Equalization of free geodetic networks by search methods for geodetic monitoring of structures in arctic areas

. While carrying out work to ensure the technical safety of buildings and structures in the Arctic regions, one of the important stages is geodetic monitoring. Provided that the complexity of ensuring the stability of reference points in the Arctic zone when conducting geodetic monitoring of engineering facilities, geodetic networks should preferably be equated as free, i.e., to equate the reference points on a par with the defined ones. The present article provides variants how to to equalize geodetic networks to carry out a search method of non-linear programming, which is given a theoretical substantiation for the method and the order of estimating accuracy of the outcomes of the equalization of the generalized Makarov method. The procedure of pseudo transformation of a matrix of normal equations of unknowns with the application of an algorithm of a search method of nonlinear programming is given. The correctness of the application of the search method to the pseudo transformation of a matrix of normal equations of unknowns by the coincidence of outcomes of the pseudo transformation of the given matrix based on skeleton expansion done in M. Mashimov's work is proved. Equalization and estimation of accuracy of free leveling network with the application of search method and generalized Makarov's method are carried out. The conclusion about the correctness of the application of a search method of nonlinear programming for the processing of free geodetic networks at the performance of geodetic monitoring of buildings and constructions in the Arctic area is made.


Introduction
As per the relevant federal legislation, state standards, and technical recommendations on the safety of buildings and structures in the Arctic regions of the Russian Federation, there exist high requirements for the safety of buildings and structures under construction and operation at all stages of a facility life cycle. In order to ensure the safety requirements of such buildings and structures it is necessary to conduct systematic observations of their technical condition. The main type of monitoring of engineering objects built in cold regions is geotechnical monitoring. Several scientific papers have already been provided concerning the importance and peculiarities of geodetic works, including geotechnical monitoring in the Arctic regions [1][2][3][4][5].
As per the code SP 305.1325800.2017 "Buildings and structures. Regulations for geotechnical monitoring during construction" geotechnical monitoring includes several works, including geodetic monitoring, that includes observations of foundation settlement of buildings and structures.
Geodetic monitoring is comprised now of the following set of works: designing the initial plan and height geodetic networks, fixing the initial points and deformation marks, stability control of the plan and height position of the initial points, conducting observations of the deformation marks, mathematical processing, and analysis of the accuracy of measurements, determination of deformation values and forecasting of deformation development.
While performing geodetic monitoring in the Arctic regions, special attention should be paid to the stability of reference points. Processes occurring in permafrost soils often lead to shifts in the position of such points. In this connection as it is outlined in several papers [4,6] there is a necessity to develop geodetic networks on the object of works with their subsequent leveling as free networks. I.e., initial points are included in the processing of results of measurements equally with defined ones.
Meanwhile, due to increasing productivity of PCs, and a significant array of measured values there exists a necessity for fast automated processing of results of geodetic measurements. Nonlinear programming methods, in particular the search method, which advantage is the possibility of measurement results equation without partial derivatives calculation, start to be applied more and more for geodetic measurements processing. Besides, the search methods possess the simplicity of realization of algorithms of problem solutions. As noted in [7,8] when solving nonlinear programming problems in the absence of constraints, methods using first and second derivatives can converge faster than methods not using derivatives. Nevertheless, when applying methods using derivatives in practice, we have to face two main obstacles. First, it is rather difficult or even impossible to obtain derivatives in the form of analytical functions needed for gradient algorithm or the algorithm using second-order derivatives in problems with a rather large number of variables [7,8]. Although the calculation of analytical derivatives can be replaced by the calculation of derivatives using difference schemes, the error arising in this case, especially in the neighborhood of an extremum, may limit the application of such approximation.
The second circumstance is that while applying optimization methods grounded upon the calculation of first and, if necessary, second derivatives, it takes quite a long time to prepare the problem for solution in comparison with search methods [7,8].
Due to the difficulties outlined above, it is also pointed out in [7] that search methods might be more user-friendly than methods using first and second derivatives.
The advantages of the application of search methods of nonlinear programming for processing results of geodetic measurements, including works on geodetic monitoring are marked in many publications by Russian and foreign authors [7][8][9][10][11].
Let us further consider the algorithm of the search method and the theoretical justification of its application for equating and assessing the accuracy of free geodetic networks.
As noted in [7,8], the search method is one of the optimization methods that does not apply derivatives to determine the minimum of the target function. Essentially, the search method per-forms a sequential change of each variable by some value Δ until the local minimum of the given function is found.
In turn, the algorithm for finding the minimum of the target function can be implemented in different ways. For example, based on the Hooke-Greeves, Nelder-Midd, Rosen Brock, etc. search algorithm [7,8]. To equalize and assess the accuracy of geodetic constructions we will use the search algorithm, which is a combination of the Powell and Davis-Swann-Kempe quadratic interpolation methods, described in detail in [7]. The software implementation of such a search algorithm in VBA is presented in the form of a block diagram in Figure 1.
The measurement results will be equated according to the least squares principle, i.e., the tar-get function is subject to minimization F(x) = V T PV = min, where V is a vector of corrections to measurement results, P and is a weight matrix of measurement results. Note that when performing the equalization, the mathematical procedure of matrix inversion is performed [12]. For example, in finding parameters for correct equalization the determination of the weight matrix P is essentially important, as it is inverse to the error covariance matrix of the observed quantities [7]. And for correct geodetic constructions accuracy estimation it is necessary to calculate inverse weight matrix Q of parameters, which is done through inversion of the normal equation coefficients matrix of unknowns N. The implementation of this mathematical procedure for matrix inversion can also be done by the algorithm of the least squares search method.
However, when performing the equation of measurement results by the search method of nonlinear programming, the question of estimation of accuracy becomes. The difficulty arises from the fact that when performing the equation by search method it is impossible to carry out the accuracy evaluation by the classical parametric method since in this case, it is impossible to make the normal equations of the unknowns. In this connection, it is proposed to evaluate the accuracy of the equation results by the algorithm of G.V. Makarov.
The information flowchart of Makarov's algorithm accuracy estimation generalized for the case of spatial geodetic network equalization is presented in Figure 2. Let us consider the application of the search method of nonlinear programming on an example of the onedimensional case-equating to a free leveling network.

Results
Consider the polygon given in the book of the geodesist M.M. Mashimov "Methods of mathematical processing of astronomical and geodetic measurements" (1990), the scheme of which is shown in Figure 3. Let independently and equally determined 5 excesses hi (i = 1, 2, ..., 5). The results of measurements are equal: Let's consider the polygon in figure 3 as a free leveling network, i.e., we will not take any point of the network as a starting point. The Makarov algorithm shown in Figure 2, but for a one-dimensional case, was used to estimate the accuracy of the equation results. In each equated value of heights of points of the leveling network, one by one and then in pairs was introduced the increment ΔH, equal to 1 cm. The coefficients of the normal equations were calculated according to the corresponding formulas in Figure 2. Thus, a matrix of normal equations of unknowns N was formed: This matrix is a degenerate square matrix, which means that it is impossible to perform its transformation to find the matrix of inverse weights Qx of the equated parameters. To find the matrix Qx, it is necessary to perform a pseudo-inversion of the matrix N, i.e., Qx=N + .
To perform a pseudo version, the calculations must be performed in the following sequence: 1) perform transposition of matrix N; 2) find matrix G as the product of matrices N and N T ; 3) calculate the inverse matrix G -1 using the search method algorithm.
The inversion of the matrix G by the search method of nonlinear programming can be done as follows. Knowing that G-G -1 =E, where G -1 and E are inverse and unit matrices, respectively, we find the evasion matrix 11 Where 11 12 , ,..., nn    -matrix elements ΨG, calculated as the difference between the corresponding element of the unit matrix E and the product of the elements of the matrix A and the matrix G -1 selected by the search method. That is, the target function in this case will be F(ΨG) = [Ψ11 2 + Ψ12 2 + ... + Ψnn 2 ] = min. Note that the resulting inverse matrix will be the only one. 4) calculate the pseudo-inverse matrix by the known formula given by the authors of [12][13][14]: As a result of performing the pseudo transformation of matrix N according to the above se-quence, the following matrix was obtained Hence, the root means a square error of the weight unit µ=0,033. Mean square error of heights of defined points mh1=mh3=0,014, mh2 = mh4=0,018.

Analysis of results
Since the above-considered leveling network was equalized as free, we have to deal with a rather complex mathematical procedure as pseudo transversion of degenerate matrices when performing the estimation of the accuracy of the equalization results. In particular, the pseudo transformation of the matrix of normal equations of the unknowns N.
Comparison of the obtained results of equalization and accuracy assessment of the given free leveling network completely coincided with the results of equalization and accuracy assessment of this network, considered in the book of Professor M.M. Mashimov's "Methods of mathematical processing of astronomical and geodetic measurements", performed with the use of the classical parametric method. Note that M.M. Mashimov used two mathematical algorithms to solve the problem -the parametric method for equalization and accuracy estimation using the skeleton decomposition method for the pseudo transformation of matrix N.
Within the framework of this work, the authors of the paper applied a single algorithm, the search method of nonlinear programming, to equalize the measurement results and pseudo trans-formation of the matrix N. Evaluation of the accuracy of the equation results was carried out by the generalized method of Professor G.V. Makarov.
As it was noted earlier in the conditions of Arctic regions it is difficult to provide stability of initial points during geodetic monitoring of constructions and operating buildings and constructions, therefore, there is a necessity to include initial points in equalization on a par with defined ones. I.e., there is a problem of equalization and assessment of the accuracy of geodetic networks as free.
The article proposes to process the results of measurements of free geodetic networks based on the algorithm of a search method of nonlinear programming and, as a result, to evaluate the ac-curacy of the results of equalization by the generalized Makarov method.
As a result, the following tasks were accomplished: -The theoretical justification of the algorithm of the search method of nonlinear programming has been performed; -The theoretical justification of the algorithm for estimating the accuracy of equation results based on the generalized method of G.V. Makarov was carried out; -the sequence of performing the pseudo transformation of the matrix of normal equations of the unknowns N based on the application of the search method of nonlinear programming is given for the first time; -the correctness of the search method for the pseudo transformation of matrix N is proved by the coincidence of the results of the pseudo transformation of this matrix based on the skeleton decomposition performed in the work of M. M. Mashimov; -the equation and evaluation of the accuracy of the free level network using the search method and the generalized Makarov method were carried out; -proved the correctness of application of considered methods for adjustment and estimation of accuracy of free leveling network by the coincidence of results of adjustment and estimation of accuracy, executed by the parametrical method in work of M.Mashimov.
Thus, we can conclude that the search method of nonlinear programming for processing free geodetic networks when performing geodetic monitoring of buildings and structures in the Arctic region is correct.