#
Study on the Pretreatment of Soil Hyperspectral and Na^{+} Ion Data under Different Degrees of Human Activity Stress by Fractional-Order Derivatives

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}

^{2−}, Cl

^{−}, SO

_{4}

^{2−}, Na

^{+}, Ca

^{2+}, Mg

^{2+}ions generated after the dissociation of sodium salts, magnesium salts and calcium salts). The characteristics of water-soluble base ions can show the migration trend of total salt to a certain extent. They are an important factor in the response of soil hyperspectral reflectivity and can be used as a characteristic factor of soil salinization. As one of the water-soluble base ions, Na+ ion is the main indicator of the degree of soil salinization [5], and the accurate estimation of its content can provide important technical support for the treatment of saline soils.

^{+}, Mg

^{2+}, Ca

^{2+}, SO

_{4}

^{2−}and Cl

^{-}ions, while the prediction performance of K

^{+}, CO

_{3}

^{2−}and HCO

_{3}

^{−}ions is poor. Srivastava et al. [8] collected soil samples from farmers’ fields in three villages in the Ganges Plain, India, and used the PLSR model to predict soil salinity information. Their simulation showed that the 1390–2400 nm band can successfully predict the saline soil information. It has strong predictive ability for electrical conductivity, Na

^{+}, and Cl

^{−}, as well as good predictive ability for Ca

^{2+}, good predictive ability for SO

_{4}

^{2−}, and poor predictive ability for CO

_{3}

^{2−}, PH, and HCO

_{3}

^{−}.

^{®}3 portable spectrometer was used to measured spectra, which was done in a dark room. The lead (Pb) and zinc (Zn) concentrations of the samples were obtained in the laboratory and PLSR and random forest (RF) were used as calibration methods. This study demonstrated that the model accuracy of RF was better than PLSR, and the RF model for predict the concentration of Pb and Zn at 0.25- and 0.5-order had the optimal prediction accuracies. Zhang et al. [13] collected soil samples in northwestern China, and soil organic matter content (SOMC) and reflectance spectra were measured in the laboratory, FOD were used to pretreat soil spectra and a partial least squares-support vector machine (PLS-SVM) model was used to estimate SOMC. The conclusions showed that the 1.05- to 1.45-order range had the highest “signal-to-noise ratio”, and it was most suitable for SOMC analysis. Wang et al. [14] used soil located at the Ebinur Lake Wetland National Nature Reserve (ELWNNR) in northwest China as research object. Three new FOD methods, the optimal remote sensing index and the subsection of spectral band method were selected to predict SOMC. Results showed that the optimal model appeared at 1.2-order, and its lowest root mean square error (RMSE) was 1.70 g/kg. Wang et al. [15] applied FOD was to the pretreatment of soil hyperspectral signals of samples collected from the Ebinur Lake basin in the Xinjiang Uighur Autonomous Region of China, and the PLSR model was used to estimate the clay content of the desert soils. Spectral reflectance and clay content were measured in the laboratory. Simulations showed that the 1.8-order FOD was effective.

^{®}3Hi-Res spectrometer in the field environment. This study provides a new method for the hyperspectral pretreatment of saline soils in different human interference areas, which has important practical significance for promoting the sustainable development of Xinjiang land resources, improving the saline soil in oases, and preventing further soil degradation.

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Soil Sample Collection

^{+}ion content by professionals. Atomic absorption spectrometry was used to determine the Na

^{+}content.

#### 2.3. Soil Hyperspectra Collection

^{®}3Hi-Res spectrometer, and the wavelength range of the spectrometer is from 350 nm to 2500 nm. We measured the soil when the local time was 12:30–15:30, and the weather is sunny with little clouds and no wind. In addition, whiteboard correction is required before each collection, and each sample point is selected at five different locations within 1 m to repeat the measurements 10 times, and the average value of the 50 sets of hyperspectral curves is the measured field hyperspectral reflectance of this sample point. At the same time, the collected field VNIR-SWIR not only needs jumping point correction and Savitzky-Golay smoothing and denoising preprocessing, but the edge wavelength (350~399 nm and 2401~2500 nm) also needs to be removed the wavelength near the moisture absorption band (1340~1420 nm and 1800~1960 nm) are deleted, leaving 1759 hyperspectral wavelengths remaining.

#### 2.4. Grünwald-Letnikov Fractional-Order Derivative

^{®}3Hi-Res spectrometer is 1 nm, and the derivative step length can be set h = 1. The difference expression of the FOD of the function can be derived from Equation (5) as follows:

#### 2.5. Correlation Coefficient

## 3. Simulation Results

#### 3.1. Statistical Characteristics of Soil Salinity Information under Different Disturbance Degrees

^{+}, K

^{+}, Ca

^{2+}, Mg

^{2+}, Cl

^{−}, SO

_{4}

^{2−}, CO

_{3}

^{2−}, HCO

_{3}

^{−}) (Table 1, Table 2 and Table 3), and give the values of the correlation coefficients in absolute values.

^{+}in Area A is better (0.738) and the correlation between total salt and SO

_{4}

^{2−}is better (0.527), while the correlation between total salt and Ca

^{2+}is lower (0.309) and total salt and the remaining five ions (K

^{+},Mg

^{2+},Cl

^{−},CO

_{3}

^{2−},HCO

_{3}

^{−}) have very low correlations. In addition, the correlation between SO

_{4}

^{2−}and Ca

^{2+}is better (0.653), and the correlation between SO

_{4}

^{2−}and Mg

^{2+}is good at 0.559, which indicates that there is more CaSO

_{4}and MgSO

_{4}in the soil of Area A. The correlation between CO

_{3}

^{2−}and Ca

^{2+},CO

_{3}

^{2−}and Mg

^{2+},CO

_{3}

^{2−}and Na

^{+},HCO

_{3}

^{−}and Ca

^{2+}is general, and its value is about 0.3, which shows that there is also a certain amount of CaCO

_{3}, MgCO

_{3}, NaCO

_{3}and CaHCO

_{3}in the soil of Area A.

^{+}in Area B is better (0.859), the correlation between total salt and Cl

^{−}is good at 0.717, the correlation between total salt and Ca

^{2+}is general at 0.433, and the correlation between total salt and Mg

^{2+}is less at 0.336, and the correlation between total salt and SO

_{4}

^{2−}is small at 0.319, and the correlation between total salt and the remaining three ions (K

^{+}, CO

_{3}

^{2−}, HCO

_{3}

^{−}) is very low. At the same time, the correlation between CO

_{3}

^{2−}and Ca

^{2+}is better (0.629), the correlation between HCO

_{3}

^{−}and Ca

^{2+}is better (0.579), the correlation between Cl

^{−}and Mg

^{2+}is better (0.598), and the correlation between Cl

^{−}and Ca

^{2+}is better at 0.523, which shows that there is a lot of CaCO

_{3}, CaHCO

_{3}, MgCl and CaCl in the soil of Area B. The correlation between CO

_{3}

^{2−}and Mg

^{2+}is general (0.376), the correlation between HCO

_{3}

^{−}and Mg

^{2+}is general (0.393), the correlation between Cl

^{−}and Na

^{+}is general at 0.390, and the correlation between SO

_{4}

^{2−}and Ca

^{2+}is general at 0.330. It shows that there is also a certain amount of MgCO

_{3}, MgHCO

_{3}, NaCl and CaSO

_{4}in the soil of Area B.

^{−}and Na

^{+}in Area C is very good, reaching 0.979 and 0.971, respectively. The correlation between total salt and SO

_{4}

^{2−}is as good as 0.697, and the correlation between total salt and Ca

^{2+}is also as good as 0.685, the correlation between the total salt and the remaining four ions (K

^{+}, Mg

^{2+}, CO

_{3}

^{2−}, HCO

_{3}

^{−}) is very low. At the same time, the correlation between Cl

^{−}and Na

^{+}is 0.990, and the correlation between SO

_{4}

^{2−}and Ca

^{2+}is 0.801, which indicates that there are a lot of NaCl and CaSO

_{4}in the soil of Area C. The correlation between Cl

^{−}and Ca

^{2+}is 0.574, and the correlation between SO

_{4}

^{2−}and Na

^{+}is 0.580, which shows that there is more CaCl and NaSO

_{4}in the soil of Area C. The correlation between HCO

_{3}

^{−}and Mg

^{2+}is 0.477, the correlation between HCO

_{3}

^{−}and Ca

^{2+}is 0.474, the correlation between CO

_{3}

^{2−}and Ca

^{2+}is 0.361, and the correlation between SO

_{4}

^{2−}and Mg

^{2+}is 0.335, so we can conclude there is a certain amount of MgHCO

_{3}, CaHCO

_{3}, CaCO

_{3}and MgSO

_{4}.

#### 3.2. Curve Characteristics of Soil Original Hyperspectral Reflectance Pretreated by FOD

^{−4}~10 × 10

^{−4}, and the range between 1.5-order and 2.0-order (Figure 1f) is −1.25 × 10

^{−4}~0.96 × 10

^{−4}. The derivative value of the original hyperspectral reflectance of Area B ranges from 0.03 to 0.43 between 0.0-order and 0.5-order, and between 0.5-order and 1.0-order is 0.001 to 0.036, and between 1.0-order and 1.5-order (Figure 2e) is −8 × 10

^{−4}~5 × 10

^{−4}, and the range between 1.5- order and 2.0-order (Figure 2f) is −5.2 × 10

^{−4}~0.9 × 10

^{−4}. The derivative value of the original hyperspectral reflectance of Area C ranges from 0.03 to 0.49 between 0.0-order and 0.5-order, and from 0.5-order to 1.0-order is 0.001 to 0.043, and between 1.0-order and 1.5-order (Figure 3e) is −1.5 × 10

^{−3}~2.23 × 10

^{−3}, and the range between 1.5-order and 2.0-order (Figure 3f) is −1.4 × 10

^{−3}~0.8 × 10

^{−3}. The reason is that as the order of the fractional-order derivative gradually increases, the peak profile of the hyperspectral reflectance curve is gradually changed, and the peak removal operation is continuously completed, which causes the reflectance derivative value of the ordinate to gradually decrease.

#### 3.3. The Correlation Coefficient between the Original Hyperspectral and Na^{+} Is Affected by FOD

^{+}is shown in Figure 5. The simulation shows that when the correlation coefficients were in the light 0.0-order to 0.6-order, moderate 0.0-order to 0.4-order and severe 0.0-order to 0.4-order, none of the correlation coefficients in any band passed the 0.01 significance level test. When the order of the FOD is increased from 0.0-order to 1.0-order, the change trend of the correlation coefficient curve shows a certain gradual form. For example, the more obvious band range of the gradual change trend of the correlation coefficient curve is: 400–700 nm, 790–900 nm and 1200–1300 nm in the Area A; 400–950 nm, 1000–1300 nm and 1450–1750 nm in the Area B; and 400–900 nm, 1120–1330 nm and 1500–1780 nm in the Area C. However, when the order increases from 1.0-order to 2.0-order, the variation of the correlation coefficient curve fluctuates greatly, which is less obvious than the gradual change from 0.0-order to 1.0-order.

^{+}. For example, in the Area A, the 0.2-order is −0.0101~0.1255, the 0.5-order is −0.2350~0.3557, the 0.7-order is −0.5105~0.4855, the 0.9-order is −0.5861~0.5848, the 1.2-order is −0.5678~0.5853, the 1.5-order is −0.5580~0.5921, and the 1.9-order is −0.5692~0.5940. In Area B, the order of 0.3 is 0.0685~0.3147, the order of 0.5 is −0.1196~0.4758, the order of 0.8 is −0.4290~0.5942, the order of 1.1 is −0.5183~0.5933, the order of 1.4 is −0.5961~0.5170, and the order of 1.7 is −0.6589~0.5262. In Area C, the order of 0.3 is −0.3966~−0.2111, the order of 0.5 is −0.5094~−0.0205, the order of 0.8 is −0.5058~0.2962, the order of 1.1 is −0.5494~0.5102, the order of 1.4 is −0.5406~0.5178, and the order of 1.7 is- 0.5075~0.5612. Therefore, with the increase of the FOD, the overall variation range of the correlation coefficient between the original hyperspectral and Na

^{+}is gradually increased.

#### 3.4. The Correlation Coefficient between Each Spectra and Na^{+} Passed the 0.01 Test under Different Derivative

^{+}passed the 0.01 test pretreated by FOD. In each spectrum, there are many FOD band numbers that exceed the integer-order, and they are concentrated in higher-order FOD derivative (1.2- to 1.6-ordder). For example, the band numbers of FOD are more than integer 1.0-order, it mainly appears in the 1.4- to1.9-order of each spectrum in the Area A, it mainly appears in the 1.2- to1.9-order of each spectrum in the Area B, while in the Area C it mainly appears in the R and $\sqrt{\mathrm{R}}$ from 1.2- to 1.6-order, in the 1/R from 0.6- to 0.9-order or 1.2- to 1.3-order.

#### 3.5. The Maximum Correlation Coefficient between Each Spectrum and Na^{+} Pretreated by FOD

^{+}, as well as its corresponding fractional-order and band (Table 4, Table 5 and Table 6). Overall, with the increase of the FOD, the absolute value of the maximum correlation coefficient of each spectrum at different orders shows a trend of first increasing and then decreasing, and the improvement of the correlation coefficient by the low-order derivative is small. While the improvement of the correlation coefficient by the higher-order derivative is great, for example, 1.4- and 1.5-order in Area A, 1.8- and 1.9-order in Area B, and 1.0- to 1.8-order in Area C.

## 4. Discussion

#### 4.1. Fractional-Order Derivative Improves the Resolution between Hyperspectral Peaks

#### 4.2. Comparison of the Fractional-order derivative and Integer-order derivative

#### 4.3. The Percentage Improvement of the Fractional-Order Maximum Correlation Coefficient Compared with the Integer-Order

^{+}is located is 1.4-, 1.5-, 1.9-orders in Area A, and 1.7-, 1.8-orders in Area B, and 1.1-, 1.3-, 1.4-, 1.7-, 1.8-orders in Area C.

^{+}in the three regions is mostly more than 3%, and the highest is more than 17%. It can be seen that the effect of the fractional-order on the correlation between hyperspectral and cation Na

^{+}is significantly better than that of the integer-order.

#### 4.4. The Advantages of Different Transform Spectra

## 5. Conclusions

^{®}3Hi-Res spectrometer is used to measure the field ground hyperspectral of the soil, and the soil Na

^{+}ion content is determined by chemical detection methods. The Grünwald-Letnikov FOD is used to preprocess the soil original spectra and its transformed spectra (R, $\sqrt{\mathrm{R}}$, 1/R, lgR, 1/lgR). The manuscript mainly focuses on the characteristics of the original hyperspectral reflectance curve of the soil pretreated by FOD, and the correlation coefficients between five transformed spectrum and soil Na

^{+}ion. In addition, the useful information hidden in the FOD of hyperspectra is examined in depth, the advantages of fractional-order and integer-order derivative to improve preprocessing are discussed, and the nonlinear characteristics and variation law of hyperspectral in the field of saline soil are revealed. The conclusions of the study could be summarized as follows: (1) With the increase of the order, the peak contour of the hyperspectral data is gradually changed, and the peak removal operation makes the original soil hyperspectral reflectivity curve pretreated by derivative gradually approach the rate of change of the slope of the curve. (2) The improvement of the correlation coefficient between the spectrum and Na

^{+}by the low-order derivative is relatively small, while the improvement of the high-order derivative is relatively large. (3) Although the change trend of the band numbers that passed the 0.01 level of significance test is different under different human disturbance areas, but in general, the hyperspectral bands that passed the 0.01 level test are mostly located at the high-order derivative. And there are many FOD bands in the original spectrum and its transformation that exceed the integer-order derivative. Therefore, this study collects the field VNIR-SWIR of soil samples in different human disturbance areas, preprocesses the VNIR-SWIR spectroscopy of the soil using FOD, and observes the details of the changes in the hyperspectral reflectance curve in different FOD. The feasibility of the application for FOD in the field hyperspectra under different degrees of human activity stress is verified.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The reflectance curve of the soil original hyperspectral calculated by FOD in the lightly disturbed area: (

**a**) 0.0-order to 0.5-order; (

**b**) 0.5-order to 1.0-order; (

**c**) 1.0-order to 1.5-order; (

**d**) 1.5-order to 2.0-order; (

**e**) 1.0-order to 1.5-order of local amplification band; (

**f**) 1.5-order to 2.0-order of local amplification band.

**Figure 3.**The reflectance curve of the soil original hyperspectral calculated by FOD in the moderately disturbed area: (

**a**) 0.0-order to 0.5-order; (

**b**) 0.5-order to 1.0-order; (

**c**) 1.0-order to 1.5-order; (

**d**) 1.5-order to 2.0-order; (

**e**) 1.0-order to 1.5-order of local amplification band; (

**f**) 1.5-order to 2.0-order of local amplification band.

**Figure 4.**The reflectance curve of the soil original hyperspectral calculated by FOD in the severely disturbed area: (

**a**) 0.0-order to 0.5-order; (

**b**) 0.5-order to 1.0-order; (

**c**) 1.0-order to 1.5-order; (

**d**) 1.5-order to 2.0-order; (

**e**) 1.0-order to 1.5-order of local amplification band; (

**f**) 1.5-order to 2.0-order of local amplification band.

**Figure 5.**The correlation coefficient between soil hyperspectral and Na+ pretreated by FOD: (

**a**) 0.0-order to 0.6-order in Area A; (

**b**) 0.7-order to 1.0-order in Area A; (

**c**) 1.1-order to 1.5-order in Area A; (

**d**) 1.6-order to 2.0-order Area A. (

**e**) 0.0-order to 0.4-order in Area B; (

**f**) 0.5-order to 1.0-order in Area B; (

**g**) 1.1-order to 1.5-order in Area B; (

**h**) 1.6-order to 2.0-order in Area B. (

**i**) 0.0-order to 0.4-order in Area C; (

**j**) 0.5-order to 1.0-order in Area C; (

**k**) 1.1-order to 1.5-order in Area C; (

**l**) 1.6-order to 2.0-order in Area C.

**Figure 6.**The number of bands where the correlation coefficient between each spectral transformation and Na

^{+}passed the 0.01 test: (

**a**) Area A, (

**b**) Area B, (

**c**) Area C.

Item | CO_{3}^{2−} | HCO_{3}^{−} | Cl^{−} | SO_{4}^{2−} | Ca^{2+} | Mg^{2+} | K^{+} | Na^{+} | Total Salt |
---|---|---|---|---|---|---|---|---|---|

CO_{3}^{2−} | 1 | ||||||||

HCO_{3}^{−} | 0.117 | 1 | |||||||

Cl^{−} | 0.241 | 0.379 * | 1 | ||||||

SO_{4}^{2−} | 0.124 | 0.164 | 0.02 | 1 | |||||

Ca^{2+} | 0.432 ** | 0.311 * | 0.136 | 0.653 ** | 1 | ||||

Mg^{2+} | 0.312 * | 0.223 | 0.04 | 0.559 ** | 0.427 ** | 1 | |||

K^{+} | 0.025 | 0.124 | 0.035 | 0.017 | 0.353 * | 0.148 | 1 | ||

Na^{+} | 0.308 * | 0.211 | 0.025 | 0.059 | 0.263 | 0.301 | 0.107 | 1 | |

total salt | 0.158 | 0.051 | 0.155 | 0.527 ** | 0.309 * | −0.063 | 0.103 | 0.738 ** | 1 |

Item | CO_{3}^{2−} | HCO_{3−} | Cl^{−} | SO_{4}^{2−} | Ca^{2+} | Mg^{2+} | K^{+} | Na^{+} | Total Salt |
---|---|---|---|---|---|---|---|---|---|

CO_{3}^{2−} | 1 | ||||||||

HCO_{3−} | 0.336 * | 1 | |||||||

Cl^{−} | 0.284 | 0.256 | 1 | ||||||

SO_{4}^{2−} | 0.029 | 0.086 | 0.036 | 1 | |||||

Ca^{2+} | 0.629 ** | 0.579 ** | 0.523 ** | 0.330 * | 1 | ||||

Mg^{2+} | 0.376 * | 0.393 * | 0.598 ** | 0.063 | 0.493 ** | 1 | |||

K^{+} | 0.09 | 0.035 | 0.03 | 0.135 | 0.158 | 0.039 | 1 | ||

Na^{+} | 0.213 | 0.098 | 0.390 * | 0.265 | 0.088 | 0.068 | 0.154 | 1 | |

total salt | 0.007 | −0.071 | 0.717 ** | 0.319 * | 0.433 ** | 0.336 * | 0.016 | 0.859 ** | 1 |

Item | CO_{3}^{2−} | HCO_{3−} | Cl^{−} | SO_{4}^{2−} | Ca^{2+} | Mg^{2+} | K^{+} | Na^{+} | Total Salt |
---|---|---|---|---|---|---|---|---|---|

CO_{3}^{2−} | 1 | ||||||||

HCO_{3−} | 0.112 | 1 | |||||||

Cl^{−} | 0.252 | 0.186 | 1 | ||||||

SO_{4}^{2−} | 0.278 | 0.448 ** | 0.573 ** | 1 | |||||

Ca^{2+} | 0.361 * | 0.474 ** | 0.574 ** | 0.801 ** | 1 | ||||

Mg^{2+} | 0.186 | 0.477 ** | 0.119 | 0.335 * | 0.461 ** | 1 | |||

K^{+} | 0.198 | 0.026 | 0.021 | 0.014 | 0.127 | 0.212 | 1 | ||

Na^{+} | 0.224 | 0.143 | 0.990 ** | 0.580 ** | 0.506 ** | 0.021 | 0.069 | 1 | |

total salt | 0.269 | 0.238 | 0.979 ** | 0.697 ** | 0.685 ** | 0.124 | 0.026 | 0.971 ** | 1 |

**Table 4.**The absolute value of the maximum correlation coefficient between the hyperspectral and Na

^{+}in Area A and its corresponding band.

Order | R | $\sqrt{\mathrm{R}}$ | 1/R | lgR | 1/lgR | |||||
---|---|---|---|---|---|---|---|---|---|---|

Band/nm | Max Value | Band/nm | Max Value | Band/nm | Max Value | Band/nm | Max Value | Band/nm | Max Value | |

0.0 | 572 | 0.1175 | 572 | 0.1253 | 584 | 0.1502 | 572 | 0.1334 | 572 | 0.1138 |

0.1 | 572 | 0.1206 | 571 | 0.1291 | 571 | 0.1565 | 571 | 0.1380 | 572 | 0.1161 |

0.2 | 2352 | 0.1255 | 571 | 0.1333 | 570 | 0.1646 | 571 | 0.1432 | 2352 | 0.1210 |

0.3 | 2352 | 0.1768 | 2352 | 0.1808 | 2352 | 0.1935 | 2352 | 0.1850 | 2352 | 0.1735 |

0.4 | 2352 | 0.2593 | 2352 | 0.2606 | 2352 | 0.2637 | 2352 | 0.2618 | 2352 | 0.2608 |

0.5 | 2352 | 0.3557 | 2352 | 0.3531 | 2352 | 0.3415 | 2352 | 0.3499 | 2352 | 0.3621 |

0.6 | 2352 | 0.4156 | 2352 | 0.4137 | 2352 | 0.3966 | 2352 | 0.4098 | 2352 | 0.4153 |

0.7 | 2307 | 0.5105 | 2307 | 0.4968 | 2303 | 0.4832 | 2303 | 0.4845 | 2307 | 0.5118 |

0.8 | 2307 | 0.5671 | 2303 | 0.5665 | 2303 | 0.5835 | 2303 | 0.5768 | 2307 | 0.5432 |

0.9 | 2307 | 0.5861 | 2303 | 0.6002 | 2303 | 0.6246 | 2303 | 0.6122 | 2307 | 0.5550 |

1.0 | 2303 | 0.5911 | 2303 | 0.6061 | 2303 | 0.6311 | 2303 | 0.6179 | 2307 | 0.5610 |

1.1 | 2307 | 0.5835 | 2303 | 0.5970 | 2303 | 0.6197 | 2303 | 0.6078 | 2307 | 0.5614 |

1.2 | 647 | 0.5853 | 498 | 0.5947 | 2303 | 0.5963 | 2303 | 0.5867 | 2307 | 0.5548 |

1.3 | 1103 | 0.5797 | 498 | 0.5934 | 1103 | 0.5937 | 1103 | 0.5942 | 1103 | 0.5570 |

1.4 | 1103 | 0.5948 | 1103 | 0.6081 | 1103 | 0.6276 | 1103 | 0.6184 | 498 | 0.5716 |

1.5 | 1103 | 0.5921 | 1103 | 0.6070 | 1103 | 0.6335 | 1103 | 0.6193 | 498 | 0.5721 |

1.6 | 2302 | 0.5888 | 2302 | 0.6022 | 1103 | 0.6201 | 2302 | 0.6097 | 498 | 0.5593 |

1.7 | 2302 | 0.5826 | 2302 | 0.5995 | 2302 | 0.6117 | 2302 | 0.6098 | 649 | 0.5576 |

1.8 | 649 | 0.5715 | 2302 | 0.5904 | 2302 | 0.6119 | 2302 | 0.6044 | 649 | 0.5531 |

1.9 | 2301 | 0.5940 | 2301 | 0.5961 | 2302 | 0.6085 | 2302 | 0.5941 | 2301 | 0.5770 |

2.0 | 2300 | 0.5912 | 2300 | 0.6014 | 2301 | 0.6016 | 2300 | 0.6134 | 2300 | 0.5712 |

**Table 5.**The absolute value of the maximum correlation coefficient between the hyperspectral and Na

^{+}in Area B and its corresponding band.

Order | R | $\sqrt{\mathrm{R}}$ | 1/R | lgR | 1/lgR | |||||
---|---|---|---|---|---|---|---|---|---|---|

Band/nm | Max Value | Band/nm | Max Value | Band/nm | Max Value | Band/nm | Max Value | Band/nm | Max Value | |

0.0 | 661 | 0.2512 | 661 | 0.2329 | 574 | 0.1660 | 591 | 0.2122 | 2311 | 0.2744 |

0.1 | 2311 | 0.2562 | 2311 | 0.2400 | 2311 | 0.1773 | 2311 | 0.2214 | 2311 | 0.2857 |

0.2 | 2311 | 0.2777 | 2311 | 0.2623 | 2311 | 0.2018 | 2311 | 0.2445 | 2311 | 0.3050 |

0.3 | 2311 | 0.3147 | 2311 | 0.3008 | 2311 | 0.2448 | 2311 | 0.2844 | 2311 | 0.3375 |

0.4 | 2311 | 0.3777 | 2311 | 0.3669 | 2311 | 0.3211 | 2311 | 0.3538 | 2311 | 0.3916 |

0.5 | 2311 | 0.4758 | 2311 | 0.4717 | 2311 | 0.4493 | 2311 | 0.4656 | 2311 | 0.4720 |

0.6 | 2311 | 0.5715 | 2311 | 0.5771 | 2311 | 0.5834 | 2311 | 0.5807 | 2311 | 0.5454 |

0.7 | 2310 | 0.5942 | 2310 | 0.6081 | 2310 | 0.6090 | 2310 | 0.6164 | 2310 | 0.5375 |

0.8 | 2310 | 0.5942 | 2310 | 0.6005 | 2310 | 0.5683 | 2310 | 0.5989 | 2310 | 0.5530 |

0.9 | 2046 | 0.5803 | 2046 | 0.5781 | 2046 | 0.5671 | 2046 | 0.5751 | 2046 | 0.5774 |

1.0 | 1983 | 0.6373 | 1983 | 0.6255 | 1983 | 0.5430 | 1983 | 0.6035 | 1983 | 0.5924 |

1.1 | 1983 | 0.5933 | 2045 | 0.5643 | 2045 | 0.5368 | 2045 | 0.5546 | 1983 | 0.5976 |

1.2 | 2045 | 0.5609 | 2045 | 0.5527 | 2045 | 0.5321 | 2045 | 0.5450 | 2045 | 0.5863 |

1.3 | 601 | 0.6064 | 601 | 0.6142 | 1603 | 0.5357 | 601 | 0.6103 | 2045 | 0.5653 |

1.4 | 601 | 0.5961 | 601 | 0.5779 | 1603 | 0.5558 | 1603 | 0.5622 | 601 | 0.6060 |

1.5 | 434 | 0.6026 | 434 | 0.6082 | 1603 | 0.5594 | 434 | 0.5887 | 434 | 0.5933 |

1.6 | 434 | 0.6371 | 434 | 0.6412 | 434 | 0.5842 | 434 | 0.6311 | 434 | 0.6300 |

1.7 | 434 | 0.6589 | 434 | 0.6589 | 434 | 0.6155 | 434 | 0.6504 | 434 | 0.6536 |

1.8 | 434 | 0.6726 | 434 | 0.6665 | 434 | 0.6113 | 434 | 0.6538 | 434 | 0.6691 |

1.9 | 434 | 0.6820 | 434 | 0.6699 | 434 | 0.5969 | 434 | 0.6516 | 434 | 0.6800 |

2.0 | 433 | 0.6801 | 433 | 0.6627 | 2311 | 0.5912 | 433 | 0.6496 | 433 | 0.6796 |

**Table 6.**The absolute value of the maximum correlation coefficient between the hyperspectral and Na

^{+}in Area C and its corresponding band.

Order | R | $\sqrt{\mathrm{R}}$ | 1/R | lgR | 1/lgR | |||||
---|---|---|---|---|---|---|---|---|---|---|

Band/nm | Max Value | Band/nm | Max Value | Band/nm | Max Value | Band/nm | Max Value | Band/nm | Max Value | |

0.0 | 1969 | 0.3514 | 1969 | 0.3588 | 1969 | 0.3694 | 1969 | 0.3643 | 522 | 0.3300 |

0.1 | 1969 | 0.3531 | 1969 | 0.3607 | 1969 | 0.3719 | 1969 | 0.3664 | 514 | 0.3311 |

0.2 | 2392 | 0.3556 | 1969 | 0.3630 | 1969 | 0.3747 | 1969 | 0.3689 | 2386 | 0.3365 |

0.3 | 2386 | 0.3966 | 2360 | 0.4012 | 2392 | 0.4114 | 2360 | 0.4059 | 2386 | 0.3832 |

0.4 | 2360 | 0.4552 | 2360 | 0.4593 | 2360 | 0.4621 | 2360 | 0.4619 | 2360 | 0.4449 |

0.5 | 2361 | 0.5094 | 2361 | 0.5130 | 570 | 0.5208 | 2361 | 0.5151 | 2361 | 0.4873 |

0.6 | 2361 | 0.5175 | 2361 | 0.5263 | 592 | 0.5493 | 2361 | 0.5322 | 2230 | 0.3660 |

0.7 | 2121 | 0.4772 | 2121 | 0.4866 | 592 | 0.5322 | 2121 | 0.4922 | 2121 | 0.3904 |

0.8 | 2121 | 0.5058 | 2121 | 0.5073 | 632 | 0.5412 | 2137 | 0.5090 | 2121 | 0.4245 |

0.9 | 886 | 0.5555 | 886 | 0.5802 | 886 | 0.6325 | 886 | 0.6017 | 886 | 0.4832 |

1.0 | 886 | 0.5521 | 886 | 0.5971 | 886 | 0.6022 | 556 | 0.6364 | 886 | 0.5098 |

1.1 | 886 | 0.5494 | 886 | 0.5544 | 1632 | 0.5686 | 556 | 0.6377 | 886 | 0.5189 |

1.2 | 886 | 0.5338 | 556 | 0.5907 | 1470 | 0.5682 | 555 | 0.6459 | 480 | 0.5068 |

1.3 | 886 | 0.5359 | 556 | 0.6204 | 1470 | 0.5925 | 555 | 0.6594 | 886 | 0.5094 |

1.4 | 886 | 0.5406 | 555 | 0.6175 | 1470 | 0.6043 | 555 | 0.6680 | 886 | 0.5146 |

1.5 | 886 | 0.5392 | 555 | 0.6146 | 555 | 0.6140 | 555 | 0.6577 | 886 | 0.5141 |

1.6 | 1694 | 0.5466 | 551 | 0.5844 | 1694 | 0.6241 | 555 | 0.6172 | 886 | 0.5045 |

1.7 | 1694 | 0.5612 | 1694 | 0.5837 | 1694 | 0.6265 | 1694 | 0.6017 | 886 | 0.4849 |

1.8 | 1694 | 0.5644 | 1694 | 0.5826 | 1694 | 0.6203 | 1694 | 0.5978 | 886 | 0.4558 |

1.9 | 1694 | 0.5542 | 1694 | 0.5698 | 1694 | 0.6061 | 1694 | 0.5836 | 1694 | 0.4748 |

2.0 | 1693 | 0.5300 | 1633 | 0.5516 | 1633 | 0.5927 | 1633 | 0.5698 | 1693 | 0.4761 |

**Table 7.**The absolute value of the fractional-order maximum correlation coefficient is greater than the absolute value of the integer-order maximum correlation coefficient (correlation coefficient improvement percentage).

Spectrum | The Fractional-Order where the Largest Correlation Coefficient Is Located | Greater Than Integer 1.0-Order | Greater Than Integer 2.0-Order | ||||
---|---|---|---|---|---|---|---|

Area A | Area B | Area C | Area A | Area B | Area C | ||

R | 1.4 order of Area A, 1.9 order of Area B,1.8 order of Area C | 0.63% | 7.01% | 2.23% | 0.61% | 0.28% | 6.49% |

$\sqrt{\mathrm{R}}$ | 1.5 order of Area A, 1.9 order of Area B, 1.3 order of Area C | 0.33% | 7.10% | 3.90% | 1.11% | 1.09% | 12.47% |

1/R | 1.5 order of Area A, 1.7 order of Area B, 1.7 order of Area C | 0.38% | 13.35% | 4.04% | 5.30% | 4.11% | 5.70% |

lgR | 1.5 order of Area A, 1.8 order of Area B, 1.4 order of Area C | 0.23% | 8.33% | 4.97% | 0.96% | 0.65% | 17.23% |

1/lgR | 1.9 order of Area A, 1.9 order of Area B, 1.1 order of Area C | 2.85% | 14.79% | 1.79% | 1.02% | 0.06% | 8.99% |

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## Share and Cite

**MDPI and ACS Style**

Tian, A.; Zhao, J.; Tang, B.; Zhu, D.; Fu, C.; Xiong, H.
Study on the Pretreatment of Soil Hyperspectral and Na^{+} Ion Data under Different Degrees of Human Activity Stress by Fractional-Order Derivatives. *Remote Sens.* **2021**, *13*, 3974.
https://doi.org/10.3390/rs13193974

**AMA Style**

Tian A, Zhao J, Tang B, Zhu D, Fu C, Xiong H.
Study on the Pretreatment of Soil Hyperspectral and Na^{+} Ion Data under Different Degrees of Human Activity Stress by Fractional-Order Derivatives. *Remote Sensing*. 2021; 13(19):3974.
https://doi.org/10.3390/rs13193974

**Chicago/Turabian Style**

Tian, Anhong, Junsan Zhao, Bohui Tang, Daming Zhu, Chengbiao Fu, and Heigang Xiong.
2021. "Study on the Pretreatment of Soil Hyperspectral and Na^{+} Ion Data under Different Degrees of Human Activity Stress by Fractional-Order Derivatives" *Remote Sensing* 13, no. 19: 3974.
https://doi.org/10.3390/rs13193974