Derivation and approximation of soil isoline equations in the red–near-infrared reflectance subspace

Abstract

This study describes the derivation of an expression for the relationship between red and near-infrared reflectances, called soil isolines, as an orthogonal concept for the vegetation isoline. An analytical representation of soil isoline would be useful for estimating soil optical properties. Soil isolines often contain a singular point on a dark soil background. Singularities are difficult to model using simple polynomial forms. This difficulty was circumvented in this work by rotating the original axis and employing a vegetation index-like parasite parameter. This approach produced a soil isoline model that could yield any desired level of accuracy based on the use of an index-like parameter. A technique is further introduced for approximating the removal of the parasite parameter from the relationship by truncating the higher-order terms during the derivation steps. Numerical experiments by PROSAIL were conducted to investigate the influence of the truncation errors on the accuracy of the approximated soil isoline equation. The numerical results showed that truncating terms of order greater than two in both bands, yielded negligible truncation errors. These results suggest that the derived and approximated soil isoline equations may be useful in other applications, such as the analysis and retrieval of soil optical properties. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Soil isoline equations in the red–NIR reflectance subspace describe a heterogeneous canopy

“…The nominal parameter sets used to describe the leaf chemical content reported in Ref. 31 were applied in the current experiments. These simulation steps are summarized in Fig.…”

“…The drawback, however, of choosing larger values for m r and m n is the increased difficulty associated with solving these equations for ρ 0 n to derive analytical formulations of the soil isoline. In a previous study, 31 we introduced a series of derivations to approximate the soil isolines by assuming full canopy coverage. In these derivations, the value of the FVC parameter ω did not appear explicitly in the study; nevertheless, it was considered as a special case in which the FVC was fixed to unity.…”

“…The details of this problem have been provided in previous studies. 30,31 In this study, the affine transformation of a vector ρ into ρ 0 is represented by the function φ, defined as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 6 ; 5 5 0…”

Analytical Relationship between Two-Band Spectral Vegetation Indices Measured at Multiple Sensors on a Parametric Representation of Soil Isoline Equations

“…One possibility is to use the analytical relationship between two reflectances of different wavelengths, known as the isoline equations [43][44][45][46]. The "soil isoline equatio"' introduced in a previous study [47] is one such choice [27,29].…”

“…The conditions assumed during the simulation comprise a fixed soil spectrum that creates a set of spectra considered to be an isoline. Soil isoline equations comprise a system of equations that describe the relationship between the red and NIR reflectances, along with a soil isoline [47,59]. A formal derivation of the soil isoline has been described in previous studies [47].…”