Grid-Based Modeling with Digital Elevation Models David G Tarboton Utah State University http://www.engineering.usu.edu/dtarb/ david.tarboton@usu.edu (several slides from David Maidment)

Outline Grid data structures Surface flow models Channel network and watershed delineation Reading: Modeling our world Chapter 9. Arc Hydro Model Chapter 4 on Drainage Systems.

Numerical representation of a spatial surface Grid TIN Contour and flowline

A grid defines geographic space as a matrix of identically-sized square cells. Each cell holds a numeric value that measures a geographic attribute (like elevation) for that unit of space.

The grid data structure Grid size is defined by extent, spacing and no data value information Number of rows, number of column Cell sizes (X and Y) Top, left , bottom and right coordinates Grid values Real (floating decimal point) Integer (may have associated attribute table)

Definition of a Grid Cell size Number of rows NODATA cell (X,Y) Number of Columns

Line as a Sequence of Cells

Polygon as a Zone of Cells

Cell Networks

Grid Zones

Floating Point Grids Continuous data surfaces using floating point or decimal numbers

Value attribute table for categorical (integer) grid data Attributes of grid zones

Spatial Surfaces used in Hydrology Elevation Surface — the ground surface elevation at each point

Topographic Slope Defined or represented by one of the following Surface derivative z Vector with x and y components Vector with magnitude (slope) and direction (aspect)

Drainage area (also called contributing area or flow accumulation) Concentrated at a point Dispersed - specific catchment area

Specific catchment area a is the upslope area per unit contour length [m2/m  m] Upslope contributing area a Stream line Contour line

Wetness index a/S or ln(a/tan) a/S evaluated at each point in the terrain Of importance in topographically based modeling of runoff generation by saturation from below with TOPMODEL

Surface Flow Models The Eight direction pour point model The D vector surface flow model

A Case Study of Hog Pen Creek Hog Pen Ck 4 km 4 km

Watershed Delineation by Hand Digitizing Drainage direction 20 ft contour 100 ft contour Stream Center Line Watershed divide Outlet

30 Meter Mesh Standard for 1:24,000 Scale Maps and the National Elevation Dataset

Digital Elevation Model (DEM) Elevations Contours 720 700 680 740

Direction of Steepest Descent 30 30 67 56 49 52 48 37 58 55 22 67 56 49 52 48 37 58 55 22 Slope:

Eight Direction Pour Point Model 32 16 8 64 4 128 1 2 ESRI Direction encoding

Eight Direction Pour Point Model D8 4 5 6 3 7 2 1 8 Band/GRASS/TARDEM Direction encoding

Grid Network

Contributing Area Grid 1 4 3 12 2 16 25 6 1 4 3 12 2 16 6 25 Drainage area threshold > 5 Cells

Filling in the Pits DEM creation results in artificial pits in the landscape A pit is a set of one or more cells which has no downstream cells around it Unless these pits are filled they become sinks and isolate portions of the watershed Pit filling is first thing done with a DEM

Watershed Draining to This Outlet

Watershed and Drainage Paths Delineated from 30m DEM Automated method is more consistent than hand delineation

Stream Segments in a Cell Network 1 3 2 4 5 6 5 5

Subwatersheds for Stream Segments Same Cell Value

Topographic Slope ? Topographic Definition Drop/Distance Limitation imposed by 8 grid directions.

The D Vector Surface Flow Model Tarboton, D. G., (1997), "A New Method for the Determination of Flow Directions and Contributing Areas in Grid Digital Elevation Models," Water Resources Research, 33(2): 309-319.) (http://www.engineering.usu.edu/cee/faculty/dtarb/dinf.pdf)

Contributing Area using D

DEM based delineation of channel networks and subwatersheds Hydrologic processes are different on hillslopes and in channels. It is important to recognize this and delineate model elements that account for this. Concentrated versus dispersed flow. Objective delineation of channel networks using digital elevation models. 500 cell theshold 1000 cell theshold

100 grid cell constant support area threshold stream delineation

200 grid cell constant support area based stream delineation

How to decide on drainage area threshold ? 3 12

Examples of differently textured topography Badlands in Death Valley. from Easterbrook, 1993, p 140. Coos Bay, Oregon Coast Range. from W. E. Dietrich

Canyon Creek, Trinity Alps, Northern California. Photo D K Hagans

Gently Sloping Convex Landscape From W. E. Dietrich

Mancos Shale badlands, Utah. From Howard, 1994.

Topographic Texture and Drainage Density Driftwood, PA Same scale, 20 m contour interval Sunland, CA

Contrasting Interpretations “landscape dissection into distinct valleys is limited by a threshold of channelization that sets a finite scale to the landscape.” (Montgomery and Dietrich, 1992, Science, vol. 255 p. 826.) “any definition of a finite channel network is arbitrary, and entirely scale dependent.” (Band, 1993, in “Channel Network Hydrology”, edited by Beven and Kirkby, p15.)

Lets look at some geomorphology. Suggestion: One contributing area threshold does not fit all watersheds. Lets look at some geomorphology. Horton’s Laws Stream Drops

Hortons Laws: Strahler system for stream ordering 1 3 most upstream is order 1 when two streams of a order i join, a stream of order i+1 is created when a stream of order i joins a stream of order i+1, stream order is unaltered 1 2 1 2 1 1 1 1 1 1 2 2 1 1 1 1 1 1

Bifurcation Ratio

Length Ratio

Area Ratio

Slope Ratio

Constant Stream Drops

Stream Drop Elevation difference between ends of stream Note that a “Strahler stream” comprises a sequence of links (reaches or segments) of the same order Nodes Links Single Stream

Break in slope versus contributing area relationship Suggestion: Map channel networks from the DEM at the finest resolution consistent with observed channel network geomorphology ‘laws’. Look for statistically significant break in constant stream drop property Break in slope versus contributing area relationship Physical basis in the form instability theory of Smith and Bretherton (1972), see Tarboton et al. 1992

Statistical Analysis of Stream Drops Threshold drainage area of 20 upwards curved grid cells to create a stream

T-Test for Difference in Mean Values 72 130 T-test checks whether difference in means is large (> 2) when compared to the spread of the data around the mean values

Statistical Analysis of Stream Drops Threshold = 10 Dd = 2.5 t = -3.5 Threshold = 15 Dd = 2.1 t = -2.08 Threshold = 20 Dd = 1.9 t = -1.03 Stream drop test for Mawheraiti River. For each upward curved support area threshold the stream drop for each stream is plotted against Strahler stream order. The large circles indicate mean stream drop for each order The weighted support area threshold, drainage density (in km-1) and t statistic for the difference in means between lowest order and all higher order streams is given.

Local Curvature Computation (Peuker and Douglas, 1975, Comput Local Curvature Computation (Peuker and Douglas, 1975, Comput. Graphics Image Proc. 4:375) 43 48 48 51 51 56 41 47 47 54 54 58

Contributing area of upwards curved grid cells only

Upward Curved Contributing Area Threshold

Constant Support Area Threshold

Curvature based stream delineation with threshold by constant drop analysis

200 grid cell constant support area based stream delineation

Channel network delineation, other options 4 5 6 3 7 2 1 8 Accumulation Area 1 2 3 Grid Order 1 4 3 12 2 16 25 6

Grid network pruned to order 4 stream delineation

Summary Concepts Grid data structures represent surfaces as an array of cells The elevation surface represented by a grid digital elevation model is used to derive surfaces representing other hydrologic variables of interest such as Slope Drainage area Wetness index Watersheds and channel networks

Summary Concepts (2) The eight direction pour point model approximates the surface flow using eight discrete grid directions. The D vector surface flow model approximates the surface flow as a flow vector from each grid cell apportioned between down slope grid cells.

Summary Concepts (3) Channel networks obey Hortons laws. Use consistency with Hortons laws to adapt support area threshold and drainage density to the natural texture of the topography. Use curvature based methods to allow channel network drainage density to be spatially variable to adapt to variable topographic texture.