U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2012–5135 

Prepared in cooperation with the Wisconsin Department of Natural Resources 

Estimation of Natural Historical Flows for the 

Manitowish River near Manitowish Waters, Wisconsin
Cover. Rest Lake Dam as viewed from Rest Lake, part of the Manitowish Chain of Lakes. (Photograph by James M. Klosiewski, Wisconsin 

Department of Natural Resources, September 22, 2002.) 
Estimation of Natural Historical Flows for 

the Manitowish River near Manitowish 

Waters, Wisconsin

By Paul F. Juckem, Paul C. Reneau, and Dale M. Robertson

Prepared in cooperation with the Wisconsin Department of Natural Resources

Scientific Investigations Report 2012–5135

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2012

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Suggested citation:

Juckem, P.F., Reneau, P.C., and Robertson, D.M., 2012, Estimation of natural historical flows for the Manitowish 

River near Manitowish Waters, Wisconsin: U.S. Geological Survey Scientific Investigations Report 2012–5135, 32 p., 

including appendix. 
iii

Acknowledgments

Local Project Coordinator

James Kreitlow, Water Resources Management Specialist, Wisconsin Department of Natural 

Resources, Rhinelander, Wis.

Field Support

Eric Dantoin, Hydrologic Technician, U.S. Geological Survey, Rhinelander, Wis.

Publishing Support

Laura Nelson, Office Assistant, U.S. Geological Survey Wisconsin Water Science Center, 

Middleton, Wis.

Michelle Greenwood, Reports Specialist, U.S. Geological Survey Wisconsin Water Science 

Center, Middleton, Wis.

Bonnie Stich, Editor, U.S. Geological Survey Science Publishing Network, Louisville, Ky.

Technical Reviewers

Robert Waschbusch, Hydrologist, U.S. Geological Survey Wisconsin Water Science Center, 

Middleton, Wis.

Glenn Hodgkins, Hydrologist, U.S. Geological Survey Maine Water Science Center, Augusta, 

Maine.

Editorial Reviewer

Elizabeth Ciganovich, Technical Writer-Editor, U.S. Geological Survey Science Publishing 

Network, Madison, Wis.

Approving Official

Kevin Breen, Bureau Approving Official, U.S. Geological Survey Office of Science Quality and 

Integrity, New Cumberland, Pa. 
iv

Contents

Acknowledgments........................................................................................................................................iii

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................2

Purpose and Scope ..............................................................................................................................3

Physical Setting.....................................................................................................................................3

Data Sources..................................................................................................................................................3

Methods...........................................................................................................................................................3

Adjusted Drainage-Area Ratio Method ............................................................................................6

Water-Budget Method.......................................................................................................................10

Natural Flows at the Rest Lake Dam ........................................................................................................14

Natural Historical Flow Durations.............................................................................................................18

Assumptions and Limitations.....................................................................................................................22

Summary and Conclusions.........................................................................................................................23

References Cited..........................................................................................................................................24

Appendix. Code for the Python-based Water-Budget Model...........................................................27

Figures

1. Map showing the Manitowish Chain of Lakes study area in north central 

Wisconsin and streamflow-gaging stations used during this study....................................4

2. Graph showing measured flows at Rice Creek (station 05357182) and 

Papoose Creek (station 05357299) during the 2-year study period ......................................5

3. Graph showing estimated flow without the effects of upstream diversions 

and measured flow for the Trout River (station 05357259).....................................................6

4. Graphs showing flows measured by the Wisconsin Department of 

Natural Resources and U.S. Geological Survey at the Manitowish River 

(station 05357302) and dam outflows at the Rest Lake Dam outlet reported 

by the dam operator.....................................................................................................................7

5. Graph showing regression relation for total water yield for gaging stations 

upstream of the Manitowish Chain of Lakes and the total water yield for the 

Bear River (station 05357335) plus the Trout River (station 05357245) .................................8

6. Graph showing measured total flow at gaging stations upstream of the 

Manitowish Chain of Lakes and computed flow into the Chain of Lakes using 

the adjusted drainage-area ratio and water-budget methods ...........................................15

7. Graph showing monthly error in estimated flow at the gaging stations 

upstream of the Manitowish Chain of Lakes for the adjusted drainage-area 

ratio and water-budget methods..............................................................................................16

8. Graph showing median daily flow at the Rest Lake Dam for each calendar day 

over the period of record available for each method...........................................................17
v

9. Graph showing dam-operator reported stage and dam outflow for the 

Manitowish Chain of Lakes and computed natural flow at the Rest Lake Dam 

from the adjusted drainage-area ratio and water-budget methods, 1973–2011..............19

10. Graph showing monthly flow-duration values for computed natural flow at 

the Rest Lake Dam based on the adjusted drainage-area ratio method for 

1991–2011 and the water-budget method for 1973–2011......................................................20

Tables

1. Generalized operation of the Rest Lake Dam...........................................................................2

2. Monthly coefficients that were multiplied by total daily flow into the 

Chain of Lakes to compute daily values of natural flow at the Rest Lake Dam 

for the adjusted drainage-area ratio method.........................................................................10

3. Target flow-duration statistics for the upstream gaging stations for the 

2-year study period, simulated equivalent values, residuals, and associated 

target weights from the water-budget model........................................................................12

4. Parameter names and descriptions, and calibrated values for the waterbudget method ............................................................................................................................13

5. Annual total measured dam outflow, total computed natural flow, and 

difference between measured dam outflows and computed natural flows 

from the water-budget and adjusted drainage-area ratio methods at the 

Rest Lake Dam.............................................................................................................................14

6. Monthly flow-duration values for computed natural flow at the Rest Lake Dam

based on the adjusted drainage-area ratio and water-budget methods..........................21
vi

Conversion Factors and Datum

Inch/Pound to SI

Multiply By To obtain

Length

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

acre 0.004047 square kilometer (km2)

square mile (mi2) 2.590 square kilometer (km2) 

Volume

cubic foot (ft3) 0.02832 cubic meter (m3) 

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

cubic foot per second per square 

mile [(ft3/s)/mi2]

0.01093 cubic meter per second per 

square kilometer [(m3/s)/km2]

inch per year (in/yr) 25.4 millimeter per year (mm/yr)

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 

(NAVD 88).

Altitude, as used in this report, refers to distance above the vertical datum.
Abstract

The Wisconsin Department of Natural Resources is 

charged with oversight of dam operations throughout Wisconsin and is considering modifications to the operating orders 

for the Rest Lake Dam in Vilas County, Wisconsin. State 

law requires that the operation orders be tied to natural low 

flows at the dam. Because the presence of the dam confounds 

measurement of natural flows, the U.S. Geological Survey, 

in cooperation with the Wisconsin Department of Natural 

Resources, installed streamflow-gaging stations and developed 

two statistical methods to improve estimates of natural flows 

at the Rest Lake Dam.

Two independent methods were used to estimate daily 

natural flow for the Manitowish River approximately 1 mile 

downstream of the Rest Lake Dam. The first method was an 

adjusted drainage-area ratio method, which used a regression 

analysis that related measured water yield (flow divided by 

watershed area) from short-term (2009 –11) gaging stations 

upstream of the Manitowish Chain of Lakes to the water yield 

from two nearby long-term gaging stations in order to extend 

the flow record (1991–2011). In this approach, the computed 

flows into the Chain of Lakes at the upstream gaging stations 

were multiplied by a coefficient to account for the monthly 

hydrologic contributions (precipitation, evaporation, groundwater, and runoff) associated with the additional watershed 

area between the upstream gaging stations and the dam at the 

outlet of the Chain of Lakes (Rest Lake Dam). The second 

method used to estimate daily natural flow at the Rest Lake 

Dam was a water-budget approach, which used lake stage and 

dam outflow data provided by the dam operator. A water-budget model was constructed and then calibrated with an automated parameter-estimation program by matching simulated 

flow-duration statistics with measured flow-duration statistics 

at the upstream gaging stations. After calibration of the waterbudget model, the model was used to compute natural flow at 

the dam from 1973 to 2011.

Daily natural flows at the dam, as computed by the 

adjusted drainage-area ratio method and the water-budget 

method, were used to compute monthly flow-duration values 

for the period of historical data available for each method. 

Monthly flow-durations provide a means for evaluating the 

frequency and range in flows that have been observed for 

each month over the course of many years. Both methods 

described the pattern and timing of measured high-flow and 

low-flow events at the upstream gaging stations. The adjusted 

drainage-area ratio method generally had smaller residual 

errors across the full range of observed flows and had smaller 

monthly biases than the water-budget method. Although it is 

not possible to evaluate which method may be more “correct” 

for estimating monthly natural flows at the dam, comparisons 

between the results of each method indicate that the adjusted 

drainage-area ratio method may be susceptible to biases at 

high flows due to isolated storms outside of the Manitowish 

River watershed. Conversely, it appears that the water-budget 

method may be susceptible to biases at low flows because of 

its sensitivity to the accuracy of reported lake stage and outflows, as well as effects of upstream diversions that could not 

be fully compensated for with this method. 

 Results from both methods are useful for understanding 

the natural flow patterns at the dam. Flows for both methods 

have similar patterns, with high median flows in spring and 

low median flows in late summer. Similarly, the range from 

monthly high-flow durations to low-flow durations increases 

during spring, decreases during summer, and increases again 

during fall. These seasonal patterns illustrate a challenge 

with interpreting a single value of natural low flow. That is, a 

natural low flow computed for September is not representative 

of a natural low flow in April. Moreover, alteration of natural 

flows caused by storing water in the Chain of Lakes during 

spring and releasing it in fall causes a change in the timing of 

high and low flows compared with natural conditions. That is, 

the lowest reported dam outflows occurred in spring and highest reported outflows occurred in fall, which is opposite the 

natural patterns. 

Estimation of Natural Historical Flows for the 

Manitowish River near Manitowish Waters, Wisconsin

By Paul F. Juckem, Paul C. Reneau, and Dale M. Robertson
2 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

Introduction

In 1887, a dam was constructed on the Manitowish River 

at the outlet of Rest Lake (the Rest Lake Dam) and formed the 

Manitowish Chain of Lakes. The Manitowish Chain of Lakes 

includes ten lakes with a total area of 4,266 acres in Vilas 

County, Wis. The Wisconsin Department of Natural Resources 

(WDNR) is charged with oversight of the dam operation, as 

outlined in a 1939 Public Service Commission hearing (Public 

Service Commission, 1939). As part of the 1939 ruling, the 

Public Service Commission aimed to balance the multiple 

needs and uses for water both upstream and downstream of the 

dam, including storage of water for downstream hydroelectricpower generation.

The current (2012) operation orders for Rest Lake Dam 

state that the minimum lake stage should be 5.0 feet (ft) above 

a specified local gage datum (1,597.9 ft above North American Vertical Datum of 1988 (NAVD 88)) and the maximum 

stage should be 8.5 ft (1,601.4 ft above NAVD 88). In addition, chap. 31 of Wisconsin State Statutes specifies that the 

minimum flow out of every dam in the State should be no less 

than 25 percent of the natural low flow (Wisconsin Statutes 

and Annotations, 2012). Dams alter the system from natural 

conditions, however, often leaving 25 percent of the natural 

low flow difficult to determine. In light of this uncertainty, the 

WDNR has equated 25 percent of natural low flow with the 

minimum 7-day mean flow that occurs once every 10 years 

(Q7,10) as described on page 3 of the “Current operation of Rest 

Lake Dam – frequently asked questions” (Wisconsin Department of Natural Resources, 2007). The WDNR has considered 

the Q7,10 for the Rest Lake Dam to be 40 cubic feet per second 

(ft3/s) because this is the flow that the dam operator has used 

in documentation of dam operation. Implementation of these 

operation orders and State statutes has resulted in management 

of the reservoir stage and minimum released flow during most 

years according to table 1.

Following a 2001 Federal Energy Regulatory Commission finding that “management of the water levels and flows 

[from the Chain of Lakes] was neither used and useful nor 

necessary or appropriate to maintain or operate for hydropower generation,” the Federal Energy Regulatory Commission relinquished its authority over the dam operation (Wisconsin Department of Natural Resources, 2011). In recognition 

of this change to the balance in competing uses of water in 

the Manitowish Chain of Lakes, and the WDNR’s obligations under the Public Trust Doctrine (Wisconsin Legislative 

Reference Bureau, 2004), the WDNR is proposing “to issue 

a new operating order to the owners of the Rest Lake Dam to 

specify water levels and flows that balance and protect public 

water resource rights as well as life, health, and property, both 

upstream and downstream of the dam” (Wisconsin Department of Natural Resources, 2011). As part of establishing new 

operating orders, the WDNR partnered with the U.S. Geological Survey (USGS) to install streamflow-gaging stations on 

the Manitowish River system and improve estimates of natural 

flow at the Rest Lake Dam.

There are several scientific challenges to estimating 

natural flow at the Rest Lake Dam, the most significant of 

which is the lack of historical data that were not affected by 

the dam operation. In addition, it is unclear if the Q7,10 is a 

good approximation of 25 percent of the natural low flow at 

the dam. As a result, this study focused on estimating flow 

durations that describe the frequency and range in natural 

flows on a monthly basis. In the Upper Wisconsin River Basin, 

which borders the Manitowish River watershed to the east, the 

Q7,10 has been estimated to approximately equal the 99.6 percent flow duration (Gebert, 1980). The 99.6 flow duration is 

the daily mean flow that is exceeded 99.6 percent of the time. 

Flow durations are typically evaluated using long-term records 

of daily mean flow, but can be computed for any recurring 

period, such as a week or month. Monthly flow-duration tables 

provide a means for evaluating the frequency and range in 

flows that are observed over the course of annual cycles that 

incorporate high-flow and low-flow conditions. It should be 

noted that the monthly 99.6 percent flow-duration values produced as part of this study are not expected to equate exactly 

with Q7,10 values, which are based on 7 consecutive days and, 

as a consequence, are computed using continuous records—

not for example, on a monthly basis. In addition, flow-duration 

estimates are a commonly used metric to describe the frequency and range in measured flows because the flow-duration 

estimates have greater resistance to extreme events (Helsel and 

Hirsch, 2002) than, for example, the Q7,10, which is a singular 

extreme event that would be difficult to estimate with indirect 

methods such as those described in this report.

Table 1. Generalized operation of the Rest Lake Dam. 

[ft, foot; in., inch; ft3/s, cubic foot per second; modified from James Kreitlow, 

Wisconsin Department of Natural Resources, written commun., May 29, 2012] 

Time period

Lake stage, in 

feet, on dam 

staff gage

Required 

flows

November 8 to spring ice breakup (when ice is 75 percent 

off of Manitowish Chain of 

Lakes; approximately 

April 20)

Minimum of 

5 ft 0 in.

Run of river

Spring ice break-up 

(approximately April 20) 

to July 1

5 ft 0 in. to 

8 ft 6 in.

40 ft3/s or more

July 1 to September 28 No lower than 

8 ft 4 in.

40 ft3/s or more

September 28 to November 8 No lower than 

5 ft 0 in.

40 ft3/s of more
Methods 3

Purpose and Scope

The purpose of this report is to describe the natural 

historical flow in the Manitowish River at State Highway 51 

near Manitowish Waters (station 05357302), as characterized 

by monthly flow-duration values. Because of a lack of direct 

flow measurements at the dam outlet, flow measured at State 

Highway 51 (05357302) was assumed to represent flow at the 

Rest Lake Dam, located approximately 1 mile (mi) upstream 

of State Highway 51. In addition, the methods used in this 

study involved computing historical average daily flows, 

but the scope of work is limited to describing the computed 

flow patterns as represented by monthly flow-duration tables. 

That is, every effort was made to compute accurate historical daily flows, but daily flows are not the focus of the study 

– the focus of the study is to produce summary statistics that 

describe broader historical flow patterns. This distinction is 

made because individual daily flow estimates are expected to 

incorporate greater uncertainty than summary statistics that 

describe the distribution of hundreds of individual daily flow 

estimates.

Physical Setting

The Manitowish River watershed upstream of State Highway 51, which covers approximately 250 square miles (mi2

), 

is part of the Northern Lakes and Forests ecoregion (Omernik 

and others, 2000). This area has low-to-moderate relief and 

contains many wetlands and headwater lakes (fig. 1). The 

surficial geology of the watershed consists of glacial deposits, including pitted outwash and other ice-contact deposits 

composed of sand, gravel, and glacial till. The depth of the 

surficial materials may be up to 100 ft over igneous and 

metamorphic bedrock (Young and Hindall, 1972; Mudrey and 

others, 1982). The groundwater-flow system is dominated by 

permeable glacial deposits, and is generally in good communication with the surface-water system (Pint, 2002; Walker 

and others, 2012). Soils of the rolling-to-undulating uplands

are primarily sandy loam, loamy sand, and sand, with permeabilities of 2.5 to 5 in/hr or more, and muck or peat soils in 

the wetlands (Hole and others, 1968; Oakes and Cotter, 1975). 

Flow into the Manitowish Chain of Lakes is primarily from 

four rivers: Manitowish River, Trout River, Rice Creek, and 

Papoose Creek.

Data Sources

Four streamflow-gaging stations were installed in October 2009 and monitored through November 2011 to provide 

flow data (Rantz and others, 1982) for evaluating 2 years of 

flow into and out of the Chain of Lakes (fig. 1). These gaging stations were located at the Manitowish River at State 

Highway 51 near Manitowish Waters (station 05357302), the 

Trout River near Boulder Junction (station 05357259), the 

Manitowish River near County Highway H near Boulder Junction (station 05357157), and Rice Creek at County Highway 

K near Boulder Junction (station 05357182). In addition, 

lake stage was also measured at the Rest Lake Dam (station 

460819089530500) using an automated digital recorder. Flow 

was measured intermittently at Papoose Creek at County 

Highway K near Manitowish Waters (station 05357299). All 

flow and lake-stage data collected as part of this study are 

available online from the USGS: http://waterdata.usgs.gov/wi/

nwis/sw.

Historical data for lake stage and flow over the Rest 

Lake Dam were provided by the dam operator. Historical flow 

data for the Bear River near Manitowish Waters (station ID 

05357335) and the Trout River at Trout Lake near Boulder 

Junction (station ID 05357245), which were used for regression analyses, were from the USGS National Water-Information system and are available online at http://waterdata.usgs.

gov/wi/nwis/sw.

Methods

Two independent methods, the adjusted drainage-area 

ratio method and the water-budget method, were used to 

estimate historical natural flow at the Rest Lake Dam. Both 

methods used the two years of measured total flow into the 

Manitowish Chain of Lakes to calibrate equations or parameters specific to each method. Then, daily natural flows at the 

Rest Lake Dam were computed using historical flow data from 

nearby streams for the adjusted drainage-area ratio method, 

and historical dam outflow data from the dam operator for 

the water-budget method. Finally, flow-duration tables were 

generated from the computed daily natural flows at the Rest 

Lake Dam.
4 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

!>

!>

!>

!> !>

!>

!>

!> 05357259

05357157

05357245

05357335

05357302

460819089530500

05357299

05357182

05357302

05357259

05357245

05357157

05357182

90 ̊0'0"W 89°55'0"W 89°50'0"W 89°45'0"W 89°40'0"W

46°10'0"N

46°5'0"N

46°0'0"N

REST

LAKE

REST 

ALDER 

LAKE

 Rice Creek

Papoose C r ee k

Trout RiverTrout River

Manitowish River Manitowish River

Bea r River Bea r River

Manitowish 

Waters

Boulder

Junction

0 1 2 4 MILES

0 1 2 4 KILOMETERS

EXPLANATION

State Highway 51

County and town roads

Cranberry bogs

Lakes and ponds

Rivers

Manitowish Chain of Lakes

Streamflow-gaging 

stations upstream of the 

Manitowish Chain of Lakes

!>

Streamflow-gaging 

stations used as part 

of the adjusted drainage-area 

ratio method

!>

Lake stage gaging station 

at the Rest Lake Dam and 

streamflow-gaging station 

at State Highway 51, 

downstream of the 

Manitowish Chain of Lakes

!>

Base map compiled and modified from digital

data sources: U.S. Geological Survey, 1:24,000

Streamflow direction

Manitowish 

 area

Figure 1. The Manitowish Chain of Lakes study area in north central Wisconsin and streamflow-gaging stations 

used during this study. 
Methods 5

The first step for both methods was to compute the total 

natural flow into the Manitowish Chain of Lakes, as measured at the four upstream gages on the major tributaries to 

the Chain of Lakes during the 2-year study (fig. 1). Given the 

relatively small flows in Papoose Creek (05357299) compared with the Manitowish River (05357157), Trout River 

(05357259), and Rice Creek (05357182), Papoose Creek was 

not continuously gaged. Instead, flow in Papoose Creek was 

measured 20 times during the study, and daily flows were 

estimated from daily flows measured in Rice Creek, using a 

regression relation developed between Papoose Creek and 

Rice Creek (fig. 2). In addition, measured daily flows for the 

Trout River near Boulder Junction (05357259) were adjusted 

to represent natural flow by removing the effects of diversions 

by cranberry bog operations (fig. 3). This was done because of 

the interest in natural flows and was accomplished by identifying rapid changes in flows caused by pump activation, which 

were visible in unpublished gaging station data recorded at 

15-minute intervals. Natural flows during times of these surface-water diversions were estimated by interpolating between 

prior and subsequent daily mean flows that appeared to be 

unaffected by the diversions. Except for an unknown amount 

of water loss due to evaporation from diverted water stored in 

holding ponds and transpiration of diverted water used for irrigation, the remainder of the diverted water at the Trout River 

gage (05357259) is expected to return to the Chain of Lakes as 

groundwater discharge to the Chain of Lakes, and is incorporated into estimated flows at the Rest Lake Dam as described 

below in “adjusted drainage-area ratio method.”

Total measured flow into the Manitowish Chain of Lakes 

( GageQmeasured ) was computed on a daily basis by summing daily 

values of adjusted natural flow at the Trout River at Boulder 

Junction (05357259), measured flow at Manitowish River near 

County Highway H near Boulder Junction (05357157), measured flow at Rice Creek at County Highway K near Boulder 

Junction (05357182), and estimated flow at Papoose Creek 

at County Highway K near Manitowish Waters (05357299). 

Total measured flow into the Manitowish Chain of Lakes was 

then used to develop a relation with nearby long-term gaging 

stations (the adjusted drainage-area ratio method) and to calibrate a water-budget model that used long-term dam operation 

records.

Figure 2. Measured flows at Rice Creek at County Highway K near Boulder Junction (station 05357182) and Papoose 

Creek at County Highway K near Manitowish Waters (station 05357299) during the 2-year study period. 

0

5

10

15

20

25

30

35

Measured streamflow in Papoose Creek, in cubic feet per second

Measured streamflow in Rice Creek, in cubic feet per second

0 5 10 15 20 25 30 35 40 45

y=2.171e 0.0573x

R2=0.75

EXPLANATION

Regression line

Measured streamflow
6 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

Lake stage and dam outflow reported by the dam operator were used in both the adjusted drainage-area ratio method 

and the water-budget method. A few adjustments were needed 

to make the lake stage and flow data reported by the dam 

operator more useful for this study. First, flows for periods 

with missing data or very questionable data (for example, 

large changes in lake stage without a corresponding change 

in dam outflow) were estimated, typically by linear interpolation between the preceding and subsequent values. Second, 

flows reported by the dam operator were known to differ from 

flows measured by the WDNR and USGS between July 2002 

and November 2011 at the downstream gage at State Highway 

51 (05357302); therefore, the results of a regression equation were used to adjust reported historical dam outflow data 

from 1973 to April 28, 2009, to flows measured by the WDNR 

from July 2002 to April 28, 2009 (fig. 4A). A second equation 

was applied to the reported dam outflow data from April 29, 

2009, to November 30, 2011, using daily gaging station data 

from December 1, 2009, to November 30, 2011 (fig. 4B). Two 

separate equations were needed because dam improvements 

performed on April 29, 2009 (Dean Steines, written commun., 

May 26, 2009) changed the relation between estimated flows 

provided by the dam operator and flows measured by the 

WDNR and USGS (fig. 4).

Adjusted Drainage-Area Ratio Method

The adjusted drainage-area ratio method involved two 

steps to estimate natural historical flow at the Rest Lake Dam. 

The first step was to estimate total flow into the Manitowish 

Chain of Lakes from nearby long-term streamflow-gaging 

stations. The second step was to adjust this computed flow into 

the Chain of Lakes into natural flow at the downstream Rest 

Lake Dam.

Estimating total flow into the Manitowish Chain of 

Lakes was accomplished by relating measured flow divided 

by watershed area (water yield) at the up-stream gages for 

the monitoring period (December 2009 to December 2011) 

and total water yield for two nearby long-term gaging stations, the Bear River near Manitowish Waters (05357335) and 

Trout River at Trout Lake near Boulder Junction (05357245). 

Two different rivers were used instead of one river to better 

describe the overall patterns in runoff throughout the area. 

Yields were used for the regression equation rather than flows 

to ensure that differences in stream size were minimized so 

that both the Bear River and Trout River gages had equal 

influence on the relation with measured flow at the gages 

( GageQmeasured ). 

-20

0

20

40

60

80

100

120

140

Streamflow, in cubic feet per second

Measured daily streamflow

EXPLANATION 

Estimated natural streamflow

12/1/2009

1/1/2010

2/1/2010

3/1/2010

4/1/2010

5/1/2010

6/1/2010

7/1/2010

8/1/2010

9/1/2010

10/1/2010

11/1/2010

12/1/2010

1/1/2011

2/1/2011

3/1/2011

4/1/2011

5/1/2011

6/1/2011

7/1/2011

8/1/2011

9/1/2011

10/1/2011

11/1/2011

11/30/2011

Date

Figure 3. Estimated flow without the effects of upstream diversions and measured flow for the Trout River near Boulder 

Junction (station 05357259). Negative flows were measured when lake water flowed into the stream channel. 
Methods 7

0

50

100

150

200

250

300

350

400

Measured streamflow, in cubic feet per second

Measured streamflow, in cubic feet per second

Dam operator reported streamflow, in cubic feet per second

Dam operator reported streamflow, in cubic feet per second

0

100

200

300

400

500

600

0 100 200 300 400 500 600

0 50 100 150 200 250 300 350 400

y=1.0158x + 24.773

R2=0.87

y=0.9656x + 19.444

R2

=0.97

EXPLANATION

Regression line

Measured streamflow

EXPLANATION

Regression line

Measured streamflow

B

A

Figure 4. Flows measured by the Wisconsin Department of Natural Resources and 

U.S. Geological Survey at the Manitowish River at State Highway 51 (station 05357302) 

and dam outflows at the Rest Lake Dam outlet reported by the dam operator. 

A, July 2002 to April 28, 2009. B, Dec. 1, 2009, to Nov. 30, 2011. 
8 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

One of two Line of Organic Correlation equations (equations 1 and 2) were used to compute yield at the upstream 

gages, dependent upon on the measured lake stage:

 GageYlow_stage = 0.3676(YBear+YTrout) + 0.2809 (1)

 GageYhigh_stage = 0.3789(YBear+YTrout) + 0.1662 (2)

where

YBear and YTrout are the daily yields for the Bear 

River near Manitowish Waters 

(05357335) and the Trout River 

at Trout Lake near Boulder 

Junction (05357245), 

respectively, and

GageYlow_stage and GageYhigh_stage are the total computed yield 

for all of the upstream gages at 

low and high lake stages, 

respectively.

Line of Organic Correlation equations (Helsel and 

Hirsch, 2002, p. 276 –278) were used for the regression instead 

of more common regression methods, such as Ordinary Least 

Squares, to better maintain the distributional properties (standard deviation or percentiles) of the estimated yield for the 

upstream gaging stations ( GageYlow_stage and GageYhigh_stage ). That 

is, Ordinary Least Squares techniques, which are the default 

method for many common spreadsheet programs, slightly 

reduce the statistical variance (square of the standard deviation) of computed values, and therefore, their use would have 

slightly underestimated the predicted peak yields and overestimated the predicted minimum yields. 

Two regression equations were used to compute yield at 

the upstream gaging stations because flow into the Chain of 

Lakes differs based upon the lake stage in the Chain (fig. 5). 

Figure 5. Regression relation for total water yield for gaging stations 

upstream of the Manitowish Chain of Lakes and the total water yield for the 

Bear River near Manitowish Waters (station 05357335) plus the Trout River 

at Trout Lake near Boulder Junction (station 05357245). Both equations 

shown in the graph used data for which the combined yield for the Bear and 

Trout Rivers was more than 3.5 cubic feet per second per square mile. 

Water yield used with both 

regressions

0.0

0.5

1.0

1.5

2.0

2.5

Water yield during falling and 

low lake stage

Water yield during rising and 

high lake stage

Regression line for yields 

at falling and low lake stage

Regression line for yields 

at rising and high lake stage

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Total water yield for the Bear and Trout Rivers, in cubic feet per second per square mile

Total water yield for streamflow-gaging stations upstream of the Manitowish Chain of Lakes,

 in cubic feet per second per square mile

y=0.3676x+ 0.2809

R2=0.96

y=0.3789x + 0.1662

R2

=0.94

EXPLANATION
Methods 9

When the lake stage is high or rising, some flow is stored in 

the channels upstream of the gages, resulting in less actual 

flow at the upstream gaging stations. It is important to note 

that acoustic velocity meters, which account for such “backwater” effects as part of the flow measurement, were used at 

the affected gages. Relatively low flows during high or rising 

lake stage suggest that this lake stage-dependent relation is 

associated with physical processes such as reduced groundwater discharge upstream of the gages and storage within 

the channels, and not measurement error or bias. To account 

for this lake stage-dependent relation with flow, equation 1 

( GageYlow_stage ) was used for estimating historical yield into the 

Chain of Lakes when the lake stage was below 1,598 ft (5.1-ft 

gage height) or when the stage was falling between 1,600.5 ft 

(7.6-ft gage height) and 1,598 ft. Equation 2 ( GageYhigh_stage ) was 

used for estimating historical yield into the Chain of Lakes 

when the lake stage was above 1,600.5 ft or rising between 

1,598 and 1,600.5 ft. The 1,598 ft and 1,600.5 ft levels were 

used to transition between equations 1 and 2 because analyses 

demonstrated that when the lake stage is between these elevations, flows into the Chain of Lakes differ depending upon 

whether the lake stage is rising or falling. Flows into the Chain 

of Lakes are not affected by rising or falling lake stages when 

lake stages are below 1,598 ft or above 1,600.5 ft. A 5-day 

moving average of lake stage was used to select between equations 1 and 2 in order to dampen changes in lake stage that 

were reported only to the nearest inch by the dam operator. 

The 5-day moving average minimized short-term fluctuations 

in lake stage and yet captured the general trend. The effect 

of high or low lake stages on measured flow into the Chain 

of Lakes is small for very large flows. Thus, both regression 

equations used data for which the combined yield for the Bear 

and Trout Rivers was more than 3.5 cubic feet per second per 

square mile [(ft3/s)/mi2] (fig. 5). 

Finally, computed yield into the Chain of Lakes was 

multiplied by the watershed area upstream of the gages 

(185.6 mi2) to compute the total daily flow into the Chain of 

Lakes ( GageQin), following equation 3:

GageQin = GageYin * GageA, (3)

where

GageQin is the total daily flow into the Manitowish 

Chain of Lakes at the upstream gaging 

stations, 

GageYin is the computed daily yield at the upstream 

gages and represents either GageYlow_stage

or GageYhigh_stage from equations 1 or 2, as 

determined from the daily measured lake 

stage, and 

GageA is the total watershed area that contributes to 

the upstream gaging stations (185.6 mi2, or 

5,174,231,000 square feet (ft2)).

Equations 1, 2, and 3 were calibrated with measured 

flows from Dec. 2009 to Nov. 2011, and then applied to all 

historical flow data for the Bear and Trout Rivers to estimate 

historical natural flow into the Chain of Lakes from Dec. 1991 

to Nov. 2011 (the period of record for the Bear and Trout 

River gages).

The second step of the adjusted drainage-area ratio 

method was to use estimated natural flows upstream of the 

Chain of Lakes ( GageQin ) to compute natural flow at the downstream dam ( DamQnat ). The simplest application of the drainagearea ratio method is to use a single coefficient that is equal to 

the area upstream of the dam divided by the monitored area 

( GageA, or the area upstream of the tributary gages, fig. 1) to 

adjust upstream flows ( GageQin ) to downstream flows at the 

dam ( DamQnat ). Such a constant coefficient approach assumes 

that the hydrologic contributions (precipitation, evaporation, groundwater, and runoff) associated with the additional 

watershed area are similar to those in the monitored area and 

are constant through time. In the adjusted drainage-area ratio 

method used in this study, however, the total natural flows at 

the gages ( GageQin ) were multiplied by a coefficient that differed for each month (table 2). Values for the monthly coefficients were computed using equation 4: 

Ki = ( DamQout +∆S ) / GageQin (4)

where

Ki is the coefficient for a particular month of the 

year (i), 

DamQout is the monthly average measured flow at State 

Highway 51 (05357302) over the 2-year 

study period and represents dam outflow,

GageQin is the monthly average computed flow into 

the Chain of Lakes at the upstream gaging 

stations over the 2-year study period, as 

computed from equation 3, and 

ΔS is the monthly average change in water 

stored in the Chain of Lakes due to 

lake stage changes measured at station 

460819089530500 over the 2-year study 

period.
10 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

This approach ensured that the monthly coefficients (Ki ) 

were computed such that the product of each coefficient (Ki ) 

and the monthly average computed flow into the Chain of 

Lakes ( GageQin ) minus the monthly average change in storage 

(∆S  ) equaled the monthly average measured flows at State 

Highway 51 ( DamQout ) for each month during the 2-year study 

period. That is, the approach ensured proper water balance on 

a monthly basis. Monthly coefficients were used, rather than 

a single constant coefficient, because monthly coefficients 

account for seasonal differences in precipitation, evaporation, 

runoff, and groundwater discharge to the Manitowish Chain 

of Lakes between the upstream gages ( GageQin ) and the dam 

( DamQnat ). Finally, the monthly coefficients were used to compute natural flow at the dam ( DamQnat ) according to equation 5:

DamQnat = Ki

 * GageQin (5)

where

Ki is the coefficient for a particular month of the 

year (i),

DamQnat is the historical daily natural flow at the Rest 

Lake Dam, and

GageQin is the historical daily computed flow into the 

Chain of Lakes at the upstream gaging 

stations, as computed from equation 3.

Differences among the monthly K values are attributed 

to changes in precipitation, evaporation, near-shore runoff, 

and groundwater discharge over the calibration period of 

Dec. 2009 to Nov. 2011. All monthly values are greater than 

1, indicating an increase in flow from the upstream gages to 

the downstream dam during all months. Much of this annual 

increase between the upstream gages and downstream dam is 

expected to derive from groundwater discharge into the Chain 

of Lakes, particularly during the winter when precipitation, 

runoff, and evaporation are expected to be near zero. The 

K values increase during spring, likely reflecting increased 

precipitation, near-shore runoff and groundwater discharge 

due to spring recharge. The K value decreases in May, likely 

due to a rising lake stage relative to groundwater levels, which 

would reduce groundwater discharge into the Chain of Lakes, 

as well as due to an increased rate of evaporation. The K

values gradually increase through summer, possibly indicating a gradual establishment of equilibrium between the lake 

stage and groundwater levels. The K values increase again in 

late fall, likely due to increased groundwater discharge due 

to lowering of the lake stage, as well as decreased evaporation. The K values steadily decrease again during the winter, 

likely as the lake stage and groundwater levels equilibrate and 

groundwater discharge becomes the primary source of water 

entering the Chain of Lakes between the upstream gages and 

the downstream dam.

As described earlier, natural flows in the Trout River 

were adjusted to account for diversion of water by cranberry 

operations near the Trout River near Boulder Junction gaging station (05357259), as shown in figure 3. Except for an 

unknown amount of water loss due to evaporation of diverted 

water stored in holding ponds and transpiration of diverted 

water used for irrigation, the remainder of the diverted water 

at the Trout River near Boulder Junction gage (05357259) is 

expected to return as groundwater discharge into the Chain of 

Lakes. This returned diversion water is accounted for by the 

monthly coefficients (K), in that the coefficient values were 

computed such that the total water into and out of the Chain of 

Lakes, as computed at the Rest Lake Dam, was balanced on a 

monthly basis. In other words, while the monthly coefficient 

(K) values do not identify the individual sources of water 

entering the Chain of Lakes between the upstream gaging 

stations and the downstream dam, the coefficients do account 

for the total amount of water entering the Chain of Lakes 

between the gaging stations and the dam. That is, returned

diversion water is incorporated into the monthly coefficients. 

Moreover, because no known cranberry operations exist in the 

Bear River (05357335) and the upstream Trout River at Trout 

Lake (05357245) watersheds, the adjusted drainage-area ratio 

method eliminates effects of diversions within the Chain of 

Lakes watershed on computed natural flow at the Rest Lake 

Dam ( DamQnat). Nonetheless, possible diversions elsewhere in 

the system, such as near Alder Lake (fig. 1), were not considered during this analysis due to a lack of data.

Water-Budget Method

The water-budget method also used two steps to estimate 

natural historical flow at the Rest Lake Dam. The first step 

was to calibrate the number of days (N-days) over which to 

average input data (lake stage, dam outflow, precipitation, and 

evaporation) and estimate groundwater discharge parameters 

by comparing computed flows with measured flows at the 

upstream gages for the 2-year study period. The second step 

was to average all historical lake stage and dam outflow data 

provided by the dam operator over the calibrated number of 

days (N-days) to compute historical natural flow at the downstream Rest Lake Dam.

Table 2. Monthly coefficients (K values) that were multiplied by total daily flow into the Chain of Lakes (GageQin) to compute daily values 

of natural flow at the Rest Lake Dam (DamQnat) for the adjusted drainage-area ratio method.

Month Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.

Coefficient 

(Ki)

1.155 1.068 1.237 1.328 1.081 1.113 1.130 1.185 1.276 1.289 1.420 1.356
Methods 11

The water budget for the Manitowish Chain of Lakes is 

described by equation 6:

ΔS = P – E + GWin + GageQin – DamQout (6)

where

ΔS is change in storage of the Manitowish Chain 

of Lakes and is equal to the sum of water 

entering the Chain of Lakes minus the sum 

of water leaving the Chain of Lakes,

P is precipitation on the lake surface,

E is evaporation from the lake surface,

GWin is the net groundwater discharge into the 

Chain of Lakes, 

GageQin is the natural flow into the Chain of Lakes at 

the upstream gaging stations, and

DamQout is the dam outflow. 

Realizing that the change in storage (∆S) accounts for 

all flows into and out of the Chain of Lakes, natural flow at 

the dam can be computed directly from lake stage and dam 

outflow data provided by the dam operator using equation 7:

DamQnat = DamQout +∆S (7)

where

ΔS is change in storage of the Manitowish Chain 

of Lakes and is positive when flows 

entering the Chain of Lakes exceed flows 

leaving the Chain of Lakes (the lake stage 

rises), and is negative when flows leaving 

the Chain of Lakes exceed flows entering 

the Chain of Lakes (the lake stage falls),

DamQout is the dam outflow, and

DamQnat is natural flow at the Rest Lake Dam.

In comparing equations 6 and 7, it is apparent that natural 

flow at the Rest Lake Dam ( DamQnat ) represents all of the 

natural sources and sinks of water (precipitation, evaporation, groundwater discharge, and flow at the upstream gages) 

to the Chain of Lakes. However, directly solving equation 7 

with daily lake stage and dam outflow data results in spurious oscillations (unrealistically high and low flows) of the 

computed natural flow at the dam. That is, because of the large 

volume of the Chain of Lakes, small changes in lake stage 

represent a large amount of water added or removed from 

storage. For example, a 1-inch decrease in lake stage releases 

14,800,000 cubic feet (ft3) of water, or 171 ft3/s, out of the 

Chain of Lakes, which is substantial given that the average 

daily flow out of the Chain of Lakes during the 2-year study 

period was 165 ft3/s. To reduce the magnitude of computed 

oscillations, or “flashiness,” daily lake stage and dam outflow 

data can be averaged over many days; however, the most 

appropriate number of days for averaging is not known in 

advance. Averaging too many days would eliminate natural 

fluctuations, while averaging too few days would not adequately reduce spurious oscillations. For example, use of daily 

data (1-day averaging) does not reduce spurious oscillations, 

whereas averaging over 365 days eliminates natural seasonal 

fluctuations.

Instead of solving equation 7 directly, the water budget 

for the Chain of Lakes (equation 6) was rewritten into a computer model (see appendix) to solve for flow at the upstream 

gages ( GageQin) to facilitate comparison with measured flow at 

the upstream gaging stations ( GageQmeasured), as shown in equation 8:

GageQin = –P + E – GWin + DamQout +∆S (8)

where the variables are the same as those defined for equation 6. Solving for flow at the upstream gages ( GageQin) was 

necessary to “tune” or calibrate properties of the model (primarily the number of days over which lake stage, dam outflow, 

precipitation, and evaporation were averaged or smoothed) by 

comparing the computed flow at the upstream gages ( GageQin) 

with measured flow ( GageQmeasured) from the 2-year study period. 

Equation 8 was solved with the computer model using the 

adjusted lake stage and dam outflow reported by the dam 

operator, precipitation data from the nearby National Weather 

Service station at Rest Lake (Coop ID 477092), and biweekly 

evaporation rates averaged over a decade for nearby (6 mi 

southeast) Sparkling Lake (Lenters and others, 2005).

The model, as represented by equation 8, was calibrated 

with the automated calibration program, PEST (Doherty, 

2011), by matching simulated flow-duration statistics with 

measured flow-duration statistics at the upstream gaging 

stations ( GageQmeasured ) for the 2-year study period (table 3). 

Flow-duration statistics describe the percentage of time that a 

particular flow is equaled or exceeded (see Natural Historical 

Flow Durations, below, for additional descriptions). A detailed 

description of model parameter estimation and calibration 

is beyond the scope of this report; Aster and others (2005), 

Doherty and Hunt (2010), and Hill (1998) provide detailed 

descriptions of model calibration methods. More generally, 

calibration of the model results (simulated values) to match 

flow-duration statistics for measured flows (target values) was 

performed by adjusting the number of days (N-days) over 

which lake stage, dam outflow, precipitation, and evaporation were averaged (centered on the date being computed), in 

addition to estimating four groundwater discharge parameters. 

N-days was the primary mechanism for matching the range 

in magnitude (minimum and maximum flows) of simulated 

flows to the range in magnitude of measured flows from the 

2-year study period. Inherently, the calibrated value for N-days 

incorporates some of the limitations of the method as well as 

the precision of the input data. Groundwater discharge to the 

Chain of Lakes was unknown and therefore estimated through 

the calibration process.
12 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

Groundwater discharge into the Chain of Lakes was 

expected to exceed the loss of lake water into the groundwater 

system, because the region has high groundwater recharge 

(Gebert and others, 2009) and the K values for the Adjusted 

Drainage-Area Ratio method (table 2), particularly those during the winter, were all greater than 1, indicating a net gain of 

water from upstream to downstream that likely included a high 

percentage of groundwater discharge. Groundwater discharge 

was assumed to increase from relatively low to relatively high 

amounts across four conditions represented by (1) rising lake 

stage, (2) high lake stage, (3) low lake stage, and (4) falling 

lake stage. This pattern of groundwater discharge is conceptualized as being driven by relative changes in the hydraulic gradient between groundwater and the Chain of Lakes caused by 

operational changes in the lake stage. That is, relatively rapid 

increases in lake stage are expected to decrease the hydraulic 

gradient between groundwater and the Chain of Lakes, during 

which time groundwater discharge into the Chain of Lakes is 

at a minimum. Conversely, relatively rapid lowering of lake 

stage would increase the hydraulic gradient between groundwater and the Chain of Lakes, thus maximizing groundwater 

discharge. Periods of stable lake stage are expected to exhibit 

transitional hydraulic gradients, during which time intermediate rates of groundwater discharge are expected. 

Groundwater discharge into the Chain of Lakes was 

estimated as a residual term during the calibration process, and 

was simulated using four adjustable parameters (table 4) based 

upon the measured lake stage: (1) GW_rising_lake_stage, 

(2) GW_high_stage_ratio, (3) GW_low_stage_ratio, and 

(4) GW_dropping_stage_ratio. The parameter representing 

the period of minimum groundwater discharge, GW_rising_lake_stage, was estimated directly during calibration. The 

remaining parameters were computed as ratios tied to GW_rising_lake_stage and then summed in sequence to ensure that 

total groundwater discharge into the Chain of Lakes increased 

incrementally from periods of rising lake stage to periods of 

high lake stage, to periods of low lake stage, and finally to 

periods of dropping lake stage (table 4). A minimum value 

was specified for each ratio as part of the calibration process to 

ensure that total groundwater discharge increased in the specified sequence. Although the calibrated values are reasonable 

based on prior groundwater studies in the area (Pint, 2002; 

Hunt and others, 2005; Hunt and others, 2008; Walker and 

others, 2012), this relationship was not tested in a physical 

framework such as by an evaluation of field measurements or 

a model incorporating known governing laws for the physics 

of groundwater movement; thus, the groundwater discharge 

estimates should not be used beyond their application in this 

report. In addition, the groundwater parameters inherently 

account for any diverted water by cranberry operations that 

seeps into the ground and returns to the Chain of Lakes.

Table 3. Target flow-duration statistics for the upstream gaging stations (GageQmeasured) for the 2-year study 

period, simulated equivalent values, residuals, and associated target weights from the water-budget 

model. 

[ft3/s, cubic foot per second; sqrt, square root; FD, flow duration, is followed by a number indicating the percentage of time 

that the target value is equaled or exceeded] 

Target description

Target value 

(ft3/s)

Simulated value 

(ft3/s)

Residual 

(target – simulated, ft3/s)

Weight (unitless)

1/sqrt (target value)

Minimum flow 66.6 52.5 14.1 0.1225

FD98 71.0 65.9 5.1 .1187

FD95 74.1 69.1 5.0 .1162

FD90 81.3 72.9 8.4 .1109

FD80 91.7 76.6 15.1 .1044

FD70 96.9 102.1 −5.2 .1016

FD60 109 117 −8 .0957

FD50 134 138 −4 .0864

FD40 148 147 1 .0821

FD30 162 158 4 .0786

FD20 174 180 −6 .0758

FD10 195 213 −18 .0716

FD5 219 232 −13 .0676

FD2 234 243 −9 .0654

Maximum flow 306 259 47 .0572
Methods 13

Following estimation of values for the N-days and 

groundwater discharge parameters through calibration of the 

water-budget model (equation 8) using measured flow into the 

Chain of Lakes ( GageQmeasured ), equation 9 was used to compute 

natural flow at the Rest Lake Dam ( DamQnat ):

DamQnat = Plow-stage – Elow-stage + GWin + GageQin (9)

where

DamQnat is natural flow computed at the Rest Lake 

Dam,

Plow-stage is the volume of water that falls as 

precipitation onto the surface area of the 

lake at low stage (1,598 ft), 

Elow-stage is the volume of water that evaporates from 

the surface area of the lake at low stage 

(1,598 ft),

GWin is the estimated groundwater discharge into 

the Chain of Lakes, as represented by 

one of the four calibrated groundwater 

discharge values tied to the measured lake 

stage, and

GageQin is the computed natural flow into the Chain of 

Lakes at the upstream gaging stations, as 

estimated from equation 8.

Precipitation and evaporation were computed using the 

low lake-stage area because large operational increases in 

lake stage would not occur under natural flow conditions. 

Natural fluctuations of lake stage are not known and were not 

estimated.

Although the first step of the water-budget method (the 

calibration step) was limited to the 2-years for which flows at 

the upstream gages were measured (2009–11), the second step 

used all available historical lake stage and dam outflow data 

provided by the dam operator to compute natural flows at the 

Rest Lake Dam. The procedure for computing daily natural 

historical flow at the dam ( DamQnat ) consisted of first solving 

for flow into the Chain of Lakes ( GageQin ) with equation 8 by 

averaging daily precipitation, evaporation, and the adjusted 

outflow and lake stage data provided by the dam operator over 

N-days (23 days). Then, the result of equation 8 ( GageQin ) was 

combined with the appropriate stage-dependent groundwater discharge parameter value (table 4) and N-day averaged 

precipitation and evaporation amounts computed with the low 

lake stage area to solve equation 9. This procedure, as implemented in the computer code provided in the appendix, was 

performed for each day that adjusted dam outflow and lake 

stage data were provided by the dam operator (Dec. 1973–

Nov. 2011). 

Some components of the water budget calculation 

(precipitation, evaporation, and groundwater) are of limited 

significance to the final calculation ( DamQnat ). This is because 

precipitation, evaporation, and groundwater were primarily 

used as a means to determine the number of days (N-days) 

over which to average input data provided by the dam operator, which acts as the primary mechanism for matching the 

range in computed and measured flows at the upstream gages. 

That is, although daily values of precipitation, evaporation, 

and groundwater discharge are important for computing 

natural flow at the upstream gages ( GageQin ) using equation 8, 

these components of the water budget are of the opposite 

sign in equation 9, and therefore are largely removed (“cancelled out”) during the process of computing natural flow at 

the dam ( DamQnat ). The only difference in the daily volume of 

precipitation and evaporation used in equations 8 and 9 is the 

difference in the lake surface area over which precipitation 

and evaporation occurs on the lake. The daily groundwater 

discharge values applied in equations 8 and 9 are identical, thus effectively eliminating the calibrated groundwater 

Table 4. Parameter names and descriptions, and calibrated values for the water-budget method.

[ft3/s, cubic foot per second; >, greater than; ft, foot; <=, less than or equal to]

Parameter name Parameter description Calibrated value

N-days Number of days over which reported lake stage, dam outflow, precipitation, 

and evaporation were averaged

23 days

GW_rising_lake_stage Simulated net groundwater discharge to the Manitowish Chain of Lakes 

(between the upstream gages and the dam) when the lake stage was rising

24.7 ft3/s

GW_high_stage_ratio A ratio multiplied by GW_rising_lake_stage to simulate net groundwater 

discharge to the Chain of Lakes when the lake stage is high (> 1,599.5 ft)

1.0, 

translates to 24.7 ft3/s 

at high stage

GW_low_stage_ratio A ratio multiplied by GW_rising_lake_stage and added to the discharge represented by GW_high_stage_ratio (24.7 ft3

/s) to simulate net groundwater 

discharge to the Chain of Lakes when the lake stage is low (<= 1,599.5 ft)

0.015, 

translates to 25.1 ft3/s 

at low stage

GW_dropping_stage_ratio A ratio multiplied by GW_rising_lake_stage and added to the discharge 

represented by GW_low_stage_ratio (25.1 ft3/s) to simulate net groundwater 

discharge to the Chain of Lakes when the lake stage is dropping

0.62, 

translates to 40.4 ft3/s 

during dropping stage
14 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

parameter values from direct computation of natural flow at 

the dam ( DamQnat ). In other words, only the low-stage lake area 

used to compute precipitation and evaporation volumes and 

the number of days (23) used to average the input data directly 

affect the conversion of lake stage and dam outflow data provided by the dam operator into computed natural flow at the 

dam ( DamQnat ). 

Natural Flows at the Rest Lake Dam

Comparison of the measured ( GageQmeasured ) and computed 

flow at the gages ( GageQin ) for the 2-year study period helps 

in evaluating the relative accuracy of the adjusted drainagearea ratio method and the water-budget method (figs. 6 and 

7). Both methods generally described the pattern and timing 

of high-flow and low-flow events at the upstream gages. The 

adjusted drainage-area ratio method, however, generally had 

smaller residual errors over the full range of observed flows 

and had smaller monthly biases than the water-budget method. 

Although a close fit with the measured flows at the upstream 

gages is important, the focus of the study is to estimate natural 

flow at the dam ( DamQnat ). Both methods produce similar total 

annual flows at the dam (table 5). Unfortunately, neither 

method can be directly evaluated against daily dam outflows, 

because dam operation alters the natural flow pattern. 

Results from the two methods can be used to aid in 

understanding how current dam operations affect water levels 

in the Chain of Lakes and flows downstream of the dam. The 

median flow computed for each day of the year using all years 

with available data is shown for each method in figure 8. 

Alteration of dam outflows caused by storing water in the 

Chain of Lakes during spring and releasing water from the 

Chain of Lakes in fall is readily apparent, with reported dam 

outflows lowest in spring and highest in fall. The computed 

natural flow hydrographs for both methods have peak flows 

in spring, minimal flows in late summer or early fall, and a 

gradual increase in flows during fall. Natural and observed 

flows for particular years vary and may not match this pattern 

exactly, but these typical patterns are useful for understanding the relationship between capturing and releasing water for 

storage in the Chain of Lakes and downstream flows.

Table 5. Annual total measured dam outflow, total computed natural flow, and difference between measured dam outflows and 

computed natural flows from the water-budget and adjusted drainage-area ratio methods at the Rest Lake Dam. 

[ft3, cubic foot] 

Time period

Total measured 

dam outflow 

(ft3)

Total computed 

natural flow 

at the dam from the 

water-budget method 

(ft3)

Total computed 

natural flow 

at the dam from 

the adjusted 

drainage-area 

ratio method 

(ft3)

Difference between 

measured dam 

outflow and 

computed 

natural flow from the 

water-budget method 

(percent)

Difference between 

measured dam 

outflow and 

computed natural 

flow from the 

adjusted drainagearea ratio method 

(percent)

Dec. 2009 – Nov. 2010 5,063,000,000 4,937,000,000 5,132,000,000 2.5 −1.3

Dec. 2010 – Nov. 2011 5,324,000,000 5,450,000,000 5,275,000,000 −2.4 .9

Dec. 2009 – Nov. 2011 10,387,000,000 10,387,000,000 10,407,000,000 .0 −.2
Natural Flows at the Rest Lake Dam 15

Figure 6. Measured total flow at gaging stations upstream of the Manitowish Chain of Lakes and computed flow into 

the Chain of Lakes using the adjusted drainage-area ratio and water-budget methods. 

1,590

1,592

1,594

1,596

1,598

1,600

1,602

0

50

100

150

200

250

300

350

Stage, in feet above North American Vertical Datum of 1988

Streamflow, in cubic feet per second

Date

Measured total streamflow into the Manitowish Chain of Lakes

Water-budget method computed streamflow into the Manitowish Chain of Lakes

Adjusted drainage-area ratio method computed streamflow into the Manitowish Chain of Lakes

U.S. Geological Survey measured lake stage

12/1/2009

1/1/2010

2/1/2010

3/1/2010

4/1/2010

5/1/2010

6/1/2010

7/1/2010

8/1/2010

9/1/2010

10/1/2010

11/1/2010

12/1/2010

1/1/2011

2/1/2011

3/1/2011

4/1/2011

5/1/2011

6/1/2011

7/1/2011

8/1/2011

9/1/2011

10/1/2011

11/1/2011

EXPLANATION
16 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

Figure 7. Monthly error in estimated flow at the gaging stations upstream of the Manitowish Chain of Lakes for the adjusted 

drainage-area ratio and water-budget methods. 

−30

−20

−10

0

10

20

30

40

Difference between monthly average computed and 

measured streamflows, in cubic feet per second

Difference between monthly average water-budget method computed 

 streamflow and measured streamflow at the upstream gages

Difference between adjusted drainage-area ratio method computed 

 streamflow and measured streamflow at the upstream gages

12/1/2009

1/1/2010

2/1/2010

3/1/2010

4/1/2010

5/1/2010

6/1/2010

7/1/2010

8/1/2010

9/1/2010

10/1/2010

11/1/2010

12/1/2010

1/1/2011

2/1/2011

3/1/2011

4/1/2011

5/1/2011

6/1/2011

7/1/2011

8/1/2011

9/1/2011

10/1/2011

11/1/2011

Date

EXPLANATION
Natural Flows at the Rest Lake Dam 17

1,590

1,592

1,594

1,596

1,598

1,600

1,602

0

50

100

150

200

250

300

350

400

Median daily lake stage, in feet above 

North American Vertical Datum of 1988 

Median daily streamflo

w, in cubic feet per second

Median reported dam outflow (adjusted) 1973–2011

Median reported dam outflow (adjusted) 1991–2011

Median computed streamflow from the water-budget 1973–2011

Median computed streamflow from the water-budget 1991–2011

Median computed streamflow from the adjusted drainage-area ratio 1991–2011

Median reported lake stage (adjusted) 1973–2011

EXPLANATION

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

Figure 8. Median daily flow at the Rest Lake Dam for each calendar day over the period of record 

available for each method. (Dashed lines represent the median values for only 1991–2011 for the waterbudget method and reported dam outflows, and are included for direct comparison with the adjusted 

drainage-area ratio method.) 
18 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

One impetus for this study was to improve understanding 

of natural flow at the dam during extreme conditions, which 

cannot be evaluated using average annual values such as those 

in table 5. Instead, a comparison of the computed long-term 

natural flow at the dam for both methods (fig. 9) allows for 

qualitative evaluation of historical extreme flows. Although 

the comparison does not allow one to evaluate which method 

may be more accurate, it is useful for identifying potentially 

problematic anomalies associated with each method. For 

example, the adjusted drainage-area ratio method indicated a 

large runoff event during the fall of 1994. Closer evaluation 

of this event, which was not computed by the water-budget 

method, indicated that the event was associated with high 

flows measured in the Bear River that were not, however, 

measured in the Trout River. This indicates that the assumption that weather patterns driving flows in the Bear, Trout, and 

Manitowish Rivers are the same may not hold at all times. 

Violation of this assumption is expected to be more important 

for high flows than low flows, because droughts are typically 

regional, whereas summer storms are often local. Conversely, 

several negative and near-negative natural flows at the dam 

were computed with the water-budget method for Oct. 1974, 

Aug.–Sept. 1976, Aug.–Oct. 1989, July–Sept. 2005, and July 

2006. Some of the near-negative natural flows that occurred in 

fall (for example, Oct. 1974) may be attributed to the precision 

of reported dam outflow data and the sensitivity of the method 

to large releases of water from the Chain of Lakes associated 

with lowering lake levels. That is, imprecision in the reported 

data can result in reported dam outflows during periods of 

falling stage that were small in comparison to the estimated 

amount of water released from the Chain of Lakes, resulting 

in a very low computed natural flow at the dam. Some nearnegative natural flows were also computed during summer for 

years with below average dam outflow (for example, Aug.–

Sept. 1976). During such instances, both the lake stage and 

reported dam outflow were declining, indicating that natural 

flow into the Chain of Lakes was likely affected by drought 

conditions. In addition, it is possible that diversions associated 

with cranberry operations in the Chain of Lakes Watershed 

could contribute to the near-negative computed flows for 

these periods (as evident in fig. 3). Natural flows at the dam 

calculated by the adjusted drainage-area ratio method are not 

expected to be affected by diversions from cranberry operations because no such operations are known to exist upstream 

of the Bear or Trout River gages. However, the water-budget 

method lacks adequate information to systematically remove 

possible effects from this type of diversion. This is because 

daily reported dam outflows inherently incorporate reductions in daily flow into the Chain of Lakes due to diversions, 

which, because the diversions were not measured cannot be 

systematically removed. Moreover, calibrating the waterbudget method with natural flow (fig. 3) into the Chain of 

Lakes ( GageQin) only accounts for effects of diversions in terms 

of the number of days (N-days) over which daily reported dam 

outflow values are averaged and in terms of the groundwater 

discharge parameters, which as described earlier, are effectively removed from direct computation of daily natural flow 

at the Rest Lake Dam ( DamQnat). As a result, to the extent that 

near-negative flows computed by the water-budget method 

are affected by diversions, the flows should not be considered 

natural.

Natural Historical Flow Durations

Daily natural flows at the dam ( DamQnat ) computed by 

equation 5 for the adjusted drainage-area ratio method and by 

the combination of equations 8 and 9, which used 23-day averaged values for the water-budget method (fig. 9), were used to 

compute monthly flow-duration values for the period of available historical data. Historical data for the Bear River near 

Manitowish Waters (05357335) and the Trout River at Trout 

Lake near Boulder Junction (05357245), which were used for 

the adjusted drainage-area ratio method, spanned from December 1, 1991, to November 30, 2011. Historical lake stage and 

dam outflow reported by the dam operator, which were used 

for the water-budget method, spanned from December 1, 1973, 

to November 30, 2011. Monthly flow-duration statistics were 

computed as described by Helsel and Hirsch (2002) by ranking all computed DamQnat values within a given month for the 

duration of data available from each method (fig. 10, table 6). 

A single flow-duration value, in cubic feet per second, 

represents the flow rate that is equaled or exceeded on the 

associated percentage of days over the period of calculation. 

For example, the Flow Duration 90 (FD90), or 90-percent 

flow duration, is the flow rate that is equaled or exceeded on 

90 percent of the days being evaluated. For this study, flowduration values were computed on a monthly basis to evaluate 

seasonal changes in flow. Thus, a FD90 value of 116 ft3/s for 

January, as computed with the adjusted drainage-area ratio 

method (table 6), means that 90 percent of all daily flows during January over the years 1992–2011 were equal to or greater 

than 116 ft3/s. Flow-duration values computed in this way are 

useful for describing natural flow patterns estimated using 

historical records, because flow-duration calculations are not 

overly influenced by extreme values, as opposed to an average 

value that may be strongly influenced by a single extreme runoff event. That is, flow-duration values, particularly the FD50 

or median, are generally resistant to outliers or infrequent 

events and, therefore, are well suited for evaluating a range of 

typical conditions. Nonetheless, flow-duration values at the 

high and low range (FD2, 10, or FD 90, 98, 99.6) are more 

susceptible to extreme flows, because by definition these flows 

are observed infrequently (0.4 percent of days for the FD99.6). 

These characteristics for flow-duration values are evident in 

the results shown in table 6 and figure 10. 
Figure 9. Dam-operator reported stage and dam outflow for the Manitowish Chain of Lakes and computed natural flow at the Rest Dam (DamQnat) from the adjusted drainage-area ratio and water-budget methods, 1973–2011. 

12/1/1973 12/1/1974 12/1/1975 12/1/1976 12/1/1977 12/1/1978 12/1/1979 12/1/1980 12/1/1981 12/1/1982 12/1/1983 12/1/1984 12/1/1985 12/1/1986 12/1/1987 12/1/1988 12/1/1989 12/1/1990 12/1/1991

−100

0

100200300400500600700800

Streamflow, in cubic feet per second Streamflow, in cubic feet per second 

900 

12/1/1992 12/1/1993 12/1/1994 12/1/1995 12/1/1996 12/1/1997 12/1/1998 12/1/1999 12/1/2000 12/1/2001 12/1/2002 12/1/2003 12/1/2004 12/1/2005 12/1/2006 12/1/2007 12/1/2008 12/1/2009 12/1/2010

−100

0

100200300400500600700800900

Date

Dam operator-reported outflow at the Rest Lake Dam

Natural streamflow at the Rest Lake Dam (DamQnat) computed by the water-budget methodNatural streamflow at the Rest Lake Dam (DamQnat) computed by the adjusted drainage-area ratio method

Dam operator-reported lake stage

EXPLANATION
20 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

Figure 10. Monthly flow-duration values for computed natural flow at the Rest Lake Dam 

based on the adjusted drainage-area ratio method for 1991–2011 and the water-budget 

method for 1973–2011. 

Natural streamflow at the Rest Lake Dam, 

in cubic feet per second −100

0

100

200

300

400

500

600

Adjusted drainage-area ratio method FD2

Adjusted drainage-area ratio method FD10

Adjusted drainage-area ratio method FD30

Adjusted drainage-area ratio method FD50

Adjusted drainage-area ratio method FD70

Adjusted drainage-area ratio method FD90

Adjusted drainage-area ratio method FD99.6

Water-budget method FD2

 Water-budget method FD10

Water-budget method FD30

Water-budget method FD50

Water-budget method FD70

Water-budget method FD90

Water-budget method FD99.6

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

EXPLANATION
Table 6. Monthly flow-duration values for computed natural flow at the Rest Lake Dam based on the adjusted drainage-area ratio and water-budget [FD, flow duration, is followed by a number indicating the percentage of time that the computed value is equaled or exceeded]

Flow 

duration

Flow-duration value (cubic feet per second)

Jan. Feb. Mar. April May June July Aug. Sept. Oct. 

Adjusted drainage-area ratio method

FD2 246 217 330 568 423 302 267 330 366 328 FD5 236 208 302 496 354 272 245 251 225 299 FD10 216 201 265 417 315 246 227 219 207 270 FD20 180 168 227 330 258 213 203 183 178 222 FD30 171 157 194 296 236 189 179 166 151 178 FD40 164 145 177 261 214 176 170 155 133 157 FD50 148 138 170 231 198 164 157 142 114 140 FD60 139 133 164 216 168 151 138 107 92.0 118 FD70 131 125 154 194 154 142 125 91.0 81.8 109 FD80 125 118 147 179 129 127 96.5 82.6 72.9 103 FD90 116 111 132 160 112 106 83.9 71.4 64.8 94.2 FD95 110 108 127 125 94.6 94.5 70.8 63.0 60.2 80.7 FD98 106 107 122 104 87.3 87.5 64.5 56.4 57.2 71.1 FD99.6 102 105 106 97.2 81.2 83.3 61.7 51.3 51.6 64.0 

Water-budget method

FD2 291 291 302 495 488 326 288 305 419 488 FD5 287 287 290 436 451 306 265 232 344 431 FD10 275 231 259 394 379 277 232 201 289 345 FD20 223 213 226 339 313 245 194 163 215 271 FD30 204 201 215 312 273 213 174 132 184 242 FD40 195 194 204 291 248 189 148 121 149 213 FD50 172 168 195 267 226 173 129 106 118 185 FD60 163 159 179 242 203 155 115 90.6 100 163 FD70 151 150 164 221 177 137 102 79.7 78.6 138 FD80 147 134 153 199 149 122 77.3 65.0 61.3 111 FD90 103 100 132 175 126 95.7 44.4 43.4 44.3 79.9 FD95 85.7 85.7 113 146 108 70.9 35.2 23.5 29.3 54.8 FD98 78.6 78.6 85.7 102 89.7 49.1 28.6 16.0 10.5 14.0 FD99.6 78.6 78.6 78.6 88.8 55.9 35.6 13.6 3.6 1.0 -8.3 
22 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

Results from both methods are useful for understanding the natural flow patterns at the dam. The monthly FD50 

(median) values in figure 10 have similar patterns to the 

median daily flows shown in figure 8, with high flows in 

spring and low flows in late summer. This seasonal pattern is 

evident in both high (FD10) and low flows (FD90) as well. 

For example, the FD10 increases from about 200 –230 ft3/s 

in February (from the adjusted drainage-area ratio and 

water-budget methods, respectively) to about 420 –390 ft3/s 

in April, decreases to about 220 –200 ft3/s in August, and 

ends the year at about 280 ft3/s (both methods) in December 

(table 6, fig. 10). Similarly, the FD90 increases from about 

110 –100 ft3/s in February to about 160 –180 ft3/s in April, 

decreases to about 70 –40 ft3/s in August, and ends the year at 

about 120 –100 ft3/s in December. Similarly, the range from 

high to low flows widens during spring, narrows during summer, and widens again during fall. 

Differences among the results of each method demonstrate the difficulty in computing natural flows from surrogate 

records. For example, although the median values (FD50) 

from each method tend to be relatively similar, values for the 

less frequent flows (for example, the FD10 and FD90) tend 

to be more variable, as would be expected given the greater 

uncertainty associated with estimating infrequent flows, 

regardless of the method. 

Identification of a representative natural low flow at the 

dam, as required by Chapter 31 of the Wisconsin State Statutes 

(Wisconsin Statutes and Annotations, 2012), is subject to 

interpretation. What is considered a low flow by one criterion 

may differ from what is considered a low flow by another criterion. For example, the effect of a particular low-flow value 

on the health of an aquatic organism can differ from the effect 

of that same flow on the utility of a water resource for economic or recreational gain. In addition, the results illustrate a 

challenge with interpreting a single value of natural low flow, 

because natural low flow varies on a seasonal basis. That is, 

a natural low flow computed for September is not representative of a natural low flow in April. The lowest natural flow 

computed by either method is –8.3 ft3/s for the FD99.6 during 

October based on the water-budget method (table 6), and that 

flow is physically impossible under natural conditions. This 

value is undoubtedly influenced by the sensitivity of the waterbudget method to small changes in lake stage, the precision of 

reported lake stage and dam outflow, and upstream diversions. 

As stated previously, natural flows computed by the adjusted 

drainage-area ratio method are expected to be less susceptible 

to biases at low flows than those computed by the water-budget method. The lowest natural flow computed by the adjusted 

drainage-area ratio method is 51.3 ft3/s for the FD99.6 during 

August. It is important to reiterate that a FD99.6 computed 

on a monthly basis is not equal to the FD99.6 computed on 

an annual basis, which Gebert (1980) equated with the Q7,10

in the nearby Wisconsin River Basin. The FD99.6 computed 

on an annual basis for natural flows estimated by the adjusted 

drainage-area ratio and water-budget methods are 57 ft3/s and 

10 ft3/s, respectively, with the water-budget method likely 

biased low.

Assumptions and Limitations

The methods used to estimate the monthly flow-duration 

values in table 6 for natural flow at the Rest Lake Dam incorporate many assumptions and limitations. It was assumed 

that the relation between reported dam outflow and measured 

flow by WDNR staff at State Highway 51 (05357302) prior 

to April 28, 2009 (fig. 4A), was stationary over time. That 

is, despite knowledge that the amount of leakage between 

stoplogs varied depending upon the stoplog conditions (use) 

and configuration (orientation and shape), no data were available for time-dependent adjustments to the reported dam 

outflows, and, therefore, the relation in figure 4A was assumed 

to be appropriate for all reported data from December 1973 

to April 28, 2009. A second assumption was that the relation 

between the yield at the upstream gaging stations and the reference gaging stations on the Bear and Trout Rivers (stations 

05357335 and 05357245, respectively) was stationary over 

time. A third assumption was that the estimated daily amount 

of water diverted by cranberry operations that subsequently 

returns to the Chain of Lakes through the groundwater system 

was accounted for on a monthly basis through the coefficients 

(K) used in the adjusted drainage-area ratio method. A fourth 

assumption was that averaging reported lake stage and dam 

outflow over 23 days, as estimated during calibration of the 

water-budget method for the 2-year study period, was appropriate for the entire period of record. 

Regardless of the assumptions, application of the results 

should recognize several limitations. First, although computed long-term median natural flows at the dam tend to be 

relatively similar between the two methods, extreme values 

(for example, the FD2, 10, 90, and 99.6) deviate substantially 

between methods (fig. 10 and table 6). These large differences 

between methods illustrate the increased uncertainty associated with estimating extreme events using secondary data 

rather than directly measured natural flows at the dam. This 

higher level of uncertainty should be considered when using 

the results. Additional uncertainty is also incorporated into the 

water-budget method in the form of the precision with which 

reported lake stages and dam outflows were measured or 

estimated. Specifically, lake stage provided by the dam operator was reported to the nearest inch and measured at approximately noon each day. As stated previously, the water-budget 

method is highly sensitive to small changes in lake-stage 
Summary and Conclusions 23

fluctuations––even when reporting to the hundredth of an 

inch––and low precision in the historical stage data. Rapidly changing stages may confound this sensitivity, although 

averaging stages over 23 days is expected to reduce the effects 

of this limited precision. Conversely, averaging over 23 days 

limits the precision with which the timing of extreme flows 

can be estimated, thus potentially affecting the month during 

which an estimated extreme event occurs. Similarly, reported 

dam outflows were estimated from lake stage and an equation for flow over a weir. The extent to which the equation is 

affected by the configuration of the stoplogs, as described previously, is incorporated into the accuracy of the reported dam 

outflows. Finally, computed near-negative flows at the dam 

from the water-budget method may incorporate biases due to 

the reported data as described previously, as well as upstream 

diversions that were not accounted for using this method.

Summary and Conclusions

The Wisconsin Department of Natural Resources is 

charged with oversight of how the dam at the outlet of Rest 

Lake in Vilas County, Wis., is operated. Chapter 31 of the 

Wisconsin State Statutes specifies that the minimum flow out 

of every dam in the State should be no less than 25 percent 

of the natural low flow (Wisconsin Statutes and Annotations, 

2012). Dams alter the system from natural conditions, however, often leaving 25 percent of the natural low flow difficult 

to determine. As part of establishing new operating orders for 

the dam, the Wisconsin Department of Natural Resources partnered with the U.S. Geological Survey to improve estimates of 

natural flow at the dam.

Two independent methods were used to estimate historical natural flow for the Manitowish River at State Highway 51 near Manitowish Waters (station 05357302), the site 

of a streamflow-gaging station located approximately 1 mile 

downstream of the Rest Lake Dam. Because of a lack of direct 

flow measurements at the dam outlet, flow measured at State 

Highway 51 (05357302) was assumed to represent flow at the 

Rest Lake Dam. The first method, the adjusted drainage-area 

ratio method, related flow divided by watershed area (yield) 

for the total measured flow into the Manitowish Chain of 

Lakes at the upstream gages ( GageQmeasured ) with total yield for 

the Bear River near Manitowish Waters (station 05357335) 

and the Trout River at Trout Lake near Boulder Junction (station 05357245) to develop an equation for computing historical flows into the Chain of Lakes ( GageQin ) using all historical 

data for the Bear and Trout River gages (1991–2011). To 

convert the computed total natural flow into the Chain of 

Lakes at the upstream gages ( GageQin ) into natural flow at the 

downstream Rest Lake Dam ( DamQnat ), computed daily flows at 

the upstream gages ( GageQin ) were multiplied by a coefficient 

that varied by month. The second method, the water-budget 

method, used a computer model to compute natural flow at the 

dam using lake stage and dam outflow data provided by the 

dam operator. The water-budget model separated the computation into two equations. The first equation computed flows into 

the Chain of Lakes at the upstream gaging stations ( GageQin ), 

and the second equation used those upstream flows to compute 

natural flows at the downstream Rest Lake Dam ( DamQnat ). 

Calculation of natural flows at the dam was separated into 

two equations so that the most appropriate number of days 

(23 days) over which to average input data (lake stage, dam 

outflow, precipitation, and evaporation) could be determined 

by comparing measured flows at the upstream gages ( GageQmeasured ) for the 2-year study period with computed flows ( GageQin ). 

Averaging input data over 23 days was the primary mechanism 

for adjusting the “flashiness” of the computed flows. Following calibration of the water-budget model to determine the 

most appropriate number of days over which to average input 

data, the model was applied using all historical lake stage and 

dam outflow data from the dam operator (1973–2011) to compute historical natural flow at the Rest Lake Dam ( DamQnat ).

Daily natural flows at the dam ( DamQnat ), computed by 

the adjusted drainage-area ratio and the water-budget methods, were used to compute monthly flow-duration values for 

the period of historical data available for each method (Dec. 

1991 to Nov. 2011 for the adjusted drainage-area ratio method 

and Dec. 1973 to Nov. 2011 for the water-budget method). 

Monthly flow durations provide a means for evaluating the 

frequency and range in flows for specific months over many 

years, including the frequency with which certain high and 

low flows occur.

Results from both methods are useful for understanding 

the natural flow patterns at the dam. The monthly natural flows 

for both methods had similar patterns, with high median flows 

in spring and low median flows in late summer. Similarly, the 

range from high flows to low flows increased during spring, 

decreased during summer, and increased again during fall. 

These seasonal patterns illustrate a challenge with interpreting a single value of natural low flow. Moreover, alteration of 

natural flows caused by storing water in the Chain of Lakes 

during spring and releasing water from the Chain of Lakes in 

fall causes a change in the timing of high and low flows compared with flows under natural conditions. That is, the lowest 

reported outflows occurred in spring and highest reported 

outflows occurred in fall. 
24 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

Both methods captured the pattern and timing of measured high-flow and low-flow events at the upstream gaging 

stations. The adjusted drainage-area ratio method generally 

had smaller residual errors than the water-budget method 

over the full range in observed flows and had smaller monthly 

biases. Unfortunately, neither method could be directly 

compared against daily dam outflow, because dam operations 

altered the natural flow patterns. Both methods produced similar total annual flows at the dam, suggesting that the computed 

natural flows at the dam were correct on an annual basis for 

both methods. Although it was not possible to definitively 

determine which method was more accurate at estimating all 

monthly natural flows, particularly at very high or very low 

flows, comparisons of the results of each method indicate 

that the adjusted drainage-area ratio method could be more 

susceptible to biases at high flows because of isolated storms 

outside of the Manitowish River watershed. Conversely, the 

water-budget method may be more susceptible to biases at low 

flows because of the sensitivity of the method to the accuracy 

of reported lake stages and dam outflows, as well as potential 

upstream diversions that could only partially be accounted 

for with this method (the adjusted drainage-area ratio method 

should not be affected by diversions by cranberry operations, 

because no such operations are known to be present in the 

Bear and Trout River watersheds that were used as part of this 

method). 

References Cited

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estimation and inverse problems: Elsevier Academic Press, 

301 p.

Doherty, J., 2011, PEST: Model-independent parameter 

estimation user manual (5th ed.), and addendum: Brisbane, 

Queensland, Australia, Watermark Numerical Computing.

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parameterized inversion—A guide to using PEST for 

groundwater-model calibration: U.S. Geological Survey 

Scientific Investigations Report 2010–5169, 59 p.

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the Upper Wisconsin River Basin, Wisconsin: U.S. Geological Survey Water Resources Investigations Open-File 

Report 80–691, 60 p.

Gebert, W.A., Walker, J.F., and Hunt, R.J., 2011, Groundwater recharge in Wisconsin—Annual estimates for 1970–99 

using streamflow data: U.S. Geological Survey Fact Sheet 

2009–3092, 4 p.

Helsel, D.R., and Hirsch, R.M., 2002, Statistical methods in 

water resources: U.S. Geological Survey Techniques of 

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calibration: U.S. Geological Survey Water-Resources Investigations Report 98–4005, 90 p.

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Hunt, R.J., Feinstein, D.T., Pint, C.D., and Anderson, M.P., 

2005, The importance of diverse data types to calibrate a 

watershed model of the Trout Lake Basin, northern Wisconsin: Journal of Hydrology, v. 321, p. 286–296.

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2008, Importance of unsaturated zone flow for simulating 

recharge in a humid climate: Ground Water, v. 46, no. 4, 

p. 551–560.

Lenters, J.D., Kratz, T.K., and Bowser, C.J., 2005, Effects of 

climate variability on lake evaporation: results from a longterm energy budget study of Sparkling Lake, northern Wisconsin (USA): Journal of Hydrology, v. 308, p. 168–195.

Mudrey, M.G., Jr., Brown, B.A., and Greenberg, J.K., 1982, 

Bedrock geologic map of Wisconsin: University of Wisconsin-Extension, Wisconsin Geological and Natural History 

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scale 1:1,000,000.

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2000, Ecoregions of Wisconsin: Transactions of the Wisconsin Academy of Science, Arts and Letters, v. 88, p. 77–103.

Pint, C.D., 2002, A groundwater flow model of the Trout Lake 

Basin, Wisconsin: Calibration and lake capture zone analysis: Madison, University of Wisconsin, M.S. thesis, 123 p.

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1939, Order for 2-WP-295: 17 p., accessed April 17, 

2012, at http://dnr.wi.gov/water/basin/upwis/restlakedam/

documents/1937-and-1939-PSCOperatingOrder.pdf.

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Wisconsin Department of Natural Resources, 2007, Current 

operation of Rest Lake––FAQ: 7 p., accessed April 17, 

2012, at http://dnr.wi.gov/water/basin/upwis/restlakedam/

documents/FAQ-RestLakeDamOperation.pdf.

Wisconsin Department of Natural Resources, 2011, Environmental overview of the Manitowish River system: 62 p., 

accessed April 17, 2012, at http://dnr.wi.gov/water/basin/

upwis/restlakedam/documents/EnvironmentalOverviewLinks.pdf.

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accessed April 17, 2012, at: http://legis.wisconsin.gov/lrb/

pubs/consthi/04consthiIV4.htm.

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sec. 34, Flow of water regulated: accessed April 18, 2012, at 

http://docs.legis.wisconsin.gov/statutes/statutes/31/34.

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Publishing support provided by the U.S. Geological Survey 

Science Publishing Network, Columbus Publishing Service Center

For more information concerning the research in this report, 

contact the

Director, Wisconsin Water Science Center

U.S. Geological Survey

8505 Research Way

Middleton, Wisconsin 53562

http://wi.water.usgs.gov/ 
Appendix 27

Appendix. Code for the Python-based Water-Budget Model

“Python” is a registered trademark of the Python Software Foundation.

Although this program has been used by the U.S. Geological Survey (USGS), no warranty, expressed or implied, is made by the 

USGS or the U.S. Government as to the accuracy and functioning of the program and related program material nor shall the 

fact of distribution constitute any such warranty, and no responsibility is assumed by the USGS in connection therewith.

# This program computes Rest Lake natural flows at upstream gages and at the dam given lake stage, outflow, and climate 

# data.

import sys

import math

import numpy as np

import matplotlib.pyplot as plt

from datetime import datetime # pull the datetime module from datetime

from datetime import timedelta # grab the time differencing module

echo = False

TARGDATES = []

DATES = []

NEWDATES = []

INFLOWS = []

OUTFLOWS = []

FLOWINS = []

FLOWINS_CFS = []

OUTF = []

LLS = []

PPT = []

EVAP = []

LEVELS = []

SIMLEVELS =[]

SIMOUTS = []

GWS = []

TF = ‘%m/%d/%Y’ # time format used in the files 

# ####################### #

# Error Exception Classes # 

# ####################### #

# -- cannot read/write/open/close file

class FileFail(Exception):

 def __init__(self,filename,filetype):

 self.filename=filename

 self.ft = filetype

 def __str__(self):

 return(‘\n\nProblem with ‘ + self.ft +’: ‘ + self.filename + ‘ \n’ +

 “Either it can’t be opened or closed, can’t be read from or written to, or doesn’t exist”) 

 

# -- wrong number of lines in cal file

class CalFail(Exception):

 def __init__(self,nlines,fn):

 self.nlines=nlines

 self.fn = fn

 def __str__(self):

 return(‘\n\nCal File: ‘ + self.fn + ‘ has wrong number of lines. \n’ +
28 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

 ‘Read ‘ + str(self.nlines) + ‘ lines in the file’)

# -- Failure parsing the input data file

class ParseFail(Exception):

 def __init__(self,offending_line):

 self.offending_line = offending_line

 def __str__(self):

 return(‘\n\nThere was a problem parsing a line in your data file. \n’ +

 ‘The offending line was:\n’ +

 ‘”’ + self.offending_line + ‘”’)

# start of code; read in the namefile from the command prompt.

# the namefile points to the files with input data.

try:

 namfile = sys.argv[1]

 lines = open(namfile,’r’).readlines()

 outfilename = lines[0].strip()

 obsfilename = lines[1].strip() 

 datfilename = lines[2].strip()

 calfilename = lines[3].strip()

 targfilename = lines[4].strip()

 outDAMfilename = lines[5].strip()

 

 # Start and end days used for cropping filler values at the beginning and end of input files that were used to help with 

 # averaging.

 STARTDAY = datetime.strptime(lines[6].strip(),TF)

 ENDDAY = datetime.strptime(lines[7].strip(),TF) 

except:

 raise(FileFail(namfile,’name (*.NAM) file’))

# open output file for computed natural flow at the upstream gages

try:

 output_file = open(outfilename,’w’)

except:

 raise(FileFail(outfilename,’output file’))

# open output file for computed natural flow results the dam

try:

 outDAM_file = open(outDAMfilename,’w’)

except:

 raise(FileFail(outDAMfilename,’DAM output file’))

# open file where re-formatted measured flow at the upstream gages will be written for processing by TSPROC (Steve 

# Westenbroek, USGS, written commun., March, 2012), which is a time-series processing utility that is used to generate 

# flow-duration values for the measured and computed natural flows at the upstream gages that are used for calibration with 

# PEST.

try:

 obs_file = open(obsfilename,’w’)

except:

 raise(FileFail(obsfilename,’obs file’))

# open file with measured flows at the upstream gage so this code can format it for interpretation by TSPROC

try:

 target_file = open(targfilename,’r’).readlines()

except:

 raise(FileFail(targfilename,’target file’))
Appendix 29

for each_line in target_file[1:]:

 DATA_TARG = each_line.split(‘,’) # split on commas

 DATE_TARG = DATA_TARG[0]

 RICE_CFS = float(DATA_TARG[1])

 MANITOWISH_CFS = float(DATA_TARG[2])

 TROUT_CFS = float(DATA_TARG[3])

 PAPOOSE = float(DATA_TARG[4])

 

 INFLOW_TARG = RICE_CFS + MANITOWISH_CFS + TROUT_CFS + PAPOOSE

 INFLOWS.append(INFLOW_TARG) # appends each INFLOW to the list INFLOWS (for plotting purposes)

 

 try:

 MONTH, DAY, YEAR = DATE_TARG.split(“/”)

 except:

 raise(ParseFail,each_line)

 MM = MONTH.zfill(2)

 DD = DAY.zfill(2) 

 OBS_outstring = (‘obs_inflow ‘ + ‘{0}/{1}/{2}’.format(MM,DD,YEAR) + ‘ 12:00:00 ‘ + 

 ‘{0:.2f}\n’.format(INFLOW_TARG))

 

 if (echo):

 print OBS_outstring

 obs_file.write(OBS_outstring) 

# open file with calibration parameters

try:

 calib_file = open(calfilename,’r’).readlines()

except:

 raise(FileFail(calfilename,’calibration parameter file’))

# make a list of just the numbers from the file

lines = []

for each_line in calib_file:

 line_stripped = each_line.strip() #removes leading and trailing whitespace

 line_split = line_stripped.split() # splits a string at whitespaces

 lines.append(line_split[0]) # appends each split character to the list: “lines”

if len(lines) != 6:

 raise(CalFail(len(lines),calfilename)) #call an error if not 6 lines in the calibration file.

# now pull out the calibration coeffs.

GW_drop = float(lines[0])

GW_low = float(lines[1])

GW_rise = float(lines[2])

GW_high = float(lines[3])

FLOAT_NSTAGE = float(lines[4])

VOLDAMP = float(lines[5])

# Pest needs to work with float numbers so that derivatives can be calculated. This code needs integer days.

# Thus, the N-days over which to average all input values calibrated by PEST is rounded and then converted to an integer.

NSTAGE = int(round(FLOAT_NSTAGE)) # round first (rounds 0.5 upwards), then convert to integer.

# open file with input data on lake stage, dam outflow, precip and evap.

try:
30 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

 data_file = open(datfilename,’r’).readlines()

except:

 raise(FileFail(datfilename,’precipitation and evaporation data file’))

for each_line in data_file[1:]:

 DATA = each_line.split(‘,’)

 DATE = DATA[0]

 PRECIP = float(DATA[1]) 

 OUTFLOW_CFS = float(DATA[2])

 LL = float(DATA[3]) #lake level

 DATET = datetime.strptime(DATE,TF) #converts to a tuple 

 DATES.append(DATET)

 OF = 86400.0 * OUTFLOW_CFS # convert outflow to ft3/day

 OUTF.append(OF) # creates a running list of outflow each day

 PI = PRECIP / 12.0 # convert the daily precipitation to feet

 PPT.append(PI) # creates a running list of precipitation each day 

 LLS.append(LL) # creates a running list of lake stage each day

 

 # For plotting only... 

 if ((datetime.strptime(DATE,TF) >= STARTDAY) & (datetime.strptime(DATE,TF) <= ENDDAY)): 

 OUTFLOWS.append(OUTFLOW_CFS) # accumulating the measured outflow for plotting purposes

 LEVELS.append(LL * 100.0 - 159500) # appends each measured lake level for plotting purposes

 # the equation simply offsets the lake level for better visualization in the plot

 

 try:

 EO = float(DATA[4])

 except:

 EO = 0.0 # assign zero evap if blank

 EVAP.append(EO) # creates a running list of evap with zero on days for which evap is not specified, such as during 

 # the winter.

# ####################### #

# Start of calculations # 

# ####################### #

 # Average stage, outflow, ppt, & evap over NSTAGE days (“N-days” in the report)

 # the multiple if, elif, and else statements are used to adjust how the stage is averaged for the beginning of the record 

 # when there’s only 1 to NSTAGE days of data.

for I, val in enumerate(DATES): # iterates through each date in the input file

 

 # for first day, sets values to first day value

 if(I == 0): 

 STAGE1 = LLS[0] 

 STAGE2 = STAGE1 

 OUTFLOW = OUTF[0] 

 PRECIP = PPT[0] 

 EV = EVAP[0]

 

 # At the start, when “I” is less than or equal to NSTAGE (or “N-day” in the report), average around the date of interest 

 # from the first day to half way to NSTAGE

 elif(I <= (NSTAGE/2.0)-1): 

 ENDAVERAGE =(2*I) 

 

 STAGE1 = np.mean(np.array(LLS[0:ENDAVERAGE]))

 OUTFLOW = np.mean(np.array(OUTF[0:ENDAVERAGE]))

 PRECIP = np.mean(np.array(PPT[0:ENDAVERAGE]))
Appendix 31

 EV = np.mean(np.array(EVAP[0:ENDAVERAGE])) 

 

 # average around the date of interest from 1/2 NSTAGE back to 1/2 NSTAGE forward of the date

 else: 

 STARTAVERAGE = int(math.floor(I-(NSTAGE/2.0-1))) # round first (math.floor rounds 0.9 down), then convert to 

 # an integer

 ENDAVERAGE = int(math.floor(I+NSTAGE/2.0)) # round first (math.floor rounds 0.9 down)

 STAGE1 = np.mean(np.array(LLS[STARTAVERAGE:ENDAVERAGE])) # averages lake stages around the current 

 # date

 OUTFLOW = np.mean(np.array(OUTF[STARTAVERAGE:ENDAVERAGE])) # averages outflow around the current 

 # date

 PRECIP = np.mean(np.array(PPT[STARTAVERAGE:ENDAVERAGE])) # averages precip around the current date

 EV = np.mean(np.array(EVAP[STARTAVERAGE:ENDAVERAGE])) # averages evap around the current date 

 

 # Calculate lake area in square feet

 AREA = (158065.01 * STAGE1 * STAGE1) - (495798355.44 * STAGE1) + 388812322721.9 # area based on stage

 LOWAREA = (158065.01 * 1597.9 * 1597.9) - (495798355.44 * 1597.9) + 388812322721.9 # area based on normal low 

 # stage only

 # Determine which of the 4 groundwater parameters (calibrated with PEST) to apply to this date based on the stage and 

 # change in stage

 if (STAGE1-STAGE2 > 0.015):

 GW = GW_rise * 86400.0 # convert to ft3/d

 elif (STAGE1-STAGE2 < -0.02):

 GW = GW_drop * 86400.0

 elif (STAGE1 > 1599.5):

 GW = GW_high * 86400.0

 else:

 GW = GW_low * 86400.0

 

 # Water balance and lake level calculations

 DSTAGE = STAGE1 - STAGE2 # subtracts yesterday’s lake stage (STAGE2) from the current lake stage STAGE1 to 

 # compute change in stage

 DVOL = AREA * DSTAGE # compute change in lake storage due to change in lake stage

 FLOWIN = DVOL + OUTFLOW + (EV * AREA) - (PRECIP * AREA) - GW # Compute natural flow into the chain of 

 # lakes at the gaging stations upstream of 

 # the lake

 FLOWIN_CFS = FLOWIN / 86400.0 # convert to ft3/s

 FLOWINDAM = FLOWIN + GW + (PRECIP * LOWAREA) - (EV * LOWAREA) # Compute natural flow at the dam. 

 # This includes inflow at the gages 

 # plus groundwater and Precip – 

 # Evap for lake area at low stage.

 FLOWINDAM_CFS = FLOWINDAM / 86400.0 # convert to ft3/s

 STAGE2 = STAGE1 # convert the current date into “yesterday” for the next iteration of this loop

 #END OF WATER BUDGET CALCULATIONS 

 

 # remove the starting and ending values that were added to simplify date averaging, and limit results to the dates of 

 # interest.

 if ((val >= STARTDAY) & (val <= ENDDAY)): 

 NEWDATES.append(val) # for plotting only 

 FLOWINS.append(FLOWIN_CFS) # for plotting only 

 SIMLEVELS.append(STAGE1 * 100.0 - 159500) # shifts lake stage for plotting and appends each stage to the list 

 # SIMLEVELS for plotting

 SIMOUTS.append(OUTFLOW/86400.0) # for plotting only 

 GWS.append(GW/86400.0) # for plotting only 

 
32 Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin

 # format the output string for natural flow at the upstream gages and at the dam

 SIM_GS_outstring = ‘SimGSin ‘ + DATES[I].strftime(TF) + ‘ 12:00:00 ‘ + ‘{0:.2f}\n’.format(FLOWIN_CFS) 

 SIM_DAM_outstring = ‘SimDAMin ‘ + DATES[I].strftime(TF) + ‘ 12:00:00 ‘ + 

 ‘{0:.2f}\n’ format(FLOWINDAM_CFS)

 if (echo):

 print SIM_GS_outstring

 print SIM_DAM_outstring

 

 output_file.write(SIM_GS_outstring) # write to the output files

 outDAM_file.write(SIM_DAM_outstring) 

# close output file 

try:

 output_file.close()

except:

 raise(FileFail(outfilename,’output file’))

 

#begin block of code to produce Matplotlib plot

plt.figure()

plt.plot(NEWDATES, LEVELS, ‘y’, NEWDATES, SIMLEVELS, ‘c’, NEWDATES, OUTFLOWS, ‘g’, NEWDATES, 

SIMOUTS, ‘m’, NEWDATES, INFLOWS, ‘b’, NEWDATES, FLOWINS, ‘r’, NEWDATES, GWS, ‘+’,

 label=[‘Levels’,’Sim_levels’,’Outflow’,’Sim_Out’,’Measured’,’Python’,’GW’])

plt.legend( (‘Levels’,’Sim_Levels’,’Outflow’,’Sim_Out’,’Measured’,’Python’,’GW’), loc=’upper left’)

plt.show() 

Juckem and others—Estimation of Natural Historical Flows for the Manitowish River near Manitowish Waters, Wisconsin—Scientific Investigations Report 2012–5135