On the Issue of Determining the Background in Hydrogeochemical Studies

P. A. Udodov, B. I. Shestakov

The interpretation of the results of field hydrogeochemical studies is still in the development stage. Various methods have been proposed for solving particular questions, but they sometimes yield results that are difficult to compare. Thus, for example, approaches differ to the solution of such fundamental questions as the determination of the hydrogeochemical background of natural waters, the identification of aqueous dispersion flows of mineralization zones, the determination of heavy metals in natural waters, and others.

At the present time, studies have already been carried out in a number of regions, and there is a need to correlate and generalize the data obtained. In doing so, however, difficulties arise owing to the use of different methods in field and laboratory work.

We believe that it is now already possible to raise the question of unifying certain hydrogeochemical methods. Therefore, for the time being we focus attention on two questions: 1) determination of the background of natural waters and 2) determination of aqueous dispersion flows of mineralization zones.

These questions, as is known, are of very great importance not only in geochemical studies, but also for the study of the geology and, especially, the hydrogeology of a region. Work experience shows that, when studying the conditions of formation of groundwater, it is always necessary to take into account the presence and content of microcomponents.

For the natural waters of mineralization zones of the mountain massifs of Western Siberia, a classification scheme has been developed, based on the content of heavy metals in the waters and their genetic associations [10].

The enrichment of natural waters with heavy metals occurs as a result of their interaction with minerals and rocks. To evaluate the concentration of metals in rocks, V. I. Vernadsky introduced the concept of the “clarke of concentrations.” Subsequently A. A. Saukov proposed the anomaly coefficient of an element within the limits of a dispersion halo, which represents the ratio of the concentration of this element within the halo \((C)\) to its concentration outside the halo, taken as the natural background \((N)\), i.e. \(K_{(\mathrm{an})}=\frac{C}{N}\) [8]. This coefficient can be applied in all geochemical methods of prospecting for mineral deposits.

For studying the content of heavy metals in natural waters, A. M. Ovchinnikov [6] proposed the modulus of chemical removal, which represents the ratio of the amount of micro-

components relative to its discharge. The enrichment of waters in mineralization areas leads to the formation of aqueous dispersion streams of mineralization zones, which usually have an elongated form in the direction of groundwater movement. They are characterized by the presence in the water of a definite complex of chemical elements, depending on the type of mineralization, with a sharply elevated content of some of them. In some areas several streams of mineralization zones are superimposed, as a result of which very complex aqueous streams sometimes form. An example of these may be the dispersion stream of ilmenite and polymetallic mineralization observed on the area of the Tom–Ya watershed [10].

In regions where there are no areas enriched in particular ore minerals, enrichment of groundwater with microcomponents occurs at the expense of the rocks. Under these conditions, waters are formed that create the regional background of the area. In some regions, owing to the distinctive character of the physical-geographical, geological, and hydrogeological conditions, as well as the geochemical setting, there are areas in which the content of individual trace elements differs from their content in the region as a whole.

We refer such areas to the local background. They are usually associated with mineralization zones.

Determination of the background content of elements in waters has been carried out in our practice by several methods. We shall now briefly describe the most frequently used of them.

1. Variation curves. Variation curves are constructed both for individual elements and for the sum of metals. From the results of concentrate analyses, graphs are compiled on which the logarithms of the contents of elements or of the sum of metals are plotted along the horizontal axis, and the corresponding percentage frequency along the vertical. The background was determined as the projection onto the horizontal axis of the point of sharp flattening of the curve. Points that lay beyond the background and adjacent to it were studied in detail with respect to the association of elements, quantitative composition, and their confinement to particular geological conditions. As a result of such study, some points were sometimes transferred from the dispersion stream to the background and vice versa.

2. Scatter diagrams. Scatter diagrams, like variation curves, are constructed both for individual elements and for the sum of metals.

The method consists in plotting, on a vertical strip 2–3 cm wide, with a scale of contents constructed on it, all results of analyses of individual elements or of the sum of metals in the form of points. In those places on the diagram where the most frequently occurring contents of elements or of the sum of metals are plotted, clusters are formed; and where contents of lower frequency are plotted, the points are more sparsely located. The background line on such diagrams is drawn above the last cluster. Points extending beyond the boundary of the background are analyzed in the same way as in the case of the variation curve.

Both variation curves and scatter diagrams suffer from subjectivity. In variation curves, the point of sharp flattening is chosen subjectively; in scatter diagrams, the line above the last cluster is chosen subjectively.

Sometimes the arithmetic mean content of an element in all samples was also taken as the background; this method is entirely non-objective, since inclusion in the calculation of samples with high contents immediately distorts the picture. Thus, for example, if there are one hundred samples with a content

in 1 µg/l and 1 sample with a content of 1000 µg/l, the background will undoubtedly be 1 µg/l; the arithmetic mean, however, gives a result of 10 µg/l.

There is also a method of determining the background in which each sample is represented by a point at a height equal, on the scale, to the magnitude of the content of the sum of metals, while around the point the complex of individual elements and their contents (by the length of the rays) is shown by rays. In assigning points to the background or to the dispersion halo, both the content of individual elements and the sums of metals were taken into account.

Somewhat different from these methods is the profile method. It consists in the following: along a certain line through an assumed area with an increased content of microelements in the waters, a graph is constructed on which the content of elements is plotted vertically and, horizontally, the distance between sampling points. The highest points of the graph are considered to belong to the dispersion halos of mineralization zones.

After considering all the previously used methods for determining the background, we came to the conclusion that it is necessary, for processing the results of hydrogeochemical investigations, to apply methods of mathematical statistics.

The content of an element in waters is a random variable, since it is affected by a multitude of factors that cannot be taken into account. There is an assumption that the contents of heavy metals in waters obey a logarithmic-normal distribution law. The essence of this law is that it is not the quantities themselves but their logarithms that are normally distributed; i.e., the integral distribution function of the random variable can be expressed by the formula:

\[ \Phi_{(z)}=\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{+\infty} e^{-\frac{(z-\bar{z})^2}{2\sigma^2}}\,dz, \]

where \(\sigma\) is the root-mean-square deviation,

\(z\) is the logarithm of the content of elements in waters [7].

This assumption was tested according to the method proposed by I. V. Dunin-Barkovskii and N. V. Smirnov [4], which has already been applied in geochemistry [3]. In order to speed up the analysis, this method was somewhat simplified. The simplification consisted in the fact that the hypothesis of lognormality was checked graphically. A rectified diagram of the element distribution was constructed. On the vertical axis, on the functional scale of the normal distribution law (for functional scales see, for example, D. L. Havra [2]), the accumulated fractions in % were plotted; on the horizontal axis, the midpoints of the classes of the logarithms of the contents. The class width was taken as 0.4 in logarithm. A distribution obeying the lognormal law will be expressed on such a graph by a straight line. Through the points obtained, a rectifying line was drawn in such a way that it nowhere went outside the 5% confidence interval. If the drawing of such a line proved possible, then we considered that the contents of the element approximately obey the lognormal law with the distribution parameters of the rectifying line (Fig. 1).

This assertion is not very rigorous, because we used spectral analysis. True, our distributions included from 300 to 1000 analyses, which should ensure a fairly good approximation to the distribution law. In addition, we adopted the requirement that the distribution be divided into classes of width no less than half the gradation of the spectral analysis, which also reduces errors. With a large number of such classes (more than 5), distributions consisting of 100 analyses were sometimes used.

As already indicated, in the case of a good approximation of the practically obtained distribution to the lognormal law, it was replaced by a theoretical distribution, whose graph is expressed by an averaging straight line in the above-mentioned coordinates (Fig. 1). This eliminates random deviations of the empirical distribution. The distribution parameters were determined as follows: the mathematical expectation was determined as the projection of the intersection of the averaging straight line with the horizontal line of 50% onto the horizontal axis, and the standard deviation was determined by the distance from the projection of the intersection of the averaging straight line with the 50% line onto the horizontal axis to the projection of the intersection of the straight line with the 84.1% line \[4\] onto the same axis (Fig. 1).

Work on testing the hypothesis of lognormality and determining the distribution parameters was carried out on tracing paper laid over a template-grid. On the grid, cumulative frequencies are plotted vertically on the functional scale of the normal distribution, and horizontally—the midpoints of the classes of the logarithms of element contents.

Fig. 1. Straightened diagram of the overall distribution of titanium and test of distribution lognormality:
a—straightening line; b—5% confidence limit; v—diagram of the variation of the 5% confidence limits; g—84.1% line; z—mathematical expectation; \(\sigma z\)—standard deviation.

The application of the lognormal distribution law of microcomponents in natural waters allows us to approach the method for determining the background in the following way. In the course of studying the distributions of various microcomponents in waters over many areas of Siberia, it has been established that, in the absence within the area under study of sectors with an increased content of microelements in the waters, there is an approximate equality \(\sigma = \Delta\), where \(\sigma\) is the root-mean-square deviation of the distribution of the logarithms of microelements in the waters, and \(\Delta\) is the logarithmic root-mean-square error of the analysis. The approximate value of \(\Delta\) is determined from the following considerations. Semiquantitative spectral analysis of concentrates gives the content of microelements within the following limits: 0.0001—0.0003; 0.0003—0.001; 0.001—0.003; 0.003—0.01; 0.01—0.03; 0.03—0.1; 0.1—0.3; 0.3—1 and more than 1%. It is natural to assume that the mean error in each gradation will tend toward half of this gradation. Taking the logarithms within the indicated limits, it is not difficult to see that the width of each gradation in logarithm is approximately equal to 0.5. Therefore, the mean error in logarithm will tend toward half the width of the gradation in logarithm, i.e., to \(0.50 : 2 = 0.25\). We roughly take this error as the logarithmic root-mean-square error of the analysis \(\Delta\). On the basis of the law of error distribution, it may be assumed that the maximum error of analysis cannot be greater than three times the root-mean-square error, i.e., greater than \(3\Delta \approx 3 \times 0.25 = 0.75\).

When, in the area of investigation, there are water flows of dispersion of mineralization zones, the picture changes sharply. The approximate equality \(\sigma = \Delta\) is violated, and the inequality \(\sigma > 0\) obtains. In this case, beyond the limits \(\bar z + 3\Delta\) there will fall all the more points, the more of them are taken in the zone of waters of the dispersion flow, since \(\bar z + 3\sigma > \bar z + 3\Delta\). This can be explained as follows. From the fundamental law of geochemical processes [3, 11] it is evident that, in the absence of high concentrations of elements, the root-mean-square deviation of the distribution of microcomponents will be comparatively small. It is entirely possible that in this case the root-mean-square deviation of the analysis will be greater than the root-mean-square deviation of the distribution of microelements in the waters. Then we record not the distribution of the microelements in the waters itself, but the distribution of errors of the spectral analysis of the concentrates. Thus, the distribution of errors of analysis, as it were, masks the distribution of microelements in the waters. This also creates the impression of equality between the root-mean-square error of analysis and the root-mean-square deviation of the contents of microelements in the waters. In the case where, within the area under study, there are water flows of dispersion of mineralization zones, the deviations of the distribution of microelements in the waters are considerably greater than the deviations resulting from errors of analysis, and therefore the actual picture of the distributions of microelements in the waters is recorded. If we superimpose on this distribution the distribution of errors of the spectral analysis of concentrates, then outside its limits there will remain precisely those points that enter into the dispersion flow of mineralization zones.

On the basis of the foregoing, we propose to take as the background value the value of the sum of the mathematical expectation in the distribution of microelements in the waters with three times the root-mean-square error of analysis, i.e., \(\bar z + 3\Delta\). A similar approach to the determination of the background has already been applied in geophysical investigations [5] and in metallometry [9]. The method for determining the mathematical expectation was described above. By adding \(\bar z\) and \(3\Delta\), we obtain the logarithm of the background, from which the background is then found. Taking into account that the root-mean-square error of analysis has been calculated inaccurately, it is necessary to exclude points lying close to the background (above and below it), ...

to take separately and analyze those individual geological, hydrogeological, and hydrogeochemical conditions in which the indicated points are found. On the basis of studying these conditions, the points are assigned either to the background or to the dispersion halo.

Let us consider the determination of the background under different geological conditions. An example of an area with no mineralization zones is the Sarakoksha intrusion in the Gorny Altai. Here, in 1954, A. G. Turov carried out geological investigations. On the basis of 87 analyses of concentrates, the distribution parameters for copper proved to be as follows: \(\bar z = -0.6\), \(\sigma = 0.25 \simeq \Delta\). In this case the background line coincided with the line \(\bar z + 3\sigma\), so that not a single point fell outside the background.

Approximately the same standard deviation is also found here for the remaining elements. Geological work carried out earlier in the area of the Sarakoksha intrusion likewise showed the complete lack of prospects of the intrusion area with respect to the presence, within its boundaries, of deposits of ore minerals. An example of an area with dispersion halos of mineralization zones is one of the sectors of the Tom–Yaya interfluve. The distribution parameters of microcomponents in the waters of this area are given in Table 1. It is evident from it that the standard deviations of the distributions of microcomponents vary within the limits 0.5–1.3. Therefore 15–20% of the points fall outside the value \(\bar z + 3\Delta\), since the waters of dispersion halos of mineralization zones were sampled here in very great detail.

As is known, the composition of background waters is formed mainly at the expense of the rocks of the area. The content of microelements in rocks varies within fairly broad limits. Therefore it may be recommended that the background content of waters be determined separately for different water-bearing lithological varieties.

After determining the background, we proceed to the identification of aqueous dispersion halos of mineralization zones. Points at which the content of microcomponents is above the background are carefully analyzed with respect to the complex of microcomponents in the waters, the presence of the sulfate ion, a lowered pH value, etc., as well as geological and hydrogeological conditions. Such an analysis makes it possible to refine and substantiate the dispersion halos of mineralization zones.

It should be noted that many previously used methods are a reflection of the logarithmic-normal distribution law. Thus, both the variation curve and the dispersion graphs are a crude representation of the graph of the differential function of the logarithmic-normal distribution law. The arithmetic mean is an attempt at an analytical determination of one of the parameters of the distribution of the element content—the mathematical expectation—and in itself entails the assumption of a normal law of distribution of elements in waters, and therefore is not applicable as a background.

For a more detailed study of the dispersion halos identified in this way, a method is proposed based on comparing the distribution parameters of the dispersion halos of mineralization zones with the distribution parameters for the area as a whole. The mathematical expectations and standard deviations are compared pairwise. That the mathematical expectations in the dispersion halo of mineralization zones are higher than the mathematical expectations for the area as a whole is quite obvious. As for the standard deviations, their comparison is based on the basic law of geochemical processes [3, 11], which states that geochemical processes leading to the accumulation of low concentrations of elements are more stable than processes leading to the accumulation of high concentrations, since the measure of the stability of a process

Table 1

Comparison of the distribution parameters of microcomponents in waters

is the standard deviation of the distribution of element contents. Therefore, we compiled the following distributions of microelements in waters: (a) general distributions for the entire study area, (b) distributions for each lithological variety of rocks, and (c) distributions by dispersion streams of mineralization zones. The parameters of these distributions were compared with one another. An increase in the standard deviation of the distribution in dispersion streams, as compared with the general distribution for the area and with that for the lithological variety in which the dispersion stream is located, indicates the possibility that, in the area of the stream, processes occurred that could have led to the accumulation of elevated and, in particular, ore concentrations of elements; a decrease or equality of the standard deviations rules out this possibility.

As an example of such processing, one may cite the materials obtained from a hydrogeochemical survey of one of the sectors located on the Tom–Yaya watershed. Geological mapping was carried out here, together with hydrogeochemical studies. In addition, the area was studied geophysically. Lithologically, the area is clearly divided into two zones. One of them is represented by Devonian sandstones and shales (the Pachinskaya and Yurginskaya suites); the other, by effusive–tuffaceous rocks (tuffs, porphyrites, diabases), also of Devonian age (the Mitrofanovskaya suite). Each of the zones occupies approximately half of the area described.

Within the Mitrofanovskaya suite, as a result of integrated work, one sector was identified. Within it there is increased tectonic stress, hydrothermally altered rocks, and also signs of an iron cap and fragments of rocks with intensified pyritization have been found. According to hydrogeochemical data, an increase in a complex of elements in the waters is noted here, as well as an increase in their contents, a decrease in pH, and locally the sulfate ion is present. For each element, three distributions were constructed—one general for the entire area, one for the Mitrofanovskaya suite, and one for the identified sector. The distribution parameters are summarized in Table No. 1, from which the following is evident. The elements Cu, Zn, Ni, Ag, Pb, and Ba increased their standard deviations and mathematical expectations in the sector. This indicates that, in the area of the sector, processes of accumulation of high concentrations of these elements were taking place (as compared with the entire study area).

The elements V and Cr slightly increased their mathematical expectations of content, but their standard deviations decreased. This means that the processes were more stable; they do not reflect a general increase in the concentration of these elements in the rocks. The remaining elements (Tl, Mn, Sr) reduced their parameters or did not change them; therefore one may assume the absence of their elevated concentrations.

The stated assumption concerning the presence of elevated contents of Cu, Zn, Ni, Ba, and Pb was recently confirmed by drilling. In one of the boreholes drilled in the area of the sector at a depth of 180 m, sphalerite crystals were encountered; another borehole at a depth of 70 m penetrated a zone of pyritization. Spectral analysis of the pyritized rocks gave the following results: Zn—up to 1%, Ba > 1%, Ni—up to 0.3, Cu—up to 0.3%, Pb—up to 0.1%.

It is very important to note the fact that the standard deviations increased the distributions only of those elements that really do have elevated contents in the rocks.

Thus, on the basis of the interpretation described, it was possible not only to establish reliably the presence of a mineralization zone in the rocks, but also to infer with sufficient accuracy the actual association of microelements in this mineralization zone. In addition, by comparing betw—

the distribution parameters of the various zones of mineralization, it is possible to establish their relative value. However, application of this method is possible only when the number of sampling points in the area of elevated microelement contents in the waters exceeds 30. In this connection, determination of microcomponents directly in the field becomes of great importance and, in particular, determination of the sum of metals by dithizone. If a sharp increase in the sum of metals is observed, several additional concentrates must immediately be collected in the area where it is detected. Similar detailing must be carried out when such indications appear as zones of hydrothermally altered rocks, increased tectonic activity, contacts of intrusions, lowered pH (if this is not connected with swamps), increased sulfate-ion content (in the absence of continental salinization), discharge from zones of tectonic disturbances, etc. This cannot greatly increase the cost of the survey, but in interpretation it will make it possible to assess the reliability of the area of elevated contents. If a mineralization zone is present, it will also be possible to infer its qualitative composition.

Despite this approach, in practice it very often happens that areas of elevated contents are represented by a number of points less than 30. In this case the reliability of assessing the credibility of the area of elevated contents drops sharply. Nevertheless, even here the methods of modern statistics help to draw certain conclusions. Here it is necessary to use the theory of small samples. In this case the question is likewise posed of the difference between mathematical expectations and root-mean-square deviations. If the conditions \(\bar z_1 > \bar z\) and \(\sigma_1 > \sigma\) are fulfilled (where \(\bar z\) and \(\sigma_1\) are the distribution parameters of the area with elevated content, and \(\bar z\) and \(\sigma\) are the distribution parameters for the district as a whole), the presence of an area with elevated mineralization is considered reliable; otherwise it is considered unreliable. Calculation of the distribution parameters for the district as a whole is carried out in the same way as usual—on a probability grid, while it is advisable to calculate the distribution parameters of the area by the formulas

\[ z_1=\frac{\sum_1^n z_n}{n_1}; \qquad \sigma_1=\sqrt{\frac{\sum_1^n (z_n-z_1)^2}{n_1-1}}; \]

where \(z_n\) are the logarithms of the contents of elements in the waters of the area of elevated contents,

\(n\) is the number of points.

It must be said that, in our case, resolving the question of the inequality of mathematical expectations with a small number of points in the area is practically impossible, since Student’s criterion, which exists for this purpose, gives such wide limits for the possible value of the mathematical expectation that comparison with the mathematical expectation of the combined distribution becomes impossible. The situation is different with the root-mean-square deviation. R. Fisher established the criterion \(F\) for resolving the question of the inequality of root-mean-square deviations. The probability that \(\sigma_1>\sigma\) is determined as follows [4]:

the ratio

\[ \frac{\sigma_1^2}{\sigma^2} \]

is calculated, and the degrees of freedom \(K_1=n-1\) and \(K=n-1\) are determined, where \(n_1\) and \(n\) are the number of observations from which \(\sigma_1\) and \(\sigma\) were calculated. From these data, in the tables of the \(F\) criterion, the probability corresponding to the probability of the condition \(\sigma_1>\sigma\) is taken.

It is considered that if this probability is above 5%, then the discrepancy between \(\sigma_1\) and \(\sigma\) should be regarded as significant, and if the probability is below 5%, then

one should assume that practically \(\sigma_1=\sigma\). Statistical handbooks usually give the 1% and 5% values of the ratio \(\frac{\sigma_1^2}{\sigma^2}\).

In the case where the practically obtained ratio is greater than the 5% value of \(\frac{\sigma_1^2}{\sigma^2}\), it may be assumed that \(\sigma_1>\sigma\). The conclusion is the more reliable, the greater the number of points and the greater the difference between the 5% ratio \(\frac{\sigma_1^2}{\sigma^2}\) and the observed ratio.

Sometimes more complete tables of the \(F\) criterion are given; in such a case the probability of the proposition \(\sigma_1>\sigma\) can be found more accurately. Let us give an example. Suppose that we have a combined distribution of zinc in rocks based on the results of 201 analyses with parameters \(\bar z=+1.5\) and \(\sigma=0.6\). There are 15 points in the sector. The parameters calculated by the formulas are \(\bar z_1=+1.9\) and \(\sigma_1=1.5\). Then \(K_1=15-1=14\), \(K=201-1=200\).

\[ \frac{\sigma_1^2}{\sigma^2}=\frac{1.25}{0.36}=3.55 \]

In the table of the \(F\) criterion, for \(K_1=14\) and \(K=200\), we find the value of \(\frac{\sigma_1^2}{\sigma^2}\) corresponding to 5% probability. The observed value is \(3.55>3.06\); consequently, we must assume that \(\sigma_1>\sigma\). In doing so it is necessary to take into account that the reliability of this conclusion is the greater, the more points there are in the sector.

The described method, owing to its sufficient simplicity and speed of execution (calculation of the parameters of one distribution for 300–500 analyses takes 25–30 min with the use of a grid), should find application in the practice of interpreting the results of hydrogeochemical investigations.

The proposed method should be applied with consideration of all geological and hydrogeological conditions. Along with it, at the first stage the comparison method \([1, 10]\) gives quite satisfactory results, and at subsequent stages, when there are deposits analogous to the one under study, the method of a known deposit \([1, 10]\). There also exist many other methods and procedures, which the authors do not dwell on, but which under certain conditions give very valuable results.

The revealed lognormal law of distribution of microcomponents in waters cannot be transferred unconditionally to any district. Under other metallogenic conditions, for example, in the complete absence of mineralization zones, significant deviations from the lognormal law of distribution are possible. In particular, it may be assumed that in districts where mineralization zones are absent and where there are no conditions promoting an increase in the contents of microelements in waters (swamps, etc.), a distribution law close to the normal one is highly probable.

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Noted Misprints