Let us introduce some designations. Let X be a factor of

environment and Y be an estimate of ecological state. We shall

classify an observation as ecologically normal if X is not

greater than some value E (which ultimately is to be determined)

called ecologically tolerable level (ETL) and as ecologically

abnormal if X is greater than E. In doing so we define the

observation as ecologically normal if Y is not greater than a

given value F and as abnormal otherwise. Let as also designate:

a is the number of observations for which Y , F and X , E;

b is the number of observations for which Y , F and X > E;

c is the number of observations for which Y > F and X , E;

d is the number of observations for which Y > F and X > E.

For simplicity, in the examples that we consider we suppose namely such a situation i.e. that we classify as abnormal the observations whose factor values are to the right from ETL. However the situation can be more complicated. For example, in the case of such a factor as concentration of oxygen diluted in water, unfavorable values of the factor are lower ones. In general, the boundary of the region of ecologically tolerable values of a factor is two-sided and is defined by two boundary values E1 and E2 called lower and upper ETL. Respectively, the quantities a,b,c, and d are defined by the conditions:

a is the number of observations for which Y,F and E1,X, E2;

b is the number of observations for which Y,F and X<E1 or

c is the number of observations for which Y>F and E1,X,E2;

d is the number of observations for which Y>F and X<E1 or

Using these designations we can define the exactness of the assertion "if X<E1 or X>E2 then Y>F" by the formula

boundaries) is our main purpose in searching for dependence of Y

upon X for a given F. We define as optimal for a given lower

boundary of exactness T that value of E for which the

completeness P is maximum.

Such an analysis which could be called "one-factor analysis" can be carried on for each factor. As an example, Fig. 3 presents the results obtained when determining the influence upon the ecological state of the factor "nitrogen concentration", one of 67 factors for which such an analysis was performed on the base of the monitoring data for Don water basins in 1975-91. The scale of estimate of ecological state is from 1 (completely normal state) to 5 (state of regress) with precision up to 0.5 (Abakumov 1991; Geletin et al. 1991). The value F=2.75 was taken as boundary value between norm and pathology and the value T=75% as lower boundary of the exactness. The maximum value of the completeness in this case

was attained for the concentration of nitrogen equal to E=0.986

mg l-1 which is so the ecologically tolerable level that was sought.

If there are several factors then their one-factor ETL may be pooled. We shall classify an observation as abnormal if at least one of the factors considered X1, ..., Xm exceeds its ETL. Correspondingly, the values of a, b, c, and d are redefined as follows:

a is the number of observations for which Y , F and X , E

b is the number of observations for which Y , F and X > E

c is the number of observations for which Y > F and X , E

d is the number of observations for which Y > F and X > E

After such redefining the formulas for T and P remain the same. The "pooled" value of the completeness P cannot be less than its one-factor values but the pooled exactness may become less than its lower boundary.

It is worthwhile to note that the expressions "for all Xi" and "at least for one Xi" mean "for all Xi having values for this observation" and "at least for one Xi having values for this observation". This permits to augment the number of observations used in the analysis that it is very important for data having many missing values.

As an alternative to the procedure of pooling, a purely multifactor procedure may be used. It consists in determining a joint set of ETL maximizing the joint completeness under condition of verifying a given constraint on minimum admissible exactness. One of advantages of this procedure is a possibility of obtaining more optimal ETL.

Unfortunately, the purely multifactor approach demands too much time for calculations even for the number of factors greater than two or three. Because of this we propose a "combined" method which proceeds as follows. Initially, by a purely two-factor method, we find the pair of factors providing the maximum completeness and verifying the constraint for the exactness (to be not less than the given lower boundary). After this, we try to add to this pair each of remaining factors calculating for each of them the optimal three-factor ETL and the corresponding three-factor completeness (these are not, however, purely three-factors parameters because the ETL found for the initial pair are not recalculated). Ultimately, we take as third factor that one which provides the maximum increase of completeness when it is joined to the initial pair of factors.

Analogously, the triplet of factors can be enlarged to a set of four factors and so on. In comparison with the procedure of pooling, this procedure guarantees verifying the constraint on minimal admissible exactness and can provide more optimal ETL (though, possibly, less optimal than for purely multifactor procedure).

The procedure of pooling described earlier could provide for 7 factors the value of P equal to 62% but only for the exactness equal to 70% whereas the combined procedure provides the exactness 75% though only for 59% of completeness.

An disadvantage of the combined procedure is its inability to work with data tables having many missing values because it demands, like the purely multifactor procedure, the presence of the values of all factors for all observations taken into account. As a more practical a procedure which could be called "stepwise" is proposed. It resembles to the combined procedure but differs from it by using only one-factor ETL. At each new step the factor providing the maximum increase of completeness is seeking but its one-factor ETL is not recalculated. At the last step we obtain the pooled completeness and exactness. The constraint on exactness again can be not fulfilled.

If there are many missing values then it is more reasonable to modify the criterion of including factors in the stepwise procedure. In place of the early defined completeness we recommend to use the next characteristic:

(N2 is the number of all observations for which Y>F) that can be

called "absolute completeness". This permits to prevent

including, especially at the first steps, those factors which

gives a great increase of P but only for a limited number of

observation.

One of results of stepwise procedure is ranking of factors in the order of their influence upon the state of the ecosystem. Another one is information about interdependences between factors. For example, if a group of factors have high values of completeness at some step but after including of one of them at this step each of remaining factors gives only a small increase of completeness at the next step then this means that their influences upon Y are correlated (this is analogous to the partial correlation analysis for quantitative variables). Information of such type is important for purposes of control: we must take into account a possibility that a factor estimated by our procedure as important may be really only correlated with another acting factor.

We already noted that the stepwise procedure, in contrast to purely multifactor and combined ones, does not guarantee fulfillment of exactness constraint. To compensate this disadvantage we propose to calculate the one-factor ETL with a more high value of the minimum admissible boundary of exactness. For example, we took as one-factor exactness boundary equal to 80% in hope to provide not least than 75% for the pooled exactness.

In addition to obtaining an estimate for ETL it is useful to assess statistical reliability of this estimate. We used for this a method of statistical simulation called "bootstrap". It proceeds as follows k samples of data of the same dimension m+1 (m of factors plus one estimate) and of the same size N are generated by random drawing observations from the original data. After this, the algorithm which was used for finding E, T, and P is applied to each of simulated data tables. As a result we obtain k sets of values for ETL, exactness, and completeness: E1, ..., Ek; Ti, ...,Tk; Pi, ...,Pk. Then we calculate for these sets their means Em, Tm, Pm, their standard deviations Es, Ts, Ps, and their coefficients of variation

The number of samples k is a compromise between the reliability of the method and the size of computation. We took k=10 and this was rather sufficient taking into account great variability of data. The coefficients of variation of ETL, exactness and completeness can be used as additional characteristics for choosing the factors the most significant from the point of view of their influence upon the ecological state (Tables 3, 4, 5).

The significant factors can be ranked in accordance with their completenesses P. This criterion may be interpreted as follows: if grace of environment defence measures the value of a factor with completeness P enter to the region confined by its ETL then P% of abnormal states will become normal (on condition that all other causes of abnormality are also eliminated).

The ETL method requires not only data for values of fac- tors but also ones for ecological estimates. In accordance with the biotic conception of environment control (Levich, 1994) the state estimates must be based upon biotic characte- ristics. The estimates used in the examples in this paper are obtained by so called method of ecological modifications and are based on such characteristics as number of species and their abundances, indices of saprobity for phytoplankton, bacterioplankton, periphyton, zoobenthos. State estimates may be also based upon another characteristics recognized adequa- te to the purposes of the study, for example, upon ichthyolo- gical characteristics of water basin (Bulgakov et al.,in press ), upon hygienic characteristics of an environment, upon physiological characteristics of indicator organisms, and so on.

We recommend to apply the principle of maximum severity when choosing between ETL corresponding to different types of state estimates of to different time lags i.e. to choose the minimum value for all upper ETL and the maximum value for all lower ETL. The user can propose some other criterion, for example, he can declare one of state estimate as the most important and then the ETL for this estimate must be preferred.

It is worth to note that when using the conception of limiting admissible concentration different norms for different purposes (medicine, fishery) are also applied.

Not only instant values of abiotic factors may be used for analysis but also their average for a year, for a month or any other average value as well their minimum and maximum values, any value lagged in time and so on. As an example in Fig. 5 a chronogram of month averages of pH for the ecological estimate based on zoobentos characteristics for lower Don is presented.

We noted already that including of a new factor in the stepwise procedure depends not only upon its own importance but also upon its correlations with other factors. We call "essentiality" of a factor with relation to a set of another factors the increase of pooled completeness when adding this factor to this set. When choosing environment defence measures we must take into account the essentiality of a factor in parallel with some others ones, for example, such as its admissibility for regulation and social impacts.

The obtained results can also be used to estimate any separate observation related to a particular factor, location and time point from the point of view of its unfavourability for ecosystem state. It is convenient for this to express the value of the factor under consideration as ratio to its ETL.

In conclusion we must note that, of course, the results obtained depend essentially upon particularities of the data used for the analysis. It may happen that the factors really influencing upon the ecosystem state are not presented among the factors we analyze and then the monitoring program must be completed. Some of factors included in the analysis must not be considered as causes of ecological disturbances being themselves influenced by some other factors. For example, indices of chemical and biological oxidation, pH, diluted oxygen are partially of this type. So our formal analysis should always be accompanied by additional expert analysis.