esa-sceva/satcom-chunk-collection huggingface dataset:
# Modeling and Control of an Underactuated System for Dynamic Body Weight Support
## 6 Simulation Research  
In terms of numerical research, some simulation tests were performed regarding the motion of the rehabilitation device on the basis of Equations (1) and (8). Tables 1 and 2 present the data applied during the simulation tests.  
The trajectory of the passive part was specified on the basis of two variants. In the first case, simulation tests were based on an analysis regarding the kinematics of the gait patterns of persons diagnosed with neurological diseases. The averaged motion along the vertical axis for one tested person was determined with (19). Therefore, this motion may be considered to be a gait template for a disabled person at a given phase of rehabilitation. This motion relates to the rope end displacement also. Pursuant to the device structure, a value of the generalized variable \(x\), which determines the chair movement within the passive part of the body weight support system, is reduced twice compared with the patient's vertical movement. Consequently, Function (20) determines the adjusted trajectory of the body weight support system's passive part.  
\[F_{p}(t)=0.016(\sin(2\pi t)-\sin(4\pi t)) \tag{19}\]  
\[x_{d}(t)=0.008(\sin(2\pi t)-\sin(4\pi t)). \tag{20}\]  
In the second case, based on literature data [43; 44], the function that determines the adjusted trajectory of the body weight support system's passive part related to vertical movement during a healthy person's gait (gait template) was defined. This function was determined with Equation (21):  
\[x_{d}(t)=0.006(\sin(4\pi t)). \tag{21}\]  
The trajectory of the body weight support system's active part is generated on an ongoing basis during device operation.  
The performed simulation tests considered three cases:  
* Variant 1: A person during rehabilitation performs movements pursuant to (19) without body weight support;
* Variant 2: During rehabilitation, a person performs movements pursuant to (19) with body weight support of 300 N;
* Variant 3: During rehabilitation, a person performs movements with body weight support of 300 N.  
The simulation tests assumed that the body weight of the person undergoing rehabilitation was 70 kg (the device has a load capacity of 150 kg).
# Research on Panorama Generation from a Multi-Camera System by Object-Distance Estimation
## 3 Proposed Method
### Object-Distance Estimation Algorithm
#### 3.3.2 Estimation of Object-Distance Pyramid Maps in Overlapping Regions
\[E_{i,j}=\rho_{1}\sum_{\varphi=0}^{\Phi-1}\frac{NCC(G_{i,j}^{N}(\varphi,0),R_{i,j}^{N}(\varphi,0))}{\Phi}+\rho_{2}\sum_{\varphi=1}^{\Phi-1}\frac{-\left|R_{i,j}^{N}(\varphi,0)-R_{i,j}^{N}(\varphi-1,0))\right|}{T} \tag{11}\]  
where \(\rho_{1}\) and \(\rho_{2}\) are the weight coefficients of similarity and smoothness, respectively. \(E_{i,j}\) is the objective function; \(\Phi-1\) is the count of rows for the \(N^{th}\) pyramid gird. Further, a group of optimum object-distance indices, denoted by \(\left(R_{i,j}^{N}(0,0),R_{i,j}^{N}(1,0),\cdots,R_{i,j}^{N}(\Phi-1,0)\right)^{*}\), can be estimated by iteratively solving Equation (12) using a dynamic programming algorithm [31]. Then, an object-distance map (ODM) is generated for the overlapping region \(A_{i,j}\) at a resolution of \(\frac{w}{2^{N}}\) by \(\frac{H}{2^{N}}\) in the \(N^{th}\) level, based on the optimal index parameters. Additionally, a corresponding index map is also created. \[\left(R_{i,j}^{N}(0,0),R_{i,j}^{N}(1,0),\cdots,R_{i,j}^{N}(\Phi-1,0)\right)^{*} =\underset{R_{i,j}^{N}(0,0),R_{i,j}^{N}(1,0),\cdots,R_{i,j}^{N}(\Phi-1,0)}{argmax}E_{i,j} \tag{12}\]  
Second, from the \(\left(N-1\right)^{th}\) to the top pyramid layer \(t^{th}\), we employ an iterative image matching procedure as follows:  
For each grid cell \(G_{i,j}^{  
parameter \(k\), represented by \(k^{*}\), is therefore extracted by searching the maximum NCC value using Equation (14), which is derived from Equations (9) and (10). \[k^{*}=\underset{k}{\mathrm{argmax}}\Big{\{}NCC\Big{(}G_{i,j}^{N}(\varphi,\psi),R_ {i,j}^{l}(\varphi,\psi)+k\zeta\Big{)}\Big{|}k=0,\pm 1,\ldots,\pm K\Big{\}} \tag{14}\]  
where \(NCC\Big{(}G_{i,j}^{l}(\varphi,\psi),R_{i,j}^{l}(\varphi,\psi)+k\zeta\Big{)}\) is computed by a pair of corresponding image patches based on Equation (10), and the image patches are generated by reprojection of each pixel in \(G_{i,j}^{l}(\varphi,\psi)\) onto \(I_{i}\) and \(I_{j}\) based on Equation (9), respectively. Therefore, the optimal object-distance is set as \(r_{R_{i,j}^{l}(\varphi,\psi)+k^{*}\zeta}\) for the grid cell \(G_{i,j}^{l}(\varphi,\psi)\). In this way, the resolution and structure of the object-distance maps, along with the index maps, are updated with the optimum object-distance for each grid cell, until reaching the top pyramid level.
# Modeling Volatility Characteristics of Epileptic EEGs using GARCH Models
## 7 Random Starting Points  
To assess whether the estimated models were dependent on the particular segment, 100 models based on randomly selected starting points within each dataset were estimated for all 66 channels and sample statistics calculated (Table 3). The average half-life for RAST3 (2.25 vs. 2.11) and RAST4  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Seizure Channels**} & \multicolumn{2}{c}{**Non-seizure**} \\  & **RAST3** & **RAST4** & **RMST4** & **Half Life** \\ \hline \multirow{4}{*}{**Segment**} & **HL** & **HL** & **HL** & **Median** \\  & **95\% CI** & **95\% CI** & **95\% CI** & **Min** \\  & **Overlapping** & **Overlapping** & **Overlapping** & **Max** \\  & **Interval Count** & **Interval Count** & **Interval Count** & **Interval Count** & **Max** \\ \hline \multirow{4}{*}{1} & 8.48 & 10.00 & 37.68 & 20.12 \\  & (6.39,12.43) & (7.73,14.02) & (22.51,112.78) & 2.20 \\  & 12 & 14 & 47 & 181.18 \\  & 17.07 & 29.08 & 23.14 & 18.25 \\  2 & (13.71,22.52) & (18.32,69.11) & (14.19,60.8) & 3.08 \\  & 24 & 44 & 46 & 85.24 \\  & 6.32 & 8.27 & 6.23 & 15.7 \\
3 & (5.29,7.81) & (6.01,13.05) & (4.50,9.91) & 0.49 \\  & 16 & 23 & 19 & 77.54 \\  & 12.14 & 181.38 & 36.23 & 5.92 \\
4 & (9.26,17.48) & (116.58,407.3) & (22.22,96.03) & 2.71 \\  & 34 & 15 & 26 & 201.51 \\  & 8.41 & 8.68 & 6.33 & 11.69 \\
5 & (6.90,10.72) & (5.82,16.44) & (4.57,10.04) & 3.09 \\  & 22 & 34 & 22 & 109.53 \\  & 18.87 & 22.62 & 10.51 & 13.49 \\
6 & (14.55,26.72) & (12.90,85.89) & (7.69,16.39) & 2.67 \\  & 37 & 48 & 33 & 78.44 \\  & 15.83 & 72.72 & 6.43 & 8.3 \\
7 & (12.10,22.75) & (43.60,216.39) & (3.91,16.19) & 2.47 \\  & 28 & 26 & 33 & 99.04 \\  & 20.3 & 380.8 & 7.92 & 17.17 \\
8 & (15.58,28.97) & (231.25,1075.45) & (4.29,37.49) & 2.49 \\  & 27 & 6 & 47 & 54.15 \\  & 10.76 & 5.94 & 6.42 & 16.65 \\
9 & (8.1,15.83) & (4.70,7.99) & (3.93,15.72) & 3.03 \\  & 25 & 15 & 25 & 59.89 \\  & 25.37 & 44.89 & 6.39 & 10.08 \\
10 & (16.88,50.32) & (29.18,96.25) & (3.60,22.43) & 2.7 \\  & 35 & 25 & 41 & 98.01 \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._(1.75 vs. 1.86) were similar to the average of the half-life from the six pre-seizure segments. The third quartile score for each signal was also less than two; all half-life estimates for both channels were lower than 75% of the estimates for the non-seizure channels. RMST4 had higher estimates than the other two channels and at least 75% of the scores fell below the median for non-seizure channels. Similar results can be seen for the awake data. For the sleep to awake and sleep data, the estimates are lower on average, but without as large or noticeable differences as seen in the pre-seizure and awake data.
# Exploring an Integrated Manure-Sewater System for Sustainable Cyanobacterial Biomass Production
## 2 Materials and Methods
### Development of Growth Media from Animal Manure and Seawater  
The manure-seawater experiment was conducted after we determined the growth performance of cyanobacterial strain HSaC at different NaCl salt concentrations. Then, inexpensive growth media for cyanobacteria were developed by integrating animal manure and seawater. Briefly, fresh animal manures, namely, cow manure, pig manure and chicken manure, were obtained from a farm in the Northern China Plain, and then, the manures were dried at room temperature to protect against nutrient loss. After the manures had dried, they were ground, and then sieved with 2 mm diameter mesh. The sieved manure was measured out based on 1 g of manure to 10 mL of distilled water, and then stirred with magnetic bars at 710 rpm for 1 h at room temperature, and autoclaved at 121 \({}^{\circ}\)C for 30 min to kill all the microbes in the manure. After autoclaving, the manures were centrifuged using 5000 rpm at room temperature for 5 min with autoclaved tubes to collect the supernatant. The collected supernatants were filtered with sterilized Whatman filter paper, and then the filtrates were sterilized with UV for 1 h and stored as stock solution at 4 \({}^{\circ}\)C. Both natural and artificial seawater were also autoclaved prior to the experiments. The first animal manure-seawater experiment was conducted in 3 replicates with CRD to identify the best manure-seawater media for the growth of cyanobacterial strain HSaC. The integrated manure-seawater treatments were: PD = pig manure with distilled water, P10 = pig manure with diluted (10 dS/m) seawater, PS = pig manure with seawater, CD = cow manure with distilled water, C10 = cow manure with diluted (10 dS/m) seawater, CS = cow manure with seawater, BGD = reference cyanobacterial growth media (BG11) with distilled water, BG10 = BG11 media with diluted (10 dS/m) seawater, BGS = BG11 media with seawater, ChD = chicken manure with distilled water, Ch10 = chicken manure with diluted (10 dS/m) seawater, and ChS = chicken manure with seawater. After we identified the best animal manure in reference to the BG11 media, we conducted a second experiment comparing both natural and artificial seawater with the selected best manure (cow manure) to investigate whether natural seawater can substitute for artificial seawater. The treatments were performed in 3 replicates using CRD as follows: DW = cow manure with distilled water, 10N = cow manure with diluted (10 dS/m) natural seawater, 10A = cow manure with diluted (10 dS/m) artificial seawater, SN = cow manure with natural seawater, and SA = cow manure with artificial seawater. For salt-sensitive cyanobacterial biomass production, we tested salt-sensitive cyanobacterial strain LC. Briefly, the treatments were performed in 3 replicates using CRD as follows: DW = cow manure with distilled water, 0.5 dS = cow manure with diluted (0.5 dS/m) seawater, and 1.5 dS = cow manure with diluted (1.5 dS/m) seawater.
## 3 Materials and Methods
### Computational Analysis
#### 3.1.3 Virtual Screening  
Due to the unknown active site, it was necessary to evaluate where the natural ligands (ATP and GTP) bind [86; 87]. Firstly, a conformational search procedure was utilized in a molecular dynamic framework implemented in Gabedit [88], and three conformers of each of these molecules were obtained. These were submitted for optimization and frequency calculations in Gaussian quantum chemistry software package (revision D.01) [89], using density functional theory method B3 LYP in 6-311++G(d,p) basis set, accompanied with GD3 dispersion correction [90] and SMD water solvent [91]. No imaginary frequency was present. The lowest one for each was chosen for the molecular docking procedure performed with Autodock Vina [92]. The water molecules were removed from the MD output and Gasteiger chargers added. Grid box size was set to cover the entire domain, and the docking procedure was started.  
Having detected the eventual binding site, the grid box was moved and resized to cover it. Afterwards, an in-house library consisting of 791 natural substances belonging to the collection of phytochemicals from the Department of Pharmacognosy and Herbal Medicines, Wroclaw Medical University, Poland was submitted for a virtual screening procedure, generating a list of the most probable inhibitors.  
The inhibition constant (_Ki_) was obtained from the binding free energy of the compound (\(\Delta\)G) using the equation:  
\[Ki=exp(\Delta G/RT)\]  
where, R is the gaseous constant (1.987 cal K\({}^{-1}\) mol\({}^{-1}\)) and T is the temperature (310.15 K).
# Sensor Fusion of GNSS and IMU Data for Robust Localization via Smoothed Error State Kalman Filter
## 3 Robust Localization via RTS Smoothing
### RTS Fundamental Design  
The RTS smoothing can be regarded as a technique for obtaining an optimal state estimate when observations are available from moment 1 to moment \(N\); it involves using previous estimates obtained through Kalman filtering in order to perform backward smoothing from moment \(k+1\) to moment \(k\) resulting in a more precise estimate. This method falls under the category of fixed interval smoothing and is considered as a fixed interval smoother, which is convenient for implementation.  
The RTS smoother involves a two\(-\)step process: a forward filter followed by a backward smoothing. The forward filter is a standard Kalman filter described by Equation (18), which maintains all the predicted and updated estimates as well as their corresponding covariances for each epoch during the entire mission. The backward smoothing procedure begins at the end of the forward filter at time \(t_{N}\), with an initial condition \(\textit{ffix}_{N,N}\) and \(t_{N,N}\). Therefore, the backward smoothing can be seen as an update to the forward filter for obtaining an enhanced estimate. The equation for the RTS algorithm can be represented in the form of  
\[\begin{array}{c}\textit{ffit}\widehat{\mathbf{x}}_{k,N}=\textit{ffit}\widehat{\mathbf{x} }_{k,k}+\mathbf{U}_{k}(\widehat{\mathbf{x}}_{k+1,N}-\widehat{\mathbf{x}}_{k+1,k})\\ \widehat{\mathbf{P}}_{k,N}=\widehat{\mathbf{P}}_{k,k}+\mathbf{U}_{k}(\widehat{\mathbf{P}}_{k+1,N}-\widehat{\mathbf{P}}_{k+1,k})\mathbf{U}_{k}^{T}\end{array} \tag{30}\]  
where delta \(\textit{ffix}_{k,N}\) is the smoothed estimate of the state vector, and \(U_{k}\) is the smoothing gain matrix, which is calculated by the following equation:  
\[\mathbf{U}_{k}=\widehat{\mathbf{P}}_{k,k}\mathbf{F}_{k+1,k}^{T}\widehat{\mathbf{P}}_{k+1,k}^{- 1}, \tag{31}\]  
where \(k=N-1,N-2,\ldots\ldots,0\). The RTS smoothing algorithm uses the results of the forward filter and backward smoothing to obtain an improved estimate. The inverse smoothing starts from the last filter result and proceeds forward one by one, so it must obtain N filter results to smooth the observations within the period [0, N]. However, if N is too large, the inverse smoothing process can be obviously lagged, which would limit the algorithm's application in real time. Therefore, a segmented RTS smoothing method is designed in this study.
## 4 Analysis of Inner Motion Loops
### The Stiffness Matrix
#### 4.1.3 Link Velocities  
In the case of link velocity control, there is a possibility of finding stiffness effects depending on the type of control. Considering the control input, \(\tau\), as a proportional-integral with gravity compensation, \(G(\dot{q})\),  
\[\tau=K_{pq}\Big{(}\dot{q}_{d}-\dot{q}\Big{)}+K_{i\dot{q}}\int\Big{(}\dot{q}_{d}- \dot{q}\Big{)}dt+G(q), \tag{25}\]  
where \(\dot{q}_{d}\) is the desired link velocity, \(K_{p\dot{q}}\) is the proportional gain matrix, and \(K_{i\dot{q}}\) is the integral gain matrix of the velocity control.  
The integral term can be analysed as a proportional term regarding the position.  
\[\tau=K_{pq}\Big{(}\dot{q}_{d}-\dot{q}\Big{)}+K_{i\dot{q}}\Big{(}q_{d}-q\Big{)}. \tag{26}\]  
Considering the robot dynamics, Equations (1) and (2) and the input in a steady state, we have  
\[K(q-\theta)=-J^{T}\!\!f, \tag{27}\]  
\[K(\theta-q)=\tau, \tag{28}\]  
\[\tau=K_{pq}\dot{q}_{d}+K_{i\dot{q}}\Big{(}q_{d}-q\Big{)}. \tag{29}\]  
Solving for \(q\) from Equation (29),  
\[q=q_{d}+K_{iq}^{-1}\Big{(}K_{pq}\dot{q}_{d}-\tau\Big{)}=q_{d}+K_{iq}^{-1}\Big{(} K_{pq}\dot{q}_{d}-J^{T}\!\!f\Big{)}. \tag{30}\]  
Ordering the equation,  
\[f=J^{-T}(K_{iq}\big{(}q_{d}-q\big{)}+K_{pq}\dot{q}_{d})\approx J^{-T}(K_{iq}J^ {-1}(x_{d}-x)+K_{pq}\dot{q}_{d}) \tag{31}\]  
Hence, the cartesian stiffness matrix of the robot will be,  
\[K_{x}=J^{-T}K_{iq}J^{-1}. \tag{32}\]  
The term \(J^{-T}K_{pq}\dot{q}_{d}\) acts like a bias. It means, for zero external force \(f\), the robot will have some deformation. Therefore, the integral control acts like active stiffness and the proportional control as a bias. If the velocity loop does not have an integrator, just like a proportional control, the control action will be.  
\[\tau=K_{pq}\Big{(}\dot{q}_{d}-\dot{q}\Big{)}, \tag{33}\]  
\[f=J^{-T}K_{pq}\Big{(}\dot{q}_{d}-\dot{q}\Big{)}=J^{-T}K_{pq}\dot{q}_{d}-J^{-T} K_{pq}\dot{q}, \tag{34}\]  
The first term acts like bias and the second one as damping. Thus, there is no active stiffness. The system will behave like a mass-damper under an external force. The final state will be achieved when the robot stops (\(\dot{q}=0\)).  
\[f=-J^{-T}K_{pq}\dot{q}_{d}, \tag{35}\]  
which means, that the reference velocities, \(\dot{q}_{d}\), should be  
\[\dot{q}_{d}=-J^{T}K_{pq}^{-1}\!\!f. \tag{36}\]  
This corresponds to an open control loop, theoretically without force error.  
In summary, if the velocity loop has an integrator, the integrator acts like active stiffness in the joint space and the proportional control as bias. In the case, it does not contain an integrator; if no stiffness term appears, then the exerted force is proportional to the reference velocity.
## 2 Materials and Methods
### Research Methods
#### 2.3.2 Stability Analysis  
The Coefficient of Variation (CV), also known as the \"Coefficient of Dispersion\", is the ratio of the standard deviation to the mean. It serves as a measure that can, to some extent, reflect the spatial distribution characteristics of the interannual stability of vegetation indices [31]. By utilizing 20 years of NDVI data in the study area, the coefficient of variation for each pixel was calculated. This process enabled the assessment of the extent of NDVI variation over the course of these two decades in the Shanxi segment of the Yellow River Basin. Its calculation formula is as follows:  
\[CV=\frac{SD_{NDVI}}{\overline{NDVI}}, \tag{5}\]  
\[SD_{NDVI}=\sqrt{\frac{\sum\limits_{i=1}^{n}{N_{i}^{2}-(\sum\limits_{i=1}^{n}{ N_{i}})^{2}/n}}{n}} \tag{6}\]  
where \(CV\) represents the _NDVI_ coefficient of variation, \(SD_{NDVI}\) is the standard deviation of the _NDVI_, \(\overline{NDVI}\) is the mean of the _NDVI_, \(N_{i}\) is the _NDVI_ value of the _ith_ year, and \(n\) is the length of the time series of the study period (here \(n\) = 20).
## 3 Results
### Effect of Filter Banks  
In this study, a filter bank is utilized to extract frequency-domain information to improve the accuracy of the proposed MDCCSP method. Here, a set of filter banks are designed with various bandwidths and various sizes of overlapping to find an optimal set of filter banks for experiments. The principle of the design of filter banks is to fully cover all the frequency range of 4-32 Hz that is considered. Therefore, the designs of filter banks are shown in Table 2.  
The number of filters included in the eight sets of filter banks is between 3 and 14. The effects of the filter bank combining traditional EEG rhythms (theta, alpha, and beta bands) are also compared in the experiments. In this experiment, \(x(t)\) and \(\Theta(t)\) are extracted to obtain AP information, and the S-CC is used in the proposed MDCCSP method. The regularization parameter in the MDCCSP is set to \(\alpha=0.001\). The number of features generated by each CCSP \(N\) is set as two. The engagement recognition results using different sets of filter banks in Table 2 are shown in Figure 3. And the recognition accuracies are shown in Table 3.  
As shown, the second filter bank achieves the highest recognition accuracy of \(87.74\pm 5.98\%\) at the significance level of \(5\%\). In the following experiments, the second filter bank with 13 filters is adopted.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{} & \multicolumn{6}{c}{**Filter Bank**} \\ \hline
**Index** & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \multirow{4}{*}{**Frequency Range** (4–32 Hz)} & 4–6 & 4–8 & 4–8 & 4–12 & 4–16 & 4–20 & 4–24 & 4–8 (theta) \\  & 6–8 & 6–10 & 8–12 & 8–16 & 8–20 & 8–24 & 8–28 & 8–13 (alpha) \\  & 8–10 & 8–12 & 12–16 & 12–20 & 12–24 & 12–28 & 12–32 & 13–32 (beta) \\  & 10–12 & 10–14 & 16–20 & 16–24 & 16–28 & 16–32 & & \\  & 12–14 & 12–16 & 20–24 & 20–28 & 20–32 & & & \\  & 14–16 & 14–18 & 24–28 & 24–32 & & & & \\  &... & 28–32 & & & & & & \\  & 30–32 & 28–32 & & & & & & \\ \hline \multirow{2}{*}{**Number of filters**} & \multirow{2}{*}{14} & \multirow{2}{*}{13} & \multirow{2}{*}{7} & \multirow{2}{*}{6} & \multirow{2}{*}{5} & \multirow{2}{*}{4} & \multirow{2}{*}{3} & \multirow{2}{*}{3} \\  & & & & & & & & & \\ \end{tabular}
\end{table}
Table 2: Filter banks validated in the experiments.
###### Abstract  
Polarimetric synthetic aperture radar (PolSAR) has unique advantages in building extraction due to its sensitivity to building structures and all-time/all-weather imaging capabilities. However, the structure of buildings is complex, and buildings are easily confused with other objects in polarimetric SAR images. The speckle noise of SAR images will affect the accuracy of building extraction. This paper proposes a novel building extraction approach from PolSAR images with statistical texture and polarization features by using a convolutional neural network and superpixel. A feature space that is sensitive to building, including G\({}^{0}\) statistical texture and PualiRGB features, is constructed and used as CNN input. Considering that the building boundary of the CNN classification result is inaccurate due to speckle noise, the simple linear iterative cluster (SLIC) superpixel is utilized to constrain the building extraction result. Finally, the effectiveness of the proposed method has been verified by experimenting with PolSAR images from three different sensors, including ESAR, GF-3, and RADARSAT-2. Experiment results show that compared with the other five PolSAR building extraction methods including threshold, SVM, RVCNN, and FFDCNN, our method without superpixel constraint, the F1-score of this method is the highest, reaching 84.22%, 91.24%, and 87.49%, respectively. The false alarm rate of this method is at least 10% lower and the F1 index is at least 6% higher when the building extraction accuracy is comparable. Further, the discussion and method parameter analysis results show that increasing the use of G\({}^{0}\) statistical texture parameters can improve building extraction accuracy and reduce false alarms, and the introduction of superpixel constraints can further reduce false alarms.  
building extraction; G\({}^{0}\) statistical texture; PolSAR; CNN; superpixel +
Footnote †: journal: _Remote Sens._**2023**, _15_, 1451. [[https://doi.org/10.3390/rs15051451](https://doi.org/10.3390/rs15051451)]([https://doi.org/10.3390/rs15051451](https://doi.org/10.3390/rs15051451))
## 4 Results and Discussion
### Mathematical Modeling
#### 4.2.1 Ergun Equation Modifying Method
To sum up, the specific calculation process of the Ergun equation modifying method is that \(U_{0\text{mf}}\) can be calculated by solving the following equation:  
\[\pi r_{0}^{2}H_{\text{mf}}(A\,U_{0\text{mf}}+BU_{0\text{mf}}^{2}\frac{r_{0}}{r_ {\text{mf}}})= 0.8995\cdot Wg \tag{27}\]  
where \(A=\)46.6\(\frac{(1-\varepsilon_{\text{mf}})^{2}}{\varepsilon_{\text{mf}}^{3}}\frac{\mu}{( \phi d_{\text{p}})^{2}}\) and \(B=\)3.25\(\frac{(1-\varepsilon_{\text{mf}})}{\varepsilon_{\text{mf}}^{3}}\frac{\rho_{g}}{ \phi d_{\text{p}}}\) (\(H_{\text{mf}}\), \(r_{\text{mf}}\), and \(\varepsilon_{\text{mf}}\) can be obtained by Equations (14)-(16), respectively). After \(U_{0\text{mf}}\) is calculated, \(\Delta P_{\text{mf}}\) can be calculated by substituting the calculated \(U_{0\text{mf}}\) into Equation (23).