ON THE STATISTICAL PROCESSING OF INDIVIDUAL INDICATORS OF THE ENGINEERING-GEOLOGICAL PROPERTIES OF HETEROGENEOUS SOILS

E. A. PISAREV

(Presented by the Council of the Research Institute of Hydrogeology and Engineering Geology)

It is well known that the greater part of the calculation schemes for assessing the stability of various structures has been developed with reference to isotropic soil masses. Nevertheless, a soil mass comparatively rarely can be regarded as homogeneous, and it usually has a layered structure. In the case when the properties of the soil layers differ, the calculation schemes used for assessing the stability of structures in isotropic soils are not unconditionally applicable and require the introduction of a corresponding correction, which in practice reduces to replacing heterogeneous layers by a single soil stratum equivalent in its properties. Usually, when assessing the stability of foundations of structures [5] or calculating the stability of slopes [6] in layered soils, the noted replacement, as a rule, reduces to the well-known operation of computing weighted-average values of indicators of the engineering-geological properties of the soil stratum using the formula:

\[ \bar{x}_{\text{w.av}} = \frac{h_1 \cdot \bar{x}_1 + h_2 \cdot \bar{x}_2 + \ldots + h_k \cdot \bar{x}_k} {\sum_{i=1}^{k} h_i}, \]

where \(\bar{x}_{\text{w.av}}\) is the weighted-average value of the indicator;
\(\bar{x}_i\) is the individual mean value of the indicator of the \(i\)-th layer;
\(h_i\) is an element of the soil mass (the thickness of the \(i\)-th layer; the length of the \(i\)-th segment of the slip line, the area of the epure of certain stresses in the \(i\)-th soil layer, etc.);
\(k\) is the number of layers.

The noted formula is applied only in the case when the characteristics of the soil properties for each of the \(k\) layers are determined in one and the same volume \(n_i\), and, moreover, no substantial divergence is observed in their random scatter, which is encountered rather rarely in the practice of engineering-geological investigations. More often the opposite case occurs, when the individual characteristics of one or another property are obtained for different soil layers in unequal volumes and with different “accuracy.” In this case, the “unequal accuracy” of the observations under consideration is caused not only by different analytical errors of measurement and by their unequal number, but also by qualitative differences in the soil layers. Below, for this case, a method of stati-

statistical processing of engineering-geological indicators of soil properties, which, as is known, is an important element in the process of obtaining design parameters necessary in assessing the stability of various structures [2, 4].

Let there be a \(k\)-layered soil mass. Within each relatively homogeneous layer, a given indicator of a soil property is distributed, with a specified degree of reliability, according to the normal law with mean value \(x_i\) and variance \(S_i^2\). In addition, the condition is observed that the particular values of this characteristic are not related to the parameters of space. The “equal accuracy” of the observations under consideration in this case can be assessed by comparing the variances \(S_i^2\). In a two-layer structure of the soil mass, the variances are compared by means of the well-known \(F\)-criterion of R. Fisher:

\[ F=\frac{S_{\max}^{2}}{S_{\min}^{2}}, \tag{1} \]

where \(S_{\max}^{2}\) is the maximum value of the variance;
\(S_{\min}^{2}\) is the minimum value of the variance.

The computed values of \(F\) must be compared with the tabular \(F_{\gamma,\,N-1}\) [1], where \(N\) is the number of determinations, and \(\gamma\) is the significance level. If \(F<F_{\gamma,\,N-1}\), the hypothesis of homogeneity of the two variances may be considered, at the adopted significance level \(\gamma\), sufficiently plausible [3]. Otherwise the variances are not equivalent, and it is already quite superfluous to speak of the “equal accuracy” of the observations. When \(k>2\), it becomes necessary to assess the “unequal accuracy” of any indicators by comparing several variances \(S_1^2, S_2^2, \ldots, S_k^2\), obtained for each layer from \(n_i\) determinations. First we compute the weighted mean variance:

\[ \bar{S}^{2}=\frac{\sum_{i=1}^{k} n_i S_i^{2}}{N}, \tag{2} \]

where \(N=\sum_{i=1}^{k} n_i\), and we find the statistical criterion

\[ B=\frac{2.303}{C}\cdot\left(N\cdot\lg \bar{S}^{2}-\sum_{i=1}^{k} n_i\cdot\lg S_i^{2}\right), \tag{3} \]

where

\[ C=1+\frac{1}{3(k-1)}\cdot\left(\sum_{i=1}^{k}\frac{1}{n_i}-\frac{1}{N}\right). \tag{4} \]

In [3] it is stipulated that the quantity \(B\) is distributed approximately as \(\chi^2\) with \(k-1\) degrees of freedom, if all \(n_i>2\). When the computed value \(B\) exceeds the tabular value \(\chi_{\gamma,\,k-1}^{2}\) at the chosen significance level \(\gamma\), we reject the hypothesis of homogeneity of the variances and consider that our particular characteristics, obtained for the different layers, are “of unequal accuracy.”

When the “unequal accuracy” of the indicators has been proved, the task consists in reducing all results to a comparable form. This will subsequently make it possible to obtain the necessary conclusions about the properties of the entire heterogeneous layered soil mass, or, more precisely, will make it possible to deter-

to determine reliable generalized and design characteristics of the physicomechanical properties of soils. Using the method of least squares, one can obtain an expression for estimating the weighted mean value of unequal-precision measurements of indicators of engineering-geological properties for a \(K\)-layer soil mass. According to this method, as the weighted mean one should take that value \(x_i\) for which the following expression would be minimized:

\[ \sum_{i=1}^{K}\left(\frac{x_i-z}{S_i/\sqrt{n_i}}\right)^2=\mathrm{minimum}, \]

where \(z\) is the estimate of the weighted mean value of the indicator. Equating to zero the first derivative of this expression with respect to \(z\), after simple transformations we obtain:

\[ z=\frac{\displaystyle\sum_{i=1}^{K}\frac{n_i}{S_i^2}\,\overline{x}_i}{\displaystyle\sum_{i=1}^{K}\frac{n_i}{S_i^2}}. \tag{5} \]

From the condition for the minimum of the expression

\[ \sum_{i=1}^{K}\left(\frac{S_i-S_z}{S_i/\sqrt{2n_i}}\right)=\mathrm{minimum} \]

one can in the same way obtain an estimate for the weighted mean standard deviation \(S_z\):

\[ S_z=\frac{\displaystyle\sum_{i=1}^{K}\frac{n_i}{S_i}}{\displaystyle\sum_{i=1}^{K}\frac{n_i}{S_i^2}}. \tag{6} \]

The weighted mean \(z\) in our case is a random variable and is characterized by its distribution law with parameters \(Mz\) (mathematical expectation) and standard deviation

\[ m_z=\frac{S_z}{\sqrt{k}}. \]

Now let us form the normalized quantity

\[ t=\frac{z-Mz}{S_z/\sqrt{k}}, \]

which, obviously, must have an asymptotic normal distribution, since all changes associated with the “unequal precision” in the structure of the indicators of soil properties have already been taken into account. Then the confidence intervals for \(Mz\) can be written in the form

\[ z+t_\gamma\cdot m_z \geq Mz \geq z-t_\gamma\cdot m_z. \tag{7} \]

They are one of the principal elements in the process of obtaining design indicators of the engineering-geological properties of soils [2, 4]. The values of \(t_\gamma\) in formula (7) are tabulated; they can be found in work [3].

For convenience, let us denote in formulas (5) and (6) the “weights” \(\dfrac{n_i}{S_i^2}\) by the quantities \(\lambda_i\). Then these expressions will have the form:

\[ z=\frac{\displaystyle\sum_{i=1}^{\kappa}\lambda_i\cdot \bar{x}_i}{\displaystyle\sum_{i=1}^{\kappa}\lambda_i}; \tag{8} \]

\[ S_z=\frac{\displaystyle\sum_{i=1}^{\kappa}\lambda_i\cdot S_i}{\displaystyle\sum_{i=1}^{\kappa}\lambda_i}. \tag{9} \]

If, for example, the “weight” \(\lambda_j\) for some \(j\)-th layer is taken as unity, i.e.

\[ \lambda_j=\frac{n_j}{S_j^2}=1, \]

then the weights for the remaining layers can be calculated from the equality:

\[ \lambda_i=\frac{n_i}{n_j}\cdot \frac{S_j^2}{S_i^2}. \tag{10} \]

All the formulas given above can be used to estimate \(Mz\) only in the case where, between the number of determinations of the property indicators \(n_i\) and the sizes of the elements of the soil layers \(h_i\), there exists a linear directly proportional relationship, i.e., the proportions of sampling for investigations must be directly proportional to the sizes of the units being sampled. Otherwise, an additional accounting is necessary for the weight contribution of each layer in estimating the weighted-average value of the indicator of some property of the soil mass. For this purpose let us denote by \(u_i\) the quantity \(\left(\sum_{i=1}^{\kappa}h_i\right)^{-1}\cdot h_i\cdot x_i\); then

the variance of the \(u_i\)-variable for the \(i\)-th layer is equal to \(S_{u_i}^2=\left(\sum_{i=1}^{\kappa}h_i\right)^{-2}\cdot h_i^2\cdot S_i^2\).

Here \(h_i\) is an element of the soil mass (the thickness of the \(i\)-th layer, the length of the \(i\)-th segment of the slip line, etc.). If the entire layered soil mass is considered, then the \(u_i\), like the \(x_i\), are “unequal-accuracy” random variables with mean value \(\mu=\frac{1}{\kappa}\cdot z\) and

standard deviation \(S_\mu=\frac{1}{\kappa}\cdot S_z\). Using the principle of the least-squares method, by analogy with the preceding it is not difficult to arrive at formulas for calculating \(\mu\), \(S_\mu\), and \(m_\mu\):

\[ \mu=\frac{\displaystyle\sum_{i=1}^{\kappa}\lambda_{u_i}\cdot \bar{u}_i}{\displaystyle\sum_{i=1}^{\kappa}\lambda_{u_i}}; \tag{11} \]

\[ S_\mu=\frac{\displaystyle\sum_{i=1}^{\kappa}\lambda_{u_i}\cdot S_{u_i}}{\displaystyle\sum_{i=1}^{\kappa}\lambda_{u_i}}; \tag{12} \]

\[ m_\mu=\frac{S_\mu}{\sqrt{\kappa}}, \tag{13} \]

where \(u_i\) and \(S_{u_i}\) are the mean value and the standard deviation of some indicator obtained for the \(i\)-th layer.

In formulas (11, 12), the quantities \(\lambda_{u_i}\), by analogy with expression (10), are equal to:

\[ \lambda_{u_i}=\frac{n_i}{n_j}\cdot\frac{S_j^2}{S_i^2}\cdot\frac{h_j^2}{h_i^2}. \tag{14} \]

Now we finally obtain

\[ z=\kappa\cdot\mu; \tag{15} \]

\[ S_z=\kappa\cdot S_\mu; \tag{16} \]

\[ m_z=\frac{\kappa\cdot S_\mu}{\sqrt{\kappa}}; \tag{17} \]

\[ \kappa\cdot(\mu+t_\gamma\cdot m_\mu)\geq Mz\geq \kappa\cdot(\mu-t_\gamma\cdot m_\mu). \tag{18} \]

In conclusion, we note that the proposed method can be used in determining various design indicators of engineering-geological properties needed in assessing the stability of structures erected in heterogeneous layered soils.

REFERENCES

1. L. N. Bolshev, N. V. Smirnov. Tables of Mathematical Statistics. “Nauka,” 1965.
2. N. V. Kolomenskii. Engineering Geology. Part II. Gosgeoltekhizdat, 1956.
3. V. V. Nalimov. The Application of Mathematical Statistics in the Analysis of Substances. Fizmatgiz, 1960.
4. E. A. Pisarev. Some applications of statistical methods to the processing of results for soil properties. In: Questions of the Methodology of Engineering-Geological Investigations, Tomsk, 1966.
5. Manual for the Design of Foundations of Buildings and Structures. Stroiizdat, 1964.
6. G. L. Fisenko. Stability of Quarry and Dump Slopes. “Nedra,” 1965.