1. 4.6.1 Upper Echelons of Surfaces

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12. 4.6.12 Ingredients of an Echelon Analysis: – Tessellation of a geographic region: –Response value Z on each cell. Determines a tessellated (piece-wise constant) surface with Z as elevation. How does connectivity (number of connected components) of the tessellation change with elevation? Echelons Description -- 1 a c b k d e f g h i j a, b, c, … are cell labels

13. 4.6.13 Think of the tessellated surface as a landform Initially the entire surface is under water As the water level recedes, more and more of the landform is exposed At each water level, cells are colored as follows: –Green for previously exposed cells (green = vegetated) –Yellow for newly exposed cells (yellow = sandy beach) –Blue for unexposed cells (blue = under water) For each newly exposed cell, one of three things happens: –New island emerges. Cell is a local maximum. Morse index=2. Connectivity increases. –Existing island increases in size. Cell is not a critical point. Connectivity unchanged. –Two (or more) islands are joined. Cell is a saddle point Morse index=1. Connectivity decreases. Echelons Description -- 2

14. 4.6.14 a c b k d e f g h i j Echelons Illustrated -- 1 Echelon Tree a a a b,c b k d e f g h i j c Newly exposed island Island grows

15. 4.6.15 Echelons Illustrated -- 2 Echelon Tree a a b,c b k d e f g h i j c Second island appears d a a b,c b k d e f g h i j c Both islands grow d e f,g New echelon

16. 4.6.16 Echelons Illustrated -- 3 Echelon Tree a a b,c b k d e f g h i j c Islands join – saddle point d e f,g h New echelon a a b,c b d e f g h i j c Exposed land grows d e f,g h k i,j,k Three echelons

17. 4.6.17 Echelons Illustrated -- 4 Each branch in echelon tree determines an echelon Each echelon consists of cells in the tessellation The echelons partition the region Each echelon determines a set of response values Z (and a corresponding set of values of the explanatory variables X, if any) Echelon Tree a a b,c b d e f g h i j c Echelon Partitioning d e f,g h k i,j,k Three echelons

18. 4.6.18 Echelons Illustrated – 5 Higher Order Echelons Receding Waterline Previous Pictures Echelon Tree labeled with echelon orders 1 1 1 1 1 2 2 (not 3) 2 3

19. 4.6.19 Echelons Illustrated – 6 Echelon Order Defined Echelon Tree labeled with echelon orders 1 1 1 1 1 2 2 (not 3) 2 3 Analogy with stream networks (Horton-Strahler order) Leaf branches have order 1 When two branches of orders p and q join, the new branch has order: Max(p, q) if p  q p + 1 if p = q

20. 4.6.20 Echelons Illustrated – 7 Echelon Smoothing Need for smoothing echelon trees Alternative to direct smoothing of surface values In complicated echelon trees, root nodes may be most indicative of noise and become prime candidates for pruning (contraction would be a better term) Criteria for pruning: Echelon relief, Echelon basal area, others? What is the corresponding smoothed surface? Echelon Tree labeled with echelon orders 1 1 1 1 1 2 2 (not 3) 2 3 Prune ?

21. 4.6.21 Ingredients – Set S whose elements are called cells or sites –Filtration of S, i.e., increasing sequence of subsets: –Neighborhood system on S, i.e., an indexed family, of subsets of S such that: Members of N t are called neighbors of cell t Echelons Mathematical Definition -- 1

22. 4.6.22 Role of the filtration –Determines the cells to be exposed as the water level drops to successively lower levels –When the water is at level n : Cells in S n-1 are previously exposed (green) Cells in S n – S n-1 are newly exposed (yellow) Cells in S – S n are unexposed (blue) Echelons Mathematical Definition -- 2

23. 4.6.23 Ways to determine a filtration – Numerical data. Let be the distinct values of Z. Then S n consists of all cells for which Z is greater than or equal to z n –Ordinal data. Same as above, except Z has only ordinal significance. –Partial ordering on cells. S n consists of the cells in the top n levels of the Hasse Diagram. Echelons Mathematical Definition -- 3

24. 4.6.24 Role of the neighborhood system – Determines the notion of connectivity among cells –Two exposed cells s, t are in the same connected component (i.e., same island) if there is a sequence of cells such that t i+1 is a neighbor of t i, i = 1,2,…,k-1. This means that Echelons Mathematical Definition -- 4

25. 4.6.25 Ways to determine a neighborhood system –From a tessellation. Two cells are neighbors if they are adjacent. As usual, one must decide if their shared boundary should be 0-dimensional (finite set of points) or 1-dimensional. We generally require a 1- dimensional shared boundary. –From a graph. Two nodes in a graph are neighbors if there is an edge joining them. Actually, this is the universal example since any neighborhood system on S determines a graph structure on S, and conversely. Echelons Mathematical Definition -- 5

26. 4.6.26 Filtration is determined by the response values Z, i.e., by the data. Neighborhood system is determined by the tessellation, i.e., by the spatial information. Echelons link the responses with the spatial information Echelons Mathematical Definition -- 6

27. 4.6.27 Echelons focus on local maxima. Dual echelons focus on local minima Initially the entire landform is exposed As the water level rises, water bodies form Dual echelons study the connectivity of the water surface For each newly flooded cell, one of three things happens: –New lake emerges. Cell is a local minimum. Morse index=0. Connectivity of the water increases. –Existing lake increases in size. Cell is not a critical point. Connectivity unchanged. –Two (or more) lakes are joined. Cell is a saddle point Morse index=1. Connectivity decreases. Echelons Duality -- 1

28. 4.6.28 Existing echelon software can be used for dual echelons: –Surface variable. Replace Z by –Z. –Filtration. Replace the filtration by the dual filtration where Echelons Duality -- 2

29. 4.6.29 Spatial Complexity with Single Response Variable Echelons Approach Issues to be addressed: Echelon trees and maps Echelon profiles and other tree metrics Noise effects and filtering Comparing echelon trees and maps Echelon stochastics: surface simulation, tree simulation, tree metric distributions

30. 4.6.30 Pre-Classification Change Detection Echelons Approach Change vector approach (cell by cell) with actual spectral data Change vector approach (cell by cell) with compressed (hyperclustered) spectral data Pattern-based approach (compressed data only): Compare segment pattern at time1 with segment pattern at time 2.

31. 4.6.31 Spatial Complexity with Multiple Indicators: Echelons Approach Compare echelon features among indicators for consistency/inconsistency: –Order –Number of ancestors (distance from root of tree) Compression by treating features as pseudo- bands