Synopsis. Index Terms AVHRR, NDVI, time series, signal processing, multi-year, phenology, harmonic series, Fourier analysis, spline, polynomial fit.

Transcription

1 Synopsis Extracting Phenological Signals from Multi-Year AVHRR NDVI Time Series: Framework for Applying High-Order Annual Splines with Roughness Damping John F. Hermance, Robert W. Jacob, Bethany A. Bradley and John F. Mustard Department of Geological Sciences, Brown University, Providence, RI Contact Information: Abstract To better understand how terrestrial vegetative ecosystems respond to climate and/or anthropogenic effects, the scientific community is increasingly interested in developing methods to employ satellite data to track changes in land surface phenology (e.g., timing and rate of green-up, amplitude and duration of growing season, timing and rate of senescence of plant classes). By increasing the inherent resolution of signal extraction procedures, while minimizing the effects of cloud cover and prolonged data gaps, such tools can significantly improve land cover classification and land cover change monitoring on multiple scales. This report describes an intuitive approach for tracking the intra-annual details and inter-annual variability of multiyear time series employing a sequence of annual high order polynomial splines (up to 14th order), stabilized by minimizing model roughness, and weighted to fit the upper data envelope to minimize cloud cover bias. The algorithm is tested using a multi-year period-of-record ( ) for three very different classes of vegetation stable agriculture, high elevation montane shrubland and semi-arid grassland with high inter-annual variability. The results accurately track both short and long-term land surface phenology, and illustrate a robust potential for extracting temporal and spatial detail from a variety of satellite-based, multi-year vegetation signals. Index Terms AVHRR, NDVI, time series, signal processing, multi-year, phenology, harmonic series, Fourier analysis, spline, polynomial fit

2 I. INTRODUCTION AND OVERVIEW In the last few years, the scientific community has shown increasing interest in developing computational tools for extracting subtle signatures of the behavior of vegetation on the earth s surface from multi-channel reflectance data recorded by earth-orbiting satellites (e.g., Azzali and Menenti, 2000; Roerink et al., 2000; Moody and Johnson, 2001; Jakubauskas et al., 2001, 2002a, 2002b; Jonsson and Eklundh; 2002, 2004; Kastens et al. 2003; Zhang et al., 2003; Chen et al., 2004; Geerken et al., 2005; Beck et al., 2006; Fisher et al., 2006). Since these observations provide a common base of self-consistent, long term time series, from local to global scales, the analysis of such data provides significant insight into the response of vegetation to short term and long term environmental forcing effects, both natural and anthropogenic. Since different plant species tend to respond differently to fluctuations of environmental factors, their phenology can be used on a site-by-site basis to identify, or discriminate among, particular land cover types, and monitor their response to typical climate and weather conditions, as well as their response to anomalous conditions of extreme drought, storms and wildfire. In other words, the nature of inter-annual (i.e. year-to-year) fluctuations in intra-annual (i.e. seasonal) variations may provide important information for identifying and discriminating among vegetation communities. The signature vegetation behaviors the phenology of specific sites over multiple years, with appropriate validation can then be used to formally classify patterns of land cover usage; and consequently to better monitor short term and long term land cover change on local, regional and global scales. Clearly, the more accurately one can track subtle variations in the timing and amplitude of plant productivity at a site, the more effective will be techniques for identification of, and discrimination among, vegetation communities. II. IMPLEMENTING A NEW SIGNAL EXTRACTION ALGORITHM A. Modeling Inter-Annual Fluctuations with High Order Splines While, in principle, either Gaussian forms (Jonsson and Eklundh; 2002, 2004) or logistic forms (Zhang et al., 2003; Beck et al., 2006) could be used as representative functionals over sub-annual intervals, we elect to employ an annual sequence of high order splines as the basis functions for our inter-annual model. Here, for a time base running from, t < we have L = 7 subintervals or years. At the join between subinterval (the knots of the spline) we explicitly enforce the condition that the value of the spline functionals and their first and second derivatives are continuous. To specifically accommodate the character of NDVI data, recognizing the intrinsic instability of high order polynomials when fitting noisy data, particularly when there are significant intervals of missing values, we need to regularize the stability of the resultant functionals. This is done using three key elements. First, we begin by computing a robust initial starting model. Next, we regularize our optimization procedure by minimizing model roughness (which is to say we apply roughness damping ). To this end, we define the total local model roughness as the sum-of-the-squared-values of the second derivative of the model function over a pre-defined - 2 -

3 interval (or intervals) of interest (Hermance, 2006). For a general function f () t, over the local time interval i start to i, this can be expressed numerically as end i end Roughness = ( ) i start 2 2 d t t pred i Minimizing this parameter with regard to the respective model parameters is invoked as a complementary minimization criterion during our least-squares optimization. This condition can be preferentially weighted more strongly during data gaps and during intervals when one would expect minimal fluctuations in NDVI values, such as during the dormant winter season. Finally, due to cloud cover and other atmospheric obscurations, the procedure is asymmetrically biased to preferentially fit the upper envelope of observed data values. This is done using residuals between the original observed NDVI values and those predicted by an initial starting model, as computed for the i-th time sample as di () = d () i d () i obs where d () i is the observed data value for i-th time sample and d ( i) is the data value obs predicted by the (prior) model, respectively. We weight the corresponding observed data value by assigning a Gaussian-type value of the form pred ( ) Weight exp sign( d / w) 2 to the diagonal elements of a weighting matrix, where sign is an operator that has a value of - 1 if d is negative, or has value of +1 if d is positive; w defines the width of the Gaussian operator; a typical value might be w = 0.01 NDVI units. B. Summary of Procedure 1) The Base Average Annual Model: As the initial step in our procedure, we construct the average annual phenological behavior for a site from multiyear time series using a nonorthogonal harmonic series with a non-linear trend. If a more refined average annual model is required, we have the option to apply roughness damping during data gaps and during intervals when one would expect minimal fluctuations in NDVI values, such as during the dormant winter season. Finally, due to cloud cover and other atmospheric obscurations, we have the option to asymmetrically bias the procedure to preferentially fit the upper envelope of observed data values (cf. Sellers et al., 1994; Jonsson and Eklund, 2002). 2) Inter-Annual High Order Spline Model: As we proceed with computing the interannual model a representation of the seasonal and year-to-year fluctuation of vegetation we apply a multistage recursive weighted least squares fit to the data using a set of annual higher-order splines with roughness damping during intervals when fluctuations in 2 pred - 3 -

4 Synopsis: Extracting Phenological Signals from NDVI with High-Order Splines photosynthetic activity are expected to be minimal. In order to bias the predicted model values toward local maxima, the algorithm uses asymmetric weighting of the residuals as described above, such that high data values are up-weighted and low data values are downweighted, tending to detect the upper envelope of observed data values. For some data types those with regular phenologies and slow, single cycle intra-annual variability lower order, computationally efficient splines are quite adequate for representing the data. For other data types those with greater inter-annual variability, perhaps characterized by abrupt green-up and senescence and/or several growth cycles in a season higher order splines may be required, with a concomitant penalty in computer time. We typically use from 8th to 14th order annual splines. Whereby such high order polynomials may be intrinsically unstable for some NDVI data, particularly when there are significant data gaps, the splined version of this application is quite stable due to the explicit continuity conditions on the functional values and their 1st and 2nd derivatives across knots (i.e., across contiguous years). Also of help in the stabilization is that we have filled data gaps with values predicted by an initial average annual model; and finally, we apply an expectation of minimum model roughness during winter dormancy. III. APPLICATION OF THE PROCEDURE A. The Study Area We illustrate these techniques using 7 years of weekly and biweekly AVHRR NDVI data from the NOAA-14 and NOAA-16 satellites for the 150 x 150 km study area of the Great Basin in west-central Nevada shown in Figure 1. Figure 1. Location map for our study area in west central Nevada, along with the names of adjacent states. Also indicated is the boundary of the Great Basin. -4-

5 B. Representative Vegetation Classes Used in this Study We apply our computational techniques to the three classes of vegetation shown in Figure 2: stable agricultural areas, mountain top (montane) shrublands, and semi-arid grasslands, respectively. Figure 2. Original NDVI time series from 3 classes of vegetation types. Top panel: Stable agricultural. Middle panel: Montane shrubland. Bottom panel: Invasive grasses (cheatgrass). The vertical gray band at yr denotes a ubiquitous data gap for this area. The top panel in Figure 2 shows the median value of 7 year time series of NDVI values simultaneously sampled at 20 widely distributed sites in areas of active cultivation (the Lovelock and Fallon, NV, agricultural districts). In selecting the original time series for the stable agricultural sites, 22,500 time series (from sites in our study area) were - 5 -

6 screened for minimum statistical variation within respective growing seasons and between years. The middle panel in Figure 2 shows the median time series for montane shrubland data from 44 contiguous sites in the Destoya Mountains, a relatively high elevation mountain range in our study area. The bottom panel in Figure 2 shows the characteristic time series for an invasive annual grassland species (cheatgrass or Bromus tectorum), which is tending to replace more productive native semi-arid grass species in the Great Basin. The cheatgrass sample is taken from 12 contiguous sites in the Type Area A described by Bradley and Mustard (2005), and is identified by on-site inspections. Due to the singular, amplified response of cheatgrass to rainfall, Bradley and Mustard (2005) use its strong inter-annual variability as a synoptic tool in classifying land cover. A principal reason for developing the present algorithms was to better characterize the unique signature of vegetation communities such as these. Here, for purposes of assessing our algorithm, we have tended to enhance the intrinsic inter-annual variability of cheatgrass due to variations in rainfall by electing to construct a time series of the maximum value of simultaneous data from the 12 adjacent sites

7 C. Results: Tracking Phenology using an Inter-Annual Spline Model The fit of the inter-annual spline model to the observed data for the three type vegetation classes is shown in Figure 3. Figure 3. Inter-annual model fits for the three classes of vegetation. The phenology of the stable agricultural sites shows very little year-to-year difference in seasonal cycles. This is largely due to intense irrigation and cultivation schedules. Sites without externally imposed controls, for example the montane shrubland in the middle panel of Figure 3, as well as the cheatgrass data in the bottom panel, show much stronger inter-annual variation

8 IV. DISCUSSION AND CONCLUSIONS The basic premise of this study is that by having better tools to identify, monitor and discriminate among plant communities, one will be better able to assess land cover response to forcing from environmental and climate factors. We are particularly interested in increasing the time resolution of procedures for extracting NDVI signals from multi-year time series, with a view toward using these phenological responses to identify and classify land cover, and to monitor land cover change on local and regional scales. The detail with which the inter-annual spline model can track representative phenological behavior is illustrated in Figure 4 which is one year s data, taken from the 7 year time series for cheatgrass in Figure 3. This example illustrates the enhanced green-up response of this invasive grassland species to an interval of enhanced rainfall in As highlighted on the figure, many attributes of the phenological signal are readily identified. Figure 4. Representative phenological markers that the inter-annual model algorithm readily recovers from an NDVI "signal". The inter-annual model tracks the phenological response of cheatgrass to a significant interval of precipitation in the Great Basin, NV, during The response of cheatgrass is an extreme example of variable phenology that is not only apparent in its strong inter-annual variability, but also apparent in the rapid fluctuations that can occur in any given year (compare different years in Figure 3). Other vegetation communities in our study area tend to be better behaved, with stronger periodic seasonal components and less pronounced impulsive aperiodic transient behavior. Based on the ability of our algorithm to track the intra-annual and inter-annual variability of cheatgrass so well, we conclude that our procedure should adapt quite well to representing a variety of characteristic phenological time scales

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