Abstract
The article presents a novel computation technique for predicting the settlement of reinforcement concrete (R.C) raft foundations based on total station and precise leveling measurements. Modified inverse distance weight (MIDW) and principal component analysis (PCA) are developed to predict the nonmeasurable settlement of raft foundations and are compared to regression techniques. Wall measurements are used to verify the proposed interpolation methods. Six observation epochs were carried out over 3 years for the R.C raft foundation of a multistory building in Baltim, Egypt, which rests on clay soil, to monitor and predict the foundation settlement. The obtained results show that PCA and MIDW models outperform other models in interpolating nonmeasurable (invisible) points, while the accuracy of MIDW is the best. The developed techniques can be used to interpolate nonmeasurable (invisible) points to understand the actual behavior of foundation settlement. The monitoring building showed that the slope of the foundation in the X direction is 7.9 and 14.5% after 5 and 26 months, respectively.
1 Introduction
Site monitoring and determination of shallow foundation settlements are essential for realtime evaluation of structure behavior. Settlement of foundation is the vertical deformation of a soil mass under the effect of compressive stress. Geotechnical and structural engineers must calculate the variability of soil properties and foundation settlement through in situ testing and the finite element method, respectively [1–4]. However, the real interaction between the structure foundation and soil is rarely performed due to its high complexity [1]. In addition, the reliability of soil laboratory tests may not be sufficient due to their timeconsuming and costly nature [3,4,5]. Therefore, to ensure the safety of existing structures, continuous monitoring of structure behaviors is required. Despite the importance of longterm monitoring of structure behavior, the cost of monitoring may affect the continuity of reporting structures’ status under different environmental effects. In addition, invisible structural members cannot be directly monitored. Thus, this study investigates the suitability of using interpolation methods for modeling and predicting the nonmeasurable settlement of raft foundations based on geodetic measurements.
Contacting and noncontacting terrestrial and aerial geodetic observations are used to measure structure behavior under normal operating conditions based on surveying accuracy specifications [2,6–10]. There are several accurate geodetic techniques, one of them is precise leveling [2,10,11]. High accuracy can be achieved by using precise leveling, with an accuracy range of 0.2 mm and 0.7 mm/km^{0.5} [10]. Mäkinen et al. [12] achieved 1 mm/km^{0.5} in the precise leveling results for measuring the vertical motion of the ground. AcostaGonzález et al. [13] used a precise leveling technique to monitor the foundation settlement of an oil tank, and they concluded that this technique can allow for adjusting other techniques in foundation monitoring. In this study, precise leveling is used to establish control points to measure the vertical displacement of the foundation through the total station instrument. The total station is used to measure accurate, with accuracies of a few millimeters or better, three directions of static and dynamic movements [14–16]. The measuring and limitation of using the total station in the monitoring were presented and discussed in Lienhart [17] and Cosser et al. [18]. Cosser et al. [18] and Lienhart et al. [19] used the total station to measure the dynamic behavior of a bridge, and the results showed the suitability of using the total station instrument in low and high frequency detection. Beshr [6] used a monitoring technique based on total station measurements to static monitor a large oil storage tank, and he found that the total station can provide accurate results in the tank movements and settlement. Threedirection static movements of a tunnel and building were also measured and evaluated using a total station, and the results approved the accuracy of it in structure monitoring [20,21].
Meanwhile, the raft foundation almost has invisible parts; this study aims to estimate the settlement of these parts. For that goal, geodetic techniques are used to measure the settlement of visible parts, and developing fitting techniques are applied to predict the settlement of nonmeasurable ones. Modified inverse distance weight (MIDW) and principal component analysis (PCA) approaches were developed and employed in the current work. PCA has been commonly used in several engineering applications, such as feature selection and data classification [22,23]. However, the PCA was applied to estimate missing data, and the results were improved [24]. PCA was also used to interpolate measured data, which gave good information for selecting input variables to get a better interpolation performance [25]. The PCA and inverse distance weight (IDW) were applied for determining the distribution of water quality, and both methods of interpolation were effective in determining the water surface quality [26]. The IDW was utilized to optimize the digital terrain model (DTM), and its performance was found high in DTM estimation [27]. Liu et al. [28] developed an adaptive IDW approach for modeling threedimensional geological models, and they concluded that the proposed method can provide an efficient, reliable, and stable spatial interpolation model for predicting large and smallscale geological maps. Other successful engineering applications for using IDW in interpolation can be found in refs [29–31].
This article proposes a development of the MIDW and PCA approaches for interpolating nonmeasurable foundation settlement. Both approaches are verified and applied to estimate the structure’s movements. The settlement of the raft foundation of a sevenfloor building is observed and predicted. Precise leveling is used to fix control points, and the total station is used to monitor the settlement of visible points of the foundation. Therefore, the main contributions of this research are: (1) we propose a new technique for monitoring invisible and nonmeasurable points through MIDW and PCA based on geodetic measurements; and (2) we develop a geodetic monitoring technique that can be used for predicting any nonmeasurable points. For that, we compare the developed methods to conventional regression models in interpolating foundation settlements through geodetic measurements to assess the accuracy of the proposed techniques.
2 Methods
2.1 Instruments and observations
The foundation settlement of a multistory building (ground floor furniture factory, four floors, two residential floors, and roof) located in Baltim city, Egypt, is measured and monitored for 26 months. The building is a reinforced concrete structure. The total height of the building is 27 m. Figure 4a presents the foundation dimensions, and the raft depth is 1.20 m. The location of Baltim and its local tectonic elements is presented in Figure 1a [32]. The soil characteristic of Baltim is presented in refs [32,33]. In summary, sand, silt, and clay intercalations are the main contents of soil for the surface layer of the study area. From the soil bore profile around the studied building, it was found that there is a layer of soft clay with poor cohesion that continues up to 12 m from the ground surface. The net bearing capacity at the foundation level from laboratory examination of soil and calculations does not exceed 0.7 kg/cm^{2}. The soil examination report recommends the necessity of supporting the building foundations using one of the appropriate structural methods to maintain the safety of the building.
Furthermore, different faults are in the study area, as presented in Figure 1a. Here, series cracks (Figure 1b) and slope were shown in the building, thus the monitoring system was designed. Because of heavy dynamic and static loads, the inside measurement for foundation settlement is impossible.
In Figure 1a, the Nile Delta is within the eastern Mediterranean basin on the Nile Delta Cone between the Herodotus abyssal plain to the west and the Levant basin to the east. The tectonostratigraphic framework is controlled by deepseated basement structures with distinct gravity and magnetic expressions and by the interaction of the NWtrending MisfaqBardawil (Temsah) and NEtrending QattaraEratosthenes (Rosetta) fault zones.
The used total station instrument, Sokkia CX103 (Figure 2a), includes the TSshield advanced security and maintenance system, which has the capability of performing all geodetic observations with reflector and reflectorless ability. The manufacturer quotes a display resolution of 0.5″ to 1″ for angle measurements with accuracy 3″ and ±(2 + 2 × 10^{−6} × D) mm for distance measurements for prisms and ±(3 + 2 × 10^{−6} × D) mm for reflectorless ability. The used automatic precise level was the Carl Zeiss NI007 and two 3m doublescale invar staves (Figure 2b and c). The used level has a builtin planeparallel plate micrometer. The manufacturer quotes a standard deviation (SD) of ±0.5 mm for a 1km double run leveling with the NI007 level. The accuracy of all instruments and the effect of systematic errors are taken into consideration during the field measurements. Laboratory tests were carried out before field observations and data collection using specified instruments to ensure the accuracy of the instruments mentioned in the specifications.
2.1.1 Experimental investigation
The precision of points that have been monitored using the discussed surveying techniques must be evaluated, and it is necessary to study the effect of the used instrument position distances and the angle of observations on the monitoring point accuracy. To achieve that goal, the monitoring of the vertical wall inside the Engineering Faculty, Egypt, is carried out. Practical measurements on a wall (12.5 m × 4.2 m) façade are represented by an extensive number of welldistributed targets. A mesh of 24 monitoring points on the wall is distributed for coordinating a building façade using sheet prisms of diameter 1 cm. The observed points are distributed randomly scuh that all points cover the whole area of the wall. A local coordinate system is selected. A local threedimensional rectangular coordinate system is assumed to calculate the spatial coordinates of any target points on the mesh. In such a system, presumably, the Xaxis is chosen as a horizontal line parallel to the wall base direction, the Yaxis is a horizontal line perpendicular to the wall base direction and positive in the direction towards the wall, and the Zaxis is a vertical line determined by the vertical axis of the instrument. The coordinates of all points and their SDs are calculated (Figure 3).
2.1.2 Site observation strategy and collection of data
To continue monitoring the building foundation for long period, a design monitoring system is developed and implemented. Figure 4 presents the foundation dimensions and the designed points for direct observation through the total station, Figure 4a, and planned points that we aim to estimate the settlement, Figure 4b. As presented in Figure 4, 18 points (M1, M2,…M18) are directly measured using the total station, whereas 14 points (M19, M20,…M32) are designed for predicting those settlements.
To measure the settlement of the foundation, six benchmark points were established first around the building using precise leveling, as shown in Figure 2c. Then, the total station is used to measure the movement of points M1–M18. The leastsquare technique was applied in each observation epoch for calculating the adjusted precise level of all monitoring points fixed on the foundation around the building.
2.2 Interpolation methods
2.2.1 Regression analysis model
In this study, three regression methods are applied to estimate the invisible (nonmeasurable) points of the foundation settlement. The linear polynomial (LP) method is used as the simplest form of approximation, which represents the required dependence Z in the form:
where X and Y are the plane coordinates of the monitoring point, and
In addition, the bilinear polynomial (BP) method is used, which can be presented as follows:
Furthermore, the plane equation (PE) method is applied to fit the foundation settlement in this study. The equation of the bestfit plane using n points (more than three points on the surface) can be determined using the least square theory as follows:
The equation of a plane can be determined using threepoint coordinates (X _{1}, Y _{1}, Z _{1}), (X _{2}, Y _{2}, Z _{2}), and (X _{3}, Y _{3}, Z _{3}) on this plane using the following formula:
The parameters of equation (3) can be determined by ref. [6]:
Thus, equation (3) can be modified and rewritten for several points on the plane as follows:
Dividing all parameters by C, then:
From the first principle of least square theory:
where V represents the model errors.
So, to get the optimum values (A, B, and C), derivatives must be employed:
2.2.2 MIDW method and its modification
IDW is a type of deterministic method for multivariate interpolation with a known scattered set of points. The assigned values to unknown points are calculated with a weighted average of the values available at the known points. IDW is the simplest interpolation method. A neighborhood around the interpolated point is identified, and a weighted average is taken of the observation values within this neighborhood. The weights are a decreasing function of distance. The user has control over the mathematical form of the weighting function, the size of the neighborhood (expressed as a radius or a number of points), in addition to other options. The suggested MIDW is a simple spatial interpolation technique and can be applied to any number of dimensions; it is also robust in estimation, does not suffer from the string effect of kriging, does not result in negative weights, i.e., no screening effect, and does not require a solving system of equations for the weights. Moreover, it provides reasonable estimates and is shown in a large number of comparative studies to be even better than geostatistical krigingbased techniques.
The simplest weighting function is inverse power [34]:
where
and
where Z is the estimated predicted value for a new point (invisible point), Z _{ i } is the value of the observed point (sample point) used for the prediction model, P is the power parameter, X and Y are the positions of the new nonmeasurable point, X _{ i } and Y _{ i } are the positions of observed points that are used for the prediction model, and n is the number of observed points.
When the observed points (sample points) are distributed regularly, the performance of the IDW technique will be better, and when the value of parameter P increases, the smoothness of the IDW output also increases [35,36]. The method of IDW interpolation faces various suspicions, especially when the observed points are clustering. Moreover, IDW interpolating technique is sensitive to the presence of outliers and suffers from discontinuities at observed points, resulting in peaks or troughs, especially where these observed points are sparse. Therefore, the IDW interpolating technique must be modified to be more accurate for all observed cases.
The suggested modified IDW method can be summarized as follows: Modified IDW is taking into consideration the overlapping effect of both relative distance and position of observed points on the interpolated new point, by adding a coefficient (K) to IDW formula (equation (11)) to adjust the distance weight of sample point according to its shielded effect in sample point positions. The coefficient (K) is based on an assumption that the observed point which is close to the prediction location (nonmeasurable point) has a shielding influence on those observed points that are farther away from the prediction location (nonmeasurable point), and the shielding influence can reduce the distance weight of the shielded observed point [37]. Combining the IDW technique formula presented in equations (9)–(11), the suggested MIDW technique can be expressed by modifying the term h _{ i }(X, Y) in equation (11) by the new modified equation, as follows:
where k _{ i } is the adjusted coefficient for distance weight and represents the comprehensive shielded influence of i(th) observed point. The value of the term k _{ i } ranges from 0 to 1. The term k _{ i } can be determined by the following formula [37]:
where θ _{ ij } is the intersection angle formed by the line that connects the observed points i(th) and j(th) and the line that passes through the prediction point and the midpoint of line 1.
As shown in equation (13), the term k
_{
i
} is expressed as a multiplicative model
where α _{ ij } is the intersection angle for two lines that connect prediction point and two observed points i(th) and j(th) separately.
The shielding influence appears when α < 360°/n and disappears when α ≥ 360°/n. To simplify, denote the term
2.2.3 PCA
PCA is a multivariate statistical method, and it is widely used to transform variables into independent principal components [26]. This method eliminated the correlation between evaluation indicators and greatly reduced the workload of indicator selection and calculation [26]. The main advantages of PCA are [24]: (i) it is an optimal (in terms of mean square error) linear scheme for compressing a set of highdimensional vectors into a set of lowerdimensional vectors; (ii) the model parameters can be computed directly from the ensemble covariance; (iii) given the model parameters, projection into and from the bases are computationally inexpensive operations. Here, the PCA used the Karhunen–Loève transform, following the classical approach for interpolating the nonmeasurable points. The objective is to find an orthogonal basis to decompose a stochastic point from the same original space, to project the nonmeasurable points. The proposed PCA steps for interpolating nonmeasurable points are presented in Figure 5.
3 Results and discussion
3.1 Verification proposed models
To verify the proposed models, vertical wall measurements are used. Here, the total station is used to measure the 24 points. First, 16point coordinates are used to build the interpolation models, and then 8 points are used to validate the resulted model parameters. In this step, the models’ parameters are determined and evaluated. To estimate the wall displacement, the following formula is used:
where
Second, several trials are carried out to test the effect of the observed point number on the accuracy of the proposed models. Seven cases are used to evaluate the proposed models with 12–18 points, as presented in Table 1. Table 1 presents the statistical evaluation (maximum [Mx], minimum [Mi], mean [M], and SD) of the model’s errors for these trials. In addition, Figure 6 demonstrates the obtained results. From Table 1, the range (maximum – minimum) of distortion for MIDW is small in all cases compared to other models. The distortion range of MIDW is 4.08, 5.52, 3.74, 3.61, 1.63, and 1.68 mm for 12, 13, 14, 15, 16, 17, and 18 points, respectively; while the distortion range for PCA is 7.39, 7.05, 6.44, 4.54, 2.58, and 2.23 mm, respectively.
Model  Distortion values (mm)  Number of input points for model building  

12 points  13 points  14 points  15 points  16 points  17 points  18 points  
LP  Max.  12.873  9.947  6.453  6.029  5.353  4.642  3.351 
Min.  −0.099  −0.614  −1.845  −0.814  −1.245  −0.914  −1.276  
Mean value  6.198  4.216  2.737  2.308  2.082  1.853  0.987  
SD  4.115  3.312  2.587  2.271  2.146  2.021  1.363  
BP  Max.  12.166  10.416  6.334  5.334  3.314  2.982  2.65 
Min.  2.518  −0.435  −1.454  −2.15  −2.048  −1.651  0.683  
Mean value  7.244  4.871  2.516  1.693  0.682  0.667  1.442  
SD  3.475  3.088  2.453  2.015  1.739  1.297  1.072  
PE  Max.  6.535  5.531  5.342  3.825  4.061  3.624  2.109 
Min.  −3.662  −2.718  −2.543  −2.718  −1.310  −1.807  −1.413  
Mean value  1.591  1.449  1.428  1.195  1.276  0.909  0.436  
SD  3.584  2.476  2.391  2.013  2.095  1.732  1.182  
MIDW  Max.  5.015  4.615  3.822  2.412  3.065  2.514  2.462 
Min.  0.930  −0.901  −1.142  −1.324  −0.541  0.882  0.781  
Mean value  1.964  1.785  1.334  0.295  1.226  1.682  1.75  
SD  2.213  1.920  1.721  1.592  1.333  0.844  0.693  
PCA  Max.  8.633  6.213  5.024  4.817  4.227  3.107  2.370 
Min.  1.245  −0.832  −1.412  0.242  −0.316  0.523  0.143  
Mean value  4.847  2.791  1.828  2.378  1.896  1.719  1.306  
SD  2.806  2.617  2.041  1.811  1.667  1.456  0.892 
In addition, from Table 1 and Figure 5, it is obviously shown that up to 16 points for designing the models, the variation in maximum distortion is high between regression and MIDW models. As well as, in all cases, the MIDW outperforms other models with low maximum distortion in all cases. The mean distortion of the MIDW and PE models is approximately constant for all cases, while over 16 points in design, the mean distortion of PE is lower than that of the MIDW model. In terms of the accuracy of the proposed models, SD term, it is clearly shown that the accuracy of MIDW is high compared to other models in all cases, followed by the PCA model. These results reveal that the accuracy of MIDW is the best to use in the interpolation of nonmeasurable points.
3.2 Foundation settlement measurement and predicting
According to the results in the previous section, MIDW and PCA models can be used to estimate the building settlement with acceptable accuracy. Here, precise leveling is used to fix the control points. Precise leveling was carried out in the mornings to eliminate the midday heat effects of the sun. For carrying out the precise leveling during the process of building monitoring, the following precautions were done: all survey lines were leveled independently in opposite directions; all survey stations were to be turning points; and back sights and foresights were to be equidistant to within 20 cm maximum. For the purpose of structural data analysis, adjustment of the resulted level network around the building was done using the least squares method to calculate the elevations of all control points. The solution was done using the least squares adjustment technique (observational equation method), assuming that the SD of observed elevation differences Δh _{ i } is proportional inversely to the route distance S _{ i }. The steps of least squares adjustment were done to calculate the adjusted elevations and their propagated accuracies for all control points.
Six observations epochs were carried out (December 2017, May 2018, October 2018, April 2019, September 2019, and February 2020), therefore five settlement values were recorded according to equation (15). In the first epoch, the monitoring points are fixed, and their positions in X and Y are recorded using the total station. Table 2 presents the collected measurements for the visible points in Figure 4a. From Table 2, it can be seen that the observed settlement with monitoring time increased. In addition, a symmetry movement of the building foundation around the axis of points M9–M16 is observed. The maximum settlement is observed at points M9 and M10, and the minimum settlement is located at points M1 and M18. The trend of mean, SD, and variance has increased from the first monitoring time to the end by 1.0, 0.25, and 0.60, respectively. This means that the building is moved, and the settlement caused a dangerous situation for the building. Herein, these values cause the building slope, since the slope is clearly shown in the direction of point M9.
Monitoring points  Displacement values for monitoring points (nm)  

Point No.  point Position  t = 5 months  t = 10 months  t = 16 months  t = 21 months  t = 26 months  
X, m  Y, m  
M1  0.5  1.5  0.95  1.75  2.85  3.4  4.5 
M2  5.5  1.5  1.6  1.9  2.7  2.85  3.95 
M3  10.7  1.5  2.25  3.05  3.25  3.95  4.65 
M4  10.7  0.5  2.2  3.2  3.45  3.65  4.7 
M5  15.2  0.5  2.65  3.3  4.05  5.35  6.75 
M6  19.7  0.5  2.9  3.75  4.8  5.55  7.1 
M7  24.2  0.5  1.95  2.8  3.6  5.75  7.6 
M8  29.2  0.5  3.35  4.65  4.5  6.1  7.9 
M9  33.85  0.5  3.7  4.45  5.2  7.25  9.5 
M10  32.55  7.6  2.9  3.95  4.25  7.35  9.35 
M11  31.2  14.7  2.75  3.8  4.71  6.7  8.1 
M12  25.9  14.7  2.85  3.25  3.6  5.2  6.05 
M13  20.8  14.7  2.9  3.4  3.95  6.05  6.7 
M14  15.7  14.7  2.05  3.75  4.05  5.65  6.75 
M15  10.6  14.7  1.8  2.6  3.55  5.5  6.8 
M16  10.6  12.3  1.65  2.8  3.9  4.05  5.8 
M17  5.5  12.3  1.6  2.75  2.9  3.65  4.55 
M18  0.5  12.3  1.45  2.25  3.55  3.55  4.8 
Sum  41.5  57.4  68.86  91.55  115.55  
Min.  0.95  1.75  2.7  2.85  3.95  
Max.  3.7  4.65  5.2  7.35  9.5  
Average  2.306  3.189  3.826  5.086  6.419  
S.D.  0.734  0.799  0.694  1.378  1.679  
Variance  0.539  0.638  0.481  1.899  2.818 
The proposed models are used to predict the invisible points, Figure 4b, for more understanding the foundation behavior under the applied loads during the monitoring period. Tables 3 and 4 present the predicted settlement of the nonmeasurable points of the raft foundation of the study building using PCA and MIDW models, respectively. In addition, Figure 6 presents the SD and variation of predicted points over the monitoring period time. From Tables 3 and 4 and Figure 6, it can be shown that the settlement of the foundation increased with the monitoring period. Meanwhile, from Tables 3 and 4, both models have the same trend for the predicted displacement. The same trend in sum, maximum, minimum, and average of all distortions for both techniques. This means both methods can be used to predict the settlement of invisible points. As the same, the trend of SD and variance of both approaches is approximately the same. However, the SD and variance of the MIDW model are lower than those for PCA. In addition, the error bar at each time monitoring period is shown to be small with the MIDW model, as presented in Figure 7. This means that the performance of the MIDW approach is better used in predicting invisible points. To fully understand the foundation movement under dynamic and static loads, a threedimensional (3D) representation of the foundation is presented in Figure 8 using the visible and invisible points. The figure shows the suitability of MIDW for use in settlement interpolation of nonmeasurable points with acceptable accuracy.
New points  Displacement values for predicted points (mm)  

Point No.  Point position  t = 5 months  t = 10 months  t = 16 months  t = 21 months  t = 26 months  
X, m  Y, m  
M19  0.5  7.5  1.28  2.05  3.1  3.3  4.44 
M20  5.5  7.5  1.57  2.38  3.04  3.32  4.43 
M21  10.7  7.5  1.93  2.9  3.5  3.96  5.2 
M22  15.2  7.5  2.31  3.31  3.94  4.9  6.19 
M23  19.7  7.5  2.56  3.42  4.07  5.53  6.79 
M24  24.2  7.5  2.64  3.47  4.01  5.9  7.27 
M25  29.2  7.5  2.86  3.84  4.15  6.6  8.3 
M26  15.2  3.75  2.52  3.33  4  5.06  6.29 
M27  19.7  3.75  2.65  3.48  4.32  5.54  7.01 
M28  24.2  3.75  2.66  3.3  3.91  5.87  7.51 
M29  29.2  3.75  3.08  4.19  4.33  6.41  8.28 
M30  15.7  11.4  2.18  3.43  4.01  5.12  6.46 
M31  20.8  11.4  2.68  3.44  3.95  5.7  6.71 
M32  25.9  11.4  2.8  3.45  3.88  5.86  6.98 
Sum  33.72  45.99  54.21  73.07  88.86  
Min.  1.28  2.05  3.04  3.3  4.43  
Max.  3.08  4.19  4.33  6.6  8.3  
Average  2.409  3.285  3.872  5.219  6.347 
New points  Displacement values for predicted points (mm)  

Point No.  Point Position  t = 5 months  t = 10 months  t = 16 months  t = 21 months  t = 26 months  
X, m  Y, m  
M19  0.5  7.5  1.63  2.48  3.32  3.83  5.02 
M20  5.5  7.5  1.79  2.68  3.34  4.03  5.17 
M21  10.7  7.5  2.01  2.93  3.58  4.41  5.68 
M22  15.2  7.5  2.23  3.15  3.79  4.86  6.12 
M23  19.7  7.5  2.44  3.33  3.95  5.32  6.67 
M24  24.2  7.5  2.58  3.46  4.04  5.67  7.08 
M25  29.2  7.5  2.81  3.77  4.22  6.42  8.06 
M26  15.2  3.75  2.42  3.22  3.85  4.86  6.1 
M27  19.7  3.75  2.56  3.43  4.23  5.37  6.84 
M28  24.2  3.75  2.45  3.37  3.97  5.71  7.32 
M29  29.2  3.75  2.99  4.05  4.35  6.18  7.94 
M30  15.7  11.4  2.13  3.25  3.85  5.12  6.34 
M31  20.8  11.4  2.58  3.36  3.92  5.61  6.64 
M32  25.9  11.4  2.71  3.46  3.94  5.66  6.81 
Sum  33.33  45.94  54.35  73.05  91.79  
Min.  1.63  2.48  3.32  3.83  5.02  
Max.  2.99  4.05  4.35  6.42  8.06  
Average  2.381  3.281  3.882  5.218  6.556 
Series settlement is detected during the monitoring period, as presented in Figure 7. The settlement of the whole foundation has happened. The rate of movement of the right side of the foundation is higher and more dangerous compared to other parts. The slope of the foundation in X direction is shown high, 7.9 and 14.5% after 5 and 26 months, respectively; and this correlated with measured slope (inclination) from the main building structure, which is 12.91 and 20.2 cm, respectively, at the same periods. Thus, we suggest that the load of the building should be decreased, and soil injection at point M9 may decrease the settlement of the foundation.
4 Conclusions
In this study, new interpolation techniques have been developed and used based on geodetic measurements. PCA and MIDW approaches have been applied based on the precise leveling and total station measurements of the settlement of building foundation. The settlement of invisible points of the raft foundation was detected and presented in the three directions to fully understand the foundation behavior under static and dynamic loads.
The comparison of developed and conventional techniques showed that the PCA and MIDW outperform other techniques. The MIDW outperforms other models with low maximum distortion in all cases of the verification stage. In terms of the accuracy of the proposed models, the SD term, it is clearly shown that the accuracy of MIDW is high compared to other models. These results reveal that the accuracy of MIDW is the best to use in the interpolation of invisible points.
The building monitoring and MIDW used in interpolating invisible points showed that the building is moved, and the settlement caused a dangerous situation for the building. The trend of mean, SD, and variance is increased from the first monitoring time to the end for the visible and invisible points by the same trend. The slope of the foundation in X direction is shown high, 7.9 and 14.5% after 5 and 26 months, respectively; and this correlated with the measured slope from the main building structure, which is 12.91 and 20.2 cm, respectively, at the same periods. Meanwhile, the suitability of MIDW to use in settlement interpolation of invisible points is shown high, and it can be used to present the settlement of the foundation in three dimensions.
Acknowledgment
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (GN: NRF2022R1I1A1A01062918).

Conflict of interest: The authors state that there is no conflict of interest.

Data availability statement: The submitted article includes all the data, models, and code generated or used during the study.
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© 2022 Ashraf A. A. Beshr and Mosbeh R. Kaloop, published by De Gruyter
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