stage–discharge relationships in open channels: practices and

STAGE–DISCHARGE RELATIONSHIPS IN OPEN

CHANNELS: PRACTICES AND PROBLEMS

Giovanni BRACA

FORALPS Project Manager: Dino Zardi

Publisher:Università degli Studi di TrentoDipartimento di Ingegneria Civile e Ambientale

Trento, March 2008

ISBN 978–88–8443–230–8

Cover design: Marco Aniello

Printed in Italy by : Grafi che Futura s.r.l.

Editorial Team: Stefano Serafi n, University of Trento, Italy

Marta Salvati, Regional Agency for Environmental Protection of Lombardia, Italy

Stefano Mariani, National Agency for Environmental Protection and Technical Services, Italy

Fulvio Stel, Regional Agency for Environmental Protection of Friuli Venezia Giulia, Italy

Partial or complete reproduction of the contents is allowed only with full citation as follows:Giovanni Braca, 2008: Stage–discharge relationships in open channels: Practices and problems. FORALPS Technical Report, 11. Università degli Studi di Trento, Dipartimento di Ingegneria Civile e Ambientale, Trento, Italy, 24 pp.

The project FORALPS has received European Regional Development Funding through the INTERREG IIIB Alpine Space Community Initiative Programme.

Stage-discharge relationships in open channels: Practices and problems Giovanni BRACA Agency for Environmental Protection and Technical Services, Rome, Italy Contact: [email protected]

mailto:[email protected]

INTERREG IIIB FORALPS

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Contents

1 Introduction 5 2 Brief literature review 6 3 Stage-discharge relationships 7

3.1 Hydraulics governing stage-discharge relationships 8 3.2 Rating curves from steady uniform flow 11 3.3 Difficulties in defining stage-discharge relationships 12

4 Operational indications 14 4.1 Disregarding measured data 14 4.2 Representation of stage-discharge relationships 14 4.3 Some issues in plotting and analysing stage discharge relationships 17 4.4 Rating curves in particular conditions 21 4.5 Uncertainty of stage-discharge relationships 22

5 References 23

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1 Introduction

The empirical, or also theoretical, relationship existing between the water-surface stage (i.e. the water level) and the simultaneous flow discharge in an open channel is known as stage-discharge relation or rating curve, or also just rating. These expressions are synonymous and they can be used interchangeably. The rating curve is a very important tool in surface hydrology because the reliability of discharge data values is highly dependent on a satisfactory stage-discharge relationship at the gauging station. Although the preparation of rating curves seems to be an essentially empiric task, a wide theoretical background is needed to create a reliable tool to switch from measured water height to discharge. The rating curve has been and is currently a extensively used tool in hydrology to estimate discharge in natural and/or artificial open channel. Since the early XIX century it is a common practice to measure the discharge of streams at suitable times, usually by a current meter or other methods (Rantz et al. 1982a; ISO 1100-1, 1998; SIMN 1998). Meanwhile, the corresponding stage is also measured; a curve of discharge against stage can then be built by fitting these data with a power or polynomial curve, looking like the one in Figure 1. The traditional and simple way to gather information on current discharge is then to measure the water level with gauges and to use the stage-discharge relationship to estimate the flow discharge. It is well known, in fact, that direct measurements of discharge in open channels is costly, time consuming, and sometimes impractical during floods.

Figure 1. Example of a stage-discharge relation (from ARPA Veneto, 2006).

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Several methods have been proposed to improve data fitting, but generally they have not adequately assessed the fundamentals of stage-discharge ratings based on fluid mechanics. As a consequence, several difficulties with stage-discharge ratings have been recognized. For example, in some cases, the relation between stage and discharge is not unique. The water surface slope, in fact, produces different discharges for the same stage. The present report does not describe all the methods for determining rating curves in detail, but it refers to the existing large body of technical literature for full descriptions. Obviously, dealing with a hydraulics subject, we widely use hydraulics concepts as, for example, uniform or normal flow, steady and unsteady flow, sediment transport and so on. Therefore we assume the reader is aware of this matter. Anyway we refer to Chow (1959) for a more detailed explanation of basic hydraulics, which are considered as understood in this report.

2 Brief literature review

In this chapter we report a brief review of existing technical and scientific literature, in order to describe the state-of-the-art about understanding, establishing, and applying stage-discharge ratings. The subject might look outdated and of low interest, but it maintains a great practical importance and a high degree of interest, particularly from the viewpoint of improving the estimation of discharge from the stage. For instance, as all evaluations of discharge are strongly dependent from stage-discharge relationships, an improvement on discharge estimates would obviously make hydrological models more reliable. Three main subjects on the topic of rating curves can be outlined: 1. the first and more deeply investigated theme is how to improve empirically the discharge

records from gauging stations for different conditions, and how to extrapolate the data beyond the measurement range;

2. the second one and less examined issue, is related to the understanding of the physical phenomena that these ratings are attempting to describe;

3. the third point includes studies on the uncertainty of stage-discharge relations. The current practice of defining and applying stage-discharge ratings can be traced back to the early XX century. At beginning of past century, for example, Jones (1916) already proposed a method to correct stage-discharge relations taking into account the surface water slope. In the same period (1914) the “Ufficio Idrografico del Magistrato alle Acque di Venezia” (Venice Water Authority) provided instructions to define the stage discharge relations for a watercourse. A great portion of the modern practices used worldwide were developed by the United States Geological Survey (USGS). The methods that are currently in use are widely described in USGS publications of Corbett et al. (1943), Dawdy (1961), Bailey and Ray (1966), Kennedy (1984), Rantz (1963), Rantz et al. (1982a, 1982b), as well as in the World Meteorological Organization (WMO) Publication n. 519, Operational Hydrology Report n.13 (1980), and in the International Organization for Standardization (ISO) Regulation n. 1100-2 1998 (ISO, 1998). Other sources are the National Engineering Handbook (Pasley et al., 1972), the “Manuale per il monitoraggio

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idrografico” (italian, Becchi et al., 1994) and the book “Streamflow Measurement” by Herschy (1995). These publications are widely referred to in the following chapters. More recent speculations on the subject can be found in scientific literature. For instance, referring to the first one of the topics listed above, several papers recently proposed new approaches in extrapolating discharge rating curves. Sivapragasam and Muttil (2005) suggest the use of Support Vector Machine (SVM, Cristianini and Shawe-Taylor, 2000) in the extrapolation of rating curves, which work on the principle of linear regression on a higher dimensional feature space. On the other hand, Bhattacharya and Solomatine (2000) already had used Artificial Neural Networks (ANNs), widely used in various areas of water-related research, for defining stage discharge relations. A comparison to a conventional statistical stage-discharge model have shown the superiority of an approach using ANNs. Deka and Chandramouli (2003) tested several modern approach to the problem: a neural network model, a modularized neural network model, a conventional curve-fitting approach and a fuzzy neural network model are compared using a case study. Overall, the fuzzy neural network model gives the best results. About topic 2, Schmidt and Yen (2001) examined the relationship of stage versus discharge in open channels based on the fundamental hydrodynamics of unsteady non-uniform flow, and identified terms in the Saint-Venant equations that should be considered in rating development. They use a tool, named Hydraulic Performance Graph, HPG, developed by Yen and Gonzalez (1994) which provides a steady-state approximation to the hydrodynamics governing open-channel flow, to develop theoretical ratings. The HPG summarizes all the backwater profile information for the channel reach in exam. Finally, different researchers have examined the uncertainties in empirical stage-discharge ratings and the resulting uncertainties in the discharge records from these stations (topic 3). Several publications deal with the uncertainty in defining rating curves or the influence of uncertainty in peak discharge evaluation (Clarke, 1999, Clark et al., 2000, Claps et al., 2003; Parodi and Ferraris, 2004; Aronica et al., 2005, 2006). Because ratings are empirical curves, the uncertainty analyses are generally limited to a statistical examination of such curves. Two further specific issues which deserve mention are (1) the use of hydraulic numerical methods to define a rating curve and its extrapolation beyond the range of measure (Franchini et al., 1999), and (2) the possibility of reconstructing a rating curve at a river section where only water level is monitored, using the discharge recorded at an upstream section (Barbetta et al., 2003).

3 Stage-discharge relationships

As repeatedly said before, continuous records of discharge at gauging stations are computed by applying a discharge rating to stage data. The discharge rating curve transforms the continuous stage data to a continuous record of stream discharge, but it is also used to transform model forecasted flow hydrographs into stage hydrographs. This is needed, for instance, to estimated the inundated areas during a flood. Discharge rating curves may be simple or complex depending on the river reach and flow regime. These relations are typically developed empirically from periodic measurements of stage and

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discharge. These data are plotted versus the concurrent stage to define the rating curve for the stream. For new gauging stations, many discharge measurements are needed to develop the stage discharge relation throughout the entire range of stream flow data. Generally, periodic measurements are needed to validate the underlying stage-discharge relationship and to track changes or shifts in the rating curve. The ISO regulation 1100-2 (ISO, 1998) recommends at least 12-15 discharge measurements during the period of analysis. Of extreme importance is the capability of the stage-discharge relation to be applicable for extreme flow conditions. Discharge measurements are usually missing in the definition of the upper and lower end of the rating curve. The extrapolation of these data are subject to serious errors that can have significant implications for flood management (upper curve) and for water resources planning (lower curve). Note that the uncertainties related to extrapolation can be reduced if indirect methods of determining unmeasured peak discharge (for example a rainfall-runoff model) are used.

3.1 Hydraulics governing stage-discharge relationships

The stage-discharge relation for open-channel flow at a gauging station is governed by channel conditions downstream from the gauge. Knowledge of the channel features that control the stage-discharge relation is therefore a crucial component in developing rating curves. Three types of controls can be recognized, depending on the channel and flow conditions: – low flows are usually influenced by a section control;

Figure 2. Example of a section control.

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– high flows are usually influenced by a channel control; – medium flows can be influenced by both type of controls. At some stages, in fact, a

combination of section and channel control can occur. A section control is a specific cross-section of a stream channel, located downstream from a water-level gauge, that controls the relation between gauge height and discharge at the gauge. A section control can be a natural feature such as a rock ledge, a sand bar, a severe constriction in the channel, or an accumulation of debris. Likewise, a section control can be a man-made feature such as a small dam, a weir, a flume, or an overflow spillway. Section controls can frequently be visually identified in the field by observing a riffle, i.e. pronounced drop in the water surface, as the flow passes over the control (Figure 2). Frequently, as gauge height increases because of higher flows, the section control will become submerged in such a way that it no longer controls the relation between gauge height and discharge. At this point, the riffle is no longer observable, and flow is then regulated either by another section control further downstream or by the hydraulic geometry and roughness of the channel downstream (i.e. channel control). A channel control consists of a combination of features throughout a reach downstream from a gauge. These features include channel size, shape, curvature, slope, and roughness. The length of channel reach that controls a stage-discharge relation can be extremely variable. The stage-discharge relation for relatively steep channels may be controlled by a relatively short channel reach, whereas the relation for a relatively flat channel may be controlled by a much longer channel reach. In addition, the length of a channel control will vary depending on the magnitude of flow. The definition of the length of a channel-control reach is usually very difficult and it could be done, for example, by numerical methods. At some stages, the stage-discharge relation may be governed by a combination of section and channel controls. This usually occurs for a short range in stage between section-controlled and channel-controlled segments of the rating (Figure 3). The development of stage-discharge curves where more than one control is effective or where control features change or where the number of measurements is limited, usually requires wide knowledge about the behavior of watercourse, in order to make a reliable interpolation between measurements and extrapolation beyond respectively the lowest or highest measurements. This is particularly true where the controls are not permanent and tend to shift from time to time, resulting in changes in the positioning of segments of the stage-discharge relation. The part of the rating where section control and channel control are both effective is commonly known as the transition zone. In other circumstances, a combination control may consist of two section controls, where each has partial controlling effect. More than two controls acting simultaneously is a generally rare condition. In any case, combination controls occur for limited parts of a stage-discharge relation and can usually be defined by plotting procedures. Transition zones, in particular, are characterized by changes in the slope or shape on a stage-discharge relation. It is easy to understand that stage-discharge relations for stable controls, such as a rock outcrop and man-made structures such as weirs, flumes, and small dams, usually present few problems in their calibration and maintenance. On the other hand, many difficulties can arise when controls are not stable and/or when variable backwater occurs. For unstable controls, segments of a stage-discharge relation may change position abruptly, corresponding to a severe flood, or even continuously.

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Figure 3. Schematic representation of controls range in rating curve.

Besides changes in section control, a further factor which can affect stage-discharge relations both for stable and unstable channels is variable backwater. Backwater effects occur when disturbances tend to propagate upstream. For example, the effect of a lake (slowing the flow down) or a waterfall (speeding the flow up) is felt upstream. Backwater effects are governed by a dimensionless number known as the Froude number:

(1)

where U is the flow velocity, h is the water depth and g is the gravity acceleration. For the case of subcritical flow with Fr < 1, backwater effects propagate upstream so that the effect of a disturbance is felt upstream as well as downstream. In the case of a supercritical flow, Fr > 1, the effect of a disturbance propagates only downstream. Sources of backwater can be downstream reservoirs, tributaries, tides, ice, dams and other obstructions that influence the flow at the upstream gauging station control. Finally, a further very important source of complexity, peculiar of some streams during unsteady flow, is hysteresis (also known as loop rating, see Figure 14) which results when the water surface slope changes due to either rapidly rising or rapidly falling water levels in a channel control reach. Hysteresis is most pronounced in flat sloped streams. On rising stages the water surface slope is significantly steeper than for steady flow conditions, resulting in greater discharge than indicated by the steady flow rating. The reverse is true for falling stages.

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Figure 4. Example of loop rating curve (hysteresis).

3.2 Rating curves from steady uniform flow

The most commonly used stage-discharge ratings treats the discharge as a unique function of the stage. These ratings typically follow a power curve of the form given by Equation 2 (Herschy, 1995; ISO 1998; Kennedy, 1984; Rantz et al., 1982b):

(2) Where Q is the discharge, h is the stage and C, a, α are calibration coefficients. C is the discharge when the effective depth of flow (h-a) is equal to 1; a is the gauge height of zero flow; α is the slope of the rating curve (on logarithmic paper); (h-a) is the effective depth of water on the control. When the exponent α approaches to 3/2 rating is also known as a Guglielmini rating curve (Ufficio Idrografico del Magistrato di Venezia, 1914). It is easy to see that Equation 2 is based on the Manning equation, which frequently is used as the governing equation for steady uniform flow problems:

(3)

where n is the Manning’s roughness coefficient, S0 is the bottom slope, A is the area and R is the hydraulic radius. However, Equation 2 is a simplification of the Manning equation, as it assumes that the conveyance function AR2/3 can be described by a simple power function of the water height as is the case, for example, of a wide rectangular cross section, where the following is valid:

(4)

For most stations, nevertheless, Equation 2 is an over-simplification. In general, the rating will be a compound curve consisting of different segments for different flow ranges. Each of these

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Figure 5. Example of a typical simple rating curve.

segments may follow the form of Equation 2, but have unique values of C, a, and α. The segments are typically connected by short transition curves. Statistical methods have been developed to fit curves in the form of Equation 2 or polynomial curves to measured stages and discharges (Herschy, 1995). However, for most natural streams, graphical fitting of a curve to the measured data is the preferred method (ISO, 1998).

3.3 Difficulties in defining stage-discharge relationships

Two conceptual approaches have been followed to justify the definition of a unique relation between the stage and discharge for an open channel: – treating the discharge as open-channel flow with a constant slope for a given stage

namely uniform (or normal) flow. Anyway, it is known that this occurs only theoretically in a prismatic channel. In the case of fixed-bed channels the term AR is constant for a given stage.

– treating the discharge as flow over a weir (where critical flow conditions occur, with a single value relation between stage and discharge).

However, in natural channels the water-surface slope varies for unsteady flow, the cross section changes with sediment deposition and erosion, and the resistance coefficient changes with bed and flow conditions. The relation between stage and discharge can be modified by a great number of factors that result in changes in the shape and position of the rating curve, or in loops in the rating curve. Principal factors that affect the rating curve include (Herschy, 1995; Kennedy, 1984; Rantz et al., 1982b): changes to the channel cross section due mainly to scour and fill; growth and decay of aquatic

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vegetation; log or debris jams (an accumulation of logs and other organic debris which blocks the flow of a stream of water); variable backwater; rapidly changing discharge; discharge to or from overbank areas; ice. Variable backwater, rapidly changing discharge, and flow to or from overbank areas all result in looped or non-unique ratings and are typically addressed through including additional parameters, such as an estimate of the water surface slope or the rate of change of the water surface at the gauge. So, when the type of flow departs significantly from the steady flow state, the simple stage-discharge relation is no longer sufficient to define the discharge. Another parameter should be included, i.e. the slope of water surface. Essentially, in these conditions the ordinary approach, i.e. using the single-valued stage-discharge rating for the computation of discharge records, is not applicable: the discharge rating under conditions of variable backwater and for highly unsteady flow cannot be defined by stage alone.

Figure 6. Typical effects of different physical processes on rating curves (from Herschy, 1995).

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4 Operational indications

4.1 Disregarding measured data

Discharge measurements, on which rating curves are based, can have different levels of accuracy depending on the conditions in which they are executed. Some rating curves would seem to fit the data more closely if one or two of the discharge measurements were eliminated from the analysis. Sometimes an outlier measurement is obviously and seriously in error, but, more often, the measurement is satisfactory and important to the rating analysis. Hydrologists, before defining rating curves, should therefore analyze the accuracy of measurements. Low and medium water measurements are, in fact, normally made by using standard procedures (ISO 4373, 1995; Boiten, 2003), and their errors rarely exceed 5 percent (Kennedy, 1984). A flood measurement, on the other hand, may be made at night or may involve road overflow or covered culvert flow, rapidly changing stage, improvised equipment, drift, or other conditions that reduce accuracy. Anyway, even these measurements are rarely in error by more than 10 percent. A stream may be so flashy that the best possible measurement may consist of only a few surface velocities, and the error may be as much as 15 percent. Indirect measurements are not usually made unless the conditions promise accuracy within about 20 percent. Such relatively large variations from the ratings are acceptable for measurements of these type. Summer flood measurements, made while inundated trees and brush are covered with leaves, tend to plot to the left of winter flood measurements, but this seasonal effect can be corrected by shifting-control adjustments and should not be confused with measuring error. Left aside those with an unaccountably high percentage difference from the rating, no discharge measurement, made either by current meter or indirectly, should be disregarded without a good reason. Disagreement with other measurements is generally not a reason good enough. If an outlier measurement is truly an error, the reason for the disparity often can be discovered. The hydrologist should check the arithmetic of an outlier measurement, compare the mean gauge height with recorded and outside gauge heights, compare the plotted cross sections and velocity distributions for several measurements made at the same site, consider the possibility of backwater, and check the equipment used. If these checks and others indicate that the outlier is a valid discharge measurement, it should be given appropriate weight in the analysis.

4.2 Representation of stage-discharge relationships

A stage-discharge relation can be represented in various format: – graphic format trough a plotted curve; – table format, using generally two columns (one for stage and one for discharge); – equation format, by mathematical expression for different stage ranges. Each kind of representation, obviously, has advantages over the others for specific purposes. The first step before making a plot of stage versus discharge is to prepare a list of discharge measurements that will be used for the plot. At a minimum this list should include from 12 to 15 14

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measurements, all made during the period of analysis (ISO 1998). These measurements should be equally distributed over the range of gauge heights experienced, but this is not always achievable because different is the frequency of the various heights (frequency of occurrence of course reduces as height increases). The list of discharge measurements should also include low and high measurements from other times that might be useful in defining the correct shape of the rating and/or for extrapolating the rating. Extreme low and high measurements should be included wherever possible. For each discharge measurement in the list, the following items should be included at least: – unique identification number. – date of measurement. – gauge height of measurement. – total discharge. – accuracy of measurement. – rate-of-change in stage during measurement, a plus sign indicating rising stage and a

minus sign indicating falling stage. Other information might be included in the list of measurements, but it is optional. If the range of gauge height or discharge is large, it may be necessary to plot the rating curve in two or more segments to provide scales that are easily read with the necessary precision. This procedure may result in separate curves for low water, medium water, and high water. Care should be taken to see that, when joined, the separate curves form a smooth, continuous combined curve. The discharge and stage measurements are plotted on either rectangular (arithmetic) coordinate or logarithmic plotting paper. Graph paper with arithmetic scales is convenient to use and easy to read. Such scales are ideal for displaying a rating curve, and have an advantage over logarithmic scales, in that zero values of gauge height and/or discharge can be plotted. However, for analytical purposes, arithmetic scales

Figure 7. Example of logarithmic scale rating curve.

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have practically no advantage. A stage-discharge relation on arithmetic scales is almost always a curved line, concave downward, which can be difficult to shape correctly if only a few discharge measurements are available. Logarithmic scales, on the other hand, have a number of analytical advantages. Therefore, most stage-discharge relations, or segments of them, are usually analyzed graphically through the use of logarithmic plotting paper. To utilize fully this procedure, gauge height should be transformed to effective depth of flow on the control by subtracting from it the effective gauge height of zero discharge. A rating curve segment, then, for a given control will tend to plot as a straight line. The slope of the straight line will conform to the type of control (section or channel), thereby providing valuable information to shape the rating curve segment correctly.In addition, this feature allows the analyst to calibrate the stage-discharge relation with fewer discharge measurements. Finally, the slope of the rating curve is easily found as the ratio of the horizontal distance to the vertical distance. Generally, a stage-discharge relation is first drawn on logarithmic plotting paper for shaping and analytical purposes, and later transferred to arithmetic plotting paper if a display plot is needed.

4.2.1 Graphic format The graphic format of a rating shows the stage-discharge relation visually and simply and is the form used for the initial rating analysis. Each point represents a stage discharge pair. Usually the discharge (dependent variable) is preferable plotted as the abscissa in such a way the concavity is downward (like in the rating in Figure 8). The discharge measurements are numbered consecutively in chronological order to facilitate the identification of time trends (Figure 8). Hand drawn curves are also used to fit the stage and discharge measurements to produce a rating curve. Considerable judgment is normally exercised to decide on the best curve. For example,

Figure 8. Example of rating curve (from Herschy, 1995).

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knowledge of the river is applied and consideration is given to such factors as the quality and magnitude of each measurement. In the simplest case, when a single control exists, these curves would be practically straight lines on logarithmic scales.

Figure 9. Example of rating curve in table format and in equation format provided in Hydrological Yearbook from National Hydrographic Service of Italy.

4.2.2 Table and analytical format As mentioned above, rating curves can be represented in a table format, that can be plotted as piecewise linear curve. But a more suitable way for hand calculation is to represent stage-discharge relation in analytical formulas like the following:

(5) The National Hydrographic Service of Italy used to provide stage-discharge relationships in both formats in its Hydrological Yearbooks. The first range of ratings, corresponding to measurements, is provided in a table format that can be represented in a piecewise linear curve. The extrapolation range is instead expressed in analytical format using Guglielmini formulas (Ufficio Idrografico del Magistrato di Venezia, 1914, Figure 9).

4.3 Some issues in plotting and analysing stage discharge relationships

In the following sections, some particular issues about the plotting and analysis of stage discharge relations are highlighted.

4.3.1 Automatic and manual plotting Completely automatic rating analysis using the curve-fitting programs normally available for computers is technically practical, but discouraged (ISO 1998). The computer programs use a least squares fitting technique, but do not allow any judgment as to the quality of individual

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measurements, especially outliers, and about the hydraulic factors that are related to bends and/or breaks in rating curves. Extrapolation of an automatically fitted curve is generally unsatisfactory. Fitting an equation to a manually drawn curve by inputting selected points from that curve rather than from the observed data to a fitting program avoids the problem and is advised wherever the equation format is needed.

4.3.2 Gauge height of zero flow One of the most significant features in the stage discharge relationship is the stage at gauging station corresponding to the discharge close to zero, named the gauge height of zero flow (GZF). The real gauge height of zero flow is the gauge height of the lowest point in the control cross-section for a section control. For natural channels, this value can sometimes be measured in the field by measuring the depth of flow at the deepest place in the control section, and subtracting this depth from the gauge height at the time of measurement. The effective gauge height of zero flow is instead a value that, when subtracted from the mean gauge heights of the discharge measurements, will cause the logarithmic rating curve to plot as a straight line (Figure 10). This is the reason why the effective gauge height of zero flow is sometimes referred to as the logarithmic scale offset.

Figure 10. The three cases of datum correction for GZF (from Herschy, 1995).

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For regular shaped section controls, the effective gauge height of zero flow will be nearly the same as the actual gauge height of zero flow. At points where the control shape changes significantly, or where the control changes from section control to channel control, the effective gauge height of zero flow will usually change. This results in a need to analyze rating curves in segments (separate logarithmic plots for each control condition) to properly define the correct hydraulic shape for each control condition. The gauge height minus the effective gauge height of zero flow is the effective depth of flow on the control. The effective gauge height of zero flow should be determined for each rating curve segment. For regular shaped controls, this value will be close to the actual gauge height of zero flow. For most controls, however, a more exact determination can be made by a trial-and-error method of plotting. A value is assumed, and measurements are plotted based on this assumed value. If the resulting curve shape is concave upward, then a somewhat larger value for the effective gauge height of zero flow should be used. A somewhat smaller value should be used if the curve plots concave downward. Usually only a few trials are needed to find a value that results in a straight fine for the rating curve segment.

4.3.3 Effects of control changes on rating curves Modifications in section or channel controls affect in various way the stage discharge relation. Understanding of these changes is fundamental in plotting and interpolate ratings. If the width of control increase, the coefficient C in Equation 5 increases and new rating will be parallel to the original rating, and shift towards the right. If the control scours, the coefficient a in Equation 5decreases and the depth (h-a) for a give gauge height increases; the new rating curve will then move to the right and will not longer be a straight line, but a curve that is concave downward. If the control becomes built up by deposition, the coefficient a in Equation 5 increases and the depth (h-a) decreases; the new rating then moves to the left and is non longer linear but concave upward.

4.3.4 Shifts in the discharge rating From the discussion above, it is apparent that the stage discharge relationship can vary over time because of changes in the physical features of the gauging station control. If a change in a rating curve becomes stable for a period of at least one month or two, then a new rating curve is prepared for the new time period over which the rating curve is effective. If the temporal change is of shorter duration, the original stage discharge relationship is kept; during the period, however, shifts or adjustments are applied to the recorded stage. This is accomplished such that the new discharge represents a recorded stage equal to the discharge from the original rating curve that corresponds to the adjusted stage. The time period over which this occurs is referred to as a period of shifting control. During periods of shifting control, frequent discharge measurements are essential to identify the stage discharge relation and the magnitude of the shifts during the period. Even with limited data, the rating curve can be estimated if available measurements are supplemented with the behavior of the shifting control. It is this behavior which we will now address.

4.3.5 Extrapolation Because measurements to cover the upper and lower ends of the rating curve often are lacking, ratings often are extrapolated to estimate flows outside the range of observations. Rating curves, for example, very often must be extrapolated beyond the range of measured discharges either for

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estimating flood discharge that are not measurable from measured stage, or to estimate the height or level corresponding to high return period floods, calculated by numerical models for flood protection. Nevertheless, a large elements of uncertainty exists in the extrapolation process; many methods of analysis anyway exist that will reduce the degree of uncertainty. The International Organization for Standardization standard recommends that, “the stage-discharge relation should not be applied outside the range of discharge measurements upon which it is based” and that “if estimates of the flow, however, are required outside the range, it may be necessary to make an extrapolation of the rating curve” (ISO, 1998). The USGS recommends that the unmeasured peak discharges should be estimated, if possible, from the peak stage, using one of the “indirect methods of peak discharge determination” (Rantz et al., 1982a, 1982b). This peak discharge should then be used in development of the rating. If, however, such peak discharge determinations are not available, several methods are available to extrapolate the rating, with the advice that “the rating should not be extrapolated beyond twice the largest measured discharge except as a last resort” (Rantz et al., 1982b). Methods for handling of extrapolation of rating curves vary widely. Some methods are only based on statistics while others consider the hydraulic conditions. The latter methods are recommended particularly if geometric data and water surface profiles of the river reach are available. Especially in such cases, numerical simulations should be performed to achieve better information on high discharges. Graphical Extrapolation A simple method for extrapolating a stage-discharge relationship is to plot all measured data as pairs of water level and discharge into one graph. The next step is to draw a curve of best fit through all plausible measured data. In general, this leads to quite good results for the range of the measured discharges but difficulties occur for lower and, especially, higher discharges. Using a logarithmic plotting paper can support the extrapolation if the measured values are placed on a straight line which can easily be extrapolated. However the result should be depicted on arithmetic plotting paper checking for plausibility and for practical application. Another aid for extrapolating the stage-discharge relationship is to plot the flow area A with water level h and the mean velocity V with h, where V is calculated as Q/A. This way, possible mistakes in measuring the flow area or the discharge will be recognized. Although the graphical extrapolation is a straight forward method and errors can easily be detected, it is not completely objective. Extrapolation based on correlation analysis A more objective way to extrapolate a stage-discharge relationship, although sometimes neglecting the authentic hydraulic conditions, is to use a generalized function with coefficients that will be optimized to achieve a good fit between the function and the measurement data. Different types of functions which can be used for fitting and extrapolation. Polynomials of second or third degree are also applied frequently. Extrapolation based on numerical modeling The most advanced way to extrapolate a stage-discharge relationship is to use a numerical model, like for example HEC RAS (USACE-HEC, 2002), and to simulate high discharges. Calculations are based on the Bernoulli equation, which is performed step by step from downstream of the modeled reach to the cross section of the gauging station. If possible the downstream section should have a specific hydraulic condition for all discharges, like normal or critical depth. The 20

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roughness coefficients of the hydraulic model should be calibrated using the measured water surface profiles as well as several water level and discharge measurements,. Low levels extrapolation Low flow extrapolation of rating curves is often required in the management of surface water supply for domestic, industrial, or agricultural uses. Low flow extrapolation is performed using rectangular coordinate graph paper since the coordinates of zero flow can not be plotted on the log graph paper. In most cases, low flow extrapolation is not very accurate. If the existing trend in the rating curve is extended to the zero discharge point, the curve will rarely pass through the zero stage point. Forcing the rating curve through the zero/zero stage-discharge point usually requires a differently shaped curve in the observed portion of the rating curve. Under these conditions, an adequate understanding of the relationship between low stage and discharge can only be achieved with additional low flow discharge measurements

Figure 11. Low flow extrapolation (from Herschy, 1995).

4.4 Rating curves in particular conditions

In this section we just make a brief mention about the problems that might arise in defining rating curves in some particular circumstances.

4.4.1 Ice-covered rivers In cold climate regions ice forms on rivers during winter season. The effect of the ice cover is to increase the hydraulic radius and the roughness, and to decrease the cross-sectional area. Like for weed growth, the stage for a given discharge is increased. The effect of ice formation and thawing is very complex and the temporary stage–discharge relation can only be determined by a series of discharge measurement using stage, temperature and precipitation records as a guide for interpolation between measurement (Herschy, 1995)

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4.4.2 Coastal rivers In coastal rivers, the water level is affected by the downstream movement of a tidal wave. In such rivers, the water level is not solely a function of the flow discharge but rather a joint function of both the discharge and tidal phenomena. The interference of the tidal backwater with the normal flow of the river causes disturbances in the normal stage-discharge relationship. There are two general approaches for obtaining a continuous discharge record: the theoretical approach involving evaluation of the equations of unsteady flow for a tide-affected reach of channel; and the empirical approach involving empirical relations whose effectiveness generally varies inversely with the degree of importance of the acceleration head. The theoretical approach is much preferred. This problem obviously is less significant in Mediterranean area where tidal oscillations are not so wide like for rivers leading in ocean. Anyway, we mention a number of empirical models have been used to develop the rating in tidal reaches. These include (Rantz, 1982b), the method of cubatures, the rating fall method , the tide correction method and the coaxial graphical correlation method (Rantz, 1963).

4.4.3 Theoretical computation When new rating curves are developed, and current-meter measurements are not available to define the relation between stage and discharge, it is generally possible to develop the relation on the basis of theoretical computations using the channel configuration, estimates of channel roughness, and assigned discharges. The approach would develop rating curves without stage or discharge measurements by using the unsteady-flow equation. Measured data will be used to verify the rating curve, instead of deriving it, allowing considerable savings in time, cost, and effort in data collection.

4.5 Uncertainty of stage-discharge relationships

As is now obvious, systematic and continuous discharge data is not actually observed; rather, records are made from converting the water level data to discharge by using a stage-discharge relationship. Obviously, uncertainty in the transformation from stage to discharge can influence significantly the development of hydrological models and the calibration of parameters based on the discharge data. In general, uncertainties in stage-discharge relations could be ascribed to different potential sources of uncertainty, listed by Yen et al. (1986): – natural uncertainties associated with the inherent randomness of natural processes; they

include the effect of processes such as turbulent fluctuations, wind, temporal changes in resistance or geometry, sediment concentration, and similar physical processes that can affect the flow but are not measured for gauges.

– knowledge uncertainty, which reflects the result of inadequate understanding of the true physical processes and can be reduced by improved knowledge of the physical processes and parameters. This includes improper assumptions in formulation of the relation between stage, discharge, and other parameters; neglection of important parameters; incorrect specification of parameters, and similar errors. This is expected to be the largest source of error in most stage-discharge relations.

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– data uncertainties. These include errors in measurement of stage, discharge, geometry, and other characteristics of the flow and channel; transcription errors, and inadequate spatial or temporal sampling.

The magnitude of uncertainty in discharge measurement is due mainly to the uncertainty in velocity measurement, in stage measurement and in cross-section geometry. On the other hand the uncertainty due to rating curves is mainly due the imperfect matching of underlying hypothesis. A great number of papers and studies have been done to try to quantify and/or reduce uncertainties in stage-discharge relations. Nevertheless the state-of-the-art in determining the uncertainty in stage-discharge ratings consists mainly of statistical analyses of the deviations of observations from a “best-fit” rating curve or equation (Herschy, 1995; ISO, 1998). The analysis is generally performed on logarithmic paper. The standard error of estimate se may be calculated from the following equation (ISO, 1998):

(6)

where Q is the measured discharge, Qc is the discharge calculated from the rating curve equation and n is the number of observations. The ISO standard recommends to perform the statistical analysis on each segment of rating curve separately. When the number of observations is large (say more then 20) then the Student’s t value used in the calculation of the 95% confidence interval (2 tailed testing) may be taken as 2. In such a way we can draw two parallel lines (in logarithmic paper) on either side of the rating curve segment at a distance of 2se from it. In other words 95% of the observations on average will be contained within this limit. This approach only considers the statistical properties of the discharge measurements used to develop the relation, ignoring the potential improvements in the relation that could be obtained from including hydraulic factors.

5 References

Aronica G. T., Candela A., Viola F., Cannarozzo M., 2006, Influenza dell’incertezza relativa alla scala delle portate sulla modellazione afflussi-deflussi a scala giornaliera, XXX° Convegno di Idraulica e Costruzioni Idrauliche - IDRA 2006, Roma Aronica G. T., Candela A., Viola F., Cannarozzo M., 2005, Influence of rating curve uncertainty on daily rainfall-runoff model prediction, In T. Wagener (ed.) Iahr IAHS Red Book – Prediction in ungauged basins 7.2 session (Model evaluation and comparison: uncertainty analysis and diagnostic). Proceedings of 7th IAHS Scientific Assembly Symposium, Brazil ARPA del Veneto, 2006, Considerazioni sulla scala di deflusso del fiume Brenta a Barziza, http://www.arpa.veneto.it/acqua/docs//interne/Considerazioni_sulla_scala_di_deflusso_del_Fiume_Brenta_a_Barziza.pdfBarbetta S., Melone F., Moramarco T., 2003, An Efficient Method for Rating Curve Estimation from Remote Discharge, Proc. Applied Simulation and Modelling ASM 2003- Marbella, Spain International Conference on Applied Simulationa and Modelling

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http://www.arpa.veneto.it/acqua/docs//interne/Considerazioni_sulla_scala_di_deflusso_del_Fiume_Brenta_a_Barziza.pdf

http://www.arpa.veneto.it/acqua/docs//interne/Considerazioni_sulla_scala_di_deflusso_del_Fiume_Brenta_a_Barziza.pdf


Bhattacharya B., Solomatine D. P., 2000, Application of artificial neural network in stage-discharge relationship, Proc. 4-th International Conference on Hydroinformatics, Iowa City, USA, July 2000 Bailey, J.F., and Ray, H.A., (1966). ”Definition of Stage-discharge Relation in Natural Channels by Step-Backwater Analysis,” U.S. Geological Survey Water-Supply Paper 1869-A, 34 p. Becchi I., Anselmo V., Saccardo I. e Pinelli P.F., 1994, Manuale per il monitoraggio idrografico, U.O. 1.31 G.N.D.C.I., ENEL –CRIS Boiten W., 2003, Hydrometry, Taylor & Francis, London Chow, V.T., 1959, Open Channel Hydraulics, McGraw-Hill, New York, NY, 680 p. Claps P., Fiorentino M., Laio F., 2003, Scale di deflusso di piena di corsi d'acqua naturali, La difesa idraulica del territorio, Trieste 10-12 settembre 2003 Clarke, R. T. (1999). Uncertainty in the estimation of mean annual flood due to rating curve indefinition. J. Hydrol. 222, 185 – 190. Clarke, R. T., Mendiondo, E. M., Brusa, L. C. (2000). Uncertainties in mean discharges from two large SouthAmerican rivers due to rating curve variability, Hydrol. Sci., 45(2), 221 – 236. Cristianini N., Shawe-Taylor J., 2000, An Introduction to Support Vector Machines and other kernel-based learning methods, Cambridge University Press, 2000 Corbett Don M. et al., 1943, Stream gauging Procedure US Geological Survey Water Supply Paper n. 888 Dawdy, D.R., 1961, Depth-Discharge Relations of Alluvial Streams--Discontinuous Rating Curves, Water-Supply Paper 1498-C, U.S. Geological Survey, 16 p. Franchini M., Lamberti P., Di Gianmarco, P., 1999, Rating curve estimation using local stages, upstream discharge data and simplified hydraulic model, Hydrology and Earth Sciences 3(4) Herschy R. W. 1995, Streamflow Measurement, Chapman & Hall, Second Edition ISO 1100-1, 1996, Measurement of liquid flow in open channels - Part 1: Establishment and operation of a gauging station ISO 1100-2, 1998, Measurement of liquid flow in open channels - Part 2: Determination of the stage-discharge relation ISO 4373, 1995, Measurement of liquid flow in open channels - Water-level measuring devices ISO 748, 1997, Measurement of liquid flow in open channels - Velocity-area methods Jones, B.E., 1916, A Method of Correcting River Discharge for a Changing Stage, Water Supply Paper 375, U.S. Geological Survey, pp. 117-130 Mosley, M. P., McKerchar A. I. (1993). Chapter 8: Streamflow. In: Handbook of Hydrology (ed.by D. R. Maidment), McGraw Hill, New York, USA. Parodi U., Ferraris, L., 2004, Influence of Stage Discharge Relationship on Annual Maximum Disharge Statistics, Natural Hazard 31: 603-611 Pasley R., Snider D., Lee O.P., Riekert E.G., 1972, Stage-discharge relationships, NEH National Engineering Handbook, Section 4 Hydrology, Chapter 14. Rantz S.E., 1963, An Empirical Method of Determining Momentary Discharge of Tide-Affected Streams, U.S. Geological Survey Water Supply Paper 1586-D

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Rantz S.E. et al., 1982a, Measurement and computation of streamflow, Volume 1 Measurement of Stage and Discharge U.S. Geological Survey Water Supply Paper 2175 Rantz S.E. et al., 1982b, Measurement and computation of streamflow, Volume 2 Computation of Discharge, U.S. Geological Survey Water Supply Paper 2175 SIMN, 1998, Norme Tecniche per la raccolta e l’elaborazione dei dati idrometeorologici (Parte II, dati idrometrici), Presidenza del Consiglio dei Ministri Dipartimento per i Servizi Tecnici Nazionali, Servizio Idrografico e Mareografico Nazionale, Roma SIMN, Annali Idrologici Parte II, Presidenza del Consiglio dei Ministri Dipartimento per i Servizi Tecnici Nazionali, Servizio Idrografico e Mareografico Nazionale, Roma Sivapragasam C, Muttil N., 2005, Discharge Rating Curve Extension – A New Approach, Water Resources Management 19: 505–520 Schmidt, A.R., Yen, B.C., 2001, Stage-Discharge Relationship in Open Channels, in Proc. 3rd Intl. Symp. on Envr. Hydr., Tempe, AZ, Dec. 5-8, 2001, ed. by D. Boyer and R. Rankin Ufficio Idrografico del Magistrato di Venezia, 1914, Norme ed istruzioni per il Servizio di Misura delle Portate, Pubblicazione n. 38, Ristampa (Venezia 1988) USACE-HEC, 2002, River Analysis System - Hydraulic Reference Manual, CPD 69 WMO, 1980, Manual on stream gauging Vol.1: Fieldwork – Vol.2: Computation of discharge, Operational Hydrology Report n.519, Geneva WMO, 1986, Level and discharge measurements under difficult conditions, Operational Hydrology Report n. 24, n.650, Geneva Yen, B.C., and Gonzalez, J.A., 1994, Determination of Boneyard Creek Flow Capacity by Hydraulic Performance Graph, Research Report No. 219, Department of Civil Engineering, Univ. of Illinois at Urbana-Champaign

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