Bayesian methods for estimating multi-segment discharge rating curves

This study explores Bayesian methods for handling compound stage–discharge relationships, a problem which arises in many natural rivers. It is assumed: (1) the stage–discharge relationship in each rating curve segment is a power-law with a location parameter, or zero-plane displacement; (2) the segment transitions are abrupt and continuous; and (3) multiplicative measurement errors are of equal variance. The rating curve fitting procedure is then formulated as a piecewise regression problem where the number of segments and the associated changepoints are assumed unknown. Procedures are developed for describing both global and site-specific prior distributions for all rating curve parameters, including the changepoints. Estimation and uncertainty analysis is evaluated using Markov chain Monte Carlo simulation (MCMC) techniques. The first model explored accounts for parameter and model uncertainties in the interpolated area, i.e. within the range of available stage–discharge measurements. A second model is constructed in an attempt to include the uncertainty in extrapolation, which is necessary when the rating curve is used to estimate discharges beyond the highest or lowest measurement. This is done by assuming that the rate of changepoints both inside and outside the measured area follows a Poisson process. The theory is applied to actual data from Norwegian gauging stations. The MCMC solutions give results that appear sensible and useful for inferential purposes, though the latter model needs further efforts in order to obtain a more efficient simulation scheme.     

Modeling discharge rating curves with Bayesian B-splines

River discharges are traditionally modeled by employing a standard power-law methodology. Recently, the Bayesian approached has successfully been applied to improve the estimates of the standard power-law. In this article, an extension to the standard power-law based on Bayesian B-splines is developed and tested on data sets from 61 different rivers. The extended model is evaluated against the standard power-law using two measures, the Deviance Information Criterion and Bayes factor. The extended model captures deviations in the data from the standard power-law but reduces to the standard power-law when that model is adequate. The standard power-law is inadequate for 26% of the rivers while the extended model provides an adequate fit in all of those cases and for the remaining 74% of the rivers the extended model and the power-law model both give adequate fit with almost identical estimates.     

Bayesian power-law regression with a location parameter,with applications for construction of discharge rating curves

This paper presents a Bayesian approach for fitting the standard power-law rating curve model to a set of stage-discharge measurements. Methods for eliciting both regional and at-site prior information, and issues concerning the determination of prior forms, are discussed. An efficient MCMC algorithm for the specific problem is derived. The appropriateness of the proposed method is demonstrated by applying the model to both simulated and real-life data. However, some problems came to light in the applications, and these are discussed.     

Uncertainty in streamflow rating curves: methods,controls and consequences

Discharge time series' are one of the core data sets used in hydrological investigations. Errors in the data mainly occur through uncertainty in gauging (measurement uncertainty) and uncertainty in determination of the stage–discharge relationship (rating curve uncertainty). Thirty‐six flow gauges from the Namoi River catchment, Australia, were examined to explore how rating curve uncertainty affects gauge reliability and uncertainty of observed flow records. The analysis focused on the deviations in gaugings from the rating curves because standard (statistical) uncertainty methods could not be applied. Deviations of greater/lesser than 10% were considered significant to allow for a measurement uncertainty threshold of 10%, determined from quality coding of gaugings and operational procedures. The deviations in gaugings were compared against various factors to examine trends and identify major controls, including stage height, date, month, rating table, gauging frequency and quality, catchment area and type of control. The analysis gave important insights into data quality and the reliability of each gauge, which had previously not been recognized. These included identification of more/less reliable periods of record, which varied widely between gauges, and identification of more/less reliable parts of the hydrograph. Most gauges showed significant deviations at low stages, affecting the determination of low flows. This was independent of the type of gauge control, with many gauges experiencing problems in the stability of the rating curve, likely as a result of sediment flux. The deviations in gaugings also have widespread application in modelling, for example, informing suitable calibration periods and defining error distributions. This paper demonstrates the value and importance of undertaking qualitative analyses of observed records. Copyright © 2012 John Wiley & Sons, Ltd.     

Trend analyses with river sediment rating curves

Sediment rating curves, which are fitted relationships between river discharge (Q) and suspended‐sediment concentration (C), are commonly used to assess patterns and trends in river water quality. In many of these studies, it is assumed that rating curves have a power‐law form (i.e. C = aQ^b, where a and b are fitted parameters). Two fundamental questions about the utility of these techniques are assessed in this paper: (i) how well to the parameters, a and b, characterize trends in the data, and (ii) are trends in rating curves diagnostic of changes to river water or sediment discharge? As noted in previous research, the offset parameter, a, is not an independent variable for most rivers but rather strongly dependent on b and Q. Here, it is shown that a is a poor metric for trends in the vertical offset of a rating curve, and a new parameter, â, as determined by the discharge‐normalized power function [C = â (Q/Q_GM)^b], where Q_GM is the geometric mean of the Q‐values sampled, provides a better characterization of trends. However, these techniques must be applied carefully, because curvature in the relationship between log(Q) and log(C), which exists for many rivers, can produce false trends in â and b. Also, it is shown that trends in â and b are not uniquely diagnostic of river water or sediment supply conditions. For example, an increase in â can be caused by an increase in sediment supply, a decrease in water supply or a combination of these conditions. Large changes in water and sediment supplies can occur without any change in the parameters, â and b. Thus, trend analyses using sediment rating curves must include additional assessments of the time‐dependent rates and trends of river water, sediment concentrations and sediment discharge. Published 2014. This article is a U.S. Government work and is in the public domain in the USA. Hydrological Processes published by John Wiley & Sons Ltd.     

The use of invariant MTS curves in deep magnetotelluric studies

Invariant amplitude curves of the phase tensor are shown to be applicable to the geoelectric model reconstruction in the lower part of the section even if its upper part includes geoelectric heterogeneities described by the tensor of galvanic distortions. The use of other types of invariant curves provides reliable constraints on the deep structure of the section only if the upper part of the section does not contain contrasting geoelectric heterogeneities.     

Existence of the frequentistic estimate for power-law regression with a location parameter, with applications for making discharge rating curves

Practical application of the power-law regression model with an unknown location parameter can be plagued by non-finite least squares parameter estimates. This presents a serious problem in hydrology, since stream flow data is mainly obtained using an estimated stage–discharge power-law rating curve. This study provides a set of sufficient requirements for the data to ensure the existence of finite least squares parameter estimates for a power-law regression with an unknown location parameter. It is shown that in practice, these requirements act as necessary for having a finite least squares solution, in most cases. Furthermore, it is proved that there is a finite probability for the model to produce data having non-finite least squares parameter estimates. The implications of this result are discussed in the context of asymptotic predictions, inference and experimental design. A Bayesian approach to the actual regression problem is recommended.     

Fitting concentration and load rating curves with generalized linear models

Generalized linear models may be used as a systematic and flexible alternative to existing practices in fitting sediment or solute concentration and load rating curves. Through widely available statistical software they offer a one‐step solution to the common and awkward problem that a bias correction is needed when estimating total loads. In particular, using a logarithmic link function has an advantage over logarithmic transformation of concentration or load, as generalized linear models return predictions directly on the original measured scale; there is thus no need for back‐transformation or subsequent correction factors. Example analyses are given for suspended sediment concentration for an upland stream in the Northern Pennines and for phosphorus load for the Illinois River. General criteria for choosing models and good practice in the use of figures of model merit and of residual plots are also discussed in detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Seismic fragility curves for greek bridges: methodology and case studies

This study focusses on the estimation of seismic fragility curves for all common bridge types found in modern greek motorways. At first a classification scheme is developed in order to classify the existing bridges into a sufficient number of classes. A total of 11 representative bridge classes resulted, based on the type of piers, deck, and pier-to-deck connection. Then an analytical methodology for deriving fragility curves is proposed and applied to the representative bridge models. This procedure is based on pushover analysis of the entire bridge and definition of damage states in terms of parameters of the bridge pushover curves. The procedure differentiates the way of defining damage according to the seismic energy dissipation mechanism in each bridge, i.e. bridges with yielding piers of the column type and bridges with bearings (with or without seismic links) and non-yielding piers of the wall type. The activation of the abutment-backfill system due to closure of the gap between the deck and the abutments is also taken into account. The derived fragility curves are subjected to a first calibration against empirical curves based on damage data from the US and Japan.     

Internal erosion of soil pipes: Sediment rating curves for soil pipes

The collapse of soil pipes due to internal erosion can result in fully mature gullies. Few studies have measured the rates of sediment detachment and transport through soil pipes in situ. The objectives of this work were to determine suspended sediment concentration (SSC) in soil pipes as a function of pipeflow rate to develop sediment rating curves (SRC) and measure the bedload transport as a function of cumulative flow per storm event. H-flumes were installed in seven discontinuous gullies formed by pipe collapse and instrumented for pipe discharge measurements and suspended sediment sampling. The typical response to pipeflow was an initial flush of high concentration of suspended sediment followed by a decrease as pipeflow increased (rising limb of hydrograph). Pipeflows were often so dynamic that it was difficult to consistently capture the initial flush of sediment, resulting in weak to non-existent SRCs. The falling limb of the hydrograph tended to have a relatively low SSC. Thus, soil pipe SRCs tended to be better represented by hysteretic SRCs, although relationships between SSC and flow rate were poorly represented by SRCs. A power law equation given by SSC = aQ^b was adopted to represent the SRC relationships. Fitting this equation to data showed a correlation between the offset, a, and the slope, b, with the slope decreasing as the offset increases. Both SRC parameters (a and b) were correlated to the contributing area of the individual pipe. Bedload appeared to be an important contributor to sediment transport, with bedload – expressed as an average event sediment concentration (mg l⁻¹) – decreasing as the volume of the event discharge (m³) increased. A significant portion (11–31%) of the bedload material was gravel and aggregates (>2 mm diameter material). While this work was the first to determine SRCs for soil pipes, refined sampling and measurement techniques are needed. © 2020 John Wiley & Sons, Ltd.     

Simple vs complex rating curves: accounting for measurement uncertainty,slope ratio and sample size

Multiple segmented rating curves have been proposed to better capture the variability of the physical and hydraulic characteristics of river–floodplain systems. We evaluate the accuracy of one- and two-segmented rating curves by exploiting a large and unique database of direct measurements of stage and discharge data in more than 200 Swedish catchments. Such a comparison is made by explicitly accounting for the potential impact of measurement uncertainty. This study shows that two-segmented rating curves did not fit the data significantly better, nor did they generate fewer errors than one-segmented rating curves. Two-segmented rating curves were found to be slightly beneficial for low flow when there were strong indications of segmentation, but predicted the rating relationship worse in cases of weak indication of segmentation. Other factors were found to have a larger impact on rating curve errors, such as the uncertainty of the discharge measurements and the type of regression method.     

An evaluation of sediment rating curves for estimating suspended sediment concentrations for subsequent flux calculations

In the absence of actual suspended sediment concentration (SSC) measurements, hydrologists have used sediment rating (sediment transport) curves to estimate (predict) SSCs for subsequent flux calculations. Various evaluations of the sediment rating‐curve method were made using data from long‐term, daily sediment‐measuring sites within large (>1 000 000 km²), medium (<1 000 000 to >1000 km²), and small (<1000 km²) river basins in the USA and Europe relative to the estimation of suspended sediment fluxes. The evaluations address such issues as the accuracy of flux estimations for various levels of temporal resolution as well as the impact of sampling frequency on the magnitude of flux estimation errors. The sediment rating‐curve method tends to underpredict high, and overpredict low SSCs. As such, the range of errors associated with concomitant flux estimates for relatively short time‐frames (e.g. daily, weekly) are likely to be substantially larger than those associated with longer time‐frames (e.g. quarterly, annually) because the over‐ and underpredictions do not have sufficient time to balance each other. Hence, when error limits must be kept under ±20%, temporal resolution probably should be limited to quarterly or greater. The evaluations indicate that over periods of 20 or more years, errors of <1% can be achieved using a single sediment rating curve based on data spanning the entire period. However, somewhat better estimates for the entire period, and markedly better annual estimates within the period, can be obtained if individual annual sediment rating curves are used instead. Relatively accurate (errors <±20%) annual suspended sediment fluxes can be obtained from hydrologically based monthly measurements/samples. For 5‐year periods or longer, similar results can be obtained from measurements/samples collected once every 2 months. In either case, hydrologically based sampling, as opposed to calendar‐based sampling is likely to limit the magnitude of flux estimation errors. Published in 2003 John Wiley & Sons, Ltd.     

Factors controlling variations in suspended sediment concentration for single-valued sediment rating curves,Fraser River,British Columbia,Canada

Functional (regression) and cause-and-effect analytical methods were employed in the investigation of factors controlling single-value sediment–discharge relationships for individual hydrological events in the Fraser River Basin. Of 1025 hydrological events identified in the 97 years of the 9-stations discharge record monitored by the Water Survey of Canada (1960–1988), 49 were associated with linear, non-linear (concave and convex) sediment rating curves. Analysis revealed that time of events offered a partial explanation for the occurrence of single-valued curves because events occurring in isolation or in quick succession did not always produce single-valued curves owing to variations in hydrograph characteristics and temporal and spatial peak attenuation factors. Hydrologically, linear and non-linear sediment rating curves were distinguished by the influence of preceding discharge or antecedent moisture, which tends to generate quick or delayed runoff, thereby causing rapid or slow increases in sediment concentration in concert with discharge changes. Hydraulically, linear curves were shown to be controlled by rates of stream bed scour (rising stage) similar to those for channel filling (falling stage); concave curves by rates of scouring (rising stage) similar to those of filling but different from those of rescouring (falling stage); and convex curves by rates of filling and scouring (rising stage) different from those of filling and rescouring (falling stages), all in relation to the timing of the exceedance of the threshold discharge for stream bed scour. Meteorologically, bed scour and fill approximately coincided with the timing of precipitation, if any, on the rising and/or falling stages under subzero (T≤0 °C), low (1≤T≤9 °C) and moderate (10≤T≤19 °C) temperature conditions. It is concluded that a combination of temporal, spatial, hydrological, hydraulic and meteorological factors control variations in sediment concentration during single hydrological events on the Fraser River in a complex fashion. Multivariate analysis of these factors should greatly improve prediction of sediment transport in the Fraser River Basin. © 1998 John Wiley & Sons, Ltd.     

Post‐wildfire changes in suspended sediment rating curves: Sabino Canyon,Arizona

Wildfire has been shown to increase erosion by several orders of magnitude, but knowledge regarding short‐term variations in post‐fire sediment transport processes has been lacking. We present a detailed analysis of the immediate post‐fire sediment dynamics in a semi‐arid basin in the southwestern USA based on suspended sediment rating curves. During June and July 2003, the Aspen Fire in the Coronado National Forest of southern Arizona burned an area of 343 km². Surface water samples were collected in an affected watershed using an event‐based sampling strategy. Sediment rating parameters were determined for individual storm events during the first 18 months after the fire. The highest sediment concentrations were observed immediately after the fire. Through the two subsequent monsoon seasons there was a progressive change in rating parameters related to the preferential removal of fine to coarse sediment. During the corresponding winter seasons, there was a lower supply of sediment from the hillslopes, resulting in a time‐invariant set of sediment rating parameters. A sediment mass‐balance model corroborated the physical interpretations. The temporal variability in the sediment rating parameters demonstrates the importance of storm‐based sampling in areas with intense monsoon activity to characterize post‐fire sediment transport accurately. In particular, recovery of rating parameters depends on the number of high‐intensity rainstorms. These findings can be used to constrain rapid assessment fire‐response models for planning mitigation activities. Copyright © 2006 John Wiley & Sons, Ltd.     

Evaluating spatial variation of suspended sediment rating curves in the middle Yellow River basin,China

Abstract The suspended sediment load in the middle Yellow River basin (YRB) cannot be well predicted by capacity‐based transport formulas because a large fraction of suspended sediment load is composed of wash load. This study evaluated the spatial variations of sediment rating curves (SRCs) in the middle YRB. Both power and linear SRCs were used to fit daily flow and suspended sediment concentration (SSC) historical data at 49 gauging stations throughout the middle YRB. The spatial variation in regression coefficients was investigated, and the relationship between regression coefficients and the physical characteristics of watersheds was discussed. The results indicate that SRC regression coefficients vary with drainage area and basin slope, but their responses to these parameters are remarkably different in watersheds with different underlying surfaces, which indicates the significance of sediment availability, erodibility, and grain size distribution. For power SRCs representing sediment transport in unsaturated flows, the regression coefficients are more closely correlated with the drainage area in loess regions and with the basin slope in rock mountain regions. For linear SRCs representing sediment transport in saturated flows, saturated SSCs vary with coarse (particle size > 0.05 mm) and fine (particle size < 0.01 mm) fractions in suspended sediment. The maximum saturated SSC among the different gauging stations is associated with the optimal grain size composition of suspended sediment, which has been proposed for loess regions in previous studies. This study provides theoretical support for estimating the regression parameters for sediment transport modelling, especially in ungauged basins.     

An investigation of suspended sediment rating curves in western and northern Greece

ABSTRACT

The suspended sediment rating curves for six stations on four rivers in western and northern Greece are investigated. For each station the suspended sediment load is a power function of the water discharge, which may be distinguished according to wet and dry seasons; the latter yields higher sediment loads for a given discharge than the former. This is due to the higher erosivity of dry season rainfall compared to wet season rainfall producing the same runoff. All rating curve exponents b lie in the range 2.5–3.5 for the wet and 2.0–3.0 for the dry season and are related to the constants a of the curves by empirical equations. The variation in a and b is explained in terms of the annual precipitation and area of the basin, the hypsometric fall, the main channel length, and the average bedslope of the river from the basin divide to the station, through empirical relationships, which also permit the prediction of rating curves for ungauged basins.     

The role of science in setting water resource use limits: case studies from New Zealand

Abstract

Water resource use limits ensure protection of environmental values and define the availability and reliability of water supply for out-of-channel use. We examined how three types of scientific tools (environmental flow setting methods, hydrological analyses for setting total allocations and spatial frameworks) have been used to define limits across jurisdictional regions comprising multiple catchments in New Zealand. We found that recently developed minimum flow and total allocation setting tools are widely used. Spatial frameworks are increasingly used to discriminate and account for variation in environmental characteristics, thereby increasing the specificity of water resource use limits. The uptake of scientific tools has enabled improvements in the clarity of water management objectives and the transparency of limits defined by regional water management plans. We argue that more integrated use of scientific tools could improve the clarity and transparency of regional limits by explicitly demonstrating the trade-off between out-of-channel use and protection of environmental values.

Editor D. Koutsoyiannis; Guest editor M. Acreman

Citation Snelder, T.H., Rouse, H.L., Franklin, P.A., Booker, D.J., Norton, N., and Diettrich, J., 2014. The role of science in setting water resource use limits: case studies from New Zealand. Hydrological Sciences Journal, 59 (3–4), 844–859.     

黄土塬区高精度重力勘探的校正技术

黄土塬区地形起伏大,地表黄土切割剧烈。地形起伏导致黄土厚度巨大变化,由于黄土密度与下伏地层之间存在较大的密度差异,由此产生了具有一定幅值的高频重力异常,并给重力勘探资料的处理、解释带来了困难。在SHJZ构造带高精度重力勘探中,综合运用了地形校正和黄土校正技术,有效地克服了由于地形起伏和黄土厚度变化对重力勘探产生的不利影响,取得了精确的重力资料。并将该重力异常,结合其它物探资料进行综合解释,对圈定局部构造发挥了积极的作用。