[Acoustics] Developing Rating Equations via Multiple Linear Regression

Mon Apr 21 11:35:14 CDT 2008

Dave,
I see that you already got he links to these reports in several replies, 
so let me just briefly describe some of what we deal with in south Florida 
- - 

As much as possible we try not to install instruments at locations that 
require complex ratings, but as you know, it is not always possible
In south Florida we have an extensive network of man-made canals with 
control structures (pumps, gates, etc.) and therefore can have the same 
stage conditions with and without flow
Complex field conditions can also result in having different ratings for 
positive and negative flow... at times including not only the Vel-X 
component, but both Vel-X and Vel-Y, or the X for one direction and the Y 
for the other... we have used these scenarios at some of our coastal sites 
where we had limited choices as to where to locate the station
I have learned that there is no one magic way to rate Index stations... if 
relocating your instruments is not a choice, you may want to try different 
variables until you get an equation that fits your data, one that can be 
verified with subsequent measurements... if you have a multi-cell unit, 
try including data for each individual cell, or the average of 2 (or more) 
for Vel-X and Vel-Y, until you get what works best for your site 
conditions

On the equation below - -
It has allowed us to improve some ratings by using stage in the equation, 
while not producing "significant" velocity when the index is zero (other 
than the value of the constant, which hopefully is near zero)... using 
stage as an independent variable at these sites can result in significant 
velocity when we have a full canal and gates closed or pumps off, 
something we want to avoid...
The equation represents a rating with a stage-dependent variable slope...  
 Vmean = Vi * (a+ b*Stage) + c, ... 

Hope this helps,
Eduardo

Hydrologist
U.S. Geological Survey
Florida Integrated Science Center - Ft. Myers
Royal Palm Square
1400 Colonial Blvd., Suite 70
Fort Myers, FL, 33907
(239) 275-8448
(239) 728-8774, cell
(239) 275-6820, FAX
epatino at usgs.gov

"Huston, Dave" <dwhuston at water.ca.gov> 
Sent by: acoustics-bounces at simon.er.usgs.gov
04/16/2008 04:47 PM

To
<acoustics at simon.er.usgs.gov>
cc

Subject
[Acoustics] Developing Rating Equations via Multiple Linear     Regression

Hey All,

First off, I'm wondering if anyone out there can send me electronic copies 
of the following USGS reports:

1) Patino and Ockerman, 1997, Computations of mean velocity in open 
channels using acoustic velocity meters:  USGS Open-File Report 97-220
2) Hittle, Patino, and Zucker, 2001, Freshwater flow from estuarine creeks 
into north-eastern Florida bay: USGS Water-Resources Investigations Report 
01-4164

I've read a couple of papers that discuss the use of multiple linear 
regression for sites where mean channel velocity may not be only function 
of index velocity but also of stage.  This might happen particularly when 
water level changes are large relative to the mean cross-sectional depth. 
Morlock et al. 2002 - USGS Report 01-4157 and Sloat & Hull, 2004, 
Sontek/YSI both reference the above papers.  We have a couple sites that 
seem to fit into this category.  Both of these papers express a common 
form of a regression model for these conditions as:

    Vmean = a*Vindex + b*Vindex*Stage + c,
        where a, b, and c are coefficients determined via regression.

We have a particular site that doesn't fit the basic regression models 
(linear or quadratic equations) nor does it fit the above model.  However, 
in a discussion with a reputable water resources statistician, he 
mentioned that if one uses "higher order" terms (e.g. Vindex*Stage) within 
a multiple linear regression, that the basic term of the higher order form 
should also be used.  So, for our troublesome site, I ran a multiple 
linear regression of the form:

    Vmean = a*Vindex + b*Vindex*Stage + c*Stage + d,
        where a, b, c, and d are coefficients determined via regression.

This provided a much better regression model for this site based on 
several measures of quality: adjusted R2, p-statistics, SSE, VIF's, the 
PRESS, Mallow's Cp, and the data are normal based on a probability plot of 
the residuals.  My last question is then, has anyone ever used this form 
of regression equation?  I've never seen it in the literature.  That's 
partly why I'm hoping to read the two papers I'm requesting.

Thanks All,
Dave 

************************************************
Dave Huston
Water Resources Engineer, P.E.
CA Dept. of Water Resources
Division of Planning and Local Assistance
Central District
901 P Street, Room 313A
Sacramento, CA. 95814
(916) 651-0762 (W)
(916) 651-0766 (Fax)
************************************************

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