ASSESSMENT OF THE ACCURACY OF MEASURED ANGLES IN TRIANGULATION DEVELOPED BY THE METHOD OF GEODETIC RESECTIONS

M. V. POSTNIKOV

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

From the theory of the method of least squares it is known that the mean square error of a series of direct measurements can be computed by the formula

\[ m = \pm \sqrt{\frac{[p v^2]}{n-g}}, \tag{1} \]

where \(m\) is the mean square error of a single measurement;
\(v\) are the most probable errors of the measurements;
\(n\) is the number of all measurements;
\(g\) is the number of measurements required to determine the unknown quantities;
\(p\) are the weights of the measured quantities.

Formula (1) gives a correct characteristic of the accuracy of a series of direct measurements if their number is very large.

In practice one most often has to deal with a small number of measurements, and therefore the question arises of the error of the mean square error computed by formula (1).

The mean square error of the quantity \(m\), found from the errors of the measurements, is computed by the formula [5]:

\[ m_m = \pm m \sqrt{\frac{1}{2(n-g)}} = \pm m \sqrt{\frac{1}{2r}}, \tag{2} \]

where the quantity \(m\) is computed by formula (1).

With the application of formula (1) for assessing the accuracy of angular observations in triangulation, for measurements of equal weight \(m\) will be computed by the following formula:

\[ m' = \pm \sqrt{\frac{[i^2]}{r}}. \tag{3} \]

The corrections \(i\) to the angles are the most probable errors of the measured angles, since they are obtained in the adjustment of the triangulation with allowance for all conditions arising in it (conditional equations of figures, horizon, poles, bases, coordinates, and others). Consequently, formulas (1) and (3) will be rigorous for solving the question of assessing the accuracy of measured angles in triangulation.

In practice, when constructing triangulation with continuous triangles, the value \(m\) is usually determined from the misclosures in the triangles, i.e.,

\[ m'_{\phi} = \pm \sqrt{\frac{-[W_{\phi}^{2}]}{3N}} . \tag{4} \]

Formula (4) is not rigorous, since it takes into account only the conditions of the figures out of the many other conditions present in a triangulation network. All the remaining conditions (of poles, azimuths, bases, coordinates, and others) do not participate in the computation of \(m_{\phi}\).

Such an approach to computing the mean square error of a measured angle in triangulation is justified by the need to obtain a simple formula for estimating the accuracy of angular observations, by which, under field conditions, without substantial computations but with a known approximation, it is possible to assess the quality of angular observations.

When measuring the angles of triangulation developed by the method of resections, it is impossible to compute the error of a measured angle from the misclosures in triangles, since in each triangle only two angles are measured.

In the existing literature there are no indications of methods for assessing the accuracy of measured angles in such networks. Below are considerations that may be adopted for assessing the accuracy of angular observations in networks developed by the method of resections.

Angular measurements may be assessed from the discrepancies of the measured values at a station (from the fluctuation of the collimation error, from deviations from the mean value of an angle or direction). The error of an angle computed from discrepancies at a station will characterize the accuracy of the measured angle only under a certain complex of conditions. As studies have shown, the error of an angle computed from misclosures in triangles is on average twice as large as the error of an angle computed from discrepancies at a station. In urban triangulations, the same angular errors are more than 1.5 times the errors computed at a station [7].

Consequently, having computed the error of a measured angle from discrepancies at all stations, and then passing to the mean square error of a measured angle for the entire triangulation network, we obtain:

\[ m_c = \pm k \sqrt{\frac{-[m_{\alpha}^{2}]}{q}}, \tag{5} \]

where \(m_{\alpha}\) is the error of an angle computed from discrepancies at a station;
\(q\) is the number of errors \(m_{\alpha}\);
\(k\) is a coefficient (for ordinary triangulations—2, for urban triangulations—1.5).

This method of determining the error of a measured angle in triangulation may be used for approximate computations.

When constructing triangulation by the method of resections, traverse lines are formed, which may be regarded as polygonometric traverses. Therefore, for judging the accuracy of a measured angle in the network, formulas used in polygonometry may be employed. To assess the accuracy of angular measurements, all polygons consisting of traverse lines must be divided into groups, depending on the class of triangulation, the accuracy of the instrument used, and the method of measuring angles.

For each group of angular measurements, the mean square error of a measured angle may be obtained from the misclosures in the angles of polygons by formula [2]:

\[ m_{\beta} = \pm \sqrt{\frac{\left[\dfrac{f_{\beta}^{2}}{n'}\right]}{N}}, \tag{6} \]

where \(m_\beta\) is the root-mean-square error of a measured angle;
\(f_\beta\) is the angular misclosure in a polygon;
\(n'\) is the number of angles in a polygon;
\(N\) is the number of polygons formed by the traverse lines.

For checking, the error \(m_\beta\) may be computed by the formula [2]:

\[ m_\beta = \pm \sqrt{\frac{[f_\beta^2]}{[n]}} , \tag{7} \]

where \([n]\) is the total number of angles in all polygons.

The error of a measured angle computed by formula (6) will characterize the accuracy of angular observations not for the entire triangulation network, since in computing \(m_\beta\) only the angles of turn and the included angles of the traverse line enter. The angles measured for determining auxiliary points do not participate in the computation of \(m_\beta\).

Consequently, \(m_\beta\), computed from the angular misclosures of traverses, is the root-mean-square error of a measured angle of turn of the traverse line. The magnitude of the mean error of the angles measured for determining auxiliary points can be calculated from the following considerations.

In the resection method, the angles of turn of the traverse line, in order to increase the accuracy of transferring directional angles, are measured in two groups independent of one another (Fig. 1).

Fig. 1.

Thus, at point \(B\), in the first group the directions \(BA\), \(BC\), \(BM\), and \(BM'\) are measured; in the second group, the directions \(BA\) and \(BC\). In each group the directions are measured by the method of circular sets, with \(n\) sets (for fourth-order triangulation, when using an OT-02 theodolite—6 sets).

Taking the number of sets as the weights of the measured angles, and allowing for formula (6), the root-mean-square error of the angles measured for determining auxiliary points will be equal to:

\[ m = \pm \sqrt{\frac{\left[\dfrac{f_\beta^2}{n'}\right]}{N}} \sqrt{\frac{P_1}{P_2}} \tag{8} \]

or

\[ m = \pm m_\beta \sqrt{\frac{P_1}{P_2}} , \tag{9} \]

where \(P_1\) is the weight of the measured turning angle of the traverse line;
\(P_2\) is the weight of the measured angle to the auxiliary points.

Similar reasoning in computing the mean square error of a measured angle can also be applied to other methods of measuring triangulation angles.

Let us compute the mean square error of a measured angle from the angular misclosures of polygons in a fourth-order triangulation constructed by the resection method in Kuzbass and observed by the author in 1953 (Table 1).

Table 1

The mean square error of the measured turning angle of the traverse line, by formula (6), will be

\[ m_3=\sqrt{\frac{2,31}{2}}=\pm 1'',07, \]

and the check by formula (7) gives

\[ m_3=\sqrt{\frac{11,57}{11}}=\pm 1'',03. \]

Taking \(P_1\)—the weight of the measured turning angle of the traverse line—as equal to 12, and \(P_2\)—the weight of the angles measured for determining the auxiliary points—as equal to 6, we compute, by formula (9), the mean square error of the angle measured to the auxiliary points:

\[ m=\pm 1'',07\sqrt{2}=\pm 1'',51. \]

However, a correct assessment of the accuracy of angular measurements in a triangulation constructed by the resection method, using formulas (6) and (9), can be made only when there is a considerable number of polygons formed by traverse lines. When the number of polygons is small, determining the mean square errors of measured triangulation angles by polygonometry formulas will be very unreliable.

It was indicated earlier that the problem of assessing the accuracy of measuring triangulation angles by the rigorous method is solved by formulas (1) and (3). The use of formulas (1) and (3) requires considerable computations, which precludes their use under field conditions.

To assess the accuracy of angular measurements in a triangulation developed by the resection method, it is necessary to obtain a formula by which, under field conditions, without substantial computational work, but with a sufficient degree of accuracy, the quality of angular measurements could be determined. Such an approximate formula for determining the mean square error of a measured angle can be derived from the following considerations.

In the resection method, each subsequent side of the triangulation traverse line is determined from two independent transmissions.

In Fig. 2, \(AB\) is the initial side (measured directly or

computed triangulation; \(BC\) is the side being determined; \(M\) is the auxiliary point of the first transfer; \(M'\) is the point of the second transfer; the angles of the first transfer are 1, 2, 3 and 4; the angles of the second transfer are \(1'\), \(2'\), \(3'\) and \(4'\). All angles 1, 2, 3, 4 and \(1'\), \(2'\), \(3'\), \(4'\) are directly observed quantities.

Fig. 2.

The first transfer gives:

\[ BC_1=\frac{AB\cdot \sin 1\cdot \sin(3+4)}{\sin(1+2)\cdot \sin 4}. \]

The second transfer gives:

\[ BC_2=\frac{AB\cdot \sin 1'\sin(3'+4')}{\sin(1'+2')\cdot \sin 4'} \]

or, as usual,

\[ w=\lg \frac{\sin 1\cdot \sin(3+4)}{\sin(1+2)\cdot \sin 4} -\lg \frac{\sin 1'\cdot \sin(3'+4')}{\sin(1'+2')\cdot \sin 4'}, \tag{10} \]

where

\(w\) is the free term of the pole conditional equation, expressed in units of the 6th decimal place of the logarithm.

From the theory of triangulation developed by the method of resections, it is known that the mean error in determining the logarithm of a side in two-figure resections, when adjusting the angles and without taking into account the error of the initial side, is expressed by the formula [1]:

\[ m_s=m\sqrt{\sum_1^n Q}, \tag{11} \]

where \(Q=2\delta_{A+B}^{\,2}+\delta_A^2-2\delta_{A+B}\cdot \delta_A\) is the quantity reciprocal to the weight of the logarithm of the side, computed for each triangle, and \(m\) is the root-mean-square error of the measured angle.

Then the mean errors of the values of the side \(BC\)—\(BC_1\) and \(BC_2\), with allowance for formula (11), will be expressed as follows:

\[ m'=m\sqrt{Q_1'+Q_2'}, \]

\[ m''=m\sqrt{Q_1''+Q_2''}. \tag{11a} \]

Hence the mean error \(m_s\) of the difference of the logarithms of the sides \(\mathrm{BC}_1\) and \(\mathrm{BC}_2\) will be equal to:

\[ m_s=\sqrt{m'^2+m''^2}. \tag{11б} \]

Assuming that the mean error \(m_s\) is equal to the value of the free term \(w\) of the pole equation (formula 10), we obtain:

\[ m_s=w=m\sqrt{Q_1' + Q_2' + Q_1'' + Q_2''}. \tag{12} \]

Denoting in formula (12) the sum \(Q_1' + Q_2' + Q_1'' + Q_2''\) by \(Q_i\), and the free term of the pole equation by \(w_i\), we obtain expression (12) in the general form:

\[ w_i=m\sqrt{Q_i}, \tag{13} \]

where

\(Q_1'\) and \(Q_2'\) are the reciprocal weights of the triangles of the first transfer;
\(Q_1''\) and \(Q_2''\) are the reciprocal weights of the triangles of the second transfer.

Suppose that in the triangulation network there are \(n\) points at which conditional pole equations have arisen, and that the angles at them have been measured with equal weight. Then, by formula (13), we have:

\[ w_1=m\sqrt{Q_1}, \]

\[ w_2=m\sqrt{Q_2}, \]

\[ w_3=m\sqrt{Q_3}, \]

\[ \ldots\ldots\ldots\ldots \]

\[ w_n=m\sqrt{Q_n}. \tag{13a} \]

Passing to squared errors after summing both parts of the equalities, we have:

\[ \sum_{1}^{n} w^2 = m^2 \sum_{1}^{n} Q, \tag{13б} \]

whence

\[ m=\pm\sqrt{\frac{[w^2]}{[Q]}}. \tag{14} \]

Formula (14) gives a fairly correct estimate of the accuracy of the measurement of angles in a triangulation developed by the method of resections, since it takes into account most of the conditions arising in the network.

Let us give the formula for computing \(m\) from the values of the free terms of the conditional horizon equations:

\[ m_r=\sqrt{\frac{[w_r^2]}{4n}}, \tag{15} \]

where

\(n\) is the number of points at which conditional horizon equations have arisen.

Formula (15) takes absolutely no account of the influence of external factors on the accuracy of angle measurement.

The computation of the mean square error of the measured angle from the free terms of the pole equations is carried out according to the scheme given below (Table 2).

Table 2

Computation
of the mean square error of a measured angle in a triangulation constructed
by the method of resections, in the coal-bearing district of Kuzbass.

\[ m=\pm \sqrt{\frac{[w^2]}{[Q]}} =\pm \sqrt{\frac{1492}{712}} =\pm 1'',45 . \]

When observing the indicated triangulation, in order to study the accuracy of the constructed network, simultaneously with the measurement of the angles necessary for determining the triangulation points by the method of resections, all angles of the triangles were measured.

The mean square error of the measured angle, computed from the misclosures in the triangles (formula 4), was found to be:

\[ m=\pm 1'',62 . \]

The results of computing $m$ by formula (4) agree well with the data obtained for the same triangulation by formula (14).

We give the values of the quantity $m$ for the triangulation of the coal-bearing district of Kuzbass, determined by different formulas (Table 3).

Conclusions and recommendations

1. The rigorous formula for assessing the accuracy of angular measurements developed by the method of resections is formula (3). However, because of the considerable amount of computational work required, this formula is not applied under field conditions.

2. The approximate formulas (7) and (9), when assessing the accuracy of angular observations, cannot give reliable results, since the number of polygons in a network composed of traverse lines is usually small. Formula (5), based on discrepancies at the station, can serve only for an approximate

Table 3

calculations and gives a considerably underestimated result. Formula (15), being approximate, does not take into account the influence of external factors in the measurement of angles. Moreover, its application is possible only when measuring angles at triangulation points, and not directions.

1. The assessment of the accuracy of angular measurements in triangulation developed by the method of resections, under field conditions, should be carried out from the magnitudes of the free terms of the conditional equations of the poles. For calculating the root-mean-square error of a measured angle, formula (14) is recommended

\[ m=\pm\sqrt{\frac{[w_n]}{[Q]}}, \tag{14} \]

where

\[ w_n \]

is the free term of the pole conditional equation, expressed in units of the 6th decimal place of the logarithm;

\[ Q=2\delta^2_{A+B}+\delta_A^2-2\delta_{A+B}\cdot\delta_A. \]

The computation of \(m\) by formula (14) is carried out under field conditions simultaneously with the determination of the side lengths of the traverse line. Formula (14), while not rigorous, gives reliable results.

1. Formula (14) may be applied for assessing the accuracy of angular observations performed in densifying the control network by the method of multiple resection.

LITERATURE

1. Durnev A. I. New systems for constructing geodetic networks. Geodezizdat, 1952.
2. Chebotarev A. S. Course of geodesy, Part II. Geodezizdat, 1949.
3. Chebotarev A. S. The method of least squares from the foundations of the theory of measurement errors. ONTI, 1936.
4. Krasovsky F. N. and Danilov V. V. Handbook of higher geodesy. Part I, Issue 2, Geodezizdat, 1939.
5. Pranis-Pranevich I. Yu. Handbook of adjustment computations for triangulation. Geodezizdat, 1941.
6. Kuzin N. A. and Lebedev N. N. Practical handbook of urban polygonometry. Geodezizdat, 1938.
7. Provorov K. L. On the construction of continuous triangulations (dissertation work), 1954.