C S I R O L A N D a nd WAT E R
Decomposition of Vegetation Cover into Woody
and Herbaceous Components Using AVHRR
NDVI Time Series
Hua Lu, Michael R. Raupach and Tim R. McVicar
CSIRO Land and Water, Canberra
Technical Report 35/01, September 2001
National Land & Water Resources Audit
A p r o g r a m o f t h e N a t u r a l H e r i t a g e T r u s t
Decomposition of Vegetation Cover intoWoody and Herbaceous Components
Using AVHRR NDVI Time Series
Hua Lu, Michael R. Raupach and Tim R. McVicar
CSIRO Land and Water Technical Report 35/01September 2001
ii
© 2001 CSIRO Australia, All Rights Reserved
This work is copyright. It may be reproduced in whole or in part for study, research ortraining purposes subject to the inclusion of an acknowledgement of the source. Reproductionfor commercial usage or sale purpose requires written permission of CSIRO Australia
Authors:
Hua Lu, Michael R. Raupach and Tim R. McVicar (2001).
E-mail: [email protected]
Phone: 61-2-6246-5923
For bibliographic purposes, this document may be cited as:
Lu, H., Michael R. Raupach and Tim R. McVicar (2001), Decomposition of VegetationCover into Woody and Herbaceous Components Using AVHRR NDVI Time Series,Technical Report 35/01, CSIRO Land and Water, PO Box 1666, Canberra, 2601, Australia
A PDF version is available athttp://www.clw.csiro.au/publications/technical2001/tr35-01
Copyright
© 2001 CSIRO Land and Water.To the extent permitted by law, all rights are reserved and no part of this publication covered by copyright maybe reproduced or copied in any form or by any means except with the written permission of CSIRO Land andWater.
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To the extent permitted by law, CSIRO Land and Water (including its employees and consultants) excludes allliability to any person for any consequences, including but not limited to all losses, damages, costs, expensesand any other compensation, arising directly or indirectly from using this publication (in part or in whole) andany information or material contained in it.
iii
Contents
Abstract .............................................................................................................................................. iv
1. Introduction................................................................................................................................. 1
2. Data Description ......................................................................................................................... 5
2.1 GAC Pathfinder AVHRR NDVI............................................................................................. 5
2.2 Ground Measurements ............................................................................................................ 7
3. Partitioning Vegetation Indices................................................................................................... 8
4. Converting Vegetation Indices to Biophysical ......................................................................... 13
5. Maps of Continental Fraction Vegetation Cover and LAI........................................................ 21
6. Discussions and Conclusions .................................................................................................... 27
Acknowledgements........................................................................................................................... 29
References ......................................................................................................................................... 30
Appendix: Methods of Time Series Decomposition and STL.......................................................... 35
iv
Abstract
A robust model is proposed to separate Normalised Difference Vegetation Index (NDVI) time
series data into woody and herbaceous NDVI using time series decomposition. The model is
capable of reducing the transient, aberrant behaviour of NDVI due to sensor errors or
atmospheric contamination and estimating temporally varying woody and herbaceous NDVI.
In this study, the separated NDVIs are used to estimate annual averaged woody cover and
monthly averaged herbaceous vegetation covers using Pathfinder AVHRR Land (PAL)
Global Area Composite (GAC) Advanced Very High Resolution Radiometer (AVHRR)
NDVI data from 1981-1994 for Australia. Empirical relationships between woody NDVI and
ground-based measurements of leaf area index (LAI) and foliage projective cover (FPC) are
derived and compared with existing empirical relationships. The new empirical relationships
are critically reviewed in relation to theoretical background and measurements. Finally, the
woody cover map is compared with a high resolution woody cover map derived from
LANDSAT Thematic Mapper for a 209,310 km2 area located in northern east Australia.
Note that the methodology and equations derived in this report update those described in a
previous technical report (Lu et al., 2001).
1. Introduction
Several parameters are commonly used to describe the structural characteristics of land
surface vegetation distributions. These vegetation parameters include leaf area index (LAI),
fraction vegetation cover (fc), foliage projective cover (FPC), and the fraction of incident
photosynthetically active radiation (PAR) absorbed by the green vegetation (fPAR). LAI is the
square meter of green leaf per square meter of ground. fc is the fractional area of the
vegetation canopy occupying a given land area. FPC is the fraction of the surface covered by
one or more layers of photosynthetic tissue vertically above that surface. For woody
vegetation, FPC is normally about 80 - 90% of the value of fc as stems and branches are
excluded in FPC measurements. fPAR is similar to FPC but not necessarily identical due to
differences in leaf and canopy geometry.
Spatial and temporal distributions of above vegetation parameters are fundamental to
many aspects of environmental science, global change detection and resource management.
The hydrological cycle, ecological health, exchanges of energy at the land surfaces and the
residence time of carbon in the terrestrial ecosystems are sensitive to vegetation structure and
in particular the partition of LAI between woody cover and herbaceous cover, such as grass,
pasture or crops. Consequently, ecosystem process, biosphere-atmosphere transfer, and
carbon accounting models all require LAI, fc, or FPC of woody and herbaceous cover as
separate inputs. Traditionally, ground observations of vegetation characteristics have provided
information concerning specific plants over a limited spatial area. Remotely sensed data
provides the means to measure broad-scale vegetation at the ecosystem level. Efforts in
vegetation mapping at continental and global scales using remotely sensed data have
increased in the recent years due to the increasing demand for up-to date information on the
Earth's land cover with respect to climate and ecosystem changes. These methods rely on the
availability of multi-temporal remotely sensed data, and the development of time series
analysis techniques.
The Normalised Difference Vegetation Index (NDVI) can be calculated from data
acquired by the Advanced Very High Resolution Radiometer (AVHRR) sensor on board
National Oceanic and Atmospheric Administration (NOAA) series of satellite. AVHRR
NDVI has provided a powerful tool to monitor the phenology of ecosystems at continental
and global scales. The NDVI = (NIR -RED)/(NIR+RED), or simple ratio SR = NIR/red,
where RED and NIR are spectral radiance measurements in the red (AVHRR Channel 1: 0.58
2
- 0.68 µm) and near-infrared (AVHRR Channel 2: 0.725 - 1.1 µm) spectral ranges,
respectively. The NDVI and SR can be inter-converted as
NDVI1
NDVI1SRor
1SR
1SRNDVI
−+=
+−= (1)
These vegetation indices are measures of the relative reflectances of the surface in the red
and NIR channels. The red reflectance is determined, in part, by the absorption by chlorophyll
in the red wavelengths, which increases with leaf chlorophyll density, while the NIR
reflectance increases with green leaf density (Tucker et al., 1985). Thus, NDVI and SR are
measures of the amount of active photosynthetic biomass, and correlates well with
biophysical parameters, such as: green leaf biomass, fraction of green vegetation cover, LAI,
total dry matter accumulation and annual net primary productivity (Asrar et al., 1985; Sellers,
1985). Accordingly, NDVI and SR have been widely used to monitor vegetation condition
and production in different environmental situations (Justice et al. 1985; Prince, 1991).
Estimating woody and herbaceous cover and LAI from NDVI is not straightforward,
because remotely sensed NDVI contains combined information affected by the following
causes:
1. Phenology: Different plants have different developmental phenologies, which influence
the temporal patterns of NDVI over single seasons or several years. Therefore, vegetation
type determines the characteristics of seasonal NDVI curve;
2. Growth: A range of relatively stable climate and environmental factors promote or
restrict vegetation growth. The climate factors include precipitation, temperature, and
radiation. The environmental factors include soil conditions, land use and management. NDVI
and vegetation patterns exhibit large interannual variability due to interannual climatic
fluctuations;
3. Disturbance: Short term, random and sometimes unrecoverable climate or environmental
disturbances impose rapid fluctuations on the NDVI signal. Climate induced disturbances
include drought, flood, freezing hazard, and violent wind. The environmental disturbances
include fire, land clearance and dramatic change in landuse and management;
4. Sensor conditions: Viewing angle, different atmospheric path lengths for different pixels,
instability of sensor response, satellites orbit, and instrument degradation affect the remotely
sensed signals; and
3
5. Signal contamination: Signals can be contaminated by clouds, aerosols, water vapour,
and background soil colour.
Of these causes, the first three relate to changes in the biophysical properties of the
surface which NDVI seeks to measure, while the last two represent extraneous contamination
(from the viewpoint of vegetation study). Because vegetation grows gradually, NDVI should
change (increase or decrease) gradually. Any sudden change of the NDVI time series (except
harvest and damage) is related to contamination. For analysis of land surface vegetation, the
factors listed in items 4 and 5 represent noise and need to be filtered from the signal. Short
term disturbances listed in item 3 can be important in many other applications but are not of
major concern for our primary goal here which is to assess the seasonal partition between long
term woody and herbaceous cover. If the noise and short term disturbance effects can be
properly filtered, the remaining NDVI should only contain information about phenology and
growth, together with the long-term consequences of disturbances.
It is logical to use time series analysis to decompose the NDVI into separate long-term
and seasonally varying components, which can be related to woody and herbaceous
vegetation. Several authors have applied time series analysis techniques to NDVI data to
characterize land-cover patterns or to relate NDVI to climate variables. The most common
techniques include Principal Component Analysis (PCA), Fast Fourier Transform (FFT) and
wavelet decomposition. Eastman and Fulk (1993) identified seasonal trends, satellite, and
orbital changes effects using PCA. To map vegetation type and changes, PCA has been used
extensively (Townshend et al., 1987; Turcotte et al., 1993; Lambin and Strahler, 1993;
Benedetti et al., 1994). Studies of inter-annual vegetation variability, and its relation to El
Ni\tilde{\rm n}o/Southern Oscillation, were performed using wavelet decomposition
(Anyamba and Eastman, 1996; Li and Kafatos, 2000). Azzali and Menenti (2000) used FFT to
decompose NDVI time series into a series of periodic components, and then related the
amplitudes and phases of the periodic functions to aridity and vegetation types. In practice,
using the above techniques with long time series of remotely sensed data introduces problems
with handling such vast time series of spatially dense imagery. Additionally it is difficult to
extract the essential information from resulting multiple components to characterize the land-
cover patterns, especially when the resulting components have vague physical meanings.
To extract information about the partition between woody and herbaceous cover,
Roderick et al. (1999) proposed a NDVI baseline to evergreen vegetation cover using a
moving average method. Their model is based on the assumption that in warm, low rainfall
4
areas, such as most of Australia, the trees and shrubs are mostly evergreen whilst the
herbaceous vegetation only seasonally green-up following rainfall or on an annual
phenological cycle. The total green vegetation NDVI at a given site should therefore oscillate
between a minimum value determined by the evergreen cover and a maximum value
determined by the seasonal peak growth of herbaceous vegetation. Their model is attractive
because of its robustness and simplicity. However, a simple moving average can only handle
time series which have no missing values and is noise free; this is rarely the case for most
NDVI. For time series NDVI which contains noise, the trend and seasonal components
produced by a moving average can be distorted by transient, aberrant behaviour in the data.
Danaher et al. (1991) proposed another method to estimate woody cover. Their analysis
involves calculating the mean and coefficient of variation of an NDVI time series for every
pixel. These two values were classified to create a woody/herbaceous mask and woody cover
was only calculated within the woody areas. However, at the spatial resolution of 1 km to 8
km at which AVHRR NDVI is sensed, mixed vegetation exists within a pixel. The
assumption of a single vegetation mosaic (either woody vegetation only or herbaceous
vegetation only) may not be appropriate.
This paper develops and applies a method for determining the partition between woody
and herbaceous cover from AVHRR NDVI, based essentially on the method proposed by
Roderick et al. (1999). The data set is the 10-day composite Pathfinder AVHRR NDVI time
series data (1981-1994) covering Australia. The plan of the paper is:
1. The method of Roderick et al. (1999) is further developed by using an advanced
seasonal-trend decomposition technique called STL (Cleveland et al., 1990). STL is used to
deal with the noise-containing NDVI and decompose it into trend, seasonal and irregular
components. The partition of woody and herbaceous vegetation indices is modelled by
combining baseline of the time series, magnitude of trend and amplitude of seasonal
component. See Section 3.
2. Empirical relationships between NDVI (or SR) and FPC (or LAI) are critically reviewed
in relation to the theoretical background. New empirical relationships between the woody
components of vegetation indices (NDVI and SR) and biophysical parameters (LAI and FPC)
are derived using data from the Murray-Darling Basin and from Queensland. An analysis is
performed to reveal the preferred linear relationships. See Section 4.
5
3. We estimate the annual average fraction of woody cover and the monthly average
herbaceous cover using empirical relationships between NDVI and vegetation fraction cover
for the Australian continent at 0.05 o spatial resolution. The woody cover map is compared
with a higher spatial resolution (30 m) woody FPC map for part of Queensland, which was
derived by using LANDSAT Thematic Mapper (TM) data. See Section 5.
Prior to discussing these three topics in turn, we briefly introduce the data sets used in
Section 2.
2. Data Description
2.1 GAC Pathfinder AVHRR NDVI
The GAC Pathfinder AVHRR NDVI time series was derived from the AVHRR sensor on
the national Oceanographic and Atmospheric Administration (NOAA) series of
meteorological satellites (NOAA-7, -9, and -11) and made available by the NASA Goddard
Space Flight Distributed Active Archive Centre (GDAAC) (James and Kalluri, 1994). The
spatial resolution is 0.08 degree (approx. 8 km) and the temporal coverage is 10-day
composite from July 1981 through to the middle of 1994. As distributed, the Pathfinder NDVI
is calibrated and corrected for change in sensor calibration, ozone absorption, Rayleigh
scattering and normalised for changes in solar zenith angle.
A subset covering the Australian continent from 1981 to 1994 was obtained by the
CSIRO Earth Observation Centre (EOC) from NASA. This NDVI data set was further noise
reduced by using a modified Best Index Slope Extraction (BISE) algorithm using a search
window of 6 decad and a NDVI change threshold of 0.1 per decad (Lovell and Graetz, 2000).
The data set was geo-re-registered and re-sampled to 0.05 degree by Damian Barrett at
CSIRO Land and Water using nearest neighbour resampling. Although the time series NDVI
has been greatly improved after these corrections, noticeable errors remain including effects
of volcanic aerosols, water vapour, background soil colour and occasional missing values
(Lovell and Graetz, 2000).
In this study, 460 Pathfinder 10-day composite NDVI images from July 1981 to April
1994 were used. Later images were excluded as large areas of southern Australia had missing
data. The information about the NDVI data set is summarised in Table 1. Recognising the
6
extensive pre-processing detailed above, this data set is the input NDVI data from the
standpoint of the present work.
Table 1: The NDVI time series used in this study (Note: ULHC is upper left hand corner,
BRHC is bottom right hand corner).
Spatial Coverage ULHC: 112o E 10o S
BRHC: 155o E 45oS
Pixel size: 0.05o
Number of columns: 860
Number of rows: 700
Temporal Coverage 10-day composite, middle July 1981 to middle
April 1994, 460 images
Original Data Source NASA Goddard Space Flight Centre
Calibration and Filtering Modified BISE (Lovell and Graetz, 2000)
Geo-re-registered and re-sampled by D. Barrett
Aimed to examine the quality of input NDVI and to test our NDVI decomposition
method described later in Section 3, we extracted the NDVI time series at some selected
locations. Up to 10 NDVI time series from seven major land cover classes from a range of
climate zones across Australia. The seven land cover classes are forest, open forest,
woodland, pasture, unmanaged land, crops, and salt lakes. These sampling locations were
chosen by using the recent Australian national land cover map, produced by the National
Land and Water Resources Audit. The selection process was also guided by authors' local
knowledge. We first visually examined those NDVI time series. It was found that the
magnitude of NDVI was around 0.75 for closed tropical rain-forest, around 0.6 for eucalypt
forest, 0.6-0.75 for crops at their normal peak greenness, 0.1-0.17 for bare soil and stubble
field after harvesting, and close to zero for salt lakes. These figures are comparable to the
synthetic NDVI found using site-specific types of land cover from literature (Wittich and
Hansing, 1995). However, it shows that the BISE filtering using fixed window has removed
the most sharp changes at the cloudy sites but retains the NDVI drops during prolonged
cloudy period. We also noticed that the revised BISE works better for removing sharp
declines in NDVI but retained most of sharp increases in NDVI due to change of view angle.
7
2.2 Ground Measurements
Point Measurements of Leaf Area Index:
The LAI measurements were obtained from a transect across the centre of the Murray-
Darling Basin (MDB) in south eastern Australia. The climate ranges from wet temperate in
eastern part to arid in western part of the MDB. LAI measurements were made on 29
relatively homogeneous vegetation sites, including closed forest, open forest, woodland and
open shrubland. The sites were accurately located by using a Trimble Pathfinder GPS. The
dominant vegetation genus was Eucalyptus. The measurements were made from March 13-21,
1990, details are found in McVicar et al. (1996b). Only LAI from woody vegetation is used
here.
Point Measurements of Foliage Projective Cover:
Overstorey FPC data were kindly provided by the Queensland Department of Natural
Resources. FPC from 73 homogeneous vegetation sites, covering a range of forest and
woodland were sampled using the vertical tube method (Wood, et al., 1996). The majority of
the sites were located between latitudes 18 o and 29 o and longitudes 136 o and 154 o. The
climate ranges from tropical in the north-east to sub-topical and temperate towards the south,
with an semi-arid climate in the west. The survey was performed during the two-year period
from January 1993 to August 1995. FPC for ten points which has same location (lon = 152.83o, lat = -27.428 o) were averaged. In addition, three points are discarded in this study due to
heterogeneous vegetation cover. The locations of those three discarded points are: 1). lon =
145.472oE, lat = 17.454oS, FPC = 92.5%; 2). lon = 143.854oE, lat = 18.549oS, FPC = 52.5%;
and 3). lon = 144.819oE, lat = 18.121oS, FPC = 55%. There are 61 data points used here.
Mapped Estimates of Foliage Projection Cover:
FPC was mapped by the Statewide Landcover and Trees Study (SLATS) project (Kuhnell
et al., 1998) using 88 LANDSAT TM scenes (30 metre spatial resolution) covering all of
Queensland. It represents the woody vegetation cover as it existed in 1991. The SLATS FPC
data were estimated using a wooded/non-wooded mask and a feature space classification
method from 1990/1991 imagery. Within the wooded mask FPC was calculated using a
multiple second-order polynomial regression with NDVI and TM band 5 as input variables. A
full description is given by Kuhnell et al.(1998). Ten scenes covering an area of 209,310 km2
8
in the north-eastern Queensland were acquired from the SLATS project team and used for
comparison in this study.
3. Partitioning Vegetation Indices
Estimating averaged annual woody canopy cover and monthly herbaceous vegetation
cover involves extracting meaningful information from the time series of vegetation indices.
Conceptually, the temporal variations of woody and herbaceous vegetation are different. The
herbaceous vegetation has a seasonal variation representing its annual growth and senescence
with its distinct repeating phenological stages, together with an interannual variation
determined by a relatively smoothly varying curve representing the influence of interannual
climatic variability on growth. In addition, irregular changes may occur at much shorter time
frames representing the short term effects of irregular rainfall. For Australian conditions,
where the evergreen Eucalyptus are the dominant species, woody vegetation is assumed to
have negligible seasonal variation. However, only a long-term phenological trends is
assumed, associated with climatic variability and constrained by local soil and geological
conditions.
Given NDVI time series can be converted to SR time series using equation (1), we use
two-step model to achieve the woody/herbaceous vegetation indices partition based on the
above differences in time varying behaviours of woody and herbaceous vegetation.
Step 1: Seasonal-Trend Decomposition:
NDVI and SR time series can be decomposed into additive trend (Ti), seasonal (Si), and
irregular (Ii) components:
NDVIi = Ti + Si + Ii (2)
or
SRi = Ti + Si + Ii (3)
where i = 1 to N denotes the time index, with N the total number of image in the time
series. The trend Ti includes the mean, so that < Ti > = <NDVIi>, <Si> = <Ii> = 0, where the
symbol <> represents taking mean value. From here after, only the procedure for NDVI
partitioning is given in this section, as the same procedure is equally applicable for SR time
series.
9
Figure 1: Example of NDVI woody/herbaceous components separation. Part (a)NDVI decomposition using STL showing the input NDVI, trend, seasonal andirregular components. Part (b) Procedure for extracting woody/herbaceous NDVIfrom trend and seasonal component, where K = minI=1 N S_i, baseline and amplitudeof seasonal component are shown.
10
Accurate estimation of the trend and seasonal components is vital and a robust
decomposition method is needed. Firstly, the method needs to be computationally efficient
and simple enough to decompose the 284229 time series to cover the whole of Australia.
Secondly, the method must deal with outliers and missing values (see Section 2). A technique
called STL (Seasonal-Trend decomposition procedure based on Loess) is employed in this
study. The details and justifications of using STL are given in the Appendix. Figure 1, part (a)
shows an example of a NDVI time series and above three components decomposed by STL.
Step 2: Extracting Woody/Herbaceous signals from Trend and Seasonal Component:
The trend Ti at a given time i is the total vegetation trend for a given location. It combines
two types of trend information, one for woody and one for herbaceous vegetation. We can
write
Ti = T wi + T gi (4)
where Twi and Tgi are the woody component trend and herbaceous (or grass) component
trend, respectively. The larger the variation in the seasonal component, the higher the
proportion of herbaceous vegetation. The larger the minimum value of NDVI, the higher the
proportion of woody vegetation. The variation of the seasonal component can be measured by
its amplitude. The minimum value of the input NDVI can be a poor indicator of physical
woody NDVI as it could be erroneous due to cloud contamination. Similar to Roderick et al.
(1999), the NDVI baseline (Bi) is a better measure, which can be estimated by shifting the
total trend Ti by a constant K = |N
i 1min
=Si|, the absolute value of minimum seasonal component
for two consecutive years when the time i is within that two years, written as:
Bi = Ti – K (5)
By assuming that the ratio between Twi and Tgi is equal to the ratio between the baseline
and the amplitude of the seasonal component, we write
i
i
gi
wi
BA
B
T
T
08.0−= (6)
where Bi is NDVI baseline, which can be calculated using equation (5) and A is the
amplitude of NDVI seasonal component. It is observed from the input NDVI time series that
the closed forest sites often have, a seasonal amplitude of around 0.08, therefore, 0.08 Bi is
used in equation(6) to account for this effect. The effect is relative large for the area where Bi
11
Figure 2: NDVI decomposition into woody NDVI and herbaceous NDVI for differentvegetation types. (a) Closed Rainforest in Queensland near Cairns (145.75o E, 17.65o
S); (b) Open Eucalyptus Forest in the eastern MDB (149.30 oE, 30.70 oS); (c)Improved Pasture near the centre of the MDB (149.63 oE, 31.65 o S); (d) UnmanagedShrubland located in the semi-arid MDB (142.15 oE, 31.75 o S); (e) Winter growingWheat in south-west Australian wheat belt (116.60o E, 30.30o S); and (f) Salt Lake inarid Western Australia (128.55o E, 22.55o S).
12
(therefore, woody vegetation) is large, but small for crops and pastures where Bi is small.
Combining equations(4), (5), and (6), the woody NDVI component can be estimated as:
AKT
KTTTNDVI
i
iiwiwi +−
−==
)(92.0
)((7)
Because
AKT
KTATTTT
i
iiwiiwi +−
−−=−=
)(92.0
)(08.0(8)
the herbaceous component can be estimated by:
iiii
iiiiigigi IS
AKT
KTATISTNDVI ωω ++
+−−−
=++=)(92.0
)(08.0(9)
where ωi is the robustness weight at time i. The robustness weights reflect how extreme Ii is.
An outlier in the data that results in a very large | Ii | will have a small or zero weight. ωi is an
output of STL. Mathematical expressions for calculating ωi are found in the Appendix.
Figure 1 (b) shows the procedure of step 2 and some relative variables and parameters.
Typical NDVI data, woody NDVI, and the herbaceous component for different land cover
types are shown in Figure 2. The forest sites have higher woody NDVI and less seasonal
oscillation. In contrast, the wheat site has a near zero woody NDVI, and a distinct herbaceous
component representing crop growth during the winter growing season. Woody NDVI
reduces as the woody vegetation reduces. Near-zero woody and herbaceous NDVI are
obtained for the salt lake, where there is no vegetation. Note that no-vegetation-related
fluctuations contained in the input NDVI time series are no longer observed in either the
woody or herbaceous NDVI components. The model was tested using the NDVI time series
sampled at selected locations mentioned in Section 2.
The main differences between the model developed here and that of Roderick et al.
(1999) is as follows. The model proposed by Roderick et al. (1999) assumes that the woody
cover NDVI tracks along a baseline of the NDVI time series defined by equation (5) and that
the herbaceous (or raingreen) NDVI is the difference between the input NDVI and that
baseline. This causes two possible weaknesses. Firstly, the herbaceous cover can be often
underestimated for those sites dominated by herbaceous vegetation. For instance, for a wheat
site, shown in the Figure 2 (e), if the baseline represents the woody cover, then Tgi would
track above the line with NDVI = 0. This effectively shelfs the Tgi in Figure 2 (e) downward
13
to NDVI = 0 by around 0.1 NDVI value. The consequence is that instead of getting NDVI
around 0.75 at peak greenness , a value around 0.65 is obtained at peak greenness. Secondly,
as their herbaceous NDVI component contains all the information of irregular components,
the resulting herbaceous NDVIs often have sharp fluctuations, which produce non-realistic
sharp variations in vegetation cover. These weaknesses are overcome by the model proposed
in this study. Instead of shifting the trend by a constant, it is shifted by the herbaceous
component trend (Tgi) at time i (equation (8)). For a forest site (Figure 2 (a) and (b)), where
the seasonal amplitude (A) is small and the total trend (Ti) is large, the shift is small. In this
case, the total trend is close to the trend of woody cover. On the other hand, for the wheat and
improved pasture sites (Figure 2 (c) and (e)), where A is large and the trend Ti is small, the
shift is relatively large and the total trend represents more about the trend of wheat or pasture.
In the mean time, low woody components are obtained. For a arid shrubland (Figure 2 (d)),
the woody and herbaceous components are small. For the case of salt lake (Figure 2 (f)), near
zero woody and herbaceous components are obtained. In all the cases in shown Figure 2, the
sharp changes in both separated NDVI components are reduced by the STL, which identifies
and filters outliers using statistical weighting functions.
4. Converting Vegetation Indices to Biophysical
In this section, we show how the partitioned vegetation indices can be used to infer
biophysical parameters.
There are primarily three approaches to estimating biophysical parameters such as fc and
LAI. The most common procedure is to establish an empirical relationship between NDVI (or
SR) and LAI (or fc) by statistically fitting ground-measured LAI values to the corresponding
remotely sensed indices. The major limitation of this empirical approach is the diversity of the
proposed equations. These equations vary not only in mathematical form, but also in their
empirical coefficients, depending primarily on vegetation type and influenced by soil
background colour.
The second approach is to assume that minimum NDVI (or SR) is related to minimum
LAI (or fc) (bare soil) and maximum NDVI (or SR) is related to maximum LAI (or fc) (fully
vegetated area). Several researchers have used this approach to relate NDVI to fc (or fPAR)
(Wittich and Hansing (1995); Gutman and Ignatov (1998); Roderick et al. (1999); Zeng et al.
14
(2000); and references therein). Under this approach with an additional assumption of fc
proportional to NDVI, the simplest formulation for fc can be written as
minmax
min
NDVINDVI
NDVINDVIfc −
−= (10)
Another less common, but probably more accurate approach, involves using bidirectional
reflectance distribution function (BRDF) models. A BRDF model is inverted and biophysical
parameters are estimated using optimisation procedures. Although the latter approach has a
theoretical basis and is potentially applicable to varying surface types, its primary limitation is
the lengthy computation time and difficulty of obtaining the required input parameters of the
model (Qi et al., 2000). In our case, with only NDVI and SR data available, we were limited
to the first approach.
Often a non-linear relationship between LAI and NDVI has been reported (Holben and
Justice, 1980; Choudhury, 1987; Qi et al., 2000). Qi et al. (2000) found that a third-order
polynomial function of LAI-NDVI and a linear LAI-NDVI relationship both fit well at low
vegetation density (LAI < 1.2). The difference between the two equations became significant
at larger LAI where vegetation overlaps vertically and NDVI becomes saturated and
insensitive to the changes of LAI.
The relationship between LAI and SR is less clear. Field data shows a scatter around a
straight line (Choudhury,1987; Holben and Justice,1980). McVicar et al. (1996a,b) used a
linear LAI-SR relationship and derived a set of empirical equations for different vegetation
types using LAI measurements from the MDB. We determined that the LAI estimated using
linear regression (McVicar et al., 1996b) for woody vegetation was too high because
uncorrected NDVI images were used. The NDVI used were about 3 times smaller than
corrected GAC Pathfinder NDVI data used here. We refitted the ground measurements of
McVicar et al. (1996b) against the Pathfinder NDVI data for 11th - 20th of March, 1990, and
found that the coefficients become similar to other studies (see Table 2).
The existing relationships between fc (or fPAR, or FPC) and NDVI (or SR) are subtle and
sometimes controversial. By a theoretical analysis and observations using in-situ spectral
radiometers on sugar beet and wheat, Kumar and Monteith (1981) provided theoretical results
showing that the SR was linearly related to fPAR, the fraction of the absorption of
photosynthetically active radiation intercepted by the canopy. Steven et al. (1983) observed
that the fraction of incident solar radiation is proportional to the logarithm of SR from
15
measurements on sugar beet. Gallo et al. (1985) obtained a polynomial fPAR-SR relationship
for a corn canopy. By a theoretical analysis Asrar et al. (1984) and Sellers (1985) showed that
the fPAR is almost linearly related to NDVI, while fPAR-SR relation is non-linear. Later, Sellers
et al. (1992) suggested that fPAR is near-linearly related to the SR. Huete et al. (1985) noted
that the vegetation indices are dependent upon the degree to which the spectral contribution of
soil can be isolated from the real vegetation signals. At higher vegetation densities they found
the SR to be most useful. They also found a strong correlation between NDVI or SR and
fractional vegetation cover, with a stronger correlation between SR and fractional vegetation
cover. Danaher et al. (1992), using a time series of 44 monthly AVHRR NDVI, adopted a
hybrid classification of mean, variation and skewness of multi-temporal NDVI, and found a
non-linear relationship between field measured woody FPC and long-term mean NDVI.
Several other studies also reported a non-linear relationship between NDVI and fc (or fPAR )
(Myneni and Williams, 1994; Carlson and Ripley, 1997). Duncan et al. (1993) evaluated a
range of greenness spectral vegetation indices, perpendicular vegetation index and brightness
indices, derived from SPOT imagery and found that the NDVI and SR have the highest
correlation with woody shrub cover in a semi-arid area. Choudhury (1987) carried out a
sensitivity analysis using a two-stream approximation to the radiative transfer equation. He
compared those relationships to a variety of observed data from homogeneous vegetated
surfaces of grasses and agricultural crops. He concluded that most of the relationships are
curvilinear and affected by the soil reflectance. The relationships between NDVI and fPAR, and
between SR and fPAR become more linear as the proportion of bare soil reflectance increases.
He also found that changes in leaf optical properties affected SR more than NDVI, suggesting
a preference of NDVI over SR. Using in-situ spectral radiometer data, Wittich and Hansing
(1995) estimate that the residual error of linear regression of fc ∝ NDVI has standard error less
than 10%, which represents the worst case error in fc. They stated that the non-linearity of fc vs
NDVI (and LAI vs SR) is beyond the detectability over a wide range of vegetation type. Table
2 shows some of the empirical relationships which have appeared in the literature. Notes that
none of the above studies have taken into account the amount of shadow in the images.
Another stream of researches has been undertaken to deal with the effect of shadows. In this
approach, the individual band reflectances in the red and near-infrared are first analysed to
infer the proportion of a pixel occupied by sunlit crown, shadows, and sunlit background, then
relate these intermediate variables to the biophysical parameters of interest (Hall et al., 1995).
Again, this approach is data intensive and out the scope of present work
Table 2: Empirical relationships between remotely sensed vegetation index and surface biophysical parameters. FPC, fPAR, and fc are in unitof %.
Relationship Regression Equation Comments References
LAI = f(NDVI) LAI = - 1.57 + 16.78 NDVI - 47.94 NDVI 2 + 55.842 NDVI 3
LAI = - 0.352 + 6.124 NDVI - 15.24NDVI 2 -18.99NDVI 3
LAI = -0.156 + 2.69 NDVI
Woody
Semiarid
Semiarid
This study
Qi et al. (2000)
Asrar et al. (1985)
LAI = f(SR) LAI = -1.51 + 1.17 SR Woody This study.
LAI = -1.47 + 1.22 SR Cropping McVicar et al. (1996a)
LAI = -0.9 + 0.72 SR Pasture McVicar et al. (1996a)
LAI = -1.15 + 0.96 SR Grasses McVicar et al. (1996a)
LAI = -4.65 + 4.2 SR Woody McVicar et al. (1996b)
FPC = f(NDVI) FPC = -12.82 + 112.41 NDVI Woody This study.
fc = f(NDVI) fc = -7.5 + 140 NDVI Pasture Carter et al. (1996)
fPAR = f(NDVI)
fc = 1.333 + 131.877 NDVI
fc = -8.3 + 208 NDVI
fc = -50.5 + 184 NDVI
fc = -18.2 + 179.5 NDVI
fPAR = 136 NDVI
Mixed
Global mixed
Mixed
Mixed
Continent
Darke et al. (1997)
Gutman and Ignatov (1998)
Ormsby et al. (1987)
Wittich and Hansing (1995)
Roderick et al. (1999)
FPC = f(SR)
fc = f(SR)
fPAR = f(SR)
FPC = -10.95 + 51.17 ln( SR)
fc = -44.3 + 22.8 SR
fPAR = -3.53 + 36.9 ln(SR)
Woody
Mixed
Sugar Beet
This study
Ormsby et al. (1987)
Steven et al. (1993)
Point measurements of LAI (McVicar et al., 1996b) and FPC (Wood et al., 1996) are
plotted against long-term averaged NDVI and SR woody components (Figure 3) which are
extracted using the partitioning method described in Section 3.
In Figure 3 (a), the LAI data suggests a non-linear relationship between NDVI and LAI.
It is found a third-order polynomial equation fits the data best. This agrees with Qi et al.
(2000), though the equation derived here estimates larger LAI comparing with other studies
(Qi et al., (2000); Asrar et al., (1985)) for the medium LAI range (LAI = 1 - 2.5). In Figure 4
Figure 3: Point measured LAI (McVicar et al., 1996b) and FPC (Wood et al., 1996)plotted against long-term averaged NDVI and SR woody components. Regressionequations are given for each case.
18
(b), a good linear relationship between LAI and SR is shown. The new equation compares
well with previous studies on other types of vegetation (e.g. crop and pasture, McVicar et al.,
1996a). In Figure 3 (c), a linear relationship between FPC and NDVI is observed. Compared
with existing equations, it predicts slightly smaller FPC at higher cover end. This is consistent
as FPC is normally smaller than fc as stems and branches are excluded in FPC measurements.
In Figure 3 (d), the data shows a non-linear relationship between FPC and SR. Our best fit for
this case is FPC proportional to ln(SR). However, shown in Figure 3 (a) and (d), those two
non-linear relationships can be reasonable replaced by a linear relationship between LAI (or
FPC) andNDVI
NDVI
−+
1
1(or
1
1
+−
SR
SR).
The vegetation indices partitioning model, described in Section 3, (Equations (2) and (3)),
assumes that NDVI and SR are additive. This assumption implies that the remotely sensed
data can be linearly related to one or more biophysical parameters we are interested in.
Although Figure 3 provides some new relationships between LAI (and FPC) and NDVI (and
SR) with two of them linear, little conclusion can be drawn about which linear relationship is
superior than another. The following analysis provides a means to find out the best suitable
linear relationships. Assuming that one of linear relationship listed in Table 3 does apply, our
selection would fall among the following four possibilities.
(a) LAI-NDVI relationship:
LAI = a1NDVI + b1 (11)
Assume LAI = 0 when NDVI = 0.15, we have b1 = -0.15 a1. From Table 2, a typical value
of a1 is around 2.7.
(b) LAI-SR relationship:
LAI = a2SR + b2 (12)
Assume LAI = 0 when NDVI = 0.15, therefore SR = 1.35, we have b2 = -1.35 a2. From
Table 2, a typical value of a2 is around 1.
19
(c) fc-NDVI relationship:
fc = a3 NDVI + b 3 (13)
Assume fc = 0 when NDVI = 0.15, we have b3 = -0.15 a3. From 2, a typical value of a3 is
around 150.
(d) fc-SR relationship:
fc = a4SR + b4 (14)
Assume fc = 0 when NDVI = 0.15, therefore SR = 1.35, we have b4 = -1.35 a4. From
Figure 4: Four linear relationships between vegetation indices (NDVI and SR) andbiophysical parameters (fc and LAI) and their general interconvertible behaviours. (a)LAI ∝ NDVI; (b) LAI ∝ SR; (c) fc ∝ NDVI; and (d) fc ∝ SR. Point measurements ofLAI and FPC are also shown.
20
Table 2, a typical value of a4 is around 22.8.
Combining equation (1) and Beer's law
)1(100 kLAIc ef −−= (15)
where k is the extinction coefficient, which is assumed to be 0.5 for uniform (or spherical)
foliage angle distribution (Choudhury, 1987), the above four possibilities become inter-
convertible.
Figure 4 shows the general behaviours of above four linear relationships when one of
them is assumed to be true. Point measured LAI and FPC data are also shown in Figure 4. To
be consistent, FPC is converted to fc by assuming fc = 1.15 FPC for Queensland FPC data.
Equation (15) is also used to convert LAI to fc and vice versa for both point measured data
sets.
Figure 4 and its comparison with the point measurements from the MDB and
Queensland, and some other existing measurements (Qi et al., 2000; Choudhury, 1987) leads
to the following conclusions:
1) LAI ∝ NDVI is a poor assumption, it gives a fc < 0.6 and LAI < 2 for NDVI = 0.8, which
significantly under-estimates both LAI and fc for higher density vegetations (Figure 4 (a) and
(b)). In general, it compares poorly with existing measurements at LAI > 1.2;
2) It is unlikely that fc ∝ SR relationship holds true as it compares poorly at low cover end
(Figure 4 (b) and (c)). Compared with other relationships, the fc ∝ SR assumption gives
slower increase in both fc and LAI for low LAI but sharp increase at higher LAI;
3) The assumption of LAI ∝ SR gives slightly larger LAI and fc compared with fc ∝ NDVI.
However these two assumptions show similar shape and are close to each other in four cases,
suggesting they could be in some sense equivalent;
4) Figure 4 (b) shows a clustered measurement points at the LAI value close to zero. It
suggests a log transformation is needed. After we re-plotting Figure 4 (b) in linear (SR)-log
relation, it reveals that SR changes little for LAI = 0.1 to 1, which is similar to Figure 4 (d),
where SR changes little for fc = 0 to 25%. This suggests that NDVI could be more stable than
SR for at lower cover end; and
5) The plot fc vs NDVI gives the least diverse estimations for all four assumptions,
suggesting that fc could be more stable than LAI for our data region.
21
From Figure 4, it can be concluded that fc ∝ NDVI and LAI ∝ SR are both suitable
descriptions of vegetation indices relating to biophysical parameters. When limited soil and
vegetation information is available, a linear relationship between fc and NDVI could be the
most preferable. These finds are consistent with that observed by Hall et al. (1995) and the
conclusion made by Roderick et al. (2000) from first principles. That is, as NDVI only
responds to the sunlit fraction of the canopy, it is generally regarded as a measure of the
fraction of PAR intercepted by green vegetation and hence is related to the potential
productivity of the vegetation. In contrast SR is more closely related to the green leaf area
index of the canopy.
5. Maps of Continental Fraction Vegetation Cover and LAI
According to the analysis in previous section, the following equations are applied to
estimate fc and LAI covering the Australia continent:
fc = 150 NDVI - 22.5 (16)
LAI = 1.11 SR - 1.43 (17)
Note that the equation (16) and (17) update those reported in Section 2.3.2 of the previous
technical report (Lu et al., 2001).
Figure 5 and 6 show the continental percentage green woody cover and monthly averaged
herbaceous cover, respectively. Those maps are produced by averaging woody NDVI across
460 images, averaging herbaceous NDVI for each calendar month and using equation (16).
Equation (17) was used to derive the continental woody LAI map and the equations given in
McVicar et al. (1996a) were used for the estimation of monthly averaged herbaceous LAI.
The LAI maps, which have similar large-scale patterns of fraction cover, are not shown here.
The cover/LAI maps show very similar patterns to the Present Vegetation map of
Australia (AUSLIG, 1990), which was largely based on visual analysis of LANDSAT MSS
1:1,000,000 false colour composites. The cover/LAI maps derived in this study give relative
finer spatial structure of the vegetation compared with the polygon based AUSLIG vegetation
map. The cover/LAI maps also highlight the human-disturbed area, such as the southern-west
wheat belt and southern-east agricultural area in greater detail than the natural vegetation map
of AUSLIG. The monthly averaged herbaceous cover/LAI maps show distinct flush of
vegetation greenness for the tropics during the summer monsoon season. However, in the
tropics in summer, it should be noted that standing water absorbs NIR light, hence causing a
22
reduction of the NDVI. This may explain, in part, why the herbaceous covers estimated for
northern Australia in summer are slightly lower than expected if compared to local in situ
measurements.
Figure 5 and Figure 6 also highlight the winter wheat growth in the south part of the
country. For those intensive agricultural area, the cover/LAI starts to increase in May, peaks
around September, decreases to near zero in November to December, and stays around zero
for next four to six months, This tracks the establishment, crop growth, harvesting, and
fallowing cycle. Very low long-term averaged fraction cover fc and LAI are obtained for
inland arid deserts, suggesting the harsh climate restricts both woody and herbaceous
vegetation growth. However, in some individual months, there were flushes of herbaceous
vegetation growth following rain events (or lateral distribution of surface water into some of
the large inland lakes) seen in the NDVI time series. Note that only long-term (1981-1994)
monthly averaged herbaceous covers are shown here.
Figure 5: Estimates of annually averaged woody fraction cover (%) for Australia.
Figure 6: Estimates of monthly averaged herbaceous fraction cover (%) in percentage for Australia.
24
We compare the percentage woody cover estimated from this study and FPC derived
from SLATS project (see Section 2.3). To be consistent with previous analysis (Section 4),
the woody cover estimated from this study is converted to FPC by using FPC = fc/1.15. We
averaged the SLATS FPC to 5 km to compare with our estimation (Figure 7). Also shown in
Figure 8, a correlation coefficient r 2 = 0.85 is obtained from the scatter plot. The histograms
plot shows a slight skewness toward a positive residual. The means and standard deviations of
the two data sets were similar to woody cover derived by this study having a slightly higher
standard deviation (see Table 3). Despite some uncertainties in both methods for estimating
woody FPC, the spatial pattern and the scatter plot show that the methods results in very
similar outputs. It, however, seems that at the 40-80% range, the woody FPC estimated in this
study is slightly higher than the SLATS FPC. We suggest the following explanations. Firstly,
the SLATS FPC was estimated in the dry seasons in 1990 to 1991. The woody cover
estimated from this study is averaged from 1981 to 1994 across all seasons. Further, between
1980 and 1990 there was considerable amount of land clearance occurring around the upper-
central Burdekin River area (Swift and Skjemstad, 2000) (located at the western side of
Townsville). Also equation (16) may result in a slight over-estimation for higher cover,
therefore, FPC.
Table 3: Comparison between woody cover estimated in this study and SLATS FPC and theirdescriptive statistics.Totoal number of points is 8698, shown in Figure 8.
Mean StandardDeviation
Mean of SLATS FPC (%) 25.54 13.48
Mean of Woody Cover (%) (this study) 26.23 16.08
Correlation coefficient 0.92
SLOPE 1.09
Standard error of SLOPE 0.005
INTERCEPT (%) -1.72
Standard error of INTERCEPT 0.15
25
Figure 7: Comparison between (a) the averaged FPC estimated in this study using AVHRRNDVI (1981-1994) and (b) that estimated by SLATS in 1991 averaged to 5 km spatialresolution. The residual (a) minus (b) is shown in (c). The original (30 m resolution) SLATS datais shown in (d). The nodata holes in original SLATS FPC image are due to water or shadows,which were unclassified.
26
Figure 8: Upper figure: Scatter plot of woody FPC estimated from this study (Figure 7a)versus SLATS data averaged to 5 km (Figure 7b). The histogram of the residual (Figure7c) is shown in lower figure.
27
6. Discussions and Conclusions
A robust method based on time series analysis of remotely sensed data is proposed for
partitioning woody and herbaceous signals from the NDVI and SR derived from the AVHRR
sensor. Temporal filtering has been applied to Pathfinder AVHRR NDVI data to obtain
broad-scale annual averaged woody cover and monthly averaged herbaceous vegetation cover
for Australia. Continental LAI maps have been similarly estimated using SR time series. This
study shows that vegetation indices estimated from time series analysis are more accurate and
stable compared with those extracted from single day or short-term composite measurements.
Although the derived maps have coarse resolution compared with LANDSAT TM-
derived products, they are much cheaper and fit the requirements of continental to global
applications where higher resolution cover (LAI) information is not necessary. Furthermore,
the time series analysis of AVHRR NDVI presented in this study gives monthly information
about herbaceous cover, which LANDSAT-TM based methods cannot provide at the present
due to the high cost of data acquisition, storage and analysis. Despite some uncertainties
behind the empirical relationships between remotely sensed vegetation indices and
biophysical parameters, the satellite-derived cover/LAI products are deemed to be a realistic
specification of the seasonally varying vegetation cover distribution. These are essential
information for many environmental related issues and models, such as regional scale
hydrologic balance, soil erosion, carbon balance and atmosphere/land surface interaction.
Another advantage of using time series analysis of NDVI is the consistency and stability of
the derived products with the simplified description of complicated vegetation layers.
Empirical relationships between woody vegetation indices and the field measurements of
LAI and FPC are obtained using field measurements from the MDB (LAI) and Queensland
(FPC). It is suggested that the linear relationship between LAI and SR and the linear
relationship between FPC and NDVI are the most suitable forms to describe the biophysical
status of woody vegetation. There are some possible problems with the method. Firstly, the
method could make false separation in regions where perennial green pasture is a significant
land cover. The cover from such regions is evergreen and shows small seasonal variation but
is not necessarily woody cover. Roderick et al. (1999) realised this problem and suggested to
use regional scale agricultural statistics to identify those potential problem areas. Secondly,
the background soil colour, and vegetation shadow effects were ignored in the analysis, which
could affect the quality of indices, though much of the noise was reduced by STL. With some
knowledge of background soil colour, the soil effect could be removed by employing the
28
method proposed by Maselli et al. (2000) before the separating. Alternatively, the semi-
empirical method of Qi et al. (2000), based on BRDF models could be used if more detailed
information such as soil reflectance, and single leaf reflectance and transmittance are known.
29
Acknowledgements
This work was supported by the National Land and Water Resources Audit. We thank
Tim Danaher at the Queensland Department of Natural Resources for making the FPC and
SLATS data available for us. Data used by the authors in this study includes that produced
through funding from the Earth Observation System Pathfinder Program of NASAs Mission
to Planet Earth, in cooperation with National Oceanic and Atmospheric Administration. These
data were provided by the Earth Observing System Data and Information System, Distributed
Active Archive Centre, at Goddard Space Flight Center where the data are archived, managed,
and distributed. Thanks to Drs. Jenny Lovell and Dean Graetz at CSIRO Earth Observation
Centre and Dr. Damian Barrett for providing us with access to the PAL GAC AVHRR NDVI
data. H. Lu appreciates support by Drs. Chris Moran, Ian Prosser and Elisabeth Bui and
fruitful discussions with them. Thanks go to Graeme Priestley for his help with GIS and
graphic design. Heinz Buettikofer helped with publishing on the Internet.
30
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35
Appendix: Methods of Time Series Decomposition and STL
There are a number of methods which can be used to perform trend analysis as stated by
equations (2) and (3). Methods range from simple moving average trend assessment (used by
Roderick et al. (1999)) to comprehensive X-11, X-12, and Auto-Regressive Integrated Moving
Average model (ARIMA). The simple moving average forms an output based on equally
weighted values within a sliding window of a time series. Although it is computationally
efficient, it has limited use as it gives the same weight to outliers and may result in a distorted
trend. Advanced methods, such as, X-11, X-12, ARIMA and SABL (Cleveland et al., (1981),
produce more accurate estimations of trend and seasonal component. However, they are either
very complex or computationally expensive. An alternative time series decomposition
approach is STL, developed by Cleveland et al. (1990). STL is a Seasonal-Trend
decomposition procedure built based on the LOESS smoother. LOESS is a LOcally wEighted
regreSsion Smooth method based on the following principle. To get a smoothing curve of
measurements y(x) for a given x, a smoothed value for each y(x) needs to be estimated.
Firstly, for a fixed x, a positive integer q, is chosen, where q ≤ N, with N is the total number
of the observations. The q values of the xi that are closest to x are selected and each is given a
neighbourhood weight based on its distance from x. Let λq(x) be the distance of the qth
farthest xi from x. Let W be the tri-cube weight function:
≥<≤−=10
10)1()(
33
u
uuuW (18)
The neighbourhood weight for any xi is
−=
)(
||)(
x
xxWxw
q
ii λ
(19)
Thus the xi close to x have the largest weights and the weights decrease as the xi increase
in distance from x and become zero at the qth farthest point. The next step is to fit a
polynomial of degree d to the data with weight wi (x) at (xi, yi). The value of the locally-fitted
polynomial at x is )(ˆ xy . It is recommended to use d = 1 (linear) for data containing gentle
curvature and d = 2 (quadratical) for data showing substantial curvature.
36
To use LOESS, d and q must be chosen. In principle, as q increases, )(ˆ xy becomes
smoother. As q tends to infinity, the wi (x) tend to one and )(ˆ xy tends to an ordinary least-
squares polynomial fit of degree d. In STL, different values of q are chosen for different
stages of data smoothing.
In STL, a robust weight iδ , is defined for each (xi, yi). Unlike the weight function
described in equation (18), this weight expresses the reliability of the observation relative to
the others. In general, large residuals cause small weights iδ while small residuals results in
large weights. Consquently, the effects of large residuals tend to be toned down or smoothed,
thereby reducing the influence of transient, aberrant behaviour of the outliers and make the
procedure robust. After replacing wi (x) by iδ wi (x), new fitted values are computed using
locally weighted regression. The determination of new weights and fitted values is repeated
for several times until it converges (the fitted values do not change).
STL consists of two recursive loops: an inner loop nested inside of an outer loop. Each
loop consists a sequence of LOESS smoothers. In each of the passes through the inner loop,
the seasonal and trend components are updated once; each complete run of the inner loop
consists of ni such passes. Each pass of the outer loop consists of the inner loop followed by a
computation of robustness weights; these weights are used in the next run of the inner loop to
reduce the influence outliers on trend and seasonal components. An initial pass of the outer
loop is carried out with all robustness weights equal to 1, and then no passes of the out loop
are carried out. It is found that 10 is sufficient for the maximum number of iterations for out
loop and 2 iterations are sufficient for inner loop.
Another input parameter is the number of observations in each cycle of the seasonal
component, n_p. It is used in the inner loop to define the seasonal cycle. In the inner loop,
each pass consists of a seasonal smoothing that updates the seasonal component, followed by
a trend smoothing that updates the trend component. Both smoothings are carried out using
LOESS. In seasonal smoothing, firstly, the time series is detrended to get a cycle-subseries.
Secondly, the cycle-subseries is smoothed using LOESS with q = ns. A diagnostic graphical
method, given in Cleveland et al. (1990), to assist in choosing this smoothing parameter (ns)
was used here. This parameter determines how much of the variation of the data, other than
trend, is placed to the seasonal component and how much in the irregular component. Thirdly,
a low-pass filter is applied to the smoothed cycle-subseries with a moving average window of
37
length np, followed by another moving average of length np, followed by a LOESS smoothing
with q = nl. Finally, the further smoothed cycle-subseries is detrended to prevent low-
frequency signal from entering the seasonal component. In trend smoothing, the first step is to
deseasonalise the time series by subtracting the seasonal component from the original input
time series. Secondly, the deseasonalised series is smoothed by LOESS with q = nt.
The outer loop calculates the robustness weights. The robustness weights reflect how
extreme the irregular component is. An outlier in the data that results in a very large |Ii| will
have a small or zero weight. The weight is calculated as
)/|(| βδ ii IB= (20)
where B is the bi-square weight function:
1
10
0
)1)(
22
≥<≤
−=
u
uuuB (21)
and |)(|6 iImedian×=β . Now the inner loop is repeated, but in smoothing the trend and
seasonal components, the neighbourhood weight for a value at time i is multiplied by the
robustness weight \delta_i. These robustness iterations of the outer loop are carried out a total
of n_o (a number defined by the user) times. STL has six parameters which determines the
degree of smoothing in each component. In this study, we follow the guidelines given in
Cleveland et al. (1990), and used the following:
np = 36 - the number of observations in each annual cycle of the seasonal component (3
input images per month; 12 month per year).
ni = 1 - the number of passes through the inner loop;
no = 8 - the number of robustness iterations of the outer loop;
nl = 37 - the smoothing parameter for the low-pass filter;
nt = 57 - the smoothing parameter for the trend component; and
ns = 35 - the smoothing parameter for the seasonal component.}
38
A Fortran version of STL was obtained from NETLIB as indicated in Cleveland et al.
(1990). The simple design of STL allows fast computation for the 860 row by 700 column
time series over the Australian continent. It also has the ability to decompose time series with
up to 5% randomly distributed missing values and outliers. STL ensures robust estimates of
the trend and seasonal components are not distorted by aberrant behaviour in the time series.
Areas such as Tasmania, where often more than 5% of the input data in continuous periods is
missing (due to satellite orbital decay), were statistically filled in before the decomposition
was performed.