Modeling risk to aquatic species 

subject to realistic, dynamic 

exposures using a generalized 

form of Haber’s law

Dr. Rick Reiss

1
Introduction

• Current exposure models in salmon 

BiOps estimate instantaneous 

concentration in streams

• These exposure estimates are compared 

to toxicity tests conducted over 48 to 96 

hours with constant exposure.

– Mismatch in exposure profile with streams

2
Salmonid Habitat

•Salmon generally reside in flowing water 

bodies

– Fast dispersion of spray drift

• Backwater pools, dammed pools and 

beaver ponds may represent 20% of 

capacity

– Still expect quicker loss than in standard 

ecotoxicity tests

3
Dye Added to Small, Low Flow 

Stream

4
Concentration vs. Time: Static Pond 

versus Stream

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50

Concentration (ppm)

Time (hours)

Static Pond

Stream

5
Simplified Example – C*T Loading

Scenario Initial 

Concentration

Dissipation Time C*T

Ecotoxicity 

Test

1 ppm None 48 hours 48 ppm-hour

Static Pond 1 ppm None 48 hours 48 ppm-hour

Stream 1 ppm T1/2 = 3 hours 48 hours 5 ppm-hour

6
Estimating Stream Concentrations 

with AGDRIFT

7
Assumed Range of Stream Velocities 

and Stream Depths

• Reviewed limited data on salmonidbearing streams

• Juvenile rearing Chinook salmon prefer 

slower-velocity streambank areas 

(USFWS, 1986):

– Migrate an average of 1.6 km/day

– Optimal stream depths of 0.3-1.2 m

– Optimal stream velocities of 0.06-0.24 m/sec

8
Assumed Range of Stream Velocities 

and Stream Depths

• Assumed stream depths of 0.1 to 0.5 

meters

• Assumed stream velocities of 0.02-0.5 

m/sec

9
Other Modeling Assumptions

• Aerial application with rate of 0.5 lb/A

• 100 feet upwind from stream (runoff 

assumed negligible)

• Fine-to-medium sized droplets

• Riparian interception factor of 0.2

•Wind direction directly from field to 

water body

• No dissipation or degradation

10
Time vs. Concentration: Averaging 

Period Has Dramatic Impact

0

5

10

15

20

25

30

35

40

45

50

0 12 24 36 48

Concentration (ppb)

Hour

Depth = 0.1 m

Velocity = 0.02 m/sec

11
Accounting for Different Toxicity 

from Different Concentration Profiles

Haber’s Law:

L=C*T

• Most risk assessments assume that this 

is true and that we can compare a timeweighted average concentration to a 

toxicity level.

•Stream dilution may present an extreme 

case

12
Generalized Haber’s Law

Generalized Haber’s Law:

where n is the toxic load exponent.

If n>1, a short pulse exposure has a 

greater effect than constant exposure 

with same C*T.

A _ c

T

n

L C t dt

0

( )

13
Generalized Haber’s Law

Consider an exponential decline in 

concentration, 

n

con

con

con nkT

nkT

C

C

1/

0

1 exp( )

n

con nkTcon Lim kT 1/

( )

C Coexp( kt)

14
Example Using Generalized Model

15
Estimating Toxic Load Exponent

Data from Legierse et al. (1999)

• 14-day lethality to guppies or pond 

snails

•Six OP insecticides

• n varied from 0.91-1.5 for five pesticides 

and was 2.7 for methidathion for 2-4 day 

exposures

16
Toxic load exponent fits for 11 OPs 

and 2 non-OP pesticides

17
Putting It All Together: Modeling 

Results

18
AGDRIFT Results Fitted to General 

Equations

0.2 5 4.75 0.023 ( )

H V

Coppb

0.7 8 0.5

0.0213 0.1

( ) 0.57 V Hk hr

19
Distribution of Fitted Rate Constants

20
Dilution Ratios

•Stream dispersion will create effective 

dilution ratios of 50-300

– Even after accounting for n>1 toxic load 

exponents

• Results may be conservative

– Salmonids move though different water 

depths to regulate body temperature

21
Recommendations

• Use AGDRIFT Stream Assessment Tool 

or equations in our paper to estimate 

stream concentrations

• Approach could be refined with:

– More steam characteristic data

– Inclusion of more observation points in 

routine ecotoxicological testing

– Probability of having different habitat 

characteristics

22
Publication

•In press* at Risk Analysis: An 

International Journal

Bogen KT; Reiss, R. Generalized Haber’s law for exponential 

concentration decline, with application to riparian-aquatic 

pesticide ecotoxicity. Risk Anal 2011 (in press). DOI: 

10.1111/j.1539-6924.2011.01663.x.

23
Acknowledgments

• Cheminova provided partial funding for 

this work as part of its comments on the 

salmonid BiOps for malathion and 

dimethoate

24