Automatic Delineation of Drainage Basins from Contour Elevation Data Using Skeleton Construction Techniques

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1 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 1/16 Automatic Delineation of Drainage Basins from Contour Elevation Data Using Skeleton Construction Techniques Giovanni Moretti and Stefano Orlandini University of Modena & eggio Emilia, Italy University Founded in 1175

2 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 2/16 Motivation b m drainage basin The drainage basin determines the area which contributes water and sediments to a given channel cross section and is therefore considered to be the fundamental unit of study of geomorphological and fluvial processes (eopold et al., 1964). a m 414 draining line segment connecting two assigned points 5 1 m divide segment perpendicular to the only upper DB1 4 DB2 39 DB When curved slope line segments are drawn manually in order to delineate a drainage basin (Maxwell, 187), human discernment is used to identify the paths that are most likely followed by imaginary drops of water falling from the upper s and drained by the lower s. On the other hand, when slope lines are drawn using a computer program, no inference on the morphology of the terrain lying within adjacent s is normally made. A curved slope line segment between two adjacent s is approximated by a (single) straight line segment, which is generally unable to satisfy perpendicularity to both s.

3 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 3/16 Fundamental Geometric Constructs (a) P 3 V 1 V 2 P 4 (b) V 1 Delaunay triangulation P 1 P2 Voronoi diagram P 1 P 2 P1 crust V 1 V 2 P 2 circle drawn to perform the incircle test skeleton V2 circle drawn to perform the incircle test A Delaunay triangulation of a set of points in the plane (P 1, P 2,...) is a triangulation with the property that no point in the set of points falls inside the circumcircle of any triangle in the triangulation. A Voronoi diagram of a set of points in the plane (P 1, P 2,...) is a subdivision of the plane into polygonal regions, where each region is that set of points closer to some input point than to any other input point. A Delaunay edge (P 1 P 2 ) belongs to the crust when a circle exists through its two endpoints (circle in dashed line) that does not contain either of its associated Voronoi vertices (V 1 and V 2 ). If this condition is not met, then the corresponding Voronoi edge (V 1 V 2 ) belongs to the skeleton. Crust and skeleton elements can be determined by performing a simple incircle test (Gold, 1999). A circle passing through the endpoints of the Delaunay edges (P 1 and P 2 ) and the closest Voronoi vertex (V 1 ) is drawn (circles in solid line) and a check is made to assess whether the second Voronoi vertex is inside the circle.

4 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 4/16 Simple Theoretical Cases (a) curve/closed (d) curve/upper Two perspectives: (b) set of (sample) points Delaunay triangulation (e) set of (sample) points curve/lower Delaunay triangulation Amenta et al. (1998) defined and proposed to use the crust as a means for reconstructing a curve from a given set of sample points. P 1 P V P 3 2 Voronoi diagram crust skeleton (branch) skeleton (branch) (f) Voronoi diagram crust skeleton (stem) crust skeleton (branch) Gold (1999) proposed to use the skeleton of the s as a means for reconstructing the most plausible morphology of the terrain lying within adjacent s or within a closed contour line.

5 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 5/16 ida Surveys Survey carried out by Helica (Italy) in the Ca ita area (Italian Apennines): Optech ATM (airborne lidar terrain mapping) system mounted on a helicopter. The flight altitude: about 8 m agl (above ground level). Average data density: about 2.9 points per square meter. Horizontal absolute accuracy:.4 m. Vertical absolute accuracy:.15 m. esolution of the gridded elevation data provided:.5 m. Survey carried out by Terrapoint (USA) in the Col odella area (Italian Alps): ATM system of Terrapoint s proprietary design mounted on a fixed-wing aircraft. The flight altitude: about 1 m agl. Average data density: about.5 points per square meter. Horizontal absolute accuracy: 1 m. Vertical absolute accuracy:.3 m. esolution of the gridded elevation data provided: 1 m. Gridded elevation data were resampled to 5-m and contour elevation data with contour intervals of 1, 5, 1, 2, and 5 m were generated.

6 2384 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 6/16 Skeleton Extraction and Drainage Basin Delineation Sampling issues are not a critical point. The program Triangle developed by Shewchuk (1996) is used to compute the Delaunay triangulation and the Voronoi diagram. a skeleton stem from adjacent contour lines representing different elevations P (n) K draining line segment connecting two assigned points m P (n) K P P (a) (a) P P 415 skeleton branches forming a tree structure from a markedly non-straight skeleton stem skeleton branch divide point drainage basin m 2417 skeleton stem skeleton branch saddle branch divide point drainage basin (n) S P (n) P K P B M type 1 saddle (n) S P 2 (n) P K P P b 2415 c S (n) (n) P P K B 5 1 m type 2 saddle 2383 P skeleton stem skeleton branch saddle branch divide point drainage basin P 2 P B K (n) P S (n) = P 1 d P skeleton stem skeleton branch divide point discarded point drainage basin skeleton stem skeleton branch characteristic branch connecting line 355 e skeleton stem skeleton branch characteristic branch connecting line 355 D 1 B 1 B 2 f P 1 2 m 1 K =K first-order skeleton branch connetcting two points of the closed 2 4 m B 6 P 1 B 5 B 4 P 2 B 3 P 3 B 2 B m B 3 D 2 B 4

7 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 7/16 Procedure DIVIDE(P,P ) equire: Contour lines, skeleton, points (P,P ) Ensure: Drainage divide through the two points (P,P Search K while K and K K do Search P (n) and P (n) if ID of the containing P (n) end if P P if a type 1 saddle is identified then Search S (n) and S (n) else if a type 2 saddle is identified then Search S (n) and S (n) end if call DIVIDE(P (n),s(n) P (n) P (n) P(n) Search K end while return S (n) and K ) ) ID of the containing P (n) then

8 Proposed Method (PM) contour 442line skeleton stem skeleton branch ± b drainage basin a m DB1 divide segments 414 perpendicular to both the upper and lower s draining line segment connecting two411 assigned points Drainage divides obtained from the proposed method conform with the general principles that would be applied by an expert hydrologist in solving the same problems manually. esults obtained using 1-m contour data are totally satisfactory and provide some improvements with respect to the results obtained by applying the current state-of-the-art method m m DB2 DB Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 8/16

9 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 9/16 Current Methods (CM) b drainage basin Current methods applied to 1-m contour elevation data (Dawes and Short, 1994; Maunder, 1999; Menduni et al., 22) m a DB divide segment perpendicular to the only upper draining line segment connecting two assigned points 5 1 m m DB2 DB

10 41 41 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 1/16 Increasing the Contour Interval a m 43 DB1 divide from 1-m contour data divide from 1-m contour data DB DB3 34 b m 4 43 DB1 divide from 1-m contour data divide from 1-m contour data DB DB3 34 The improvements offered by the new method are even more significant for the contour elevation data with contour intervals of 5, 1, 2, and 5 m. c m 4 DB1 divide from 2-m contour data divide from 1-m contour data 42 DB2 38 DB3 34 d m 4 DB1 divide from 2-m contour data divide from 1-m contour data 42 DB2 38 DB3 34 One can note that the gain in accuracy is not only due to the skeleton stems as a sort of contour data enrichment, but also to the methodological advance allowed by a profitable use of the entire skeleton structure (stems and branches).

11 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 11/16 Flow Net Construction a automatically identified ridge flow line connecting line flow line automatically identified group of depressions b automatically identified saddle automatically identified peak 1 2 m J C P 1 P 2 E 1 S 2 4 m E 2 P 3 S 2 1 D 1 D 2 E 3 E 4 automatically identified group of peaks S 4 S 3 34 The methodology described in Moore et al. (1988), Moore and Grayson (1991), and Gallant (2) is extended in order to process groups of peaks and/or depressions lying between adjacent s. The ordering of skeleton elements allows skeleton stems and skeleton branches to be recognized and intersected at right angles or followed, respectively, by flow lines and drainage divides.

12 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 12/16 Procedure FOWINES(E 1, E 2 ) equire: Contour lines, connecting lines, points (E 1, E 2 ) Ensure: Insertions or terminations of flow lines between E 1 and E 2 CID = ID of the containing E 1 while E 2 is not found do Travel all elements of the CID Accumulate the distance along the CID if a connecting line is found then Perform insertions or terminations on the CID Travel the connecting line until the connected is found C = point of intersection between the connecting line and the connected contour line call FOWINES(C, E 2 ) end if end while Perform insertions or terminations on the CID return

13 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 13/16 Type 1 and Type 2 Errors in Drainage Areas A 1 DBD1 P A 2 11 P A 3 DBD2 drainage basin determination DBD1 obtained from given contour elevation data and method drainage basin determination DBD2 obtained by varying either the elevation contour data or the method draining line segment connecting two assigned points (P,P ) E 1 = A 1 A 3 A 1 + A 2 E 2 = A 1 + A 3 A 1 + A 2 E 1 E 2 E 2 E 1 One can note that error E 1 is just an indicator of the absolute difference between the drainage areas of DBD1 and DBD2, independently on the location of these two basin determinations, whereas error E 2 also accounts for the nonoverlapping between the two basin determinations.

14 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 14/16 Numerical Analysis Errors relative to the solution obtained from the PM and 1-m contour data. relative error (-) relative error (-) a d DB1 DB b DB2 DB5 E 1, CM E 2, CM E 1, PM E 2, PM e c f DB3 DB contour interval (m) contour interval (m) contour interval (m) elative errors generally increase as the contour interval increases. Using the PM and contour intervals of 5 5 m, relative errors are normally less 2% and rarely greater than 5%.

15 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 15/16 Computational Efficiency The computational burden imposed by the proposed method can be evaluated by considering separately the two contributions to total central processing unit (CPU) time required for (1) skeleton extraction from s and (2) drainage basin delineation or flow net construction. The CPU time required to extract the skeleton from s rapidly decreases as contour interval increases. Using an Intel Core 2 Duo 67 processor, the CPU time required to extract the skeleton from s representing the areas shown in this presentation decreases form 2.43 h to 5. min as the contour interval increases from 1 to 1 m. The CPU time required to delineate a simple drainage basin is in the order of a few seconds for both proposed and current state-of-the-art methods. However, if 1-m contour data are used and specific checks are made in order to attempt to resolve complex topographic structures using the current state-of-the-art method, the CPU time may go beyond 1 h (and accurate results are not always achieved). In this context, the computational burden imposed by skeleton extraction from s appears quite bearable. In scientific studies, CPU times in the order of a few hours appear acceptable in exchange for (accurate) solutions from fine resolution elevation data. In technical applications, CPU times in the order of a few minutes appear acceptable in exchange for accurate solutions from coarse resolution elevation data.

16 Summer School in New York, USA, Polytechnic University, July 21 25, 28 p. 16/16 Conclusions The proposed method allows fully-automated delineations of drainage basins and constructions of flow nets from contour elevation data, even when critical topographic structures such as ridges, saddles, and peaks are present. For any given set of s, the proposed method provides more accurate solutions than previously proposed methods, the gain in accuracy normally increasing as the contour interval increases. Skeleton construction techniques allow the morphological information implicitly present in contour elevation data to be explicitly revealed and correctly processed by a computer program, and are a useful means for improving the accuracy with which physiographic features of drainage basins are determined. The description of the true topography of a real terrain is an ideal process that can be reached only by using infinitely accurate instruments and techniques along with infinitesimal elevation model resolutions. The methods developed in this study are intended to allow the maximum exploitation of elevation data having a given accuracy and resolution.