the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 05 Nov 2024
Technical note: Quadratic solution of the approximate reservoir equation (QuaSoARe)
Abstract. This paper presents a method to solve the reservoir equation, a special type of scalar ordinary differential equation controlling the dynamic of conceptual reservoirs found in most hydrological models. The method called “Quadratic Solution of the Approximate Reservoir Equation” (QuaSoARe) is applicable to any reservoir equation regardless of its non-linearity or the number of fluxes entering and leaving the reservoir. The method is based on a piecewise quadratic interpolation of the flux functions, which lead to an analytical and mass conservative solution. It is applied to two routing models and two rainfall-runoff stores that are representatives of hydrological model components and evaluated on six catchments located in Eastern Australia that experienced one of the most extreme floods in recent Australian history. A comparison of the method against two standard numerical schemes, the Radau fifth order implicit and Runge-Kutta of order 5(4) explicit schemes suggests that it can reach similar accuracy while reducing runtime by a factor of 10 to 50 depending on the model considered. At the same time, the model code is simple enough to be presented as a short pseudo-code included in our paper. Beyond solving a given reservoir equation, the method constitutes a promising avenue to define flexible models where flux functions are defined as piecewise quadratic functions, which can be solved exactly with QuaSoARe.
- Preprint
(1682 KB) - Metadata XML
- BibTeX
- EndNote
Status: final response (author comments only)
-
RC1: 'Comment on egusphere-2024-3184', Anonymous Referee #1, 22 Dec 2024
This is a very useful article reminding modellers of the numerical issues that need to be considered.It should be of interest to all modellers.
As noted in the paper, numerical methods are needed in situations where an analytical solution is not possible. For example, linear ODEs that are commonly used in unit hydrograph components of hydrological models (e.g. IHACRES) can be solved analytically, so a numerical solution is not necessary. Some non-linear ODEs can also be solved analytically (e.g. the drainage equation in the IHACRES CMD module). Many non-linear ODEs cannot be solved analytically, particularly when combined into a system of ODE equations. It is models that use such ODEs that will benefit greatly from using QuaSoARe.
The issue with hydrological models is that typically, the time step employed is dictated by the available data, not what is needed to ensure a sufficiently accurate estimation of the solution of the ODEs included in the model. The use of a coarse time step in the input data means a loss of information about what is happening at finer time scales, leading to uncertainty in the model output.
In Appendix A, an alternative approach would be to take the Taylor series approximation about the centre point of the interval. This would not ensure a match at the ends of the interval, leading to a likely discontinuity between intervals that would not be desirable. The approach taken ensures a continuous approximation of the function (noting that it will be discontinuous in the first derivative) and is effective providing the function is sufficiently close to a quadratic form. This will depend on the width of the interval and the variability of the function within that interval. The illustrative example used is S3, so a quadratic approximation would work well providing that the interval is not so large that the value of S ranges from near zero to a large enough value within the interval (see discussion of the illustrative example on page 8). This then defines the acceptable interval width that should be used. In general applications, this will depend on the form of the function f.
Overall, the paper is scientifically very sound and relevant to the general modelling community. It is suitable for publication once the minor errors noted below are fixed.
Minor comments
- Line 142: should be “m-1 intervals”
- Figure 5: readers might not notice the scale factor on the top right of each panel. May be better to have this in the axis label on the right side – e.g. (1e-3 mm)
- Figure 6: the use of a shaded white font for the median values is difficult to read at times.
Typographical/grammatical errors
- Line 26: “a history”
- Line 33: “to bridge”
- Line 45: comma after “example”
- Line 58: “requires significant”
- Line 77: “, for example, the Saint-Venant …”
- Line 97: “large systems” or “a large system “
- Line 100: “that” instead of “which”
- Line 112: “Sections 2.2 and 2.3, which an accompanying Python …”
- Line 123: delete second “given”
- Line 149: “functions”
- Line 154: comma after “analytically”
- Line 165: “requirement”
- Line 207: comma before “such”
- Line 210: “as blue lines” – delete “in”
- Line 240: “If this is the case”
- Line 263: “that do not exist”
- Line 270: “conditions”
- Line 287: “outflow”
- Line 351: “For each catchment”
- Line 372: “quadratic functions”
- Line 411: “(500) that lead to”
- Line 421: “reach an error”
- Line 430: “polynomials”
- Line 445: “is satisfactory”
- Line 450: comma before “which” and another after “work” on the next line
- Line 454: “that” rather than “which”
- Line 470: “context, like most rainfall-runoff models, remain arbitrary"
- Line 473: “currently being explored”
- Line 483: insert a comma before “which”
- Line 484: “method’s” and “reservoirs”
- Line 486: “... is an order of magnitude smaller than the typical ..."
Citation: https://doi.org/10.5194/egusphere-2024-3184-RC1 -
RC2: 'Comment on egusphere-2024-3184', Anonymous Referee #2, 08 Jan 2025
The technical note proposed by J. Lerat introduces a new method for the integration of a scalar Ordinary Differential Equation (ODE), describing the evolution of a conceptual component such as a reservoir in a lumped hydrological model. The rationale for the development of the method is that, given this ODE formulated in state-space, it is difficult to compute together both the evolution of the single state variable (denoted S) over a timestep, as well as the different outputs from the system integrated over the timestep. The QuaSoARe method uses analytical (quadratic) substitutes for the instantaneous fluxes entering/leaving the reservoir.
The paper is written in a very straightforward way and is relatively easy to follow with some familiarity in analytical and numerical integration (apart from the very first part where continuity assumptions are stated, see comments below). I am overall supportive of the approach which has a great potential for speeding up current formulations of many lumped models, but I still have questions regarding the kind of solutions which should be put forward to cure the problems raised by Clark and Kavetski about the numerical soundness of many such models, in the case of multistate model structures.
Please see PDF in attachment for detailed comments.
Viewed
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 163 | 44 | 10 | 217 | 3 | 1 |
- HTML: 163
- PDF: 44
- XML: 10
- Total: 217
- BibTeX: 3
- EndNote: 1
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1