[Fwd: [Acoustics] Developing Rating Equations via Multiple Linear Regression]

[Arthur R. Schmidt, P.E., Ph.D.]

Hi, Dave:
these are available on-line
1) Patino and Ockerman, 1997, Computations of mean velocity in open 
channels using acoustic velocity meters:  USGS Open-File Report 97-220
http://pubs.er.usgs.gov/usgspubs/ofr/ofr97220
2) Hittle, Patino, and Zucker, 2001, Freshwater flow from estuarine 
creeks into north-eastern Florida bay: USGS Water-Resources 
Investigations Report 01-4164
http://fl.water.usgs.gov/Abstracts/wri01_4164_hittle.html

Problem you're seeing is that streams never went to any of our fine University or USGS courses to learn that they are supposed to follow a linear rating relation.  It is well established that some relation between the distance from the streambed and the relative velocity (e.g., v / vmax) can be defined.  The format of this relation is not as clearly understood, but the log-law or power law are commonly used.  When we do a conventional measurement (six-tenths or two- and eight-tenths method or similar methods) we are assuming that stream is following a well-defined relation in the vertical.  In many cases where we use velocity indexing the stream is strongly influenced by backwater and tends to have a relatively small range of stages but a wide range in flows.  In this case we are in a fairly small range of the vertical velocity distribution and a relation of Vmean = a + b*Vindex tends to work fairly well.  In some cases, particularly where there is a larger range of stage, we are moving along the vertical-velocity distribution so the above simplem relation starts to show more error than we like.  In a case such as this, a second term as a function of stage often gives excellent results.  What this essentially does is change the mean velocity from a constant times the Vindex to allowing the constant to change at different depths--essentially allowing some tracking of the velocity relation in the vertical.  

In more complex cases (e.g., compound channels, channel/floodplain systems) this is still not enough.  In a channel with variable roughness/resistance around the cross section, the simple explanation I gave falls apart, because the theoretical vertical distribution of velocity is altered by the changes in resistance around the perimeter.  Likewise, vegetation can introduce a time-varying component to the rating.   In some floodplain situations, the index-velocity rating can actually cuve back on itself because of the effect of the floodplain.  

WHile a multiple regression (e.g., polynomial or other explanatory variables) will improve the goodness of fit statistics, it may not reflect the physics of your site.  I personally am a big fan of trying to understand what processes at your site are affecting the relation between the mean velocity and the index velocity you measure in a subset of the cross section.  Then you can try to define a more physically based (and hopefully more robust to conditions outside the range measured) rating relation.

I hope this is helpful to you.
Sincerely
Art SChmidt

-- 
*************************************************************************
Arthur R. Schmidt, P.E.
Research Assistant Professor
Department of Civil & Environmental Engineering
University of Illinois at Urbana-Champaign         voice: (217) 333-4934
2535a HydroSystems Lab, MC-250                       fax: (217) 333-0687
205 N. Matthews Ave                             email: aschmidt at uiuc.edu
Urbana, IL 61801           http://cee.uiuc.edu/people/aschmidt/index.htm


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