Relationship Between Indirect and Conditional Observations

V. S. Nuvaryev

First of all, let us dwell on the very distinction between indirect and conditional measurements. From the general meaning of these definitions it may be assumed that the name “method of indirect measurements” is intended to emphasize that, in the given case, what were measured were not the sought quantities themselves, but quantities connected with them. A distinctive feature of this method is the so-called error equations, which connect the direct observations \(l_i\) with \(m\) functions of them \(X,Y,Z\ldots\) and have the form

\[ a_iX+b_iY+c_iZ+\ldots-l_i=v_i\quad (i=1,2,\ldots,n);\ n>m. \tag{a} \]

The method of conditional measurements is characterized by equations that connect only direct observations (or corrections to them), or only corrections to functions of observations¹. These equations have the form

\[ [a_iL_i]+a_0=0;\quad [b_iL_i]+b_0=0\ldots [r_iL_i]+r_0=0 \tag{b} \]

or, with respect to corrections to observations or their functions,

\[ [a_iv_i]+w_1=0;\quad [b_iv_i]+w_2=0\ldots [r_iv_i]+w_r=0 \tag{b} \]

\[ (i=1,2\ldots n);\ (n>r). \]

We find that system \((a)\) can be represented in a form that gives us greater scope in investigations², namely:

\[ a_iX+b_iY+c_iZ+\ldots+m_iM-L_i=0;\ (i=1,2\ldots n);\ n>m \tag{a}' \]

If there exists a system of equations \((a)\), then there also exists a system \((a)'\), in which \(m\) true values of the functions of direct observations \(X,Y,Z\ldots\) are connected with \(n\) values of the quantities \(L_i\) \((n>m)\) subject to measurement. We may assert that between the quantities \(X,Y,Z,\ldots M\) there exist exact conditions \((a)'\), i.e. conditional equations. If their form differs from the usual conditional equations \((b)\), their meaning is the same: the exact values of the quantities entering systems \((a)'\) and \((b)\) are connected by exact conditions.

Both systems are solved under the condition

\[ [v^2]=minimum\quad \text{or}\quad [pv^2]=minimum \tag{c} \]

As follows from mathematical analysis, finding the minimum of the function \((c)\), in which the \(v_i\) are defined by equations \((a)\) and are not related to one another, is solved by the method of an absolute minimum; but if the variables in the function \((c)\) are related by equations of the form \((b)'\), then one seeks the relative minimum of this function, i.e., by means of undetermined Lagrange multipliers (correlates). What, then, is to be understood by the name “indirect” measurements? It is customary to think that in this method the functions (or corrections to them) of direct observations are found by seeking the absolute minimum of the function \((c)\). But, as has already been indicated, one can also, and quite successfully, seek corrections to the functions of direct observations by applying correlate adjustment. Thus “indirect” measurements are not at all connected with the method of solution. In mathematical analysis it is proved that, in the case when the variables in the function \((c)\) are related by \(r\) conditions \((b)\), one can pass from the method of the relative minimum to the method of the absolute minimum, i.e., using the existing terminology, from the method of conditional adjustment to the method of indirect adjustment. To do this, it is necessary to solve the system \((b)'\) with respect to, say, the first \(r\) corrections and substitute their values into the function \((c)\), after which the absolute minimum is sought for the new function, in which the remaining \(n-r\) corrections are no longer connected by any conditions.

In other words, from the point of view of mathematical analysis, one should not present proofs of the connection between indirect and conditional observations in the form in which this question is set forth in textbooks, since in such proofs there is no transition from the method of conditional measurements to the method of indirect measurements, but one and the same problem is solved by different devices. We believe that it is necessary to pose another problem, namely: to pass from the method of indirect measurements to the method of conditional measurements, i.e., from the absolute minimum to pass to the relative minimum. Such a problem is not considered in mathematical analysis, which is why we consider it necessary to carry out an investigation of this question. It was indicated above that what has been proved is the coincidence of the results of solving one and the same problem, and not the connection between the methods. We believe that between the systems \((a)\) and \((b)\) there is a sharp difference, consisting in the fact that system \((a)\) includes both direct observations and their functions, whereas system \((b)\) includes only direct observations. The transition from the system of indirect observations \((a)\) to the system of conditional observations could be proved only on the basis of the form of writing the equations \((a)'\) proposed by us.

Let us solve the system \((a)'\) with respect to the unknowns \(X, Y, Z \ldots\), using for this purpose, say, the first \(m\) equations; then, denoting the determinant of this system by \(D\), and the determinants for \(X, Y, \ldots, M\) respectively by \(D_1, D_2, \ldots, D_m\), we obtain

\[ X=\frac{D_1}{D};\quad Y=\frac{D_2}{D};\ldots M=\frac{D_m}{D}; \]

substituting the values \(X, Y, \ldots\) into the system of the remaining \(n-m\) equations \((a)\) and multiplying by \(D\), we obtain

\[ a_{m+1}D_1+b_{m+1}D_2+\cdots+m_{m+1}D_m-DL_{m+1}=0 \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ a_nD_1+b_nD_2+\cdots+m_nD_m-DL_n=0 \tag{d} \]

The equations \((d)\) will have the form of the conditional equations \((b)\), with the difference that in them \(a_0=b_0=\cdots=0\).

For simplicity, let us take a system of four equations:

\[ \begin{array}{ll} a_1X+b_1y-L_1=0, & a_3X+b_3y-L_3=0,\\ a_2X+b_2y-L_2=0, & a_4X+b_4y-L_4=0. \end{array} \tag{e} \]

We solve the first two equations of the system \((e)\) and substitute the values found into the 3rd and 4th equations; then we obtain a system of equations of the form

\[ \begin{aligned} D_{23}L_1-D_{13}L_2+D_{12}L_3&=0,\\ D_{24}L_1-D_{14}L_2+D_{12}L_4&=0, \end{aligned} \tag{f} \]

where \(D_{ik}\) are the second-order minors of the matrix of the system, corresponding to the pairs of equations \((e)\) with numbers \(ik\).

If in the systems \((e)\) and \((f)\) the true values \(L_i\) are replaced by the measured values \(l_i\), then we obtain the systems customary for the method of indirect and conditional observations.

Our task consists in showing that the result of adjustment will not depend on which system we choose for the solution, i.e. it is necessary to show that the adjusted value, say, \(l_1^\circ\), by the methods of indirect and conditional observations will coincide.

It is known that the value \(l_1^\circ\), adjusted by the method of conditional observations, may be represented in the form

\[ l_1^\circ=[Fl], \]

where \(F\) is determined by the formulas of the method of least squares (37), or by the formulas (41), (42), and (43) derived by us.\(^1\) Since the latter formulas will lead to the goal more quickly, we shall use them, taking into account that in the system \((f)\), in place of the coefficients \(a\) and \(b\), there stand determinants composed from the coefficients of equations \((e)\); in equations (41), (42), and (43), however, the determinants must be composed from the coefficients of equations \((f)\).

Then

\[ F_1=\frac{1}{D'}\left\{D_{12}^2D_{14}^2+D_{13}^2D_{12}^2+D_{12}^4\right\} =\frac{D_{12}^2}{D'}(D_{12}^2+D_{13}^2+D_{14}^2); \]

\[ F_2=-\frac{1}{D'}\left\{-D_{12}^2D_{24}D_{14}-D_{12}^2D_{23}D_{13}\right\} =\frac{D_{12}^2}{D'}(D_{13}D_{23}+D_{14}D_{24}); \]

\[ F_3=\frac{1}{D'}\left\{(-D_{14}D_{23}+D_{13}D_{24})D_{12}D_{14}-D_{12}D_{23}D_{12}^2\right\} =\frac{D_{12}^2}{D'}(D_{14}D_{34}-D_{12}D_{23}); \]

\[ F_4=\frac{1}{D'}\left\{(D_{14}D_{23}-D_{13}D_{24})D_{12}D_{13}-D_{12}^2D_{24}\right\} =-\frac{D_{12}^2}{D'}(D_{12}D_{24}+D_{13}D_{34}); \]

\[ \begin{aligned} D'=[D'^2]&=(-D_{14}D_{23}+D_{13}D_{24})^2+D_{12}^2D_{24}^2+D_{23}^2D_{12}^2+D_{14}^2D_{12}^2\\ &\quad +D_{12}^2D_{13}^2+D_{12}^4 =D_{12}^2(D_{12}^2+D_{13}^2+D_{14}^2+D_{23}^2+D_{24}^2+D_{34}^2) =D_{12}^2D; \end{aligned} \]

or

\[ \left. \begin{array}{ll} F_1=\dfrac{1}{D}(D_{12}^2+D_{13}^2+D_{14}^2); & F_3=\dfrac{1}{D}(D_{14}D_{34}-D_{12}D_{23});\\[1.2em] F_2=\dfrac{1}{D}(D_{13}D_{23}+D_{14}D_{24}); & F_4=\dfrac{1}{D}(D_{12}D_{24}+D_{13}D_{34}); \end{array} \right\} \tag{g} \]

\(^1\) V. S. Nuvar’ev. “Theorems on the most probable value of an adjusted quantity,” Izvestiya Tomskogo politekhnicheskogo instituta, vol. 67, issue 2.

Thus the solution of the conditional equations \((f)\), obtained from the system \((e)\), gives the adjusted value

\[ l_1^0=[Fl], \tag{h} \]

where \(F_i\) is determined by equations \((g)\).

Let us show that the solution of the system of error equations \((e)\), if \(L_i\) is replaced by \(l_i\), will lead to the result \((h)\).

The normal equations corresponding to the system \((e)\) will have the form

\[ [aa]x+[ab]y-[al]=0 \]

\[ [ab]x+[bb]y-[bl]=0, \]

whence

\[ x=\frac{1}{D}\bigl([bb][al]-[ab][bl]\bigr), \]

\[ y=\frac{1}{D}\bigl([aa][bl]-[ab][al]\bigr). \]

The determinant of the system of normal equations, on the basis of (40) and (41) (see the cited work), is equal to

\[ D=D_{12}^{2}+D_{13}^{2}+\cdots+D_{34}^{2}=[D_{kj}^{2}]. \]

Then

\[ l_1^0=a_1x+b_1y, \]

\[ l_1^0=\frac{1}{D}\left(a_1^2[bb]+b_1^2[aa]-2a_1b_1[ab]\right)l_1 +\frac{1}{D}\left(a_1a_2[bb]-a_1b_2[ab]+b_1b_2[aa]-a_2b_1[ab]\right)l_2+\cdots . \]

Denoting the coefficient of \(l_1\) by \(F_1\) and carrying out the multiplication, we have

\[ F_1=\frac{1}{D}\left(a_1^2b_2^2+a_2^2b_1^2-2a_1a_2b_1b_2+a_1^2b_3^2+a_3^2b_1^2-2a_1a_3b_1b_3+\right. \]

\[ \left.+a_1^2b_4^2+a_4^2b_1^2-2a_1a_4b_1b_4\right); \]

\[ F_1=\frac{1}{D}\left(D_{12}^{2}+D_{13}^{2}+D_{14}^{2}\right). \tag{i} \]

In exactly the same way we can obtain

\[ F_2=\frac{1}{D}\left(D_{13}D_{23}+D_{14}D_{24}\right)\quad \text{and so on.} \tag{i'} \]

Comparing the values \(F_i\) from \((i)\) and \((g)\), we see that the adjusted values \(l_i^0\) from the systems of equations \((e)\) and \((f)\), by the method of indirect and conditional observations, are identical.

Solution of One and the Same Example by the Method of Indirect and Conditional Observations

Let there be the system of equations:

\[ \begin{aligned} -2X+3Y-L_1&=0\\ 3X-2Y-L_2&=0\\ 6X+Y-L_3&=0\\ -X+2Y-L_4&=0. \end{aligned} \tag{k} \]

Let the measured values be

\[ l_1=6;\quad l_2=28;\quad l_3=125;\quad l_4=13. \]

Then from the normal equations

\[ 50x-8y-809=0 \]

\[ -8x+18y-113=0 \]

we obtain

\[ x=18.5;\quad y=14.5, \]

and the corrections to the observed values, obtained by the method of indirect observations, will be equal to

\[ v_1=+0.50;\quad v_2=-1.50;\quad v_3=+0.50;\quad v_4=-2.50. \]

Let us solve the system \((k)\) by the method of conditional observations. We compute the determinants of the system \((k)\).

Then the conditional equations on the basis of \((f)\) will have the form

\[ 15L_1+20L_2-5L_3=0 \]

\[ 4L_1+L_2-5L_4=0. \tag{l} \]

From the normal equations of the correlates

\[ 650k_1+80k_2+25=0 \]

\[ 80k_1+42k_2-13=0, \]

in which the free terms have been computed after substituting into the system \((l)\) the measured values \(l_i\), we obtain

\[ k_1=-0.1,\quad k_2=+0.5, \]

and the corrections to the observed values \(l_i\)

\[ v_1=+0.5;\quad v_2=-1.5;\quad v_3=+0.5;\quad v_4=-2.5; \]

i.e., the corrections obtained from solving system \((l)\) by the method of conditional equations are equal to the corrections found earlier by the method of indirect observations for system \((k)\).

Conclusions

1. Any system of \(n\) equations connecting direct observations with \(m\) functions of them \((n > m)\) can be reduced to \(n - m\) exact equations with respect to the direct observations.

2. The result of the adjustment does not depend on which of these systems is chosen for solution.

1. The possibility of introducing into the system of conditional equations \((b)\) corrections to functions of direct observations was proved in the work of V. S. Nuvaryev, A New Method for Adjusting Triangulations, Irkutsk, OGIZ, 1932. ↩

2. In considering the question of the relationship between the methods of “indirect” and “conditional” observations, Prof. Idelson (see his work The Method of Least Squares, Moscow, 1947, § 27, p. 195) uses the notation \(v_i=0\), which cannot be considered successful, since \(v_i\) are obviously not equal to zero. Nor can the notation of equations (according to A. A. Markov) in the form of approximate equalities be considered successful. ↩