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: hua.lu@cbr.clw.csiro.au

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.

Important Disclaimer

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.