GEOGG121: Methods Inversion I: linear approaches Dr. Mathias (Mat) Disney UCL Geography Office: 113, Pearson Building Tel:

Linear models and inversion –Least squares revisited, examples –Parameter estimation, uncertainty –Practical examples Spectral linear mixture models Kernel-driven BRDF models and change detection Lecture outline

Linear models and inversion –Linear modelling notes: Lewis, 2010 –Chapter 2 of Press et al. (1992) Numerical Recipes in C (online version – – Reading

Linear Models For some set of independent variables x = {x 0, x 1, x 2, …, x n } 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 = {r , r , … r m, 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)

Reflectance Band 1 Reflectance Band 2 11 22 33 r

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 xx1x1 x2x2

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

P0 P1 RMSE

P0 P1 RMSE

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, fAPAR Many do not –E.g. LAI Need to (at least) understand impact of scaling

Crop Mosaic LAI 1LAI 4LAI 0

Crop Mosaic 20% of LAI 0, 40% LAI 4, 40% LAI 1. ‘real’ total value of LAI: –0.2x0+0.4x4+0.4x1=2.0. LAI1LAI1 LAI4LAI4 LAI0LAI0 visible: NIR

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: RossThin kernel Similar approach for RossThick

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

Kernels 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

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 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) FROM: 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

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 54 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,