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The central hypothesis of a nonlinear geophysical flood theory postulates that, given space-time rainfall intensity for a rainfall-runoff event, solutions of coupled mass and momentum conservation differential equations governing runoff generation and transport in a self-similar river network produce spatial scaling, or a power law, relation between peak discharge and drainage area in the limit of large area. The excellent fit of a power law for the destructive flood event of June 2008 in the 32,400-km2 Iowa River basin over four orders of magnitude variation in drainage areas supports the central hypothesis. The challenge of predicting observed scaling exponent and intercept from physical processes is explained....
Categories: Publication; Types: Citation; Tags: Climate, floods, scaling theory
Natural river channel networks have been shown in empirical studies to exhibit power-law scaling behavior characteristic of self-similar and self-affine structures. Of particular interest is to describe how the distribution of distance to the outlet changes as a function of network size. In this paper, networks are modeled as random self-similar rooted tree graphs and scaling of distance to the root is studied using methods in stochastic branching theory. In particular, the asymptotic expectation of the width function (number of nodes as a function of distance to the outlet) is derived under conditions on the replacement generators. It is demonstrated further that the branching number describing rate of growth of...
The spatial variability of two fundamental morphological variables is investigated for rivers having a wide range of discharge (five orders of magnitude). The variables, water-surface width and average depth, were measured at 58 to 888 equally spaced cross-sections in channel links (river reaches between major tributaries). These measurements provide data to characterize the two-dimensional structure of a channel link which is the fundamental unit of a channel network.The morphological variables have nearly log-normal probability distributions. A general relation was determined which relates the means of the log-transformed variables to the logarithm of discharge similar to previously published downstream hydraulic...
A statistical framework is introduced that resolves important problems with the interpretation and use of traditional Horton regression statistics. The framework is based on a univariate regression model that leads to an alternative expression for Horton ratio, connects Horton regression statistics to distributional simple scaling, and improves the accuracy in estimating Horton plot parameters. The model is used to examine data for drainage area A and mainstream length L from two groups of basins located in different physiographic settings. Results show that confidence intervals for the Horton plot regression statistics are quite wide. Nonetheless, an analysis of covariance shows that regression intercepts, but...
Key results in the last 20 years have established the theoretical and observational foundations for developing a new nonlinear geophysical theory of floods in river basins. This theory, henceforth called the scaling theory, has the explicit goal to link the physics of runoff generating processes with spatial power-law statistical relations between floods and drainage areas across multiple scales of space and time. Published results have shown that the spatial power law statistical relations emerge asymptotically from conservation equations and physical processes as drainage area goes to infinity. These results have led to a key hypothesis that the physical basis of power laws in floods has its origin in the self-similarity...
We examine the appearance of power-law behavior in rooted tree graphs in the context of river networks. It has long been observed that the tails of statistical distributions of upstream areas in river networks, measured above every link, obey a power-law relationship over a range of scales. We examine this behavior by considering a subset of all links, defined as those links which drain complete Strahler basins, where the Strahler order defines a discrete measure of scale, for self-similar networks with both deterministic and random topologies. We find an excellent power-law structure in the tail probabilities for complete Strahler basin areas, over many ranges of scale. We show analytically that the tail probabilities...
Categories: Publication; Types: Citation
Recent work has demonstrated that the topological properties of real river networks deviate significantly from predictions of Shreve's random model. At the same time the property of mean self-similarity postulated by Tokunaga's model is well supported by data. Recently, a new class of network model called random self-similar networks (RSN) that combines self-similarity and randomness has been introduced to replicate important topological features observed in real river networks. We investigate if the hypothesis of statistical self-similarity in the RSN model is supported by data on a set of 30 basins located across the continental United States that encompass a wide range of hydroclimatic variability. We demonstrate...
The spatial variability of two fundamental morphological variables is investigated for rivers having a wide range of discharge (five orders of magnitude). The variables, water-surface width and average depth, were measured at 58 to 888 equally spaced cross-sections in channel links (river reaches between major tributaries). These measurements provide data to characterize the two-dimensional structure of a channel link which is the fundamental unit of a channel network.The morphological variables have nearly log-normal probability distributions. A general relation was determined which relates the means of the log-transformed variables to the logarithm of discharge similar to previously published downstream hydraulic...
A field experiment consisting of geophysical logging and tracer testing was conducted in a single well that penetrated a sand-and-gravel aquifer at the U.S. Geological Survey Toxic Substances Hydrology research site on Cape Cod, Massachusetts. Geophysical logs and flowmeter/pumping measurements were obtained to estimate vertical profiles of porosity phi, hydraulic conductivity K, temperature, and bulk electrical conductivity under background, freshwater conditions. Saline-tracer fluid was then injected into the well for 2 h and its radial migration into the surrounding deposits was monitored by recording an electromagnetic-induction log every 10 min. The field data are analyzed and interpreted primarily through...
River networks in the landscape can be described as topologic rooted trees embedded in a three-dimensional surface. We examine the problem of embedding topologic binary rooted trees (BRTs) by investigating two space-filling embedding procedures: Top-Down, previously developed in the context of random self-similar networks (RSNs), and Bottom-Up, a new procedure developed here. We extend the concept of generalized Horton laws to interior sub catchments and create a new set of scaling laws that are used to test the embedding algorithms. We compare the two embedding strategies with respect to the scaling properties of the distribution of accumulated areas Aω and network magnitude Mω for complete order streams ω. The...
A field experiment consisting of geophysical logging and tracer testing was conducted in a single well that penetrated a sand-and-gravel aquifer at the U.S. Geological Survey Toxic Substances Hydrology research site on Cape Cod, Massachusetts. Geophysical logs and flowmeter/pumping measurements were obtained to estimate vertical profiles of porosity phi, hydraulic conductivity K, temperature, and bulk electrical conductivity under background, freshwater conditions. Saline-tracer fluid was then injected into the well for 2 h and its radial migration into the surrounding deposits was monitored by recording an electromagnetic-induction log every 10 min. The field data are analyzed and interpreted primarily through...
In the mid-1990s, the Colorado Division of Water Resources (CDWR) adopted rules governing measurement of tributary ground-water pumpage for the Arkansas River Basin. The rules allowed ground-water pumpage to be determined using one of two approaches?power conversion coefficient (PCC) or totalizing flowmeters (TFM). In addition, the rules allowed a PCC to be applied to the electrical power usage up to 4 years in the future to estimate ground-water pumpage. As a result of concerns about potential errors in applying the PCC approach forward in time, a study was done by the U.S. Geological Survey, in cooperation with CDWR and Colorado Water Conservation Board, to evaluate the variability in differences in pumpage between...
Categories: Publication; Types: Citation
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...
A wide variety of stochastic modeling procedures has been applied to rainfall, landforms, and streamflow. A representative selection of these procedures is reviewed in this article. Emphasis is placed on interesting features that pose a challenge to modeling, such as the intermittency of rainfall in both space and time, the discrete branching structure of channel networks, and small streamflow measurement samples necessitating regional analysis methods. In all three areas, traditional modeling notions are discussed and modern scaling approaches introduced. Dependence of streamflow on rainfall and landforms is discussed in the final section, and it is demonstrated how stochastic and physical process ideas come together...
Categories: Publication; Types: Citation