Quantitative estimates of a drainage network model Rahul Roy Indian - PowerPoint PPT Presentation

Quantitative estimates of a drainage network model Rahul Roy Indian Statistical Institute, New Delhi joint work with Kumarjit Saha and Anish Sarkar.

Hurst exponent A river basin network is an anisotropic system defined by a longitudinal length L ∥ = L and a typical perpendicular length L ⊥ ≤ L . The history of the Hurst exponent in fractional Brownian motion comes from when Harold Hurst studying the Nile river postulated the scaling L ⊥ = L H , 0 ≤ H ≤ 1 . The basin is said to be self affine if H < 1 and self-similar if H = 1 1 . 1 For Nile, Hurst obtained H = 0 . 90 .

Hack’s law The Hurst exponent can be viewed as a measure of the meandering of the river and the tributaries it gathers in the process. Hack’s law studies the length, l , of the longest stream vis-` a-vis the drainage area, a , of the basin, i.e. the area of the land which collects the precipitation contributing to the network. John Hack observed that l ∼ a h where h is the Hack exponent 2 . 2 For the Shenandoah valley in Virginia, Hack calculated h = 0 . 571 .

Fella river basin Figure : Fella river network in Northern Italy. From Maritan et al , Phy. Rev. E 1996

River network ‘A river network is a spanning tree defined in a lattice of arbitrary size and shape 3 .’ We study the Howard’s model of ‘headward growth and branching in a random fashion’ 4 . There are various river network models proposed by geologists. We first discuss a few of them. Rodriguez-Iturbe and Rinaldo [1997] (Fractal River Basins) presents a survey of the development of this field. 3 Maritan et al , Phy. Rev. E 1996 4 Rodriguez-Iturbe and Rinaldo (1997)

Scheidegger Model Scheidegger (1967) introduced a drainage model consisting of a two dimensional grid where each square, representing a unit area, is to be drained, to one of its two neighbours in a preferred direction. Here, the flow can happen along the chosen direction. The preferred direction is assumed to be direction of the slope. Thus, the drainage network forms an oriented network.

Part of the 2 -dimensional oriented lattice ( L ): ✉ ✉ ✉ ✉ ✉ ✉ � ❅ � ❅ � ❅ � ❅ � ❅ � � ❅ � ❅ � ❅ � ❅ � ❅ � � ✉ ✉ ✉ ✉ ✉ ✉ ❅ ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ❅ ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ✉ ✉ ✉ ✉ ✉ ❅ ✉ � ❅ ❅ � � ❅ � ❅ � ❅ � ❅ � ❅ ❅ � � ❅ � ❅ � ❅ � ❅ � ✉ ✉ ✉ ✉ ✉ ✉ ❅ � ❅ ❅ � � ❅ � ❅ � ❅ ❅ ❅ � ❅ ❅ Direction � � ❅ � ❅ � ❅ ❅ ✉ ❅ ✉ ✉ ✉ ✉ ❅ ✉ � ❅ � � � ❅ � ❅ � ❅ ❅ � ❅ � � � ❅ � ❅ � ❅ ❅ � ✉ ✉ ✉ ✉ ✉ ✉ ❅ � ❅ � � � ❅ � ❅ ❅ ❅ ❄ ❅ � ❅ ❅ � � � � ❅ ❅ ❅ ✉ ✉ ✉ ✉ ✉ ❅ ✉ � ❅ � ❅ ❅ � � � � ❅ ❅ � ❅ � ❅ � ❅ � � � ❅ ❅ � ✉ ✉ ✉ ✉ ✉ ✉ ❅ � ❅ � ❅ � ❅ � � ❅ ❅ ❅ � ❅ ❅ � � ❅ � � ❅ ✉ � � ❅ ✉ � ❅ ✉ � ❅ ✉ � ❅ ❅ ✉ ❅ ❅ ✉

Each vertex chooses one of the two downward edges with equal probability. ✉ ✉ ✉ ✉ ✉ ✉ � ❅ � � ❅ ❅ ✠ � ✉ ✉ ❅ ❘ � ✠ ✉ � ✉ ✠ ✉ ❘ ❅ ✉ ❅ ❘ � � � ❅ ❅ � � ✠ ✠ ✉ � � ✠ ✉ ✉ ❅ ❘ ✉ ❘ ❅ ✠ ✉ � ✉ � ❅ � � ❅ � ✉ � ✠ ✉ ❘ ❅ ✠ � ✉ ✉ ✠ � ✉ ❅ ❘ ✠ � ✉ ❅ � ❅ ❅ ❅ � ❘ ❅ ✠ � ✉ ✉ ❅ ❘ ✉ ❘ ❅ ✉ ❘ ❅ � ✠ ✉ ✉ Direction ❅ � � ❅ ❅ � ✉ ❘ ❅ � ✉ ✠ ✠ � ✉ ✉ ❘ ❅ ✉ ❘ ❅ ✠ ✉ � ❅ � � ❅ ❅ � ❘ ❅ ✉ ✠ � ✠ � ✉ ✉ ❘ ❅ ✉ ❅ ❘ ✠ � ✉ ✉ ❄ � � ❅ � ❅ ❅ ✉ � ✠ ✠ � ✉ ✉ ❅ ❘ ✠ ✉ � ✉ ❘ ❅ ✉ ❘ ❅ ❅ � � � ❅ � ❘ ❅ ✠ � ✉ ✉ � ✠ � ✉ ✠ ✉ ❘ ❅ � ✠ ✉ ✉

The Graph � ❅ � � ❅ ❅ � ✠ ❘ ❅ ✠ � � ✠ ❅ ❘ ❅ ❘ � � � ❅ ❅ � � ✠ ✠ � ✠ � ❅ ❘ ❅ ❘ � ✠ � ❅ � � ❅ � � ✠ ❘ ❅ ✠ � � ✠ ❅ ❘ � ✠ ❅ � ❅ ❅ ❅ � ❅ ❘ ✠ � ❘ ❅ ❅ ❘ ❅ ❘ ✠ � ❅ � � ❅ ❅ � ❅ ❘ ✠ � ✠ � ❅ ❘ ❅ ❘ � ✠ ❅ � � ❅ ❅ � ❅ ❘ ✠ � ✠ � ❅ ❘ ❅ ❘ ✠ � � � ❅ � ❅ ❅ � ✠ ✠ � ❘ ❅ � ✠ ❅ ❘ ❘ ❅ ❅ � � � ❅ � ❅ ❘ � ✠ ✠ � ✠ � ❅ ❘ � ✠

Observations From a vertex there is exactly one edge going down. So no loops are possible. The graph is either ▶ a connected infinite tree ▶ each component a connected infinite tree Questions : ▶ How many trees are there? ▶ Are they bi-infinite in both the direction?

Howard’s Model Howard [1971] removed the restriction of drainage to a neighbouring square and modelled a network to include “headward growth and branching in a random fashion”. This was a model for a region where the gradient is very high and not all points are sources. The flow happens between the sources.

Howard’s Model - Source (black) Points Declare points open with probability 0 < p < 1 , independently of each other. Open points act as the sources. ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

Every open point connects to the closest open point in the next level. If there is a choice of points, one among the choices, is chosen with equal probability. Note, unlike the Scheidegger Model, the edges are not connecting nearest neighbours.

has a choice ✕ ✁ ✁ ✁ ✓✏ ✁ ✁ ✉ ✉ ✉ ✒✑ ❄ ❄ ✉ ✉ ✉ ✉ ❅ ❅ ❅ ❅ ❘ ❅ ❄ ❄ ❄ ✘ ✟ ✘ ✉ ✉ ✉ ✘ ✉ � ✟ ✘ ✘ ✟ ✘ ✘ � ✟ ✘ ✘ ✟ ✘ ✘ � ✟ ✘ ✘ ✟ ✘ ✘ � ✟ ✘ ✘ ✟ ✘ � ❄ ✙ ✠ ✾ ✘ ✟ ✉

Each source has exactly one child, but some may not have any ancestor. ✉ ✉ ✉ � � � � ✠ � ❄ ❄ ✉ ✉ ✉ ✉ ❅ ❅ ❅ ✓✏ ❅ ❅ ❘ ❄ ❄ ❄ ✘ ✟ ✘ ✉ ✉ ✉ ✘ ✉ ✒✑ � ❙ ✟ ✘ ✘ ✟ ✘ ✘ � ✟ ❙ ✘ ✘ ✟ ✘ ✘ � ✟ ❙ ✘ ✘ ✟ ✘ ✘ � ✟ ✘ ❙ ✘ ✟ ✘ ✘ ✾ � ✙ ✟ ✠ ❄ ❙ ✇ ✉ has no ancestors

Directed Spanning Forest (DSF) Baccelli and Bordenave (2007) introduced a model which they called directed spanning forest. DSF appears as an essential tool for the asymptotic analysis of the Radial Spanning Tree. We start with a homogeneous Poission point process of unit intensity on R 2 . Each vertex in the point process connects to the nearest point of the process having a strictly smaller y co-ordinate.

Geometry of the network All these models have two common geometric properties: (i) For d = 1 and d = 2 , the graph consists of a single tree almost surely and, for d ≥ 3 , the graph has infinitely many disjoint trees. (ii) There is no bi-infinite path almost surely for any d ≥ 1 .

Howard’s Model: d = 1 We study Howard’s model for d = 1 only.

Notation For an open vertex ( u , t ) let h ( u , t ) denote the open vertex on the line { y = t + 1 } such that < ( u , t ) , h ( u , t ) > is an edge of the random graph. Thinking of h ( u , t ) as the progeny of ( u , t ) , h k ( u , t ) := h ( h k − 1 ( u , t ) is the k th generation progeny of ( u , t ) on the line { y = t + k } . For an open vertex ( v , s ) set C k ( v , s ) := { ( u , s − k ) ∈ V : h k ( u , s − k ) = ( v , s ) } , the k -th generation ancestors of ( v , v ) , and, C ( v , s ) := ∪ k ≥ 0 C k ( v , s ) . the set of all ancestors of ( v , s ) .

Notation In the terminology of drainage networks, | C ( v , s ) | represents the amount of water that is drained through the points ( v , s ) . We define, H ( v , s ) := inf { k ≥ 1 : C k ( v , s ) = ∅} , the height to the mouth of the river. Since there are no bi-infinite paths, H ( v , s ) < ∞ almost surely.

Notation On the event, { H ( x , t ) > n } , define, for 0 ≤ k ≤ n , R k ( x , t ) = max { y : ( y , t − k ) ∈ C k ( x , t ) } and L k ( x , t ) = min { y : ( y , t − k ) ∈ C k ( x , t ) } . Clearly R k ( x , t ) ≥ L k ( x , t ) . Set D k = ( D k ( x , t ) =) R k ( x , t ) − L k ( x , t ) . Define, a process in C [ 0 , 1 ] as follows: for s ∈ [ 0 , 1 ] , ( s ) = D [ ns ] + ( ns − [ ns ])( D [ ns ]+ 1 − D [ ns ] ) W ( x , t ) . √ n n