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GEOGG121: Methods Inversion I: linear approaches
Dr. Mathias (Mat) Disney UCL Geography Office: 113, Pearson Building Tel: 7670 0592 Email: [email protected] www.geog.ucl.ac.uk/~mdisney
• Linear models and inversion
– Least squares revisited, examples – Parameter estimation, uncertainty – Practical examples
• Spectral linear mixture models • Kernel-driven BRDF models and change detection
Lecture outline
• Linear models and inversion
– Linear modelling notes: Lewis, 2010 – Chapter 2 of Press et al. (1992) Numerical Recipes in C (online
version http://apps.nrbook.com/c/index.html) – http://en.wikipedia.org/wiki/Linear_model – http://en.wikipedia.org/wiki/System_of_linear_equations
Reading
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 ρ Fr ⋅= ρ
Linear Mixture Modelling
• r = {rλ0, rλ1, … rλm, 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
PPP
r
rr
!""
!!!!""
!λρλρλρλρ
λρλρλρλρ
λρλρλρλρ
λ
λ
λ
Fr Η=
Linear Mixture Modelling
• n=(m+1) – square matrix
• Eg n=2 (wavebands), m=2 (end-members)
Fr Η=
rF 1−Η=
Reflectance
Band 1
Reflectance
Band 2
ρ1
ρ2
ρ3 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 +Η=
ee ⋅
( ) ( ) ( ) eeFrFrFr ml
l⋅=⋅−=Η−⋅Η− ∑
−=
=
21
0 λλ ρ
Method of Least Squares (MLS)
Error Minimisation
• Set (partial) derivatives to zero
( )( ) ( )
02 1
0
21
0=⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
⋅∂⋅−=
∂
⎥⎦⎤
⎢⎣⎡ ⋅−∂
∑∑
−=
=
−=
= ml
lii
ml
l
FF
FrP
Frλ
λλ
λλ ρρ
ρ
( ) ( ) ( ) eeFrFrFr ml
l⋅=⋅−=Η−⋅Η− ∑
−=
=
21
0 λλ ρ
( ) ( )iiFF λρρλ
=∂⋅∂
( ) ( )( )
( )( ) ( ) ( )( )∑∑
∑
−=
=
−=
=
−=
=
⋅=
⋅−=
1
0
1
0
1
020
ml
l iml
l i
ml
l i
Fr
Fr
λρρλρ
λρρ
λλ
λλ
Error Minimisation
• Can write as:
PMO =
( )( ) ( ) ( )( )∑∑−=
=
−=
=⋅=
1
0
1
0
ml
l iml
l i Fr λρρλρλλ
( )( )
( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
−=
=
−−−−
−
−
−=
=
−
∑∑
1
1
0
1
0
111110
111110
0101001
0
1
1
0
n
ml
l
nlnlnllnll
lnlllll
lnlllll
ml
l
nll
ll
ll
F
FF
r
rr
!"
!!!""
!λρλρλρλρλρλρ
λρλρλρλρλρλρ
λρλρλρλρλρλρ
λρ
λρ
λρ
Solve for P by matrix inversion
e.g. Linear Regression mxcy +=
PMO = ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛∑∑−=
=
−=
= mc
xxx
xyy nl
l ll
lnl
l ll
l1
02
1
0
1
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛
mc
xxx
yxy
2
1
( )x
xyy
xx
xy
xx
xyxx2
2
2
22
σ
σ
σ
σσ+
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−=−
11 2
21
xxxM
xxσ
222 xxxx −=σ
RMSE
( )( )∑−=
=
+−=1
0
22nl
lii mxcye
mnRMSE
−=
2ε
y
x x x1 x2
Weight of Determination (1/w)
• Calculate uncertainty at y(x)
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⎟⎟⎠
⎞⎜⎜⎝
⎛=⋅=
mc
xPQxy
1
QMQw
T 11 −=
we 1
=ε
( )2
2
11
xx
xxw σ
−+=
P0
P1 RMSE
P0
P1 RMSE
Issues
• Parameter transformation and bounding • Weighting of the error function • Using additional information • Scaling
Parameter transformation and bounding
• Issue of variable sensitivity – E.g. saturation of LAI effects – Reduce by transformation
• Approximately linearise parameters • Need to consider ‘average’ effects
Weighting of the error function
• Different wavelengths/angles have different sensitivity to parameters
• Previously, weighted all equally – Equivalent to assuming ‘noise’ equal for all
observations ( ) ( )( )[ ]
∑
∑=
=
=
=
−= Ni
i
Ni
imeasured ii
RMSE
1
1
2modelled
1
ρρ
Weighting of the error function
• Can ‘target’ sensitivity – E.g. to chlorophyll concentration – Use derivative weighting (Privette 1994)
( ) ( )( )
∑
∑=
=
=
=
⎥⎦
⎤⎢⎣
⎡∂
∂
⎥⎦
⎤⎢⎣
⎡ −∂
∂
=Ni
i
Ni
imeasured
P
iiPRMSE
1
21
2
modelled
ρ
ρρρ
Using additional information
• Typically, for Vegetation, use canopy growth model – See Moulin et al. (1998)
• Provides expectation of (e.g.) LAI – Need:
• planting date • Daily mean temperature • Varietal information (?)
• Use in various ways – Reduce parameter search space – Expectations of coupling between parameters
Scaling
• Many parameters scale approximately linearly – E.g. cover, albedo, fAPAR
• Many do not – E.g. LAI
• Need to (at least) understand impact of scaling
Crop Mosaic
LAI 1 LAI 4 LAI 0
Crop Mosaic
• 20% of LAI 0, 40% LAI 4, 40% LAI 1. • ‘real’ total value of LAI:
– 0.2x0+0.4x4+0.4x1=2.0.
LAI 1
LAI 4
LAI 0
)2/exp())2/exp(1( LAILAI s −+−−= ρωρ
visible: NIR
1.0;2.0 == sρω3.0;9.0 == sρω
canopy reflectance
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
LAI
refle
ctan
ce
visible
NIR
canopy reflectance over the pixel is 0.15 and 0.60 for the NIR. • If assume the model above, this equates to an LAI of 1.4. • ‘real’ answer LAI 2.0
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
136
143
150
157
164
171
178
185
192
199
206
218
226
233
240
247
254
261
268
275
282
Julian Day
ND
VI
original NDVI MVC BRDF normalised NDVI
AVHRR NDVI over Hapex-Sahel, 1992
Linear BRDF Model
• of form: ( ) ( ) ( ) ( ) ( ) ( )Ωʹ′Ω+Ωʹ′Ω+=Ωʹ′Ω ,,,, geogeovolvoliso kfkff λλλλρ
Model parameters:
Isotropic
Volumetric
Geometric-Optics
Linear BRDF Model
• of form: ( ) ( ) ( ) ( ) ( ) ( )Ωʹ′Ω+Ωʹ′Ω+=Ωʹ′Ω ,,,, geogeovolvoliso kfkff λλλλρ
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 −− +−ʹ′+
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛ −+
=Ωʹ′Ω ρµµ
γπ
γγ
πω
ρ 12
cossin
32
,1 µµ
µµ
ʹ′
ʹ′+=2L
X
Volumetric Scattering
• If LAI small:
Xe X −≈− 1
( ) ( ) Xs
Xl ee −− +−ʹ′+
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛ −+
=Ωʹ′Ω ρµµ
γπ
γγ
πω
ρ 12
cossin
32
,1 µµ
µµ
ʹ′
ʹ′+=2L
X
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ʹ′
ʹ′+−+⎟⎟
⎠
⎞⎜⎜⎝
⎛ʹ′
ʹ′+ʹ′+
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛ −+
=Ωʹ′Ωµµµµ
ρµµµµ
µµ
γπ
γγ
πω
ρ2
12
2cossin
32
,1 LLs
l
( ) sl L
ρµµµµ
µµ
γπ
γγ
πω
ρ +⎟⎟⎠
⎞⎜⎜⎝
⎛ʹ′
ʹ′+ʹ′+
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛ −+
=Ωʹ′Ω2
2cossin
32
,1
Volumetric Scattering
• Write as:
( ) sl L
ρµµµµ
µµ
γπ
γγ
πω
ρ +⎟⎟⎠
⎞⎜⎜⎝
⎛ʹ′
ʹ′+ʹ′+
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛ −+
=Ωʹ′Ω2
2cossin
32
,1
( ) ( ) ( ) ( )Ωʹ′Ω+=Ωʹ′Ω ,,, 10 volthin kaa λλλρ
( )2
2cossin
, πµµ
γπ
γγ−
ʹ′
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛ −+
=Ωʹ′Ωvolk
( ) slL
a ρω
λ +=60
( )πω
λ31
lLa =
RossThin kernel
Similar approach for RossThick
( )LBL−≈⎟⎟
⎠
⎞⎜⎜⎝
⎛ʹ′
ʹ′+− exp2
expµµµµ
Geometric Optics
• Consider shadowing/protrusion from spheroid on stick (Li-Strahler 1985)
h
b
r
θ
A(θ)
Projection (shadowed)
Sunlit crownshadowed crown
shadowed ground
h
b
r
θ
A(θ)
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
kern
el v
alue
RossThick LiSparse
Retro reflection (‘hot spot’)
Volumetric (RossThick) and Geometric (LiSparse) kernels for viewing angle of 45 degrees
Kernel Models
• Consider proportionate (α) mixture of two scattering effects ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )Ωʹ′Ω+Ωʹ′Ω−+
++−=Ωʹ′Ω
,,11,,
11
00
geogeovolvol
multgeovol
kakaaa
λαλα
λρλαλαλρ
Using Linear BRDF Models for angular normalisation • Account for BRDF variation • Absolutely vital for compositing samples
over time (w. different view/sun angles) • BUT BRDF is source of info. too!
MODIS NBAR (Nadir-BRDF Adjusted Reflectance MOD43, MCD43) http://www-modis.bu.edu/brdf/userguide/intro.html
MODIS NBAR (Nadir-BRDF Adjusted Reflectance MOD43, MCD43) http://www-modis.bu.edu/brdf/userguide/intro.html
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 ⋅=Ωʹ′Ω,,λρ
( )( )( )⎟
⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
=
λ
λ
λ
geo
vol
iso
fff
P ( )( )⎟
⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
Ωʹ′Ω
Ωʹ′Ω=
,,1
geo
vol
kkK QMQ
wT 11 −=
And uncertainty via
Linear BRDF Models to track change
• Examine change due to burn (MODIS)
FROM: http://modis-fire.umd.edu/Documents/atbd_mod14.pdf
220 days of Terra (blue) and Aqua (red) sampling over point in Australia, w. sza (T: orange; A: cyan).
Time series of NIR samples from above sampling
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
1122
+
−=
+
−=
ρρ
ε
ρρχ
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
So BRDF source of info, not JUST noise! • Use model expectation of angular reflectance
behaviour to identify subtle changes
54 54 Dr. Lisa Maria Rebelo, IWMI, CGIAR.
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
http://modis-fire.umd.edu/Burned_Area_Products.html Roy et al. (2005) Prototyping a global algorithm for systematic fire-affected area mapping using MODIS time series data, RSE 97, 137-162.