905490001
 ESTIMATED LOADINGS FROM SEVEN MICHIGAN

   TRIBUTARIES AND RECOMMENDATIONS FOR

      TRIBUTARY SAMPLING STRATEGIES


                  by
             Robert M.  Day
     Surface Water Quality Division
Michigan Department of Natural Resources
        Lansing, Michigan 48909
            Project Officer

            David Rockwell
       Project Nos. S-005741-01-2,
       S-005741-02-1, S-005741-03
   GREAT LAKES NATIONAL PROGRAM OFFICE
               REGION V
  U.S.  ENVIRONMENTAL PROTECTION AGENCY
        CHICAGO, ILLINOIS   60605

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                                ABSTRACT









Annual load estimates of twelve parameters from seven Michigan




tributaries were calculated from 1984 to 1986.  Estimates were




calculated by dividing sample concentrations into high and low strata and




applying Scale's Ratio Estimator.  The greatest annual loads of the




twelve parameters usually came from the St. Joseph followed by the Black




or Rouge rivers and the lowest annual loads came from either the Pere




Marquette or Ontonagon rivers.









Monte Carlo studies indicate that flow stratified sampling strategies




yield unbiased and relatively precise total phosphorus load estimates




when the samples were selected randomly.  Strategies that confine




sampling to the first half of the year or neglect either the rising-area




or falling area of the hydrograph will yield biased load estimates.  A




systematic sampling strategy will insure that each sample within each




strata has an equal probability of being selected and usually yields




unbiased total phosphorus load estimates.









Sample sizes necessary to estimated total phosphorus loads were calculated




for four of the seven Michigan tributaries studied using load average and




variance predicted by flow variability versus load variability regression




equations.  This method can be used to provide sample size estimates for




many tributaries with little or no prior information about total phosphorus




concentrations but is not reliable for the most event responsive rivers.

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                            TABLE OF CONTENTS


                                                                   Page

ABSTRACT 	   i

LIST OF FIGURES	ill

LIST OF TABLES	iv

ACKNOWLEDGEMENTS 	   v

CONCLUSIONS  	   1

INTRODUCTION 	   4

PROJECT DESIGN
  -Site Descriptions 	   6
  -Flow Measurements and Estimations 	  11

ANNUAL LOAD ESTIMATES
  -Load Estimation Methods 	  11
  -Annual Loads from Seven Tributaries 	  13

SAMPLE STRATEGY DEVELOPMENT
  -Determining the Necessity of Event Sampling Strategies	18
  -Testing for Bias Introduced by Event Sampling Strategies. ...  20
  -Flow Stratified Systematic Sampling 	  25

SAMPLE SIZE ESTIMATION
  -Sample Size Estimation Method 	  28
  -Predicting Load Variability for Sample Size Calculation ....  31
  -Sample Size Estimates for Michigan Tributaries	37

LITERATURE CITED 	  42

APPENDICES
  -Appendix 1.  Project Watershed Maps	43
  -Appendix 2.  Average Daily Flow vs. Average Daily Load 	  51
                                   ii

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                             LIST OF FIGURES
Number                                                            Page

  1       River mouth stations samples by the Michigan
          Department of Natural Resources	7
          The relationship between the coefficient of
          variation (CV) of total phosphorus daily loads and
          average daily flow variability (CVLF5) 	 35

          The relationship between the coefficient of
          variation (CV) of total phosphorus daily loads and
          average daily flow variability (CVLF5) in the high
          flow strata.  .	36

          The relationship between the coefficient of
          variation (CV) of total phosphorus daily loads and
          average daily flow variability (CVLF5) in the low
          flow strata	36
                                   111

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                             LIST OF TABLES


Number                                                           Pag«

  1       Drainage area of the project watersheds 	  8

  2       Land use (%) by watershed	9

  3       Soil types (%) by watershed	9

  4       Drainage area ratios (DAR) and U.S. Geological Survey
          gaging station location by river. . .  	 12

  5       Annual loads from seven Michigan tributaries
          (metric tons per year)	14

  6       Average daily flow (cfs) , 20th percentile cut-off
          flow (cfs) and the number of days in each high flow
          and low flow strata for the Black, Clinton, Rouge
          and Huron rivers 1984-1986, and the Ontonagon,
          Pere Marquette and St. Joseph rivers,  1984	17

  7       Total phosphorus average daily load estimates
          (Kg/day) and average percent estimated bias from
          Monte Carlo-analyses for the Sandusky, Grand and
          Raisin rivers 	 23

  8       Low, high and median 95% confidence intervals
          calculated on load estimates (Kg/day), from
          Monte Carlo subsampling, for the Sandusky, Grand and
          Raisin rivers	 29

  9       Comparison between estimated sample sizes, at
          various levels of precision, calculated using
          variance and average estimates from the complete data
          set and using the CV predicted from the regression
          equation	38

 10       Predicted number of samples required per year to
          estimate total phosphorus loads with 95% confidence
          intervals less than or equal to the indicated
          precision of the estimate	40
                                   IV

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                            ACKNOWLEDGEMENTS









Funding for this project was provided by the Environmental Protection




Agency, Great Lakes National Program Office.  Event sampling programs




are labor intensive and many Michigan Department of Natural Resources




personnel helped with data collection.  Sampling was coordinated and




conducted by Rick Lundgren, Frank Horvath and Bob Wood.  Additional




assistance was provided by Jim Rossio, Greg Goudy, Dave Kenaga,




Chris Hull, Jim Young, Amy Peterson, Bruce Rabe and Rob MacLean.









Colleagues who provided comments on earlier drafts include Jim Rossio,




Dave Kenaga, Greg Goudy, Rick Lundgren and Bob Wood.  Also, R. Peter




Richards (Heidelberg College) and David M. Dolan (International Joint




Commission) provided comments on an earlier draft of this report.

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                           CONCLUSIONS









     Annual load estimates of total phosphorus, suspended solids,




ammonia, total Kjeldahl nitrogen, nitrate, calcium, sodium, silica,




sulfate, magnesium and potassium were calculated by dividing sample




concentrations into two groups, or strata, and applying Beale's Ratio




Estimator.  The sample concentrations were divided into high and low




strata with the cut-off being the historical upper 20th percentile of




flow.









     Annual load estimates calculated from 1984 to 1986 on the Black,




Clinton, Rouge and Huron rivers were variable from year to year.  Annual




loads seemed to be related to the magnitude of the average annual flows




or to the actual number of high flow days in a year.   In 1984 annual




loads were also estimated from the St. Joseph, Pere Marquette, and




Ontonagon rivers.  The greatest annual loads of the twelve parameters




usually came from the St. Joseph followed by the Black or Rouge rivers.




The lowest annual loads came from either the Pere Marquette or Ontonagon




rivers.









     Event sampling strategies yield excellent load estimates regardless




of the relationship between load and flow.  However,  event sampling is




resource intensive and not required unless loads of the constituent




increase with increasing flow.  Plots of daily average load versus daily




average flow indicate a positive relationship between loads and flow for




all twelve parameters in the Black, Clinton, Rouge, Huron, Ontonagon, and




St. Joseph rivers.  Suspended solids and ammonia in the Pere Marquette




                                    1

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River did not increase with flow and therefore loads could be estimated




with a fixed interval sampling program instead of an event sampling




program.









     The results of Monte Carlo studies indicate that random sampling is




the only way to insure that load estimates will be unbiased.  Strategies




that confine sampling to the first half of the year or neglect either the




rising arm or falling arm of the hydrograph will yield biased total




phosphorus load estimates.









     Although it is difficult to develop a completely random sampling




strategy, systematic sampling insures that each sample within each strata




has an equal probability of being selected. The results of Monte Carlo




studies indicate that in most cases systematic sampling yields unbiased




total phosphorus load estimates.









     The number of samples required to estimate loads will not be the




same for each river.  Sample size estimates were calculated using load




average and variance estimates obtained directly from the complete data




sets.  Load estimates from the Monte Carlo studies were usually within




the precision specified by the sample size estimation formula.









     The variance of total phosphorus loads can be predicted from flow




variability.  The load variability can then be used to predict the number




of high and low flow samples required from each river.  Estimated sample




sizes were calculated using load average and variance predicted by flow




variability versus load variability regression equations.  This method




                                    2

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can be used to provide sample size estimates for many tributaries with




little or no prior information about the constituent of concern but is




not reliable for the most event responsive rivers.









     Sample size estimates were calculated for four of the seven Michigan




tributaries studied using the load variability versus flow variability .




relationship.  Sample size predictions were good for the Ontonagon, Huron




and Clinton rivers but poor for the most event responsive Black river.




Low intensity sampling on the more stable Pere Marquette and St. Joseph




rivers yielded relatively precise loads supporting the contention that




rivers with stable flows generally require less intensive sampling




programs.

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                          INTRODUCTION









     The Great Lakes are the largest body of freshwater in the world and




support a variety of human activities.  They receive wastes from point




sources such as municipal and industrial facilities as well as from




non-point sources including combined sewer overflows, urban and




rural runoff, and atmospheric deposition.  Although complete mass




balances have not been conducted for all the Great Lakes, tributaries




are known contribute large amounts of certain chemical constituents.




Many of these constituents are present and required in trace amounts for




the existence of aquatic life but if present in excess can cause nuisance




conditions or toxicity problems.




    The Michigan Department of Natural Resources (MDNR) has monitored




several Great Lakes tributaries for numerous chemical constituents for




more than 30 years.  This monitoring has been used to describe trends,




identify emerging problems, document existing conditions for waste




discharge permits and estimate tributary loadings.  Tributary Loadings




have historically been calculated by multiplying average monthly flows by




a single monthly sample concentration, but indications are that although




existing monitoring was sufficient for most purposes, it was poorly




suited for calculating loads of most constituents.




     Many tributary systems are characterized by loads that are dominated




by non-point sources.  Concentrations of some parameters tend to increase




or remain relatively constant with increased flow.  Yaksich and Verhoff




(1983) reported that in several Ohio rivers, the greatest loadings




occurred during periods of high flow or high flow runoff events and




Richards and Holloway (1987) stated that in some Lake Erie tributaries as




                                    4

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much as 80% of the annual load of certain constituents was delivered




past a monitoring point during the 20% of the time that the highest flows




occurred.  In these cases where most of the annual load occurs during the




20% of the time with the highest flows the distribution of loads is




usually highly skewed.  Load estimates from monthly monitoring often




underestimate the true load from these "event responsive" rivers by 15%




to 30% (Yaksich and Verhoff 1983) .




     One sampling design that can substantially reduce load estimate




errors is flow stratified sampling.  Stratified sampling is performed by




dividing the flow into subgroups or strata and sampling from each strata.




This procedure breaks the flow into groups that are less variable than




the complete flow record.  Strata with highly variable loading rates can




be sampled more intensively than the less variable strata so that




estimate errors within each strata are minimized and precision of the




overall estimate is increased (Bierman et al 1988).  Also, precision can




be gained by forming strata so that a heterogenous flow record is divided




into fairly homogenous parts (Snedecor and Cochran 1980).




     Richards and Holloway (1987) conducted Monte Carlo studies to test




various sampling strategies and load estimation techniques using large




data sets from three Lake Erie tributaries.  Based on these studies, they




recommended flow stratified sampling with proportionately more samples




collected during periods of high flow.  They found that for event




responsive streams in Ohio, less than 50 samples per year provided




strongly biased and imprecise load estimates.  Yaksich and Verhoff (1983)




and Bierman et al (1988) also concluded that sampling strategies should




be stratified by flow in these event responsive rivers.  Yaksich and




Verhoff (1983) recommended an event sampling strategy for Lake Erie




                                    5

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tributaries that included 15 to 20 grab samples over two or three of the



largest events with 5 to 10 additional steady flow samples.  They stated



that their strategy would yield a load estimate "with a 10% to 20%



standard error.                    v



     The MDNR presently monitors the water quality of several rivers



throughout the state with a fixed station monthly monitoring program.  As



previously mentioned, one goal of that program is to provide data to



calculate annual pollutant loads to the Great Lakes from the tributaries.



This project was undertaken to sample several rivers more extensively



during high flow periods in order to obtain better load estimates and to



develop a load estimation sampling strategy that would be applicable to



Michigan rivers.







                         PROJECT DESIGN "  .



Site Descriptions and Sampling Methods



     Seven tributaries were selected for study including the Black,



Clinton, Huron, Rouge, Ontonagon, Pere Marquette and St. Joseph rivers


                                                         2             2
(Figure 1).  River watersheds ranged in size from 1201 Km  to 12,124 Km



with the smallest being the Rouge and the largest being the St. Joseph



(Table 1).  Land use and soil types varied among watersheds.  The



Ontonagon and Pere Marquette drainage basins are mostly forested and



wetlands; the Black, Huron, Clinton and St.  Joseph watersheds are



primarily agricultural; and the Rouge watershed is primarily urban and



suburban (Table 2).  All the watersheds are predominately loam soils



except for the Pere Marquette,  which is mostly sandy soils (Table 3).



Maps of the tributaries are included in Appendix 1.



                                    6

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ONTONAGON
                        PERE MARQUETTE
                                                                                 ROUGE
                                                                               HURON
                          ST. JOSEPH
 Figure 1.  River mouth stations sampled by the Michigan Department of Natural Resources.

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Table 1.  Drainage area of the project watersheds
Tributary




Rouge




Black




Pere Marquette




Clinton




Huron




Ontonagon




St. Joseph
sq.  mi




 467




 711




 740




 760




 908




1390




4681
sq.  km




 1210




 1842




 1917




 1968




 2352




 3600




12123

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Table 2.   Land use (%)  by watershed.
Watershed
Ontonagon
Pere Marquette
Black
Huron
Clinton
Rouge
St. Joseph
Urban &
Suburban
0.
0.
2.
8.
25.
73.
3.
1
6
4
2
6
4
1
Agricultural Forest &
& Range Wetlands
13
33
82
67
62
23
72
.0
.2
.8
.4
.8
.6
. 7
83
64
14
22
9
2
22
.8
.8
.8
.4
.9
.8
.9
- Inland-
Waters
3
1
<0
2
1
0
1
.1
.4
.1
.1
.7
.2
.3
Table 3.  Soil types (%) by watershed,
Watershed
Ontonagon
Pere Marquette
Black
Huron
Clinton
Rouge
St. Joseph
Clay
36.4
15.8
18.2
10.1
17.4
28.6
8.6
Loam
46.0
7.9
75.0
85.3
71.8
48.4
81.5
Sand
17.6
77.0
6.9
4.6
10.8
23.0
10.0

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     The rivers were sampled over a 3-year period during 1984 through




1986.  The study was conducted on all seven rivers in 1984, but only the




Black, Clinton, Huron and Rouge were sampled in 1985 and 1986.  Three




sampling strategies were used including (1) monthly throughout each year,




(2) weekly during the spring when flows are typically highest (scheduled




samples), and (3) twice daily during periods of high flow caused by




precipitation and/or snow melt (event samples).  Water samples collected




using monthly and scheduled strategies were analyzed for suspended




solids, total phosphorus, ammonia, total Kjeldahl nitrogen, nitrates,




chlorides, calcium, sodium, silica, sulfate, magnesium and potassium




while event samples were only analyzed for total phosphorus, total




Kjeldahl nitrogen, nitrates, ammonia, chlorides and suspended solids.




The number of high and low flow samples collected varied among rivers,




years and parameters.




     Event sampling was initiated based on weather forecasts and daily




telephone monitoring of river stage heights, measured by the United




States Geological Survey (U.S.G.S.), at gaging stations on each river.




When it was determined that an event was starting, sampling was initiated




focusing primarily on the rising arm, the peak, and initial falling slope




of the event hydrograph.  Samples were usually collected twice a day for




seven consecutive days during the high flow event.




     Surface water grabs were taken as close to the mouth of each




tributary as possible but upstream of areas influenced by seiches.




Samples were collected with a can sampler at about '30 cm below the




surface in the center of the stream in an area of high flow.  Sample




collection, handling and preservation procedures are described in "




Quality Assurance Manual for Water and Sediment Samples" (MDNR 1982




edition).





                                   10

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Flow Measurement and Estimation




     Since river sampling stations were downstream of gaging stations, it




was necessary to adjust the' flow value obtained at the gaging stations to




reflect the additional drainage areas between the gages and the sampling




stations.  To estimate the discharge at the sampling site near the




tributary mouth, a single gage reading or sum of gage readings was




multiplied by a correction factor.  The correction factor used was the




drainage area ratio (DAR), calculated by dividing the drainage area above




the sampling station by the drainage area above the gaging station.  The




estimated discharge at sampling stations in the Pere Marquette,




Ontonagon, Huron, Clinton and Black rivers were obtained directly by




multiplying the DAR by the appropriate gage reading (Table 4).  Since




there was no gage in the St. Joseph River downstream of the confluence of




either the Dowagiac or the Paw Paw rivers, the flow .at the mouth was




estimated by multiplying the DAR by the sum of the flows at the three




gages.  In the Rouge River there were no gages below the confluence of




either the Middle Rouge or Lower Rouge Rivers.  Also, the Ford Rouge




Plant continuously discharges 784 cfs of water, drawn from the Detroit




River, to the Rouge River downstream of the gaging station but approxi-




mately 2.5 miles (4 km) upstream of the water sampling station.  The Rouge




River flow estimate at the mouth was obtained by multiplying the DAR by




the sum of the three gage flows (Rouge, Middle Rouge and Lower Rouge) and




adding the 784 cfs discharged from the Ford Rouge plant.









                      ANNUAL LOAD ESTIMATES




Load Estimation Methods




     After samples have been collected with a flow stratified sampling




                                   11

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Table 4.  Drainage Area Ratios (DAR) and U.S. Geological Survey gaging
          station locations by river.
River

Pere Marquette

Ontonagon

Huron

Clinton

Black

St.  Joseph



Rouge
Gages

04122500   at Scottville

04040000   near Rockland

04174800   at Ann Arbor

04165500   at Mt.  Clemens

04159500   near Fargo

04101500   St. Joseph R. at Niles
04101800   Dowagiac R. at Summerville
04102500   Paw Paw R. at Riverside

04166500   River Rouge at Detroit
04167000   Middle  Rouge at Garden City
04168000   Lower Rouge at Inkster
DAR

1.05

1.04

1.21

1.04

1.48

1 .09



1.24
                                   12

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strategy, a variety of methods are available for calculating the annual




load.  Dolan et al (1981) tested several load estimation methods,




including means of loads' over time, regression estimators and a ratio




estimator (Beale's ratio estimator) using Monte Carlo studies with a




large data set from the Grand River in Michigan.  They found that Beale's




ratio estimator (BRE)- consistently yielded estimates with the least




bias and best precision and concluded that the BRE was the best estimator




for systems with complete daily flow records and relatively little




concentration information.  Richards and Holloway (1987) tested Beale's




ratio estimator against other estimators and also concluded that the BRE




provided the most precise and unbiased estimates.




     For calculation purposes the flows were divided into two strata.




Although the number of flow strata can be more than two, Dolan et al




(1981) and Richards and Holloway (1987) also divided the flows into two




strata.  Dolan et al divided the strata at two times the median flow




while Richards and Holloway divided the flow into the upper 20th




percentile and bottom 80th percentile.  In this study the flows were




divided into high and low flow strata with the cut-off being the




historical upper 20th percentile of flow.  In other words, the cutoff




flow was exceeded by 20% of the recorded flows and was greater than 80%




of the recorded flows.  Percentiles of flow are available (in five




percentile intervals) from U.S.G.S. flow duration analyses.









Annual Loads from Seven Tributaries




     The total annual loads of twelve constituents from seven tributaries




were calculated using the BRE and dividing the samples into high and low




flow strata  (Table 5).  In some cases, estimates of annual loads varied




                                   13

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Table 5.   Annual loads froi seven Michigan tributaries (netric tonnes per year).
River and Total
Year Phos,
Black River
1984
Load 184.0
+/-35X C.I. 32.6
nh 47
nl 25
1985
Load 225.4
+/-95X C.I, 90.3
nh 41
nl 48
1988
Lead :39,5
+/-35X C.I. 21.4
nh 45
nl 48
Clinton River
1984
Load" 100.1
+/-95X C.I. 23.5
nh 32
nl 39
1985
Load 167.4
+/-95X C.I. 36.8
nh 48
nl 41
1986
Load '.20.0
+ /-95X C.I. 20.4
nh 75
nl 27
Rouge River
1984
Load 114.2
»/-95X C.I. 16.2
nh 33
nl 36
Load 186.1
t/-35X C.L 36.2
nh 50
nl 45
1986
Load 128.6
+/-95X C.I. 13.4
nh 56
al 37

Total
Suspended
Solids Auonia


79490
22420
48
25

100900
54190
41
49

81S90
26700
43
48


25560
8242
32
41

50120
16060
48
42

36400
11890
73
27


24210
5979
44
26
41920
19930
50
46

25300
6632
54
37



226,1
163.4
7
14

170.6
45.2
41
49

257,4
57.2
45
48


97.67 •
20,84
6
14

130.1
38.3
48
42

.123.5
18.3
75
27


389.9
113.5
4
15
640.1
82.5
50
46

488.8
47.5
56
37

Total
Kjeldahl
Nitrogen


1397
46
47
25

1229
301
41
49

1371
77
48
48


671.4
,56.6
32
40

1387
131
48
42

909.8
56.5
76
27


1110
82
33
36
1858
173
50
45

1328
103
55
37

Total
Nitrate


1462
371
7
14

1317 '
2"9
41
49

1111
173
45
48


1108
218
5
14

1432
157
48
42

1184
230
76
27 -


520.2
211.7
4
15
942.8
109.1
50
46

562.7
44.2
56
37

Total
Calciai


38310
11070
7
14

32250
11260
11
13

•13350
12180
10
17


31110
1820
5
15

51820
5830
4
10

44700
5790
16
12


40830
2770
3
16
54530
3600
12
12

46630
4140
15
11
14
Total
Sodiui


8693
3165
7
14

5945
1993
11
14

7572
3180
10
17


28990
5170
5
15

44450
10920
14
11

49030
15640
16
12


29730
4700
3
16
56170
15880
12
13

42240
11490
15
11

Total
Silica


1197
153
7
14

1331
396
10
14

1521
519
1C
18


'1112
140
6
13

2093
301
i 0
11

1576
2100
17
12


1253
544
4
15
1932
209
11
13

1314
167
16
11

Total
Sulfate


36480
13420
7
14

31780
13310
12
14

28670
9860
11
18


24120
5110
5
14

42180
4930
• r
1 1
i i

27010
6200
19
12


30220
3010
4
15
44430
4180
13
13

32160
3850
18
10

Total
Cloride


16590
6460
•j
14

14530
4770
12
14

15660
2020
45
48


45540
11700
6
14

78S70
17200
15
1 1
i *

55760
3770
74
27


58000
18150
4
15
100400
23860
13
13

79560
9970
55
37

Total
Hagnesiua


11390
3430
7
14

3035
3221
11
14

10000
3570
10
17


9642
598
c
15

14560
2150
14


130SO
1860
16
12


10580
570
0
j
16
13580
900
12
13

12280
1260
15
11

Tctai
Patassiuj


3172
267
7
14

3290
414
11
14

3794
289
10
17


1875'
103
5
:s

2902
2C5
14
11

2553
109
16
12


2242
234
3
16
2637
180
12
13

3119
651
15
11


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Table 5.  continued.
River and Total
Tear Phos.
Huron River
1984
Load 35.49
+/-95X C.I. 2.48
nh 36
nl 35
1985
Load 64.39
t/-95X C.I. 7.67
nh 42
nl 53
1986
Load 47.90
+/-95X C.I. 2.85
nh 38
nl 53
Ontonagon River
1984
Load 110.2
+/-95X C.I. 29.6
nh 31
nl 80
Pere Harqaette River
1984
Load 34.00
I/-95X C.I. 3.35
nh 17
nl 8
'St. Joseph River
1984
Load 317.5
t/-95X C.I. 44.9
nh 10
ni 14
Totil
Suspended
Solids


10840
1944
36
36

23680
5386
42
54

15000
1943
38
53


• -104200
505,20
31
80


5900
1554
18
9


80850
22410
10
14
Amonia


38.34
39.89
6
15

120.8
25.5
42
54

78.39
11. §5
38
53


40.18
10.68
3
18


137.5
40.3
15
9


620.1
201.2
9
14
Total
Kjeldahl
Hitrogen


521.3
20.7
36
35

770.8
50.2
42
54

727.0
22.8
33
53


671.3
47.9
31
80


475.2
in o
J U . J
17
8


3972
297.6
10
14
Total
Nitrate


450,3
68.0
5
15

760.7
53.8
42
54

'SO. 5
43.3
38
53


154.7
42.0
7
18


143.2
19.5
15
3


7803
913
10
14
Total
Calciui


29260
1500
5
16

46260
2910
12
12

45460
2050
9
15


18550
580
10
18


3C510
1370
16
3


273900
12130
9
14
Total
Sodiui


14720
1200
5
18

22990
4050
12
13

24520
3100
3
15


4079
365
10
18


6915
645
15
3


50970
3980
9
14
Total
Silica


874,8
294.4
6
15

1714
355
11
13

1375
375
,9
16


4424
288
3
18


2750
312
15
3


11450
2750
3
14
Total
Suifate


25430
1130
6
15

43510
4020
13
13

34080
1080
10
15


8421
933
8
18


13580
705
' 5
8


204800
9800
3
14
Total
Cioride


27240
1970
6
15

43230
6500
13
13

4184C
1710
38
53


3186
378
8
18


10630
1050
16
3


93500
5520
3
14
Total
Xagnesiui


8309
510
5
16

12990
S40
12
13

12920
300
9
15


5156
378
8
18


10500
520
15
0


31720
5170
9
14
Total
Pctassiua


1258
77
5
16

1310
113
12
13

1907
111
9
15


1.712 -
423
8
18


i53,2
51.:
15
Q


3534
286
3
14
nh= umber  of high flow saiples
nl= niiiber  of low Flow saaples
                                                         15

-------
substantially among different years.  For example, the annual total




phosphorus load in the Huron River was approximately 83% greater in 1985




than 1984.  Most of the estimated loads from the Rouge, Clinton, and




Huron rivers were highest in 1985, intermediate in 1986 and lowest in




1984. This is probably related to annual average daily flows that were




also highest in 1985, intermediate in 1986 and lowest in 1984 (Table 6).




     Estimated loads from the Black River did not follow this pattern




even though the annual average daily flows did.  In the Black River, from




1984 to 1986, the lowest annual average daily flow occurred in 1984




followed by 1986 and 1985.  However, more high flow days, or days in




which the daily average flow exceeded the historical upper 20th




percentile cut-off flow of 347.8 cfs, occurred in 1984 followed by 1986




and 1985.  Estimated annual loads of total phosphorus and suspended




solids were highest in 1985 and lowest in 1986 following the pattern of




the relative magnitude of annual daily average flows.  At the same time,




estimated annual loads of total Kjeldahl nitrogen, nitrate, calcium,




sodium, sulfate and chloride were highest in 1984, the year with the most




high flow days.  It may be that the number of high flow days was more




important than the magnitude of flow on high flow days.  In such a case,




loads of some parameters may be less in years with a few large events




than in years with many smaller events.




     Estimates of annual loads from the Ontonagon, Pere Marquette and St.




Joseph rivers are available for 1984.  Fewer samples were collected in




the Pere Marquette and St. Joseph rivers compared to the other




tributaries but sampling effort on the Ontonagon River was relatively




intense.  Scheduled samples from the Ontonagon were collected by an MDNR




Conservation Officer stationed at White Pine and event samples were




                                   16

-------
Table  6.  Average daily flow  (cfs), 20th percentile cut-off  flow
           (cfs) and the number of days in each high flow and  low
           flow strata for the Black, Clinton, Rouge and Huron
           rivers, 1984-1986, and the Ontonagon, Pere Marquette and
           St. Joseph rivers, 1984.
Flow
River Year Cut-off
Black
1984
1985
1986
Clinton
1984
1985
1986
Rouge
1984
1985
1986
Huron
1984
1985
1986
Ontonagon
1984

347
347
347

713
713
713

1143
1143
1143

904
904
904

1518

.8
.8
.8

.1
.1
.1





.9
.9
.9


High
Flow
Days

143
109
129

65
140
140

67
129
117

62
99
71

62
Low
Flow
Days

223
256
236

301
225
225

299
236
248

304
266
294

304
Ave .
High
Flow

1633
2927
2195

1472
1971
1323

1731
1877
1654

1187
1671
1408

3602
Ave .
Low
Flow

140
97
131

342
356
439

932
953
979

350
444
552

924
Annual
Ave .
Daily
Flow

.0
.84
.0

.5
.0
.8

.6
.9
.0

. 7
.0
. 7

. 1

723
942
860

543
975
778

1079
1280
1195

492
776
719

1378

.3
.7
.5

. 1
.5
.6





.4
.8
. 1


Pere Marquette
    1984       879.9
St. Joseph
    1984
5927
         161
91
        205
275
         1097
7842
 705.6


3841
                   877.8
                                                           4836
                                  17

-------
collected by park rangers from Porcupine Mt. State Park.  The precision

estimates for these three rivers were comparable to the other four

tributaries, but if annual load estimates are as variable as those from

the four eastern tributaries, then caution should be used in extra-

polating estimates to other years.

     In 1984 the greatest annual loads of twelve parameters usually came

from the St. Joseph River followed by the Black or Rouge rivers.  Large
                                                          •K.
loadings from the St. Joseph river are not surprising since the average

daily flow in the St. Joseph river was more than 3.5 times the daily

average flow in any of the other rivers and nearly 73% of the watershed

is developed for agriculture.  Also, both the Black and Rouge river

watersheds have large urban or agricultural areas that may be non-point

sources of some constituents.  The smallest annual loadings of each

constituent came from either the Pere Mar-quette or Ontonagon rivers.

Both of these watersheds are relatively undeveloped and dominated by

forests and wetlands.



                   SAMPLE STRATEGY DEVELOPMENT

Determining the Necessity of Event Sampling Strategies

     Yaksich and Verhoff (1983) found that event sampling strategies

yielded excellent load estimates in all cases regardless of the relation-

ship between concentration and flow.  However event sampling is more

resource intensive than other sampling strategies and is not required

unless loads of the constituent increase with increasing flow.   When

concentration and flow are not related annual loads can be estimated with

fixed interval sampling programs.

                                   18

-------
     To determine which constituents in each tributary actually required

a flow stratified sampling strategy, daily average loads were plotted

against the corresponding daily average flow.  Daily average loads were

calculated on each day samples were collected by multiplying daily

average flow (cfs) by parameter concentration (mg/1) and by a conversion

factor of 2.45 (cfs x mg/1 x 2.45 = Kg/day).  Least squares linear

regression estimates were calculated and a t-test was used to demonstrate
                                               •v
that the loads of some parameters were related to flow.

     Plots of loads versus flow of twelve parameters in Seven tributaries

are included in Appendix 2 and the regression of loads on flow in the

Black, Rouge, Clinton, Huron, Ontonagon and St. Joseph rivers indicate a

statistically significant (alpha 0.05) positive relationship between

loads and flows for all parameters.  The variability tended to increase

with increased flow which is otie.reason for taking a proportionately

higher number of samples during high flow periods.

     In the Clinton and Rouge rivers, sodium and chloride concentrations

tended to increase with flows but loads were highly variable at all

flows.  This may have been due to chloride uses not related to flow, such

as seasonal use of road deicers.  Urban areas, where large amounts of

deicers are used, are much more extensive in the Clinton and Rouge

watersheds than in the other five tributaries monitored.

     In the Pere Marquette River, loads of suspended solids and ammonia

were not related to flow.  Therefore, load estimates of suspended solids

and ammonia from the Pere Marquette River do not require event sampling

and could be obtained with a less labor intensive fixed interval sampling

program.  All other constituent loads require event sampling to obtain

reliable estimates.


                                   19

-------
Testing for Bias Introduced by Event Sampling Strategies




     Another objective of the project was to develop a sampling strategy




that would yield relatively precise and accurate load estimates (in this




case total phosphorus loads) from Michigan tributaries.  To test several




different high flow sampling strategies, Monte Carlo runs were conducted




on three large data sets of total phosphorus concentrations.  The data




sets selected were the Sandusky River (1834 total phosphorus samples




collected during calendar years 1982-1985) the Raisin River (1237 total




phosphorus samples collected during water years 1983 to 1986) and the




Grand River (361 samples collected between March 1, 1976 and March 1,




1977).  These data sets were obtained through STORET, the Environmental




Protection Agency's data storage and retrieval computer system.  All




three had duplicate samples on some days, missing values on some days, or




both.  Therefore, the data sets were adjusted so that there was a single -




total phosphorus concentration recorded for each day.  Average




concentrations were calculated on days with multiple samples and




concentrations were estimated by linear interpolation on days when no




samples were.collected.  A "known" annual load and average daily load




were calculated using the adjusted daily total phosphorus concentrations




and daily average flows recorded at U.S.G.S. gaging stations.  Although




the true load cannot be "known", these data sets were the most complete




available.




     The next step was to assess different methods of stratified random




sampling by drawing subsamples from the complete data sets and comparing




estimates to the "known" load.  The data sets were broken down into




individual years within each tributary so that there were a total of nine




complete data sets (four each from the Sandusky and Raisin and one from




                                   20

-------
 the  Grand).   Subsamples  were  drawn and average daily loads  were




 calculated  two  hundred and  fifty times for each of the nine different




 adjusted data sets  and each of  the five different  sampling  strategies.




 The  number  of samples  selected  for each subsample  was calculated using  a




 sample  size  estimation formula  that requires  an estimate  of average and




 variance of  the load as  well  as a specified confidence interval.




      A  precision of +/-  50% was selected for  the estimates  of average




 load calculated from the Monte  Carlo runs.  This precision  was selected




 for  the Monte Carlo runs because the data sets were adjusted to have one




.sample  per  day  and  the estimated sample size  could not exceed the number




 of days in  each strata.  Also,  a relatively small  estimated sample size




 provides a  larger number of random combinations of subsamples.




 Investigators estimating tributary loads generally require  more precise




 estimates than  +/-  50% and  to achieve a greater precision more




 samples are  required.  However,  conclusions about  the best  sampling




 strategy, based on  Monte Carlo  studies,  will  be independent of the




 precision of load estimates.




      Daily  average  loading  rates and 95% confidence intervals were




 calculated  for  each estimate  along with bias,  which was the percent




 difference  between  the "known"  load and the estimated load.   Each group




 of runs was  tested  for deviations from a normal distribution using a test




 for  skewness and a  test  for kurtosis.   The  average estimated load from




 each group  of runs  that  was normally distributed,  or could  be transformed




 to a normal  distribution, was tested for deviation from the known load




 using a t-test.   Median  estimates were calculated  for all subsets with




 distributions that  could not  be transformed to normal.
                                    21

-------
     The first flow stratified sampling strategy tested on each data set


was random sampling from each strata.  Individual daily total phosphorus


samples were selected using a Lotus 123 random number generator.  One


data set was non-normal and had a median bias of -4.05%; but the average


bias of the other seven estimates of average daily loads (Kg/day) ranged


from -0.605% to 0.266% and none of the averages of daily loads were


significantly different than the known load (Table 7).  This indicates
              -^

that "random" sampling of high and low flow days is a strategy that would


usually provide estimates that, on average, are unbiased.  Unfortunately


it is impossible to use random numbers to select sampling days without


having prior knowledge of the number and date of occurrence of high and


low flow days in an upcoming year.  Therefore, several flow stratified


sampling programs were tested to document any bias introduced by


non-random sampling and to propose an alternative program.


     Previous studies have indicated that daily load estimates may be


influenced by season.  In spring 1977, MDNR personnel collected total


phosphorus samples from several southeastern Michigan tributaries during


high flow conditions.  They found the highest total phosphorus concen-


trations during the first event of the season despite relatively low


flows during the event (Schroeter 1978).  Peak total phosphorus


concentrations tended to decline with each of the first three successive


events, regardless of flow, in each of the tributaries sampled.  These


data suggest that if a disproportionate number of high and low flow


samples were to be collected in the spring, the resulting annual load


estimate may be biased.


     To test this hypothesis, Monte Carlo runs were conducted using


subsamples drawn randomly from the first half of the available high and


low flow days to simulate a sampling strategy that concentrates sampling



                                   22

-------
Table ?,   Total  phosphorus  average  dailj  load  estimates  Ug/sayl and average percent estlntsd
          bias fro« Honte  Carlo  analyses  for the  Sandusky, Grand and Ra.sin rivers.

•up ling
Strategy
"known Load"
Randot
•«*
Seasonal
No Rising Hydro
No Failing Hydro
Ranaoi Gr:u?s
"known Load"
lUadoa
Seasonal
No Rising Hydro
No Falling Hydro
Randoj Groups
"known Load"
Randoi
Seasonal
No Rising Hydro
No tailing Hydro
Randoa Groups
Average Estimated Percent Eias
Sandusky River
1332 1983
1670 998.5
1668 -0.119X 998,9 0.04COX
1522 ** -7.0"X 1123 ** 12. 5X
1738 ** 4.075 10C1 G.250X
17.19 ** 4, 135 1107" 10.35
1E55 H 0.39SX 333.: D.C30IX
Grand River
3/1/76-2/28/7?
1734
1734
150" ** -7.32»
1583 " -2.34X
1745 0.534X
1727 -0.402X
Raisin ?.iver
1983 1384
589,8 452.2
591.1 0.220X 453.1 0.199X
4?2.5 ** -19. 9X 465.0 ** 2. 335
573.1 ** -2.33X 423.3 ** -8.39
5?;. 1 ** 14. 2X 541.9 ** 19. 8X
591.3 K 0.333X 441.0 M -2.48X

1384 1985
1581 1133
1517 K -4.05X M 1136 0.262X
1383 «* -12. 2X 1229 ** 3.39X
1550 i -1.96X K 1151** -2.70X
1351 ** 17. IX 1353 ** H.8X
1542 H 2.35X M 11S6 M 1.1CX
1385 1936
563.3 396.2
554.2 C.266X 393.8 -0.505*,
601.2 ** 5.73X 339.3 ** -14.15
521.7 ** -7.29 397.' 0.3"35
630.1 ** 22. EX 431,3 ** 3.C1X
555.3 * 0.5345 397.8 0.404X
 *  significantly  different  than  :he known load (alpha=0.05l
 ** significantly different  than the known load  lalpha=0.01)
 U: Hedian value  was  calculated  due to non-norial distribution of the averages.
                                                23

-------
effort during the first part of a year.  The sample sizes remained the


same even though samples from days toward the end of the year were


excluded from all subsets.  All of the average estimates calculated were


significantly biased, but the direction of the bias was not predictable.


The range of bias estimates was from -19.9% to 12.5% with four of the


average bias estimates positive and five negative (Table 7). Therefore,


sampling should not be unproportionately concentrated in one season or
                                             •^

bias may be introduced.  Data collected from the Rouge and Clinton rivers


indicate that this type of seasonal variability may be especially


pronounced with chloride concentrations.


     Schroeter (1978) also compared total phosphorus concentrations from


the rising and falling arms of event hydrographs and concluded that total


phosphorus concentrations were generally higher during the rising


hydrograph.  This indicates that.if a sampling strategy were to


systematically neglect either the rising arm or the falling arm, then


bias may be introduced by the sampling strategy.


     To test this hypothesis, high flow samples were divided into three


categories.  The first category included samples on the rising arm of the


hydrograph, the second category included samples at the peak of an event


hydrograph while the third category included samples from the falling arm


of the event hydrograph.  To test a strategy that consistently missed the


beginning of an event hydrograph, high flow subsamples were selected


exclusively from high flow days in categories two and three and low flow


samples were selected randomly from the entire number of low flow days.


This type of sampling strategy may be implemented accidentally if field


crews are unable to respond fast enough to an event and subsequently miss


the beginning of each event.


                                   24

-------
     Six of the nine average daily load estimates were significantly


biased, two were not significantly biased, and one set of estimates was


non-normal and not tested but had a median bias of -1.96%.  Average bias


ranged from -7.39% to 4.07% and of the average biases that were


significantly different than zero, five were negative and one was


positive (Table 7).  Although this strategy did not introduce bias in all


cases, two-thirds of the average estimates were biased and one was
                                  «^

untested indicating the potential for this strategy to yield inaccurate


estimates.


     To assess the effects of missing samples at the end of an event,


Monte Carlo runs were made with subsets that included high flow samples


randomly selected from high flow days in categories one and two and low


flow samples selected randomly from the entire low flow data set.  Again,


this type of strategy may be implemented accidentally if field crews are


consistently unable to continue sampling a site for the duration of an


event, and the falling arm of an event hydrograph is sampled less often


than earlier portions of the hydrograph.  All the average daily load


estimates from the nine sets of runs were significantly different than


the known load except for the Grand River average estimate.  The average


bias estimates were positive and ranged from 0.634% to 22.5% (Table 7).


Therefore, it appears that strategies that consistently miss samples on


the falling arm of the hydrograph may be introducing bias.






Flow Stratified Systematic Sampling


     It appears that the key to developing an unbiased stratified


sampling strategy is to insure that within each strata, each sample has


the same probability of being selected.  However, it is difficult to


                                   25

-------
develop a plan to sample randomly without prior knowledge of future




flows.  One method would be to use a systematic sampling strategy where




samples are drawn at regular intervals based on the percentage of




subsamples desired (See Snedecor and Cochran 1980 for further information




on systematic sampling).  For example, if the historical upper 20th




percentile of flow is used as a cut-off between high and low flow strata,




then an average year will have 72 high flow days (365 x 20% = 72).  If




the estimated high flow strata sample size is 24 then it is necessary to




collect samples on one third of the high flow days.  If the high flow




days are arranged chronologically and "counted off" into groups of three,




then in an average year there would be three groups of 24 samples each.




A number between one and three could be randomly selected and would




designate a particular group of samples.




     For example, assume that in an average year the number one was




selected randomly, high flow sampling would be initiated on the first




high flow day of the year and samples would be collected on high flow




days 1,4,7,10,	64,67,70 for a total of 24 samples.  However, most




years would not have exactly 72 high flow days and 72 is not divisible by




all estimated sample sizes.  If 72 is not divisible by the estimated




sample size then the investigator will have to use a rounded "real




number" spacing interval.   For example, if the estimated sample size is




30 then a sample should be collected every 2.4 days.  If the sampling




started on day one then the sampling interval should be 1,  3.4, 5.8, 8.2,




10.6,....65.8, 68.2,  70.6   but samples would actually be collected on




days 1,3,6,8,11	66,68,71.




     To test a systematic  sampling strategy, high flow days in each of




the data sets from the Sandusky,  Raisin and Grand rivers were arranged in




                                   26

-------
chronological order, counted off based on the estimated sample size, and




divided into groups.  Each subsample used to estimate a load consisted of




a randomly selected group of high flow samples combined with low flow




samples selected randomly from the entire low flow data set.




Distributions of estimates from five of the Monte Carlo runs were




non-normal and not transformable, usually due to bimodal distributions




based on the high flow group randomly selected to calculate an individual




estimate.  Four of the average estimates were tested against the "known"




and one was significantly biased (alpha 0.05).  Median bias ranged from




3.86% to -2.48% and average bias ranged from -0.403% to 0.534% with




0.534% being statistically different than zero (Table  7).




     If a monitoring program were conducted over many years the average




number of samples per year could be predicted but for any one year the




number of samples collected could be highly variable.  During a wet year




with more than the average number of high-flow days additional samples




would need to be collected since indications are that discontinuing




sampling before the end of the year would introduce bias to the sampling




strategy.  However, a potential benefit of a systematic sampling strategy




is that additional samples are collected during years with more than




average high flow days and this should improve estimates by sampling more




often during more variable years.  Also, fewer samples would be collected




during dry years without a loss in precision.




     The assumption was made that low flow samples will be collected




"randomly".  Less attention is paid to low flow sampling because




generally the low flow strata contributes less to the annual load and is




less variable.  Although no strategies for random sampling in the low




flow strata were tested, collecting the samples over the entire year (i.e




regular fixed interval sampling) should be sufficient.




                                   27

-------
                     SAMPLE SIZE ESTIMATION


Sample Size Estimation Method


     The number of samples required to estimate loads is an important


aspect of any sampling strategy.  Generally sample sizes can be predicted


by using the following formula:


                    n    =(t2 x S2)/(D2 x X2)
 ^                   CO U

                    where:


                    n    = estimated sample size
                     est                v
                     2
                    t = student's t value squared

                     2
                    S = variance of the daily load

                     2
                    D = precision, as a percentage of the


                        average, squared (ie. D=0.5 would


                        indicate a precision of +/- 50%)

                     2
                    X = average daily load squared


The approach taken was to estimate the sample size within each flow


strata, so an estimate of the daily average load and variance of the


daily average load within both the high and low flow strata was required.


     Again, sample size estimates for Monte Carlo studies were predicted


using average and variance estimates calculated from the complete data


sets.  If predictions of variance and average are accurate then the


estimated 95% confidence interval should have been within +/- 50% of the


estimate.  Confidence intervals for loads to the Sandusky River,


calculated from random sampling, ranged from +/- 8.29% to +/- 51.3% of


the estimate (Table 8).  Confidence intervals for loads to the Grand and


Raisin rivers,  ranged from 7.62% to 40.9%,  and 7.30% to 47.0%,


                                   28

-------
Table 8.  Low, high and ledian 95X confidence intervals calculated on lead estimates
          (Kg/day), froi Honte Carlo subsanpling,  for the Sandusky,  Grand and
          Raisin rivers.
River and
Year Strategy
Sandusky River
ftandoi
1982
1983
1984
1385
Seasonal
1982
1983
1984
1985
No Rising
1982
1983
1984
1985
No Palling
1982
1983
1984
1985
Systenatic
1982
1982
1984
1385
Grand River
3/1/77 to Randoi
3/1/78
Seasonal

No Rising

Ho Falling

Systeiatic

Saiple
Size


66
65
56
66

56
66
56
66
Hydro.
56
66
56
66
fljdro.
"6
66
56
66
Saiphng
35-96
75
71
77


13

13
Hydro.
13
Hydro.
13
saipling
16-17
+/- 95X
low


106.3
67.42
131.2
98.39

151.3
86.53
170.6
134.5

•5. 07
76.31
135.5
99.99

:59,5
122.0
159.5
134.7

399.6
118.5
225.4
130.0


127.3

116.3

151.0

123.1

159,6
confidence
high


906.4
203.9
778.0
223.1

754.0
171.8
243.4
230.3

357.3
205.1
792.3
207.4

933.3
203.2
730.9
207.6

472.0
156.1
539.8
154,5


824.1

322.5

702.3

748.8

3:i,5
interval
ledian


584,0
138.9
302.3
176.0

595.3
159.9
215.5
203.5

552.2
149.0
320.1
157.0

?46.1
190.0
718.9
189,4

407.5
153.9
534,3
157.2


316.7

352.?

336.9

23:. 3

303.1
I/- 95X
X low


9.43X
8.29X
9.28X
9.27X

14.6)1
3.64X
12. 6X
11. 3X

?.81X
3.91X
10. 2X
9.02X

14. IX
12. OX
10. OX
10.31

24. 4X
11. 3X
14. 5X
11. 2X


7.62X

8.08X

8. SOX

7.54X

3.51X
confidence
X high


51.25
20. 4X
47. OX
18.81

49. 5X
16, OX
18. 5X
18, 5X

4S.2X
20. IX
48. 3X
18. OX

53, 3X
18. 8X
41. 5X
15. 3X

17 o*
u ( * Cti*
:5.6X
32, 'X
•3.CX


40. 9X

iQ QV
'to • OS

*
-------
Table 8.   Continued,
River and
Year Strategy
Raisin River
Randan
iff 1983
»Y 1334
Iff 1935
mr 1986
Seasonal
Iff 1983
»Y 1984
KY 1985
BY 1986
No Rising
KY 1983
»Y 1984
»Y 1335
»Y 1986
Mo Falling
Iff 1933
VY 1984
»Y 1985
Iff 1986-
Systenatic
»Y 1983
WY 1984
Iff 1985
Iff 1986
Saiple
Size


39
39
39
39

39
39
39
39
Hydro.
29
i 0
33
33
Hydro,
39
39
39
39
Saapling
67-68
60-61
42-43
56
•/- 35X
low


43.39
47.50
75.84
24.56

30.54
51.29
100.4
21.34

40.33
15.32
43.49
29.23

55.18
50.81
130.4
23.44

124.5
55.8
108.5
40,0
confidence
high


301.6
115.5
151.1
104.2

92.35
91.58
153,3
53.13

305.3
105.4
:35.c
110.6

278.3
106.4
147,5
104.0

171.1 ,
73.1
132,3
80.0
interval
aedian


177.7
81.11
121,2
52.94

56.44
78.24
144,5
23.31

157.3
73. 15
112.7
68,74

234.7
84.84
132.7
87.30

141.4
67,46
115,1
68.23
+/- 95S
X low


9.80X
11. 3S
14. OX
7, SOX

7.24X
11. OX
2C-.SX
5."4X

10. 3X
11. 3X
1D.7X
8.20X

11. IX
10.2!
18. IX
7.83X

20. 2X
11. «
18. 6X
10.1!!
confidence
X high


47. OX
23. 4X
28. 7X
25. 8X

18. Sv,
20. 2X
25. IX
17. K

47. 7X
26. OX
21. 3X
27. IX

41. IX
19. 5X
21. 2X
23. 7X

27. 4X
17. 7X
2G.8X
20. OX
interval
X ledian


29. 6X
17. 9X
21. 4X
16. 2X

11.9J
15. 4X
23. 3X
3.83X

32. 3X
17, 4X
21. 5X
17.51

31. 8X
15.7*
19. 3X
19. 3X

23. 3X
15, 5X
20. 3X
17. OX
                                             30

-------
respectively.  Calculated 95% confidence intervals from the systematic




sampling strategy ranged from 11.2% to 37.2% of Sandusky River loads,




9.51% to 43.0% of Grand River loads, and 10.1% to 27.4% of the Raisin




River loads.  Confidence intervals rarely exceeded +/- 50% and were




usually much less than +/- 50%.




     In some cases there was a relatively large range in confidence




intervals.  Data from a tributary, year and subsample that by chance




yield a low confidence interval, should not be taken as an indication




that fewer samples could be collected in the next year.  For example, one




combination of samples taken from the Sandusky River in 1982 yielded an




estimate with a 95% confidence interval of +/- 9.48%, while another




combination of samples from the same tributary and year yielded an




estimate with a 95% confidence interval of +/- 51.3%.  A decision to




change the sample size based on either of these confidence intervals may




lead to wasted resources on unnecessary precision or an estimate less




precise than desired.  A better way to adjust sample size estimates after




one year of sampling would be to use improved estimates of average daily




load and variability in the sample size estimation formula.




     Sample size estimates of 13 for the more stable Grand River would




not be considered intensive event sampling.  However, the precision of




the calculated estimates was always less than the desired +/- 50%




indicating that the sample size was adequate.  This method of sample size




estimation may save investigators from making arbitrary decisions about




the intensity of flow stratified sampling.









Predicting Load Variability for Sample Size Calculations




     Many tributaries will not have extensive concentration data on each




parameter of concern-.  While an approximation of the average load in each





                                   31

-------
strata may be available from monthly monitoring data, variance estimates



typically require more information.   Richards (in press) found that one



way to predict the variance of the daily load of a particular parameter



was to relate it to the variance of the average daily flow.  He quantified



a relationship between the daily suspended solids load variance and the



average daily flow variance using complete, or nearly complete, suspended



solids and flow data sets from 11 tributaries to lakes Michigan, Erie and



Ontario.  These tributaries ranged from event responsive systems in which



flow was highly variable, to stable systems with less variable flows and


                                            2         2
the watersheds ranged in size from 16,395 Km  to 44 Km .



     Richards used the coefficient of variation (standard deviation



divided by the average) of the logs of the set of percentiles of flow



(5%,10%,15%,20%	80%,85%,90%,95%) (CVLF5) to quantify flow



variability.  The CVLF5 is provided in U.S.G.S. flow duration analyses.



All flows were in cfs and Richards pointed out that CVLF5 is affected by



the units used to measure flow.  The variability of the daily suspended



solids load was quantified by calculating the coefficient of variation



(CV) of the logarithms of daily suspended solids loads.  A least-squares



linear regression between the CVLF5 and the CV of the log of daily loads



for nine of the eleven tributaries yielded a predictor equation of Y=



0.0482 + 0.7197(CVLF5) with R= 0.99.   Two tributaries were excluded as



outliers because Richards felt that the watersheds were small and the



period of flow record was short.   This equation enabled him to predict



the CV of the log of suspended solids loads on any tributary which had a



complete flow record.
                                   32

-------
     A similar type of relationship was developed, for this project,




between flow variability and total phosphorus daily load variability.




Data sets from six Great Lakes tributaries with complete or nearly




complete records of daily total phosphorus concentrations were used.  The




six tributaries were the  Sandusky River, Honey Creek, (classified by




Richards as event responsive) Raisin River, Maumee River, Cuyahoga River




(variable responsive) and the Grand River (stable responsive).  As stated




earlier, the total phosphorus concentrations used from the Sandusky River




were recorded during calendar years 1982-1985 and included 1834 total




phosphorus measurements.  From the Raisin River, 1237 total phosphorus




samples were collected during water years 1983 to 1986 and 361 total




phosphorus samples were collected from the Grand River between March 1,




1976 and March 1, 1977.  The additional three data sets included 2359




total phosphorus samples from the Cuyahoga Rj.ver, 2431 total phosphorus




samples from Honey Creek and 2564 total phosphorus samples from the




Maumee River collected between January 1, 1982 and December 31, 1986.




     To quantify load variability the.CV of the total phosphorus load was




estimated for each of these rivers by calculating the antilog of the




standard deviation of the logs of daily loads.  For example, the standard




deviation of the logs of daily average loads in the Grand River was 0.305




and the antilog of 0.305 is 2.018.  A standard deviation of +/-0.305 in




the logarithm of the load can be transformed back to a non-logarithmic




standard deviation equal to the mean load multiplied (or divided) by




2.018.  So if the standard deviation of the logarithms equaled 0.305 then




the standard deviation of the geometric mean would equal +/- 1.018% of




the mean.  Standard deviations expressed as percentages of the mean are




coefficients of variation (CV).




                                   33

-------
     A Plot of the CV of total phosphorus daily loads versus CVLF5


indicates that there is a linear relationship between flow variability


and load variability and that Honey Creek seems to be an outlier  (Figure


2).  The least squares linear regression equation, based on five  points,


was CV=-2.505+(CVLF5*32.2797) with R2= 0.872.  Honey Creek was the


smallest and most variable of the six watersheds.  The relationship may


not be linear at the upper range of CVLFS's or the size of the watershed


may influence the relationship between load and flow variabilities.


     In order to estimate sample sizes in each strata, it is necessary to


know the parameter variability within each flow strata, not just  the


parameter variability for the range of flow conditions within a given


year.  Therefore, the same type of parameter variability versus flow


variability relationship was developed within each of the two flow


strata.

                                                               2
     In the high flow strata there was a linear relationship (R =0.967)


between CVLF5 and' the CV of daily high flow loads excluding Hone}' Creek


(Figure 3).  In the low flow strata there was also a linear relationship

  2
(R =0.847 when Honey Creek was excluded) between daily load and daily


flow variability (Figure 4).  The CV of daily total phosphorus load in


the high flow strata could be estimated by using the high flow predictor


equation CV=0.08381 +(CVLF5*7.0478) and the CV of the low flow strata can


be estimated by CV=-0.8528 +(CVLF5*11.8341).


    Estimated sample sizes were calculated for the high and low flow


strata of each of the five rivers using variance estimates from the


complete data sets and estimates calculated from the predictor equations.


These two different sample size estimates were not always close and the


magnitude of the absolute difference between the two estimates increased


                                   34

-------
i
         O.OS
0.45
Figure 2.  The Relationship  between the Coefficient of  Variation (CV)
           of Total Phosphorus Daily Loads and  Average  Daily Flow
           Variability  (CVLtfS).
                                    35

-------
      1.S -
       1 -
      as -
                                    CVLF5
Figure  3.  The Relationship between the Coefficient of Variation (CV)
           of Total  Phosphorus Daily Loads and Average Daily Flow
           Variability (CVLF5) in the High Flow Strata.
.x
2
        o.os
                                                               0.45
 Figure 4.   The Relationship between the Coefficient of Variation  (CV)
            of Total Phosphorus Daily Loads and Average Daily Flow
            Variability (CVLFB) in the Low Flow Strata.
                                     36

-------
with increased flow variability.  At a precision of +/-50% the absolute




difference between estimated Grand River sample sizes was one and the




absolute difference between Sandusky River sample sizes was seventy-eight




(Table 9).  This method of predicting sample size requirements is one




procedure that can be used when sampling programs are desired on




tributaries with little or no prior information about the constituent of




concern but it seems that estimates are less reliable for the more event




responsive rivers.




      Flow variability versus load variability relationships may be




quantified for other constituents that vary with flow.  Parameters that




tend to be more variable than total phosphorus will require more sampling




effort to achieve the same precision while parameters that are less




variable will require fewer samples.









Sample Size Estimates for Michigan Tributaries




     Sampling requirements were estimated for four of the seven Michigan




tributaries using the predictor equation and methods described above.




The predictor equations were developed using five tributaries with




CVLF5's ranging from 0.08469 to 0.26547 so the CV should only be




estimated for rivers with CVLFS's in this range since the slope or




relationship may change outside of this range. The CVLF5 at the mouth of




each tributary was 0.03092 for the Rouge, 0.04679 for the Pere Marquette,




0.05724 for the St. Joseph, 0.08245 for the Ontonagon, 0.09867 for the




Huron, 0.13991 for the Clinton, and 0.28044 for the Black rivers.   The




CVLFS's for the Clinton and Huron Rivers fell within this range and




sample size estimates were calculated.  The CVLF5 for the Ontonagon River




was near the lower end of the predictor equations and the Black River was




                                   37

-------
Table 9.  Coiparison between estiiated saaple sizes, at various levels of precision,
          calculated using variance and average estiaates froa the coaplete data set
          and using the CV predicted froa the regression equation.
Tributary
precision
(X of estiaate) Sandusky
A.

+/-
+/-
t/-
*/-
+/-
t/-
High Flow

SOX
40X
30X
25X
20X
10X
Strata
C*
32
48
84
120
185
732

P**
61
94
166
238
371
1476
Kauaee

C
21
32
54
77
119
467

P
37
56
98
140
218
865
Raisin

C
30
45
77
no
173
670

P
29
43
74
:os
163
647
Cuyahoga

C
37
56
98
140
218
865

P
24
37
63
90
140
551
Grand

C
8
12
13
26
33
147

P
10
14
22
31
47
180
B.  Lo» Flow Strata

+/- 50X
+/- 40X
+/- 30X
t/- 25X
+/- 20X-
t/- 10X
 C
34
        P
       83
 51    128
 88    226
126    324
195    506
775   2013
  C
 21
 32
 55
 P
37
57
98
 77   140
120   218
468   364
  C     ?
  9    24
 13    35
 20    54
 23    89.
 42   138
181   546
 C
11
16
27
37
 P
18
27
46
86
                                         58   101
                                        221   398
C.  Total Nuaber of Saaples
  C
  5
  6
  9
 11
"16
 57
 P
 2
 2
 4
 4
 5
11

+/-
+/-
t/-
+/-
+/-
*/-
D.
*/-

SOX
40X
30X
25X
20X
10X
Absolute
50X
C
56
99
172
246
380
1507
Difference
78
P
144
222
392
562
877
3489


C
42
64
109
154
239
935


P
74
113
196
280
436
1729

32
C
39
58
97
138
212
331


P
53
73
138
194
301
1193

14
C
48
72
125
1"7
276
1086


P
42
64 •
103
156
241
349

6
C
13
18
28
37
55
204


P
12
16
26
35
52
191

1
*  Saaple size eatiiates calculated using variance and  average  estimates  froa  the
   coaplete data sets.

** Saaple size estiaates calculated using the CVLF5  and  the  predictor  equations
                                                  38

-------
near the upper end of the range so sample size estimates were also




calculated for these tributaries.  Although sample size predictions should




not be made for rivers with a CVLF5 below 0.08469, sampling requirements




will decrease with a decrease in flow variability and the corresponding




load variability.




     The CVLF5 was lowered dramatically in the Rouge River by the Rouge




Ford plant diversion.  This constant addition of 784 cfs, to a median




upstream flow of 129 cfs, increased the flow at the mouth without




influencing the magnitude of the range between high and low flows.  The




Rouge River CVLF5 calculated using flows above the diversion was 0.19993




and would indicate flow variability greater than all study rivers except




the Black River.  At the mouth the CVLF5 was 0.03092 indicating flow




variability less than all of the rivers studied.   Actual Rouge River




sampling yielded relatively  precise 95% confidence intervals of +/-14%




of the estimate in 1984 (n=69), +/-19% in 1985 (n=95) and +/-!!% in 1986




(n=93).  Although, these estimates are relatively precise, the results of




Monte Carlo studies indicate that large ranges of confidence intervals,




using the same sample size, are common.  A calculated precision from any




single subset, within the group, may not be a good indicator of the




required sample size.  On the other hand there is no evidence indicating




that additional sampling would have been beneficial.




     The estimated sample sizes are presented in Table 10 so that the




predicted precision can be contrasted with the actual precision estimate.




Comparing the estimated number of samples (171) to the actual number of




samples collected in the Black River indicates that this tributary should




have been sampled more intensely to insure a precision of at least +/-




50%.  However, 95% confidence intervals ranged from 10.8% (n=93) to 40.1%




                                   39

-------
Table 10.   Predicted number of samples required per year to
           estimate total phosphorus loads with 95% confidence
           intervals less than or equal to the indicated precision
           of the estimate.
Tributary
Precision
(% of estimate) Ontonagon
A. High Flow Strata
+/- 50%
+/- 40%
+/- 30%
+/- 25%
+/- 20%
+ /- 10% 1
B. Low Flow Strata
+/- 50%
+/- 40%
+/- 30%
+/- 25%
+/- 20%
+ /- 10%
C. Total Number of
+/- 50%
+/- 40%
+/- 30%
+/- 25%
+/- 20%
+/- 10% 1

10
13
21
30
45
72

2
3
4
4
5
9
Samples
12
16
25
34
50
81
Huron

12
17
28
40
61
236

4
5
7
9
12
40

16
22
35
49
73
276
Clinton

20
30
51
73
112
449

13
18
30
42
65
250

33
48
81
115
177
699
Black

70
107
191
275
430
1719

101
156
275
395
615
2457

171
263
466
670
1045
4176
                                 40

-------
(n=89) and it is likely that, as in the case of the Sandusky River, the




predicted sampling requirements are more rigorous than necessary.




     Comparing the estimated total phosphorus sample size for the Clinton




River to the actual sample size indicates that precision should have been




within 40% in 1984 and 30% in 1985 and 1986.  The actual precision was




better than the predicted precision in all three cases and was +/- 23.4%




(n-71), +/- 12.0% (n-89) and +/- 17.0% (n-103) in 1984, 1985 and 1986,




respectively.  Sampling conducted on the Huron River should have yielded




95% confidence intervals within approximately +/- 25% of the estimate and




actual confidence intervals ranged from +/- 6.0% (n=91) in 1986 to +/-




11.8% (n=95) in 1985.




     Sampling effort in the Ontonagon River should have yielded a total




phosphorus estimate with a 95% confidence interval of within approxi-




mately +/- 25% and the actual 95% confidence interval was just outside




the predicted range at 26.9%.  This could be related to problems




extending the predictor line past the lowest point, variability of the




estimate derived from the predictor equation, higher than normal total




phosphorus load variability in the Ontonagon River in 1984 or a




combination of these factors.




     No sample size estimates were calculated for the Pere Marquette or




St. Joseph rivers, but relatively low intensity sampling on both rivers




(25 samples from the Pere Marquette and 24 samples from the St. Joseph




rivers) yielded 95% confidence intervals of +/- 14.2% and +/- 9.8% for




the Pere Marquette and St. Joseph rivers respectively.  As mentioned




earlier, calculated confidence intervals are variable and are not always




good  indicators of sampling requirements, but these estimates were




relatively precise and support the contention that rivers with




stable-flows generally require less intensive sampling programs.




                                   41

-------
                            LITERATURE CITED

Bierman, V.J. Jr., S.J. Preston and S.E. Silliman.  1988.
     Development of estimation methods for tributary loading
     rates of toxic chemicals.  Dept. Civil Eng., Notre Dame
     Univ. Tech. Rept. No. 183

Dolan, D.M., A.K. Yui, and R.D. Geist.  1981.  Evaluation of
     River Load Estimation Methods for Total Phosphorus.  J.
     Great Lakes Res.  7(3):  207-214.

Richards, R.P.  (in press).  Measures of Flow Variability
     and a New Classification of Great Lakes Tributaries.   Water
     Quality Laboratory, Heidelberg College, 310 E. Market Street
     Tiffin, Ohio.

Richards, R.P. and J. Holloway.  1987.  Monte Carlo Studies of
     Sampling Strategies for estimating Tributary Loads.  Water
     Res. Res. 23:  1939-1948.

Schroeter, J.  1978.  Total Phosphorus-Flow Relationship in
     Southeastern Michigan Tributaries.  Mich. Dept. Nat.  Res.,
     Environmental Services Division, Pub No. 4833-5082.

Snedecor, G.W. and W.G. Cochran.  1980.  Statistical Methods.
     Seventh ed.  Iowa St. Univ. Press.  Ames, Iowa.

Yaksich, S.M., and F.H. Verhoff.  1983.  Sampling Strategy for
     River Pollutant Transport. J. of the Envir. Eng. Div., ASC
     109 (No.  EE1):  3-8.
                                   42

-------
Appendix 1.  Watershed Maps and Locations in Michigan
                           43

-------
   \
0
                  Black
                   ver  Basn
                Sanilac Co
                Lapeer Co,
10
 Scale of Miles
                          St, Cla.ir Co,
                                        vvvvvvvvvvvvwvv



                                        v  Lake Huron


                                         \ vvvvvvvvvvv
                                          \
                                          •\
                                           \
                                           \ vvvvvw
                                           H
                              44

-------
                            r\
                            \J
        \ !
ve
Scale of Miles
                             Lapeer Co
                                   r>. I  ni ,   p
                                   it LiQir  Co,
Oakland Co,
                      Wayne Co,
                                                                 \Haconb  Co,
                                                                       vvvvvvvvvv
                                                                       vvvvvvvvvv
                                                              vvvvvvvvvvvvvvvvvvvv
                                       45

-------
9*7

-------
Huron   Rver
                                       c~
r
!
Inghan Co,
Jackson Co,
                                /Oakland Co,
                                            Scale of Miles
                    Vash-fcenaw Co
                             Monroe Co,
     vvvvvvvvvvvvvvvv
                                                vvvvvvvvvvvvvvwv
    Lake  Er
                                            vvvvvvvvvvvvvvvvvvvvvvvvv
                              47

-------
.40
                             AAAAAAAAAA





                           AAAAAAAAAAAAA

-------
vvwvvvvw
vvvvvvvvvvvvv
                   Dere
                   Hasoii Co,
v
           \_
/
               Lake Co,
                  ' Dceana Co,
               Neroygo Co,
                                                    11
                                       Scalp of Hiles
                                                               \
                                         49

-------
 CD
 H5
 O
CO
 O
 in

-------
Appendix 2.  Relationship between Average' Daily Load
             and Average Daily Flow.
                           51

-------
700
                    a
                 a  a  a
                           468
                                 (Thouiand*)
                           Avarag* Daly How (cf •)
                                                           10
                                                                      12
     Average Daily Load  of Calcium versus Average Daily Flow in the
     Black River {1984-1986).
300
280
260
240
220
200
180
160
140
120
100
 ao
 60
 40
 20
  0
                                     a a
                                  n
                                  0
                                  a    a
                                                a
                                                a
           a
                    a
                    a
                                                           10
                                                                     12
                                 Daly
     Average Daily Load of  Chloride versus Average Daily Flow in the
     Black River (1984-1986).
                                 52

-------



••
"5
1
**
*i
i'
•*• 9
si
*
1
5






90 -
ao -


70 -
60 -
so -
4O -

3D -


Q


a
a
a

a
a

a
a
a
20 -\

a
10 -I QpO*3
Q l^r 	 . 	 , 	 . 	 . 	 . 	 . 	 , 	 , 	 , 	
                     4         6
                            (Thousand*)
                     Axcrag* Ocily Row (cf»)
10
                                                               12
Average Daily  Load of Potassium versus Average Daily Flow in the
Black River  (1984-1986).




•^
^
£
11
^ a
2|
t
1
4







iao -
170 -
160 -
1SO -
140 -
130 -
120 -
11O -
100 -
so -
ao -
70 -
60 -
so -
° 0


a °



0 a
a a



a
cm
40 -I QlQ
1 PT^
3D -\ £
20 -\Sa
10 -If
0 -I
9

0 2 4 68 10 1
CThouKirid*)
Average Daily Row (cf»)
Average Daily  Load of Magnesium versus Average Daily Flow in the
Black River  (1984-1986).
                           53

-------




f
^
tl
.98
^ a
"0*
I
9
|
<











I
^v
9
Vjf
g "J
J !
ti
8
0
1







no -
130 -

120 -
no -
100 -
so -
80 -
70 -

60 -
SO -

4O -

30 -

a
a




0 a Q
a
a
a ^
i?
a
D
a a
B>
rT0 0
20 -1 rffi&
\jffn
10 -jpu
0 2 4 6 8 10 1
CThouiand*)
Avorag* Ddly Plow (cf»)
Average Daily Load of Sodium versus Average Daily Plow in th
Black River (1984-1986).
10 -
9 -
8 -

7 -

6 •«

s -
4 -

3 -

2 -

i _
i —

o -
a D
B° B .
a a °
a

a D B
a
a a
a

B
8
a a D
a B 0
B a D
^Q_ D

00°° m °
^HtaBrn ^"
                                         8
10
                                                             12
                         Plow (cf»)

Average Daily Load of Ammonia versus Average Daily Plow in  the

Black River  (1984-1986).
                            54

-------
35 -
Avorao* Ddfcr Load 
(TKoiMK)nd»)
0 W 8 M 8
0 H
C
a
a a
a
a ° a °
a ° o a a
if a ° °
eft a a
a a
a
Q g a
a a
_ o a a
a g
a
^=
^^^^^l" 1 1 1 I 1 1 1 1 I I
) 246 8 10 13
1. 1 rmimanamj
Avarag* Daly Plow (cf»)
45 -
35 -
1 »-
*.^
"8"§ 25 -
tj 20-
Qt
1 1S "
10 -
S -
n -
Average Daily Load of Nitrate versus Average Daily Flow in the
Black River (1984-1986).
a
a
a n

a
a
a
a
j^t,n
                     46
                            ^ThouK
                     A\«raa» Daly Flow
                                                     10
                                                               12
Average Daily  Load of Silicate versus  Average Daily Flow  in the
 Black River  (1984-1986).
                           55

-------
600
500 -
«,-
200 -
100
                          4         6
                                 CThouKind*)
                          Avwog* Deily Flow («f
1O
12
      Average Daily .Load of Sulfate  versus Average Daily Plow  in  the
      Black River (1984-1986).

8 -
7 -
1 *~
I? 5"
H 4-
a
| 3 -
2 -
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                                56

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                           57

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Average Daily Load of Calcium versus Average Daily Flow in the
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                           58

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                                      59

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                                     60

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Average Daily Load of Nitrate versus Average Daily Flow in th«
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                          61

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                                62
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                                    63

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                              64

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                               65

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                               66

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Average Daily Load of Nitrate versus Average Daily Flow in the
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                              67

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                                        68

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                                      70

-------
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                                       71

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                             72

-------



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                               73

-------
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                                 74

-------
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                                       75

-------
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St. Joseph River  (1984-1986).






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                           76

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                           77

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                           78

-------

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                          79

-------
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                               80

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                              82

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                              86

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-------