2.2 Spatial Structure
Variography is initiated by the grouping of the available pair-values into a number of lags or distance classes in accordance with their in between distances. Variograms provide a means of quantifying the commonly observed relationship where by samples close together will tend to have more similar values than samples farther apart. The variogram ?(h) is defined as:
g(h)=(1/2)Var[Z(x)-Z(x+h)]
where h is the lag distance separating pairs of data points, Var is the variance of the argument. Z(x) is the value of the regionalized variable of interest at location x, and Z(x + h) is the value at the location x + h.
An isotropic experimental variogram
g*(h), is given by:
where
g*(h) is variogram for interval lag distance class h, n(h) is the number of pairs separated by the lag distance h.
An anisotropic experimental variogram is defined as:
where
q is the angle of a principal axis.
For anisotropic analyses the principal axis is the base axis from which the offset angles are calculated.
Offset angles are 0
°, 45
°,
90
°
and 135
° clockwise from the base axis (Gamma Design, 1995).
The axis of 0
° is defined from the north-south axis.
The points aligned sufficiently close to one or another of these angles with
22.5
° tolerance are included in the anisotropic analysis for that angle.
The main features of a typical variogram are three-fold: (1) range, (2) sill, and (3) nugget effect. Range is the distance at which the variogram reaches its maximum value. Paired samples whose in-between distance is greater than the range is uncorrelated. This means that range is regarded as a measure of the spatial continuity of the investigated variable. Sill, as the upper limit of the variogram which tends to level off at large distances, is a measure of the population variability of the investigated variable generally, the higher the sill, the greater the variability in the population. The nugget effect is exhibited by the apparent jump in the variogram at the origin, a phenomenon that may be attributed to the small-scale variability of the investigated process and/or to measurement errors.
3. Results and Discussion
3.1 Statistics concerning Aggregation sizes
The aggregation processes were based on the 12.5m, 25m, 37.5m, and 50m NDVI resolutions of three different land cover sites. Table 1 summarizes the descriptive statistics relating to the aggregate results. Table1 presents the statistics on the aggregation of the pure grass, mixed grass-shrub and broad forest sites. The statistics, mean, standard deviation, minimum and maximum, of the NDVI data were almost identical to those of the above four resolutions at the pure grass site as listed in Table 1. Meanwhile, The statistics, mean, standard deviation, minimum and maximum, of the NDVI data were different to those of the above four resolutions at the mixed grass-wood and wood sites as listed in Table 1. Comparison revealed the statistics of the aggregated NDVI data in the above four resolutions at the pure grass site display more consistently than those at the mixed grass-shrub and broad forest sites.
3.2 Experimental Variogram
Experimental variograms of the above four resolution NDVI at the three different land cover sites were calculated at same active lag and lag interval. The experimental variograms are isotropic experimental variograms displayed in Fig. 2. Fig. 2(a) displays the experimental variograms of the aggregated NDVI data at the pure grass site in 12.5m, 25m, 37.5m and 50m resolutions. These experimental variogram show that the spatial structures of the above four NDVI resolutions displayed almost identical pattern in isotropic formation. The experimental variograms of the 25 m and 50 m resolutions of the wood site displayed a similar tendency. Moreover, the 12.5 m and 37.5 m resolutions NDVI data exhibit a similar pattern on their variograms, as illustrated in Fig. 2(b). Meanwhile, the four resolutions of NDVI data at the mixed grass-shrub site displayed different tendencies and variations on their experimental variograms, as displayed in Fig. 2(c). These results illustrated that the aggregation size effect on the isotropic spatial structure varies on different land covers in this study area.
The anisotropic formation analysis results illustrated that the experimental variograms of the four resolutions on the pure grass site at 0o, 45o and 135o directions displayed a similar tendency, as presented in Figs. 3(a), (b) and (c). Similar to the isotropic formation analysis Figs. 4(a), (b) and (c) indicated that the experimental variograms at 0o, 45o and 135o directions on the 25 m and 50 m resolutions of the broad forest site displayed a similar tendency. Moreover, on this broad forest site the 12.5 m and 37.5 m resolutions NDVI data exhibit a similar pattern on their variograms at 0o, 45o and 135o directions, as illustrated in Figs. 4(a), (b) and (c). Meanwhile, the four NDVI resolutions data at the mixed grass-shrub site displayed different tendencies and variations on their experimental variograms at 0o, 45o and 135o directions, as displayed in Figs. 5(a), (b) and (c). These results indicated that the aggregation size effect on the anisotropic spatial structure varies on different land covers in this study area.
4. Conclusion
This study has demonstrated isotropic and anisotropic spatial structures in 12.5m, 25m, 37.5m, and 50m resolutions NDVI data from a Spot image at three different 0.1407 km2 land-cover sites that were an almost pure grass, a mixed grass and shrub, and a broad forest within the Yang Ming Shan National Park in Taiwan to evaluate the impact of aggregation on the spatial structure at different land-cover sites. The results indicated that the experimental variograms of the NDVI data in these four different resolutions displayed identical spatial structure in both isotropic and anisotropic directions at the grass site of Yang Ming Shan National Park. At the mixed grass-shrub site of this study area the NDVI experimental variograms in the 12.5m, 25m, 37.5m, and 50m resolutions roughly displayed a similar spatial tendency but different spatial variations. The experimental variograms of NDVI of these four resolutions at the broad-forest site of this study area displayed different spatial patterns and spatial variations in both isotropic and anisotropic directions. The experimental variograms of the broad forest site in the 25 m and 50 m resolutions displayed a similar tendency. Moreover, the 12.5 m and 37.5 m resolutions NDVI data exhibit a similar pattern on their variograms. However, the aggregation effect displayed significantly on the mixed grass-shrub and the broad forest sites in this study.
