Dr. Bruce H. Stanley DuPont Crop Protection Stine-Haskell Research Center Newark, Delaware

1. The Moment Generating Function As A Useful Tool in Understanding Random Effects on First-Order Environmental Dissipation Processes Dr. Bruce H. Stanley DuPont Crop Protection Stine-Haskell Research Center Newark, Delaware

2. The Moment Generating Function As A Useful Tool in Understanding Random Effects on First-Order Environmental Dissipation Processes Abstract Many physical and, thus, environmental processes follow first-order kinetics, where the rate of change of a substance is proportional to its concentration. The rate of change may be affected by a variety of factors, such as temperature or light intensity, that follow a probability distribution. The moment generating function provides a quick method to estimate the mean and variance of the process through time. This allows valuable insights for environmental risk assessment or process optimization.

3. Agenda • First-order (FO) dissipation • The moment generating function (MGF) • Relationship between FO dissipation and MGF • Calculating the variance of dissipation • Other “curvilinear” models • Half-lives of the models • References • Conclusions

4. - First-Order Dissipation -

5. Model: First-Order Dissipation Rate of change: Model: Transformation to linearity: Constant half-life:

6. Example: First-Order Dissipation

7. Some Processes that Follow First-Order Kinetics • Radio-active decay • Population decline (i. e., “death” processes) • Compounded interest/depreciation • Chemical decomposition • Etc…

8. - The Moment Generating Function -

9. Definition: Moment Generating Function

10. Example: Moment Generating Function X ~ Gamma(,)

11. Relationship Between – First-Order Dissipation –and the Moment Generating Function

12. Random First-Order Dissipation where r ~ PDF Constant

13. Conceptual Model:Distribution of Dissipation Rates dCt1/dt = r1.Ct1 dCt2/dt = r2.Ct2 dCt3/dt = r3.Ct3 dCt4/dt = r4.Ct4 r < 0

14. Transformation of r or t? r < 0 X = -r It’s easier to transform t, I.e.,  = -t = -t so substitute t = - And treat r’s as positivewhen necessary r = -1.X fr(r) = fX(-r) E(rn) = (-1)n.E(Xn)

15. Typical Table of Distributions(Mood, Graybill & Boes. 1974. Intro. To the Theory of Stats., 3rd Ed. McGraw-Hill. 564 pp.)

16. Some Possible Dissipation Rate Distributions • Uniform r ~ U(min, max) • Normal r ~ N(r, 2r) • Lognormal r ~ LN(r= e+ 2/2,2r = r2.(e 2-1)) = ln[r /(1+ r2/2r)],; 2 = ln[1+ (r2/2r)] • Gamma r ~ (r= /,2r = /2) = r2/2r;  = r/2r(distribution used in Gustafson and Holden 1990) * Where r and 2r are the expected value and variance of the untransformed rates, respectively.

17. Application to Dissipation Model: Uniform No need to make  = -t substitution

18. Application to Dissipation Model: Normal No need to make  = -t substitution Note: Begins increasing at t = -r/r2, and becomes >C0 after t = -2.r/r2.

19. Application to Dissipation Model: Lognormal Note: Same as normal on the log scale.

20. Application to Dissipation Model: Gamma(Gustafson and Holden (1990) Model) Make  = -t substitution

21. Distributed Loss Model

22. Key Paper: Gustafson & Holden (1990)

23. - Calculating the Variance -

24. Example: Variance for the Gamma Case Make  = -t substitution

25. - Random Initial Concentration -

26. Variable Initial Concentration:Product of Random Variables Delta Method Delta Method

27. - Other “Non-Linear” Models -

28. Other “Non-linear” Models • Bi- (or multi-) first-order model ………..…... • Non-linear functions of time, …………..…… e.g., t = degree days or cum. rainfall (Nigg et al. 1977) • First-order with asymptote (Pree et al. 1976).. • Two-compartment first-order……………….. • Distributed loss rate…………………….…… (Gustafson and Holden 1990) • Power-rate model (Hamaker 1972)………..…

29. First-order With Asymptote

30. Two Compartment Model

31. Distributed Loss Model

32. Power Rate Model

33. - Half-lives -

34. Half-lives for Various Models (p = 0.5) • First-order*………………………. • Multi-first-order*………………… • First-order with asymptote ……… • Two-compartment first-order …… • Distributed loss rate …………….. • Power-rate model ………………. * Can substitute cumulative environmental factor for time, i.e.,

35. - References -

36. References Duffy, M. J., M. K. Hanafey, D. M. Linn, M. H. Russell and C. J. Peter. 1987. Predicting sulfonylurea herbicide behavior under field conditions Proc. Brit. Crop Prot. Conf. – Weeds. 2: 541-547. [Application of 2-compartment first-order model] Gustafson, D. I. And L. R. Holden. 1990. Nonlinear pesticide dissipation in Soil: a new model based upon spatial variability. Environ. Sci. Technol. 24 (7): 1032-1038. [Distributed rate model] Hamaker, J. W. 1972. Decomposition: quantitative aspects. Pp. 253-340 In C. A. I. Goring and J. W. Hamaker (eds.) Organic Chemicals in the Soil Environment, Vol 1. Marcel Dekker, Inc., NY. [Power rate model] Nigg, H. N., J. C. Allen, R. F. Brooks, G. J. Edwards, N. P. Thompson, R. W. King and A. H. Blagg. 1977. Dislodgeable residues of ethion in Florida citrus and relationships to weather variables. Arch. Environ. Contam. Toxicol. 6: 257-267. [First-order model with cumulative environmental variables] Pree, D. J., K. P. Butler, E. R. Kimball and D. K. R. Stewart. 1976. Persistence of foliar residues of dimethoate and azinphosmethyl and their toxicity to the apple maggot. J. Econ. Entomol. 69: 473-478. [First-order model with non-zero asymptote]

37. Conclusions • Moment-generating function is a quick way to predict the effects of variability on dissipation • Variability in dissipation rates can lead to “non-linear” (on log scale) dissipation curves • Half-lives are not constant when variability is present • A number of realistic mechanisms can lead to a curvilinear dissipation curve (i.e., model is not “diagnostic”)

38. Questions?

39. - Thank You! -