General Form of Constraint Relations in the Adjustment of Surface Ray Networks and the Problems Arising from Them

B. F. Krutoi

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

§ 1. Statement of the Question

In my article [1], the principal types of surface ray networks were indicated, their properties were considered, and a general method was given for the unique construction of such networks from one of the necessary sets of initial data. The next task is the adjustment of surface ray networks, which arises in the case when the set of initial data is extended, i.e., exceeds what is necessary for the unique construction of these networks.

The indicated adjustment problem naturally splits into the following three stages.

Preparatory stage—the compilation, for a ray network, of certain primordial constraint relations, which include either only the extended initial data, or else also functions of these initial data.

Intermediate stage—the reduction of the primordial constraint relations to a plane form, as a result of which they pass into the corresponding initial equations of corrections to directly or indirectly observed quantities.

Final stage—the solution of the plane initial equations of corrections compiled above under some additional requirement, for the purpose of obtaining adjusted data of the extended set or of obtaining their functions.

In this and a number of my following articles, all attention will be concentrated on the first two stages of the adjustment of surface ray networks. In doing so, we shall consider the primordial constraint relations and the initial equations of corrections for all three principal methods of adjustment of the indicated networks: a) mixed, b) direct, c) indirect.

Our aim will be:

1) to establish the most important types of primordial constraint relations arising in the named three methods of adjustment of surface ray networks;

2) to present these primordial constraint relations, having-

which in the general case have a complex structure, by simpler planar initial correction equations;

3) the search for more efficient ways of composing the initial correction equations in surface ray networks of general form.

For a better clarification of all these questions and to reveal their internal interconnection, in the present introductory article we shall examine the principal features of the enumerated methods of adjustment and shall write, in general form, the corresponding relations of connection—the primitives and derivatives. In conclusion we shall indicate in somewhat greater detail our further tasks in developing the range of questions outlined here.

§ 2. Primitive relations in the three principal methods of adjusting observations

Let us consider the nature of those primitive relations of connection which arise in the three principal methods of adjusting observations mentioned above [2].

Mixed method. In the mixed method of adjusting observations, the primitive relations of connection, numbering $\mathcal M$, include, in addition to the enlarged initial data, also $s$ of their functions, and each of these relations contains no fewer than two observed quantities¹.

The number $\mathcal M$ of the primitive relations of connection themselves in the indicated method of adjustment is always less than the number $n$ of observed quantities entering into them. Consequently, from these primitive relations it is impossible to determine each separately taken observed quantity independently of the remaining observed quantities.

As for the number $s$ of functions contained in the primitive relations of connection under the mixed adjustment of observations, it exceeds the number $\mathcal M$ of primitive relations, or else, in the limiting case, is equal to this number. The latter occurs when the enlarged set of initial data in the ray network is expressed in the required form and when, consequently, the need for adjusting observations disappears of itself.

It follows from what has been said that, in the mixed adjustment of observations, the primitive relations of connection have the following form:

\[ F_e(\bar{\nu}_1,\bar{\nu}_2,\ldots,\bar{\nu}_s;\bar{\xi}_1,\bar{\xi}_2,\ldots,\bar{\xi}_n;\bar{\chi}_1,\bar{\chi}_2,\ldots,\bar{\chi}_q)\equiv \bar{F}_e=0 \tag{1} \]

\[ (e=1,2,\ldots,\mathcal M;\ n>\mathcal M\geqslant s), \]

where $\bar{\xi}_1,\bar{\xi}_2,\ldots,\bar{\xi}_n$ are the adjusted values of the observed quantities $\xi_i$, of the directions $\beta_j$, or of the angles $\kappa_l$;

$\bar{\chi}_1,\bar{\chi}_2,\ldots,\bar{\chi}_q$ are fixed quantities;

$\bar{\nu}_1,\bar{\nu}_2,\ldots,\bar{\nu}_s$ are the adjusted values of the functions $\nu_s$ of the observed $\xi_i$ and fixed $\chi_\mu$ quantities.

Direct method. The direct method of adjusting observations is obtained from the mixed method considered above by means of its appropriate transformation. For this purpose, using any $s$ equalities (1), let us express the $s$ functions $\nu_g$ entering into them through the remaining adjusted-observed $\bar{\xi}_i$ and fixed $\bar{\chi}_\mu$ quantities. We shall then have

\[ \bar{\nu}_g=\varphi_g(\bar{\zeta}_1,\bar{\zeta}_2,\ldots,\bar{\zeta}_n;\bar{\chi}_1,\bar{\chi}_2,\ldots,\bar{\chi}_q)=\bar{\varphi}_g=0 \tag{2} \]

\[ (g=1,2,\ldots,s). \]

If we then substitute the values (2) found for the \(s\) functions \(\bar{\nu}_g\) into the remaining \(m-s=r\) equations of the set (1), we arrive at the transformed constraint relations of the form

\[ \Phi_j(\bar{\zeta}_1,\bar{\zeta}_2,\ldots,\bar{\zeta}_n;\bar{\chi}_1,\bar{\chi}_2,\ldots,\bar{\chi}_q)=\bar{\Phi}_j=0 \tag{3} \]

\[ (j=1,2,\ldots,r;\ n>r=m-s), \]

which contain only the adjusted-observed \(\bar{\zeta}_i\) and fixed \(\bar{\chi}_\mu\) quantities.

The set of transformed constraint relations (3) will be the desired one. This set serves as the basis for the aforementioned direct method of adjusting observations, in which the original data of extended composition are corrected directly—namely, their observed part.

The indirect method. This method of adjusting observations has the special feature that, in it, each observed quantity \(\bar{\zeta}_i\) is expressed explicitly through certain known functions \(\bar{\vartheta}_h\) of all the original data. The number \(m\) of the indicated functions \(\bar{\vartheta}_h\) is less than the number \(n\) of observed quantities \(\zeta_i\), or, in the limiting case, equal to the number of these quantities. The latter occurs when the extended set of original data degenerates into the necessary set.

From what has been said we conclude that, under the indirect adjustment of observations, the transformed constraint relations will have the following form:

\[ \bar{\zeta}_i=\Psi_i(\bar{\vartheta}_1,\bar{\vartheta}_2,\ldots,\bar{\vartheta}_m;\bar{\chi}_1,\bar{\chi}_2,\ldots,\bar{\chi}_q)=\bar{\Psi}_i \tag{4} \]

\[ (i=1,2,\ldots,n;\ m\leq n), \]

where \(\bar{\vartheta}_1,\bar{\vartheta}_2,\ldots,\bar{\vartheta}_m\) are known functions of the adjusted original data, so that

\[ \bar{\vartheta}_h=f_h(\bar{\zeta}_1,\bar{\zeta}_2,\ldots,\bar{\zeta}_n;\bar{\chi}_1,\bar{\chi}_2,\ldots,\bar{\chi}_q)=\bar{f}_h \tag{5} \]

\[ (h=1,2,\ldots,m;\ m\leq n). \]

§ 3. Reduction to the finite-difference planar form

For the purposes of the subsequent adjustment of observations, the transformed constraint relations (1), (3), (4) must be reduced to a finite-difference planar form. This can easily be done if the adjusted quantities \(\bar{\zeta}_i\), \(\bar{\nu}_g\), \(\bar{\vartheta}_h\) contained in them are represented as follows:

\[ \bar{\zeta}_i=\zeta_i+\Delta \zeta_i=\zeta_i+v_i \qquad \bar{\nu}_g=\nu_g^{(0)}+\Delta \nu_g^{(0)}=\nu_g^{(0)}+\varepsilon_g \tag{6} \]

\[ \bar{\vartheta}_h=\vartheta_h^{(0)}+\Delta \vartheta_h^{(0)}=\vartheta_h^{(0)}+\pi_h, \]

where

\[ \zeta_i,\ v_i \text{ are the observed quantities and their corrections;} \]

\[ \nu_g^{(0)},\ \vartheta_h^{(0)} \text{ and } \varepsilon_g,\ \pi_h \text{ are the approximate (supporting) values of the functions } \nu_g,\ \vartheta_h \text{ and their corrections.} \]

The observed quantities \(\zeta_i\) entering here are obtained by us sufficiently accurately directly from the observations themselves. The reference values \(\nu_g^{(0)}, \vartheta_h^{(0)}\) of the functions \(\nu_g, \vartheta_h\) can be found with good accuracy from relations (2) and (5) by replacing \(\bar{\zeta}_i\) in them by \(\zeta_i\). Hence it follows that the corrections \(v_i, \varepsilon_g, \pi_h\) entering into (6) are always small in magnitude. Taking this essential circumstance into account, we arrive at the conclusion that relations (1), (3), (4) may be quite reliably represented by the following Taylor expansions:

\[ \sum_{g=1}^{s} a_{eg}\varepsilon_g+\sum_{i=1}^{n} b_{ei}v_i+w_e=0 \qquad (e=1,2,\ldots,M;\; n>M\geq s) \tag{7} \]

\[ \sum_{i=1}^{n} c_{ji}v_i+\omega_j=0 \qquad \left( \begin{array}{c} j=1,2,\ldots,r\\ n>r=M-s \end{array} \right) \tag{8} \]

\[ \sum_{h=1}^{m} d_{ih}\pi_h-l_i=v_i, \qquad (i=1,2,\ldots,n;\; m<n) \tag{9} \]

or, in matrix notation,

\[ a_{(Ms)}\varepsilon_{(s)}+b_{(Mn)}v_{(n)}=0_{(M)} \qquad (n>M\geq s) \tag{7.1} \]

\[ c_{(rn)}v_{(n)}+\omega_{(r)}=0_{(r)} \qquad (r=M-S<n) \tag{8.1} \]

\[ d_{(nm)}\pi_{(m)}-l_{(n)}=v_{(n)} \qquad (n>m) \tag{9.1} \]

The coefficients and free terms of the reductions of plane relations (7)—(9) obtained by us are determined, of course, as follows:

\[ \left\{ \begin{aligned} a_{eg}&=\left(\frac{\partial F_e}{\partial \nu_g}\right)_0, & b_{ei}&=\left(\frac{\partial F_e}{\partial \zeta_i}\right)_0, \\[4pt] w_e&=F_e\bigl(\nu_1^{(0)},\nu_2^{(0)},\ldots,\nu_s^{(0)};\, \zeta_1,\zeta_2,\ldots,\zeta_n;\, \bar{\chi}_1,\bar{\chi}_2,\ldots,\bar{\chi}_q\bigr) =F_e^{(0)} \end{aligned} \right. \tag{10} \]

\[ \left\{ \begin{aligned} c_{ji}&=\left(\frac{\partial \Phi_j}{\partial \zeta_i}\right)_0, \\[4pt] \omega_j&=\Phi_j\bigl(\zeta_1,\zeta_2,\ldots,\zeta_n;\, \bar{\chi}_1,\bar{\chi}_2,\ldots,\bar{\chi}_q\bigr) =\Phi_j^{(0)} \end{aligned} \right. \tag{11} \]

\[ \left\{ \begin{aligned} d_{ih}&=\left(\frac{\partial \Psi_i}{\partial \vartheta_h}\right)_0, \\[4pt] l_i&=\zeta_i-\Psi_i\bigl(\vartheta_1^{(0)},\vartheta_2^{(0)},\ldots,\vartheta_m^{(0)};\, \bar{\chi}_1,\bar{\chi}_2,\ldots,\bar{\chi}_q\bigr) =\zeta_i-\Psi_i^{(0)}. \end{aligned} \right. \tag{12} \]

Here the superscript \({}^{\circ}\) indicates that the corresponding partial derivatives are computed for the observed \(\zeta_i\) and the reference quantities \(\nu_g^{(0)}, \vartheta_h^{(0)}\).

§ 4. Names and Subsequent Problems

Names. The reductions of the derived plane relations (7)—(9) shall be called the initial equations of corrections. They are the desired plane representation of the original reductions of per-

imaginary relations (1), (3), (4). These initial systems (1), (3), (4) and the corresponding plane systems (7)—(9) will henceforth be called:

a) mixed — (1) and (7);
b) direct — (3) and (8);
c) indirect — (4) and (9).

Subsequent problems. For surface ray networks, the most difficult question is that of the principal types of primary and finite-difference connection relations in the mixed and direct adjustment of these networks. Especially difficult here will be the problem of finding the most advantageous ways of composing such relations in ray networks of general structure. Therefore, on this large and important question we shall concentrate our principal attention, devoting the next two articles to it.

As for the simpler problem of obtaining indirect connection relations in surface ray networks, we shall deal with this in the last two articles. The most essential point here will be the general method I have proposed for determining the coefficients of the indirect equations of corrections in ray networks on an arbitrary regular surface.

For the sake of brevity of exposition, in this collection of my articles a single sequence of numbering has been introduced for the mathematical expressions obtained in them.

REFERENCES

1. Krutoi B. F. Surface adjusted ray networks and a general method of their construction. Izvestiya TPI, vol. 93, 1958.

2. Jordan V. Handbook of Geodesy, vol. I, Redburo GUGK. Moscow, 1939.

1. The case in which each mixed relation of connection of primitive form contains only one observed quantity must be considered separately (see further, “indirect method”). ↩