GEOGG 121 Methods Inversion I linear approaches Dr

GEOGG 121: Methods Inversion 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

Linear Models • For some set of independent variables x = {x 0, x 1, x 2, … , 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 = {rl 0, rl 1, … 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)

r 2 Reflectance Band 2 r r 3 r 1 Reflectance 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

RMSE

y x 2 x x 1 x

Weight of Determination (1/w) • Calculate uncertainty at y(x)

RMSE P 1 P 0

RMSE P 1 P 0

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

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

Scaling • Many parameters scale approximately linearly – E. g. cover, albedo, f. APAR • Many do not – E. g. LAI • Need to (at least) understand impact of scaling

Crop Mosaic LAI 1 LAI 4 LAI 0

L A I 1 Crop Mosaic • 20% of LAI 0, 40% LAI 4, 40% LAI 1. • ‘real’ total value of LAI: – 0. 2 x 0+0. 4 x 4+0. 4 x 1=2. 0. visible: NIR L A I 4 L A I 0

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

AVHRR NDVI over Hapex-Sahel, 1992

Linear BRDF Model • of form: Model parameters: Isotropic Volumetric Geometric-Optics

Linear BRDF Model • of form: 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

Volumetric Scattering • If LAI small:

Volumetric Scattering • Write as: Ross. Thin kernel Similar approach for Ross. Thick

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 – Li. Sparse – Li. Dense

Kernels Retro reflection (‘hot spot’) Volumetric (Ross. Thick) and Geometric (Li. Sparse) 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 MOD 43, MCD 43) http: //www-modis. bu. edu/brdf/userguide/intro. html

MODIS NBAR (Nadir-BRDF Adjusted Reflectance MOD 43, MCD 43) 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 And uncertainty via

Linear BRDF Models to track change • Examine change due to burn (MODIS) 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 FROM: http: //modis-fire. umd. edu/Documents/atbd_mod 14. pdf

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 Discrepency 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. 54

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.