geogg121: methods inversion i: linear approachesmdisney/teaching/teachingnew/...• linear models...

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


•  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=


•  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


•  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

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


•  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


•  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


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


DOY 275

MODIS Channel 5 Prediction

DOY 277


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.