Implementation of Standardized Uncertainty Analysis for Streamflow Measurements Acquired with Acoustic Doppler Velocimeters

The GUM framework is based on uniform use of mathematical statistics principles for propagating elemental sources of errors to final results. With the assumption that the reader is broadly familiar with UA, we provide herein only essential elements of GUM implementation (see Fig. 1). More details about GUM framework implementation can be found in Muste et al. (2012) and Muste (2017b). Details on the estimation of individual sources of uncertainties associated with acoustic instruments such as ADVs and acoustic doppler current profilers (ADCPs) can be found in Muste et al. (2004), Gonzalez-Castro and Muste (2007), Muste et al. (2010), and Lee et al. (2014). The UA calculations presented in this paper are facilitated by GUM software package QMSys Enterprise via the QMSys Calculator, a customized version of the software for hydrometric measurements (QualiSyst 2019).

The UA robustness depends on the efforts and details placed in Step 1 (definition of the measurement process) and Step 2 (identification of all uncertainties associated with the functional relationship of the measurement process and making decisions on what type of estimation methods are used for the evaluation of these uncertainties, i.e., Type A or B). Type A uncertainties are evaluated by statistical means applied to data collected during the current measurements; Type B uncertainties are determined by other means (i.e., previous experiments or technical judgement). Computer scripts for Type A and B evaluations are available in sections “Type A Method for Uncertainty Assessment of Repeated Measurements”–“Type B Method for Uncertainty Assessment by the Law of Propagation of Uncertainties” of Bertrand-Krajewski et al. (2021). Steps 1 and 2 are complex, tedious, and costly, as they entail identification and assessment of all possible uncertainties affecting the data acquisition chain, from the probe sensing the flow to the data display (including signal conversion, conditioning, and internal processing), and uncertainties induced by methods, operations, and environmental conditions. Steps 3, 4, and 5 are merely algebraic calculations that can be automated using simple processing scripts.

Strictly speaking, each new hydrometric measurement is characterized by its own uncertainty. Even if the instrument protocols and methods are the same, measurements are taken at specific sites and times characterized by a unique measurement environment in terms local factors and influences. From this perspective, robust (fully fledged) UAs need rigorous organization and execution, with considerations of all sources of errors affecting the measurement functional relationship. If the measurements for all the input quantities in the functional relationship are repeated (ideally more than 10 times), the total uncertainty can be evaluated by Type A method. However, because this is rarely possible in practice due to limited time and resources, the uncertainty of a measurement result can be executed at various levels of rigor that need to be documented. When limited or no resources are available to conduct Type A evaluations, the JCGM (100:2008) accepts uncertainties estimated by the Type B evaluation method. Because the mathematical model of the measurement process may be incomplete, all relevant quantities should be varied to the fullest practical extent so that the evaluation of the uncertainties replicates as much as possible the potential range of the observed data. This is easily done with numerical simulations via the Monte Carlo method (MCM) tempered by engineering judgement (JCGM 101:2008).

The measurement of streamflow benefits from extensive past developments and the continuous adoption of new measurement technologies (Rennie et al. 2017). Historically, the most popular method to directly measuring streamflow in natural channels of all sizes is based on point-velocity measurements used in conjunction with the velocity-area (VA) discharge estimation method. There are multiple published resources detailing VA alternatives and their practical implementation for a variety of instruments (e.g., Rantz 1982; Herschy 2009); therefore, we will not repeat that information here. Provided below are the terminology and notations along with information about the measurement method, instrumentation, and protocol that are relevant for the UA presented in the next sections. It should be noted upfront that the VA method applied in this case study is customized to reflect the changes brought about by the recently revised hydrometry uncertainty guidance (ISO 2020).

The measurement of discharge for a stream ensues from integration over the cross section of the mean velocity for a point velocity, $u_x(y,z,t)$, multiplied by the elemental area, $dA$, for which this mean velocity is representative

The mean cross-sectional velocity is obtained by sampling multiple-point velocities over the cross-section extent (see the measurement grid defined by indices $j$, $k$ in Fig. 2). Specifically, mean point velocities, $u_{j,k}$ ($y_j$, $z_{j.k}$), and the location of their measurement—i.e., the position of the vertical from the bank, $y_i$, and the vertical location $z_{j,k}$ ($y_j,z_k$)—are measured across the section. Elemental discharges are determined by the mean velocity, $u_{j,k}$, associated with and elemental cross section, $\mathrm{\Delta }{y}_j\mathrm{\Delta }{z}_{j,k}$. Multiple verticals are collected over the cross section with each vertical containing multiple point velocity measurements. As correctly pointed out in HUG (ISO 2020), instruments cannot measure at (or near) the bed and free surface, leaving portions of the cross section unmeasured (see Fig. 2). Accounting for this realization and using notations in Fig. 2, the total stream discharge, $Q$, is obtained as the sum of all measured elemental discharges, $Q_{m(j,k)}$, complemented by the unmeasured area discharges, $Q_p=Q_{\mathrm{top}}+Q_{\mathrm{bottom}}+Q_{\mathrm{edges}}$ (ISO 2020), as follows:where $F_y$ and $F_z$ = factors accounting for the discrete summation of the elemental discharges in the $y$ and $z$ directions, respectively. Obviously, the more points sampled within the measured area and the closer the measurements near free-surface and boundaries, the better the estimate of the cross-sectional (bulk flow) velocity. The impact of the $F_y$ and $F_z$ factors prescribed by HUG (ISO 2020) diminishes if the density of points is increased.

Eq. (2) is a generic expression for the VA midsection approach recommended in HUG (ISO 2020). The equation can be applied for a common measurement situation when just one point is sampled in the vertical. Eq. (2) serves well for defining the functional relationship of the measurand (Step 1 in Fig. 1) for a wide range of instruments and methods. This new formulation for the functional relationship provided in HUG (ISO 2020) departs from previous standards on VA method (e.g., ISO 748) by explicitly including the unmeasured areas of the cross section. Eq. (2) can be customized to accommodate various VA measurement protocols [e.g., velocity measurements from fixed verticals or moving boat, midsection or mean-section VA, index-velocity, or entropy-based bulk velocity estimation, as described by Chiu and Chen (1998)] and instruments (e.g., mechanical and electromagnetic meters, ADV, ADCP, and large-scale particle image velocimetry). Such customization is illustrated in the case study presented below.

Acoustic instruments have proven to be reliable and efficient alternatives for measurement of velocities in open channel flows. Their use continues to expand due to the lack of moving parts, nonintrusive nature, and ease of deployment and operation compared with previous generations of velocimeters. Consequently, ADVs have experienced considerable growth in the last three decades, becoming leading tools for field studies in several countries. A representative probe from this category is the ADV produced by SonTek/YSI (San Diego) that will be used herein to illustrate UA implementation. Various aspects of ADV configuration, operation, and performance are covered by instrument producers (e.g., SonTek 1997) and other published references (e.g., Kraus et al. 1994; Goring and Nikora 1998; Voulgaris and Trowbridge 1998; McLelland and Nicholas 2000; Dombroski and Crimaldi 2007). The schematic of the SonTek ADV configuration is shown in Fig. 3(a). The specifications for the 16-MHz MicroADV used in this study are shown in Fig. 3(b) (Sontek 2017).

Assuming familiarity of the readers with ADV-related resources, we will only focus on instrument aspects related to UA. ADVs are instruments that measure velocities of small particles suspended in the flow, assuming that these particles travel at the same velocity as the water (Rehmel 2007). The cylinder-like ADV measurement volume is located a short distance from the acoustic sensor [see Fig. 3(a)]. When the ADV is fully submersed, the sound pulses emitted from transmitters are reflected in all directions by particles contained in the instrument measurement volume. A portion of the reflected energy is directed toward the receivers and captured as a voltage proportional to the instantaneous backscattered pressure (Lemmin 2017). If the particles are moving with respect to the probe, the backscattered sound will have a different acoustic frequency. The change in frequency “sensed” by each receiver is subsequently used to estimate the flow velocity through the following functional expression:where $V_B$ = flow velocity along the bistatic axis; $c$ = speed of sound in water; and $f_D$ = difference in the acoustic frequency between the emitted ($f_0$) and return ($f_B$) pulses due to doppler shift ($f_0-f_B$). The bistatic axis is the line dividing the angle between the transmitter and receiver whose vertex is at the center of the sampling volume (Kraus et al. 1994). One transmitter and three receivers resolve three velocity components. The bistatic velocities acquired by the three acoustic sensors can be transformed into a cartesian coordinate system by the instrument software using the following relationship (Kraus et al. 1994):where $u$, $v$, and $w$ = velocity components expressed in instrument coordinate system; $f_{D1}$, $f_{D2}$, and $f_{D3}$ = frequency differences sensed by the three receivers; and $G_3$ = geometrical transformation matrix that relates to the arrangement of the three receivers with respect to the central emitter. These equations are applied to each acoustic pulse, leading after internal or external processing to time-averaged velocity components, fluctuating components, and higher-order correlations.

The measurement protocol is critical for UA implementation because it dictates the structure of the measurement definition and the association of the elemental uncertainty sources to the measurement process components. As each experiment is unique from multiple perspectives, we illustrate in the next sections the UA-GUM implementation for the measurement protocol employed in a case study. This protocol follows the HUG (ISO 2020) formulation recommended for the midsection VA method in conjunction with point-velocity measurements [see Eq. (2)]. The presented protocol differs slightly from the typical midsection VA measurement method described by ISO 748 (ISO 2007a) that is widely popular in national hydrologic services for both traditional and emerging instruments (WMO 2017). The difference is minor, entailing the estimation of the discharges in the unmeasured area around the boundaries, i.e., $Q_p$ in Eq. (2) that are not explicitly refer to in ISO 748 (ISO 2007a). Description of the midsection method including explicitly edge discharges is also used in Herschy (2009) and in the section by section discharge measurements with ADCPs (Mueller et al. 2007).

In the present case study, velocities and water depths are sampled in verticals distributed throughout the section, as shown in Fig. 4. For each velocity measurement point, the following variables are recorded: distance of vertical $j$ from the start bank, $b_j$, water depth at the $j$th vertical location, $d_j$, and the mean point-velocity values, $u_{j,k}$, measured at a point, $d_{j,k}$. Illustration of the point-velocity measurements taken in the panel centered on vertical $j=12$ are visualized in Fig. 4.

To reconcile differences between HUG generic equation [see Eq. (2)] and the typical VA implementation (Turnipseed and Sauer 2010; WSC 2015), we associate the velocities in the unmeasured areas near the top and bottom of the verticals to the depth-averaged velocity (see ISO 748, 2007, Section 8.3.2). Using this approach facilitates access to a wealth of information on the uncertainty related to the sampling of the point velocities in the verticals. The depth-averaged velocity, $U_j$, is defined as the value of the spatially averaged mean point-velocities acquired in a vertical extrapolated to the top and bottom with canonical vertical velocity models such as logarithmic or exponential distribution laws (e.g., Nezu and Nakagawa 1993; Nystrom et al. 2002). Depth-averaged velocity determined in this way is considered representative for the panel where the point velocities were acquired. The panel geometry is defined by the local depth measured in the vertical and the half distance between the verticals neighboring the ones where the point velocity measurements are taken, $(b_{j+1}-b_{j-1})/2$ (see illustration of the definitions for panel associated with vertical 12 in Fig. 4). As a consequence, the total discharge in the area with point-velocity measurements is estimated by summing up discharges through the panels (a.k.a., sub-sections) extending from the free surface to the riverbed rather than applying the generic Eq. (2) to small surface elements associated with the point measurements. Another reconciliation is needed for the inclusion of the near-river bank areas (edges) that are not accounted for in the discharge calculation with ISO 748 (ISO 2007a). For this purpose, HUG (ISO 2020) introduces the edge discharges, recognizing that in some situations (e.g., small streams) the edge areas might become important in the determination of the total discharge (i.e., representing more than 5% of the total flow). The panels near the river banks (edges) contain only one neighboring (the first and last) vertical (see Fig. 4). Various formulas are available for computing edge discharges (e.g., Fulford and Sauer 1986; Le Coz et al. 2012).

Accounting for the above considerations and using notations in Fig. 4, the total discharge, $Q_t$, is obtained from the following equation:

According to GUM, the most rational approach to systematically tracking uncertainty sources is to group them around the variables in the functional relationship. This grouping also facilitates the organization of the experiments for UA at the elemental level to ensure that each estimation is made for only one source of error acting in isolation, if possible at all. Inspection of the functional relationship defined by Eq. (5) reveals the following main variables: depth-averaged velocity, $U_j$, depth in the verticals, $d_j$, and distance between verticals, ($b_{j+1}-b_{j-1}$). According to the abovementioned considerations, we associate the $Q_{\mathrm{top}}$ and $Q_{\mathrm{bottom}}$ uncertainties to the depth-averaged velocity. In addition to the above variables, there are uncertainties associated with the models used for determining the discharge in the measured area, $Q_{MO}$, and those associated with the unmeasured areas close to the edges, $Q_{Re}$, and $Q_{Le}$. Ensuing from Steps 3 and 4 of GUM framework, and using Eq. (5) as functional relationship for the measurement of total discharge, the analytical expression for the combined standard uncertainty of the total discharge iswhere $q_{n,j}$ = elemental discharge for subsection $j$. The terms $u(Uj)$, $u(d_j)$, and $u(b_j)$ are summing up all the elemental uncertainties affecting the measurements of the depth-averaged velocities, vertical depths, and distances from the start bank, respectively. These elemental uncertainties are cumulated using the root-sum-square (see Step 2 in Fig. 1) as further described in the discussion of the UA case study. The partial derivatives in Eq. (6), labeled sensitivity coefficients in GUM terminology, describe how the estimate of the output quantity will be influenced by small changes in the estimates of the input quantities. The sensitivity coefficients are determined numerically as second-order approximations (e.g., UKAS 2007; Bertrand-Krajewski et al. 2021). Terms 4, 5, and 6 in Eq. (6) capture the contribution of the correlated uncertainties. Terms 4 and 5 characterize correlated uncertainties created by the fact that the total discharge is the sum of discharges whereby velocities, depths, and subsection widths are measured with the same instruments in consecutive subsections. The three input variables (velocity, depth, and width) are affected by correlated uncertainties regardless of the velocity-area method used for determining the total discharge.

The seventh term in Eq. (6), $u(Q_{MO})$, is primarily related to the discharge estimation model used for discharge calculation. Based on engineering judgment and experience with the measurement method, it should consider all the measurements uncertainty sources that cannot be directly attributed to the abovementioned variables (see also Table 3). Consequently, the $u(Q_{MO})$ term aggregates uncertainties induced by the discharge model used for computation, $u(Q_{mo})$ and sources of uncertainties associated with the measurement protocols, and the conditions during data acquisition over the measured area, i.e., the number of verticals where velocities are acquired, $u(Q_{nv})$, and influence factors (e.g., weather, operator skills, etc.) affecting the measurement protocol, $u(Q_{op})$, respectively. The last two terms are associated with the model used for calculating the right and left edge discharges, $u(Q_{Re})$ and $u(Q_{Le})$. Given that the last three terms are directly affecting the determination of the total discharge (i.e., the measurand), their corresponding sensitivity coefficients are unity. Care should be taken to not double count operational errors affecting individual variables (e.g., water salinity directly affects the velocity data) and those affecting the measurement protocol (e.g., operator skills).

Eq. (6) assumes that the probability distribution associated with the measurement results is approximatively normal (Gaussian), which is the case for many practical measurements even if some of the elemental sources of uncertainties are determined from other type of probability distributions [JCGM 101 (JCGM 2008b)]. Eq. (6) is fully compliant with GUM and HUG (ISO 2020) formulations and it is different from those in ISO 748 (ISO 2007a) and other related publications through several aspects. First, it strictly follows the generic uncertainty propagation equation used by GUM (see Step 3 in Fig. 1) that is dimensional expressed in discharge units (e.g., ${\mathrm{m}}^3/\mathrm{s}$). Second, each of the independent variables are associated with their uncertainty sources and sensitivity coefficients (that can be positive or negative and account for the variation of the variables across the section). A direct consequence of the above factors is that it allows to distinguish the individual contribution of each uncertainty to the total budget, rather than lumping them in compounded uncertainties. Third, the equation allows to separate correlated and uncorrelated sources of uncertainty in a systematic way. Finally, the final result of a measurement, $U(Q_t)$, is expressed as the best estimate for the measurand along with its uncertainty and a confidence interval in reporting the uncertainty. This interval is expected to contain a large fraction of the distribution of values that can be reasonably attributed to the measurand. Such an interval, labeled expanded uncertainty in Fig. 1, is obtained using a coverage factor that multiplies the combined standard [JCGM 100 (JCGM 2008a)], as follows:

The coverage factor, $k$, in this equation requires knowledge of the probability distribution for the input and output quantity [JCGM 100 (JCGM 2008a)]. The simplest, and often adequate, approach is to assume $k=2$ corresponding to an interval of 95% level of confidence.

The three KICT-REC outdoor experimental channels used river water for feeding the flumes. The channel bed and banks were made of local natural material including native vegetation (see Fig. 5). The flumes were equipped with multiple monitoring devices for controlling the flow stages and discharges. Customized measurement platforms were available to accommodate experiments geared toward assessment of instrument performance and their uncertainties. With a maximum discharge of $10{\mathrm{m}}^3/\mathrm{s}$, the KICT-REC flumes can realistically replicate low to moderate flows occurring in small natural streams. The KICT-REC measurement environment also enables repetition of the measurements in close-to-ideal conditions that ensures a robust assessment of individual sources of uncertainties. In such environment, measurements can be repeated by “freezing” all the sources of uncertainties except the one that is subject to the evaluation. The Type A evaluations carried out in this case study used this approach.

At the time of the experiment, the channel cross section had slightly vegetated channel banks and bottom growing on a coarse sand substrate. The bed was stable during the experiments as shown by the pre- and postsurveys of the cross section in the test area. Essential aspects of the facility and flow conditions for the case study are provided in Tables 1 and 2, respectively. The three SonTek microADVs used in the study were installed on mobile traverse set across the channel, as shown in Fig. 5. Velocities in points were acquired with an ADV attached to a mechanically operated rod with finest graduation of 0.0008 m. The ADV data were acquired in 208 fixed locations arranged in a preestablished grid, shown in Fig. 4. The distance between the verticals was 0.25 m to ensure a high-density of measurements across the stream section for a robust estimation of the reference discharge and good sample for testing the sensitivity of the uncertainty associated with the number of verticals. This spacing also ensures that each panel conveys less than 10% of the total discharge as recommended by ISO 1088 (ISO 2007b) standard. The overall duration of the experiment was 32 h. The flow steadiness and uniformity were tracked 9 h before the experiment and over the whole duration of the experiment with dedicated sensors (Kim et al. 2018).

Given that UA studies require a reference for uncertainty evaluations, a benchmark for the study was created. Ideally, such a benchmark should be acquired with a high-quality instruments different from the ones subject to UA. The controlled measurement environment offered by the KICT-REC facility equipped with high-precision, high-resolution instruments, and well-established measurement protocols, offers favorable conditions for the creation of a benchmark data set that is close enough to being considered as reference for uncertainty assessments. A similar approach is used by Bertrand-Krajewski et al. (2021).

Taking advantage of the high-spatial density of ADV velocities, the mean streamwise point velocities acquired over 90-s sampling duration (Muste et al. 2021) could be reliably extrapolated to the top and bottom of the vertical using a logarithmic (or similar) velocity model applied to all the ADV measurements in individual verticals. For our case, the “reference” velocity distribution is the “log law,” as illustrated in Fig. 6 for vertical #12. Assuming negligible integration errors, the equation for the regression lines in each vertical is considered as reference velocity. Using log-law velocity profiles in verticals #1 to #23, the cross-section geometry surveyed by multiple methods (see section on the depth instrument accuracy in the Appendix), and the edge discharge obtained with additional measurements in the edge areas (Kim 2021; Muste et al. 2021) enabled to determine the “reference” discharge reported in Table 2.

Table 3 provides the summary of the identified uncertainty sources and their estimates as documented by direct measurements conducted in the case study (Type A) or prior knowledge and expert judgement (Type B). The previous standards [especially ISO 748 and ISO 1088, also referred to in HUG (ISO 2020)] are helpful in providing some of the Type B elemental uncertainties. Short descriptions for individual error sources and considerations on the estimates are provided in the Appendix. Type A uncertainty estimates contained in Table 3 were obtained through three repeated field measurement campaigns carried out in the Andong experimental channel, each lasting more than 30 h. The flow conditions for the repeated measurements were practically the same, with discharge differences less than 7%. The small difference between the bulk flow parameters for various experiments ensures that the values of estimated uncertainties are transferable among the three experiments. During the uncertainty assessments, it was aimed at capturing the effect of the individual uncertainty sources acting in isolation, hence qualifying for the “freezing” approach desired in the evaluation of the individual sources of errors.

Type B uncertainties were selected from various sources as specified in the last column of Table 3. All these data sources were deemed rigorous with respect to uncertainty assessment procedures. However, given that the information on some elemental uncertainty might be different in various sources, critical judgement was used to select estimates that are most relevant to our specific measurement situation. The elemental uncertainties associated with each group are aggregated using the root-sum-square (see Step 2 in Fig. 1) to obtain the standard uncertainty for each variable in Eq. (6).

There are several notable aspects about the information contained in Table 3, as follows:

The total uncertainty of the discharge using the ADV point-velocity measurements is obtained following the algebraic calculations shown in Steps 3 and 4 of Fig. 1. The total uncertainty (expressed by the combined uncertainty) in the flow discharge is obtained with Eqs. (6) and (7) using a coverage factor of $k=2$. The combined uncertainties associated with the main measured variables (mean velocity in the verticals, $u(U_j)$, depth in verticals, $u(d_j)$, distance between verticals, $u(b_j)$, and with the discharge estimation model, $u(Q_{MO})$, are aggregated through the following relationships:

Notations and essential specifications for all elemental sources of uncertainties are provided in Table 3 and the Appendix, respectively. Noting the absence of estimates for the correlated uncertainties in Table 3, the corresponding terms in Eq. (6) are also dropped in the evaluation of the uncertainty for the final result.

The calculation of the total discharge and the propagation of the elemental uncertainties generated in the measured and unmeasured areas of the cross section were conducted using the QMsys Enterprise (Qualisyst 2019). A dedicated interface, labelled QMSys Calculator (Qualisyst 2019), was used for providing the functional relationship and the values for elemental uncertainties in Table 3. This GUM-compliant software is executing Step 3 in Fig. 1 using interactive graphical user interfaces (GUI) that aid users with the data reduction process, as well as visualizing the UA results and summaries in a user-friendly format (Muste and Lee 2011). The software is initiated by typing in the software Eq. (5) for the functional relationship of the measurement process. Eqs. (6) and (7) are numerically determined by the software. A confidence level of 95% was uniformly applied to the elemental sources and total uncertainty.

The QMsys Enterprise software is also capable of estimating uncertainties in results using the MCM, an approach often used as a means of “validation” for the GUM conventional method (Bertrand-Krajewski et al. 2021). The reason for choosing MCM as reference is that it does not require special provisions for Type B of uncertainties. If both methods produce close results using the same inputs for the variance of elemental sources of uncertainties, then the law of propagation of uncertainty can be applied, which may be more convenient, especially with repetitive calculations. If the results are not equivalent, then the MCM should be used (Bertrand-Krajewski et al. 2021). The results of the QMsys produced for this case study obtained with the GUM protocol for propagation of variances (labelled GUF in the software) and MCM simulations are shown Table 4.

The graphical comparison of the two GUM-complaint approaches is illustrated in Fig. 7. The data in the table illustrate that the expanded standard uncertainties estimated by the conventional GUM and MCM are in good agreement. The MCM validation, with respect to the GUM uncertainty framework (GUF), was also verified by calculating a tolerance interval. An absolute difference between the endpoints of the two coverage intervals ($d_{\mathrm{low}}$ and $d_{\mathrm{high}}$ in columns 7 and 8 of Table 4) is less than the numerical tolerance value for $\delta$ in column 9, leading to the conclusion that GUF estimated was validated for this case study.

The value of 3.28% for the expanded uncertainty of the measured discharge (estimated at 95% confidence level) is well within the 5% value stated as the acceptable uncertainty for qualifying a measurement of satisfactory quality. However, given the overall good quality of the experiments (i.e., flow in the facility, measurement environment, instruments, and controlled execution of the experiments), the value of the expanded uncertainty may be deemed as large considering that, at natural-scale sites, multiple uncertainty sources may be simultaneously present that can easily exceed the 5% accepted threshold for discharge estimation. Another important UA outcome is the uncertainty budget illustrated graphically in Fig. 8. The uncertainty budget plays multiple roles if the identification of the uncertainties considered in the analysis is complete (see the “Discussion” section).

This paper, as well as previous case studies conducted by the authors in open-channel flows (e.g., Muste and Lee 2011; Muste 2017b; Muste et al. 2012, 2021) and in urban drainage and stormwater management systems (e.g., Bertrand-Krajewski et al. 2021), illustrate that the full-fledged GUM-UA framework can be executed if proper resources are secured. These illustrations highlight the capability of the framework to adapt to diverse measurement situations, including typical streamflow measurements. While the results of the study are not readily usable for typical in situ measurements, the contribution of this study is identification of the sources of uncertainties and reporting details on how uncertainties at all levels are rigorously estimated as specified by rigorous protocols.

The analysis conducted in this study highlights that the execution of UA requires not only a robust understanding of the UA fundamentals, but also cross-disciplinary knowledge on the flow-related and instrument-sensing processes, instrument configuration and components, and measurement methods and their execution in various environments. Typically, these areas are mastered by different specialization areas or sub-areas, therefore collaboration among various actors is paramount. This paper also demonstrates that the burdensome UA computations are rapidly being (more than) compensated for by the availability of computer processing power. Currently, there are more than 50 GUM-compliant software packages available (Muste 2017a).

Selecting and implementing a generic UA framework such as GUM in the practice of a professional community are actions that require long-term commitment and effort. This paper, ensuing from more than a decade of sustained effort to adopt UA in hydrometry, intends to provide readers with an overview of the benefits of using a judiciously selected UA framework and demonstrating that it can be successfully applied for assessment of measurement uncertainty. According to Thomas (2002), the following steps are essential in the adoption of a community standard: evaluation, prioritization, implementation, planning, accessing various resources, and maintaining the drive. The effort made by the WMO’s Project, “Assessment of the Performance of Flow Measurement Instruments and Techniques,” is an example in this regard (Pilon et al. 2010). Similar efforts are emerging in other water-related communities (Wahlin et al. 2005; Bertrand-Krajewski et al. 2021).

Besides its main role of quantifying the quality of the data, UA implementation can bring several additional benefits (e.g., NAP 2013; Kline 1985). Convincing arguments by McMillan et al. (2017) reveal the critical roles played by UA in water management applications (from reducing project costs to making robust and publicly acceptable decisions at a time of increased uncertainties produced by stresses on water resources from natural and human-induced causes). It is beyond the scope of this paper to detail all these benefits. Instead, we will focus on UA benefits ensuing from the present analysis.

The ranges of variation for the three uncertainty sources plotted in Fig. 9 are strictly valid for measurements with micro-ADV acquired in turbulent channel flows similar to those tested in the KICT-REC experiments (i.e., streams of similar aspects ratios without prominent changes in the boundary geometry). It should be noted that these uncertainty ranges are slightly narrower and sensibly lower than values currently provided in the ISO 748 (ISO 2007a) hydrologic standard. These differences are due to the highly favorable measurement environment, the superior capabilities of the instrumentation, and the careful measurement design and execution in the KICT-REC case study in comparison with those typically encountered in field studies. From this perspective, it can be stated that the uncertainty estimates illustrated in this study are closer to the lower range of the uncertainties for this kind of measurements. The insights of the above evaluations are, however, valid for other in situ measurements with the VA method. The information is particularly important because the guidelines for sampling point velocities across the section vary widely among various national hydrometric agencies (Le Coz et al. 2012; Despax et al. 2016b).

The present embodiment of the GUM framework applied to a set of experiments conducted with acoustic point-velocity meters and VA method highlights the complexity required for implementing GUM or any rigorous UA method. This realization is noted upfront in the HUG (ISO 2020): “For practitioners of hydrometry and for engineers, the GUM is not a simple document to refer to” as the framework was developed by professionals with a working knowledge of statistical and mathematical backgrounds. The one-time analysis presented here for one measurement method and one instrument makes it also obvious that UA can easily exceed the resources available for a typical hydrometric project. Additional costs are incurred to implement such analyses within routine data acquisition and training personnel programs of the monitoring agencies.

UA implementation for practical situations is a long-term target due to cost, complexity of uncertainty assessments, institutional resistance, and the additional data acquisition and processing procedures needed (e.g., customized experimental protocols, tools for processing and storing data). Notable progress along the UA technical side is the release of the HUG (ISO 2020) guide that strives to adapt the GUM framework for practical needs of hydrometric engineers and data managers. Further efforts for changing the UA implementation paradigm rely on: (1) providing UA implementation examples using data from routine hydrometric measurements; (2) sharing the data relevant to UA in well-structured databases to enable access to the much-needed Type B uncertainty evaluations; and (3) developing computational and operational tools embedded in software packages that are readily operable by users not familiar with the UA framework.

The pressing need to include UA in practice in the hydrometric communities is partially due to a bewildering proliferation of uncertainty methods published in the literature (e.g., Hall and Solomatine 2008; Kiang et al. 2009). Ideally, a unique procedure applied across a wide variety of instruments and methods is preferable to using multiple specific methodologies. One such procedure is the GUM framework. Attaining this target requires coordinated efforts to rally around generic and rigorous UA protocols and making concerted community efforts for their adoption and gradual implementation. Community and individual trainings are needed to fill the gaps in the required UA knowledge (Coelho et al. 2019) and to converge on how to conduct UA in a more practical manner that enables addition of UA when taking in situ measurements. At this time, however, the data necessary for UA are not always available, so new data collection campaigns (perhaps including time-consuming expert elicitation exercises) may need to be commissioned (Hall and Solomatine 2008). Given the high cost of the UAs, data exchange and sharing are essential for community adoption and sustained UA implementation. The data archiving and exchange should be sensitive to all parties involved with the data: producers, providers, and users [see Chapter 10 in Bertrand-Krajewski et al. (2021)].