Lumped Parameter Modelling

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


• 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


• 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


• 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


• 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


• 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,,,


dkfkff geogeovolvoliso

1

0,,,

KP ,

dk

dkK

geo

vol

1

0

1

0

,

,

1

d

1

0,,,


• 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


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


DOY 275

MODIS Channel 5 Prediction

DOY 277


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


• Gilabert et al.– Linear mixture model

• Soil and canopy: f = exp(-CL)

• Parametric model of multiple scattering

CLs exp

sff )1(

BlA


• 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

Documents

Lumped Parameter Modelling