Fundamentals of Cellular Automata Theory and Applica7ons to Hydrological Modeling. M. Verdecchia ICTP Trieste May,

Transcription

1 Fundamentals of Cellular Automata Theory and Applica7ons to Hydrological Modeling M. Verdecchia ICTP Trieste May,

2 Two observa7ons: Nature DOES NOT use complex algorithms to produce complex behavior Many biological systems made with many singular parts seem to have a collec>ve behavior Fundamentals of Cellular Automata Theory

3 The crucial experiment: What happen if we create a program without any specific task in mind? Fundamentals of Cellular Automata Theory

4 Defini7on of simple Cellular Automata System Fundamentals of Cellular Automata Theory

5 A first experiment with rule 254 Fundamentals of Cellular Automata Theory

6 Fundamentals of Cellular Automata Theory Rule 250

7 Fundamentals of Cellular Automata Theory Rule 90

8 Fundamentals of Cellular Automata Theory Rule 90

9 A first conclusion: star7ng from a very simple configura7on, the different rules produce regular o periodic paserns False! Fundamentals of Cellular Automata Theory

10 Fundamentals of Cellular Automata Theory Rule 30

11 Fundamentals of Cellular Automata Theory Rule 30

12 Fundamentals of Cellular Automata Theory Rule 110

13 A first conclusion: star7ng from a very simple configura7on and simple rules also complex behavior emerges Fundamentals of Cellular Automata Theory

14 A first conclusion: star7ng from a very simple configura7on and simple rules also complex behavior emerges Fundamentals of Cellular Automata Theory

15 A very ambi7ous ques7on: when we solve a differen7al equa7on in a numerical model, we simply discre7ze an equa7on wrisen for a con7nuous system A t + Q x = q c Can this approach be considered the best one? Or we may inves7gate to find the Best Rule giving the discharge in a cell of the model from the status of the closed cells? The ques7on is indeed very good the answer not yet Fundamentals of Cellular Automata Theory

16 Introduc7on to Cellular Automata Theory End of Part 1 Fundamentals of Cellular Automata Theory

17 Part 2. Applica7ons of CA theory Formal definition of 2D Cellular Automata ü A cellular automaton is a discrete dynamical system ü Space, Time and States of the System automaton are discrete quan77es ü Each point in a regular la^ce, called cell, can have anyone of a finite number of states ü The state of the cells in the la^ce are updated according to a local rule ü All the cells are updated synchronously Applica7ons of Cellular Automata Theory in CHyM model

18 The game of life Life rules by Chris G. Langton Ø The status of each CA can be ON or OFF Ø If more than 3 CA in the neighborhood are ON CA became OFF Ø If less than 2 CA in the neighborhood are ON, CA became OFF Ø Otherwise CA became ON Applica7ons of Cellular Automata Theory in CHyM model 18

19 What happen when you run CHyM script?

20 What happen when you run CHyM script?

21 A first set of drainage network Applica7ons of Cellular Automata Theory in CHyM model

22 DEM singulari7es Applica7ons of Cellular Automata Theory in CHyM model

23 CA concepts to correct DEM singulari7es ü CHyM grid is considered an aggregate of cellular automata ü The status of a cell corresponds to the value of a CHyM matrix (DEM) ü The state of the cells in the lakce is updated according to following rule + 8 hi hi α β j ( hj hi ) j ü All cells on the lakce are updated synchronously ü Update ends when flow scheme is OK Applica7ons of Cellular Automata Theory in CHyM model

24 CA concepts to correct DEM singulari7es Applica7ons of Cellular Automata Theory in CHyM model

25 CA concepts to correct DEM singulari7es Applica7ons of Cellular Automata Theory in CHyM model

26 Recipe for DEM pits and flat areas correc7on q Smooth DEM using CA rules un7l FD can be obtained for all the cells q Generate streamflow network using smoothed DEM q Use true DEM and modify ONLY the cells draining toward an higher cell Applica7ons of Cellular Automata Theory in CHyM model

27 CA concepts to correct DEM singulari7es Applica7ons of Cellular Automata Theory in CHyM model

28 CA concepts to merge different precipita7on data Applica7ons of Cellular Automata Theory in CHyM model

29 CA concepts to merge different precipita7on data Applica7ons of Cellular Automata Theory in CHyM model

30 CA concepts to merge different precipita7on data ü CHyM grid is considered an aggregate of cellular automata ü The status of a cell corresponds to the value of precipitation ü The state of the cells in the lattice is updated according to following rule + 8 hi hi α β j ( hj hi ) j But cells corresponding to rain gauges or defined in a previous Module are not updated ü All cells on the lattice are updated synchronously ü Update ends when a stable state is reached Applica7ons of Cellular Automata Theory in CHyM model

31 CA concepts to merge different precipita7on data Applica7ons of Cellular Automata Theory in CHyM model

32 CA concepts to merge different precipita7on data Applica7ons of Cellular Automata Theory in CHyM model

33 CA concepts to merge different precipita7on data Applica7ons of Cellular Automata Theory in CHyM model

34 CA concepts to merge temperature data Temperature field from MM5 Temperature field after CA algorithm Applica7ons of Cellular Automata Theory in CHyM model