CALENDARIO DE APORTACION DEL GOBIERNO DEL ESTADO DE GUERRERO

It has been shown that analyzing the derivatives of spectral reflectance measurements can provide valuable insight into underlying physical processes. For example, Kokaly et al have used spectral derivative analysis of airborne imagery in order to accurately map vegetation in Yellowstone to 74% accuracy. [53] In addition to being useful in vegetation analysis, spectral analysis can be invaluable in studying the directional reflectance of soils and sands.

For our purposes, we have used finite approximation in order to obtain metrics of numerical derivatives. In our laboratory and field studies, central difference formulas are employed and approximations of the first- and second-order derivatives are made to within a truncation error of O(∆λ2), where ∆λ is defined as the spectral distance in terms of wavelength units between the bands of interest to the study. The first order derivative of the reflectance spectra of interest, S(λ), can be numerically approximated by the following equation:

S0(λ) = S(λ + ∆λ) − S(λ − ∆λ)

2∆λ (2.42)

derivative is currently being calculated. The second order derivative at a given λ can be calculated according to the equation:

S00(λ) = S(λ + ∆λ) − 2S(λ) + S(λ − ∆λ)

2∆λ2 (2.43)

In order to facilitate visual comparison between different orders of derivatives as the value ∆λ increases, higher order derivatives can be ”enhanced” by replacing the denom- inator of higher order derivatives with a normalizing factor of 2∆λ. [9] By using this normalizing factor, derivatives of different orders can be compared on a common axis for easy comparison. Because this convention makes spectral detail of the derivatives more apparent to analysts, this convention is adopted in our laboratory studies when considering suitable values of ∆λ for soil absorption features of interest.

Noise artifacts in reflectance measurements can be detrimental to the use of numerical derivatives in spectral analysis. Noise can be removed from signals through the use of smoothing filters. However, one must use caution when applying filtering techniques in order to avoid suppressing spectral absorption features in the process of smoothing. For this study, the Savitsky-Golay filter was employed based on its ability to resolve relatively weak spectral absorption features. [54] The only drawback of using this technique is that noise is assumed constant across the spectrum, while in reality noise is a function of factors such as signal strength and frequency. [12] Detailed analysis of optimal parameters for the smoothing filters can additionally be used to improve the smoothing capabilities of the system. In order to ensure that the smoothing filters are being applied correctly, visual inspection and error metrics must be used to determine ability to remove noise without changing the magnitude of identified spectral features.

A critical step in the use of spectral derivatives is the selection of suitable values of ∆λ. [9] Features with widths that are smaller than the chosen value of ∆λ will be undetected by numerical derivatives, while features that are at the scale of ∆λ will be magnified by numerical derivatives. In this way, one can use the average width of a spectral absorption feature of interest to isolate the relevant features, while also smoothing over noise that is finer than the scale of the feature of interest. A downside of using larger values of ∆λ is that it will not be possible to calculate derivatives near the beginning or ending of the spectrum, due to the fact that central difference formulas are computed at the middle point of a wavelength range. [9] An example of this is shown in Figure 2.14, where two different values of ∆λ are applied to the same spectrum with different results in terms of the analyst’s ability to resolve spectral detail.

Figure 2.14: An example of the effect of using different values of ∆λ in the use of spec- tral derivatives. The top image shows the original reflectance spectrum, the middle the derivative using a small value of ∆λ, and the bottom the derivative using a large value of ∆λ. Credit: [9]

Methods

After GRIT-T was fully operational, there was a significant amount of testing and calibra- tion necessary to ensure that it achieved optimal elevation measurements, and directional reflectance measurements. I played a major role in the development of an on-board Li- DAR capability, the investigation into light-source fluctuations, and an investigation into polarization sensitivity of the instrument. In addition, I have developed numerous soft- ware tools that can be used for the calculation of roughness metrics. In this Section, I will describe these methods in great detail to provide an understanding of how they relate to our research group’s modeling goals.

3.1

Custom LiDAR System

GRIT-T is capable of rotating its azimuth chassis, its pointing head, and its pointing arm simultaneously. Additionally, it is equipped with a laser-ranging unit that can provide distance measurements. The potential combination of these features for use in developing elevation models propelled us to develop a LiDAR mode for the goniometer. This mode operates the goniometer with a nadir look angle to the surface of interest. I developed and characterized this capability for use in the field and laboratory. The system’s algorithm and achieved results will be discussed in the following sections.