05 Feb 2021
05 Feb 2021
Numerical daemons of hydrological models are summoned by extreme precipitation
- 1Hydrology and Quantitative Water Management Group, Wageningen University, Wageningen, Netherlands
- 2Geography, College of Life and Environmental Sciences, University of Exeter, Exeter, UK
- 3Coldwater Laboratory, University of Saskatchewan, Canmore, Alberta, Canada
- 1Hydrology and Quantitative Water Management Group, Wageningen University, Wageningen, Netherlands
- 2Geography, College of Life and Environmental Sciences, University of Exeter, Exeter, UK
- 3Coldwater Laboratory, University of Saskatchewan, Canmore, Alberta, Canada
Abstract. Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on approximate numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation like that observed during hurricanes Harvey and Katrina impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation events. In this experiment, a large number of hydrographs is generated with the modular modeling framework FUSE, using eight numerical techniques across a variety of forcing datasets. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational expense and numerical error associated with each hydrograph were recorded. It was found that numerical error (root mean square error) usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both low numerical error and low computational cost. A basic literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be sub-optimal. We conclude that relatively large numerical errors might be common in current models, and because these will likely become larger as the climate changes, we advocate for the use of low cost, low error numerical methods.
Peter T. La Follette et al.
Status: open (until 11 Apr 2021)
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RC1: 'Comment on hess-2021-28', Jasper Vrugt, 23 Feb 2021
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The present paper is another attempt to make the surface hydrologic community aware of the problems associated with the use of poor numerics. This is always important, and the present hook is climate change (increased precipitation). The work presented in this paper is standard material for a numerics class - how to solve differential equations. I certainly appreciate the topic and second its importance, yet, do like to articulate that a need for more publications on this topic highlights a fundamental flaw in hydrologic education. Proper awareness/knowledge of mathematics, numerics and, also, statistics.
The paper is well written and generally well prepared. Nevertheless, the paper would benefit from a careful read by a native speaker. I will give some examples in my review below. What is more, at a number of different places in the manuscript, I believe the authors do not use the 'right' technical language. I will also provide some examples of this. Altogether, I believe this paper should be published after some moderate, perhaps major revision, if they deem some of the comments important enough and/or worthy of further exploration. I list my comments in random order - as I wrote them down when scrolling through the pdf.
1. Is it possible to express the numerical error as percentage of the rainfall error? If you make some simple assumptions about the rainfall error. Then you can use this as metric and show how it increases with rainfall intensity/duration. Same question about discharge measurement error. If one makes an assumption about 10% error (heteroscedastic) then one can express the numerical error as percentage of the discharge measurement error. One can show how this error evolves with time.
2. Then on a related note, can you investigate the relationship between numerical error and flow level (=discharge) ? May be interesting to see - as this, I believe, is not explicitly addressed in earlier studies. This should show that numerical errors are relatively large during rainfall events, and these errors dissipate during a subsequent drying period. This, dissapation is one reason numerics has not got the attention it deserves from the community. I will revisit this point in a later comment.
3. Did you consider midpoint methods? Why/why not? Similar question for Runge-Kutta methods by the Carl Runge and Wilhelm Kutta? They developed a whole toolbox of explicit/implicit/fixed/variable time step integration methods. What about backward Euler? I know this implies more work, nevertheless, maybe there was a reason not to include these methods in your analysis - then, it would be good to know the reasons.
4. When it comes to Heun and Euler some papers are cited but the original inventors of these solution methods are not mentioned! I would include a reference to Leonhard Euler (Institutionum calculi integralis) and Karl Heun, among others.
5. One reason proper numerics receives little attention among surface hydrologists is the nature of hydrologic systems. Negative feedback loops ameliorate differences in initial states and promote convergence to a stable state. Indeed, for such systems one can simply use a spin-up period (as you use in the paper) to prepare the initial states of the model for subsequent simulation. In systems with positive feedback, numerical errors will have a devastating effect on long term behavior as model runs will diverge rapidly and suggest a very different system behavior later on. Thus, one reason that numerical errors have historically received relatively little attention is the nature of hydrologic systems, that is, negative feedback loops induce stable attractor states - hence, why we can solve poor knowledge of the initial states with a spin-up period. Note, in some fields, differential equations are so incredible sensitive to numerics that these small errors can induce chaos (example: two-predator-two-prey systems).
6. The paper of Schoups et al: doi:10.1029/2009WR008648 draws similar conclusions as herein, that is, the use of a second-order integration methods is preferred. I believe the text should address this earlier paper and those possibly related to it more properly. The paper is cited, but the text does not address that similar findings have been reported elsewhere. On a related note, there are more surface hydrologic model codes that use proper numerics. For example, the hmodel of Schoups et al. (same paper) has been used in various publications. This model uses 2nd order adptive Heun for its numerical solution.
7. You may want to emphasize in the paper that poor numerics not only affects the simulated discharge, but compromises tasks such as parameter estimation, prediction, simulation of related state variables (groundwater table, soil moisture), etc. Those familiar with proper numerical procedures are aware, but not all those others reading this manuscript.
8. I do not see anything else that is wrong with this paper (see my written comments on pdf), except that the questions listed above may help find a second hook to 'sell' this work. Climate change is an interesting hook, yet, one may argue that the precipitation intensities as used herein are a bit exaggarated. Maybe a more detailed investigation into how numerical errors behave in a simulation may be interesting, their dependence on simulated flow level and simulated state variables. Perhaps even better, can you pinpoint which process in the model contribute most to the numerical error during precipitation extremes. This must be the most nonlinear process - or that process (its flux) that changes most rapidly in a time step.
Please see the attached supplement for other comments.
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AC1: 'Reply on RC1', Peter La Follette, 23 Feb 2021
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Dear Prof. Vrugt, On behalf of all of the authors, I would like to thank your for your insightful and detailed review; I am certain it will lead to an improved manuscript. We anticipate that we will respond this or next week. Sincerely, Peter La Follette
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AC2: 'Reply on RC1', Peter La Follette, 04 Mar 2021
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The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess-2021-28/hess-2021-28-AC2-supplement.pdf
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AC1: 'Reply on RC1', Peter La Follette, 23 Feb 2021
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Peter T. La Follette et al.
Data sets
Data for La Follette et al (Numerical Daemons and extreme precipitation) Peter La Follette https://doi.org/10.4211/hs.acf9a56e93bc4863b972e1e8af36dc0c
Peter T. La Follette et al.
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