Information on the relationships between pairs of wavelength bands is useful when analyzing multispectral sensor data. The soil isoline is one such relationship that is obtained under a constant soil spectrum. However, numerical determination of the soil isoline in the red and NIR reflectance subspace is problematic because of singularities encountered during polynomial fitting. In our previous work, this difficulty was effectively overcome by rotating the original red-NIR subspace by an angle identical to a soil line slope. In the context of hyperspectral data analysis, the applicability of this approach should be investigated thoroughly for band combinations other than red-NIR. The objective of the present study was to expand the applicable range of band combinations to 400–2500 nm by conducting a set of numerical simulations of radiative transfer. Soil isolines were determined numerically by varying soil reflectance and biophysical parameters. The results demonstrated that, as shown previously for the red-NIR band combination, singularities can be avoided for most band combinations through use of the rotation approach. However, for some combinations, especially those involving the shortwave infrared range, the rotation approach gave rise to a further numerical singularity. The present findings thus indicate that special caution should be exercised in the numerical determination of soil isoline equations when one of the chosen bands is in the shortwave infrared region.
The remotely sensed reflectance spectra of vegetated surfaces contain information relating to the leaf area index (LAI) and the chlorophyll-a and -b concentrations (Cab) in a leaf. Difficulties associated with the retrieval of these two biophysical parameters from a single reflectance spectrum arise mainly from the choice of a suitable set of observation wavelengths and the development of a retrieval algorithm. Efforts have been applied toward the development of new algorithms, such as the numerical inversion of radiative transfer models, in addition to the development of simple approaches based on the spectral vegetation indices. This study explored a different approach: An equation describing band-to-band relationships (vegetation isoline equation) was used to retrieve the LAI and Cab simultaneously from a reflectance spectrum. The algorithm used three bands, including the red edge region, and an optimization cost function was constructed from two vegetation isoline equations in the red-NIR and red edge-NIR reflectance subspaces. A series of numerical experiments was conducted using the PROSPECT model to explore the numerical challenges associated with the use of the vegetation isoline equation during the parameter retrieval of the LAI and Cab. Overall, our results indicated the existence of a global minimum (and no local minima) over a wide swath of the LAI-Cab parameter subspace in most simulation cases. These results suggested that the use of the vegetation isoline equation in the simultaneous retrieval of the LAI and the Cab provides a viable alternative to the spectral vegetation index algorithms and the direct inversion of the canopy radiative transfer models.
This study introduces derivations of the soil isoline equation for the case of partial canopy coverage. The derivation relied on extending the previously derived soil isoline equations, which assumed full canopy coverage. This extension was achieved by employing a two-band linear mixture model, in which the fraction of vegetation cover (FVC) was considered explicitly as a biophysical parameter. A parametric form of the soil isoline equation, which accounted for the influence of the FVC, was thereby derived. The differences between the soil isolines of the fully covered and partially covered cases were explored analytically. This study derived the approximated isoline equations for nine cases defined by the choice of the truncation order in the parametric form. A set of numerical experiments was conducted using coupled leaf and canopy radiative transfer models. The numerical results revealed that the accuracy of the soil isoline increased with the truncation order, and they confirmed the validity of the derived expressions.
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.
Differences in spectral response function among sensors have known to be a source of bias error in derived data products such as spectral vegetation indices (VIs). Numerous studies have been conducted to identify such bias errors by comparing VI data acquired simultaneously by two different sensors. Those attempts clearly indicted two facts: 1) When one tries to model a relationship of two VIs from different sensors by a polynomial function, the coefficients of polynomial depends heavily on region to be studied: 2) Although increase of the degree of polynomial improves the translation accuracies, this improvement is very limited. Those facts imply that a better functional form than a simple polynomial may exist to model the VI relationships, and also that the coefficients of such a relationship can be written as a function of variables other than vegetation biophysical parameters. This study tries to address those issues by deriving an inter-sensor VI relationship analytically. The derivation has been performed based on a relationship of two reflectances at different wavelengths (bands), called soil isoline equation. The derived VI relationship becomes a form of rational function with the coefficients that depend purely on the soil reflectance spectra. The derived relationship has been demonstrated numerically by a radiative transfer model of canopy, PROSAIL. It is concluded that a rational function is a good candidate to model inter-sensor VI relationship. This study also shows the mechanism of how the coefficients of such a relationship could vary with the soil reflectance underneath the canopy.
Retrieval of biophysical parameters from remotely sensed reflectance spectra often involves algebraic manipulations,
e.g. spectral vegetation index, to enhance pure signals from a target of one‘s interest. An underlying
assumption of those processes is an existence of high correlation between an obtained value from the manipulations
and amount of the target object. These correlations can be seen in scatter plots of reflectance spectra as
isolines that represent a relationship between two reflectances of different wavelengths (bands) under constant
values of physical parameters. Therefore, modeling the isolines would contribute to better understanding of
retrieval algorithms and eventually to improve their accuracies. The objective of this study is to derive one such
relationship observed under a constant spectrum of soil surfaces, known as soil isolines, in red-NIR reflectance
space. This work introduces a parametric representation of the soil isolines (soil isoline equation) with the parameter
obtained by rotating the red-NIR reflectance space by approximately a quarter of pi radian counter
clockwise. The accuracy in the soil isoline equation depends on the order of polynomials used for the representations:
It was investigated numerically by conducting experiments with radiative transfer models for vegetation
canopy. The results showed that when the first-order approximation were employed for both bands, the accuracy
of the parametric representations/approximations of the soil isolines is approximately 0.02 in terms of mean
absolute difference from the simulated spectra (with no approximation). The accuracies improved dramatically
when one retains the polynomial terms up to the second-order or higher for both bands.
KEYWORDS: Vegetation, Reflectivity, Error analysis, Sensors, Detection and tracking algorithms, Climate change, Climatology, Near infrared, Information science, Information technology
Fraction of vegetation cover (FVC) has been used for environmental studies of both regional and global scale,
and data products of similar kinds have been generated from several agencies. Although there are differences
in sensors/datasets used and algorithms employed among those products, many of those use spectral mixture
analysis either directly or indirectly, and/or assume an essence of spectral mixture in their models. In the
FVC estimations, noises in reflectance spectra of both target and endmember are propagated into the estimated
FVC. Those propagation mechanisms such as patterns and degree of influences need to be clarified analytically,
where this study tries to contribute. The objective of this study is to investigate characteristics of the noise
propagation into the estimated FVC based on one of the linear mixture models known as VI-isoline based LMM.
In order to facilitate analytical discussions, the number of endmember spectra is limited into two. In addition,
a band-correlated noise is assumed in both reflectance spectrum of a target pixel and endmember spectra of
vegetation and non-vegetation surfaces. The propagated error in FVC from those spectra is analytically derived.
The derived expressions indicated that the characteristic behavior of the propagated errors exists such that there
are certain conditions among the band correlated noises which result in the cancellations of propagated errors
on FVC value (it looks as if the spectra are noise-free). Findings of this study would reveal unknown behavior
of the propagated noise, and would contribute better understanding of FVC retrieval algorithms of this kind.
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