*4.1. Di*ff*erential Calculation of Root Mean Square and Logarithm Reciprocal*

To study the effects on spectral data by fractional differentials in detail, starting differential order is 0, termination differential order is 2, and order interval is 0.2. The results of differential calculation in the bands 1450 nm and 1650 nm of soil ground hyperspectral curve for root mean square transformation and logarithm reciprocal transformation are shown in Figure 3. Differential values of two spectral transformations gradually approach 0, as the order slowly ascends from 0-order to 1-order, fractional differential curve gradually approximates the first-order differential curve. When the order is gradually increased from 1-order to 2-order, fractional derivative curve slowly approaches the 2-order differential curve, which verifies the sensitivity of fractional derivative to some extent. In addition, it can be also seen in Figure 3c,d that the derivative value in the band 1450–1550 nm fluctuates greatly, while the derivative value in the band 1550–1650 nm is less fluctuating.

**Figure 3.** Fractional differential calculation of <sup>√</sup> R and 1/lgR reflectance at 1450–1650 nm: (**a**) 0-order to 1-order of root mean square; (**b**) 0-order to 1-order of logarithm reciprocal; (**c**) 1-order to 2-order of root mean square; and (**d**) 1-order to 2-order of logarithm reciprocal.
