the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 22 Apr 2024
An evaluation of flow-routing algorithms for calculating contributing area on regular grids
Abstract. Calculating contributing area (often used as a proxy for surface water discharge) within a Digital Elevation Model (DEM) or Landscape Evolution Model (LEM) is a fundamental operation in geomorphology. Here we document that a commonly used multiple-flow-direction algorithm for calculating contributing area, i.e., D∞ of Tarboton (1997), is sufficiently biased along the cardinal and ordinal directions that it is unsuitable for some standard applications of flow-routing algorithms. We revisit the purported excess dispersion of the MFD algorithm of Freeman (1991) that motivated the development of D∞ and demonstrate that MFD is superior to D∞ when tested against analytic solutions for the contributing areas of idealized landforms and the predictions of the shallow-water-equation solver FLO-2D for more complex landforms in which the water-surface slope is closely approximated by the bed slope. We also introduce a new flow-routing algorithm entitled IDS (in reference to the iterative depth-and-slope-dependent nature of the algorithm) that is more suitable than MFD for applications in which the bed and water-surface slopes differ substantially. IDS solves for water flow depths under steady hydrologic conditions by distributing the discharge delivered to each grid point from upslope to its downslope neighbors in rank order of elevation (highest to lowest) and in proportion to a power-law function of the square root of the water-surface slope and the five-thirds power of the water depth, mimicking the relationships among water discharge, depth, and surface slope in Manning’s equation. IDS is iterative in two ways: 1) water depths are added in small increments so that the water-surface slope can gradually differ from the bed slope, facilitating the spreading of water in areas of laterally unconfined flow, and 2) the partitioning of discharge from high to low elevations can be repeated, improving the accuracy of the solution as the water depths of downslope grid points become more well approximated with each successive iteration. We assess the performance of IDS by comparing its results to those of FLO-2D for a variety of real and idealized landforms and to an analytic solution of the shallow-water equations. We also demonstrate how IDS can be modified to solve other fluid-dynamical nonlinear partial differential equations arising in Earth-surface processes, such as the Boussinesq equation for the height of the water table in an unconfined aquifer.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Journal article(s) based on this preprint
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RC1: 'Comment on egusphere-2024-1138', Anonymous Referee #1, 29 Jul 2024
The work deals with a comparison of flow-routing algorithms for calculating surface flow contributing area on regular grids.
As it is the manuscript does not represent an interesting contribution. It fails in the state of the art description of overland flow models avalaible and concentrates in a sort of comparison between the aprporximate algorithms IDS and D8 in order to show the superiority of IDS. At the same time the hydraulic shallow water based FLO-2D is used as a reference solution. Taking into account the many contribiutions that overland flow simulation has received in the last20 years the scope of the present work is very narrow.
The formal aspects are also worth mentioning as the authors do not follow a clear structure in the presentation of the methods and results.
Even though the superiority of the IDS technique is clear from tthe tests presented the overall manuscipt is far from having the requiere quality to be recommended for publication.
Citation: https://doi.org/10.5194/egusphere-2024-1138-RC1 -
CC1: 'Reply on RC1', Alexander Prescott, 01 Aug 2024
We thank the referee for their comments and we appreciate this opportunity to clarify some points as we work to improve the manuscript in response to their concerns.
The referee states that our paper “fails in the state of the art description of overland flow models avalaible”. However, our paper was not focused on overland flow models. Instead, the goal of our paper was to evaluate existing methods (and propose an improved method) for calculating contributing area on regular grids. Prior to our paper, proposed methods for calculating contributing area on regular grids did not involve surface-water hydraulics. As noted in lines 475-484 of our paper, Tarboton’s (1997) leading D∞ algorithm (>3000 citations) was based on the concept that contributing area should be a function of topography alone (i.e., it should not involve any aspect of hydraulics such as flow depth, dispersion, etc.). A novel aspect of our paper is that we propose that methods for calculating contributing area should involve some hydraulic variables and equations (e.g., Manning’s equation in the partitioning of the incoming contributing area to each pixel to the neighboring pixels downslope, as in equation (7)). One outcome of this approach is a new method (IDS) that performs better than existing methods (MFD and D∞) for a variety of test cases, as demonstrated in Table 2. Crucially, our paper also documents a grid-orientation bias in the leading algorithm (D∞) that is large enough to result in errors of a factor of 2 for some applications (Figure 1). This is important information that anyone who computes contributing area on regular grids should be aware of.
The referee states that our paper “concentrates in a sort of comparison between the aprporximate algorithms IDS and D8 in order to show the superiority of IDS”. In fact, our paper focuses not on D8 but on the multiple-flow-direction algorithms MFD and D∞ (e.g., Table 2, which compares those methods to IDS). Results of the D8 algorithm are presented in only one place in the manuscript (Figure 4a) as a means of visually demonstrating the inadequacy of D8 in reproducing flow paths (and thus contributing area) over real topography. The result of our work is a novel method for calculating contributing area on regular grids that maintains the computational efficiency of methods such as D∞ that are widely used in landscape evolution models due to their simplicity (e.g., they do not require any type of precipitation time series or any similar time-dependent input of the kind required for surface water hydraulic models) but that simultaneously achieves the accuracy of hydraulic models such as FLO-2D.
Because our paper draws from two distinct portions of the scientific literature that have previously been considered separately (i.e., methods for 1) calculating contributing area of regular grids and 2) surface water hydraulic modeling), we can understand how a reader may have misinterpreted our goal. We will aim to more clearly state in the revision that our goal is to improve methods for calculating contributing area, not evaluating state-of-the-art models for surface water hydraulics. We will document the line-by-line changes we propose to make when all of the comments have been submitted and we can consider all of the potentially different recommendations made by the referees and commenters in deciding how best to improve the manuscript during the revision phase.
Citation: https://doi.org/10.5194/egusphere-2024-1138-CC1
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CC1: 'Reply on RC1', Alexander Prescott, 01 Aug 2024
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RC2: 'Comment on egusphere-2024-1138', Anonymous Referee #2, 12 Nov 2024
Dear editor, dear authors,
The submitted article entitled "An evaluation of flow-routing algorithms for calculating contributing area on regular grids" looks at the problem of flow-routing algorithms.for calculating contributing area. The theme of this article is perfectly in line with the scope of the target journal, as shown by the article on the same theme (including the use of a modified MFD algorithm) recently published in ESurf: Coatléven, J. and Chauveau, B.: Large structure simulation for landscape evolution models, Earth Surf. Dynam., 12, 995–1026, https://doi.org/10.5194/esurf-12-995-2024, 2024.
The proposed article compares existing flow routing algorithms (MFD and D∞) with a new one (IDS). The new method takes into account hydraulic elements, which is not the case with conventional approaches. The algorithms are applied to several test cases. These demonstrate the superiority of the proposed method. The proposed method is of interest to the community.
However, the authors don not do enough to point out what already exists and the difficulties and scientific obstacles that exist. I suggest a state of the art/review of existing routing algorithms (and to complete the bibliography), giving the advantages and limitations of the algorithms mentioned, and justifying the choice of MFD and D∞ as reference algorithms among other existing algorithms. This would highlight the noveties and achievements obtained with the new proposed algorithm (both in introduction and conclusion).
Here is an example of bibliographical references that can be cited. This list is by no means complete, and authors are free to cite other references that may be more relevant.
1) Rieger, W. (1998). A phenomenon‐based approach to upslope contributing area and depressions in DEMs. Hydrological Processes, 12(6), 857-872. https://doi.org/10.1002/(SICI)1099-1085(199805)12:6<857::AID-HYP659>3.0.CO;2-B
2 ) Qiming Zhou & Xuejun Liu (2002) Error assessment of grid-based flow routing algorithms used in hydrological models,
International Journal of Geographical Information Science, 16:8, 819-842, DOI:10.1080/136588102101494253) Erskine, R. H., Green, T. R., Ramirez, J. A., & MacDonald, L. H. (2006). Comparison of grid‐based algorithms for computing upslope contributing area. Water Resources Research, 42(9).https://doi.org/10.1029/2005WR004648
4) Wilson, J. P., Lam, C. S., & Deng, Y. (2007). Comparison of the performance of flow‐routing algorithms used in GIS‐based hydrologic analysis. Hydrological Processes: An International Journal, 21(8), 1026-1044. https://doi.org/10.1002/hyp.6277
5) Seibert, J., & McGlynn, B. L. (2007). A new triangular multiple flow direction algorithm for computing upslope areas from gridded digital elevation models. Water resources research, 43(4). https://doi.org/10.1029/2006WR005128
6) Wilson, J. P., AGGETT, G., Yongxin, DENG., & LAM, C. S. (2008). Water in the landscape: a review of contemporary flow routing algorithms. Advances in digital terrain analysis, 213-236.
7) Xiong, L., Tang, G., Yan, S., Zhu, S., & Sun, Y. (2014). Landform‐oriented flow‐routing algorithm for the dual‐structure loess terrain based on digital elevation models. Hydrological Processes, 28(4), 1756-1766. https://doi.org/10.1002/hyp.9719
8) Coatléven, J. (2020). Some multiple flow direction algorithms for overland flow on general meshes.ESAIM: Mathematical Modelling and Numerical Analysis, 54(6), 1917-1949. https://doi.org/10.1051/m2an/2020025
The paragraphs dealing with the analytical solution in section 2.1 is not clear, it should be improved.
Some details are necessary concerning the way the equations (4) and (5) are solved (numerical method, discretization/scheme, ...).
There are a few typos to correct and things to clarify.
Line 90 : description of figure 1, I have the feeling that there is no description of subfigure (a), it has to be checked. All figures' legends need to be checked. On right part of b and c, axis might be added to help to understand the orientation of the graphics.
Line 104, concerning FLO-2D, if possible it would be fine to cite an article in addition to the reference manual.
Line 105 It is good to quote Delestre et al.'s article, which describes the SWASHES library of analytical solutions (a kind of review of analytical solutions for free-surface flows), but the authors should also quote MacDonald et al.'s article, which is the source of the solution.
9) I. MacDonald, M. J. Baines, N. K. Nichols, and P. G. Samuels. Analytic benchmark solutions for open-channel flows. Journal of Hydraulic Engineering, 123(11):1041–1045, November 1997
Line 115 "algorithmic performance in in cases ..." I think that the sentence needs to be checked.
Line 147 "Sections 2.3₃.3 ..." spaces need to be added.
Lines 152-153 "verified to be sufficient time for ..." I think that the sentence should be reformulated.
Line 171 "... Eqs. (4)&(5) ..." spaces need to be added. Some other spaces need to be added lines 254, 255, 282, 290, 321, 328 and 421.
Would it be able to apply the algorithm to unsteady flow?
Best regards.
Citation: https://doi.org/10.5194/egusphere-2024-1138-RC2 - AC1: 'Author response to referees for egusphere-2024-1138', Jon Pelletier , 10 Dec 2024
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2024-1138', Anonymous Referee #1, 29 Jul 2024
The work deals with a comparison of flow-routing algorithms for calculating surface flow contributing area on regular grids.
As it is the manuscript does not represent an interesting contribution. It fails in the state of the art description of overland flow models avalaible and concentrates in a sort of comparison between the aprporximate algorithms IDS and D8 in order to show the superiority of IDS. At the same time the hydraulic shallow water based FLO-2D is used as a reference solution. Taking into account the many contribiutions that overland flow simulation has received in the last20 years the scope of the present work is very narrow.
The formal aspects are also worth mentioning as the authors do not follow a clear structure in the presentation of the methods and results.
Even though the superiority of the IDS technique is clear from tthe tests presented the overall manuscipt is far from having the requiere quality to be recommended for publication.
Citation: https://doi.org/10.5194/egusphere-2024-1138-RC1 -
CC1: 'Reply on RC1', Alexander Prescott, 01 Aug 2024
We thank the referee for their comments and we appreciate this opportunity to clarify some points as we work to improve the manuscript in response to their concerns.
The referee states that our paper “fails in the state of the art description of overland flow models avalaible”. However, our paper was not focused on overland flow models. Instead, the goal of our paper was to evaluate existing methods (and propose an improved method) for calculating contributing area on regular grids. Prior to our paper, proposed methods for calculating contributing area on regular grids did not involve surface-water hydraulics. As noted in lines 475-484 of our paper, Tarboton’s (1997) leading D∞ algorithm (>3000 citations) was based on the concept that contributing area should be a function of topography alone (i.e., it should not involve any aspect of hydraulics such as flow depth, dispersion, etc.). A novel aspect of our paper is that we propose that methods for calculating contributing area should involve some hydraulic variables and equations (e.g., Manning’s equation in the partitioning of the incoming contributing area to each pixel to the neighboring pixels downslope, as in equation (7)). One outcome of this approach is a new method (IDS) that performs better than existing methods (MFD and D∞) for a variety of test cases, as demonstrated in Table 2. Crucially, our paper also documents a grid-orientation bias in the leading algorithm (D∞) that is large enough to result in errors of a factor of 2 for some applications (Figure 1). This is important information that anyone who computes contributing area on regular grids should be aware of.
The referee states that our paper “concentrates in a sort of comparison between the aprporximate algorithms IDS and D8 in order to show the superiority of IDS”. In fact, our paper focuses not on D8 but on the multiple-flow-direction algorithms MFD and D∞ (e.g., Table 2, which compares those methods to IDS). Results of the D8 algorithm are presented in only one place in the manuscript (Figure 4a) as a means of visually demonstrating the inadequacy of D8 in reproducing flow paths (and thus contributing area) over real topography. The result of our work is a novel method for calculating contributing area on regular grids that maintains the computational efficiency of methods such as D∞ that are widely used in landscape evolution models due to their simplicity (e.g., they do not require any type of precipitation time series or any similar time-dependent input of the kind required for surface water hydraulic models) but that simultaneously achieves the accuracy of hydraulic models such as FLO-2D.
Because our paper draws from two distinct portions of the scientific literature that have previously been considered separately (i.e., methods for 1) calculating contributing area of regular grids and 2) surface water hydraulic modeling), we can understand how a reader may have misinterpreted our goal. We will aim to more clearly state in the revision that our goal is to improve methods for calculating contributing area, not evaluating state-of-the-art models for surface water hydraulics. We will document the line-by-line changes we propose to make when all of the comments have been submitted and we can consider all of the potentially different recommendations made by the referees and commenters in deciding how best to improve the manuscript during the revision phase.
Citation: https://doi.org/10.5194/egusphere-2024-1138-CC1
-
CC1: 'Reply on RC1', Alexander Prescott, 01 Aug 2024
-
RC2: 'Comment on egusphere-2024-1138', Anonymous Referee #2, 12 Nov 2024
Dear editor, dear authors,
The submitted article entitled "An evaluation of flow-routing algorithms for calculating contributing area on regular grids" looks at the problem of flow-routing algorithms.for calculating contributing area. The theme of this article is perfectly in line with the scope of the target journal, as shown by the article on the same theme (including the use of a modified MFD algorithm) recently published in ESurf: Coatléven, J. and Chauveau, B.: Large structure simulation for landscape evolution models, Earth Surf. Dynam., 12, 995–1026, https://doi.org/10.5194/esurf-12-995-2024, 2024.
The proposed article compares existing flow routing algorithms (MFD and D∞) with a new one (IDS). The new method takes into account hydraulic elements, which is not the case with conventional approaches. The algorithms are applied to several test cases. These demonstrate the superiority of the proposed method. The proposed method is of interest to the community.
However, the authors don not do enough to point out what already exists and the difficulties and scientific obstacles that exist. I suggest a state of the art/review of existing routing algorithms (and to complete the bibliography), giving the advantages and limitations of the algorithms mentioned, and justifying the choice of MFD and D∞ as reference algorithms among other existing algorithms. This would highlight the noveties and achievements obtained with the new proposed algorithm (both in introduction and conclusion).
Here is an example of bibliographical references that can be cited. This list is by no means complete, and authors are free to cite other references that may be more relevant.
1) Rieger, W. (1998). A phenomenon‐based approach to upslope contributing area and depressions in DEMs. Hydrological Processes, 12(6), 857-872. https://doi.org/10.1002/(SICI)1099-1085(199805)12:6<857::AID-HYP659>3.0.CO;2-B
2 ) Qiming Zhou & Xuejun Liu (2002) Error assessment of grid-based flow routing algorithms used in hydrological models,
International Journal of Geographical Information Science, 16:8, 819-842, DOI:10.1080/136588102101494253) Erskine, R. H., Green, T. R., Ramirez, J. A., & MacDonald, L. H. (2006). Comparison of grid‐based algorithms for computing upslope contributing area. Water Resources Research, 42(9).https://doi.org/10.1029/2005WR004648
4) Wilson, J. P., Lam, C. S., & Deng, Y. (2007). Comparison of the performance of flow‐routing algorithms used in GIS‐based hydrologic analysis. Hydrological Processes: An International Journal, 21(8), 1026-1044. https://doi.org/10.1002/hyp.6277
5) Seibert, J., & McGlynn, B. L. (2007). A new triangular multiple flow direction algorithm for computing upslope areas from gridded digital elevation models. Water resources research, 43(4). https://doi.org/10.1029/2006WR005128
6) Wilson, J. P., AGGETT, G., Yongxin, DENG., & LAM, C. S. (2008). Water in the landscape: a review of contemporary flow routing algorithms. Advances in digital terrain analysis, 213-236.
7) Xiong, L., Tang, G., Yan, S., Zhu, S., & Sun, Y. (2014). Landform‐oriented flow‐routing algorithm for the dual‐structure loess terrain based on digital elevation models. Hydrological Processes, 28(4), 1756-1766. https://doi.org/10.1002/hyp.9719
8) Coatléven, J. (2020). Some multiple flow direction algorithms for overland flow on general meshes.ESAIM: Mathematical Modelling and Numerical Analysis, 54(6), 1917-1949. https://doi.org/10.1051/m2an/2020025
The paragraphs dealing with the analytical solution in section 2.1 is not clear, it should be improved.
Some details are necessary concerning the way the equations (4) and (5) are solved (numerical method, discretization/scheme, ...).
There are a few typos to correct and things to clarify.
Line 90 : description of figure 1, I have the feeling that there is no description of subfigure (a), it has to be checked. All figures' legends need to be checked. On right part of b and c, axis might be added to help to understand the orientation of the graphics.
Line 104, concerning FLO-2D, if possible it would be fine to cite an article in addition to the reference manual.
Line 105 It is good to quote Delestre et al.'s article, which describes the SWASHES library of analytical solutions (a kind of review of analytical solutions for free-surface flows), but the authors should also quote MacDonald et al.'s article, which is the source of the solution.
9) I. MacDonald, M. J. Baines, N. K. Nichols, and P. G. Samuels. Analytic benchmark solutions for open-channel flows. Journal of Hydraulic Engineering, 123(11):1041–1045, November 1997
Line 115 "algorithmic performance in in cases ..." I think that the sentence needs to be checked.
Line 147 "Sections 2.3₃.3 ..." spaces need to be added.
Lines 152-153 "verified to be sufficient time for ..." I think that the sentence should be reformulated.
Line 171 "... Eqs. (4)&(5) ..." spaces need to be added. Some other spaces need to be added lines 254, 255, 282, 290, 321, 328 and 421.
Would it be able to apply the algorithm to unsteady flow?
Best regards.
Citation: https://doi.org/10.5194/egusphere-2024-1138-RC2 - AC1: 'Author response to referees for egusphere-2024-1138', Jon Pelletier , 10 Dec 2024
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Alexander B. Prescott
Jon D. Pelletier
Satya Chataut
Sriram Ananthanarayan
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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