lumped parameter modelling
DESCRIPTION
Lumped Parameter Modelling. P. Lewis & P. Saich RSU, Dept. Geography, University College London, 26 Bedford Way, London WC1H 0AP, UK. Introduction. introduce ‘simple’ lumped parameter models Build on RT modelling RT: formulate for biophysical parameters LAI, leaf number density, size etc - PowerPoint PPT PresentationTRANSCRIPT
Lumped Parameter Modelling
P. Lewis & P. Saich
RSU, Dept. Geography, University College London, 26 Bedford Way,
London WC1H 0AP, UK.
Introduction
• introduce ‘simple’ lumped parameter models • Build on RT modelling• RT: formulate for biophysical parameters
– LAI, leaf number density, size etc
– investigate eg sensitivity of a signal to canopy properties • e.g. effects of soil moisture on VV polarised backscatter or Landsat
TM waveband reflectance
– Inversion? Non-linear, many parameters
Linear Models
• For some set of independent variables
x = {x0, x1, x2, … , xn}
have a model of a dependent variable y which can be expressed as a linear combination of the independent variables.
110 xaay
22110 xaxaay
ni
iii xay
0
xay
Linear Models?
ni
iiiii xbxaay
10 cossin
ni
iiii bxaay
10 sin
nn
ni
i
ii xaxaxaaxay 0
202010
00 ...
xaeay 10
xay
Linear Mixture Modelling
• Spectral mixture modelling:– Proportionate mixture of (n) end-member spectra
– First-order model: no interactions between components
11
0
ni
i iF
1
0
ni
i ii Fr Fr
Linear Mixture Modelling
• r = {r, r, … rm, 1.0} – Measured reflectance spectrum (m wavelengths)
• nx(m+1) matrix:
1
2
1
0
112111101
11210101
10201000
1
1
0
0.10.10.10.10.1 n
nmmmm
n
n
m
P
P
P
P
r
r
r
Fr
Linear Mixture Modelling
• n=(m+1) – square matrix
• Eg n=2 (wavebands), m=2 (end-members)
Fr
rF 1
Reflectance
Band 1
Reflectance
Band 2
1
2
3
r
Linear Mixture Modelling
• as described, is not robust to error in measurement or end-member spectra;
• Proportions must be constrained to lie in the interval (0,1)
– - effectively a convex hull constraint;
• m+1 end-member spectra can be considered;• needs prior definition of end-member spectra;
cannot directly take into account any variation in component reflectances
– e.g. due to topographic effects
Linear Mixture Modelling in the presence of Noise
• Define residual vector
• minimise the sum of the squares of the error e, i.e.
eFr
ee
eeFrFrFrml
l
21
0
Method of Least Squares (MLS)
Error Minimisation
• Set (partial) derivatives to zero
021
0
21
0
ml
lii
ml
l
F
FFr
P
Fr
eeFrFrFrml
l
21
0
iiFF
1
0
1
0
1
020
ml
l i
ml
l i
ml
l i
Fr
Fr
Error Minimisation
• Can write as: PMO
1
0
1
0
ml
l i
ml
l i Fr
1
1
0
1
0
111110
111110
010100
1
0
1
1
0
n
ml
l
nlnlnllnll
lnlllll
lnlllll
ml
l
nll
ll
ll
F
F
F
r
r
r
Solve for P by matrix inversion
e.g. Linear Regressionmxcy
PMO
m
c
xx
x
xy
y nl
l ll
lnl
l ll
l1
02
1
0
1
m
c
xx
x
yx
y2
1
x
xyy
xx
xy
xx
xyxx
2
2
2
22
1
1 2
2
1
x
xxM
xx
222 xxxx
RMSE
1
0
22nl
lii mxcye
mnRMSE
2
y
xx x1x2
Weight of Determination (1/w)
• Calculate uncertainty at y(x)
m
c
xPQxy
1
QMQw
T 11
we
1
2
2
11
xx
xx
w
Lumped Canopy Models
• Motivation– Describe reflectance/scattering but don’t need
biophysical parameters• Or don’t have enough information
– Examples• Albedo• Angular normalisation – eg of VIs• Detecting change in the signal• Require generalised measure e.g cover• When can ‘calibrate’ model
– Need sufficient ground measures (or model) and to know conditions
Model Types
• Empirical models– E.g. polynomials
– E.g. describe BRDF by polynomial
– Need to ‘guess’ functional form
– OK for interpolation
• Semi-empirical models– Based on physical principles, with empirical linkages
– ‘Right sort of’ functional form
– Better behaviour in integration/extrapolation (?)
Linear Kernel-driven Modelling of Canopy Reflectance
• Semi-empirical models to deal with BRDF effects– Originally due to Roujean et al (1992)– Also Wanner et al (1995)– Practical use in MODIS products
• BRDF effects from wide FOV sensors– MODIS, AVHRR, VEGETATION, MERIS
Satellite, Day 1 Satellite, Day 2
X
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
13
6
14
3
15
0
15
7
16
4
17
1
17
8
18
5
19
2
19
9
20
6
21
8
22
6
23
3
24
0
24
7
25
4
26
1
26
8
27
5
28
2
Julian Day
ND
VI
original NDVI MVC BRDF normalised NDVI
AVHRR NDVI over Hapex-Sahel, 1992
Linear BRDF Model
• Of form:
,,,, geogeovolvoliso kfkff
Model parameters:
Isotropic
Volumetric
Geometric-Optics
Linear BRDF Model
• Of form:
,,,, geogeovolvoliso kfkff
Model Kernels:
Volumetric
Geometric-Optics
Volumetric Scattering
• Develop from RT theory– Spherical LAD– Lambertian soil– Leaf reflectance = transmittance– First order scattering
• Multiple scattering assumed isotropic
Xs
Xl ee
12
cossin
3
2,1
2
LX
Volumetric Scattering
• If LAI small: Xe X 1
Xs
Xl ee
1
2cossin
3
2,1
2
LX
2
12
2cossin
3
2,1 LL
sl
sl L
2
2cossin
3
2,1
Volumetric Scattering
• Write as:
sl L
2
2cossin
3
2,1
,,, 10 volthin kaa
2
2cossin
,
volk
slL
a
60
31
lLa
RossThin kernel
Similar approach for RossThick
LBL
exp2
exp
Geometric Optics
• Consider shadowing/protrusion from spheroid on stick (Li-Strahler 1985)
h
b
r
A()
Projection (shadowed)
Sunlit crownshadowed crown
shadowed ground
h
b
r
A()
Projection (shadowed)
Sunlit crownshadowed crown
shadowed ground
Geometric Optics
• Assume ground and crown brightness equal
• Fix ‘shape’ parameters
• Linearised model– LiSparse– LiDense
Kernels
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
-75 -60 -45 -30 -15 0 15 30 45 60 75
view angle / degrees
ke
rne
l va
lue
RossThick LiSparse
Retro reflection (‘hot spot’)
Volumetric (RossThick) and Geometric (LiSparse) kernels for viewing angle of 45 degrees
Kernel Models
• Consider proportionate () mixture of two scattering effects
,,1
1,,
11
00
geogeovolvol
multgeovol
kaka
aa
Using Linear BRDF Models for angular normalisation
BRDF Normalisation
• Fit observations to model
• Output predicted reflectance at standardised angles – E.g. nadir reflectance, nadir illumination
• Typically not stable
– E.g. nadir reflectance, SZA at local mean
KP ,,
geo
vol
iso
f
f
f
P
,
,
1
geo
vol
k
kK QMQw
T 11
And uncertainty via
Linear BRDF Models for albedo
• Directional-hemispherical reflectance– can be phrased as an integral of BRF for a given
illumination angle over all illumination angles.
– measure of total reflectance due to a directional illumination source (e.g. the Sun)
– sometimes called ‘black sky albedo’.
– Radiation absorbed by the surface is simply 1-
,
,
d
1
0,,,
Linear BRDF Models for albedo
dkfkff geogeovolvoliso
1
0,,,
KP ,
dk
dkK
geo
vol
1
0
1
0
,
,
1
d
1
0,,,
Linear BRDF Models for albedo
• Similarly, the bi-hemispherical reflectance
– measure of total reflectance over all angles due to an isotropic (diffuse) illumination source (e.g. the sky).
– sometimes known as ‘white sky albedo’
dd
1
0
1
0,,
KP
Spectral Albedo
• Total (direct + diffuse) reflectance– Weighted by proportion of diffuse illumination
KP
KDKDP
KPDKPD
)1(
)1(
Pre-calculate integrals – rapid calculation of albedo
Linear BRDF Models to track change
• E.g. Burn scar detection
• Active fire detection (e.g. MODIS) – Thermal– Relies on ‘seeing’ active fire– Miss many– Look for evidence of burn (scar)
Linear BRDF Models to track change
• Examine change due to burn (MODIS)
MODIS Channel 5 Observation
DOY 275
MODIS Channel 5 Observation
DOY 277
Detect Change
• Need to model BRDF effects
• Define measure of dis-association
wee
predictedobservedpredictedobserved
11
22
MODIS Channel 5 Prediction
DOY 277
MODIS Channel 5 Discrepency
DOY 277
MODIS Channel 5 Observation
DOY 275
MODIS Channel 5 Prediction
DOY 277
MODIS Channel 5 Observation
DOY 277
Single Pixel
Detect Change
• Burns are:– negative change in Channel 5– Of ‘long’ (week’) duration
• Other changes picked up– E.g. clouds, cloud shadow– Shorter duration – or positive change (in all channels)– or negative change in all channels
Day of burn
Other Lumped Parameter Optical Models
• Modified RPV (MRPV) model – Multiplicative terms describing BRDF ‘shape’– Linearise by taking log
Other Lumped Parameter Optical Models
• Gilabert et al.– Linear mixture model
• Soil and canopy: f = exp(-CL)
• Parametric model of multiple scattering
CLs exp
sff )1(
BlA
Other Lumped Parameter Optical Models
• Water Cloud model– Attema & Ulaby (1978)
– Microwave scattering from vegetation (and soil)
h
Sh
P2
exp2
exp12
scattering
attenuation
Water Cloud model
• Lump terms:
• Empirical additional dependency on LAI • Champion et al (2000)
2
ah
b2
h
Sh
P2
exp2
exp12
bLbLo Seea 1log10
bLbLeo SeeaL 1log10
Water Cloud Model
• Soil scattering:– Simple function of moisture
– Calibrate for particular roughness, texture– For each frequency & polarisation
vDmCS
Water Cloud Model
• resulting model mimics variations in observed backscatter dependencies on soil moisture and LAI.
• model parameters (a, b, C, D, e) vary for different canopies– canopy backscatter depends on more terms than just LAI– soil backscatter on more than moisture.
• model uses ‘calibration’ of the lumped parameter terms to hide fact that biophysical parameters will be correlated – e.g. LAI and leaf size, number density etc.
Water Cloud Model
• Use of the model:– Localised applications
• Known crop, soil properties, so use calibration terms
– Examine relative contributions of veg/soil
– Inversion (?)• Not from single channel (eg ERS SAR)
– Unless fix one term
• Potential (for localised) applications from multi-channel– E.g ASAR on ENVISAT
Conclusions
• Developed ‘semi-empirical’ models– Many linear (linear inversion)– Or simple form
• Lumped parameters – Information on gross parameter coupling– Few parameters to invert
Conclusions
• Uses of models– E.g. linear, kernel driven– When don’t need ‘full’ biophysical
parameterisation• Eg albedo, BRDF normalisation, change detection
• Forms of models– Similar forms (from RT theory)
• For optical and microwave