Keywords: Polynomial regression; GPS elevation fitting; Quadric surface fitting; Plane fitting; Geodetic height. Palabras clave: Regresión polinómica; Ajuste de elevación GPS; Ajuste de superficie cuadrática; Montaje en plano; Altura geodésica. How to cite item Wang, J., & Xie, D. (2020). GPS elevation fitting study based on ternary polynomial regression. Earth Sciences Research Journal, 24(2), 201-205. DOI: https://doi.org/10.15446/ esrj.v24n2.87228 For the traditional GPS elevation fitting method, the accuracy has not been significantly improved in recent years, and the method has become increasingly complicated. This paper proposes to insert the geodetic height ‘H’ into the calculation system and use a ternary polynomial regression function to fit the GPS elevation anomaly. The feasibility of the ternary polynomial regression method in GPS elevation fitting was verified by an example, and compared with the results of the traditional plane fitting and quadric surface fitting method, it was proved that the proposed method is suitable for terrain with large terrain fluctuations. The fitting residual error and the standard deviation are smaller, and through example calculations, it is concluded that the ternary polynomial regression method under the seven parameters has the highest fitting accuracy. ABSTRACT GPS elevation fitting study based on ternary polynomial regression Estudio de ajuste de elevación GPS basado en regresión polinómica ternaria ISSN 1794-6190 e-ISSN 2339-3459 https://doi.org/10.15446/esrj.v24n2.87228 Para el método tradicional de ajuste de elevación GPS, la precisión no se ha mejorado significativamente en los últimos años, y el método se ha vuelto cada vez más complicado. Este documento propone insertar la altura geodésica "H" en el sistema de cálculo y utilizar una función de regresión polinómica ternaria para ajustar la anomalía de elevación del GPS. La viabilidad del método de regresión polinómica ternaria en el ajuste de elevación GPS se verificó mediante un ejemplo y, en comparación con los resultados del método tradicional de ajuste de plano y ajuste de superficie cuadrática, se demostró que el método propuesto es adecuado para terrenos con grandes fluctuaciones. El error residual de ajuste y la desviación estándar son menores, y a través de cálculos de ejemplo, se concluye que el método de regresión polinómica ternaria bajo los siete parámetros tiene la mayor precisión de ajuste. RESUMEN Record Manuscript received: 13/11/2019 Accepted for publication: 07/04/2020 EARTH SCIENCES RESEARCH JOURNAL Earth Sci. Res. J. Vol. 24, No. 2 (June, 2020): 201-205 Jianmin Wang*, Dongping Xie College of Surveying and Mapping and Geographic Sciences, Liaoning Technical University, Fuxin, 123000, China * Corresponding author: furuiteng123456@126.com GP S M EA SU RE M EN TS A PP LI ED T O GE OP HY SI CA L P RO BL EM S https://doi.org/10.15446/esrj.v24n2.87228 https://doi.org/10.15446/esrj.v24n2.87228 https://doi.org/10.15446/esrj.v24n2.87228 mailto:furuiteng123456@126.com 202 Jianmin Wang, Dongping Xie Introduction In recent years, China’s BeiDou navigation satellite system (BDS) has made great progress and is expected to achieve the goal of global coverage by 2020. The coordinate system used is the BeiDou coordinate system (BDCS) is a coordinate system which is defined in the framework of the international terrestrial reference frame (ITRS) as the current world geodetic coordinate system (WGS84) (Li & Huang, 2016). Therefore, the research on elevation fitting of global positioning system (GPS) is helpful for the effective conversion between Beidou height and normal height in the future. At present, through the research of some scholars, the methods of fitting GPS elevation data can be roughly divided into two categories. One is the traditional function fitting method, and the other is the new model fitting method. The traditional methods include weighted average method, plane fitting method, quadratic polynomial surface fitting method, moving quadratic polynomial surface fitting method and cubic polynomial surface fitting method, and the new models include neural network fitting method, support vector machine fitting method and curve cloud model fitting method (Zhang et al., 2015; Ren et al., 2015; Guo, 2018; Tang et al., 2016). In the actual operation, the new fitting method is not suitable for large-scale popularization because of its complex model, long data processing cycle and high requirements for hardware and software. However, the traditional fitting method only considers the relationship between the elevation anomaly and the plane coordinates x and y of the point, and the fitting accuracy is low, even in the practical engineering application, it can not achieve the experimental accuracy. In this paper, based on the traditional GPS elevation fitting method, the geodetic height is included in the solution system, and a GPS elevation fitting model based on ternary polynomial regression is proposed (Zhou et al., 2016; Ding & Sun, 2016; Liu et al., 2016; Yang & Zie, 2017; Sun et al., 2014). Ternary polynomial regression method The traditional elevation fitting method only considers that the elevation anomaly varies with the position of the plane. However, by understanding the elevation system (positive height, normal height, and geodetic height) and deducing the formula, it can be known that the elevation anomaly has a certain relationship with geodetic height. The ternary polynomial regression method can fully consider the relationship between elevation anomalies and x, y, H, so as to better fit the real elevation anomaly of ground points, and realize the conversion between the geodetic height and the normal height (Zhao et al., 2011; Wang et al., 2016; Chen et al., 2014; Song et al., 2014). Interpretation of elevation anomaly and geodetic height In Figure 1, the point P and the point ′P are on the same vertical line and have the same plane position. Then there are H N Hg + = (1) H N Hg  + ′ = ′ (2) In the formulas, Hg is the positive height lPQof point P, N is the geoid undulation lQV, H is the geodetic height lPU; Hg is the positive height lP Q′ of point ′P , ′N is the geoid undulation lQV, ′H is the geodetic height lP W′ . It can be seen from Figure 1 that N N= ′, so there is ∇ − − ∇H H H H H Hg g g= ′ = ′ = (3) That is, the positive elevation difference between the P point and the ′P point is equal to the geodetic height difference. In the formula: Hg is the positive elevation difference between point P and point ′P , and H is the geodetic height difference between point P and point ′P . According to the literature (figure 1), the difference between the elevation anomaly  and the geoid N meets N g Hm m m g= + ⋅η −γ γ (4) In the formula, gm is the real average gravity value on the vertical line between the geoid and the earth’s surface, and m is the average gravity value from the reference ellipsoid along the normal line to the approximate earth’s surface (Li & Huang, 2016). For point ′P , there is ′ = ′ + ′ ′ ′ N gm m m η −γ γ (5) For formula (4) and (5), there are H Hg g≠ ′ ,g gm m≠ ′ and γ ≠ γm m′ . According to the complexity of the leveling surface and the earth’s gravity field, g H g Hm m m g m m m g −γ γ −γ γ ⋅ ≠ ′ ′ ′ ⋅ ′ and combined with the formula N N= ′, Figure 1. Elevation system diagram 203GPS elevation fitting study based on ternary polynomial regression parameters is carried out respectively. The fitting accuracy is shown in Table 5, and the trend of fitting accuracy is shown in Figure 2. From the results in Table 3, it can be seen that the smallest value of the three-parameter ternary polynomial regression fitting residual is 0.17 cm, the average residual is 0.89 cm, and the smallest value of the plane fitting residual is 0.23 cm, the average residual is 2.61 cm. It is concluded that the ternary polynomial regression fitting method has strong feasibility in the actual GPS elevation fitting.  ≠ ′ can be obtained (Shijun & Li, 2015; Gong et al., 2014; Škrekovski et al., 2019; Zhao et al., 2017). Looking back at the point P and point ′P , they have the same plane position, but the difference is their positive height. Therefore, it can be seen that the elevation anomaly  is related to positive elevation Hg. Formula (3) shows that the change of positive elevation is equal to that of geodetic height. Therefore, the elevation anomaly  is related to geodetic height H. Ternary polynomial regression function If x, y and H are taken as three independent variables and elevation anomaly  is taken as dependent variable, then there are functions 1) Three parameters f a x a y a H3 0 1 2= = + + (6) 2) Four parameters f a a x a y a H4 0 1 2 3= = + + + (7) 3) Seven parameters f a a x a y a H a xy a yH a xH7 0 1 2 3 4 5 6= = + + + + + + (8) 4) Eight parameters f a a x a y a H a xy a yH a xH a xyH8 0 1 2 3 4 5 6 7= = + + + + + + + (9) 5) Eleven parameters f a x a y a H a xy a yH a xH a x a y 11 0 1 2 3 4 5 6 7 2 8 = + + + + + + + + η= α 22 9 2 10+ +a H a (10) By calculating the known data set (x, y, H, ), the corresponding polynomial parameters can be obtained and the functional relationship can be determined. Among them, three-parameter function requires at least three known point data, four-parameter function requires at least four known point data, seven-parameter function requires at least seven known point data, eight-parameter function requires at least eight known point data, and eleven-parameter function requires at least eleven known point data. If there is more redundant known data, it can be calculated according to the least square criterion (Liu et al., 2016; Wang et al., 2016; Wu & Miao, 2010). X A PA A PLT T= ( ) ⋅1 (11) In the formula, A is a coefficient matrix composed of known data x, y, H and xy, P is a weight matrix, generally a unit matrix, L is an observation matrix composed of known data , and X is the parameter matrix obtained. Case analysis In order to analyze the feasibility of the ternary polynomial regression fitting method proposed in this paper, the first 12 points are taken as the reference points and the last 7 points are taken as the points to be fitted by using the data provided in literature (Li, 2013). Planar fitting and three-parameter ternary polynomial regression fitting are performed separately. The known data is shown in Table 1 and Table 2. After calculation, the parameters of plane fitting function are respectively a0 34 1044= . , a e1 8 0703 6=  . , a e2 3 1533 5=  . and, and the parameters of three-parameter ternary polynomial fitting are respectively a e0 3 6290 7=  . , a e1 4 4750 5=  . , a e2 2 3390 4=  . and. The fitting results are shown in Table 3. By using the above data, quadric surface fitting and seven-parameter ternary polynomial regression fitting are used respectively. The fitting results are shown in Table 4. Using the above data, the ternary polynomial regression fitting of three parameters, four parameters, seven parameters, eight parameters and eleven Table 1. Data of reference points (unit: m) Point number Ordinate x Abscissa y Geodetic height H 1 3564231.786 499937.723 30.442 2 3565858.080 499248.000 31.225 3 3564029.592 499613.378 30.134 4 3565549.066 498813.558 31.110 5 3563826.318 499348.917 30.059 6 3566091.401 499632.434 30.709 7 3564312.203 500321.498 31.206 8 3566375.346 499179.740 26.424 9 3564827.161 500392.773 29.487 10 3566324.251 498659.474 25.790 11 3564001.762 500035.270 31.338 12 3567660.247 499189.334 31.128 Table 2. Data of points to be fitted (unit: m) Point number Ordinate x Abscissa y Geodetic height H 13 3566814.699 499080.199 30.989 14 3567961.396 498691.018 30.160 15 3566854.849 498567.506 30.741 16 3567538.805 499757.287 31.074 17 3566774.076 499599.745 30.621 18 3567524.909 500219.017 31.183 19 3568016.260 499235.369 30.332 Table 3. Plane fitting and three-parameter ternary polynomial regression fitting results. Point number Elevation anomaly/m Plane fitting Three-parameter ternary polynomial regression fitting Fitting value/m Residual/ cm Fitting value/m Residual/cm 13 21.0273 21.0566 2.93 21.0322 0.49 14 20.9978 21.0351 3.73 21.0146 1.68 15 21.0076 21.0401 3.25 21.0093 0.17 16 21.0639 21.0721 0.8 21.0622 -0.17 17 21.0710 21.0733 0.23 21.0555 -1.55 18 21.0516 21.0868 3.52 21.0829 3.13 19 21.0138 21.0518 3.8 21.0389 2.51 204 Jianmin Wang, Dongping Xie From the results in Table 4, it can be seen that the fitting effect of seven- parameter ternary polynomial regression is better than that of three-parameter ternary polynomial regression. The average residual is reduced to 0.27 cm, and the overall residual is mostly within 1 cm. Similarly, the fitting effect of quadric surface is better than that of plane fitting. The minimum residual is reduced to 0.05 cm, and the average residual is reduced to 1 cm. It can be seen that the overall effect of seven-parameter ternary polynomial regression fitting is better than that of quadric surface fitting. Conclusion In this paper, in the process of GPS elevation fitting, the geodetic height is included in the calculation system, and the ternary polynomial regression method is used to solve the elevation anomaly. The method takes into account the influence of the geodetic height on the elevation anomaly and avoids the phenomenon of distortion caused by the omission of the geodetic height. The feasibility of the proposed method and its superiority in fitting the global elevation anomaly in a large range are verified by the experimental results. In addition, it can be seen from Table 5 that in the actual engineering application, the fitting effects of the seven-parameter and eight-parameter are the best. The internal fitting accuracy is less than 0.2 cm, and the external fitting accuracy is about 1.5 cm. 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