Scale invariance in sub-grid variability

The limited effect of the topographically corrected downscaling means that FPAR may not be scale invariant during the winter season in this study area (Eq. 4.3 and 4.4). A simple test of scale invariance of NDVI between Landsat and MODIS scales may give an idea how different NDVI values are affected by different spatial aggregation methods. It also may provide a possible explanation of the discrepancy of maximum and minimum NDVI values between Landsat and MODIS scales. The scale invariance of NDVI between Landsat and MODIS scales can be tested by comparing between the mean NDVI (NDVIavg) of sub-grid Landsat NDVI within a single

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MODIS pixel and the lumped NDVI (NDVIlump) from aggregated radiance at the MODIS scale. In the same way, the scale invariance of FPAR can be checked by comparing Eq. 4.3 and 4.4 values. If α parameters (Eq. 4.5) are assumed to be constant at each sub-grid pixel (homogeneous land cover and biome types), this test results in the comparison between NDVIavg (Eq. 4.3) and the weighted mean of sub-grid NDVI (NDVIi) with respect to sub-grid incoming radiation (IPARi). It can be rewritten from Eq. 4.4 and 4.5 without the time function (DOY and t) as

(

)

∑

∑

⋅

=

IPAR

IPAR

NDVI

NDVI

_(4.8)

Figure 4.13a shows a scatter plot between NDVIavg and NDVIlump, and the temporal patterns of relative difference between them in the study site (n = 369). NDVI is definitely not scale invariant in the middle of winter season, when NDVIavg underestimates NDVIlump. This test presents a possibility that the mean Landsat NDVI can be slightly lower than the lumped MODIS NDVI in the middle of winter in the study site. Note that systematic decreases of the mean Landsat NDVI in the middle of winter are observed in this study (Figure 4.5), but not recognized in the filtered MODIS NDVI (Figure 4.2).

Figure 4.13b shows a scatter plot between NDVIavg and NDVIwgt, and the temporal patterns of relative difference between them in the study site (n = 369). They shows similar pattern to the relation between NDVIavg and NDVIlump, so FPAR is not scale invariant during the winter season. Note that larger αsimple values (Eq. 4.6) than αtopo_corrected values (Eq. 4.7) in the winter (Figure 4.10) already analytically supports the overestimation of NDVIwgt compared to NDVIavg.

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Figure 4.13: Temporal patterns of relative differences (left column) and scatter

plots (right column) between (a) NDVIavg and NDVIlump, (b) NDVIavg and NDVIwgt,

and (c) NDVIlump and NDVI

radiance at the MODIS scale. NDVIavg is the averaged NDVI at MODIS scale

from sub-grid Landsat NDVI values. NDVIwgt is the weighted averaged NDVI with

respect to sub-grid incoming radiance (Eq. 4.8). Horizontal and vertical lines

represent 5th and 95th percentiles of the spatial NDVI values within the WS08

watershed (n = 8654; Figure 4.1).

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The temporal patterns of relative difference between NDVIavg and NDVIlump (Figure 4.13a), and between NDVIavg and NDVIwgt (Figure 4.13b) show that these relative differences are very similar on corresponding dates. Therefore, there seems little relative difference between NDVIlump and NDVIwgt (Figure 4.13c) even in winter season, when only increased spatial variances are observed. Therefore, we can derive this relationship from Eq. 4.4 and 4.5.

lump wgt NDVI NDVI

FPAR=

α

α

(4.9)

where FPAR represents the integrated FPAR at the MODIS scale. Eq. 4.9 indicates that the same α parameter between FPAR and NDVI is applied both at the sub-grid scale (Eq. 4.5) and at the MODIS scale. Therefore, the linear NDVI-FPAR relationship may be scale invariant in the study site, though FPAR and NDVI is not scale invariant.

Previous studies of the scale invariance of NDVI show conflicting results whether the lumped NDVI calculated from aggregated reflectance is larger than the averaged NDVI (Hall et al. 1992; Friedl et al. 1995). Hu and Islam (1997) proved that the difference between lumped and averaged NDVI was dependent on the variances of the red and near-infrared band radiances and the covariance between two radiances using a Taylor series approximation. Eq. 4.4 shows that the scale invariance of FPAR is only dependent on the covariance between IPARi,DOY and NDVIi,DOY in Eq. 4.8, but not on each variance term. The similarity between NDVIlump and NDVIwgt (Figure 4.13c) indicates the scale invariance of NDVI is only dependent on the covariance between IPARi,DOY and NDVIi,DOY within a coarse pixel in the regions with homogeneous land cover and biome types.

In this study, the linear NDVI-FPAR relationship is estimated by matching 1-km MODIS NDVI (MOD13A2) and FPAR (MOD15A2) in the study area. This relationship is used to derive 250-m MODIS FPAR from 250-m MODIS NDVI (MOD13Q1). Tian et al. (2002a) examined the scale-dependent property of the MODIS NDVI and LAI algorithms and they found that MODIS

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LAI is not scale invariant between 250 m and 1 km while the mean of MODIS NDVI changed little with different spatial resolutions. Tian et al. (2002b) also found that LAI retrieval errors at coarse resolutions are proportional to sub-pixel heterogeneity in land cover especially mixed with non-forest biome types. However, Friedl et al. (1995) pointed out that the linear NDVI-FPAR relationship was little affected by aggregating NDVI and FPAR unlike the non-linear NDVI-LAI relationship due to its linearity. In this sense, several studies using three-dimensional radiation transfer models pointed out that the linear NDVI-FPAR relationship is scale invariant by comparing estimated relationships from homogeneous and heterogeneous canopy (Myneni and Williams 1994; Myneni et al. 1995). Moreover, the study area is represented as relatively homogeneous land cover as deciduous broadleaf forests with closed canopy and well mixed colluvial soils. Therefore, the assumption of scale invariance of the linear NDVI-FPAR relationship may be valid in the study area.