Control Computations in the Adjustment of Insertions of New Points into a Fixed Triangulation Network by the Method of Intermediate (Indirect) Measurements
Abstract Generated abstract
This paper examines control computations for adjusting newly inserted points in a fixed mine triangulation network by the method of indirect measurements, with particular attention to transformations of direction error equations according to Schreiber’s rules. It argues against the view that the standard least squares check is inapplicable after such transformations, deriving reduced normal equations and showing that the check remains valid when the free terms at each observation point sum to zero, as occurs when the initial direction angle is taken as an arithmetic mean. The author specifies what this check can and cannot verify, extends the formulas to insertions of two points, and compares them with examples from Romanov’s textbook. The paper concludes that three checks should be applied in sequence: solving the reduced normal equations, verifying the transformed adjustment computation, and only then performing the final comparison of adjusted directions.
Full Text
CHECK COMPUTATIONS IN THE ADJUSTMENT OF INSERTIONS OF NEW POINTS INTO A FIXED TRIANGULATION NETWORK BY THE METHOD OF INTERMEDIATE (INDIRECT) MEASUREMENTS
V. I. AKULOV
(Presented by the scientific seminar of the Departments of Mine Surveying and Geodesy).
The development of a mine triangulation network, as well as its reconstruction in connection with the undermining of individual points, is usually carried out by inserting one or several points into a fixed network.
The adjustment of insertions of new points into a fixed mine triangulation network, taking into account the assessment of the accuracy of the coordinates of the new points, is usually performed by the method of intermediate observations.
The adjustment of point insertions by the method of intermediate observations consists of the following main stages of computation:
1) computation, for the inserted points, of approximate coordinate values;
2) compilation of the error equations and normal equations;
3) solution of the normal equations;
4) computation of the adjusted coordinates of the points being determined.
The critical stage of adjustment processing is the compilation of the error equations.
The form of the error equations is determined by the scheme of angular measurements at the triangulation points (fixed and inserted).
Thus, when one point is inserted, the error equations have the form:
a) when angles are measured,
\[ \varepsilon_i = (a_{i+1} - a_i)\delta x + (b_{i+1} - b_i)\delta y + v_i, \tag{1} \]
b) when directions are measured,
\[ \varepsilon_i = -\delta z + a_i\delta x + b_i\delta y + v_i, \tag{2} \]
where \(\varepsilon\) is the correction to the measured angle (direction);
\(\delta x, \delta y\) are corrections to the approximate coordinates;
\(\delta z\) is the orientation correction;
\(a, b\) are coefficients of the unknown corrections \(\delta x\) and \(\delta y\);
\(v\) is the absolute term of the equation;
\(i+1, i\) are the numbers of the following and preceding directions enclosing the angle with number \(i\).
In geodetic and mine-surveying practice at the present time, directions are measured at the points of a mine triangulation network (fixed and inserted). Therefore, insertions of points into a mine triangula-
...tion network are adjusted by directions, using the error equation (2).
The number of unknowns (orientation corrections $\delta z$ and corrections $\delta x$ and $\delta y$ to the approximate coordinates of the inserted points) in adjustment by directions is determined by the formula
\[ N = n + 2k, \tag{3} \]
where $n$ is the number of points (fixed and inserted) at which directions were measured;
$k$ is the number of inserted points.
If one proceeds from the error equations (2), then in order to determine all $N$ unknowns it is necessary to form $N$ normal equations arising from the solution of the system of error equations (2) under the condition $[P\varepsilon^2]=\min$, where $P$ is the weight of the measured direction.
To reduce the number of normal equations, the system of error equations (2) is usually transformed according to Schreiber’s rules. As a result of transforming the error equations, the number of normal equations is reduced from $(n+2k)$ to $2k$.
The transformation of the error equations (2) according to Schreiber’s rules requires accuracy and great care.
An error committed in transforming the error equations is not revealed by the checks of adjustment computation known in the literature.
The current check of adjustment computation by the method of indirect measurements known in the literature,
\[ [P\varepsilon] - [Pvv] = [Pav]\,\delta x + [Pbv]\,\delta y, \tag{4} \]
in the opinion of Prof. V. A. Romanov, “has no place in the problem of inserting a point... since the error equations (errors—author) are subjected to transformation”1.
Our investigations have shown that check (4) is applicable when transforming the error equations (2) according to Schreiber’s rules and is a reliable check of the correctness of their transformation.
Let us prove the validity of our propositions for the insertion of point 3 (Fig. 1).
In the diagram (Fig. 1), points 1, 2, 4, 5, and 6 are fixed; point 3 is the inserted point.
Suppose that at the fixed points (1, 2) and at the inserted point (3), directions have been measured with unequal accuracy, with weights respectively $P_1, P_2, P_3$.
The error equations for the individual points have the form:
a) for point 1
\[ \begin{aligned} \varepsilon_1 &= -\delta z_1 + v_1,\\ \varepsilon_2 &= -\delta z_1 + v_2,\\ \varepsilon_3 &= -\delta z_1 + a_3\delta x + b_3\delta y + v_3; \end{aligned} \tag{5} \]
b) for point 2
\[ \begin{aligned} \varepsilon_4 &= -\delta z_2 + v_4,\\ \varepsilon_5 &= -\delta z_2 + a_5\delta x + b_5\delta y + v_5; \end{aligned} \tag{6} \]
c) for point 3
\[ \begin{aligned} \varepsilon_6 &= -\delta z_3 + a_6 \delta x + b_6 \delta y + v_6,\\ \varepsilon_7 &= -\delta z_3 + a_7 \delta x + b_7 \delta y + v_7,\\ \varepsilon_8 &= -\delta z_3 + a_8 \delta x + b_8 \delta y + v_8, \end{aligned} \tag{7} \]
where \(\delta z_1, \delta z_2, \delta z_3\) are the orientation corrections for points 1, 2, and 3, respectively.
Fig. 1. Diagram of the insertion of a point into a triangulation network.
Composing, on the basis of the systems of error equations 5, 6, and 7, the normal equations for each point, we obtain
a) for point 1
\[ \begin{aligned} \underline{P_1 n_1 \delta z_1} - P_1 [a]_1 \delta x - P_1 [b]_1 \delta y - P_1 [v]_1 &= 0,\\ \underline{[Paa]_1 \delta x} + [Pab]_1 \delta y + [Pav]_1 &= 0,\\ \underline{[Pbb]_1 \delta y} + [Pbv]_1 &= 0; \end{aligned} \tag{8} \]
b) for point 2
\[ \begin{aligned} \underline{P_2 n_2 \delta z_2} - P_2 [a]_2 \delta x - P_2 [b]_2 \delta y - P_2 [v]_2 &= 0,\\ \underline{[Paa]_2 \delta x} + [Pab]_2 \delta y + [Pav]_2 &= 0,\\ \underline{[Pbb]_2 \delta y} + [Pbv]_2 &= 0; \end{aligned} \tag{9} \]
c) for point 3
\[ \begin{aligned} \underline{P_3 n_3 \delta z_3} - P_3 [a]_3 \delta x - P_3 [b]_3 \delta y - P_3 [v]_3 &= 0,\\ \underline{[Paa]_3 \delta x} + [Pab]_3 \delta y + [Pav]_3 &= 0,\\ \underline{[Pbb]_3 \delta y} + [Pbv]_3 &= 0, \end{aligned} \tag{10} \]
where \(n_1, n_2, n_3\) is the number of measured directions at points 1, 2, and 3.
Summing the coefficients of equations (8, 9, and 10) for like unknown corrections, we obtain the general system of normal equations of the form
\[ \begin{aligned} P_1 n_1 \delta z_1 - o \cdot \delta z_2 - o \cdot \delta z_3 - P_1 [a]_1 \delta x - P_1 [b]_1 \delta y - P_1 [v]_1 &= 0,\\ o \cdot \delta z_1 + P_2 n_2 \delta z_2 - o \cdot \delta z_3 - P_2 [a]_2 \delta x - P_2 [b]_2 \delta y - P_2 [v]_2 &= 0,\\ o \cdot \delta z_1 + o \cdot \delta z_2 + P_3 n_3 \delta z_3 - P_3 [a]_3 \delta x - P_3 [b]_3 \delta y - P_3 [v]_3 &= 0,\\ - P_1 [a]_1 \delta z_1 - P_2 [a]_2 \delta z_2 - P_3 [a]_3 \delta z_3 + [Paa]\delta x + [Pab]\delta y + [Pav] &= 0,\\ - P_1 [b]_1 \delta z_1 - P_2 [b]_2 \delta z_2 - P_3 [b]_3 \delta z_3 + [Pab]\delta x + [Pbb]\delta y + [Pbv] &= 0, \end{aligned} \tag{11} \]
where
\[ \begin{aligned} [Paa] &= [Paa]_1 + [Paa]_2 + [Paa]_3,\\ [Pbb] &= [Pbb]_1 + [Pbb]_2 + [Pbb]_3,\\ [Pav] &= [Pav]_1 + [Pav]_2 + [Pav]_3,\\ [Pbv] &= [Pbv]_1 + [Pbv]_2 + [Pbv]_3. \end{aligned} \tag{12} \]
From the first three normal equations of system (11), we have
\[ \delta z_1 = \frac{[a]_1}{n_1}\delta x + \frac{[b]_1}{n_1}\delta y + \frac{[v]_1}{n_1}, \tag{13} \]
\[ \delta z_2 = \frac{[a]_2}{n_2}\delta x + \frac{[b]_2}{n_2}\delta y + \frac{[v]_2}{n_2}, \tag{14} \]
\[ \delta z_3 = \frac{[a]_3}{n_3}\delta x + \frac{[b]_3}{n_3}\delta y + \frac{[v]_3}{n_3}. \tag{15} \]
Substituting the orientation corrections \(\delta z_1\), \(\delta z_2\), and \(\delta z_3\) from (13, 14, and 15) into the last two normal equations of system (11), we obtain
\[ \begin{aligned} [PAA]\delta x + [PAB]\delta y + [PAv] &= 0,\\ [PAB]\delta x + [PBB]\delta y + [PBv] &= 0, \end{aligned} \tag{16} \]
where
\[ [PAA] = [Paa] - \frac{P_1 [a]_1^2}{n_1} - \frac{P_2 [a]_2^2}{n_2} - \frac{P_3 [a]_3^2}{n_3}, \tag{17} \]
\[ [PBB] = [Pbb] - \frac{P_1 [b]_1^2}{n_1} - \frac{P_2 [b]_2^2}{n_2} - \frac{P_3 [b]_3^2}{n_3}, \tag{18} \]
\[ [PAB] = [Pab] - \frac{P_1 [a]_1 [b]_1}{n_1} - \frac{P_2 [a]_2 [b]_2}{n_2} - \frac{P_3 [a]_3 [b]_3}{n_3}, \tag{19} \]
\[ [PAV] = [Pav] - \frac{P_1 [a]_1 [v]_1}{n_1} - \frac{P_2 [a]_2 [v]_2}{n_2} - \frac{P_3 [a]_3 [v]_3}{n_3}, \tag{20} \]
\[ [PBV] = [Pbv] - \frac{P_1 [b]_1 [v]_1}{n_1} - \frac{P_2 [b]_2 [v]_2}{n_2} - \frac{P_3 [b]_3 [v]_3}{n_3}. \tag{21} \]
The normal equations (16) are called reduced.
When forming the normal equations on the basis of the transformed system of error equations according to Schreiber’s rules, we likewise obtain the system of reduced normal equations (16).
Indeed, transforming the systems of error equations (5, 6, 7) according to Schre оiber’s first rule, we obtain
a) for point 1
\[ \varepsilon'_3=a_3\delta x+b_3\delta y+v_3 \quad \text{with weight } P_1, \]
\[ \varepsilon_{1\phi}=P_1[a]_1\delta x+P_1[b]_1\delta y+P_1[v]_1 \quad \text{with weight } -\frac{1}{n_1P_1}; \tag{5a} \]
b) for point 2
\[ \varepsilon'_5=a_5\delta x+b_5\delta y+v_5 \quad \text{with weight } P_2, \]
\[ \varepsilon_{2\phi}=P_2[a]_2\delta x+P_2[b]_2\delta y+P_2[v]_2 \quad \text{with weight } -\frac{1}{n_2P_2}; \tag{6a} \]
c) for point 3
\[ \varepsilon'_6=a_6\delta x+b_6\delta y+v_6 \quad \text{with weight } P_3, \]
\[ \varepsilon'_7=a_7\delta x+b_7\delta y+v_7 \quad \text{with weight } P_3, \tag{7a} \]
\[ \varepsilon'_8=a_8\delta x+b_8\delta y+v_8 \quad \text{with weight } P_3, \]
\[ \varepsilon_{3\phi}=P_3[a]_3\delta x+P_3[b]_3\delta y+P_3[v]_3 \quad \text{with weight } -\frac{1}{n_3P_3}. \]
From the transformed systems of error equations (5a, 6a and 7a) we immediately obtain the reduced system of normal equations (16).
The most probable corrections \(\delta x\) and \(\delta y\) to the approximate values of the coordinates of the inserted point are determined from the system of reduced normal equations (16).
Let us write the check (4) for the general system of normal equations (11); we obtain
\[ [P\varepsilon\varepsilon]-[Pvv] = -P_1[v]_1\delta z_1-P_2[v]_2\delta z_2-P_3[v]_3\delta z_3 +[Pav]\delta x+ \]
\[ +[Pbv]\delta y, \tag{22} \]
where \([P\varepsilon\varepsilon]\) is the sum of the products of the squares of the most probable corrections by the corresponding weights for all measured directions in the network;
\([Pvv]\) is the sum of the products of the squares of the free terms of the untransformed system of error equations by the corresponding weights for all measured directions in the network;
\([v]_1\) is the sum of the free terms of the system of error equations (5) for point 1;
\([v]_2\) is the sum of the free terms of the system of error equations (6) for point 2;
\([v]_3\) is the sum of the free terms of the system of error equations (7) for point 3.
For
\[ [v]_1=[v]_2=[v]_3=0 \tag{23} \]
the check (22) will have the form
\[ [P\varepsilon\varepsilon]-[Pvv]=[Pav]\delta x+[Pbv]\delta y, \tag{24} \]
For the case when the sum of the free terms of the systems of error equations for each point is equal to zero, i.e. when there is...
equality (23), the free terms of the reduced system of normal equations (16), proceeding from (20, 21), will be equal to
\[
[PAV]=[Pav],
\]
\[
[pBV]=[Pbv].
\tag{25}
\]
Therefore
\[ [P\varepsilon\varepsilon]-[Pvv]=[PAV]\delta x+[PBV]\delta y. \tag{26} \]
For observations of equal weight, check (26) has the form
\[ [\varepsilon\varepsilon]-[vv]=[AV]\delta x+[BV]\delta y, \tag{26a} \]
where \([AV]\) and \([BV]\) are the free terms of the reduced system of normal equations.
Thus, check (4) holds when the equations of errors are transformed according to Schreiber’s rule for the case when at each point the sum of the free terms of the systems of equations of errors is equal to zero.
The sum of the free terms of the system of equations of errors for a point is equal to zero when, as the approximate value of the direction angle of the initial direction, its arithmetic mean is taken.
Indeed, for example, for point 3 (Fig. 1), we have
\[
(z_1)_3=(\alpha_6)-r_6,
\]
\[
(z_2)_3=(\alpha_7)-r_7,
\tag{27}
\]
\[
(z_3)_3=(\alpha_8)-r_8,
\]
where \((\alpha_6),(\alpha_7),(\alpha_8)\) are the approximate, exactly computed direction angles of the directions \(3—1\), \(3—2\), and \(3—6\);
\(r_6,r_7,r_8\) are the measured directions;
\((z_1)_3,(z_2)_3,(z_3)_3\) are the approximate values of the direction angle of the initial direction for point 3.
Taking as the approximate value of the direction angle of the initial direction for point 3 the arithmetic mean, we obtain
\[ (z)_3=\frac{(z_1)_3+(z_2)_3+(z_3)_3}{3}. \tag{28} \]
The free terms of the equations of errors for point 3 are equal to
\[
v_6=(r_6)-r_6,
\]
\[
v_7=(r_7)-r_7,
\tag{29}
\]
\[
v_8=(r_8)-r_8,
\]
where \((r_6),(r_7),(r_8)\) are the computed values of the directions.
Since
\[
(z_6)=(\alpha_6)-(z)_3,
\]
\[
(z_7)=(\alpha_7)-(z)_3,
\tag{30}
\]
\[
(z_8)=(\alpha_8)-(z)_3,
\]
then
\[
v_6=(\alpha_6)-r_6-(z)_3,
\]
\[
v_7=(\alpha_7)-r_7-(z)_3,
\tag{29a}
\]
\[
v_8=(\alpha_8)-r_8-(z)_3.
\]
Taking (27) into account, we obtain
\[ \begin{aligned} v_6 &= (z_1)_3 - (z)_3,\\ v_7 &= (z_2)_3 - (z)_3,\\ v_8 &= (z_3)_3 - (z)_3. \end{aligned} \tag{29б} \]
Hence
\[ v_6 + v_7 + v_8 = (z_1)_3 + (z_2)_3 + (z_3)_3 - 3(z)_3. \]
Taking (28) into account, we have
\[ v_6 + v_7 + v_8 = 0. \tag{31} \]
The procedure for computing the absolute terms of the error equations by formulas (29б) is recommended by Prof. V. A. Romanov; therefore his conclusion that check (26) is inapplicable when transforming the error equations by Schreiber’s rules does not correspond to reality.
When transforming the error equations by Schreiber’s rules, when the absolute terms \(v\) are computed by formulas (29б), check (26) is obligatory.
Check (26) checks:
a) the correctness of transforming the system of error equations by Schreiber’s rules;
b) the correctness of forming the coefficients of the reduced system of normal equations;
c) the correctness of solving the reduced system of normal equations;
d) the correctness of computing the most probable corrections to the measured directions;
e) the correctness of computing the sums \([P\varepsilon\varepsilon]\) and \([Pvv]\).
Check (26) does not check:
a) the correctness of computing the coefficients \(a\) and \(b\) for the unknowns \((\delta x,\delta y)\);
b) the correctness of determining the absolute terms of the error equations.
Before check (26), it is necessary to perform a check on the correctness of solving the reduced system of normal equations (16) by the formula
\[ [PAV]\delta x + [PBV]\delta y = -\frac{[PAV]^2}{[PAA]} -\frac{[PBV\cdot 1]}{[PBB\cdot 1]}. \tag{32} \]
When two points are inserted into a fixed triangulation network, the reduced system of normal equations (16) has the form
\[ \begin{aligned} \underline{[PAA]\delta x_1} + [PAB]\delta y_1 + [PAC]\delta x_2 + [PAD]\delta y_2 + [PAV] &= 0,\\ \underline{[PBB]\delta y_1} + [PBC]\delta x_2 + [PBD]\delta y_2 + [PBV] &= 0,\\ \underline{[PCC]\delta x_2} + [PCD]\delta y_2 + [PCV] &= 0,\\ \underline{[PDD]\delta y_2} + [PDV] &= 0, \end{aligned} \tag{33} \]
where \(\delta x_1,\delta y_1,\delta x_2,\delta y_2\) are the most probable corrections to the approximate coordinate values of the inserted points.
Checks (32) and (26) are written as follows:
\[ [PAV]\delta x_1 + [PBV]\delta y_1 + [PCV]\delta x_2 + [PDV]\delta y_2 = \]
\[ = -\frac{[PAV]^2}{[PAA]}-\frac{[PBV\cdot 1]^2}{[PBB\cdot 1]}-\frac{[PCV\cdot 2]^2}{[PCC\cdot 2]}-\frac{[PDV\cdot 3]^2}{[PDD\cdot 3]}. \tag{34} \]
and
\[ [P\varepsilon\varepsilon]-[Pvv]=[PAV]\delta x_1+[pBV]\delta y_1+[pCV]\delta x_2+[PDV]\delta y_2. \tag{35} \]
The final check of the adjustment computation is performed by the formulas
\[ r^\circ=r+\varepsilon; \tag{36} \]
\[ r^\circ=\alpha-z, \tag{37} \]
where \(r, r^\circ\) are the measured and adjusted values of the direction;
\(\alpha\) is the adjusted directional angle of the direction, computed from the adjusted coordinates of the inserted points;
\(z\) is the adjusted directional angle of the initial direction for the point.
The value \(z\) for the point is determined by the formula
\[ z=(z)+\delta z, \tag{38} \]
where \((z)\) is the approximate directional angle of the initial direction;
\(\delta z\) is the most probable correction to the directional angle of the initial direction.
The equality of the adjusted values of the measured directions, computed by formulas (36) and (37), is the final check of the correctness of the adjustment computation.
It is necessary to note that the final check, with correctly computed coefficients \(a\) and \(b\) and the free terms of the error equations, will agree for arbitrary values of the corrections \(\delta x\) and \(\delta y\).
Therefore, check (26), which checks the adjustment computation, must precede the final check.
In conclusion, to confirm all our conclusions, let us verify check (26) on two examples considered by Prof. V. A. Romanov in the textbook Theory of Errors and the Method of Least Squares, Ugletekhizdat, pp. 231 and 251, 1952.
Example 1. Insertion of one point (p. 231).
For the indicated example we have (p. 239)
\[ [PAV]\delta x+[PBV]\delta y=-1184,5, \tag{I} \]
\[ -\frac{[PAV]^2}{[PAA]}-\frac{[PBV\cdot 1]^2}{[PBB\cdot 1]}=-1184,5, \tag{II} \]
\[ [P\varepsilon\varepsilon]=118,0, \]
\[
[Pvv]=1302,7
\]
(computed by the author from the vertical column 7, Table 49).
Substituting the sums \([P\varepsilon\varepsilon]\) and \([Pvv]\) into the right-hand side of check (26), we obtain
\[ [P\varepsilon\varepsilon]-[Pvv]=118,0-1302,7=-1184,7. \tag{III} \]
Comparing equalities (I) and (III), we note that check (26) is satisfied within the accuracy of the computations.
Example 2. Insertion of two points (p. 251)
For the example cited we have (p. 268)
\[ [PAV]\delta x_1+[PBV]\delta y_1+[PCV]\delta x_2+[PDV]\delta y_2=-455,6; \tag{IV} \]
\[ -\frac{[PAV]^2}{[PAA]}-\frac{[PBV\cdot 1]^2}{[PBB\cdot 1]}-\frac{[PCV\cdot 2]^2}{[PCC\cdot 2]}-\frac{[PDV\cdot 3]^2}{[PDD\cdot 3]}=-455,7; \tag{V} \]
\[ [P\varepsilon\varepsilon]=115,6; \]
\[
[Pvv]=571,4
\]
(computed by the author from vertical column (6), Table 63).
The right-hand side of check (35) is equal to
\[ [P\varepsilon\varepsilon]-[Pvv]=115,6-571,4=-455,8. \tag{VI} \]
Comparing the right- and left-hand sides of check (35), we note their good agreement.
Example 3. Insertion of one point (p. 231).
Suppose that in the second transformed error equation (Table 53, p. 237) the absolute term was incorrectly computed (instead of \(V=-8,93\), \(V'=+8,93\) was found).
With the subsequent adjustment, we obtain
\[ [PAV]\delta x+[PBV]\delta y=-1008,8; \tag{A} \]
\[ -\frac{[PAV]^2}{[PAA]}-\frac{[PBV\cdot 1]^2}{[PBB\cdot 1]}=-1008,8; \tag{B} \]
\[ [P\varepsilon\varepsilon]=514,8; \]
\[ [Pvv]=1207,4; \]
\[ [P\varepsilon\varepsilon]-[Pvv]=514,8-1207,4=-692,6; \tag{C} \]
\[ \delta x'=0,067\ \text{m}, \qquad \text{instead of } \delta x=-0,011\ \text{m}, \]
\[ \delta y'=-0,028\ \text{m}, \qquad \text{instead of } \delta x=-0,102\ \text{m}. \]
The final check, with the erroneous corrections \(\delta x'=0,067\ \text{m}\) and \(\delta y'=-0,028\ \text{m}\), is satisfied.
Therefore, if check (26) is not carried out, as Prof. V. A. Romanov recommends, then the adjustment must be regarded as having been performed correctly.
In reality, however, a gross error has occurred in the adjustment computation, which follows from check (26), whose right-hand side is equal to \(-629,6\), while the left-hand side is \(-1008,8\). The gross failure of check (26) indicates the presence of an error in the adjustment computation.
Conclusions
- In adjusting insertions of points by directions with transformation of the error equations according to Schreiber’s rules, the following current checks are present:
a) the first
\[ [PAV]\delta x+[PBV]\delta y=-\frac{[PAV]^2}{[PAA]}-\frac{[PBV\cdot 1]^2}{[PBB\cdot 1]}; \]
b) the second
\[ [P\varepsilon\varepsilon]-[Pvv]=[PAV]\delta x+[PBV]\delta y; \]
c) the third (final)
\[ r+\varepsilon=\alpha-z. \]
- The first check verifies only the correctness of solving the system of normal equations.
-
The second check verifies the correctness of transforming the systems of error equations according to Schreiber’s rules, the correctness of forming the coefficients of the normal equations, and their solution.
-
The third (final) check, only when the second check has been carried out, verifies the correctness of adjusting the insertion of a point by the method of indirect observations.
-
The first check must precede the second, and the second must precede the third (final) check.