On the Solution of Dependent Equations in Adjustment by the Method of Conditional Observations

A. I. VOLKOV

The success of adjustment computations depends first of all on the correct determination of the number of conditions and on their proper formulation. If determining the number of conditions presents no difficulty owing to the existence of the corresponding formulas, then their proper formulation depends mainly on the experience of the computer.

Errors in the formulation of conditions may be of the most varied kinds. Among other errors, the inclusion of consequences in adjustment computations occupies a special place, since it is an error of a fundamental character.

In the present work it is considered in what manner and at what stage of the computations consequences erroneously included among the conditions can be detected. In the mine-surveying and geodetic literature there are no concrete instructions on this point, apart from warnings as to the inadmissibility of including consequences in adjustment computations. Thus, on this question, in the course in higher geodesy by Prof. F. N. Krasovskii and V. V. Danilov (ed. 1939), we read: “In every network one can set up figure conditions or side conditions in a number considerably exceeding that determined by the corresponding formula... This circumstance sometimes leads to the erroneous inclusion among the conditions of some which are consequences of conditions already introduced ... the introduction of such conditions, dependent on the others, is equivalent to the removal from the equations of the network of the same number of independent conditions.”

In the course Mine Surveying (special part, ed. 1932), Prof. I. M. Bakhurin says: “There are not infrequently cases in which the computer ... in setting up conditional equations erroneously does not obtain them as independent, as a result of which he sets up an extra equation and omits a necessary one. As a result, of course, all the computational labor of adjusting the figure, sometimes long and persistent, is wasted.”

More precise indications on this question are given in V. V. Vitkovskii’s Practical Geodesy (ed. 1898), where it is said: “... the introduction of an extra one (a consequence is meant) leads to corrections of the form \( \frac{0}{0} \), i.e., to indeterminacy, and then all the computational work is wasted.”

Before proceeding to the exposition of the present question, it is necessary to note that the statements cited above on this subject are not exhaustive, at least from the practical point of view.

Criteria of dependence in the joint solution of normal equations¹

Since consequences are a function of conditions, their inclusion in an adjustment computation leads to the solution of a system of dependent equations. In the theory of determinants it is proved that, when such a system is solved, the unknowns are obtained in an indeterminate form.

Let us analyze the following system of equations

\[ \begin{aligned} a_1 E_1 + a_2 E_2 + a_3 E_3 + \ldots + a_n E_n + \omega_1 &= 0,\\ b_1 E_1 + b_2 E_2 + b_3 E_3 + \ldots + b_n E_n + \omega_2 &= 0, \tag{1}\\ c_1 E_1 + c_2 E_2 + c_3 E_3 + \ldots + c_n E_n + \omega_3 &= 0, \end{aligned} \]

to which correspond the normal equations

\[ \begin{aligned} [aa]\,k_1 + [ab]\,k_2 + [ac]\,k_3 + \omega_1 &= 0,\\ [ab]\,k_1 + [bb]\,k_2 + [bc]\,k_3 + \omega_2 &= 0, \tag{2}\\ [ac]\,k_1 + [bc]\,k_2 + [cc]\,k_3 + \omega_3 &= 0. \end{aligned} \]

If the third equation of system (1) is a consequence of the first two, then the coefficients and the free term of this equation can be expressed in terms of the coefficients and free terms of the first two equations as follows²:

\[ c_i = \alpha a_i + \beta b_i, \]

\[ \omega_3 = \alpha \omega_1 + \beta \omega_2, \tag{3} \]

and, correspondingly, we shall have

\[ [ac] = \alpha[aa] + \beta[ab], \qquad [bc] = \alpha[ab] + \beta[bb], \]

\[ [cc] = \alpha[ac] + \beta[bc]. \tag{4} \]

Let us solve the system of normal equations (2), using the substitution method adopted in adjustment practice.

After eliminating \(k_1\), we shall have

\[ [bb.1]\,k_2 + [bc.1]\,k_3 + \omega_{2.1} = 0 \]

\[ [bc.1]\,k_2 + [cc.1]\,k_3 + \omega_{3.1} = 0. \tag{5} \]

After eliminating \(k_2\), we obtain

\[ [cc.2]\,k_3 + \omega_{3.2} = 0 \tag{6} \]

Taking (4) into account, we transform the coefficients and free terms of equations (5) and (6)

\[ [bc.1] = [bc] - \frac{[ab][ac]}{[aa]} = \alpha[ab] + \beta[bb] - \frac{[ab]}{[aa]}\bigl(\alpha[aa] + \beta[ab]\bigr) \]

or

\[ [bc.1] = \beta[bb.1]. \]

Analogously we find

\[ [cc.1]=\beta^{2}[bb.1], \qquad \omega_{3}.1=\beta(\omega_{2}.1), \]

\[ [cc.2]=\beta^{2}[bb.1]-\frac{\beta^{2}[bb.1]^{2}}{[bb.1]}=0, \]

\[ \omega_{3}.2=\beta(\omega_{3}.1)-\frac{\beta[bb.1]\cdot(\omega_{2}.1)}{[bb.1]}=0. \]

Substituting the transformed values of the coefficients and free terms into (5) and (6), we obtain

\[ [bb.1]\,k_{2}+\beta[bb.1]\,k_{3}+\omega_{2}.1=0, \]

\[ \beta[bb.1]\,k_{2}+\beta^{2}[bb.1]\,k_{3}+\beta(\omega_{2}.1)=0, \tag{5'} \]

\[ 0k_{3}+0=0, \tag{6'} \]

whence

\[ k_{3}=\frac{0}{0}. \tag{7} \]

From the analysis of this example it follows that the coefficients and the free term of the normal equation corresponding to an equation-consequence become zeros after eliminating the correlates of the equations whose function is the given consequence.

In the example considered, we assumed that the third, i.e. the last, equation of system (1) was the consequence. However, such a position of the consequence among the other equations is not obligatory if, besides the dependent equations, there are also independent ones.

Let, for a complete geodetic quadrilateral (Fig. 1), five correction equations be composed, among which one is superfluous, i.e. a consequence,

Fig. 1

\[ \begin{aligned} E_{1}+E_{2}+E_{3}+E_{4} \qquad &+\omega_{1}=0,\\ E_{3}+E_{4}+E_{5}+E_{6} \qquad &+\omega_{2}=0,\\ E_{5}+E_{6}+E_{7}+E_{8}+\omega_{3}&=0,\\ E_{1}+E_{2} \qquad &+E_{7}+E_{8}+\omega_{4}=0,\\ a_{1}E_{1}-a_{2}E_{2}+a_{3}E_{3}-a_{4}E_{4}+a_{5}E_{5}-a_{6}E_{6}+a_{7}E_{7}-a_{8}E_{8}+\omega_{5}&=0 \end{aligned} \tag{8} \]

and the corresponding normal equations

\[ \begin{aligned} 4k_{1}+2k_{2}+0k_{3}+2k_{4}+[aa]\,k_{5}+\omega_{1}&=0,\\ 2k_{1}+4k_{2}+2k_{3}+0k_{4}+[ba]\,k_{5}+\omega_{2}&=0,\\ 0k_{1}+2k_{2}+4k_{3}+2k_{4}+[ca]\,k_{5}+\omega_{3}&=0,\\ 2k_{1}+0k_{2}+2k_{3}+4k_{4}+[da]\,k_{5}+\omega_{4}&=0,\\ [aa]\,k_{1}+[ba]\,k_{2}+[ca]\,k_{3}+[da]\,k_{4}+[aa]\,k_{5}+\omega_{5}&= \end{aligned} \tag{9} \]

In equation (9), by \(a, b, c, d\) are denoted the coefficients of the first four equations of corrections (8), which, as is evident, are respectively equal to unity or zero; and consequently,

\[ \begin{aligned} [aa]&=a_1-a_2+a_3-a_4,\\ [ba]&=a_3-a_4+a_5-a_6,\\ [ca]&=a_5-a_6+a_7-a_8,\\ [da]&=a_1-a_2+a_7-a_8. \end{aligned} \tag{10} \]

Let us solve the normal equations (9), as in the preceding example, by the method of substitution.

Eliminating \(k_1\) from the system (9) by determining it from the first equation, we obtain

\[ k_1=-\frac{1}{2}k_2-\frac{1}{2}k_4-\frac{[aa]}{2}k_5-\frac{\omega_1}{2} \tag{11} \]

\[ \begin{aligned} 3k_2+2k_3-k_4+[ba.1]\,k_5+\omega_2.1&=0,\\ 2k_2+4k_3+2k_4+[ca.1]\,k_5+\omega_3.1&=0,\\ -k_2+2k_3+3k_4+[da.1]\,k_5+\omega_4.1&=0,\\ [ba.1]\,k_2+[ca.1]\,k_3+[da.1]\,k_4+[aa.1]\,k_5+\omega_5.1&=0. \end{aligned} \tag{12} \]

After eliminating \(k_2\), we find

\[ k_2=-\frac{2}{3}k_3+\frac{1}{3}k_4-\frac{[ba.1]}{3}k_5-\frac{\omega_2.1}{3} \tag{13} \]

\[ \begin{aligned} \frac{8}{3}k_3+\frac{8}{3}k_4+[ca.2]\,k_5+\omega_3-\frac{2}{3}\omega_2-\frac{1}{3}\omega_1&=0,\\ \frac{8}{3}k_3+\frac{8}{3}k_4+[da.2]\,k_5+\omega_4+\frac{1}{3}\omega_2&=0,\\ [ca.2]\,k_3+[da.2]\,k_4+[aa.2]\,k_5+\omega_5.2&=0. \end{aligned} \tag{14} \]

After eliminating \(k_3\), we shall have

\[ k_3=-k_4-\frac{3}{8}[ca.2]\,k_5-\frac{3}{8}\omega_3+\frac{1}{4}\omega_2-\frac{1}{8}\omega_1 \tag{15} \]

\[ \left(\frac{8}{3}-\frac{8}{3}\right)k_4+[da.3]\,k_5+\omega_4-\omega_3+\omega_2-\omega_1=0 \tag{16} \]

\[ [da.3]\,k_4+[aa.3]\,k_5+\omega_5.3=0; \]

the coefficient of \(k_4\) from the first equation of the system (16) is equal to zero; the absolute term of this equation, on the basis of (8), is likewise equal to zero; consequently, the coefficient of \(k_5\) is also equal to zero. The latter could also be proved by expanding the symbol \([da.3]\). Expanding \([da.3]\), we arrive at the expression

\[ [da.3]=[da]+[ba]-[ca]-[aa], \]

which, on the basis of (10), is equal to zero.

Substituting the true values of the quantities found into system (16), we obtain

\[ 0k_1 + 0k_5 + 0 = 0 \]

\[ 0k_4 + [\alpha\alpha . 3]k_5 + \omega_5 \cdot 3 = 0. \tag{17} \]

The circumstance that the coefficients and the free term of the first equation (17) are equal to zero shows that the fourth equation of corrections (8) is a consequence of the first three equations.

Excluding from (8) the third correction equation—the consequence—and everything connected with it from the remaining equations, we obtain expressions corresponding to a complete geodetic quadrilateral, namely:

\[ \begin{aligned} &E_1 + E_2 + E_3 + E_4 &&+ \omega_1 = 0\\ &\phantom{E_1 + E_2 + {}}E_3 + E_4 + E_5 + E_6 &&+ \omega_2 = 0\\ &\phantom{E_1 + E_2 + E_3 + E_4 + {}}E_5 + E_6 + E_7 + E_8 &&+ \omega_3 = 0\\ &\alpha_1 E_1 - \alpha_2 E_2 + \alpha_3 E_3 - \alpha_4 E_4 + \alpha_5 E_5 + \alpha_6 E_6 + \alpha_7 E_7 - \alpha_8 E_8 + \omega_5 = 0 \end{aligned} \tag{8'} \]

\[ \begin{aligned} &4k_1 + 2k_2 + 0k_3 + [a\alpha]k_5 + \omega_1 = 0\\ &2k_1 + 4k_2 + 2k_3 + [b\alpha]k_5 + \omega_2 = 0\\ &0k_1 + 2k_2 + 4k_3 + [c\alpha]k_5 + \omega_3 = 0\\ &[a\alpha]k_1 + [b\alpha]k_2 + [c\alpha]k_3 + [\alpha\alpha]k_5 + \omega_5 = 0 \end{aligned} \tag{9'} \]

\[ k_1 = -\frac{1}{2}k_2 - \frac{[a\alpha]}{2}k_5 - \frac{\omega_1}{2} \tag{11'} \]

\[ \begin{aligned} &3k_2 + 2k_3 + [b\alpha . 1]k_5 + \omega_2 . 1 = 0\\ &2k_1 + 4k_3 + [c\alpha . 1]k_5 + \omega_3 . 1 = 0\\ &[b\alpha . 1]k_2 + [c\alpha . 1]k_3 + [\alpha\alpha . 1]k_5 + \omega_5 . 1 = 0 \end{aligned} \tag{12'} \]

\[ k_2 = -\frac{2}{3}k_3 - \frac{[b\alpha . 1]}{3}k_5 \tag{13'} \]

\[ \frac{8}{3}k_3 + [c\alpha . 2]k_5 + \omega_3 . 2 = 0 \]

\[ [c\alpha . 2]k_3 + [\alpha\alpha . 2]k_5 + \omega_5 . 2 = 0 \tag{14'} \]

\[ k_3 = -\frac{3[c\alpha . 2]}{8}k_5 - \frac{\omega_3 . 2}{8} \tag{15'} \]

\[ [\alpha\alpha . 3]k_5 + \omega_5 . 3 = 0 \tag{16'} \]

Thus, if a consequence has been formed in excess of the necessary number of conditions, then its participation in the adjustment is not reflected in the result and does not entail any recomputations. In the same case, when a consequence is included in the adjustment in place of some condition, then after detecting and excluding it, it is necessary to formulate the omitted condition and to perform a recomputation in the solution of the normal equations only with respect to this equation and the quantities connected with it, leaving everything else unchanged.

Criteria of Dependence in the Two-Group Method of Adjustment

The examples considered above relate to the simultaneous solution of all equations; in practice, however, two-group adjustment is often applied. Let us clarify at what stage of the computations the consequences are revealed in this case.

In adjustment by the two-group method, the following three cases of distribution of dependent equations among groups are possible:

1. All dependent equations are assigned to one group (the first or the second).
2. One part of the dependent equations is assigned to one group, the other to the second. In this case the consequence will always be in the second group.
3. Only one dependent equation—the consequence—is assigned to the second group.

In the first and second cases, the consequence is revealed when solving the normal equations of the first or second group, as was explained above. In the last case, the consequence is revealed when transforming the coefficients and absolute terms of the correction equations of the second group, which, for the consequence equation, become zero.

To confirm the last conclusion, take a geodetic quadrilateral (Fig. 1), and write for it the following equations:

\[ \begin{aligned} &E_1+E_2+\ [[\unclear: continuation of first equation]] &&-E_5-E_6 &&+\omega_1=0,\\ &[[\unclear: beginning of second equation]]+E_3+E_4 &&&&-E_7-E_8+\omega_2=0,\\ &E_1+E_2+E_3+E_4+E_5+E_6+E_7+E_8+\omega_3=0,\\ &E_1+E_2+E_3+E_4+ &&&&+\omega_4=0. \end{aligned} \tag{18} \]

Here the fourth equation is a consequence of the first three. Assigning the first three equations to the first group, we obtain the following expressions for the system of normal equations and for the correlates themselves:

\[ \begin{aligned} 4k_1+\omega_1&=0,\\ 4k_2+\omega_2&=0,\\ 8k_3+\omega_3&=0, \end{aligned} \tag{19} \]

\[ \begin{aligned} k_1&=-\frac{\omega_1}{4},\\ k_2&=-\frac{\omega_2}{4},\\ k_3&=-\frac{\omega_3}{8}. \end{aligned} \tag{20} \]

The auxiliary correlates in this case will be equal to

\[ \rho_1=-\frac{1}{2} \]

\[ \rho_2=-\frac{1}{2} \]

\[ \rho_3=-\frac{1}{2} \]

The transformed values of the coefficients and of the free term of the equation of the second group are found from the expressions

\[ A_1=A_2=1+\rho_1+\rho_2;\quad A_3=A_4=1+\rho_2+\rho_3;\quad A_5=A_6=-\rho_1-\rho_3;\quad A_7=A_8=-\rho_2+\rho_3, \]

\[ W_1=\omega_4+\rho_1\omega_1+\rho_2\omega_2+\rho_3\omega_3. \]

Substituting the values of \(\rho_i\) from (21) and taking into account, on the basis of (18), that

\[ \omega_4=-\frac{1}{2}\omega_1+\frac{1}{2}\omega_2+\frac{1}{2}\omega_3, \]

we obtain

\[ A_1=A_2=A_3=A_4=A_5=A_6=A_7=A_8=0;\quad W_1=0. \]

Conclusions

1. It must be remembered that the success of adjustment computations depends first of all on the correct determination of the number of conditions and on their correct formulation, since only in this case will the expenditure of labor and time on solving the problem be minimal.

2. In the examples considered above we had only one consequence equation each; however, the conclusions following from these examples remain valid when any number of consequences is included in the processing.

3. When solving the normal equations jointly, the consequence is detected in the process of solving these equations. The indication of the presence of a consequence in this case is the circumstance that the coefficients and the free term of the normal equation corresponding to the consequence turn into zeros (in practice these quantities will only be close to zero) after the elimination of the correlates of the equations of which the given consequence is a function.

4. In the two-group method of adjustment, the following three cases of distributing the correction equations among groups are possible:

a) all dependent correction equations are assigned to one group (the first or the second);

b) one part of the dependent equations is assigned to the first group, the other to the second group; in this case the consequence always falls into the second group;

c) only one dependent correction equation—the consequence—is assigned to the second group.

In the first and in the second case, the dependent consequence equation is detected in the process of transforming the equations of the second group; the transformed values of the coefficients and of the free term of the dependent consequence equation are equal to zero.

1. If the consequence equation has been drawn up in excess of the required number of conditions, then its participation in the processing is not reflected in the result and does not entail any recalculations. In the case where the consequence has been included in the processing in place of some condition, after detecting and eliminating it one must write the omitted condition and carry out the recalculations connected only with this equation, leaving everything else unchanged.

2. If, when formulating the conditions, difficulty arises in determining the independence of some equation, then such an equation must be included in the processing, in the course of which the question of its independence will be resolved automatically.

3. Since, when solving the normal equations, the entire group of dependent equations is detected, any one of these equations may be excluded from the processing, and not only the last one. This circumstance should be borne in mind when sinusoidal dependent equations are present.

1. Since any one of the dependent equations is a consequence of the other equations, in the present work by “consequence” we shall understand the last dependent equation in order. ↩

2. “Handbook of Geodesy,” W. Jordan, ed. 1939. ↩