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Current address: Department of Natural Resources and Environmental Management, University of Hawaii at Manoa, 1910 East West Road, Sherman 101, Honolulu, HI 96822, USA.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

The spectral unmixing of a linear mixture model (LMM) with Normalized Difference Vegetation Index (NDVI) constraints was performed to estimate the fraction of vegetation cover (FVC) over the earth’s surface in an effort to facilitate long-term surface vegetation monitoring using a set of environmental satellites. Although the integrated use of multiple sensors improves the spatial and temporal quality of the data sets, area-averaged FVC values obtained using an LMM-based algorithm suffer from systematic biases caused by differences in the spatial resolutions of the sensors, known as scaling effects. The objective of this study is to investigate the scaling effects in area-averaged FVC values using analytical approaches by focusing on the monotonic behavior of the scaling effects as a function of the spatial resolution. The analysis was conducted based on a resolution transformation model introduced recently by the authors in the accompanying paper (Obata

Time series analysis of biophysical and climatological parameters is indispensable for studies of the Earth’s climate, land, and hydrology systems and the interactions among these systems. Such analysis often relies on the information retrieved from reflectance spectra measured by satellite instruments [

Long-term observations of biophysical quantities spanning several decades generally require integrated use of an ensemble of satellite sensors [

Conceptual approaches and measurement models related to the scale issues have been introduced by Strahler

Scaling effects have been investigated extensively in the context of biophysical parameter retrievals, such as calculations of the vegetation index [

Spectral unmixing under NDVI constraints (the NDVI-isoline-based LMM) has been used to estimate FVC values. The technique has the advantages of both the VI and LMM approaches [

The structure of this work is as follows. After the brief explanation of theoretical background, we clarify the monotonicity of scaling effects in FVC (Section 3). We then explain a geometric relationship between the monotonic trend and endmember spectra by numerical experiments using simulated endmember spectra in Section 4, followed by a validation exercise in Section 5. Derivation of the maximum error bounds of the scaling effects and numerical simulation will be explained in Section 6. The discussion and conclusion sections (Sections 7 and 8, respectively) follow.

A reflectance spectrum from a target pixel may be represented as a linear sum of the spectra corresponding to a set of representative surface types (endmember spectra) in a LMM. We assume here that a target field consists of vegetation and non-vegetation classes. Red and near-infrared (NIR) bands are considered. Under these assumptions, a modeled spectrum, _{r}_{n}_{v}_{v,r}_{v,n}_{s}_{s,r}_{s,n}_{r}_{n}_{v}_{s}

Under the two-endmember assumption, if an endmember spectra assumed in a model rigorously corresponds to the true endmember spectra over a target field, the spatially averaged FVC will be independent of the spatial resolution (the averaged FVC values must be true values) [

The analytical form of the biased FVC under the two-endmember assumption (the target field is modeled under the same assumption) can be derived according to the following steps. First, we assume that a measured spectrum, _{v}_{v,r}_{v,n}_{s}_{s,r}_{s,n}

Scaling effects in the calculation of an area-averaged FVC can be seen in the data from Landsat7-ETM+. In order to clearly show the scaling effects, we conducted the following experiment using the data acquired on 7 July 2001 (Path:109, Row:36), which covers both urban and suburban regions in the Aichi prefecture in Japan. The spectral data of the size of 64 × 64 pixels (extracted from the original scene) was used in this experiment (^{i}^{i}

The averaged FVC values (

The FVC estimate, _{v}_{v}_{s}_{s}

The reflectance spectrum for a pixel _{j,k}_{j,k}_{v}_{s}_{j,k}_{1,1}) is expressed as a function of the endmember spectra corresponding to the vegetation and non-vegetation coverage over a target field (_{v}_{s}_{1,1}). The FVC estimate at the 1st resolution level (_{1,1} =
_{1,1} and the endmember spectra assumed in the model (_{v}_{s}_{2,k}) are functions of the FVC for the pixel (_{2,k}) and the endmember spectra (_{v}_{s}_{2,k}) can be obtained by unmixing the target spectra _{2,k} based on the endmember spectra assumed in the model, _{v}_{s}_{2,1} and _{2,2} with weights

The averaged FVC estimate at the

The variable
_{v}_{s}_{v}_{s}_{2,k}) or the partition ratio (

The sign and dependence of the inputs on
_{2,1}, using _{2,1}. The derivative terms on the right-hand side of _{2,k} (for _{2,k},
_{2,2} with respect to _{2,1} can be expressed as
_{1,1} can be expressed as
_{2,2} can be written as
_{2,1} can be obtained as

The behavior of
_{2}_{,k}_{v}_{s}_{v}_{s}

First, the factor _{2,1},_{2,2}),
_{2,1} = _{2,2}, the function _{2,1},_{2,2}),

For _{2,1} ≠ _{2,2} and _{2,1}_{2,2}) is not equal to zero, and the sign of the function depends on all variables. In this case, _{2,1},_{2,2}) as follows. The sign of _{2,1},_{2,2}) can be readily determined once the following information is known. The range of (_{2,1} + _{2,2})/2 (the last factor in _{2,1},_{2,2}), namely the sign of
_{2,1}.
_{2,1} because the sign of the last factor in _{2,1}). Finally, if

The sign and dependence of the input variables on
_{2,k}) or on the fractional areas by which one pixel is divided into two (_{v}_{s}_{v}_{s}

As derived in the previous subsection, the sign of the factor

When this is the case,

On the one hand, the average FVC values can change non-monotonically for

The variable

If

Next, we present a set of numerical examples that demonstrate the relationship between the monotonicity (_{v}_{s}_{v}_{s}_{v}_{s}_{v}_{s}

_{r}_{n}_{v}_{s}_{v}

In this section, we introduce a set of numerical experiments in an effort to validate the results derived here in a practical application.

The major difficulties associated with this validation lie in the difficulty of estimating

The satellite data presented in _{s}_{s}_{s}

_{s}

The magnitude of the scaling effects depend on the true FVC value within a target area under a fixed pair of assumed endmember spectra. In this section, we focus on the maximum scaling effects in the FVC as a function of the true FVC. The magnitude of the scaling effects is measured as the difference between the two extreme resolutions (the lumped and distributed cases). This difference can be considered to provide the bounds on the errors resulting from the scaling effects when the assumed endmember spectra meet the conditions for monotonicity, as described in Theorem 1 (

The focus of this discussion is on the differences between the FVC values at the coarsest and finest resolution. Note that the finest resolution corresponds to the case in which all pixels are composed of only one type of surface. In this case, the spectrum of each pixel may be represented by a single endmember spectrum (either vegetation or non-vegetation). Comparing the two extremes, the scaling effects in the FVC may be defined by
_{(v)} and _{(s)} represent the FVC estimates for the vegetation and non-vegetation endmember spectra, expressed by

In order to derive the maximum of
_{v}_{s}_{v}_{s}_{v}_{s}_{v}_{s}_{max}^{t}

The results derived to express the maximum variations in FVC in terms of the scaling effects were validated by numerical experiments. The variables _{max}

Simulations were conducted according to the following steps. First, the values of
_{max}_{max}_{max}

Previous studies of the scaling effects attempted to derive appropriate error bounds and to develop algorithms for correcting scaling effects [

This study was conducted as an extension of our previous work, in which we analyzed the scaling effects in the calculation of area-averaged NDVI (the monotonicity of the NDVI) [

In light of the findings associated with the error bounds discussed above, we further derived the maximum difference between FVC values at the two extreme resolution levels as a function of a true FVC value for a fixed set of true and assumed endmember spectra. We derived the expression for the maximum difference as a function of both the “true” endmember spectra and the “assumed” spectra in the algorithm.

The scope of practical applications that lend themselves to the findings of this study is limited due to the difficulties associated with accurately estimating the “true” endmember spectra in a target area. Nevertheless, the estimations of the error bounds in FVC calculations are only one type of application; any technique may be used to measure the ranges over which the true endmember spectra vary across the red–NIR reflectance space. Such approaches can lead to uncertainty in the estimates of the averaged FVC values across spectral data collected at multiple resolutions. One often encounters this type of application in the context of long-term observations of biophysical variables by multiple sensors, where the inter-sensor calibration between sensors of two different resolutions plays an important role. Further studies are needed to explore this possibility.

This work investigated the mechanism underlying the scaling effects in an fraction of vegetation cover (FVC) retrieval algorithm using an NDVI-isoline-based linear mixture model (LMM) in an extension of our previous analysis (which treated the scaling effects on NDVI). The analysis was performed by focusing on the monotonicity of area-averaged FVC calculations as a function of the spatial resolution. The assumption of a two-endmember LMM facilitated the analytical treatment, which was found to be consistent with our previous investigations of NDVI scaling effects. Interestingly, the monotonic behavior of the FVC was somewhat different from that observed in NDVI calculations, even though the FVC algorithm used NDVI as a condition. The NDVI changes monotonically within a resolution class under the two-endmember LMM, whereas the FVC computed by the NDVI-isoline-based LMM does not necessarily change monotonically. This non-monotonic behavior occurs when the endmember spectra satisfies a certain condition. In other words, the NDVI and FVC may behave differently regarding their monotonic aspect, which is one of the findings of this study.

The condition of monotonicity was determined by the factor

If FVC varies monotonically (

The mechanism underlying the scaling effects observed in FVC calculations was analyzed in terms of the function monotonicity and the error bounds. The treatment developed here provides a theoretical basis for the scaling effects, which is strength of this work. The findings of this study can contribute to development of a scale-invariant algorithm for FVC retrieval under scenarios in which multiple datasets collected at different spatial resolutions are integrated in a single analysis.

Since the analyses have been performed based on two-endmember linear mixture model, the number of endmember spectra might be the major limitations of this work. For instance, increase of the number of endmember spectra would cause differences in the monotonic behavior to some extent. This point still remains unclear from this work, which should be solely investigated in the future. In addition, further validations of the findings with actual satellite data will also be needed as future efforts.

This work was supported by The Circle for the Promotion of Science and Engineering (KO), a NASA grant NNX11AH25G (TM), and JSPS KAKENHI 21510019 (HY).

(a) A false-color image of a target field used to simulate the scaling effects in an area-averaged FVC calculation; (b) the reflectance spectra of the field and endmember spectra assumed in the model. One vegetation endmember spectrum assumed in the algorithm was (0.05, 0.35), denoted by a green circle. Two non-vegetation spectra were assumed: (0.12, 0.14) and (0.18, 0.22), that is, “soil endmem.-1” and “soil endmem.-2”, indicated by the dark and light brown circles, respectively.

False-color images of the spectral data at several resolution levels used to simulate the scaling effect in the calculation of an averaged FVC.

Scaling effects of area-averaged FVC, calculated based on the NDVI-isoline-based LMM as a function of the spatial resolution, using two pairs of endmember spectra. The averaged FVC derived using the vegetation endmember and the soil endmem.-1 spectra is indicated by the filled circles. The averaged FVC derived using the vegetation endmember and soil endmem.-2 spectra is indicated by the filled squares. Increasing and decreasing trends in the averaged FVC depend on the choice of endmember spectra assumed in the model. (The trend also depends on the endmember spectra present over the target fields).

Illustration of the variables, the partitioning process, and the unmixing process used to implement a FVC estimate using the isoline-based LMM.

Illustration of the relationship between the vectors describing the true endmember spectra and the assumed endmember spectra (Δ

A geometrical interpretation of the variable

An example numerical demonstrations of the relationships between the endmember spectra and the monotonic behavior of the area-averaged FVC. Δ_{v}_{s}

Results of numerical demonstration of the relationship between the endmember spectra and the monotonic behavior of the area-averaged FVC. Nine vector variations corresponding to different endmember spectra Δ

Scaling effects in the FVC, determined for various endmember spectra corresponding to the non-vegetation surface (

Numerical demonstration of _{max}_{max}

Summary of the major FVC-equivalent products.

Source : | CNES, |
ESA | EUMETSAT | NOAA | UMD |

Product : | fCover [ |
fCover [ |
FVC [ |
GVF [ |
VCF [ |

Sensor : | VEGETATION | MERIS | SEVIRI | AVHRR | MODIS |

Period : | 1998– | 2002– | 2005– | 1981– | 2000–2001 |

Resolution : | 1 km | 300 m | 3 km | 1.1 km | 500 m |

Scale : | Global | Global | Europe and Africa | Global | Global |

Endmember spectra in the red and NIR, corresponding to vegetation (_{v,r}_{v,n}_{s,r}_{s,n}_{r}_{n}

_{v,r} |
0.03 | 0.03 | 0.03 | 0.04 | 0.04 | 0.04 | 0.05 | 0.05 | 0.05 |

_{v,n} |
0.24 | 0.24 | 0.24 | 0.32 | 0.32 | 0.32 | 0.4 | 0.4 | 0.4 |

_{s,r} |
0.05 | 0.15 | 0.30 | 0.05 | 0.15 | 0.30 | 0.05 | 0.15 | 0.30 |

_{s,n} |
0.06 | 0.18 | 0.36 | 0.06 | 0.18 | 0.36 | 0.06 | 0.18 | 0.36 |

Δ_{r} |
−0.02 | −0.12 | −0.27 | −0.01 | −0.11 | −0.26 | 0 | −0.10 | −0.25 |

Δ_{n} |
0.18 | 0.06 | −0.12 | 0.26 | 0.14 | −0.04 | 0.34 | 0.22 | 0.04 |

Red and NIR reflectances of the assumed non-vegetation endmember spectra in the validation experiment.

Red reflectance | 0.12 | 0.13 | 0.14 | 0.15 | 0.16 | 0.17 | 0.18 |

NIR reflectance | 0.14 | 0.16 | 0.17 | 0.18 | 0.19 | 0.20 | 0.22 |

True endmember spectra for the vegetation (_{v}_{s}_{v}_{v}

_{v} |
_{s} |
_{v} |
_{s} | |
---|---|---|---|---|

Red reflectance | 0.05 | 0.10 | 0.05 | 0.03∼0.38 at 0.05 (8 cases) |

NIR reflectance | 0.40 | 0.12 | 0.36 | NIR=1.2Red |