On the expected width function for topologically random channel networks
No abstract available.
Journal of Applied Probability↗
Geology topics
Brent M. Troutman
Publications and source records attributed to Brent M. Troutman.
Scaling of peak flows with constant flow velocity in random self-similar networks
A methodology is presented to understand the role of the statistical self-similar topology of real river networks on scaling, or power law, in peak flows for rainfall-runoff events. We created Monte Carlo generated sets of ensembles of 1000 random self-similar networks (RSNs) with geometrically distributed interior and exterior generators having parameters p i and p e , respectively. The parameter values were chosen to replicate the observed topology of real river networks. We calculated flow hydrographs in each of these networks by numerically solving the link-based mass and momentum conservation equation under the assumption of constant flow velocity. From these simulated RSNs and hydrographs, the scaling exponents β and φ characterizing power laws with respect to drainage area, and corresponding to the width functions and flow hydrographs respectively, were estimated. We found that, in general, φ > β, which supports a similar finding first reported for simulations in the river network of the Walnut Gulch basin, Arizona. Theoretical estimation of β and φ in RSNs is a complex open problem. Therefore, using results for a simpler problem associated with the expected width function and expected hydrograph for an ensemble of RSNs, we give heuristic arguments for theoretical derivations of the scaling exponents β (E) and φ (E) that depend on the Horton ratios for stream lengths and areas. These ratios in turn have a known dependence on the parameters of the geometric distributions of RSN generators. Good agreement was found between the analytically conjectured values of β (E) and φ (E) and the values estimated by the simulated ensembles of RSNs and hydrographs. The independence of the scaling exponents φ (E) and φ with respect to the value of flow velocity and runoff intensity implies an interesting connection between unit hydrograph theory and flow dynamics. Our results provide a reference framework to study scaling exponents under more complex scenarios of flow dynamics and runoff generation processes using ensembles of RSNs.
Nonlinear Processes in Geophysics↗
Testing statistical self-similarity in the topology of river networks
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 that the generators of the RSN model obey a geometric distribution, and self-similarity holds in a statistical sense in 26 of these 30 basins. The parameters describing the distribution of interior and exterior generators are tested to be statistically different and the difference is shown to produce the well-known Hack's law. The inter-basin variability of RSN parameters is found to be statistically significant. We also test generator dependence on two climatic indices, mean annual precipitation and radiative index of dryness. Some indication of climatic influence on the generators is detected, but this influence is not statistically significant with the sample size available. Finally, two key applications of the RSN model to hydrology and geomorphology are briefly discussed.
Journal of Geophysical Research F: Earth Surface↗
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.
Water Resources Research↗
An analysis of input errors in precipitation-runoff models using regression with errors in the independent variables
Errors in runoff prediction caused by input data errors are analyzed by treating precipitation-runoff models as regression (conditional expectation) models. Independent variables of the regression consist of precipitation and other input measurements; the dependent variable is runoff. In models using erroneous input data, prediction errors are inflated and estimates of expected storm runoff for given observed input variables are biased. This bias in expected runoff estimation results in biased parameter estimates if these parameter estimates are obtained by a least squares fit of predicted to observed runoff values. The problems of error inflation and bias are examined in detail for a simple linear regression of runoff on rainfall and for a nonlinear U.S. Geological Survey precipitation-runoff model. Some implications for flood frequency analysis are considered. A case study using a set of data from Turtle Creek near Dallas, Texas illustrates the problems of model input errors.
Texas↗
Asymptotic distribution of the maximum deficit with correlated, partially regulated outflows
The asymptotic distribution of the maximum accumulated deficit with partially regulated, Markov-dependent net outflows having a Bernoulli distribution is derived, and the distribution for independent, continuous outflows is presented. It is demonstrated that under partial regulation the maximum deficit behaves as log n , where n is the length of the series; this is to be contrasted with the n ½ behavior exhibited by fully regulated outflows. Also, as would be expected, the presence of correlation tends to increase the magnitude and variability of the maximum deficit.
Water Resources Research↗
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.
Water Resources Research↗
Generalizing a nonlinear geophysical flood theory to medium-sized river networks
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-km 2 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. We show scaling in mean annual peak discharges, and briefly discuss that it is physically connected with scaling in multiple rainfall-runoff events. Scaling in peak discharges would hold in a non-stationary climate due to global warming but its slope and intercept would change.
Geophysical Research Letters↗
The influence of topology on hydraulic conductivity in a sand-and-gravel aquifer
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 ϕ, 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 the use of Archie's (1942) law to investigate the role of topological factors such as pore geometry and connectivity, and grain size and packing configuration in regulating fluid flow through these coarse‐grained materials. The logs reveal no significant correlation between K and ϕ, and imply that groundwater models that link these two properties may not be useful at this site. Rather, it is the distribution and connectivity of the fluid phase as defined by formation factor F , cementation index m , and tortuosity α that primarily control the hydraulic conductivity. Results show that F correlates well with K , thereby indicating that induction logs provide qualitative information on the distribution of hydraulic conductivity. A comparison of α, which incorporates porosity data, with K produces only a slightly better correlation and further emphasizes the weak influence of the bulk value of ϕ on K .
Massachusetts↗
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 , α, β), where α parameterizes the link (channel segment) length distribution, and β 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 , α, β) 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 t p = (2 N ) ½ α * /β * where α * is mean link length, and β * is a scalar hydraulic parameter (usually average celerity). This asymptotic IUH is identical for all linear routing schemes.
Water Resources Research↗
Errors and parameter estimation in precipitation-runoff modeling: 1. Theory
Errors in complex conceptual precipitation-runoff models may be analyzed by placing them into a statistical framework. This amounts to treating the errors as random variables and defining the probabilistic structure of the errors. By using such a framework, a large array of techniques, many of which have been presented in the statistical literature, becomes available to the modeler for quantifying and analyzing the various sources of error. A number of these techniques are reviewed in this paper, with special attention to the peculiarities of hydrologic models. Known methodologies for parameter estimation (calibration) are particularly applicable for obtaining physically meaningful estimates and for explaining how bias in runoff prediction caused by model error and input error may contribute to bias in parameter estimation.
Water Resources Research↗
Errors and parameter estimation in precipitation-runoff modeling: 2. Case study
A case study is presented which illustrates some of the error analysis, sensitivity analysis, and parameter estimation procedures reviewed in the first part of this paper. It is shown that those procedures, most of which come from statistical nonlinear regression theory, are invaluable in interpreting errors in precipitation-runoff modeling and in identifying appropriate calibration strategies.
Water Resources Research↗
Variability of differences between two approaches for determining ground-water discharge and pumpage, including effects of time trends, Lower Arkansas River Basin, southeastern Colorado, 1998-2002
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 the two approaches, including the effects of time trends. This report compared measured ground-water pumpage using TFMs to computed ground-water pumpage using PCCs by developing statistical models of relations between explanatory variables, such as site, time, and pumping water level, and dependent variables, which are based on discharge, PCC, and pumpage. When differences in pumpage (diffP) were computed using PCC measurements and power consumption for the same year (1998-2002), the median diffP, depending on the year, ranged from +0.1 to -2.9 percent; the median diffP for the entire period was -1.5 percent. However, when diffP was computed using PCC measurements applied to the next year's power consumption, the median diffP was -0.3 percent; and when PCC measurements were applied 2, 3, or 4 years into the future, median diffPs were +1.8 percent for a 2-year forward lag and +5.3 percent for a 4-year forward lag, indicating that pumpage computed with the PCC approach, as generally applied under the ground-water pumpage measurement rules by CDWR, tended to overestimate pumpage as compared to pumpage using TFMs when PCC measurement was applied to future years of measured power consumption. Analyses were done to better understand the causes of the time trend; an estimate of the overall trend with time (uncorrected for pumping water-level changes) yielded a trend of about 2.2 percent per lag year for diffP. A separate analysis that incorporated a surface-water diversion term in the statistical model rendered the time-trend term insignificant, indicating that the time trend in the models served as a surrogate for other variables, some of which reflect underlying hydrologic conditions. A more precise explanation of the potential causes of the time trend was not obtained with the available data. However, the model results with the surface-water diversion term indicate that much of the trend of 2.2 percent per lag year in diffP resulted from applying a PCC to estimate pumpage under hydrologic conditions different from those under which the PCC was measured. Although there is no evidence to conclude that the upward time trend determined in the data for this 5-year period would hold in the future, historical static ground-water levels in the study area generally have exhibited small variations over multidecadal time scales. Therefore, the approximately 2 percent per lag year trend determined in these data is expected to be a reasonable guideline for estimating potential errors in the PCC approach resulting from temporally varying hydrologic conditions between time of PCC measurement and pumpage estimation. Comparisons also were made between total, or aggregated, pumpage for a network of wells as computed by the PCC approach and the TFM approach. For 100 wells and a lag of 4 years between PCC measurement and pumpage estimation, there was a 95-percent probability that the difference between total network pumpage measured by the PCC approach and that measured using a TFM would be between 5.2 and 14.4 percent. These estimates were based on a bias of 2.2 percent per lag year estimated for the period 1998-2002 during which hydrologic conditions were known to have changed. Using the same assumptions, the estimated d
Scientific Investigations Report↗
Regional flood probabilities
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.
Water Resources Research↗
Comparison of two approaches for determining ground-water discharge and pumpage in the lower Arkansas River Basin, Colorado, 1997-98
In March 1994, the Colorado Division of Water Resources (CDWR) adopted “Rules Governing the Measurement of Tributary Ground Water Diversions Located in the Arkansas River Basin” (Office of the State Engineer, 1994); these initial rules were amended in February 1996 (Office of the State Engineer, 1996). The amended rules require users of wells that divert tributary ground water to annually report the water pumped monthly by each well. The rules allow a well owner to report the pumpage measured by a totalizing flowmeter (TFM) or pumpage determined from electrical power data and a power conversion coefficient (PCC) (Hurr and Litke, 1989). Opinions by representatives of the State of Kansas, presented before the Special Master hearing a court case [State of Kansas v. State of Colorado, No. 105 Original (1996)] concerning post-Compact well pumping, stated that the PCC approach does not provide the same level of accuracy and reliability as a TFM when used to determine pumpage. In 1997, the U.S. Geological Survey (USGS), in cooperation with the CDWR, began a 2-year study to compare ground-water pumpage estimates made using the TFM and the PCC approaches. The study area was along the Arkansas River between Pueblo, Colorado, and the Colorado-Kansas State line (fig. 1). The two approaches for estimating ground-water discharge and pumpage were compared for more than 100 wells completed in the alluvial aquifer of the Arkansas River Basin. The TFM approach uses an inline flowmeter to directly measure instantaneous discharge and the total volume of water pumped at a well. The PCC approach uses electrical power consumption records and a power conversion coefficient to estimate the pumpage at ground-water wells. This executive summary describes the results of the comparison of the two approaches. Specifically, (1) the differences in instantaneous discharge measured with three portable flowmeters and measured with an inline TFM are evaluated, and the statistical differences in paired instantaneous discharge between the two approaches are determined; (2) short- and long-term variations in the PCC’s are presented; (3) differences in pumpage between the two approaches are evaluated, and the statistical differences in pumpage between the two approaches are determined; (4) potential sources of discrepancy between pumpage estimates are discussed; and (5) differences in total network pumpage using the two approaches are presented. During the irrigation seasons of 1997 and 1998, instantaneous discharge and electrical power demand were measured at randomly selected wells to determine PCC’s. At more than 100 wells, the PCC’s determined during the 1998 season were applied to total electrical power consumption data that was recorded between the initial and final readings at each network well site in 1998 to estimate total ground-water pumpage. At each site, an inline TFM was installed in a full-flowing, acceptable test section of pipe on the discharge side of the pump where the measurement of discharge was made. Measurements of instantaneous ground-water discharge also were made using three different types of portable flowmeters. The average velocity multiplied by the cross-sectional area of the discharge pipe was used to compute the discharge in gallons per minute. Whenever possible, discharge measurements were made at each network site using all three types of portable flowmeters.
Colorado↗
Estimation of Renyi exponents in random cascades
We consider statistical estimation of the Re??nyi exponent ??(h), which characterizes the scaling behaviour of a singular measure ?? defined on a subset of Rd. The Re??nyi exponent is defined to be lim?????0 [{log M??(h)}/(-log ??)], assuming that this limit exists, where M??(h) = ??i??h(??i) and, for ??>0, {??i} are the cubes of a ??-coordinate mesh that intersect the support of ??. In particular, we demonstrate asymptotic normality of the least-squares estimator of ??(h) when the measure ?? is generated by a particular class of multiplicative random cascades, a result which allows construction of interval estimates and application of hypothesis tests for this scaling exponent. Simulation results illustrating this asymptotic normality are presented. ?? 1999 ISI/BS.
Bernoulli↗
Evaluation of process errors in bed load sampling using a Dune Model
Reliable estimates of the streamwide bed load discharge obtained using sampling devices are dependent upon good at-a-point knowledge across the full width of the channel. Using field data and information derived from a model that describes the geometric features of a dune train in terms of a spatial process observed at a fixed point in time, we show that sampling errors decrease as the number of samples collected increases, and the number of traverses of the channel over which the samples are collected increases. It also is preferable that bed load sampling be conducted at a pace which allows a number of bed forms to pass through the sampling cross section. The situations we analyze and simulate pertain to moderate transport conditions in small rivers. In such circumstances, bed load sampling schemes typically should involve four or five traverses of a river, and the collection of 20–40 samples at a rate of five or six samples per hour. By ensuring that spatial and temporal variability in the transport process is accounted for, such a sampling design reduces both random and systematic errors and hence minimizes the total error involved in the sampling process.
Water Resources Research↗
A note on subtrees rooted along the primary path of a binary tree
Let F n denote the set of rooted binary plane trees with n external nodes, for given T ∈ F n let u i ( T ) be the altitude i node along the primary path of T , and let δ i ( T ) denote the number of external nodes in the induced subtree rooted at u i ( T ). We set δ i ( T ) = 0 if i is greater than the length of the primary path of T . We prove lim n →∞ ∑ i ≤ x / n E n { δ i }/∑ i <∞ E n { δ i } = G ( x ), where E n denotes the average over trees T ∈ F n and where the distribution function G is determined by its moments, for which we present an explicit expression.
Discrete Applied Mathematics↗