C S I R O L A N D a n d W AT E R
Using AVHRR data and meteorological surfaces as covariates to spatially interpolate moisture availability in the Murray-Darling Basin
Tim R. McVicar and David L.B. Jupp
CSIRO Land and Water, Canberra Technical Report 50/99, November 1999
Using AVHRR data and meteorological surfaces as covariates to spatially interpolate moisture availability in the Murray–Darling Basin
Tim R. McVicar and David L.B. Jupp
CSIRO Land and Water Technical Report 50/99
November 1999
© 1999 CSIRO Australia, all rights reserved This work is copyright. It may be reproduced in whole or in part for study, research, or training purposes subject to the inclusion of an acknowledgement of the source. Reproduction for commercial usage or sale requires written permission of CSIRO Australia.
Authors: McVicar, Tim R.1* and Jupp, David L.B.2 1
CSIRO Land and Water, PO Box 1666, Canberra, ACT 2601, Australia
[email protected] Ph: 61-2-6246-5741 Fax: 61-2-6246-5800 2
CSIRO Earth Observation Centre, PO Box 3023, Canberra, ACT 2601, Australia
[email protected]
Ph: 61-2-6216-7203 Fax: 61-2-6216-7222
* corresponding author
For bibliographic purposes, this document may be cited as: McVicar, T.R., and Jupp, D.L.B. (1999). Using AVHRR data and meteorological surfaces as covariates to spatially interpolate moisture availability in the Murray– Darling Basin, CSIRO Land and Water Technical Report 50/99, Canberra, Australia. 45 pp.
A .PDF version is available at http://www.clw.csiro.au/publications/
ISBN 0 643 06074 X
SUMMARY 1 INTRODUCTION
1
2 THEORETICAL BACKGROUND
6
2.1 Spline Interpolation 2.2 Background to the Normalised Difference Temperature Index (NDTI)
3 METHODS
6 10
11
3.1 Selecting the Spatial Covariates
11
3.2 Developing the Spatial Covariates 3.2.1 Air Temperature (Ta ) 3.2.2 Vapour Pressure (e a ) 3.2.3 Atmospheric Transmittance (τ0) 3.2.4 Surface Temperature (Ts) 3.2.5 Broadband Albedo (α) 3.2.6 Vegetation Cover (VegCov) 3.2.7 Net Radiation (Rn)
12 14 14 16 17 18 18 18
3.3 Applying the Spatial Covariates
20
4 RESULTS
21
4.1 Selecting the Spatial Covariates
21
4.2 Developing the Spatial Covariates 4.2.1 Air Temperature (Ta ) 4.2.2 Vapour Pressure (e a ) 4.2.3 Atmospheric Transmittance (τ0) 4.2.4 Surface Temperature (Ts) 4.2.5 Broadband Albedo (α) 4.2.6 Vegetation Cover (VegCov) 4.2.7 Net Radiation (Rn)
24 24 27 29 30 30 32 33
4.3 Applying the Spatial Covariates
33
5 DISCUSSION
37
5.1 Selecting the Spatial Covariates
37
5.2 Developing the Spatial Covariates
37
5.3 Applying the Spatial Covariates
37
6 CONCLUSIONS
39
7 ACKNOWLEDGEMENTS
40
8 REFERENCES
41
SUMMARY
Moisture availability is estimated in the 1.1 million km 2 Murray–Darling Basin (MDB) in south-east Australia. Remotely sensed data from the Advanced Very High Resolution Radiometer (AVHRR) are combined with meteorological data to estimate the Normalised Difference Temperature Index (NDTI). The NDTI provides a measure of the moisture availability – the ratio of actual to potential evapotranspiration. Surface temperature minus air temperature, percent vegetation cover, and net radiation explained 85% of variation in the modelled NDTI. Using these three covariates across the network of meteorological stations allows NDTI images, which map moisture availability across the MDB, to be calculated. This method uses a ‘calculate then interpolate’ (CI) approach that uses the per-pixel variation present in the AVHRR data as the backbone for the spatial interpolation. Combining this CI approach with the spatially dense AVHRR-based covariates avoids errors between measurement points when interpolating variables for regional hydrologic modelling, most significantly the spatial pattern of rainfall. The NDTI provides a link into regional water-balance modelling which does not require daily rainfall to be spatially interpolated. Assessing spatial and temporal interactions between the NDTI and the Normalised Difference Vegetation Index (NDVI) provides useful information about regional hydro-ecological processes, including agricultural management, within the context of Australia’s highly variable climate and sparse network of meteorological stations. This report fully documents the development of the covariates and results, which are summarised in a companion invited journal paper that will appear in a special issue of Remote Sensing of Environment to be published next year.
1 INTRODUCTION In order to categorise a landscape, studies of regional hydrology frequently classify remotely sensed data into patches of homogeneous landcover. Hinton (1996) reviewed the integration of remote sensing in a Geographical Information Systems (GIS) framework for environmental applications and, while recognising that remotely sensed data offer more than input data to thematic (discontinuous) maps, the review exclusively focussed on classification of remotely sensed data and how subsequent vectors are imported into a GIS. Remote sensing can provide regional hydrology much more than the means to classify vegetation to stratify the study area. One benefit remote sensing offers to regional hydrology is its high spatial density. Many hydrologic and plant growth models operate at a single point. To extend these models to regions, an approach frequently used is to interpolate all input parameters and driving variables and then calculate at each location (e.g. Cole et al., 1993; Kittel et al., 1995; Carter et al., 1996; Nalder and Wein, 1998; Thornton et al., 1997). Depending on the complexity of the model, and the spatial and temporal resolution and extent of the modelling, this can be a daunting task. Others have used some remotely sensed variables with interpolated meteorological variables and then performed the calculations (e.g. Moran et al., 1996; Pierce et al., 1993; Zhang et al., 1995). Raupach et al. (1997) used output from a general circulation model (GCM) as input to a coupled carbon, water, and energy flux model. In this approach the GCM is viewed as an ‘interpolator’ of the required input meteorological variables for the spatially distributed process model. Prince et al. (1998) use high-frequency Advanced Very High Resolution Radiometer (AVHRR) data to provide estimates of some of the variables needed for a regional plant growth model, which is run at every point. The approach used in all of these methods can be summarised as ‘interpolate then calculate’ (IC). Another less frequently used approach is to ‘calculate then interpolate’ (CI). Stein et al. (1991) first introduced the concept, and compared CI to IC procedures for simulating the moisture deficit for a 404-ha area in the Netherlands. They used 399 observations in the modelling framework, where 7 procedures (4 CI and 3 IC) were performed. Subsequently the results for each of the 7 were compared with another 100 observations, and a mean squared error (MSE) for each was calculated. Overall the CI procedures provided lower MSE results than the IC procedures, which lead Stein et al. (1991) to state “in short, CI procedures are to be preferred over IC procedures”. Bosma et al. (1994) simulated the 3-dimensional flow of a heavy metal contaminant and used both IC and CI methods using ordinary point kriging, to predict a parameter for a field of
1
20 m 2. The influence of sample size was analysed and for the smallest samples it was concluded that CI procedures perform better than IC procedures. These previous papers provide support for the use of CI in comparison to IC. However, there are several underlying issues that must be considered for individual cases (Addiscott and Tuck, 1996). These include: (i) the degree of spatial autocorrelation; (ii) whether it is parameter(s) and/or variable(s) which are being spatially interpolated; and (iii) whether the parameter(s) and/or variable(s) are used in a linear or non-linear fashion in the final model. In the field of regional hydrology there are numerous examples of the IC approach; however there seems to be no examples of the CI approach. In related disciplines Wahba and Wendelberger (1980) noted that satellite radiance data could be used as a covariate to interpolate point observations of meteorological variables. Reynolds (1988) and Reynolds and Smith (1994) ‘blended’ qualitycontrolled drifting-buoy estimates of sea surface temperature (SST) with AVHRR SST estimates using optimal interpolation techniques (Gandin, 1963). Recently Walker and Wilkin (1998) used 8 years of cloud-interrupted AVHRR data with spatial statistical interpolation coupled with optimal time-averaging techniques to produce uninterrupted SST maps for the Indo-Australian region. This report provides an example of CI for regional hydrology using thermal remotely sensed data linked with a resistance energy-balance model to produce rasters of moisture availability for the Murray–Darling Basin. The surface-energy balance describes the partitioning of net short- and long-wave radiation into latent and sensible heat fluxes and changes in heat storage. A description of this partitioning is given by a resistance energy-balance model (REBM: Monteith and Unsworth, 1990), which is ‘closed’ by specified meteorological inputs defined at a reference height above the surface and some assumptions or models defined at or below the earth’s surface. It describes the energy fluxes between the soil, plant canopies, and the atmosphere in terms of ‘resistances’. With appropriate models for the resistances and other terms, energy-balance methods can provide estimates of the sensible and latent heat fluxes at a specific time-of-day in the surface layer. Resistance energy-balance models require adequate meteorological data and information about the land-surface type and structure. The resistances can be estimated from information such as the surface and canopy roughness, wind speed, canopy Leaf Area Index (LAI), and cover fraction. With sufficient assumptions, the model can be solved with the provision of surface temperature to estimate moisture availability. For the specific choices and assumptions used here see Jupp et al. (1998). To solve a resistance energy-balance model at the time that the surface is remotely sensed, several surface and near-surface meteorological variables are required. These include air temperature, relative humidity or vapour pressure, 2
solar radiation, and wind speed at a reference height above the surface and over a time limit of sufficient length for equilibrium to be assumed. The effectiveness of estimating these variables at the specific time-of-day when remotely sensed data are acquired using standard daily meteorological data has been documented by McVicar and Jupp (1999). This allows development of the ‘dual’ approach to the well known methods developed by Jackson et al. (1983) in which daily evapotranspiration is estimated from one-time-of-day remotely sensed data. This current study integrates data types with very different spatial and temporal scales. AVHRR data are spatially dense, with a 1.1-km2 resolution at nadir, and are recorded over large areas in a matter of seconds at a specific time for specific wavelengths. This means, for the extent of the image, remotely sensed data are a ‘census’ at a particular spatial scale recorded at a specific time. Depending on the amount of cloud coverage and the satellite repeat characteristics, optical remotely sensed data may only be available weekly or monthly. On the other hand, meteorological data are recorded sparsely, with the points often separated by hundreds of kilometres. The variables measured at these points represent a certain area. However, the exact area being represented by a given point measurement is unknown because the spatial autocorrelation is unknown. Meteorological data from standard, long-term, large-area, surface networks are usually daily integrals (e.g., rainfall and wind-run totals), or daily extremes (e.g., maximum and minimum air temperatures) acquired regularly over decades. This study uses data from Australian Bureau of Meteorology stations that record minimum and maximum air temperature and rainfall daily. Thus, remotely sensed data are spatially dense but temporally sparse, while meteorological data are spatially sparse but temporally dense. To spatially interpolate the Normalised Difference Temperature Index (which is introduced and defined below), covariates are derived by combining spatially dense AVHRR-derived variables with spatially interpolated meteorological variables. This is very similar to using elevation, in the form of a digital elevation model, as a covariate to interpolate surfaces of air temperature. To spatially interpolate point data, many different numerical approaches exist (Lam, 1983). Two of the most popular methods are splines and kriging. Recently several papers have compared the outputs from splines and kriging (e.g. Dubrule, 1983; Dubrule, 1984; Hutchinson, 1993; Hutchinson and Gessler, 1994; Laslett, 1994; Borga and Vizzaccaro, 1997). When both spline and kriging models are tested with careful selection of interpolation parameters, there is generally little difference between the two methods. In this research, splines are used. Incorporating regular grided data (usually a digital elevation model, but in this case AVHRR-based variables) as a covariate can be performed routinely using ANUSPLIN (Hutchinson, 1997), a commercially available software package. Any spline or kriging package that can utilise covariates could have been used to spatially interpolate the Normalised Difference Temperature Index. A glossary of commonly used abbreviated symbols is presented in Table 1.
3
Table 1. Common abbreviations used throughout the report.
Symbol
Description
General
IC
interpolate then calculate
CI
calculate then interpolate
DEM
digital elevation model
MDB
Murray–Darling Basin
AVHRR
(subscript) specific time-of-day that remotely sensed data is acquired
PT
(subscript) measurement made at ‘a point’
GD
(subscript) measurement, or interpolated output, which is a continuous ‘grid’
DAY
(subscript) daily integrals (e.g., actual evapotranspiration, all-wave net radiation, rainfall, or wind run) or daily extremes (e.g., maximum and minimum air temperatures)
Advanced Very High Resolution Radiometer (AVHRR) based
Ts
surface temperature (K)
VegCov
vegetation cover (%)
α
broadband albedo (%)
NDTI
normalised difference temperature index (unitless)
CWSI
crop water stress index (unitless)
NDVI
normalised difference vegetation index (unitless)
LAI
leaf area index (unitless, m2 leaf / m2 ground)
Australian Bureau of Meteorology (ABM) based
Tn
daily minimum air temperature (K)
Tx
daily maximum air temperature (K)
P
daily rainfall (mm day–1)
Ta
air temperature (K)
ea
vapour pressure (hPa)
u
wind speed (m s –1)
Td
dew-point temperature (K)
Resistance Energy-Balance Model (REBM) based
λEa
actual evapotranspiration (W m–2)
λEp
potential evapotranspiration (W m–2)
ma
moisture availability, the ratio of λEa to λEp (unitless)
rs
composite surface resistance (s m–1)
T0
REBM surface temperature when rs = 0 s m–1 (i.e., λEa = λEp)
T∞
REBM surface temperature when rs = ∞ s m–1 (i.e., λEa = 0 W m–2)
4
Radiation modelling based
Rn
all-wave net radiation (W m–2)
Rs
shortwave (0.15 – 4 µm) radiation (W m–2)
Rs_down
downward shortwave radiation (W m–2)
Rs_up
upward (or reflected) shortwave radiation by the surface (W m–2)
Rl
longwave (> 4 µm) radiation (W m–2)
Rl_down
longwave radiation emitted from the atmosphere to the surface (W m–2)
Rl_up
longwave radiation emitted from the surface to the atmosphere (W m–2)
Shortwave radiation modelling based
τt
total daily atmospheric transmittance (%)
τz
effective beam transmittance at altitude z m (%)
τ0
effective beam transmittance at altitude 0 m (%)
p
transmission path length or relative airmass (unitless)
cos θs
cosine of the solar zenith angle (radians)
Q 0'
exoatmospheric normal solar irradiance modified for sun–earth distance (W m–2)
Longwave radiation modelling based
εa
atmospheric emissivity
εs
surface emissivity
εg
ground (or soil) emissivity
εv
vegetation emissivity
Spline based
TPS
thin-plate spline
PTPS
partial thin plate spline, a TPS which incorporates the constant linear dependence on a covariate
λ
smoothing parameter, which controls the trade-off between data fidelity and the spline-generated surface roughness
m
order of the partial derivative, which also influences the spline-generated surface roughness
σ$ ε
estimate of error variance
T ( m , λ)
expected true mean-square error
GCV (m , λ)
generalised cross-validation error
tr(I – A)
residual degrees of freedom, where A is the influence matrix and I is the identity matrix
n
number of observations
M
number of lower-order monomials
2
5
2 THEORETICAL BACKGROUND 2.1 Spline Interpolation This notation and mathematical development of the thin-plate spline (TPS) model follows Hutchinson and Gessler (1994). The relationship between the observed data values and the trend, which is estimated by the TPS, and error component of those observations is given by: y( x i ) = z (x i ) + ε( xi ) i = 1, 2,..., n
(1)
where: y (x i ) the observed data values; z( xi ) the trend to be estimated, consider this signal component in the observations; ε( xi ) the discontinuous error term, comprising both random measurement error and microscale variations (termed the nugget effect in kriging analysis) which has only short range correlations below the limits of normal observations. This error term is assumed be independent having zero mean and a variance σε2 , which is assumed to be constant across all the data points; x i is the coordinates in two dimensional (or higher) Euclidean space; n is the number of observations; and i is the observation number. The TPS estimate of z( xi ) is calculated by a suitably smooth function f which minimises: n
∑
2
yi − f ( xi ) + λJ m ( f )
(2)
i =1
where: yi is the observed data value for observation i ; f ( xi ) is the spline function which is being estimated; λ is a positive number called the smoothing parameter; Jm ( f ) is a measure of the roughness of the function f which is defined in terms of m-th order partial derivatives; n is the number of observations; and i is the observation number. The smoothing parameter λ controls the trade-off between the amount of data fidelity and surface roughness. This is usually determined by minimising the generalised cross validation (GCV), which is given below. Wahba (1990, Ch. 4) provides an overview on the importance of selecting an appropriate value of λ . The smoothing parameter has also been denoted ρ (Laslett, 1994; Dubrule, 1983; Dubrule, 1984; Hutchinson, 1993). Hutchinson (1993) introduces the spline models with some weighting, whereas Hutchinson and Gessler (1994) do not discuss weighting. The solution to the above function f can be expressed explicitly as:
6
M
n
j =1
i =1
f ( x) = ∑ a j φ j ( x) + ∑ bi Ψ( ri )
(3)
where: M is the number of low order monomials; this depends of the number of independent variables (the dimension of x), the number of covariates (in the case of partial thin plate splines(PTPS)) and the m-th order partial derivates (see Table 2 below for the number of terms for a bi-variate and tri-variate TPS with m varying from 2 to 4); j is the monomial number; a j is a set of unknown coefficients; φ j is the set of M low-order monomials; bi is a set of unknown coefficients; ri is the Euclidean distance between x and xi ; Ψ is a scalar function of ri ; n is the number of observations; and i is the observation number. For a PTPS, covariates must be provided as a continuous grid at the same scale as the required output surface. Between the isolated data points where the dependent variable is measured (or modelled) a constant linear dependency between the dependent variable and the variation in the covariate is developed. This modifies the estimate of the dependent variable based on the independent variables, which usually represent geographical position, for example longitude and latitude. A classic example of this is using a DEM as a covariate when interpolating surfaces of Ta. The coefficients bi must satisfy the boundary condition: n
∑ b φ (x ) = 0 i
j
i
j = 1, 2..., M
(4)
i =1
where: n is the number of observations; i is the observation number; bi is the set of unknown coefficients; φ j is the set of M low-order monomials; j is the monomial number; and M is the number of low-order monomials. The number of low-order monomials, M, is determined by expanding the Pascal triangle for a given number of independent variables and m (Table 2). If a PTPS is used (that is, covariates are used), then add the number of covariates to the number of monomial terms for a given number of independent variables and m (see bi-variate spline and tri-variate partial in Table 2 and Table 5). The influence matrix, A, is an n × n matrix which is defined by: f ( x1 ) y ( x1 ) M = A M f (xn ) y(xn )
7
(5)
The signal degrees of freedom, denoted tr(A), represents the number of degrees of freedom that the spline model has used to fit the data. The residual degrees of freedom is denoted tr(I – A), where I is the identity matrix defined by: tr(A) + tr(I – A) = n
(6)
where: tr(A) is the signal degrees of freedom; tr(I – A) is the residual degrees of freedom; and n is the number of observations.
Table 2. Number of lower order monomial terms, M, for bi-variate and tri-variate TPS with m varying from 2 to 4. The expansion of the Pascal triangle is also shown, where c is a constant and the variables are represented as x, y, and z. No covariates are assumed; if covariates are used then the number of monomial terms increases by the number of covariates.
Nr of Independent Variables
m
M
Expanded Terms
2
2
3
c, x, y
2
3
6
c, x, y, xy, x2, y2
2
4
10
c, x, y, xy, x2, y2, x2y, xy2, x3, y3
3
2
4
c, x, y, z
3
3
10
c, x, y, z, xy, yz, zx, x2, y2, z2
3
4
20
c, x, y, z, xy, yz, zx, x2, y2, z2 x2y, xy2, y2z, yz2, z2x, zx2, x3, y3, z3, xyz
Hutchinson and Gessler (1994) generally recommend that n – M > tr(I – A) > n / 2. For many applications, if tr(I – A) is smaller than n / 2 then this suggests that the data are too sparse to support spline interpolation. If tr(I – A) reaches its maximum possible value, n – M, then this indicates that the fitted spline is a least-squares regression of the data using the M low-order monomials in the set φ j . $ 2ε is given by: An estimate of the error variance σ ( y − Ay) T ( y − Ay) σ$ 2ε = tr ( I − A) 2 $ ε is an estimate of the error variance; where: σ y denotes the n-dimensional vector of observations y (x i ) ; T denotes a transpose of the matrix; A is the influence matrix; I is the identity matrix; and tr(I – A) is the residual degrees of freedom.
8
(7)
$ 2ε may be used to calculate an expected true mean-square error The value σ T ( m, λ) which is defined by: $ 2ε tr(I − A) / n + σ $ 2ε T ( m, λ) = ( y − Ay) T ( y − Ay) / n − 2σ
(8)
where: T ( m, λ) is the expected true mean-square error, which is dependent on the order of the partial derivates (m) and the smoothing parameter ( λ) ; σ$ 2ε is an estimate of the error variance; y denotes the n-dimensional vector of observations y (x i ) ; T denotes a transpose of the matrix; A is the influence matrix; I is the identity matrix; tr(I – A) is the residual degrees of freedom; and n is the number of observations. T ( m, λ) is regarded as an optimistic measurement of error, since the estimate of $ 2ε ) has been removed from the analysis. the error variance (σ The generalised cross-validation GCV ( m, λ) statistic is given by: GCV (m, λ) =
( y − Ay) T ( y − Ay) / n tr ( I − A) / n
2
(9)
where: GCV ( m, λ) is the generalised cross-validation, which is dependent on the order of the partial derivates (m) and the smoothing parameter ( λ) ; y denotes the n-dimensional vector of observations y (x i ) ; T denotes a transpose of the matrix; A is the influence matrix; I is the identity matrix; tr(I – A) is the residual degrees of freedom; and n is the number of observations. Comparing GCV ( m, λ) with T ( m, λ) it is recognised that GCV ( m, λ) can be regarded as a pessimistic, or conservative, measurement of error. This is because $ 2ε , is inherently included in the calculation of the estimate of the error variance, σ GCV ( m, λ) , whereas it is explicitly removed from the calculation of T ( m, λ) . GCV ( m, λ) is often used to find the ‘best’ model among competitors (Davis, 1987). The mean squared residual (MSR) is given by: MSR =
1 n ( yi − f i ) 2 ∑ n i= 1
(10)
where: n is the number of observations; yi is the data recorded for the i-th observation; and f i is the spline-fitted estimate for the i-th observation. This provides a measure of how close the fit is between the observations and the spline-fitted estimate. MSR is closely related to the bias (or mean error);
9
however in the calculation of the bias (or mean error) the differences between the observation and model are not squared. 2.2 Background to the Normalised Difference Temperature Index (NDTI) Jackson et al. (1977) developed an empirical model to estimate λEa_DAY in which λEa_DAY and Ts are related through λEa _DAY − Rn_DAY = A − B (Ts − Ta ) , where A and B are empirical coefficients. This approach has been extended by Seguin et al. (1982a, 1982b) , and Lagouarde (1991) describes an implementation of this approach. The problematic effect of changes in surface-cover type and amount (especially roughness length) on the coefficient B must be addressed (Lagouarde, 1991). Courault et al. (1996) have extended this work and provided alternative parameterisations for B. In principle, A and B should be consistent over areas with similar land-cover structure. The CSWI was originally developed at the agricultural field scale and at the daily time scale (Jackson et al., 1981). Relating λEa_AVHRR to λEa_DAY is generally performed using the approximation of Jackson et al. (1983), which has been extended by Xie (1991). Both are in the context of agricultural field-scale experiments. Rather than using the spatial variance in remotely sensed data to map λEa_DAY , there exists the opportunity to link Ts_AVHRR with meteorological variables Ta_AVHRR , ea_AVHRR , Rs_AVHRR , and u_AVHRR. These variables are usually recorded during intensive field campaigns for small, well-instrumented catchments. Recently, McVicar and Jupp (1999) concluded that these meteorological variables can be adequately estimated from daily meteorological data for use with a REBM_AVHRR. This allows the development of a ‘dual’ approach to the well-known method developed by Jackson et al. (1983, 1977, 1981) to derive a spatially varying index of ma from daytime Ts. There are many ABM stations that measure daily data and hence the development by McVicar and Jupp (1999) also avoids the constraint of only using well-instrumented small catchments for the time of intensive field campaigns. Consequently, large regions in an operational framework can be monitored. To undertake this ‘dual’ approach Jupp et al. (1992) and McVicar et al. (1992) jointly developed the NDTI which is defined as: (T − T ) NDTI = ∞ s (11) (T∞ − T0 ) where: T0 is the modelled surface temperatures using the REBM assuming zero surface resistance (or fully available moisture); T∞ is the modelled surface temperatures using the REBM assuming infinite surface resistance (or no available moisture); and Ts is the remotely sensed derived surface temperature; in this case it is derived from the AVHRR sensor, but it could be derived from other space- or air-borne thermal sensor data which are acquired during the day. The temperature T0 is that of the surface when λEa = λEp , whereas T∞ is the temperature corresponding to zero λEa . The two bounding temperatures are 10
derived by inverting a REBM. The methods and assumptions used here are discussed fully in Jupp et al. (1998). If the REBM and Ta_AVHRR , ea_AVHRR , Rs_AVHRR , and u_AVHRR are all well-defined, a time series of Ts should fall within the envelope defined by the limits T0 and T∞. This relationship has been presented for 5 years of AVHRR Ts data recorded at Cobar, Australia (McVicar and Jupp, 1998). The NDTI can be regarded as a specific time-of-day version of the CWSI. The NDTI brings the concept of mapping ma from Ts onto a regional basis, and is generic across different land surfaces by using the per-pixel variation present in the AVHRR data as the backbone for the spatial interpolation. To achieve this aim, suitable spatial covariates, which incorporate AVHRR per-pixel variation, must be developed. From the model of Jackson et al. (1977), both Ts – Ta and Rn are potential covariates. To include changes in surface cover type and amount, VegCov is also considered a potential covariate.
3 METHODS Our requirement for daily meteorological data are modest: only Tn , Tx , and P are essential. However, in the 1.1 million km 2 MDB there are only 63 ABM stations that have measured this data continuously from 1980 until the present (Fig. 1). Where available, daily wind run data (km day–1) was used to estimate u_AVHRR (McVicar and Jupp, 1999) for the REBM. To incorporate the influence of the north-west cloudband, a synoptic scale feature of the Australian region (Colls and Whitaker, 1990), 13 additional ABM stations to the north and west of MDB are included in the meteorological data base (Fig. 1). Within the extended NDTI study area (the area used to develop the covariates for the spatial interpolation of the NDTI), there are 70 ABM stations; 57 of these are in the MDB and an additional 13 are to the north and west (Fig. 1). Remotely sensed data was acquired by the AVHRR sensor onboard the NOAA-9 and NOAA-11 satellites. There are five channels recorded by the AVHRR sensor. These are: (1) 580–680 nm; (2) 725–1100 nm; (3) 3.55–3.93 µ m; (4) 10.5–11.3 µm; and (5) 11.5–12.5 µm. Cracknell (1997) provides an extensive overview of the AVHRR sensor, the NOAA series of satellites, and some previous applications of AVHRR data. The data archive, focussing on the MDB, consists of 97 AVHRR single overpass afternoon images from June 1986 until January 1994, recorded at a near-monthly time interval. Extensive preprocessing of the entire archive included rectification, validating the rectification accuracy (McVicar and Mashford, 1993), visual cloud clearing, and ‘salt and pepper’ removal. 3.1 Selecting the Spatial Covariates At Cobar (Fig. 1), daily meteorological data Tn , Tx , P and wind run (km day–1) were linked with the procedures described in McVicar and Jupp (1999) to estimate Ta_AVHRR , ea_AVHRR , Rs_AVHRR , and u_AVHRR. These data were used with variables routinely derived from AVHRR data (Ts , VegCov, and α; see below) in 11
the REBM (Jupp et al., 1998) to model NDTI and λEa. This was only performed for cloud-free AVHRR data. The utility of each of the three potential covariates (Ts – Ta , VegCov, and Rn) is evaluated by linearly regressing, using multiple linear regression as required, the three potential covariates against the REBM outputs of NDTI or λEa. Regression-model estimates of the NDTI and λEa, denoted NDTI' and λEa' respectively, have been calculated and then cross-plotted against the respective REBM NDTI or REBM λEa. Results from this analysis are presented in Section 4.1 below. o
-23 LEGEND Bureau of Meteorology station Extended NDTI area Focused NDTI area
QUE E NS L A ND
BRISBANE
o
-30
S O UT H A U ST R A L I A er
Cobar
iv
g lin
R
ar
N E W S O UT H WA L E S
D
SYDNEY ADELAIDE
Mu
rra
yR
CANBERRA A.C.T.
ive
r
o
-36 AUSTRALIA MURRAY-DARLING BASIN
V I CT O R I A
MELBOURNE o
o
138
o
o
142
148
o
150
-39 o 154
Figure 1. Location of the Murray–Darling Basin, Australia. The sites of the 70 ABM stations are indicated. The focus and extended NDTI study areas are also shown.
3.2 Developing the Spatial Covariates To interpolate the NDTI, the CI approach is used; that is, the NDTI is calculated at the ABM stations (considered points in the MDB) and then the NDTI is spatially 12
interpolated. Ts – Ta , VegCov, and Rn were selected as covariates to spatially interpolate the NDTI (see Section 4.1). Continuous grids, to be used as covariates or in the modelling of Rn , were developed from: (i) interpolated meteorological data (Ta , ea , and τ0); (ii) remotely sensed data (Ts and α ); (iii) supervised classification of AVHRR reflective data used as a GIS stratum with the time series of AVHRR reflective data (VegCov); and (iv) modelling Rn using all of the above six variables as inputs.
Parton and Logan (1981)
T a_AVHRR_PT
tested in McVicar and Jupp (1999) Base Meteorological Daily Data Tx, T n and P
McVicar and Jupp (1999)
e a_AVHRR_PT
extension to Kimball et al. (1997)
τ
Bristow and Campbell (1984),
o_AVHRR_PT
non-linear solution and Leckner (1978)
SPATIAL INTERPOLATION
Fig. 2 illustrates the development of the 6 input variables required to generate Rn and Fig. 3 shows how Rn is calculated. From some of the 6 variables, Ts – Ta and VegCov are developed. Methods used to develop the spatial covariates, and intermediate variables, are outlined below.
Surface temperature estimated using AVHRR
T a_AVHRR_GD
e a_AVHRR_GD
τ
o_AVHRR_GD
T s_AVHRR_GD
Channels 4 and 5 with the Split Window Algorithm Base AVHRR Data Decade, Near Monthly
Broadband albedo estimated using AVHRR
α_AVHRR_GD
Channels 1 and 2 with the method of Saunders (1990)
Vegetation cover estimated using AVHRR
VegCov_AVHRR_GD
Channels 1 and 2, GIS strata, and empirical relationships
Figure 2. Flowchart of the development of the required variables at the specific time-of-day when AVHRR data is acquired. The six variables are derived from two main sources. Daily meteorological data (Tn , Tx , and P) are used to estimate Ta , ea , and τ0. Then these 3 variables are spatially interpolated. AVHRR data is used to estimate Ts, α, and VegCov. Ts – Ta and VegCov are used as covariates. All six variables are used in modelling the third covariate Rn (see Fig. 3). The subscript ‘AVHRR’ refers to the specific time-of-day of the AVHRR overpass. The subscript ‘PT’ refers to point measurements (see Fig. 1 for the locations of the 70 ABM stations) and the subscript ‘GD’ refers to a continuous grid (a raster) covering the entire study area.
13
3.2.1 Air Temperature (Ta) Continuous grids of Ta for the specific times of AVHRR data acquisition were calculated in a two -step process. The values of Tn and Tx are temporally interpolated to Ta at the 70 ABM stations, denoted Ta_AVHRR_PT in Fig. 2. For use with remotely sensed data McVicar and Jupp (1999) showed that the Parton and Logan (1981) model for temporally interpolating Ta is satisfactory, the bias between the modelled and measured Ta is within 0.2°C with an r2 of 0.99 (McVicar and Jupp, 1999). The output of this step is denoted Ta_AVHRR_PT in Fig. 2. Secondly, the values of Ta_AVHRR_PT are spatially interpolated to generate continuous grids of Ta for the extended NDTI study site (Fig. 1), denoted Ta_AVHRR_GD in Fig. 2. Ta_AVHRR_GD is required for the Ts – Ta covariate, and also to calculate atmospheric emissivity (εa) which is used in the calculation of Rl_down (see Section 3.2.7 and Fig. 3). To select the spline parameters, detailed analysis was undertaken using Ta_AVHRR_PT supporting the AVHRR overpass acquired by NOAA-9 Orbit Number 14301 recorded on 22 September 1987 at 1629 local time. The influence of elevation was incorporated into the TPS by fitting the data with four variations; these are similar to the variations described by Hutchinson (1995). Here the four variations explored are: (a) bi-variate TPS function of longitude and latitude; (b) tri-variate PTPS incorporating a bi-variate TPS function of longitude and latitude and a constant linear dependence on elevation; (c) tri-variate TPS function of longitude, latitude and elevation in metres; and (d) tri-variate TPS function of longitude, latitude and elevation in kilometres. Scaling does not influence the output statistics when using elevation, or any other variable, as a constant linear dependent (or covariate) in a PTPS. However, when using elevation (or any other variable) as an independent spline variable in TPS, the units can influence the output statistics. Hutchinson (1995) explored this issue when interpolating mean monthly rainfall and found that using elevation data expressed as kilometres to be both convenient and in the range of units that provided a minimal validation residual. The range of all independent variables should be approximately equal. When using a tri-variate TPS, elevation was either in metres (case (c) above) or in kilometres (case (d) above). m was varied from 2 to 4 (three cases) for each of the above four cases ((a) to (d) above). In total, 12 options were explored for the spatial interpolation of Ta, with the results presented in Section 4.2.1. 3.2.2 Vapour Pressure (e a) Calculating continuous grids of ea for the specific times of AVHRR data acquisition is a two -step process. This variable is required to calculate εa, which is used in the calculation of Rl_down (Section 3.2.7 and Fig. 3). The first step
14
involves temporally estimating ea. Recently, Kimball et al. (1997) estimated Td based on empirical regressions between Tx and Tn, and an introduced term ‘EF’, which is the ratio of Priestly–Taylor λEp_DAY divided by annual precipitation. The assumption that ea remains constant throughout the day was improved upon by linearly interpolating ea between the times of sunrise for consecutive days, denoted K97 ‘interpolation’ in McVicar and Jupp (1999). The K97 interpolation method was used to develop ea_AVHRR_PT (Fig. 2) estimates at the 70 ABM stations (Fig. 1). Secondly, the values of ea_AVHRR_PT are spatially interpolated to generate continuous grids of ea for the extended NDTI study site (Fig. 1), denoted ea_AVHRR_GD in Fig. 2. To test the sensitivity of m and the use of and units of elevation as a covariate (Jackson et al., 1985) in developing ea_AVHRR_GD , the same 12 options explored for Ta (above) were also explored for ea. Results are presented in Section 4.2.2.
T a_AVHRR_GD
R l_down _AVHRR_GD
e a_AVHRR_GD
εa_AVHRR_GD
VegCov_AVHRR_GD
εs_AVHRR_GD
R l_up _AVHRR_GD
T s_AVHRR_GD
τ
R n _AVHRR_GD
o_AVHRR_GD
R s_down _AVHRR_GD
α_AVHRR_GD
R s_up _AVHRR_GD
Figure 3. Flowchart of the development of Rn . (Rl and Rs refer to longwave and shortwave radiation respectively.) The subscript ‘down’ refers to radiative fluxes from the atmosphere towards the land surface and the subscript ‘up’ refers to radiative fluxes from the land surface toward the atmosphere. Refer to Fig. 2 for the development of the six input variables. The subscript ‘AVHRR’ refers to the specific time-of-day of the AVHRR overpass, and the subscript ‘GD’ refers to a continuous grid (a raster) covering the entire study area.
15
3.2.3 Atmospheric Transmittance ( τ 0) Calculating continuous grids of τ0 for the specific days of AVHRR data acquisition is a two -step process. The first step is to calculate τ0_AVHRR_PT (Fig. 2) for each of the 97 AVHRR days at each of the 70 ABM stations. This is made of three sub-steps. The first sub-step is based on the approximation to estimate total daily atmospheric transmittance, τt, at the 70 ABM stations. This uses the Bristow and Campbell (1984) method, which is an approximation based on daily Tn , Tx , and P. The estimate of τt is based on a function of the daily temperature extreme, denoted ∆T. This is calculated as: ∆T( J ) = Tx ( J ) − ( Tn ( J ) + Tn ( J +1 ) ) / 2
(12)
where J is the day number. Two simple empirical rules, termed rainfall adjustment, are used to modify ∆T to ∆TRA (Bristow and Campbell, 1984, p. 161). Firstly, if there was rain on the current day then ∆TRA(J) = 0.75 ∆T(J). Secondly, if on the day before rain began ∆T(J–2) – ∆T(J–1) > 2°C, then ∆TRA(J–1) = 0.75 ∆T(J–1). Their model expresses τt as:
τ t = ATc (1 − e − b ∆TRA ) c
(13)
The regional coefficients used for the MDB are ATc(0) = 0.807, b = 0.175, and c = 0.849 (McVicar and Jupp, 1999). The second sub-step takes into account changes due to transmission path length, which are a function of time of day, day of year, and geographical position; τt is modified to an effective beam transmittance at altitude z m, denoted τz. The time trace of total downward shortwave irradiance, Rs_down (W m –2), is modelled by assuming a constant atmosphere with τz and expressing the radiation components between sunrise and sunset as in Hungerford et al. (1989): ' Rs_down = τ zp / 2 cos θs Q0
(14)
where: p is the transmission path length or relative airmass; cosθ s is the cosine of the solar zenith angle; and Q0' is the exoatmospheric normal solar irradiance modified for the sun– earth distance. The formula of Kasten (1966), which allows for earth curvature and refraction, has been used to estimate p, but other options are being investigated. By solving for τz from τt for each day it is possible to provide a disaggegrated model that is consistent with the daily total solar radiation. That is, τz is the solution to the nonlinear equation: τt =
R s,day Q ′0,day
=
∫
s set
τ zp / 2 cos θ s ( s ) ds
s rise
∫
s set
s rise
16
cos θ s ( s ) ds
(15)
where the sun rise and set times (s rise , sset ) and the sun zenith angle ( θs(s) ), on which p depends, are functions of time of day (s) and depend on season and location. In the final sub-step, an altitudinal lapse-rate model can be applied to τz to estimate an effective beam transmittance at altitude 0 m, denoted τ0. This is given by
τ z = τ 0 Pz / P0
(16)
where P is the total pressure and the subscripts denote the altitude above a zeroheight surface. Leckner (1978) described the variation of atmospheric pressure by: Pz = e − c1 z P0
(17)
where c1 = 0.118 km –1. Rearranging Eq. (16) and Eq. (17), ensuring that c1 is converted into metres (the same units as the site elevation), and that τz is expressed as a fraction (not a percentage), provides: τ0 = τz
[ 1 /( e −0.000118 * z )]
(18)
The value of τ0 has been calculated for the 70 ABM stations for each of the 97 dates in the AVHRR data base and is denoted τ0_AVHRR_PT in Fig. 2 using these three sub-steps. The second step is to spatially interpolate τ0_AVHRR_PT to provide continuous grids of τ0, denoted τ0_AVHRR_GD (Fig. 2) using a TPS. Results for the bi-variate TPS function of longitude and latitude with m varying from 2 to 4 for the development of τ0_AVHRR_GD are presented in Section 4.2.3 below. Elevation was not incorporated in this spline model as the relationship between τ0 and elevation is implicitly defined by using a DEM in the calculation of Rs_down. The value of τ0_AVHRR_GD is assumed constant over the day, the specific time-of-day, day-ofyear, and geographical position of each resampled AVHRR pixel centre are used to determine p and cos θs , which are used in modelling Rs_down (Section 3.2.7). 3.2.4 Surface Temperature (Ts) The value of Ts_AVHRR_GD is calculated using a split window algorithm, which takes advantage of the differential atmospheric absorption observed in AVHRR channels 4 and 5. The radiance measured by AVHRR channels 4 and 5 are converted to brightness temperatures, denoted T4 and T5 respectively, using Plank’s law. An estimate of Ts has then been developed using the formula Ts = T5 + 3 ∆T4,5 – 0.5 (Jupp, 1989) where ∆T4,5 is T4 – T5. Other AVHRR coefficients for the development of land surface Ts have been presented (Czajkowski et al., 1998); these have been evaluated, but the resulting Ts has been consistently higher than the REBM T∞. Generating Ts_AVHRR_GD does not require any spatial interpolation. This variable is required for the Ts – Ta covariate, and is also used to calculate Rl_up (see Fig. 3 and Section 3.2.7). 17
3.2.5 Broadband Albedo ( α ) The value of α_AVHRR_GD was calculated using the Saunders (1990) model with AVHRR channels 1 and 2 (Fig. 2). Sensor drift degradation was taken in account using the methods provided by Mitchell (1999). This provided an effective reflectance factor used in the Saunders model to estimate α_AVHRR_GD for all 97 AVHRR images; results are discussed below (Section 4.2.5). This is required to calculate Rs_up (Fig. 3 and Section 3.2.7). 3.2.6 Vegetation Cover (VegCov) The value of VegCov _AVHRR_GD is calculated as continuous grids for the specific days of AVHRR data acquisition in a two -step process. This is used as a covariate and is also used to calculate εs, which is in turn used to calculate Rl_up (Fig. 3 and Section 3.2.7). Firstly, a stratum of woody and non-woody vegetation was defined. This involved classifying reflective AVHRR data from several dates throughout 1987 into a woody/non-woody vegetation stratum (Jupp et al., 1990). Secondly, empirical relationships between in situ LAI and AVHRR channels 1 (red) and 2 (NIR) were developed. For non-woody vegetation, LAI = –1.15 + 0.96 * (NIR/red) (McVicar et al., 1996a), and for woody vegetation, LAI = – 4.65 + 4.22 * (NIR/red) (McVicar et al., 1996b) were used to develop AVHRR-based estimates of LAI. Finally, LAI_AVHRR_GD were converted into estimates of VegCov using the relationship VegCov = 100.0 * (1.0 − exp ( − LAI / 2 ) ) , assuming random distribution of foliage above the soil and uniform leaf-angle distribution (Choudhury, 1989). Results are presented in Section 4.2.6 below. 3.2.7 Net Radiation (Rn) Net radiation can be expressed as the sum of its four main components: Rn= Rs_down – Rs_up + Rl_down – Rl_up
(19)
where: Rs_down is the downward shortwave radiation (0.15–4 µm) from the sun and atmosphere; Rs_up is the upward (or reflected) shortwave radiation by the surface; Rl_down is the longwave radiation (> 4 µm) emitted from the atmosphere toward the surface; and Rl_up is the longwave radiation emitted from the surface into the atmosphere. The four components can be calculated from the six variables introduced above (refer to Fig. 3). For Rn an IC approach is used. This is because Rn is strongly influenced by α and Ts, which can be discontinuous due to land use (for example, when agricultural land is adjacent to remnant wooded vegetation). Information about α, Ts , and VegCov can be obtained directly from the spatially dense AVHRR data (Fig. 2). This means that changes in α, Ts , and VegCov will be accounted for in space at 1/64-th of a degree (the spatial resolution of the
18
resampled AVHRR data), and, in time, near-monthly (the temporal resolution of the AVHRR data base). Downward shortwave radiation (Rs_down) is modelled by substituting τ0 for τz from Eq. (16) into Eq. (14). Hence the model for Rs_down becomes: Pz
Rs_down = τ0P0
p/ 2
cos θs Q0'
(20)
In this model the input variables required for the extended NDTI study site (Fig. 1) are: 1. a continuous raster of τ0 (see Section 3.2.3 above); 2. the specific time-of-day, day-of-year, and geographical position of the pixel centre of resampled AVHRR data are used to determine p and cos θs ; 3. a continuous raster of height obtained from AUSTDEM Ver 4.0 at a 1/60-th of a degree, purchased from the Australian National University’s Centre for Resource and Environmental Studies, was used. This data was resampled to a 1/64-th of a degree to be used with the MDB AVHRR data base established for this study; and 4. slope and aspect, derived from the 1/64-th of a degree DEM, are used to modify the calculation (Iqbal, 1983). Upward shortwave radiation (Rs_up) is calculated as Rs_up = α Rs_down
(21)
where: Rs_up is the upward (or reflected) shortwave radiation (W m –2); α is the albedo of the surface; and Rs_down is the downward shortwave radiation (W m –2). Downward longwave radiation (Rl_down) is given by Rl_down = εa σTa4
(22)
where: Rl_down is the longwave downward radiation (W m –2); εa is the effective atmospheric emissivity; σ is the Stefan–Boltzmann constant (5.67 x 10–8 W m –2 K–4); and Ta is air temperature at reference height (K). The value of εa is given by (Brutsaert, 1975) as:
ε a = 1.24 ( e a/ Ta)1 / 7
(23)
where: ea is the vapour pressure (hPa or millibars; these units are equivalent); and Ta is air temperature (K) at reference height. The value of εa_AVHRR_GD is calculated using ea_AVHRR_GD and Ta_AVHRR_GD in Eq. (23), which has subsequently been used in Eq. (22) to produce Rl_down_AVHRR_GD. Upward longwave radiation (Rl_up) is given by Rl_up = εs σ Ts4
(24) 19
where: Rl_up is the longwave upward radiation (W m –2); εs is the surface emissivity; σ is the Stefan–Boltzmann constant (5.67 x 10 –8 W m –2 K–4); and Ts is the surface temperature (K). The relatively small component of outgoing longwave radiation by reflected sky radiation is ignored (Jackson et al., 1985, p. 156). The value of εs is calculated as εs = f v εv + (1 − f v ) εg
(25)
where: εs is the surface emissivity; εv is the vegetation emissivity; εg is the ground (or soil) emissivity; and f v is the fraction of vegetation, which is derived from VegCov _AVHRR_GD (see Section 4.2.6 above). The value of εs_AVHRR_GD is calculated using VegCov _AVHRR_GD , with 0.98 used for εv and 0.96 for εg (Wan and Dozier, 1996) in Eq. (25). Values of ε s_AVHRR_GD are subsequently used with Ta_AVHRR_GD in Eq. (24) to produce Rl_up_AVHRR_GD.
3.3 Applying the Spatial Covariates The CI approach is used to develop the 97 surfaces of the NDTI for the focus area of the MDB (Fig. 1). Firstly, the NDTI is calculated at the ABM stations in Fig. 1, denoted NDTI_AVHRR_PT . See Jupp et al. (1998) for the specific choices and assumptions of the REBM used to invert T0 and T∞. At the 70 ABM stations these two bounding temperatures are combined with AVHRR Ts to calculate NDTI_AVHRR_PT (Eq. 11). Secondly, the output surfaces NDTI_AVHRR_GD are spatially interpolated using the three covariates ((i) Ts – Ta ; (ii) VegCov ; and (iii) Rn) and possibly elevation with the input data being NDTI_AVHRR_PT . For each of the 97 AVHRR overpasses there is usually less than the potential maximum of 70 points (each ABM station is considered a ‘point’) due to cloud contamination in different parts of each of the 97 AVHRR overpasses. To decide on which spline model to use, several PTPS models have been tested; they are: (a) quint-variate PTPS incorporating a bi-variate TPS function of longitude and latitude with constant linear dependencies on Ts – Ta , VegCov, and Rn ; (b) sext-variate PTPS incorporating a bi-variate TPS function of longitude and latitude with constant linear dependencies on Ts – Ta , VegCov, Rn and elevation; (c) sext-variate PTPS incorporating a tri-variate TPS function of longitude, latitude, and elevation (in units of metres) with constant linear dependencies on Ts – Ta , VegCov, and Rn ; and (d) sext-variate PTPS incorporating a tri-variate TPS function of longitude,
20
latitude, and elevation (in units of kilometres) with constant linear dependencies on Ts – Ta , VegCov, and Rn. As above, scaling does not influence the output statistics when using elevation, or any other variable, as a constant linear dependent (or covariate) in a PTPS. However, when using elevation (or any other variable) as an independent spline variable, the units can influence the output statistics. To account for this, when using elevation as an independent spline variable, elevation is used in units of metres and kilometres (see cases (c) and (d) above). The value of m was varied from 2 to 4 (three cases) for each of the above four cases ((a) to (d) above). In total, 12 options were explored for the interpolation of NDTI (see Section 4.3 for the results).
4 RESULTS 4.1 Selecting the Spatial Covariates There are 97 AVHRR overpasses in the data base; after visual cloud clearing there are 73 valid overpasses at Cobar. The three potential covariates, (i) Ts – Ta ; (ii) VegCov; and (iii) Rn, were linearly regressed against the REBM estimates of the NDTI or λEa. Results are presented in Table 3. The REBM is fully detailed in Jupp et al. (1998). Only data with a composite surface resistance (rs) (Jupp et al., 1998) greater than 0 s m–1 and Rn greater than 125 W m–2 are used in this analysis. Results describing the selection of this data filter are provided in Table 4. When regression analysis between the REBM NDTI and one potential covariate in turn is performed, Ts – Ta has the largest r2 and lowest SEY (Table 3). Using multiple linear regression shows that Ts – Ta and Rn are able to explain 79% of the variance within the NDTI, while Ts – Ta and VegCov are only able to explain 60% (Table 3). When all three potential covariates are used, 85% of the variance within the NDTI is explained and the SEY is less than 0.1. This illustrates the worth of using all three covariates to spatially interpolate the NDTI. It is interesting to note that, when regressing λEa against each potential covariate in turn, Rn provides the highest r2 and lowest SEY (Table 3). When both Ts – Ta and Rn are used, the resulting r2 and SEY are almost identical to when Ts – Ta , Rn , and VegCov are used (Table 3). This indicates that, if interpolating λEa rather than ma , VegCov is not as important as Ts – Ta and Rn. When only observations that have a specific time-of-day modelled Rn > 125 W m–2 are used, Ts – Ta, Rn , and VegCov are able to explain 85% of the variance in REBM NDTI (Table 4). Relaxing the Rn threshold reduces the r2 statistic but increases the number of observations in the analysis, and vice versa. Using the threshold of Rn > 125 W m–2 balances the strength of the relationship (between the potential covariates Ts – Ta , Rn , and VegCov and the REBM NDTI) and the number of observations used. Note from Table 4 that using a threshold based upon Rn has no influence on the statistics between the three potential covariates of the REBM λEa.
21
Table 3. Regression results, coefficient of determination (r2), standard error of the estimate of Y on X (SEY), and degrees of freedom (df) using REBM NDTI or REBM λEa as the dependent variable and the potential covariable(s) as the independent variable(s). There are 49 observations.
Dependent Variable REBM λEa
REBM NDTI Independent Variable(s)
r2
SEY
df
r2
SEY
df
Ts – Ta
0.42
0.17
47
0.01
67.1
47
Rn
0.00
0.23
47
0.38
53.0
47
VegCov
0.31
0.19
47
0.21
59.7
47
Ts – Ta and Rn
0.79
0.11
46
0.95
15.4
46
Ts – Ta and VegCov
0.60
0.14
46
0.21
60.3
46
Ts – Ta , Rn , and VegCov
0.85
0.09
45
0.97
12.5
45
Table 4. Multiple linear regression results, r2, SEY, and n, using different filters. REBM NDTI or REBM λEa is the dependent variable and Ts – Ta , Rn , and VegCov are the independent variables used in the multiple linear regression. If Rn was greater than the threshold, it also had to have rs > 0 s m–1 before it was used. df = n – 4.
Dependent Variable REBM NDTI REBM λEa Data Condition –1
rs > 0 s m
–2
Rn > 50 W m
r2
SEY
n
r2
SEY
n
0.77
0.10
56
0.97
12.1
56
0.82
0.09
53
0.97
12.2
53
–2
0.83
0.09
51
0.97
12.3
51
–2
Rn > 125 W m
0.85
0.09
49
0.97
12.5
49
Rn > 150 W m–2
0.85
0.09
45
0.97
12.9
45
Rn > 200 W m–2
0.87
0.09
34
0.97
13.6
34
Rn > 100 W m
Regression model estimates of the NDTI and λEa, denoted NDTI' and λEa' respectively, based on one covariate (Ts – Ta), two covariates (Ts – Ta and Rn) and three covariates (Ts – Ta , Rn , and VegCov) have been calculated. These have then been cross-plotted against the REBM NDTI or REBM λEa , respectively (see Fig. 4). Using three covariates to model the NDTI' explains the most variance of the REBM NDTI (Fig. 4 (a) to (c)). This confirms that all three potential covariates should be developed as spatial surfaces to interpolate the NDTI. If interested in spatially interpolating λEa from comparing λEa' with REBM λEa (Fig. 4 (d) to (f)), it is possible that this can be performed adequately using only Ts – Ta and Rn.
22
Figure 4. Crossplots of REBM NDTI with regression model NDTI' [(a) to (c)]; and REBM λEa with regression model λEa' [(d) to (f)]. The regression models are based on Ts – Ta [(a) and (d)]; Ts – Ta and Rn [(b) and (e)]; Ts – Ta , Rn , and VegCov [(c) and (f)].
23
4.2 Developing the Spatial Covariates 4.2.1 Air Temperature (Ta) Detailed results for the spatial interpolation of Ta_AVHRR_GD , using Ta_AVHRR_PT as the input data (Fig. 2), for the 12 options described in Section 3.2.1, are presented in Table 5. The output statistics reported as square roots are in the units of the dependent variable (Hutchinson, 1997). For all 12 spline models, λ was automatically selected by minimising GCV (m, λ ) . Introducing elevation into the spline model, either as a covariate or as an independent variable, reduces the measures of error: compare GCV (m, λ ) and T (m, λ ) between all tri-variate models to the bi-variate model (Table 5). On these grounds the bi-variate TPS is not considered further. For the tri-variate TPS with elevation in metres for m = 2, 3, and 4, tr(I – A) reaches, or is very close to, its maximum possible value, n – M (refer to Table 5). This indicates that the fitted spline is a least-squares regression of the data using the M low-order monomials in the set φ j . Consequently the tri-variate TPS with elevation in metres is not considered further. For the remaining 6 models (tri-variate PTPS, m = 2, 3, and 4, and tri-variate TPS with elevation in kilometres, m = 2, 3, and 4), the value of GCV (m, λ ) ranges from 0.792 to 0.863ºC and
T (m, λ ) ranges from 0.342 to 0.424ºC, (Table 5). There is
very little difference between the 6 remaining models. Having relatively low values of GCV (m, λ ) and T (m, λ ) , the tri-variate PTPS, m = 4, and the tri-variate TPS with elevation in kilometres, m = 3, are the contenders for consideration (Table 5). Values of Ta_AVHRR_GD were calculated for both spline models and the differences compared. Descriptive statistics presented in Table 6 indicate that the two models are essentially identical with most statistics, except for the first percentile, being within 0.5ºC of each other. These differences are within measurement error of Ta. Analysing the differences on a per-pixel basis (calculated as tri-variate PTPS, m = 4, Ta_AVHRR_GD minus tri-variate TPS, m = 3, elev km, Ta_AVHRR_GD) reveals that the mean and median Ta differences are within –0.15ºC. The central 98% of data is different by less than 1.05ºC. The largest per-pixel difference between the two spline models of Ta is 1.79ºC (Table 6). It was decided to use a tri-variate TPS, m = 3, elevation in km, to spatially interpolate the 97 surfaces of Ta_AVHRR_GD from the input data Ta_AVHRR_PT based on the following reasons: 1. Using spline models (or any other model for that point) with less terms (10 monomial terms rather than 11, see Table 5) potentially provides more stability away from the observations; 2. Independent of m, MSR and σ ε2 are consistently 0.1 ºC smaller for the tri-variate TPS, elevation in km, when compared to the tri-variate PTPS (Table 5);
24
Table 5. Model output exploring the influence of the spline, m, and scaling of elevation (if used) for the spatial interpolation of Ta.
Model
M
tr(I – A)
tr(A)
σ ε2
GCV ( m, λ )
T (m, λ )
MSR
(°C)
(°C)
(°C)
(°C)
Bi-variate TPS m=2
3
54.2
15.8
1.11
0.463
0.857
0.974
m=3
6
55.8
14.2
1.12
0.452
0.895
1.00
m=4
10
54.6
15.4
1.15
0.476
0.897
1.02
m=2
4
52.5
17.5
0.860
0.372
0.646
0.745
m=3
7
54.8
15.2
0.842
0.348
0.659
0.745
m=4
11
54.7
15.3
0.829
0.342
0.649
0.733
m=2
4
66.0
4.0
1.04
0.241
0.979
1.01
m=3
10
60.0
10.0
0.908
0.318
0.778
0.841
m=4
20
49.2
20.8
0.880
0.402
0.618
0.738
m=2
4
41.3
28.7
0.863
0.424
0.509
0.662
m=3
10
50.5
19.5
0.792
0.355
0.571
0.673
m=4
20
45.6
24.4
0.837
0.399
0.545
0.675
Tri-variate PTPS
Tri-variate TPS (elev m)
Tri-variate TPS (elev km)
Table 6. Descriptive statistics for the two spline models used to interpolate Ta and the perpixel differences between the two output images. Units are ºC.
Ta_AVHRR_GD
Min
Max
1%
99%
Mean
Median
1: Tri-Variate PTPS, m = 4
15.69
29.55
19.89
28.86
25.03
25.17
2: Tri-Variate TPS, m = 3 (elev km)
15.22
29.61
20.72
28.90
25.17
25.30
Grid 1 – Grid 2
–1.50
1.79
–1.05
0.27
–0.144
–0.1
25
3. The results of the two spline models with the lowest similar (Table 6); and
GCV (m, λ ) are very
4. Relative to Ta interpolation, the lapse-rate inherent in the tri-variate PTPS is a fixed function of altitude, whereas in the tri-variate TPS the lapse-rate can vary with elevation, longitude, and latitude. Allowing the environmental lapse-rate to vary as a function of altitude is probably more realistic (Hutchinson, pers. comm.). Fig. 5(a) shows the output using a tri-variate TPS, m = 3, elevation in km, for 22 September 1987; Ta gradually increases from the south-east to the north-west. Modifying this general trend are local changes in elevation. The north-west portion is considered semi-arid and in the south-east in September there is a spring growth flush of cereals crops (wheat, barley, and oats). For Ta_AVHRR_GD (Fig. 5(a)) there is no small-scale variation due to landcover (compare with Fig. 5(f)). This is a result of the low spatial density of ABM stations. For the other 96 AVHRR dates/times, Ta_AVHRR_GD was spatially interpolated using a tri-variate TPS, m = 3, elevation in km. There were several cases when tr(I – A) was smaller than 35.0. Hutchinson and Gessler (1994) generally recommend that if tr(I – A) < n/2 the data may be too sparse to support spline interpolation. However, n/2 is only a ‘rule-of-thumb’. This threshold was relaxed to determine what constitutes an ‘adequate’ spline model compared with a ‘doubtful’ spline model, which may need to be reprocessed (Table 7). A threshold of 25.0 was used to define spline models as ‘doubtful’ (Table 7). For the 15 ‘doubtful’ spline models, ANUSPLIN was rerun and the λ determined by minimising the T (m, λ ) , rather than λ being defined by automatically minimising the GCV (m, λ ) as with the other 82 cases. To minimise the T (m, λ ) , the error standard deviation estimate was supplied to the algorithm; this was calculated as the mean of
σ ε2 for the 82 ‘adequate’ cases. For these cases σ ε2 ranged from 0.422ºC to 1.66ºC and averaged 0.784ºC. The 97 Ta_AVHRR_GD surfaces were combined with Ts_AVHRR_GD to develop the covariate Ts – Ta which is subsequently used as a covariate for the spatial interpolation of the NDTI. An example of this output is illustrated in Fig. 7(a), below. The variable Ta_AVHRR_GD is also used in calculation of εa. Both Ta and εa are used subsequently to calculate Rl_down (Fig. 3 and Fig. 6(c)). This is in turn used in the calculation of Rn (Fig. 3 and Fig. 7(c)). Table 7. Influence of the threshold to define the number of ‘doubtful’ spline models for the spatial interpolation of Ta.
Thresh- Nr of times when Nr of times when Nr of times when Nr of ‘doubtful’ old tr(I – A) < threshold tr(A) = n – M tr(I – A) = n – M spline models 35.0 30.0 25.0 20.0
27 18 9 4
4 4 4 4
26
2 2 2 2
33 24 15 10
4.2.2 Vapour Pressure (ea) Detailed results for the spatial interpolation of ea_AVHRR_GD , using ea_AVHRR_PT as the input data, for the 12 options described in Section 3.2.2 are presented in Table 8. Again, as with Ta above, for all 12 spline models λ was automatically selected by minimising GCV (m, λ ) . Since ea changes with elevation, it was decided not to further consider the bi-variate TPS model. Independent of m, the tri-variate TPS, elevation in metres, the GCV (m, λ ) value is much worse than other remaining 6 cases (Table 8). Consequently, the tri-variate TPS, elevation in metres, spline model will not be considered further. The remaining 6 models (tri-variate PTPS, m = 2, 3, and 4, and trivariate TPS with elevation in kilometres, m = 2, 3, and 4) are essentially equivalent. The two models, tri-variate PTPS, m = 2, and tri-variate TPS, elevation in km, m = 2, have the lowest GCV (m, λ ) (Table 8). It was decided to calculate ea_AVHRR_GD for these two spline models and analyse the differences. Descriptive statistics presented in Table 9 confirm that the two models are essentially identical with all statistics being within 0.2 hPa of each other; most are within 0.1 hPa. These differences are within measurement error of ea. Calculating the differences on a per-pixel basis as tri-variate PTPS, m = 2, ea_AVHRR_GD minus trivariate TPS, m = 2, elev km, e a_AVHRR_GD shows that the mean and median ea differences are within –0.04 hPa (Table 9). The central 98% of data has estimates of ea that are different by less than 0.22 hPa. The largest per-pixel difference between the two models is 0.34 hPa (Table 9). Since the two spline models are essentially identical, and as it is assumed that ea will change with a fixed altitudinal lapse-rate, it was decided to use a tri-variate PTPS, m = 2. Fig. 5(b) shows the output for 22 September 1987. The area of high ea in the northern portion of Fig. 5(b) approximately conforms to the area visually identified as cloud in the AVHRR data (Fig. 5 (d) to (f)). For the other 96 AVHRR dates/times, ea_AVHRR_GD was spatially interpolated using a tri-variate PTPS, m = 2. There were several cases when tr(I – A) was smaller than 35.0. Again, as with Ta above, we explored relaxing this ‘rule-of-thumb’ threshold to determine what constitutes an ‘adequate’ and a ‘doubtful’ spline model for the spatial interpolation of ea_AVHRR_GD (Table 10). A threshold of 30.0 was used to define spline models as ‘doubtful’ (Table 10). For the 9 ‘doubtful’ spline models, ANUSPLIN was rerun and the λ determined by minimising the T (m, λ ) , rather than the λ being defined by automatically minimising the GCV (m, λ ) as was done with the other 88 cases. To minimise the T (m, λ ) the error standard deviation estimate was supplied to the algorithm; this was calculated as
σ ε2 for the 88 ‘adequate’ cases. For these cases σ ε2 ranged from 0.405 hPa to 1.74 hPa and averaged 0.896 hPa. The 97 ea_AVHRR_GD surfaces were subsequently used in the calculation of εa , which in turn is used in the calculation of Rl_down (Fig. 3 and Fig. 6(c)). This is in turn used in the calculation of Rn (Fig. 3 and Fig. 7(c)).
the mean of
27
Table 8. Model output exploring the influence of the spline, m, and scaling of elevation (if used) for the spatial interpolation of ea.
Model
M
tr(I – A)
tr(A)
σ ε2
GCV ( m, λ )
T (m, λ )
(hPa)
(hPa)
(hPa)
(hPa)
MSR
Bi-variate TPS m=2
3
41.7
28.3
1.12
0.549
0.667
0.864
m=3
6
57.7
12.3
1.15
0.437
0.947
1.04
m=4
10
45.3
24.7
1.14
0.546
0.740
0.920
m=2
4
46.2
23.8
1.11
0.527
0.734
0.904
m=3
7
56.0
14.0
1.13
0.452
0.904
1.01
m=4
11
46.5
24.5
1.15
0.546
0.747
0.925
m=2
4
63.2
6.8
1.30
0.384
1.17
1.23
m=3
10
53.8
16.2
1.31
0.552
1.01
1.15
m=4
20
50.0
20.0
1.22
0.551
0.871
1.03
m=2
4
35.1
34.9
1.12
0.561
0.563
0.794
m=3
10
49.2
20.8
1.17
0.534
0.821
0.980
m=4
20
36.6
33.4
1.21
0.603
0.631
0.873
Tri-variate PTPS
Tri-variate TPS (elev m)
Tri-variate TPS (elev km)
Table 9. Descriptive statistics for the two spline models used to interpolate ea and the per-pixel differences between the two output images. Units are hPa.
ea_AVHRR_GD
Min Max
1%
99%
Mean
Median
1: Tri-Variate PTPS, m = 2
6.38
12.16
7.32
11.92
8.9167
8.50
2: Tri-Variate TPS, m = 2 (elev km)
6.57
12.01
7.27
11.82
8.9363
8.55
–0.28
0.34
–0.17
0.22
–0.0196
–0.04
Grid 1 – Grid 2
28
Table 10. Influence of the threshold to define the number of ‘doubtful’ spline models for the spatial interpolation of ea.
Thresh- Nr of times when Nr of times when Nr of times when Nr of ‘doubtful’ old tr(I – A) < threshold tr(A) = n – M tr(I – A) = n – M spline models 35.0
20
0
1
21
30.0
8
0
1
9
25.0
4
0
1
5
20.0
2
0
1
3
4.2.3 Atmospheric Transmittance (ττ0) Detailed results for the spatial interpolation of τ0_AVHRR_GD , using τ0_AVHRR_PT as the input data, for the bi-variate TPS described in Section 3.2.3 are presented in Table 11. Again, as with Ta and ea above, λ was automatically selected by minimising GCV (m, λ ) . From Table 11 it can be seen that the bi-variate TPS, m = 2, has the lowest GCV (m, λ ) ; however all the models have very similar output statistics and are very similar. It was decided to use the model with the lowest
GCV (m, λ ) , which also has
the lowest number of monomial terms, to interpolate surfaces of τ0 for the other 96 times. Fig. 5(c) shows the output for 22 September 1987. Lower τ0 are located where there is higher ea (Fig. 5(b)), and these areas approximately conform to the area visually identified as cloud in Fig. 5 (d) to (f). For the other 96 AVHRR dates/times τ0 was spatially interpolated using a bi-variate TPS, m = 2. Again, as with Ta and ea above, there were several cases when tr(I – A) was smaller than 35.0. Again, as with Ta and ea above, relaxing this ‘rule-of-thumb’ threshold to determine what constitutes an ‘adequate’ and a ‘doubtful’ spline model for the spatial interpolation of τ0_AVHRR_GD was explored (Table 12). A threshold of 30.0 was used to define spline models as ‘doubtful’ (Table 12). For the 9 ‘doubtful’ spline models ANUSPLIN was rerun and the λ determined by minimising the T (m, λ ) (rather than the λ being defined by automatically minimising the GCV (m, λ ) , as with the other 88 cases). To minimise the T (m, λ ) , the error Table 11. Model output exploring the influence of m for the spatial interpolation of τ0.
Model
M
tr(I – A)
tr(A)
GCV (m, λ )
(%)
T (m, λ )
MSR
(%)
(%)
σ ε2 (%)
Bi-variate TPS m=2
3
60.7
9.3
0.0157
0.00533
0.0136
0.0146
m=3
6
60.8
9.2
0.0160
0.00539
0.0139
0.0149
m=4
10
59.3
10.7
0.0163
0.00586
0.0138
0.0150
29
standard deviation estimate was supplied to the algorithm; this was calculated as the mean of
σ ε2 for the 88 ‘adequate’ cases and was 0.0223%. For the ‘adequate’
cases σ ε2 ranged from 0.0113% to 0.0517%. The 97 τ0_AVHRR_GD surfaces were subsequently used in the calculation of Rs_down (Fig. 3 and Fig. 6(c)), which is in turn used in the calculation of Rn (Fig. 3 and Fig. 7(c)). Table 12. Influence of the threshold to define the number of ‘doubtful’ spline models for the spatial interpolation of τ0.
Thresh- Nr of times when Nr of times when Nr of times when Nr of ‘doubtful’ old tr(I – A) < threshold tr(A) = n – M tr(I – A) = n – M spline models 35.0 30.0 25.0 20.0
12 7 4 1
0 0 0 0
2 2 2 2
14 9 6 3
4.2.4 Surface Temperature (Ts) The value of Ts was calculated using the split window algorithm outlined in Section 3.2.4 above with AVHRR channels 4 and 5, denoted Ts_AVHRR_GD (Fig. 2). This has been performed for the 97 AVHRR overpasses. Fig. 5(d) shows the increase in Ts toward the semi-arid area in the north-west; lower Ts values to the south-east are associated with an increase in ma ; the increases in λEa are associated with increases in VegCov due to the spring growth flush of cereal crops (Fig. 5(f)). Visual cloud clearing means that AVHRR data are only available where there was no cloud. The 97 Ts_AVHRR_GD surfaces were combined with Ta_AVHRR_GD to develop the covariate Ts – Ta (Fig. 7(a)). Ts is also used to calculate Rl_up (Fig. 6(d)), which is used to calculate Rn (Fig. 3). 4.2.5 Broadband Albedo (α α) Values of α were calculated directly from AVHRR channels 1 and 2 (Fig. 2). This output is denoted α_AVHRR_GD. This has been performed for the 97 AVHRR overpasses; these are used to calculate Rs_up which is used to calculate Rn (Fig. 3). Fig. 5(e) give examples of α; the high albedo area to the west is Lake Frome, a salt lake. Remnant forests and woodlands have the lower albedos. ____________________________________________________________________ Figure 5 (opposite). Continuous grids of the six required variables: (a) Ta (°C); (b) ea (hPa); (c) τ0 (%); (d) Ts (°C); (e) α (%); and (f) VegCov (%) for the extended NDTI area shown in Fig. 1. The three variables (a), (b), and (c) are derived from spatial interpolation based on data recorded at the 70 ABM stations. The three variables (d), (e), and (f) are derived from AVHRR data. Some areas in the AVHRR data are cloud-affected and are not reliable; these have been nulled and are the black areas internal to the data shown in (d), (e), and (f). Note the difference in the smoothness of the output based on the source of the data. Refer to Fig. 2 for the development of the six variables.
30
35
45
15
0
(a) Air Temperature (°C)
(d) Surface Temperature (°C) 14
25
4
0
(b) Vapor Pressure (hPa)
(e) Broadband Albedo (%) 72
100
60 (c) Atmospheric Transmittance (%)
0 (f) Vegetation Cover (%)
Figure 5. Development of the 6 variables (detailed caption opposite).
31
4.2.6 Vegetation Cover (VegCov) VegCov was calculated using a GIS strata of wooded/non-woody vegetation with empirical relationships between in situ LAI and AVHRR channels 1 and 2 for the 97 AVHRR overpasses. Again, visual cloud clearing means that this data in only available where there was no cloud influencing the observation. Fig. 5(f) illustrates this output for 22 September 1987. The areas of higher amounts of VegCov are remnant forests and woodlands and the large expanse of moderate values across the southern portion of the image (Fig. 5(f)) is due to the spring growth flush of cereal crops. The large expanse with 0% VegCov toward the north-west is a semi-arid region (Fig. 5(f)) and corresponds to the area of high Ts in Fig. 5(d). VegCov is used as a covariate and is also used to calculate εs (Eq. 25), which is used to calculate Rl_up (Eq. 24 and Fig. 6(d)), which in turn is used to calculate Rn (Fig. 3 and Fig. 7(c)). 500
400
150
250 (c) Downward Longwave Radiation (W m–2)
(a) Downward Shortwave Radiation (W m–2) 100
550
0
300
(b) Upward Shortwave Radiation (W m–2)
(d) Upward Longwave Radiation (W m–2)
Figure 6. Development of the 4 Rn components (detailed caption opposite).
32
4.2.7 Net Radiation (Rn) The four components of Rn have been developed using the methods described in Section 3.2.7 and in Fig. 3. For NOAA 9 Orbit 14301, the resulting four components – R s_down (Fig. 6(a)), Rs_up (Fig. 6(b)), Rl_down (Fig. 6(c)), and Rl_up (Fig. 6(d)) – are illustrated. In Fig. 6(a), the longitudinal banding of Rs_down is due to changes in p and cos θs when the AVHRR data was acquired at 1629 h local time. Modification by elevation, slope, and aspect are also shown. For Rs_up (Fig. 6(b)), the influence of α and of the banding of Rs_down are illustrated. The spatial pattern of Rl_down, shown in Fig. 6(c), is dominated by Ta. Variations in Rl_up (Fig. 6(d)) follow the spatial variability present in Ts. The components have been added together (Eq. 19) to provide continuous rasters of Rn , denoted Rn_AVHRR_GD , which is used as a covariate. Values of Rn_AVHRR_GD for 22 September 1987 are illustrated in Fig. 7(c). 4.3 Applying the Spatial Covariates Continuous surfaces of the NDTI, denoted NDTI_AVHRR_GD , have been developed for each of the 97 single AVHRR afternoon overpasses using a CI method outlined above. For the AVHRR data acquired on 22 September 1987, the three covariates Ts – Ta , Rn , and VegCov (see Fig. 7 (a), (b), and (c), respectively) have been used in this process. Detailed results for the spatial interpolation of NDTI_AVHRR_GD , using NDTI_AVHRR_PT as the input data and for the 12 options described in Section 3.3, are presented in Table 13. Again, as with Ta , ea , and τ0 above, λ was automatically selected for all 12 spline models by minimising GCV (m, λ ) . The three models with the lowest
GCV (m, λ ) statistics are the sext-variate PTPS
(elevation in metres) with 3 covariates for m = 2 and 3, and the quint-variate PTPS with 3 covariates, m = 2 (Table 13). The GCV (m, λ ) statistic for the quint-variate PTPS with 3 covariates, m = 2 (0.143, Table 13) is only slightly higher than for the other two models (0.136 and 0.137, respectively, Table 13). T (m, λ ) for the quintvariate PTPS with 3 covariates, m = 2, is 0.0449 which is relatively much lower than the values of 0.0671 and 0.0685 for the sext-variate PTPS (elevation in metres) with 3 covariates for m = 2 and 3, respectively (Table 13). The quint-variate PTPS with 3 covariates, m = 2, also warrants further detailed analysis as elevation is not used, either as an independent spline variable or a covariate, in this spine model. For the quint-variate PTPS with 3 covariates, m = 2, tr(I – A) reaches its maximum possible value, n – M (Table 13), which indicates that the fitted spline is a least-squares regression of the data using the M low-order monomials in the set φ j . _____________________________________________________________________ Figure 6 (opposite). Continuous grids of the four radiation components: (a) R s_down (W m–2); (b) Rs_up (W m–2); (c) Rl_down (W m–2); and (d) Rl_up (W m–2) for the extended NDTI area shown in Fig. 1 used to calculate Rn. Two variables (b) and (d) require input variables from AVHRR data (see Fig. 3). Consequently the areas cloud affected in the AVHRR data (see Fig. 5) are nulled and are shown as black for these two variables. Refer to Fig. 3 for the development of the 4 components.
33
10
350
–10
50
(a) Ts – Ta (°C)
(c) Net Radiation (W m–2) 100
Figure 7. An example of the 3 covariates. Continuous grids of the three covariates: (a) Ts – Ta (°C); (b) VegCov (%); and (c) Rn (W m–2) for the extended NDTI area shown in Fig. 1. All three covariates use AVHRR data, consequently areas cloud affected in the AVHRR data (see Fig. 5 and Fig. 6) are nulled and are shown as black for each of the three covariates.
0 (b) Vegetation Cover (%)
The other nine spline models, quint-variate PTPS with 3 covariates, m = 3 and 4; sextvariable PTPS with 4 covariates, m = 2, 3, and 4; sext-variable PTPS (elevation in km) with 3 covariates, m = 2, 3, and 4; and sext-variate PTPS (elevation in metres) with 3 covariates, m = 4, are discarded due to the magnitude of GCV (m, λ ) (Table 13). Consequently, NDTI surfaces were generated for the 3 remaining spline models: 1. quint-variate PTPS with 3 covariates, m = 2. This incorporates a bi-variate TPS function of longitude and latitude with constant linear dependencies on Ts – Ta , VegCov, and Rn ; 2. sext-variate PTPS (elevation in metres) with 3 covariates, m = 2. This incorporates a tri-variate TPS function of longitude, latitude, and elevation (in units of metres) with constant linear dependencies on Ts – Ta , VegCov, and Rn; and
34
Table 13. Model output exploring the influence of the spline, m, and scaling of elevation (if used) for the spatial interpolation of NDTI.
tr(A)
GCV ( m, λ )
T (m, λ )
σ ε2
Model
M
tr(I – A)
Quint-variate PTPS (3 covariates) m=2 m=3
6 9
48.0 45.0
6.0 9.0
0.143 0.149
0.0449 0.0557
0.127 0.125
0.135 0.136
m=4
13
41.0
13.0
0.163
0.0698
0.124
0.142
m=2 m=3
7 10
47.0 44.0
7.0 10.0
0.146 0.152
0.0491 0.0592
0.127 0.124
0.136 0.138
m=4
14
40.0
14.0
0.167
0.0732
0.124
0.144
Sext-variate PTPS (elev m) 3 covariates m=2 m=3 m=4
7 13 23
31.3 28.8 23.5
22.7 25.2 30.5
0.136 0.137 0.155
0.0671 0.0685 0.0672
0.0788 0.0733 0.0766
0.104 0.100 0.102
Sext-variate PTPS (elev km) 3 covariates m=2 m=3 m=4
7 13 23
47.0 41.0 31.0
7.0 13.0 23.0
0.146 0.162 0.200
0.0491 0.0692 0.0989
0.127 0.123 0.115
0.136 0.141 0.152
MSR
Sext-variate PTPS (4 covariates)
3. sext-variate PTPS (elevation in metres) with 3 covariates, m = 3. This incorporates a tri-variate TPS function of longitude, latitude, and elevation (in units of metres) with constant linear dependencies on Ts – Ta , VegCov, and Rn. The overall descriptive statistics for the three spline models are presented in Table 14. Descriptive statistics presented in Table 14 illustrate that the three models have somewhat similar minimum, mean, median, and 1% values. However, the 99% value increases from 0.62 to 0.73. Fig. 8 shows the resulting NDTI surface for the three spline models. While the main structure for each surface is relatively similar, in both the sext-variate PTPS (elev m) 3 covariates (m = 2 and m = 3), the NDTI increases as a function of elevation in the south-east (Fig. 8 (b) and (c)). There are more isolated hills having a higher NDTI value for the model with m = 3 (Fig. 8(c)) than for the model with m = 2 (Fig. 8 (b)). This has resulted in the increase of the 99% and maximum value for the sext-variate PTPS (elev m) 3 covariates, m = 3 (Table 14).
35
Table 14. Descriptive statistics for the spline models used to interpolate NDTI.
NDTI_AVHRR_GD
Min
Max
1%
99%
Mean
Median
Quint-variate PTPS 3 covariates, m = 2 Sext-variate PTPS (elev m) 3 covariates, m = 2 Sext-variate PTPS (elev m) 3 covariates, m = 3
–0.19
1.07
–0.07
0.62
0.2715 0.27
–0.21
0.99
–0.07
0.67
0.2859 0.28
–0.27
1.29
–0.10
0.73
0.2675 0.27
It is unlikely that the NDTI would increase as a function of elevation, especially considering the elevation range over this image area is from 27 to 922 m. It may be more physically realistic if areas with high elevation have lower NDTI values. This is because areas with higher relative elevation – hills – are usually associated with higher local slopes, and surface water should be laterally redistributed. This means that these areas may be drier, and hence the resulting NDTI could be expected to be lower. It is expected that this type of feedback would be incorporated as Ts increases, and so would not have to be modelled explicitly in the spline model. It was decided to use quint-variate PTPS with 3 covariates, m = 2, to develop the time series of NDTI surfaces. This has been performed on the other 96 AVHRR images in the MDB data base.
0 (a) Quint-variate PTPS 3 covariates, m = 2
1 NDTI (b) Sext-variate PTPS (elev m) 3 covariates, m = 2
(c) Sext-variate PTPS (elev m) 3 covariates, m = 3
Figure 8. Three different spline models tested. Resulting NDTI images for 22 September 1987 for the focus NDTI area shown in Fig. 1 for: (a) quint-variate PTPS with 3 covariates, m = 2. This incorporates a bi-variate TPS function of longitude and latitude with constant linear dependencies on Ts – Ta, VegCov, and Rn; (b) sext-variate PTPS (elevation in metres) with 3 covariates, m = 2. This incorporates a tri-variate TPS function of longitude, latitude and elevation (in metres) with constant linear dependencies on Ts – Ta, VegCov, and Rn; and (c) sext-variate PTPS (elevation in metres) with 3 covariates, m = 3. This incorporates a trivariate TPS function of longitude, latitude, and elevation (in metres) with constant linear dependencies on Ts – Ta, VegCov, and Rn.
36
5 DISCUSSION 5.1 Selecting the Spatial Covariates To adequately describe the variance present in the REBM NDTI, results presented confirmed that all three potential covariates (Ts – Ta , VegCov, and Rn) were required to spatially interpolate the NDTI. Consequently, spatial surfaces of these three covariates were developed. If λEa is to be interpolated, only Ts – Ta and Rn are required as covariates. 5.2 Developing the Spatial Covariates To produce the NDTI spatially, the CI approach is used. The NDTI was calculated, using a REBM, at ABM stations (considered points in the MDB). To support the spatial interpolation of the NDTI the three covariates were developed as spatial surfaces. A variety of data sources are used to develop the covariates, or the required intermediate variables; these are: (i) interpolated meteorological data (Ta , ea , and τ0); (ii) remotely sensed data (Ts and α ); (iii) supervised classification of AVHRR reflective data used as a GIS stratum with the time series of remotely sensing data (VegCov); and (iv) modelling Rn using as inputs all of the six variables above. At the specific time of AVHRR data acquisition, the interpolated meteorological surfaces (Ta , ea , and τ0) are fairly smooth, whereas those derived from AVHRR data (Ts , α , and VegCov) are spatially discontinuous, responding to changes in land cover and rainfall. By combining these data sources we are underpinning the spatial interpolation of the NDTI, using a CI approach, with the high spatial density of AVHRR data. Rather than interpolating the meteorological variables, an opportunity exists to use meteorological surfaces generated from a GCM. This has not been explored in this current research. 5.3 Applying the Spatial Covariates The NDTI was spatially interpolated using a quint-variate PTPS with 3 covariates, m = 2. This incorporates a bi-variate TPS function of longitude and latitude with constant linear dependencies on Ts – Ta , VegCov, and Rn. The high spatial density of AVHRR data is seen in the resulting NDTI images. Fig. 9 (a) shows the NDTI image for spring (22 September 1987) and Fig. 9 (b) shows the NDTI image for summer (25 December 1987). Changes in the NDTI between the two dates are shown in Fig. 9 (c), where blue is an increase, green fairly constant, and red a decrease between the two dates. Fig. 9 (a) shows that the Menindee Lake system is the only area where the NDTI > 0.8, with the southern portion of the wheat belt in the south-east having a NDTI of about 0.5. Some areas around the Menindee Lakes also have a NDTI of about 0.5, and these are assumed to have received scattered rainfall; the larger extent of this pattern, confirmed by a decrease in Ts , is seen in Fig. 5 (d). In Fig. 9 (b) some 37
1
100
0
0
(a) NDTI 22 Sept 1987
(d) VegCov (%) 22 Sept 1987 1
100
0
0
(b) NDTI 25 Dec 1987
(e) VegCov (%) 25 Dec 1987 0.5
30
–0.5
–30
(c) Change in NDTI (b) – (a)
(f) Change in VegCov (%) (e) – (d)
Figure 9. NDTI, VegCov, and difference between each for two dates. Images of the focus NDTI area (shown in Fig. 1) for: (a) NDTI for 22 September 1987; (b) NDTI for 25 December 1987; and (c) the difference of NDTI between these two dates, calculated as (b) – (a). (d) to (f): as for (a) to (c), except that VegCov is shown.
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areas with a NDTI > 0.8 are lakes and some are associated with flood irrigation of crops and remnant river red gum forests (Barham and Gulpa State Forests) along the Murray River. On the Cobar peneplain there are scattered areas with NDTI values of approximately 0.5, which correspond to areas where the NDTI has increased between the two dates (Fig. 9 (c)); these areas have probably received scattered rainfall. The AVHRR-derived estimates of VegCov are shown for spring (Fig. 9 (d)) and summer (Fig. 9 (e)). In Fig. 9 (d) and (e) the remnant deep-rooted forests are identified by their having a stable VegCov between the two dates. This is especially seen for the remnant river red gum forests along the Murray River. In the woodlands of the Cobar peneplain, VegCov is mainly stable. However, there are some areas on the Cobar peneplain (blue areas in Fig. 9 (f)) where VegCov has increased between the two dates. These areas are associated with an increase in the NDTI (Fig. 9 (c)). This increase in VegCov is probably associated with an increase in the amount of grass cover. These grasses have adapted to respond to short-term changes in available moisture and will be observable through the relatively open overstorey canopy (10–20%). These changes in NDTI between the two dates, shown in Fig. 9 (c), and in VegCov, shown in Fig. 9 (f), are similar, though not identical. For example, along the northeastern border of the two images there is a large negative change in VegCov due to harvesting of cereal crops, whereas in this area the NDTI remains fairly constant (it is less than 0.2 for both dates). To assist in regional agricultural management, including drought assessment, the resulting NDTI images and reflective-based images, such as VegCov or the NDVI, and their interactions, need to be analysed in a spatial–temporal context. If the NDTI is viewed as an indicator of moisture availability, and the NDVI is thought of as an indicator of moisture utilisation, then an opportunity exists to separate a climate-induced variability from management-induced variability (McVicar and Jupp, 1998). For example, in cropping or pasture agricultural systems, if the NDTI decreases during a growing season while the NDVI increases this would indicate that some of the decrease in moisture availability was due to transpiration. However, there may be cases where the NDTI declines during the growing season and the NDVI fails to show any response. Such a case could result from disease or insect damage early in the crop growing season and the resulting decrease in moisture availability may only be attributable to soil evaporation. During crop growing seasons these interactions may be best analysed by calculating the integrals under a time series of NDTI and NDVI images. The timing of the maximum NDTI and NDVI during crop growth periods would probably need to be included in this analysis.
6 CONCLUSIONS The NDTI, which can be considered a specific time-of-day version of the CSWI, is calculated using a REBM at the ABM stations where meteorological data are acquired. We have used a CI approach to spatially interpolate the NDTI from these isolated stations to generate a continuous surface in the MDB. To perform this, three covariates were required: Ts – Ta , VegCov , and Rn. These were obtained from a 39
combination of data sources, primarily directly from AVHRR data and spatial interpolation of selected meteorological variables, with Rn being modelled using an IC approach. We are not advocating CI in preference to IC; each is useful depending on the underlying issues present (Addiscott and Tuck, 1996). We are advocating, and have presented, a CI method that inherently uses the high spatial density of AVHRR data as the backbone for the spatial interpolation. The NDTI provides a link into regional water-balance modelling which does not require spatial interpolation of daily rainfall. Assessing spatial and temporal interactions between the NDTI and VegCov or NDVI will provide useful information about regional hydro-ecological processes, including agricultural management, within the context of Australia’s highly variable climate.
7 ACKNOWLEDGEMENTS This research has been supported in part with contributions by CSIRO, ACIAR (Australian Centre for International Agricultural Research), and LWRRDC (Land and Water Resources Research Development Commission). Thanks to Isabelle Balzer, Guy Byrne, James Davidson, Li Lingtao, Kate Mashford, Elizabeth McDonald, and Nicole Williams who, at various times, helped maintain the MDB AVHRR archive. Thanks to staff members from ABM (Garry Moore) and Queensland Department of Natural Resources (John Carter and Keith Moodie) for assistance developing the daily meteorological data set used in this study. Thanks to the many unknown individuals involved in collecting the daily meteorological data and also to those involved in designing, constructing, launching, and maintaining the NOAA series of satellites. Rama Nemani of University of Montana helped greatly with provision of the HOURLY code. Thanks to Michael Hutchinson (Australian National University, Centre of Resource and Environmental Studies), Warrick Dawes and Lu Zhang (both from CSIRO Land and Water), and Dean Graetz (CSIRO Earth Observation Centre) for helpful discussions while conducting this research and preparing this technical report. Thanks to Andrew Bell, from Exact Editing, for his services and to Heinz Buettikofer for the cover design. Thanks also to anyone we may have overlooked. The photo on the cover is from the collection of Dean Graetz.
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