scholarly journals Study on the Pretreatment of Soil Hyperspectral and Na+ Ion Data under Different Degrees of Human Activity Stress by Fractional-Order Derivatives

2021 ◽  
Vol 13 (19) ◽  
pp. 3974
Author(s):  
Anhong Tian ◽  
Junsan Zhao ◽  
Bohui Tang ◽  
Daming Zhu ◽  
Chengbiao Fu ◽  
...  
Keyword(s):  
Correlation Coefficient ◽  
Fractional Order ◽  
Human Activity ◽  
Saline Soil ◽  
Water Soluble ◽  
Order Derivative ◽  
Continuous Increase ◽  
Integer Order ◽  
Disturbed Area ◽  
Soluble Base

Soluble salts in saline soil often exist in the form of salt base ions, and excessive water-soluble base ions can harm plant growth. As one of the water-soluble base ions, Na+ ion, is the main indicator of the degree of soil salinization. The pretreatment of visible, near infrared and short-wave infrared (VNIR-SWIR) spectroscopy data is the key to establishing a high-precision inversion model, and a proper pretreatment method can fully extract the effective information hidden in the hyperspectral data. Meanwhile, different degrees of human activity stress will have an impact on the ecological environment of oases. However, there are few comparative analyses of the data pretreatment effects for soil water-soluble base ions on the environment under different human interference conditions. Therefore, in this study, the difference in the degree of soil disturbance caused by human activities was used as the basis for dividing the experimental area into lightly disturbed area (Area A), moderately disturbed area (Area B) and severely disturbed zone (Area C). The Grünwald-Letnikov fractional-order derivative (FOD) was used to preprocess the VNIR-SWIR spectroscopic data measured by a FieldSpec®3Hi-Res spectrometer, which could fully extract the useful information hidden in the FOD of the VNIR-SWIR spectroscopy results and avoid the loss of information caused by the traditional integer-order derivative (1.0-order, 2.0-order) pretreatment. The spectrum pretreatment was composed of five transform spectra (R,R, 1/R, lgR, 1/lgR) and 21 FOD methods (step size is 0.1, derivative range is from 0.0- to 2.0-order). In addition, this manuscript compares and analyzes the pretreatment advantages between fractional-order and integer-order. The main results were as follows: (1) Grünwald-Letnikov FOD can reveal the nonlinear characteristics and variation laws of the field hyperspectral of saline soil, namely, due to the continuous performance of the order selection, the FOD accurately depicts the details of spectral changes during the derivation process, and improves the resolution between the peaks of the hyperspectral spectrum. (2) There is a big difference in the shape of the correlation coefficient curve between the original hyperspectral and Na+ at different FOD. The correlation coefficient curve has a clear outline in rang of the 0.0- to 0.6-order, and the change trend is gentle, which presents a certain gradual form. With the continuous increase of the order of the FOD, the change range of the correlation coefficient curve is gradually increased, and the fluctuation is greater between the 1.0-order and the 2.0-order. (3) Regardless of the transformation spectrum and different interference regions, the improvement effect of the FOD on the correlation between hyperspectral and Na+ is significantly better than that of the integer-order derivative. Comparative analysis shows that he percentage of increase of the former is more than 3%, and the highest is more than 17%.

Dynamic response of Euler–Bernoulli beam resting on fractionally damped viscoelastic foundation subjected to a moving point load

2020 ◽  
Vol 234 (24) ◽  
pp. 4801-4812 ◽  
Author(s):  
Rajendra K Praharaj ◽  
Nabanita Datta
Keyword(s):  
Dynamic Response ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Point Load ◽  
Order Derivative ◽  
Viscoelastic Foundation ◽  
Bernoulli Beam ◽  
Integer Order ◽  
Foundation Model ◽  
Euler Bernoulli Beam

The dynamic behaviour of an Euler–Bernoulli beam resting on the fractionally damped viscoelastic foundation subjected to a moving point load is investigated. The fractional-order derivative-based Kelvin–Voigt model describes the rheological properties of the viscoelastic foundation. The Riemann–Liouville fractional derivative model is applied for a fractional derivative order. The modal superposition method and Triangular strip matrix approach are applied to solve the fractional differential equation of motion. The dependence of the modal convergence on the system parameters is studied. The influences of (a) the fractional order of derivative, (b) the speed of the moving point load and (c) the foundation parameters on the dynamic response of the system are studied and conclusions are drawn. The damping of the beam-foundation system increases with increasing the order of derivative, leading to a decrease in the dynamic amplification factor. The results are compared with those using the classical integer-order derivative-based foundation model. The classical foundation model over-predicts the damping and under-predicts the dynamic deflections and stresses. The results of the classical (integer-order) foundation model are verified with literature.

Geometric interpretation of fractional-order derivative

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Geometry ◽  
Fractional Order ◽  
Infinite Series ◽  
Fractional Derivatives ◽  
Geometric Interpretation ◽  
Order Derivative ◽  
Jet Bundles ◽  
Fractional Order Derivative ◽  
Integer Order ◽  
Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

scholarly journals Impact of Fractional Calculus on Correlation Coefficient between Available Potassium and Spectrum Data in Ground Hyperspectral and Landsat 8 Image

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 488 ◽  
Author(s):  
Chengbiao Fu ◽  
Shu Gan ◽  
Xiping Yuan ◽  
Heigang Xiong ◽  
Anhong Tian
Keyword(s):  
Correlation Coefficient ◽  
Fractional Order ◽  
Saline Soil ◽  
Hyperspectral Data ◽  
Potassium Content ◽  
Landsat 8 ◽  
Data Pretreatment ◽  
Spectral Transformations ◽  
Fractional Differential ◽  
New Perspective

As the level of potassium can interfere with the normal circulation process of biosphere materials, the available potassium is an important index to measure the ability of soil to supply potassium to crops. There are rarely studies on the inversion of available potassium content using ground hyperspectral remote sensing and Landsat 8 multispectral satellite data. Pretreatment of saline soil field hyperspectral data based on fractional differential has rarely been reported, and the corresponding relationship between spectrum and available potassium content has not yet been reported. Because traditional integer-order differential preprocessing methods ignore important spectral information at fractional-order, it is easy to reduce the accuracy of inversion model. This paper explores spectral preprocessing effect based on Grünwald–Letnikov fractional differential (order interval is 0.2) between zero-order and second-order. Field spectra of saline soil were collected in Fukang City of Xinjiang. The maximum absolute of correlation coefficient between ground hyperspectral reflectance and available potassium content for five mathematical transformations appears in the fractional-order. We also studied the tendency of correlation coefficient under different fractional-order based on seven bands corresponding to the Landsat 8 image. We found that fractional derivative can significantly improve the correlation, and the maximum absolute of correlation coefficient under five spectral transformations is in Band 2, which is 0.715766 for the band at 467 nm. This study deeply mined the potential information of spectra and made up for the gap of fractional differential for field hyperspectral data, providing a new perspective for field hyperspectral technology to monitor the content of soil available potassium.

Edge enhancement of magnetic data using fractional-order-derivative filters

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. J7-J17 ◽  
Author(s):  
Muzaffer Özgü Arısoy ◽  
Ünal Dikmen
Keyword(s):  
Fractional Order ◽  
Order Derivative ◽  
Magnetic Data ◽  
Frequency Noise ◽  
Edge Enhancement ◽  
Traditional Use ◽  
Data Set ◽  
Fractional Order Derivative ◽  
Detection Techniques ◽  
Integer Order

Edge enhancement and detection techniques are fundamental operations in magnetic data interpretation. Many techniques for edge enhancement have been developed, some based on profile data and others designed for grid-based data sets. Methods that are traditionally applied to magnetic data, such as total horizontal derivative (THD) and analytic signal (AS), require the computation of integer-order horizontal and vertical derivatives of the magnetic data. However, if the data set contains features with a large variation in amplitude, then the features with small amplitudes may be difficult to outline. In addition, because most edge enhancement and detection filters are derivative-based filters, they also amplify high-frequency noise content in the data. As a result, the accuracy of derivative-based filters is restricted to data of high quality. We suggested the modification of the THD and AS filters by combining the amplitude spectra of fractional-order-derivative filters with ad hoc phase spectra, particularly designed for edge detection in magnetic data. We revealed the capability of the proposed algorithm on synthetic magnetic data and on aeromagnetic data from Turkey. Compared with the traditional use of THD and AS (with integer-order derivatives), we developed the method based on fractional-order derivatives that produced more effective results in terms of suppressing noise and delineating the edges of deep sources.

scholarly journals Numerical Solution of the Bagley-Torvik Equation Using the Integer-Order Derivatives Expansion

2018 ◽  
Vol 6 (2) ◽  
Author(s):  
Afrah Sadiq Hasan
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Numerical Solution ◽  
Fractional Order ◽  
Infinite Series ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Integer Order

Numerical solution of the well-known Bagley-Torvik equation is considered. The fractional-order derivative in the equation is converted, approximately, to ordinary-order derivatives up to second order. Approximated Bagley-Torvik equation is obtained using finite number of terms from the infinite series of integer-order derivatives expansion for the Riemann–Liouville fractional derivative. The Bagley-Torvik equation is a second-order differential equation with constant coefficients. The derived equation, by considering only the first three terms from the infinite series to become a second-order ordinary differential equation with variable coefficients, is numerically solved after it is transformed into a system of first-order ordinary differential equations. The approximation of fractional-order derivative and the order of the truncated error are illustrated through some examples. Comparison between our result and exact analytical solution are made by considering an example with known analytical solution to show the preciseness of our proposed approach.

scholarly journals Sinogram Restoration for Low-Dosed X-Ray Computed Tomography Using Fractional-Order Perona-Malik Diffusion

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Shaoxiang Hu ◽  
Zhiwu Liao ◽  
Wufan Chen
Keyword(s):  
Fractional Order ◽  
Anisotropic Diffusion ◽  
Order Derivative ◽  
X Ray ◽  
Integer Order ◽  
Staircase Effect ◽  
X Ray Computed ◽  
Nonlinear Anisotropic Diffusion ◽  
Sinogram Restoration ◽  
Mean Square Errors

Existing integer-order Nonlinear Anisotropic Diffusion (NAD) used in noise suppressing will produce undesirable staircase effect or speckle effect. In this paper, we propose a new scheme, named Fractal-order Perona-Malik Diffusion (FPMD), which replaces the integer-order derivative of the Perona-Malik (PM) Diffusion with the fractional-order derivative using G-L fractional derivative. FPMD, which is a interpolation between integer-order Nonlinear Anisotropic Diffusion (NAD) and fourth-order partial differential equations, provides a more flexible way to balance the noise reducing and anatomical details preserving. Smoothing results for phantoms and real sinograms show that FPMD with suitable parameters can suppress the staircase effects and speckle effects efficiently. In addition, FPMD also has a good performance in visual quality and root mean square errors (RMSE).

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