Analysis of using the modified centring plates with eccentric points for geodetic measurements

Many engineering structures require high measurement accuracy. Their displacement and deformation are determined from the results of special measurements. For the measurements to be accurate, a properly constructed and marked survey network is necessary. The long-term stability of survey points can be ensured by marking (installing) them on solid rock or special triangulation pillars. Accurate and repeatable instrument positioning and premarking over the points is ensured by centring plates. Centring plates with eccentric points can be used when a survey involves several instruments. The article presents the results of measurements and computations done using centring plates with eccentric points. The measurements were conducted in a metrology laboratory. The sought points were premarked with prisms and reflective targets. The measuring methods were angular intersection, linear intersection, and linear-angular measurements. We computed coordinates for the measured points, and the results were compared to their known directory values. The results demonstrate that centring plates with eccentric points can be employed in engineering surveys.


Introduction
Geodetic measurements provide information on various types of objects.In popular belief, surveyors measure the surface of the Earth, most often parcels.Nevertheless, the true geodetic challenge is to measure various engineering objects, such as hydraulic structures (Avella 1993;Chrzanowski et al. 1992) or tall and thin constructions like factory chimneys (Breuer et al. 2002;Zheng et al. 2012;Kregar et al. 2015).It is critical to determine their location, shape, or dimensions (Juraev et al. 2020;Makuch and Gawronek 2020;Sztubecki et al. 2022;Gawronek and Makuch 2019), but to measure and determine any changes in these parameters over time is an even greater feat (U.S. Army 2002;Tretyak et al. 2015;Mrówczyńska et al. 2021).Ground deformation monitoring in mining areas also requires high levels of accuracy and special surveying networks (Kwinta and Gradka 2018;Hejmanowski and Kwinta 2001).As these measurements require top-notch accuracy, the right instruments and methods need to be employed (Chrzanowski et al. 1992;Nurpeisova et al. 2015).The stability of the survey control stations is crucial (Amiri-Simkooei et al. 2016).
Classical measurement methods and basic geodetic instruments and equipment are sufficient for simple surveys.Still, as the required accuracy increases, the necessary accuracy capability of instruments and equipment follows and measurement methods grow more complex (Schofield and Breach 2007).
A component of measuring methodology that significantly affects measurement accuracy is the positioning of the instrument and the premarking of tie (reference) points and sought points (Ruiz-Armenteros et al. 2013).For simple surveys, instruments are placed on wooden tripods over the station.Depending on the instrument's plummet, centring accuracy can significantly affect results (Lambrou and Nikolitsas 2017;Nestorović 2014;Mijic-Vasic 2015;Garcia-Balboa et al. 2018).In the case of short measurement intervals, the tripod can be installed "permanently" (a non-foldable tripod concreted on a construction site, for example).Another solution is to erect a concrete pillar that is demolished after construction is completed.Still, when high-level accuracy is required for long-term measurements, special instrument installation methods need to be applied (Gil et al. 2005;Kalkan et al. 2010;Acosta et al. 2018;Stewart and Tsakiri 2001).One of the primary, widely employed methods is to use the so-called centring plates.Such centring plates are installed on reinforced-concrete pillars of suitable dimensions sitting on the solid rock (Mijic-Vasic 2015;Močnik et al. 2020;Lambrou et al. 2011;Kalkan 2014;Casaca et al. 2015).Such stations have two-fold functions: instrument stations and tie points for measurements from other stations.However, when the survey interval is extensive, traditional centring plates with a single instrument seat may be insufficient (Pantazis et al. 2016).If a 'rapid' measurement is necessary (because of variable parameters of the objects due to sunlight impact, for example) and observations are to be made from several stations simultaneously, parallel installation of instruments and premarking of the stations may prove problematic.
The problem can be resolved with a centring plate with auxiliary eccentric holes with which both instruments and premarking (e.g.prisms) can be installed together.Such a rectifiable plate is a utility model registered with the Polish Patent Office (Ru.071751) (Kwinta 2021).This article presents the advantages of the solution with angular, linear, and angular-linear measurements in a metrology laboratory.

Centring plate
Tripods fail to ensure sufficient measuring conditions when very high accuracy is required (Garcia-Balboa et al. 2018).Measurement accuracy can be improved with structures that provide both instrument stability during measurement and repeatable positioning at a specific point in space.Such conditions can be ensured with forced centring plates installed on reinforced-concrete pillars or monolithic rock, if possible (U.S. Army 2002; Kalkan 2014).The plates are fixed to the substrate (pillar, rock) with steel anchors.
It is often necessary to conduct geodetic measurements over short periods.Tying can be problematic when centring plates already have instruments set up on them.The situation can be improved with additional eccentric seats with which the plate can be premarked, while an instrument is installed on it.This solution is employed in the present research.
The rectifiable centring plate is copyright protected and registered with the Polish Patent Office (Ru.071751) (Kwinta 2021).The plate has been designed to provide high instrument centring accuracy, premarking capabilities, and rectifiability throughout long and intensive use.The plate was developed for the metrology laboratory at the University of Agriculture in Kraków.
The device consists of a plate with an installation and rectification mechanism.During use, the centring plate should provide constant conditions for positioning geodetic instruments (such as total stations).The plate should be made of abrasion-and weather-resistant materials (such as stainless steel).The plate has seats for instruments and mount holes.Figure 1 shows an example of a centring plate.The plate (1) has three mount holes with set screws (2).
There is the central socket (3) in the centre of the plate with which the instrument is set up using a centring sphere (such as the Freiberg sphere (Rüeger 2006)).Apart from the main measurement socket, the plate has several auxiliary threaded holes (eccentric holes, 4) where additional surveying equipment can be installed (tribrachs, prisms for distance measurement, targets, staffs, rules).Figure 2 shows pictures of example embodiments of centring plates.
Two types of centring plates were made for the test.The smaller model A plate (400 mm in diameter, Fig. 2(1)) has a diameter smaller than that of the pillar and three eccentric points equidistant to the central point.The larger plate (535 mm in diameter, Fig. 2(2)) extends over the pillar's footprint and has eccentric points on two circles.In the other case (B), the outermost points can be used to install premarking targets below the plate (Fig. 3).
The exact coordinates of all the points on the plates had been determined from repeated geodetic measurements (both the centring socket and eccentric holes, Fig. 1).As the eccentric holes are precisely bored, the equipment (such as a prism) does not have to be set up on the tribrach after the plate is set up in the horizontal plane (adjustable during installation).The centring plates were installed in the laboratory over ten years ago.Measurements made over the time (and adjustments of the entire test network) demonstrate significant stability of the arrangement despite a rather intensive use during academic courses and instrument check measurements.Only one plate from among the fourteen installed had to be rectified after ten years.

Measurement arrangement
The measurements were completed in the metrology laboratory in a building of the University of Agriculture in Kraków.The laboratory is designed to accommodate course delivery and research on geodetic instrument metrology.The facility is 37.5-m long, 6.8-m wide, and 4.2-m high.The laboratory has various structures that facilitate research, including a cast-in-place reinforcedconcrete slab sitting on a foundation independent of the building's foundation ( 33 In addition, the laboratory has the following: • 22 precision Leica prisms • 20 targets with reflective film The experimental measurement involves three pillars, three mini-targets, and two prisms.The locations of the measured points and instrument stations are shown in Fig. 5. The central sockets that are instrument stations are marked with black triangles (points 12, 13, and 14).The distances between these points (12-13, 13-14) are 8 m.The auxiliary eccentric points are marked with black squares.These points are selected in such a way that an instrument on the centre point sees two additional points on each plate (points 1, 2, 3, 4, 5, 7, and 8).The following points are the sought points: 60, 61, 63, 65, and 67.The points marked with blue circles (points 63, 65, and 67) are premarked with reflective targets (see Fig. 6(1)).The points premarked with L-bar mini prisms are marked with red circles (points 60 and 61) (see Fig. 6(2)).All central points used in the experiment are assumed to have coordinates as determined in the past (considered reference values in the case of the sought points).
The points premarked with targets are selected to hinder aiming; the selected points are located near the ceiling, at about 30 gons vertical to horizontal aiming line of the instrument.This selection of the points forced the instrument to measure in approximate mode rather than precise mode as the aiming line in the horizontal plane is not perpendicular to the face of the target.Regarding the mini prisms, the points are slightly above the horizontal aiming line, and the prisms face the measuring instrument.The eccentric points are marked with prisms installed on carriers directly in holes in the centring plates.

Measurement methods
Three measurement methods are selected to compare them regarding the applicability of eccentric points and classical point arrangement (measurement from two points that are stations and tie points at the same time): angular intersection, linear intersection, and angular-linear intersection (Blachut et al. 1979;Brinker and Minnick 1994).The intersections can be done from more than two stations (Mulyani and Tampubolon 2020; Uren and Price 2010).The classical measurements involve the central sockets of the centring plates.The diagram of the classical measurements is presented in Fig. 7.
The angular measurement from two known stations is a classical geodetic problem of intersection.As shown in Fig. 7(1), horizontal angles are measured on two given stations (A and B).Assuming labels as in Fig. 7(1), the procedure for determining the coordinates of point C (and accuracy analysis) involves three steps: -Determination of the A BC azimuth and its mean error: where, (1) -Determination of distance d BC and its mean error: where, -Determination of the coordinates of the target (determined) point and its positioning error using values from Eqs.
(1) to (4): The second case (Fig. 7(2)) is a linear intersection: Distances from two known points A and B to a sought point C are measured.The calculation of the coordinates of the point is best started with the law of cosines: (5)

BA
where the mean error for the baseline segment is determined with the calculations in Eq. ( 4).
Next, the area of triangle ABC is quadrupled to aid further computations: Now, we seek to determine the coordinates of the target point: The positioning error of the point can be calculated from ( 12): ( 8) The third case (Fig. 7(3)) will be tackled with adjustment computations (Blachut et al. 1979;Brinker and Minnick 1994;Denli 2008;Gargula 2021b;Setan and Singh 2001).The first two cases involve two unknowns (coordinates of the sought point) and two observations (two equations)-no redundant observations.Now, we seek to determine two coordinates, but there are four observations (two angles and two distances).Here, one should employ adjustment of independent observations in line with the least squares method condition (13) (Gargula 2021a): where V is the matrix of the calculated corrections (deviations between observed and theoretical values) and P is the observation weights matrix.The coordinates of the sought point are determined by solving an overdetermined system of linear equations: where A is the matrix of observation equation coefficients, x is the vector of unknowns (two unknowns in this case), and l is the vector of absolute terms.Linear equations can be solved by expanding a nonlinear equation into a Taylor series.Therefore, we need to approximate the unknowns for (13) ⋅ ⋅ = minimum (14) ⋅ = point C (x o ,y o ) using one of the methods presented above (linear or angular intersection).The absolute term vectors are the differences between observations of the geometric elements and their values calculated from the approximate coordinates.Since the equations are linearized, adjustments require that increments of coordinates dx C and dy C be determined and the then unknowns calculated: A system of normal equations is obtained from Eq. ( 14) and the P weights matrix: The system of Eq. ( 16) is solved as follows: The accuracy analysis is based on a Q theoretical variance-covariance matrix determined during the adjustment of the unknowns ( 17): and using the mean (unit) error of the adjusted observations: where, u -the number of redundant observations (u = 2).If two diagonal elements of matrix Q are labelled Q xx and Q yy , the mean errors of the coordinates are described as follows: If mean errors of the network (stations) should be included in the computations, the equations grow slightly more complex because the variance-covariance matrix (18) should contain information about the errors.To achieve this, all measurement points (stations, tie points, and target points) should be taken into consideration when building the system of observation equations.
Linearized observation equations for data as in Fig. 7(3) are as follows: dx C , dy C -sought increments of the coordinates to the approximate values (as in Eq. ( 15)) In light of the above, matrix of observation equations A can be presented as follows: (21) where the last four rows of the matrix make it possible to account for coordinate errors.
Weights matrix P can be considered classically in relation to observation errors (diagonal matrix with inverted squares of mean observation errors).For station points and tie points, on the other hand, we introduce mean coordinate errors or a variance-covariance matrix.
The measurements employing eccentric points on centring plates are schematically shown in Fig. 8.There is no difference between the classical method and eccentric linear intersection measurements when all network points are determined.A measurement using eccentric points can, in such a case, be used as a check for the relative positions of the points.
According to Fig. 8, the difference concerning the classical measurement in both cases involves tying the measurements to eccentric points A1 and B1.In the second case (angular-linear measurements, Fig. 8(2)), there is no difference in relation to the classical measurement.Only the observation matrix and weights matrix grow larger to accommodate elements for the eccentric points.
If just angular measurements are made (Fig. 8(1)), it is necessary to calculate the intersecting angles in triangle ABC and then apply the same equations as for the classical measurement: Eqs. ( 1) to ( 6).One can apply the law of cosines and find eccentric corrections for the intersecting angles or calculate the corrections based on the azimuth difference.As shown in Fig. 8, we calculate intersecting angles α and β from azimuths, coordinates, and the measured angles: and The measurement involves known points, so azimuths are easily determined, and measurements of angles α 1 and β 1 let us calculate the sought unknown angles.
Mean errors of angles determined this way depend on the mean errors of the measured angles and mean errors of azimuths as in Eqs. ( 1) and (2).
The article presents the results of measurements in the metrology laboratory and the results of computations completed with the equations presented in this section.

Data analysis
The test measurements are performed with a single total station: Leica (WILD) TC2003.Horizontal angles are measured with the directional method, and distances are measured in the high-precision mode.Having preprocessed the results, we complete the computations as outlined in the previous section.We determine coordinates and errors for each of the five sought points using all the methods.In classical measurements, each point is determined thrice with each method.Angular and linear measurements are performed for each pair of centring plates, points 14-13, 13-12, and 14-12 (Fig. 5).Concerning eccentric measurements, each point is determined twelve times in total.The instrument is positioned both in the central socket and in two eccentric points for each centring plate.For example, when the instrument is positioned on stations (s) 14 and 13, the tie points (t) are 1, 2, 3, and 4.This means that the following combinations of intersections are calculated for angular measurements: 14(s)-3(t) and 13(s)-1(t), 14(s)-4(t) and 13(s)-1(t), 14(s)-3(t) and 13(s)-2(t), and 14(s)-4(t) and 13(s)-2(t).
The resulting coordinates are compared with directory coordinates.Distances between directory points and measured points are then determined.With directory values of the coordinates marked with superscript T and measured and computed values marked with superscript P, we have the following: These differences are juxtaposed in Tables 1 and 2 and illustrated in Figs. 9 and 10.The mean values-in the tables and Figs. 9 and 10 (black Xs)-are based on mean values of the computed coordinates of the points: where n is the number of independently determined coordinates of the point.
The results in Table 1 and Fig. 9 confirm that the angular measurements offer better accuracy than linear measurements, especially when a reflective film is used instead of prisms.Another important factor here is that the aiming line is not perpendicular to the target's face.The point arrangement geometry also affects accuracy.This confirms that the angle measured at the intersection point (between aiming lines) should be as close to 100 gons as possible.The worst results for point 63 (Fig. 9) come from measurements from stations 13-14.For point 67, the worst results are recorded for stations 12-13.
A comparison of the computations for angular and linear measurements (Table 1, Fig. 9) shows greater errors (distances to directory values) for linear measurements, particularly for points 63, 65, and 67 premarked with reflective film targets (Fig. 6(1)).Therefore, we introduce weights matrix P into the calculations of coordinates from the angular-linear measurement.The assumed direction error is 10 cc, and the distance measurement error for prisms (points 60 and 61) is 1.5 mm and for targets, (29) 2.5 mm.With these accuracy assumptions, the results of angular-linear measurements generally yield better accuracies than for each measurement type separately.Results worse than for the angular measurement occur only for point 65.Note that the linear measurement error for this point from stations 12-13 exceeds 6 mm, which affects the end result.The coordinates of eccentric points are determined as presented above.The results are summarised in Table 2 and Fig. 10.Table 2 shows the mean differences between the directory and sought values for each method.
The results for the eccentric measurements confirm the effectiveness of the solution.In the case of angular measurements, the worst results (largest distance) for point 63 are obtained from stations 13-14 with eccentric points 2 and 3: 2.62 mm.The worst result for point 67 (2.79 mm) is obtained from stations 13-12 and eccentric points 4 and 7.In these cases, the large deviations are also related to poor geometry of the network in relation to the sought points.The worse results of angular-linear measurements are for point 67 from stations 12 and 13.
A comparison of results for classical and eccentric measurements unravels very similar end results, which suggests that the measurements are equivalent while reducing measurement duration by employing several instruments simultaneously.

Conclusions
Geodetic measurements of displacements and deformations require significant attention to detail at various stages of the survey.The required accuracy is oftentimes nearly identical to the maximum accuracy of the measurement method and instruments.The survey preparation (planning) stage is of significance.The right design of the network and measurement process is also important.The stability of tie points and stations and the precise placement of premarking and instruments over points are key factors.
The article presents an embodiment of a centring plate with the main socket and eccentric holes.Such plates are installed in the metrology laboratory and have been successfully used for years.Using three stations on triangulation pillars with centring plates, we measured five points premarked with targets (three points) and prisms (two points).Based on angular and linear measurements, we calculated the coordinates of points from classical intersections and using the eccentric points.The results are presented in tables and figures.They demonstrate the following: • Centring plates with eccentric points should yield results similar to or sometimes even better than classical centring plates (repeating aiming to several points).
• The unfavourable geometric arrangement of occupied stations in relation to the sought point increases the coordinate error.
• Angular measurements offer greater accuracy than distance measurements.
• No targets with a reflective film should be used in displacement and deformation measurements.
• It is important to assume correct mean errors in the weights matrix when analysing angular and linear measurements.
The experiment was performed under laboratory conditions.Such circumstances (short aiming lines) are even more challenging than field conditions.The solution will be employed in field measurements.

Fig. 1
Fig. 1 Diagram of a centring plate (1 steel plate; 2 set screw; 3 central instrument socket; 4 eccentric measurement holes) .00 × 0.60 × 0.75 m) and two rows of triangulation pillars.The rows are 4.5 m apart.Each row has seven pillars (inter-pillar distances: 1-2-4-8-8-8 m).The pillars are 1.42-m high and 0.46 m in diameter.Figure 4 is a photograph of the laboratory.The pillars have two types of centring plates installed as described in the previous section.

Fig. 5
Fig. 5 Diagram of point positioning

Fig. 10
Fig. 10 Measured coordinate deviations in relation to directory coordinates: eccentric measurement , v d AC , v d BC -corrections to observations, with condi- tion that sum of vv is minimal.

Table 2
Measured coordinate deviations in relation to directory coordinates: classical measurement