GEOGG121: Methods Inversion I : linear approaches
description
Transcript of GEOGG121: Methods Inversion I : linear approaches
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GEOGG121: MethodsInversion I: linear approachesDr. Mathias (Mat) DisneyUCL GeographyOffice: 113, Pearson BuildingTel: 7670 0592Email: [email protected]/~mdisney
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• 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
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• 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
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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
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Linear Models?
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202010
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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
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ni
i iiFr Fr
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Linear Mixture Modelling
• r = {rl0, rl1, … rlm, 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
llll
llllllll
l
l
l
Fr
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Linear Mixture Modelling
• n=(m+1) – square matrix
• Eg n=2 (wavebands), m=2 (end-members)
Fr
rF 1
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Reflectance
Band 1
Reflectance
Band 2
1
2
3
r
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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
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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
eeFrFrFr ml
l
21
0 ll
Method of Least Squares (MLS)
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Error Minimisation
• Set (partial) derivatives to zero
02 1
0
21
0
ml
lii
ml
l
F
FFr
P
Frl
ll
ll
eeFrFrFr ml
l
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0 ll
iiFF ll
1
0
1
0
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020
ml
l iml
l i
ml
l i
Fr
Fr
ll
l
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ll
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Error Minimisation
• Can write as:
PMO
1
0
1
0
ml
l iml
l i Fr llll
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
FF
r
rr
llllll
llllllllllll
l
ll
Solve for P by matrix inversion
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e.g. Linear Regression
mxcy
PMO
mc
xxx
xyy nl
l ll
lnl
l ll
l1
02
1
0
1
mc
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yxy
2
1
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xyy
xx
xy
xx
xyxx2
2
2
22
11 2
21
xxxM
xx
222 xxxx
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RMSE
1
0
22nl
lii mxcye
mnRMSE
2
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y
xx x1x2
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Weight of Determination (1/w)
• Calculate uncertainty at y(x)
mc
xPQxy
1
QMQw
T 11
we 1
2
2
11
xx
xxw
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P0
P1RMSE
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P0
P1RMSE
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Issues
• Parameter transformation and bounding• Weighting of the error function• Using additional information• Scaling
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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
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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
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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
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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
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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
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Crop Mosaic
LAI 1 LAI 4 LAI 0
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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
visible: NIR 1.0;2.0 s
3.0;9.0 s
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canopy reflectance
0
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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
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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
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Satellite, Day 1 Satellite, Day 2
X
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0
0.05
0.1
0.15
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Julian Day
ND
VI
original NDVI MVC BRDF normalised NDVI
AVHRR NDVI over Hapex-Sahel, 1992
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Linear BRDF Model
• of form: ,,,, geogeovolvoliso kfkff llll
Model parameters:
Isotropic
Volumetric
Geometric-Optics
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Linear BRDF Model
• of form: ,,,, geogeovolvoliso kfkff llll
Model Kernels:
Volumetric
Geometric-Optics
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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
32
,1
2L
X
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Volumetric Scattering
• If LAI small:
Xe X 1
Xs
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Volumetric Scattering
• Write as:
sl L
2
2cossin
32
,1
,,, 10 volthin kaa lll
2
2cossin
,
volk
slL
a
l 60
l31
lLa
RossThin kernel
Similar approach for RossThick
LBL
exp2
exp
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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
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Geometric Optics
• Assume ground and crown brightness equal• Fix ‘shape’ parameters• Linearised model
– LiSparse– LiDense
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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
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Kernel Models
• Consider proportionate (a) mixture of two scattering effects
,,11,,
11
00
geogeovolvol
multgeovol
kakaaa
lalallalal
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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
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MODIS NBAR (Nadir-BRDF Adjusted Reflectance MOD43, MCD43)http://www-modis.bu.edu/brdf/userguide/intro.html
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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 ,,l
lll
geo
vol
iso
fff
P
,,
1
geo
vol
kkK QMQ
wT 11
And uncertainty via
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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
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MODIS Channel 5 Observation
DOY 275
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MODIS Channel 5 Observation
DOY 277
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Detect Change
• Need to model BRDF effects• Define measure of dis-association
wee
predictedobservedpredictedobserved
1122
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MODIS Channel 5 Prediction
DOY 277
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MODIS Channel 5 Discrepency
DOY 277
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MODIS Channel 5 Observation
DOY 275
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MODIS Channel 5 Prediction
DOY 277
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MODIS Channel 5 Observation
DOY 277
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So BRDF source of info, not JUST noise!• Use model expectation of angular reflectance
behaviour to identify subtle changes
5454Dr. Lisa Maria Rebelo, IWMI, CGIAR.
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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
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Day of burn
http://modis-fire.umd.edu/Burned_Area_Products.htmlRoy et al. (2005) Prototyping a global algorithm for systematic fire-affected area mapping using MODIS time series data, RSE 97, 137-162.