Regression Analysis of Phosphorous Loading Data for the Maumee River, Water Years Charlie Piette David Dolan Pete Richards Department of Natural and Applied Sciences University of Wisconsin Green Bay National Center for Water Quality Research, Heidelberg College

Phosphorus and the Great Lakes Water Quality Agreement Goal for reduction Initial targets Secondary targets

Maumee River Watershed 5

Maumee River Facts Size Contribution

Data Source USGS NCWQR Used data from WY

Purpose of Our Research ECOFORE 2006: Hypoxia Assessment in Lake Erie Estimate TP loads to Lake Erie using data from Heidelberg College and effluent data from permitted point sources Constructing a daily time series of phosphorus loading (Maumee River)

Problems in Constructing a Time Series for the Maumee Missing data All three years missing some data No major precipitation events were missed in water years 2003 and ……..

Water Year 2005 Data Overview Missing an important time period December 2004-January 2005, moving the lab Very significant period of precipitation 32.8 inches of snow in January ’05 Third wettest January on record Warm temps- 52˚F on New Year’s Day

Importance of WY 2005 Fifth largest peak flow in 73 year data record- 94,100 cfs Orders of magnitude larger than average flows for the same time period in WY ’03 and ’04 3,437cfs and 10,039 cfs respectively Need to model the missing data to complete the time series

Objectives Use statistical analysis to develop a model for predicting missing T.P. for the Maumee in WY 2005 Calculate an annual load for WY 2005 using measured and predicted data Compare estimated regression load to estimated load from another method Assess effectiveness of final regression model on other Lake Erie Tributaries

Reconstructing the Missing Concentration Data Multiple regression w/ SAS Producing an equation that can be used to model for the missing phosphorus concentrations

Basic Regression Equation Y=ß о + ß 1 X 1 + ß 2 X 2 + ……… ß p X p + E The terms…..

Basic Assumption of Regression Linear relationship between dependent and independent variables

Basic Assumptions: Continued Normal distribution of residuals

So, the data is suitable for regression analysis. What makes for a strong model? Hypothesis for model significance Hypothesis for parameter estimate significance P-values- <.05 R 2 value M.S.E.

Beale’s Equation

Beale’s Ratio Estimator Daily load for sampled days Mean daily load Flow-adjusted mean daily load Bias-corrected X 365 = annual load estimate DateFlowP_Concentration 10/1/ /2/ /3/ /4/ /5/ /6/ /7/ /8/ /9/ /10/ /11/

Beale Stratified Ratio Estimator Stratification- flow or time More accurate estimation “It’s an art!”

Beale Vs. Regression Both a means to the same end- annual load estimate Both relying on one main assumption- a linear relationship Big difference- Beale is not good for reconstructing a time series

Regression Analysis

Data Analysis Step 1 Transforming the data to log space

Regression Model 1 Log P-Conc = b 0 + b 1 (Log Flow) + error Most simple model Historical use

Regression Model 2 Log P-Conc = b 0 + b 1 (Log Flow) + b 2 (Season) + error Addition of second independent variable “Season” Dual Slope Analysis

Purpose of adding “Season”

Regression Model 3 Log P-Conc = b 0 + b 1 (Log Flow) + b 2 (Season) + b 3 (Season Effect) + error Addition of “Season Effect” Interaction variable

Purpose of adding “Season Effect” Interaction b/w two independent variables Slope adjustment Change in log TP concentration per unit flow during the winter season

Results of Regression Models for the Maumee, WY 2005

Selecting the Best Model for WY 2005 Model 1 Results InterceptLog Flow Overall ModelMean Square Estimate R² SignificanceError P-Value<.0001

Selecting the Best Model for WY 2005 Model 2 Results InterceptLog FlowSeason Overall ModelMean Square Estimate R²SignificanceError P-Value< <.0001

Selecting the Best Model for WY 2005 Model 3 Results InterceptLog FlowSeasonSeas. Effect Estimate R²Mod. SigMSE P-Vals.< <.0001

Results of Regression Model 3 for the Maumee, WY

Model 3: Viable Option? Looked like a good choice for WY 2005 Ran with WY data

WaterInterceptLog FlowSeasonSeason Effect Mod. YearEstimate R²Sig P-values< < P-values< P-values< <.0001

Estimating an Annual TP Load Using Regression Results

Estimating an Annual Load With Regression Used Model 3 Need to bring the log TP concentrations out of log-space (back-transforming) Back-transforming bias and estimated concentrations

Bias Correction To make up for the low bias…. Total Phosphorus Concentration (ppm) = Exp[LogPredicted P Concentration + (Mean Square Error *.5)] Estimating annual TP load from both measured and estimated data Couple conversion factors……Annual Estimated Load in metric tons/year

What did We Find???

Major Purpose of Our Research The main objective- developing a daily time series for accurately estimating an annual load for the Maumee in 2005

How did the Regression Estimates Compare to the Beale Estimate? 95% Confidence Intervals WaterRegression EstimateBeale Estimate95% Confidence Year(Metric Ton/Year) Interval

The Discrepancy

Problem with Regression Under-prediction Low-flow bias

Future Directions Improving the regression model Other independent variables More years

Thank You Any Questions?