UNITED STATES 

DEPARTMENT OF COMMERCE 

Carlos M. Gutierrez 

Secretary 

NATIONAL OCEANIC AND 

ATMOSPHERIC ADMINISTRATION 

Conrad C. Lautenbacher, Jr. 

Administrator 

RESOURCE SHED DEFINITIONS AND COMPUTATIONS 

Thomas E. Croley II 

NOAA, Great Lakes Environmental Research Laboratory, Ann Arbor, MI 48105 

Chansheng He 

Western Michigan University, Department of Geography, Kalamazoo, MI 49008 

Joseph F. Atkinson 

University at Buffalo, Great Lakes Program, Buffalo, NY 14260 

David F. Raikow 

EPA, National Exposure Research Laboratory, Cincinnati, OH 45268 

Great Lakes Environmental Research Laboratory 

Ann Arbor, Michigan 

NOAA Technical Memorandum GLERL-141 
ii

NOTICE 

Mention of a commercial company or product does not constitute an endorsement 

by the NOAA. Use of information from this publication concerning proprietary 

products or the tests of such products for publicity or advertising purposes is not 

authorized. This is GLERL Contribution No. 1425. 

This publication is available as a PDF file and can be downloaded from GLERL’s 

web site: www.glerl.noaa.gov. Hard copies can be requested from GLERL, Information services, 2205 Commonwealth Blvd., Ann Arbor, MI 48105. 

pubs@glerl.noaa.gov. 
iii

TABLE OF CONTENTS 

ABSTRACT.....................................................................................................................................1 

1. Introduction ...............................................................................................................................1 

2. Resource Sheds .........................................................................................................................3 

 2.1 Definitions ..........................................................................................................................3 

 2.2 A Classical Example...........................................................................................................5 

3. Material Densities .....................................................................................................................6 

4. Estimating Resource Sheds .......................................................................................................8 

 4.1 Hydrological Resource Sheds.............................................................................................8 

 4.2 Material Fractions...............................................................................................................9 

 4.3 Lake Resource Sheds..........................................................................................................9 

5. Example Maumee Resource Sheds .........................................................................................12 

 5.1 Hydrological Model..........................................................................................................12 

 5.2 Example Resource Sheds..................................................................................................15

 5.3 Average Resource Shed vs. Resource Shed of Average Meteorology.............................15 

6. Computation Reduction...........................................................................................................17 

 6.1 Estimation, Interpolation, and Resolution ........................................................................17 

 6.2 Ordering Computations ....................................................................................................19 

7. Linking Resource Sheds..........................................................................................................20 

 7.1 Definitions for the Joint Resource Shed...........................................................................20 

 7.2 Example Calculation Algorithm.......................................................................................23 

 7.3 Example Lake Erie—Maumee River Linked Resource Sheds.........................................25 

8. Summary .................................................................................................................................26 

9. References ...............................................................................................................................29 

Appendix of Notation ....................................................................................................................31 

Appendix of Proper Gray Scale Rendering of Figure Pages for Printing......................................35 
iv

LIST OF TABLES 

Table 1. , , 25, ..., 31, 25, ..., 31 i j β i j = = , May, Sandusky ......................................................23 

Table 2. 25,31, , 25, ..., 31 L

n γ n = , May, Sandusky............................................................................23 

Table 3. 25, ,1, 25, ..., 31 W

j z j = , May, Sandusky .............................................................................23 

Table 4. Daily Watershed Outflow W

j g , j = 25, ..., 31, May, Sandusky......................................24 

Table 5. 25, ,1 25, ,1 / , 25, ..., 31 W WW

j jj q z gj = = , May, Sandusky .........................................................24 

Table 6. , 25, ,1, 25, ..., 31, 25, ..., W

mn m β qn m n = = , May, Sandusky and additional computations..25 

Table 7. Watershed—Lake Resource Shed Distribution Convolution Algorithm Pseudo Code. 26 

LIST OF FIGURES 

Figure 1. Example Locus (Lines) of Selected Y t (,) τ ..................................................................3 

Figure 2. Example Resource Sheds for Figure 1 (δ =1). ............................................................5 

Figure 3. Isochronal Travel Time Map and Resultant Time-Area Histogram. ............................6 

Figure 4. Example Resource Shed Distribution, ( ) 4,6 z ω ∀ 4,6 ω ∈S , for Figure 2. ...................7 

Figure 5. DLBRM Schematic for One Cell................................................................................10 

Figure 6. Example Estimating Resource Shed Distribution by Tracking Particles Backward. .13 

Figure 7. Maumee River Watershed Model Animation for August 31—September 3, 1950....14 

Figure 8. Maumee Resource Sheds on January 1, 1950 from Prior Days of Loading...............15 

Figure 9. Maumee River Six-Day Average Resource Sheds for Selected Times of the Year ...16 

Figure 10. Resource Shed Distribution Map Computation Resolutions ......................................17 

Figure 11. Western Maumee Resource Shed Distribution Map Interpolations and 

Resolutions .................................................................................................................18 

Figure 12. Example Lake Resource Shed Just Touching Watershed Mouth 3δ Ago.................21 

Figure 13. Maumee—Erie Linked Resource Shed for Average August 31 Conditions for Site 

835..............................................................................................................................27 

Figure 7. Alternate Figure 7 Page (14) Rendered in Proper Gray Scale for Printing ................36 

Figure 8. Alternate Figure 8 Page (15) Rendered in Proper Gray Scale for Printing ................37 

Figure 9. Alternate Figure 9 Page (16) Rendered in Proper Gray Scale for Printing ................38 

Figure 13. Alternate Figure 10 Page (27) Rendered in Proper Gray Scale for Printing ..............39 
1

Resource Shed Definitions and Computations 

Thomas E. Croley II, David F. Raikow, Chansheng He, and Joseph F. Atkinson 

ABSTRACT. When we consider a location with a material (e.g., water, pollutant, 

sediment) passing through it, we can ask: Where did the material come from and 

how long did it take to reach the location? We can quantify the answer by defining 

the areas contributing to this location during various time periods as resource sheds. 

Various kinds of resource sheds and their source material distributions are rigorously defined and properties derived. For watershed hydrology, we compute resource sheds and their source material distributions with a spatially distributed hydrology model by tracing material departing from a cell (say 1 km2

) over one time 

interval and arriving at the watershed mouth in another time interval. This requires 

modeling all cells, but only tracing contributions from one at a time. By then combining these simulations for all cell loadings, we construct a map of the contributions over the entire watershed for specific departure and arrival time intervals. We 

then combine results of several sets of simulations to determine the source distribution for any time period and infer resource sheds from these mappings. For lake 

circulation, we discuss the construction of resource sheds and their source distributions in the lake, by using lake circulation models to drive particle tracers in reverse 

time, and subsequent correction. We present examples for the Maumee River watershed in northern Ohio, discuss methods of computation reduction, discuss linkage with a lake circulation model to construct joint resource sheds in Lake Erie, and 

suggest areas of extension for the future. 

1. INTRODUCTION 

Accurately defining the spatial boundary of resource distribution and transport is essential to the 

understanding and management of resource dispersion over multiple spatial and temporal scales. 

Since the early twentieth century the watershed (land draining into a stream or lake at a given 

location) has been widely used as a basic unit in hydrologic research (Chow et al. 1988; U.S. Environmental Protection Agency 1995). A watershed is a hydrologic system that consists of a 

boundary, an internal structure (drainage areas, vegetation, river channels, lakes, reservoirs, etc.), 

inputs (water and materials), and outputs (water, services, and pollutants, etc) (Chow et al. 1988). 

As a hydrologic system, the boundary, structures, inputs (precipitation, pollutants, etc.), outputs 

(evapotranspiration, outflow, sediments, etc.), and processes of the watershed change over time 

and space. Its boundary, for example, is a continuous surface in space enclosing the watershed 

structure of interest (Chow et al. 1988), ranging from a small tributary (with an area of a few 

hectares) to a continental river basin (such as the Mississippi River Basin of a few million km2

). 

Since the goal of water resources management is to develop and implement policies, processes, 

technologies, and organizations for understanding, distributing, and improving the movement 

and characteristics of water resources to meet the multiple needs of human societies and ecosystems in a socially responsible, economically viable, and environmentally sustainable way (He et 

al. 2005), the watershed is recognized as a natural unit for managing the water resources and associated ecosystem (U.S. Environmental Protection Agency 1995; National Research Council 

1999; Bruckhorst and Reeve 2006). 
2

Over the years, researchers have extended the watershed concept to water resources and atmospheric studies. Michel (2000), for example, applied the concept of hydrocommons (defined as 

hybrid basins created by linking water sending and receiving basins by conveyance systems such 

as storage reservoirs and aqueducts) in analyzing the linkages between transbasin water transfers, 

water quality, and watershed ecosystems. Creation of the hydrocommons removes the natural 

boundaries of both sending and receiving basins, results in altered hydrology, water quality, ecosystems, economies and land use patterns in both the sending and receiving watersheds (Michel 

2000). Others have used hydroregion in understanding the role of regional hydrologic influences 

in fish species richness (Oswood et al. 2000; Santoul, et al. 2004). In atmospheric research, the 

airshed has been used to define the areas in which sources of pollutant emissions are located, 

which are transported and deposited in metropolitan areas or regions, for monitoring and forecasting air quality and implementing management programs such as ozone action days (Tullar 

and Suffet 1975; Chang and Cardelino 2000; Morawska et al. 2002; Ellis et al. 2006). 

Similarly, ecologists have sought to spatially delimit otherwise intuititve ecological units such as 

ecosystems (Cousins 1990). Where changes in ecological conditions are abrupt, such as at the 

land-water interface of a coast for example, ecosystem boundaries appear distinct. Yet the phenomenon of ecological subsidy, or donor-controlled supply of resources supporting food webs 

spatially distinct from source areas, blurs otherwise distinct ecosystem boundaries (Polis et al. 

1997). Embracing this common feature of food webs, Power and Rainey (2000) proposed the 

explicit delineation of ecological subsidy in space through the resource shed, or “source areas for 

resources consumed by individuals during their lifetimes”. Hence processes, rather than physical 

landscape features alone, can define geographic areas of ecological relevance to systems under 

study. Indeed, Power and Rainey (2000) observed that landscape features such as topography, 

coupled with physical forcing variables, can determine the shape of resource sheds. In the present study, resource sheds are determined by the movement of water, and hence relevant to water-borne materials. Moreover, we generalize the definition of resource shed to encompass 

source areas from which materials are derived for an individual, population, or location, over a 

specified time period, during a specified season. In this definition materials can include nutrients, organic matter, sediments, organism propagules, prey items, or pollutants; i.e. anything 

transportable by water. Hence a resource shed is a geographic area from which originate materials that subsidize food webs, or that are donated to a location, at a specific time over a specific 

time period. Applying this concept, Ben-David et al. (2005) investigate the variable resource 

sheds created by the movements and behavior of river otters and their impacts on nutrient cycling, changes in productivity and in community structure and function, and landscape heterogeneity of terrestrial communities. 

While the concepts of resource shed and watershed are similar, they have some differences as 

well. First, the boundary of a watershed is delineated by elevation and flow direction, following 

the gravity principle (water flows downhill), and therefore are relatively more stable (e.g. the 8-

digit hydrologic unit codes defined by the U.S. Geological Survey for the U.S. watersheds were 

developed in the 1970s). The boundary of a hydrological resource shed, however, is delineated 

by the contributing sources of water and materials to a river or lake (landscape features) during 

hydrologic events (physical forcing variables). Thus hydrological resource sheds have a more 

dynamic border, or rather a border relevant to a specific time period during which physical forcing variables operate (e.g. the resource sheds of water or sediments in a watershed change from 

one storm event to another). Second, the concept of watershed emphasizes temporal distribution 
3

of water and materials within a given space, and the concept of resource shed focuses on both 

temporal and spatial distribution of water and materials within a changing space. Third, the watershed concept has been in existence for over 100 years and is understood, accepted, and used 

worldwide while the resource shed concept is relatively new and not yet well developed. For 

example, what are the theoretical principles for defining resource sheds? What are appropriate 

spatial and temporal scales of resource sheds? How do we integrate geographic information systems and spatial modeling techniques with currently available computing technology to define 

and model resource sheds? Despite these challenges, we believe the concept of resource sheds 

provides a new way of displaying, understanding, and discovering the transport and distribution 

of water and materials and has the potential of helping resource managers better track and manage source loadings in a study area. 

In this paper, we define resource sheds and propose a set of analysis procedures to determine 

them. We first describe various resource shed definitions mathematically and then derive one 

from the other. We also describe various resource shed distributions (source density of material 

of interest over the resource shed) and then derive one from the other. We present methodologies for computing hydrological resource sheds within watersheds. We apply the methods to 

several examples in the Maumee River watershed and discuss estimation issues. Finally we 

show how to extend resource sheds and resource shed distributions defined for a location in a 

lake into contributing watersheds about the lake by linking results from the different settings. 

2. RESOURCE SHEDS 

2.1 Definitions 

Consider a location with material passing through it (e.g., water, pollutant, sediment). Where did 

the material come from and how long did it take to reach the location? We can quantify the answer by computing the areas contributing to this location during various time periods. We define 

such areas as resource sheds. More specifically, let ( , ) Y t τ represent the set of all locations 

where materials departing at time τ arrive at a location of interest at time t ; see Figure 1. (Note, 

upper case letters are used exclusively to denote sets and lower case letters are used to represent 

Figure 1. Example Locus (Lines) of Selected Y t (,) τ . 

Location of 

Y ( ) 3,6 Y ( ) 4,6 Y ( ) 5,6 interest 
4

functions or scalars.) Figure 1 shows ( , ) Y t τ as simple non-intersecting lines for a given time, t , 

for clarity. 

We now can define a resource shed a little more rigorously as: 

( ) , ,, (,)

a bctd

R abcd Y t ττ ≤ < ≤<= ∪ ∪ (1) 

where the operator, ∪ , represents the union of two or more sets. R( ) abcd ,,, is the set of all 

locations (resource shed) where materials departing during time interval [,) a b arrive at the location of interest during time interval [, ) c d . While the definition applies for all time intervals, 

note that for a b > or c d > or a d > , R = ∅ (the empty set). 

For convenience, consider only discrete time in intervals of δ and define 

, ( ) () ( ) 1,, 1, V Ri i j j i j =− − δδ δ δ (2) 

Then Vi j , is the resource shed where materials departing during the time interval of length δ

prior to time i (the i th time interval) arrive at the location of interest during the j th time interval. 

Also define 

, ( ) () ( ) 1, , 1, i j S Ri j j j =− − δδ δδ (3) 

, ( ) () () 1, , 1, T Ri j i j i j =− − δδ δδ (4) 

where i j , S is the resource shed where materials departing during time intervals , ..., i j {corresponding to time interval ⎡( ) i j −1 , δ δ ) ⎣ } arrive at the location of interest during the j th time interval and Ti j , is the resource shed where materials departing during time intervals , ..., i j arrive 

at the location of interest also during time intervals , ..., i j . Figure 2 illustrates some of these 

definitions. Figure 2 and all remaining figures portray mutually exclusive Vi j , , i j = , 1 j − , 

j − 2 , ... for clarity. However, all resource shed definitions and derivations herein apply without loss of generality for the general case of overlapping Vi j , . Note that 

, ,

,

ij mj

mij

S V == ∪ (5) 

, ,

,

i j im

mij

T S == ∪ (6) 

We expect that resource sheds for any time period to fully enclose (contain) those for included 

time periods, for a given location of interest and time. Since 
5

V6,6 

V4,6 

V5,6 

Figure 2. Example Resource Sheds for Figure 1 (δ =1). 

S4,6 = V4,6 ∪ V5,6 ∪ V6,6

S5,6 = V5,6 ∪ V6,6

S6,6 = V6,6

 

, ,

, ,

, mj mj

mij mkj

V V ki = =

∪ ∪ ⊂ ≤ (7) 

where “ A ⊂ B ” represents the statement “set A is contained in set B ;” then by (5) 

, , , ij kj S S ki ⊂ ≤ (8) 

Equations (8) and (6) indicate 

,, ,

,, ,

, ,

,

,

im km km

mij mij mkj

ij kj

S S S ki

T T ki

== =

⊂⊂ ≤

⇒ ⊂ ≤

∪∪ ∪

 (9) 

where “⇒” denotes “implies.” 

2.2. A Classical Example 

Note that the classical travel-time isochronal map for a watershed (Linsley et al. 1982 p280) is an 

example of our definitions. It is built to estimate a watershed’s time-area histogram, which is 

then used further to estimate the unit hydrograph for a watershed. For example, consider the 

mouth of a watershed at time 0 with outflow resulting from the application of a unit depth of water over the entire watershed. If we determine the travel times from all locations in the watershed 

to the mouth and plot them, we have the classical hydrological travel time isochronal map shown 

on the left side of Figure 3 for an arbitrary watershed. Each isochrone represents ( ,0) Y τ ; for 

example, the 3δ isochrone represents ( 3,0) Y − . Then, resource sheds ( ) Vi,0 for water departing 

during the i th time interval 0, 1, 2, ... i = −− , and arriving at the watershed mouth during the 0th

time interval are those areas with travel times within [− −+ i i δ δ ,( 1) ) ; the shaded area in Figure 3 

shows V−3,0 . By identifying similar resource sheds for other time intervals from the isochronal 

map, we can build the time-area histogram of classical hydrology, shown on the right side of 
6

 

Figure 3. Isochronal Travel Time Map and Resultant Time-Area Histogram. 

1δ

0 

2δ 6δ 5δ 4δ 3δ

7δ

1δ 2δ 3δ 4δ 5δ 6δ 7δ

Area 

Figure 3, which can be transformed into an estimate of a unit hydrograph by routing through a 

hypothetical reservoir. The shaded bar in the time-area histogram corresponds to the shaded resource shed in the isochronal map. 

3. MATERIAL DENSITIES 

While these definitions refer to the spatial extent of a contributing area, they do not address the 

differences between parts of a contributing area. Some parts of an area may supply more material to our location of interest than other parts. Represent the material’s areal density rate of 

change with departure and arrival times (mass/area/time/time) of material departing location ω

at time τ and arriving at the location of interest at time t as 

h t Yt ( ) () ωτ ω τ ,, , ∀ ∈ (10) 

where the expression,“∀ ω ∈ A,” represents the statement “for all ω within set A .” Integrating 

over departure and arrival times, we have the spatial density of material (mass/area), f , at location ω departing during [,) a b and arriving during [, ) c d , 

( ) () ( ) ,,,, ,, ,,,

b d

a c

f ω ωτ τ ω a b c d h t dt d R a b c d = ∀∈ ∫∫ (11) 

Note that for physically relevant situations, 

( ) ()

( )

,,,, 0 ,,,

0 ,,,

f abcd R abcd

R abcd

ω ω

ω

>∀∈

=∀∉ (12) 
7

(We omit the second line of the above equation in the following function definitions, presuming 

it is understood.) Again for convenience, consider only discrete time intervals of δ and define 

, () ( ) ( ) ( ) , , 1,, 1, i j i j x ω ω δδ δ δ ω = − − ∀∈ f i ij j V (13) 

Then ( ) i j , x ω is the areal density of material at location ω departing during the i th time interval 

and arriving during the j th time interval. Define 

, () ( ) ( ) ( ) , , 1, , 1, i j i j z f i jj j S ω ω δδ δδ ω = − − ∀∈ (14) 

, () ( ) ( ) ( ) , , 1, , 1, w f i ji j T i j ω ω δδ δδ ω = − − ∀∈ i j (15) 

where ( ) i j , z ω is the density of material at location ω departing during time intervals , ..., i j and 

arriving during the j th time interval and ( ) wi j , ω is the density of material at location ω departing during time intervals , ..., i j and arriving also during time intervals , ..., i j . Figure 4 illustrates an example resource shed distribution based on Figure 2; notice there are variations within 

the resource shed on how much material is contributed. A resource shed boundary can be discerned from a resource shed mapping. It corresponds to the border between areas with positive 

contributions and those with zero contributions. Note that 

() () ,, ,

,

ij mj ij

mij

zx S ω ωω == ∀∈ ∑ (16) 

() () ,, ,

,

i j im i j

mij

wz T ω ωω == ∀∈ ∑ (17) 

Similar to resource sheds, we expect resource shed distributions for any time period to enclose 

(be greater than) those for included time periods, for a given location of interest and time. Since 

Figure 4. Example Resource Shed Distribution, ( ) 4,6 z ω ∀ 4,6 ω ∈S , for Figure 2. 

Location of 

interest at 

time 6

less dense 

more dense 
8

( ) i j , x ω , ( ) i j , z ω , and ( ) wi j , ω are strictly positive within their encompassing resource sheds, 

then similar to (7) and (8), for all ω

() ()

() ()

, ,

, ,

, ,

,

,

mj mj

mij mkj

ij kj

x x ki

z z ki

ω ω

ω ω

= =

≤ ≤

⇒ ≤ ≤

∑ ∑

 (18) 

Equations (18) and (17) indicate for all ω

() () ()

() ()

,, ,

,, ,

, ,

,

,

im km km

mij mij mkj

ij kj

z z z ki

w w ki

ωω ω

ω ω

== =

≤≤ ≤

⇒ ≤ ≤

∑∑ ∑

 (19) 

4. ESTIMATING RESOURCE SHEDS 

4.1 Hydrological Resource Sheds 

In a watershed, resource sheds can be computed from spatially distributed watershed hydrology 

models; material placed anywhere in the watershed will appear at the watershed outlet (mouth) 

over a period of time. We calculate the material appearing at the mouth at any time, contributed 

by a specific area. We consider every cell c (say 1 km2) of a watershed surface by tracing the 

material departing the cell over time interval i and computing the amount arriving at the mouth 

in time interval j , ijc , , x

( )

( )

,, , ,

, ,

,

, 1, ..., /

c

c

ijc ij c ij

A

ij ij

A

x x d AV

x d c va

ω ω

ω ω

= ∀∈

= =

∫

∫ (20) 

where Ac is the set of all locations comprising cell c , i j , v is the area of Vi j , , and a is the area of 

Ac , , 1, ..., / i j c va = (all cells have the same area); the units of ijc , , x are mass. This requires modeling all cells, but tracing contributions from only one at a time. Since it involves no more computation, we simulate material movement from each cell c during time intervals , ..., i j to the 

mouth in time interval j ijc , , z ; by (16) 

() ()

( )

,, , , ,

,

, ,

,

,, ,

,

,, ,

,

,

,

,

, 1, ..., /

c c

c

ijc ij mj c ij

A A mij

mj c ij

mij A

mjc c ij

mij

mjc ij

mij

z z d x d AS

x d AS

x AS

x c sa

ωω ω ω

ω ω

=

=

=

=

⎡ ⎤ = = ∀∈ ⎢ ⎥ ⎣ ⎦

= ∀∈

= ∀∈

= =

∫ ∫ ∑

∑ ∫

∑

∑

 (21) 
9

where i j , s is the area of i j , S . Similarly we can combine results of several sets of simulations as 

in (17) to determine the source distribution of departing material in the watershed going through 

the mouth in any specified time period 

( ) ,, , ,

,, ,

,

,

, 1, ..., /

c

ijc ij c ij

A

imc i j

mij

w w d AT

z c ta

ω ω

=

= ∀∈

= =

∫

∑ (22) 

where i j , t = area of Ti j , . (We infer resource sheds from these mappings as described previously). 

4.2 Material Fractions 

We can also calculate relative fractions as well as absolute amounts; they will be useful later as 

source density estimates. Represent the material fraction arriving in time interval j that departed in time interval i from an area Ac as i jc , , p . Represent the material fraction arriving in 

time interval j that departed during time intervals ,..., i j from area Ac as i jc , , q . Finally, represent the material fraction arriving during time intervals ,..., i j that departed during time intervals 

i j ,..., from area Ac as i jc , , u . Note that 

,, ,, ,

,, ,, ,

,, ,, ,

,

, 1, ..., /

, 1, ..., /

, 1, ..., /

ijc ijc j ij

ijc ijc j ij

ijc ijc m ij

mij

p x g c va

q z g c sa

u w gc ta =

= =

= =

= = ∑

 (23) 

where j g is the total material arriving at the location of interest in time interval j . Note that 

,, ,, ,, ,,

, ,,

,, , , , , ,, , ,

, ,,

, ,

1 1

1 1

ijc mjc j mjc mjc

mij mij mij j j

i j c im c m im c i jm im c

m m mij mij mij

mij mij

q x gp p

g g

u z gq q

g g

γ

= ==

= ==

= =

== =

== =

∑ ∑∑

∑ ∑∑ ∑ ∑

 (24) 

where , ,

,

i jm m k

k ij

γ g g == ∑ , which is the fraction of all material arriving at the location of interest 

during time intervals ,..., i j that arrived in time interval m ( ) imj ≤ ≤ . 

4.3 Lake Resource Sheds 

In a lake, resource sheds may be computed with a lake circulation model in which particles (tracers) are defined at a given location of interest and the model is run backward in time from a 

specified time. The locations of the particles at earlier times define the extent of the resource 
10

shed. By releasing the same number of particles each day in the backward simulation, one can 

estimate resource sheds for various prior periods to the date in question. The density of the particle distribution over the resource shed defines the fraction of material contributed. The example in Figure 5 shows only a small sampling for clarity. 

The total material arriving at the location of interest in time interval j is 

, ,

1, ,

j m jk

k Nm j

g x = =−∞= ∑ ∑ (25) 

where N = number of cells in the lake (watershed). By rewriting the first line of (23) after substituting (25), 

, ,

, ,

, ,

1, ,

i jc

i jc

m jk

k Nm j

x

p

x = =−∞

=

∑ ∑ (26) 

We have been using (26) in resource shed developments to this point. However, the estimate of 

the material fraction arriving in time interval j that departed in time interval i from an area Ac

that is determined from this backward time particle tracing is: 

, ,

, ,

, ,

1,

ˆ i jc

i jc

i jk

k N

x

p

x =

=

∑ (27) 

So backward time particle tracing where the same number of particles are released each day in 

the backward simulation gives only the spatial distribution of each time interval’s contributions 

to the location of interest during the last time interval, but not the spatial-temporal distribution. 

Therefore, 

Figure 5. Example Estimating Resource Shed Distribution by Tracking Particles Backward. 

Location of 

interest at 

time 0 
11

,, ,,

, , 1,

,, , ,,

ˆ

ˆ

i jc i jk

i jc k N

i jc i j i jc

j j

p x

x

p p

g g

φ = == =

∑

 (28) 

where φi j , = fraction of all material arriving during time interval j that originated over all cells 

during time interval i . The use of the estimate in (27) for i jc , , p in (24) implies the use of φi j , = 1 

(constant); the same number of particles arrive on day j originating on day i for all i . For example, the backward time particle tracing with the same number of particles released each day in 

the backward simulation uses ,

1

7 φi j = for each day of a 7-day simulation. But constant φi j , is 

unreasonable. We expect that , 0 φi j → as i → −∞ and that φ φ ij mj , , < for imk j < << where k

is the date/time of the peak contribution. Also note that 

,

,

1 i j

i j

φ j =−∞

∑ = ∀ (29) 

We would have information on φi j , indirectly in a forward simulation that uses a loading pattern 

with specified dynamic inputs and outputs from all rivers and channels. If we use the backward 

simulation, then we have to supply φi j , either from additional small forward simulations or other 

considerations. For example, we could use the reversed 2-parameter gamma distribution as a 

function of time lag only, 

( )

1

1

,

1 x j i

i j j i

x e dx

α δ

δ β φ δ βαβ

− − + −

−

⎡ ⎤ = ⎢ ⎥ Γ ⎣ ⎦ ∫ (30) 

where δ = time increment length, Γ( )i is the gamma function, and α and β are parameters. Or 

we could use the reversed 1-parameter gamma distribution ( β = δ ), 

( )

1 1

,

j i 1 x

i j j i

x e dx α φ

α

− + − −

− = Γ ∫ (31) 

where α is taken as a function of time (season?). 

Furthermore, 

,, ,, , ,,

, ,

,, ,, ,, ,, , ,,

, ,,

ˆ

ˆ

ijc mjc mj mjc

mij mij

i j c i jn in c i jn mn mn c

n i j n i j m in

qp p

uq p

φ

γ γφ

= =

= ==

= =

= =

∑ ∑

∑ ∑∑ (32) 

With information on φi j , , we can calculate resource shed densities for material originating over 

time interval ( ) i j , and arriving over the j th time interval. With information on i jn , , γ and φi j , , 
12

we can calculate resource shed densities for material originating over time interval ( ) i j , and arriving over the same time interval. 

5. EXAMPLE MAUMEE RESOURCE SHEDS 

5.1 Hydrological Model 

We use the Great Lakes Environmental Research Laboratory’s Distributed Large Basin Runoff 

Model (DLBRM) to model watershed outflow. It represents each cell (1 km2

) of a watershed’s 

area as a cascade of moisture storages or tanks, each modeled as a linear reservoir, where tank 

outflows are proportional to tank storage; see Figure 6. Snow melt is proportional to snow accumulation and air temperature each day when temperatures are above freezing and infiltration 

into the top soil tank is a function of tank saturation (variable area infiltration where zero area 

corresponds to complete saturation). It computes potential evapotranspiration from heat available during the day, which is proportional to the exponential of air temperature, and computes 

actual evaporation or evapotranspiration for each tank from the potential and the moisture content of the tank; potential and actual evapotranspiration are therefore non-complementary, appropriate for small areas. Each tank has lateral flows between cells: an upstream flow into the tank 

and a downstream flow out of the tank for all moisture storages: surface zone, upper soil zone, 

lower soil zone, and groundwater zone. Each cell’s inflow hydrographs must be known before 

its outflow hydrograph can be modeled and the DLBRM arranges calculations by flow network 

to assure this. It is implemented to minimize the number of pending hydrographs in storage and 

the time required for them to be in storage. It uses the same routing network for lateral flows 

between all surface, upper soil zone, lower soil zone, and groundwater zone storages. 

Each cell’s sub model (shown in Figure 6) has 15 parameters. It is calibrated for all cells by systematically searching the 15 spatial-average-parameter space by using gradient search techniques 

to minimize the root mean square error between modeled and actual basin outflow. The spatial 

variations of each parameter is the same as a function of selected watershed characteristics; for 

example the upper soil zone capacity from cell to cell is proportional to measurements taken 

from the field and the percolation coefficient (linear reservoir coefficient) varies proportional to 

measured permeability of the upper soil zone in the field. See Croley and He (2005, 2006) and 

Croley et al. (2005) for DLBRM details and Croley (2002) for the earlier lumped-parameter 

model which preceded the DLBRM. To speed up calibrations, we preprocess all meteorology 

for all watershed cells and preload it into computer memory. 

Figure 7 shows example animation frames for the Maumee watershed in northwest Ohio. The 

watershed outlet is to the northeast; see the right map column in Figure 7 showing surface flows. 

In Figure 7, a storm peaks on September 1, 1950 and its pattern and intensity are obvious from 

the 1st column in Figure 7. The figure also shows that evapotranspiration is much higher on September 1st than on other days (see the middle column) and that surface response is higher too (see 

the right column). The surface outflow network is seen as most extensive during the storm peak. 

The Maumee is a very flashy watershed that responds quickly to surface supply. Note that while 

the spatial structure of the precipitation is revealed in the 1st column of Figure 7, the 2nd column 

reveals the spatial structure of the watershed as well as of the precipitation supply. These spatial 

variations mimic upper soil zone permeability. Likewise, the 3rd column reveals the spatial 

structure of the drainage network. 
13

Figure 6. DLBRM Schematic for One Cell.

Insolation

Precipitation

Temperatur e

Snow Rain

Melt, m

Supply, s

Snow Pack, P

Upper Soil Zone

Moisture, U

Lower Soil Zone

Moisture, L

Groundwater Zone

Moisture, G

Surface Storage,

S

Runoff

Surface

Runoff

s U

( ) C

Evapotranspiration

( ) βuepU

Capacity, C

Evapotranspiration

βl ( ) epL

Evapotranspiration

Evaporation

Interflow

( ) αiL

Ground

Water

( ) αgG

Basin Outflow

( ) αsS

Percolation

( ) αpU

Deep

Percolation

( ) αdL

( ) βgepG

βs ( ) epS
14

Figure 7. Maumee River Watershed Model Animation for August 31—September 3, 1950
15

5.2 Example Resource Sheds 

Figure 8 shows example resource sheds for the Maumee River on January 1, 1950 from 1, 7, and 

31 days of previous loading. In Figure 8, the brightest areas correspond to cells contributing 

about 0.015% of the total flow on January 1, 1950. The darkest areas are close to zero. Note 

several things about Figure 8. The southern and western ridgelines are prominent as is a line to 

the north that marks the boundary between Ohio and Michigan (AB in Figure 8). This boundary 

reflects the differences in the two States’ definitions of some soil properties and so is an artifact 

of data standard differences. Point C in Figure 8 identifies the mouth of the watershed. The first 

map in Figure 8 shows a little response from the previous day’s light rain near the mouth of the 

watershed. The second and third maps show most response along the edges of the watershed furthest from the mouth. Inspection of rainfall maps shows there is not much rainfall over the prior 

4 days but there is a large amount 5 days prior in the southwest area. Also, spatially uniform 

rainfall fell over the entire watershed 7 days prior, 11-13 days prior, 15 days prior, 21-22 days 

prior, and 29 days prior. We can see the bright spot corresponding to the large peak 5 days earlier; the area closer to the mouth is relatively dark in the second and third maps because the only 

supplies there (seven or more days earlier) had already run off and are not part of the flow on this 

day. Note also that what happens prior to 7 days changes the picture very little (compare the last 

two maps). This is because the response of the watershed to supply is quick, on the order of 1 to 

6 or 7 days, depending on location within the watershed. Most all supplies falling more that 6 or 

7 days ago have already runoff and do not form a part of the flow on this day. 

5.3 Average Resource Shed vs. Resource Shed of Average Meteorology 

There are 17,541 1-km2

 cells in the Maumee River watershed. The DLBRM requires 0.2—0.4 

seconds on today’s desktop computers to simulate 1 day’s hydrology from all of these cells, for 

Figure 8. Maumee Resource Sheds on January 1, 1950 from prior days of loading: 

 1-day: January January c 1,1950, 1,1950, q , c = 1, ..., 17,541; 

 7-day: December January c 26,1949, 1,1950, q , c = 1, ..., 17,541; 

31-day: qDecember Januar 2,1949, y1,1950, c , c = 1, ..., 17,541. 

A 

B 

N 

C 

1-Day 7-Day 31-Day 

0 0.015% 
16

any loading pattern. With 17,541 loading patterns (one tracing each cell’s contribution), a oneday simulation of loadings requires 1—2 hours of computation. For 1-, 2-, ..., 10-day simulations for one date there are 55 simulation days, which require 2—5 days of computation. 

A parallel study requires average resource sheds for May 31 and August 31 for each contribution 

period from 1 through 31 days for linking to resource sheds in a lake that extend back into its 

watersheds. To calculate Maumee resource sheds over the historical record of 1950—1999 for 

averaging for these two dates and 31 contribution periods would require 49,600 simulation days 

or 5.5—11 years of computation, which is clearly impractical. Rather than calculate resource 

sheds for historical meteorology and then averaging them, we averaged the meteorology over all 

years for each day of the year and then calculated resource sheds based on this (1 year of) average meteorology. This required only 2% of the above: 992 simulation days or 40—80 days of 

computation. This is practical but begs the question: how well does the resource shed of average 

meteorology approximate the average resource shed (or how linear is the watershed model)? 

To answer this question, we calculated resources sheds for selected dates for each year of record 

(1950—1999) for various contribution periods using about 20 machines to reduce the computation period. We averaged over the historical record and compared the resource sheds; Figure 9 

shows selected average resource sheds and the resource sheds of average meteorology throughout the annual cycle. Figure 9 reveals two non-linearities in the model with respect to air temperature. The first occurs over winter, when air temperatures are close to or below freezing; the 

average resource shed (top Figure 9) reflects that snowmelt occurs in some years. However, average air temperature over the winter is zero and resource sheds calculated for an averaged winter date will have no snow melt (bottom Figure 9). This is because the model calculates snowFigure 9. Maumee River Six-Day Average Resource Sheds for Selected Times of the Year. 
17

melt proportional to snow accumulation and temperature each day when temperatures are above 

freezing and zero when below freezing. There is a consequent over-accumulation of snow in the 

model applied to the average meteorology resulting in overly high snowmelt; see bottom Figure 

9 for March 1. The second non-linearity occurs mostly in the summer. Then exponent of the 

average air temperature (used to calculate heat available for evapotranspiration) is less than the 

average exponent of the air temperatures and there is less potential (and actual) evapotranspiration in Figure 9 bottom than top. Therefore more of the moisture in the watershed ends up as 

outflow in the resource shed of the average meteorology (Figure 9 bottom) than in the average 

resource shed (Figure 9 top). However, the differences are slight so that the resource shed of the 

average meteorology approximates the average resource shed for April—October. 

6. COMPUTATION REDUCTION 

6.1 Estimation, Interpolation, and Resolution 

To reduce computations we can simulate for a sample of cells distributed uniformly over the watershed; see Figure 10. In Figure 10, the three-digit codes beneath each resolution example refer 

to the horizontal and vertical offsets and the spacing, respectively. For example in the first resolution in the upper left corner of Figure 10, the code refers to a 4-cell spacing with 0 cell offsets 

in both the horizontal and vertical directions. For a selected resolution we interpolate between 

sampled cells to estimate values at the other cells. As computer resources become available, we 

can simulate again for a separate sample of cells and add them to the previous sample to increase 

resolution, reducing interpolation errors. Alternatively, we could simulate for all cells of the watershed at a coarser resolution. However, the disadvantage of that approach is that we would not 

be able to combine with another simulation to improve resolution; instead we would have to 

simulate for all cells of the watershed at a finer resolution. The third example in the upper right 

corner of Figure 10 has every 4 out of 16 cells computed, or 1 of 4, which is used in Figure 11. 

Figure 10. Resource Shed Distribution Map Computation Resolutions. 

004 004, 224 004, 224, 024, 204 

004, 224, 024, 204 

334, 114, 314, 134

004, 224, 024, 204, 334, 114, 314, 134 

324, 104, 304, 124, 234, 014, 214, 034 
18

Figure 11. Western Maumee Resource Shed Distribution Map Interpolations and Resolutions. 

D-1 D-2

TIN THS

16 / 16 

(no interpolation)

We considered several interpolation schemes, as shown in Figure 11 for the western Maumee 

watershed. At all resolutions, the inverse distance (D-1) and inverse distance squared (D-2) interpolations produce artifacts at the scale of the resolution (which are barely discernable at the scale 
19

used in Figure 11). The inverse distance squared gives better definition to the features than the 

inverse distance at all resolutions. The triangular irregular network with linear interpolation 

(TIN) does not produce the artifacts of the two inverse distance methods and gives better definition to features in the watershed. It also gives the most detailed picture of structure of the three 

methods at all resolutions. Finally, the Thiessen method appears probably the most aesthetically 

pleasing since the contrast is better at all resolutions, giving the illusion of more detail. It also 

paints a true picture of the resolution being used. However, it can be argued that even though the 

Thiessen interpolation appears crisper than the corresponding TINs, it may not represent the actual surface as well. That appears to be the case here: see the bright spot in the Southwest section of the watershed; the first three interpolations appear to capture the double tail lying to the 

southeast of the bright spot, while the Thiessen does not (compare with the right-most picture 

where all cells are present and no interpolation is required). On the other hand, the northwest 

head of the bright spot seems best represented in the Thiessen picture. 

6.2 Ordering Computations 

Computation reduction may also be achieved through computation ordering. Consider the calculation of a resource shed, i jc , , q for specific values of i and j for all c , , 1, ..., / i j c sa = . There 

are 1 j − +i days to simulate. (For the DLBRM Maumee example, recall that each simulation 

day requires 1—2 hours of computation on today’s desktop computers.) If we simply repeat the 

calculation for every prior period from one to 31 days, there are 1 + 2 + ··· + 31 = 496 simulation 

days required for one end date, j . For 365.25 end dates in an average year, there are 

496×365.25 = 181,164 simulation days required per year. However, we could utilize more information from one simulation with the hydrology model. Consider that one simulation from 

day i to day j actually gives us (for all c ) i jc , , q , ij c , 1, q − , ij c , 2, q − , ···, iic , , q . Therefore, if we do 

365.25×31 = 11,323 simulations days per year, we can calculate the same information as above 

with only 6.25% of the effort. 

Finally, there is interest in resource shed calculations in near real time to aid in detection of watershed source areas responsible for pollutant transport to receiving waters on an event basis. 

This would aid in the management of watershed runoff for limiting harmful algal blooms or in 

the closing of beaches in the receiving waters. It would also enable more effective land use 

management within a watershed. Consider a watershed application where it is deemed sufficient 

each day to estimate the 31 resource sheds corresponding to prior periods of 1 through 31 days. 

This would require 496 simulation days each day by the first method outlined previously (or 

about 496—992 hours of computation each day) which is clearly impractical in near real time. 

The second method outlined previously does not save computation since one 31-day simulation 

yields 31 resource sheds all ending on different days; we would again require 496 simulation 

days to calculate the 31 resource sheds of different durations all ending on the same day. 

However, by saving all internal model moisture and tracer storages at the end of the day from the 

model for each of the first 30 durations (1-day through 30-day), the next day we could make a 

one-day simulation (for the 1-day resource shed) and use the saved internal storages (from yesterday’s 1-day through 30-day simulations) as initial conditions in 30 one-day simulations to calculate the 2-day through 31-day resource sheds. By again saving all internal storages at the end 

of the day, we could repeat for the next day and so forth. We need only compute 31 1-day simu-
20

lations (31 simulation days or about 31—62 hours of computation per day). There are several 

ways to reduce computations further: change the hydrological model cell size from 1 km2

 to 2 

km × 2 km (4 km2

) to reduce the computations to one sixteenth of the 1 km2 model, use the 1 

km2

 model cell size but use less than full resolution and interpolate (e.g., reducing resolution by 

half and interpolating would reduce computations by half), and use multiple processors (e.g., a 

ten-processor machine would do one-tenth of the total computations on each processor in parallel, reducing computations to one tenth). Any or a combination of these methods could reduce 

computations; for example, using a 10-processor machine at half resolution and interpolating for 

1-km2

 model cell size reduces computation time by 20. That comes to about 1.5—3 hours of 

computation per day, which is practical in near real time. 

7. LINKING RESOURCE SHEDS 

7.1 Definitions for the Joint Resource Shed 

We desire to (eventually) extend a lake’s resource shed back into a contributing watershed, requiring the joining of resource sheds, estimated with different techniques, through a point (the 

mouth of a watershed). See Figure 12 where the superscripts denote Lake (L) and Watershed 

(W). The joint resource shed for material departing in time interval i and arriving at the location 

of interest (in the lake) in time interval j , ,

C Vi j , is 

, ,

, ,

,

,

j

C L

ij ij j

L W

i j im j

m ik

V V ik

V V ik =

= >

= ≤ ∪ ∪ (33) 

where j k is the last time interval for material leaving the watershed arriving at the location of 

interest in the lake in time interval j . Since , j

W Vi k = ∅ ∀ j i k > , we can write (33) simply as 

,, ,

, j

CL W

i j i j im

m ik

VV V == ∪ ∪ (34) 

Summing (34), 

,, ,, ,

, , ,, , , ,

, ,

, ,,

j jj

j

C L WL W

n j n j nm n j nm

n i j n i j n i j m nk n i j n ik m nk

L W

n j nm

n i j m ik n im

VV VV V

V V

= = == = = =

= ==

= =

=

∪ ∪

∪

∪ ∪ ∪∪ ∪ ∪∪

∪ ∪∪ (35) 

Then by (5), we can construct the resource shed with material departing during time intervals 

i j , ..., and arriving during time interval j , ,

C

i j S (noting that , j

W

i k S = ∅ ∀ j i k > ) 

,, ,

, j

CL W

i j i j im

m ik

SS S == ∪ ∪ (36) 
21

Lake 

Boundary 

Location 

of interest 

at time 6 

5,6

L V

4,6

L V

2,3

W V

3,6

L V

2,6

L V

3,3

W V

6,6

L

2,2 V

W V

Figure 12. Example Lake Resource Shed Just Touching Watershed Mouth 3δ Ago. 

Likewise, summing (36) 

,, ,

, , ,, n

CL W

in in im

n i j n i j n i j m ik

SS S = = ==

∪ ∪ ∪∪ = ∪ (37) 

or 

,, ,

, , n

CL W

i j i j im

n i j m ik

TT S = == ∪ ∪ ∪ (38) 

The joint resource shed distributions for material departing in time interval i and arriving in time 

interval j are 

,, ,, ,

,, , , , ,

,

, 1, ..., / ,

1, ..., / ,

j

CL C

ijc ijc ij j

LW C

i j c m j im c i j j

m ik

x x c v a ik

x α x c v a ik =

= =>

=+ = ≤ ∑ (39) 
22

where α m j , (calculated with the lake circulation model) is the fraction of material leaving the watershed in time interval m that arrives at the location of interest in the lake in time interval j . 

(Since α m j , is applied to , ,

W

im c x , , 1, ..., / W

i m c va = , j i k ≤ , the assumption is that watershed outflow 

is fully mixed.) Note that the watershed and lake do not overlap (intersect only at a point); therefore one of the two terms on the right side of (39) will be zero for each , 1, ..., / C

i j c va = . Because 

, 0 α m j = ∀ m k > j , we write simply 

,, ,, , , , ,

,

1, ..., / CL W C

i j c i j c m j im c i j

mij

x x x c va α

=

=+ = ∑ (40) 

By summing (40) over consecutive time intervals , ..., i j , analogous to (35), one can demonstrate 

,, ,, , , , ,

,

1, ..., / CL W C

i j c i j c m j im c i j

mij

z z z c sa α

=

=+ = ∑ (41) 

Similarly, summing (41), analogous to (37), gives 

,, ,, , , , ,

, ,

1, ..., / CL W C

i j c i j c mn im c i j

n i j m in

w w z c ta α = ==+ = ∑ ∑ (42) 

Finally, looking at fractions of material moved, by (23) and (40) 

,, ,,

,, , , ,

,

,

,, , ,

,

,, , , , ,

,

1

, 1, ..., /

C L

C W ijc ijc

i j c m j im c L LL

j jj mij

L WW m j

i j c m im c L

mij j

LW C

i j c m j im c i j

mij

x x

p x

g gg

p gp

g

p p c va

α

α

β

=

=

=

==+

= +

=+ =

∑

∑

∑

 (43) 

where , ,

W L

mj mj m j β α= g g which is the fraction of material arriving at the location of interest in 

the lake in time interval j that came from the watershed mouth in time interval m . Likewise 

,, ,, , , , ,

,

1, ..., / CL W C

i j c i j c m j im c i j

mij

q q q c sa β

=

=+ = ∑ (44) 

,, ,, ,, , , , ,

, ,

1, ..., / CL L W C

i j c i j c i jn mn im c i j

n i j m in

u u q c ta γ β = ==+ = ∑ ∑ (45) 

where , ,

,

LL L

i jn n k

k ij

γ g g == ∑ , which is the fraction of all material arriving at the location of interest 

in the lake during time intervals ,..., i j that arrived in time interval n ( ) inj ≤ ≤ . 
23

7.2 Example Calculation Algorithm 

Suppose we want to calculate source material distributions for the 7-day resource shed for May 

31 for Lake Erie and the Sandusky River watershed. [We define this resource shed distribution 

as meaning the areal distribution of all material (water) that moved through our location of interest in the lake in the 7-day period ending May 31, (May 25—31).] Suppose that lake circulation 

modeling gave us the information in Table 1 on the fraction of material arriving at the location of 

interest in the lake in time interval j that came from the watershed mouth in time interval i , βi j , . 

Furthermore, the fraction of the 7-day flow volume arriving at the location of interest in the lake 

each day of May 25—31 was supplied by the lake circulation modelers in Table 2. 

Table 2. 25,31, , 25, ..., 31 L

n γ n = , May, Sandusky 

n 25 26 27 28 29 30 31 

25,31,

L

n γ 0.241 0.217 0.193 0.157 0.096 0.060 0.036 

From watershed model calculations, we have the resource shed distributions over the Sandusky 

watershed for material arriving at the watershed mouth in time interval j that came from cell 1 

in time intervals i j , ..., , , ,1

W

i j z in Table 3. 

Table 3. 25, ,1, 25, ..., 31 W

j z j = , May, Sandusky 

j 25 26 27 28 29 30 31 

25, ,1

W

j z 2.10e-5 3.22e-5 2.11e-5 1.86e-5 2.09e-5 1.59e-5 2.66e-5 

The total arriving at the watershed mouth in time interval j , W

j g , is given in Table 4. 

Table 1. , , 25, ..., 31, 25, ..., 31 i j β i j = = , May, Sandusky 

i \ j 25 26 27 28 29 30 31 

25 0 0 0 0.01 0.07 0.09 0.10 

26 0 0 0 0.02 0.03 0.06 

27 0 0 0 0.05 0.08 

28 0 0 0 0.04 

29 0 0 0 

30 0 0 

31 0 

Shaded cells are produced by the lake circulation model. 

Un-shaded cells are zeroes. 
24

Table 4. Daily Watershed Outflow W

j g , j = 25, ..., 31, May, Sandusky. 

j 25 26 27 28 29 30 31 

W

j g 9.80e-2 1.12e-1 1.05e-1 9.42e-2 9.21e-2 8.29e-2 9.61e-2 

As in (23), divide each column in Table 3 by the corresponding entry in Table 4 to calculate the 

fraction of material departing cell 1 in time intervals , ..., i j arriving at the watershed mouth in 

time interval j , , ,1

W

i j q , shown in Table 5. 

Table 5. 25, ,1 25, ,1 / , 25, ..., 31 W WW

j jj q z gj = = , May, Sandusky 

j 25 26 27 28 29 30 31 

25, ,1

W

j q 2.14e-4 2.88e-4 2.01e-4 1.97e-4 2.27e-4 1.92e-4 2.77e-4 

The convolution of watershed resource shed distributions with lake resource shed distributions, 

as shown in (45), gives the total resource shed distribution over both the watershed and the lake 

for water departing its origin in time intervals , ..., i j and arriving at the location of interest in 

the lake during the same period for cell c , , ,

C

ijc u , as either the resource shed distribution over the 

lake, , ,

L

ijc u , when c is a cell in the lake, or a convolution of watershed resource shed distributions 

arriving at the watershed mouth in time interval m , , ..., mi j = departing its origin in time intervals , ..., i m, , ,

W

imc q , when c is a cell in the watershed. For c = 1 (in the watershed), the 7-day 

total resource shed distribution (within the watershed) ending on May 31 (i = 25, j = 31) is 

found by first computing the terms, , 25, ,1

W

mn m β q , 25, ..., 31 n = , 25, ..., m n = . From Tables 1 and 

5, we build the shaded part of Table 6. Then we sum each column to calculate , 25, ,1

25,

W

mn m

m n

β q =

∑ , 

n = 25, ..., 31 in the next row. We place 25,31,n γ entries from Table 2 in the next row for n = 25, 

..., 31, multiply these two rows together to calculate 25,31, , 25, ,1

25,

W

n mn m

m n

γ β q =

∑ , n = 25, ..., 31 in the 

final row, and finally sum the final row for n = 25, ..., 31, to get 25,31, , 25, ,1

25,31 25,

W

n mn m

n mn

γ β q = =

∑ ∑ in 

the bottom row right column, in black. Thus 6.86e-6 or 0.000686% of the flow through the location of interest in the lake during May 25—31 came from cell 1 in the Sandusky watershed during the same period. 

To summarize, for a given location in the lake and a given month, the procedure to calculate 

, ,

C

ijc u over the watershed ( 1, ..., W c n = where Wn = the number of cells in the watershed) for i = 

1, 25, and 31 and j = 31 (the 31-day resource shed, the 7-day resource shed, and the 1-day resource shed, respectively for the 31st) given , , , 1, ..., , , ..., W

imc W z c n mi j = = , , , ..., W

m g mi j = , 

, , , ..., , , ..., m n β n i jm i j = = , and , , , , ..., L

ijn γ ni j = is shown with pseudo code given in Table 7. 
25

7.3 Example Lake Erie—Maumee River Linked Resource Sheds 

We can do an example coupling of Lake Erie and Maumee River resource sheds and their distributions by using (44) and (45) with , ,

W

ijc q calculated as i jc , , q in (32) and with φi j , in (32) calculated with (31) for α = 30 days. Shown in Figure 13 are coupled resource shed densities for material originating over the indicated periods ending on August 31 and passing through the location of interest over that same period. They were calculated by using the hydrology model for 

the Maumee watershed and the Princeton Ocean Model applied to Lake Erie circulation at the 

Great Lakes Program of the University of Buffalo. (Note that both the lake and watershed resource sheds are calculated from daily average meteorology over 1983—2002). Lake Erie is depicted with a constant color not to be confused as part of the spectrum of resource shed values. 

The location of interest is Western Basin Lake Erie Site 835 just adjacent to the tip of the arrow 

in Figure 13. There we see that on August 31, the 1-day resource shed does not extend into the 

watershed (flow from Maumee actually takes about two days to reach the location of interest at 

this time of year). The latter resource sheds do extend into the Maumee watershed and intensify 

as we look over longer prior periods. 

Table 6. , 25, ,1, 25, ..., 31, 25, ..., W

mn m β qn m n = = , May, Sandusky and additional computations. 

m \ n 25 26 27 28 29 30 31 n=25,31

∑

25 0 ×

2.14e-4 

0×

2.14e-4 

0×

2.14e-4 

0.01×

2.14e-4 

0.07×

2.14e-4 

0.09×

2.14e-4 

0.10×

2.14e-4 

26 0×

2.88e-4 

0×

2.88e-4 

0 ×

2.88e-4 

0.02×

2.88e-4 

0.03×

2.88e-4 

0.06×

2.88e-4 

27 0×

2.01e-4 

0 ×

2.01e-4 

0×

2.01e-4 

0.05×

2.01e-4 

0.08×

2.01e-4 

28 0 ×

1.97e-4 

0×

1.97e-4 

0×

1.97e-4 

0.04×

1.97e-4 

29 0×

2.27e-4 

0×

2.27e-4 

0×

2.27e-4 

30 0×

1.92e-4 

0×

1.92e-4 

31 0×

2.77e-4 

 

, 25, ,1

25,

W

mn m

m n

β q =

∑ 0 0 0 2.14e-6 2.07e-5 3.80e-5 6.26e-5 

25,31,n γ 0.241 0.217 0.193 0.157 0.096 0.060 0.036 

25,31, , 25, ,1

25,

W

n mn m

m n

γ β q =

∑ 0 0 0 3.36e-7 1.99e-6 2.28e-6 2.25e-6 6.86e-6 
26

8. SUMMARY 

A resource shed is defined as the area contributing material, over one time interval, passing 

through a location of interest over another time interval. We looked at definitions and concepts 

that preceded the resource shed concept and discussed interpretations in several fields. We also 

contrasted the concept with watersheds. Following this introduction, we provided rigorous general definitions of resource sheds and derived expressions for: 1) resource sheds where materials 

departing during one time interval arrive at the location of interest during a later time interval, 2) 

resource sheds where materials departing during a sequence of time intervals arrive at the location of interest during the last of those time intervals, and 3) resource sheds where materials departing during a sequence of time intervals arrive at the location of interest during that same sequence of time intervals. We also expressed each of these resource sheds in terms of each other 

and showed that definition 2 resource sheds are contained in others with a longer departing time 

for the same arrival interval. Likewise, definition 3 resource sheds are contained in others with 

earlier start times and the same end times. These definitions of resource sheds as areas of source 

materials were illustrated with the classical travel-time isochronal map for a watershed, used to 

build a watershed’s time-area histogram for use in deriving a synthetic unit hydrograph. 

We then provided rigorous general definitions of resource shed distributions and derived expressions for the distributed amount of source material within a resource shed, for all three definitions of same. We again expressed each of the resource shed distributions in terms of each other 

and showed that resource shed distributions are contained by others with earlier start times, 

analogous to resource sheds themselves, for both definitions 2 and 3. We discussed the estimation of resource shed distributions (corresponding to all three definitions) within a watershed 

Table 7. Watershed—Lake Resource Shed Distribution Convolution Algorithm Pseudo Code. 

! FROM LAKE CIRCULATION MODEL OUTPUT FOR GIVEN LOCATION & MONTH: 

! INPUT BETA(M, N), M = I,...,J, N = I,...,J 

! INPUT OR CALCUATE GAMMA(I, J, N), N = I,...,J 

! FROM WATERSHED HYDROLOGY MODEL OUTPUT FOR GIVEN MONTH: 

! INPUT ZW(I, M, C), I = 1,...,31, J = 1,...,31, C = 1,...,NW 

! INPUT GW(M), M = I,...,J 

J = 31 

DO I = 1, 25, 31 

 DO C = 1, NW 

 A1 = 0

! The 31-day, 7-day, & 1-day resource sheds 

! NW = number of watershed cells 

! dummy accumulator variable

 DO N = I, J 

 A2 = 0 ! dummy accumulator variable 

 DO M = I, N 

 A2 = A2 + BETA(M, N) * ZW(I, M, C) / GW(M) 

 END DO 

 A1 = A1 + A2 * GAMMA(I, J, N) 

 END DO 

 UC(I, J, C) = A1 

 END DO 

 END DO

Shaded denotes non-executable comments. 
27

Figure 13. Maumee—Erie Linked Resource Shed for Average August 31 Conditions for Site 835

1 day 

2 weeks 

1 week 

3 weeks 

4 weeks 
28

through the use of a spatially distributed hydrology model, in which the watershed area consists 

of discrete cells, by first defining resource shed distributions in terms of discrete space and showing their interrelationships with each other, presenting the distributions in terms of relative fractions of the material passing through the location of interest, introducing a specific hydrologic 

model for use on the Maumee River watershed in Ohio, and calculating example resource shed 

distributions for material passing through the watershed outlet on January 1, 1950 for 3 different 

periods of previous loadings. 

Next, we used resource shed distributions to observe the non-linearity of the hydrology model 

and its relative impact in different seasons of the year. We did this by comparing resource sheds 

computed from average meteorology to the average of resource sheds calculated from actual meteorology. We found two non-linearities associated with air temperature: a major one associated 

with snow melt during the winter and a lesser one associated with evaporation during the summer and fall. We concluded that resource sheds calculated from average meteorology are only 

slightly different from average resource sheds for April—October and may be used in studies 

linking average resource sheds between a watershed and receiving lake waters, for example. 

The large amount of computation sometimes required in the construction of resource sheds and 

their distributions encourages methods to reduce calculations. We can calculate resource shed 

distributions over only a sample of watershed cells and then interpolate for the other cells. We 

explored alternate resolution trade-offs and a few spatial interpolation technique trade-offs. 

Also, we can order resource shed computations, when more than one resource shed is computed, 

to save on computations. Ordering makes sense when contiguous dates and durations are considered or in near real time where yesterday’s computations are extended one day in today’s. 

Finally, we considered the joining of a resource shed beginning in a lake and progressing (as we 

consider longer prior periods of flow for a given date) into the adjoining watersheds that contribute to the lake. We derived definitions and relations for the combined resources shed and the 

combined resource shed distribution for all three definition types identified above. We depicted 

the joint resource shed distribution for the Lake Erie—Maumee watershed system in an example 

application. 

It should be possible to extend this work in several ways. Currently, we are linking hydrological 

watershed resource sheds and their distributions with lake resource sheds generated via different 

methods, as exemplified here. We plan to produce, for the World Wide Web, a dynamic linking 

of several Lake Erie watersheds with the lake to produce joint resource sheds and their distributions associated with about 35 locations of interest in the lake and extending into the tributary 

watersheds. Also, as we add material transport capabilities to our hydrologic model, we will 

produce resource sheds and their distributions for nutrients, sediment, insecticides, and microbes 

that may be of more direct use in prediction of harmful algal blooms and beach closings than 

simply water transport. Finally, in support of these latter predictions, we are also building a 

daily near real time generator of resource sheds and their distributions and plan to move to 

hourly as the model develops further. 
29

9. REFERENCES 

Ben-David, M., G. M. Blundell, J. W. Kern, J. A. K. Maier, E. D. Brown, and S. C. Jewett, 2005. 

Communication in river otters: creation of variable resource sheds for terrestrial communities.

Ecology, 86(5), 1331. 

Brunckhorst, D. and I. Reeve, 2006. A geography of place: principles and application of defining ‘eco-civic’ resource governance regions. Australian Geographer 37(2):147-166. 

Chang, M., and C. Cardelino, 2000. Application of the urban airshed model to forecasting nextday peak ozone concentrations in Atlanta, Georgia. Journal of the Air and Waste Management 

50(11):2010-2024. 

Chow, V. T., D. R. Maidment, and L.W. Mays, 1988. Applied Hydrology. McGraw-Hill Inc., 

New York. 

Cousins, S. H., 1990. Countable ecosystems deriving from a new food web entity. Oikos,

57:270-275.C. 

Croley, T. E., II, 2002. Large basin runoff model. In Mathematical Models in Watershed Hydrology (V. Singh, D. Frevert, and S. Meyer, Eds.), Water Resources Publications, Littleton, 

Colorado, 717-770. 

Croley, T. E., II, and C. He, 2005. Distributed-parameter large basin runoff model I: model development. Journal of Hydrologic Engineering, 10(3):173-181. 

Croley, T. E., II, and C. He, 2006. A watershed surface and subsurface spatial intraflows model. 

Journal of Hydrologic Engineering, 11(10):12-20. 

Croley, T. E., II, C. He, and D. H. Lee, 2005. Distributed-parameter large basin runoff model II: 

application. Journal of Hydrologic Engineering, 10(3):182-191. 

Ellis, A. W., D. M. Brommer, and R. C. Balling, 2006. Climatic conditions linked to high PM10 

concentration in a bi-national airshed: Nogales (Arizona, USA, and Sonora, Mexico). Climate 

Research 30(2): 113-124. 

He, C., S. Cheng, and Y. Luo, 2005. Desiccation of the Yellow River and the South Water 

Northward Diversion Project. Water International (SCI) 30(2):261-268. 

Linsley, R. K., Jr., M. A. Kohler, and J. L. H. Paulhus, 1982. Hydrology for Engineers, Third Edition, McGraw-Hill, New York, New York, 508 pp. 

Michel, S. M., 2000. Defining hydrocommons governance along the border of the Californias: A 

case study of transbasin diversions and water quality in the Tijuana-San Diego metropolitan region. Natural Resources Journal 40:931-972. 
30

Morawska, L., D. Vishvakarman, K. Mengersen, and S. Thomas, 2002. Spatial variation of airborne pollutant concentrations in Brisbane, Australia and its potential impact on population exposure assessment. Atmospheric Environment 36(21):3545-3555. 

National Research Council, 1999. New Strategies for America’s Watersheds. National Academy 

Press, Washington, D. C. 

Oswood, M. W., J. B. Reynolds, J. G. Irons, and A. M. Milner, 2000. Distribution of freshwater 

fishes in ecoregions and hydroregions of Alaska. Journal of the North American Benthological 

Soceity 19(3):405-418. 

Polis, G. A., W. A. Anderson, and R. D. Holt, 1997. Toward an integration of landscape and 

food web ecology. Annual Review of Ecology and Systematics, 28:289-316. 

Power, M. E., and W. E. Rainey, 2000. Food webs and resource sheds: towards spatially delimiting trophic interactions, pp. 291-314 in Hutchings, M.J., E.A. John, and A.J.A. Stewart (eds), 

The Ecological Consequences of Environmental Heterogeneity, Blackwell Science, Oxford, U.K. 

Santoul, F., A. Soulard, J. Figuerola, R. Cẻrẻghino, and S. Mastrorillo, 2004. Environmental factors influencing local fish species richness and differences between hydroregions in SouthWestern France. International Review of Hydrobiology 89(1):79-87. 

Tullar, I. V. and I. H. Suffet, 1975. Fate of vanadium in an urban airshed-Lower Delaware River 

Valley. Journal of The Air Pollution Control Association 25(3):282-286. 

U. S. U.S. Environmental Protection Agency, 1995. Watershed Protection: A Statewide Approach. Office of Water, EPA 841-R-95-004. Washington, D. C. 
31

APPENDIX OF NOTATION 

The following symbols are used in this paper (upper case letters denote sets and lower case letters denote functions or scalars): 

Ac = set of all locations comprising cell c ; 

a = area of Ac , , 1, ..., / i j c va = (all cells have the same area); 

c = watershed surface cell number; 

f ( ) ω,,,, abcd = spatial density of material (mass/area) at location ω departing during [ , ) a b

and arriving at LOI during [ , ) c d ; 

j g = total material arriving at LOI in time interval j ; 

W

j g = j g , defined over watershed for LOI at mouth of watershed; 

h t ( ) ω τ, , = areal density rate of change with departure and arrival times of material 

(mass/area/time/time) departing location ω at time τ and arriving at LOI at 

time t ; 

j k = last time interval for material leaving watershed to arrive at LOI in lake in 

time interval j ; 

 LOI = denotation of “location of interest;” 

i jc , , p = material fraction arriving at LOI in time interval j that departed in time interval i from area Ac ; 

, ,

C

ijc p = ijc , , p defined over watershed and lake for LOI in lake; 

, ,

L

ijc p = ijc , , p defined over lake for LOI in lake; 

, ,

W

ijc p = ijc , , p defined over watershed for LOI at mouth of watershed; 

, , ˆi jc p = material fraction arriving at LOI in time interval j that departed in time interval i from area Ac , determined from backward time particle tracing where the 

same number of particles are released each day in the backward simulation; 

i jc , , q = material fraction arriving at LOI in time interval j that departed during time 

intervals ,..., i j from area Ac ; 

, ,

C

ijc q = ijc , , q defined over watershed and lake for LOI in lake; 

, ,

L

ijc q = ijc , , q defined over lake for LOI in lake; 

, ,

W

ijc q = ijc , , q defined over watershed for LOI at mouth of watershed; 
32

R( ) abcd ,,, = set of all locations (resource shed) where materials departing during time interval [,) a b arrive at LOI during time interval [, ) c d ; 

i j , S = set of all locations (resource shed) where materials departing during time intervals , ..., i j arrive at LOI during time interval j ; 

,

C

i j S = i j , S defined over watershed and lake for LOI in lake; 

,

L

i j S = i j , S defined over lake for LOI in lake; 

,

W

i j S = i j , S defined over watershed for LOI at mouth of watershed; 

i j , s = area of i j , S ; 

,

C

i j s = area of ,

C

i j S ; 

Ti j , = set of all locations (resource shed) where materials departing during time intervals , ..., i j arrive at LOI also during time intervals , ..., i j ; 

,

C Ti j = Ti j , defined over watershed and lake for LOI in lake; 

,

L Ti j = Ti j , defined over lake for LOI in lake; 

,

W Ti j = Ti j , defined over watershed for LOI at mouth of watershed; 

t = time; 

i j , t = area of Ti j , ; 

,

C

i j t = area of ,

C Ti j ; 

i jc , , u = material fraction arriving at LOI during time intervals ,..., i j that departed 

during time intervals ,..., i j from area Ac ; 

, ,

C

ijc u = ijc , , u defined over watershed and lake for LOI in lake; 

, ,

L

ijc u = ijc , , u defined over lake for LOI in lake; 

, ,

W

ijc u = ijc , , u defined over watershed for LOI at mouth of watershed; 

Vi j , = set of all locations (resource shed) where materials departing during time interval i arrive at LOI during time interval j ; 

,

C Vi j = Vi j , defined over watershed and lake for LOI in lake; 

,

L Vi j = Vi j , defined over lake for LOI in lake; 

,

W Vi j = Vi j , defined over watershed for LOI at mouth of watershed; 
33

i j , v = area of Vi j , ; 

,

C

i j v = area of ,

C Vi j ; 

( ) wi j , ω = areal density of material at location ω departing during time intervals , ..., i j

and arriving at LOI also during time intervals , ..., i j ; 

wijc , , = mass of material in watershed surface cell c departing during time intervals 

i j , ..., and arriving at LOI also during time intervals , ..., i j ; 

, ,

C wijc = wijc , , defined over watershed and lake for LOI in lake; 

, ,

L

wijc = wijc , , defined over lake for LOI in lake; 

, ,

W wijc = wijc , , defined over watershed for LOI at mouth of watershed; 

( ) i j , x ω = areal density of material at location ω departing during time interval i and 

arriving at LOI during time interval j ; 

ijc , , x = mass of material in watershed surface cell c departing during time interval i

and arriving at LOI during time interval j ; 

, ,

C

ijc x = ijc , , x defined over watershed and lake for LOI in lake; 

, ,

L

ijc x = ijc , , x defined over lake for LOI in lake; 

, ,

W

ijc x = ijc , , x defined over watershed for LOI at mouth of watershed; 

 (,) Y t τ = set of all locations where materials departing at time τ arrive at LOI at time 

t ;

( ) i j , z ω = areal density of material at location ω departing during time intervals , ..., i j

and arriving at LOI during time interval j ; 

ijc , , z = mass of material in watershed surface cell c departing during time intervals 

i j , ..., and arriving at LOI during time interval j ; 

, ,

C

ijc z = ijc , , z defined over watershed and lake for LOI in lake; 

, ,

L

ijc z = ijc , , z defined over lake for LOI in lake; 

, ,

W

ijc z = ijc , , z defined over watershed for LOI at mouth of watershed; 

α = gamma distribution parameter; 

α m j , = fraction of material leaving watershed in time interval m that arrives at LOI 

in lake in time interval j ; 
34

β = gamma distribution parameter; 

β m j , = fraction of material arriving at LOI in lake in time interval j that came from 

watershed mouth in time interval m ; 

Γ( )i = gamma function; 

i jm , , γ = fraction of all material arriving at LOI during time intervals ,..., i j that arrived 

in time interval m ; 

, ,

L

i jm γ = fraction of all material arriving at LOI in lake during time intervals ,..., i j that 

arrived in time interval m ; 

δ = length of time interval used in discrete-time resource shed definitions; i.e., 

time interval i is [i - δ , i ); 

τ = time; 

φi j , = fraction of all material arriving at LOI during time interval j that originated 

over all cells during time interval i ; 

∀ = denotation of range; e.g., “∀ ω ∈ A,” denotes “for all ω within set A ;” 

∪ = operator representing union of sets; 

∅ = the empty set; 

⊂ = denotation of inclusion; i.e., “ A ⊂ B ” denotes “set A is contained in set B ;” 

∈ = denotation of inclusion; e.g., “∀ ω ∈ A,” denotes “for all ω within set A ;” 

and 

⇒ = denotation of implication; e.g., “ A B ⊂ and B ⊂ C ⇒ A C ⊂ ” denotes “set 

A is contained in B and B is contained in C implies A is contained in B .” 
35

APPENDIX OF PROPER GRAY SCALE RENDERING OF FIGURE PAGES FOR PRINTING 

The following pages correspond to pages in this report that appear in color but are re-rendered in 

proper gray scale for those requiring them. Some color printers may not show the detail in the 

color figures but may show it properly in gray scale. For best results, print the following gray 

scale figure pages in color to preserve the 24-bit grayscale. Direct conversion of the color figures to gray scale will distort the maps and usually convert to 8-bit grayscale. 
14

Figure 7. Maumee River Watershed Model Animation for August 31—September 3, 1950
15

5.2 Example Resource Sheds 

Figure 8 shows example resource sheds for the Maumee River on January 1, 1950 from 1, 7, and 

31 days of previous loading. In Figure 8, the brightest areas correspond to cells contributing 

about 0.015% of the total flow on January 1, 1950. The darkest areas are close to zero. Note 

several things about Figure 8. The southern and western ridgelines are prominent as is a line to 

the north that marks the boundary between Ohio and Michigan (AB in Figure 8). This boundary 

reflects the differences in the two States’ definitions of some soil properties and so is an artifact 

of data standard differences. Point C in Figure 8 identifies the mouth of the watershed. The first 

map in Figure 8 shows a little response from the previous day’s light rain near the mouth of the 

watershed. The second and third maps show most response along the edges of the watershed furthest from the mouth. Inspection of rainfall maps shows there is not much rainfall over the prior 

4 days but there is a large amount 5 days prior in the southwest area. Also, spatially uniform 

rainfall fell over the entire watershed 7 days prior, 11-13 days prior, 15 days prior, 21-22 days 

prior, and 29 days prior. We can see the bright spot corresponding to the large peak 5 days earlier; the area closer to the mouth is relatively dark in the second and third maps because the only 

supplies there (seven or more days earlier) had already run off and are not part of the flow on this 

day. Note also that what happens prior to 7 days changes the picture very little (compare the last 

two maps). This is because the response of the watershed to supply is quick, on the order of 1 to 

6 or 7 days, depending on location within the watershed. Most all supplies falling more that 6 or 

7 days ago have already runoff and do not form a part of the flow on this day. 

5.3 Average Resource Shed vs. Resource Shed of Average Meteorology 

There are 17,541 1-km2

 cells in the Maumee River watershed. The DLBRM requires 0.2—0.4 

seconds on today’s desktop computers to simulate 1 day’s hydrology from all of these cells, for 

Figure 8. Maumee Resource Sheds on January 1, 1950 from prior days of loading: 

 1-day: January January c 1,1950, 1,1950, q , c = 1, ..., 17,541; 

 7-day: December January c 26,1949, 1,1950, q , c = 1, ..., 17,541; 

31-day: qDecember Januar 2,1949, y1,1950, c , c = 1, ..., 17,541. 

A 

B 

N 

C 

1-Day 7-Day 31-Day 

0 0.015% 
16

any loading pattern. With 17,541 loading patterns (one tracing each cell’s contribution), a oneday simulation of loadings requires 1—2 hours of computation. For 1-, 2-, ..., 10-day simulations for one date there are 55 simulation days, which require 2—5 days of computation. 

A parallel study requires average resource sheds for May 31 and August 31 for each contribution 

period from 1 through 31 days for linking to resource sheds in a lake that extend back into its 

watersheds. To calculate Maumee resource sheds over the historical record of 1950—1999 for 

averaging for these two dates and 31 contribution periods would require 49,600 simulation days 

or 5.5—11 years of computation, which is clearly impractical. Rather than calculate resource 

sheds for historical meteorology and then averaging them, we averaged the meteorology over all 

years for each day of the year and then calculated resource sheds based on this (1 year of) average meteorology. This required only 2% of the above: 992 simulation days or 40—80 days of 

computation. This is practical but begs the question: how well does the resource shed of average 

meteorology approximate the average resource shed (or how linear is the watershed model)? 

To answer this question, we calculated resources sheds for selected dates for each year of record 

(1950—1999) for various contribution periods using about 20 machines to reduce the computation period. We averaged over the historical record and compared the resource sheds; Figure 9 

shows selected average resource sheds and the resource sheds of average meteorology throughout the annual cycle. Figure 9 reveals two non-linearities in the model with respect to air temperature. The first occurs over winter, when air temperatures are close to or below freezing; the 

average resource shed (top Figure 9) reflects that snowmelt occurs in some years. However, average air temperature over the winter is zero and resource sheds calculated for an averaged winter date will have no snow melt (bottom Figure 9). This is because the model calculates snow 

Figure 9. Maumee River Six-Day Average Resource Sheds for Selected Times of the Year. 
27

Figure 13. Maumee—Erie Linked Resource Shed for Average August 31 Conditions for Site 835

1 day 

2 weeks 

1 week 

3 weeks 

4 weeks