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Changes in the spatial distributions of vegetation across the globe are routinely monitored by satellite remote sensing, in which the reflectance spectra over land surface areas are measured with spatial and temporal resolutions that depend on the satellite instrumentation. The use of multiple synchronized satellite sensors permits long-term monitoring with high spatial and temporal resolutions. However, differences in the spatial resolution of images collected by different sensors can introduce systematic biases, called scaling effects, into the biophysical retrievals. This study investigates the mechanism by which the scaling effects distort normalized difference vegetation index (NDVI). This study focused on the monotonicity of the area-averaged NDVI as a function of the spatial resolution. A monotonic relationship was proved analytically by using the resolution transform model proposed in this study in combination with a two-endmember linear mixture model. The monotonicity allowed the inherent uncertainties introduced by the scaling effects (error bounds) to be explicitly determined by averaging the retrievals at the extrema of the resolutions. Error bounds could not be estimated, on the other hand, for non-monotonic relationships. Numerical simulations were conducted to demonstrate the monotonicity of the averaged NDVI along spatial resolution. This study provides a theoretical basis for the scaling effects and develops techniques for rectifying the scaling effects in biophysical retrievals to facilitate cross-sensor calibration for the long-term monitoring of vegetation dynamics.

Satellite remote sensing provides a historical record of the biophysical parameters that may be used to model the global vegetation dynamics [

Differences in the sensor characteristics, including the spatial, radiometric, or spectral resolution, often introduce systematic biases in the arithmetically averaged (area-averaged) biophysical parameters [

This study focuses on the systematic bias present in retrievals (parameters) due to differences in the spatial resolution of the data [

In practice, biophysical parameter such as leaf area index, fraction of vegetation cover, and biomass is often derived from spectral vegetation index with empirical regression models. Therefore, the bias error in the NDVI caused by the scaling effect will be propagated into such parameters. For instance, a practical case study can be seen in [

The scaling effects arise from the uncertainty caused by surface heterogeneity and nonlinearity in algorithms for retrieving pixel scale reflectance data [

A key approach to improving our understanding of the scaling effects in the calculation of NDVI has been to analytically clarify the mechanisms underlying the scaling effects. Raffy proved that the error bounds associated with the scaling effects were associated with changes in the spectra at any point within a target pixel (for constant pixel-scale reflectance spectra) [^{∧} and ^{∨}, which are obtained using retrieval algorithms and pixel-scale reflectances [

To investigate the behavior of the area-averaged NDVI as a function of the spatial resolution, we developed a theoretical framework [

Several studies have implicitly or explicitly examined the monotonic behavior of the area-averaged NDVI over a given area as a function of the spatial resolution [

The non-monotonic behavior of the average NDVI can be easily demonstrated using numerical simulations under two-endmember assumptions (vegetation and non-vegetation endmembers), as shown in

Although the monotonicity of the area-averaged NDVI is central to determining the error bounds, it is not always guaranteed. Hence, monotonic and non-monotonic behaviors are expected to arise in the calculation of an area-averaged NDVI as a function of the spatial resolution. In this study, we try to clarify the mechanisms underlying the scaling effects in terms of the monotonicity of the average NDVI, particularly under conditions in which the average NDVI changes monotonically or non-monotonically. Models of the transformation of the spatial resolution and target spectra will be discussed further in the next section.

In this study, a target spectrum was modeled using LMM under the constraint that the sum of the endmember spectral weights is unity [_{v}_{v,r}_{v,n}_{s}_{s,r}_{s,n}_{v}_{s}_{v}_{s}

To proof the monotonicity, we should focuses on a comparison among the area-averaged NDVI values obtained at different spatial resolutions. For this purpose, the area of a target region was fixed at a given size selected to be larger than or equal to the size of the lowest sensor spatial resolution. The number of pixels contained in a target region depended on the spatial resolution of each sensor, that is, the ‘resolution level’ was given by the number of pixels within a fixed area, indicated by the index

The resolution transformation from one level to another was modeled by first defining a simple rule for partitioning pixels, that is, each pixel was partitioned into two at a given time point, as illustrated in

A fixed target region contained a total of _{R,j,k}_{N,j,k}

Because the target region was divided into _{j,k}_{j,k}_{j,k}

The NDVI can be obtained for each pixel, _{j,k}

The variables, parameters, and indices are illustrated in

Consider a sequence of images obtained by repeating the partitioning process described in the previous subsection (_{j}_{−1}) and the difference Δ

The above equation suggests that, for a resolution sequence, the area-averaged NDVI should change monotonically if the sign of Δ

In this section, we analyze Δ_{2,1}, which indicates the FVC for a pixel after the partitioning process. The partial derivative of Δ_{2} − _{1} with respect to _{2,1} is equal to the partial derivative of _{2} because _{1} is independent of _{2,1}.

The value of the VI obtained by averaging over an area after the partitioning process can be expressed as
_{2,1} = (_{R}_{,2,1}, _{N}_{,2,1}), _{2,2} = (_{R}_{,2,2}, _{N}_{,2,2})) are modeled by a two-endmember LMM,

Because the value of the FVC in the original target pixel (_{1,1}) must be conserved after the partitioning process, the following relationships among the FVCs holds:
_{2,2} and substituting the result into _{2,2} in terms of _{2,1},

_{2,1} and _{2,2} can be expressed in terms of a single parameter, _{2,1} for a given pair of endmember spectra and for a fixed value of _{1,1} (the total fraction of vegetation over a given target region). Consequently, _{2,1} and _{2,2} can be expressed as functions of _{2,1}.

Although _{2,1} is independent of _{1,1}, its range depends on _{1,1} such that

It will be useful to calculate the partial derivative of _{2} with respect to _{2,1}. From _{2,1} and _{2,2}.

The partial derivative of _{2,}_{k}_{2,}_{k}_{2,2} with respect to _{2,1} can be obtained from

Again, from _{2,2} can be written by _{2,1},
_{2,2} with respect to _{2,1}, then, becomes

Finally, the partial derivative of _{2} with respect to _{2,1} becomes
_{1} and _{2} are defined by

The parenthetical equation given above becomes positive, as indicated in following equation derived from

This indicates that the sign of _{2}_{2,1} (_{v}_{1} and ||_{s}_{1}, and _{2,1}. We define the parameter

Using the above definition, the sign of the partial derivative behaves as follows:

Additionally, when _{2}/_{2,1} = 0, indicating that the area-averaged NDVI does not depend on the spatial resolution [

Using _{2,1} = _{1,1}. Therefore, from

These results indicate that the sign of the difference between the area-averaged NDVI values at resolutions 1 and 2 is determined by _{j,k}

The results of the previous section were applied to determine the spatial resolution at which the average NDVI transformed monotonically with the spatial resolution.

According to _{2} is smaller than _{1}, the value before partitioning, if the level-1 norm of the vegetation endmember in a given area exceeds that of the non-vegetation endmember (_{2} is larger than _{1} if the level-1 norm of the vegetation endmember is less than that of the non-vegetation endmember (_{2} is less than _{1}. When the resolution is transformed from level 2 to level 3, one of the two rectangular pixels (of level 2) is divided into two. The average NDVI after partitioning the pixel,
_{3} is certainly smaller than _{2}. Therefore, the average NDVI decreases monotonically over a resolution sequence as the spatial resolution increases. The resolution sequences define a ’resolution class’. These results are summarized in the following theorems.

This condition is sufficient for NDVI to vary monotonically with the resolution. The following theorem relating to the trends in the averaged NDVI can be proven based on

Two example resolution classes are illustrated in _{j}_{j}_{j}_{j}_{j}

As a result, VI at the extreme resolutions (the coarsest and finest resolutions) should reach a maximum or minimum because (1) the average NDVI varies monotonically within a resolution class, (2) extreme resolutions belong to the same resolution class, and (3) any type of resolution certainly belongs to a resolution class. These results indicate that the error bounds on the average NDVI associated with a given spatial resolution are determined by the two extreme resolution cases.

Numerical simulations were conducted to verify the monotonic behavior of the area-averaged NDVI. To this end, we considered two areas of the same size that included both vegetation and non-vegetation endmembers, as shown in

The target field was assumed to be acquired at the _{+} represents the set of positive integers. Then all possible members of the _{N}_{N}_{N}_{1}, _{2}, _{3}, ···, _{n}_{+} and

In this simulation, we assumed _{40000} becomes
_{1} ⊂ _{40000} and _{2} ⊂ _{40000} were assumed to be
_{1} and _{2} can be referred to as ‘class-1’ and ‘class-2’, respectively, as illustrated in _{40000}. On the other hand, the area-averaged NDVI varied monotonically over the resolution class sets (_{1} and _{2}).

A resolution transform model with a certain partitioning rule was proposed to analyze the area-averaged NDVI for a fixed area described under different spatial resolutions. To elucidate the error bounds on the calculated NDVI, the resolution class (condition for monotonicity) was introduced. The resolution class indicates the values that are interconnected under the forward and reverse application of the partitioning rule. Within a class, the area-averaged NDVI certainly vary monotonically with the spatial resolution. The trends in the average NDVI (decreasing or increasing) within a resolution class can be determined based on the factor

A major limitation of this work is the number of endmember spectra assumed in the LMM. This limitation may cause a loss of practicality in some extent, since in general one encounters more numbers of distinct surfaces within a satellite imagery. This assumption, however, is reasonable at this stage of investigation for the following reason. The monotonicity of NDVI in the framework of scaling issue has not been fully and thoroughly investigated to date. As an initial step of tackling the theme, it is appropriate to start with the simplest case. Moreover, this assumption enables us to analyze the influence of spatial resolution analytically from the beginning to the end. And, owing to this simplicity, one can even reach the understanding of deep insight of the complex phenomena with no ambiguity. Therefore, the assumption brings us both advantage and disadvantage.

The assumption is indeed not practical knowing the fact that pixel size of sensors with middle to lower resolution (with wide field-of-view) is most likely large enough to include three or more numbers of endmember species. Nevertheless, considering the trend of sensor technology, spatial resolution of future sensor will tend to be higher. As a consequence, the pixel size eventually becomes fine enough to the level such that region of interest can be considered as a two-endmember case. In this sense, the results and findings from this study would serve as theory that reasonably explains behavior of scaling effect with some degrees of practicality.

We analytically investigated the mechanisms underlying the scaling effects present in the calculation of NDVI under a two-endmember LMM, and we related the scaling effects to the monotonic behavior as a function of the spatial resolution. The scaling effects depended on the spectral features, and this relationship was clarified to show that the trend (increasing or decreasing) depended on the vegetation and non-vegetation endmember spectra. The proof of the monotonicity indicates that the error bounds may be derived deterministically using the area-averaged values calculated from the data at the extreme resolutions.

This study establishes a theoretical framework for describing the scaling effects in the calculation of biophysical or climatological parameters retrieved from remotely sensed data. In this framework, remotely sensed data is modeled using a linear mixture of endmember spectra. The applicability of the results and the findings of this study are mostly restricted by the assumption of two endmembers. Further investigations are required to expand the discussion to (1) the analysis of a greater number of endmembers, and (2) the analysis of other biophysical parameters, such as LAI or FAPAR. Nevertheless, the fundamental behavior of the scaling effects form a theoretical basis for similar investigations.

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).

Non-monotonic behavior on an area-averaged NDVI as a function of the number of pixels (spatial resolution) over a fixed area. (

Elements of the resolution transformation process based on a simple partitioning procedure.

Illustration of the resolution transform using the partitioning rule. The upper panel shows an example of the resolution transform from levels 1 to 4, and the lower panel shows the transform from levels 1 to 9.

The endmember abundances of the vegetated surface (fraction of vegetation cover) for pixel _{j,k}

Illustration of the variables used in this study. _{j}_{j,k}_{j,k}_{v}_{s}_{j,k}

The fraction of a pixel after the partitioning process, represented by

Monotonicity of the area-averaged NDVI in consecutive resolution sequences. A trend increases or decreases depending on the factor

Illustration of the resolution transform from resolution levels 1 to 4. (

Numerical demonstration of the scaling effects in the calculation of the area-averaged NDVI. (_{r}_{n}_{40000}); (c) Variations within class-1 (subset _{1}) only, or (d) for class-2 (subset _{2}) only. (

Two resolution classes used in the experiments. The upper sequence (class-1, _{1}) presents a different class than the lower sequence (class-2, _{2}).