measure of the hydrologic response is less solidly defined because of the possible loss of irretrievable information induced by the dynamics of runoff processes therefore the question posed in the title cannot be solved in general although many elements for a general theory are seemingly established

homotopy continuation for spatial interference alignment in arbitrary mimo x networks in this paper we propose an algorithm to design interference alignment ia precoding and decoding matrices for arbitrary mimo x networks the proposed algorithm is rooted in the homotopy continuation techniques commonly used to solve systems of nonlinear equations homotopy methods find the solution of a target system by smoothly deforming the solution of a start system which can be trivially solved unlike previously proposed ia algorithms the homotopy continuation technique allows us to solve the ia problem for both unstructured i e generic and structured channels such as those that arise when time or frequency symbol extensions are jointly employed with the spatial dimension to this end we consider an extended system of bilinear equations that include the standard alignment equations to cancel the interference and a new set of bilinear equations that preserve the desired dimensionality of the signal spaces at the intended receivers we propose a simple method to obtain the start system by randomly choosing a set of precoders and decoders and then finding a set of channels satisfying the system equations which is a linear problem once the start system is available standard prediction and correction techniques are applied to track the solution all the way to the target system we analyze the convergence of the proposed algorithm and prove that for many feasible systems and a sufficiently small continuation parameter the algorithm converges with probability one to a perfect ia solution the simulation results show that the proposed algorithm is able to consistently find solutions achieving the maximum number of degrees of freedom in a variety of mimo x networks with or without symbol extensions further the algorithm provides insights into the feasibility of ia in mimo x networks for which theoretical results are scarce

multifractal formalism for fractal signals the structure function approach versus the wavelet transform modulus maxima method several attempts have been made recently to generalize the multifractal formalism originally introduced for singular measures to fractal signals we report on a systematic comparison between the structure function approach pioneered by parisi and frisch in 2 proceedings of the international school on turbulence and predictability in geophysical fluid dynamics and climate dynamics edited by m ghil r benzi and g parisi north holland amsterdam 1985 p 84 to account for the multifractal nature of fully developed turbulent signals and an alternative method we have developed within the framework of the wavelet transform analysis we comment on the intrinsic limitations of the structure function approach this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases we demonstrate that our method based on the wavelet transform modulus maxima representation works in most situations and is likely to be the ground of a unified multifractal description of self affine distributions our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations

scale invariance properties of the peak of the width function in topologically random networks some scaling properties of the topological width function for an infinite population of networks obeying the random model are analyzed a monte carlo procedure is applied to generate width functions according to the hypothesis of topological randomness the probability distributions of both peak and distance to peak of the topological width functions conditioned 1 on the network diameter and 2 on and parameter log 2 l log are studied the parameter can be considered a shape factor of the network indeed low values describe elongated networks while high values refer to fan like networks scale invariance for both random variables is established in the first case by using as a scale parameter also in the second case scale invariance is observed for both the peak and the distance to peak of the topological width function in particular the invariance property for the peak involves a scaling function which is directly related to the shape factor allowing determination of the statistical similarity between random networks indexed by the same then a coarse graining procedure is applied to a set of 15 000 width functions with 512 a scaling behavior of peaks of the original width function and aggregated ones is observed over a wide range of aggregation scales consequently a statistical self similarity for the peaks is also observed which involved the same related scaling function finally possible implication of the present results on the hydrologic response at the basin scale is discussed

waveguide a joint wavelet based image representation and description system data representation and content description are two basic components required by the management of any image database a wavelet based system called the waveguide which integrates these two components in a unified framework is proposed in this work in the waveguide system images are compressed with the state of the art wavelet coding technique and indexed with color texture and object shape descriptors generated in the wavelet domain during the encoding process all the content descriptors are extracted by machines automatically with a low computational complexity and stored with a low memory space extensive experiments are performed to demonstrate the performance of the new approach

statistical texture characterization from discrete wavelet representations we conjecture that texture can be characterized by the statistics of the wavelet detail coefficients and therefore introduce two feature sets 1 the wavelet histogram signatures which capture all first order statistics using a model based approach and 2 the wavelet co occurrence signatures which reflect the coefficients second order statistics the introduced feature sets outperform the traditionally used energy best performance is achieved by combining histogram and co occurrence signatures

probabilistic structure of the distance between tributaries of given size in river networks we analyze the distribution of the distances between tributaries of a given size or of sizes larger than a given area draining along either an open boundary or the mainstream of a river network by proposing a description of the distance separating prescribed merging contributing areas we also address related variables like mean or bankfull flow rates and channel and riparian area widths which are derived under a set of reasonable hydrologic assumptions the importance of such distributions lies in their ecological hydrologic and geomorphic implications on the spreading of species along the ecological corridor defined by the river network and on the propagation of infections due to water borne diseases particularly in view of exact theoretical predictions explicitly using the alongstream distribution of confluences carrying a given flow use is made here of real river networks suitably extracted from digital elevation models optimal channel networks and exactly solved tree like constructs like the peano and the scheidegger networks the results obtained redefine theoretically in a coherent and general manner and verify observationally the distribution function of the above distances and thus provide the general probabilistic structure of tributaries in river networks specifically we find that the probability of exceedence of the alongstream distance d of tributaries of size larger than a has the explicit form p d exp cd a h 1 h where c is a constant that depends on the choice of boundary conditions and h 1 is the hurst exponent

minimum energy and fractal structures of drainage networks this paper explores the similarities of digital elevation maps dems of natural river basins and optimal channel network ocn configurations obtained minimizing the total rate of energy expenditure in the system as a whole and in its parts striking similarities are observed for natural and optimal networks in their fractal aggregation structure and in certain multifractal structures found to be characteristic of river basins our results suggest upon critical assessment of the reliability of the identification of the attractor of the underlying dynamics implied by our optimality concepts that fractal structures are indeed possibly a product of least energy dissipation power laws emerging in the description of the distribution of aggregated quantities from both dems and ocns suggest a link with the framework of self organized criticality in the dynamics of natural channel network formation also the geomorphological description of ocns reveals surprising analogies with well known empirical or experimental results a comparison of peano s basins with ocns suggests that nature seems to reject the type of strict self similarity exhibited by peano s construct in favor of different shapes implying statistical self similarity not only because of chance acting through random conditions but also because of necessity as reflected by least energy expenditure considerations

reexamining the determination of the fractal dimension of river networks in analyzing the literature on the fractal nature of river networks one can recognize several points that require clarification or additional investigations for instance many interpretations with different conclusions have been given to the important empirical results found in the previous literature on river geomorphology similarly the argument concerning a possible more likely value of the fractal dimensions of river networks has not yet found a convincing answer in an attempt to shed more light on the topic particularly on the latter point we have reexamined the basic aspects related to selfsimilarity in river networks using uniquely the fundamental concepts of fractal geometry this approach leads to relations for the determination of the fractal dimensions which hold exactly for deterministic self similar trees and are compatible with related findings reported in literature results obtained from application of these relations to several river networks in southern italy indicate very low variability around the average values of about 1 7 1 5 and 1 1 for the fractal dimensions of the whole river network of its topological equivalent and of its individual streams respectively this outcome in addition to comparable results taken from the literature provides new arguments to the hypothesis that natural networks tend to have the same fractal dimensions

fast extraction of wavelet based features from jpeg images for joint retrieval with jpeg2000 images in this paper some fast feature extraction algorithms are addressed for joint retrieval of images compressed in jpeg and jpeg2000 formats in order to avoid full decoding three fast algorithms that convert block based discrete cosine transform bdct into wavelet transform are developed so that wavelet based features can be extracted from jpeg images as in jpeg2000 images the first algorithm exploits the similarity between the bdct and the wavelet packet transform for the second and third algorithms the first algorithm or an existing algorithm known as multiresolution reordering is first applied to obtain bandpass subbands at fine scales and the lowpass subband then for the subbands at the coarse scale a new filter bank structure is developed to reduce the mismatch in low frequency features compared with the extraction based on full decoding there is more than 72 reduction in computational complexity retrieval experiments also show that the three proposed algorithms can achieve higher precision and recall than the multiresolution reordering especially around the typical range of compression ratio

allometric relationships between traveltime channel networks convex hulls and convexity measures 1 the channel network s is a nonconvex set while its basin c s is convex we remove open end points of the channel connectivity network iteratively to generate a traveltime sequence of networks sn the convex hulls of these traveltime networks provide an interesting topological quantity which has not been noted thus far we compute lengths of shrinking traveltime networks l sn and areas of corresponding convex hulls c sn the ratios of which provide convexity measures cm sn of traveltime networks a statistically significant scaling relationship is found for a model network in the form l sn a c sn 0 57 from the plots of the lengths of these traveltime networks and the areas of their corresponding convex hulls as functions of convexity measures new power law relations are derived such relations for a model network are cm sn and cm sn in addition to the model study these relations for networks derived from seven subbasins of cameron highlands region of peninsular malaysia are provided further studies are needed on a large number of channel networks of distinct sizes and topologies to understand the relationships of these new exponents with other scaling exponents that define the scaling structure of river networks

a new monte carlo simulation for two models of self avoiding lattice trees in two dimensions by means of a new monte carlo sampling of a grand canonical ensemble we verify universality for the critical exponents and of two models of lattice trees constrained to be self avoiding on sites or on bonds the attrition constants are also obtained this algorithm a generalization of that recently proposed by berretti and sokal for random walks appears to optimize the critical slowing down in the scaling region systematic and statistical errors are carefully estimated

inference for a generaliaed gibbsian distribution channel networks we introduce a two parameter 0 and 1 gibbsian probability model to characterize the spatial behavior of channel networks this model is defined for trees draining basins on a square lattice the probability of a tree s is proportional to exp 0h s 1 where h s 1 is taken to be a summation over lattice points in the basin of a s 1 letting a s be the area upstream from but not including point procedures are developed for estimating the parameters given an actual network obtained for example from digital elevation data and for using bootstrapping to obtain confidence intervals for these estimates we also develop several goodness of fit tests for the model these procedures are applied to a set of data for 50 subnetworks from willow creek in montana the estimation algorithm converged successfully for 34 of these subnetworks for these 34 subnetworks the two parameter model gives in most cases a fit significantly better than a one parameter model with 1constrained to be 1 studied in a previous paper there is still however statistically significant lack of fit for as many as 13 subnetworks depending on which test is applied the parameter 1 averages 0 75 for the 34 subnetworks the function h s 1 in the gibbsian model may be given a statistical mechanical interpretation of energy associated with a configuration s other researchers have employed a similar definition of energy expenditure in drainage systems but have given physical arguments that the exponent 1 should be 0 50 we finally discuss the idea of defining optimal channel networks as those which minimize free energy in the gibbsian model it is shown that minimization of energy rather than free energy as other researchers have proposed is valid only under the assumption of a low temperature i e large 0 system our results indicate that this assumption would not be justified for willow creek we devote some discussion to the discrepancies between our approach and results and those of other researchers

exact analysis of the peano basin the peano basin is analysed as a deterministic model for river networks the fundamental distributions characterizing the real basins morphology can be explicitly calculated the recently proposed finite size scaling ansatz is tested apart from oscillatory amplitudes typical of deterministic fractals

horton s laws and the fractal nature of streams stream networks and catchment basins are characterized by numerous fractal dimensions the traditional one being the mainstream length as a function of length scale which we denote by d we consider an ideal horton system and define fractal dimensions based on 1 the subcatchment length area relationship which gives d1 2 the total length of streams as a function of scale which gives a network similarity dimension d2 3 the mainstream length area relationship d3 and 4 the total area of streams as a function of areal scale d4 these fractal dimensions are related to the values of the bifurcation ratio the stream length ratio and the stream area ratio their value should be indirectly estimated from these horton parameters an improved understanding of the relationships between the fractal dimension and horton s laws is obtained by defining an eigenarea of a stream of a particular horton order this is the area of the drainage basin which is not drained by streams of immediately lower order the variation of eigenareas with scale controls the value of d1 which equals d3 for hydrologically realistic systems in such cases d4 2 this paper assumes that river systems are hortonian in nature and that measured departures from hortonianity are due to statistical variability estimates of the horton parameters and of the fractal dimensions are therefore subject to uncertainty two sources of statistical variability are considered herein the first source of statistical variability is due to the fluctuations in stream numbers stream lengths and stream areas these fluctuations affect the values of the horton parameters which are traditionally estimated using a graphical technique we show that an analytical technique which corresponds to a constrained graphical technique is preferable the second source of variability that we consider arises from the fact that estimates of horton s parameters and fractal dimensions are always made on truncated systems because the particular scales of measurement determine the order of the catchment in a horton system it is straight forward to estimate d1 which is defined exactly at all scales direct attempts to estimate d2 and d3 as opposed to indirect attempts via the horton parameters are difficult because their true value is approached asymptotically in the limit of infinite order catchments

a note on fractal channel networks this paper studies the relation between the structure of river networks and the features of their geomorphologic hydrologic response a mathematical formulation of connectivity of a drainage network is proposed to relate contributing areas and the network geometry in view of the connectivity conjecture horton s bifurcation ratio r b tends for high values of strahler s order omega of the basin to the area ratio r a and horton s length ratio r l equals in the limit the single order contributing area ratio r a the relevance of these arguments is examined by reference to data from real basins well known empirical results from the geomorphological literature melton s law hack s relation moon s conjecture are viewed as a consequence of connectivity it is found that in hortonian networks the time evolution of contributing areas exhibits a multifractal behavior generated by a multiplicative process of parameter 1 r b the application of the method of the most probable distribution in view of connectivity contributes new inroads toward a general formulation of the geomorphologic unit hydrograph in particular generalizing its width function formulation a quantitative example of multifractal hydrologic response of idealized networks based on peano s construct for which r b r a 4 r l 2 closes the paper

river network fractal geometry and its computer simulation the hierarchical ordinal and statistical models of river networks are proposed their investigation has been carried out on the basis of river networks computer simulation as well as on empirical data analysis the simulated river networks display self similar behavior on small scales the fractal dimension d 1 52 and hurst s exponent h 1 0 and self affine behavior on large scales the lacunary dimension dg 1 71 h 0 58 similar behavior is also qualitatively characteristic for natural river networks for catchment areas from 142 to 63 700 km2 we obtained dg 1 87 and h 0 73 thus in both cases one finds a region of scales with self affine behavior h 1 and with dg 2 proceeding from fractal properties of the river networks the theoretical basis of scaling relationships l a and ae widely used in hydrology are given l and a denote the main river length the total length of the river network and catchment area respectively 1 1 h and e dg 2

area and width functions of river networks new results on multifractal properties 1 this paper investigates the multiscale statistical structure of the area and width functions of simulated and real river networks via state of the art wavelet based multifractal mf formalisms first several intricacies in performing mf analysis of these signals are discussed and a robust framework for accurate estimation of the mf spectrum is presented second it considers the following three questions 1 does the topology of river networks leave a unique signature on the mf spectrum of area and width functions 2 how different are the mf properties of commonly used simulated trees and those of real river networks and 3 are there differences between the mf properties of width and area functions and what can these tell us about the topology of hillslope versus channelized drainage patterns in a river basin the results indicate discrepancies between the statistical scaling of the area functions of real networks found to be multifractal with a considerable spread of local singularities and the most prevailing singularity ranging from 0 4 to 0 8 and that of several commonly used stochastic self similar networks found to be monofractal with a single singularity exponent h in the range of 0 5 0 65 moreover differences are found between the mf properties of width and area functions of the same basin these differences may be the result of distinctly different branching topologies in the hillslope versus channelized drainage paths and need to be further investigated