1. Li-Strahler Model Li-Strahler model is the most commonly used model in remote sensing application because it is invertible. The radiative transfer model that we discussed before are not directly invertible. This type of model is called forward model, meaning modeling radiation transfer from the target surface to the sensor. It provides us the spectral properties given the surface conditions, but not the other way around. The primary goal of remote sensing is to invert surface characteristics from remotely sensed data. A primary advantage of Li-Strahler model is that we can use it the extract surface structure from remotely collected signals over vegetation canopies.

2. Li-Strahler Model Assumptions: (1) Tree crowns are assumed to be solid cones (2) plant canopy is an assemblage of large solid 3-D objects. (3) Pixel size is sufficiently fine so that the spectral signals of a pixel interact with the size and placement of the cones, i.e. the pixel is several times greater than the average size of the trees. Note: all models are a simplified mathematical representation of the real world, thus it always developed based on some assumptions. We should be always aware of these assumptions when we use a model and understand the implications of the assumption to the application in hand.

3. Geometry of a Tree Crown h  r Coefficient of variation of h: Coefficient of variation of r: If  is fixed, then

4. Geometry for a Pixel Pixel size = A Tree No. = n Average of squared radii Treeness factor: Trees randomly distributed

5. 22 r   O A B C Side view Nadir view  If  > , a shadow will be cast.

6. r   OA B C   AREA R2R2  Area for the sunlit crown Area for the shadowed crown Shadowed area on the ground = ABCO-ACO Total covered area =

7. Areas for a Pixel A=area of a pixel A g =illuminated background A- A g =covered area A c =illuminated crowns A z = illuminated background A t =shadowed crowns K g = A g /a K c = A c /(A- A g ) K t = A t /(A- A g ) K z =Az/(A- A g ) (A- A g )= A c + A t + A z

8. Spectral signature for the four components: G=reflectance for sunlit background C=reflectance for sunlit crown Z=reflectance for shadowed background T=reflectance for shadowed crown Reflectance for a pixel is the area weighted average of the four Components in the pixel Where X is the average spectral signature for the covered area. Because Kg and X depends on viewing and illumination directions, the Li-Strahler model can be used to simulate BRDF effects in the forward mode. The shape of BRDF is characteristic of surface structure which is what MISR is after.

9. Substituting Kg into the equation: In the brightness/greenness space, the four components look like in the figure on the right. The pixel reflectance is G when there is no tree. As trees add on, the pixel reflectance S move from G towards X. Because of overlapping, Kz will decrease and Kt increase, making the trajectory of S diverge from X G C T Z XX X S Brightness Greenness Rearranging: Because (G-S) lies on (G-X), therefore, the above equation is actually scalar This is possible in theory, but in reality we have to choose G that is very different from X to minimize noise effects.

10. Inverting Tree Size for a Stand Where E(R2) is the expected value of R2 for a stand. V(m) is the variance of m for all pixels within a stand, and M is the is the mean value of m. Therefore, the Li-Strahler model inversion is driven by the interpixel variation in m. Intuitively, the larger the trees, the more variable the number of trees to be contained in a pixel. This inversion only applies to sparse canopy where overlapping is minimal. We will talk about overlapping cases in the next lecture.