To verify the performance of the proposed algorithms in solving the parameter estimation problem for a MAAMRE model with inequality constraints and to facilitate a discussion of their application value in geodesy, this paper presents two sets of examples designed for verification and analysis: global positioning system (GPS) elevation fitting model examples and DTM model examples. To better highlight the effectiveness and advantages of the proposed algorithms, the existing least squares solution, weighted least squares iterative solution and bias-corrected weighted least squares iterative solution for MAAMRE models given in Xu et al. (2013) are used for comparison. For convenience, the methods corresponding to each of the six schemes referred to in the examples below are listed in Table 1.

Table 1 Six schemes and their corresponding methods

GPS elevation fitting model

To preliminarily verify the feasibility and advantages of the proposed algorithms, Examples 1 and 2 will use a GPS elevation fitting model example for verification and analysis. Since the two inequality constraint algorithms proposed in this paper are derived for the cases of non-ill-posed and ill-posed coefficient matrices, Examples 1 and 2 are based on simulations of non-ill-posed and ill-posed GPS elevation fitting problems disturbed by MAAMREs, respectively.

Example 1

Example 1 is based on the observation equation for a non-ill-posed GPS elevation fitting model given in Chen (2017), and the function model is as follows:

$$ y = 10 + x + 2x^{2} + 2x^{3} $$

(19)

where (x) represents the distance between the elevations of a ground points and the origin of the coordinate system divided by 100, with a value range of 0–300 m in intervals of 10 m, (y) represents the true value of the ground elevation point obtained based on the value (x) in Eq. (19), and the parameters to be estimated are the five coefficients of Eq. (19), namely, ({tilde{varvec{beta }}} = left[ {begin{array}{*{20}c} {10} & 1 & 2 & 2 \ end{array} } right]^{{text{T}}}).

Example 2

Example 2 is based on the observation equation for an ill-posed GPS elevation fitting model given in Wang and Chen (2021a), and the function model is as follows:

$$ y = 10 + x + x^{2} + x^{3} + 0.2x^{4} $$

(20)

where (x) represents the distance between the elevations of a ground point and the origin of the coordinate system divided by 100, with a value range is 0–400 m in intervals of 10 m, (y) represents the true value of the ground elevation point obtained based on the value (x) in Eq. (20), and the parameters to be estimated are the five coefficients of Eq. (20), namely, ({tilde{varvec{beta }}} = left[ {begin{array}{*{20}c} {10} & 1 & 1 & 1 & {0.2} \ end{array} } right]^{{text{T}}}).

Result analysis

According to Wang and Chen (2021a), the corresponding observation equation for a GPS elevation fitting model disturbed by MAAMREs can be expressed as follows (Wang and Chen 2021a):

$$ {varvec{Y}} = {varvec{y}},{ odot }({varvec{1}} + {{varvec{upvarepsilon}}}_{m} ) + {{varvec{upvarepsilon}}}_{a} $$

(21)

where ({varvec{Y}}) represents the observation vector of ground elevation points disturbed by MAAMREs, ({varvec{y}}) represents the vector of the true values of the ground elevation points, and ({1}) represents a unit column vector.

As can be seen from Eq. (21), the MAAMREs affects the observations in both additive and multiplicative forms, which is clearly inconsistent with traditional theories and methods; thus, further research is needed. In accordance with Wang and Chen (2021a), for this example, the standard deviations of the multiplicative and additive random errors are set to 0.05 and 0.15 m, respectively, and the random errors are generated by the mvnrnd function in MATLAB. To illustrate the extent to which the elevations of the ground elevation points are affected by the random errors, the elevations before and after disturbance by random errors are plotted in Figs. 4 and 5, respectively. Then, following the practice of prior inequality constraint conditions between the parameters of a linear equation system defined in Xie (2014), the prior inequality constraint conditions shown in Tables 2 and 3 are added in Examples 1 and 2, respectively. At this time, the physical meaning of the inequality constraint conditions can be expressed as a prior conditions that must be satisfied by the parameter estimates. In actual data processing, these prior conditions may be derived from previous observations or the requirements to be satisfied in the practical situation.

Fig. 4
figure 4

Elevations of the ground elevation points before and after disturbance by MAAMREs in the first example

Fig. 5
figure 5

Elevations of the ground elevation points before and after disturbance by MAAMREs in the second example

Table 2 Inequality constraint data for the first example
Table 3 Inequality constraint data for the second example

In Figs. 4 and 5, the curves and circles represent the true values and observations, respectively, of the elevations of the ground elevation points. The circles greatly deviate from the curves, indicating that the elevations have suffered severe deviations due to the effect of the MAAMREs. Therefore, further research is needed regarding a GPS elevation fitting model disturbed by MAAMREs.

The unit weight error (sigma_{0}) is set to 0.3 (Wang and Chen 2021a). For Example 1, the LS method, the WLS method, the bcWLS method and Algorithm 1 are used for calculation, and for Example 2, the LS method, the WLS method, the bcWLS method and Algorithms 2 and 3 are used for calculation. The parameter estimates ({hat{varvec{beta }}}) and the 2-norms (left| Delta varvec{hat{beta} } right|) between the parameter estimates and the true values are listed in Tables 4 and 5, respectively.

Table 4 Parameter estimates and the 2-norm results between the parameter estimates and true values for the first example
Table 5 Parameter estimates and the 2-norm results between the parameter estimates and true values for the second example

As can be seen from the results in Table 4, the parameter estimates obtained with Algorithm 1 are closer to the true values than the three existing solutions given by Xu et al. (2013), indicating that considering the prior inequality constraint information during adjustment can further improve the results of parameter estimation and that applying the exhaustive search method to solve the MAAMRE model with inequality constraints is feasible and effective. Meanwhile, the parameter estimates obtained with Algorithm 1 and the inequality constraint data show that the three constraints in the optimal solution are all effective constraints.

By comparing the results of the LS method, the WLS method, the bcWLS method and Algorithms 2 and 3 in Table 5, it can be seen that the parameter estimates obtained with the three existing solutions given by Xu et al. (2013) seriously deviate from the true values; this is because when deriving the formulas for the three existing solutions, the ill-posed coefficient matrix was not considered. Meanwhile, by comparing the results of Algorithms 2 and 3, it can be seen that the parameter estimates obtained with Algorithm 3 are closer to the true values than those of Algorithm 2, which further shows the effectiveness and advantages of using the exhaustive search method to solve the ill-posed MAAMRE model with inequality constraints. In addition, the parameter estimates obtained with Algorithm 3 and the inequality constraint data show that the four constraints in the two sets of optimal solutions are all effective constraints. However, since the second set of inequality constraint conditions is closer to the true values than the first set, the parameter estimates obtained are also closer to the true values.

Digital terrain model

In actual data processing, a model may easily be ill-posed; thus, Example 3 considers the parameter estimation problem for an ill-posed DTM model with inequality constraints. DTMs, which realize the general expression of natural surface morphologies in numerical form, have been widely used in many information engineering constructions (Shekhar and Xiong 2008; He 2014). In this example, we will imitate the practice in Wang and Chen (2021a, 2021b) and use the interpolation method to simulate the observation equation for an ill-posed DTM model disturbed by MAAMREs, which can be expressed as follows:

$$ {varvec{H}}(x,y) = {varvec{h}}(x,y),{ odot, (}{varvec{1}} + {{varvec{upvarepsilon}}}_{m} ) + {{varvec{upvarepsilon}}}_{a} $$

(22)

where ({varvec{H}}(x,y) in {varvec{R}}^{1681 times 1}) represents the observation vector of the DTM model elevations disturbed by MAAMREs, ({varvec{1}} in {varvec{R}}^{1681 times 1}) represents a unit column vector, and ({varvec{h}}(x,y) in {varvec{R}}^{1681 times 1}) represents the vector of the true values of the DTM model elevations, expressed as ({varvec{h}}(x,y) = sumlimits_{i = 1}^{4} {beta_{i} } f_{i} (x,y)). The true values of the parameter estimates are ({tilde{varvec{beta }}} = [1.5{ 20 10 – 4]}^{{text{T}}}), and the functions (f_{i} (x,y)(i = 1,2,3,4)) are as follows (Wang and Chen 2021a, 2021b):

$$ left{ begin{gathered} f_{1} (x,y) = exp left{ { – [(x – 22)^{2} + (y – 22)^{2} ]/500} right} hfill \ f_{2} (x,y) = exp left{ { – [(x – 28)^{2} + (y – 28)^{2} ]/500} right} hfill \ f_{3} (x,y) = exp left{ { – [(x – 25)^{2} + (y – 25)^{2} ]/500} right} hfill \ f_{4} (x,y) = exp left{ { – [(x – 20)^{2} + (y – 20)^{2} ]/500} right} hfill \ end{gathered} right. $$

(23)

where the value ranges of the ground points on both the abscissa (x) and the ordinate (y) are 0–80 m with intervals of 2 m.

At present, observations that exhibit MAAMREs have not received sufficient attention, and existing research on MAAMREs does not give the specific standard deviations of the additive and multiplicative random errors. Therefore, following the practice of the existing studies on MAAMRE models (Xu et al. 2013; Shi 2014; Shi and Xu 2021; Wang and Chen 2021a, 2021b), in this example, the standard deviations of the MREs are set to 0.005, 0.05 and 0.1, and the standard deviation of the AREs is set to 0.15 m, based on the theoretical and practical tests on LiDAR data reported by Flamant et al. (1984), Wang and Pruitt (1992), Kraus and Pfeifer (1998), Berg and Ferguson (2000), Hill et al. (2003), Kobler et al. (2007), Leigh et al. (2010) and Veneziano et al. (2010). Meanwhile, to illustrate the extent to which the DTMs are affected by MAAMREs, the DTMs before and after disturbance by random errors are plotted in Figs. 6, 7, 8 and 9.

Fig. 6
figure 6

DTM model without random errors in the third example

Fig. 7
figure 7

DTM model disturbed by MREs with (sigma_{m} = 0.005) in the third example

Fig. 8
figure 8

DTM model disturbed by MREs with (sigma_{m} = 0.05) in the third example

Fig. 9
figure 9

DTM model disturbed by MREs with (sigma_{m} = 0.1) in the third example

By comparing the DTM model elevations in Figs. 6, 7, 8 and 9, it can be seen that the elevations in Fig. 6 are smooth, while the elevations in Figs. 7, 8 and 9 exhibit folds that become increasingly severe as the MRE increases, indicating that the random errors have a greater impact on the DTM model elevations. After calculation, the condition number of the normal equation matrix is (7.1054 times 10^{5}), indicating that the model is severely ill-posed. Therefore, detailed research is needed on ill-posed DTMs disturbed by MAAMREs.

To further explain the influence of the range of the inequality constraint conditions on parameter estimation, two sets of inequality constraint conditions are designed for this example, as shown in Table 6. For the case in which the prior unit weight error (sigma_{0}) is set to 0.3 (Xu et al. 2013; Shi 2014; Shi and Xu 2021; Wang and Chen 2021a, 2021b) and the first set of inequality constraint conditions is used for calculation, the results of the parameter estimates and the 2-norms between the parameter estimates and the true values are listed in Table 7. As seen from the inequality constraint data in Table 6, the second set of inequality constraint conditions is closer to the true values than the first set; therefore, the parameter estimates obtained using the second set of inequality constraint conditions also should theoretically be closer to the true values. For further verification, the results obtained in the case of (sigma_{m} = 0.05) when the second set of inequality constraint conditions is used for calculation are listed in Table 8.

Table 6 Inequality constraint data for the third example
Table 7 Parameter estimates and the 2-norm results between the parameter estimates and true values for the third example
Table 8 Parameter estimates and the 2-norm results between the parameter estimates and true values in the case of (sigma_{m} = 0.05) for the third example

As seen from the 2-norm results in Table 7, the parameter estimates obtained with the three existing solutions given by Xu et al. (2013) greatly deviate from the true values. Meanwhile, by comparing the results for the three sets of MREs, it can be seen that the degree of elevation deviation gradually increases as the MRE becomes larger. Since the ill-posed nature of the coefficient matrix was considered when deriving the formulas for the proposed algorithms, the parameter estimates obtained are closer to the true values. Among them, the 2-norm results obtained with Algorithm 2 are reduced by 68.32%, 92.25%, and 93.5% compared to the results of the bcWLS method, and the 2-norm results obtained with Algorithm 3 are reduced by 65.2%, 62.01% and 81.5% compared to the results of Algorithm 2, indicating that the proposed algorithms are affected by the size of the MRE; specifically, the greater the MRE is, the more obvious the advantages of the proposed algorithms. In addition, as seen from the parameter estimates obtained with Algorithm 3 in Table 8 and the inequality constraint data in Table 6, the four constraints in both sets of optimal solutions are all effective constraints. However, since the second set of inequality constraint conditions is closer to the true values than the first set, the parameter estimates obtained with the second set of inequality constraint conditions are closer to the true values. This finding is consistent with the conclusion obtained in Example 2 and further illustrates the feasibility and advantages of the proposed algorithms.

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