A Differential Equation for Specific Catchment Area

Transcription

1 Proeedings of Geomorphometry Zurih, Sitzerland, 3 ugust - 2 September, 2009 Differential Equation for Speifi Cathment rea J. C. Gallant, M. F. Huthinson 2 CSIRO Land and Water, GPO Box 666, Canberra CT 260, ustralia Telephone: Fax: John.Gallant@siro.au 2 Fenner Shool of Environment and Soiety, ustralian National University, Canberra CT 0200, ustralia Telephone: Fax: Mihael.Huthinson@anu.edu.au. Introdution Speifi athment area is one of the key land-surfae parameters used in the fields of hydrology, geomorphology, pedology and eology and many methods have been devised to estimate it from grid DLSMs. n aurate referene is required to test these methods, but speifi athment area has only been determined analytially for simple surfaes suh as inlined planes and ones. To assess the results on the omplex surfaes of natural terrain developers and users have resorted to omparisons beteen the results from different methods and visual inspetion of the patterns of estimated speifi athment area. This paper presents a differential equation that desribes the rate of hange of speifi athment area along a flo line. The equation an be solved numerially along a flo line that is derived numerially from a grid digital elevation model, alloing preise values of speifi athment area to be obtained at any loation on a omplex terrain surfae. The development of the onepts and exploration of the results and impliations are addressed in more detail in Gallant and Huthinson, (in preparation). 2. Stream Tubes and Speifi Cathment rea Speifi athment area, ontributing area (or total athment area, TC) and flo idth are all defined in terms of a stream tube (Onstad and Brakensiek, 968) onsisting of to adjaent flo lines terminating at the donstream end on the opposite ends of a ontour segment (Fig. ). Speifi athment area a is defined as a = lim () 0 here is ontributing area, the area of the stream tube, and is the idth of the loer end of the stream tube. 28

2 Proeedings of Geomorphometry Zurih, Sitzerland, 3 ugust - 2 September, 2009 (a) (b) Figure. (a) stream tube of area defined by a ontour segment of length. (b) segment of stream tube shoing hange in ontour length determined by ontour line urvature. The area at distane l from the hilltop is the integral of flo idth ith respet to flo length: or equivalently l ( l) = ( u) du (2) 0 d = (3) Consideration of the geometry of a short segment of stream tube here the radius of urvature of the ontour lines is r leads to an expression for the rate of hange of ontour segment idth : d = = K (4) r here K = is the plan urvature (or ontour urvature), defined here to be positive r for divergent regions (as shon in Fig. (b)) and negative for onvergent regions. d d The equations for and allo a solution to the variation of speifi athment area a along a flo line starting from the definition of a at a point: da a = lim (5) 0 d = lim (6) 0 29

3 Proeedings of Geomorphometry Zurih, Sitzerland, 3 ugust - 2 September, 2009 = lim 0 d 2 d (7) = lim 0 2 K 2 (8) K = lim 0 (9) = K lim (0) 0 da = K a () Equation desribes the rate of hange of speifi athment area a along the flo path. It is a non-linear differential equation that an be integrated numerially along a flo path onstruted starting from a hilltop ith a = 0. Under suitable onditions it an be solved analytially, although most of the analytial results an be obtained more easily by solution from first priniples. This equation has some interesting properties. The to terms on the right hand side relate to the to soures of hange in speifi athment area: the onstant term represents the inrement due to inreasing length of the flo line, hile the seond term aptures the effets of onvergene and divergene. In divergent terrain ( K > 0 ) the to terms ompete and da depending on the size of a and K. If may inrease or derease along the flo path a = the to terms balane exatly and speifi athment area remains onstant along the flo path. In onvergent terrain the to terms are both positive leading to an exponential inrease in speifi athment area. The use of a single flo line to determine speifi athment area beomes untenable in strongly onvergent areas and in hannels, here total athment area is a more relevant quantity, but the results of Equation are appliable over most of the landsape. robust method to determine here Equation eases to be appliable; the ondition a > 5l is suggested in Gallant and Huthinson (in preparation). 3. Numerial Solution The solution of () on a grid DLSM requires: an interpolation method to reate a smooth surfae so that first and seond derivatives an be omputed at any point a method to onstrut flo lines on that surfae a method to integrate () along a flo line We use a biquadrati interpolation method (de Boor, 978) to provide a ontinuous surfae ith ontinuous first derivatives. The flo lines are onstruted as short K 30

4 Proeedings of Geomorphometry Zurih, Sitzerland, 3 ugust - 2 September, 2009 straight line segments using a midpoint method ith adaptive step size. t the start of eah step, the loal flo diretion (aspet) is determined using the first derivatives of the surfae, a trial half-step is taken, the diretion is re-omputed at the half-step point then a full step is taken in that ne diretion. The step size is adjusted so that the diretions at the beginning and end of eah step are quite lose (dot produt of unit vetors is greater than 0.99). Integration of () is ahieved by analytial solution for eah segment of the flo line ith appropriate assumptions on the variation of K along the segment. In most ases the form: is used, yielding the solution: a ( l) K ( l) = l + (2) l l = + a0 (3) provided and 0 and that K does not hange sign along the line segment. If any of those onditions are not satisfied, an alternate solution ith onstant K is used: a K l ( l) a e = K 0 K (4) The integration ommenes at the top of the stream line ith a = 0 and proeeds segment by segment to the end of the line. The auray of the numerial solution ill depend on the auray ith hih the flo line is onstruted and the auray of the integration of Equation along the line. The hoie of the interpolation method ill also have an impat on the results, although this is more a matter of hoie than auray. In our experiments, e have used onservative parameters for the onstrution of the line so that many short line segments are onstruted and further refinement of the line does not produe notieable hanges to either the path of the line or the integration of Equation. Gallant and Huthinson (in preparation) demonstrate that the numerial results are indistinguishable from full analytial solutions on surfaes here analytial solutions are available (planes and ones). 4. Comparisons ith Conventional Methods The most valuable appliation of Equation is as a referene against hih the approximate (and muh more effiient) methods of alulating speifi athment area an be ompared. Figure 2 shos a flo path and stream tube alulated from a 20 m resolution DLSM in the Brindabella Ranges near Canberra in southeastern ustralia (35 2 S E). Figure 3 shos speifi athment area alulated along the entral flo line using Equation along ith estimates from the most ommonly used onventional methods: D8 (O Callaghan and Mark, 984); slope-eighted multiple flo diretion here referred to as M8 (Freeman, 99; Quinn et al., 99, 995); DEMON (Costa-Cabral and Burges, 994); and D (Tarboton, 997). 3

5 Proeedings of Geomorphometry Zurih, Sitzerland, 3 ugust - 2 September, 2009 Figure 2. Flo lines derived from a 20 m resolution DLSM Figure 3. Speifi athment area along the flo line of Figure 2 omputed using Equation and four onventional methods. long the divergent ridgeline (flo length of m) all the onventional methods over-estimate speifi athment area by about a fator of to. This is largely due to the ay in hih flo idth is estimated in these methods; flo idths range from to.44 times the ell size, hereas on a ridgeline flo leaves the ell over muh of the perimeter of the ell. long the planar hillslope ( m) all methods perform quite ell ith D8 giving the losest results to Equation. This result reflets the orientation of the flo 32

6 Proeedings of Geomorphometry Zurih, Sitzerland, 3 ugust - 2 September, 2009 path in a ardinal diretion, here the knon defiienies of the D8 method do not our. Beteen 400 and 600 m the terrain is gently divergent then onvergent. The D8 method is unable to apture this subtle variation, M8 performs better but still poorly hile DEMON and D behave quite ell. D appears to best apture the deline in speifi athment area around 450 m. In the more onvergent area beyond 600 m the result from Equation inreases exponentially, hih is an unrealisti result that an be traed bak to the assumption in Equation that flo alays responds to loal urvature. 5. Conlusions Equation provides a method for aurately alulating speifi athment area at a speified point on a grid DEM ithout having to separately alulate athment area and flo idth. This method provides for the first time a means of testing approximate grid-based methods in omplex terrain. brief omparison ith existing methods shos that the D method best aptures the variations in speifi athment area along the single flo line examined, although all methods over-estimate speifi athment area on divergent ridge lines. Referenes Costa-Cabral M and Burges S, 994, Digital Elevation Model Netorks (DEMON): model of flo over hillslopes for omputation of ontributing and dispersal areas. Water Resoures Researh 30(6): de Boor C, 978, pratial guide to splines. Springer-Verlag, Berlin. Freeman T, 99, Calulating athment area ith divergent flo based on a regular grid. Computers and Geosienes 7(3): Gallant J and Huthinson M, in preparation. Exat alulation of speifi athment area on grid DEMs, to be submitted to Water Resoures Researh. O'Callaghan J and Mark D, 984, The extration of drainage netorks from digital elevation data. Computer Vision, Graphis and Image Proessing 28(3): Onstad C and Brakensiek D, 968, Watershed simulation by the stream path analogy. Water Resoures Researh 4(5): Quinn P, Beven K, Chevallier P and Planhon O, 99, The Predition of Hillslope Flo Paths for Distributed Hydrologi Modelling using Digital Terrain Models. Hydrologial Proesses 5(): Quinn P, Beven K and Lamb R, 995, The ln( a tan β ) index: Ho to alulate it and ho to use it ithin the TOPMODEL frameork. Hydrologial Proesses 9(2):6-82. Tarboton D, 997. ne method for the determination of flo diretions and upslope areas in grid digital elevation models. Water Resoures Researh 33(2):