# Real-Time DTM Generation with Sequential Estimation and OptD Method
## 2 Materials and Methods
### Sequential Estimation in the Context of DTM Generation
The principles governing the parameters used during measurements depend on the type of measurement and the technology employed [40; 41]. The first acquired measuring strip is subjected to reduction by OptD and undergoes a filtering process using adaptive TIN models [42]. The OptD method can be used before  
Figure 2: Rules for acquiring datasets. or after filtering [10]. During processing using the OptD method, all points located in the measuring strip are considered. In the X0Z plane (or Y0Z depending on the X and Y axes setting), the reduction process is performed based on the principles of generalization using the Douglas-Peucker method. The following parameters characterizing the dataset after the OptD method are obtained: extremal coordinates (min, max), range (R = Zmax - Zmin), standard deviation (SD), number of points after reduction, different densities of points in different places of the object, number of iterations, width of the measuring strip, and tolerance (which determines the degree of reduction [42]). Parameters obtained from the reduction process can be directly utilized in sequential processing workflows, improving overall efficiency, and some of them can be used during the sequential estimation process. After the reduction and filtration process, the point cloud representing ground points constitutes the dataset for Sequence I of the calculations. In this step of the proposed concept, appropriate matrices are determined and corrections calculated according to Equation (2) and a polynomial model is adopted \(Z_{i}(X_{i}Y_{i})\). The parameters of the polynomial model are determined, which describes the fragment of the terrain represented by points from the first measuring strip. The second measuring strip is acquired during the processing of the first measuring strip. After the DTM parameters are determined, the data from the second measuring strip are included in sequence I. The dataset from sequence II is also reduced before it enters the computation of sequence II. It is important to consider the sequence overlap (sequence overlap I + II, II + III, i - 1 + i). Overlap is also considered for blocks (block overlap 1 + 2, 2 + 3,..., k - 1 + k). After the reduction and filtration process in each sequence, the decision should be made regarding the selection of a mathematical model that will be used to build the DTM. This model will be the most favorable approximation of the dataset of observation results. Of the many possibilities, a polynomial model is most often used:  
\[Z_{i}=a_{0}+a_{1}X_{i}+a_{2}Y_{i} \tag{1}\]  
where \(Z_{i}\), \(X_{i}\), and \(Y_{i}\) are the orthogonal coordinates of points obtained from LiDAR measurements; \(Z_{i}\) is the height, and \(X_{i}\) and \(Y_{i}\) are the rectangular coordinates of the points in the point cloud (\(i=1\ldots n\); \(n\) is the number of points in the dataset), and \(a_{0},\ a_{1},\ a_{3}\) are parameters. To fit the area obtained from Equation (1) to the point cloud using the least squares method of the correction equation \(v_{i}\) with respect to the height \(Z_{i}\), the following equation is used:  
\[\left[ \begin{array}{c}v_{1}\\ v_{2}\\ \ldots\\ v_{n}\end{array} \right]=\left[ \begin{array}{c}1X_{1}Y_{1}\\ 1X_{2}Y_{2}\\ \ldots\\ 1X_{n}Y_{n}\end{array} \right]\left[ \begin{array}{c}a_{0}\\ a_{1}\\ a_{2}\end{array} \right]+\left[ \begin{array}{c}-Z_{1}\\ -Z_{2}\\ \ldots\\ -Z_{n}\end{array} \right] \tag{2}\]  
\(\mathbf{v}=\mathbf{A}\mathbf{X}+\mathbf{L}\)  
where: \(\mathbf{v}\) is the correction matrix, \(\mathbf{A}\) represents a coefficient matrix, \(\mathbf{X}\) is a vector of parameters, \(\mathbf{L}\) is the matrix of free terms, and consistently \(\mathbf{X}^{T}=\left[a_{0}\ a_{1}\ a_{2}\right]^{T}\).