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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

Email: mdisney@ucl.geog.ac.uk

www.geog.ucl.ac.uk/~mdisney

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

Reading

- 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

- 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.

- Spectral mixture modelling:
- Proportionate mixture of (n) end-member spectra
- First-order model: no interactions between components

Constraint

- r = {rl0, rl1, … rlm, 1.0}
- Measured reflectance spectrum (m wavelengths)

- nx(m+1) matrix:

- n=(m+1) – square matrix
- Eg n=2 (wavebands), m=2 (end-members)

r1

r2

Reflectance

Band 2

r

r3

Reflectance

Band 1

- 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

- Define residual vector
- minimise the sum of the squares of the error e, i.e.

Method of Least Squares (MLS)

- Set (partial) derivatives to zero

- Can write as:

Solve for P by matrix inversion

x

x1

x2

y

x

- Calculate uncertainty at y(x)

P1

RMSE

P0

P1

RMSE

P0

- Parameter transformation and bounding
- Weighting of the error function
- Using additional information
- Scaling

- Issue of variable sensitivity
- E.g. saturation of LAI effects
- Reduce by transformation
- Approximately linearise parameters
- Need to consider ‘average’ effects

- Different wavelengths/angles have different sensitivity to parameters
- Previously, weighted all equally
- Equivalent to assuming ‘noise’ equal for all observations

- Can ‘target’ sensitivity
- E.g. to chlorophyll concentration
- Use derivative weighting (Privette 1994)

- 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 (?)

- Need:
- 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

- 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

AVHRR NDVI over Hapex-Sahel, 1992

Model parameters:

Isotropic

Volumetric

Geometric-Optics

- of form:

Model Kernels:

Volumetric

Geometric-Optics

- of form:

- Develop from RT theory
- Spherical LAD
- Lambertian soil
- Leaf reflectance = transmittance
- First order scattering
- Multiple scattering assumed isotropic

- If LAI small:

Similar approach for RossThick

- Write as:

RossThin kernel

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

- 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

- Consider proportionate (a) mixture of two scattering effects

- 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

And uncertainty via

- 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

- E.g. nadir reflectance, nadir illumination

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

FROM: http://modis-fire.umd.edu/Documents/atbd_mod14.pdf

DOY 275

DOY 277

- Need to model BRDF effects
- Define measure of dis-association

DOY 277

DOY 277

DOY 275

DOY 277

DOY 277

Use model expectation of angular reflectance behaviour to identify subtle changes

Dr. Lisa Maria Rebelo, IWMI, CGIAR.

51

- 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

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.

- 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