Guidelines for the Assessment of Uncertainty for Hydrometric Measurement

Transcription

1 WEATHER CLIMATE WATER Gidelines for the Assessment of Uncertainty for Hydrometric Measrement WMO-No. 1097

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3 WMO-No World Meteorological Organization, 017 The right of pblication in print, electronic and any other form and in any langage is reserved by WMO. Short extracts from WMO pblications may be reprodced withot athorization, provided that the complete sorce is clearly indicated. Editorial correspondence and reqests to pblish, reprodce or translate this pblication in part or in whole shold be addressed to: Chairperson, Pblications Board World Meteorological Organization (WMO) 7 bis, avene de la Paix Tel.: +41 (0) P.O. Box No. 300 Fax: +41 (0) CH-111 Geneva, Switzerland pblications@wmo.int ISBN NOTE The designations employed in WMO pblications and the presentation of material in this pblication do not imply the expression of any opinion whatsoever on the part of the Secretariat of WMO concerning the legal stats of any contry, territory, city or area, or of its athorities, or concerning the delimitation of its frontiers or bondaries. Opinions expressed in WMO pblications are those of the athors and do not necessarily reflect those of WMO. The mention of specific companies or prodcts does not imply that they are endorsed or recommended by WMO in preference to others of a similar natre which are not mentioned or advertised. The findings, interpretations and conclsions expressed in WMO pblications with named athors are those of the athors alone and do not necessarily reflect those of WMO or its Members.

4 CONTENTS FOREWORD... PREFACE... Page iii v 1. REVIEW OF UNCERTAININTY ANALYSIS FRAMEWORK Introdction General UA (GUA) approaches.... SYNTHESIS OF RECOMMENDED UA APPROACH Recommendation Rationale Smmary of the GUM (JCGM, 100:008) Basic Concepts and Terminology (JCGM, 100:008) Uncertainty Analysis Implementation The measrement process Type A evalation of standard ncertainty Type B evalation of standard ncertainty Determining the combined standard ncertainty Determining expanded ncertainty Practical Considerations APPENDIX A Uncertainty Analysis Pblications... 5 APPENDIX B Discharge Uncertainty Example: weighing and timing measrements of discharge... 9 APPENDIX C Discharge Uncertainty Example: wading measrements of discharge sing a point velocity meter and the velocity-area method... 4 APPENDIX D Discharge Uncertainty Example: wading measrements of discharge sing a point velocity meter and the velocity-area method... 54

5 FOREWORD In view of the increasing pressre on water resorces, National Hydrological Services (NHSs) worldwide are faced with the challenge to deliver hydrological services of high qality, timeliness and proven credibility, to assist the decision making process of water and natral resorces managers. Nowadays, it is commonly expected that these services mst be based on information that incldes an accrate assessment of its ncertainty. Stream discharge, the basic hydrological variable, is no exception, yet most measrements are still reported as a vale withot any information on its associated ncertainty. It is therefore very appropriate that the WMO Commission for Hydrology has ndertaken the project for the Assessment of the Performance of Flow Measrement Instrments and Techniqes, with the participation of experts from the International Association of Hydrological Sciences (IAHS), the International Association for Hydro-Environment Engineering and Research (IAHR), the Association of Hydro- Meteorological Eqipment Indstry (HMEI) and International Organization for Standardization (ISO). The present report, one of the initial otpts of the project, constittes the first step towards establishing an internationally agreed recommended practice, for operational hydrologists to follow, in estimating the ncertainty associated with discharge measrements. On behalf of WMO, I look forward to the ftre developments, especially to the planned dissemination of easy-to-se tools to operationalize the procedres described in the crrent pblication, and I wish to commend the athors, reviewers and all those experts who contribted their time and knowledge to the preparation of these Gidelines. (Petteri Taalas) Secretary-General

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7 PREFACE At its twelfth session in 004, the Commission for Hydrology (CHy) recognized that significant advances had occrred in stream gaging instrmentation and methodologies dring the preceding few decades. In considering these advances, it reached a general consenss that information on the appropriate se of sch instrmentation and methodologies was rgently needed. Accordingly, the Commission initiated the development of a project proposal to assess the performance of flow measrement instrments and techniqes against WMO standards. At its thirteenth session, the project proposal was reviewed and approved by the Commission. In addition to lanching the project, CHy-13 also adopted a thematic focs entitled Qality Management Framework Hydrology (QMF-H). With the growing need within National Hydrological Services (NHSs) to design, develop and deliver prodcts and services effectively and efficiently, the QMF-H theme area provided a means to establish the approaches by which the ncertainty of hydrometric measrements cold be assessed. Sch activities are of fndamental importance in assessing data reliability and for ensring that the instrmentation and techniqes being sed by NHSs operate at the reqired accracy. A project Management Committee, chaired first by Mr Pal Pilon (Canada) and, from 014 by Mr Jean-François Cantin (Canada), was established and charged with developing, pdating and carrying ot the project work plan. In ndertaking its responsibilities, the Management Committee recognized the need for gidelines specifying how the hydrological commnity shold approach assessing the ncertainty of hydrometric measrements. It reviewed varios analytical frameworks and recommended the se of the Joint Committee for Gides in Metrology (JCGM) docment entitled Gide to the Expression of Uncertainty of Measrements. In Febrary 009, the Advisory Working Grop of CHy adopted the ncertainty analysis framework for hydrometric measrements as developed by the project. This report describes in detail the approaches that shold be taken when estimating ncertainty and otlines step-by-step procedres for its practical implementation in hydrometry. Three detailed examples are inclded to provide a clear illstration of how the concepts can be applied in order to facilitate their adoption by the broader hydrological commnity. The main report was prepared by Mr Marian Mste (IAHR). Mrs Janice Flford (USA) prepared two examples demonstrating the application of the framework in hydrometry, while Messrs Marian Mste and Jean-Lc Bertrand-Krajewski (France) prepared a third. The draft text was reviewed by Messrs Jérôme Le Coz (IAHS), Patrick McCrry (Canada), Pal Pilon (Canada), Tom York (ISO) and Cladio Caponi (WMO). Project activities were condcted in association with the Commission for Hydrology s Open Panel of CHy Experts (OPACHE) on Basic Systems (Hydrometry and Hydralics). I extend my deep appreciation to the members of the Management Committee who have helped gide the process and oversee the preparation of the manscript. I also wish to extend my sincerest gratitde to the athors and reviewers for their invalable contribtions to the advancement of hydrometry throgh the pblication of these Gidelines for the Assessment of Uncertainty of Hydrometric Measrements. (Harry Lins) President, Commission for Hydrology

8 1. Review of Uncertainty Analysis Frameworks 1.1 Introdction Althogh measrement systems are sbject to ncertainty, they are often planned, designed, operated and managed withot acconting for it. Uncertainty in measrements arises from the randomness and complexity of physical phenomena and errors in observations and/or processing of the measred data. With the considerable increase in data, information, prodcts and services, the National Hydrological Services (NHSs) are seeking ways of expressing ncertainty and it is essential to provide gidance on how best to provide this information. Users of one discipline also need to nderstand the ncertainty of the data and prodcts from another discipline prior to sing them, and the methodology sed in ncertainty estimation mst be consistent. Finally, varios commnities sch as the pblic also benefit when they see the ncertainty expressed for data and prodcts of varios disciplines within NHSs. Uncertainty analysis (UA) is a rigoros and scientifically robst techniqe to estimate the interval abot a measred variable or determined reslt within which the tre vale is thoght to lie with a certain degree of confidence (Coleman and Steele, 1999). In typical measrement sitations, several physical parameters (e.g., flow velocity, depth, and channel width) are physically measred to obtain a derived qantity (e.g., stream discharge). The individal physical measrements are then sed in a data-redction eqation (e.g., velocity area method) to obtain the targeted vale. Conseqently, the two major steps involved in the ncertainty analysis are: (a) identification and estimation of the ncertainties associated with the measrement of the individal variables, and (b) propagation of the individal measrement ncertainties in the final reslt. While the methods sed by varios commnities for estimating the elemental sorces of ncertainty are qite similar (statistical analysis or se of previos experience, expert opinion, and manfactrer specifications), the methods of determining how those sorces of ncertainty are acconted for in the final reslt have differed widely (TCHME 003). In addition, variations can even occr within a given methodology. Coleman and Steele (1999) discss six different variations of the Taylor-series expansion estimation method (which is the most widely sed ncertainty-estimation approach for the propagation of ncertainties). UA is tilized in varios forms from the initial planning of the experiment to the actal testing and data analysis. Given its significance, Kline (1985) points ot that it is ltimately more important to perform some (any!) ncertainty analysis than to se a particlar methodology. Uncertainty analysis is a critical component for assessing the performance of flow measrement and techniqes for both conventional and newer instrmentation and methodologies. These analyses are of fndamental importance to the application of risk management procedres and sstainable water resorces management, by ensring that the methodology and instrmentation selected for a task will deliver the reqired accracy. These analyses also help to ensre that investments in hydrological instrmentation are costeffective. Given the vast nmber of pblications on the topic, the task of reviewing the frameworks, gidelines, and standards related to ncertainty analysis of flow measrement is danting and well beyond the scope of the present project. In order to illstrate the magnitde of sch an exhastive literatre review, we refer to a recent overview of the flow measrements standards issed by the International Standards Organization (ISO the most athoritative standards instittion). The review by Reader-Harris (007) lists abot 160 standards related to the scope of or work issed by varios ISO technical committees. Two thirds of the standards that make reference to flid flow measrement are prodced by

9 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT for technical committees: TC 8 - Petrolem prodcts and lbricants, TC 30 - Measrement of flid flow in closed condits, TC Hydrometry, and TC Flid power systems. The Technical Committee on Hydrometry (the ISO TC closest to or area of interest) alone prepared 9 standards on measrement of liqid flow in open channels between 1973 and 1983 (ISO, 1983). Rather than reviewing or even listing this vast body of literatre, we will focs on the approaches, standards, gidelines, and relevant references that have come to the fore in the most recent advancements in the UA field and gained visibility within the international ser commnity. The sbseqent discssion will distingish three types of Uncertainty Analysis (UA) pblications (i.e. frameworks, standards, gidelines or references): (a) (b) (c) General UA (GUA) Flow Measrement UA (FMUA), and Specific Flow Measrement UA (SFMUA). Examples of pblications from each category are provided below: (a) GUA: Gide to the Expression of Uncertainty in Measrement (JCGM 100:008) (b) FMUA: Measrement of Flid Flow Procedres for the evalation of ncertainties (ISO, 005), and (c) SFMUA: Manal on Stream Gaging (WMO, 1980). Appendix A lists selected pblications for each of the above-mentioned types. Given their importance to the present discssion, the GUA and FMUA frameworks will be discssed below in more detail. 1. General UA (GUA) approaches UA has always concerned scientists, engineers, and practitioners; for several decades they have arged abot how best to condct UA (Abernethy and Ringhiser, 1985). The sbject has been plaged by controversy, argment, confsion and even emotion (Abernethy and Ringhiser, 1985). Despite several decades of extensive research and development on procedres for estimating ncertainty, ntil recently there has been no consenss on a single approach or even a single nifying vision. Starting in the 1950s, the American Society of Mechanical Engineers (ASME), throgh its Performance Test Codes Committee (PTC) and the Measrement of Flid Flow in Closed Condits Sbcommittee Determination of Uncertainties (MFC-), considerably stepped p efforts to write a standard measrement-ncertainty docment. Althogh the ASME approach was based on the landmark paper by Kline and McClintock (1953), there were many different methods in se and heated disagreements for many years. The ASME efforts finally achieved consenss in 1986, with the adoption of the newly-developed ASME- PTC 19.1 Measrement Uncertainty standard (ASME, 1986). In addition to ASME and the American National Standards Institte (ANSI), this standard was accepted by the Society of Atomotive Engineers (SAE), the American Institte of Aeronatics and Astronatics (AIAA). The standard was also recognized by ISO, the Instrment Society of America crrently the Instrmentation, Systems, and Atomation Society (ISA), the US Air Force, and the Joint Army Navy NASA Air Force (JANNAF). A detailed description of the terminology, principles and procedres in the ASME (1986) standard is provided in Coleman and Steele (1989). In 1978, while the US efforts were in fll swing, the lack of international consenss on the expression of ncertainty in measrements prompted the world s highest athority in

10 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 3 metrology, the Comité International des Poids et Mesres (CIPM) to take action. The CIPM reqested the Brea International des Poids and Mesres (BIPM) to address the problem in conjnction with the national standards laboratories and to make a recommendation. BIPM prepared a detailed qestionnaire covering the isses involved and distribted it to 3 national metrology laboratories known to have an interest in the sbject. All responses revealed that it was important to arrive at an internationally accepted procedre for expressing measrement ncertainty and for combining individal ncertainty components into a single total ncertainty. However, there was no consenss on the method to be sed. BIPM then convened a working grop of experts from 11 national standards laboratories to formlate an approach for the classification, estimation, and expression of experimental ncertainties to be sed in the gide. The fndamental reqirements for the gide were: to be niversal (applicable to all kinds of measrements), internally consistent (directly derivable from the components that contribte to the actal qantity), and transferable (sable for both primary and derived qantities). The task of drafting a detailed gide based on the BIPM grop recommendations was referred to ISO, on the gronds that it cold better reflect the needs arising from the broad interests of indstry and commerce. ISO assembled a joint grop of international experts representing seven organizations: BIPM, ISO, International Electrotechnical Commission (IEC), International Federation of Clinical Chemistry (IFCC), International Union of Pre and Applied Chemistry (IUPAC), International Union of Pre and Applied Physics (IUPAP), and International Organization of Legal Metrology (OIML). This elaborate and extensive effort led to the Gide to Expression of Uncertainty in Measrement (GUM, 1993) which is the first set of widely internationally recognized gidelines for the condct of ncertainty analysis. Since its pblication, the GUM (the name sally associated with this methodology) has been revised several times and is increasingly accepted by the metrology services of varios professional organizations. Since 000, the Joint Committee for Gides in Metrology (JCGM) has been responsible for GUM pdates and distribtion. Crrently, JCGM members are entitled to pblish the GUM, hence the citation for the GUM in the present material is JCGM (100:008). In the present text GUM and JCGM (100:008) are sed interchangeably as 95 per cent of their content is identical. The GUM framework is based on the most recent advancements and principles in mathematical statistics for the propagation of the elemental sorces of errors to the final reslts. GUM offers general rles for evalating and expressing ncertainty in measrement rather than providing detailed and specific instrctions tailored to any specific field of stdy. GUM differs from previos ncertainty assessment methodologies with respect to terminology, more specifically the classification of errors and procedres (Herschy, 00). The main distinction between GUM and previos methods is that there is no inherent difference between an ncertainty arising from a random effect and one arising from a correction for a systematic effect (an error is classified as random if it contribtes to the scatter of the data; otherwise, it is a systematic error). Conseqently, JCGM (100:008) departs from the traditional engineering categories of systematic (bias) and precision (random) ncertainties, sing instead a classification based on how the ncertainties are estimated, i.e. type A (evalated statistically) and type B (evalated by other means). GUM provides a realistic vale of ncertainty based on the fndamental principle that all components of ncertainty are of the same natre and are to be treated identically. In many sitations, bias ncertainties reqire type B evalations and random ncertainties reqire type A evalations. However, there is no simple correspondence between random or systematic and type A and type B ncertainties [ASME (1998) p. 16; UKAS (007) p. 11)]. The ncertainty from a known systematic effect may in some cases be obtained by a type A evalation bt in other cases by a type B evalation. This can also be the

11 4 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT case for an ncertainty characterizing a random effect (JCGM 100:008; p. 6). Irrespective of the terminology, it is generally obvios which approach has to be sed in evalating the varios components of ncertainty in a measrement. Despite these minor departres from conventional ncertainty practice, JCGM (100:008) methodology is recognized today as being the most athoritative framework for a rigoros ncertainty assessment. Its terminology and methodology for assessing the qality of measrements have been adopted by several scientific and engineering areas [e.g., (NIST TN 197, 1994), (NF ENV 13005, 1999), (ISO 5168, 005), (UKAS, 007)]. JCGM (100:008) methodology, bt not the complete terminology, has also been adopted by the newly revised ASME Standard PTC 19.1 (005) and AIAA Standard S-071 (1999). The latter standards broght their original methodologies developed in the 1970s and 1980s into line with the GUM by adding the assmptions necessary to achieve a less complex large sample methodology reqired by GUM and retaining the se of traditional engineering concepts of systematic (bias) and precision (random) ncertainties. The harmonization of the AIAA and ASME standards with the GUM has been achieved by combining the technology of the ISO gide with historical gropings of elemental ncertainties sed in earlier versions of engineering ncertainty standards (ASME, 1986), i.e., systematic (bias) and precision (random) ncertainties. The AIAA (1995) standard was bilt on previos docments drafted by the NATO Advisory Grop on Aerospace Research and development (AGARD). ASME (1998) bilds on the ASME Measrement Uncertainty standard. Given the historical realities of the varios ncertainty classifications, ASME (005) recommends for the first time the se of the dal classification for the elemental ncertainties that blends the two different terminologies. The hydrometric commnity has been continosly seeking to improve procedres for the assessment of measrement errors bt, ntil recently, has not agreed on one particlar framework. Moreover, the few pblications dealing with ncertainty analyses in the civil engineering field are focsed on reliability analyses pertinent to the design and safety of hydralic strctres rather than on ncertainty analyses for hydralic measrements (TCHME, 003). Recognizing the limitations of the stats qo in this area, new international and national initiatives have recently been lanched (TCHME, 003; UNESCO, 007). The Technical Committee on Hydralic Measrements and Experimentation (TCHME) of the American Society of Civil Engineers (ASCE) Environmental & Water Resorces Institte (EWRI) set p a Task Committee on Experimental Uncertainty and Measrement Errors in Hydralic Engineering in 003. The overall mission of this Task Committee is to provide information and gidance on the crrent practices sed to describe and qantify measrement errors and experimental ncertainty in hydralic engineering and experimental hydralics. Similarly, in 004 the UNESCO International Hydrology Program (IHP-VI) began to compile gidelines for integrated rban water management that also inclde sections dedicated to UA. Both of these working grops opted to adopt existing frameworks for condcting UA rather than developing a specific standard for the hydrometry commnity. The grops conclded that the existing methodologies, sch as JCGM (100:008) or AIAA (1995), can be sccessflly applied for assessing the measrement ncertainty in laboratory and field measrements in hydralics research (Bertrand-Krajewski and Bardin, 00; Mste and Stern, 000; Mste et al., 004; Kim et al., 005; Kim et al., 007; Gonzalez-Castro and Mste, 007; UNESCO, 007, Mste et al., 01). An alternative approach to the widespread UA based on Taylor-series expansion for propagation of the elemental errors to the final reslts, is to se Monte Carlo simlation techniqes (e.g., JCGM 101:008). In this approach, the first inpt is the assmed tre vale of each variable in the data redction eqation. Then the estimated random ncertainty for each variable and the estimated elemental systematic ncertainties for each variable are determined. A Gassian random nmber generator is sed to prodce scaled random and

12 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 5 systematic elemental errors for individal rnning tests. Using the inpt vales for all the variables, the final reslt is calclated. This process is repeated N times (10,000 to 50,000 iterations are needed), after which the mean and standard deviations of the distribtions of the reslts are calclated. With this approach it is also relatively simple to incorporate non- Gassian error distribtions for the elemental sorces and degrees of confidence other than 95 per cent can be specified. The Monte Carlo simlation is sefl for investigating complex UA cases. Appendix A lists selected GUA pblications. 1.3 Flow Measrement UA (FMUA) and Specific FMUA (SFMUA) approaches A review of the literatre on UA of the flow measrements is a core aspect of the present project. It is a difficlt task becase the sbject is addressed in a variety of pblications, sing a mltitde of analytical approaches, with most of the pblished analyses reflecting conventional instrments or techniqes. The sccinct literatre review condcted as part of this project fond minimal gidance available on UA for flow measrements (WMO, 007; Pilon et al., 007). Existing WMO gidance and standards docments do not sfficiently address the comptation of ncertainty for newer hydrological measrements. A recent edition of the Gide to Hydrological Practices (WMO, 008) reviews new instrmentation and technologies to prodce hydrological information bt does not address the aspects of ncertainty analysis of data and information. A concerted effort of the international commnity as reflected in this project cold greatly advance or crrent state of practice in the assessment of flow-measrement instrments. Early discssions by the WMO project grop on aspects of the UA for flow measrements were initiated in parallel with the formlation of gidelines for instrment operation, data acqisition and processing of data collected with varios hydrological instrments. These efforts can be traced back to the early days of hydrometry, as the area always attracted the attention of the civil engineering commnity (Kolpaila, 1961). In the early 1950s, awareness of UA for flow measrement was increasingly apparent throghot Western Eropean contries. Efforts to develop a standard for flow measrements were sally spearheaded by government-spported agencies sch as the British Standards Instittion (BSI) in United Kingdom, Association Française de Normalisation (AFNOR) in France, and Detsches Institt fr Normng (DIN) in Germany. Despite the many athoritative docments on flow measrement that were available (e.g., ASME, 1971), the first effort at developing a national standard for flow measrement in the United States was initiated in 1973 (Abernethy et al., 1985). The first standard on flow measrement developed by ISO was Measrement Flow by Means of Orifice Plates and Nozzles [ISO (1967)] and is based on compromises between procedres in se in the United States and those sed throghot Western Erope. All of these efforts addressed the accracy of flow measrement with varying degrees of thoroghness. However, each of the reslting pblications proposed personalized procedres for estimating ncertainty, often biased by the jdgement of the individals involved in developing the procedre (Abernethy et al, 1985). The present discssion will be limited to pblications issed by ISO, the most athoritative international sorce for standards. Manals, gidelines, and monographs drafted by national standardization agencies (e.g., EU, DIN, NIST) and international, professional organizations (e.g., WMO, UNESCO, IAHR, IAHS) are not reviewed here. A specially designed web srvey will be distribted by WMO in the near ftre as part of this project. It will reqest information from National Hydrological Services (NHSs) on these latter type docments that are sed in specific contries, together with other standards, gidelines, and manals issed in the varios contries. Appendix A lists selected FMUA and SFMUA pblications.

13 6 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT As pointed ot by Thomas (00), the task of reviewing jst the existing ISO standards on flow instrments and techniqes is overwhelming. His paper gives gidance on selecting an open-channel flow measrement method and selecting one of the 64 ISO standards relevant to flow measrement in open channels. Becase of the diversity and large nmber of available standards on flow measrements, he concldes that there is a need for an organization to have a standards programme (SP). The SP s role wold be to provide gidance on the varios types of standards (how they can be sed and when), the decision process for implementing standards, the varios key access points for information abot the standards and their availability. ISO/TR 8363 [ISO (1997)] is recommended as being the standard of standards for flow measrement as it offers the most qalified gidance for selecting an open-channel flow measrement method and an applicable standard. The first criterion that the ISO (1997) ses to select a specific flow measrement instrment or techniqe is the reqired or expected level of ncertainty of the measrement. The first pblication of a standard devoted entirely to estimating the ncertainty of a flow rate measrement was ISO/DIS 5168, pblished in This standard was drafted by the ISO/TC 30/sc 9 General Methods Sbcommittee which considers both ncertainties and calibration techniqes, and it proved very sefl for many years. The standard was sbseqently pdated in 1998 (as ISO/TR 5168) and later on in 005 [as ISO (005.a)] in light of JCGM (100:008), which provides a general framework for expressing the ncertainties in all areas of measrement. ISO (005.a) was the first standard to provide gidance on how to apply the GUM for establishing the ncertainty of flow measrements. ISO (005.a) is referenced in another standards developed by five other ISO committees (Reader-Harris, 007). Other related ISO standards that adopted JCGM 100:008 are ISO 1088 (007) and ISO (005.b). As noted in Sections 1. and 1.3, the general and the more specific flowmeasrement standards on UA (GUA and FMUA) are methodologically aligned, while not all FMUA ses the terminology of JCGM (100:008). Following years of disagreement and lack of consenss on nifying approaches, there is now a broad consenss that JCGM (100:008) is the most athoritative docment on all aspects of ncertainty assessment.. Synthesis of Recommended UA Approach.1 Recommendation Rationale The review of the UA general frameworks condcted in Section 1. illstrates that from the first attempt to develop a modern standard approach to UA (Kline and McClintock, 1953) to the most recent and comprehensive effort (JCGM, 100:008), many other methodologies were developed, reflecting the varios schools of thoght on the matter. Systematic stdies condcted to compare varios recent UA methodologies (i.e. Steele et al., 1994; Coleman and Steele, 1999) conclde that the ncertainty models presented in the JCGM (100:008) are more appropriate for determining ncertainty intervals than other existing models. JCGM (100:008), however, provides general rles for evalating and expressing ncertainty in measrement rather than providing detailed, scientific- or engineering-specific instrctions. Moreover JCGM (100:008) does not discss how the ncertainty of a particlar measrement reslt, once evalated, may be sed for different prposes sch as, for example, to draw conclsions abot the compatibility of that reslt with other similar reslts, establish tolerance limits in a manfactring process, or decide whether a certain corse of action may be safely ndertaken. Given its general formlation, JCGM (100:008) acknowledges that it might be necessary to develop particlar standards based on the

14 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 7 framework of the gide to address the problems specific to individal fields of measrement. Sch relevant otgrowths of the generalized gide are the sbseqently developed standards of the American Institte of Aeronatics and Astronatics (AIAA, 1999) and the American Society of Mechanical Engineers (ASME, 005). These latter standards have kept their terminology while harmonizing the methodology with GUM specifications and procedres. This adoption is an important step forward in the area of UA implementation as it had been speclated that the complexity of the GUM wold likely prevent its application in normal engineering practice (Coleman and Steele, 1989). From the practical point of view, the most distinct and important difference that JCGM (100:008) brings into the discssion is its emphasis on the categorization of ncertainty into type A and B standard ncertainties, rather than advocating the widely sed systematic and random ncertainty terminology fond in the AIAA and the ASME engineering commnities. type A and type B define how an ncertainty estimate was made, while bias and precision ncertainties indicate their case. Type A ncertainties are those evalated by the statistical analysis of series of observations, while type B ncertainties are evalated by means other than the statistical analysis of series of observations. For many practising engineers the most commonly sed classification criterion is related to their effect rather than the method of their evalation. That is, if an ncertainty sorce cases random scatter in a test reslt, it is termed a precision (random) ncertainty, and it has been cased by random errors. If the scatter is not random, it is termed systematic ncertainty or bias and is considered to be cased by systematic errors, for which a correction factor may be applied to the measrements. In this traditional terminology, accracy is viewed as the combination of precision and bias. Argments can be made for both sets of nomenclatres. Type A and type B nambigosly define how an ncertainty estimate was evalated, and the GUM typically assmes that all bias has been removed from the measrements. GUM terminology for ncertainty components is directly related to the se made of the corresponding qantity, namely how that qantity appears in the mathematical model describing the measrement process. On the other hand, the systematic/random categorization is deeply rooted in the applied research and engineering vocablary. The latter classification is particlarly sefl dring the debgging phase of an experiment for determining the expected dispersion of reslts for a particlar experimental sitation (Coleman and Steel, 1999). In principle, an ncertainty component arising from a systematic effect may in some cases be evalated by method A and in others by method B, and the same applies to an ncertainty component arising from a random effect. The correspondence between the commonly sed classification and the GUM approach is not straightforward. For example, a systematic calibration ncertainty can become a sorce of scatter (and ths a random ncertainty) if a new calibration is performed before each reading of the sample. In fact, NIST (1994) recommends that the terms random ncertainty and systematic ncertainty be avoided becase, while the adjectives random and systematic are appropriate modifiers for the word error, they are not appropriate modifiers for the word ncertainty. The JCGM (100:008) ncertainty classification is made for convenience only. The propagation and final ncertainty are evalated in the same way as in the engineering-based ncertainty standards. The GUM states (JCGM 100:008, p. 6): The prpose of the type A and type B classification is to indicate the two different ways of evalating ncertainty components and is for convenience of discssion only; the classification is not meant to indicate that there is any difference in the natre of the components reslting from the two types of evalation. Both types of evalation are based on probability distribtions and the ncertainty components reslting from either type are qantified by variances or standard deviations. In other words, no convention for classifying ncertainties affects the estimation of the total ncertainty. It has been shown that for most engineering applications, when 10 or

15 8 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT more readings can be obtained for each variable in an experimental reslt and the systematic ncertainties have degrees of freedom of 9 or more, the engineering standards (sch as ASME, 005) and JCGM (100:008) are sbstantially the same (UKAS, 007, pp ). Where there are small degrees of freedom in the reslts, the GUM standard (JCGM 100:008) is recommended (Steele et al., 1994). JCGM (100:008) has fond wide acceptance in the United States by prominent specialized organizations ( sch as the National Conference of Standards Laboratories, North American Collaboration in Measrement Standards, National Volntary Laboratory Accreditation Program, and the American Association for Laboratory Accreditation. Moreover, the GUM has been adopted by the National Institte of Standards and Technology (NIST) and most of NIST's sister national metrology instittes throghot the world, sch as the National Research Concil (NRC) in Canada, the National Physical Laboratory (NPL) in the United Kingdom, and the Physikalisch- Technische Bndesanstalt in Germany. The GUM has been adopted by the American National Standards Institte (ANSI) as an American National Standard (ANSI, 1997). GUM methods have been widely adopted abroad by varios metrology and related organizations inclding: Eropean Collaboration in Measrement Standards (EUROMET), Eropean Cooperation for Accreditation (EA) and the Eropean Union; adopted by CEN and pblished as EN ISO pblished the French translation of the GUM in 1995; German and Chinese translations were also pblished in 1995, and an Italian translation was pblished in Translations of the GUM into Estonian, Hngarian, Japanese, Spanish, and Rssian have been completed or are well nder way. Most recently, the Joint Committee for Gides in Metrology (JCGM) has assmed responsibility for the maintenance and revision of the GUM and therefore is listed here as JCGM 100:008. The JCGM members are the seven international organizations listed above: BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, and OIML, together with the International Laboratory Accreditation Cooperation (ILAC). The series of JCGM 100:008 companion standards inclde protocols for vocablary, the Monte Carlo method, and the application of the GUM to any nmber of otpt qantities ( As there are several near-eqivalent options for performing a UA, it is beneficial and necessary to recommend a single method to help simplify the isses srronding UA implementation for se in hydrometry. While aware of the tradition in engineering of the systematic/random ncertainty categorization and its seflness in engineering experimentation, we recommend that the hydrological commnity adopt JCGM (100:008). The standard is widely accepted and sed, and we recommend herein that it be adapted for or commnity.. Smmary of the GUM (JCGM, 100:008) Environmental systems can be generically described by a set of variables X 1,, X k related in p fnctional forms (Singh et al., 001) f j ( ) = 0, j = 1,,..., p (1) X 1,...,X k ;α 1,...,α l depending on α q parameters (q = 1,, l). If the variables are random, the fnctional relation redces to a strctral relation, which in environmental systems involves time-space distribted variables. The problem posed in environmental sciences is to estimate α q from a set of observations. If we were able to observe vales of X withot error, there wold be no statistical problem, and the problem wold merely mathematical. However, the variables that we observe (measre) are prone to errors that need to be estimated as ncertainties. The vexing problem of ncertainty in the decision-making process is that it is impossible to deal

16 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 9 with every kind of ncertainty. In the present section, only the kind of ncertainty that can be qantitatively measred, at least in principle, will be considered. This is termed the measrement ncertainty. The measrement process is the act of assigning a vale to some physical variable, by operating sensors and instrments in conjnction with data acqisition and redction procedres. In the ideal measrement, the vale assigned by the measrement wold be the actal vale of the physical variable to be measred. However, the measrement process and environmental errors introdce ncertainty into the measrement. To give some measre of confidence to the measred vale, measrement errors mst be identified, and their probable effect on the reslt estimated. Uncertainty is simply an estimate of the magnitde of the error in the reported vale of a measrement. The process of systematically qantifying the estimate of the magnitde of error is known as ncertainty analysis (UA). The UA methodology presented below is adapted from JCGM (100:008). Definitions of specific terms are consistent with the International Vocablary of Basic and General Terms in Metrology (ISO, 1993)...1 Basic Concepts and Terminology Uncertainty analysis is a rigoros methodology, combining statistical and engineering concepts, for estimating ncertainties in measrements and in the reslts calclated from them. The objective of a measrement is to determine the vale of a measrand, that is the vale of the particlar qantity to be measred. The term measrand in this gide is eqivalent to the term tre vale sed by the engineering standards. A measrement for a specified measrand therefore entails the measrement methods and procedres along with the effect of the inflence qantities (environmental factors). Collectively, these components form the measrement system. In general, a measrement has imperfections that give rise to an error in the measrement reslt. Conseqently, the reslt of a measrement is only an approximation or estimate of the vale of the measrand and ths is complete only when accompanied by a statement of the ncertainty of that estimate. In practice, the reqired specification or definition of the measrand is dictated by the reqired accracy of measrement. The accracy of a measrement indicates the closeness of the match between the reslt of a measrement and the vale of the measrand. The measrement error is defined as the reslt of a measrement mins the tre vale of the measrand. Neither the tre vale nor the vale of the measrand can ever be known exactly becase of the ncertainty arising from varios effects. Traditionally, these effects are assigned to random and systematic errors. Random errors presmably arise from npredictable or stochastic temporal and spatial variation of factors that inflence the reslts of the measrement (factors, termed qantities, that are not measred bt that affect the reslt of a measrement). These errors give rise to variations in repeated observations of the measrand. Althogh it is not possible to compensate for the random error of a measrement reslt, it can sally be redced by increasing the nmber of observations. Freqently in engineering practice, a nmber of measrements are sed to establish a conventional tre vale (see Figre 1). The systematic error (also termed bias), like the random error cannot be eliminated, bt it can be redced. If a systematic error arises from a recognized effect of an inflence qantity, if the effect can be qantified, and if it is significant in size relative to the reqired accracy of the measrement, then a correction can be estimated and applied to the measrement to compensate for the effect. It is assmed that after sch a correction has been applied, the expected vale of the error arising from the particlar effect is zero. The effect of systematic and random errors on repeated measrements is shown in Figre.

17 10 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT Figre 1 - Errors in the measrement of an inpt qantity (µ is the mean of the measrement poplation). (Coleman & Steele, 1995) Figre - Illstration of measrement error effect (AIAA, 1995) A distinction shold be made between error and ncertainty. Uncertainty is an estimate of the error. The ncertainty of the reslt of a measrement reflects the lack of exact knowledge of the tre vale of the measrand. The ncertainty arises from random effects and imperfect correction of the reslts for systematic effects. In practice, there are many possible sorces of ncertainty in a measrement, inclding incomplete definition of the measrand, imperfect realization of the definition of the measrand, non-representative sampling, inadeqate knowledge of the effects of environmental conditions, personal bias in reading analoge instrments, finite instrment resoltion or discrimination threshold, inaccrate vales of measrement standards and reference materials, inaccrate vales of constants and other parameters obtained from external sorces and sed in the dataredction algorithm, approximations and assmptions incorporated in the measrement method and procedre, and variations in repeated observations of the measrand nder apparently identical conditions.

18 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 11 By defining ncertainty as the dispersion of the vales that cold reasonably be attribted to the measrand, the nderlying concept of the GUM is that there is no inherent difference between the ncertainty components arising from random and systematic effects. Both effects are assmed to exist as dispersions abot the measred vale. Conseqently, ncertainties are always estimated sing probability density fnctions or freqency distribtions; their classification shold therefore be based on the method sed to estimate their nmerical vales, i.e. type A evalated by statistical methods, or type B by other means. Type A ncertainties are evalated by statistical analysis of repeated observations to obtain statistical estimates. Type B ncertainties are evalated by other means, i.e. assmed probability distribtions based on scientific jdgements and consideration of a pool of comparatively reliable information that may inclde previos measrements, calibrations and experience, or general knowledge of the behavior and properties of relevant instrments and measrement procedres. Another way of interpreting the GUM classification is that it distingishes between information that comes from sorces local to the measrement process (type A) and information from other sorces (type B). For the final ncertainty, it makes no difference how the components are classified, becase the GUM treats type A and type B evalations in the same manner. Typically, type A ncertainty captres the randomness of the measrement process generated by varios sorces when repeated measrements (more than one) are available, while type B is necessary when there is one (single) measrement or no measrements are available. Conseqently, previos knowledge or engineering jdgement is reqired. As an example of type A evalation of a systematic ncertainty, consider the sitation of a measring instrment calibrated against a standard. The calibration process normally involves taking a nmber of readings. The elements of the ncertainty associated with the calibration cased by random effects will then be evalated statistically (type A). When the calibrated measring instrment is sed in a measrement process, the evalation of the ncertainty mst inclde the ncertainty in the calibration. However, the errors obtained throgh the calibration will make a systematic contribtion to the new measrements. The effect of random errors in the calibration process will have become fossilized into an effect that is systematic and is hence classed as type B ncertainty. As another example of the se of type B evalation for a random ncertainty, consider a measrement made with an instrment that displays the reading to jst three digits and is performed jst once (UKAS 007). This will introdce a random error de to the limited resoltion of the reading. The tre vale of the measrand can lie anywhere in the range of ±0.5x (vale of the least significant digit) with eqal probability. The fndamental definition for measrement ncertainty is the standard deviation of the probability distribtions, which is termed standard ncertainty. It is defined as the positive sqare root of the variance of the probability density fnction obtained from an observed freqency distribtion (for type A) or assmed probability density fnction (for type B). The standard ncertainty of the reslt of a measrement, when that reslt is obtained from the vales of a nmber of other qantities, is termed combined standard ncertainty. It is the standard deviation associated with the reslt and it is obtained sing the positive sqare root (RSS) of all variance and covariance components, however evalated. The combined ncertainty takes all elemental sorces and components of ncertainties into accont. To meet the needs of some area-specific applications, an expanded ncertainty is obtained by mltiplying the combined standard ncertainty by a coverage factor. The intended prpose of the expanded ncertainty is to provide an interval abot the reslt of a measrement that may be expected to encompass a large fraction of the distribtion of vales that cold reasonably be attribted to the measrand. The choice of the coverage factor, sally in the range to 3, is based on the coverage probability or level of confidence reqired of the interval.

19 1 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT The implementation of the UA assmes that the ncertainties involved are small compared with the measred vales, except when the measrements are close to zero. For this to be tre, the following mst strictly apply (AIAA, 1995, GUM, 1993): (a) (b) (c) (d) (e) (f) The measrement process is nderstood, critically analysed, and well defined The measrement system and process are controlled All appropriate calibration corrections have been applied The measrement objectives are specified The instrment package and data-redction procedres are defined Uncertainties qoted in the analysis of a measrement are obtained employing fll intellectal honesty and professional skills... Uncertainty analysis implementation The ncertainty analysis methodology presented below closely follows the JCGM (100:008) gide. To ensre proper application, the standard is reprodced roghly verbatim, with only minor changes....1 The measrement process In most cases a measrand (physical qantity sbject to measrement) Y is not measred directly, bt is determined from N other qantities throgh a fnctional relationship f: Y = f (X 1, X,..., X N ) () For economy of notation, the same symbol is sed for the physical qantity and for the random variable that represents the possible otcome of an observation of that qantity. In a series of observations, the k th observed vale of X i is denoted by X i,k. The estimate of X i (strictly speaking, of its expectation) is denoted by x i. The inpt qantities pon which the otpt qantity Y depends may themselves be viewed as measrands and may themselves depend on other qantities, inclding corrections and correction factors for systematic effects. The fnction f of Eqation () shold express not simply a physical law bt a measrement process, and in particlar, it shold contain all qantities that can contribte a significant ncertainty to the measrement reslt. Thereby, the fnctional relationship f may never be written down explicitly so it needs to be determined experimentally or as an algorithm evalated nmerically. The set of inpt qantities may be categorized as: qantities whose vales and ncertainties are directly determined in the crrent measrement. These vales and ncertainties may be obtained from, for example, a single observation, repeated observations (similar to precision errors), or jdgement based on experience, and may involve the determination of corrections to instrment readings and corrections for inflence qantities, sch as ambient temperatre, barometric pressre, and hmidity. Alternatively, vales of the qantities of the ncertainties are broght into the measrement from external sorces, sch as qantities associated with calibrated measrement standards, certified reference materials, and reference data obtained from handbooks (similar to bias errors). An estimate of the measrand Y, denoted by y, is obtained from eqation () sing inpt estimates for the vales of the N qantities. Ths the otpt estimate y, which is the reslt of the measrement, is given by.

20 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 13 y = f (x 1, x,..., x N ) (3) The estimated standard deviation associated with the otpt estimate or measrement reslt y, termed combined standard ncertainty and denoted by, is determined from the estimated standard deviation associated with each inpt estimate ncertainty and denoted by., termed standard Each inpt estimate and its associated standard ncertainty are obtained from a distribtion of possible vales of the inpt qantity,. This probability distribtion may be freqency based, that is, based on a series of observations of or it may be from an a priori distribtion. Type A evalations of standard ncertainty components are fonded on freqency distribtions derived from data, while type B evalations are made sing a priori or assmed distribtions. It mst be recognized that in both cases the distribtions are models that are sed to represent the state of or knowledge.... Type A evalation of standard ncertainty Type A evalation of standard ncertainty may be based on any valid statistical method for treating data, i.e., calclating the standard deviation of the mean of a series of independent observations, applying the method of least sqare to fit a crve to data, or sing variance analysis sch as ANOVA. In most cases, the best available estimate of the expected vale of a random variable q measred by n independent observations, obtained nder the same measrement conditions is the arithmetic mean or average of the n observations: q = 1 n q n k (4) k =1 Ths for an inpt qantity estimated from n independent repeated observation, the arithmetic mean obtained from eqation (4) is sed as the inpt estimate in eqation (3) to determine the measrement reslt y; that is,. Those inpt estimates not evalated from repeated observations mst be obtained by other methods (external sorces). The individal observations differ in vale becase of random variations in the inflence qantities, or random effects. The experimental variance of the observations, which estimates the variance of the probability distribtion of q, is given by This estimate of variance and its positive sqare root s (q k ) = 1 n (q n 1 k q ) (5) deviation, characterize the variability of the observed vales dispersion abot their mean. k =1, termed the experimental standard, or more specifically, the The best estimate of, the variance of the mean, is given by

21 14 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT The best estimate of, the variance of the mean, is given by s (q) = s (q k ) n (6) The experimental variance of the mean and the experimental standard deviation of the mean qantify how well estimates the expectation of q, and either may be sed as a measre of the ncertainty of. Ths, for an inpt qantity determined from n independent repeated observations, the standard ncertainty of its estimate is with calclated according to eqation (6). For convenience, and standard ncertainty Type B evalation of standard ncertainty are sometimes called a type A variance and a type A A type B evalation of standard ncertainty is sally based on scientific jdgement sing all the relevant information available, which shold inclde at least the following error sorces (UKAS, 007): the reported calibration ncertainty assigned to reference standards and any drift or instability in their vales or readings, the calibration of the measring eqipment, the resoltion and any instability of the measring eqipment, the operational procedre, the variability indced by the operator and environmental conditions. For an estimate x of an inpt qantity that has not been obtained from repeated observations, the associated estimated variance or the standard ncertainty is evalated by scientific jdgement based on all of the available information on the possible variability of. The pool of information may inclde previos measrement data, experience with or general knowledge of the behavior and properties of relevant materials and instrments, manfactrer s specifications, data provided in calibration and other certificates, and ncertainties assigned to reference data taken from handbooks. For convenience, evalated in this way are sometimes called a type B variance and a type B standard ncertainty, respectively. Type B standard ncertainty is based on the expected dispersion of measrements and the assmed probability distribtion. The dispersion,, is the estimated semi-range of a component of ncertainty associated with an inpt estimate,, as defined in Table 1. The probability distribtion can take a variety of forms bt it is generally acceptable to assign welldefined geometric shapes for which the standard ncertainty can be obtained from a single calclation (see Table 1). Typical examples of rectanglar probability distribtions inclde (ISO, 005): maximm instrment drift between calibrations, error de to limited resoltion of an instrment s display or digitizer, manfactrers tolerance limits. A normal probability distribtion can also be sed in association with calibration certificates qoting a confidence level (or coverage factor) with the expanded ncertainty. The trianglar probability distribtion is sed where the only information available abot a qantity is the maximm bonds within which all vales of the qantity are assmed to lie. In some measrement sitations the pper and lower bonds for an inpt qantity are not symmetrical with respect to the best estimate de, for example, to a drift in the instrment. For sch sitations the asymmetric distribtion wold be appropriate for estimating the standard ncertainty. and

22 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 15 When the sorce of ncertainty information is well defined, sch as a calibration certificate or a manfactrer s tolerance, the choice of probability distribtion will be clear-ct. However, when the information is less well defined, for example when assessing the impact of a difference between the conditions of the calibration and se, the choice of a distribtion becomes a matter of professional jdgement. Table 1 - Probability distribtion sed to estimate type B ncertainties Distribtion Standard ncertainty of a measred vale x i Rectanglar Normal a i = U = expanded ncertainty; k = coverage factor Trianglar Asymmetric...4 Determining the combined standard ncertainty Uncorrelated inpt qantities. This sitation refers to the case where all inpt qantities and their ncertainties are independent. The case where two or more inpt qantities are related is labelled as correlated inpt qantities. The standard ncertainty of y, where y is the estimate of the measrand Y and ths the reslts of the measrement, is obtained by appropriately combining the standard ncertainties of the inpt estimates. This combined standard ncertainty of the estimate y is denoted by. The combined standard ncertainty is the positive sqare root of the combined variance, which is given by N f c (y) = (x x i ) (7) i =1 i

23 16 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT where f is the fnction given in eqation (). Each evalated as described above. The combined standard ncertainty is a type A or B standard ncertainty is an estimated standard deviation and characterizes the dispersion of the vales that cold reasonably be attribted to the measrand Y. Eqation (7) is based on a first-order Taylor series approximation of. The partial derivatives in eqation (7) are eqal to evalated at. These derivatives, called sensitivity coefficients, describe how the otpt estimate y varies with changes in the vales of the inpt estimates. Strictly speaking, partial derivatives are evalated at the expectations of the. However, in practice, the partial derivatives are estimated by f = f (8) x i X i x 1,x,,x N Developing the general ncertainty eqation for a given data-redction eqation often reqires the determination of nmeros partial derivatives. When the data-redction eqation is complex, this can be a very tedios task. An alternative is to se nmerical approximations of the partial derivatives (UKAS, 007). They can be compted from: f f (x i + δx i ) f (x i δx i ) (9) x i δx i with the vales of all other variables in the data-redction eqation held constant while varying. To implement this, one shold choose a very small variation for for the qantity (typically 1000 times smaller than the vale of ) and compte the derivative, then redce to one-half of the starting vale and compte the derivative again. Contine redcing the estimated vale of the derivative converges. This method can be easily implemented in commercial spreadsheet or mathematical-analysis software. Correlated inpt qantities. Eqation (7) is valid only if the inpt qantities their ncertainties are independent or ncorrelated. If some of the ntil and are significantly correlated, the correlation mst be taken into accont. There may be significant correlation between two inpt qantities if the same measring instrment, physical measrement standard, or reference datm having a significant standard ncertainty is sed in their determination. The appropriate expression for the combined variance reslts of a measrement when some of the inpt qantities are correlated is associated with the c ( y) = N i =1 f x i (x i ) + N 1 i =1 N f j =i +1 x i f x j (x i, x j ) (10)

24 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 17 where and are estimates of and and is the estimated covariance associated with and....5 Determining expanded ncertainty Althogh can be niversally sed to express the ncertainty of a measrement reslt, in many applications it is often necessary to give a measre of ncertainty that defines an interval abot the measrement reslt that may be expected to encompass a large fraction of the distribtion of vales that cold reasonably be attribted to the measrand. The additional measre of ncertainty that meets the reqirement of providing an interval of the kind indicated above is termed expanded ncertainty, denoted by U. The expanded ncertainty U is obtained by mltiplying the combined standard ncertainty by a coverage factor k: U = k c (y) (11) The reslt of a measrement is then conveniently expressed as, which is interpreted as the best estimate of the vale attribtable to the measrand, and that y to is an interval that may be expected to encompass a large fraction of the distribtion of vales that cold reasonably be attribted to Y. Sch an interval is also expressed as. Ideally, one wold like to be able to choose a specific vale of the coverage factor k that wold provide an interval corresponding to a particlar level of confidence p, sch as 95 or 99 per cent; eqivalently, for a given vale of k, one wold like to be able to state neqivocally the level of confidence associated with that interval. The expanded ncertainty that has an approximate level of confidence p can be written as U p = k p c (y) (1) However, this is not easy to do in practice becase it reqires extensive knowledge of the probability distribtion characterized by the measrement reslt y and its combined standard ncertainty. Althogh these parameters are of critical importance, they are by themselves insfficient for the prpose of establishing intervals having exactly known levels of confidence. The simplest approach, often adeqate in measrement sitations, is to assme that the probability distribtion characterized by y and is approximately normal and the effective degrees of freedom of is of significant size (as many as 30 repeated observations). When this is the case, which freqently occrs in practice, one can assme that taking k = prodces an interval having a level of confidence of approximately 95 per cent, and that taking k = 3 prodces an interval having a level of confidence of approximately 99 per cent. In general, the distribtion of y and may be approximated by a t-distribtion with an effective degrees of freedom,, obtained from the Welch-Satterthwaite formla: ν eff = N ( ) ( ) 4 c y 4 i y i =1 ν i (13)

25 18 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT If is obtained from a type A evalation, is determined as, with n being the nmber of repeated measrements for each X i. If is obtained from a type B evalation and it can be treated as exactly known, which is often the case in practice, ; otherwise estimate from eqation ( x i ) ( ) ν i = 1 σ x i [ ] 1 ( ) Δ x i ( x i ) (14) The qantity in sqare brackets is the relative ncertainty of....6 Practical Considerations If all of the qantities on which the reslt of a measrement depends are varied, its ncertainty can be evalated by statistical means (type A evalation method). However, becase this is rarely possible in practice de to limited time and resorces, the ncertainty of a measrement reslt is sally evalated sing a mathematical model of the measrement and the law of propagation of ncertainty. Ths implicit in JCGM (100:008) is the assmption that a measrement can be modelled mathematically to the degree imposed by the reqired accracy of the measrement. Becase the mathematical model may be incomplete, all relevant qantities shold be varied to the fllest practical extent so that the evalation of the ncertainty can be based as mch as possible on observed data. The steps to be followed for evalating and expressing the ncertainty of the reslt of a measrement can be smmarized as follows (see Figre 3). Figre 3 - Propagation of errors into experimental reslts

26 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 19 For each measrement sitation, the fnctional relationship () is determined first. The fnction shold contain every qantity, inclding all corrections and correction factors that can contribte a significant component of ncertainty to the reslt of the measrement. Then a block diagram of the measrements is constrcted to help organize the individal measrement systems and illstrate the ncertainties involved in the measrement process. Determine and the standard ncertainties for each inpt estimate sing type A and/or B evalation procedres. Evalate the covariances associated with any inpt estimates that are correlated. Once the ncertainties have been identified, their relative significance shold be established based on order of magnitde estimates. A rle of thmb recommended by AIAA (1995) is that those ncertainty sorces that are smaller than 1/4 or 1/5 of the largest sorces are sally considered negligible. Calclate the reslt of the measrement with Eqation (). Determine the combined standard ncertainty by propagating the standard ncertainties to the final reslts sing Eqations (7) or (10). The expanded ncertainty U can then be determined for a specific coverage factor, k. It is recommended that the reslt of the measrement together with its combined standard ncertainty or expanded ncertainty be reported. JCGM (100:008) provides the smmary of procedres for evalating and expressing ncertainty (Section 8, page 7). For practical prposes, we propose here a UA development strctre that compresses the eight detailed steps in for procedres. We will develop the companion examples sing this strctre. The end-to-end procedres that are recommended for developing a fll-fledged UA are: (1) Define the measrement process (short title: Measrement Process). The direct or mltivariate measrement process throgh which the physical qantity vale is estimated needs to be modelled throgh a fnctional relationship between the inpt and otpt of the measrement (sally a mathematical expression) and other factors involved in the measrement process. At this stage of the analysis, it is also sefl to briefly describe the measrement setp, environmental conditions, and technical information abot the instrments to help identify the measrement process errors, inclding errors not associated with the variable in the modelled measrement. This step incldes the data acqisition for the final measrement reslt as well as the cstomized measrements for spporting the condct of ncertainty analysis. () Identify the error sorces and estimate the corresponding ncertainties (short title: Assessment of Uncertainty Sorces). Once the sorces of errors in the measrement process are identified, the ncertainty estimates are developed sing measred (type A) or assmed probability distribtions (type B). The standard ncertainty of the elemental error sorces is described by the sqare root of the variance of the measrement error distribtion. This step also incldes the estimation of the correlated ncertainty sorces between any of the inpt variables. A smmary table of all sorces of error (simple or correlated) is provided in a tablar from. (3) Propagate the ncertainties to the final reslt (Combined Standard Uncertainty). This step is accomplished by sing the variance addition rle (a direct method is sed in the Monte-Carlo Method simlation). For one inpt variable, the addition to the variance of varios sorces of ncertainties of that variable is applied. For a mltivariate measrement, in addition to considering the variance of the individal inpt variables, the possible correlations between the measrement process errors need to be considered. For mltivariate analysis, it is also important that the inpt variable ncertainties are weighted by the appropriate sensitivity coefficients. The

27 0 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT degrees of freedom for each ncertainty component, as well as that of the combined ncertainty obtained throgh the propagation of elemental ncertainties to the final reslt, are then determined. (4) Report the analysis reslt (Reslt Reporting and Uncertainty Bdget). The reports for ncertainty estimates shold present the vale of the qantity of interest, its combined total ncertainty, and the expanded ncertainty at a 95 per cent level of confidence. Also sefl for the analysis is the reporting of the ncertainty bdget, the list of the measrement process ncertainties and associated degrees of freedom for each component and applicable cross-correlated ncertainties, and sensitivity coefficients. Note that the above recommended procedres grop the detailed steps recommended in Section 8 (JCGM, 100:008) as follows: Procedre 1 above incldes steps 1,, and 5 Procedre above incldes steps 3 and 4 Procedre 3 above incldes step 6 Procedre 4 above incldes steps 7 and 8 The implementation of the Gide assmes that the reslt of a measrement has been corrected for all recognized significant systematic effects and that every effort has been made to identify sch effects. In some cases, the ncertainty of a correction for a systematic effect need not be inclded in the evalation of the ncertainty of a measrement reslt. Althogh the ncertainty has been evalated, it may be ignored if its contribtion to the combined standard ncertainty of the measrement reslt is insignificant. In order to decide if a measrement system is fnctioning properly, the experimentally observed variability of its otpt vales, as measred by their observed standard deviation (end-to-end approach in the AIAA, 1005 terminology), is often compared with the predicted standard deviation obtained by combining the varios ncertainty components that characterize the measrement. In sch cases, only those components (whether obtained from type A or type B evalations) that cold contribte to the experimentally observed variability of these otpt vales shold be considered. It is recommended that a preliminary ncertainty analysis be done before measrements are taken. This procedre allows corrective action to be taken prior to acqiring measrements to redce ncertainties. The pre-test ncertainty analysis is based on data and information that exist before the test, sch as calibration, histories, previos tests with similar instrmentation, prior measrement ncertainty analysis, expert opinions, and, if necessary, special tests. Pre-test analysis determines if the measrement reslt can be measred with sfficient accracy, to compare alternative instrmentation and experimental procedres, and to determine corrective action. Corrective action reslting from pre-test analysis may inclde: (a) (b) (c) Improvements to instrment calibrations if systematic ncertainties are nacceptable Selection of a different measrement method to obtain the parameter of interest Repeated testing and/or increased sample sizes if ncertainties are nacceptable Cost and time may dictate the choice of the corrective action. If corrective action cannot be taken, there may be a high risk that test objectives will not be met becase of the large ncertainty interval, and cancellation of the test shold be a consideration. Post-test

28 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 1 analysis validates the pre-test analysis, provides data for validity checks, and provides a statistical basis for comparing test reslts. Mistakes in recording or analysing data can introdce a significant error in the reslt of a measrement. Major mistakes can sally be identified by a proper review of the data. Measres of ncertainty are not intended to accont for sch mistakes. Althogh JCGM (100:008) provides a framework for assessing ncertainty, it cannot sbstitte for critical thinking, intellectal honesty, and professional skill. The qality and tility of the ncertainty qoted for the reslt of a measrement therefore ltimately depend on the nderstanding, critical analysis and integrity of those who contribte to the assignment of its vale. For each measred reslt the ncertainty shold be reported as indicated in Section 7 of JCGM (100:008). When reporting the reslt of a measrement and its ncertainty, it is preferable to err on the side of providing too mch information rather than too little. Details of the ncertainty analysis shold be docmented either in an appendix to the primary test report or in a separate docment.

29 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT References Abernethy, A.B., Benedict, R.P. and Dowdell, R.B., (1985): ASME Measrement Uncertainty, Jornal of Flids Engineering, Transactions of the ASME. Abernethy, A.B and Ringhiser, B., (1985): The History and Statistical Development of the New ASME-SAE-AIAA-ISO Measrement Uncertainty Methodology, AIAA/SAE/ASME/ASEE Proplsion Conference, 1985 (available on-line at: AIAA (1995). Assessment of Wind Tnnel Data Uncertainty, American Institte of Aeronatics and Astronatics Standard S , AIAA, Washington, DC. AIAA (1999). Assessment of Experimental Uncertainty with Application to Wind Tnnel Testing, American Institte of Aeronatics and Astronatics Standard S-071A-1995, AIAA, Reston, VA. ANSI (1997). American National Standard for Expressing Uncertainty--U.S. Gide to the Expression of Uncertainty in Measrement, ANSI/NCSL Z , NCSL International, Bolder, CO. ASME (1986). Measrement Uncertainty, ASME PTC 19.1., ASME New York, NY. ASME (1998). Test Uncertainty, PTC , American Society of Mechanical Engineering, New York, NY (revision of ANSI/ASME (1986). ASME (005). Test Uncertainty, PTC , American Society of Mechanical Engineering, New York, NY (revision of ASME PTC ). ASME (1971). Flid Meters, Their Theory and Application, American Society of Mechanical Engineering, 6 th edition. Bertrand-Krajewski J.-L. and Bardin, J.-P. (00): Uncertainties and Representativity of Measrements in Stormwater Storage Tanks, Proceedings 9th International Conference on Urban Drainage, Portland, OR. Coleman, H.W. and Steele, W.G. Jr. (1989): Experimentation and Uncertainty Analysis for Engineers, 1 st Edition, John Wiley & Sons, Inc., New York. Coleman, H. W. and Steele, W.G. Jr. (1998): Handbook of Flid Dynamics, CRC Press, LLC, New York, NY. Coleman, H. W. and Steele, W.G. Jr. (1999): Experimentation and Uncertainty Analysis for Engineers, nd Edition, John Wiley & Sons, Inc., New York. Gonzalez-Castro, J. and Mste, M. (007): Framework for Estimating Uncertainty of ADCP Measrements from a Moving Boat Using Standardized Uncertainty Analysis, Special Isse on Acostic Velocimetry for Riverine Environments, J. Hydr. Engrg., 133(1), pp Herschy, R.W. (00): The Uncertainty in a Crrent Meter Measrement, Flow Measrement and Instrmentation, 13, pp HUG (007): Hydrometric Uncertainty Gidance, ISO/TS , International Organization for Standardization, Geneva, Switzerland. GUM (1993): Gide to the Expression of Uncertainty in Measrement, ISBN , BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, International Organization for Standardization, Geneva, Switzerland. ISO (1967). ISO/R541: Measrement of Flid Flow by Means of Orifice Plates and Nozzles, International Organization for Standardization, Geneva, Switzerland. ISO (1983): Measrement of Liqid Flow in Open Channels, ISO Standards Handbook 16, International Organization for Standardization, ISBN , Geneva, Switzerland. ISO (199): Measrement Uncertainty, ISO/TC 69/SC 6 (draft dated April 199). ISO (1993): International Vocablary of Basic and General Terms in Methodology, ISBN , International Organization for Standardization, Geneva, Switzerland. ISO (1997), ISO/TR 8363: Measrement of Liqid Flow in Open Channels General Gidelines for Selection of Method, International Organization for Standardization, Geneva, Switzerland. ISO (1998), ISO/DIS 5168: Estimation of the ncertainty of a flow-rate measrement, International Organization for Standardization, Geneva, Switzerland.

30 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 3 ISO (005.a). ISO 5168: Measrement of Flid Flow Procedres for the Evalation of Uncertainties, International Organization for Standardization, Geneva, Switzerland. ISO (005.b): Measrement of liqid flow in open channels Rotating element crrent meters, International Organization for Standardization, ISO 537, Geneva, Switzerland. ISO (007): Hydrometry-Velocity-area Methods Using Crrent-meters-Collection and Processing of Data for Determination of Uncertainties in Flow Measrement. International Organization for Standardization. ISO 1088, Geneva. Switzerland. JCGM (100:008): Evalation of measrement data Gide to the expression of ncertainty in measrement (GUM 1995 with minor corrections) Joint Committee for Gides in Metrology. JCGM (101:008): Evalation of measrement data Spplement 1 to the Gide to the expression of ncertainty in measrement Propagation of distribtions sing a Monte Carlo method, Joint Committee for Gides in Metrology. Kim, D., Mste, M., Gonzalez-Castro, J.A. and Ansar, M. (005): Graphical User Interface for ADCP Uncertainty Analysis, Proceeding ASCE World Water & Environmental Resorces Congress, Anchorage, AK. Kim, Y., Mste, M., Haet, A., Bradley and Weber, L. (007): Uncertainty Analysis for LSPIV In-sit Velocity Measrement, Proceedings 3 nd IAHR Congress, Venice, Italy. Kline, S. J. (1985): 1983 Symposim on Uncertainty Analysis Closre, J. Flids Engineering, 107, pp Kline, S.J. and McClintock, F.A. (1953): Describing Uncertainties in Single-Sample Experiments, Mechanical Engineering, 75. pp Kolpaila, S. (1961): Bibliography of Hydrometry, University of Notre Dame Press, Notre Dame, IN. Mste, M. and Stern, F. (000): Proposed Uncertainty Assessment Methodology for Hydralic and Water Resorces Engineering, Proceedings of ASCE 000 Joint Conference on Water Resorces Engineering and Water Resorces Planning & Management, Minneapolis, MN (CD-ROM). Mste, M., Y, K., Gonzalez-Castro, J. and Starzmann, E. (004): Methodology for Estimating ADCP Measrement Uncertainty in Open-Channel Flows, Proceedings World Water & Environmental Resorces Congress 004 (EWRI), Salt Lake City, UT. Mste, M., Lee, K. and Bertrand-Krajewski, J.L. (01): Standardized ncertainty analysis for hydrometry: a review of relevant approaches and implementation examples, Hydrological Sciences Jornal, 57 (4), NF ENV (Norme Francaise Eropaische VorNorm) (1999): Gide por L Expression de L incertitde de Mesre, AFNOR, Paris, France (in French). NIST (1994): Gidelines for Evalating and Expressing Uncertainty of NIST Measrement Reslts, nd Edition (prepared by Taylor B.N and C.E. Kyatt), National Institte of Standards and Technology, Gaithersbrg, MD. Pilon, P.J., Flford, J.M., Kopaliani, Z., McCrry, P.J., Ozbey, N. and Caponi, C. (007): Proposal for the Assessment of Flow Measrement Instrments and Techniqes, Proceedings XXXII IAHR Congress, Venice, Italy. Reader-Harris, M. (007): ISO flow measrement standards Report on the ISO/TC 30 meeting in November 006, Flow Measrement and Instrmentation 18, pp Singh, V.P., Strpczewski, W.G., Weglarczyk, S. (001): Uncertainty in Environmental Analysis, Proceedings NATO Advance Research Workshop on Integrated Technologies for Environmental Monitoring and Information Prodction (Harmanciogl, N.B., S.D. Ozkl, O. Fistikogl. and P. Geerders, Eds), NATO Service Series, Klwer Academic Press, Dordrecht, The Netherlands.

31 4 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT Steele, W.G., Fergson, R.A., Taylor, R.P., Coleman, H.W. (1994): Comparison of ANSI/ASME and ISO Models for Calclation of Uncertainty, ISA Transactions, 33, pp TCHME (003): Annotated Bibliography on Uncertainty Analysis, Task Committee on Experimental Uncertainty and Measrement Errors in Hydralic Engineering, EWRI, ASCE, Available on line at: Thomas, F. (00): Open Channel Flow Measrement Using International Standards: Introdcing a Standards Programme and Selecting a Standard, Flow Measrement and Instrmentation, 13, pp UKAS (007): The Expression of Uncertainty and Confidence in Measrements, United Kingdom Accreditation Service, Middlesex, UK. UNESCO (007): Data Reqirements for Integrated Urban Water Management, Eds. T.D. Fletcher and A. Deletic, Taylor and Francis, Oxfordshire, UK. WMO (1994): Gide to Hydrological Practices, 5th Edition, World Meteorological Organization No. 168, Geneva. WMO (1980): Manal on Stream Gaging, Operational Hydrology Report No. 13, World Meteorological Organization, No. 519, Geneva. WMO (004): Commission for Hydrology 1th Session, Abridged Final Report with Resoltions and Recommendations, World Meteorological Organization No. 979, Geneva. WMO (007): Exploratory Meeting on CHy s Proposal for the Assessment of the Performance of Flow Measrement Instrments and Techniqes, 04/5-7/007 Meeting Final Report, (available at: World Meteorological Organization, Geneva, Switzerland. WMO (008): Gide to Hydrological Practices, Commission for Hydrology, WMO-No 168, Sixth edition, World Meteorological Organization, Geneva, Switzerland, available on line at:

32 APPENDIX A UNCERTAINTY ANALYSIS PUBLICATIONS General Uncertainty Analysis Gides, texts and articles Brown, K.K., Coleman, H.W., Steele, W.G., and Taylor, R.P. (1996). Evalation of Correlated Bias Approximations in Experimental Uncertainty Analysis. AIAA Jornal, 34(5), Brown, K.K. Coleman, H.W., and Steele, W.G. (1998). A methodology for determining experimental ncertainties in regressions. J. of Flids Engineering, 10, Castrp, H. (000). Estimating bias ncertainty. Integrated Sciences Grop, Bakersfield, CA. Coleman, H.W and Steele, W.G. (1999). Experimentation and Uncertainty Analysis for Engineers, nd Ed., John Wiley & Sons, New York, 75 pp. Coleman, H.W. and Steele, W.G. (1995). Engineering application of experimental ncertainty analysis. AIAA Jornal, 33(10), Holman, J.P. (1989). Experimental methods for engineers. Fifth edition, McGraw-Hill, New York, NY. Kline, S.J. and McClintock, F.A. (1953). Describing ncertainties in single-sample experiments. Mechanical Engineering, 75, 3-7. Kline, S.J. (1985). The prposes of ncertainty analysis. Jornal of Flids Engineering, Transactions of the ASME, Jne Kline, S.J. (1985). Closre to 1983 Symposim on Uncertainty Analysis. Jornal of Flids Engineering, Transactions of the ASME, Jne Lassahn, G.D. (1985). Uncertainty definition. Jornal of Flids Engineering, Transactions of the ASME, Jne 1985 Mandel, J. (1984). The statistical analysis of experimental data. Corier Dover Pbs. 448 pg. Mandel, J. and Nanni, L. (1986). Measrement evalation. National Brea of Standards Special Pblication 700-, Indstrial Measrement Series. Moffat, R.J. (1985). Using ncertainty analysis in the planning of an experiment. Jornal of Flids Engineering, Transactions of the ASME, Jne Moffat, R.J. (198). Contribtions to the theory of single-sample ncertainty analysis. Transactions of the ASME, 104, Moffat, R.J. (1988). Describing the ncertainties in experimental reslts. Experimental Thermal and Flid Science, 1, Nielsen, H.S. (1997). Using the ISO Gide to the Expression of Uncertainty in Measrements to determine calibration reqirements. National Conference of Standards Laboratories Workshop & Symposim. Phillips, S.D, and Eberhardt, K.R. (1997). Gidelines for expressing the ncertainty of measrement reslts containing ncorrected bias. J. of Research of the National Institte of Standards and Technology, 10, Taylor, B.N., and Kyatt, C.E. (1994). Gidelines for Evalating and Expressing the Uncertainty of NIST Measrement Reslts. NIST Technical Note 197. Taylor, J.R. (198). An Introdction to Error Analysis. University Science Books, Mill Valley, California. Taylor, J.R. (1997). An introdction to error analysis: the stdy of ncertainties in physical measrements. University Science Books, Sasalito, CA. Standards Abernethy, A.B., Benedict, R.P. and Dowdell, R.B. (1985). ASME measrement ncertainty. Jornal of Flids Engineering, Transactions of the ASME, Jne AIAA (1999). Assessment of Experimental Uncertainty with Application to Wind Tnnel

33 6 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT Testing, Standard S-071A American Institte of Aeronatics and Astronatics. 84 pp. Eisenhart, C., K, H.H. and Colle, R. (1983). Expression of the ncertainties of final measrement reslts. Reprints. National Brea of Standards, Washington DC. Eropean Co-operation for Accreditation (1999). Expression of the ncertainty of measrement in calibration. Pblication Reference EA-4/0. ISO (1988). ISO 7066-:1988. Assessment of ncertainty in the calibration and se of flow measrement devices -- Part : Non-linear calibration relationships. ISO (1993). Gide to the expression of ncertainty in measrement (GUM). ISO/IEC Gide 98, Geneva. ISO (1995). Gide to the expression of ncertainties in measrement. (ISO, IEC, IFCC, IUPAC, IUPAP, OIML). ISO (1997) ISO/TR :1997. Assessment of ncertainty in calibration and se of flow measrement devices -- Part 1: Linear calibration relationships. NIST (1997) American National Standard for Expressing Uncertainty--U.S. Gide to the Expression of Uncertainty in Measrement. ANSI/NCSL Z Flow Measrement UA Gides, texts and articles Pelletier, P. M. (1988). Uncertainties in the single determination of river discharge: a literatre review. Canadian Jornal of Civil Engineering, vol. 15, no. 5, p Saer, V. B. and Meyer, R. W. (199). Determination of error in individal discharge measrements. U.S. Geol. Srvey Open-File Report 9-144, 1 p. Herschy, R. W. (1985). Accracy-streamflow measrement. Elsevier Applied Science Pblishing, Chapter 14, p Herschy, R.W. (1978). Accracy. Hydrometry: Principles and practices. Ed. Herschy, R.W.: John Wiley and Sons. Whalley, N., Iredale, R.S. and Clare, A.F. (001). Reliability and ncertainty in flow Standards measrement techniqes some crrent thinking. Phys. Chem. Earth (C), 6(10-1), Abernethy, A.B. (1985). Flid Flow Measrement Uncertainty. ISO/DIS 5168, 10th draft. 54 pp. ISO (005).Measrement of flid flow -- Procedres for the evalation of ncertainties. ISO 5168, Geneva. ISO (007). Measrement of liqid flow in open channels Velocity-area methods. ISO 748, Geneva. ISO (007). Hydrometry Velocity-area methods sing crrent-meters Collection and processing of data for determination of ncertainties in flow measrement. ISO 1088, Geneva, Switzerland. ISO (007). ISO/CEN PDTS 5377 (007). Hydrometric Uncertainty Gide (HUG).

34 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 7 Specific Flow Measrement UA Acostic profilers Changjiang Water Resorces Commission, Brea of Hydrology (004). River discharge measrement on the Yangtze River with acostic Doppler crrent profiler, Field Measrement Verification Test and Development Stdy. Brea of Hydrology, Ministry of Water Resorces, Whan, China. 9pp. Gaeman, D. and Jacobson, R.B. (005). Aqatic habitat mapping with an acostic crrent profiler: Considerations for data qality. U.S. Geological Srvey Open-file Report González-Castro, J. A. and Mste, M. (007). Framework for Estimating Uncertainty of ADCP Measrements from a Moving Boat by Standardized Uncertainty Analysis. Special Isse on Acostic Velocimetery for Riverine Environments, J. Hydr. Engrg., 133(1), pp ISO (005). Hydrometry Measring river velocity and discharge with Acostic Doppler profilers. PDTS ISO (in press) ISO/TR Gide to the application of acostic Doppler crrent profiler for measrement of discharge in open channels. Morlock, Scott, E. (1996). Evalation of acostic Doppler crrent profiler measrements of river discharge. U. S. Geol. Srvey Water-Resorces Inv. Report Mste, M., Y, K, Gonzalez-Castro, J. and Starzmann, E. (004). Methodology for estimating ADCP measrement ncertainty in open channel flows. Proc. World Water & Environmental Research Congress (EWRI) Salt Lake City, UT. Mste, M., Y,K., Pratt, T., and Abraham, D. (004). Practical aspects of ADCP data se for qantification of mean river flow characteristics: Part 11 Fixed-vessel measrements. J.Flow Meas. And Instr 15(1) pp17-8. Simpson, M. R. and Oltmann, R. N. (1993). Discharge measrement system sing an acostic Doppler crrent profiler with applications to large rivers and estaries. U. S. Geological Srvey Water-Spply Paper 395, 3 p. Simpson, M.R. (001). Discharge measrements sing a broad-band acostic doppler crrent profiler. USGS Open-File Report 01-1, Sacramento, CA. PDF. SonTek/YSI (000). ADP acostic doppler profiler. Principles of operation. SonTek/YSI. San Diego, California 5pp. SonTek/YSI (003). Principles of River Discharge Measrements. San Diego, California. US Geological Srvey (00). Policy and Technical Gidance on Discharge Measrements sing Acostic Doppler Crrent Profilers. Office of Srface Water Technical Memorandm, pp. Yorke, T.H. and Oberg, K.A. (00). Measring river velocity and discharge with acostic Doppler profilers. Flow Measrement and Instrmentation, Vol. 13, Nmber 5-65pp. Mechanical Meters Carter, R.W. and Anderson, I.E. (1963). Accracy of crrent meter measrements. Am. Soc. Civil Engineers Jor., V. 89, No. HY4, pp Dickinson, W. T. (1967). Accracy of discharge measrements. Hydrology Papers, Colorado State University, Fort Collins, CO, No. 0, Jne, p Herschy, R. W. (1971). The magnitde of errors at flow measrement stations. Water Resorces Board, Reading Bridge Hose, Reading, Berkshire, England, TN11, (Revised), 30 p. Smoot, G.F. and Carter, R.W. (1968). Are individal crrent meter ratings necessary? American Society of Civil Engineers Jornal of the Hydralics Division, 94 (HY), p

35 8 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT Flmes Abt, S.R., Florentin, C.B., Genovez, A., and Rth, B.C. (1995). Settlement and sbmergence adjstments for Parshall flme. Jornal of Irrigation and Drainage Engineering, Vol. 11, No. 5. Clemmens, A. (001). Water measrement with flmes and weirs. International Institte for Land Reclamation and Improvement, Pblication 58, Wageningen, The Netherlands. Dodge, R.A. (1990). Effects of Montain Stream Topography on the Accracy of Small Parshall Flmes. U.S. Dept. of the Interior, Brea of Reclamation, Research Report R Jones, R.W. (00). A method for comparing the performance of open channel velocity-area flow meters and critical depth flow meters. Flow Measrement and Instrmentation, 13, Peck, H. (1988). Sbmerged flow in Parshall flmes. Model-Prototype Correlation of Hydralic Strctres, International Symposim, Colorado Springs, CO Ag. 9-11, Thomas, C.W. (1959). Errors in measrement of irrigation water. ASCE transactions, 14, Orifice Meters Clark, W.J. (1965). Flow measrement by sqare-edged orifice plate sing corner tappings. Pergamon Press, Oxford, UK. International Organization for Standardization (1991). Measrement of flid flow by means of pressre differential devices Part 1: Orifice plates, nozzles and Ventri tbes inserted in circlar cross-section condits rnning fll. ISO , Geneva, Switzerland.

36 APPENDIX B DISCHARGE UNCERTAINTY EXAMPLE: WEIGHING AND TIMING MEASUREMENTS OF DISCHARGE B.1 Introdction This example demonstrates the application of the GUM to a laboratory measrement of discharge. Discharge measrements, like all measrements, are an approximation of the tre vales. The ncertainty mst be qantified to give a complete measrement reslt (Taylor and Kyatt, 1994). The ncertainty of a discharge measrement is dependent on the instrmentation, method, environmental conditions, and care with which the measrement is made. Several different types of instrmentation and techniqes can be sed to measre discharge in a laboratory. Depending on the instrmentation, measrement techniqes, and flow conditions, the comptation of discharge and its associated ncertainty can be relatively simple or complex. The measrement of discharge by timing of a flow into a container and weighing the flow captred by the container is a relatively simple method of measring a laboratory discharge. For this simple discharge measrement method, the estimation of the measrement ncertainty will be presented for three measred discharges, demonstrating the application of methods recommended by GUM for compting ncertainty. The example demonstrates that ncertainty can be compted sing a combination of measrement data and information from instrmentation specifications and laboratory conditions. It also illstrates that repeated measrements (increased sampling) can be sed to redce the ncertainty of the final (or averaged) measrement. This example broadly follows the general steps of the GUM: (1) nderstanding the measrement process and expressing the fnctional relationship between the measrand and the measred inpt qantities; () identifying and estimating the contribting sorces of ncertainty to the measrand, inclding those that may not be explicitly expressed in the fnctional relationship; (3) combining the standard ncertainties into the combined ncertainty by determining sensitivity coefficients and combining the standard ncertainties sing a first-order Taylor series approximation of the fnctional relationship; and (4) reporting the expanded ncertainty and measrement reslt sing one of the recommended GUM formats. The steps are presented herein as: (1) the fnctional relationship and the measrement process for a weighing and timing measrement of discharge, () the identification and estimation of the ncertainty sorces, (3) the formlation of the combined ncertainty eqation, and (4) the comptation (and reporting) of the measrement ncertainty for single and mltiple measrements of discharge. B. Estimating Measrement Uncertainty As presented in previos sections, GUM classifies standard ncertainties either as type A, compted from measred data, or type B, estimated by methods not sing measred data. Generally, random effects are captred by a type A ncertainty and systematic effects are captred by a type B ncertainty. Exceptions to the generalization can occr, and random effects may be estimated sing a type B ncertainty, while systematic effects may be estimated from the statistical analysis of measrement data (type A). Depending on the availability of measrement data, an ncertainty estimate might consist of only type B ncertainties.

37 30 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT Sorces of ncertainty can change from being either random or systematic, depending on the measrement process. For example, if a grop of discharge measrements is made by an individal technician, the technician contribtes a systematic ncertainty to the discharge measrements, bt if the grop of discharge measrements is made by many different technicians, the technicians contribte a random ncertainty to the discharge measrements. Standard ncertainties assme that ncertainties can be modelled by probability distribtions qantified by variances and standard deviations (UKAS, 007). Regardless of the sorce of ncertainty, the standard deviation is defined as the standard ncertainty. If data are available from which a standard deviation can be compted (type A ncertainty), a normal distribtion is typically assmed. If other sorces of information are sed to determine the standard ncertainty (type B ncertainty), other probability distribtions may be sed. This may reqire additional processing to convert the initial information for type B ncertainties into a form that is eqivalent to one standard deviation for the poplation. The processing reqired is dependent on the type of probability distribtion that is assmed sch as, trianglar, rectanglar or normal. The combined standard ncertainty, c, is compted from the individal standard ncertainties sing eqation 7 (in section ) or eqation 10 (in section ). Similar to a standard deviation, it covers approximately 68 per cent of the possible measrement otcomes. The expanded ncertainty, U, compted by eqation 1 (in section ) typically covers 95 per cent of the possible measrement otcomes and is the combined standard ncertainty mltiplied by a coverage factor that is a fnction of sample size. Applying the GUM to a UA withot any other gidance can be a danting task, especially for novices to UA. Several national standardization and accreditation agencies have written sccinct gides to assist in the application of the GUM to determining measrement ncertainty. Two sch gides that novices to UA will find helpfl are: The Expression of Uncertainty and Confidence in Measrement (UKAS, 007) and Gidelines for Evalation and Expressing the Uncertainty of NIST Measrement Reslts (NIST, 1994). B.3 Fnctional Relationship and the Measrement Process The measrement of discharge by weighing and timing a constant flow is often sed to calibrate discharge measring devices, sch as small flmes, in hydralic laboratories. The measrement of discharge by timing and weighing a captred amont of water is compted from: ( w w ) 1 Q = (B.1) γ t where Q is the discharge in volme per nit time, w is the weight of the container and water, t is the elapsed time, and γ is the specific weight of water. The sbscripts 1 and indicate the initial and final weight measrement, respectively. The specific weight of water can be compted from γ = ρg where g is the acceleration de to gravity and ρ is the density of water. Becase weight is defined as a force in the technical setting of the hydralic laboratory (Thompson and Taylor, 008), specific weight is defined as force per volme.

38 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 31 The ncertainty of a discharge measrement by weighing and timing is dependent on the data redction eqation, what instrmentation is sed and how the measrement is made. Eqation B.1 is the data redction eqation. From this eqation, sensitivity factors for ncertainty sorces can be compted. Information on the accracy and resoltion of the instrmentation sed to make the measrements, the vales sed for the acceleration de to gravity and water density and the process sed in the laboratory to make the measrement are necessary to compte the ncertainty of the measrement. The discharge is measred in an indoor hydralic laboratory at the exit of the small flme test stand. The small flme test stand allows the flow exiting from the flme to be diverted by a laboratory technician into a large drm. The measrement process starts with the empty drm being weighed by laboratory staff to determine the initial weight. The initial weight is recorded by hand in a log book. The flow is then diverted into the drm and timed manally by laboratory staff ntil the flow is diverted away from the drm. The drm is again weighed by laboratory staff to determine the final weight. The elapsed time and final drm weight are recorded in the log book. A valve on the drm is then opened to allow it to empty. The data recorded for the measrement are the initial weight, final weight and elapsed time. The discharge is compted sing eqation B.1 and assming γ = kn/m 3 (6.43 lb/ft 3 ). A hand-held stop watch is sed to time the filling of the drm. An electronic floor scale is sed to weigh the drm and water. The flow throgh the flme is spplied by a pmp system that ses a constant head tank. The discharge measrements are made in a laboratory that is heated in the winter bt is not air conditioned. Air temperatres in the laboratory range annally from abot 15 to 40 0 C and water temperatre at the time of the experiment is not measred. B.4 Identification and Estimation of Sorces of Uncertainty Sorces of ncertainty for the discharge measrement inclde the accracy and resoltion of the floor scale, the accracy and resoltion of the stop watch, the laboratory staff, the steadiness of the flow, and the accracy of the vales sed for the acceleration of gravity and water density. Temperatres of the water and the air are not measred dring the discharge measrement and may affect the ncertainty of the instrments sed and the discharge measred. B.4.1 Time and weight A stop watch is sed to measre the time, t, in seconds to fill the drm. It has a manfactrer s accracy of ± 5 s/day, a resoltion of 0.01 s and an operating temperatre range of 1 to 59 C. An indstrial floor scale with a digital readot is sed to weigh the empty and filled drm and has a manfactrer s accracy of ± 0. 1 kg ( ± 0. lbs) and a display resoltion of 0.1 kg (0. lbs). The scale s digital display shows a ronded representation of the analoge measrement reslting in a measrement resoltion of one-half of the digits displayed (display resoltion), 0.5 X 0.1 kg = 0.05 kg. The scale has an operating temperatre range of -10 to 40 C. As the same scale is sed to make both weight measrements, the ncertainties for w 1 and w are correlated. Table B.1 smmarizes the available manfactrer information on the stop watch and floor scale sed in the discharge measrement.

39 3 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT Table B.1 Information from manfactrer specifications for ncertainty analysis. Instrment Operating temperatre range C Range Accracy Display Resoltion Floor Scale -10 to kg 0.1 kg 0.1 kg Stop Watch 1 to 59 9h 59m 59.99s % 0.01 s B.4. Specific weight The specific weight of water sed to compte water volme from weight, γ = kn/m 3 (6.43 lb/ft 3 ) is based on the standard acceleration of gravity of m/s (3.174 ft/s ) and the density of water at 4 C. Local gravity is related to latitde and can be estimated from the NOAA srface gravity prediction map ( The hydralic laboratory is 5 ft above mean sea level and is located at latitde and longitde Using the NGS, NOAA calclator, local acceleration of gravity is estimated as m/s ( ft/s ); abot per cent difference with the standard vale. The laboratory water spply is pmped from an otdoor constant-head tank. Water temperatre measrements are not made dring the discharge measrement. It is estimated that the laboratory water temperatre cold range from 10 to 30 C over a year. The ncertainty on the volme of water measred de to not correcting for temperatre effects on water density range from per cent to per cent. This reslts in nder measring the volmetric discharge. The estimated mean vale of the specific weight dring discharge measrements is kn/m 3 ( lb/ft 3 )and varies from kn/m 3 to kn/m 3 (6.39 to lb/ft 3 ) The ncertainty de to sing a specific weight of water of kn/m 3 (6.43 lb/ft 3 ) is smmarized in table B.. Table B. Percentage difference in specific weight of water from sing standard gravity and water density at 4 C o C Specific Weight, standard gravity (kn/m 3 ) Percentage difference de to temperatre Specific weight, local gravity (kn/m 3 ) (std vale) Percentage difference with standard vale B.4.3. Repeatability of discharge measrements Data from previos discharge measrements are available and can be sed to estimate the standard ncertainty associated with the repeatability of the discharge measrement. Repeatability of the measrement is dependent on the steadiness of the flow and the skill of the laboratory staff operating the stop watch and diverting the flow. Standard deviations compted from prior experiments with replicated discharge measrements are plotted with mean flow rate in figre B.1. Figre B.1 contains discharge measrements from 109 discharge measrements that were obtained for 11

40 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 33 flme calibrations. The measred discharges range from.83 x 10-4 m 3 /s to x 10-4 m 3 /s (0.01 to 0.55 ft 3 /s). STANDARD DEVIATION IN M 3 /S MEAN DISCHARGE IN M 3 /S Figre B.1 - Standard deviations from previosly repeated measrements plotted verss mean discharge B.5 Formlating the Combined Uncertainty Estimate The combined ncertainty estimate ses a first-order Taylor series approximation to combine the individal sorces of ncertainty. Before sqaring the standard ncertainties, they are mltiplied by sensitivity coefficients. Sensitivity coefficients, c i, for each inpt in eqation B.1 can be compted by taking partial derivation with respect to each inpt. Q w = c w = 1 ρgt (B.) Q w 1 Q t = c = c w1 t 1 = ρgt = ( w w ) ρgt 1 (B.3) (B.4) Q = c γ γ = ( w w ) γ t 1 (B.5) Each standard ncertainty is mltiplied by the appropriate sensitivity coefficient. This ensres that all the standard ncertainties will be expressed in a common nit. For this example the common nit is discharge in m 3 /s.

41 34 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT The estimated combined ncertainty is a fnction of the discharge and the dration of the test as well as the standard ncertainties of the inpt variables in eqation B.1. From previos measrements, it is also known that the flme discharge measrements have significant variability at a fixed flow rate that reslts from nintentional operating errors, splashing water ot of the drm when diverting the flow and timing the filling of the drm. Using eqation 10 from section...4 with the inpt variables from the data redction eqation B.1, and the ncertainty from the measrement repeatability that captres the nintentional operator errors, the estimated combined ncertainty eqation is: ') ( 1), ( ) ( ) ( 1) ( ) ( ) ( 1 1 q c w w c c c t c w c w c y q w w t w w c = γ γ (B.6) The first three terms on the right-hand side of the eqation can be expanded to inclde the small accracy and resoltion effects from the instrments sed. ) ( ) ( ) ( r a w w w + = (B.7) ) 1 ( ) 1 ( 1) ( r a w w w + = (B.8) ) ( ) ( ) ( a t r t t + = (B.9) The ncertainty terms sbscripted with a represent the accracy (or systematic) effects of the instrments and terms sbscripted with r represent the ncertainty from the resoltion of the instrment. The forth term on the right-hand side of the eqation, the ncertainty of the vale sed for specific weight, incldes the effects of sing a vale of specific weight that is too large and ignoring the effects of water temperatre variation on water density. It can be expanded as: ) ( ) ( ) ( γ γ γ + = (B.10) where γ is the difference between the average tre specific weight and the vale sed, and γ is the ncertainty de to the water temperatre variation from the average temperatre of the laboratory water. Eqation B.6 can be rewritten sing the expanded eqations for the for terms as: ) 1, ( ') ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( w w c c q c c c t c w c w c t c w c w c y w w q r t r w r w a t a w a w c = γ γ γ γ (B.11) The last term in eqation B.11 is the correlated ncertainty reslting from sing the same scale for measring the initial and final weight. The first six right-hand terms are type B ncertainties estimated from the manfactrer instrment specification for accracy and resoltion. The ncertainty contribted by the repeatability of the measrement, ) (q', is the only standard ncertainty estimated sing measred data (type A). This term incldes effects from technician skill, flow steadiness, and the process of diverting the flow dring the measrement that are not represented in the other terms and are important contribtions to the measrement ncertainty.

42 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 35 Eqation B.11 can be simplified. Becase the same scale is sed for all the weighing measrements, the weighing terms can be rearranged and combined. The standard ncertainties for weighing are eqal: ( wa ) = ( w1 a ) and (w r ) = (w r ) = (w1 r ). The correlated ncertainty (covariance) can be rewritten as = ( w ) ( w1 ) = ( w ). Using eqations B. and B.3, the sensitivity ww1 a a a coefficients for weighing can be written as, cw = cw = cw 1 and cwcw 1 = cw. The correlation of the instrment accracy (or systematic) ncertainty between the two measrements is negative, and reslts in the accracy ncertainties from weighing not contribting to the combined ncertainty of the discharge measrement. The random ncertainties of the two measrements are additive. Eqation B.11 can been rearranged as, c ( y) = cw' ( wr ) + ct ( ta ) + ct ' ( tr ) + c ( ) c ( γ γ + γ γ ) + cq ( q') (B.1) In eqation B.1, each standard ncertainty, i, is compted by: ai i = (B.13) Divisor where a i is assmed to be either the limits within which the tre vale will lie or a standard deviation. The Divisor is the vale by which a i can be divided to yield the standard deviation for the probability distribtion assmed for the i-th sorce of ncertainty. A rectanglar distribtion is assmed for the resoltion of weighing and timing and for the accracy specification of the scales and stop watch. In the absence of information on the probability distribtion, the manfactrer s specifications are assmed to be the limits of their respective probability distribtions. Similarly, the range of specific weight de to the effect of variable water temperatre is also assmed to have a rectanglar probability distribtion. The ncertainty de to the systematic error in the specific weight is assmed to have a trianglar distribtion, becase the estimated vale of the tre specific weight is most likely to be very near the tre vale. A normal distribtion is sed for the discharge repeatability term, becase this vale is based on measred data having a normal probability distribtion. B.6 Compting the Uncertainty of a Discharge Measrement Determined by a Single Sample The estimated ncertainties for discharge measrements of 3.681x10-4 m 3 /s, x 10-4 m 3 /s, and x 10-4 m 3 /s (0.013 ft 3 /s, ft 3 /s, and 0.50 ft 3 /s) obtained from a single weighing and timing of the flow are compted sing eqation B.1. Instrment specifications, information on specific weight variation, and existing discharge measrement data are sed to estimate the individal standard ncertainties in eqation B.1. The existing discharge data is typical of what arises in practice. Many older measrements are available bt differ slightly from the crrent measrement. For any given measrement condition only a small nmber of repeated measrements are made nder exactly the same conditions. The data collected at approximately the same discharges, listed in table B.3, are sed to compte pooled statistics for the standard deviation de to measrement repeatability.

43 36 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT Table B.3 - Measred data sed for compting pooled statistics for estimating repeatability ncertainties. [n, nmber of samples; std. dev., standard deviation] Discharge measrement in m 3 /s 3.681x x x 10-4 Pooled data in m 3 /s Pooled data in m 3 /s Pooled data in m 3 /s n Mean std. dev. n mean std. dev. n mean std. dev x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x10-5 For each discharge, the estimated standard deviation of the repeatability at a given discharge can be compted from a pooled variance: M ' ( nk 1) ( qk ) ' k= 1 ( q p ) = M (B.14) ( n 1) k= 1 k where M is the nmber of different conditions, ( q ' p ), is the pooled variance and ( q ' k ) is a variance for the kth condition. The pooled standard deviation compted from the sqare root of the variance, is listed for each discharge for which the ncertainty is to be compted in table B.4 along with the pooled degrees of freedom and the stdent s t vale for a 95 per cent confidence interval.

44 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 37 Table B.4 - Pooled statistics for the ncertainty de to repeatability Discharge m 3 /s Degrees of Pooled standard Stdent s t freedom deviation m 3 /s 3.681x x x x x x Tables B.5 throgh B.7 smmarize the ncertainty comptations for a single measrement (one sample) in m 3 /s for three different discharges: 3.681x10-4, x 10-4 m 3 /s, and x 10-4 m 3 /s (0.013 ft 3 /s, ft 3 /s, and 0.50 ft 3 /s). Table rows 1 throgh 6 correspond to each term on the right-hand side of eqation B.1. The rightmost colmn, except in the last two rows, contains the contribtion of each term to the combined ncertainty. The combined ncertainty, in table row 7, is compted sing eqation 10 (in section ). The ncorrected specific weight vale contribtes ncertainty from the effects of temperatre variation and from the bias in the vales sed. The ncorrected specific weight mean bias is the difference between the nsed vale and the most likely tre vale for specific weight, kn/m 3. The coverage factor, k, sed to compte the expanded ncertainty, is dependent on the effective sample size. A coverage factor of is appropriate when the ncertainties are based on large sample sizes ( n 30) and a 95 per cent confidence interval. For small sample sizes, stdent s t distribtion is sed to determine the coverage factor. The type A standard ncertainty for measrement repeatability has a known sample size. For the type B standard ncertainties sed, the sample size is assmed to be infinite. For the largest discharge, all standard ncertainties sed to compte the combined ncertainty are based on large sample sizes. For the two smaller discharges, the effective sample size for the ncertainty contribtion from the repeatability term is smaller than 30. The type B standard ncertainties contribte vales of 0 to the smmation in the denominator. For these sitations the effective degrees of freedom are compted sing eqation 1 (in section ), the Welch-Satterthwaite formla. The only non-zero term in the denominator is compted from the pooled statistics from repeated discharge measrements, which are listed in table B.4. For a discharge of 3.681x10-4 m 3 /s the effective degrees of freedom from the Welch-Satterthwaite formla are: v eff 6 4 ( ) = 0 = and for x 10-4 m 3 /s the degrees of freedom are: 5 4 ( ) = 3 v =. eff

45 38 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT For the effective degrees of freedom, 0 and 3, the stdent s t vales for a 95 per cent confidence interval give the coverage factors of.09 and.07 respectively. The expanded ncertainties are compted by mltiplying the combined ncertainty with a coverage factor to yield a 95 per cent confidence interval. The expanded ncertainties, in table row 8, are reported to significant figres in nits of m 3 /s and per cent. The qality of the data sed to compte expanded ncertainty rarely jstifies reporting more than significant figres (UKAS, 007, p. 1). Sorce of ncertainty Table B.5 Example of compting the ncertainty for: Q=3.681x10-4 m 3 /s, t= 63.3 s, W -W 1 =( )kg a i,vale Probability distribtion Divisor c i, Sensitivity coefficient Contribtion to ncertainty (m 3 /s) Correlated resoltion of 0.05 kg rectanglar x x10-8 weights Stop watch accracy % rectanglar x x10-9 Stop watch resoltion 0.01 sec rectanglar x x10-9 Uncorrected specific weight N/m 3 trianglar x x10-8 mean bias Uncorrected specific weight variation de to temperatre N/m 3 rectanglar x x10-8 Repeatability 3.897x10-6 m 3 /s normal x10-6 Combined ncertainty 3.89 x10-6 m 3 /s Expanded ncertainty m 3 /s, k= x10-6 (.%)

46 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 39 Table B.6 - Example of compting ncertainty for: Q= x10-4 m 3 /s, t=34.6s, W -W 1 =( )kg Sorce of ncertainty a i,vale Probability distribtion Divisor c i, Sensitivity coefficient Contribtion to ncertainty (m 3 /s) Correlated resoltion of 0.05 kg rectanglar x x10-7 weights Stop watch accracy % rectanglar x x10-8 Stop watch resoltion 0.01 sec rectanglar x x10-8 Uncorrected specific weight N/m 3 trianglar x x10-7 mean bias Uncorrected specific weight variation de to temperatre N/m 3 rectanglar x x10-7 Repeatability 1.688x10-5 m 3 /s normal x10-5 Combined 1.69x10-5 ncertainty m 3 /s Expanded ncertainty m 3 /s, k = x10-5 (0.65%) Table B.7 Example of compting ncertainty for: Q= x10-4 m 3 /s, t=1.49s, W -W 1 =( )kg Sorce of ncertainty a i,vale Probability distribtion Divisor c i, Sensitivity coefficient Contribtion to ncertainty (m 3 /s) Correlated resoltion of 0.05 kg Rectanglar x x10-7 weights Stop watch accracy % Rectanglar 3 1.0x x10-8 Stop watch resoltion 0.01 sec Rectanglar 3 1.0x x10-7 Uncorrected specific weight N/m 3 Trianglar x x10-6 mean bias Uncorrected specific weight variation de to temperatre N/m 3 Rectanglar x x10-6 Repeatability 1.485x10-4 m 3 /s Normal x10-4 Combined 1.48x10-4 ncertainty m 3 /s Expanded ncertainty m 3 /s, k = x10-4 (.0%)

47 40 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT The combined ncertainty for a single discharge measrement, compted in tables 3 throgh 5, is plotted in figre B. as a fnction of discharge. The plot shows that the ncertainty in m 3 /s of an individal discharge measrement increases with discharge. The combined ncertainty is dominated by the ncertainty contribtion from the repeatability of the discharge measrement UNCERTAINTY IN M 3 /s DISCHARGE IN M 3 /S Figre B. - Uncertainty for laboratory discharge by weighing. B.7 Compting the Uncertainty of a Discharge Measrement Determined by Three Samples Mltiple samples or measrements at the same discharge can be sed to redce the ncertainty of the discharge measrement when there is significant variation in repeated measrements. When mltiple samples are sed, the measred discharge is the average of the repeated discharge measrements. The standard ncertainty contribted by the repeatability to the discharge measrement is redced by a factor of 1 where n is the nmber of individal measrements made at a set discharge n (UKAS, 007, p. 19). Table B.6 lists the compted ncertainties of the discharge measrement when three measrements (or samples) are made at a set discharge. Uncertainty of the measred discharge is redced by approximately 30% when three repeated measrements are averaged (figre B.3). Table B.6 - Effect of repeating the discharge measrement three times on discharge ncertainty Discharge (m 3 /s) Nmber of measrements Standard ncertainty for repeatability (m 3 /s) Combined ncertainty (m 3 /s) Expanded ncertainty (per cent) 3.681x x x x x x x x x

48 GUIDELINES FOR THE ASSESSMENT OF UNCERTAINTY FOR HYDROMETRIC MEASUREMENT 41 Expanded Uncertianty in Percent measrement 3 measrements Discharge in m 3 /s Figre B.3 - Effect of increasing nmber of measrements made at a discharge on expanded ncertainty B.5 Smmary The ncertainty analysis can be sed to estimate the ncertainty of a laboratory discharge measrement sing the weighing and timing method. The GUM method ses type A and type B classifications of ncertainty instead of random and systematic. However, the different terminology does not alter the methodology or reslt that organizations sch as AIAA and ASME have adopted. To determine the individal standard ncertainties, information on the measrement process and instrmentation sed is needed. Repeated measrement data is also helpfl in determining the ncertainty de to random effects bt can be estimated by other means if necessary. The discharge measrement example demonstrates that ncertainty can be compted sing a combination of measrement data and information from instrmentation specifications and laboratory conditions. It also illstrates that repeated measrements can be sed to redce the ncertainty of the final (or averaged) measrement. B.6 References Coleman, H. W. and Steele, W.G. Jr. (1999). Experimentation and Uncertainty Analysis for Engineers. nd Edition, John Wiley & Sons, Inc., New York. Thompson, A. and Taylor, B.N. (008). Gide for the Use of the International System of Units (SI), NIST Special Pblication 811, pages 3-4. Taylor B.N. and Kyatt, C.E. (1994). Gidelines for Evalation and Expressing the Uncertainty of NIST Measrements, NIST Technical Note 197, United States Department of Commerce, National Institte of Standards and Technology, 0 pp. United Kingdom Accreditation Service (007). M3003: The Expression of Uncertainty and Confidence in Measrement, Edition, 8 pp.

49 APPENDIX C DISCHARGE UNCERTAINTY EXAMPLE: WADING MEASUREMENTS OF DISCHARGE USING A POINT VELOCITY METER AND THE VELOCITY-AREA C.1 Introdction This example demonstrates the application of the GUM to a single discharge measrement made with a point velocity meter in the ncontrolled environment of a natral stream. In a previos section, an example of compting the ncertainty of a weighing discharge measrement made in the controlled conditions of a hydralic laboratory was presented. The general steps to estimate ncertainty sing the GUM for a measrement inclde: (1) nderstanding the measrement process and expressing the fnctional relationship between the measrand and the measred inpt qantities; () identifying and estimating the contribting sorces of ncertainty to the measrand, inclding those that may not be explicitly expressed in the fnctional relationship; (3) combining the standard ncertainties into the combined ncertainty by determining sensitivity coefficients and combining the standard ncertainties sing a first-order Taylor series approximation of the fnctional relationship; and (4) reporting the expanded ncertainty and measrement reslt sing one of the recommended GUM formats. The GUM approach for compting the measrement ncertainty of a single discharge measrement made with a point velocity meter and the velocity-area method, broadly following the order of the general steps, is investigated and illstrated herein. The fnctional relationship sed to compte discharge and the discharge measrement process for a typical wading discharge measrement are presented. The contribting sorces of ncertainty to the measrand and the information available for estimating the ncertainties are discssed and presented. The eqation for combining the varios contribting sorces of ncertainty for a typical wading discharge measrement is formlated and presented. The formlated eqation for the combined ncertainties is sed to estimate the ncertainty of a single discharge measrement at for different stream locations. The reslting estimated ncertainties are compared with estimated ncertainties compted from repeated discharge measrements made at the stream locations to determine if the formlated eqation adeqately explains the measrement ncertainty. C. Fnctional Relationship and the Measrement Process The U.S. Geological Srvey and many other national hydrologic services se the mid-section method to compte discharge from velocity measrements made with a point velocity meter. The fnctional relationship or eqation sed to compte the discharge is: 1 = (C.1) N 1 Q i= 1 1 ( x1 xlew ) dlew vlew + ( xi+ 1 xi 1) di vi + ( xrew xn ) d rew vrew where x is the distance from the horizontal datm, d is the depth and v i is the depthaveraged velocity for a vertical profile. N is the total nmber of velocity measrements. Note that at i=0 and i=n the measrement is located at the left edge of water (lew) and right edge of water (rew) looking downstream, respectively. The velocity and depth are sally assmed to be zero at the edge of water, allowing the first and third terms on the right-hand side of the eqation to be ignored. This eqation expresses the fnctional relationship between the measrand, total discharge Q, and the inpt qantities that are physically measred: width, depth and velocity.

50 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT 43 The U.S. Geological Srvey measrement process reqires discharge measrements to have at least 0 sb-area measrements and frther restricts a sbarea to contribte no more than 10 per cent to the total measred discharge. This restriction is to ensre that the measrement adeqately samples the spatial variation in the depth and velocity in the measred section. The mean velocity is measred in the profile by assming a logarithmic velocity profile and by measring at the 0. and 0.8 depths and averaging the two measrements, or by measring at the 0.6 depth. Becase stream velocities have plsations over several seconds to a few mintes, the U.S. Geological Srvey reqires that velocity measrements be sampled (or measred) for a dration of at least 40 seconds to give a temporal average velocity. Typical instrmentation sed for a discharge measrement made while wading incldes a tagline that is marked in intervals, a wading rod on which the velocity meter is fixed and a velocity meter. C.3 Identifying and Estimating Sorces of Uncertainty Sorces of ncertainty for the discharge measrement inclde the accracy and resoltion of the instrments sed to make the measrement, shallow depths, spatial resoltion of the measrement (the nmber of sb-areas sed to measre the longitdinal variability of velocity and depth), the steadiness of the flow (water level is constant dring the measrement), cyclical flow plsations de to trblence, and the skill of the operator performing the measrement. C.3.1 Accracy and Resoltion of Instrments The width of the measrement section and the location of each velocity and depth measrement are measred with a tagline marked in nits of feet. Taglines sed for the width measrements are sally marked at 1 ft (30.5 cm) intervals or larger. Accracy of tagline markings is at least 3 per cent, and is considerably more accrate than the observation based on the resoltion of the tagline. It is left to the stream gager to improve the resoltion of the measrement by eye. It is cstomary to record tagline distances to the nearest 0.1 ft (3 cm) even thogh the actal marking interval gives a 0.5 ft (15. cm) resoltion. Depth measrements are typically made with a wading rod marked in nits of feet at 0.1 ft (3 cm) intervals. Similarly to the tagline, the markings on the rod are considerably more accrate than the resoltion of the markings. Other contribtions to the ncertainty of the depth measrement relate to the skill of the operator and inclde not holding the rod plmb dring the measrement and poorly estimating the location of the mean water level in the presence of water srface waves (or oscillations) or water rn p on the rod de to the deceleration of the water on the rod. Point velocity measrements are typically made in the U.S. Geological Srvey with a mechanical meter (Price AA or Price pygmy) or Flowtracker acostic meter. For this paper, only the Price AA and Price pygmy mechanical meter will be considered. However the accracy and precision of other point velocity meters cold be sbstitted for the Price meters in the ncertainty comptation for a single discharge measrement presented herein. The accracy of either Price meter is a fnction of the measred velocity and the magnitde and dration of flow plsations. Typically, the meter measres in nits of feet per second (ft/s). At the lowest observable velocities the Price AA has accracy of ±6

51 44 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT per cent and at velocities higher than. ft/s (67.1 cm/s) has an accracy of ±1.5 per cent. Table 1 lists the accracy for the Price AA and pygmy meters over a range of velocities (Thibodeax, 007). Based on the nmber of digits displayed on the standard USGS rating tables and on the typical display sed to time and cont the revoltions of the Price AA meter, the resoltion of the velocity meter is 0.01 ft/s (3 mm/s). Table C.1 - Accracy of Price mechanical velocity meters as per cent. Operating range in ft/s (cm/s) >0 Price AA (3 - > 600) Pygmy >10 (3 - > 300) 0.5 (7.6) 0.50 (15.0) Velocity in ft/s (cm/s) (.8) (33.5) (45.7) >.0 (67.1) ±6.0% ±3.4% ±.5% ±.0% ±1.5% ±1.5% ±6.0% ±3.4% ±.5% - ±1.8% ±1.5% C.3. Shallow Depths The accracy of the velocity measred by either the Price AA meter or the Price pygmy meter is adversely affected by shallow water. Rantz (198) stated that the Price pygmy meter accracy was adversely affected when depths were less than 9 cm (0.3 ft) and the Price AA meter was adversely affected when depths were less than 15 cm (0.5 ft). Laboratory flme investigations by Pierce (1941) fond discharges made by Price meters in shallow depths to vary between and 14 per cent from reference measrements made by a pitot tbe. In practice, depths may be too shallow at only a few measrement locations in the cross section. The ncertainty from shallow depths can be estimated by: a 0 05 (C.) shallow =. qshallow where q shallow are the sbarea discharges measred in depths of less than 15 cm for the Price AA meter and 6 cm for the Price pygmy meter. C.3.3 Spatial Resoltion of the Measrement The accracy of the discharge measrement is affected by the nmber and spacing of the depth and velocity measrements made. Kiang and others (009) developed the Interpolated Variance Estimator (IVE) to help determine whether there is a need for additional sampling. This method estimates the ncertainty contribted by the spatial resoltion of the depth and velocity measrements in the cross section to the discharge measrement ncertainty; and all of the random sorces of ncertainty are acconted for sing IVE. A similar idea to IVE is to compte only the contribtion of spatial resoltion to the ncertainty. The spatial resoltion ncertainty contribtion is estimated by compting the difference between the discharge measrement and a discharge compted from sing a redced nmber of velocity and depth measrement locations. This techniqe is freqently employed by nmerical modellers to explore whether their models have reached nmerical convergence. The bonds or range of the ncertainty de to spatial resoltion of the discharge measrement, a res, can be estimated by:

52 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT 45 a res = Q Q (C.3) N n / where Q N is the discharge compted with all the measred locations and Q n/ is the discharge compted with every other measrement location omitted. C.3.4 Steadiness of the Flow Ideally, discharge measrements are made while the discharge is not changing. If the flow is either increasing or decreasing dring the corse of making the measrement, then an additional ncertainty occrs. The ncertainty de to nsteady flow can be estimated from an existing plot of water level and measred discharges (a rating crve) for the measrement site or by repeating a sb-area measrement at one of the first measrement locations in the cross section. Using an existing rating, the limits for the ncertainty de to increasing or decreasing discharge can be estimated by: a nstready = Q ( y ) Q( y ) 0 Q measred f (C.4) Using repeated velocity and depth measrements at one of the initially measred locations, the limits for the ncertainty de to increasing or decreasing discharge can be estimated by: a nsteady = vi ( t0 ) di ( t0 ) vi ( t f ) di ( t f ) [ v ( t ) d ( t ) + v ( t ) d ( t )] i 0 i 0 i f i f (C.5) where sbscripts 0 and f denote vales at the start and end of the measrement respectively. C.3.5 Flow Plsations Flow plsations can contribte ncertainty to a discharge measrement becase the time periods over which the plsations occr determine the sampling time needed to measre an accrate time average of the velocity. These flow plsations are determined by the channel roghness, slope, srface wind direction, and geometry pstream of and in the measrement cross section. Pelletier (1988) smmarized several stdies on the time period needed for single velocity measrement and fond that the recommendations varied from 60 to 600 seconds for a measrement ncertainty of 4 per cent. However, discharge measrements are rarely composed of only one velocity measrement. For typical discharge measrements, each velocity measrement is sampled not only at a different location bt at a slightly different time. As a reslt of the sampling method, the ncertainty of the discharge measrement de to the plsations is a fnction of the time period over which each velocity measrement is made and the nmber of individal velocity measrements taken. Anderson (1961) investigated a sample of 3 streams and fond that for individal velocity measrements over 45 seconds at N verticals the per cent ncertainty (or error) de to flow plsations for the total discharge, plse, can be estimated sing: plse = (C.6) N

53 46 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT C.3.6 Skill of the Operator The operator making the discharge measrement affects the ncertainty of the measrement. Slight bt repeatable differences in where an operator stands in relation to the wading rod, how careflly they hold the rod vertically, and the size of the physical flow distrbance created by standing in the stream near the crrent meter vary among skilled operators. The ncertainty de to the operator can be estimated from repeated measrements made by several operators for steady flow in a niform cross section. Earlier work comparing different meter models (Flford et. al., 1994) fond that operators cold contribte an ncertainty of ± to ±.5 per cent, regardless of the type of point velocity meter sed. Table lists standard deviations compted for discharge measrements made dring steady flow conditions at the otdoor flme at the U.S. Geological Srvey Hydrologic Instrmentation Facility at Stennis Space Center, MS. The variance in the measrements is primarily de to operator differences. The flow was steady dring the measrements. The measrement section was mowed grass with a smooth even bottom. Table C. - Relative standard deviations for measrements made by eight or nine operators with six different models of velocity meters for steady flow in a half trapezoid cross section Meter model Nmber of measrements Nmber of operators Compted standard deviation in per cent All six models Price AA 8 8. Price pygmy Valeport BFM Marsh McBirney Ott C Swoffer C.4 Formlating the Combined Uncertainty Estimate The combined ncertainty estimate ses a first-order Taylor series approximation of the fnctional relationship to combine the individal sorces of ncertainty. In practice, this is simply the sqare root of the sm of the sqared standard ncertainties. The standard ncertainties for the identified ncertainty sorces are the variances of each sorce and any covariances between sorces that are estimated from the information presented in the previos section. Before sqaring the standard ncertainties, the individal ncertainties are mltiplied by sensitivity coefficients. For a sb-area discharge at the i th measrement, the fnctional relationship is: 1 = ( xi+ 1 xi ) di vi (C.7) qi 1 Sensitivity coefficients, c, for each inpt are fond by taking the partial derivatives for each independent variable in the above eqation:

54 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT 47 q v i = c vi 1 = ( x i + 1 x i 1 ) d i (C.8) q d i = c di = 1 ( x i + 1 x i 1 ) v i (C.9) q x i+ 1 = c xi+ 1 1 = vid i (C.10) q x i 1 = c xi 1 1 = vid i (C.11) The sensitivity coefficients ensre that all the standard ncertainties will be expressed in a common nit. For discharge, the common nit is either m 3 /s or ft 3 /s. If the standard ncertainty is already in the common nit, the sensitivity coefficient is 1. For a sb-area, the combined ncertainty is: ( v ) + c ( d ) + c ( x ) + c ( x ) c c ( x ) ( x ) (C.1) q = cvi i di i xi+ 1 i+ 1 xi 1 i 1 + xi+ 1 xi 1 i+ 1 i 1 Each ncertainty term,, in the above eqation contribtes ncertainty from the accracy, a, and the resoltion, r, of the instrment and may be expanded. ( v ) ( v ) + ( v ) i i, a i, r = (C.13) ( d ) ( d ) + ( d ) i i, a i, r = (C.14) ( x ) ( x ) + ( x ) i i, a i, r = (C.15) The accracies of the two width measrements in the sb-area are correlated becase the same tagline is sed. Expanding the standard ncertainties to inclde the contribtions from instrment accracy and resoltion and the correlated ncertainty, the sb-area combined ncertainty is: q + c = c xi 1 vi [ ( vi, a ) + ( vi, r )] + cdi [ ( di, a ) + ( di, r )] + cxi+ 1[ ( xi+ 1, a ) + ( xi+ 1, r )] [ ( x ) + ( x )] + c c ( x ) ( x ) i 1, a i 1, r xi+ 1 xi 1 i+ 1, a i 1, a The correlated width terms in the previos eqation can be rearranged to give: [ ( v ) + ( v )] + c [ ( d ) + ( d )] + c ( x ) q cvi i, a i, r di i, a i, r xi+ 1 i+ 1, r (C.16) = (C.17) Becase the resoltion of the wading rod markings is mch coarser than the accracy of the marking placement on the rod, the ncertainty de to the accracy of the wading rod and the correlated ncertainty from sing the same wading rod throghot the measrement can be ignored. When smming the sb-areas, the correlated ncertainty contribted by sing the same velocity meter shold be considered. Smming the sb-

55 48 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT areas and inclding the correlated ncertainties from the velocity meter, the combined ncertainty for the discharge measrement is: U Q + c + v0 N i= 0 N 1 [ cvi { ( vi, a ) + ( vi, r )} + cvicvi+ 1( vi, a ) ( vi+ 1, a ) + cdi ( di, r )] i= 1 [ ( v ) + ( v )] + c [ ( v ) + ( v )] + c ( d ) + c ( d ) = c xi 0, a ( x ) + c ( Q ) + c ( Q ) + c ( Q ) + c ( Q ) i, r Qp 0, r p vn Qt N, a t N, r Qshallow d 0 shallow o, r Qres dn N, r res (C.18) The last three terms inclde ncertainties de to plsation of flow, p, nsteadiness of the flow, t, and the measrement resoltion, res. In the above eqation, each standard ncertainty, j, is compted by: a j j = (C.19) Divisor where a j is assmed to be either the limits within which the tre vale will lie or a standard deviation. The Divisor is the vale by which a j, can be divided to yield the standard deviation for the probability distribtion assmed for the j-th sorce of ncertainty. C.5 Compting the Uncertainty of a Discharge Measrement The combined ncertainty eqation for a discharge measrement can be sed to estimate the ncertainty in a single discharge measrement. Repeated discharge measrement data collected for a previos meter stdy (Flford et. al., 1994) were available for the Gibbon River and the Hot River in Yellowstone National Park. Additional repeated measrements were available for the Jocko River above and below J canal near Ravalli, Montana. The standard deviation of the repeated measrements can be compted to se as an estimate of the tre ncertainty of the measrement. The reslting estimate can be compared with the ncertainty estimated by the combined ncertainty eqation. The discharge measrements in Yellowstone National Park were made by several operators sing varios models of velocity meters inclding an electromagnetic meter, and horizontal- and vertical-axis mechanical meters. The measrements for the Gibbon River were made by for different operators sing six different models of velocity meters. The measrements for the Hot River were made by for different operators sing eight different models of velocity meters. The ncertainty analyses are presented for the discharge measrements made with a Price AA meter. A smmary of the discharge measrements is presented in table C.3. No change in water level was noted dring either series of measrements, and discharge was estimated to be constant. The discharge measrements in Jocko River near Ravalli Montana were made by eight different operators. Except for a single measrement made with a Price AA meter, all measrements were made with a Price pygmy meter. The cross sections were located pstream and downstream of a canal on the river. A smmary of the discharge measrements is presented in table C.3. No change in water level was noted dring the measrements.

56 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT 49 Table C.3 - Discharge measrements for the Gibbon River and the Hot River in Yellowstone National Park, Wyoming and Jocko River, pstream and downstream sections, near Ravalli, Montana Site Gibbon River Hot River Jocko Canal US Jocko Canal DS No. of sbareas Width (m) Mean Depth (cm) Mean velocity (cm/s) Price AA discharge (m 3 /s) No. of discharge measrements Mean discharge (m 3 /s) The estimated ncertainty for a discharge measrement was compted sing the previos eqation, spreadsheet software (Excel), and the estimates of the a j vales for sorces of ncertainty. The ncertainty for a single discharge measrement made with a Price AA meter was compted for the Gibbon and the Hot River. The ncertainty for a single discharge measrement made with a Price pygmy meter was compted for the Jocko River pstream and downstream of J canal. Tables C.4 and C.5, for the Gibbon and Hot Rivers respectively, and tables C.6 and C.7 for the Jocko River pstream and downstream respectively of J canal, smmarize the ncertainty sorces, the a j vales, the assmed probability distribtions, the Divisor sed and the estimate of the sqared standard ncertainty for each term in the eqation for the combined ncertainty of the individal discharge measrements. The tables inclde the expanded ncertainty as well as the combined standard ncertainties. The combined standard ncertainties are the sqare root of the sm of the sqared standard ncertainties. The expanded combined ncertainty is the combined standard ncertainties mltiplied by the k factor to give the estimate for a 95 per cent confidence interval. For most cases the se of k= is accrate for an ncertainty estimate and that vale is sed herein. The a j vales are all based on previos information except for the resoltion ncertainty and are not compted sing statistical methods. The first five sorces in the tables are compted for a sb-area and smmed. The final five sorces are estimates that are compted as a fraction of the total discharge.

57 50 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT Table C.4 - Smmary of ncertainty sorces and vales sed for a single discharge measrement, Gibbon River in Yellowstone National Park, Wyoming Sorce of a j, Vales Probability Divisor c j j, in (m 3 /s) ncertainty distribtion Meter accracy Table 1 normal x10-5 Meter resoltion 3 mm/s rectanglar x10-6 Meter correlation Table 1 normal x10-5 Wading rod 3 cm rectanglar x10-5 Tagline 15. cm rectanglar x10-4 Shallow depth 5 % Q normal x10-5 shallow Spatial resoltion 73.6 x10-3 m 3 /s rectanglar x10-3 Operator % Q measred normal x10-3 Flow plsation 8.0 x10-3 m 3 /s normal x10-4 Unsteady discharge Combined ncertainty Expanded ncertainty k= m 3 /s (.5%) m 3 /s (5.0%) Table C.5 - Smmary of ncertainty sorces and vales sed for a single discharge measrement, Hot River in Yellowstone National Park, Wyoming Sorce of aj, Vales Probability Divisor c j j, in (m 3 /s) ncertainty distribtion Meter accracy Table 1 normal 1.05 x10-6 Meter resoltion 3 mm/s rectanglar x10-7 Meter correlation Table 1 normal x10-6 Wading rod 3 cm rectanglar x10-6 Tagline 15. cm rectanglar x10-5 Shallow depth 5 % Q normal x10-5 shallow Spatial resoltion 17.7 x10-3 m 3 /s rectanglar x10-5 Operator % Q measred normal 1.77 x10-4 Flow plsation x10-3 m 3 /s normal x10-5 Unsteady discharge Combined ncertainty (m 3 /s) Expanded ncertainty k= 0.03 m 3 /s (.8%) m 3 /s (5.6%)

58 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT 51 Table C.6 - Smmary of ncertainty sorces and vales sed for a single discharge measrement, Jocko Canal, pstream measrement section, Montana Sorce of aj, Vales Probability Divisor c j j, in (m 3 /s) ncertainty distribtion Meter accracy Table 1 normal x10-4 Meter resoltion 3 mm/s rectanglar x10-5 Meter correlation Table 1 normal 1.00 x10-4 Wading rod 3 cm rectanglar x10-4 Tagline 15. cm rectanglar x10-4 Shallow depth 5 % Q normal x10-6 shallow Spatial resoltion x10-3 m 3 /s rectanglar x10-5 Operator % Q measred normal x10 - Flow plsation.191 x10-3 m 3 /s normal x10-3 Unsteady discharge Combined ncertainty (m 3 /s) Expanded ncertainty k= m 3 /s (.%) 0.74 m 3 /s (4.4%) Table C.7 - Smmary of ncertainty sorces and vales sed for a single discharge measrement, Jocko Canal, downstream measrement section, Montana Sorce of aj, Vales Probability Divisor c j j, in (m 3 /s) ncertainty distribtion Meter accracy Table 1 normal x10-5 Meter resoltion 3 mm/s rectanglar x10-6 Meter correlation Table 1 normal x10-4 Wading rod 3 cm rectanglar 3.00 x10-4 Tagline 15. cm rectanglar x10-4 Shallow depth 5 % Q shallow normal x10-5 Spatial resoltion 3.73 x10-3 m 3 /s rectanglar x10-4 Operator % Q measred normal x10-3 Flow plsation.191 x10-3 m 3 /s normal x10-3 Unsteady discharge Combined m 3 /s (.%) ncertainty (ft 3 /s) Expanded ncertainty k= 0.37 m 3 /s (4.5%)

59 5 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT The compted ncertainties for the for discharge measrements did not vary mch and ranged from.8 to. per cent. The relative contribtion of each standard ncertainty is plotted in figre C.1. The ncertainty contribted by the operator is the largest sorce of ncertainty for the Hot, Gibbon, and Jocko River discharge measrements. Uncertainty from the spatial resoltion of the measrement is the second largest contribtor to the combined ncertainty for the Hot and Gibbon River. For the Jocko River measrements, the ncertainty from flow plsations is the second largest contribtor to the combined ncertainty. Spatial resoltion was an important ncertainty sorce for the Jocks River downstream of the J canal, bt was not as great as the ncertainty contribtions from the crrent meter for the Jocko River pstream of the J canal. The comparison of the combined ncertainty with the standard deviations compted from the repeated discharge measrements is listed in table C.6. The estimated combined ncertainty for the Gibbon River is very nearly the same as the standard deviation compted from the repeated measrements. The ncertainty estimates for the Jocko River discharges are abot the same vale and are within 1 per cent of the standard deviations compted from the repeated measrements. The measrements for the Gibbon River and the two locations on the Jocko River have relatively small ncertainties. Figre C.1 - The relative contribtion of the standard ncertainties in each discharge measrement The vale of the ncertainty estimate compted for the Hot River is similar to those compted for the other discharge measrements. It does not compare as well to the ncertainty estimated from the repeated measrements. The estimated ncertainty for the Hot River is abot half of the compted standard deviation for the repeated measrements. Obviosly, the estimated ncertainty for the Hot River does not captre all the significant sorces of ncertainty in the discharge measrements. The ncertainty analysis did not inclde any ncertainty de to operators obstrcting the flow dring the measrement. Operators wading the section in the Hot River significantly obstrcted the cross sectional flow area by abot 5 per cent of the cross sectional width. Additionally, the high temperatres of 43.3 C (110 F) and the large nmber of velocity measrements made near the bondary may have had adverse effects on meter performance.

60 GUIDELINES FOR THE ASSESSMENT OF UNCERTIANTY FOR HYDROMETRIC MEASUREMENT 53 Table C.6 - Comparison of combined ncertainty estimate with compted standard deviation of repeated measrements Site Combined ncertainty for single measrement Discharge (m 3 /s) Per cent No. of measrements Repeated measrements Discharge (m 3 /s) Per cent Gibbon River Hot River Jocko Canal US Jocko Canal C.6 Smmary The GUM ncertainty analysis method was sed to develop a combined ncertainty eqation for a single wading discharge measrement sing a point velocity meter. For discharge measrements were sed to demonstrate the se of the ncertainty eqation. For the Gibbon River the combined ncertainty estimate was similar to the standard deviation compted from repeated discharge measrements at the site. For the Hot River the combined ncertainty estimate was half the size of the standard deviation compted from the repeated discharge measrements. The poor estimate is likely de to the combined ncertainty eqation not inclding the effect of the operator obstrcting the flow dring measrement. The combined ncertainty estimates compted for the Jocko River pstream of J canal and downstream of J canal were the same and were bracketed by the standard deviations compted from the repeated discharge measrements at the two cross sections. C.7 References Anderson, I.E. (1961). Errors in streamflow measrement. U.S. Geological Srvey Professional Paper 44-C p 37. Flford, J.M., Thibodeax, K.G. and Kaehrle, W.R. (1994). Comparison of Crrent Meters Used for Stream Gaging. In: Fndamentals and Advancements in Hydralic Measrements and Experimentation. Proceedings of Symposim, Hydralic Division, Agst 1-5, 1994, American Society of Civil Engineers, New York, NY, pp Kiang, J.E., Cohn, T.E. and Mason, R.R. (009). Qantifying ncertainty in discharge measrements: a new approach. Proceedings World Environmental and Water Resorces Congress, American Society of Civil Engineers, pp 1-8. Pierce, C.H. (1941). Investigations of Methods and Eqipment Used in Stream Gaging Part 1. Performance of Crrent Meters in Water of Shallow Depth. U.S. Geological Srvey Water-Spply Paper 868-A 35 pp. Pelletier, P.M. (1988). Uncertainties in the single determination of river discharge: a literatre review. Vol. 15, no. 5, Canadian Jornal of Civil Engineering. Rantz, S.E. and others (198). Measrement and Comptation of Streamflow: Volme 1. Measrement of Stage and Discharge, U.S. Geological Srvey Water-Spply Paper 175, p 13. Thibodeax, K.G. (007).Testing and evalation of inexpensive horizontal-axis mechanical crrent meters. USCID Forth International Conference pp (p. 196).

61

62 APPENDIX D DETERMINATION OF THE DISCHARGE IN A CIRCULAR SEWER PIPE The ncertainty in the discharge measred in a circlar sewer pipe is calclated sing the GUM framework (JCGM 100:008). For the sake of brevity, only salient calclations are presented. D.1 Measrement Process Consider a circlar sewer pipe of radis R (m), eqipped with a piezoresistive sensor (MilltronicsNivBar) providing the water level h (m) and a Doppler sensor (Milltronics DEK-B EX/30) for determining the flow velocity U (m/s) throgh the pipe cross section S (m ), as shown in Fig. D.1. It is assmed that the pipe is circlar and that there are no sediment deposits or debris on the pipe invert. The discharge Q (m 3 /s) is then given by the following fnctional relationship (Bertrand-Krajewski and Mste 008) (D.1) Figre D.1 - Schematic view of the circlar pipe D. Assessment of Uncertainty Sorces The radis, R, is an inpt variable for which repeated measrements were acqired in sit. No repeated measrements are available for the water level or the mean flow velocity in the sewer pipe. Described below are the procedres sed to estimate the standard ncertainties in R, h and U, respectively (R), (h) and (U) sing the available information. A more rigoros ncertainty analysis wold reqire calibration against certified primary or secondary standards sing standardized protocols. This approach can be applied to instrments for distance measrement, bt not for velocity as there is no standard for velocity measrement. Conseqently, in the present paper we have sed ncertainty estimates available from prior measrements for velocity measrement instrments of known/proven accracy as sbstittes for more accrate calibration protocols. Estimation of standard ncertainty in the pipe radis. The average pipe radis R was calclated sing n = 4 measrements of the diameter D carried ot randomly at varios radial positions in the pipe cross section. The measrements were respectively: 1.00, 1.000, 0.997, and 1.00 m. The mean vale of mm and ths the mean radis R = = m. The standard deviation s of the for vales of D is eqal to m, so (R) = ( )/ m. Estimation of standard ncertainty in h. The in-sit water-level measrements were measred with a piezoresistive sensor. The sensor was previosly tested in laboratory conditions (Bertrand-Krajewski and Mste 008). In essence, the tests sed a perspex colmn with a class II certified metallic meter as reference, with a standard ncertainty of 0.5 mm as