Rainfall composition in Minnesota: integrating the chemistry, synoptic meteorology and numerical modelling

ENVIRONMENTAL POLLUTION

Environmental Pollution 104 (1999) 477±483

Rainfall composition in Minnesota: integrating the chemistry, synoptic meteorology and numerical modelling S. Krupa a,*, M. Nosal b a Department of Plant Pathology, University of Minnesota, St. Paul, MN 55108, USA Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada

b

Received 15 May 1998; accepted 24 July 1998

Abstract ÿ Clear quantitative di€erences (particularly with NH+ 4 and NO3 ) were observed in the chemical composition of rain collected by an in situ, refrigerated event sampler versus using a non-refrigerated, composite, weekly sampling protocol, at a site in the upper ÿ Midwestern USA. There was a non-linear, parabolic relationship between pH and SO2ÿ 4 or NO3 concentrations. Such relationships could be strati®ed by the occurrences of three di€erent classes of rainfall chemistry, governed by three di€erent types of branching air trajectories preceding them. These overall processes were in¯uenced by source-receptor issues. A Fourier modulated, three-way ÿ parabolic spline regression model was used to explain the non-linear relationships between H+ and SO2ÿ 4 or NO3 . This model performed much better (R2: 0.48±0.61) than the linear (R2: 0.008±0.31) and non-modulated three-way parabolic spline (R2: 0.22± 0.36) regression analyses. Overall, the study allowed an integration of rainfall chemistry with synoptic meteorology and numerical modelling. # 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Rainfall composition; Non-linear relationships; Synoptic meteorology; Fourier modulated, three-way spline model; Process integration

1. Introduction The occurrence of acidic precipitation and ion loading through wet deposition continues to be of major environmental concern in many parts of the world (Howells, 1995). Within the north central region of the USA, parts of Minnesota are considered to represent geographic areas with low geochemical capacity for neutralizing acidic deposition (National Research Council, 1983). Previously, several studies were conducted to characterize the chemical composition of precipitation in the region (Munger, 1982; Pratt et al., 1983, 1984; Verry, 1983; Krupa et al., 1987). Pratt et al. (1984) found a non-linear, parabolic relationship between the SO2ÿ 4 concentrations and the pH of summer rainfall (May±September) in Minnesota and west central Wisconsin. Krupa et al. (1987) showed that this parabolic relationship was governed by three distinct classes of rainfall chemistry and branching air trajectories preceding those individual classes of rain events. This conclusion was based on data gathered from daily rain samples. In contrast, the most frequently used * Corresponding author.

sampling method in the USA is to collect composite, weekly precipitation (Krupa, in press). There is evidence to show that at a given location, composition of daily samples can be di€erent from the weekly composite, rain samples (Sisterson et al., 1985; Lamb and Comrie, 1993). Independent of this, a weekly composite sampling protocol in Minnesota also resulted in the parabolic relationships between pH and various major ions in precipitation. 2. Results and discussion 2.1. Chemistry of daily versus weekly, composite rain samples Most types of rain samplers currently in use (excluding continuous samplers) may not provide a complete seal between the collected sample and the ambient environment. Under these conditions, absorption and desorption of water-soluble gases such as CO2 should be expected, particularly when the samples are allowed to remain in the ®eld, from days to a month alternately warming and cooling. Although this aspect is generally

0269-7491/98/$Ðsee front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S0269 -7 491(98)00148 -1

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S. Krupa, M. Nosal / Environmental Pollution 104 (1999) 477±483

ignored in most networks, as previously noted, sampling duration has an important in¯uence on the chemical composition of rain (Sisterson et al., 1985; Lamb and Comrie, 1993). The loss of water molecules by evaporation from composite samples during the summer months can be important. This issue is considered to be site-speci®c and has not been fully examined. Nonetheless, it is believed by many that it is not a major factor. It is generally accepted that any di€erences in the measured precipitation depth between a standard rain gauge and a rain sampler are strictly due to the collection eciency of each sampler. Any measurable loss of water from the sample would naturally result in higher concentrations of some of the measured ions (mass to volume ratio). However, as shown in Table 1, this was not the case. Several independent studies have shown similar results (de Pena et al., 1985; Sisterson et al., 1985). In these studies the precipitation volume-weighted concentrations of several major inorganic ions were all higher when daily or event rain sample data were appropriately converted for comparison with composite weekly samples and such an e€ect was pronouncedly more so, when the rain samples were refrigerated in situ (Table 1). Table 1 An example comparison of volume weighted precipitation chemistry between co-sampled unrefrigerated weekly and in situ refrigerated event samples cumulatively converted to weekly valuesa Variable pH Ca2+ Mg2+ K+ Na+ NH+ 4 NOÿ 3 Clÿ SO2ÿ 4

Unrefrigerated weekly samplesb

Refrigerated event samplesb

5.49c (4.92±6.87)d 11.62 (4.99±57.88) 3.53 (1.40±30.77) 0.83 (0.33±9.41) 1.64 (0.52±17.05) 24.92 (0.56±78.90) 15.07 (6.29±50.33) 1.49 (0.28±16.08) 22.83 (3.12±54.94)

5.28 (4.3±8.02) 18.83 (1.7±90.8) 4.92 0.3±35.9) 1.32 (0.3±13.8) 2.33 (0.4±21.0) 38.03 (0.6±220.6) 78.61 (2.3±79.1) 4.74 (0.1±23.2) 29.32 (5.7±163.5)

Source: Krupa and Pratt (1982). a All values except pH in meq lÿ1. Samples were collected during warm months only, at Lamberton, Minnesota. Range of long-term average daily air temperatures at Lamberton, Minnesota (between 1961 and 1995), during May±September: daytime: 17.1±30.2 C; nighttime: 3.7±17.5 C (M.W. Seeley, University of Minnesota, personal communication). b Unrefrigerated sampler of Aerochemetrics, Pompano Beach, FL; refrigerated sampler of Coscio et al. (1982) respectively. c Average value. d Range.

de Pena et al. (1985) were unable to o€er a speci®c reason for the lower ion concentrations quanti®ed in weekly precipitation compared to event samples at a Pennsylvania site. In comparison, at an Illinois site, Sisterson et al. (1985) observed signi®cantly less NH+ 4 and higher laboratory pH during all seasons in weekly versus event samples and more SO2ÿ 4 during fall, winter and spring. Weekly samples had signi®cantly more Ca2+ and Mg2+ during seasons with little precipitation, while weekly and event NOÿ 3 concentrations were never signi®cantly di€erent. Sisterson et al. (1985) attributed the observed di€erences to chemical degradation of the weekly samples while left in the ®eld before retrieval and during shipment from the ®eld site to the analysis laboratory. They also suggested possible biological ÿ conversion of NH+ 4 to NO3 . It is equally possible that 2+ the dissolution of Ca and Mg2+ from the insoluble particulate fraction into the soluble fraction and the consequent increase in pH could have contributed to some conversion of NH+ 4 to NH3 and its volatilization. Vesely (1990) found that the concentration of free H+ in Czechoslovakia was a€ected after deposition by several processes, the most important being biological + consumption of NH+ 4 leading to an increase in the H level, depending on the length of the sampling interval, the time of the year and the way the samples were stored prior to their analysis. Similarly in Kansas, Ramundo and Seastedt (1990) compared NH+ 4 concentrations in weekly composite rain samples split for analysis between two independent laboratories (Kansas State University and the Central Analytical Laboratory of the National Atmospheric Deposition Program, NADP). They found that the NH+ 4 concentrations were lower in the NADP analysis, exhibiting a strong seasonallydependent di€erence from the results of the Kansas State University laboratory. According to the authors, the losses were not likely to be due to volatilization; microbial immobilization of NH+ 4 probably occurred during transport of the samples from the collection site to the NADP laboratory. Precipitation chemistry data derived from more than 6 years of concurrent sampling involving weekly composite versus daily samples, were analyzed by Lamb and Comrie (1993). The most notable bias occurred for NH+ 4 concentration in the weekly samples, a ®nding consistent with all the other studies. The authors concluded that this bias may be related to the relatively long time that the weekly samples remained in the ®eld, thus strengthening the arguments in favor of daily sampling protocols. The studies of Krupa and Pratt (1982) (Table 1) at an agricultural site in Minnesota provide a strong sitespeci®c argument for the role of microbes. In this study, comparisons were made between in situ refrigerated, event (Coscio et al., 1982) and weekly non-refrigerated, composite samples (Table 1). According to Ridder et al.

S. Krupa, M. Nosal / Environmental Pollution 104 (1999) 477±483

479

(1985), in the Netherlands, storage of rainwater in the dark at 4 C resulted in a satisfactory sample preservation in comparison to room temperature. Almost all of the ions included in Table 1 are essential for microbial growth. In general, microbes do not grow at any measurable rate at refrigeration (+5 C) temperatures and most microorganisms parasitic on vegetation are normally grown in the laboratory at 20±25 C (Ainsworth and Sussman, 1965). Overall, microbial populations in the atmosphere are known to be high in agricultural regions during the crop growth season (Agrios, 1988) and there are unpublished reports that composite rain samples can be turbid and slimy at some agricultural sites during the summer months. Turbidity, light scattering or absorption can be used as measures of microbial growth in dilute solutions (Ainsworth and Sussman, 1965). However, we are not aware of any precipitation sampling networks where these types of variables are measured prior to the ®ltration of the samples, to exclude the insoluble fraction. Nevertheless, such measurements should be made in the future to determine the importance of microbial activity in any observed changes in rain composition. 2.2. Relationships between various ions and pH of rain in Minnesota Fig. 1(a) shows the parabolic relationships between the measured ion concentrations and pH in rain (May± September) and as an example [Fig. 1(b)], the modeled (three-way parabolic spline) relationship between SO2ÿ 4 and pH. Table 2 provides a summary of the three different classes of rainfall chemistry governing the parabolic relationships and Fig. 2 shows examples of branching air trajectories for the time periods preceding the three individual classes. These classes are represented in Fig. 1(b) by: (1) left branch of the parabolaÐ + ÿ concentrations, relatively high SO2ÿ 4 , NO3 and H with air trajectories passing through the high SOx and NOx emission regions of the Ohio River Valley [Fig. 2(a)]; (2) central branch of the parabolaÐrelatively ÿ + concentrations, with air lower SO2ÿ 4 , NO3 and H trajectories from the north (low emission regions of Canada; short atmospheric residence times and conse+ quently low neutralization of any acids, e.g., NH+ 4 :H ratios) [Fig. 2(b)]; and (3) right branch of the pa+ and NOÿ conrabolaÐhigh SO2ÿ 4 3 , but very low H centrations (most common in the summer), with air trajectories originating directly from the southern USA and passing through the central plains (agricultural areas and long atmospheric residence times; conse+ ratio) quently neutralization of the acids, e.g., NH+ 4 :H (Table 2 and Fig. 2(c)). As previously stated, analysis of the weekly composite samples also showed the parabolic relationship (Table 3). However, these results cannot be directly

Fig. 1. (a) Best ®t lines from measured data, showing the parabolic relationships between various major cations or anions and pH of rain in Minnesota (Source: Krupa and Pratt, 1982). (b) The three-way parabolic spline showing the relationship between SO2ÿ and pH of 4 rain, as an example. I: (y=a0+a1x+a2x2); II: (y=b0+b1x+ b2x2) and III: (y=c0+c1x+c2x2). For the de®nitions of the model terms, see Table 4 and the text. Table 2 Relationships between selected ions in Lamberton, Minnesota rainfall (daily, refrigerated rain samples, May±September)a Parameter x pH x H+ meq lÿ1 ÿ1 x NOÿ 3 meq l ÿ1 x SO2ÿ meq l 4 R2 H+/SO2ÿ 4 R2 H+/NOÿ 3 + x NH+ 4 :H

Class (Branch) Ib

Class (Branch) II

Class (Branch) III

4.20 63 54 125 0.10 0.31 1.44

4.44 36 21 39 ÿ0.04 ÿ0.04 0.76

5.80 2 64 100 0.14 ÿ0.02 168

Source: modi®ed from Krupa et al. (1987). a All values except pH were initially weighted by rain volume. b See Fig. 1 for additional information.

compared to those presented in Table 2. While the results shown in Table 2 are represented by discrete, refrigerated, daily or event sampling (Coscio et al., 1982), the data on composite, weekly samples were obtained using a non-refrigerated sampler of a di€erent type (the bucket sampler of Aerochemetrics Inc., Pampano Beach, FL). Furthermore, the weekly composite

480

S. Krupa, M. Nosal / Environmental Pollution 104 (1999) 477±483

Fig. 2. Examples of the branching air trajectories for the time periods preceding the three individual classes of rain chemistry in Minnesota (see Table 2). The individual arrows on the trajectories represent three hour travel time intervals of the same air mass from its geographic location of origin, until its arrival in Minnesota (Source: Krupa et al., 1987).

Table 3 Relationships between selected ions in Lamberton, Minnesota rainfall (weekly, composite rain samplesa, May±September)b

Y=log SO4, has a general convex shape with three distinct subsets delineated as follows:

Parameter

Branch I : x4x1 ; Branch II : x1 < x < x2 ;

x pH x H+ meq lÿ1 ÿ1 x NOÿ 3 meq l ÿ1 x SO2ÿ meq l 4 R2 H+/SO2ÿ 4 R2 H+/NOÿ 3 + x NH+ 4 :H

Class (Branch) Ic

Class (Branch) II

Class (Branch) III

4.23 59 94 132 ÿ0.08 ÿ0.11 1.75

4.47 34 49 23 ÿ0.50 0.17 0.58

6.23 0.59 91 179 0.05 ÿ0.15 325

a

Weekly, non-refrigerated composite samples of one or more rain events. Raw data from the National Atmospheric Deposition Program (NADP). b All values except pH were initially weighted by rain volume. c See Fig. 1 for additional information.

and Branch III : x2 4x A corresponding three-way parabolic spline regression model can be written, satisfying the following relationships: a0 ‡ a1 x1 ‡ a2 x21 ˆ G; b0 ‡ b1 x1 ‡ b2 x21 ˆ G; b0 ‡ b1 x2 ‡ b2 x22 ˆ H; c0 ‡ c1 x2 ‡ c2 x22 ˆ H a0 ˆ G ÿ a1 x1 ÿ a2 x21 ; b1 …x1 ÿ x2 † ‡ b2 …x21 ÿ x22 † ˆ GÿH c0 ˆ H ÿ c1 x2 ÿ c2 x22

samples were not represented by any single class of rain (Table 2), but by a random combination of all three types. In addition, the average values of the ion concentrations shown in Table 3, were from a di€erent sampling period than those in Table 2. Nevertheless, to maintain data uniformity with the largest precipitation monitoring network in the USA, the National Acid Deposition Program (NADP), composite, weekly ion concentration data from three NADP sites (Marcell, Lamberton and Fernberg Road, Ely) in Minnesota were used in the numerical modelling. 2.3. Descriptions of the numerical models and their outputs For de®nitions of the mathematical terms used in the models, the reader is referred to Table 4. 2.3.1. Three-way parabolic spline, regression model First assume, that the scattergram of observations of a pair of random variables (X,Y), e.g., X=pH and

b1 ˆ

G ÿ H ÿ b2 …x21 ÿ x22 † x1 ÿ x2

b0 ˆ G ÿ

G ÿ H ÿ b2 …x21 ÿ x22 † x1 ÿ b2 x21 x1 ÿ x2

The spline equations can be expressed as: Branch I : y ˆ G ‡ a1 …x ÿ x1 † ‡ a2 …x2 ÿ x21 †     x ÿ x1 x1 ÿ x ‡H ‡ Branch II : y ˆ G 1 ‡ x1 ÿ x2 x1 ÿ x2   x2 ÿ x22 …x ÿ x1 † b2 …x2 ÿ x21 † ÿ 1 x1 ÿ x2 Branch III : y ˆ H ‡ c1 …x ÿ x2 † ‡ c2 …x2 ÿ x22 † The corresponding sum of squares for regression can be written as:

S. Krupa, M. Nosal / Environmental Pollution 104 (1999) 477±483

In order to simplify the derivations, the following notation is introduced:

Table 4 De®nitions of numerical terms in the models Numerical term

De®nition Intercept of Branch I of the spline (Fig. 1(b)) Intercept of Branch II of the spline (Fig. 1(b)) Intercept of Branch III of the spline (Fig. 1(b)) Regression coecient of the linear term, Branch I Regression coecient of the linear term, Branch II Regression coecient of the linear term, Branch III Regression coecient of the quadratic term, Branch I Regression coecient of the quadratic term, Branch II Regression coecient of the quadratic term, Branch III x-Coordinate of the intersection of Branch I and Branch II x-Coordinate of the intersection of Branch II and Branch III y-Coordinate of the intersection of Branch I and Branch II y-Coordinate of the intersection of Branch II and Branch III Distance along axis x, between intersections of the spline branches Auxiliary variable Auxiliary variable Number of observations under Branch I Number of observations under Branch III Fundamental frequency of the Fourier model Sum of Squares for the spline regression Sigma notation for sums Summation within the splineÐBranch I Summation within the splineÐBranch II Summation within the splineÐBranch III i-th Observation of pH i-th Observation of log SO2ÿ 4 corresponding to xi Seven normal equations N1, N2, ... N7 Partial derivative of the sum of squares Intercept of the Fourier Modulated Parabolic Spline (FMPS) Regression coecient of the linear term of FMPS Regression coecient of the quadratic term of FMPS Auxiliary transformation variable Auxiliary transformation variable Auxiliary transformation variable Auxiliary transformation variable Auxiliary transformation variable See Fourier model See Fourier model

a0 b0 c0 a1 b1 c1 a2 b2 c2 x1 x2 G H k ` Hi m1 m3 w SSQ  1 2 3 xi yi N1±N7  0 1 2

ti ui vi wi zi ak bk

SSQ ˆ

X xi
yi ÿ G ÿ a1 …xi ÿ x1 † ÿ

X 

a2 …x2i

ÿ

2 x21 † ‡

  xi ÿ x1 x1 ÿ x yi ÿ G 1 ‡ ÿ ÿH x x ÿ x 1 2 1 ÿ x2 x1
481

xi ÿ x1 ˆ ti x2i ÿ x21 ˆ Hi

x21 ÿ x22 ˆ `

x1 ÿ x2 ˆ k

x2i ÿ x22 ˆ wi

xi ÿ x2 ˆ vi ` ui ÿ t i ˆ z i k

X

ˆ

xi x2

X

1

X x1
ˆ

X

2

X

ˆ

X

3

xi x2

Then, X SSQ ˆ 1‰yi ÿ G ÿ a1 ti ÿ a2 ui Š2 ‡  2 h h ti X  ti i ` i 2 yi ÿ G 1 ‡ ÿ H ÿ ÿ b2 ui ÿ ti k k k X ‡ 3‰yi ÿ H ÿ c1 vi ÿ c2 wi Š2 The corresponding seven normal equations are derived for the regression (N1, N2...N7), X SSQ ˆ ÿ2 1‰yi ÿ G ÿ a1 ti ÿ a2 ui Šti ˆ 0 a1 X X X X 1yi ti 1t2i ‡ a2 1ui ti ˆ ‡ G 1ti ‡ a1 N1

X SSQ ˆ ÿ2 1‰yi ÿ G ÿ a1 ti ÿ a2 ui Šui ˆ 0 a2 X X X X 1ui ti ‡ a2 1u2i ˆ ‡ 1yi ui G 1ui ‡ a1 N2

h X h SSQ ti i ˆ ÿ2 2 yi ÿ G 1 ‡ ÿ b2 k i h ti i H ÿ ÿ b2 z i z i ˆ 0 k X h ti i X h ti i 2zi ÿ ‡ G 2zi 1 ‡ ‡ H kX Xk 2yi zi 2z2i ˆ ‡ b2 N3

X SSQ ˆ ÿ2 3‰yi ÿ H ÿ c1 vi ÿ c2 wi Švi ˆ 0 c1 X X X X 3yi vi 3v2i ‡ c2 3vi wi ˆ ‡ H 3vi ‡ c1 N4

X SSQ ˆ ÿ2 3‰yi ÿ H ÿ c1 vi ÿ c2 wi Šwi ˆ 0 c2 X X X X 3yi wi H 3wi ‡ c1 3vi wi ‡ c2 3w2i ˆ ‡ N5

482

S. Krupa, M. Nosal / Environmental Pollution 104 (1999) 477±483

h ti i ti tti ˆ 1 ‡ tti ˆ ÿ k k

X SSQ ˆ ÿ2 1‰yi ÿ G ÿ a1 ti ÿ a2 ui Š‡ G h ih h i h X ti ti i ti i ˆ0 ÿ2 2 y i ÿ G 1 ‡ ÿ H ÿ ÿ b2 z i 1 ‡ k k k   X h X h ti i2 ti ih ti i 2 1‡ ‡H 2 1‡ ÿ ‡ G m1 ‡ k k k X X X h ti i 1ti ‡ a2 1ui ‡ b2 2 1 ‡ zi ˆ a1 k hX X h ti ii 2yi 1 ‡ ‡ 1yi ‡ k N6

All program steps correspond to Table 5 and the computer program is available in Array Programming Language (APL). 2.3.2. Fourier modulated, three-way parabolic spline, regression model In the case of the relationship between sulfate and acidity, for example, the three-way parabolic spline described in the previous section captures the general trends, but is not sensitive enough to express the overall variability in the data. For this reason the spline residuals can be further analyzed using Fourier series expansions. The spline coecients can be denoted as:

h X h SSQ ti i ˆ ÿ2 2 yi ÿ G 1 ‡ k H ih t i h ti i i ÿH ÿ ÿ b2 zi ÿ k k X ÿ2 3‰ yi ÿ H ÿ c1 vi ÿ c2 w i Š ˆ 0  X h ti i2  X h ti ih ti i 2 ÿ G 2 1‡ ‡ ÿ ‡ H m3 ‡ k k k X h ti i X X 2zi ÿ ‡ c1 3vi ‡ c2 3wi ˆ b2 k hX h t i X i i 3yi ‡ 2yi ÿ ‡ k N7

0 …x1 ; x2 ; x†

1 …x1 ; x2 ; x†

2 …x1 ; x2 ; x†;

where, 0 …x1 ; x2 ; x†

ˆ a0 for x  x1 b0 for x1 < x < x2 c0 for x  x2

1 …x1 ; x2 ; x†

The matrix of coecients of the normal equations are summarized in Table 5. For the sake of simplicity, the program uses the following additional variables:

ˆ a1 for x  x1 b1 for x1  x  x2 c1 for x2  x

Table 5 Normal equation coecients for three-way parabolic spline Variable/Eq. #

N3

G P 1ti P 1ui  P  2zi 1 ‡ tki

N4

0

N5

0

N1 N2

N6 N7

H 0 0 P  ti  2zi ÿ k P 3vi P 3wi   P  2 1 ‡ tki ÿ tki P  2 m3 ‡ 2 ÿ tki

2 P  m1 ‡ 2 1 ‡ tki   P  2 1 ‡ tki ÿ tki

a1 a2 P 2 P 1uiti 1ti P 2 P 1ui 1uiti

b2

c1

c2

Ð

0

0

0

0

0

0

Ð P 2 2zi

0

0

0

0 P 1ti

0 P 1ui

0

0

0  P  2 1 ‡ tki zi P  ti  2zi ÿ k

0 0 P 2 P 3vi 3viwi P 2 P 3wi 3viwi 0 P 3vi

P

0 3wi

Right-hand side P + 1yiti P + 1yiui P + 2yizi P + 3yivi P + 3yiwi P  P  ‡ 1yi ‡ 2yi 1 ‡ tki P  ti  P  ‡ 2yi ÿ k ‡ 3yi

Table 6 ÿ Numerical relationships of H+ versus SO2ÿ 4 or NO3 for Minnesota rainfall R2

Sampling site a

Linear

Fernberg Road (Ely) Marcell Lamberton a

b

pH b

Parabolic spline

Fourier modulated spline

Optimal spline knotsb

SO2ÿ 4

NOÿ 3

SO2ÿ 4

NOÿ 3

SO2ÿ 4

NOÿ 3

SO2ÿ 4

NOÿ 3

0.31 0.1 0

0.1 0.23 0

0.36 0.22 0.28

0.33 0.26 0.24

0.57 0.51 0.54

0.61 0.53 0.48

4.5±6.1 4.5±5.9 4.5±7.2

4.6±6.2 4.5±5.9 4.5±6.8

Modi®ed results from refrigerated, event sample analysis of Krupa et al. (1987). These results were better than the values from composite samples. For information from weekly, composite rain samples at one of the sites, see Table 3. b Data from weekly, composite samples collected by the National Atmospheric Deposition Program (NADP).

S. Krupa, M. Nosal / Environmental Pollution 104 (1999) 477±483 2 …x1 ; x2 ; x†

Thanks are due to Leslie Johnson and Sid Simms for their valuable assistance in the preparation of the manuscript.

ˆ a2 for x  x1 b2 for x1 < x < x2 c2 for x2  x

Then, the three-way parabolic spline equation can be written as: yˆ

0 …x1 ; x2 ; x†

‡

1 …x1 ; x2 ; x†

‡

2 2 …x1 ; x2 ; x†x

Finally, the Fourier modulated spline model is given as follows: Yˆ

0 …x1 ; x2 ; x†

‡

1 …x1 ; x2 ; x†x n X

‡

ak sin kwx ‡

kˆ1

483

2 2 …x1 ; x2 ; x†x ‡ n X

bk cos kwx

kˆ1

The `n' and `w' values used for ®tting the NADP data at the three Minnesota sites were respectively: Marcell, 40 and 10; Lamberton, 40 and 10 and Fernberg Road, 25 and 10. Table 6 provides the results of the model application. Among the three models tested, the linear regression analysis (even with refrigerated, event samples) provided poor results. This is consistent with similar general observations of others using event (Pratt et al., 1984) or composite, weekly samples (Munger, 1982; Verry, 1983) and Table 3. The best results were provided by the Fourier modulated, spline model (Table 6). These observations are not surprising, since the parabolic spline would be expected to provide better numerical explanation of the non-linear relationships between or NOÿ H+ (pH) and SO2ÿ 4 3 compared to the linear model. Similarly, the Fourier modulation accounts for the power or dynamic structure of the variability in the data. This is important, because climatic and atmospheric processes that govern the chemical composition of precipitation events are fundamentally stochastic in nature. Therefore, future e€orts need to include the biologically (e.g. physiological phenology of terrestrial plants) meaningful time element in the occurrences of the three classes of rain composition described here. This is particularly relevant, since the relationships between rain chemistry, synoptic meteorology and the corresponding numerical explanation have been coupled as a ®rst order e€ort. In addition, this work can be extended to the dynamics of the occurrences of other phytotoxic air pollutants in examining their joint e€ects on vegetation. Acknowledgements The senior author was supported in kind by the University of Minnesota Agricultural Experiment Station.

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