Geogg121 methods inversion i linear approaches
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GEOGG121: Methods Inversion I : linear approaches. Dr. Mathias (Mat) Disney UCL Geography Office: 113, Pearson Building Tel: 7670 0592 Email: /~ mdisney. Lecture outline. Linear models and inversion Least squares revisited, examples

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GEOGG121: Methods Inversion I : linear approaches

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GEOGG121: MethodsInversion I: linear approaches

Dr. Mathias (Mat) Disney

UCL Geography

Office: 113, Pearson Building

Tel: 7670 0592


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

  • 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 Models?

Linear Mixture Modelling

  • Spectral mixture modelling:

    • Proportionate mixture of (n) end-member spectra

    • First-order model: no interactions between components


Linear Mixture Modelling

  • r = {rl0, rl1, … rlm, 1.0}

    • Measured reflectance spectrum (m wavelengths)

  • nx(m+1) matrix:

Linear Mixture Modelling

  • n=(m+1) – square matrix

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




Band 2




Band 1

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

  • Define residual vector

  • minimise the sum of the squares of the error e, i.e.

Method of Least Squares (MLS)

Error Minimisation

  • Set (partial) derivatives to zero

Error Minimisation

  • Can write as:

Solve for P by matrix inversion

e.g. Linear Regression







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

  • 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

Weighting of the error function

  • Can ‘target’ sensitivity

    • E.g. to chlorophyll concentration

    • Use derivative weighting (Privette 1994)

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

  • 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


AVHRR NDVI over Hapex-Sahel, 1992

Model parameters:




Linear BRDF Model

  • of form:

Model Kernels:



Linear BRDF Model

  • of form:

Volumetric Scattering

  • Develop from RT theory

    • Spherical LAD

    • Lambertian soil

    • Leaf reflectance = transmittance

    • First order scattering

      • Multiple scattering assumed isotropic

Volumetric Scattering

  • If LAI small:

Similar approach for RossThick

Volumetric Scattering

  • Write as:

RossThin kernel

Geometric Optics

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

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

  • Consider proportionate (a) mixture of two scattering effects

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)

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

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


MODIS Channel 5 Observation

DOY 275

MODIS Channel 5 Observation

DOY 277

Detect Change

  • Need to model BRDF effects

  • Define measure of dis-association

MODIS Channel 5 Prediction

DOY 277

MODIS Channel 5 Discrepancy

DOY 277

MODIS Channel 5 Observation

DOY 275

MODIS Channel 5 Prediction

DOY 277

MODIS Channel 5 Observation

DOY 277

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 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

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

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