Inversion I linear approaches Dr Mathias Mat Disney UCL Geography Office 113 Pearson Building Tel 7670 0592 Email mdisneyuclgeogacuk wwwgeoguclacuk mdisney Linear models and inversion ID: 547954
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Slide1
GEOGG121: MethodsInversion I: linear approaches
Dr. Mathias (Mat) Disney
UCL Geography
Office: 113, Pearson Building
Tel: 7670 0592
Email:
mdisney@ucl.geog.ac.uk
www.geog.ucl.ac.uk
/~
mdisneySlide2
Linear models and inversion
Least squares revisited, examples
P
arameter estimation, uncertaintyPractical examplesSpectral linear mixture modelsKernel-driven BRDF models and change detection
Lecture outlineSlide3
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_modelhttp
://
en.wikipedia.org
/wiki/System_of_linear_equations
ReadingSlide4
Linear ModelsFor 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.Slide5
Linear Models?Slide6
Linear Mixture Modelling Spectral mixture modelling:
Proportionate mixture of (n) end-member spectra
First-order model: no interactions between componentsSlide7
Linear Mixture Modelling
r
= {r
l0, rl1, … rlm, 1.0} Measured reflectance spectrum (m wavelengths)nx(m+1) matrix:Slide8
Linear Mixture Modelling
n=(m+1) – square matrix
Eg n=2 (wavebands), m=2 (end-members)Slide9
Reflectance
Band 1
Reflectance
Band 2
r
1
r
2
r
3
rSlide10
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 Slide11
Linear Mixture Modelling in the presence of Noise Define residual vector
minimise
the
sum of the squares of the error e, i.e.
Method of Least Squares (MLS)Slide12
Error MinimisationSet (partial) derivatives to zeroSlide13
Error MinimisationCan write as:
Solve for
P
by matrix inversionSlide14
e.g. Linear RegressionSlide15
RMSESlide16
y
x
x
x
1
x
2Slide17
Weight of Determination (1/w)Calculate uncertainty at y
(
x
)Slide18
P0
P1
RMSESlide19
P0
P1
RMSESlide20
IssuesParameter transformation and boundingWeighting of the error functionUsing additional information
ScalingSlide21
Parameter transformation and boundingIssue of variable sensitivity
E.g. saturation of LAI effects
Reduce by transformation
Approximately linearise parametersNeed to consider ‘average’ effectsSlide22Slide23
Weighting of the error functionDifferent wavelengths/angles have different sensitivity to parametersPreviously, weighted all equally
Equivalent to assuming ‘noise’ equal for all observationsSlide24
Weighting of the error functionCan ‘target’ sensitivity
E.g. to chlorophyll concentration
Use derivative weighting (Privette 1994)Slide25
Using additional informationTypically, for Vegetation, use canopy growth model
See
Moulin et al. (1998
)Provides expectation of (e.g.) LAINeed:planting dateDaily mean temperatureVarietal information (?)Use in various waysReduce parameter search spaceExpectations of coupling between parametersSlide26
ScalingMany parameters scale approximately linearlyE.g. cover, albedo, fAPARMany do not
E.g. LAI
Need to (at least) understand impact of scalingSlide27
Crop Mosaic
LAI 1
LAI 4
LAI 0Slide28
Crop Mosaic20% 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
visible:
NIR
Slide29
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 2.0Slide30
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 productsBRDF effects from wide FOV sensorsMODIS, AVHRR, VEGETATION, MERISSlide31
Satellite, Day 1
Satellite, Day 2
XSlide32
AVHRR NDVI over Hapex-Sahel, 1992Slide33
Linear BRDF Modelof form:
Model parameters:
Isotropic
Volumetric
Geometric-OpticsSlide34
Linear BRDF Modelof form:
Model Kernels:
Volumetric
Geometric-OpticsSlide35
Volumetric ScatteringDevelop from RT theorySpherical LADLambertian soil
Leaf reflectance = transmittance
First order scattering
Multiple scattering assumed isotropicSlide36
Volumetric ScatteringIf LAI small: Slide37
Volumetric ScatteringWrite as:
RossThin
kernel
Similar approach for
RossThickSlide38
Geometric OpticsConsider shadowing/protrusion from spheroid on stick (Li-Strahler 1985)Slide39
Geometric OpticsAssume ground and crown brightness equalFix ‘shape’ parametersLinearised model
LiSparse
LiDenseSlide40
Kernels
Retro reflection (‘hot spot’)
Volumetric (RossThick) and Geometric (LiSparse) kernels for viewing angle of 45 degreesSlide41
Kernel ModelsConsider proportionate (a) mixture of two scattering effectsSlide42
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.htmlSlide43
MODIS NBAR (Nadir-BRDF Adjusted Reflectance MOD43, MCD43)
http://www-
modis.bu.edu
/
brdf
/
userguide
/
intro.
htmlSlide44
BRDF NormalisationFit observations to modelOutput predicted reflectance at standardised angles
E.g. nadir reflectance, nadir illumination
Typically not stable
E.g. nadir reflectance, SZA at local mean
And uncertainty viaSlide45
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 samplingSlide46
MODIS Channel 5 Observation
DOY 275Slide47
MODIS Channel 5 Observation
DOY 277Slide48
Detect ChangeNeed to model BRDF effectsDefine measure of dis-association Slide49
MODIS Channel 5 Prediction
DOY 277Slide50
MODIS Channel 5 Discrepency
DOY 277Slide51
MODIS Channel 5 Observation
DOY 275Slide52
MODIS Channel 5 Prediction
DOY 277Slide53
MODIS Channel 5 Observation
DOY 277Slide54
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.Slide55
Detect ChangeBurns are:negative change in Channel 5
Of ‘long’ (week’) duration
Other changes picked up
E.g. clouds, cloud shadowShorter duration or positive change (in all channels)or negative change in all channelsSlide56
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.