Regardless of the method (LSPIV or velocity radars) used to meas= ure surface-water velocities, computing a discharge requires:

- Mean-channel velocity
- Cross-sectional area

This post offers methods for translating surface-water velocities into a=
mean-vertical (*u*_{vertical}) or mean-channel (* u*) velocity either directly (USGS Surface-water M=
ethod, Probability Concept) or indirectly (Index Velocity Rating). Future p=
osts will address steps for (1) assessing the quality of surface-waters sca=
tterers, (2) correcting for wind drift, which can bias measurements and alt=
er surface-water velocities, (3) schemes for filtering instantaneous veloci=
ty measurements, (4) computing area, and (5) computing real-time discharge.=

It is important that when reporting *u _{avg}*, the method should account for the velocity distribution that exists at =
the transect or cross-section-of-interest. For example, if the maximum velo=
city occurs at the water surface, a logarithmic or or power law can be assu=
med; however, if the maximum velocity occurs below the water surface, a non=
-standard velocity distribution equation (e.g., Chiu velocity equation) sho=
uld be used.

__Direct Measurement:__

*USGS Surface-water Method for estimating the mean-vertical v=
elocity*

If surface-water velocities (*u _{D}*) are measured direct=
ly (LSPIV or velocity radars) and at multiple stations (25-30) from the lef=
t edge of water (LEW) to the right edge of water (REW),

*u*_{vertical}=3D u_{D}x coefficient (typ= ically ranging from .84 to .90) &n= bsp; =*Eq. 1*

This assumes the vertical-velocity profile can be characterized by a log=
arithmic or 1/6^{th} or 1/7^{th} power law (Muell=
er, 2013). Rantz et al. (1982) and Turnipseed and Sauer (2010) recommend a =
coefficient is necessary to convert a surface-water velocity to a *=
u _{vertical}*; however, these coef=C2=ADficients are generally =
difficult to determine reliably because they may vary with stage, depth, an=
d position in the measur=C2=ADing cross section. Experience has shown that =
the coefficients generally range from about 0.84 to about 0.90, depending o=
n the shape of the vertical-velocity curve and the proximity of the vertica=
l to channel walls, where secondary currents may develop causing the maximu=
m velocity to occur below the water surface. During these conditions, the c=
oefficient can exceed unity (1.0). Larger coefficients are generally associ=
ated with smooth streambeds and normally shaped vertical-velocity curves; w=
hereas, smaller coefficients are associated with irregular streambeds and i=
rregular vertical-velocity curves.

In many instances, the velocity distribution is non-standard or the maxi=
mum velocity occurs below the water surface. In these cases, an alternative=
velocity distribution equation is needed to translate a surface-water velo=
city into a * u_{avg}* (Chiu, 1989; Chiu a=
nd Tung, 2002; Fulton and Ostrowski, 2008) or

*Probability Concept Method for estimating the mean-channel v=
elocity*

The Probability Concept was pioneered Chiu (1989) and offers an eff=
icient platform for computing * u_{avg}* a=
t a cross-section-of-interest. Two parameters,

*u*_{D}=3D u_{max}/M x ln [1+ (e^{M}-1)= x 1/(1-h/D) x exp(1- 1/(1- h/D)] &= nbsp; = Eq. 2

*=CF=95 =3D**u*u_{avg / }_{max &nb= sp; &= nbsp; = &nbs= p; &n= bsp; Eq. 3}

* *

*Where =CF=95 =3D function of M (2 to 5.6) and gen=
erally ranges from .58 to .82 and*

*u _{max} =3D maximum in-stream velocity*

*u _{avg} =3D mean-channel velocity*

*M =3D entropy parameter and is related to =CF=95 =3D e^{M }/ (e^{M}-1) -1/M<=
/em>*

*u _{D} =3D surface-water velocity*

*h/D =3D location of u _{max} below the water surface at =
the y-axis/water depth at the y-axis*

*Index Velocity Rating for estimating the mean-channel veloci=
ty*

The protocol for establishing index velocity ratings are described by Le=
vesque and Oberg (2012) where an index such as *u _{D}*&=
nbsp;can be paired to a measured discharge for a variety of flow conditions=
.

Chiu, C.-L., and Tung, N.C., 2002. Velocity and regularities in open-cha= nnel flow. Journal of Hydraulic Engineering, 128 (94), 390-398.

Chiu, C.-L., Tung, N.C., Hsu, S.M., and Fulton, J.W., 2001, Comparison a= nd assessment of methods of measuring discharge in rivers and streams, Rese= arch Report No. CEEWR-4, Dept. of Civil & Environmental Engineering, Un= iversity of Pittsburgh, Pittsburgh, PA.

Chiu, C.-L., 1989, Velocity distribution in open channel flow, Journal o= f Hydraulic Engineering, 115 (5), 576-594.

Fulton, J.W., Cederberg, J.R., Best, H.R., Fulford, J.M., Mills, J.T., J=
ones, M.E., and Bjerklie, D.M., *in preparation*, SWOT-based Ri=
ver Discharge: Ground-truthing using the Probability Concept and Continuous=
-wave Velocity Radars.

Fulton, J.W. and Ostrowski, J., 2008, Measuring real-time streamflow usi= ng emerging technologies: Radar, hydroacoustics, and the probability concep= t, Journal of Hydrology 357, 1=E2=80=9310.

Guo, J. and Julien, P.Y., 2008, Application of the Modified Log-Wak= e Law in Open-Channels, Journal of Applied Fluid Mechanics, 1 (2), 17-23.

Jarrett, R.D., 1991, Wading measurements of vertical velocity profiles, = Geomorphology, 4, 243-247.

Kundu, S. and Ghoshal, K., 2012, Velocity Distribution in Open Channels:= Combination of Log-law and Parabolic-law, World Academy of Science, Engine= ering and Technology, International Journal of Mathematical, Computational,= Physical, Electrical and Computer Engineering, 6 (8), 1234-1241.

Levesque, V.A. and Oberg, K.A., 2012, Computing discharge using the inde=
x velocity method: U.S. Geological Survey Techniques and Methods 3=E2=80=93=
A23, 148 p.

(Available online at http://pubs.usgs.gov/tm/3a=
23/).

Mueller, D., 2013, extrap, Software to assist the selectin of extrapolat= ion methods for moving-boat ADCP streamflow measurements, Computers & G= eosciences, 54, 211-218.

Rantz, S.E. and others, 1982. Measurement and computation of streamflow:= Volume 1. Measurement of stage and discharge. Water-Supply Paper 2175, U.S= . Geological Survey.

Turnipseed, D.P. and Sauer, V.B., 2010, Discharge measurements at gaging= stations: U.S. Geological Survey Techniques and Methods book 3, chap. A8, = 87 p. (Also available at http://pubs.usgs.gov/tm/tm3-a8/= ).

Wiberg, P.L. and Smith, J.D., 1991, Velocity distribution and bed = roughness in high-gradient streams, Water Resources Research, 27 (5), 825-8= 38.

Yang, S.-Q., Xu, W.-L., and Yu, G.-L., 2006, Velocity distribution in gr= adually accelerating free surface flow, Advances in Water Resources, 29, 19= 69-1980.