Dec., 2010 J. Resour. Ecol. 2010 1(4) 361-367 DOI:10.3969/j.issn.1674-764x.2010.04.009 www.jorae.cn Journal of Resources and Ecology Vol.1 No.4 Article Scale-dependent Spatial Relationships between NDVI and Abiotic Factors LI Shuangcheng*, YANG Zhuoxiang and GAO Yang College of Urban and Environmental Sciences, Peking University; The Key Laboratory for Earth Surface Processes, Ministry of Education, Beijing 100871, China Abstract: Wavelet transform demonstrates that abiotic factors impact change with spatial scale, confirming a scale-dependent relationship between NDVI and factors that influence it. To elaborate these scale effects, NDVI transect data and abiotic variables climatic and topographic at the 32.5 degree north latitude on the Qinghai-Tibet Plateau in China, were analyzed at different spatial scales by using wavelet transform. The results show that climatic variables such as precipitation and temperature are not dominant factors of NDVI patterns at the less than 80 km scale, while significant wavelet coherency is observed at the more than 80 km scale in some ecoregions. As a differentiating factor, elevation affects NDVI patterns only at the local level in longitudinal range-gorge region at certain specific scales. Wavelet transform is an alternative approach to examining multiscale relationships between NDVI and abiotic factors. Key words: NDVI; abiotic factors; scale-dependence; wavelet transform; Qinghai-Tibet Plateau 1 Introduction Terrestrial ecosystems exhibit spatial variability at scales ranging from centimeters to kilometers. Spatial complexity results from both underlying patterns of the physical environment and complex biotic interactions (Logan et al. 1998; Nelsona et al. 2007; Crawford 2008). Ecological patterns and processes are dependent on the spatial scale at which they are investigated, and a process at any particular scale may be influenced by factors at other scales (Allen and Starr 1982; Menge and Olson 1990; Levin 1992; Bunnell and Huggard 1999; Bugmann et al. 2000; Elkie and Rempel 2000; Thompson et al. 2000; Whittaker et al. 2001; Schneider 2001; Blackburn and Gaston 2002; Chase and Leibold 2002; Urban 2005; Tylianakis et al. 2006; Field et al. 2008; Parviainen et al. 2010). Developing a full understanding of the spatial scales at which abiotic conditions impinge on ecological patterns and processes therefore demands a multiscale approach (Wiens 1989). Concerns about the detection of scale-dependent phenomena and modeling ecological processes across scales have increased significantly in recent years. Bian and Walsh et al. (1993) examined the effects of spatial scale on the relationships between vegetation biomass and three topographic variables with regression, semivariance and fractal analyses, and then identified a characteristic length of variables interaction. Moody et al. (1995) investigated the scale-dependence of the relationship between NDVI variability and variability in land cover for a complex mountainous landscape. Walsh et al. (1997) further conducted an elaborate research to develop the hypotheses that the relationships between NDVI, cover types, and elevation are scale-dependent. Foody (2004) used geographical weighted regression to study the relationship between species richness and a set of perceived environmental determinants, comprising temperature, precipitation, and normalized difference vegetation index (NDVI), and confirmed the existence of scale-dependent relationships between species richness and abiotic factors. By introducing waveletcoefficient regression, Keitt and Urban (2005) formalized scale-specific relationships between vegetation and environmental factors. Saunders et al. (2005) examined the utility of three techniques (lacunarity, spectral, and wavelet analyses) for detecting scales of pattern of ecological data and concluded that the appropriate technique for assessing scales of pattern depends on the type of data available, the question being asked, and the detail of information desired. Received: 2010-10-05 Accepted: 2010-11-12 Foundation: National Key Research Development Plan No. 2010CB951704 and National Natural Science Foundation of China No. 40771001 * Corresponding author: LI Shuangcheng. Email: scli@urban.pku.edu.cn.
362 The main goal of this research is to further examine the large-scale spatial dependence of statistical relationships between a group of selected abiotic factors and the spatial structure of NDVI at multiple spatial scales by using wavelet transform. We seek to address: (i) the effective range of spatial scales within which NDVI and abiotic variables were spatially dependent; (ii) optimum spatial scales for representing the relationships between abiotic factors and NDVI; and (iii) regional differentiation of spatial dependence of these relationships in different ecoregions of the study area. This paper is organized as follows: the methodology and data, especially the theoretical background and algorithm of wavelet transform technique as applied to multiscale analysis are explained in section 2, including the introduction of Morlet wavelet and wavelet coherence. In section 3, the geographical features of the study area are briefly described. In section 4, we present results of the investigation of spatial dependent relationships between NDVI and abiotic factors. Section 5 is conclusions and discussions. 2 Methods and Data 2.1 Wavelet transform Wavelet transform (WT) is a merited technique for analyzing localized variations of power within a time series. Compared with Fourier analysis, the main advantage of wavelet transform is the use of varying window size, being wide for low frequencies and narrow for high frequencies, leading to an optimal time-frequency resolution in all the frequency ranges. Continuous wavelet transform (CWT) can decompose a signal into a set of finite basis functions, so it can uncover transient characteristics in the signal. Wavelet coefficients W x (a,τ) are produced through the convolution of a mother wavelet function ψ (t) with the analyzed signal x(t), as a function of both time t and frequency (scale) a, it is: where a and b denote the dilation (scale factor) and translation (time shift parameter) respectively. ψ is called a mother wavelet, a smooth and quickly vanishing oscillating function. W are wavelet coefficients, which are a function of scale and position. The symbol * indicates the complex conjugate. By adjusting the scale a, a series of different frequency components in the signal can be obtained. The Morlet wavelet is a symmetric and periodic wavelet that results from the superposition of a sine and a Gaussian. In complex notation this can be written as: Journal of Resources and Ecology Vol.1 No.4, 2010 where π -1/4 is a normalization term, η is the dimensionless time parameter, ω 0 is the dimensionless frequency parameter (taken as ω 0 =6 for this work), and ω is the frequency parameter. Because of its smoothness and periodicity, Morlet wavelet is a good choice for data that is varying continuously in time and is periodic or quasiperiodic. Another advantage of Morlet wavelet transform (MWT) is that the phase information of signal can be obtained; therefore this tool can also be used to study the phase synchronization between two signals. A graph of this function is presented in Fig.1, in the left, the real part, and on the right, the imaginary part. 2.2 Wavelet coherency Wavelet coherency analysis is a powerful tool to measure intensity of the covariance of two series in time-frequency space that provides a perfect view of linear and nonlinear correlation between the two time series. Wavelet coherence is defined as correlation coefficient between the wavelet transform coefficients representing two time series in timefrequency domain (Popinski and Kosek 1994). Wavelet coherency is computed using the wavelet power spectrum of the two time series. To measure wavelet coherency, following Torrence and Webster (1998) we define it as < > indicates smoothing in both time and frequency, W x (a, b) is the wavelet transform of series x(t), W y (a, b) is the wavelet transform of y(t), and W xy (a, b) is the cross-wavelet transform. The smoothing is performed, as in Fourier spectral approaches, by a convolution with a constant-length window function both in the time and frequency directions. Detailed information on the smoothing procedure can be found in Torrence and Webster (1999). Wavelet coherence ranges from 0 to 1, with a value of 1 indicating maximum coherency. It gives a measure of the dependency and synchrony between the two time series as a function of both scale (or period) and time. Wavelet phase measures the phase difference between the complex Fig. 1 Real (left) and imaginary (right) part of the Morlet wavelet with ω 0 =6.
LI Shuangcheng, et al.: Scale-dependent Spatial Relationships between NDVI and Abiotic Factors 363 wavelet transforms and indicates whether two time series tend to oscillate simultaneously, rising and falling together with the same period. Wavelet coherence and significance levels for wavelet power spectra were computed following the methods of Grinsted et al. (2004). CWT has edge artifacts because a wavelet is not completely localized in time. It is therefore useful to introduce a Cone of Influence (COI), where edge effects may become important because of the finite duration of the time series. Here we take COI as the area in which the wavelet power caused by a discontinuity at the edge has dropped to e 2 of the value at the edge (Grinsted et al. 2004). We conducted all wavelet transforms using Matlab (Fig. 2). 2.3 Data and transect sampling 2.3.1 Data For this study, we used 18 years (1982 2000, 1994 not used because of the sensor failure) NDVI monthly data produced by the Global Inventory Monitoring and Modeling Studies (GIMMS) group from measurements of the advanced very high resolution radiometer (AVHRR) onboard the NOAA 7, NOAA 9, NOAA 11, and NOAA 14 satellites. After data preprocessing for improving navigation, sensor calibration, and atmospheric corrections, we extracted spatial pattern of mean annual NDVI value during 1982 2000 by using Calculator in ArcGIS (ESRI Inc. 1999 2009). For the period 1982 2000, we obtained monthly meteorological data of 104 stations in the study area, including air temperature, precipitation, maximum temperature, minimum temperature, surface temperature, and hours of sunshine, from the China Meteorological Administration (CMA). Unlike the NDVI data that are raster images, meteorological variables are point data. After they were aggregated into mean annual values, all meteorological factors were interpolated to create raster data layers using Kriging method in ArcGIS, whose grid size equals to that of NDVI. Basic topographic data files of the study area are digital elevation model (DEM) data, which were produced by the National Geomatics Center of China at a scale of 1:250000. The terrain attribute chosen to perform the analysis is elevation, which is directly calculated using built-in functions in ArcGIS. To elaborate the spatial scale-dependent relationship between NDVI and environmental variables, we subsampled the NDVI and abiotic factors spaced at 5000m intervals along the north 32.5 degree latitude by running the Zonal Statistics function in Spatial Analyst of ArcGIS. 3 Study area The Qinghai-Tibet Plateau, the largest geomorphologic unit on the earth with an area of 2.5 million km 2 and average 4500 m elevation above sea level, is an important part of the global terrestrial ecosystem. From southeast to northwest, four types of ecosystems can be identified on the Plateau, namely, montane forest, alpine shrub/meadow, alpine steppe, and alpine desert. The transect along the 32.5 degree latitude runs across 4 ecoregions from the east to the west, i.e. western Sichuaneastern Tibet montane coniferous forest zone, Golog- Nagqu high-cold shrub-meadow zone, Qiangtang highcold steppe zone, and Ngari montane desert-steppe and desert zone (Zheng 1996). The mean values of NDVI and abiotic factors of each ecoregion are listed in table 1. 4 Results 4.1 NDVI and precipitation Visual inspection of wavelet coherency between NDVI and precipitation reveals three scales of variation across the transect: 0 to 40 km, 40 to 80 km, and >80 km (Fig. 2). For the less than 40 km scale, the wavelet coherency is relatively small, especially in the 0 800 km part of the transect, which crosses the western Sichuan-eastern Tibet montane coniferous forest zone. The phase relationship between NDVI and precipitation changes significantly at the 1250 km location of the transect. In-phase relationship appear at 0 to 1250 km of the transect, which corresponds to the ecoregions of montane coniferous forest and highcold shrub-meadow, while anti-phase relationships are observed at greater than1250 km locations in the transect, where ecoregions are montane desert-steppe and desert. For the 40 to 80 km scale, there are two high wavelet coherency regions with the centre of 1250 km and 1500 Table 1 NDVI and abiotic factors of different ecoregion on the Qinghai-Tibet Plateau. NDVI Annual temperature ( ) Annual precipitation (mm) Elevation (m) Eco-region* Mean Min Max Mean Min Max Mean Min Max Mean Min Max I 174 125 204 6.8 1.0 15.2 252 676 1651 4112 1219 7213 II 161 137 189 0.1 6.9 5.6 331 555 826 4506 3136 6166 III 140 121 161 0.2 7.3 8.4 32 234 623 5023 1890 6836 IV 137 126 154 1.6 0.2 4.2 33 73 187 4933 3054 7410 * I. western Sichuan-eastern Tibet montane coniferous forest zone, II. Golog-Nagqu high-cold shrub-meadow zone, III. Qiangtang high-cold steppe zone, IV. Ngari montane desert-steppe and desert zone.
364 km respectively, and the former has in-phase relationship while the latter has anti-phase relationship between NDVI and precipitation. For the more than 80 km scale, regions of high wavelet coherency are centered along the transect at 200 500 km and 1700 1900 km locations with in-phase relationships, which are located in the montane coniferous forest region and shrub-meadow region, and characterized by high elevation and rugged landform. 4.2 NDVI and temperature Figure 4 shows the patterns of wavelet coherency between the NDVI and the temperature series. For the less than 40 km scale, six wavelet coherency regions are observed with the centers at 175 km, 780 km, 1175 km, 1280 km, 1750 km and 2000 km along the transect. The phase relationship between NDVI and temperature also changes around 1250 km, and shows in-phase relationship at the less than 1250 km part and in anti-phase at the greater than 1250 km part. For the 40 80 km scale, only two dispersed regions with centers at 1300 km and 2100 km have relatively high wavelet coherency values. For the more than 80 scale, three significant wavelet coherency regions are found along the transect. In the montane coniferous forest ecoregion, a high coherency region spans 100 km and centered at 250 km of the transect, which corresponds to the spatial scale of 100 200 km. The outof-phase relationship appears from 1000 1500 km in the high-cold shrub-meadow ecoregion, which corresponds to Journal of Resources and Ecology Vol.1 No.4, 2010 the spatial scale of 250 300 km. For the more than 400 km scale, there is a significant coherency region, with in-phase relationship that range from 600 km to 1350 km along the transect. 4.3 NDVI and elevation From Figure 5 we can see that the patterns of wavelet coherency between NDVI and elevation are much simpler than that between NDVI and climatic factors. For the less than 40 km scale, only small scattered wavelet coherency regions with centers at 450 km, 1200 km, and 1700 km are shown. For the 40 to 80 km scale, there is a significant wavelet coherency region with anti-phase relationship between NDVI and elevation from 0 to 900 km along the transect, which corresponds to the montane coniferous forest ecoregion. There is another very significant coherency region with in-phase relationship ranging from 1800 km to 1900 km along the transect, which is located in the ecotone between Qiangtang high-cold steppe and Ngari montane desert-steppe and desert zone, indicating that elevation dominates the pattern of NDVI. For the more than 80 km scale, a significant wavelet coherency region with spans of 750 km from 0 to 750 km along the transect is observed in the western Sichuan-eastern Tibet montane coniferous forest ecoregion and transitional region between montane coniferous forest and Plateau high-cold scrub meadow, which are dominated by anti-phase relationship. Fig. 2 Matlab toolbox window of Morlet wavelet transform.
LI Shuangcheng, et al.: Scale-dependent Spatial Relationships between NDVI and Abiotic Factors 365 Fig. 3 Morlet squared wavelet coherence between NDVI and precipitation along the 32.5 degree north transect (the values are coherency coefficient that ranges from 0 to 1). The 5% significance level against red noise is shown as a thick contour, and the cone of influence (COI) where edge effects might distort the picture is shown with lighter shade. The relative phase relationship is shown as arrows. The direction of the arrows in the coherence spectrum indicates the phase between the two series involved: horizontal right is 0 and corresponds to an in-phase situation, horizontal left is 180 and corresponds to an anti-phase situation, and both vertical up (90 ) and vertical down (270 ) correspond to an out-of-phase situation. 5. Conclusions and discussions 5.1 Conclusions From the results of wavelet coherency analysis between precipitation and NDVI, we can conclude that precipitation is not the dominant factor of NDVI at the less than 40 km scale, but it controls the spatial pattern of NDVI at the more than 80 km scale, especially at the scale of 200 km. The phase relationships between NDVI and precipitation varies in different eco-geographic regions at the less than 80 km scale, and it shows an in-phase relationship in the forest and scrub regions and anti-phase relationship in the high-cold steppe, desert and semi-desert regions. For the more than 80 km scale, phase relationship shows as inphase. Consistent with the coherency between NDVI and precipitation, temperature affects the patterns of NDVI insignificantly at the less than 80 km scale but significantly at the more than 80 km scale, and this is observed along almost the entire transect. Again, phase conversion also occurs around the 1250 km location of the transect at the less than 80 km scale, and in-phase dominates the relationship between NDVI and temperature at this scale. The topographic factor, elevation, is not a global variable that dominates NDVI pattern along the whole 2300 km sample transect. The significant NDVI-elevation wavelet coherency regions cannot be found at the less than 25 km scale, indicating that elevation-related NDVI pattern does not exist below this scale. Elevation affects NDVI pattern at the 25 km to 600 km scale in this study, and significant elevation-ndvi coherency is only localized in regions with high variations of relief, especially in Fig. 4 Morlet squared wavelet coherence between NDVI and temperature along the transect (all other parameters are the same as described in the caption of Fig. 3).
366 Journal of Resources and Ecology Vol.1 No.4, 2010 Fig. 5 Morlet Squared wavelet coherence between NDVI and elevation along the transect (all other parameters are the same as described in the caption of Fig. 3). longitudinal range-gorge region, i.e., along the 0 1000 km part of the transect. In conclusion, climatic variables such as precipitation and temperature are not dominant factors of NDVI patterns at the less than 80 km scale. Moreover, phase relationships often reverse around the 1250 km location along the transect. At the more than 80 km scale, the significant wavelet coherency regions appear in some ecoregions along the transect, indicating that macro-patterns of NDVI are correlated with climatic factors. Unlike the climatic variables, the topographic factor, elevation, only affects NDVI pattern in some ecoregions and at a specific range of scales. Elevation has the most prominent effect on the variation of NDVI in longitudinal range-gorge zone with high variations of relief and surface fragmentation. 5.2 Discussions By multiscale analysis on the transect data using wavelet transform, the research confirms that there is a spatial scale-dependent relationship between NDVI and climatic factors, which changes with different eco-geographical regions. The scale-dependent phenomena suggest that we should be more prudent when predicting or scaling up / scaling down only using single-scale regression relationship. Besides, multiscale examination of the interaction of NDVI and climatic factors is helpful for ascertaining the effective scale of environmental factor s impacts on ecosystem, which is useful for prediction and ecological regionalization. Acknowledgements This research was funded by a grant from the National Key Research Development Plan (grant no. 2010CB951704) and the National Natural Science Foundation of China (NSFC 40771001). The authors thank anonymous reviewers for their helpful comments. References Allen T F H and T B Starr. 1982. Hierarchy: perspectives for ecological complexity. Chicago: The University of Chicago Press, 310. Bian L and S J Walsh. 1993. Scale dependencies of vegetation and topography in a mountainous environment of Montana. The Professional Geographer. 45:1 1. Blackburn T M and K J Gaston. 2002. Scale in macroecology. Global Ecol. Biogeog. 11:185 189. Bugmann H, M Lindner, P Lasch, M Flechsig, B Ebert and W Cramer. 2000. Scaling issues in forest succession modelling. Climatic Change, 44:265 289. Bunnell F L and D J Huggard. 1999. Biodiversity across spatial and temporal scales: problems and opportunities. Forest Ecology and Management, 115:113 126. Chase J M and M A Leibold. 2002. Spatial scale dictates the productivitybiodiversity relationship. Nature, 416:427 430. Crawford J. 2008. Multi-scale investigations of alpine vascular plant species in the San Juan Mountains of Colorado, USA GLORIA target region. Scientifica Acta, 2(2):65 69. Elkie P C and R S Rempel. 2000. Detecting scales of pattern in boreal forest landscapes. Forest Ecology and Management, 147:253 261. Field R, et al. 2008. Spatial species-richness gradients across scales: A metaanalysis. Journal of Biogeography. doi:10.1111/j.1365-2699.2008.01963.x Foody G M. 2004. Spatial nonstationarity and scale-dependency in the relationship between species richness and environmental determinants for the sub-saharan endemic avifauna. Global Ecology and Biogeography, 13: 315 320. Grinsted A, J C Moore and S Jevrejeva. 2004. Application of cross wavelet transform and wavelet coherence to geophysical time series. Nonlinear Processes in Geophysics, 11: 561 566. Keitt T H and D L Urban. 2005. Scale-specific inferences using wavelets. Ecology, 86:2497 2504. Levin S A. 1992. The problem of pattern and scale in ecology. Ecology, 73:1943 1967. Logan J A, P White, B J Bentz and J A Powell. 1998. Model analysis of spatial patterns in mountain pine beetle outbreaks. Theoretical Population Biology, 53: 236 255. Menge B A and A MOlson. 1990. Role of scale and environmental factors in regulation of community structure. Trends Ecol. Evol., 5:52 57. Moody A, S J Walsh, T R Allen and D G Brwon. 1995. Scaling properties of NDVI and their relationship to land-cover spatial variability. Proc. 15th Int. Geosci. and Remote Sens. Symp., Firenze, Italy, 10 14 July 1995, vol. 3, pp. 1962 1964. Nelsona A, T Oberthrb, S Cookb. 2007. Multi-scale correlations between topography and vegetation in a hillside catchment of Honduras. International Journal of Geographical Information Science, 21(2): 145
LI Shuangcheng, et al.: Scale-dependent Spatial Relationships between NDVI and Abiotic Factors 367 174. Parviainen M, M Luoto, R K Heikkinen. 2010. NDVI-based productivity and heterogeneity as indicators of plant-species richness in boreal landscape. Boreal Environmental Research, 15:301 318. Popinski W and W Kosek. 1994. Wavelet transform and its application for short period earth rotation a analysis. Artificial Satellites, Planetary Geodesy, 29(2):75 86. Saunders S C, Chen J, T D Drummer, E J Gustafson and K D Brosofske. 2005. Identifying scales of pattern in ecological data: A comparison of lacunarity, spectral and wavelet analysis. Ecological Complexity, 2:87 105 Schneider D C. 2001.The rise of the concept of scale in ecology. Bioscience, 51:545 553. Thompson F R, S K Robinson, T M Donovan, J R Faaborg, D R Whitehead and D R Larsen. 2000. Biogeographic, landscape, and local factors affecting cowbird abundance and host parasitism levels. In: J. N. M. Smith J N M, T L Cook, S I Rothstein, S K Robinson and S G Sealy (eds.). Ecology and Management of Cowbirds and Their Hosts. Austin, TX: University of Texas Press, 271 279. Torrence C and P J Webster. 1999. Interdecadal changes in the ENSO- Monsoon system. J. Climate, 12: 2679 2690. Torrence C and P J Webster. 1998. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79(1): 61 78 Tylianakis J M, A-M Klein, T Lozada and T Tscharntke. 2006. Spatial scale of observation affects α, β and γ diversity of cavity-nesting bees and wasps across a tropical land-use gradient. Journal of Biogeography, 33:p1295 1304. Urban D L. 2005. Modeling ecological processes across scales. Ecology, 86:1996 2006. Walsh S J, A Moody, T R Allen and D G Brown. 1997. Scale dependence of NDVI and its relationship to mountainous terrain. In: Quattrochi D A and M F Goodchild (eds.). Scale in Remote Sensing and GIS, Lewis Publishers, 27 55. Whittaker R J, K J Willis and R Field. 2001. Scale and species richness: towards a general, hierarchical theory of species diversity. Journal of Biogeography, 28: 453 470. Wiens J A. 1989. Spatial scaling in ecology. Functional Ecology, 3:385-397. Zheng D. 1996. The system of physio-geographical regions of the Qinghai- Xizang (Tibet) Plateau. Science in China (Series D), 39:410 417. 基于小波分析的 NDVI 与环境因子空间尺度依存关系研究 李双成, 杨卓翔, 高阳 北京大学城市与环境学院地表过程分析与模拟教育部重点实验室, 北京 100871 摘要 : 本文通过多尺度分解途径分析了 NDVI 与环境因子如地形与气候之间的空间尺度依存关系 为了揭示两者关系的尺度效应, 选取青藏高原北纬 32.5 度作为研究样带, 应用小波变换分析了不同空间尺度下的小波一致性和相位关系 研究结果表明 : 在青藏高原小于 80km 空间尺度上, 气候变量如降水和气温不是控制 NDVI 的主导因素 ; 而大于这个尺度, 在一些生态区可以发现 NDVI 和气候因子具有显著的小波一致性 作为一个分异因子, 海拔高度在青藏高原东南缘的纵向岭谷区对 NDVI 有着显著影响 通过这一研究发现, 小波变换是研究 NDVI 与影响因素之间多尺度关系的一个有力途径 关键词 :NDVI; 环境因子 ; 尺度依存 ; 小波变换 ; 青藏高原