Linear layout of multiple flow-direction networks for landscape-evolution simulations

We present an algorithm that is well suited to find the linear layout of the multiple flow-direction network (directed acyclic graph) for an efficient implicit computation of the erosion term in landscape evolution models. The time complexity of the algorithm varies linearly with the number of nodes in the domain, making it very efficient. The resulting numerical scheme allows us to achieve accurate steady-state solutions in conditions of high erosion rates leading to heavily dissected landscapes. We also establish that contrary to single flow-direction methods such as D8, D$\infty$ multiple flow-direction method follows the theoretical prediction of the linear stability analysis and correctly captures the transition from smooth to the channelized regimes. We finally show that the obtained numerical solutions follow the theoretical temporal variation of mean elevation.

[1]  Joe Celko.  Graphs, Trees, and Hierarchies , 2012 .

[2]  W. Culling.  Soil Creep and the Development of Hillside Slopes , 1963, The Journal of Geology.

[3]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[4]  Michael F. Goodchild,et al.  Gis and Environmental Modeling: Progress and Research Issues , 1996 .

[5]  Jaan Kiusalaas,et al.  Numerical methods in engineering with Python , 2005 .

[6]  Kyungrock Paik,et al.  Simulation of landscape evolution using a global flow path search method , 2012, Environ. Model. Softw..

[7]  Yuan Guo-xing,et al.  Verification and Validation in Scientific Computing Code , 2010 .

[8]  W. Culling,et al.  Analytical Theory of Erosion , 1960, The Journal of Geology.

[9]  Qian-Ping Gu,et al.  Efficient parallel and distributed topological sort algorithms , 1997, Proceedings of IEEE International Symposium on Parallel Algorithms Architecture Synthesis.

[10]  Christopher J. Roy,et al.  Review of code and solution verification procedures for computational simulation , 2005 .

[11]  Z. Lou,et al.  A new algorithm to automatically extract the drainage networks and catchments based on triangulation irregular network digital elevation model , 2014 .

[12]  Günter Blöschl,et al.  On the definition of the flow width for calculating specific catchment area patterns from gridded elevation data , 2005 .

[13]  Interbasin and Intrabasin Competitions Control Drainage Network Density , 2019, Geophysical Research Letters.

[14]  Ali Shokoufandeh,et al.  A new rapid watershed delineation algorithm for 2D flow direction grids , 2018, Environ. Model. Softw..

[15]  P. Moin.  Fundamentals of Engineering Numerical Analysis , 2001 .

[16]  D. Tarboton.  A new method for the determination of flow directions and upslope areas in grid digital elevation models , 1997 .

[17]  Jon D. Pelletier,et al.  Minimizing the grid-resolution dependence of flow-routing algorithms for geomorphic applications , 2010 .

[18]  Michael F. Hutchinson,et al.  A differential equation for specific catchment area , 2011 .

[19]  T. G. Freeman,et al.  Calculating catchment area with divergent flow based on a regular grid , 1991 .

[20]  S. Willett,et al.  On steady states in mountain belts , 2002 .

[21]  M. Costa-Cabral,et al.  Digital Elevation Model Networks (DEMON): A model of flow over hillslopes for computation of contributing and dispersal areas , 1994 .

[22]  Philip M. Morse,et al.  Methods of Mathematical Physics , 1947, The Mathematical Gazette.

[23]  Björn Birnir,et al.  The scaling of fluvial landscapes , 2001 .

[24]  Gregory J. McCabe,et al.  Comparison of Single and Multiple Flow Direction Algorithms for Computing Topographic Parameters in TOPMODEL , 1995 .

[25]  Nicole M. Gasparini,et al.  The Channel-Hillslope Integrated Landscape Development Model (CHILD) , 2001 .

[26]  Hans Petter Langtangen,et al.  Finite Difference Computing with PDEs: A Modern Software Approach , 2017 .

[27]  Robert E. Tarjan,et al.  Edge-disjoint spanning trees and depth-first search , 1976, Acta Informatica.

[28]  W. Dietrich,et al.  Controls on the spacing of first-order valleys , 2008 .

[29]  J. Roering,et al.  How well can hillslope evolution models “explain” topography? Simulating soil transport and production with high-resolution topographic data , 2008 .

[30]  K. A. Stroud,et al.  Engineering Mathematics , 2020, Nature.

[31]  John F. O'Callaghan,et al.  The extraction of drainage networks from digital elevation data , 1984, Comput. Vis. Graph. Image Process..

[32]  W. Dietrich,et al.  Longitudinal Profile Development into Bedrock: An Analysis of Hawaiian Channels , 1994, The Journal of Geology.

[33]  Nicole M. Gasparini,et al.  Creative computing with Landlab: an open-source toolkit for building, coupling, and exploring two-dimensional numerical models of Earth-surface dynamics , 2016 .

[34]  Paul H. J. Kelly,et al.  A dynamic topological sort algorithm for directed acyclic graphs , 2007, ACM J. Exp. Algorithmics.

[35]  J. Kirchner,et al.  Branching geometry of valley networks on Mars and Earth and its implications for early Martian climate , 2018, Science Advances.

[36]  Hamida Aktara Hoque,et al.  Trees in Disemigraphs , 2016 .

[37]  J. Morel,et al.  Landscape evolution models: A review of their fundamental equations , 2014 .

[38]  A. Howard.  A detachment-limited model of drainage basin evolution , 1994 .

[39]  Richard Barnes,et al.  Accelerating a fluvial incision and landscape evolution model with parallelism , 2018, Geomorphology.

[40]  G. Tucker,et al.  Dynamics of the stream‐power river incision model: Implications for height limits of mountain ranges, landscape response timescales, and research needs , 1999 .

[41]  A. Porporato,et al.  Channelization cascade , 2018, 1812.03696.

[42]  R. Bras,et al.  Vegetation-modulated landscape evolution: Effects of vegetation on landscape processes, drainage density, and topography , 2004 .

[43]  J. Pelletier.  Fluvial and slope‐wash erosion of soil‐mantled landscapes: detachment‐ or transport‐limited? , 2012 .

[44]  Tom J. Coulthard,et al.  Landscape evolution models: a software review , 2001 .

[45]  G. Parker,et al.  Inception of channelization and drainage basin formation: upstream-driven theory , 1995, Journal of Fluid Mechanics.

[46]  Becker Thorsten,et al.  Numerical Modeling of Earth Systems: An introduction to computational methods with focus on solid Earth applications of continuum mechanics , 2015 .

[47]  Elizabeth R. Jessup,et al.  A Technique for Accelerating the Convergence of Restarted GMRES , 2005, SIAM J. Matrix Anal. Appl..

[48]  K. Beven,et al.  THE PREDICTION OF HILLSLOPE FLOW PATHS FOR DISTRIBUTED HYDROLOGICAL MODELLING USING DIGITAL TERRAIN MODELS , 1991 .

[49]  A. Porporato,et al.  Channelization cascade in landscape evolution , 2020, Proceedings of the National Academy of Sciences.

[50]  P. O. Koons,et al.  The topographic evolution of collisional mountain belts; a numerical look at the Southern Alps, New Zealand , 1989 .

[51]  Patrick J. Roache,et al.  Verification and Validation in Computational Science and Engineering , 1998 .

[52]  Harish Garg,et al.  Proceedings of the Third International Conference on Soft Computing for Problem Solving - SocProS 2013, Volume 2, Greater Noida Extension Centre, IIT Roorkee, India, December 26-28, 2013 , 2014, International Conference on Soft Computing for Problem Solving.

[53]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[54]  R. Kirk,et al.  Rain, winds and haze during the Huygens probe's descent to Titan's surface , 2005, Nature.

[55]  A-Xing Zhu,et al.  An efficient method for applying a differential equation to deriving the spatial distribution of specific catchment area from gridded digital elevation models , 2017, Comput. Geosci..

[56]  J. Perron,et al.  The root of branching river networks , 2012, Nature.

[57]  Luca Bergamaschi,et al.  Spectral preconditioners for the efficient numerical solution of a continuous branched transport model , 2019, J. Comput. Appl. Math..

[58]  A. Maritan,et al.  Evolution and selection of river networks: Statics, dynamics, and complexity , 2014, Proceedings of the National Academy of Sciences.

[59]  A. Porporato,et al.  On the theory of drainage area for regular and non-regular points , 2018, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[60]  I. Rodríguez‐Iturbe,et al.  A coupled channel network growth and hillslope evolution model: 2. Nondimensionalization and applications , 1991 .

[61]  M. Malin,et al.  Meter-Scale Characteristics of Martian Channels and Valleys , 2000 .

[62]  Gerard Govers,et al.  Keeping the edge: A numerical method that avoids knickpoint smearing when solving the stream power law , 2015 .

[63]  A. Jan Pahl,et al.  Mathematical Foundations of Computational Engineering: A Handbook , 2001 .

[64]  Peter Jan Pahl,et al.  Mathematical Foundations of Computational Engineering , 2001 .

[65]  Jean Braun,et al.  A very efficient O(n), implicit and parallel method to solve the stream power equation governing fluvial incision and landscape evolution , 2013 .

[66]  Christopher J. Roy,et al.  Verification and Validation in Scientific Computing: Preface , 2010 .

[67]  Alessandro Flammini,et al.  Scaling, Optimality, and Landscape Evolution , 2001 .

[68]  F. Bretherton,et al.  Stability and the conservation of mass in drainage basin evolution , 1972 .

[69]  Chenghu Zhou,et al.  An adaptive approach to selecting a flow‐partition exponent for a multiple‐flow‐direction algorithm , 2007, Int. J. Geogr. Inf. Sci..

[70]  P. Holmgren.  Multiple flow direction algorithms for runoff modelling in grid based elevation models: An empirical evaluation , 1994 .

[71]  G. Tucker,et al.  Modelling landscape evolution , 2010 .

[72]  Ali Shokoufandeh,et al.  Development of a data model to facilitate rapid Watershed Delineation , 2019, Environ. Model. Softw..

[73]  A. B. Kahn,et al.  Topological sorting of large networks , 1962, CACM.

[74]  William E. Dietrich,et al.  The Problem of Channel Erosion into Bedrock , 1992 .

[75]  Petter Pilesjö,et al.  A Triangular Form‐based Multiple Flow Algorithm to Estimate Overland Flow Distribution and Accumulation on a Digital Elevation Model , 2014, Trans. GIS.