Lumped Parameter Modelling P Lewis P Saich RSU

Lumped Parameter Modelling P. Lewis & P. Saich RSU, Dept. Geography, University College London, 26 Bedford Way, London WC 1 H 0 AP, 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 = {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 1. as described, is not robust to error in measurement or end-member spectra; 2. Proportions must be constrained to lie in the interval (0, 1) 1. - effectively a convex hull constraint; 3. m+1 end-member spectra can be considered; 4. needs prior definition of end-member spectra; cannot directly take into account any variation in component reflectances 1. 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)

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 – – 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: 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

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: – 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

Other Lumped Parameter Optical Models • Water Cloud model – Attema & Ulaby (1978) – Microwave scattering from vegetation (and soil) scattering attenuation

Water Cloud model • Lump terms: • Empirical additional dependency on LAI • Champion et al (2000)

Water Cloud Model • Soil scattering: – Simple function of moisture – Calibrate for particular roughness, texture – For each frequency & polarisation

Water Cloud Model • resulting model mimics variations in observed backscatter dependencies on soil moisture and LAI. • model parameters (a, b, C, D, e) vary for different canopies – canopy backscatter depends on more terms than just LAI – soil backscatter on more than moisture. • model uses ‘calibration’ of the lumped parameter terms to hide fact that biophysical parameters will be correlated – e. g. LAI and leaf size, number density etc.

Water Cloud Model • Use of the model: – Localised applications • Known crop, soil properties, so use calibration terms – Examine relative contributions of veg/soil – Inversion (? ) • Not from single channel (eg ERS SAR) – Unless fix one term • Potential (for localised) applications from multi-channel – E. g ASAR on ENVISAT

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

Conclusions • Uses of models – E. g. linear, kernel driven – When don’t need ‘full’ biophysical parameterisation • Eg albedo, BRDF normalisation, change detection • Forms of models – Similar forms (from RT theory) • For optical and microwave