Framework for Estimating Uncertainty of ADCP ...

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Framework for Estimating Uncertainty of ADCP Measurements from a Moving Boat by Standardized Uncertainty Analysis Juan A. González-Castro, A.M.ASCE1; and Marian Muste, A.M.ASCE2 Abstract: In spite of the extensive use of acoustic-Doppler current profilers 共ADCP兲 for measurement of velocity and discharge in open-channel and riverine environments, a rigorous methodology for estimating ADCP discharge measurement uncertainty that follows current engineering standards for uncertainty analysis is not yet available. In this paper, we apply the broadly accepted engineering standard for uncertainty analysis put forth by the American Institute of Aeronautics and Astronautics in 1995 共AIAA兲 to develop a framework for the estimation of uncertainty in ADCP measurements from a moving boat. First, we summarize the terminology and methodology of measurement uncertainty analysis and review the data reduction equations used by ADCPs to estimate the total discharge in measurements from a moving boat. Second, we discuss briefly the various elemental error sources that contribute to the uncertainties of the ADCP measured variables, which in turn contribute to the total uncertainty of ADCP discharge measurements. In discussing the elemental errors, we look into what determines their uncertainties and whether they can be evaluated using available information. We then apply the guidelines of the AIAA standard to develop an analytical framework for propagating the uncertainties from the elemental sources to obtain the total uncertainty of ADCP discharge measurements from a moving boat. DOI: 10.1061/共ASCE兲0733-9429共2007兲133:12共1390兲 CE Database subject headings: Open channel flow; River flow; Flow measurement; Uncertainty principles; Small crafts; Velocity.

Introduction Acoustic Doppler current profilers 共ADCPs兲 are highly efficient and reliable instruments for flow measurements in riverine and open-channel environments. In hydrometry, their primary use is the measurement of river discharge and channel bed survey from a moving boat 共Simpson 2001兲. Other applications of ADCPs that have been, or are currently being evaluated include: 共1兲 characterization of streamwise velocity distribution 共e.g., GonzálezCastro et al. 1996; Muste et al. 2004兲; 共2兲 estimation of turbulence quantities 共e.g., Droz et al. 1998; Stacey et al. 1999; Lu and Lueck 1999a,b; Schemper and Admiraal 2002; Nystrom et al. 2002; Howarth 2002; Kawanisi 2004; Kostaschuk et al. 2004兲; 共3兲 sediment transport 共e.g., Deines 1999; Rennie et al. 2002兲; 共4兲 characterization of the spatial distribution of velocities in riverine habitats and bed survey 共e.g., Shields et al. 2003; Gaeuman and Jacobson 2005兲; and 共5兲 estimation of the longitudinal dispersion coefficient for river flow 共e.g., Carr and Rehmann 2005; Carr et al. 2005兲. 1

South Florida Water Management District, 3301 Gun Club Rd., West Palm Beach, FL 33406 共corresponding author兲. E-mail: jgonzal@ sfwmd.gov 2 Iowa Institute of Hydraulic Research-Hydroscience & Engineering, The Univ. of Iowa, Iowa City, IA 52242. E-mail: marian-muste@ uiowa.edu Note. Discussion open until May 1, 2008. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on June 22, 2006; approved on June 21, 2007. This paper is part of the Journal of Hydraulic Engineering, Vol. 133, No. 12, December 1, 2007. ©ASCE, ISSN 0733-9429/2007/12-1390–1410/ $25.00.

The current and emerging applications of ADCPs have prompted the need for identifying sources of measurement errors, assessing the impact of these errors on the quality and reliability of the measurements, and developing good measurement practices. Notable in this regard are the studies by Oberg and Mueller 共1994兲, Morlock 共1996兲, Shih et al. 共2000兲, Mueller 共2002a,b兲, Gartner and Ganju 共2002兲, González-Castro et al. 共2002兲, Marsden and Ingram 共2004兲, Abad et al. 共2004兲, Mueller 共2004兲, Schmidt and Espey 共2004兲, Gaeuman and Jacobson 共2005兲, and Rennie and Rainville 共2006兲. Estimates of total ADCP measurement uncertainty are necessary to properly report ADCP discharge data collected for the calibration of water control structures used for indirect real-time flow monitoring 共GonzálezCastro, 2002, Accuracy of ADCP Discharge Measurements for Rating of Flow-Control Structures, Statement of Work for Cooperative Agreement between South Florida Water Management District and IIHR-Hydroscience & Engineering, SFWMD, West Palm Beach, Fla.兲, validation of numerical simulations 共Cheng et al. 2005兲, and insuring that quality assurance programs become acceptable to the scientific community and stand legal challenges 共González-Castro, 2002, Accuracy of ADCP Discharge Measurements for Rating of Flow-Control Structures, Statement of Work for Cooperative Agreement between South Florida Water Management District and IIHR-Hydroscience & Engineering, SFWMD, West Palm Beach, Fla.兲. However, the estimation of the total uncertainty of discharge measurements by ADCP has received limited attention 共Simpson and Oltman 1993; Simpson 2001兲. Moreover, a framework for estimating ADCP measurement uncertainty based on recent engineering and scientific uncertainty analyses standards accepted by specialized monitoring agencies and research institutes is not yet available. Currently, methods for estimating and reporting uncertainty for

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special applications in water resources are rather uncommon in the hydraulic and hydrologic engineering communities. Over the last decade, the American Society of Civil Engineers’ Environmental and Water Resources Institute has formed several task committees to address this need. The goals of these committees have been to review existing methodologies for uncertainty analysis and indicate the best practices for quantifying measurement errors that can be adapted by the ASCE Hydraulic Measurement and Experimentation Community. The current task committee is developing guidelines for estimating measurement uncertainty that stem from measurement uncertainty standards currently accepted by the scientific and engineering communities 共Wahlin et al. 2005兲. Pressed by the need for estimating ADCP measurement uncertainty, the South Florida Water Management District 共SFWMD兲 launched a collaborative project with the Iowa Institute of Hydraulic Research-HydroScience & Engineering 共IIHR-HS&E兲 in 2002. The main goal of this project was to develop a framework for assessing ADCP measurement uncertainty according to accepted standardized uncertainty analysis 共UA兲 used by the scientific and engineering communities 共GGonzález-Castro, 2002, Accuracy of ADCP Discharge Measurements for Rating of FlowControl Structures, Statement of Work for Cooperative Agreement between South Florida Water Management District and IIHR-Hydroscience & Engineering, SFWMD, West Palm Beach, Fla.兲. There are three broadly accepted standards available for estimating measurement uncertainty, the International Standards Organization standard 共ISO 1993兲, the American Institute of Aeronautics and Astronautics standard S-071-1995 共AIAA 1995兲, and the American Society of Mechanical Engineers performance test code ANSI/ASME PTC 19.1-1998 共ASME 1998兲. The ISO standard or so-called “Guide to the expression of uncertainty in measurement” includes the latest statistical and mathematical advancements relevant to UA and is the preferred standard by a number of scientific communities 共Taylor and Kuyatt 1994; AFNOR 1999; Bertrand-Krajewski and Bardin 2002兲. The AIAA standard and the ANSI/ASME standard are used in the aeronautics and astronautics community and the mechanical engineering community, respectively. The three standards rely on the same statitistical principles for the propagation of uncertainties. The differences are in terminology and implementation procedures; the ISO standard classifies errors based on the approach to evaluate them, while the AIAA and ANSI/ASME standards use the traditional engineering approach of classifying errors based on how they affect the result. The AIAA UA standard has been applied to report experimental uncertainty in hydraulic engineering 共Stern et al. 1999兲 and water resources 共Muste and Stern 2000兲. In the next section, we describe the application of the AIAA UA standard for the development of a framework for estimating the uncertainty in ADCP discharge measurements from a moving boat applicable to the broadband, four-transducer ADCPs with Janus configuration manufactured by RD Instruments 共Use of trade, product or firm names in this paper is solely for descriptive purposes. It does not imply endorsement by SFWMD or IIHR-HS&E.兲 We then discuss the ADCP data reduction equations and identify the elemental error sources in ADCP measurements. We also discuss how the uncertainties from the fundamental sources propagate into the final result, whether they are accounted for in the data reduction equations, what factors they depend upon, and possible ways to estimate them. Afterwards, we describe the analytical framework we developed for UA of ADCP discharge measurements from a moving boat and show how it allows us to propagate the uncer-

tainties from the elemental sources of quantities measured or calculated by ADCPs into the total uncertainty of the measured discharge. Finally, we provide some conclusions and recommendations for assessing the uncertainties from sources for which uncertainty estimates are not available and describe a plausible way to incorporate UA into common field ADCP measurements practice.

Uncertainty Analysis Terminology and Procedures The error in a measurement is the difference between the true and measured value of the result. In most situations, we do not know the true value; hence, we need to estimate the limits that bound the possible measurement error. The engineering approach for error classification is based on the effect of the error on the measured, namely bias 共fixed兲 and precision 共random兲 errors, respectively. Bias errors are systematic, hence, difficult to detect and remove. Precision errors are random, thus we can estimate them from the scatter of repeated measurements. In the terminology of the AIAA UA standard 共AIAA 1995兲, estimates of bias and precision errors are called bias limit, B, and precision limit, P, respectively. The estimate of the total error is the total uncertainty, U. It combines both bias and precision uncertainties as described below. In the standard, the errors from all the variables in a measurement are assumed normally distributed. Hence, the mean value is expected to be within the interval defined by two standard deviations of the mean with odds of 19 out of 20, or equivalently, with a 95% confidence level. In general, the uncertainty limits of a measured quantity define the interval about its average value that has a prescribed probability 共or confidence level兲 of containing the true value. Here, as in most of the UA standards applied in engineering, the confidence level prescribed for establishing the uncertainty limits in a measurement is 95%. Usually, a measurement result, r, is expressed through a data reduction equation 共DRE兲 that combines measurements of J individual variables, Xi r = r共X1,X2,X3, . . . ,X j兲

共1兲

In uncertainty analysis it is important to distinguish between multiple and single measurements. Multiple measurements are ideal for conducting full uncertainty analysis. Multiple measurements are obtained by repeating all the measurements in DRE with the same instrumentation, measurement environment, and procedures. In multiple tests, an average result ¯r is determined from M sets of measurements r共X1 , X2 , . . . , XJ兲k as M

¯r =

1 rk M k=1

兺

共2兲

When the measurements are complex it becomes both difficult and expensive to replicate them for uncertainty assessment purposes. In such instances, the result, r, is determined from a single set of measurements of the individual variables 共X1 , X2 , . . . , XJ兲. According to the AIAA methodology, this case is called the single test result. Further, a test is a single test if we perform the entire test only once, even if we measure one or more of the variables using many samples. For example, when we make boat-mounted ADCP discharge transect measurements at a river station, some of the variables are estimated from single measurements 共e.g., salinity, temperature, velocities in individual bins兲 while others are obtained from multiple measurements 共e.g., distance to channel banks兲. Thus, in standard UA terminology, each ADCP transect is

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / DECEMBER 2007 / 1391

␪i =

Fig. 1. Steps for uncertainty assessment implementation

a single test result, whereas discharge estimates from several transects is a multiple test result.

⳵r ⳵ Xi

共6兲

Bi⫽bias limit in Xi; and Bik⫽correlated bias limit in Xi and Xk. Correlated biases occur when different variables are measured with the same sensor or when different sensors or parameters are calibrated against the same standard. The covariance of correlated variables increases or decreases the total bias limits, depending on the sign of the correlation. Correlated biases are zero for statistically independent variables. The sensitivity coefficients, evaluated at the variables’ measured mean value, weigh the contribution of the respective bias to the total bias. The bias of a sensor can be estimated from its manufacturers’ specification limits, typically reported at the 95 or 66% confidence level. These limits are often interpreted as the bias limits of normally distributed biases from many identical sensors 共Coleman and Steele 1995兲. When the limits are known to be for asymmetrically distributed data, the bias limits can be established as suggested by Coleman and Steele 共1999兲. Biases not reported in the instrument’s specifications can be estimated from the best available information, e.g., expert’s opinions, intercomparison with instrument traceable to standards, and end-to-end calibrations. Estimation of Precision Limits

Total Uncertainty of Measured Result The steps to estimate the total uncertainty are summarized in Fig. 1. Identification of all elemental error sources associated with the individual variables and assessing their uncertainties are key steps. It requires tracking all steps in the measurement cycle, from sensor to data display, and understanding the hypotheses and conditions of the measurement procedures. Uncertainty assessment only accounts for the uncertainty from errors “active” during measurements and apply only over the range of conditions under which the test is conducted. Therefore, in ADCP measurements, the hydrographer must document thoroughly the site conditions, measurement procedures, flow and measurement environment conditions, and the data acquisition and reduction procedures used 共e.g., water and bottom tracking modes, bin size, blanking distance, extrapolation model, ADCP depth and alignment, and distances to banks兲. The result of a measurement, is reported as 共3兲

¯r ± Ur

with ¯r representing an estimate of the mean from single test result or the average from multiple tests; and Ur⫽magnitude of the total uncertainty limits at a 95% confidence level. The total uncertainty in the result from single and multiple tests is computed as the root sum square of the bias and precision limits for the result, that is Ur = 共Br2 + Pr2兲1/2

共4兲

Estimation of the Bias Limits For both single test and multiple tests, the general expression in the AIAA standard for estimating the bias limit of the result, Br, is based on the first-order Taylor series Approximation for propagation of uncertainty, i.e.

冉兺 J

Br=

i=1

J−1

␪2i B2i

+2

J

兺兺 ␪i␪kBik i=1 k=i+1

where ␪i⫽sensitivity coefficients defined by

冊

1/2

共5兲

Here, we only present the estimation of the precision limits for the multiple tests case, because our UA framework applies to discharge measurements based on multiple ADCP transect measurements from a moving boat. Estimation of the precision limits for single test measurements should follow the guidelines in the AIAA standard. If a set of M repeated measurements is available, the precision limit of the result is Pr = P¯r =

tSr

共7兲

冑M

where t, coverage factor at a 95% confidence level,⫽Student’s t value with M-1 degrees of freedom 共the AIAA standard recommends taking t as 2 for M ⱖ 10兲; and Sr⫽standard deviation of the test’s estimates

冋兺册 M

Sr =

k=1

共rk − ¯兲 r2 M−1

1/2

共8兲

Readers interested in the justification for adopting a coverage factor of 2 when M ⱖ 10 are referred to Coleman and Steele 共1995兲 and ISO 共1993兲.

Data Reduction Equations for ADCP Discharge Measurements from Moving Boat The elemental error sources that contribute to the total uncertainty of ADCP discharge measurements should be identified from the DREs, data processing models, instrument configuration, operation, and conditions of the measurement environment. In this section, we review the DREs used by the RDI software to estimate the total discharge from data collected by four-beam, broadband ADCPs with Janus configuration during measurements from a moving boat. These DREs are essentially the algorithms in the RDI software to compute the discharge from quantities measured by the ADCP and to estimate the discharges through areas where the ADCP cannot measure. Through the review, we assume that readers are familiar with the operating principles, configuration,


冕冕 T

Qm =

0

Fig. 2. Schematic of measurable and unmeasurable areas of river cross section in ADCP discharge measurements from moving boat

and operation of ADCPs. Readers interested in these issues are referred to Gordon 共1996兲, Simpson 共2001兲, and RDI 共2003兲. Data Reduction Equation for Total Discharge The total discharge, Q, through an arbitrary area s can be defined as

Q=

冕

V f · n ds

共9兲

s

where V f ⫽water velocity vector; and n⫽unit vector normal outward to the differential area ds. In open-channel flow, s⫽total cross-sectional area. Constraints of the ADCPs architecture and operating principles, renders them unable to measure near solid boundaries or the free surface 共see Fig. 2兲. Thus, the total discharge in ADCP transect measurements is estimated as the summation of: 共1兲 the discharge computed from quantities directly measured by the ADCP and external devices, Qm; 共2兲 the discharge estimated by the ADCP postprocessing algorithms when part of the directly measured data are missing, Qem; 共3兲 the discharge estimated in the unmeasurable area near the free surface, Qet; 共4兲 the discharge estimated in the unmeasurable area near the channel bed, Qeb; and 共5兲 the discharge estimated in unmeasured areas near the left and right channel edges, Qel and Qer, respectively Qt = Qm + Qem + Qet + Qeb + Qel + Qer

共10兲

Eq. 共10兲 is the DRE of ADCP transect measurements. Below, each term in Eq. 共10兲 is treated as a separate DRE to facilitate the identification of error sources and the UA. Data Reduction Equations for ADCP Directly Measured Discharge In ADCP measurements from a moving boat, the flow, Qm, through the portion of the cross-sectional area defined by the vertical plane along the ADCP’s path is estimated from the measured water and boat velocities. Following Christensen and Herrick 共1982兲, since: 共1兲 ds = 兩Vb兩dz dt, where dz⫽differential depth and 兩Vb 兩 dt⫽differential length along the ADCP path; 共2兲 Vb ⫻ k = 兩Vb 兩 n, where k= vertical unit vector, positive upwards; and 共3兲 the scalar triple product V f · 共Vb ⫻ k兲 = 共V f ⫻ Vb兲 · k, we can express Eq. 共9兲 as

zU共t兲

共共V f ⫻ Vb兲 · k兲dz dt

共11兲

zL共t兲

where both V f = 共u f , v f 兲⫽water velocity vector and Vb = 共ub , vb兲 ⫽boat velocity vector are in the same coordinate system; T⫽transect time; and zL共t兲 and zU共t兲⫽lower and upper limits of the water column at time t during the transect where the water velocity is measured. An ADCP measures both water and boat velocities along the path of the acoustic beams and outputs these data in beam, instrument, or ship coordinates. In beam coordinates, the positive direction is towards the transducers. In instrument coordinates, x and y align with Transducers 1 and 2, and Transducers 4 and 3, respectively, whereas in the ship coordinates, x aligns with the ship’s aft-forward axis and y with the port-starboard axis, respectively 共see Fig. 3兲. These velocities can be transformed to earth coordinates using the heading measured by an internal compass. In earth coordinates, positive x and y point towards the east and north, respectively. ADCPs actually measure the water velocity with respect to the ADCP, Va, and the boat velocity with respect to a fixed reference by acoustical bottom tracking 共tracking of the channel bed with respect to the ADCP measured by Doppler shift of signals sent by each transducer and reflected from the channel bed兲 or differential global positioning system 共DGPS兲. The actual water velocity vector is V f = Va + Vb. Introducing this identity into Eq. 共11兲, simplifying, and expanding into pseudoscalar form with Va = 共ua , va兲 yields

冕冕 T

Qm =

0

zU共t兲

共共uavb − ubva兲兲dz dt

共12兲

zL共t兲

The scalar triple product in Eq. 共12兲 accounts for the directionality of the water flux, so in moving ADCP measurements, it actually gives the net discharge measured in a transect. Eq. 共12兲 cannot be expressed in terms of the quantities directly measured by a boat-mounted ADCP using acoustical bottom tracking because the measured Doppler shifts, gating times, and tilts are variables not reported in the output files. However, using the transformations described in Appendix I, it can be expressed in terms of: 共1兲 the bin area, given by the distance along the ADCP path between times ti and ti+1, and the distance between depths z j and z j+1; 共2兲 the radial water velocities with respect to the ADCP, v1, v2, v3, v4; 共3兲 the radial beam bottom-tracking velocities, vb1, vb2, vb3, vb4; and 共4兲 the roll and pitch angles, r and p. When the boat velocity is determined by DGPS tracking, Eq. 共12兲 should be expressed in terms of DGPS latitude-longitude positions converted to distance using an ellipsoid transformation 共Rennie and Rainville 2006兲 and the times between DGPS position updates. The flow through a measuring cell in ADCP discharge measurements from a moving boat, is Qmi,j = 共uavb − ubva兲i+1,j共z j+1 − z j兲共ti+1 − ti兲 ⬅ 共共Va ⫻ Vb兲 · k兲i+1,j共z j+1 − z j兲共ti+1 − ti兲

共13兲

where subscripts i and j⫽ping 共transmitted acoustic pulse兲 and bin 共depth-measuring cell兲 counting indexes in ADCP measurements. ADCPs actually estimate both water and bottom velocities from measurements of the frequency Doppler shifts between transmitted pulses and the signals backscattered by particles along the path of the acoustic beams and the channel bed using


Fig. 3. Velocities measured by ADCP: 共a兲 isometric showing boat, beam, and water velocities; 共b兲 top view of instrument, ship, and earth coordinate systems in ADCP output data

v = FD

C 2FS

共14兲

where FS and FD⫽frequency of the acoustic wave transmitted by the ADCP and the Doppler frequency shift in Hertz, respectively; and C⫽sound speed in water at the transducer’s face. The socalled water modes and bottom tracking modes used by RDI’s ADCPs are a set of proprietary algorithms to measure FD. Clearly, the uncertainty of radial beam velocities, v, depends not only upon the uncertainties of C and FS, but on the uncertainties of various quantities measured by the ADCP and other parameters in the algorithms used by the ADCP’s to estimate FD. Because ADCPs geometrically map the depth and distance to the transducers for each measuring cell based on time gating and estimates of C, the uncertainties in measuring time, C, and geometric cellmapping also contribute to the total uncertainties of the radial velocities. Measurement uncertainties of data collected by external devices 共depth sounders, external gyroscopes, and DGPS兲 while making ADCP discharge measurements will contribute to the total ADCP discharge measurement uncertainty, and need to be accounted for in the DRE. The DREs for discharge measurements from a moving ADCP in terms of the variables directly measured by the ADCP and acoustical bottom tracking or DGPS are Qmi,j = fFD1,FD2,FD3,FD4,FS,C,FDb1,FDb2,FDb3,FDb4,␪,r,p,␦ta,␦z 共15a兲 Qmi,j = fFD1,FD2,FD3,FD4,FS,C,␦Pg or vbg,␪,r,p,h,␦ta,␦tg,␦z 共15b兲 where subscripts i and j are as defined above; subscripts 1–4 and b1 – b4 refer to the radial water velocities with respect to the ADCP and of the radial bottom tracking velocities, respectively; ␪⫽ADCP beam angle; r, p, and h⫽roll, pitch, and heading

angles; ␦Pg⫽DGPS tracking distance between consecutive positions; vbg⫽boat velocity by DGPS tracking; ␦ta, ␦tg⫽time intervals between consecutive ADCP pings and DGPS tracking pings, respectively; and ␦z⫽depth of measuring cell. The actual form of Eqs. 共15a兲 and 共15b兲 and some of the measured variables are not available. Therefore, Eq. 共13兲 in its pseudoscalar form is the DRE used by RDI for computing the discharge through a measuring cell, Qmi,j, in ADCP measurements from a moving boat. The discharge, Qm, in Eq. 共10兲 can be expressed as the summation of the flow through the measured bins, the discharge estimated in the top and bottom unmeasured areas at each bin, and the estimate of discharge through measurable bins where some of the ADCP data is missing Ns−1 m

Qt =

兺兺 i=1 j=1

Ns−1

Qmi,j +

兺 i=1

K

共Qet + Qeb兲i +

Qem + Qel + Qer 兺 k=1 k

共16兲 with i and j as defined above; m⫽number of measured velocities in an ensemble; Ns⫽number of ensembles; and K⫽number of bins in the measurable area with missing data. The uncertainty analysis of ADCP discharge measurements from a moving boat we describe below relies on the DRE for flow through a measuring cell based in its triple-scalar-product form, i.e., the right side of the identity given by Eq. 共13兲. This form of the DRE allows us to: 共1兲 account for the uncertainty of the ADCP measured quantities; 共2兲 obtain analytical expressions for the sensitivity coefficients needed for the propagation of uncertainty; 共3兲 use available manufacturer’s specifications to estimate the bias limits according to standards; and 共4兲 evaluate the sensitivity of the total uncertainty of ADCP measurement to variations in the uncertainty estimates of the independent error sources. Although ADCPs now account for small, off-plane manufacturing transducer misalignments through instrument-specific cor-


Fig. 4. Depth and water layers for discharge computation in ADCP measurements

rections to transform radial velocities to instrument coordinates, in the UA we present below, we consider ADCPs with identical, same-plane misalignments. Data Reduction Equations to Estimate Discharge through Unmeasured Areas ADCPs estimate the discharge through the unmeasured top and bottom areas by extrapolation or using some approximate models. Currently, ADCPs can estimate the discharge through the top unmeasured area with one of the three approaches below: 1. The velocity follows a power distribution throughout the water column; 2. The velocity is equal to the velocity measured in the top measurable cell; and 3. The slope of the velocity profile at the top follows the slope of the three top measurable cells. ADCPs can estimate the discharge through the bottom unmeasured area with one of the three following approaches: 1. The velocity follows a power distribution throughout the water column; 2. The velocity follows a power distribution in the lowest 20% of the depth and the no-slip condition applies at the bed. When the ADCP cannot measure in the lowest 20% of the depth, the power distribution parameters are estimated with the velocity measured in the lowest good bin along with the no-slip condition—this is known as the “no-slip method” in RDI’s software; and 3. The velocity throughout the bottom layer is assumed constant and equal to the velocity in the lowest measurable bin. The depths for estimating the discharge through the top and bottom layers in ADCP measurements are defined in Fig. 4; the formulas to compute them are summarized in Appendix III. The discharge through the unmeasured top and bottom layers are most commonly estimated using option 共1兲 with the power set

to the default one-sixth-power option or a user-defined power. This implicitly assumes that the nonslip condition applies at the bed. RDI’s software does not do the extrapolation on the streamwise velocities, but on the scalar triple product in its pseudoscalar form. Moreover, it does not do it by traditional least squares, but by equating the definite integral of the power law with the discreet integral of the scalar triple product in the measured area, i.e. m

Da共b + 1兲 f共Z兲 = a⬙Z = b

共uavb − ubva兲 j 兺 j=1

b+1 共Zb+1 2 − Z1 兲

Zb

共17兲

where Da⫽cell length 共bin size兲; b⫽user-defined power; 共ua , va兲 and 共ub , vb兲⫽water velocity and boat velocity vectors; m⫽number of cells in the measurable area collected during the respective ping; Z⫽distance to channel bed; and Z1 and Z2⫽distance from bed to bottom and top of measurable area, respectively as defined in Fig. 4 and summarized in Appendix III. The DREs for estimating the discharge through the top and bottom unmeasured layers based on power distribution are m

b+1 Da共Zb+1 3 − Z2 兲共ti+1 − ti兲

Qeti =


b+1 共Zb+1 2 − Z1 兲

共18兲

m

DaZb+1 1 共ti+1 − ti兲 Qebi =


b+1 共Zb+1 2 − Z1 兲

共19兲

where Z3⫽distance from bed to the free surface; ti and ti+1⫽times at pings i and i + 1, respectively; and other variables as defined above.


ADCPs can estimate the discharge, Qe, through the unmeasured left and right edges of triangular or square shapes as Qe = KVeLeZe

共20兲

where Ve and Ze⫽mean velocity and depth at the edge; Le⫽distance to the riverbank; and K⫽coefficient set to 0.35 and 0.91 for triangular and rectangular shapes, respectively. This formula is an extension of the one proposed by Fulford and Sauer 共1986兲 to estimate the discharge through unmeasured edges of triangular shape. Eq. 共20兲 is the DRE for estimating the flow through the unmeasured areas near the channel banks Qel and Qer, in Eq. 共16兲. The flow through cells in the measurable area that cannot be directly computed from ADCP measured data, Qem, is typically estimated by postprocessing interpolation algorithms. These algorithms depend upon the option chosen for extrapolation in the bottom layer and whether single-cell, ping, or bottom tracking data are missing. Currently, the discharge in a missing cell that is not a top or bottom cell can be estimated by two methods. In one, used when extrapolation is based on the power distribution, the missing value is interpolated from this distribution. In the other, applied along with the RDI no-slip method for bottom layer extrapolation, the missing discharge is linearly interpolated from the data measured in the two nearest cells directly above and below the cell with missing data. Bad or missing data in measurable cells near the top and bottom layers are estimated by extrapolation. Ping data lost due to missed bottom tracking, decorrelation, or low backscatter, are estimated using the scalar triple product in the next good ensemble and the time interval between the two good bounding pings. The DREs presented above are the basis for estimating the total discharge in ADCP measurements from a moving boat. In summary, 1. Eq. 共10兲 and its expanded form Eq. 共16兲 are the DREs for the total discharge; 2. Eq. 共13兲 is the DRE for computing discharge in the measured cells; 3. Eqs. 共18兲 and 共19兲 are the DREs for estimating the discharge through the top and bottom unmeasured areas based on velocity extrapolation assuming power law distribution; and 4. Eq. 共20兲 is the DRE for estimating the discharge through the near-bank unmeasured areas. These DREs are also the basis for the propagation of uncertainties from the various sources in ADCP discharge measurements from a moving boat presented below.

Identification of ADCP Elemental Measurement Errors Measurement errors are the result of imperfect calibrations, data acquisition system, data reduction techniques, sampling protocols and measuring methods, and natural variability of the measuring environment. Typical errors in ADCP discharge measurement due to incorrect ADCP draft setting and mounting, poor choice of data collection mode, moving bed, poor boat navigation, strong flow shear, small end subsections, shallow depth, flow unsteadiness, wind shear, sediment concentration, and large turbulence intensity can be kept inactive or minimized during a measurement by following good measuring practices. Technical guidelines for good measuring practices and quality assurance on discharge measurements by ADCP can be found in the technical memoranda issued by the Office of Surface Water of the U.S. Geological Survey

共e.g., USGS-OSW 2002a; b兲 and other reports 共e.g., Simpson 2001; Oberg et al. 2005兲. Below, we provide a brief description of the elemental error sources in velocity and discharge measurements by broadband ADCPs that builds upon that in Simpson 共2001兲. Most of these elemental errors contribute to the total uncertainty of ADCP measurements even when good measuring practices are followed. Further details on these errors and a summary of available information to assess their uncertainty can be found in Muste et al. 共2007兲. Spatial Resolution An ideal geometric arrangement for multicomponent profilers should sample point velocities throughout the water column 共e.g., Lemmin and Rolland 1997兲. RDI’s ADCPs, however, have a geometric arrangement that measures radial water velocities with respect to the ADCP with four monostatic, diverging transducers. Velocities are then transformed into an orthogonal coordinate system by assuming that the flow is horizontally homogeneous. This transformation results in spatial averaging that acts as a low-pass filter of the flow structure, biases estimates of mean flow in highly three-dimensional flows, and limits the ADCP’s ability to characterize turbulence 共Nystrom et al. 2002兲. RDI’s ADCPs collect redundant data that are used to assess whether the flow is horizontally homogenous 共RDI 2003兲. Doppler Noise Noise in the Doppler-shift measured by ADCPs includes both noise in the return signal and noise added to the first-pulse return signal by reflections from scatterers near the path of the first return signal as it intercepts the second pulse. ADCPs resolve the Doppler shift by two or more short identical pulses in phase with each other 共pulse-to-pulse coherent兲 and adjust the time between pulses to minimize ping-to-ping interference. The uncertainty of radial velocities measured by pulse-to-pulse coherent ADCPs is directly proportional to the acoustic wavelength and Doppler bandwidth. It is also affected by the scatterers’ residence time in the measuring volume, turbulence, and acceleration during the averaging period 共Brumley et al. 1990兲. Velocity Ambiguity Error ADCPs determine velocity by measuring the phase-angle difference between pulse pairs. These phase shifts are subject to ambiguity errors because the reference yardstick is half a cycle at the transmitted frequency. If the velocity exceeds the expected velocity range, a phase shift outside of the expected −180 to 180° range occurs. The processing algorithms in RDI’s ADCPs correct velocity ambiguity error based on lag-spacing measurements; however, this technique increases the ping or ensemble measuring cycle, thus resulting in variable ping rate. Side-Lobe Interference Error Transducers have parasitic side lobes at 30 and 40° angles with the main acoustic beam 共Simpson 2001兲. Side lobe signals reflect from a solid boundary before signals from the main beam. Signals returning from solid boundaries are stronger than returns from scatterers in the water, travel the shortest path to the surface, and add the “velocity of the boundary” to the water velocities mea-


sured by the main lobe inducing so-called side-lobe interference. Signals from 20°-beam ADCPs are typically affected by side-lobe interference in the lowest 6% of the depth. Timing Errors ADCPs sample data through the water column at equal distances during each ping. Timing errors contribute to the measurement uncertainty in cell mapping, Doppler shift, and ultimately in velocity and discharge computations. In measuring turbulenceaveraged flow features such as mean velocity and Reynolds stresses, the sampling should be at as high a frequency as possible; as a result, the uncertainty in timing contributes to the total uncertainty of turbulence quantities more than to the uncertainty of ADCP discharge measurements. Sound Speed Error Sound speed error contributes to the uncertainties in range gating for cell mapping, Doppler shift, and ultimately, water and boat velocities. ADCPs calculate range gating and Doppler shifts using the near-transducer water temperature measurements and assuming constant salinity and temperature in the water column. As a result, refraction of the acoustic waves in waters stratified due to gradients in salinity or temperature or both cannot be accounted for. Beam-Angle Error Beam-angle errors are due to manufacturing imperfections; its bias limits are bound by fabrication tolerances. Boat Speed Error Boat accelerations may force the ADCP compass to swing out of its vertical position and induce compass errors 共Gaeuman and Jacobson 2005兲. Thus, high boat-to-flow speed ratios may result in systematic errors in heading, DGPS positioning, and bottom tracking. Even in measurements with acceptable boat-to-flow speed ratios, the precision uncertainty increases with boat speed. Sampling Time Error Insufficient sampling time results in biased estimates of the variance of the mean flow. This bias can be minimized by sampling long enough at the highest sampling frequency so that a statistically large sample of flow-specific, large-scale, low-frequency flow structures is captured—the frequency of large-scale flow structures can be characterized using, for example, the universal Strouhal number 共González-Castro and Chen 2005兲. The uncertainty from this source biases turbulence-averaged quantities more than discharge estimates because in the latter, spatial averaging partly compensates for time variability. Limited sampling also biases the discharge estimates through the unmeasured layers because the ADCP extrapolation algorithms apply to turbulenceaveraged, boundary-layer velocity distribution models, not to instantaneous velocities.

ers after transmitting an acoustic pulse. It depends on the ADCP frequency, transducer characteristics, and signal-processing algorithm. ADCP-flow interactions induce disturbances that bias the flow field near the ADCP with respect to the undisturbed flow field. It depends on ADCP geometry, draft, channel geometry, and Reynolds number, and decreases with distance to ADCP transducers. Reference Boat-Velocity Error ADCPs measure water velocities relative to the ADCP. In ADCP transects, to calculate the water velocity, the velocity of the ADCP relative to the channel bed is measured with respect to a fixed reference by acoustical bottom tracking or DGPS tracking. Moving bed, bed sediment transport, uneven bed, high sediment concentration, and boat operation bias the boat velocity measurements by acoustical bottom tracking. High boat-to-water speed ratios, DGPS precision, satellite reception, signal loss, and signal multipath due to riverbank vegetation and other obstructions bias the boat velocity estimates by DGPS tracking. Depth Error This error is associated with bottom-tracking profiling. The transmit time for the bottom-tracking profiling is longer than that used for the water profiling and the echo is processed in a different way. Bias errors in depth measurement stem from errors in measuring the distance of the ADCP transducers to the free surface or draft, sampling errors due to limitations of the acoustic beams and bin size, from sound speed errors, and from random errors due to echoes reflected from the bed. Uncertainties in depth depend on uncertainties in transmit-pulse length, blank beyond transmit, cell size, average measured depth, and ADCP draft. Cell Mapping Error The position of the top first cell is determined by range gating and the sound speed, transmit pulse length, blank beyond transmit, bin size, transducer beam angle, transmit frequency, and, when the data are collected using some of the RDI’s water modes, the lag between transmit pulses, or correlation lag. In salinity or temperature stratified flows, ADCPs map cells inaccurately. Rotation „Pitch, Roll, and Heading… Error Errors in pitch and roll affect the water velocity estimates through the transformations from radial to instrument coordinates. Heading errors propagate through the transformation from ship to earth coordinates. Rotation errors directly depend on the ADCP configuration. ADCPs are equipped with internal tilt sensors to measure pitch and roll, and an internal compass to measure heading. Compass errors due to incorrect magnetic declination do not effect ADCP discharge measurements by bottom tracking. However, errors in compass calibration bias ADCP discharge measurements by DGPS tracking. Edge Estimation Error

Near-Transducer Error This error lumps two biases in measuring velocities near the ADCP transducers, one due to ADCP ringing, and the other due ADCP-flow interaction. Ringing is the resonance of the transduc-

The distance to shore from the end points of ADCP transects to estimate edge discharges is not measured by the ADCP. The uncertainty in edge discharge depends on the uncertainty in measuring distances to shore, the environmental and operational


conditions when measuring velocities at the edges, the shape of the edges, and the model for computing edge discharge. Vertical-Velocity Distribution Error The effect of this source of error depends on whether the ADCP collects data from a moving or fixed boat and the velocity distribution model used for extrapolation. The random uncertainty on velocity profiles from long sampling records collected at fixed locations is substantially smaller than in profiles estimated from short records. The bias from this source must be quantified as the deviation of short-time velocity profiles with respect to long-term profiles. We recommend the use of the power law with free parameters as reference velocity distribution, with the neartransducer bias removed. Although Barenblatt 共1993兲 has shown that in turbulent boundary layers the power law follows incomplete similarity scaling laws, in which both parameters vary with Reynolds number, we suspect that in open-channel flow the parameters also depend upon the pressure and acceleration gradients as well as on the geometry and roughness of the channel. Discharge Model Error Model errors in the methods used by ADCPs to compute the total discharge from a moving boat influence the uncertainty in ADCP measurements. The optimal model should use actual turbulenceaveraged point estimates of the water velocity with high spatial resolution. In ADCP measurements, too many pings induce correlated errors, and too few resolution errors. The ADCPs’ spatial averaging induces correlated errors between velocities measured in contiguous cells. Estimates of these errors are available only for contiguous cells in the same ping. Finite Summation Error The error in discharge computation in ADCP measurements stemming from finite summation is similar to the finite summation error in measurements with conventional mechanical current meters 共Pelletier 1988; ISO 1985兲. The uncertainty from this source is a direct function of the spatio-temporal averaging strategies specified by the cell size, number of pings per ensemble, sampling frequency, and water and bottom tracking modes. Measuring Environment and Operational Errors This error group lumps errors due to poor use of good ADCPmeasurement practices as those cited in the literature 共RDI 2003; USGS-OSW 2002a, b; Simpson 2001; Oberg et al. 2005兲. Secondary currents, hydraulic structures, small channel aspect ratios, and other factors that might induce considerable threedimensional characteristics of the channel flow, result in measurement conditions that may violate the assumptions of both the ADCP operational principles and good measurement practices. The channel roughness and the level of turbulence are factors that also affect the velocity distribution models and sampling procedures used for estimating velocity and discharge. This error group depends so much on the site and operation mode that it is difficult to assess. However, we list it here for completeness. Not all the sources of uncertainty listed above can be directly propagated into the total discharge measurement uncertainty through the variables in the DREs 共e.g., spatial resolution兲; moreover, some of the error sources lump uncertainties from several elemental sources of a different nature. For example, the noise

error, associated with the measurement of the water velocity, lumps the effect of noise in Doppler-shift measurement, selfnoise, finite bin size, and nonuniform signal absorption 共Simpson 2001兲. In Table 1 we summarize 20 sources of uncertainty in ADCP measurements from a moving boat. In the table, we indicate what intermediate variables are biased by each source, whether they are accounted for in the ADCP DREs and how, and what factors they depend upon 共instrument characteristics, operating conditions, environment, etc.兲. The uncertainties from elemental sources in a measurement process can be assessed from manufacturer’s specifications, critical evaluations of prior published information or, when prior information is not available, from direct calibration/measurements specifically designed for this purpose. The last column of Table 1 indicates possible ways to estimate the systematic uncertainty from the identified sources. Muste et al. 共2007兲 reviewed and compiled available information on individual error sources and proposed procedures to estimate the uncertainty from sources not yet documented.

Propagation of Uncertainties from Elemental Sources into Total ADCP Discharge Measurement Uncertainty In this section, we summarize the analytical expressions we derived for propagating the uncertainties from the fundamental error sources into the total uncertainty of ADCP discharge measurements from a moving boat. The total systematic uncertainty in moving-boat ADCP discharge measurements that accounts for the uncertainties from all the sources in the DRE is obtained by applying Eq. 共5兲 to Eq. 共10兲. Treating the terms in Eq. 共10兲 as uncorrelated yields an expression for the total bias limit that is simply the root-sum-square of the bias limit from each term BQ2 t = BQ2 m + BQ2 em + BQ2 et + BQ2 eb + BQ2 el + BQ2 er

共21兲

As discussed above, Eq. 共13兲 is the DRE for computing the total discharge in the measured cells. The power-distribution extrapolation and no-slip condition are the most commonly used approaches for estimating the discharge through the top and bottom unmeasured areas. Other options available for this purpose have a theoretical basis only applicable to very particular cases. Hence, we use Eqs. 共18兲 and 共19兲 as the DREs for estimating the discharge through the top and bottom unmeasured areas to develop the proposed framework. Eq. 共20兲 is the DRE for estimating the discharge through the near-bank unmeasured areas. Here, we have limited our attention to the total uncertainty of ADCP measurements by acoustical bottom tracking. Expressions for estimating the total uncertainty of ADCP discharge measurements by DGPS tracking can be derived in a similar manner. The expressions for estimating the bias limit of each term in Eq. 共10兲 should be estimated by propagating the bias limits of the contributing elemental sources applying Eq. 共5兲 to each term’s DRE. Below, we summarize the expressions for the bias limits BQm, BQem, BQet, BQeb, BQel, and, BQer. The expressions are given in terms of the partial derivatives with respect to the elemental variables to avoid lengthy equations. The analytical expressions for the partial derivatives or so-called sensitivity coefficients are summarized in Appendix II


Table 1. Biases from Elemental Sources in ADCP Measurements Biases estimation of

Accounted in reduction equations througha

Water and boat velocities, depths Water and boat velocities

† B va, B vb

Water and boat velocities Discharge through unmeasured areas High frequency velocity components Water and boat velocities, depths

† * † BC

Water and boat velocities, depths Water and boat velocities, depths Long-term means

B␪ † †

Velocities near the ADCP

Bnt

e11: Reference boat velocity e12: Depth

Water and boat velocities, depths Discharge through unmeasured areas

B vb BDa, BDp, BDp, BDo,BDavge

e13: Cell positioning e14: Rotation

Measured and unmeasured discharge Water and boat velocities, depths and geographic orientation Distances by gating and discharge Discharges through channel edges

Bt, BDa, BDp, BDo, BDavg B p, B r, B h

ADCP, setting, water properties ADCP, setup, site

Bt B ␪, B L

ADCP, speed of sound, gating time ADCP settings, bathymetry, cross section, edge distances Velocity distribution model, turbulence intensity Discharge model

Source e1: Spatial resolution e2: Doppler noise e 3: e 4: e 5: e 6:

Velocity ambiguity Side-lobe interference Temporal resolution Sound speed

e7: Beam angle e8: Boat speed e9: Sampling time e10: Near transducer


e15: Timing e16: Edge

BQ1f

Depends upon

Can be estimated from

ADCP, mode, settings, boat speed ADCP frequency, mode, settings, speed of sound, gating time Mode, settings Beam angle, settings, bathymetry Settings Water properties

End-to-end calibrationb UA of signal processing algorithms, instrument intercomparison End-to-end calibration End-to-end calibration End-to-end calibration UA of C共salinity, temperature兲 with data from reference meter Manufacturer’s specifications End-to-end calibration Instrument intercomparison based on long data records under steady conditions Experimental measurements and CFD modeling Manufacturer’s specifications UA of depths as f共C and gating time兲 and BC, Bt and BDADCP and concurrent depth range measurements 冑 Manufacturer’s specifications

ADCP Site, flow, boat operation Frequency of large-scale flow structuresc ADCP, draft, settings, velocity, flow depth Sediment concentration, flowd ADCP, settings, draft, bathymetry, water properties, time gating

Manufacturer’s specifications Manufacturer’s specifications

Field and laboratory experiments with reliable CFD-LES modeling Highly resolved data/End-to-end BQ2f e18: Discharge model calibration ADCP settings, boat velocity 冑 Discharge through measured area BQ3f e19: Finite summation Total discharge † Site, boat operation Concurrently measured data e20: Site conditions & operation a 共†兲 refers to biases that cannot be accounted for through the reduction equations; 共*兲 indicates that the bias is somewhat minimized by the ADCP processing algorithm. b In end-to-end calibrations, bias and precision limits are estimated from one or multiple sources of error at a time using repeated measurements and analyzing the statistics of the result. Unlike systematic errors that can be estimated from large samples, measurement bias must be estimated using a reference instrument of accuracy traceable to standards. c The characteristic frequency of large-scale flow structures is discussed in the text. d When boat velocity is estimated by DGPS, bias is due to DGPS positioning or DGPS velocity errors. e Biases that can be estimated by UA of time gating algorithms. f Model biases which relative contribution to total uncertainty cannot be accounted through UA based on the DREs. e17: Vertical profile model

Discharge through unmeasured top and bottom areas Discharge through measured area

冢冤 n

Ns−1

BQ2 m =

兺 i=1

兺 j=1

n−1

+

兺 j=1

冉冊冉冊冉冊兺冉冊冉冊冉冊冉冊冉冊 ⳵ Qmi,j

B2p

⳵p

⳵ Qmi,j

+

冉

⳵ Qmi,j

2

⳵ Qmi,j

Bz2i+1 +

4

兺 r=1

冉

Br2

⳵r

2

⳵ z j+1

0.15B2va

+

⳵ Qmi,j

2

+

Bz2i +

⳵ var,j ⳵ var,j+1

B2va

⳵ Qmi,j

Bt2i+1 +

4

2

⳵ var,j

r=1

2

⳵ ti+1

冊冊

⳵ Qmi,j ⳵ Qmi,j

+

⳵ Qmi,j

2

⳵ zj

B2␪

⳵␪

⳵ Qmi,j

4

2

+

B2vb

兺 r=1

冉冊 ⳵ Qmi,j

2

⳵ vbr,j

2

Bt2i

⳵ ti

Ns−1

BQ2 em

=

冤冢

⳵ Qemi,j

兺兺 i=1 j=L+1

B2p +

⳵p

+

+

B2va

⳵ Qemi,j

2

Br2 +

⳵r

BQ2 eb =

兺 i=1

r=1

冉冊 ⳵ Qemi,j

⳵ var,j−1

2

⳵ z j+1

Bz2i+1 +

⳵ Qemi,j

4

2

+

⳵ Qebi

BQ2 el =

兺 i=1

2

B2p +

⳵p

+

+

BQ2 et =

+

⳵ var,j+1

r=1

⳵ Qebi

B2vb

⳵ vbr,j−1

r=1

⳵ Qemi,j

2

Bz2i +

⳵ zj

⳵ Qemi,j

+

⳵ Qemi,j

2

Bt2i+1 +

⳵ ti+1

⳵ Qemi,j

4

2

冉冊冉冊冉冊

⳵ ti+1

⳵ Qeti ⳵p

+

+

BD2 a +

2

⳵ vbr,j+1

r=1

2

Bt2i

⳵ ti

⳵ Qeti

⳵ ti+1

+

⳵ Qeti

BD2 a +

2

Bt2i+1

Bt2i +

+

j=1

n

2

⳵r

⳵ D0

Br2

+

j=1

⳵ Qeti

2

BD2 p +

⳵ Dp

⳵ Qeti ⳵ ti

Bt2i

⳵ Qeti ⳵ D0

+

j=1

+

BD2 0 +

B2vb

⳵ Qeti

2

␦i,j

共23兲

BD2 b

j

⳵ Qeti

r=1

r=1

+

⳵ vbr,j

⳵␪

j

⳵ Qeti

B2Da g +

⳵ Db

v


⳵ Qeti

2

j

2

BD2 b

冣冣

B2␪

2

⳵ Db

2

⳵ D avg

BQ2 er =


⳵␪

j

⳵ Qeti ⳵ Qeti

4

共26兲

+

v

4

2

2

0.15B2va

⳵ Qebi

2

⳵ Qebi

B2Da g +


r=1

⳵ var,j

r=1

⳵ Qel 2 2 ⳵ Qel 2 2 ⳵ Qel 2 2 BVel + B Le + BDavg ⳵ Vel ⳵ Le ⳵ Davg ⳵ Qel 2 2 ⳵ Qel 2 2 BDADCP + B␪ ⳵ DADCP ⳵␪

冣冥

⳵ Qebi ⳵ Qebi

⳵ Qeti

4

B2va

n−1

2

i

j i

⳵ vbr,j

2

⳵ D avg

4

0.15B2va

r=1

⳵ Qebi

BD2 0 +

n−1

2

⳵ ti

2

⳵ Da

⳵ Qeti

⳵ Qebi

Bt2i+1 +

B2p

BD2 p +

+ B2vb

2

⳵ Qebi

4

2

⳵ var,j

r=1

⳵ Qebi

2

⳵ Dp

2

2

j=1

⳵ Qebi

⳵ Qebi

4

B2va

Br2 +

共22兲

computing the systematic uncertainty of the discharge estimated in cells with missing data when the power-distribution interpolation approach is used. The derivation of the sensitivity coefficients of the DRE for Qme based on power-distribution interpolation is similar to the derivation of the expressions for top and bottom power-distribution extrapolation given in Appendix II

n

2

⳵r

2

⳵ Da

⳵ Qebi

⳵ Qebi

冉冊冉冊冉冊冉冊冉冊

+

⳵ Qemi,j

4

2

冉冊冉冊兺冉兺冉冊兺冉冊冊冉冊冉冊冉冊冉冊冉冊冉冊冉冊冉冊兺冉兺冉冊冊冉冊冉冊兺冉兺冉冊兺冉冊冊冉冊冉冊冉冊冉冊冉冊冉冊冉冊冉冊兺冉兺冉冊冊

冢冢

Ns−1

B2␪

⳵␪

where L and H⫽lowest and highest bins with measured data for ping i, and ␦i,j = 0 if measured data are available for bin 共i , j兲, otherwise, ␦i,j = 1. Eq. 共23兲 applies when missing bin data are estimated by linear interpolation of the scalar triple product from the neighboring upper and lower bins, which is used in the RDI’s algorithm when extrapolation towards the bottom layer is based on the no-slip condition. A different expression should be used for

Ns−1

2

冉兺冉冊兺冉冊冊冉兺冉冊兺冉冊冊 ⳵ Qemi,j

4

H−1

⳵ Qemi,j

2

␦i,j

j

where ␦i,j = 1 if measured data are available for bin 共i , j兲, otherwise, ␦i,j = 0

冉冊冉冊冉冊

冣冥

2

B2␪

共24兲

i

i

冉冊冉冊冉冊冉冊冉冊 ⳵ Qer 2 2 ⳵ Qer 2 2 ⳵ Qer BVer + BLer + ⳵ Ver ⳵ Ler ⳵ Davgr

+

⳵ Qer 2 2 ⳵ Qer 2 2 BDADCP + B␪ ⳵ DADCP ⳵␪

共25兲

2

BD2 avg

r

共27兲

The bias limits in Eqs. 共22兲–共27兲 stem from variables measured or calculated in ADCP transect measurements. In deriving these equations, we assume that the biases due to spatial sampling of the water velocities measured in contiguous cells within a ping have a correlation of 0.15 共Simpson 2001; Gordon 1996兲. The water and boat velocities in Eqs. 共22兲–共27兲 are in radial velocities 共beam coordinates兲. In boat-mounted ADCP measurements, users rarely chose the option of radial coordinates. Fortunately, velocities can be easily transformed from earth, ship, or instrument coordinates to radial velocities as described in Appendix I. The systematic uncertainties of water and boat radial velocities with respect to the ADCP can be estimated by using the manufacturer’s specifications for horizontal water and boat velocities, because the specifications are given for measuring conditions in which pitch and roll are negligible. The systematic uncertainties of other ADCP measured data can be estimated from the manufacturer’s specifications 共RDI 2005兲. Eqs. 共22兲–共27兲 provide estimates of the systematic uncertainties in each ADCP transect measurement around the measured values in each ping and cell. The systematic uncertainty of ADCP measurements estimated from several transects can be obtained as the average of the estimates of the systematic uncertainties for each transect. Eq. 共20兲 accounts for the systematic uncertainties from elemental sources propagated through the DREs used by the ADCP. Uncertainties from sources not directly accounted for in the DREs need to be estimated from available uncertainty information, specially designed field and laboratory experiments, or quantified separately through rigorous end-to-end calibrations. These uncertainties can then be combined based on RSS with the total uncertainty of the elemental variable at the level at which the bias is estimated. For example, the near-transducer bias induced by ADCP-flow interaction biases the velocities measured in cells near the transducers and varies with mean flow 共V兲 and distance to transducer 共D兲, i.e., Bnt = B共V , D兲. Hence, if the discharge in the top unmeasured areas were to be estimated with data measured by the ADCP in the near-transducer disturbed region, the total bias of the top discharge will depend upon both the near-transducer bias, Bnt, and the bias in measuring the water velocities with respect to the ADCP, Bva. These two uncertainties combined through RSS, BvaT共V , D兲2 = B2v + Bnt共V , D兲2, should replace Bva, in Eq. 共25兲. a The total precision limit, PQt, for an ADCP discharge measurement from a moving boat can be estimated with the standard deviation, SQt, of the sample of M replications using Eq. 共7兲 with t = 2 for M ⱖ 10 and t⫽student’s t for 95% confidence level with M − 1 degrees of freedom, i.e. P Qt =

tSQt

冑M

共28兲

The total uncertainty of ADCP discharge measurements from a moving boat is then computed as the root sum square of the total bias and precision limits as

冑

UQt = BQ2 t + PQ2 t

共29兲

The equations provided here suffice for estimating the ADCP discharge measurements within a 95% confidence level, provided that estimates of the uncertainties from the elemental error sources are available. The analytical framework we propose is able to account for the directionality of the flow and is suitable to properly include correlated biases. It can be applied to: 共1兲 estimate the total uncertainty in moving-boat ADCP discharge measurements by ac-

counting for the uncertainties from the source represented through the variables in the DREs using available bias-limit estimates until better estimates become available; 共2兲 assess the sensitivity of the total uncertainty to variations in the uncertainties of the elemental error sources; and 共3兲 estimate the uncertainty percentage contribution from each uncertainty source.

Conclusions In this paper we introduced a standardized framework for uncertainty analysis of discharge measurements in open channels from a moving boat by a four-transducer, broadband ADCP with Janus configuration. The framework, developed following the sound analytical, engineering, and statistical techniques recommended in the AIAA UA standard, applies to the DREs given by the algorithms most commonly used in moving-boat ADCP discharge measurements by acoustical bottom tracking. We identified relevant elemental uncertainty sources in the ADCP measurement process and discussed their propagation into the total uncertainty of the discharge measurement. The framework is presented in the form of analytical expressions derived by propagating the uncertainties from the fundamental sources into the total ADCP measurement and the necessary analytical expressions for computing the sensitivity coefficients. The systematic uncertainties from the various sources in the ADCP measuring process identified in the DREs can be estimated from available manufacturer’s specifications, existing data, and by propagating the uncertainties from more fundamental variables measured by ADCPs into intermediate variables. We hope that since the framework accounts for the uncertainties of all the quantities measured and computed by the ADCP system, manufacturers may feel motivated to provide specifications for all the quantities directly measured by ADCPs so that users have a reliable source to estimate bias limits of the measured quantities for estimating total ADCP measurement uncertainty. Estimates of ADCP measurement uncertainty will help ascertain the total uncertainty of practical applications of ADCP discharge data that range from validation, calibration, and verification of hydrodynamic models to indirect flow monitoring necessary for water resource management. For example, propagating the uncertainty of ADCP discharge measurements into the total uncertainty of flow ratings calibrated with this data, will eventually allow us to estimate the uncertainty of flow monitoring records. The sensitivity of the total uncertainty of ADCP discharge measurements to the uncertainties from elemental error sources as well as their relative contribution can also be assessed with the proposed framework. Estimates of the relative contributions from independent sources to the total uncertainty will help identify the instrument components, measurement procedures, or measurement environment conditions that need special attention to help minimize their contribution to the total ADCP measurement uncertainty. Estimating the uncertainty from sources that contribute to the total ADCP discharge measurement uncertainty by end-to-end tests must be done in well-controlled measurement environments observing experimental repeatability. A full-fledged, comprehensive uncertainty assessment comprising all uncertainty sources in ADCP measurements requires the collaboration of manufacturers and specialized ADCP users. The collective effort will build upon existing ADCP discharge measurements quality-assurance guidelines by including a robust approach for reporting ADCP measurement uncertainty. The reliability of ADCP measurement un-


Fig. 5. Definition of ADCP instrument coordinate system with respect to its transducers

共30兲 certainty estimates will improve as uncertainty estimates from the various sources assessed through well-controlled field and laboratory end-to-end calibrations and experiments, and a more comprehensive set of manufacturer’s specifications for the quantities measured by ADCPs become available. The framework we presented can be programmed as a software package with a format similar to those for earlier ADCP UA efforts 共e.g., Simpson 2001; Kim et al. 2005兲. The software should be user-friendly to make the UA computations easy and provide graphical and numerical outputs that efficiently summarize the uncertainties in the measurement process.

Acknowledgments This work is the result of cooperative efforts by the South Florida Water Management District and the IIHR-Hydroscience & Engineering, The University of Iowa led by the writers. Special thanks are due to Dr. Kwonkyu Yu and Dongsu Kim, former and current students at the University of Iowa for their involvement in earlier versions of the ADCP uncertainty framework, and to Rodrigo Musalem from the South Florida Water Management District for assisting in preparing Appendix I. The first writer gratefully acknowledges Robb Startzman and Matahel Ansar from the SFWMD’s Scada and Hydro Data Management Department for their encouragement and support to work on ADCP measurement uncertainty. He also thanks his former colleagues at the USGS Illinois District for introducing him into hydroacoustics and its applications. The technical support by RDI, particularly that provided by Dr. Peter Spain in answering questions on instrument specifications, is gratefully acknowledged.

in which ␪⫽beam angle, typically 20°; v1, v2, v3, and v4⫽radial velocities; ui and vi⫽velocities in the instrument’s x and y directions, respectively; and w12 and w34⫽vertical velocity estimates independently obtained from the radial velocities along Beams 1 and 2, and Beams 3 and 4, respectively. ADCPs also calculate a so-called error velocity, e, with w12 and w34, as

共31兲 This “error velocity” is orthogonal to u, v, and w and has a magnitude equal to the mean of the magnitudes of u and v. Because of this normalization, in horizontally homogeneous flows, the ‘variance of the error velocity is thought to represent the portion of the variance attributable to the random error induced by instrument noise. However, it might also include the variance induced by low-frequency turbulence in the measurement environment. After applying Te above, ADCPs transform the velocity from instrument coordinates to ship coordinates by correcting for pitch, p, and roll, r, as

Appendix I. Coordinate Systems in ADCP Discharge Measurements from Moving Boat and Transformation Matrices

共32兲

ADCPs measure the water and boat velocities with respect to the ADCP in so-called beam or radial velocities 共see Fig. 5兲. Radial velocities are positive when pointing to the transducers. ADCPs apply the following transformation to transform from radial velocities to instrument coordinates, i.e.

By default, ADCPs output velocities in ship coordinates. The last transformation rotates the horizontal velocities in ship coordinates to earth coordinates due to heading, h. In earth coordinates, the x axis is positive towards the east and the y axis is positive towards the north. ADCP systems do this transformation by applying


共36兲

共33兲

These transformations apply to both water or bed velocities.

Appendix II. Sensitivity Coefficients to Propagate Uncertainties from Elemental Variables into Total Uncertainty of ADCP Discharge Measurements

Transformation from Ship to ADCP Beam Coordinates ADCP standard output velocity data in ship coordinates can be transformed back to radial coordinates sequentially because the inverse forward transformations are nonsingular. This can be done by first transforming velocities from ship coordinates to instrument coordinates using the inverse transformation 关PR兴−1

In this appendix, we summarize the sensitivity coefficients for propagating the systematic uncertainties from elemental variables into the uncertainty of ADCP discharge measurements. We use matrix and vector notation to give the coefficients in a concise manner. Sensitivity Coefficients for Qm The sensitivity coefficients of the DRE for computing discharge through a measuring cell in ADCP measurements from a moving boat are given by the partial derivatives of Eq. 共13兲 with respect to pitch, p, roll, r, beam angle, ␪, and water and boat velocities with respect to the ADCP in radial velocities as summarized below

⳵ Qmi,j

=

⳵␳ 共34兲

Second, we append the error velocity, e, described above, to the velocity vector in instrument coordinates. Third, we transform the extended four-component velocity vector in instrument coordinates to radial velocities as

冉冉

冊冊

⳵ Va ⳵ Vb ⫻ Vb + Va ⫻ ·k ⳵␳ ⳵␳

共z j+1 − z j兲共ti+1 − ti兲 i+1,j

共37兲 where ␳ represents pitch, p, roll, r, measured during ping i + 1, or beam angle, ␪

⳵ Qmi,j ⳵ v ar ⳵ Qmi,j ⳵ v br

=

=

冉冉冉冉

冊冊冊冊

⳵ Va ⫻ Vb · k ⳵ v ar

i+1,j

⳵ Vb ·k ⳵ v br

i+1,j

Va ⫻

共z j+1 − z j兲共ti+1 − ti兲

共38兲

共z j+1 − z j兲共ti+1 − ti兲

共39兲

where subscript r refers to the radial components 1–4

⳵ Qmi,j ⳵ ti+1

=−

⳵ Qmi,j ⳵ ti

= 共共Va ⫻ Vb兲 · k兲i+1,j共z j+1 − z j兲 ⬅

Qmi,j 共ti+1 − ti兲共40兲

⳵ Qmi,j ⳵ z j+1

=−

⳵ Qmi,j ⳵ zj

= 共共Va ⫻ Vb兲 · k兲i+1,j共ti+1 − ti兲 ⬅

Qmi,j 共z j+1 − z j兲共41兲

The derivatives of the velocities are

共35兲

If the velocities are in geographic coordinates, we need to apply the inverse transformation from earth to ship coordinates as follows

⳵ Vs 1 ⳵ P = RTVr ⳵p 4 ⳵p

共42兲

⳵ Vs 1 ⳵ R = P TVr ⳵r 4 ⳵r

共43兲

⳵T ⳵ Vs 1 = PR Vr ⳵␪ ⳵␪ 4

共44兲


⳵ Vs 1 = PRTi ⳵ vri 4

共45兲 T=

where V⫽water velocity or boat velocity and the subscripts s and r refer to ship and radial coordinates, respectively; T⫽submatrix of Te obtained by dropping the last row, the subscript i refers to radial velocities 1–4; and Ti represents column i of matrix T. The matrices ⳵P / ⳵ pRT, P ⳵ R / ⳵rT, PR ⳵ T / ⳵␪; and PRTi can be obtained by multiplying the following matrices

冤

cos r 0 sin r PR = sin p sin r cos p − sin p cos r − cos p sin r sin p cos p cos r

⳵T 1 = ⳵␪ 4

冥

冤

0 0 0 ⳵P R = cos p sin r − sin p − cos p cos r ⳵p sin p sin r cos p − sin p cos r

共46兲

冤

1 4

−

冤

2 2 − sin ␪ sin ␪ 0

0

1 cos ␪

1 cos ␪

冤

− sin r 0 cos r sin p cos r 0 sin p sin r − cos p cos r 0 − cos p sin r

0

sin ␪ cos2 ␪

sin ␪ cos2 ␪

D p + D0 2 Dp Z2 = Davg − Db − 2

冥

共47兲

⳵␳

=

兺

冥

Qebi =

0

0

2 cos ␪ 2 cos ␪ − 2 sin ␪ sin2 ␪ sin ␪ sin ␪ 2 cos ␪ cos2 ␪

冥

共50兲

− t i兲


b+1 共Zb+1 2 − Z1 兲

共48兲

m

DaZb+1 1 共ti+1 − ti兲

冉冊冉冊

for Water Modes 5, 8, and 11

冊冊

⳵ Va ⳵ Vb ⫻ Vb + Va ⫻ ·k ⳵␳ ⳵␳

共共Va ⫻ Vb兲 · k兲 j 兺 j=1

共51兲

b+1 共Zb+1 2 − Z1 兲

for Water Modes 1 and 12 共52兲 for Water Modes 5, 8, and 11

where ␳ represents either pitch, p, or roll, r, measured at time ti+1

for Water Modes 1 and 12

冉冉

冥

m

where n⫽number of cells with bad data directly above the last bin that would be unaffected by side lobe interference. Using these equations, we can obtain the partial derivatives of Qti with respect to pitch, p, roll, r, beam angle, ␪, water and boat velocities with respect to the ADCP in radial coordinates, times ti and ti+1, transmit pulse length, D p, lag between transmit pulses or correlation lag, D0, depth cell 共bin兲 size, Da, average measured depth from ADCP transducers, Davg, and blank beyond transmit, Db. These partial derivatives are

⳵ Qebi

1 cos ␪

共49兲

The reduction equations for bottom extrapolation by power fit and for estimating Z1 and Z2are

⳵ Qebi ⳵␪

共53兲

m DaZb+1 1 共ti+1 − ti兲 b+1 共Zb+1 2 − Z1 兲 j=1

2 sin ␪

Sensitivity Coefficients for Qeb

共D p + D0兲 1 + n − Da 2 2 1 Dp + n − Da Z1 = Davg共1 − cos ␪兲 + 2 2

Z2 = Davg − Db −

2 sin ␪

1 cos ␪

0

⬅

Z1 = Davg共1 − cos ␪兲 +

−

2 cos ␪ 2 cos ␪ sin2 ␪ sin2 ␪

DaZb+1 1 共ti+1

⳵R = P ⳵r

0

0

=

DaZb+1 1 共ti+1 − ti兲 b+1 共Zb+1 2 − Z1 兲

+

冉

m

兺 j=1

冉冉

⳵ DaZb+1 1 共ti+1 − ti兲 b+1 ⳵ ␪ 共Zb+1 2 − Z1 兲

冊冊

⳵ Va ⳵ Vb ⫻ Vb + Va ⫻ ·k ⳵␪ ⳵␪

冊兺

j

m

共共Va ⫻ Vb兲 · k兲 j

共55兲

j=1

The second term of Eq. 共55兲 can be further simplified to yield

⳵ Qebi ⳵␪

j

共54兲 1404 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / DECEMBER 2007

=

DaZb+1 1 共ti+1 − ti兲 b+1 共Zb+1 2 − Z1 兲

m

兺 j=1

冉冉

+ QebiDavg sin ␪共b + 1兲

冊冊

⳵ Vb ⳵ Va ⫻ Vb + Va ⫻ ·k ⳵␪ ⳵␪

冉

Zb2

1 + b+1 Z1 共Zb+1 2 − Z1 兲

冊

j

共56兲

⳵ Qebi ⳵ vai j

⳵ Qebi ⳵ vbi j

=

=

冊冊

冉冉冉冉

DaZb+1 ⳵ Va 1 共ti+1 − ti兲 ⫻ Vb · k b+1 b+1 ⳵ vai j 共Z2 − Z1 兲

The derivatives of the water and boat velocities can be computed as previously indicated

共57兲

⳵ Qebi

冊冊

= Qebi共b + 1兲

⳵ Da

DaZb+1 ⳵ Vb 1 共ti+1 − ti兲 Va ⫻ ·k b+1 b+1 ⳵ vbi j 共Z2 − Z1 兲

共58兲

⳵ Qebi ⳵ Dp

⳵ Qebi ⳵ Do ⳵ Qebi ⳵ Do

⳵ Qebi ⳵ Davg

= Qebi共b + 1兲

⳵ Qebi ⳵ Db

冉

= Qebi

冉

共b + 1兲 1 共Zb2 + Zb1兲 + b+1 2 Z1 共Z2 − Zb+1 1 兲

冉

Zb2 共Zb+1 2

1 2

n−

Z1

冉冊

1 n − Zb1 1 2 + + b+1 Da 共Zb+1 2 − Z1 兲

冉

共b + 1兲 1 共Zb2 + Zb1兲 + b+1 2 Z1 共Z2 − Zb+1 1 兲

⳵ ti+1

=−

⳵ Qebi ⳵ ti

=

−

Zb+1 1 兲

冊

冊

共62兲

⳵ Qeti

=

⳵␪

b+1 Da共Zb+1 3 − Z2 兲共ti+1 − ti兲 b+1 共Zb+1 2 − Z1 兲

⫻

共63兲

共64兲

共ti+1 − ti兲

⳵ Qeti ⳵ vai j

=

Sensitivity Coefficients for Qet The reduction equations for top extrapolation by power fit and for estimating Z3 are

兺 j=1

冉冉

Qeti =

⳵ Qeti ⳵ vbi j

=

共uavb − ubva兲 j

冉冉


⬅

共共Va ⫻ Vb兲 · k兲 j 兺 j=1

b+1 共Zb+1 2 − Z1 兲

Va ⫻

⳵ Qeti ⳵ Da

共65兲

⳵ Qeti 共66兲

⳵ Dp

Using these equations and the DREs for estimating Z2 and Z1 given by Eqs. 共52兲 and 共53兲, we can obtain the partial derivatives of Qti with respect to the fundamental variables as

⳵ Qeti

Z3 = Davg + DADCP

⳵ Qeti

=

b+1 Da共Zb+1 3 − Z2 兲共ti+1 − ti兲 b+1 共Zb+1 2 − Z1 兲

m

⫻

兺 j=1

冉冉

⳵ Do ⳵ Qeti

冊冊

⳵ Vb ⳵ Va ⫻ Vb + Va ⫻ ·k ⳵␳ ⳵␳

冉

j

Zb2 b+1 共Zb+1 2 − Z1 兲

冊

共68兲

冊冊

共69兲

冊冊

共70兲

b+1 Da共Zb+1 ⳵ Va 3 − Z2 兲共ti+1 − ti兲 ⫻ Vb · k b+1 ⳵ vai 共Zb+1 j 2 − Z1 兲

b+1 Da共Zb+1 ⳵ Vb 3 − Z2 兲共ti+1 − ti兲 ·k b+1 ⳵ vbi 共Zb+1 j 2 − Z1 兲

b+1 共Zb+1 2 − Z1 兲 m

⳵␳

冉冉

冊冊

⳵ Va ⳵ Vb ⫻ Vb + Va ⫻ ·k ⳵␪ ⳵␪

The derivatives of the water and boat velocities with respect to pitch, p, and roll, r, measured at time ti+1, beam angle, ␪, and radial velocity components are given by Eqs. 共42兲–共45兲

m

兺 j=1

共60兲

共61兲

m

Qebi,


共59兲

for Water Modes 1 and 12

+ Qeti,共b + 1兲共Davg sin ␪兲

⳵ Qebi

冊

冣

for Water Modes 5, 8, and 11

=0

共1 − cos ␪兲 Zb2 − 共1 − cos ␪兲Zb1 − b+1 Z2 共Zb+1 2 − Z1 兲

= Qebi共b + 1兲

冊

= Qebi

冢

冉冊

⳵ Db 共67兲 j

where, again, ␳ represents either pitch, p, or roll, r, measured at time ti+1

⳵ Qeti ⳵ Davg

= Qeti

冢

1 + Da

冉冊

共b + 1兲 n −

1 b Z 2 1

b+1 共Zb+1 2 − Z1 兲

冣

冉

冊

共72兲

冉

冊

共73兲

冊

共74兲

= Qeti

共b + 1兲共Zb2 + Zb1兲 Zb2 + b+1 b+1 2 共Zb+1 共Zb+1 3 − Z2 兲 2 − Z1 兲

= Qeti

共b + 1兲共Zb2 + Zb1兲 Zb2 + b+1 b+1 2 共Zb+1 共Zb+1 3 − Z2 兲 2 − Z1 兲

= Qeti共b + 1兲

= Qeti共b + 1兲

冉

共71兲

冉

Zb2 共Zb+1 3

Zb3 b+1 共Z3

−

Zb+1 2 兲

− Zb2 −

Zb+1 2 兲

−

+

Zb2 共Zb+1 2

− Zb+1 1 兲

Zb2 − 共1 − cos ␪兲Zb1 b+1 共Zb+1 2 − Z1 兲

冊共75兲


⳵ Qeti ⳵ DADCP

= Qeti共b + 1兲

⳵ Qeti ⳵ ti+1

=−

⳵ Qeti ⳵ ti

=

Zb3 b+1 共Zb+1 3 − Z2 兲

Qeti 共ti+1 − ti兲

共76兲

共77兲

The DRE for estimating the flow Qem through cell共s兲 in the middle area with ADCP missing data by linear interpolation of data measured directly above and below the cell共s兲 is

Qem j = 共z j−1 − z j+1兲共ti+1 − ti兲

冉

共uavb − ubva兲 j−1 + 共uavb − ubva兲 j+1 2

⬅ 共z j−1 − z j+1兲共ti+1 − ti兲 Sensitivity Coefficients for Qel and Qer

⫻

The reduction equation for estimating discharge through channel edges is Qe = KVeLeZe

共78兲

where the coefficient K depends on the channel edge shape; K = 0.35 for a triangular and K = 0.91 for a rectangular edge shapes, respectively; Ve⫽mean velocity near the channel edge; Le⫽distance to the channel bank; and Z3 = Davg + DADCP⫽total depth at the transect’s edge. The partial derivatives of Qe with respect to Ve, Le, Davg, and DADCP are

⳵ Qe = KZe ⳵ Ve

共79兲

⳵ Qe = KVeZe ⳵ Le

共80兲

⳵ Qem j ⳵␳

= 共z j−1 − z j+1兲共ti+1 − ti兲 +

冊

共82兲

冉冉

冉冉冉

冊冊

⳵ Va ⳵ Vb ⫻ Vb + Va ⫻ ·k ⳵␳ ⳵␳

冊冊冊

⳵ Va ⳵ Vb ⫻ Vb + Va ⫻ ·k ⳵␳ ⳵␳

j−1

共83兲

j+1

where ␳ represents either p, or roll, r, measured at time ti+1, or beam angle, ␪. The derivates of Qem j with respect to times ti and ti+1, and depths z j−1 and z j+1 are

⳵ Qem j 共81兲

=−

⳵ Qem j ⳵ ti

= 共z j−1 − z j+1兲 ⬅

Sensitivity Coefficients for Qem The method for estimating flows through cells in the middle measurable area with ADCP missing data depends on whether singlecell, ping, or bottom-tracking data are missing and varies according to the option chosen for bottom-layer extrapolation. Flow through a cell with missing data that is not a top or bottom cell can be estimated based on: 共1兲 power distribution using the data measured in all the cells above and below the cell with missing data; or 共2兲 linear interpolation between the data measured in the cells directly above and below the cell共s兲 with missing data. Linear interpolation is applied when bottom-layer extrapolation is based on the no-slip condition. The flow through cells with missing data near the top and bottom layers is estimated by extrapolation, treating these cells as if they were part of the unmeasured layer. Flow through cells comprised of a whole ping lost due to missed bottom tracking, decorrelation, or low backscatter, are estimated using the scalar triple product in the next good ensemble and the time interval between the two good bounding pings. The sensitivity coefficients for Qem based on power interpolation are not included here because of space limitations. These coefficients can be easily derived following an approach similar to that followed in deriving sensitivity coefficients of the DRE’s for top and bottom power extrapolation, Qet and Qeb. Here, we only give the sensitivity coefficients that apply to the Qem DRE that relies on linear interpolation.

共共Va ⫻ Vb兲 · k兲 j−1 + 共共Va ⫻ Vb兲 · k兲 j+1 2

The partial derivatives of Qem j with respect to pitch, roll, and beam angle are

⳵ ti+1 ⳵ Qe ⳵ Qe = = KVe ⳵ Davg ⳵ DADCP

冉

冊

⳵ Qemi,j ⳵ z j+1

冉共共

Va ⫻ Vb兲 · k兲 j−1 + 共共Va ⫻ Vb兲 · k兲 j+1 2

Qem j

共84兲

共ti+1 − ti兲

=−

⳵ Qemi,j ⳵ zj

= 共ti+1 − ti兲 ⬅

冊

冉

共共Va ⫻ Vb兲 · k兲 j−1 + 共共Va ⫻ Vb兲 · k兲 j+1 2

Qemi,j

冊共85兲

共z j−1 − z j+1兲

The derivatives of Qem water and boat velocities with respect to the ADCP in radial components are


⳵ Qem j ⳵ varl

⳵ Qem j ⳵ vbrl

= 共z j−1 − z j+1兲共ti+1 − ti兲

冉冉

⳵ Va ⫻ Vb · k ⳵ var

冉冉

Va ⫻

= 共z j−1 − z j+1兲共ti+1 − ti兲

冊冊

l

冊冊

l

⳵ Vb ·k ⳵ vbi

共86兲

共87兲

where the subscript r refers to the radial components 1–4, and the subscript l represents both cells j − 1 and j + 1 in ping i. The derivatives of the water and boat velocities with respect to pitch, p, and roll, r, measured at time ti+1, beam angle, ␪, and radial velocity components are given by Eqs. 共42兲–共45兲.

Appendix III. Expressions to Determine Depths and Layers in RDI ADCP Measurements The expressions summarized here define the depths and layers in measurements by broadband, four-transducer ADCPs manufactured by RDI 共RDI 2003兲. These depths and layers are shown schematically in Fig. 4. The depth to the center of the first bin in the measurable layer is 共88兲

Dtop = DADCP + DB

where DADCP⫽depth of the transducer face from the water surface, and DB is given as DB = Db +

D p + D0 + Da 2

共89兲

for measurements in Water Modes 1 or 12, or by DB = Db +

D p + Da 2

共90兲

for measurements in Water Modes 5, 8, and 11. In these expressions Db⫽blank beyond transmit; D p⫽transmit pulse length; D0⫽lag between transmit pulses or correlation lag; and Da⫽cell depth or bin size. The depth of the last possible bin, DLG max, where ADCPs can measure without signal contamination due to side lobe reflections is given by DLG max = 共Davg cos ␪ + DADCP兲 −

D p + D0 2

共91兲

for measurements in Water Modes 1 or 12, or by DLG max = 共Davg cos ␪ + DADCP兲 −

Dp 2

共92兲

for measurements in Water Modes 5, 8, and 11, where Davg⫽average measured depth from center of the ADCP transducers and ␪⫽beam angle. The distance to the bottom, Dtotal, is Dtotal = Davg + DADCP. The distance from the bed to the end of the measurable layer is Z1 = Dtotal − DLG −

Da 2

共93兲

where DLG⫽depth of the last bin with good data above DLG max. The distance from the bed to the free surface is Z3 = Dtotal, and the distance from the bed to the top of the measurable are is Z2 = Dtotal − Dtop +

Da 2

共94兲

The top unmeasured, middle measurable, and bottom unmeasured layers are between Z3 and Z2, Z3 and Z2, and, Z1 and the channel bed, respectively. The expression for Z1 given by Eq. 共52兲 can be obtained be expanding Eq. 共93兲and accounting by the n cells with bad data directly above the last bin expected to be unaffected by side lobe interference. The expression for Z2 given by Eq. 共53兲 can be easily obtained from Eq. 共94兲 by expanding Dtotal and Dtop as defined above.

Notation The following symbols are used in this paper: B ⫽ bias limit; b ⫽ user-specified exponent for power fit extrapolation into unmeasured areas; C ⫽ sound speed in water; Da ⫽ cell depth, bin size or length of range gate; DADCP ⫽ ADCP draft 共depth of ADCP transducers face from water surface兲; Davg ⫽ average measured depth from center of transducers. DB ⫽ distance from ADCP transducers to center of first measuring cell or bin; Db ⫽ blank beyond transmit; DLG ⫽ depth of last good bin 共i.e., last good bin in middle兲; DLG max ⫽ depth of last possible good bin 共i.e., last bin potentially free of side-lobe interference兲; D p ⫽ transmit pulse length 共time span of ADCP acoustic pulse兲; Dtop ⫽ depth of center of first bin; D0 ⫽ lag between transmit pulses or correlation lag in RDs, ds = 兩Vb 兩 dzdt ⫽ differential area along ADCP transect path; dz ⫽ differential depth; e ⫽ error velocity FD ⫽ Doppler frequency shift; Fs ⫽ frequency of transmitted acoustic wave; i ⫽ velocity profile 共pings兲 ensemble index; j ⫽ cell depth index; K ⫽ coefficient used in estimating discharge through channel edges; 0.35, and 0.91 for triangular and rectangular shapes, respectively; k ⫽ unit vector in vertical direction, positive upwards; L ⫽ distance from ADCP transect edge to channel riverbank; Ns ⫽ number of ensembles in transect; n ⫽ unit vector normal outward to differential area ds; n ⫽ number of measured velocities in ensemble; number of cells with bad data above last bin unaffected by side lobe interference; P ⫽ pitch correction matrix for transforming velocities from instrument to ship coordinates; P ⫽ precision limit; Pg共x , y兲 ⫽ DGPS position available at time ADCP collects ping i after converting from lat-long to distances;


Q ⫽ total discharge through arbitrary area s; Qeb ⫽ discharge estimated for area between ADCP measurable area and channel bed; Qem ⫽ discharge estimated from quantities directly measured by ADCP; Qer, Qel ⫽ discharge through left and right edges of channel cross-sectional area; Qet ⫽ discharge estimated for area near free surface on top of ADCP measurable area; Qm ⫽ discharge measured by ADCP; Qt ⫽ total discharge estimate from ADCP transect measurements; R ⫽ roll correction matrix for transforming velocities from instrument to ship coordinates; R ⫽ general functional representation of data reduction equation; ¯r ⫽ average result from M tests; r, p, and h ⫽ roll, pitch, and heading angles, respectively; Sr ⫽ standard deviation of distribution of single estimates r; T ⫽ submatrix of Te obtained by dropping Te’s last row; T ⫽ total transect time; Te ⫽ transformation matrix to convert velocities in radial coordinates to instrument coordinates; t ⫽ Student’s t for M − 1 degrees of freedom at two-sided 95% confidence level; U ⫽ total uncertainty; Ur ⫽ total uncertainty estimate for average result ¯; r Va = 共ua , va兲 ⫽ water velocity vector with respect to ADCP; Vb = 共ub , vb兲 ⫽ boat velocity vector; 兩Vb 兩 dt ⫽ differential length along ADCP transect path; Ve ⫽ mean velocity at channel edge e; V f = 共u f , v f 兲 ⫽ water velocity vector; Vr ⫽ water velocity or boat velocity in beam 共radial兲 coordinates; Vs ⫽ water velocity or boat velocity in ship coordinates; vb1 , vb2 , vb3 , vb4 ⫽ bottom-tracking velocities in ADCP radial coordinates; v1 , v2 , v3 , v4 ⫽ water velocities in ADCP radial coordinates 共along-beam coordinates兲; w12, w34 ⫽ vertical velocities estimated from radial velocities along Beams 1 and 2, and Beams 3 and 4, respectively; ¯X ⫽ average of sample measured values; Xi ⫽ measured variables; Ze ⫽ mean depth at channel edge e; Z1 ⫽ distance from channel bed to free surface during ADCP ping or ensemble; Z2 , Z3 ⫽ distances from bed to bottom and top of measurable area in ADCP ping or ensemble; zL共t兲 and zU共t兲 ⫽ lower and upper limits of portion of water column at time t; ␦Pg ⫽ distance interval between consecutive positioning updates in DGPS tracking;

␦ta , ␦tg ⫽ time intervals between two consecutive ADCP and DGPS pings, respectively; ␦z ⫽ depth interval defined by ADCP cell or bin height; ␪ ⫽ ADCP beam angle; and ␪i = ⳵r / ⳵Xi␪i, ⫽ sensitivity coefficients for reduction equation r with respect to Xi.

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