SELECTION OF THE SCALE OF SPECIAL (DEPARTMENTAL) TOPOGRAPHIC SURVEYS

G. N. Gushchin

(Presented by the scientific seminar of the Departments of Mine Surveying and Geodesy)

General Information

Topographic surveys at scales of 1 : 5000—1 : 500 are intended to obtain graphic-analytical material used in solving various kinds of problems of the national economy. The accuracy of this material depends mainly on the scale of the survey. At the same time, the cost and the time required for its execution change sharply when moving to adjacent scales on the scale range (see Tables 1, 2, 3). For example, when surveying \(1\ \mathrm{km}^2\) of terrain of medium category and with a contour interval of \(0.5\ \mathrm{m}\), the difference in the cost of work for scales 1 : 500 and 1 : 1000 amounts to 21,000 rubles, and the difference in time is 51.9 working days. This circumstance deserves the most serious attention, since the correct choice of the topographic-survey scale is important not only for a sufficiently accurate solution of problems of exploration and design, but also for saving funds and time in carrying out the topographic survey itself.

However, in the specialized literature the technical and economic justifications for choosing the most appropriate scale of departmental topographic surveys are covered insufficiently fully and purposefully (see [1], [2], [3], [4], [5]). Design departments formulate their requirements for topographic-geodetic work in the most general terms. And this very important question is still practically resolved in different ways: as a rule, on the basis of accidentally established traditions, the experience and intuition of the designer. Experience in choosing survey scales is not generalized; as a result, in production, when choosing it, they often proceed not from the need to have sufficiently accurate survey material, but from convenience of design. Such an incorrect approach leads to the fact that, despite the considerable effort in carrying out a large volume of special topographic surveys, excesses are permitted in them.

The appearance of excesses, in our opinion, is explained by a number of reasons.

First of all, the circumstance has an effect that design departments do not sufficiently delve into the economic side of the matter and po-

thereby underestimating the sharp difference in the cost of the work and in the time required for compiling topographic plans of adjacent scales.

Further, cases are common in which the designer, not knowing the permissible errors for the individual stages of survey work (since these errors have in most cases not been established) and playing it safe, strives for exaggerated accuracies, while the geodesist unquestioningly carries out his order.

Finally, it is quite often observed that, when choosing the scale of a topographic survey, the requirements of its necessary accuracy and the convenience of design are not separated, but are based on the simultaneous satisfaction of both these requirements. Designers not infrequently require a larger-scale depiction of the terrain than that on the available topographic plan, but without increasing the accuracy or the degree of detail, explaining this by saying that otherwise the plan will be cluttered and it will be impossible to show all the necessary design data. For example: “In the technical design submitted by a division of Giprokkommunstroi, for the planning and land-management arrangement of one of the cities, a survey over an area of 300 km² at a scale of 1:5000 was envisaged. On the recommendation of a division of Gosgeonadzor, Giprokkommunstroi, without detriment to the quality of the design work, used the available plans at a scale of 1:10000, enlarging them to the scale 1:5000. As a result, the planned expenditures were reduced by more than 1 million rubles” (see [4], p. 33).

Similar data confirming the above-mentioned reasons for overstating the scales of topographic surveys can be cited for many other agencies as well. Thus, in the practice of geological investigations, geologists require a topographic map even at a larger scale than the scale of the geological survey; in detailed exploration of mineral deposits, topographic surveys at scales of 1:1000 and 1:500 are widely used. There are cases when, in detailed geological surveying, geologists specify the scale of the topographic survey of the entire territory under study as the one required for surveying areas of landslides, karst, etc.^1

It should be noted that the current instructions of the Ministry of Geology and Protection of Mineral Resources define the scales of surveys for geological exploration work imprecisely and proceed rather from the convenience of design, recommending that the contours of the calculated reserves be geometrized on plans and sections at a scale larger than the scale of the topographic base used in the calculation. For example (see [6], p. 13, § 24): “The reserves are calculated on a topographic base at a scale of 1:1000 or 1:2000 and are geometrized on sections of the same scale, and, where a larger scale is necessary, with indication of the lithologically and qualitatively different beds composing the useful stratum.” Or ([6], § 27): “…The area of distribution and the thickness of the useful stratum may be established approximately on the basis of at least rare workings or outcrops and geological considerations. For the calculation of reserves, a topographic base of any scale may be used, provided it permits geometrization of the presumed contours of distribution of the useful mineral.” In the instructions for the classification of reserves of manganese ores and for placer deposits of graphite, tin, and tungsten, the scales of topographic surveys are altogether—

^1 Data were obtained by familiarization in 1956–1958 with materials of the engineering department of GIPROGOR (Moscow, chief project engineer N. P. Terenetskii), the West Siberian Geological Administration (Novokuznetsk, deputy head of the production department N. G. Chetvertukhin), Zheldorproekt (Tomsk, head of the construction-materials parties I. T. Yachmenev), the complex geological-prospecting expedition (Tomsk, head of the topographic party Yu. Stezhenko, I. V.), and others.

are not indicated at all. For each separate stage of exploration, the instructions provide for two scales. There are no explanations as to when one scale or the other should be used, or, in general, whether one of these two scales is necessary, and perhaps a third could be used as well.

Such an approach to the choice of the scale of a topographic survey is not economically justified, as is shown by Tables 1, 2, and 3.

Studying the tables, we see that the cost and the time required for producing surveys adjacent on the scale series differ by a factor of 2–3. Moreover, the difference in cost and time required for producing surveys of adjacent scales is the greater, the larger the scales are.

Table 1

Cost of a plan for 1 km² in one copy, depending on the scale and method of survey (according to [9])

In order to save money and time, the scale of the survey should be no larger than the scale required by the accuracy of the design work.

When choosing the survey scale, one should take into account the purpose for which the plan will subsequently be used, and approach the choice of scale

differentiated, from the general to the particular, taking into account the stage of the investigations. At the same time, there is no reason to make a topographic survey of the entire investigated territory at the same scale at which it is necessary to survey individual (critical) areas.

Table 2

Time requirement in days for topographic surveying of 1 km², depending on the scale and method of surveying (according to [10])

In our opinion, the principal requirements determining the scale of the survey are:

1. The necessary accuracy of design data graphically determined from the plan.
2. Convenience of design.
3. The detail of depiction of special elements that must be indicated in the assignment.

When choosing the scale, the indicated requirements should be divided according to their importance, subordinating the second and third to the first, i.e., produc—

carry out the survey at such a scale whose accuracy corresponds to the necessary accuracy of the design data determined by the appropriate plan.

Table 3

Time required and cost of preparing a line plan for 100 km², in one copy on a rigid base, without contours, from aerial photography materials (according to [10])

As for the convenience of design, it is achieved by subsequently enlarging the image, or else the result of the survey is plotted at once on a tablet at a scale convenient for design.

If, for some considerations, the accuracy of determining the position of individual terrain objects must be higher than the accuracy of the original, then in these cases, in addition to the graphical plan, analytical data should be provided (coordinates, angles, distances, elevations) characterizing the position of such terrain objects. It is more convenient to determine analytical data simultaneously with the development of the working control.

If necessary, the original for the most critical sections may be compiled in greater detail.

Such a procedure makes it possible to accelerate the obtaining of the plan by two times or more, and also to save by a factor of 2–3 the funds spent on surveying. This is confirmed by Tables 4 and 5.

Comparing in the same way the methods of obtaining plans of other adjacent scales, we become convinced that, when choosing the scale of a special topographic survey, it is inadvisable to strive for the simultaneous satisfaction of the requirements of the necessary accuracy and convenience of design (as has been done up to now). It will be more correct to satisfy, first of all, the requirement for the necessary accuracy of the survey. Such an approach fully ensures the solution of the tasks of investigation and design and will provide a significant saving of funds and time.

From all that has been said above it follows, therefore, that when choosing the scale one must proceed from the basic requirement that the root-mean-square error in the position of a point on the plan, \(M_{\text{п}}\), should not exceed-

attain the specified accuracy (rms error) of the design data \(M_{\mathrm{pr}}\) determined from the plan, i.e.,

\[ M_{\mathrm{p}} \leq M_{\mathrm{pr}}. \tag{1} \]

Table 4

Relative cost of plans obtained by different methods (in percent)

Table 5

Relative time required to obtain plans by different methods (in percent)

The rms error \(M_{\mathrm{p}}\) entering into (1) depends on a number of factors, the most important of which are:

1) the error of the initial geodetic data \(m_1\);
2) the error of the working geodetic control \(m_2\);
3) the error in determining the position of a picket (spot-height) point \(m_3\);
4) the error of graphical construction (determination) \(m_4\);
5) the error due to deformation of the paper \(m_5\).

All the errors listed will act independently of one another. Therefore their combined effect in general form may be expressed as follows:

\[ M_{\mathrm{p}}^{2}=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}+m_{5}^{2}. \tag{2} \]

Then the basic requirement for choosing the scale (formula 1) will take the form:

\[ M_{\mathrm{pr}}^{2} \geq m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}+m_{5}^{2} \tag{3} \]

or

\[ m_{4}^{2}\leq M_{\mathrm{pr}}^{2}-m_{1}^{2}-m_{2}^{2}-m_{3}^{2}-m_{5}^{2}. \tag{4} \]

Now expressing the error of graphical construction on the plan \(m_4\) through the denominator of the numerical scale of the plan \(M\), from relation (4) we can further determine the denominator of the numerical scale of the survey that satisfies the stated requirement:

\[ M \leq \sqrt{\frac{M_{\mathrm{pr}}^{2}-m_{1}^{2}-m_{2}^{2}-m_{3}^{2}-m_{5}^{2}}{b^{2}}}. \tag{5} \]

Here $\delta$ denotes the smallest distance on the plan visible to the unaided eye.

Example.

On the basis of the general considerations given above, let us now examine the choice of the scale of a topographic survey used in prospecting for mineral deposits.

In all cases of prospecting it is entirely inexpedient to require the scale of the topographic plan to be larger than the scale of the geological map being compiled. This is explained by the fact that the number of geological survey points determined for one and the same area is many times smaller than the number of topographic survey points at the same survey scale. Thus, in a survey at a scale of 1 : 25000, geologists can make do with only 3 points located per 1 km², and moreover not in characteristic places, but to a considerable extent in random places of the geological structure, which has not yet been studied during the period of geological surveying; in a survey at a scale of 1 : 10000, the instruction requires only 17 geological workings per 1 km². It is therefore quite clear that the results of a geological survey whose scale is equal to the scale of the topographic plan are obtained with clearly less accuracy and detail than the results of the corresponding topographic base (see [2]).

The instruction to take as the basis for a geological survey a topographic map of a larger scale than the intended scale of the geological map makes sense only for small-scale surveys (1 : 500000, 1 : 200000, 1 : 100000). However, it is often observed that this same requirement is, quite unjustifiably, transferred purely mechanically to large-scale surveys as well, where there is no longer any need for it, since the latter illuminate the locality in sufficient detail.

The scale of the plan must be such that the elements determined graphically from it are sufficiently accurate. In determining a sufficient degree of accuracy, let us use the production tolerance for topographic-geodetic work in prospecting for mineral deposits, namely $\pm 1.0$ m (see [7]).

Setting out in the field the points of geological workings with the indicated error is permissible in all types of prospecting, since the indicated error $M_{\mathrm{pr}} = \pm 1.0$ m does not exceed the limits of accuracy of calculations for determining reserves of the mineral. Moreover, the correctness of the geological characteristic will be equally probable regardless of whether the working coincides in accuracy with the designed point or deviates from it by 1–2 m.

Let us now determine the scale of the topographic plan whose accuracy corresponds to the above-mentioned production tolerance.

According to the GUGK instruction [8], the accuracy of plans at scales of 1 : 5000 and 1 : 2000 must satisfy the following requirements:

a) the mean error in the position on the plan of points of the planimetric survey control relative to triangulation or traverse points must not exceed $\pm 0.1$ mm;

b) the mean error in the position on the plan of objects and contours of the locality relative to the nearest points of the planimetric survey control must not exceed $\pm 0.4$ mm.

With the indicated tolerances for the mean square error $m$ in the position of a point on the plan relative to the points of the main geodetic control, we obtain:

$$ m = \pm \sqrt{(0.1)^2 + (0.4)^2} = \pm 0.4 \text{ mm}. $$

Taking further into account the requirements of the instruction and the production tolerance for topographic-geodetic work, one may with full justification

to assert that in all cases of prospecting for mineral deposits it is inadvisable to carry out topographic surveys at a scale larger than 1:2000. This follows from the fact that a topographic plan at a scale of 1:2000 ensures the accuracy of graphical preparation of the design for the location of workings with a mean error of \(\pm 0.8\) m \((0.4\ \text{mm} \times 2000)\), with a tolerance of \(\pm 1.0\) m.

Thus, summarizing the considerations set forth above, we arrive at the conclusion that, in detailed exploration of mineral deposits, topographic surveys should expediently be carried out at scales of 1:5000—1:2000, with subsequent enlargement of the image, if necessary, to the required size. A topographic survey at a scale of 1:2000 should be performed in the exploration of mineral deposits of high categories, when the graphical tying-in of workings is carried out.

LITERATURE

1. Glotov G. F. Geodetic Work on the Construction Site of an Industrial Enterprise. Geodezizdat, Moscow, 1955.

2. Sirotkin M. P. Geodetic Work in Geological and Hydrological Investigations for Hydraulic Engineering Construction. Geodezizdat, Moscow, 1955.

3. Sirotkin M. P. On the choice of the scale of topographic survey for designing a drainage network. Geodesy and Cartography, No. 12, 1958.

4. Nalivkin A. N., Spassky N. N. Tasks of the departments of State Geodetic Supervision in reducing the cost of topographic-geodetic work. Geodesy and Cartography, No. 3, 1959.

5. Kurushin A. M. On topographic surveys for estimating reserves of mineral resources. Geodesy and Cartography, No. 10, 1959.

6. Instruction on the Application of the Classification of Reserves to Deposits of Cement Raw Materials. Gosgeoltekhizdat, Moscow, 1954.

7. Fialkov D. N. Production tolerance for geodetic work in the exploration of mineral deposits. Collection of Scientific Papers of TISI, vol. V, 1959.

8. Instruction on Topographic Survey at Scales of 1:5000 and 1:2000, GUGK, Ministry of Internal Affairs of the USSR. Geodezizdat, Moscow, 1955.

9. Handbook of Consolidated Cost Indicators for Design and Survey Work, Moscow, 1956.

10. Unified Output Standards for Geodetic and Topographic Work. Geodezizdat, Moscow, 1954.

11. Krumenis V. A. Large-scale survey of built-up territories by the graphical method. Geodesy and Cartography, No. 9, 1957.