Lumped Parameter Modelling UoL MSc Remote Sensing Dr Lewis plewis@geog.ucl.ac.uk

Introduction introduce ‘simple’ lumped parameter models Build on RT modelling RT: formulate for biophysical parameters LAI, leaf number density, size etc investigate eg sensitivity of a signal to canopy properties e.g. effects of soil moisture on VV polarised backscatter or Landsat TM waveband reflectance Inversion? Non-linear, many parameters

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)

r2 Reflectance Band 2 r r3 r1 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 x2 x x1 x

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

Lumped Canopy Models Motivation Describe reflectance/scattering but don’t need biophysical parameters Or don’t have enough information Examples Albedo Angular normalisation – eg of VIs Detecting change in the signal Require generalised measure e.g cover When can ‘calibrate’ model Need sufficient ground measures (or model) and to know conditions

Model Types Empirical models Semi-empirical models E.g. polynomials E.g. describe BRDF by polynomial Need to ‘guess’ functional form OK for interpolation Semi-empirical models Based on physical principles, with empirical linkages ‘Right sort of’ functional form Better behaviour in integration/extrapolation (?)

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 (a) mixture of two scattering effects

Using Linear BRDF Models for angular normalisation

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 for albedo Directional-hemispherical reflectance can be phrased as an integral of BRF for a given illumination angle over all illumination angles. measure of total reflectance due to a directional illumination source (e.g. the Sun) sometimes called ‘black sky albedo’. Radiation absorbed by the surface is simply 1-

Linear BRDF Models for albedo

Linear BRDF Models for albedo Similarly, the bi-hemispherical reflectance measure of total reflectance over all angles due to an isotropic (diffuse) illumination source (e.g. the sky). sometimes known as ‘white sky albedo’

Spectral Albedo Total (direct + diffuse) reflectance Weighted by proportion of diffuse illumination Pre-calculate integrals – rapid calculation of albedo

Linear BRDF Models to track change E.g. Burn scar detection Active fire detection (e.g. MODIS) Thermal Relies on ‘seeing’ active fire Miss many Look for evidence of burn (scar)

Linear BRDF Models to track change Examine change due to burn (MODIS)

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

Single Pixel

Detect Change Burns are: Other changes picked up 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

Other Lumped Parameter Optical Models Modified RPV (MRPV) model Multiplicative terms describing BRDF ‘shape’ Linearise by taking log

Other Lumped Parameter Optical Models Gilabert et al. Linear mixture model Soil and canopy: f = exp(-CL) Parametric model of multiple scattering

Conclusions Developed ‘semi-empirical’ models Lumped parameters Many linear (linear inversion) Or simple form Lumped parameters Information on gross parameter coupling Few parameters to invert

Conclusions Uses of models Forms of models Applications: E.g. linear, kernel driven When don’t need ‘full’ biophysical parameterisation Forms of models Similar forms (from RT theory) Applications: BRDF normalisation Albedo Change detection