GEOGG121: Methods Inversion I : linear approaches

1. GEOGG121: MethodsInversion 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

2. 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

3. 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

4. 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.

5. Linear Models?

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

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

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

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

10. 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

11. 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)

12. Error Minimisation • Set (partial) derivatives to zero

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

14. e.g. Linear Regression

15. RMSE

16. x x1 x2 y x

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

18. P1 RMSE P0

19. P1 RMSE P0

20. Issues • Parameter transformation and bounding • Weighting of the error function • Using additional information • Scaling

21. 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

23. 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

24. Weighting of the error function • Can ‘target’ sensitivity • E.g. to chlorophyll concentration • Use derivative weighting (Privette 1994)

25. 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

26. 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

27. 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

28. Satellite, Day 1 Satellite, Day 2 X

29. AVHRR NDVI over Hapex-Sahel, 1992

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

31. Model Kernels: Volumetric Geometric-Optics Linear BRDF Model • of form:

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

33. Volumetric Scattering • If LAI small:

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

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

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

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

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

39. 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) http://www-modis.bu.edu/brdf/userguide/intro.html

40. MODIS NBAR (Nadir-BRDF Adjusted Reflectance MOD43, MCD43) http://www-modis.bu.edu/brdf/userguide/intro.html

41. And uncertainty via 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

42. Linear BRDF Models to track change 220 days of Terra (blue) and Aqua (red) sampling over point in Australia, w. sza (T: orange; A: cyan). • Examine change due to burn (MODIS) Time series of NIR samples from above sampling FROM: http://modis-fire.umd.edu/Documents/atbd_mod14.pdf

43. MODIS Channel 5 Observation DOY 275

44. MODIS Channel 5 Observation DOY 277

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

46. MODIS Channel 5 Prediction DOY 277

47. MODIS Channel 5 Discrepancy DOY 277

48. MODIS Channel 5 Observation DOY 275

49. MODIS Channel 5 Prediction DOY 277

50. MODIS Channel 5 Observation DOY 277