Relationships Between Errors Propagated in Fraction of Vegetation Cover by Algorithms Based on a Two-Endmember Linear Mixture Model

Abstract

Remotely sensed reflectance spectra may be biased by several intervening factors, and the biases are propagated into estimations of the fraction of vegetation cover (FVC) by algorithms based on a linear mixture model (LMM). The errors propagated in FVCs depend on the retrieval algorithm used, due to differences in the assumptions of the model as well as constraints employed in the algorithm. These differences should be fully understood prior to algorithm selection for practical applications. Although numerous studies have investigated the relationships between errors propagated by different algorithms, these relationships have not been fully understood from a deterministic perspective. This study introduces a technique for deriving the analytical underpinnings of error propagation in FVC based on several LMM-based algorithms. The derivation assumes that measurement noise is band-correlated additive noise. The bias errors propagated in FVC depended on the endmember spectra assumed in the algorithm, the target spectrum, and the coefficients of the spectral vegetation index, which were employed as constraints, as well as magnitude of the input error. It was found that the relationships among the propagated errors assume asymmetric elliptical forms with coefficients that are determined by the input variables. These results suggest that the relationships depend heavily on the choice of endmember spectra as well as the spectrum of the target pixel and the vegetation index employed as a constraint. The present findings should assist in the selection of an optimum algorithm based on prior knowledge of the target field.

Comparison of the Noise Robustness of FVC Retrieval Algorithms Based on Linear Mixture Models

“…The choice of endmember variable (endmember model) and constraints on the algorithms are summarized in Table 1. Below, we briefly introduce the three algorithms for the two-band case, as assumed in previous studies [28,29]. Table 1.…”

“…The errors propagated in the FVC by each of the three algorithms were derived previously [29]. The errors in algorithm-1 are represented in terms of the band-correlated noise and the endmember spectra (independent of the target spectrum) as…”

“…Biased errors in the spectral measurements are modeled as band-correlated noise in the red-NIR reflectance space [29]. The magnitude of the noise spectrum is represented by σ t , and the direction of the band correlation in the error is represented by the directional vector (unit vector) in the red-NIR reflectance space, e e e = (cos θ, sin θ).…”

“…Here, the relationship between FVC values derived from different algorithms is expressed analytically. Errors in the calculated FVC, which were propagated from errors in the reflectance spectra, are investigated [29], and the relationships among errors propagated in different algorithms are described. The results show that the relationships are described by an asymmetric ellipse with coefficients that depend on the target spectrum, assumed endmember spectra, and vegetation indices (VIs) used as constraints.…”

“…This study is performed as a series of investigations [28,29] about the three LMM-based FVC algorithms, which are widely used in practical applications. In these studies, a two-endmember LMM assuming two multispectral bands was used throughout to facilitate analytical derivations.…”