Kenta Taniguchi

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.

Investigation of inter-sensor NDVI relationships based on analytical representation of soil isolines

This study introduces an analytical approach to derive a relationship between values of Normalized Difference Vegetation Index (NDVI) obtained from datasets of two sensors with different band passes in the context of inter-sensor cross-calibration of such vegetation index (VI) products. Derivation of the relationship has been performed based on a concept of soil isolines. Starting from the soil isoline equations, an inter-sensor NDVI relationship was derived and represented by a system of equation with a single common parameter. In the derived form of NDVI relationship, all the coefficients were written by the soil reflectances (independent of canopy layer.) The functional form of the relationship becomes rational function of the first-order polynomials, when the soil isoline equations are approximated by the form of first-order polynomials. Those results indicate a functional form suitable to model a relationship of NDVI from two sensors.

Validity of soil isoline equation for a system of canopy and soil layers

For retrievals of soil optical properties, relationships between the two reflectances of different wavelengths become a key information. This relationship is known as soil isolines from the fact that the reflectances of two different wavelengths form a contour line (isoline) in a reflectance subspace when the soil properties are set to be constant. In our previous work, a relationship between two reflectances of different wavelengths has been derived under a constant soil optical properties. Although accuracy of the derived relationship has been demonstrated numerically, its validity had not been fully investigated from fundamental and physical points of view. The objective of this study is to provide theoretical validity of the derived soil isoline equation and numerically demonstrate its validity by a radiative transfer model of a canopy-soil system of layers. Specifically, the polynomial fittings employed during the derivation will be validated by conducting an analysis based on a theory of radiative transfer. As a result, a similar form of the previously derived soil isoline equation has been derived successfully based on the analytical form of TOC reflectance.

Application of vegetation isoline equations for simultaneous retrieval of leaf area index and leaf chlorophyll content using reflectance of red edge band

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.

Evaluation of BIAS reduction in cross-calibration of NDVI based on soil isoline equations: Comparison with error estimated from signal-to-noise ratio

Intercalibration among remotely-sensed data products such as spectral vegetation indices (VIs) from sensors onboard different satellites has been investigated for integrational use of multiple datasets. To facilitate intercalibration technique of VIs, the relationships among the VI values from different sensors need to be fully understood. This study evaluated the accuracies of NDVI translation technique based on soil iso-line equations. The evaluation was performed by comparing the errors in intersensor relationships of NDVI simulated by radiative transfer model and propagted error in NDVI from the sensor specific signal-to-noise ratio (SNR) of each band. The numerical results indicated that the translation equation showed enough accuracy relative to the propagated error originated from SNR, if the soil brightness underneath the canopy layer can be estimated prior to the translation.

Optimization technique of asymmetric-order vegetation isoline equations

This study investigated an optimization technique for improving the inter-band reflectance relationship known as the vegetation isoline equations. A single factor was introduced into the recently developed vegetation isolines [Miura et al. 2015] to decompose the higher-order interaction terms of photons between the canopy layers and soil surfaces. A set of numerical experiments based on the canopy radiative transfer (RT) model revealed that the error in the optimized isoline equation could be reduced by approximately 80 percent relative to the error calculated for the asymmetric-order isoline without optimization. This accuracy improvement generally held for combinations of red and near-infrared wavelengths. These results indicated that the optimized form of the vegetation isoline could be used as an emulator for the RT model in the context of inverse problems for parameter retrievals.

Analysis of the scaling effect present in the relative differences between NDVIs obtained from multiple sensors, based on the soil isoline equation

Differences among the normalized difference vegetation indices (NDVIs) simulated from multiple sensors were analyzed using soil isoline equations (SIEs). The upscaled NDVI image was obtained by generating images of red and near-infrared (NIR) bands using an upscaling factor through (1) convolution of a spectral response function (SRF) and hyperspectral data from the EO-1 HYPERION, and (2) adaptation of an area-averaging window. Per-pixel classification was conducted as a solution to the inverse problem of the SIEs, which emulated the spectral behaviors in the red and NIR reflectance subspaces. We compared the NDVIs obtained using three combinations of SRFs that described the existing sensors and identified different trends in the NDVI biases within each class for all sensor combinations, regardless of the scaling factor. These results suggested the potential utility of the SIE-based map for separating clusters of NDVI differences using to the sensor characteristics of the SRF and the spatial resolution.

Improved Accuracy of the Asymmetric Second-Order Vegetation Isoline Equation over the RED–NIR Reflectance Space

The relationship between two reflectances of different bands is often encountered in cross calibration and parameter retrievals from remotely-sensed data. The asymmetric-order vegetation isoline is one such relationship, derived previously, where truncation error was reduced from the first-order approximated isoline by including a second-order term. This study introduces a technique for optimizing the magnitude of the second-order term and further improving the isoline equation’s accuracy while maintaining the simplicity of the derived formulation. A single constant factor was introduced into the formulation to adjust the second-order term. This factor was optimized by simulating canopy radiative transfer. Numerical experiments revealed that the errors in the optimized asymmetric isoline were reduced in magnitude to nearly 1/25 of the errors obtained from the first-order vegetation isoline equation, and to nearly one-fifth of the error obtained from the non-optimized asymmetric isoline equation. The errors in the optimized asymmetric isoline were compared with the magnitudes of the signal-to-noise ratio (SNR) estimates reported for four specific sensors aboard four Earth observation satellites. These results indicated that the error in the asymmetric isoline could be reduced to the level of the SNR by adjusting a single factor.

Soil isoline equation for the range of visible to shortwave infrared in a context of hyperspectral data analysis

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.

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

Abstract. 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.