Probabilistic Regularities in Unfavorable Hydrochemical Phenomena

Water Resources, Vol. 32, No. 4, 2005, pp. 410–416. Translated from Vodnye Resursy, Vol. 32, No. 4, 2005, pp. 452–458. Original Russian Text Copyright © 2005 by Dolgonosov, Korchagin.

WATER QUALITY AND PROTECTION: ENVIRONMENTAL ASPECTS

Probabilistic Regularities in Unfavorable Hydrochemical Phenomena B. M. Dolgonosov and K. A. Korchagin Water Problems Institute, Russian Academy of Sciences, ul. Gubkina 3, GSP–1, Moscow, 119991 Russia Received September 30, 2004

Abstract—The balance equation for a substance washed out in a river basin is analyzed under the assumption that the runoff of this substance and its reserves in the watershed are directly proportional. The proportionality factor is perturbed by a random component, which accounts for the effect of atmospheric precipitation. The balance equation is transformed into a stochastic differential equation with a multiplicative white noise, which is used to construct a Fokker–Plank equation for the probability density of chemical flow. A stationary solution containing a power function is found for this equation. Because of the proportionality of the concentration and chemical flow, the concentration distribution also obeys the power law. Statistical treatment of empirical data on some water quality characteristics and water flow showed that the power law adequately describes the probability of unfavorable hydrochemical events. The parameters of this law for turbidity, color index, permanganate oxidability, and ammonia concentration are evaluated.

INTRODUCTION Unfavorable hydrochemical phenomena, which manifest themselves in an abrupt rise in the concentrations of some components in river water, may have various causes. Of great importance in this context are processes on watersheds involving the leaching and removal of some components from soils by rain and snowmelt water, especially under the conditions of anthropogenic transformations of landscapes and chemical pollution of the area. Forecasts of such phenomena based on the evaluation of the values of their characteristics with given exceedance probability (quantiles) are of great use for developing the strategies of water protection measures, designing water treatment facilities, and planning the reserves of reagents at water stations now in operation. The probability of high values of water flow and its turbidity can be adequately described by the power law [2]. This result was obtained by purely empirical methods based on the analysis of histograms constructed with the use of time series. The objective of this study is the theoretical substantiation of the application of the power law and its extension to various characteristics of water quality. To solve this problem, we use purely phenomenological analysis without involvement of data on the structure of the watershed area. This approach is sufficient only for establishing the functional form of the distribution law and does not allow us to correlate model parameters with watershed characteristics. In this study, we restrict ourselves to that minimum statement of the problem.

CHEMICAL FLOW Water quality in a river forms at the expense of substances washed out from the watershed surface by rain and snowmelt water and under the effect of polluted groundwater discharging into the river. We describe the chemical balance of a substance washed out from a watershed by the equation dM/dt = P – R – V ,

(1)

where M is the reserve of the substance in the watershed upstream of a control section of the river (it has the dimension of mass); P, R, and V are the rates of supply, decay, and removal of the substance from the watershed into the river (mass per unit time). The rate of removal (chemical flow) is proportional to the available reserve of the substance V = kM,

(2)

where k is a proportionality factor directly depending on the frequency and intensity of atmospheric precipitation. Isolating the random component of k, we obtain k = k + σξ ( t ),

(3)

where k is the mean value, ξ(t) is a white noise with the intensity of σ. The approximation of white noise is, essentially, an idealization of the real noise with a small correlation time. Considering (2) and (3), equation (1) can be represented in the form dM = ( P – R – kM )dt + σMdW t ,

(4)

where Wt is a standard Wiener process (Brownian motion). Fluctuations of the variable P – R are small as compared with the reserves of the substance in the basin; therefore, they can be neglected.

0097-8078/05/3204-0410 © 2005 MAIK “Nauka /Interperiodica”

PROBABILISTIC REGULARITIES IN UNFAVORABLE HYDROCHEMICAL PHENOMENA

Following equations (2), let us introduce partially averaged rate of removal V˜ = kM, which, in contrast to V, does not include fluctuations of k, but still remains a random variable, since it depends on a random variable M. Let us pass in equation (4) from variable M to V˜ and next introduce the dimensionless variables kM V˜ X = ------------- = ------------- , P–R P–R

t* = kt,

σ σ = ---, k

where X will be considered as a dimensionless chemical flow. Let us change variables to transform equation (4) into the stochastic differential equation (SDE) (for the sake of brevity, we will write t instead of t*) d X t = ( 1 – X t )dt + σ X t dW t˙.

(5)

Here we have a nonlinear effect of interaction between the noise and the random process considered (a model of multiplicative noise). Two interpretations have been suggested for stochastic differential equations—according to Ito and Stratonovich [7]. When choosing an interpretation of SDE (5), one should take into account that this equation idealizes the real noise with a finite correlation time by replacing it with uncorrelated white noise. As shown in [7], stochastic integrals in this case must be interpreted according to Stratonovich. In this interpretation, Fokker–Plank equation for the probability density p(x, t) can be drawn from SDE (5) [7] ∂ 2 ∂p –1 –1 ∂ 2 ------ = – ------ ( 1 – x + α x ) p + α --------2 x p, α = -----. (6) 2 ∂x ∂t ∂x σ 2

The general properties of Fokker–Plank equations are discussed in detail in [6]. A stationary solution of equation (6) has the form α

α – α – 1 – α/ x p ( x ) = ------------ x e . Γ(α)

(7)

When x
α, we obtain p( x ) ∼ x

–α–1

(8)

.

Thus, the probabilities of the high values of chemical flow follow the power law. Considering that the concentration of a substance in river water is proportional to the chemical flow, the asymptotic relationship (8) should hold for concentrations as well. This means that the probability density of unfavorable hydrochemical events (high concentrations) can be described by a power law. This theoretical result will be checked below against empirical data. WATER RESOURCES

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WATER RUNOFF In [4], SDE (5) was used to describe the probabilities of catastrophic floods. In that study, stochastic integrals were interpreted according to Ito [7], presumably because catastrophic floods are rare discrete events, while SDE (5) was obtained in this case as a continuous limit of long-term intervals. The density that was found as a result differs from equation (7) in that x–α – 1 is replaced by x–α – 2. It is of interest to try to extend equation (5) to higher values of water flow, which are not as large as those for catastrophic floods, but occur much more frequently. Such extension is possible, but the interpretation of SDE (5) should be changed from Ito to Stratonovich, since in Stratonovich interpretation, the passage to the limit to white noise starts from a real noise with a small correlation time, which in our case describes atmospheric precipitation. Thus, the result (7)–(8) obtained here can be applied not only to chemical flow but to water flow as well (river runoff), for which balance equation of type (1) is also true. In this case, variables in (1) means the following: M is water reserve in the basin, V, P, and R are river runoff, water supply with precipitation, and water losses (for evaporation, chemical and biological bounding, infiltration into aquifers from which water cannot return into the river, and economic withdrawal). As applied to river runoff, assumption (2) means that water runoff is proportional to water reserves in the river basin. This assumption differs from the hyperbolic relationship V ~ M/(M∗ – M) proposed in [3]; however, when water reserves are small relative to the critical level (M  M∗), the difference between the hyperbolic and linear (V ~ M) dependencies become negligible. Since catastrophic events, corresponding to near-critical water, are not considered here, this approximation appears acceptable. TREATMENT OF EMPIRICAL DATA The initial data used in calculations included time series of water flow in a water intake section (the Moskva River, the upper pool of Rublevo dam) and a number of water quality characteristics presented in [1]. The characteristics considered are as follows: water turbidity M, mg/l according to FTU; color index, CI, degree at Pt–Co scale; permanganate oxidability, PO, mg O/l; ammonia A, mg/l; and phytoplankton population, î, cell/ml. The probability densities of the variables examined, estimated from their time series, are given in Fig. 1). Plotted along the abscissa are the relative values x = X/Xav, where X is water flow in the river (Q, m3/s) or one of quality characteristics (M, CI, PO, A, Φ), Xav is the mean value of the respective variable (Table 1). Values along the ordinate axis represent the probability density p(x) of the variable considered. To unify values of dif-

412

DOLGONOSOV, KORCHAGIN p(x)

p(x) 2

(a)

(b)

2

1 1

0

0 2

(d)

(c) 2

1 1

0 4

0 (e)

(f)

0.8

2 0.4

0

2

x 0

2

x

Fig. 1. Probability density of (a) water flow in a river and (b–f) water quality characteristics: turbidity, color index, permanganate oxidability, ammonia, and phytoplankton, respectively here and in Fig. 2.

ferent type with the aim to compare their distributions, we used a dimensionless form of representation. The tails of distributions are shown separately in Fig. 2. The application of double logarithmic coordinates is convenient for rectifying the power approximation p ( x ) = Ax

–α–1

,

(9)

which was chosen in accordance with the asymptotic approximation (8). The statistical characteristics of distributions and the values of approximation parameters, obtained by the least-squares method, are shown in Table 1. The common features of the distributions are as follows: the main maximum of each distribution lies in the domain x < 1; the distributions have extended tails lying

in the domain x > 1; two pairs of correlated variables with similar distributions can be found; these are water flow–turbidity (the correlation coefficient r = 0.67) and color index–oxidability (r = 0.83). The probability density of water flow is bimodal (Fig. 1a), which is due to the regulation of releases through Rublevo dam, the upper reach of which contains water intake. The first peak lies in the neighborhood of zero and reflects the regime of water accumulation, and the second peak (x = 0.65) corresponds to normal drawdown of water level. The high asymmetry of the distribution (Table 1) suggests the existence of a long tail, which is shown separately in Fig. 2a. The points in double logarithmic coordinates are seen to allow a good approximation by a straight line, which WATER RESOURCES

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Table 1. Statistical characteristics of distributions (A and α are parameters of power approximation (9), R2 is the level of its reliability, dash means no data available) Mean

Skewness

A

α

R2

20.8 m3/s 4.20 mg/l 27.3 deg 5.90 mg O/l 0.161 mg/l 8900 cell/ml

12.1 5.98 1.89 1.64 1.91 2.50

0.454 0.486 1.70 2.44 1.34 –

1.62 1.78 4.04 5.81 2.84 –

1.00 0.99 0.99 0.96 0.99 –

Characteristic Flow Turbidity Color index Oxidability Ammonia Phytoplankton

corresponds to the power dependence (9) with an exponent α = 1.62. The most probable value for turbidity corresponds to x = 0.45. The tail of the distribution is described by p(x) 1

the relationship (9) with an exponent α = 1.78. Figures 1b and 2b suggest certain similarity between the distributions of turbidity and flow, especially in their tail parts, which is confirmed by the closeness of their paramep(x) 1

(a)

0.1

0.1

0.01

0.01

0.001 1

10

1

0.001 1 1

(c)

0.1

0.1

0.01

0.01

0.001 1 1

10 (e)

(b)

0.001 1 1

10 (d)

10 (f)

0.1

0.1

0.01 0.01 0.001 0.001 1

10 x

0.0001 1

10

100 x

Fig. 2. Tails of probability densities for (a) water flow and (b–f) water quality characteristics. Markers are measured values, straight lines are power approximation (9), and the curve is an exponential approximation (10). WATER RESOURCES

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DOLGONOSOV, KORCHAGIN

ters. This demonstrates the interrelationship between these variables, the nature of which consists in the washing out of weathering products from the watershed surface and their supply to the river in the water mass. The character of this relationship (with respect to mean values) was studied in [2]. The distributions of the color index and oxidability have modes at x = 0.80–0.85 (Figs. 1c, 1d). The higher values of the exponents α = 4.04 and 5.81 demonstrate that the probability of large overshoots in the color index and turbidity is not as large as in the case of flow and turbidity (Fig. 2). A similarity in the distributions of these variables is due to their common nature: both variables reflect the presence in water of organic matter, mostly, terrigenous humus. The most probable value for ammonia is x = 0.7 (Fig. 1e). The main peak is wider than in other cases. In terms of decrease rates in the probability of large overshoots, ammonia occupies an intermediate position between the pairs of flow–turbidity and color index– oxidability, which corresponds to an intermediate value of the exponent α = 2.84 (Fig. 2e). The behavior of phytoplankton essentially differs from that of other water quality characteristics. Phytoplankton population is controlled not only by processes on the watershed that form the reserve of biogenic substances in the aquatic environment but also by the local factors, such as water temperature and water illumination. This radically changes the character of the distribution, both in its main and tail parts (Figs. 1f, 2f). The peak of phytoplankton population is very narrow and shifted to the domain of small values of x (the maximum at x = 0.06), and the tail extends to x = 12 (the domain x > 3 is not shown in Fig. 1f but it can be seen in Fig. 2f), which reflects a certain probability that a large population of cells appears as is the case in the period of phytoplankton blooming. Because of its specific features, the distribution of phytoplankton is described by exponential rather than power relationship p ( x ) = B exp ( – βx ),

(10)

where B = 0.307, α = 0.568; the reliability of the approximation R2 = 0.985. The analysis made above shows that the shape of the probability distribution depends on the genesis of the water quality characteristic examined. The power distribution adequately describes the behavior of the variables that form on the watershed and for which the stochastic equation (5) is valid. However, when local factors dominate, the character of the distribution changes. In the case of phytoplankton, for example, it changes into the exponential distribution. Phytoplankton is considered here as an example of alternative behavior, which cannot be described by equation (5).

PROBABILITIES OF UNFAVORABLE EVENTS Given the probability density, we can find the probability of the unfavorable event P{X > x}, which is taken to mean that random variable X exceeds threshold x. In the case of a sufficiently large x, the probability density has the power form (9) and the probability of the event becomes ∞

P{ X > x } =

∫ p ( x' ) dx' =

–1 –α

Aα x .

(11)

x

Note that the probability decreases with increasing x according to the power law P ~ x-α. By writing P{X > x} = q and using (11) we find the quantile xq for the random variable X, which corresponds to the probability q x q = Kq

– 1/α

(12)

,

where K = (A/α)1/α. The probability that the random variable exceeds xq is equal to q. Unlike other variables, phytoplankton has an exponential distribution. Using (10), we find –1 –β x

P { X > x } = Bβ e

,

(13)

x q = β ln [ B/ ( βq ) ]. –1

The dependence of quantiles on the exceedance probability for different variables (water flow and quality characteristics) estimated using equations (12) and (13), is plotted in Fig. 3 in double logarithmic coordinates. Clearly, extremely unfavorable events (corresponding to large values of quantiles) have a very low probability and correspond to small q values. For example, a 20-fold exceedance of the mean value of turbidity has a probability of 10–3. The exceedance probability can be interpreted as the proportion of time q within which the random variable exceeds the threshold defined by the quantile X > xq. When the period considered is a year, the duration of unfavorable (above-threshold) periods within this period will be equal to T = 365q day. Unfavorable events with different exceedance probability are characterized (in the absolute values of the variables) in Table 2. The results given above should be considered as mean annual. In specific implementations, deviations from these values in both directions are possible. The maximum values of the variables (mean daily values) recorded during the seven-year observational period (1997–2003 [1]), are as follows 3

Q = 748 m /s, M = 73 mg/l, CI = 86 deg, (14) PO = 17 mg é/l, A = 0.80 mg/l, î = 98 thousand cell/ml. Let us consider the lower row in Table 2, which describes the extremely unfavorable events corresponding to quantiles with the exceedance probability q = WATER RESOURCES

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0.001. Their comparison with the attained maximums shows that some of these quantiles have not been exceeded within the observational period, although this had to take place. This refers to the turbidity (the threshold is equal to 98, and the attained maximum is 73 mg/l), color index (120 and 86 deg), and ammonia (1.4 and 0.8 mg/l). Therefore, the power law is in fact an intermediate asymptotic approximation, and the distribution needs some type of truncation in the domain of very large values. This fact was also mentioned in [4] for the distribution of water flow in a river. General ideas regarding the limitations of the power law are formulated in [5], where a scaling function f(y) is introduced, which is approximately constant at y < ~1 and rapidly decreases at y ∞. With the use of this function, the power law (8) can be transformed to the form p( x ) ∼ x

–α–1

f ( x/x c ),

where xc is the characteristic scale of the variable, the exceedance of which results in a rapid drop of the probability density. This reflects the fact that events with a scale larger than xc have a low probability. The characteristic values of xc for water flow and various characteristics of water quality can be estimated from the absolute maximum (14) attained during the seven-year period of observations. With respect to the mean values of variables (Table 1), we have xc ≈ 36.0 (water flow), 17.4 (turbidity), 3.2 (color index), 2.9 (oxidability), 5.0 (ammonia). It can be seen that the flow and turbidity have extended intervals within which the power law is valid, whereas in the case of ammonia and, especially, for color index and oxidability, these intervals are narrow. It is of interest to consider the obtained results from the viewpoint of stable distributions [5, 6] for which it is characteristic that the sum of any number of random variables has the distribution, in a certain sense similar to that of each individual summand. Stable distributions are used in the probabilistic description of catastrophic events. Parameter α controls the asymptotic behavior of the distribution, as in the asymptotic dependence (8) and can take values in the interval of 0 < α ≤ 2. The boundary condition α = 2 corresponds to the normal distribution. The less α, the more probable large over-

415

xq 100

1

10

2 6 3 4 5

1 0.001

0.01

0.1 q

Fig. 3. Quantiles depending on the exceedance probability for the following characteristics: (1) water flow, (2) turbidity, (3) ammonia, (4) color index, (5) permanganate oxidability, (6) phytoplankton.

shoots of the random variable. This is especially important in the case where α ≤ 1, when the mathematical expectation is infinite. At α < 2, the distribution has an infinite variance. The results of processing time series (Table 1) show that the values of α for water flow and turbidity fall within the interval of 1 < α < 2; therefore, the respective distributions have a finite expectation but an infinite variance. The time series of these characteristics demonstrate catastrophic overshoots [1]. Ammonia, color index, and oxidability have α > 2; therefore, their distributions are not stable. The first two moments for these distributions are finite. Ammonia has no third moment, which corresponds to an infinite skewness. The value given in Table 1 is finite because of the small sample size, and the asymmetry will increase with increasing size. The color index has finite moments up to the fourth, and oxidability, to the fifth order inclusive. The behavior of these three indices does not have catastrophic character: their absolute maximums are only 3–5 times greater than the mean

Table 2. Conventional duration T of unfavorable periods within the year and quantiles with given exceedance probability q for water flow and different quality characteristics q

T, day/year

Qq , m3/s

Mq , mg/l

CIq , deg

ΠOq , mg O/l

Aq , mg/l

Φq, thousand cell/ml

0.1 0.05 0.01 0.005 0.001

36 18 3.6 1.8 0.4

39 60 160 250 670

7.4 11 27 40 98

39 46 69 82 120

7.6 8.5 11 13 17

0.28 0.35 0.63 0.80 1.4

26 37 62 73 98

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values (recall that the mean values of flow are exceeded by a factor of 36). When there is no catastrophicity, the only common feature of the power laws is the absence of characteristic scales of the processes. CONCLUSIONS The analysis is based on the balance equation of substances transferred from a watershed into a river. It was assumed that the flow of a substance and its reserves in the watershed are directly proportional. A random component that accounts for the effect of precipitation was isolated in the coefficient of proportionality. The balance equation was transformed into a stochastic differential equation with a multiplicative white noise. The obtained SDE was used to construct a Fokker– Plank equation for the probability density of chemical flow. The stationary solution of the Fokker–Plank equation is shown to contain a power component which becomes predominant at larger values of chemical flow. The proportionality of concentration and the chemical flow allowed us to conclude that the power law is applicable to the distribution of substance concentration. Statistical treatment of empirical data on some characteristics of water quality (in addition to water flow) show that the power law adequately describes the probability of occurrence of unfavorable hydrochemical events. The parameters of the power law for water turbidity, color index, permanganate oxidability, ammonia concentration, and water flow are evaluated. It is shown that the quality characteristics the formation of which is largely dependent on the processes proceeding within

the water body have a distribution other than power. In particular, an exponential distribution was obtained for phytoplankton. Quantiles of distributions with different exceedance probability levels are estimated. These quantiles are necessary for solving the prediction problems. REFERENCES 1. Dolgonosov, B.M., Vlasov, D.Yu., Dyatlov, D.V., et al., Statistical Characteristics of Variations in Waters Quality Supplied to a Water Station, Inzh. Ekologiya, 2004, no. 3, pp. 2–20. 2. Dolgonosov, B.M. and Korchagin, K.A., Statistical Assessment of Relationships between Water Flow in a River and Water Turbidity in Water Intakes, Vodn. Resur., 2005, vol. 32, no. 2, pp. 196–204 [Water Resour. (Engl. Transl.), vol. 32, no. 2, pp. 175–182]. 3. Naidenov, V.I. and Kozhevnikova, I.A., On the Power Law of Catastrophic Floods, Dokl. Akad. Nauk, 2002, vol. 386, no. 3, pp. 338–344 [Dokl. (Engl. Transl.), vol. 386, no. 3]. 4. Naidenov, V.I., Shveikina, V.I., and Vikhrova, M.A., Probabilistic Regularities in Catastrophic Floods, Meteorol. Gidrol., 2003, no. 6, pp. 81–95. 5. Upravlenie riskom: Risk. Ustoichivoe razvitie. Sinergetika (Risk Control: Risk. Sustainable Development. Synergetics), Moscow: Nauka, 2000. 6. Feller, V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya (Introduction to the Theory of Probabilities and Its Applications), Moscow: Mir, 1967. 7. Khorstkhemke, V. and Lefevr, R., Indutsirovannye shumom perekhody (Noise-Induced Transitions), Moscow: Mir, 1987.

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