bayesian estimation of uncertainty in runoff prediction and the value of data an application of the glue approach this paper addresses the problem of evaluating the predictive uncertainty of topmodel using the bayesian generalised likelihood uncertainty estimation glue methodology in an application to the small ringelbach research catchment in the vosges france the wide range of parameter sets giving acceptable simulations is demonstrated and uncertainty bands are presented based on different likelihood measures it is shown how the distributions of predicted discharges are non gaussian and vary in shape through time and with discharge updating of the likelihood weights using bayes equation is demonstrated after each year of record and it is shown how the additional data can be evaluated in terms of the way they constrain the uncertainty bands

homeomorphic conjugates of fuchsian groups by a fuchsian group we mean a discrete subgroup of m it may contain also orientation reversing elements usually a fuchsian group is thought to act on i but this is not a problem since the action on s can be uniquely extended to an action on 1 and later we might also consider a fuchsian group to act on 1 depending on the situation we are here concerned with the problem when is a group g ofhomeornorphisms of s topologically conjugate to a fuchsian group we propose that this happens if and only if g is a convergence group of s which means that it satisfies the convergence property this means that whenever gi g are distinct we can pass to a subsequence in such a way that for some a b c s

on constraining the predictions of a distributed model the incorporation of fuzzy estimates of saturated areas into the calibration process distributed hydrological models are generally overparameterized resulting in the possibility of multiple parameterizations from many areas of the parameter space providing acceptable fits to observed data in this study topmodel parameterizations are conditioned on discharges and then further conditioned on estimates of saturated areas derived from ers i synthetic aperture radar sar images combined with the in tan topographic index and compared to ground truth saturation measurements made in one small subcatchment the uncertainty associated with the catchment wide predictions of saturated area is explicitly incorporated into the conditioning through the weighting of estimates within a fuzzy set framework the predictive uncertainty associated with the parameterizations is then assessed using the generalized likelihood uncertainty estimation glue methodology it is shown that despite the uncertainty in the predictions of saturated area the methodology can reject many previously acceptable parameterizations with the consequence of a marked reduction in the acceptable range of a catchment average transmissivity parameter and of improved predictions of some discharge events

a hybrid method for calculation of ruelle pollicott resonances we present a numerical method for calculation of ruelle pollicott resonances of dynamical systems it constructs an effective coarse grained propagator by considering the correlations of multiple observables over multiple time steps the method is compared to the usual approaches to the example of the perturbed cat map and is shown to be numerically efficient and robust

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

jsj splittings for finitely presented groups over slender groups we generalize the jsj splitting of rips and sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups such classes include the class of all 2 ended groups and the class of all virtually z z groups the approach called track zipping is relatively elementary and differs from the rips sela approach in that it does not rely on the theory of r trees but rather on an understanding of certain embedded 1 complexes called patterns in a presentation 2 complex for the ambient group

comparison between weibull and gamma distributions to derive synthetic unit hydrograph using horton ratios the two parameter weibull distribution 2pwd similar to an instantaneous unit hydrograph iuh is parameterized in terms of horton catchment ratios on the basis of a geomorphologic model of catchment response for this the shape and scale parameters of the weibull distribution are expressed analytically in terms of horton s catchment ratios the two parameters of the iuh derived using nash s model which is a two parameter gamma distribution 2pgd are also expressed analytically in terms of horton ratios the performance of the proposed methods is tested for describing a synthetic unit hydrograph suh under limited data conditions a comparison is made with the unit hydrographs derived from the event data of two real catchments and with the existing geomorphological based 2pgd for developing suh given by rosso 1984 the sensitivity analysis of the 2pwd to the nondimensional parameter of the uh a product of peak discharge and time to peak shows to be more sensitive to the shape parameter a than the scale parameter b further examination to find any similarity between the behavior of 2pwd and 2pgd showed that a in 2pwd corresponds to the shape parameter n in the 2pgd and b behaves similar to the scale parameter k in the 2pgd finally practical applicability of the proposed approach to ungauged catchments is tested using field data less

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

random self similar river networks and derivations of generalized horton laws in terms of statistical simple scaling recent data analyses of a dozen or so large networks support the presence of simple scaling in the probability distributions of many topologic geometric and hydraulic geometric variables we have labeled them generalized horton laws in contrast to traditional horton laws which are stated only in terms of means these data sets also show that horton s bifurcation ratios are significantly larger than 4 which can not be predicted by the well known random model motivated by the need to understand these data analyses of large river networks within a common theoretical framework a new class of models with statistically self similar network topology is introduced the network construction process can be viewed as a statistical generalization of the type of a recursive process which generates many fractal geometrical objects unlike the well known random model our random channel networks do not belong to the class of binary galton watson stochastic branching processes the generalized horton laws are predicted in the limit of large horton strahler order as a scale parameter for a subclass of random self similar network models a derivation of horton s law of stream numbers as a consequence of the simple scaling of distributions of basin magnitude gives it a new theoretical significance a further subclass also exhibits tokunga self similarity which is an average property of the side tributary structure in river networks comparisons with empirical observations for three large basins analyzed here show that the generalized horton laws are well predicted by this new class of random self similar network models and analytically predicted relationships between topologic and geometric horton ratios are also seen in data

definition of new equivalent indices of horton strahler ratios for the derivation of the geomorphological instantaneous unit hydrograph 1 the horton strahler concept has been important in river basin geomorphology it describes scaling properties and identifies giuh but has also attracted criticisms because it depends on the threshold area s which is used to extract the channel network from digital elevation models dem on the position of the outlet and on the few number of ratios used to identify the scaling laws to overcome these limitations this paper proposes new indices independently of s and which have similar properties than the horton strahler ratios applications were conducted on seven french basins the methodology consists on analyzing the morphometric properties such as the drained area the number of sources and the length of the channel network as a function of s the results show that horton strahler ratios vary considerably with s particularly while the order length l is a staircase function of s results show that the channel network is articulated around two main nodes independent of s which are descriptors of l then we compared the properties of horton strahler laws to those obtained from self similarity analysis which enables to define new indices independent of s these indices are calculated automatically from dem have similar properties as the horton strahler ratios and are used to calculate the giuh independently of s the four new descriptors rbe rle rae and le which have similar properties as rb rl ra and l respectively are equivalent to those of horton strahler s laws and can be used as indicators of hydrological similarity for catchment comparison and regionalization

invariant morphometric properties of headwater subcatchments 1 the distinction between the channel network the headwater subcatchments and the lateral subcatchments plays an important role in distributed hydrological and ecohydrological applications this paper presents some newly found invariance properties of headwater and upstream subcatchments and shows that the invariant morphometric properties characterize only natural networks and virtual networks verifying optimal channel networks ocn properties but are not verified for virtual non ocns a model based on self affine properties was developed in order to calculate the number of headwater catchments n and the total upstream area of headwater catchments u as a function of the cutoff area a used to delineate streams for 18 french catchments between 43 and 116 450 km2 and for 4 virtual ocns results show that u a a0 with a0 being the catchment area is independent of a for 0 5 a 5 km2 and seems to be constant 0 29 0 03 for various shapes and sizes of channel networks and consequently can be considered as an invariant general descriptor of natural channel and virtual ocn networks on the contrary this is not the case when the approach is applied on six virtual non ocns moreover results show that the knowledge of six morphometric indices enable us to calculate both functions n a and u a for all values of a a0 these indices can be considered as geometric and topological properties of headwater and upstream subcatchments and are useful for studying the effects of cutoffs on self affine river networks or as similarity indices for channel network comparison

feasible optimality implies hack s law we analyze the elongation the scaling properties of drainage area with mainstream length in optimal channel networks ocns obtained through different algorithms searching for the minimum of a functional computing the total energy dissipation of the drainage system the algorithms have different capabilities to overcome the imprinting of initial and boundary conditions and thus they have different chances of attaining the global optimum we find that suboptimal shapes i e dynamically accessible states characterized by locally stationary total potential energy show the robust type of elongation that is consistently observed in nature this suggestive and directly measurable property is not found in the so called ground state i e the global minimum whose features including elongation are known exactly the global minimum is shown to be too regular and symmetric to be dynamically accessible in nature owing to features and constraints of erosional processes thus hack s law is seen as a signature of feasible optimality thus yielding further support to the suggestion that optimality of the system as a whole explains the dynamic origin of fractal forms in nature

a deterministic gamma type geomorphologic instantaneous unit hydrograph based on path types a geomorphological instantaneous unit hydrograph is obtained by dividing the whole catchment into subcatchments according to the different path types which water follows to the outlet of catchment each subcatchment produces at its own outlet an impulse response which is described by a two parameter n k gamma function in traveling to the outlet of whole catchment each impulse response undergoes a further pure translation with delay time thus producing at the outlet of the catchment a partial impulse response described by three parameters n k and summation of these partial impulse responses with area weights w forms the impulse response of whole catchment the parameters k and are estimated from catchment characteristics and the flow velocity of each flood event area weights w can be calculated from maps after defining the subcatchments the parameter n is estimated by optimization the proposed model is tested in two catchments in china and shows encouraging results

a similarity approach to predict landscape saturation in catchments 1 an investigation into landscape saturation has been carried out analytically using similarity parameters derived from topographic soil and climatic attributes three parameters an input index q z l ks convergence ratio cr and profile factor b define the extent of hillslope saturation from these a single dimensionless parameter has been derived to describe the occurrence of saturation in planar concave and convex profile hillslope shapes and with parallel convergent and divergent planforms the relationship can be represented by a single dimensionless curve the theory derived in this paper provides a basis for combining the individual parameters into a similarity criterion for saturation in landscapes the theory and ensuing similarity hypothesis are applied to three natural catchments near canberra australia and are tested against observed rainfall runoff data the results support the validity of the similarity approach there is a clear link between the distribution of hillslope similarity parameters and catchment saturation behavior

planar groups and the seifert conjecture we describe a number of characterisations of virtual surface groups which are based on the following result let g be a group and f be a field we show that if g is fp2 over f and if h2 g f thought of as a k vector space contains a 1 dimensional ginvariant subspace then g is a virtual surface group i e contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane in particular this applies to rational poincare duality groups we also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group we recover the result of mess which characterises such groups as groups which are quasiisometric to complete riemannian planes we also give a cohomological version of the seifert conjecture from which the topological seifert conjecture proven by tukia mess gabai casson and jungreis can be recovered via work of zieschang and scott

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

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

unified view of scaling laws for river networks scaling laws that describe the structure of river networks are shown to follow from three simple assumptions these assumptions are 1 river networks are structurally self similar 2 single channels are self affine and 3 overland flow into channels occurs over a characteristic distance drainage density is uniform we obtain a complete set of scaling relations connecting the exponents of these scaling laws and find that only two of these exponents are independent we further demonstrate that the two predominant descriptions of network structure tokunaga s law and horton s laws are equivalent in the case of landscapes with uniform drainage density the results are tested with data from both real landscapes and a special class of random networks

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

a reformulation of horton s laws for large river networks in terms of statistical self similarity the well known horton s laws are empirical observations on how the means of measurements for river networks and basins vary with horton strahler order it is now known that these laws are a consequence of an average sense self similarity in the bifurcation structure of river networks in this paper we present a reformulation of horton s laws which generalizes the familiar scaling of first moments or means to scaling of entire distributions we also present extensive data analysis which supports this reformulation and show that this feature is also exhibited by shreve s well known random topology model

locating pollicott ruelle resonances in chaotic dynamical systems a class of numerical schemes a class of numerical methods to determine pollicott ruelle resonances in chaotic dynamical systems is proposed this is achieved by relating some existing procedures that make use of the pade approximants and interpolating exponentials to both the memory function techniques used in the theory of relaxation and the filter diagonalization method used in the harmonic inversion of time correlation functions this relationship leads to a theoretical framework in which all these methods become equivalent and which allows for new and improved numerical schemes

ruelle resonances in quantum many body dynamics we define a quantum perron frobenius master operator over a suitable normed space of translationally invariant states adjoint to the quasi local c algebra of quantum lattice gasses e g spin chains whose spectrum determines the exponents of decay of time correlation functions the theoretical ideas are applied to a generic example of kicked ising spin 1 2 chains we show that the chaotic eigenmodes corresponding to leading eigenvalue resonances have fractal structure in the basis of local operators

geometry of river networks i scaling fluctuations and deviations this paper is the first in a series of three papers investigating the detailed geometry of river networks branching networks are a universal structure employed in the distribution and collection of material large scale river networks mark an important class of two dimensional branching networks being not only of intrinsic interest but also a pervasive natural phenomenon in the description of river network structure scaling laws are uniformly observed reported values of scaling exponents vary suggesting that no unique set of scaling exponents exists to improve this current understanding of scaling in river networks and to provide a fuller description of branching network structure here we report a theoretical and empirical study of fluctuations about and deviations from scaling we examine data for continent scale river networks such as the mississippi and the amazon and draw inspiration from a simple model of directed random networks we center our investigations on the scaling of the length of a subbasin s dominant stream with its area a characterization of basin shape known as hack s law we generalize this relationship to a joint probability density and provide observations and explanations of deviations from scaling we show that fluctuations about scaling are substantial and grow with system size we find strong deviations from scaling at small scales which can be explained by the existence of a linear network structure at intermediate scales we find slow drifts in exponent values indicating that scaling is only approximately obeyed and that universality remains indeterminate at large scales we observe a breakdown in scaling due to decreasing sample space and correlations with overall basin shape the extent of approximate scaling is significantly restricted by these deviations and will not be improved by increases in network resolution

splittings of finitely generated groups over two ended subgroups we describe a means of constructing splittings of a one ended finitely generated group over two ended subgroups starting with a finite collection of codimension one two ended subgroups in the case where all the two ended subgroups have two ended commensurators we obtain an annulus theorem and a form of jsj splitting of rips and sela the construction uses ideas from the work of dunwoody sageev and swenson we use a particular kind of order structure which combines cyclic orders and treelike structures in the special case of hyperbolic groups this provides a link between combinarorial constructions and constructions arising from the topological structure of the boundary in this context we recover the annulus theorem of scott and swarup we also show that a one ended finitely generated groups which contains an infinite order element and such that every infinite cyclic subgroup is virtually codimension one is a virtual surface group

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

toward a generalization of the topmodel concepts topographic indices of hydrological similarity preliminary studies of the application of topmodel to the 36 ha ringelbach catchment suggested that the original form of exponential transmissivity function leading to the in a tan topographic index and first order hyperbolic base flow recession curve is not appropriate to this catchment two alternative forms of topographic index and soil topographic index are developed based on parabolic and linear transmissivity functions leading to the more frequently observed second order hyperbolic and exponential recession curves respectively it is shown how these can be used in the same way as the original to relate catchment average water table depths to local water table depths so that patterns of saturation can be evaluated two companion ambroise et al this issue freer et al this issue papers show how the new parabolic index is used in the prediction of ringelbach discharges and how the limitations of the model are reflected in the estimated predictive uncertainties using the generalised likelihood uncertainty estimation glue approach

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

leading pollicott ruelle resonances for chaotic area preserving maps recent investigations in nonlinear sciences show that not only hyperbolic but also mixed dynamical systems may exhibit exponential relaxation in the chaotic regime the relaxation rates which lead the decay of probability distributions and correlation functions are related to the classical evolution resolvent perron frobenius operator pole logarithm the so called pollicott ruelle resonances in this brief report the leading pollicott ruelle resonances are calculated analytically for a general class of area preserving maps besides the leading resonances related to the diffusive modes of momentum dynamics slow rate we also calculate the leading faster rate related to the angular correlations the analytical results are compared to the existing results in the literature

cyclic splittings of finitely presented groups and the canonical jsj decomposition the classification of stable actions of finitely presented groups on trees has found a number of applications perhaps one of the most striking of these applications is the theory of canonical jsj decompositions of hyperbolic groups developed by sela in his seminal paper se

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

classical nonlinear response of a chaotic system ii langevin dynamics and spectral decomposition the spectrum of a strongly chaotic system consists of discrete complex ruelle pollicott rp resonances we interpret the rp resonances as eigenstates and eigenvalues of the fokker planck operator obtained by adding an infinitesimal diffusion term to the first order liouville operator we demonstrate how the deterministic expression for the linear response is reproduced in the limit of vanishing noise for the second order response function we establish an equivalence of the spectral decomposition in the limit of vanishing noise and the long time asymptotic expansion in the deterministic case

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

the algebraic torus theorem a structure theorem is proved for a finitely generated group with a finitely generated virtually polycyclic codimension one subgroup

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

power law distribution of discharge in ideal networks data for several river networks show algebraically decaying distribution functions for the mean annual discharge and the dissipation rate rodriguez iturbe et al 1992 we derive a relation between the exponent a in the integrated mean annual discharge distribution p q q q and the topological dimension dt of the network 1 1 dt using the experimentally determined value dt 1 8 tarboton et al 1988 we find 0 45 in good agreement with the data of rodriguez iturbe et al 1992 for the random model shreve 1967 1969 smart and werner 1974 we find an exponent of 1 2

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

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

leading ruelle resonances of chaotic maps the leading ruelle resonances of typical chaotic maps the perturbed cat map and the standard map are calculated by variation it is found that excluding the resonance associated with the invariant density the next subleading resonances are approximately the roots of the equation z 4 ensuremath gamma where ensuremath gamma is a positive number that characterizes the amount of stochasticity of the map the results are verified by numerical computations and the implications to the form factor of the corresponding quantum maps are discussed

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

analogues of cayley graphs for topological groups we define for a compactly generated totally disconnected locally compact group a graph called a rough cayley graph that is a quasi isometry invariant of the group this graph carries information about the group structure in an analogous way to the ordinary cayley graph for a finitely generated group with this construction the machinery of geometric group theory can be applied to topological groups this is illustrated by a study of groups where the rough cayley graph has more than one end and a study of groups where the rough cayley graph has polynomial growth

geometry of river networks ii distributions of component size and number the structure of a river network may be seen as a discrete set of nested subnetworks built out of individual stream segments these network components are assigned an integral stream order via a hierarchical and discrete ordering method exponential relationships known as horton s laws between stream order and ensemble averaged quantities pertaining to network components are observed we extend these observations to incorporate fluctuations and all higher moments by developing functional relationships between distributions the relationships determined are drawn from a combination of theoretical analysis analysis of real river networks including the mississippi amazon and nile and numerical simulations on a model of directed random networks underlying distributions of stream segment lengths are identified as exponential combinations of these distributions form single humped distributions with exponential tails the sums of which are in turn shown to give power law distributions of stream lengths distributions of basin area and stream segment frequency are also addressed the calculations identify a single length scale as a measure of size fluctuations in network components this article is the second in a series of three addressing the geometry of river networks