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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] /~ mdisney. Lecture outline. Linear models and inversion Least squares revisited, examples

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Geogg121 methods inversion i linear approaches

GEOGG121: MethodsInversion I: linear approaches

Dr. Mathias (Mat) Disney

UCL Geography

Office: 113, Pearson Building

Tel: 7670 0592

Email: [email protected]

Lecture outline

  • Linear models and inversion

    • Least squares revisited, examples

    • Parameter estimation, uncertainty

    • Practical examples

      • Spectral linear mixture models

      • Kernel-driven BRDF models and change detection


  • Linear models and inversion

    • Linear modelling notes: Lewis, 2010

    • Chapter 2 of Press et al. (1992) Numerical Recipes in C (online version



Linear models
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.

Linear mixture modelling
Linear Mixture Modelling

  • Spectral mixture modelling:

    • Proportionate mixture of (n) end-member spectra

    • First-order model: no interactions between components


Linear mixture modelling1
Linear Mixture Modelling

  • r = {rl0, rl1, … rlm, 1.0}

    • Measured reflectance spectrum (m wavelengths)

  • nx(m+1) matrix:

Linear mixture modelling2
Linear Mixture Modelling

  • n=(m+1) – square matrix

  • Eg n=2 (wavebands), m=2 (end-members)




Band 2




Band 1

Linear mixture modelling3
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
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)

Error minimisation
Error Minimisation

  • Set (partial) derivatives to zero

Error minimisation1
Error Minimisation

  • Can write as:

Solve for P by matrix inversion






Weight of determination 1 w
Weight of Determination (1/w)

  • Calculate uncertainty at y(x)








  • Parameter transformation and bounding

  • Weighting of the error function

  • Using additional information

  • Scaling

Parameter transformation and bounding
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
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

Weighting of the error function1
Weighting of the error function

  • Can ‘target’ sensitivity

    • E.g. to chlorophyll concentration

    • Use derivative weighting (Privette 1994)

Using additional information
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


  • Many parameters scale approximately linearly

    • E.g. cover, albedo, fAPAR

  • Many do not

    • E.g. LAI

  • Need to (at least) understand impact of scaling

Linear kernel driven modelling of canopy reflectance
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


Satellite, Day 1

Satellite, Day 2


Linear brdf model

Model parameters:




Linear BRDF Model

  • of form:

Linear brdf model1

Model Kernels:



Linear BRDF Model

  • of form:

Volumetric scattering
Volumetric Scattering

  • Develop from RT theory

    • Spherical LAD

    • Lambertian soil

    • Leaf reflectance = transmittance

    • First order scattering

      • Multiple scattering assumed isotropic

Volumetric scattering1
Volumetric Scattering

  • If LAI small:

Volumetric scattering2

Similar approach for RossThick

Volumetric Scattering

  • Write as:

RossThin kernel

Geometric optics
Geometric Optics

  • Consider shadowing/protrusion from spheroid on stick (Li-Strahler 1985)

Geometric optics1
Geometric Optics

  • Assume ground and crown brightness equal

  • Fix ‘shape’ parameters

  • Linearised model

    • LiSparse

    • LiDense


Retro reflection (‘hot spot’)


Volumetric (RossThick) and Geometric (LiSparse) kernels for viewing angle of 45 degrees

Kernel models
Kernel Models

  • Consider proportionate (a) mixture of two scattering effects

Using linear brdf models for angular normalisation
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)

MODIS NBAR (Nadir-BRDF Adjusted Reflectance MOD43, MCD43)

Brdf normalisation

And uncertainty via

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

Linear brdf models to track change
Linear BRDF Models to track change

220 days of Terra (blue) and Aqua (red) sampling over point in Australia, w. sza (T: orange; A: cyan).

  • Examine change due to burn (MODIS)

Time series of NIR samples from above sampling


Detect change
Detect Change

  • Need to model BRDF effects

  • Define measure of dis-association

Modis channel 5 discrepancy
MODIS Channel 5 Discrepancy

DOY 277

So brdf source of info not just noise
So BRDF source of info, not JUST noise!

Use model expectation of angular reflectance behaviour to identify subtle changes

Dr. Lisa Maria Rebelo, IWMI, CGIAR.


Detect change1
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
Day of burn

Roy et al. (2005) Prototyping a global algorithm for systematic fire-affected area mapping using MODIS time series data, RSE 97, 137-162.


  • Linear models & inversion

    • Form of linear models

    • Linear least squares revisited

    • Phrase problem as optimisation i.e. minimisation of some objective/cost function e2 == (obs-modelled)2

    • Applications

      • Linear mixture modelling, issues, constraints etc.

      • Kernel-driven BRDF modelling for remote sensing applications: angular normalisation, change detection / burned area mapping

  • Be able to:

    • Identify form of a linear model

    • Understand how the LLS problem phrased

    • Understand the strengths and limitations of the approach + applications