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mond, D.; Hartmann, B.; Maynard, V. Geochim. Cosmochim. Acta 1979,43, 1075-1090. (9) Tessenow, U.; Baynes, Y. Naturwissenschaften 1975,62, 342-343. (10) Robbins, J. A.; Callender, E. Am. J. Sci. 1975,275,512-533. (11) Cornwell, J. C. Can. J. Fish. Aquat. Sci. 1985,42,809-814. (12) Whalen, S. C.; Cornwell, J. C. Can. J. Fish. Aquat. Sci. 1985, 42,797-808. (13) Evans, R. D.; Rigler, F. H. Enuiron. Sci. Technol. 1980,14, 216-218. (14) Medlin, J. H.; Suhr,N. H.; Bodkin, J. B. At. Absorpt. Newsl. 8, 25-29. (15) Stainton, M. P. J . Fish. Res. Board Can. 1973, 30, 1441-1445. (16) Krauskopf, K. B. Geochim. Cosmochim. Acta 1957, 12, 61-68. (17) Giovanoli, R.; Burki, P.; Giaffredi, M.; Stumm, W. Chimica 29, 517-520. (18) Stumm, W.; Morgan, J. J. “Aquatic chemistry”;WileyInterscience: New York, 1981. (19) Hem, J. D. Adu. Chem. Ser. 1980,189,45-72. (20) Davis, J. A.; Leckie, J. 0. Enuiron. Sci. Technol. 1978,12, 1309-1315. (21) Lerman, A. “Geochemical Processes: Water and Sediment Environments”;Wiley: New York, 1979. (22) Li, Y.-H.; Gregory, S. Geochim. Cosmochim. Acta 1974,38, 703-713.
(23) Tsunogai, S.; Yonemaru, I.; Kusakabe, M. Geochem.J. 1979, 13, 239-252. (24) Kadko, D.; Heath, G. R. J. Geophys. Res. 1984, 89, 6567-6570. (25) Graybeal, A. L.; Heath, G. R. Geochim. Cosmochim.Acta 1984, 48, 965-975. (26) Wetzel, R. G. “Limnology”;Saunders: Philadelphia, 1975. (27) Tipping, E. Chem. Geol. 1981, 33, 81-89. (28) Jones, B. F.; Bowser, C. J. In “Lakes: Chemistry, Geology, Physics”;Lerman, A., Ed.; Springer-Verlag: New York, 1978; p 179-235. (29) Berrang, P. G.; Grill, E. V. Mar. Chem. 1974,2, 125-148. (30) Tipping, E.; Heaton, M. J. Geochim. Cosmochim.Acta 1983, 47, 1393-1397. (31) Edgington, D. N.; Robbins, J. A. Enuiron. Sci. Technol. 1976,10,266-274. (32) Hamilton-Taylor, J. Environ. Sci. Technol. 1979, 13, 6a3-697. (33) Nriagu, J. 0.; Wong, H. K. T.; Coker, R. D. Environ. Sci. Technol. 1982,16,551-560. (34) Gorham, E.; Swaine, D. J. Limnol. Oceanogr. 1965, I O , 268-279. (35) Carignan, R.; Flett, R. J. Limnol. Oceanogr. 1981,361-366. (36) Carignan, E.; Nriagu, J. 0. Geochim. Cosmochim. Acta 1985, 49, 1753-1764.
Received for review May 20, 1985. Accepted October 7, 1985.
CORRESPONDENCE 19, 552-557.
Comment on “Acidification of Southern Appalachian Lakes”
Alan W. Katzenstein
SIR: R. W. Talbot and A. W. Elzerman, writing on the acidification of southern Appalachian lakes (I),note that “S042-concentrations have been increasing over the past decade (in streams) in this region of the United States”, and comment that “because this trends seems to be similar historically to regional atmospheric emission patterns of SOz and NO,, the increased S042-concentrations may be related to inputs of acidic atmospheric deposition.” The authors do not make clear whether it is the acidity or the sulfate content of the “acidic atmosphere deposition” that might account for the increase in sulfate concentrations in surface waters. However, the data in their report shows no relationship between sulfate and acidity in the lake waters surveyed. The correlation coefficient for the data on sulfate and acidity, transformed from pH to hydrogen ion concentration, in the 10 lakes covered in their Table I, is r = -0.26. Initially, this might suggest that acidity and sulfate are negatively correlated. However, the correlation coefficient is not statistically significant, from which it must then be inferred that acidity and sulfate are not related a t all in this data set. The same lack of relationship-widely varying coefficients,both positive and negative-have been noted for other data sets on lake water acidity, leading to the broader conclusion that the alleged linkage between sulfate and acidity does not generally exist. Literature Cited (1) Talbot, R. W.; Elzerman, A. W. Environ. Sci. Technol. 1985, 302
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51 Rockwood Drive Larchmont. New York 10538
SIR: A. W. Katzenstein’s correspondence concerning our note ( I ) raises the point that there is a complicated relationship between atmospheric deposition and surface water concentrations of acidity and sulfate, as others have observed and we discuss in the note. Correlations between acidity and sulfate concentrations in atmospheric deposition have been observed (2). Temporal trends in sulfate concentrations in surface waters of the region are less well-established, but as referenced in our note ( I ) , there is evidence they have increased (3). Katzenstein’s conclusion that sulfate and acidity are not related in our data set (I) relies on the lack of a correlation between H+and Sot- concentrations in various surface waters. Clearly, such a simple relationship would not be expected, as H+ and S042-exhibit different “reactivities” in natural systems. The work of Cronan et al. ( 4 ) ,Burns et al. (5), and Galloway et al. ( 6 ) ,for example, indicates the local hydrogeology and interaction of rain water H+and S O P with watershed rocks and soils profoundly influence the resultant surface water composition (e.g., the concentrations of H+, Sod2-, HC03-, and base cations). As discussed in our note ( I ) , our data for several aquatic systems in the southeastern United States support this hypothesis and also suggest, like other available research results, that specific characteristics of the watershed are very important in determining surface water composition. Moreover, the model of Galloway et al. (6), describing surface water
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chemistry responses to inputs of acidic deposition, predicts a temporally varying SO-: concentration in surface water dependent largely on the degree of SO-: retention or loss from watershed soils. Thus, acidic deposition most likely results in a suite of changes in surface water composition, with relative changes in a species, such as Sod2-, being determined by complex biogeochemical processes and the history of inputs.
Var(s) separately, by uncertainty analysis if necessary, and use regression methods based on that knowledge. Literature Cited (1) Halfon, E. Enuiron. Sci. Technol. 1985, 19, 747. (2) Acton, F. S. “Analysis of Straight Line Data”; Dover: New York, 1966; pp 129ff.
Rane L. Curl
Literature Cited (1) Talbot, R. W.; Elzerman, A. W. Enuiron. Sci. Technol. 1985, 19, 552-557. (2) Gorham, E.; Martin, F. B.; Litzan, J. T. Science 1984,225, 407-409. (3) Smith, R. A.; Alexander, R. B. “Evidence for Acid-Precipitation-Induced Trends in Stream Chemistry at Hydrologic Bench-Mark Stations”; U.S.Geological Survey Circular 910, 1983. (4) Cronan, C. S.;Reynolds, R. C., Jr.; Lang, G. E. Science 1978, 200, 309-311. (5) Burns, D. A.; Galloway, J. N.; Hendrey, G. R. Water Air Soil Pollut. 1981, 16, 277-285. (6) Galloway, J. N.; Norton, S. A.; Church, M. R. Enuiron. Sci. Technol. 1983, 17, 541A-545A.
Robert W. Talbot Atmospheric Sciences Division NASA Langley Research Center Hampton, Virginia 23665
H. H. Dow Building Department of Chemical Engineering University of Michigan Ann Arbor, Michigan 48 109-2 136
SIR: Curl’s comments (1)to my paper (2) refer to the case where both y and x are subject to measurement error only (no natural variability). As Lindley (3) points out and Curl (1) reaffirms, to compute the slope, b, of the linear regression line, we should first estimate, through carefully planned laboratory experiments, the ratio, w (eq 1in Curl’s paper), of the true variance of y at a given (true) value of x to the true value of x given y. In this case, the slope, b, is computed as b=
c y 2 - w Ex2
+ [(wE x 2 - Ey2)2 + 4w(Cxy)2]1/2 2cxy
(1)
Alan W. Elzerman Environmental Systems Engineering Clemson University Clemson, South Carolina 29634-09 19
Comment on “Regresslon Method in Ecotoxicology: A Better Formulation Using the Geometric Mean Functional Regression” SIR: The recommendation of Halfon (1) to use the “geometric mean functional regression” when there is variance in both y and x in a regression is only valid when the error variances in y and x , Var(t) and Var(s), respectively, meet the criterion
Equation 1 reduces to C y 2 / C x y , the reciprocal of the predictive regression of x on y when w = 0 [Var(t) = 01, and it reduces to the predictive regression of y on x when w becomes infinitely large as the error in x approaches zero [Var(s) = 01. Thus, if eq 1 is used to compute the slope of the functional regression, then the two predictive regression estimates, y on x and x on y, set the extreme limits to the functional regression. Ricker (4) states that “In practice, the ratio w as required in eq 1 is rarely available. . . , it [the computation of the ratio w]requires special observations on y when x is accurately known and on x when y is accurately known.” While these experiments can surely be performed, generally authors, seeking functional relationships, perform linear regression analysis on literature data from many sources. Ricker (4) performed a complete analysis of all alternative options to estimate w (section C30 in his paper) and concluded that the best estimate of w,when no educated guess is possible, is
w = Ey2/Cx2
(2)
If expression is substituted in eq 1,eq 1reduces to the GM regression line or the ratio of standard deviations of t and s is the same as the measured slope. Equation 1follows from maximum likelihood parameter estimation based upon bivariate normality and the other statistical assumptions stated by Halfon. As Acton (2)states, “It is necessary to know any two of the three quantities”, Var(t), Var(s), and pst (the true error correlation coefficient), in order to obtain a maximum likelihood regression, with error in both y and x . Halfon provides no guidance for determining criteria for the applicability of the very special (and improbable) relation, eq 1. It would, in general, be better to estimate Var(t) and
b=
(CY~/CX~)’/~
(3)
Ricker then concludes [see also Dent (5)and Sprent (S)] that “The GM regression is the best available estimate of the functional relationship for the situation [raised by Curl (I)] where all the variability of both variates is due to measurement error and there is no supplementary information concerning the relative point errors in x and y.” After reading Dr. Curl’s comments (1) and discussing the matter with him on the telephone, I concur that ecotoxicologists should dedicate much effort to the estimate of w through the measurement of chemical characteristics and their respective variances [Var(t) and Var(s) in ( I ) ]
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