ON THE ACCURACY OF DETERMINING THE VOLUME OF A DUMP FROM A CONTOUR-LINE PLAN BY VARIOUS METHODS

A. I. VOLKOV, P. A. MARCHENKO

(Presented by the scientific seminar of the departments of mine surveying and geodesy)

The question of the accuracy of determining the volume of dumps has not at present been sufficiently studied. Work by VNIMI has been devoted to the study of this question. In [1] data are given on the root-mean-square error in determining the volume of dumps, which includes the error due to the method of surveying the dump, the compilation of graphs, and the error of the method of calculating the dump volume from the graph.

In the present work the question is considered of the accuracy of various methods for calculating the volume of a dump represented by contour lines. The most widespread calculation methods were investigated: the volumetric grid method, and the methods of vertical and horizontal sections. Since reliable conclusions about the accuracy of individual calculation methods can be made only on the basis of mass determinations of volumes, senior students were brought into the work.

Fig. 1. Plan of a coal dump in contour lines.
Legend:
— contour lines of the dump surface,
--- contour lines of the ground surface,
-.- boundary of the dump.

To determine the volume, two variants of dumps were adopted, represented on a plan in contour lines at a scale of 1:500 (Fig. 1). In order to increase the number of independent determinations, the contour interval in each variant was taken equal to 0.2; 0.25; 0.3;

Appendix 1

Table No. 1
for determining \(A_1\), \(A_2\), and \(A_3\)

TABLE No. 2
for determining the value “\(a^2 b^2\)”

TABLE No. 2

for determining the quantity \(Q^{2}b^{2}\)

Appendix 2

0.4; 0.5; 1.0 m. Since the subvariants were common in the plans of the dumps and differed only in the contour interval, the true volumes of the dumps in the subvariants differed from one another by a certain amount proportional to the ratio of the contour intervals. This made it possible, when processing the experimental material, to reduce all volume values to a single interval for each variant. The total number of volume determinations by the various methods was 336, of which: variant I—165, variant II—171. For both variants, for each of the methods, the mean volume values were determined (Table 1).

Table 1

When determining volumes by grid, the side of a square of the latter was taken equal to 1 cm; when determining volumes by the method of vertical sections, the distances between sections were taken equal to 2 cm. In the further processing of the material, the volume determined by grid was accepted as the most probable value, since the number of determinations by grid exceeds the number of determinations by the other methods several times over, and, with a large number of superpositions of the grid, the overall density of the points at which thicknesses were determined was extremely high, which made it possible to take sufficiently accurate account of the character of the variation in the thickness of the dump. From the deviations from the most probable value (from the grid method), the errors of a single determination and of the mean values were determined (Table 2).

Table 2

Since in practice the determination of volume by any method is carried out twice, the error of the mean of two determinations was taken to be \(\sqrt{2}\) times smaller than the error of one determination. The errors obtained in this way are given in column 7 of Table 2. Taking into account all determinations (the two variants together), for each method the errors of one determination, of the mean of \(n\), and of the mean of two are given in Table 3.

Table 3

It follows from Tables 2 and 3 that the volumetric-grid method gives the most accurate results in calculating volume. The methods of vertical and horizontal sections give approximately identical results. The tables also show that the error of the mean of 2 determinations by the volumetric-grid method is about 2%. The methods of vertical and horizontal sections give coarser results, and this error lies within the range of 2.5–3%.

The quality of the volume determinations can also be characterized by the frequency of occurrence of errors of different magnitudes and by the range of their scatter. Such a characteristic is given by us in Table 4.

It is seen from this table that the frequency of occurrence of errors equal in magnitude for individual variants and methods shows fairly close agreement. In addition, it follows from the table that the main body of determinations (80%) for the volumetric-grid method has an error within 3%, while for the methods of vertical and horizontal sections this quantity fluctuates within 4–5%.

On the basis of Table 4 (columns 14, 15, 16), a table of the accumulated frequency of occurrence of errors was compiled (Table 5).

In this table, for comparison with the actual accumulated frequency, the theoretically calculated frequency is given (columns 2, 4, 6). Comparison of the theoretical data with the actual data, above all, confirms the good quality of the initial material, as well as the conclusions made above from the analysis of Table 4.

In order to reveal the influence of the size of the grid cell on the result of calculating the volume, determinations of the volumes of the same dumps were made with a grid-cell size of \(2 \times 2\) cm. Altogether, with this grid-cell size, 99 volume determinations were made. Of these: variant I—49, variant II—50. The mean volume values and the mean errors of one determination, by deviations from the most probable value adopted for variant I as \(6413\ \text{m}^3\) and for variant II as \(5051\ \text{m}^3\) (the means for the grid method with a cell size of \(1 \times 1\) cm), are given in Table 6.

Comparison of the data in Tables 2 and 6 shows exceptionally good agreement between the mean volume values and the mean errors of one determination.

Table 7 gives the theoretical and actual distribution of errors (accumulated frequency) for the case of determining volume by a grid with a cell size of \(2 \times 2\) cm.

Table 4

Table 5

Table 6

Table 7

The data in Table 7 indicate good agreement between the theoretical and actual distribution of errors, and also show that for the bulk of the volume determinations (78%) the magnitude of the error does not exceed 3%, and that the limiting error does not exceed three times the mean.

All this shows that, for the given shape of the dump, a grid cell size of \(2 \times 2\) cm is quite permissible, since it provides the same accuracy in determining the volume as a grid with a cell size of \(1 \times 1\) cm, while reducing the amount of work by a factor of 4.

Conclusions

The investigations presented above make it possible to draw the following conclusions:

1. The volumetric grid method, although more labor-intensive in comparison with the other methods studied, provides the best results in determining volume, since the error for the main mass of determinations (80%) does not exceed 3%.

2. The methods of vertical and horizontal sections may be considered equivalent. These methods give cruder results than the volumetric grid method. The error in determining volume for the main mass of cases (83–85%) does not exceed 5%.

3. In view of the variety of dump forms, the size of the cell of the volumetric grid cannot be specified in advance. Its optimal dimensions should be established experimentally for each enterprise separately, taking into account the established form of the dump.

4. According to data from VNIMI [1], in tacheometric surveying of a dump at a scale of 1:500 (with a volume from 3000 to 6000 m³), the error in determining the volume is ±3.5%; in the case of tacheometric surveying, ±4%. These errors are overall errors and include the errors of field measurements, the preparation of graphic materials, and the error of the computation method.

Excluding from the total error the error of computation by the volumetric grid method, which according to our data averages 2%, we find that the error in determining volume due to field measurements and the preparation of graphic material should not exceed 2.9% for plane-table surveys and 3.5% for tacheometric surveys. The data obtained make it possible to plan field and graphic work in the most rational way.

Literature

1. Handbook of Mine Surveying. Ugletekhizdat, 1955.