Lecture 7. Spectral Unmixing. Summary. Mixtures in Remote Sensing

Transcription

1 Lecture 7 Spectral Unmixing Summary This lecture will introduce you to the concepts of linear spectral mixing. This methods is sometimes also called: Spectral Mixture Analysis (SMA: Wessman et al 1997) Or Mixture Modeling (MM: Drake et al 1999) Suggested Reading: Drake, N.A., Mackin, S. and Settle, J.J., 1999, Mapping vegetation, soils, and geology in semiarid shrublands using spectral matching and mixture modelling of SWIR AVIRIS imagery, Remote Sensing of Environment, 67, pp Wessman, C.A., Bateson, C.A. and Benning, T.L., 1997, Detecting fire and grazing patterns in tallgrass prairie using spectral mixture analysis, Ecological Applications, 7, 2, Mixtures in Remote Sensing Pixels rarely consist of just one component. 1

2 Common Sub-pixel Analysis Techniques Linear Spectral Unmixing (SMA or MM) Fuzzy Classification Additional Hyperspectral Sub-Pixel Techniques Linearly Constrained Minimum Variance Automatic Subpixel Detection Orthogonal Subspace Projection (OSP) Linear Spectral Unmixing Each surface component within a pixel is sufficiently large enough such that no multiple scattering exists between the components (Singer and McCord, 1979). The linear scattering approximation is valid when the size of the pixel is smaller than the typical patch or component being sensed - i.e. linear mixing occurs at the macroscopic scale. The Horwitz (1971) Linear Mixture Model For an image with: N bands, C different cover types, X i = {x 1, x 2,, x N } T = observed image values in the i th band, and f i = {f 1, f 2,, f c } T = proportion of each pixel within each cover type c. The Linear Mixture Model is defined as: Where, M is an {n x c} matrix. x = Mf + e The columns of this matrix represent the spectra of the different endmembers. Source: Drake et al

3 If M and N are known, then the fraction term, f, can be estimated using a least squares solution. This least squares approach seeks to estimate f by minimizing the following: (x-mf) T N -1 (x-mf) Subject to the following constraints: <= f i <= 1 CONSTRAINT 1 f 1 + f 2 + f n = 1 CONSTRAINT 2 Researchers have noted that 2 is easy to implement, but 1 is not. As a result, most studies only implement the 2nd Constraint in the model and then apply the 1st constraint to the results. Assumptions Using linear spectral unmixing relies on four assumptions (Settle and Drake, 1993), which are: There is no significant occurrence of multiple scattering between the different surface components. Each surface component within the image has sufficient spectral contrast to allow their separation. In each pixel the total land cover is unity. Each surface component (endmember) is known. Steps The steps involved in using linear spectral unmixing: 1. Convert the imagery to reflectance (preferably ground) 2. Endmember Collection 3. Chose your Constraints 4. Normalize the Fraction maps 5. Interpret your Results 3

4 Endmember Collection In order to use a linear mixture model there is a need to measure the spectral reflectance of the pure endmembers. Ideally ground-based spectra would be acquired to produce accurate end-members, since endmembers taken from even very high spatial-resolution imagery may contain multiple surface components. However, such errors can be minimized through sampling the image end-members from within the center of known features (lakes, burned areas, grasslands, ploughed fields, etc) or from know locations visited during fieldwork. Image Endmember Collection Using Principal Components Analysis Johnson et al. (1985) and Smith et al. (1985) demonstrated that principal component analysis could be used to identify the individual end-members of multiple surface components. They observed that for a mixture of three substances the scatter-plot of the first two principle components produced a triangle in which the pure end-members were located at the corners Constraints In ENVI you can ONLY chose to either use or not use the 2nd Constraint: i.e. Note: Idrisi allows both constraints. f 1 + f 2 + f n = 1 In ENVI you get fraction maps where the TOTAL fraction in each pixel meets the constraint BUT each fraction can be +,, or even negative! Example Fraction Maps for 3 Different Endmembers In order to use these maps you need to implement the 1st constraint - i.e. each fraction can only be between and 1. 4

5 Relative Fraction Maps To produce a relative fraction map, you either set all - values to ZERO; all >1 values to 1. Part B Non-Linear Spectral Unmixing Non-Linear Spectral Unmixing Linear spectral unmixing is all very well - but unfortunately in reality most scenarios are non-linear. In non-linear mixing the light incident on a small surface components interacts (or scatters) with multiple components before being detected by the sensor. Non-linear Vertical Mixing e.g. Light in tree canopies? Non-linear Horizontal Mixing e.g. granular mixtures 5

6 Example 1 - Soil Mixtures Mixture of soils or minerals generally exist as intimate mixtures. An intimate mixture occurs when different surface components are in intimate contact on a scattering surface. Such that light that reflects off one of the grains is likely to hit another grain before reaching the sensor. Scenario - Light and Dark Grains: 1. Light first hits light grain 2. If the light then scatters onto: a light grain -> scatter a dark grain -> absorbed Result -> It appears that more dark grains are present than is the case. -> i.e. Preferential Absorption Intimate Mixtures: Particle Size Dependence The effect of preferential absorption is highly dependent on the respective sizes of the two particle types within the mixture. 1.8 D 2 /D 1 =1.6 ro.4 D 2 /D 1 =1.2 D 2 /D 1 = M 1/(M 1+M 2) Example 2 - Ash Mixtures The combustion products from wildfires typically consist of a mixture of black and white ash, which exhibit spectral reflectances at the opposite extremes. Checkerboard - Classic Linear Mixing Scenario Result - Mixing is nearly Linear 6

7 Example 2 - Ash Mixtures cont. Intimate Mixture - Classic Non-Linear Mixing Scenario How is this modeled? Non-Linear Spectral Unmixing: Hapke Theory A Few Assumptions: Particle size >> wavelength of the light measuring it (e.g. very fine white ash = 5 microns >>.5-.7 microns of VIS light) Isotropic Scatterers within the mixture Defined angle of source and sensor from vertical F r = the azimuthalangle? i = the angle of incidence (zenith angle)? r = the angle of reflectance 9 -? i = the elevation angle Hapke Theory II In a Lab, we generally measure either the: A. Bi-directional Reflectance i.e. Collimated source and Sensor only measuring reflected light at one angle -> What we measure in a lab. B. Hemispherical Reflectance i.e. Collimated Source and Integrating Sphere or BRDF The Aim is to be able to use the Diffusive Reflectance. r. A and B can be approximated as r o in the following cases. R A (i=6; r=6; porosity of reference panel <=.25) = r R B (i=6) = r 7

8 Hapke Theory III Hapke (1993) describes the parameter r o as called the relative reflectance. This is defined as the reflectance relative to that of a standard surface consisting of an infinitely thick particulate medium of non-absorbing, isotropic scatters, with low porosity, and illuminated and viewed at the same geometry as the sample. i.e. A Spectralon Panel The reason we like r o is that it allows us to make use of a very useful theoretical measure called the single scattering albedo, w: w = 1 γ 2 γ ( Γ) = Hapke Theory III Where, The parameter? of the combined mixture can then be obtained using (Hapke 1993): / 2 [( µ + µ ) Γ + (1+ 4µ µ Γ)(1 Γ) ] ( µ + µ ) 1 + 4µ µ Γ Γ The single scattering albedo is the probability that, given an interaction between the photon and an individual particle that the particle will be scattered rather than absorbed Hapke Theory IV The reason we use w is because the non-linear mixing effects of the reflectance can be linearly modeled with w, using equations developed by Hapke (1981, 1993): Hapke developed several equations for different mixing scenarios. These include, intimate mixtures, horizontal and vertical layers, etc For our intimate mixture example, we have a BINARY mixture and can use the following equations to model the combined mixture reflectance: M = bulk density w + ε = 1 w2 1 + ε M ρ ε =. D w o M 2 ρ1 D1? = solid density (Assumed by Hapke (1993) to be typically equal) D = diameter of the two particles 8

9 Hapke Theory V The bi-directional reflectance, G (what we measure in the lab), can be measured via the following equation: 2 1 γ Γ( µ, µ, g) = (1 + 2γµ )(1+ 2γµ ) Where, µ = cos (e) and µ= cos (i), i and e denote the angles of incidence and reflection respectively 9