Scaling properties of Canadian flood flows

This study investigated the spatial scaling properties of Canadian flood flows, namely, annual maximum mean 1‐, 5‐ and 7‐day flows using both the product moments (PMs) and probability weighted moments (PWMs). Both approaches demonstrate that flood flows in climatic regions 1 (Pacific), 2 (South British Columbia mountains), 3 (Yukon and northern British Columbia), 6 (Northeastern forest), 7 (Great Lakes and St. Lawrence rivers), 8 (Atlantic), and 10 (Arctic tundra) exhibit simple scaling with scaling exponent θ/H close to 0·90, while flood flows in regions 4 (Prairie provinces), 5 (Northwestern forest), and 9 (Mackenzie) does not with scaling exponent θ/H close to 0·50. The plots of coefficient of variations of flood flows versus drainage area indicate that Cv remains almost constant in regions 1, 2, 3, 6, 7, 8, and 10, while it decreases as drainage area increases in regions 4, 5, and 9. These results demonstrate that the index flood method is applicable in climatic regions 1, 2, 3, 6, 7, 8, and 10, while it is not in climatic regions 4, 5, and 9. The physical backgroud of the simple scaling of flood flows in most Canadian climatic regions is that snowmelt or rain‐on‐snow runoff is a dominant flood‐generating mechanism across the country. Copyright © 2008 John Wiley & Sons, Ltd.     

Scaling of Canadian low flows

Scaling properties of Canadian low flows, namely annual minimum mean 1-, 5- and 7-day flows, are evaluated across Canada and in its sub-climatic regions. Across the entire country, the log relationship between the kth product moments (PMs, E[Q_i^k]) of low flows and drainage area (A_i) can be represented by: ln(E[Q_i^k])=a_k+b_kln(A_i)and b_k=k, with = 0.86, 0.94 and 0.93 for annual minimum mean 1-, 5- and 7-day flows, respectively. The log linear relationships between the kth probability weighted moments (PWMs, ) and A_i are ln()=c_k+Hln(A_i), in which H is constant and is independent of k. The values of H are 0.87, 0.97, and 0.96 for annual minimum mean 1-, 5- and 7-day flows, respectively, which are almost the same as the values. The coefficients of variation (Cv) are almost independent of drainage area. These results demonstrate that Canadian low flows generally exhibit simple scaling and drainage area alone describes most of the variability in the moments of the low flows. Low flows in each of the sub-climatic regions also obey a simple scaling law. The values of , H and Cv are different in each region, which may stem from physiographical and climatological differences among these regions. The finding lays a basis for applying the index flood method to conduct regional low flow frequency analysis as simple scaling is equivalent to the index flood method.Acknowledgements The authors thank Prof. Thian Yew Gan of University of Alberta, Canada for providing additional pristine data sites for regions 4 and 10. A constructive comments provided by an anonymous reviewer improved the quality of the paper.     

Spatial scaling properties of annual streamflow in the United States

Abstract

The spatial scaling properties of annual average streamflow is examined using records from 1 433 river basins across the continental United States. The log-linear relationship ln(E[Q^r _i]) = a + b_r ln(A_i) is representative throughout the United States, where E[Q^r _i] represents the expectation of the rth moment of annual streamflow at site i, and A_i represents drainage area. The scaling model parameters a_r and b_r follow nearly perfect linear relationships a_r = rα and b_r = rβ throughout the continental United States. We conclude that the probability distribution of annual streamflow follows simple scaling relationships in all regions of the United States. In temperate regions where climate is relatively homogeneous, scale alone describes most of the variability in the moments of annual streamflow. In the more climatically heterogeneous regions, such as in the Upper Colorado and Missouri river basins, scale alone is a poor predictor of the moments of annual flow.     

On the use of partial probability weighted moments in the analysis of hydrological extremes

The use of partial probability weighted moments (PPWM) for estimating hydrological extremes is compared to that of probability weighted moments (PWM). Firstly, estimates from at‐site data are considered. Two Monte Carlo analyses, conducted using continuous and empirical parent distributions (of peak discharge and daily rainfall annual maxima) and applying four different distributions (Gumbel, Fréchet, GEV and generalized Pareto), show that the estimates obtained from PPWMs are better than those obtained from PWMs if the parent distribution is unknown, as happens in practice. Secondly, the use of partial L‐moments (obtained from PPWMs) as diagnostic tools is considered. The theoretical partial L‐diagrams are compared with the experimental data. Five different distributions (exponential, Pareto, Gumbel, GEV and generalized Pareto) and 297 samples of peak discharge annual maxima are considered. Finally, the use of PPWMs with regional data is investigated. Three different kinds of regional analyses are considered. The first kind is the regression of quantile estimates on basin area. The study is conducted applying the GEV distribution to peak discharge annual maxima. The regressions obtained with PPWMs are slightly better than those obtained with PWMs. The second kind of regional analysis is the parametric one, of which four different models are considered. The congruence between local and regional estimates is examined, using peak discharge annual maxima. The congruence degree is sometimes higher for PPWMs, sometimes for PWMs. The third kind of regional analysis uses the index flood method. The study, conducted applying the GEV distribution to synthetic data from a lognormal joint distribution, shows that better estimates are obtained sometimes from PPWMs, sometimes from PWMs. All the results seem to indicate that using PPWMs can constitute a valid tool, provided that the influence of ouliers, of course higher with censored samples, is kept under control. Copyright © 2007 John Wiley & Sons, Ltd.     

Alternative PWM-estimators of the Gumbel distribution

Probability weighted moments (PWM) are widely used in hydrology for estimating parameters of statistical distributions, including the Gumbel distribution. The classical PWM-approach considers the moments β_i=E[XFⁱ] with i=0,1 for estimation of the Gumbel scale and location parameters. However, there is no reason why these probability weights (F⁰ and F¹) should provide the most efficient PWM-estimators of Gumbel parameters and quantiles. We explore an extended class of PWMs that does not impose arbitrary restrictions on the values of i. Estimation based on the extended class of PWMs is called the generalized method of probability weighted moments (GPWM) to distinguish it from the classical procedure. In fact, our investigation demonstrates that it may be advantage to use weight functions that are not of the form Fⁱ. We propose an alternative PWM-estimator of the Gumbel distribution that maintains the computational simplicity of the classical PWM method, but provides slightly more accurate quantile estimates in terms of mean square error of estimation. A simple empirical formula for the standard error of the proposed quantile estimator is presented.     

A probability-weighted moment test to assess simple scaling

We present a statistically robust approach based on probability weighted moments to assess the presence of simple scaling in geophysical processes. The proposed approach is different from current approaches which rely on estimation of high order moments. High order moments of simple scaling processes (distributions) may not have theoretically defined values and consequently, their empirical estimates are highly variable and do not converge with increasing sample size. They are, therefore, not an appropriate tool for inference. On the other hand we show that the probability weighted moments of such processes (distributions) do exist and, hence, their empirical estimates are more robust. These moments, therefore, provide an appropriate tool for inferring the presence of scaling. We illustrate this using simulated Levystable processes and then draw inference on the nature of scaling in fluctuations of a spatial rainfall process.     

A probability-weighted moment test to assess simple scaling

We present a statistically robust approach based on probability weighted moments to assess the presence of simple scaling in geophysical processes. The proposed approach is different from current approaches which rely on estimation of high order moments. High order moments of simple scaling processes (distributions) may not have theoretically defined values and consequently, their empirical estimates are highly variable and do not converge with increasing sample size. They are, therefore, not an appropriate tool for inference. On the other hand we show that the probability weighted moments of such processes (distributions) do exist and, hence, their empirical estimates are more robust. These moments, therefore, provide an appropriate tool for inferring the presence of scaling. We illustrate this using simulated Levystable processes and then draw inference on the nature of scaling in fluctuations of a spatial rainfall process.     

Comparison of scaling methods for waveform inversion

Waveform inversion can lead to faint images for later times due to geometrical spreading. The proper scaling of the steepest-descent direction can enhance faint images in waveform inversion results. We compare the effects of different scaling techniques in waveform inversion algorithms using the steepest-descent method. For the scaling method we use the diagonal of the pseudo-Hessian matrix, which can be applied in two different ways. One is to scale the steepest-descent direction at each frequency independently. The other is to scale the steepest-descent direction summed over the entire frequency band. The first method equalizes the steepest-descent directions at different frequencies and minimizes the effects of the band-limited source spectrum in waveform inversion. In the second method, since the steepest-descent direction summed over the entire frequency band is divided by the diagonal of the pseudo-Hessian matrix summed over the entire frequency band, the band-limited property of the source wavelet spectrum still remains in the scaled steepest-descent directions. The two scaling methods were applied to both standard and logarithmic waveform inversion. For standard waveform inversion, the method that scales the steepest-descent direction at every frequency step gives better results than the second method. On the other hand, logarithmic waveform inversion is not sensitive to the scaling method, because taking the logarithm of wavefields automatically means that results for the steepest-descent direction at each frequency are commensurate with each other. If once the steepest-descent directions are equalized by taking the logarithm of wavefields in logarithmic waveform inversion, the additional equalizing effects by the scaling method are not as great as in conventional waveform inversion.     

How long is a hillslope?

Hillslope length is a fundamental attribute of landscapes, intrinsically linked to drainage density, landslide hazard, biogeochemical cycling and hillslope sediment transport. Existing methods to estimate catchment average hillslope lengths include inversion of drainage density or identification of a break in slope–area scaling, where the hillslope domain transitions into the fluvial domain. Here we implement a technique which models flow from point sources on hilltops across pixels in a digital elevation model (DEM), based on flow directions calculated using pixel aspect, until reaching the channel network, defined using recently developed channel extraction algorithms. Through comparisons between these measurement techniques, we show that estimating hillslope length from plots of topographic slope versus drainage area, or by inverting measures of drainage density, systematically underestimates hillslope length. In addition, hillslope lengths estimated by slope–area scaling breaks show large variations between catchments of similar morphology and area. We then use hillslope length–relief structure of landscapes to explore nature of sediment flux operating on a landscape. Distinct topographic forms are predicted for end‐member sediment flux laws which constrain sediment transport on hillslopes as being linearly or nonlinearly dependent on hillslope gradient. Because our method extracts hillslope profiles originating from every ridgetop pixel in a DEM, we show that the resulting population of hillslope length–relief measurements can be used to differentiate between linear and nonlinear sediment transport laws in soil mantled landscapes. We find that across a broad range of sites across the continental United States, topography is consistent with a sediment flux law in which transport is nonlinearly proportional to topographic gradient. © 2016 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd.     

Interpreting sediment yield scaling

Terrestrial sediment yield – often measured as suspended sediment load in stream channels – commonly scales with drainage area within homogeneous land surface regions. But the effect of drainage area has not usually been recognized in comparative sediment yield analyses, rendering most comparisons of sediment yield from disparate source areas invalid. The procedure to discount scale differences for comparative purposes is presented. Mathematical scaling varies according to landscape condition and provides a physical interpretation of that condition. The results open the way for rational construction of a ‘sediment delivery ratio’. Copyright © 2017 John Wiley & Sons, Ltd.     

Effects of hillslope dynamics and network geometry on the scaling properties of the hydrologic response

This paper investigates the specific contributions of river network geomorphology, hillslope flow dynamics and channel routing to the scaling behavior of the hydrologic response as function of drainage area. Scaling relationships emerged from the observations of geomorphological and hydrological data and were reproduced in previous works through mathematical models, for both idealized self-similar networks and natural basins. Recent literature highlighted that scale invariance of hydrological quantities depends not only on the metrics of the drainage catchment but also on effective flow routing. In this study we employ a geomorphological width function scheme to test the simple scaling hypothesis adopting more realistic dynamic conditions than in previous approaches, specifically taking into account the role of hillslopes. The analysis is based on the derivation of the characteristic distributions of path lengths and travel times, inferred from DEM processing and measurements of rainfall and runoff data. The study area is located in the Tiber River region (central Italy).Results indicate that, while scaling properties clearly emerge when the hydrologic response is defined on the basis of the sole geomorphology, scale invariance is broken when less idealized flow dynamics are taken into account. Lack of scaling appears in particular as a consequence of the catchment to catchment variability of hillslope velocities.     

Scaling and physiographic controls on streamflow behaviour on the Precambrian Shield, south-central Ontario

Relationships explaining streamflow behaviour in terms of drainage basin physiography greatly assist efforts to extrapolate streamflow metrics from gauged to ungauged basins in the same landscape. The Dorset Environmental Science Centre (DESC) has monitored streamflow from 22 small basins (3.4–190.5 ha) on the Precambrian Shield in south-central Ontario, in some cases since 1976. The basins exhibit regional coherence in their interannual response to precipitation; however, there is often a poor correlation between streamflow metrics from basins separated by as little as 1 km. This study assesses whether inter-basin variations in such metrics can be explained in terms of basin scale and physiography. Several characteristics (annual maximum, minimum and average flow) exhibited simple scaling with basin area, while magnitude, range and timing of annual maximum daily runoff showed scaling behaviour consistent with the Representative Elementary Area (REA) concept. This REA behaviour is partly attributed to convergence of fractional coverage of the two dominant and hydrologically-contrasting land cover types in the DESC region with increasing basin size. Three Principal Components (PCs) explained 82.4% of the variation among basin physiographic properties, and several runoff metrics (magnitude and timing of annual minimum daily runoff, mean number of days per year with 0 streamflow) exhibited significant relationships with one or more PC. Significant relationships were obtained between basin quickflow (QF) production and the PCs on a seasonal and annual basis, almost all of which were superior to simple area-based relationships. Basin physiography influenced QF generation via its control on slope runoff, water storage and hydrologic connectivity; however, this role was minimized during Spring when QF production in response to large rain-on-snow events was relatively uniform across the DESC basins. The PC-based relationships and inter-seasonal changes in their form were consistent with previous research conducted at point, slope and basin scales in the DESC region, and perceptions of key hydrological processes in these small basins may not have been as readily obtained from scaling studies using streamflow from larger basins. This process understanding provides insights into scaling behaviour beyond those derived from simple scaling and REA analyses. The physiography of the study area is representative of large portions of the Precambrian Shield, such that basin streamflow behaviour could potentially be extended across much of south-central Ontario. This would assist predictions of streamflow conditions at ungauged locations, development and testing of hydrological models for this landscape, and interpretation of inter-basin and intra-annual differences in hydrochemical behaviour on the southern Precambrian Shield.     

Probability distribution type of Canadian annual minimum streamflow / Type de distribution de probabilité du débit minimum annuel au Canada

Abstract

The method of L-moment ratio diagrams along with the averaged weighted distance (AWD) is applied to identify a probability distribution of annual minimum streamflow, namely annual minimum daily streamflow in 11 climatic regions of Canada. Across the entire country, the Pearson type III probability distribution is an acceptable distribution for describing annual minimum streamflow with the 3-parameter lognormal and log Pearson type III distributions as potential candidates. Some minor differences in the probability distribution type among different climatic regions are also observed, which may be taken into account in the selection of the distribution type of annual minimum streamflow.     

Quaternary drainage network reorganization in the Colombian Eastern Cordillera plateau

Dramatic drainage reorganization from initial longitudinal to transversal domains has occurred in the Eastern Cordillera of Colombia. We perform a regional analysis of drainage basin geometry and transformed river profiles based on the integral form of the slope-area scaling, to investigate the dynamic state of drainage networks and to predict the degree of drainage reorganization in this region. We propose a new model of drainage rearrangement for the Eastern Cordillera, based on the analyses of knickpoint distribution, normalized river profiles, landforms characteristic of river capture, erosion rates and palaeodrainage data. We establish that the oldest longitudinal basin captured by the Magdalena River network was the Suárez Basin at ≈409 ka, inferring the timing of abandonment of a river terrace using in situ produced cosmogenic beryllium-10 (¹⁰Be) depth profiles and providing a first estimation of incision rate of 0.07 mm/yr. We integrate published geochronologic data and interpret the last capture of the Sabana de Bogotá, providing a minimum age of the basin opening to the Magdalena drainage at ≈38 ka. Our results suggest that the Magdalena basin Increased its drainage area by integrating the closed basins from the western flank of the Eastern Cordillera. Our study also suggests that the Magdalena basin is an aggressor compared to the basins located in the eastern flank of the orogen and provides a framework for examining drainage reorganization within the Eastern Cordillera and in similar orogenic settings. The results improve our understanding of headward integration of closed basins across orogenic plateaux. © 2020 John Wiley & Sons, Ltd.     

Scaling and regionalization of flood flows in British Columbia,Canada

A regionalization of flood data in British Columbia reveals a common scaling with drainage area over the range 0·5×10²<A_d<10⁴ km². This scaling is not a function of flood return period, which implies that simple scaling—consistent with a snowmelt‐dominated flow regime—applies to the province. The observed scale relation takes the form , similar to values reported in previous studies. The scaling relation identified was used to define the regional pattern of hydroclimatic variability for flood flows in British Columbia after discounting the effect of drainage area. The pattern was determined by kriging a scale‐independent runoff factor k for the mean annual flood, 5 year flood and 20 year flood. The analysis permits quantification of uncertainty of the estimates, which can be used in conjunction with the mapped k‐fields to calculate a mean and range for floods with the identified return period for ungauged basins. Owing to the sparsity of data, the precision is relatively poor. The standard error is generally less than 75% of the estimate in the southern half of the province, whereas in the northern half it is often between 75 and 100%. Examination of the relative increase in flood magnitude with increasing return period reveals spatially consistent but statistically insignificant differences. Flood magnitude tends to increase more rapidly in the western regions, where rain events may contribute to flood generation. The relative increase in flood magnitude with return period is consistently lower in the eastern mountain ranges, where snowmelt dominates the flood flow regime. Copyright © 2002 John Wiley & Sons, Ltd.     

Backus averaging,scattering and drift1

Differences between traveltimes from sonic to seismic frequencies, commonly known as drift, can be attributed to a combination of multiple scattering and absorption. The portion due to scattering can be estimated directly by calculating synthetic seismograms from sonic logs. A simple alternative approach is suggested by the long-wave equivalent averaging formulae for the effective elastic properties of a stack of thin layers, which gives the same traveltime delays as the low-frequency limit of the scattering dispersion. We consider the application of these averaging formulae over a frequency-dependent window with the hope of extending their use to frequencies higher than those allowed by the original validity conditions. However, comparison of the time delay due to window-averaging with the scattering dispersion predicted by the O'Doherty-Anstey formula reveals that it is not possible to specify a form of window that will fit the dispersion across the spectrum for arbitrary log statistics. A window with a width proportional to the wavelength squared matches the behaviour at the low-frequency end of the dispersive range for most logs, and allows an almost exact match of the drift across the entire spectrum for exponential correlation functions. We examine a real log, taken from a hole in nearly plane-layered geology, which displays strong quasi-cyclical variations on one scale as well as more random, smaller-scale fluctuations. The details of its drift behaviour are studied using simple models of the gross features. The form of window which gave a good theoretical fit to the dispersion for an exponential log correlation function can only fit the computed drift at high or low frequencies, confirming that there are at least two significant scale-lengths of fluctuation. A better overall fit is obtained for a window whose width is proportional to the wavelength. The calculated scattering drift is significantly less than that observed from a vertical seismic profile, but the difference cannot be wholly ascribed to absorption. This is because the source frequency of the sonic tool is not appropriate for its resolution (receiver spacing) so that the scattering drift from sonic to seismic frequencies cannot be fully estimated from the layer model derived from the log.     

A BRIEF INTRODUCTION TO THE NEW METHOD FOR RIVER PROFILE ANALYSIS: Integral Approach

The topography and geomorphology of active orogens result from the interaction of tectonics and climate. In most orogens, a fluvial channel is most sensitive to the coupling between tectonics, lithology, and climate. Meanwhile, the related signals have been recorded by both the drainage geometry and channel longitudinal profile. Thus, how to extract tectonic information from fluvial channels has been a focused issue in geologic and geomorphologic studies. The well known stream-power river incision model bridges the gap between tectonic uplift, river incision and channel profile change, making it possible to retrieve rock uplift pattern from river profiles. In this model, the river incision rate depends on the rock erodibility, contributing drainage area and river gradient. The steady-state form of the river incision model predicts a power-law scaling between the drainage area and channel gradient. Via a linear regression to the log-transformed slope-area data, the slope and intercept are channel concavity and steepness indices, respectively. The concavity relates to lithology, climatic setting and incision process while the channel steepness can be used to map the spatial pattern of rock uplift. For its simple calculation process, the slope-area analysis has been widely used in the study of tectonic geomorphology during past decades. However, to calculate river slope, the coarse channel elevation data must be smoothed, re-sampled, and differentiated without any reasonable smooth window or rigid mathematical fundamentals. One may lose important information and derive stream-power parameters with high uncertainties. In this paper, we introduce the integral approach, a procedure that has been widely used in the latest four years and demonstrated to be a better method for river profile analysis than the traditional slope-area analysis. Via the integration to the steady-state form of the stream-power river incision equation, the river longitudinal profile can be converted into a straight line of which the independent variable is the integral quantity χ with the unit of distance and the dependent variable is the relative channel elevation. We can calculate the linear correlation coefficient between elevation and χ based on a series of concavity values and find the best linear fit to be the reasonable channel concavity index. The slope of the linear fit to the χ value and elevation is simply related to the ratio of the uplift rate to the erodibility. Without calculating channel slope, the integral approach makes up for the drawback of the slope-area analysis. Meanwhile, via the integral approach, a steady-state river profile can be expressed as a continuous function, which can provide theoretical principle for some geomorphic parameters (e.g., slope-length index, hypsometric integral). In addition, we can determine the drainage network migration direction using this method. Therefore, the integral approach can be used as a better method for tectonogeomorphic research.     

Estimating restorable wetland water storage at landscape scales

Globally, hydrologic modifications such as ditching and subsurface drainage have significantly reduced wetland water storage capacity (i.e., volume of surface water a wetland can retain) and consequent wetland functions. While wetland area has been well documented across many landscapes and used to guide restoration efforts, few studies have directly quantified the associated wetland storage capacity. Here, we present a novel raster‐based approach to quantify both contemporary and potential (i.e., restorable) storage capacities of individual depressional basins across landscapes. We demonstrate the utility of this method by applying it to the Delmarva Peninsula, a region punctuated by both depressional wetlands and drainage ditches. Across the entire peninsula, we estimated that restoration (i.e., plugging ditches) could increase storage capacity by 80%. Focusing on an individual watershed, we found that over 59% of restorable storage capacity occurs within 20 m of the drainage network, and that 93% occurs within 1 m elevation of the drainage network. Our demonstration highlights widespread ditching in this landscape, spatial patterns of both contemporary and potential storage capacities, and clear opportunities for hydrologic restoration. In Delmarva and more broadly, our novel approach can inform targeted landscape‐scale conservation and restoration efforts to optimize hydrologically mediated wetland functions.     

Scaling relations for seismic events induced by mining

The values of seismic moment andS-wave corner frequency from 1575 seismic events induced in South African, Canadian, Polish, and German underground mines were collected to study their scaling relations. The values ofP-wave corner frequency from 649 events were also available. Seismic moments of these events range from 5*10³ to 2*10¹⁵ N·m (moment magnitude is from –3.6 to 4.1), theS-wave corner frequency ranges from 0.7 to 4438 Hz, and theP-wave corner frequency is between 5 and 4010 Hz. The slope of a regression line between the logarithm ofS- andP-wave corner frequencies is equal to one, and the corner frequencies ofP waves are higher than those ofS waves on the average by about 25 percent. In studies of large and moderate earthquakes it has been found that stress drop is approximately independent of the seismic moment, which means that seismic moment is inversely proportional to the third power of corner frequency. Such a behavior was confirmed for most of the data considered here. A breakdown in the similarity betwen large and small events seems to occur for the events with moment magnitude below –2.5. The average values of seismic moment referred to the same range of corner frequency, however, are vastly different in various mining areas.     

Hack's law of debris-flow basins

Hack's law was originally derived from basin statistics for varied spatial scales and regions.The exponent value of the law has been shown to vary between 0.47 and 0.70,causing uncertainty in its application.This paper focuses on the emergence of Hack's law from debris-flow basins in China.Over 5,000 debris-flow basins in different regions of China with drainage areas less than 100km2 are included in this study.Basins in the different regions are found to present similar distributions.Hack's law is derived fi'om maximum probability and conditional distributions,suggesting that the law should describe some critical state of basin evolution.Results suggest the exponent value is approximately 0.5.Further analysis indicates that Hack's law is related to other scaling laws underlying the evolution of a basin and that the exponent is not dependent on basin shape but rather on the evolutionary stage.A case study of a well known debris-flow basin further confirms Hack's law and its implications in basin evolution.