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In: Wagner W., Székely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Voi. XXXVIII, Part 7B 
buffer of surrounding area is covered by 1298 AVHRR pixel 
on a 8 km spatial resolution and each of these GIMMS time 
series consist of 612 data points, covering 25 !4 years from 
July 1981 until December 2006 with a scan frequency of 2 
data / month. This work was done with the updated GIMMS 
data that were release in 2007. 
3.2 Decomposition of NDVI time series 
Provided that the NDVI value represents the photosynthetic 
active vegetation amount, the dynamics of vegetation cover 
can be characterised by the temporal behaviour of the NDVI 
value. Thus, NDVI values for a specific pixel over the period 
from July 1981 to December 2006 will be considered as a time 
series. 
For an analysis of the statistical characteristics - mean, 
variance and auto-correlation function (acf) - a given time 
series needs to be stationary. The GIMMS NDVI doesn't 
fulfill this constraint, as they contain significant cyclic 
(seasonal and/or multi-annual) components. To derive 
stationary time series, each NDVI series x t is decomposed into 
the following components: 
(a) long-term mean x, 
(b) Cyclical Component c, 
(c) Seasonal Component s, 
(d) Irregular Component i, 
Where the cyclical component consists of a (linear) trend m t 
and long term (multi-annual) anomalies a t . The latter model 
variations that last over more than 1 year, or that are even not 
periodically. Variations with periodicities shorter than 1 year 
are modelled within the seasonal component. The last 
component i t describes short term anomalies and allows 
therefore an interpretation of alterations of the variance of the 
NDVI signal. It is supposed in the context of this work that all 
components superimpose, so as to the time series can be 
written as: 
x,=m,+a t + s,+ i, (1) 
W(f—f) = weighting function for the 
observed time series 
the filtered estimation of the spectral density X T {f) fora 
time series with finite length. The Fourier transform Y(f) of a 
filtered signal then results from 
(3) 
where H(f) = filter transfer function 
A moving average (MA), with window size 24, could serve as 
a simple realisation of such a low-pass filter. As MA-filter 
show significant side lobes in their step response, these kind 
of filter produce a leakage for the filtered time series. 
Furthermore, the negative values around the odd side lobes 
result in a phase shift of 180° for the filtered signal in these 
parts. Both disadvantages can be avoided by using a Raised 
Cosine filter as FIR. 
Figure 3. normalised step response of the used FIR filter, due 
to a sampling rate of 2 / month the value 24 
represents a frequency of 1 year -1 
3.4 Determination of the Seasonal Component s t 
extracting significant frequencies 
Seasonal variations in the NDVI signal can be modelled with 
the phase mean or stack method in the time domain. This 
algorithm estimates the seasonal component by calculating 
mean values for each observation date of a year as given in (4) 
This decomposition of a NDVI time series aims the 
differentiation of long-term and seasonal dynamics as well as 
an interpretation of alterations from these periodical 
behaviour. 
3.3 Determination of the Cyclical Component c t 
As c t models long-term components of the NDVI signal, it can 
be separated by filtering the time series with a low-pass filter. 
To design an appropriate filter and to apply filtering 
efficiently, the time series was transformed into the frequency 
domain with a Discrete Fourier Transformation (DFT). 
According to (Meier and Keller 1990) describes 
X T (f)=S X(f)W(f-f)df (2) 
where X(f) = Fourier transform 
where r = date of observation within the year (r = 1, ..., 24) 
n = year of observation (n = 1, ..., 25 - [1981 - 2006]) 
\ x r i n )~ c r( n )\ = time series adjusted for c t 
s t belongs to r) 
r \0,else ) 
While this approach gives direct access to time related 
information such as the date of the annual max./min NDVI 
value or the temporal run of the NDVI curve, an analysis in 
the frequency domain provides information about the 
frequencies / periodicities of NDVI dynamics. But due to the 
great number of data points also the no. of frequencies goes 
usually beyond the scope of interpretation for longer time 
series. Reducing the number of frequencies should preserve 
the information (Z of power for all frequencies) of a time 
series as much as possible.