A probability model for acid rain data

War. Res. Vol. 24, No. II, pp. 1335-1339, 1990 0043-1354/90 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright ~ 1990 Pergamon Press pk

A PROBABILITY MODEL FOR ACID RAIN DATA

A. H. EL-SHAARAWI and A. NADERI National Water Research Institute, Canada Centre for Inland Waters, Burlington,

Ontario, Canada L7R 4A6

(First received July 1989; accepted in revised form May 1990)

Almraet--A class of probability models is proposed for making inferences about the regional distribution of lake alkalinity. This class includes the three-parameter Iognormal distribution which is frequently used as a regional model for lake chemistry. The adequacy of the lo~ormal distribution can be assessed by testing the significance of a single parameter in the proposed model. It is shown that the three-parameter lognormal does not fit the majority of data sets from Eastern Canada. Furthermore, the pH-alkalinity relationship developed by Small and Sutton is used to yield a class of probability models representing the derived distributions for pH. It is shown that there is an improved agreement between the pH-alkalinity relationship and the observations when the water colours are less than or equal to 30 Hazen units.

Key words--alkafinity, pH, pH-alkalinity relationship, probability distributions, statistical inference, Box-Cox Power transformation

INTRODUCTION

Acidification models permit the prediction of changes in lake or stream chemistry that correspond to changes in acid deposition and allow for the identification of important acidification factors,, thus giving guidance to decision makers in the formulation of policy for combating the deleterious effects of acid rain. Although the prediction of pH is almost always required for evaluating biological impacts, most models predict the changes in alkalinity which are then used to predict the corresponding changes in pH. Recently, Small and Sutton (1986a) derived an explicit equation relating pH and alkalinity of natural waters and tested its applicability using both regional and temporal lake acidification data. In a second paper, Small and Sutton (1986b) used a three-parameter lognormal to represent the regional distribution of lake alkalinity and developed a model for predicting the impacts of lake acidification on fish survival. Small et aL (1988) provided further evidence of the applicability of the pH--alkalinity relationship and distributions for a number of regions and dis- cussed reasons why they may not work in certain cases. Since the lognormality of the regional distribution of alkalinity and the pH-alkalinty relationship are among the critical assumptions involved in generating the predictions of the model, this paper examines the adequacy of those assumptions for the pH-alkalinity data from Canadian lakes in New Brunswick, Newfoundland, Nova Scotia and Quebec. The main objective of the paper, however, is to develop a probability model to represent the lak~ alkalinity distributions. This model is based on the Box--Cox (1964) power transformation and covers the family of lognormal distributions.

pH-ALKALINITY RELATIONSHIP

Small and Sutton (1986a) expressed the relationship between pH and alkalinity (Alk) by the equation:

p H - a + b arcsinh ( - ~ ) , (l)

where a, c and d are unknown constants and b -- I/In I0.

This equation was fitted to the pH-Alk data from Canadian lakes in New Brunswick, Newfoundland, Nova Scotia, and Quebec using the International Mathematical and Statistical Library (IMSL, 1987) subroutines for nonlinear regression methods. The data were collected by Environment Canada using a non-random sampling method and was compiled at the Canada Centre for Inland Waters. The surface area of the lakes range from 0.I to 18,330 ha with the exception of one lake with a surface area of 66,000 ha. Information regarding the data can be found in Howell and Brooksbank (1987). The data set used here is not representative of all the lakes and includes only those lakes for which the measurements for all three parameters (Alk, pH and water colour) were available. The estimates of a, c and d are given in Table I, and the fitted and observed pH values are compared in Figs I--4. The results show that the fit is poor for the coloured lakes in Nova Scotia. Howell and Brooksbank (1987) indicates that when dissolved organic carbon concentrations increase above 5 mg/l and/or water colours exceed 30 Hazen units, organic acidity will become more important to the overall acidification process. In order to determine whether the lack of fit is due to the mixed population of coloured and clear lakes, the data were classified into high colour (> 30 Hazen units) and low colour

1335

1336 A. H. EL-SHAARAWI and A. NADERI

Table 1. Estimates of the p,-i-Alk parameters

No. of Degrees of freedom Mean square pH-Alk observations d a~ ~ for error error

New Brunswick 117 5.264 0.217 0.569 114 0.036 New Brunswick

(water colours ~<30 Hazen units) 86 5.302 0.199 0.560 83 0.027 New Brunswick

(water colours >30 Hazen units) 31 5•161 0.228 0.778 28 0.028 Newfoundland 323 5.345 0.106 0.536 320 0.050 Newfoundland

(water colours 430 Hazen units) 168 5.329 -0.056 0.459 165 0.043 Newfoundland

(water colours >30 Hazen units) 155 5.326 0.238 0.590 152 0.043 Nova Scotia 196 5.291 0.220 0.560 193 0.071 Nova Scotia

(water colours g30 Hazen units) 95 5.370 0.163 0.521 92 0.034 Nova Scotia

(water colours >30 Hazen units) 101 5.186 0.202 0.707 98 0.075 Quebec 177 5.014 0.039 0.219 174 0.313

(~< 30 Hazen units) groups and then equat ion (1) was fitted to each group. The results are given in Table 1 and Fig. 5. These results indicate an improved agreement between the observed and fitted values when the water colours are ~< 30 Hazen units.

A PROBABILITY MODEL FOR LAKE ALKALINITY

Consider the Box and Cox (1964) transformation g(x) defined by:

Llog(x + k2), k I ---- 0, X'> --'~2"

Let X b e a random variable. I f g ( X ) is normal (/~, a2), then the probability density function (lxlf) of X is given by:

i f (x ) = ~/~0 • (x + k2) x' - '

x exp{-(g(2"~/1)2} . (2)

This form generates a wide class of distributions for X d e p e n d i n g on the values o f ;t t a n d /1. 2. For

example, the normal and lognormal distributions are obtained by setting 2, to 1 and 0, respectively. When ,~, = 0 but ~-2 # 0, the three-parameter lognormal distribution is obtained. Let X,, X2 . . . . . X, be a sequence of independent and identically dis- tributed random variables with I x l f f ( x ) . Then the likelihood function is

L (2,, :.2, I~, a 2) = c(2n~2) -'/2 " ~ (xl + 22) ~'- l

t - I

x (g (x,)

where c is a constant. Assuming that Alk is a random variable, a

probabili ty distribution model for the lake Alk is defined by

= Prob(Alk ~< x) = f ~ f ~ ( y ) F x) dy,

where fro, is the Ixtf of Alk defined by (2). F~ak is called the cumulative distribution function (cdf) o f A l L The parameters o f f ~ are estimated as follows. Consider a populat ion o f n lakes and let Alk,. be the Alk value

9 -

8

7 'S"

7 m s ~

5

FitteO

4 ObserveO

3 I I I I I - 5 2 0 4 5 70 9 5 120

ALK (meq/L)

Fig• I. Observed and fitted pH for the New Brunswick data•

"7"

8

7 ~"."

6 :'"

4

I _3 15

Fitted

O~erve~

I I 35 55 75

AL.K (meq/Ll

95

Fig. 2. Observed and fitted pH for the Newfoundland data.

A probability model for acid

8

7 -

?

4 / "

3 -10

~tted Observed

20 50 80

ALK (maq/L)

110

Fig. 3. Observed and fitted pH for the Nova Scotia data.

9

3 - 3

Fitted Observed

I I 7 17 27

ALK (meq / t }

Fig. 4. Observed and fitted pH for the Quebec data.

for the ith lake (i = I, 2 . . . . . n). The estimates 12;,.; 2 a n d ' 2 o f ~ a n d ~ ~ o~,.~ z are obtained from the sample mean and variance of the transformed data:

l~,.~ 2 = ~ g(Alki)/n, i - I

#2.;~ = ~" (g(Alki) - :.h.~2)2/n. i-I

.I- Q.

rain data 1337

i

a Fitted Observed

I I I I I -3o Io 3o ~,o 7o so

ALK (maq/ t )

Fig. 5. Observed and fitted pH for the Nova Scotia data with water colours ~ 30 Hazen units.

The estimates 'fl and ~f2 o f ~-i and ~.2 are obtained so that L(J~ t , ~2,/2~,.I2, 8t,.x2) ;~ L(~q, ,1. 2 ,/2~,.~ 2, 8~,.~) for all possible values o f ,l~ and ,12. Let

R = In L(,I I, 42,/~;,.;2, ~ , . ; , )

- In L 0f,, ,f2,12x,.~,, # ],.z,)

Then- 2R is assymptotically X2(2), i.e. chi-square distribution with 2 degrees of freedom. Hence an approximate I00(I -a)% confidence region for ~t~, and 42 may be obtained by:

{(~, :.2):-2R< C,}={0.,,2.2):R,-~}. where Ca is determined from Prob0f2(2)...<Co) >_ - l -a .

The estimated parameters o f the Canadian lakes alkalinity distributions are given in Table 2. Figure 6 presents the 95% confidence regions for the New- foundland and Quebec data. The confidence region for the Quebec data includes ,t, = 0 and hence the loguormal distribution is adequate for this data set. This confidence region also indicates that the values of :.t and :.2 for the Quebec data are more precisely determined than those of the Newfoundland data. Figures 7 and 8 show good agreement between the

Table 2. Estimates of the AIk distribution parameters

AIk data 'fJ f.2 /i 8

New Brunswick -0 .7 3.95 1.030 0.172 New Brunswick

(water colours ~30 Hazen units) -0 .2 0.40 0.835 1.142 New Brunswick

(water colours >30 Hazen units) -0 .6 4.25 !.057 0.212 Newfoundland -0 .7 2.55 0.974 0.156 Newfoundland

(water colours 430 Hazen units) -0 .6 2.20 1.036 0.220 Newfoundland

(water colours >30 Hazen units) -0 .8 2.96 0.921 0.106 Nova Scotia :_ - 1.9 24.80 0.525 0.0003 Nova Scotia

(water eolours ~ 30 Hazen units) - 1.4 24.00 0.707 0.002 Nova Scotia

(water colours > 30 Hazen units) - 1.3 7.55 0.720 0.02 Quebec 0.1 - 0.01 0.061 1.362

WR24ill~

1338 A . H . EL-SHAARAWl and A . NADERI

)'2

5.0

3.9

2.8

1.7

06

NewfoundLand . . . . Quebec

-O5 I I t I -1.5 -1.0 -0.5 O0 0.5

X1

Fig. 6. 95% confidence regions for the transformation parameters of the Newfoundland and Quebec data.

LI,.. Q L,}

I0~

08

06

04

0 2 - -

/

Fitted Observed

00_3 = I I I 7 17 27

ALK ( m e q / L ]

Fig. 8. Observed and fitted distribution functions of the Quebec alkalinity data.

observed and fitted distributions of the Newfound- land and Quebec data, and further indicate the applicability of the optimal transformation method.

pH D I S T R I B U T I O N

The pH distribution may be obtained directly by the methods described in the previous section. How- ever, it is often desirable to allow predicted shifts in the Alk distribution to be directly translated into shifts in the pH distribution. Hence, assuming (1), the pH distribution may be derived directly from the corresponding Alk distribution:

Fpa(x) = Prob(pH ~< x)

= Prob(Alk ~ d + c sinh(b))

The corresponding pdf may be obtained by:

c

The derived pH distributions as well as the corresponding observed distributions of the Newfoundland and New Brunswick data are shown in Figs 9 and 10.

C O N C L U D I N G REMARKS

The Small and Sutton (1986b) model to predict the regional distribution of lake acidification has the form: All% ffi A l l % - c ( l - F), where All% and All% are the current and zero-deposition lake alkalinity respectively, c is a deposition concentration factor and F is a random variable representing neutraliz- ation rates in different watersheds. The distribution of both All% and All% in the above model are assumed to be lognormal. The difficulty with such a model is that the distribution of All% must be the convolution of All% and - c ( l - F ) and hence, in

if. e~ L~

1.0

0.8

0.6

0.4

02

0.0 -5

/

$ !

i

i

i t t l I i I

15

Fitted Observed

I I 35 55 75

ALK [ m e q / L ]

I 95

Fig. 7. Observed and fitted distribution functions of the Newfoundland alkalinity data.

1.0

081

0.6

0 4

/ /

/ f

Q2 . j - - Derive0

/ Observed

J J i t 0.0 4 5 6 7 8

pH

Fig. 9. Observed and derived distr ibution funct ions o f the N e w f o u n d l a n d pH data.

i /

A probability model

1.0-

0 . 8 - -

0.6 b_ t~ ¢.3

0.4

0.2 . / . . . ' " " Observed

0.0 ~ ' ' " "1 I I I 5 6 7 8

pH Fig. 10. Observed and derived distribution functions of the

New Brunswick pH data.

general, the distributions of AI~ and Alk0 cannot be the same. To maintain the same distributional form for alkalinity, one possibility is to replace F in the model by the mean/~F of F. As a result, Alk~ and Alk0 both have the same distribution. Nevertheless, the distributional form of Alk, and the pH-AIk relationship are essential assumptions involved in developing predictions using the model that corresponds to changes in acid deposition. The distributional model

for acid rain data 1339

presented here allows more flexibility and extends the scope of applications to a wider class of distributions than what is presently available. Furthermore, when the pH-Alk relationship does not hold for a par- ticular set of data, the data set, if possible, may be divided into groups so that those groups for which the pH-Alk relationship is adequate are identified. This has been illustrated in this paper by dividing the Nova Scotia data into two groups according to water colours. The results indicate an improved agreement between the pH-Alk relationship and the observations when the water colours are ~< 30 Hazen units.

REFERENCF~

Box G. E. P. and Cox D. R. 0964) An analysis of transformations (with discussion). JI R. Statist. Soc. Ser. B 39, 211-252.

Howell G. D. and Brooksbank P. (1987) An assessment of LRTAP acidification of surface waters in Atlantic Canada. IW/L-AR-WQB-87-121, Environment Canada.

IMSL (1987) MATH~LIBRARY and STAT/LIBRARY. IMSL, Houston, Tex.

Small M. J. and Sutton M. C. (1986a) A regional pH-alkalinity relationship. Wat. Res. 20, 335-343.

Small M. J. and Sutton M. C. (1986b) A direct distribution model for regional aquatic acidification. Wat. Resour. Res. 22, 1749-1758.

Small M. J., Sutton M. C. and Milke M. W. (1988) Parametric distribution of regional lake chemistry: fitted and derived. Envir. Sci. Teclmol. 22, 196-204.

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A probability model for acid rain data