the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Sep 2021
Quantifying time-variant travel time distribution by multi-fidelity model in hillslope under nonstationary hydrologic conditions
Abstract. The travel time distribution (TTD) is a lumped representation of groundwater discharge and solute export responding to rainfall. It reflects the mixing process of water parcels and solute particles of different ages and characterizes reactive transport progress in hillslope aquifers. As a result of the mixing process, groundwater leaving the system at a certain time is an integration of multiple water parcels of different ages from different historical rainfall events. Under nonstationary rainfall input condition, the TTD varies with transit groundwater flow, leading to the time-variant TTD. Most methods for estimating time-variant TTD are constrained by requiring either the long-term continuous hydrogeochemical data or the intensive computations. This study introduces a multi-fidelity model to overcome these limitations and evaluate time-variant TTD numerically. In this multi-fidelity model, groundwater age distribution model is taken as the high-fidelity model, and particle tracking model without random walk is taken as the low-fidelity model. Non-parametric regression by non-linear Gaussian process is applied to correlate the two models and then build up the multi-fidelity model. The advantage of the multi-fidelity model is that it combines the accuracy of high-fidelity model and the computational efficiency of low-fidelity model. Moreover, in groundwater and solute transport model with low P\'eclet number, as the spatial scale of the model increases, the number of particles required for multi-fidelity model is reduced significantly compared to random walk particle tracking model. The correlation between high and low-fidelity models is demonstrated in a one dimensional pulse injection case. In a two dimensional hypothetical model, convergence analysis indicates that the multi-fidelity model converges well when increasing the number of high-fidelity models. Error analysis also confirms the good performance of the multi-fidelity model.
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RC1: 'Comment on hess-2021-430', Anonymous Referee #1, 06 Oct 2021
Comments on “Quantifying time-variant travel time distribution by multi-fidelity model in hillslope under nonstationary hydrologic conditions” by Mao et al.
The authors presented a computationally efficient method for estimating travel time distributions (TTDs), residence time distributions (RTDs), and the StorAge Selection(SAS) functions using process-based models that account for the diffusion (or hydrodynamic dispersion) process. They argued that there are two ways of estimating those distributions and functions, but those methods are computationally burdensome. One way is solving the advection-diffusion (or advection-dispersion) equation (ADE) for each rainfall event, and another way is using the particle tracking model considering the random walk. Their proposed method (called multi-fidelity model) estimates the TTDs, RTDs, and the SAS functions by combining the result of the particle tracking model without the random walk component (so-called low-fidelity model) and the result of the ADE simulations for selected rainfall events (so-called high-fidelity model). They argued that the multi-fidelity model reproduces the result of the ADE model closely (especially when about 10% of all rainfall events are simulated using the ADE model) with a potential reduction of computational time.
My recommendation for the current manuscript is “rejection” due to the following reasons. First, the need for this method is not convincing especially because the proposed method reduces the accuracy of TTDs, RDTs, and the SAS function estimation (see major comment 1). Second, the test is not well designed, and the method’s performance is questionable (see my major comment 2). Third, some model results seem to be inaccurate (see major comment 3). I will also provide other comments in the hope that those comments will be helpful to the authors.
Major comments
1. Not convincing motivation and unclear computational efficiency
The manuscript does not provide a sufficient explanation of the motivation for developing the multi-fidelity model. Reducing computation time seems to be the primary motivation (e.g., L84-86), but there are many other ways to reduce the computation time (especially when sacrificing accuracy for reduced computation time is allowed). We can run a full model (e.g., a model that solves the ADE for each rainfall event or the particle tracking model with the random walk component) in a computationally efficient manner. For example, utilizing a parallelized code (e.g., ParFlow), using a coarser grid, or using a coarser time step is an option. Utilizing parallelized codes to solve Richards’ equation or tracking particles would not reduce the accuracy of the results but significantly reduce the computation time. Using a coarser grid or a coarser time step could reduce the model’s accuracy, but the proposed method, using the multi-fidelity model, could also reduce the accuracy. We can also estimate a BTC conditional to a set of consecutive events rather than to a single event which then can be used to estimate the TTD. We can probably define the injection time of the set as the center of mass of those consecutive events (and, if needed, perform a bit of correction to consider negative travel time). This method would not be able to estimate fast time variabilities of the TTDs, RTDs, and the SAS functions, but there is also no guarantee that the proposed method, the multi-fidelity model, could estimate such fast time variability (see also my major point 2). The result generated using this method, estimating TTDs condition to a set of events, would actually be easier to interpret since we know what information is averaged out, while interpreting the result of the multi-fidelity model would not be very easy. How should a user of the multi-fidelity model interpret the result of the multi-fidelity model? In other words, what should a user expect to be the difference between the high-fidelity model result and the multi-fidelity model result?
Also, when running the ADE model for every single rainfall event (without the simplifications discussed above) or running the particle tracking model with the random walk component is practically feasible, the proposed method would not be useful. For example, let’s say running a full model (e.g., running the ADE model for every single rainfall event) takes ten days. Running the multi-fidelity model would take one day + additional time to run a low fidelity model, e.g., the particle tracking model without the random walk component. In this case, do we need to use the multi-fidelity model to save less than nine days even though the method reduces the accuracy (since the multi-fidelity model result is not the same as the high-fidelity model result)? In what case do we really need this multi-fidelity model?
In addition, computational efficiency is not discussed in detail in this manuscript. Since the method was developed for computational efficiency, the efficiency must be discussed in more detail.
2. Performance of the multi-fidelity model and design of the test
Figure 4 shows that the model error does not converge to zero as the rainfall interval decrease. When the rainfall interval of 5 is used, the mean error is a bit larger than the mean error for the case with the rainfall interval of 10. If the mean error is not expected to converge to zero as the rainfall interval decreases, it needs to be discussed in more detail so that readers know what to expect from the multi-fidelity model.
Furthermore, Figures 4 and 5 show that the multi-fidelity model has a large variance of error, meaning that the time variability of hydrologic transport is still not well captured. Even if I assume that the error is bearable for this test case (i.e., the test using the homogeneous 2D hillslope domain), the error could become larger if the method is applied to a hydrologic system with larger time variability of hydrologic transport dynamics.
The temporal variability of hydrologic transport dynamics in the homogeneous 2D hillslope domain is not large, resulting in the SAS function with a limited time variability (Figure 9b). In many real-world studies, the SAS function time variability is much larger than what is presented in Figure 9b (see many studies that applied the SAS Function). Could this multi-fidelity method be useful when the time variability of hydrologic transport dynamics is large?
3. Model results
Some results do not seem to be accurate. Figure 9 shows the SAS function, and the shape of the SAS function is very spiky especially for the low fidelity model. The SAS function, in general, is expected to have a much smoother shape (and that is one reason why the SAS function becomes popular). Also, see Kim et al. (2020), where they showed that the SAS functions derived for a similar hydrologic domain are smooth.
This incorrect result is perhaps due to the insufficient number of particles used in the model for the low-fidelity model. Or maybe there is a problem in the estimation of the RTDs and the SAS functions. If the former is the case, this incorrect result may have resulted in the non-zero bias I pointed out in major comment 2.
[Reference]
Kim. M, T. H. M. Volkmann, Y. Wang, C. J. Harman, & P. A. Troch (2020), Direct observation of hillslope scale StorAge Selection functions in an experimental hydrologic system: Geomorphologic structure and the preferential discharge of old water, https://doi.org/10.1002/essoar.10504485.1
4. Section 2.3
It is unclear what “correlation” means here, and how (15) leads to the argument in L177-178 ("Theoretically, a mapping is capable of being setting up between ... if the diffusion parameter D is known. This is the theoretical foundation of the multi-fidelity model") and L198-199 is also unclear. What authors show in this section using (10) – (15) is trivial---As the diffusion coefficient decreases, the transport is dominated by advection. How does it support the argument in L177-178? Also, the 1-D configuration used in this section is not enough to support the argument. There is no consideration of transversal diffusion or dispersion. Furthermore, spatial and temporal variations of velocity are not considered.
Minor comments
Needs less focus on the hillslope scale: In many places throughout the manuscript, the authors talked about hillslope (e.g., in the title, L3, L18, and so on). However, I am not sure if the multi-fidelity model could be useful at the hillslope scale, which is relatively small. For a small hydrological domain like a hillslope, running the full ADE simulations or the particle tracking model with the random component will be feasible.
Diffusion process: It is unclear what the diffusion process simulated in the 2D hillslope model is. The authors may have assigned a specific concentration for the rainfall event of interest while the concentration of all other water is set to zero. Since the diffusion is controlled by the concentration gradient and the diffusion coefficient, it needs to be more precise what the gradient in the model (which depends on the assigned value for the rainfall event of interest) means and what the diffusion coefficient means.
L5: “transient ground water flow” is not the only process that could result in time-variant TTDs.
L11-13: In is not clear how significantly the number of particles can be reduced.
L29: “yearly averaged” does not seem to describe the time-invariant TTD correctly.
L67-69 and L146-148: While I agree, this sentence is not enough to motivate this study. I think the authors need to cite studies that show that considering diffusion or dispersion requires significantly more particles (if such a study exists) or need to show it clearly somewhere in the manuscript.
L76: Not clear what message the authors are trying to convey.
L83: Not clear if the authors solved the full 5D equation or just combine the results of the high fidelity model and the low fidelity model.
L152: No evidence is provided to support the “drastic” decrease of the number of particles
L175: While I agree with the “no matter what boundary conditions” argument, what the authors show in this section is for a specific boundary condition used to get (14).
L265: There are many studies where RTDs are estimated using BTCs. Most of the experimental studies of the SAS function and many the SAS function studies that utilize process-based model estimated RTDs using BTCs.
L280-281: This sentence is unnecessary for the purpose of this paper.
L291: Lower concentration of which solute?
L305: “quasi-steady state” means something different.
L313: Each single rainfall event at daily time step?
L320-321: Richards’ equation-based model has many assumptions. Also, perhaps better to avoid the term “real physical process”.
Figure 3: Please clarify what the particles at x = 100 m are.
L373-376: A model result of solute concentration cannot be lower during the whole time than the concentration estimated using another model.
Citation: https://doi.org/10.5194/hess-2021-430-RC1 - AC1: 'Reply on RC1', Rong Mao, 30 Nov 2021
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RC2: 'Comment on hess-2021-430', Anonymous Referee #2, 16 Oct 2021
The aim of this study was to propose a multi-fidelity model is to quantify the time-variant travel time distribution (TTD. This multi-fidelity model undergoes four main steps of (i) decomposing the advection-diffusion equation (ADE) into a series of ADE’s, (ii) solving these ADEs through particle tracking without random walk (low-fidelity model), (iii) solving several ADE’s by numerical simulation (high-fidelity model), (iv) establishing the relationship relation between the low fidelity and high-fidelity models, and (v) calculating the time-variant TTD from the breakthrough curves of the multi-fidelity model. The manuscript is well written and one can follow the proposed theoretical framework smoothly. However, the superiority of the proposed multi-fidelity model over the very well-known approaches like the storage section function framework is not clear. Also, the impact on model accuracy of some major simplifications in both the low- and high-fidelity models has not been demonstrated. As such, I recommend major revision at this stage of review. Further major and minor comments are presented below.
Major comments
- Introduction
Line 49-53: This statement of is not correct because the influence of permeability architecture and preferential flow is already projected in the flow velocities and residence times, which in turn are reflected in the time-variant TTD. Therefore, the age maser equation can indeed be employed to explore the above issues by studying hillslopes with a wide range of physical conditions. Some other former studies have already explored them, too.
- Groundwater age distribution model
- The two main parts of this section is (i) the groundwater age distribution model and (ii) the approach to solve the so-called 5D equation (1). Both of them have already been contextualized in the work of Ginn (1999) and Gomez and Wilson (2013), respectively. The tone of material presentation in the section (e.g., in Line 95) looks like the authors are proposing a novel approach of treating the original equations by Ginn (1999), but it does not seem so.
- Line 137-140: This part is written very concisely and is not clear to a reader. In particular, four questions are unknown: (1) Why should the high-fidelity model be run multiple times and in a limited number of runs? (2) Why should not the high-fidelity model required be run for all input pulses, but only some of them? (3) What does the “trend” of the variation in age distributions imply? And (4) what does make the multi-fidelity model biased? To clarify the above issue, the authors must first describe the elements of the multi-fidelity models and then explain how those elements work together to build the multi-fidelity model with sufficient elaboration on its properties/advantages/caveats.
- Particle-tracking model
- The claimed lower computational cost of the proposed low-fidelity model seems to be obtained at the cost of ignoring the molecular diffusion and mechanical dispersion of solutes. As such, the superiority/accuracy of the low-fidelity model as compared to the other competing particle-tracking models like EcoSILM is not clear. Also, please note the following studies in which some approaches to diminish the computational of particle-tracking are discussed:
- Yang et al. (2021), Accurate load balancing accelerates Lagrangian simulation of water ages on distributed, multi-GPU platforms, https://doi.org/10.1002/essoar.10507899.1.
- Yang et al. (2021), Accelerating the Lagrangian particle tracking of residence time distributions and source water mixing towards large scales, Computers & Geosciences, https://doi.org/10.1016/j.cageo.2021.104760.
- Maxwell, RM, Condon, LE, Danesh-Yazdi, M, Bearup, LA. Exploring source water mixing and transient residence time distributions of outflow and evapotranspiration with an integrated hydrologic model and Lagrangian particle tracking approach. Ecohydrology. 2019; 12:e2042. https://doi.org/10.1002/eco.2042.
- It is also noted that injecting a sufficient number of particles into the particle-tracking models is not required to only reduce the random noise of diffusion and mechanical dispersion, but also to reduce bias in the tracked evapotranspired water. Since the present study highlights its superiority in terms of employing a much smaller number of particles, the model accuracy on tracking fluxes like evapotranspiration in not clear and discussed.
- Analytical theorem for one dimensional scenario
The presented theoretical foundation of the multi-fidelity model in 1D seems to hold upon the assumption of constant advection velocity in the entire domain. But this is not certainty true in real-world cases and is against the goal of this study.
- Unclear advantage of the multi-fidelity model and the limited experiments
According to the motivation of this study, the proposed multi-fidelity model is expected to ease the estimation of TTD by skipping the complex process of SAS function quantification proposed by former studies (e.g., Botter et al., 2011). However, determining the mathematical form of the nonlinear mapping of equation (17) is said to be strongly dependent on the properties of a hydrogeological system under study. Therefore, it seems that the challenge of SAS function calibration in the formerly developed framework has only switched to quantifying another function of nonlinear type, i.e., g in equation (17), in the present study. From the designed experiments, the relative efficiency of the multi-fidelity model performance as compared to the SAS function calibration approach is not clear and appears to be exaggerated.
Minor comments
- Abstract, line 1: TTD is not necessarily representative of groundwater, but also surface runoff and any the processes that compose streamflow. If the focus of this work is to consider the travel time of flow pathways in groundwater, the authors should clarify it first in the manuscript.
- Abstract, line 5: The non-stationarity of TTD is not only because of non-stationarity n rainfall, but all the processes closing the water balance in a hydrological system.
- The abstract is a little bit confusing and the study storyline and objectives are not clear.
- Line 20: mixing process and also the heterogeneity of flow pathways and transient pore water velocity.
- Line 36: RTD was not defined earlier in the manuscript.
- Line 54 and 58: Is “periodic” the best word for this condition?
- Line 86: In comparison to… Please fix.
- Line 92: groundwater flow pathways… Please fix.
Citation: https://doi.org/10.5194/hess-2021-430-RC2 - AC2: 'Reply on RC2', Rong Mao, 30 Nov 2021
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RC3: 'Comment on hess-2021-430', Anonymous Referee #3, 31 Oct 2021
Review of Mao et al
This manuscript present work comparing travel time distribution formulations in a simple hillslope experiment. The work has some interesting aspects, comparing different approaches for simulating water residence time.
General Comments:
The manuscript is generally well written and on a topic of interest to the readership of HESS. The terminology of high-to-low fidelity is confusing or perhaps even misleading. I have some comments and suggestions below that might help clarify some aspects of the work as presented and help to better place it in context of the literature, which would help better establish the novelty of the current approach. It is also unclear the the proposed mixed approach is “better” in a computational sense, the authors jump to this conclusion seemingly without comprehensive evaluation.
Specific Comments:
1. S2.2 / lines 141-154. The approach presented has similarities and other approaches (e.g. Maxwell 2019, Maxwell 2016, Remondi 2018, Wilusz, 2020), the statements about inefficiency of the particle tracking (low-fidelity) approach are not well justified.
2. Subsurface heterogeneity can impart significant differences on macro scale behavior (e.g. Benson 2019; Danesh-Yazdi 2018, Engdahl 2014)
3. Effects beyond the hilslope are important (e.g. Kollet 2008; Maxwell 2016; Remondi 2018; Wilusz 2020) and should be discussed.
4. Some references included below that the authors may want to read / consider. This is not a comprehensive list.
References
Benson, D. Schmidt, M. Bolster, D. Harman, C. Engdahl, N Aging and mixing as pseudo-chemical-reactions between, and on, particles: Perspectives on particle interaction and multi-modal ages in hillslopes and streams, Advances in Water Resources,132,2019, https://doi.org/10.1016/j.advwatres.2019.103386.
Kollet, S.J. and Maxwell, R.M. Demonstrating fractal scaling of baseflow residence time distributions using a fully-coupled groundwater and land surface model. Geophysical Research Letters 35, L07402, doi:10.1029/2008GL033215, 2008.
Engdahl, N.B. and Maxwell, R.M. Approximating groundwater age distributions using simple streamtube models and multiple tracers. Advances in Water Resources, 66, 19-31, doi:10.1016/j.advwatres.2014.02.001, 2014.
Maxwell, R.M., Condon, L.E., Danesh-Yazdi, M. and Bearup, L.A. Exploring source water mixing and transient residence time distributions of outflow and evapotranspiration with an integrated hydrologic model and Lagrangian particle tracking approach. Ecohydrology, 12(1):e2042, doi:10.1002/eco.2042, 2019.
Maxwell, R.M., Condon, L.E. , Kollet, S.J., Maher, K., Haggerty, R., and Forrester, M.M. The imprint of climate and geology on the residence times of groundwater. Geophysical Research Letters, 43, doi:10.1002/2015GL066916, 2016.
Remondi, F., Kirchner, J. W., Burlando, P., & Fatichi, S. (2018). Water flux tracking with a distributed hydrological model to quantify controls on the spatio-temporal variability of transit time distributions. Water Resources Research, 54, 3081– 3099. https://doi.org/10.1002/2017WR021689
Wilusz, D. C., Harman, C. J., Ball, W. B., Maxwell, R.M. and Buda, A. R. Using particle tracking to understand flow paths, age distributions, and the paradoxical origins of the inverse storage effect in an experimental catchment. Water Resources Research, 56, e2019WR025140, doi:10.1029/2019WR025140, 2020.
Citation: https://doi.org/10.5194/hess-2021-430-RC3 - AC3: 'Reply on RC3', Rong Mao, 30 Nov 2021
Status: closed
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RC1: 'Comment on hess-2021-430', Anonymous Referee #1, 06 Oct 2021
Comments on “Quantifying time-variant travel time distribution by multi-fidelity model in hillslope under nonstationary hydrologic conditions” by Mao et al.
The authors presented a computationally efficient method for estimating travel time distributions (TTDs), residence time distributions (RTDs), and the StorAge Selection(SAS) functions using process-based models that account for the diffusion (or hydrodynamic dispersion) process. They argued that there are two ways of estimating those distributions and functions, but those methods are computationally burdensome. One way is solving the advection-diffusion (or advection-dispersion) equation (ADE) for each rainfall event, and another way is using the particle tracking model considering the random walk. Their proposed method (called multi-fidelity model) estimates the TTDs, RTDs, and the SAS functions by combining the result of the particle tracking model without the random walk component (so-called low-fidelity model) and the result of the ADE simulations for selected rainfall events (so-called high-fidelity model). They argued that the multi-fidelity model reproduces the result of the ADE model closely (especially when about 10% of all rainfall events are simulated using the ADE model) with a potential reduction of computational time.
My recommendation for the current manuscript is “rejection” due to the following reasons. First, the need for this method is not convincing especially because the proposed method reduces the accuracy of TTDs, RDTs, and the SAS function estimation (see major comment 1). Second, the test is not well designed, and the method’s performance is questionable (see my major comment 2). Third, some model results seem to be inaccurate (see major comment 3). I will also provide other comments in the hope that those comments will be helpful to the authors.
Major comments
1. Not convincing motivation and unclear computational efficiency
The manuscript does not provide a sufficient explanation of the motivation for developing the multi-fidelity model. Reducing computation time seems to be the primary motivation (e.g., L84-86), but there are many other ways to reduce the computation time (especially when sacrificing accuracy for reduced computation time is allowed). We can run a full model (e.g., a model that solves the ADE for each rainfall event or the particle tracking model with the random walk component) in a computationally efficient manner. For example, utilizing a parallelized code (e.g., ParFlow), using a coarser grid, or using a coarser time step is an option. Utilizing parallelized codes to solve Richards’ equation or tracking particles would not reduce the accuracy of the results but significantly reduce the computation time. Using a coarser grid or a coarser time step could reduce the model’s accuracy, but the proposed method, using the multi-fidelity model, could also reduce the accuracy. We can also estimate a BTC conditional to a set of consecutive events rather than to a single event which then can be used to estimate the TTD. We can probably define the injection time of the set as the center of mass of those consecutive events (and, if needed, perform a bit of correction to consider negative travel time). This method would not be able to estimate fast time variabilities of the TTDs, RTDs, and the SAS functions, but there is also no guarantee that the proposed method, the multi-fidelity model, could estimate such fast time variability (see also my major point 2). The result generated using this method, estimating TTDs condition to a set of events, would actually be easier to interpret since we know what information is averaged out, while interpreting the result of the multi-fidelity model would not be very easy. How should a user of the multi-fidelity model interpret the result of the multi-fidelity model? In other words, what should a user expect to be the difference between the high-fidelity model result and the multi-fidelity model result?
Also, when running the ADE model for every single rainfall event (without the simplifications discussed above) or running the particle tracking model with the random walk component is practically feasible, the proposed method would not be useful. For example, let’s say running a full model (e.g., running the ADE model for every single rainfall event) takes ten days. Running the multi-fidelity model would take one day + additional time to run a low fidelity model, e.g., the particle tracking model without the random walk component. In this case, do we need to use the multi-fidelity model to save less than nine days even though the method reduces the accuracy (since the multi-fidelity model result is not the same as the high-fidelity model result)? In what case do we really need this multi-fidelity model?
In addition, computational efficiency is not discussed in detail in this manuscript. Since the method was developed for computational efficiency, the efficiency must be discussed in more detail.
2. Performance of the multi-fidelity model and design of the test
Figure 4 shows that the model error does not converge to zero as the rainfall interval decrease. When the rainfall interval of 5 is used, the mean error is a bit larger than the mean error for the case with the rainfall interval of 10. If the mean error is not expected to converge to zero as the rainfall interval decreases, it needs to be discussed in more detail so that readers know what to expect from the multi-fidelity model.
Furthermore, Figures 4 and 5 show that the multi-fidelity model has a large variance of error, meaning that the time variability of hydrologic transport is still not well captured. Even if I assume that the error is bearable for this test case (i.e., the test using the homogeneous 2D hillslope domain), the error could become larger if the method is applied to a hydrologic system with larger time variability of hydrologic transport dynamics.
The temporal variability of hydrologic transport dynamics in the homogeneous 2D hillslope domain is not large, resulting in the SAS function with a limited time variability (Figure 9b). In many real-world studies, the SAS function time variability is much larger than what is presented in Figure 9b (see many studies that applied the SAS Function). Could this multi-fidelity method be useful when the time variability of hydrologic transport dynamics is large?
3. Model results
Some results do not seem to be accurate. Figure 9 shows the SAS function, and the shape of the SAS function is very spiky especially for the low fidelity model. The SAS function, in general, is expected to have a much smoother shape (and that is one reason why the SAS function becomes popular). Also, see Kim et al. (2020), where they showed that the SAS functions derived for a similar hydrologic domain are smooth.
This incorrect result is perhaps due to the insufficient number of particles used in the model for the low-fidelity model. Or maybe there is a problem in the estimation of the RTDs and the SAS functions. If the former is the case, this incorrect result may have resulted in the non-zero bias I pointed out in major comment 2.
[Reference]
Kim. M, T. H. M. Volkmann, Y. Wang, C. J. Harman, & P. A. Troch (2020), Direct observation of hillslope scale StorAge Selection functions in an experimental hydrologic system: Geomorphologic structure and the preferential discharge of old water, https://doi.org/10.1002/essoar.10504485.1
4. Section 2.3
It is unclear what “correlation” means here, and how (15) leads to the argument in L177-178 ("Theoretically, a mapping is capable of being setting up between ... if the diffusion parameter D is known. This is the theoretical foundation of the multi-fidelity model") and L198-199 is also unclear. What authors show in this section using (10) – (15) is trivial---As the diffusion coefficient decreases, the transport is dominated by advection. How does it support the argument in L177-178? Also, the 1-D configuration used in this section is not enough to support the argument. There is no consideration of transversal diffusion or dispersion. Furthermore, spatial and temporal variations of velocity are not considered.
Minor comments
Needs less focus on the hillslope scale: In many places throughout the manuscript, the authors talked about hillslope (e.g., in the title, L3, L18, and so on). However, I am not sure if the multi-fidelity model could be useful at the hillslope scale, which is relatively small. For a small hydrological domain like a hillslope, running the full ADE simulations or the particle tracking model with the random component will be feasible.
Diffusion process: It is unclear what the diffusion process simulated in the 2D hillslope model is. The authors may have assigned a specific concentration for the rainfall event of interest while the concentration of all other water is set to zero. Since the diffusion is controlled by the concentration gradient and the diffusion coefficient, it needs to be more precise what the gradient in the model (which depends on the assigned value for the rainfall event of interest) means and what the diffusion coefficient means.
L5: “transient ground water flow” is not the only process that could result in time-variant TTDs.
L11-13: In is not clear how significantly the number of particles can be reduced.
L29: “yearly averaged” does not seem to describe the time-invariant TTD correctly.
L67-69 and L146-148: While I agree, this sentence is not enough to motivate this study. I think the authors need to cite studies that show that considering diffusion or dispersion requires significantly more particles (if such a study exists) or need to show it clearly somewhere in the manuscript.
L76: Not clear what message the authors are trying to convey.
L83: Not clear if the authors solved the full 5D equation or just combine the results of the high fidelity model and the low fidelity model.
L152: No evidence is provided to support the “drastic” decrease of the number of particles
L175: While I agree with the “no matter what boundary conditions” argument, what the authors show in this section is for a specific boundary condition used to get (14).
L265: There are many studies where RTDs are estimated using BTCs. Most of the experimental studies of the SAS function and many the SAS function studies that utilize process-based model estimated RTDs using BTCs.
L280-281: This sentence is unnecessary for the purpose of this paper.
L291: Lower concentration of which solute?
L305: “quasi-steady state” means something different.
L313: Each single rainfall event at daily time step?
L320-321: Richards’ equation-based model has many assumptions. Also, perhaps better to avoid the term “real physical process”.
Figure 3: Please clarify what the particles at x = 100 m are.
L373-376: A model result of solute concentration cannot be lower during the whole time than the concentration estimated using another model.
Citation: https://doi.org/10.5194/hess-2021-430-RC1 - AC1: 'Reply on RC1', Rong Mao, 30 Nov 2021
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RC2: 'Comment on hess-2021-430', Anonymous Referee #2, 16 Oct 2021
The aim of this study was to propose a multi-fidelity model is to quantify the time-variant travel time distribution (TTD. This multi-fidelity model undergoes four main steps of (i) decomposing the advection-diffusion equation (ADE) into a series of ADE’s, (ii) solving these ADEs through particle tracking without random walk (low-fidelity model), (iii) solving several ADE’s by numerical simulation (high-fidelity model), (iv) establishing the relationship relation between the low fidelity and high-fidelity models, and (v) calculating the time-variant TTD from the breakthrough curves of the multi-fidelity model. The manuscript is well written and one can follow the proposed theoretical framework smoothly. However, the superiority of the proposed multi-fidelity model over the very well-known approaches like the storage section function framework is not clear. Also, the impact on model accuracy of some major simplifications in both the low- and high-fidelity models has not been demonstrated. As such, I recommend major revision at this stage of review. Further major and minor comments are presented below.
Major comments
- Introduction
Line 49-53: This statement of is not correct because the influence of permeability architecture and preferential flow is already projected in the flow velocities and residence times, which in turn are reflected in the time-variant TTD. Therefore, the age maser equation can indeed be employed to explore the above issues by studying hillslopes with a wide range of physical conditions. Some other former studies have already explored them, too.
- Groundwater age distribution model
- The two main parts of this section is (i) the groundwater age distribution model and (ii) the approach to solve the so-called 5D equation (1). Both of them have already been contextualized in the work of Ginn (1999) and Gomez and Wilson (2013), respectively. The tone of material presentation in the section (e.g., in Line 95) looks like the authors are proposing a novel approach of treating the original equations by Ginn (1999), but it does not seem so.
- Line 137-140: This part is written very concisely and is not clear to a reader. In particular, four questions are unknown: (1) Why should the high-fidelity model be run multiple times and in a limited number of runs? (2) Why should not the high-fidelity model required be run for all input pulses, but only some of them? (3) What does the “trend” of the variation in age distributions imply? And (4) what does make the multi-fidelity model biased? To clarify the above issue, the authors must first describe the elements of the multi-fidelity models and then explain how those elements work together to build the multi-fidelity model with sufficient elaboration on its properties/advantages/caveats.
- Particle-tracking model
- The claimed lower computational cost of the proposed low-fidelity model seems to be obtained at the cost of ignoring the molecular diffusion and mechanical dispersion of solutes. As such, the superiority/accuracy of the low-fidelity model as compared to the other competing particle-tracking models like EcoSILM is not clear. Also, please note the following studies in which some approaches to diminish the computational of particle-tracking are discussed:
- Yang et al. (2021), Accurate load balancing accelerates Lagrangian simulation of water ages on distributed, multi-GPU platforms, https://doi.org/10.1002/essoar.10507899.1.
- Yang et al. (2021), Accelerating the Lagrangian particle tracking of residence time distributions and source water mixing towards large scales, Computers & Geosciences, https://doi.org/10.1016/j.cageo.2021.104760.
- Maxwell, RM, Condon, LE, Danesh-Yazdi, M, Bearup, LA. Exploring source water mixing and transient residence time distributions of outflow and evapotranspiration with an integrated hydrologic model and Lagrangian particle tracking approach. Ecohydrology. 2019; 12:e2042. https://doi.org/10.1002/eco.2042.
- It is also noted that injecting a sufficient number of particles into the particle-tracking models is not required to only reduce the random noise of diffusion and mechanical dispersion, but also to reduce bias in the tracked evapotranspired water. Since the present study highlights its superiority in terms of employing a much smaller number of particles, the model accuracy on tracking fluxes like evapotranspiration in not clear and discussed.
- Analytical theorem for one dimensional scenario
The presented theoretical foundation of the multi-fidelity model in 1D seems to hold upon the assumption of constant advection velocity in the entire domain. But this is not certainty true in real-world cases and is against the goal of this study.
- Unclear advantage of the multi-fidelity model and the limited experiments
According to the motivation of this study, the proposed multi-fidelity model is expected to ease the estimation of TTD by skipping the complex process of SAS function quantification proposed by former studies (e.g., Botter et al., 2011). However, determining the mathematical form of the nonlinear mapping of equation (17) is said to be strongly dependent on the properties of a hydrogeological system under study. Therefore, it seems that the challenge of SAS function calibration in the formerly developed framework has only switched to quantifying another function of nonlinear type, i.e., g in equation (17), in the present study. From the designed experiments, the relative efficiency of the multi-fidelity model performance as compared to the SAS function calibration approach is not clear and appears to be exaggerated.
Minor comments
- Abstract, line 1: TTD is not necessarily representative of groundwater, but also surface runoff and any the processes that compose streamflow. If the focus of this work is to consider the travel time of flow pathways in groundwater, the authors should clarify it first in the manuscript.
- Abstract, line 5: The non-stationarity of TTD is not only because of non-stationarity n rainfall, but all the processes closing the water balance in a hydrological system.
- The abstract is a little bit confusing and the study storyline and objectives are not clear.
- Line 20: mixing process and also the heterogeneity of flow pathways and transient pore water velocity.
- Line 36: RTD was not defined earlier in the manuscript.
- Line 54 and 58: Is “periodic” the best word for this condition?
- Line 86: In comparison to… Please fix.
- Line 92: groundwater flow pathways… Please fix.
Citation: https://doi.org/10.5194/hess-2021-430-RC2 - AC2: 'Reply on RC2', Rong Mao, 30 Nov 2021
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RC3: 'Comment on hess-2021-430', Anonymous Referee #3, 31 Oct 2021
Review of Mao et al
This manuscript present work comparing travel time distribution formulations in a simple hillslope experiment. The work has some interesting aspects, comparing different approaches for simulating water residence time.
General Comments:
The manuscript is generally well written and on a topic of interest to the readership of HESS. The terminology of high-to-low fidelity is confusing or perhaps even misleading. I have some comments and suggestions below that might help clarify some aspects of the work as presented and help to better place it in context of the literature, which would help better establish the novelty of the current approach. It is also unclear the the proposed mixed approach is “better” in a computational sense, the authors jump to this conclusion seemingly without comprehensive evaluation.
Specific Comments:
1. S2.2 / lines 141-154. The approach presented has similarities and other approaches (e.g. Maxwell 2019, Maxwell 2016, Remondi 2018, Wilusz, 2020), the statements about inefficiency of the particle tracking (low-fidelity) approach are not well justified.
2. Subsurface heterogeneity can impart significant differences on macro scale behavior (e.g. Benson 2019; Danesh-Yazdi 2018, Engdahl 2014)
3. Effects beyond the hilslope are important (e.g. Kollet 2008; Maxwell 2016; Remondi 2018; Wilusz 2020) and should be discussed.
4. Some references included below that the authors may want to read / consider. This is not a comprehensive list.
References
Benson, D. Schmidt, M. Bolster, D. Harman, C. Engdahl, N Aging and mixing as pseudo-chemical-reactions between, and on, particles: Perspectives on particle interaction and multi-modal ages in hillslopes and streams, Advances in Water Resources,132,2019, https://doi.org/10.1016/j.advwatres.2019.103386.
Kollet, S.J. and Maxwell, R.M. Demonstrating fractal scaling of baseflow residence time distributions using a fully-coupled groundwater and land surface model. Geophysical Research Letters 35, L07402, doi:10.1029/2008GL033215, 2008.
Engdahl, N.B. and Maxwell, R.M. Approximating groundwater age distributions using simple streamtube models and multiple tracers. Advances in Water Resources, 66, 19-31, doi:10.1016/j.advwatres.2014.02.001, 2014.
Maxwell, R.M., Condon, L.E., Danesh-Yazdi, M. and Bearup, L.A. Exploring source water mixing and transient residence time distributions of outflow and evapotranspiration with an integrated hydrologic model and Lagrangian particle tracking approach. Ecohydrology, 12(1):e2042, doi:10.1002/eco.2042, 2019.
Maxwell, R.M., Condon, L.E. , Kollet, S.J., Maher, K., Haggerty, R., and Forrester, M.M. The imprint of climate and geology on the residence times of groundwater. Geophysical Research Letters, 43, doi:10.1002/2015GL066916, 2016.
Remondi, F., Kirchner, J. W., Burlando, P., & Fatichi, S. (2018). Water flux tracking with a distributed hydrological model to quantify controls on the spatio-temporal variability of transit time distributions. Water Resources Research, 54, 3081– 3099. https://doi.org/10.1002/2017WR021689
Wilusz, D. C., Harman, C. J., Ball, W. B., Maxwell, R.M. and Buda, A. R. Using particle tracking to understand flow paths, age distributions, and the paradoxical origins of the inverse storage effect in an experimental catchment. Water Resources Research, 56, e2019WR025140, doi:10.1029/2019WR025140, 2020.
Citation: https://doi.org/10.5194/hess-2021-430-RC3 - AC3: 'Reply on RC3', Rong Mao, 30 Nov 2021
Model code and software
Codes and dataset used in the manuscript entitled "Quantifying time-variant travel time distribution by multi-fidelity model in hillslope under nonstationary hydrologic conditions" Rong Mao https://doi.org/10.5281/zenodo.5195801
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