Identification of water sources using normalized chemical ion balances: A laboratory test

Identification of water sources using normalized chemical ion balances: A laboratory test

Journal of Hydrology, 76 (1985) 205--231 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands 205 [21 IDENTIFICATION OF WATER S...

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Journal of Hydrology, 76 (1985) 205--231 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

205

[21 IDENTIFICATION OF WATER SOURCES USING NORMALIZED CHEMICAL ION BALANCES: A L A B O R A T O R Y TEST

DAVID A. WOOLHISER, WILLIAM E. EMMERICH and EDWARD D. SHIRLEY

Southwest Rangeland Watershed Research Center, U.S.D.A., A.R.S., Tucson, AZ 85719 (U.S.A.) (Received February 21, 1984 ; revised and aceepted August 4, 1984 )

ABSTRACT Woolhiser, D.A., Emmerich, W.E. and Shirley, E.D., 1985. Identification of water sources using normalized chemical ion balances: a laboratory test. J. Hydrol., 76: 205--231. A normalized chemical ion balance (NCIB) technique, that has been proposed to identify multiple surface and groundwater inflows to a reach of a stream, is tested using two sets of mixtures prepared in the laboratory. The first set consisted of 64 mixtures of from 2--8 natural waters selected from 10 source waters. A second set of 25 mixtures was prepared using 5 source waters. Each of the source waters and each mixture was analyzed for at least 8 ions. Proportions of source waters in each sample were estimated using a quadratic programming solution to minimize the sum of squared error terms of normalized chemical ionbalance equations. Because PO~- was present in very low concentrations, and was subject to significant error, it was found that better estimates could be obtained by omitting it from the analyses. For the first mixture set, the probability of failure to detect a source when it was present in the mixture was 0.23, and the probability of identifying a source when it was not present was 0.29. For the second set of mixtures, the probabilities were 0.0 and 0.33, respectively. Identification was improved by dividing each normalized ion-balance equation by the error variance for that ion.

1. INTRODUCTION To design o p t i m a l strategies to c o n t r o l surface or g r o u n d w a t e r p o l l u t i o n from p o i n t and n o n p o i n t sources, authorities need models that relate observed c o n c e n t r a t i o n s o f d i s s o l v e d i n o r g a n i c or o r g a n i c species i n t h e s t r e a m o r i n t h e g r o u n d w a t e r t o t h e s o u r c e s o f t h e s e species. N o n p o i n t p o l l u t i o n m o d e l s , b a s e d o n t h e p h y s i c a l , b i o l o g i c a l a n d c h e m i c a l p r o c e s s e s i n v o l v e d in t h e t r a n s p o r t o f v a r i o u s c h e m i c a l s f r o m s o u r c e areas t o s t r e a m s o r g r o u n d w a t e r , h a v e b e e n d e v e l o p e d i n t h e p a s t d e c a d e (cf. K n i s e l , 1 9 8 0 ) . H o w e v e r , t h e s e m o d e l s are v i r t u a l l y i m p o s s i b l e t o v e r i f y f o r a n y b u t t h e s m a l l e s t s i n g l e - u s e watersheds because of the diffuse n a t u r e of m u c h of the inflow to the s t r e a m . B e c a u s e o f large u n c e r t a i n t i e s i n t h e s o u r c e - b a s e d m o d e l s , t h e r e is a n e e d for t e c h n i q u e s which utilize the chemical characteristics of surface or

206 groundwater to identify contributions from various sources, and thus can serve as a check of model consistency. The water present in a reach of stream at any time is generally considered to be a mixture of waters that have followed different pathways to the stream. During a surface r u n o f f event, for example, water in the stream might include direct rainfall, r u n o f f from rainfall on saturated areas near the stream, classical Hortonian overland flow, interflow and baseflow. The chemical characteristics of each water source depend upon the chemical quality of the rainfall, the physical and chemical characteristics of vegetation and plant litter, and the physical and chemical composition of the geologic material that it has come in cont act with as it moved through the watershed. Although the true n u m b e r of pathways is very large, we can usually select a rather small n u m b e r of sources that have unique absolute and relative concentrations of principal and minor anions and cations. Furthermore, it is often assumed that the stream water (or groundwater) is a mixture of solutions derived from different origins with no loss of dissolved species occurring after mixing. In groundwater studies, the Piper (1944) trilinear diagram has been widely used to graphically test for apparent mixtures of waters from different sources (Morris et al., 1983). More recently, numerical models of chemical transport in groundwater flow have been developed to generate chemical patterns and assist in interpreting field data (Schwartz and Domenico, 1973). The substantial differences between the dissolved solids c o n t e n t (or electrical conductivity) of surface r u n o f f and groundwater (baseflow) have long been used in a mass-balance approach to estimate the p r o p o r t i o n of stream discharge contributed from each source (Pinder and Jones, 1969; Hall, 1970; Visocky, 1970). The same technique has been used with some of the environmental isotopes to differentiate surface and groundwater sources (Sklash et al., 1976). This approach is adequate when only two sources of inflow are considered, but fails when there are more than two sources. Schwartz (1980) apparently used a trial-and-error ion-balance technique to estimate the quantities of surface water contributed from three sources: groundwater, direct precipitation and drainage from a muskeg. Tsurumi (1982) utilized the multiple ion-balance approach to estimate n o t only the relative proportions of sources present in a mixture, but also the ranges of chemical composition of the sources of chemical constituents. His m e t h o d requires several mixtures of the source waters at different proportions and uses an iterative least-squares solution. The source compositions are subject to an anion--cation balance constraint. Chemical element balances (CEB's) have also been used to identify sources of particulates for many elements which can be associated with specific types of air-pollution sources (Kowalczyk et al., 1982). According to the CEB model, the composition of particles at a receptor is a linear combination of co n cen tr atio n patterns of particles from contributing sources. Kowalczyk et al. (1982) determined source-strength coefficients by a least-squares fit to the observed concentrations of several " m a r k e r elements".

207

Woolhiser et al. (1979, 1982) developed a technique to estimate multiple inflows to a stream reach based on a quadratic programming solution to find the u n k n o w n positive inflow quantities that minimized the sum of squares of normalized errors for up to eight ion-balance equations. T hey used Monte Carlo techniques to investigate the sensitivity of the solution to errors in the chemical analysis and errors in measurements of streamflow. This analysis showed conclusively that the use of the normalized ion-balance equations led to more accurate solutions than those obtained by minimizing the sum of squared errors in the original ion-balance equations. The quadratic programming algorithm constrained all inputs to be equal to or greater than zero. Gorelick et al. (1983) proposed a m e t h o d of identifying groundwater pollution sources that includes a linear programming approach to the minimization of errors in ion-balance equations and can a c c o m m o d a t e both nonconservative tracers and transient cases. The technique presented by Woolhiser et al. (1979, 1982) may have practical applications in detecting sources of surface water or groundwater pollution when several potential sources are so located that it should be physically possible for a pollutant to have traveled to the point in question and where each potential source uses the suspected pollutant, but has different relative proportions of other constituents. If water-quality samples are obtained at regular intervals, it may also be possible to estimate the contribution of each source to the annual runoff. The key assumption in all of these models, except those that include numerical models of chemical transport (Schwartz and Domenico, 1973; Gorelick et al., 1983) is that all elements or dissolved ionic species considered are conservative within the system under investigation. There must be no chemical exchange, deposition, solution, gaseous transport and no uptake by organisms. Woolhiser et al. (1982) listed the following errors that can affect the accuracy of the calculated inflows: (1) Errors in the chemical analyses (including sampling and storage errors). (2) Errors in determining the discharge rates at the upper and lower ends of the stream reach. (3) Nonrepresentative samples of the inflow waters. (4) Omission of a significant inflow. Th ey also pointed out that, if the chemical characteristics of two or more inflow sources are nearly identical, it should be very difficult to distinguish between them. In their empirical sensitivity analysis, they examined the implications of error categories (1), (2) and (4), above, using "theoretical m i xt ures" perturbed by adding normally distributed, zero mean error terms. The "theoretical mi xt ur e s " were obtained by using ion-balance equations to calculate the concentrations of each ion in a mixture of 3--7 water sources sampled near a surface coal mine site in western Colorado, U.S.A. T h e y concluded that the technique appears to be promising as a m e t h o d of estimating surface and groundwater inflows from several sources in a reach of a stream, but that further testing is required to determine the limitations.

208

If the normalized chemical ion-balance technique is to be used to identify possible pollution sources, it would be desirable to know something about the probability of making incorrect decisions (i.e. the probability of identifying a source if it is n o t present, or not identifying a source if it is present) under controlled laboratory conditions. One might infer that, under field conditions, the identification ability would be somewhat worse. In this paper, we examine the ability of the technique proposed by Woolhiser et al. (1982) to identify the c om ponent s of laboratory mixtures of waters and the implications of potential interference between sources with small differences in chemical characteristics. The effect of different weighting factors for each ion is also briefly examined. 2. MATERIALS AND METHODS

The first set of 64 mixtures prepared in the laboratory included from 2 to 8 source waters selected from a set of 10 source waters. The 10 source waters included 8 samples that had been collected as part of water-quality studies of the Agricultural Research Service on the Walnut Gulch Experimental Watershed, near To mbs t one, Arizona, U.S.A. The other two samples were from the Tucson and Tombstone, Arizona, municipal supplies. A description of the source waters is given in Table I, and the measured ion concentrations TABLE I Description of source waters Source No.

Description

Firs t set: groundwater sample from Walnut Gulch Experimental Watershed L.G. Ranch well groundwater sample from Walnut Gulch Experimental Watershed well 63 2 groundwater sample from Walnut Gulch Experimental Watershed well 77 3 groundwater sample from Walnut Gulch Experimental Watershed well 91 4 sample from shallow well in alluvium of Walnut Gulch; alluvium saturated by 5 sewage plant effluent surface runoff from main stem of Walnut Gulch 6 surface runoff from main stem of Walnut Gulch 7 rain-water sample, Walnut Gulch Experimental Watershed 8 Tucson, Arizona, municipal water supply (groundwater) 9 Tombstone, Arizona, municipal water supply (groundwater) I0 1

S e c o n d set:

1 2 3 4

+ + + +

6 7 8 9

5 + 10

Tombstone, Arizona, municipal water supply (groundwater) groundwater sample from Walnut Gulch Experimental Watershed well 75 rain-water sample, Walnut Gulch Experimental Watershed sample from shallow well in alluvium of Walnut Gulch; alluvium saturated by sewage plant effluent Tucson, Arizona, municipal water supply (groundwater)

209

T A B L E II

C h e m i c a l c h a r a c t e r i s t i c s o f first a n d s e c o n d s e t o f s o u r c e s a m p l e s Source No.

I o n c o n c e n t r a t i o• n PO 3( ~ g l i)

$1

CI-

NO:3

5.2 11.4 4.2 1.3 68.8 2.5 6.1 1.4 32.5 3.0

6.4 0.0 0.0 2.8 32.5 1.6 0.0 0.5 3.1 3.2

45.0 4.4 1.4 48.0 8.0 42.0 4.4 1.4 52.0 9.0

10.8 4.2 0.0 3.2 3.8 10.6 4,2 0,0 3.0 3.7

SO 2-

I o n .2 balance ( m e q 1 1)

HCO:3

Ca 2+ Mg 2+

Na +

K+

3.5 8.5 7.2 9.8 57.3 9.5 5.2 10.2 151.5 6.9

*3 ----72.0 35.1 ----

29.2 19.5 21.6 29.1 68.7 24,7 12.2 3.6 73.7 43.0

10.2 6.5 9.3 5.2 10.8 1.3 0.6 0.5 14.1 17.6

30.5 10.5 20.5 8.6 63.1 2.1 2.8 0.3 65.6 9.4

2.3 3.2 3.6 3.7 13.3 4.5 4.4 0.2 2.7 2.5

-----0.13 0.08 ----

43.5 14.0 1.8 39.8 28.4 41.6 13.6 1.3 38.0 27.2

244.0 164.7 6.1 67.1 146.4 244.0 164.7 6.1 183.0 176.9

76.3 44.9 1.3 57.1 42.2 76.0 44.9 1.3 56.4 42.7

20.1 8.8 0.4 7.6 4.6 20.1 8.8 0.4 7.5 4,7

32.3 7.7 0.1 41.9 22.9 31.9 7.8 0.0 41.4 23.3

1.7 1.8 0.0 6.5 0.6 1.4 1.8 0.0 6.5 0.6

0.56 0.16 -- 0.08 2.01 0.19 0.64 0.17 --0.07 -- 0.02 -- 0.24

Firs t set." 1 2 3 4 5 6 7 8 9 10

0 0 0 0 12,680 6 63 27 5 0

Second set. 1 2 3 4 5 6 7 8 9 10

0 0 0 3,700 0 0 55 5 3,650 185

*1 m g 1- 1 u n l e s s o t h e r w i s e i n d i c a t e d . ,2 m i l l i - e q u i v a l e n t s c a t i o n s m i n u s a n i o n s . *3 = not analyzed.

are s h o w n in Table II. Surface water samples were centrifuged to r e m o v e the s e d i m e n t , and t h e o t h e r samples were n o t treated before use. 1 0 - m l m i x t u r e s were prepared by c o m b i n i n g t h e sources in the p r o p o r t i o n s given in Table III. Each p r o p o r t i o n used to m a k e t h e m i x t u r e was m e a s u r e d to the nearest 0.01 ml. A s e c o n d set o f 25 m i x t u r e s was prepared using 5 different source waters similar to the first 10, e x c e p t t h e y w e r e c o l l e c t e d at a later date. Mixtures in the s e c o n d set were 1 0 0 m l in v o l u m e , and were prepared by c o m b i n i n g sources in t h e p r o p o r t i o n s given in Table III. S o u r c e s 1, 2, 3, 4 and 5 are t h e same as 6, 7, 8, 9 and 10, respectively, e x c e p t t h e y were filtered t h r o u g h a 0.45-pro filter b e f o r e t h e y were used to m a k e the mixtures. T h e first five s o u r c e s were n o t treated in a n y w a y , and all p r o p o r t i o n s were m e a s u r e d to 0.01 ml. During the preparation and analysis o f the s e c o n d set o f mixtures, e x t r e m e care w a s t a k e n at each step to m i n i m i z e error.

210 TABLE III Actual and calculated proportions of first and second set of source waters in laboratory mixtures along with an error index Mixture No.

Source No.

Error index

1

2

3

4

5

6

7

8

9

10

A .1 C1.2 C2 .3

0.25 0.00 0.13

0.10 0.30 0

0 0.02 0.04

0 0.68 0.25

0 0 0

0.25 0 0

0.40 0 0.51

0 0 0.04

0 0 0 0 0.00 0.04

0.10 0.525

A C1 C2

0.50 0.42 0.36

0 0.05 0

0 0.11 0

0 0.15 0

0 0 0

0 0.29 0

0.50 0 0.47

0 0 0.08

0 0 0

0 0 0.10

0.40 0.825

A C1 C2

0.50 0.36 0.39

0 0.17 0

0 0 0

0 0.47 0

0 0 0

0 0 0

0 0 0.05

0.50 0 0.50

0 0 0

0 0 0.06

0.36 0.89

A C1 C2

0.50 0.73 0.48

0 0.19 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0.06

0.50 0 0.47

0 0.08 0

0.50 0.945

5

A C1 C2

0.50 0 0 no c o n v e r g e n c e .4 0.39 0 0

0

0

0

0

0

0

0.50

0

0.01

0

0

0.12

0

0.49

0.88

6

A C1 C2

0 0 0

0.50 0.98 0.44

0 0 0.03

0 0.02 0

0 0 0

0.50 0 0

0 0 0.41

0 0 0.11

0 0 0

0 0 0

0.50 0.45

A C1 C2

0 0 0

0.50 0.67 0.28

0 0 0.03

0 0 0

0 0 0

0 0 0

0.50 0.33 0.67

0 0 0.02

0 0 0

0 0 0

0.83 0.78

A C1 C2

0 0 0

0.50 0.97 0.43

0 0 0

0 0.03 0

0 0 0

0 0 0

0 0 0.33

0.50 0 0.24

0 0 0

0 0 0

0.50 0.67

A C1 C2

0 0.21 0

0.50 0.64 0.47

0 0.05 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0.05

0.50 0 0.45

0 0.10 0.02

0.50 0.93

A C1 C2

0 0 0

0.50 0.48 0.44

0 0 0

0 0.10 0

0 0 0

0 0 0.05

0 0 0

0 0 0.07

0 0 0

0.50 0.41 0.43

0.89 0.88

A C1 C2

0 0 0

0 0.50 0.66 0.34 0.10 0.43

0 0 0

0 0 0

0.50 0 0

0 0 0.47

0 0 0

0 0 0

0 0 0

0.34 0.43

A C1 C2

0 0.04 0

0 0.52 0.40

0.35 0.43 0.49

0 0 0

0 0 0

0 0 0

0.25 0 0

0 0 0

0.40 0 0.10

0 0.01 0

0.35 0.45

A C1 C2

0 0.17 0

0 0.66 0

0.50 0 0.39

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0.13

0.50 0 0.47

0 0.17 0.01

0.00 0.86

First set:

i

2

3

4

7

8

9

10

11

12

13

211 T A B L E III ( c o n t i n u e d ) Mixture No.

Source No. 1

Error index

2

3

4

5

6

7

8

9

10

First set (cont.): A C1 C2

0 0 0.02

0 0.03 0

0.50 0.47 0.44

0 0.18 0.01

0 0 0

0 0 0

0 0 0

0 0 0.16

0 0 0.01

0.50 0.31 0.37

0.78 0.81

A C1 C2

0 0 0

0 0.02 0

0 0 0

0.50 0.98 0.42

0 0 0

0.50 0 0.33

0 0 0

0 0 0.24

0 0 0

0 0 0.01

0.50 0.75

A C1 C2

0 0 0 0 0.07 0 no c o n v e r g e n c e

0.50 0.93

0 0

0 0

0.50 0

0 0

0 0

0 0

0.50

A C1 C2

0 0 0

0 0.06 0

0 0 0.05

0.50 0.94 0.26

0 0 0

0 0 0.02

0 0 0

0.50 0 0.66

0 0 0

0 0 0

O.50 0.76

A C1 C2

0 0.27 0

0 0.70 0

0.10 0 0

0.25 0.03 0

0 0 0.01

0 0 0.12

0 0 0

0.40 0 0.57

0.25 0 0 0 0.22 0.06

0.03 0.63

A C1 C2

0 0 0.03

0 0 0

0 0 0

0.50 0.55 0.42

0 0 0

0 0 0

0 0 0

0 0 0.12

0 0 0

0.50 0.45 0.43

0.95 0.85

A C1 C2

0.025 0 0

0.05 0.125 0 0.17 0.19 0.13

0 0 0.07

0.125 0.12 0.11

0.125 0.14 0.26

0.225 0.21 0

0.125 0.17 0.13

0 0 0

0.20 0.18 0.12

0.89 0.65

A C1 C2

0 0 0.01

0 0 0

0 0 0

0 0 0

0.50 0.45 0.42

0 0.08 0.11

0.50 0.39 0.42

0 0.06 0

0 0 0.02 0 0.04 0

0.84 0.84

A C1 C2

0 0 0

0 0 0

0 0 0

0 0 0

0.50 0.40 0.37

0 0 0

0 0.13 0.23

0.50 0.46 0.37

0 0.01 0.03

0 0 0

0.86 0.74

A C1 C2

0 0 0

0 0 0

0 0 0

0 0 0

0.50 0.44 0.40

0 0 0

0 0 0.10

0 0.11 0.02

0.50 0.45 0.48

0 0 0

0.89 0.88

A C1 C2

0 0 0

0 0 0

0.033 0 0

0.067 0 0

0.167 0.16 0.15

0 0.24 0.24

0.30 0.11 0.14

0.267 0.31 0.28

0.00 0.01 0.01

0.167 0.18 0.18

0.44 0.59

A C1 C2

0.10 0 0.05

0 0.15 0

0.05 0 0.08

0 0.85 0

0 0 0

0.40 0 0.33

0 0 0.02

0.45 0 0.51

0 0 0

0 0 0

0.00 0.89

26

A C1 C2

0.20 0 0 no c o n v e r g e n c e no convergence

0

0

0

0.80

0

0

0

27

A C1 C2

0.20 0.11 0.18

0 0.61 0

0 0 0

0 0 0

0 0 0

0.80 0 0.82

0 0 0

0 0 0

14

15

16

17

18

19

2O

21

22

23

24

25

0 0.28 0

0 0 0

0.11 0.98

212

T A B L E III ( c o n t i n u e d ) Mixture No.

Source No. 1

Error index

2

3

4

5

6

7

8

9

10

First set ( c o n t . ) : 28

29

30

31

32

33

34

35

36

37

38

39

40

41

A C1 C2

0.20 0.53 0.17

0 0.32 0

0 0 0

0 0 0.15

0 0 0

0 0 0

0 0 0

0 0 0

0.80 0 0.51

0 0.14 0.17

0.20 0.68

A C! C2

0.10 0.05 0.06

0.05 0.05 0

0 0 0

0 0.08 0

0 0 0.01

0 0 0.02

0 0 0.01

0 0 0.08

0 0 0

0.85 0.82 0.82

0.92 0.88

A C1 C2

0 0 0

0.20 0.77 0

0 0 0.06

0 0.23 0

0 0 0

0.80 0 0.08

0 0 0.59

0 0 0.22

0 0 0

0 0 0.04

0.20 0.08

A C1 C2

0 0 0

0.20 0.70 0.17

0 0 0

0 0.30 0

0 0 0

0 0 0

0 0 0.02

0.80 0 0.81

0 0 0

0 0 0

0.20 0.97

A C1 C2

0 0.36 0.01

0.20 0.11 0

0 0.45 0.11

0 0 0.34

0 0 0

0 0 0

0 0 0

0 0 0.10

0.80 0 0.18

0 0.08 0.26

0.11 0.18

A C1 C2

0 0 0

0 0.29 0

0.20 0.56 0.26

0 0.15 0

0 0 0

0.80 0 0.01

0 0 0.24

0 0 0.50

0 0 0

0 0 0

0.20 0.21

A C1 C2

0 0 0

0 0 0

0.20 0.08 0.O7

0 0 0

0 0 0

0 0 0

0.80 0.79 0.81

0 0.13 0.12

0 0 0

0 0 0

0.87 0.87

A C1 C2

0 0 0.03

0 0.04 0.01

0.20 0.14 0.12

0 0.08 0

0 0 0

0 0 0

0 0 0.02

0 0 0.13

0 0 0

0.80 0.74 0.68

0.88 0.8!

A C1 C2

0 0 0

0 0.24 0

0 0 0.11

0.20 0.76 0

0 0 0

0.80 0 0.43

0 0 0.08

0 0 0.38

0 0 0

0 0 0

0.20 0.43

A C1 C2

0 0 0

0 0 0

0 0.09 0.07

0.20 0 0

0 0 0

0 0.09 0

0.80 0 0.07

0 0.83 0.85

0 0 0

0 0 0

0.00 0.07

A C1 C2

0 0.22 0.09

0 0.15 0.26

0 0 0

0.10 0 0

0.05 0.02 0.05

0 0.08

0 0 0

0 0 0.04

0.40 0.45 0.41 0.12 0.38 0.17

0.54 0.60

A C1 C2

0 0 0

0 0 0

0 0 0

0.20 0.32 0.24

0 0 0

0 0 0

0 0 0

0 0 0.07

0 0 0

0.80 0.68 0.68

0.88 0.88

A C1 C2

0 0 0

0 0.02 0.02

0 0 0

0 0 0

0.20 0.13 0.12

0.80 0.23 0.28

0 0.48 0.47

0 0 0.08

0 0.03 0.03

0 0 0

0.36 0.40

A C1 C2

0 0 0.02

0 0 0

0 0.01 0

0 0 0

0.20 0.15 0.14

0 0.35 0.38

0.80 0.31 0.32

0 0.17 0.13

0 0 0

0 0 0

0.46 0.46

213 T A B L E III ( c o n t i n u e d ) Mixture No.

Source No. I

Error index

2

3

4

5

6

7

8

9

I0

First set (cont.): 42

A CI C2

0 0 0

0 0 0

0 0 0

0 0 0

0.20 0.08 0.08

0 0 0

0 0.30 0.31

0.80 0.58 0.57

0 0.04 0.04

0 0 0

0.66 0.65

A C1 C2

0 0 0

0 0 0

0 0 0

0 0 0

0.20 0.09 0.10

0 0 0

0 0.19 0.16

0 0.05 0.09

0.80 0 0.67 0 0.65 0

0.76 0.75

A C1 C2

0 0 0

0 0.34 0.31

0 0 0

0 0 0

0.20 0.12 0.13

0 0.03 0

0 0.08 0.12

0 0.12 0.14

0 0.80 0.02 0.28 0.02 0.29

0.40 0.42

A C1 C2

0.10 1.00 0

0 0 0

0 0 0.07

0 0 0

0 0 0

O.9O 0 0.57

0 0 0.13

0 0 0.22

0 0 0

0 0 0.01

0.10 0.57

A C1 C2

0.10 0 0.07

0 0.27 0

0 0 0

0 0.73 0.01

0 0 0

0 0 0

0 0 0

0.90 0 0.91

0 0 0

0 0 0

0.00 0.97

A C1 C2

0.10 0.42 0

0 0.55 0

0 0 0

0 0 0

0 0 0.01

0 0 0

0 0 0.11

0 0 0.20

0.90 0 0 0.03 0.69 0

0.10 0.69

A C1 C2

0.10 0.03 O.06

0 0.02 0

0 0 0

0 0.14 0

0 0 0

0 0 0.05

0 0 0

0 0 0.09

0 0 0

0.90 0.81 0.79

0.84 0.85

A C1 C2

0 0 0

0.10 1.00 0

0 0 0.01

0 0 0

0 0 0

0.90 0 0

0 0 0.97

0 0 0.02

0 0 0

0 0 0

0.10 0.00

A C1 C2

0 0 0

0.10 0 0

0 0 0

0 0 0

0 0 0

0 0 0

O.90 1.00 0.98

0 0 0.02

0 0 0

0 0 0

0.90 0.90

A C1 C2

0 0 0

0.10 0.09 0.06

0 0 0

0 0.01 0

0 0 0

0 0.04 0

0 0 0.O2

0.90 0.86 0.91

0 0 0

0 0 0.01

0.95 0.96

52

A C1 C2

0 0.10 0 no c o n v e r g e n c e 0 0 0

0

0

0

0

0

0.90

0

0

0

0

0.07

0.15

0.77

0

0.77

53

A C1 C2

0 0 0

0 0.06 0

0 0 0

0.90 0 0

0 0 0.63

0 0 0.27

0 0 0

0 0 0

0.00 0.10

54

A C1 C2

0 0 0.10 no c o n v e r g e n c e 0 0 0

0

0

0

0

0

0.90

0

0

0

0

0.05

0.09

0.86

0

0.86

55

A C1 C2

0 0 0.05

0 0.20 0.05

0 0 0

0 0 0

0 0 0.01

0 0 0.15

0 0 0

0.90 0.77 0.74

0.77 0.74

43

44

45

46

47

48

49

50

51

0 0.93 0

0 0.03 0

0.10 0 0.10

0.10 0 0

214 T A B L E III ( c o n t i n u e d ) Mixture No.

S o u r c e No. I

Error index

2

3

4

5

6

7

8

9

10

0 0.33 0

First set ( c o n t . ) : 56

57

58

59

6O

61

62

63

64

A C1 C2

0 0 0

0 0 0.06

0.10 0.67 0

0 0 0

0.90 0 0.27

0 0 0.18

0 0 0.48

0 0 0

0 0 0

0.10 0.27

A C1 C2

0 0 0 0 0 0.01 no convergence

0.10 0

0 0

0 0

0.90 0.29

0 0.69

0 0

0 0

0.29

A C1 C2

0 0 0

0 0.07 0.03

0 0 0.01

0.10 0 0

0 0 0

0 0.07 0.01

0 0 0.02

0.90 0.86 0.92

0 0 0

0 0 0

0.86 0.90

A C1 C2

0 0.37 0

0 0.40 0.04

0 0 0

0.10 0 0

0 0 0.01

0 0 0

0 0 0.02

0 0 0.14

0.90 0 0.79

0 0.23 0

0.00 0.79

A C1 C2

0 0 0.02

0 0.05 0

0 0 0

0.10 0.35

0 0 0

0 0 0.07

0 0 0

0 0 0.27

0 0 0

0.90 0.60 0.63

0.70 0.63

A C1 C2

0 0 0

0 0 0.02

0 0 0

0.10 0.02 0.02

0.90 0.10 0.18

0 0.84 0.75

0 0 0

0 0.03 0.03

0 0.01 0

0.12 0.20

A C1 C2

0 0 0

0 0 0

0 0 0

0.10 0.04 0.03

0 0.06 0.13

0.90 0.72 0.68

0 0.16 0.14

0 0 0.01

0 0.01 0.01

0.76 0.71

A C1 C2

0 0 0

0 0 0

0 0 0

0.10 0.07 0.08

0 0 0

0 0.06 0.01

0.90 0.86 0.90

0 0.01 0

0 0 0

0.93 0.98

A C1 C2

0 0 0

0 0.28 0

0 0 0

0.10 0.02 0

0 0 0

0 0 0.14

0 0 0

0.90 0.70 0.86

0 0 0

0.72 0.86

S e c o n d set: I

2

3

4

A C1 C2

O.5 0.46 0.46

0.5 0.46 0.46

0 0 0

0 0 0

0 0.08 0.08

0.92 0.92

A C1 C2

0.5 0.42 0.42

0 0 0

0.5 0.43 0.43

0 0 0

0 0.15 0.15

0.93 0.93

A C1 C2

O.05 0.47 0.48

0 0 0

0 0 0

0.5 0.47 0.45

0 0.06 O.07

0.97 0.93

A C1 C2

0.5 0.45 0.45

0 0 0

0 0 0

0 0 0

0.5 0.55 0.55

0.95 0.95

215 T A B L E IIl ( c o n t i n u e d ) Mixture No.

Source No. 1

2

Error index 3

4

5

6

7

8

9

10

Second set ( c o n t . ) : 5

6

7

8

9

10

11

12

13

14

15

16

17

18

A C1 C2

0 0 0

0.5 0.52 0.52

O.5 0.48 0.48

0 0 0

0 0 0

0.98 0.98

A C1 C2

0 0.01 0

0.5 0.46 0.46

0 0 0

0.5 0.50 0.50

0 0.03 0.03

0.96 0.97

A C1 C2

0 O.5 0 no c o n v e r g e n c e 0.01 0.51 0

0

0.5

0

0.48

0.98

A CI C2

0 0.01 0.01

0 0 0

0.5 0.45 0.45

0.5 0.49 0.48

0 0.05 0.06

0.94 0.93

A CI C2

0 0 0

0 0.03 0.02

0.5 0.49 0.49

0 0 0

0.5 0.48 0.48

0.97 0.98

A C1 C2

0 0.01 0.01

0 0 0

0 0 0

0.5 0.51 0.51

0.5 0.48 0.48

0.98 0.98

A C1 C2

0.25 0.19 0.19

0.25 0.31 0.31

0.25 0.16 0.16

0.25 0.24 0.25

0 0.09 0.08

0.85 0.86

A C1 C2

0.25 0.21 0.21

0.25 0.26 0.26

0.25 0.20 0.20

0 0 0

0.25 0.33 0.33

0.91 0.91

A C1 C2

0.25 0.24 0.24

0.25 0.26 0.26

0 0 0

0.25 0.24 0.24

0.25 0.26 0.26

0.98 0.98

A C1 C2

0.25 0.21 0.22

0 0.06 0.07

0.25 0.19 0.18

0.25 0.24 0.23

0.25 0.30 0.31

0.89 0.88

A C1 C2

0 0 0

0.25 0.27 0.27

0.25 0.23 0.23

0.25 0.26 0.25

0.25 0.25 0.25

0.98 0.98

A C1 C2

0.5 0.5 0.5

0.5 0.5 0.5

0 0 0

0 0 0

0 0 0

1.00 1.00

A C1 C2

0.5 0.45 0.45

0 0.01 0.02

0.5 0.42 0.42

0 0 0

0 0.12 0.11

0.93 0.93

A C1 C2

0.5 0.46 0.48

0 0 0

0 0 0

0.5 0.50 0.47

0 0.04 0.05

0.96 0.95

216 T A B L E III ( c o n t i n u e d ) Mixture No.

Source No. I

2

Error index 3

4

5

6

7

8

9

10

A C1 C2

0.5 0.56 0.46

0 0.07 0.07

0 0.15 0

0 0 0

0.5 0.22 0.47

0.72 0.93

A C1 C2

0 0.05 0

0.5 0.27 0.50

0.5 0.69 0.50

0 0 0

0 0 0

0.77 1.00

A C1 C2

0 0 0

0.5 0.47 0.47

0 0.01 0

0.5 0.49 0.47

0 0.04 0.07

0.96 0.94

A C1 C2

0 0.25 0

0.5 0.05 0.51

0 0.71 0

0 0 0

0.5 0 0.49

0.05 0.99

A C1 C2

0 0 0

0 0.01 0.01

0.5 0.46 0.45

0.5 0.48 0.47

0 0.05 0.07

0.94 0.92

A C1 C2

0 0.14 0

0 0.02 0.02

0.5 0.83 0.51

0 0 0

0.5 0 0.48

0.51 0.98

A C1 C2

0 0.02 0.03

0 0.02 0.01

0 0 0

0.5 0.46 0.44

0.5 0.50 0.52

0.96 0.94

Second sel (cont.): 19

20

21

22

23

24

25

• 1 P r o p o r t i o n s p r e s e n t in l a b o r a t o r y m i x t u r e . • 2 Calculated using 8 ions. • 3 Calculated o m i t t i n g PO~-. Wolfe's (1959) algorithm did n o t converge a f t e r 400 iterations.

All source samples and mixtures were analyzed for eight different ions, and two of the sources in set 1, and all sources and mixtures in set 2, were analyzed for HCO%. Concentrations of the cations Na+, K ÷, Mg 2÷ and Ca 2÷ were determined by atomic absorption spectrometry. A Technicon ® Autoanalyzer Industrial II System* was used to determine the concentrations of Cl-, NO%, SO42- and PO4a-. HCO% ion was determined by potentiometric titration. 3. A N A L Y S E S 3.1. Mathematical The normalized

model ion-balance equations

can be written in the form:

* C i t a t i o n o f b r a n d n a m e s is for t h e readers' i n f o r m a t i o n only, and does n o t imply e n d o r s e m e n t by t h e U.S. D e p a r t m e n t o f Agriculture.

217 cT,

CT2

...

C'~n - - 1

X1

G1

C~l

C~2

,..

C~n -- 1! X2

C2

Vii

Ct2... Ci~...Ci% - - 1

C~,

* Cm2

..,

*n Cm

-

xj

=

ci

(1)

1

w h e r e C~ is the c o n c e n t r a t i o n of the ith ion in the j t h i n f l o w divided b y the c o n c e n t r a t i o n o f the ith ion in the m i x t u r e ; x j , j = 1, 2, . . . , n, is t h e d e c i m a l f r a c t i o n o f the j t h inflow in the m i x t u r e ; and el, i = 1, 2 . . . . , m, is the normalized e r r o r t e r m . F r o m mass balance:

~

xj = 1

(2)

j=l

Eq. 2 can be solved f o r xn and s u b s t i t u t e d into eq. 1, resulting in the expression: CI 1

CI 2

...

Cln_ 1

Ctn -- I

Xl

1

C21

C22

.,.

C2n-1

C~n - - 1

X

2

Vii

Ci2

Cij

Cin- 1 C*,, - - 1

xi

..,

Cmn- 1 C*.

1

Cml

Cm2

-- 1

=

i

(3)

m

w h e r e Ci.i = C ~ - C*,. Thus, we have m e q u a t i o n s w i t h ( n - 1) u n k n o w n s w h e r e m ~> n (usually). Because negative x i ' s are n o t p h y s i c a l l y realistic (in m o s t cases) we wish to have the x y / > 0. U n d e r these c i r c u m s t a n c e s , it is c o n v e n i e n t to f o r m u l a t e this as a q u a d r a t i c p r o g r a m m i n g p r o b l e m b y finding t h e xj, j = 1, 2, . . . , n - - 1 t h a t m i n i m i z e t h e o b j e c t i v e f u n c t i o n : E =

~

(ffi)2

(4)

i=l

Squaring each e q u a t i o n in eq. 3, s u m m i n g a n d neglecting the c o n s t a n t , we obtain a quadratic form: E = DTX+ XTAX

(5)

218 w h e r e D is a v e c t o r o f c o e f f i c i e n t s o f the linear t e r m s and X T indicates t r a n s p o s e . Thus, E is to be m i n i m i z e d subject to the n o n n e g a t i v i t y c o n s t r a i n t , x~ ~ 0; j = 1, 2 . . . . , n - - 1. It should be n o t e d t h a t , in this f o r m , x , is n o t c o n s t r a i n e d to be ~ 0, so s o m e care m u s t be used in setting up the d a t a so t h a t xn is the m o s t likely source. C o n d i t i o n s f o r e x i s t e n c e and u n i q u e n e s s of s o l u t i o n s can be f o u n d in H a d l e y (1964). In o u r p r o b l e m , the m a t r i x A is positive semi-definite. A n y local m i n i m u m will be a global m i n i m u m , b u t it m i g h t n o t be unique. Since the x j ' s are u n b o u n d e d , there m i g h t be no solution. In t h e latter case, a s o l u t i o n m i g h t be f o u n d b y eliminating a n o t h e r variable. A will be positive d e f i n i t e w h e n its d e t e r m i n a n t is non-zero, and in this case t h e r e is a u n i q u e solution. A version o f the Wolfe ( 1 9 5 9 ) q u a d r a t i c p r o g r a m m i n g a l g o r i t h m was used to find the best linear c o m b i n a t i o n o f s o u r c e waters f o r each m i x t u r e . T h e c o m p l e t e source set was c o n s i d e r e d in the a l g o r i t h m , and t w o runs were m a d e f o r each m i x t u r e , o n e with 8 ions c o n s i d e r e d a n d a s e c o n d with PO~o m i t t e d . In the s e c o n d source a n d m i x t u r e set, m i x t u r e s 1--15 and 16--25 were f r o m sources 1--5 and 6--10, respectively, and were calculated on t h a t basis. Because sources 1--5 and 6--10 were essentially the s a m e e x c e p t for filtration, all m i x t u r e s m a d e f r o m b o t h source groups were c o n s i d e r e d p a r t o f t h e s e c o n d m i x t u r e set unless o t h e r w i s e n o t e d . HCO~ was o m i t t e d f r o m the analyses f o r b o t h sets. T h e results o f these analyses are s h o w n in T a b l e III. I t should be n o t e d t h a t in set 1, the n u m b e r o f c a n d i d a t e sources (10) is greater t h a n t h e n u m b e r of ion- and w a t e r - b a l a n c e e q u a t i o n s (9 with PO~i n c l u d e d and 8 w i t h o u t ) . It is a p p a r e n t t h a t , in s o m e c i r c u m s t a n c e s , this s i t u a t i o n can lead to p r o b l e m s o f n o n u n i q u e n e s s . T o e x a m i n e this factor, we c o n s t r u c t e d a t h e o r e t i c a l m i x t u r e consisting o f 0.8 parts o f source 2 a n d 0.2 parts o f source 4. T w o trials were run, the first with sources I - - 9 as c a n d i d a t e s , a n d t h e s e c o n d with sources 1--10 as candidates. T h e a l g o r i t h m c o n v e r g e d to the c o r r e c t answers in b o t h cases. T o e x a m i n e the e f f e c t of errors on the q u a d r a t i c m a t r i x a n d a l g o r i t h m c o n v e r g e n c e , i n d e p e n d e n t , n o r m a l l y d i s t r i b u t e d e r r o r terms, with zero m e a n and c o e f f i c i e n t o f v a r i a t i o n o f 0.05 were a d d e d to the c o n c e n t r a t i o n o f each ion in every c a n d i d a t e s o u r c e and in t h e m i x t u r e . T e n o f these s i m u l a t e d runs were e x a m i n e d f o r each case. F o r case 1 (9 c a n d i d a t e s and 9 e q u a t i o n s ) , the a l g o r i t h m c o n v e r g e d to c o r r e c t answers in 7 runs, and did n o t c o n v e r g e in 4 0 0 i t e r a t i o n s f o r 3 runs. F o r case 2 (10 c a n d i d a t e s and 9 e q u a t i o n s ) , the a l g o r i t h m did n o t c o n v e r g e 3 times, and c o n v e r g e d to the a p p a r e n t l y c o r r e c t answers 7 times. F o r each run in which the a l g o r i t h m did n o t converge, t h e values o f the xi, a f t e r 400 iterations, were v e r y close to the c o r r e c t values. A l t h o u g h e v e r y e f f o r t s h o u l d be m a d e to r e d u c e the n u m b e r o f c a n d i d a t e sources to f e w e r t h a n the n u m b e r of ions used in the analyses, it a p p e a r s t h a t useful results can be o b t a i n e d if n is slightly larger t h a n (m + 1), p r o v i d e d t h a t t h e r e are f e w e r t h a n (rn + 1) sources a c t u a l l y present. F o r e x a m p l e , in m i x t u r e 20 in T a b l e III, 8 sources were included in the

219 mixture;

thus, when PO~- was omitted

from the analysis, there were as many

equations as s o u r c e s i n t h e m i x t u r e . The algorithm identified 7 nonzero s o u r c e s , a n d all e r r o r s , e i , w e r e e x a c t l y z e r o . T h i s s o l u t i o n is p r o b a b l y n o t u n i q u e . I t is a l s o p o s s i b l e t h a t s o m e o f t h e o t h e r s o l u t i o n s f o r t h e f i r s t s e t o f mixtures are not unique; however, it should be noted that in each case, the s u m o f s q u a r e d e r r o r t e r m s i n e q . 1, c a l c u l a t e d b y u s i n g t h e x i i d e n t i f i e d the algorithm, was less than if the "correct" xj were used. The question u n i q u e n e s s is e x p l o r e d i n m o r e d e t a i l i n t h e A p p e n d i x . 3.2. Interactions

between

by of

sources

To determine if a solution with the chemical characteristics of any of the sources could be approximated by a linear combination of any of the other sources, the analyses program was run with each source considered as a T A B L E IV Best linear c o m b i n a t i o n s of first a n d s e c o n d set o f sources Mix- P r o p o r t i o n o f s o u r c e i n m i x t u r e ture No. 1 2 3 4 5 6

R2

Intercept

Slope

7

8

9

10

0 0 0 0 0 0.54 0.04

0 0.51 0.72 1.32 0.75 *3 0 0 0.69 0.26 1.08 *2 0 0 0.69 2.14 0.64 *2 0 0.55 0.88 1.27 1.10 .4 0.44 0 0.15 28.7 .1 - - 0 . 0 0 2 0 0 0.59 2.61 0.58 *2 0 0 0.07 8.28 .1 - - 0 . 0 8 0 0 0.86 2.15 1.43 .4 0.36 0 . 1 0 8 7.7 0.06 0 -0.87 1.76 0.59 *4

First set:

1 2 3 4 5 6 7 8 9 10

0 0 0

0.01 0.48 -1.0 0.99 - 0 0.45

0 0 0.01 -

0 0 0 0

0 0 0 0

0 0 0 0

0.56 0 0 0

0 0 0.96 0

0 0 0 0

0 0.46 0 0

-0 0 0

0 0 0.34

0 0 -0.66

--

0.33 0.30 0 0.16 0.10 0

0 0.74

0 0

0 0

0 0

0 0

-

-

-

-

S e c o n d set:

1 2 3 4 5 6 7 8 9 10

-0 0 0.26 0.03 0.56 0 no c o n v e r g e n c e 0.15 0 0 -0.21 0.18 0.61 0

0.74 0.41

0.87 5.75 0.77 - - 0.75

1.40 *3 1.72 *3

0.85 --

1.16 *2 1.77 *3 1.43 *3 1.78 *2 1.12 *2 1.70 *3

0 0.03 - -

0 0.19 0.56 0

0.81 0.41

0,73 7.74 0.82 0.92 0.87 5.72 0.76--1.83

0.12 0 0.22 0.19

0 ' -0.59 0

0.88 --

0.67 9.86 0.83--0.75

-

-

no c o n v e r g e n c e

• 1 I n t e r c e p t significantly d i f f e r e n t f r o m zero at 5% level. • 2 Slope significantly d i f f e r e n t f r o m zero at 5% level. • 3 Slope significantly d i f f e r e n t f r o m zero at 1% level. • 4 Slope significantly d i f f e r e n t f r o m zero at 0.1% level.

220 m i x t u r e and the o t h e r sources c o n s i d e r e d as candidates. The results o f this analysis are s h o w n in Table IV. This shows, for e x a m p l e , t h a t source 1 o f the first set is best a p p r o x i m a t e d (according to the objective f u n c t i o n given by eq. 3) by a m i x t u r e of 0.01 parts source 2, 0.48 parts source 3, and 0.51 parts source 10. Simple linear regressions were run b e t w e e n the concent r a t i o n o f each ion as calculated f r o m the p r o p o r t i o n s shown in Table IV and the measured c o n c e n t r a t i o n s o f each ion for each source. The coefficients of d e t e r m i n a t i o n , R 2, the intercept, and the slope for these regressions are shown in the last three c o l u m n s of Table IV. An e x a m i n a t i o n of the coefficients o f d e t e r m i n a t i o n , slopes and intercepts reveals several cases when one o f the sources can be r a t h e r closely approxim a t e d by linear c o m b i n a t i o n s of o t h e r sources. F o r example, source 4 of the first set can be a p p r o x i m a t e d by a c o m b i n a t i o n o f 0.45 parts source 3 plus 0.55 parts source 10 with R 2 = 0.88. The i n t e r c e p t is not significantly d i f f e r e n t f r o m zero, and the slope is close to one. T h e r e f o r e , one w o u l d e x p e c t t h a t it w o u l d be difficult to i d e n t i f y these sources, or t h a t t h e y could be identified w h e n t h e y are actually n o t present.

4. RESULTS AND DISCUSSION PO~- ion was present in very low c o n c e n t r a t i o n s (pg 1- l range) in most of the sources used to make the m i x t u r e s (Table II). Handling and analyzing samples with low c o n c e n t r a t i o n s p r e s e n t e d analytical p r o b l e m s t h a t b e c a m e i m p o r t a n t in this study. The t w o problems that were observed, with P O ~ - i n particular, were c o n t a m i n a t i o n and dilution below the d e t e c t i o n limit. C o n t a m i n a t i o n was c o n f i r m e d in the second set o f source samples where PO~- ion was f o u n d in sources 7, 8 and 10 after filtration. In the first set of sources and mixtures, b o t h c o n t a m i n a t i o n and dilution could a c c o u n t for the observed results o f PO~- ion in a source, b u t n o t f o u n d in a m i x t u r e c o n t a i n i n g t h a t source. C o n t a m i n a t i o n of the analysis sample used to analyze sources 6, 8 and 9 were suspected in that, when m i x t u r e s c o n t a i n i n g these sources were c o m b i n e d with o t h e r sources t h a t did n o t contain PO~-, the resulting m i x t u r e c o n t a i n e d P O ~ - i n o n l y t w o cases. C o n t a m i n a t i o n and dilution problems were n o t observed with the o t h e r ions, although t h e y p r o b a b l y were present. At higher c o n c e n t r a t i o n s , small a m o u n t s o f cont a m i n a t i o n are n o t detectable. If the NCIB m e t h o d is to be used simply as a d e t e c t i o n t e c h n i q u e to i d e n t i f y the presence or absence of a particular source in a m i x t u r e , it is useful to e x a m i n e the p r o b a b i l i t y of n o t d e t e c t i n g a source w h e n it is p r e s e n t ( t y p e d error) and the p r o b a b i l i t y of d e t e c t i n g a source when it is n o t p r e s e n t (type-II error). The probabilities o f type-I and -II errors for each source, and f o r all sources c o m b i n e d , are shown in Table V. A l t h o u g h these probabilities apply o n l y to these particular sets o f sources and are a f f e c t e d by the mixing strategy used, t h e y do reveal some i m p o r t a n t points. In general the type-I e r r o r was r e d u c e d and -II was increased when PO~- was o m i t t e d in

221 TABLE V Probability of type-I and -II errors Source

First set

Second set

No. type I

all ions 1 2 3 4 5 6 7 8 9

0.29 0.20 0.50 0.40 0.0 0.80 0.36 0.47 0.69 10 0.07 All sources 0.38

t y p e II

tyl)e I

t y p e II

P043omitted

all ions

PO 3omitted

all PO~ions omitted

all ions

PO~omitted

0.21 0.47 0.21 0.73 0.07 0.33 0.29 0.0 0.0 0.0 0.23

0.14 0.57 0.16 0.39 0.0 0.20 0.16 0.18 0.17 0.20 0.20

0.18 0.18 0.24 0.14 0.10 0.22 0.64 0.82 0.17 0.26 0.29

0.0 0.13 0.0 00 0.13 0.0 0.0 0.0 0.0 0.50 0.08

0.43 0.29 0.0 0.0 0.86 0.70 0.83 0.50 0.0 0.70 0.43

0.43 0.29 0.0 0.0 0.86 0.17 0.83 0.0 0.0 0.70 0.33

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

the first m i x t u r e set, suggesting t h a t including PO~- has some merit in reducing incorrect d e t e c t i o n , but this occurs at the expense o f increasing the chances of failure to identify a w a t e r source t h a t is, in fact, present. F o r the s e c o n d set of mixtures, o m i t t i n g PO43- r e d u c e d b o t h type-I and -II errors, Overall, the s e c o n d set of m i x t u r e s had a m u c h lower incidence o f t y p e - I errors than the first set, b u t had a larger n u m b e r of t y p e - I I errors. The larger n u m b e r of t y p e - I I errors was p r o b a b l y due to similarity of the sources, as indicated by Table IV. A l t h o u g h there were m o r e t y p e - I I errors (Table III), the size o f the errors was generally m u c h smaller for the s e c o n d set o f mixtures. The t y p e - I and -II errors are very small for source 5 o f the first set and sources 4 and 9 o f the s e c o n d set. These sources are quite d i f f e r e n t f r o m the others, as indicated b y the data in Table II and the results o f the linear c o m b i n a t i o n test s h o w n in Table IV. It should be n o t e d t h a t the presence or absence o f sources such as these, which indicate water-quality changes due to m a n ' s activities, are of p r i m a r y interest in p o l l u t i o n studies. The surface r u n o f f samples, sources 6 and 7 in set 1, had rather large type-I and -II errors, p r o b a b l y because t h e y are i n t e r m e d i a t e in c o m p o s i t i o n b e t w e e n the rainfall sample and the Walnut Gulch g r o u n d w a t e r samples. It is s o m e w h a t surprising t h a t the errors were so low for water 10, w h i c h has a high simple c o r r e l a t i o n with a linear c o m b i n a t i o n o f sources 1, 2 and 4. A n o t h e r way of p o r t r a y i n g results of this l a b o r a t o r y test o f the NCIB t e c h n i q u e is to c o n s t r u c t an error index t h a t a c c o u n t s for differences in actual and predicted p r o p o r t i o n s of each source in each mixture. The following index was used for each m i x t u r e : Ns

I~ = 1 - 0.5 ~_ t~k --x~1~l j=l

(6)

222 1,0 . . . . .

FIRST

---

- -

SET

SEGOND S E T

0.8 /

I

i/

1I ~-~

0.6

ii

J

i

VI

EIGHT IONS ~

1/

\. O_

I

///

I'l

z

0.4

/

0.2

//

s-

/

----

-

--~'1

O0"

/

--PC) 4 0 M I T T E

0.2

-

t/ f

- --C-E IGHT I O N S J L

0.4 0.6 E R R O R INDEX

~

J

~

0.8

~J 1.0

i

Fig. 1. Empirical distribution f u n c t i o n of error index.

where xjk is the p r o p o r t i o n of mixture k made up of source j; xjk is the estimated proportion; and Ns is the n u m b e r of sources considered. If the agreement is perfect, I k = 1. These error indices are tabulated in the last c o l u m n of Table III, and the sample distribution f u n c t i o n s are s h o w n in Fig. 1. Fig. 1 clearly shows that errors in the PO 3- analyses led to substantial prediction errors, especially in the first mixture set. It is apparent that, although the normalizing procedure compensates for the relative abundance o f ions, a set of weighting factors, w i, should be applied to each e q u a t i o n in eq. 3 to a c c o u n t for uncertainties and accuracy of chemical analysis for each ion. K o w a l c z y k et al. ( 1 9 8 2 ) weighted c o n c e n t r a t i o n s by 1/o/2 where oi was the estimated standard deviation of the error in c o n c e n t r a t i o n for the ith ion. Error statistics were calculated for each ion for each of the mixtures s h o w n in Table VI, according to the equation: Cik

~-

(CTi k -- CMik)/CTi

k

(7)

where Cwik is the theoretical c o n c e n t r a t i o n of the ith ion in the kth mixture and CMik is the measured concentration. The theoretical c o n c e n t r a t i o n is a linear c o m b i n a t i o n of sources included in the mixture: Ns CTik

=

~ j=l

CsijXjk

(8)

223 TABLE VI Error statistics Ion

Na+ K÷ Ca2+ Mg2+ NO3 po~-(*) SO~-

CI-

First set

Second set

mean error (%)

standard deviation

21.3 9.8 12.4 14.3 22.7

15.8 7.5 11.1 11.1 52.8

. 5.3

15.3

mean error (%)

standard deviation

- 0.42 - 0.52 0.11 1.32 3.44

1.63 7.10 0.76 3.53 7.45

11.9

5.72

7.00

25.0

7.06

5.70

.

.

.

*PO~- omitted from calculations because of a very large percentage error. where Cs~j is the c o n c e n t r a t i o n of the ith ion in source j; and xj~ is the p r o p o r t i o n of source i in the k t h prepared mixture. The mean and s t a n d a r d deviation o f the errors for each ion are s h o w n in Table VI. These errors for the first m i x t u r e set were m u c h larger than anticipated, and show a significant bias in t h a t the average theoretical c o n c e n t r a t i o n is greater than the measured c o n c e n t r a t i o n for all ions. The bias ranges f r o m a low o f + 5.3% for SO42- to + 22.7% for NO~. The high value for NO~ is w h a t one c o u l d e x p e c t f r o m a n o n c o n s e r v a t i v e ion: however, n o such bias w o u l d be a n t i c i p a t e d for the o t h e r ions. There are six possible sources o f bias in water analyses: u n r e p r e s e n t a t i v e sampling, instability o f samples b e t w e e n sampling and analyses, interference effects, biased calibration, a biased blank c o r r e c t i o n , and inability to d e t e r m i n e all f o r m s of the d e t e r m i n a n t (Cheeseman and Wilson, 1972). F o r the first set o f m i x t u r e samples, it appears t h a t instability, interference effects, and possibly biased calibration m i g h t a c c o u n t for the errors s h o w n in Table VI. Also, mistakes in preparing the mixtures and dilution mistakes are, o f course, reflected in these data, b u t are difficult to identify. I d e n t i f y i n g the source o f these errors was m a d e m o r e difficult because HCO~ was m e a s u r e d o n l y for sources 6 and 7 (Table II) in set 1, so we could n o t c h e c k the ion balances for the o t h e r sources or mixtures. These large unexplainable errors served as the i m p e t u s to prepare the s e c o n d set of mixtures and d e t e r m i n e if the errors c o u l d be reduced. The results indicated t h a t all errors generally were r e d u c e d for the s e c o n d m i x t u r e set (Tables III, V and VI). The m e t h o d of m i x t u r e p r e p a r a t i o n was the likely source o f m o s t of the error in the first set o f mixtures. In the p r e p a r a t i o n of the first set o f mixtures, the small a m o u n t s o f source samples used to m a k e the m i x t u r e s c o u l d a c c o u n t for s o m e of the errors, and this was c o r r e c t e d in the s e c o n d set of m i x t u r e s by using larger samples. The

224

m e t h o d s of analysis were the s a m e f o r b o t h sets of m i x t u r e s , a l t h o u g h d i f f e r e n t t e c h n i c i a n s p e r f o r m e d the analyses, and this c o u l d a c c o u n t f o r s o m e e r r o r or bias. T h e s e c o n d set o f m i x t u r e s still h a d an overall bias, with the t h e o r e t i c a l c o n c e n t r a t i o n being greater t h a n the m e a s u r e d c o n c e n t r a t i o n , b u t the e r r o r was significantly smaller, as m e a s u r e d b y the m e a n error and s t a n d a r d d e v i a t i o n ( T a b l e VI). Because of the H C O 3 analysis of the sources and m i x t u r e s , the errors in the s e c o n d set were investigated f u r t h e r by calculating the ion balance using the m a j o r cations and anions (Table II). T h e d i f f e r e n c e in milliequivalents per liter b e t w e e n the cations and anions was less t h a n 1 m e q for all sources e x c e p t source 4. Source 4 was f r o m a shallow well i n f l u e n c e d by sewage effluent, a n d m i c r o b i a l activity in the sample was s u s p e c t e d to have shifted the H C O 3 e q u i l i b r i u m a n d a f f e c t e d the ion balance. T h e influence of source 4 on t h e m i x t u r e s was evident in t h a t a l m o s t all the m i x t u r e s c o n t a i n i n g it h a d the higher ion-balance errors ( d a t a n o t p r e s e n t e d ) . S o u r c e 9, which was the same as source 4, b u t filtered t h r o u g h a 0.45-pro filter t h a t should r e m o v e m i c r o b i a l activity, had low ion-balance errors t h r o u g h o u t . Since ionbalance errors were relatively low and e m b o d i e d all the analysis errors, a specific source o f e r r o r c o u l d n o t be d e t e r m i n e d . T h e errors in the c o m p u t e d p r o p o r t i o n s of source waters, ex, for all m i x t u r e s , were c a l c u l a t e d using the relationship: f

1.00

i

r

I

,

t

, t~lt:~

~]

, DO

, O

O

D'

F

0.80

o SET

I

o SET

2

0.60

LL 0.40

0.20

0.00

o

o rm rm I

-I.00

I

-0.60

-4

' -0120

0'.0

i

0120

J

e x

Fig. 2. D i s t r i b u t i o n of p r o p o r t i o n errors,

x]

fcj.

0 60

I

I

1.00

T A B L E VII C o m p a r i s o n o f p r e d i c t i o n s b e t w e e n variance w e i g h t e d e q u a t i o n s and equally w e i g h t e d equations Mixture No.

Source No.

Error index

1

2

3

4

5

6

7

8

9

10

A .1 C2 .2 W*3

0.25 0.13 0.21

0.10 0 0

0 0.04 0

0 0.25 0

0 0 0

0,25 0 0.13

0.40 0.51 0.55

0 0.04 0.03

0 0 0

0 0.04 0.07

0.53 0.75

A C2 W

0.5 0.36 0.46

0 0 0

0 0 0

0 0 0.04

0 0 0

0 0 0

0.5 0.47 0.40

0 0.08 0.07

0 0 0

0 0.10 0

0.83 0.90

A C2 W

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0

0 0.05 0.02

0.5 0.5 0.5

0 0 0

0 0.06 0.03

0.89 0.95

A C2 W

0.5 0.39 0.45 0.5 0.48 0.28

0 0 0

0 0 0

0 0 0

0 0.06 0.14

0,5 0.47 0.46

0 0 0.12

0.95 0.74

A C2 W

0 0.5 0 0 0.44 0.03 no c o n v e r g e n c e

0 0

0 0

0.5 0

0 0.41

0 0.11

0 0

0 0

0.45

16

A C2 W

0 0 0 no c o n v e r g e n c e no c o n v e r g e n c e

0.5

0

0

0.5

0

0

0

26

A C2 W

O.2 0 0 no c o n v e r g e n c e no c o n v e r g e n c e

0

0

0

0.8

0

0

0

57

A C2 W

0 0 0 no c o n v e r g e n c e 0 0 0

0.10

0

0

0.9

0

0

0

0

0

0.23

0.52

0.26

0

0

0.52

A C2 W

0.25 0.19 0.24

0.25 0.31 0.24

0.25 0.16 0.24

0.25 0.25 0.24

0 0 0.03

0.86 0.97

A C2 W

0.25 0.21 0.24

0.25 0.26 0.25

0.25 0.20 0.25

0 0 0

0.25 0.33 0.27

0.91 0.99

A C2 W

0.25 0.22 0.25

0 0.07 0.03

0.25 0.18 0.24

0.25 0.23 0.22

0.25 0.31 0.26

0.88 0.96

A C2 W

0.5 0.45 O.48

0 0.02 0

0.5 0 0.42 0 0.48 0

A C2 W

0 0 0

0.5 0.50 0.49

0.5 0 0.50 0 0.51 0

A C2 W

0 0 0.01

0 0.01 0

0.5 0.45 0.50

First set:

i

2

3

4

6

S e c o n d set: 11

12

14

17

20

23

• 1 A = actual. • 2 C2 -- equal weight, PO~- o m i t t e d . • 3 W = variance weighted, PO~- o m i t t e d .

0.5 0.47 0.49

0 0.11 0.03 0 0 0 0 0.07 0

0.93 0.97 1.00 0.99 0.92 0.99

226 ex = x j - - d e i

(9)

where xj is the actual p r o p o r t i o n o f source j in the m i x t u r e ; and ~ is the a m o u n t calculated with PO 3- o m i t t e d . T h e empirical distribution f u n c t i o n s o f these errors are s h o w n for b o t h sets o f m i x t u r e s in Fig. 2. Because o f the mass-balance constraint, given by eq. 2, the mean error is zero. T h e r e is also a positive p r o b a b i l i t y mass of e x = 0, c o r r e s p o n d i n g to the sources n o t p r e s e n t in a m i x t u r e that were c o r r e c t l y identified as being n o t present. The falsely identified sources are reflected in the negative errors in Fig. 2, and the t e n d e n c y to u n d e r e s t i m a t e the p r o p o r t i o n s of sources present is shown in the positive errors. T h e r e d u c t i o n in the dispersion of these errors by the i m p r o v e d p r o c e d u r e s for m i x t u r e set 2 is remarkable. A set of weighting factors, wi = 1/oi 2 where o; 2 is the error variance for the ith ion, was calculated, and the analysis was r e p e a t e d for 14 mixtures. T h e results are c o m p a r e d with the actual p r o p o r t i o n s and the p r o p o r t i o n s calculated with the equally weighted e q u a t i o n s in Table VII. These m i x t u r e s were selected to r e p r e s e n t those for which the estimates, using equally weighted equations, were quite good, and those which were poor, or for which the Wolfe algorithm did n o t converge. F o r the first set of mixtures, the variance weighting i m p r o v e d the estimates for m i x t u r e s 1, 2 and 3, and allowed convergence for 57. T h e algorithm still did n o t converge for 16 and 26, and the weighted estimate for m i x t u r e 4 was inferior to the u n w e i g h t e d estimate. The algorithm did n o t converge for the variance weighted e q u a t i o n s for m i x t u r e 6. For the second set o f mixtures, the variance weighting i m p r o v e d the estimates for m i x t u r e s 11, 12, 14, 17 and 23, and had little e f f e c t on m i x t u r e 20. It also r e d u c e d the p r o b a b i l i t y of i d e n t i f y i n g a source when it was n o t present.

5. SUMMARY AND CONCLUSIONS A n o r m a l i z e d chemical ion-balance t e c h n i q u e (NCIB), to i d e n t i f y water sources in a m i x t u r e , was tested using two sets o f m i x t u r e s p r e p a r e d in the l a b o r a t o r y . T h e first set consisted o f 64 m i x t u r e s of f r o m 2--8 natural waters, selected f r o m 10 sources (surface, g r o u n d w a t e r and precipitation). A s e c o n d set o f 25 m i x t u r e s was p r e p a r e d using 5 source waters. Each o f the source waters and each m i x t u r e was a n a l y z e d for at least 8 ions. P r o p o r t i o n s o f source waters in each sample were e s t i m a t e d using a t e c h n i q u e described b y Woolhiser et al. ( 1 9 7 9 , 1982). F o r the first set o f mixtures, the p r o b a b i l i t y of failure to d e t e c t a source when it was present in the m i x t u r e ( t y p e d error) was 0.23, and the p r o b a b i l i t y o f i d e n t i f y i n g a source when it was n o t present (type-I! error) was 0.29. F o r the second set o f mixtures, the probabilities were 0.0 and 0.33, respectively. It is possible t h a t some o f the solutions for m i x t u r e set 1 are n o t unique. However, the values o f the objective f u n c t i o n s attained with the e s t i m a t e d xj were always b e t t e r t h a n those attained with the t h e o r e t i c a l l y " c o r r e c t " xj.

227 E v e r y e f f o r t s h o u l d be m a d e to include at least as m a n y ions in t h e analyses as t h e r e are c a n d i d a t e sources. T h e linear d e p e n d e n c e b e t w e e n sources was investigated and f o u n d to be significant f o r several sources. Significant errors, including bias as well as large e r r o r variances, were f o u n d in the c h e m i c a l analyses f o r the first set of mixtures. T h e s e errors were substantially r e d u c e d for the s e c o n d set of m i x t u r e s , p r o b a b l y b e c a u s e larger s a m p l e v o l u m e s were used and greater care was t a k e n in p r e p a r i n g and analyzing the samples. T h e sensitivity o f the N C I B t e c h n i q u e to errors in c h e m i c a l analyses dictates t h a t great care should be t a k e n in s a m p l e c o l l e c t i o n and analyses. T h e NCIB t e c h n i q u e , w i t h each n o r m a l i z e d ion-balance e q u a t i o n weighted b y the inverse of the e r r o r variance, s h o w e d i m p r o v e d e s t i m a t e s a n d a r e d u c t i o n in the p r o b a b i l i t y o f the t y p e - I I error. It a p p e a r s t h a t there is little chance of failing to d e t e c t a source if it is p r e s e n t in t h e m i x t u r e in relatively large p r o p o r t i o n s (>~ 0.25), unless t h e r e is significant i n t e r f e r e n c e b e t w e e n sources. H o w e v e r , sources d e t e c t e d with an e s t i m a t e d p r o p o r t i o n smaller t h a n 0.10 are quite likely to be i n c o r r e c t l y identified. A d d i t i o n a l investigations w o u l d be required to verify or disprove t h e i r presence.

ACKNOWLEDGEMENTS T. E c o n o p o u l y assisted with c o m p u t e r p r o g r a m m i n g . L a b o r a t o r y analyses were p e r f o r m e d b y D o n n a H a n s e n and V. Weiss.

APPENDIX - UNIQUENESS OF QUADRATIC PROGRAMMING SOLUTIONS T h e n o r m a l i z e d ion-balance e q u a t i o n s can be w r i t t e n in the f o r m :

ei = ~ c~xy - - 1 : (C*X)i--1,

1 ~< i ~< rn

(A-I)

j=l

w h e r e C* = (c~.), a rn × n m a t r i x ; c*- = c o n c e n t r a t i o n of the ith ion in t h e j t h inflow divided b y the c o n c e n t r a t i o n of the ith ion in the m i x t u r e ; X = (xi), a n × 1 c o l u m n m a t r i x ; xj = f r a c t i o n o f the j t h i n f l o w in t h e m i x t u r e ; a n d ei : n o r m a l i z e d e r r o r t e r m (due to m i s m e a s u r e m e n t , etc.). T h e n o r m a l i z e d inflows satisfy: x i >~ 0

and

)_

xi = 1

]=1

so we can establish values f r o m t h e m b y m i n i m i z i n g :

F(X) = ~ (ei) 2 - - r n i=l

(A-2)

228 on D = {X

~.~=lXj=l;xi~O,l~j'.~n}

F can be written in m a t r i x n o t a t i o n as:

F(X)

(A-3)

2BT X

= XT AX-

where A = C* T C*, a n × n matrix;

and B = a n × I m a t r i x with:

m

Bj = E cij i=l

F is c o n t i n u o u s , and D is a closed and b o u n d e d set. Thus, F has a m i n i m u m on D, t h o u g h it m a y n o t be unique. F is a quadratic form, and:

i=1

j=l

so A is a positive semi-definite. Thus, the long f o r m of Wolfe's ( 1 9 5 9 ) algorithm can be used to find a m i n i m u m . If IIAII~ 0, t h e n A is positive definite and, as n o t e d by Wolfe (1959), F has a u n i q u e m i n i m u m . When the d e t e r m i n a n t is zero, as w h e n m < n , f u r t h e r analysis is n e e d e d to decide if a m i n i m u m f o u n d by Wolfe's algorithm is unique. S u p p o s e X 0 is a m i n i m u m o f F on D. If there is a n o t h e r m i n i m u m , say X1, t h e n consider X0 + t V , 0 ~ < t ~ < l where V = X 1 - - X 0 . This is the line segment joining Xo and X1. Since X0 and X1 are in D, and D is convex, this segment lies in D. T h e c o n v e x i t y of F gives:

F(Xo + tV) = F[tXI + ( 1 - t ) X o ] ~< tF(X 1) + ( 1 - t)F(Xo) and since F(Xo) = F(X, ) is the m i n i m u m value of F on D, we get:

F(Xo) <~ F(Xo + tV) <~ tF(X~) + ( 1 - - t ) F(Xo) = tF(Xo) + (1 - t) f ( X o )

= f(Xo)

This reasoning gives rise to necessary and sufficient c o n d i t i o n s for F to have multiple minima: F has multiple m i n i m a if there is a v e c t o r V such that: (1)

V :~ 0

(2)

Xo + t V E D

(3)

F(X o + tV) is c o n s t a n t for small t ~> 0

for small t ~> 0

(A-4)

F o r (2) o f expressions (A-4) to hold, we need 1 = E ( X 0 + t V ) = ZXo + tEV, and Xo + tV>~ 0 for small t. Since EXo = 1 a n d X 0 ~> 0, this h a p p e n s just w h e n Z V = 0, and Vj ~> 0 for every j c o r r e s p o n d i n g t o zero c o o r d i n a t e s o f Xo. Since:

229 F(Xo + tV) = (XoTAXo

--

2BTX0) ~- t(XoTA V + VTAXo -- 2BT V)

+ t2(VTAV) for (3) of expressions (A-4) to hold, we need X oTA V + VTAXo - 2 B T V = 0 and VTAV = 0. As noted by Wolfe (1959), VTAV = O implies AB -- 0 and, by taking the transpose, VTA -- 0. Thus, the two equalities reduce to BTV =- 0 and A V = 0. In summary, we get: F has multiple minima if there is a vector V such that:

(1)

V ~ 0

(2)

Vj ~ 0

(3) Z

for

j C K = { j l ( X o ) j = 0}

(A-5)

Vj = 0

j=l

(4)

AV

(5)

B TV = 0

= 0

N o t e t h a t if V satisfies expressions (A-5), t h e n so does a n y scalar multiple o f V. In particular, in view o f (1) o f expressions (A-5), we m a y divide V by the m a x i m u m o f the absolute values o f the c o o r d i n a t e s o f V to get an a d d e d condition: (6)

--1

~< Vj ~< 1

for

1 ~< j ~< n

A change of variables, V/ = Vj + 1, and letting b = ( 1 , . . . , 1) T gives: F has a multiple m i n i m a if there is a v e c t o r V such that: (1)

V ¢

(2)

Vj >t l f o r j

(3)

L

b E K

Vj = n

j=l

(A-6)

(4)

A V = Ab

(5)

BT v

(6)

Vi ~< 2

for

l <~ j <~ n

(7)

Vi ~ 0

for

l <~ j <~ n

= BT b

We are n o w in a position to express o u r p r o b l e m in terms o f a n o t h e r quadratic p r o g r a m m i n g p r o b l e m ; this time we are only c o n c e r n e d with the value o f a m a x i m u m , n o t its uniqueness. (2)--(7) o f expressions (A-6) can be p u t in a n o r m a l f o r m b y adding slack and surplus variables as described in H a d l e y (1964). T h a t is, there are variables Wi, which include the V/and matrices A* and b*, which allow us to write (2)--(7) o f expressions (A-6) as A* W = b* and W ~ 0. N o w consider the q u a d r a t i c f o r m :

230

f(W)

:

~ ( V i - - 1) 2 - - / / j=l

(A-7)

T h e m a x i m u m o f f o n D * = { W I A * W = b *, W~>0} will b e - - n j u s t w h e n W in D * gives V = b. T h u s , w e a r r i v e a t t h e c o n c l u s i o n : F h a s a u n i q u e m i n i m u m o n D if t h e m a x i m u m o f f o n D * is g r e a t e r t h a n - - n. W e "also n o t e t h a t , b e c a u s e o f ( 6 ) a n d ( 7 ) o f e x p r e s s i o n s ( A - 6 ) , a n d t h e m a n n e r in w h i c h s l a c k a n d s u r p l u s v a r i a b l e s a r e a d d e d , t h a t D * is c l o s e d a n d b o u n d e d . T h u s , f will h a v e a m a x i m u m o n D * .

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