RMS errors for all sections before and after DEM error modeling.
Abstract
The Digital Elevation Model (DEM) can be created using airborne Light Detection And Ranging (LIDAR), Image or Synthetic-Aperture Radar (SAR) mapping techniques. The direct georeferencing of the DEM model is conducted using a GPS/inertial navigation system. The airborne mapping system datasets are processed to create a DEM model. To develop an accurate DEM model, all errors should be considered in the processing step. In this research, the errors associated with DEM models are investigated and modeled using Principal Component Analysis (PCA) and the least squares method. The sensitivity analysis of the DEM errors is investigated using PCA to define the significant GPS/inertial navigation data components that are strongly correlated with DEM errors. Then, the least squares method is employed to create a functional relationship between the DEM errors and the significant GPS/inertial navigation data components. The DEM model errors associated with airborne mapping system datasets are investigated in this research. The results show that the combined PCA analysis and least squares method can be used as a powerful tool to compensate the DEM error due to the GPS/inertial navigation data with about 27% in average for DEM errors produced by the direct georeferenced airborne mapping system.
Keywords
- sensitivity
- PCA
- least squares
- DTM errors
- navigation
1. Introduction
The Digital Elevation Model (DEM) can be created using airborne Light Detection And Ranging (LIDAR), Image or Synthetic-Aperture Radar (SAR) mapping techniques. The direct georeferencing of DEM model is conducted using GPS/inertial navigation system. The accuracy of the developed DEM model is strongly dependent on the mapping system and the georeferencing system grades. The selection of the mapping system and the georeferencing system grades is carried out in the planning stage based on the accuracy requirements of the required DEM model.
The previous literatures focused and heavily investigated the DEM model generation based on LIDAR, Images, and SAR data cleaning and filtering techniques such as Triangulated Irregular Network (TIN)-based filtering, slope-based filtering, mathematical morphological filtering, interpolation-based filtering, and Machine-learning-based filtering. The original TIN-based filtering was developed based on the classical progressive TIN densification (PTD) and was implemented effectively in the commercial software TerraScan [1, 2]. Then, the revised PTD was investigated and reduced the total errors by about 8% when compared with classical PTD method [3]. Afterwards, a Parameter-Free PTD (PFPTD) algorithm was developed and outperforms the classical and revised PTD methods [4]. The original slope-based filtering was derived based on height differences in the training dataset [5, 6]. Then, adaptive slope-based filtering algorithm was developed to improve the accuracy in urban applications when compared with the original slope-based filtering algorithm [7]. The original mathematical morphological filtering was proposed to filter LIDAR data [8]. Then, the progressive morphological filtering algorithm was developed to improve the original method by applying threshold condition based on the elevation differences and proposed increasing gradually the filtering window size [9]. Afterward, the spline iteration method was introduced to improve the morphological filtering algorithm [10]. All mathematical morphological filtering methods outputs are strongly dependent on adapting the filtering window size. The original interpolation-based filtering method was proposed to deal with the steep areas [11]. Then, the interpolation-based DEM generation method was developed where one of the Inverse Distance Weighted, Kriging, and Natural Neighbor (NN) can be employed for DEM generation [12]. The Natural Neighbor (NN) method was proven to provide most efficient results. Finally, Machine-learning-based filtering was investigated where this method depends on topographic characteristics of the areas under investigation [13]. The deep convolutional neural network (CNN) was proposed for develop accurate DEM model [14]. The Machine learning method was optimized using windowing method to improve the DEM model generation [15].
However, the above-mentioned methods are mainly dependent on the data cleaning and filtering techniques of the heights to develop the DEM model. These methods are associated with errors that are not considered in the DEM modeling and could be corelated with system navigation data. Therefore, this research investigated the sensitivity analysis modeling of DEM errors that are potentially correlated with navigation data to improve the overall accuracy of the DEM model.
2. Combined PCA and least squares method
The sensitivity and modeling of DEM errors are investigated in this research using PCA and least squares function modeling. The system navigation data (position, velocities, attitudes, accelerations and dopplers) are considered the inputs to the model and the DEM height error is considered the desired output of the model.
PCA is a numerical technique used to study multidimensional processes that can be used to (1) reduce the dimensionality of a dataset and (2) identify relationships between the underlying variables of the process. PCA is based on eigen or singular value analysis of the process correlation or covariance matrix. The goal of PCA is to determine the minimum number of eigenvectors that best describe the key features of the process correlation matrix. This results in a reduced-dimensionality model for the matrix which can be used for data analysis, reduction, and model synthesis. Singular value decomposition (SVD) is fundamental to PCA. More details on PCA and SVD can also be found in Jolliffe [16].
2.1 PCA analysis
Let
where
where
Note that
Several relationships can be derived. The SVD equation for
which is a linear combination of the right singular values {
which is a linear combination of the left singular values {
where
We can define the
In PCA,
2.2 Least squares functional modeling with PCA optimization
We ultimately seek a model that relates the inputs (navigation parameters) to the height error. The least squares method is used to model DEM errors. More details on the least squares method can also be found in Ghilani [17]. Assuming a linear forward model:
The set of navigation data parameters is used to form the
Using the matrix form, the
Shifting the mean and scaling the vector elements affects the performance of the estimate. Assuming we have enough realizations, we can write a least-squares empirical estimate of
While we can use
The singular value spectrum of
Note that while the order of the model parameters or error values within the vectors is not important, the range of cases included does matter. The model may be unable to adequately represent a case not included in the training set. Also, the model can only represent linear functions of the input model parameters. (Recall, that we can produce non-linear responses by including non-linear transforms of model parameters in the model parameter vector.)
Let
The error e is subtracted from the height map to remove the height error. Note that due to our formulation of the original model and height vectors that this is done “block-wise”.
In this research, the sensitivity analysis is carried out using PCA and the modeling is investigated the least squares function modeling where the system navigation data are considered the inputs to the model and the DEM height error is considered the desired output of the model.
3. Data and methodology
The DEM model errors associated with airborne mapping system datasets are investigated in this research. Figure 2 shows an example of DEM errors case study. It can be seen that the DEM errors map shows large errors in the left side associated with low depression angles and small errors in the right side associated with high depression angles. The across-track DEM errors sections at three different depression angles are investigated (sections 1 to 3) as shown in Figure 2 where the DEM errors are the highest in these locations.
The methodology for the proposed combined PCA analysis and least squares method is shown in Figure 3. The PCA analysis is utilized to identify the significant inputs from multiple navigation data and the least squares method is implemented to estimate the DEM errors models. The root mean squares (RMS) errors are used to quantify the accuracy of the developed DEM errors models.
4. Results and discussion
Three across-track sections (1 to 3) as shown in Figure 2 have been investigated to test the performance of combined PCA analysis and least squares method. The navigation data (3 positions, 3 velocities, 3 attitudes, 3 accelerations, 3 attitude rates, 3 attitudes accelerations, 1 doppler, and 1 doppler rate) represent the input parameters for DEM errors modeling were investigated using PCA analysis method. Out of 20 inputs, 10 inputs were found significant (2 positions, 2 accelerations, 2 attitudes, 2 attitude rates, 1 doppler and 1 doppler rate) in the three across-track sections.
Then the least squares were employed to model the functional relationship between the 10 significant navigation inputs and the targeted DEM error output. The RMS errors were estimated before and after modeling to test the performance of the developed model to compensate the DEM errors. Figure 4 shows the targeted DEM errors and the modeled DEM errors where Figure 5 shows the differences between the targeted and modeled DEM errors for section (1) case. In section (1) case, the estimated RMS error before modeling is 0.59 m and after modeling is 0.44 m which means that about 25% of DEM errors can be compensated in section (1) case. Figure 6 shows the targeted DEM errors and the modeled DEM errors where Figure 7 shows the differences between the targeted and modeled DEM errors for section (2) case. In section (2) case, the estimated RMS error before modeling is 0.45 m and after modeling is 0.33 m which means that about 27% of DEM errors can be compensated in section (2) case. Figure 8 shows the targeted DEM errors and the modeled DEM errors where Figure 9 shows the differences between the targeted and modeled DEM errors for section (3) case. In section (3) case, the estimated RMS error before modeling is 0.35 m and after modeling is 0.25 m which means that about 28% of DEM errors can be compensated in section (3) case.
Table 1 and Figure 10 summarize the RMS errors and the percentage of DEM error compensation from the three sections where the overall percentage of DEM error compensation is about 27% on average. The results show that the combined PCA analysis and least squares method can be used as a powerful tool to compensate the DEM error produced by the navigation data with about 27% in average for the case study investigated in this research.
Section | RMS error before modeling (m) | RMS error after modeling | Percentage of DEM error compensation (%) |
---|---|---|---|
Section (1) | 0.59 | 0.44 | 25 |
Section (2) | 0.45 | 0.33 | 27 |
Section (3) | 0.35 | 0.25 | 28 |
5. Conclusions and recommendation
The sensitivity analysis of DEM errors to the GPS/inertial navigation data was investigated in this research. It was concluded that the sensitivity analysis of the DEM errors can be performed using the PCA to identify the significant GPS/inertial navigation data components that are strongly correlated with DEM errors. Also, it is concluded that the least squares method can be rigorously utilized to establish the functional relationship between the DEM errors and the significant GPS/inertial navigation data components. The combined PCA and least squares method were validated using the DEM model errors associated with airborne mapping system datasets using three across-track sections of the DEM errors datasets. The results show that the combined PCA analysis and least squares method can be used as a powerful tool to compensate for the DEM error produced by the GPS/inertial navigation data with about 27% on average. Therefore, it is recommended to use the combined PCA analysis and least squares method to reduce the DEM errors associated with the DEM model produced by the direct georeferenced airborne mapping system.
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