1. Lumped Parameter Modelling
P. Lewis & P. Saich
RSU, Dept. Geography, University College London, 26 Bedford Way, London WC1H 0AP, UK.

2. 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

3. 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.

4. Linear Models?

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

6. Linear Mixture Modelling
r = {rl0, rl1, … rlm, 1.0}
Measured reflectance spectrum (m wavelengths)
nx(m+1) matrix:

7. Linear Mixture Modelling
n=(m+1) – square matrix
Eg n=2 (wavebands), m=2 (end-members)

8. r2
Reflectance
Band 2 r r3 r1
Reflectance
Band 1

9. 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

10. 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)

11. Error Minimisation
Set (partial) derivatives to zero

12. Error Minimisation
Can write as:
Solve for P by matrix inversion

13. e.g. Linear Regression

14. RMSE

15. y x2 x x1 x

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

17. 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

18. 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 (?)

19. 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

20. Satellite, Day 1
Satellite, Day 2 X

21. AVHRR NDVI over Hapex-Sahel, 1992

22. Linear BRDF Model Of form: Model parameters: Isotropic Volumetric
Geometric-Optics

23. Linear BRDF Model
Of form:
Model Kernels:
Volumetric
Geometric-Optics

24. Volumetric Scattering
Develop from RT theory
Spherical LAD
Lambertian soil
Leaf reflectance = transmittance
First order scattering
Multiple scattering assumed isotropic

25. Volumetric Scattering
If LAI small:

26. Volumetric Scattering
Write as:
RossThin kernel
Similar approach for RossThick

27. Geometric Optics
Consider shadowing/protrusion from spheroid on stick (Li-Strahler 1985)

28. Geometric Optics Assume ground and crown brightness equal
Fix ‘shape’ parameters
Linearised model
LiSparse
LiDense

29. Kernels Retro reflection (‘hot spot’)
Volumetric (RossThick) and Geometric (LiSparse) kernels for viewing angle of 45 degrees

30. Kernel Models
Consider proportionate (a) mixture of two scattering effects

31. Using Linear BRDF Models for angular normalisation

34. 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

35. 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-

36. Linear BRDF Models for albedo

37. 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’

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

39. 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)

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

41. MODIS Channel 5 Observation
DOY 275

42. MODIS Channel 5 Observation
DOY 277

43. Detect Change Need to model BRDF effects
Define measure of dis-association

44. MODIS Channel 5 Prediction
DOY 277

45. MODIS Channel 5 Discrepency
DOY 277

46. MODIS Channel 5 Observation
DOY 275

47. MODIS Channel 5 Prediction
DOY 277

48. MODIS Channel 5 Observation
DOY 277

49. Single Pixel

50. 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

51. Day of burn

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

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

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

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

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

57. 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.

58. 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

59. 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

60. Conclusions Uses of models Forms 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