HAND, a new terrain descriptor using SRTM-DEM - DPI - Inpe

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Table 1 Δ Function definition Flow direction Δ (Flow direction) N h0,−1i NE h1,−1i E h1,0i SE h1,1i S h0,1i SW h−1,1i W h−1,0i NW h−1,−1i Null h0,0i Montgomery and Dietrich (1988) discussed the criteria to predict the point where channels should begin (headwater). One of the first procedures for delineating drainage networks (O'Callaghan and Mark, 1984) used a minimum contributing area threshold to identify channel beginning. The drainage network is thus defined by those grid points that have a contributing area greater than a given threshold. The contributing area is computed by counting the number of points whose flow paths converge to the considered point. Considering a given point h i,j i, we can determine its neighborhood as Nðhi; jiÞ ¼ fhk; liaGjk ¼ fi1; i; i 1g; l ¼ fj1; j; j þ 1g; hk; liphi; jig ð2Þ The points in N(hi,ji) are called the neighbors of h i,j i.Ifpointh i,j i is not on the border grid (i≠1, i≠c, j≠1 andj≠r), then it has 8 neighbors, that is Nðhi; jiÞ ¼ fhi1; j 1i; hi 1; ji; hi 1; j þ 1i; hi; j 1i; hi; j þ 1i; hi þ 1; j 1i; hi þ 1; ji; hi þ 1; j þ 1ig otherwise, it will have less than 8 neighbors. Based on the D8 method, the point hi,ji can be hydrologically connected to only one of its neighbors. If F(hi,ji) is the point to which hi,ji flows, it can be defined as: Fðhi; jiÞ ¼hi; jiþDðLDDðhi; jiÞÞ ð4Þ where Δ is a function that represents the relative position hΔi,Δji between two hydrologically connected points. Table 1 shows the correspondence between the flow directions and the relative positions. Note that if point hi, ji is a pit, then F(hi, ji)=hi, ji. The contributing area of point hi, ji, defined as A(hi,ji), is computed through an iterative process. Let A t (hi, ji) be the contributing area of the point hi, ji in the t th iteration. In the first iteration (t=1) A 1 ðhi; jiÞ ¼ 1 ifhi; jigFðGÞ 0 otherwise where F(G) represents the set of all upward points connected to any given point, that is FG ð Þ ¼ fhk; liaGjahi; jiaNðhk; liÞjFðhi; jiÞ ¼ hk; lig: ð6Þ In this first iteration, the contributing area of all points that initiate a flow path is equal to one. For the other iterations (tN1) A t A ðhi; jiÞ ¼ t 1ðhi; jiÞ if At 1ðhi; jiÞp0 1 þ P F−1ðhi;jiÞAt 1ðhk; liÞ if 0gAt 1 F 1 8 < ðhi; jiÞ : 0 otherwise ARTICLE IN PRESS 4 C.D. Rennó et al. / Remote Sensing of Environment xxx (2008) xxx-xxx ð3Þ ð5Þ ð7Þ where F − 1 (hi, ji) represents a set of all neighbors of point hi,ji that flow to it, that is F 1 ðhi; jiÞ ¼ fhk; liaNðhi; jiÞjFðhk; liÞ ¼ hi; jig ð8Þ The iteration process stops (t=n) when, for any point hi,ji, A t −1 (hi,ji)≠0, that is, A(hi,ji)=A n (hi,ji). The method based on the contributing area is very easy to implement and therefore widely used. Tarboton (2003) pointed out that a significant question with this method is the choice of contributing area threshold. The basic hypothesis is that channel heads, where there is a transition from convex to concave profiles, is also where concentrated fluxes begins to dominate over diffusive fluxes (Tarboton et al., 1991, 1992). Some authors have called for objective criteria to define channel heads, such as the relationship between slope and contributing area (Tarboton et al., 1991). Others use local curvatures to account for spatially variable drainage densities (Tarboton and Ames, 2001). Montgomery and Foufoula-Georgiou (1993) compared the constant contributing area with the slopedependent contributing area. Although these approaches represent progress towards the automatic determination of channel heads, the tests we conducted for Central Amazonia showed that the application of these methodologies to SRTM data did not estimate drainage density properly. We suspect that vegetation masking ground topography may be the explanation. For this reason, we decided to use the contributing area threshold as one of the main criteria to define the drainage network, validating channel heads with field data. To define the channel heads we used an additional criteria based on simplifications of horizontal and vertical geomorphic curvatures. Firstly, the channel head element should represent a convergent point, meaning it should have two or more overland flux paths converging to it (horizontal curvature). Secondly, the channel profile should be concave, i.e. where the channel head profile point has a smaller change of elevation than the mean of the elements located uphill and downhill of it (vertical curvature). 2.2. The height above the nearest drainage terrain descriptor To quantify relevant parameters that could uniquely identify generalizable spatial properties of hill slopes there was the need to have a local frame of variable topographic reference that should be more useful than the all encompassing and generic height ASL, or the third dimension in the DEM. Horizontal distances of DEM grid points to connected drainage channels (slope length) have been computed by a number of approaches (e.g. Tucker et al., 2001). As the initial interest in this study was to predict potential hydrological properties for each DEM grid point, especially the depth to the permanently saturated zone, this approach wasn't useful. The gravitational potential energy difference between any given grid point and the other extremity of the hill slope flow path, at the functional stream outlet (explicit in the LDD), defines a unique and permanent property of that grid point that we call draining potential. The vertical distance of a given grid point to its drainage outlet (which is hydrologically important) can in most cases be expressed as a relative height, or the height difference between those points. Thus there will be no gravitationally driven water movement between two hydrologically connected points that share the same height. Classifying all grid points according to their respective draining potentials allows them to be grouped into classes of equipotential (equivalent draining gravitational potential), defining environments or zones with inferred similar hydrological properties. The linking of every grid point to its outlet on the drainage system allows for the whole DEM to be normalized for the drainage network (adjusting point heights in relation to the drainage), which in this way becomes a distributed frame of topographic reference. If D is a set of drainage network points, identified by a number through a bijective function in order that each point of the drainage Please cite this article as: Rennó, C. D., et al., HAND, a new terrain descriptor using SRTM-DEM: Mapping terra-firme rainforest environments in Amazonia, Remote Sensing of Environment (2008), doi:10.1016/j.rse.2008.03.018
network be identified by a unique identification number k a N⁎. We define that Dðhi; jiÞ ¼ k if hi; ji is a drainage network point 0 otherwise and its inverse function as D 1 ðÞ¼ k fhi; jiaGjDðhi; jiÞ ¼ kg: ð10Þ Following respective flow paths, each and every point in the grid is necessarily connected to a drainage point. Let I(hi,ji) be the function that identifies the drainage point connected to the point hi,ji. This function is computed through an iterative process and the result is a grid that associates the identification number of the drainage point that each point is connected to. If I t (hi,ji) is the drainage identification number of the point hi, ji in the t th iteration. In the first iteration (t=1), I 1 ðhi; jiÞ ¼ Dðhi; jiÞ: ð11Þ For the other iterations (tN1) I t I ðhi; jiÞ ¼ t 1ðhi; jiÞ if It 1ðhi; jiÞp0 It 1ðFðhi; jiÞÞ if It 1 8 < : 0 ðFðhi; jiÞÞp0 otherwise: ð9Þ ð12Þ The iteration process stops (t=m) when, for any point hi,ji, I t (hi,ji)≠0, that is, Iðhi; jiÞ ¼ I m ðhi; jiÞ: ð13Þ Assuming that all points belong to a flow path and that all flow paths are associated to respective drainage points, we define the HAND value of any given point hi, ji as HANDðhi; jiÞ ¼ H hi; ji ð Þ HDðhi; jiÞ if HDðhi; jiÞbHðhi; jiÞ 0 otherwise ARTICLE IN PRESS C.D. Rennó et al. / Remote Sensing of Environment xxx (2008) xxx-xxx Fig. 3. Procedure to calculate the HAND grid. BBBBlue squares represent grid points belonging to the drainage network. Only black arrows are considered as flow paths. ð14Þ where H(hi,ji) represents the height of the point hi, ji given by the original DEM and H D(hi, ji) is the height of drainage point hydrologically connected to point hi,ji following the flow path, that is, HDðhi; jiÞ ¼ HD 1 ðIðhi; jiÞÞ : ð15Þ It is important to note that if the point i,j is a point belonging to the drainage, then Iðhi; jiÞ ¼ Dðhi; jiÞ ð16Þ D 1 ðIhi; jiÞ ¼ hi; ji ð17Þ HDðhi; jiÞ ¼ Hðhi; jiÞ ð18Þ and so HANDðhi; jiÞ ¼ Hðhi; jiÞ Hðhi; jiÞ ¼ 0: ð19Þ In other words, all grid points belonging to the drainage network are zeroed in height, which implies that the draining potential (according with the HAND definition) along the stream channel is disregarded. Although the drainage channel is also a flow path, which effectively drains, the HAND descriptor uses the whole channel as a flat relative topographic reference, the end of all non channel flow paths. By definition, the HAND drainage outlet grid point is the nearest draining point to itself, therefore it can only be subtracted from itself. It is important to note that the breaching process, required to reach a topologically sound and accurate drainage network, introduces occasional canyon like artifacts into the DEM, as a result of aberrant height differences adjacent to the drainage network. These artifacts would be transferred to the HAND grid if it were computed from the corrected DEM. When using original SRTM data for the HAND grid computation, some associations prescribed by the corrected drainage connection grid result in small negative differences that could be interpreted as points Please cite this article as: Rennó, C. D., et al., HAND, a new terrain descriptor using SRTM-DEM: Mapping terra-firme rainforest environments in Amazonia, Remote Sensing of Environment (2008), doi:10.1016/j.rse.2008.03.018 5
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