Feature scale and identifiability: How much information do point hydraulic measurements provide about heterogeneous head and conductivity fields?

Hansen, Scott K.; O'Malley, Daniel; Hambleton, James P.

doi:https://doi.org/10.5194/egusphere-2024-337

Preprints

https://doi.org/10.5194/egusphere-2024-337

Preprints

11 Mar 2024

| 11 Mar 2024

Feature scale and identifiability: How much information do point hydraulic measurements provide about heterogeneous head and conductivity fields?

Scott K. Hansen, Daniel O'Malley, and James P. Hambleton

Abstract. We systematically investigate how the spacing and type of point measurements impacts the scale of subsurface features that can be identified by groundwater flow model calibration. To this end, we consider the optimal inference of spatially heterogeneous hydraulic conductivity and head fields based on three kinds of point measurements that may be available at monitoring wells: of head, permeability, and groundwater speed. We develop a general, zonation-free technique for Monte Carlo (MC) study of field recovery problems, based on Karhunen-Loève (K-L) expansions of the unknown fields whose coefficients are recovered by an analytical, continuous adjoint-state technique. This allows unbiased sampling from the space of all possible fields with a given correlation structure and efficient, automated gradient-descent calibration. The K-L basis functions have a straightforward notion of period, revealing the relationship between feature scale and reconstruction fidelity, and they have an a priori known spectrum, allowing for a non-subjective regularization term to be defined. We perform automated MC calibration on over 1100 conductivity-head field pairs, employing a variety of point measurement geometries and evaluating the mean-squared field reconstruction accuracy, both globally and as a function of feature scale. We present heuristics for feature scale identification, examine global reconstruction error, and explore the value added by both the groundwater speed measurements and by two different types of regularization. We find that significant feature identification becomes possible as feature scale exceeds four times measurement spacing and identification reliability subsequently improves in a power law fashion with increasing feature scale.

Received: 04 Feb 2024 – Discussion started: 11 Mar 2024

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this preprint. The responsibility to include appropriate place names lies with the authors.

Download & links

Preprint (PDF, 972 KB)

Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
Preprint (972 KB)

Download & links

The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.

Journal article(s) based on this preprint

24 Mar 2025

Feature scale and identifiability: how much information do point hydraulic measurements provide about heterogeneous head and conductivity fields?

Scott K. Hansen, Daniel O'Malley, and James P. Hambleton

Hydrol. Earth Syst. Sci., 29, 1569–1585, https://doi.org/10.5194/hess-29-1569-2025,https://doi.org/10.5194/hess-29-1569-2025, 2025

Short summary

Scott K. Hansen, Daniel O'Malley, and James P. Hambleton

Interactive discussion

Status: closed

CC1:
'Comment on egusphere-2024-337', Giacomo Medici, 13 Mar 2024

General comments
Good theoretical research with implication on groundwater flow modelling and the engineering of the reservoirs where the geological flow heterogeneities are of paramount importance. Please, follow my guidance to improve the manuscript.

Specific comments
Line 6. I suggest “This technique allows unbiased”. Add the word “technique”.
Lines 20-22. Mini-permeameter, slug, packer and pumping tests can be also used to identify flow heterogeneities and determine the hydraulic conductivity. Specify this point.
Line 22. “Point-to-point tracer tests” to detect flow heterogeneities. Please, add recent literature on the topic:
- Deleu, R., Frazao, S. S., Poulain, A., Rochez, G., & Hallet, V. (2021). Tracer Dispersion through Karst Conduit: Assessment of Small-Scale Heterogeneity by Multi-Point Tracer Test and CFD Modeling. Hydrology, ((4)
- Lorenzi, V., Banzato, F., Barberio, M. D., Goeppert, N., Goldscheider, N., Gori, F., Lacchini A., Manetta M., Medici G., Rusi S., Petitta, M. (2024). Tracking flowpaths in a complex karst system through tracer test and hydrogeochemical monitoring: Implications for groundwater protection (Gran Sasso, Italy). Heliyon, 10(2)
- Poulain, A., Rochez, G., Van Roy, J.P., Dewaide, L., Hallet, V. and De Sadelaer, G., 2017. A compact field fluorometer and its application to dye tracing in karst environments. Hydrogeology Journal, 25
Lines 48-88. The literature on the topic is much broader.
Line 84. Disclose the 3 to 4 specific objectives of your research by using numbers (e.g., i, ii and iii).
Line 90. “Feature scale”. This expression is difficult to understand. Do you mean “observation scale”?
Line 199. “theoretical observations”. Can you re-call the key equations instead?
Line 501. “Other geophysical fields”. (i) remind to the reader that the principal implications are in the calibration of groundwater flow models, (ii) which other implications/applications in geophysics? You can look at my general comments.
Line 514. Please, add relevant literature on the topic.

Figures and tables
Figure 1. Very nice figure, but it needs some changes. (i) Make the rectangles closer, (ii) make words and numbers larger, and (iii) thicker the black nodes.
Figures 2-4. Make words and figures larger on x and y axes.

Citation: https://doi.org/10.5194/egusphere-2024-337-CC1
- AC3: 'Reply on CC1', Scott K. Hansen, 18 Jul 2024
  
  We thank the commenter for his positive overall assessment.
  He makes many well-founded specific comments for improving the textual and graphic clarity, as well as the referencing. We will take them to heart.
  
  Citation: https://doi.org/10.5194/egusphere-2024-337-AC3
RC1:
'Comment on egusphere-2024-337', Anonymous Referee #1, 19 Apr 2024

The authors present an approach for using Karhunen-Loeve explanations to estimate spatial variability of properties and system states on observations of head, permeability, and velocity magnitude, both with and without regularization. They apply the adjoint approach for efficiently determining gradients of the objective function, allowing for more efficiency and, presumably, high-order approximations. The methods are explained thoroughly, including the assumptions and simplifications. I just have a few minor comments.
1. A substantial amount (about 20%) of the paper was devoted to developing the adjoint equation and the adjoint-based form of the derivatives of the objective function. Since that was just a tool to be used in the analysis, the detailed development seemed to detract from the main focus of the paper. Could Section 2 be moved to an appendix?
2. Figure 1 is very helpful as an example of the recovery of spatial distributions of head and permeability through the approach presented in the paper. All other figures show "error" between measures and fits, so Figure 1 is very useful as means of showing the reader the intermediate step. However, Figure 1 is out of place - it appears on p. 11, but it isn't mentioned in the text until page 20. Also more explanation can be provided to make the link between what appears in Figure 1 and how that is related to the data point plotted in the other figures. I wonder also why the two subplots of Figure 1 use different true ln K fields. If subplot a has regularization and subplot b does not, it would be more informative to see that results from the same true ln K field so that the reader can see the benefit of regularization. It would also be helpful to see the numerical value of "error" (the quantities that are plotted in the other figures) to get a sense for where these fits fall in those plots, compared to all other realizations that appear in the plots in Figures 2 and above.
3. I would like to see some explanation of the practicality of this method. What measurement density is needed? Is it different for measurements of different quantities?

Citation: https://doi.org/10.5194/egusphere-2024-337-RC1
- AC2: 'Authors' reply to RC1', Scott K. Hansen, 18 Jul 2024
  
  We thank the reviewer for their helpful comments.
  We agree with point 1: the paper would be well served by moving much of the adjoint-state derivation into an appendix.
  Regarding the comments on figures in point 2: we agree that it would be more useful for the regularized and unregularized examples to share the same ground truth; it is no problem to do this. We also agree about adding more linkage between these diagrams and the rest of the text: both by qualitative discussion and by reporting the various error metrics that correspond the fields shown, both before and after calibration. The figure should also be moved closer to its point of first reference.
  Regarding point 3: in our view one of the most interesting conclusions of our work is that measurements must be at least 4x as dense as the scale of contiguous features one wishes to delineate, with denser measurements providing increasingly better performance. And although some combinations of measurements performed somewhat better than others, this basic pattern was shared by all the different combinations of measurement types we considered, as shown in Fig. 4. Though this information was contained in the original draft, we think it could benefit from being communicated more clearly, as it is so important.
  
  Citation: https://doi.org/10.5194/egusphere-2024-337-AC2
RC2:
'Comment on egusphere-2024-337', Anonymous Referee #2, 17 Jun 2024

This is a well written paper (until the discussion section) on the study of the worth of different types of data for inverse modeling in hydrogeology. Its aim is to try to identify the necessary sampling density in order to identify the underlying hydraulic conductivity structure. While I enjoyed reading the paper and the detailed derivations of the different equations, I had a hard time trying to find the novelty of the work. This question has been studied as early as in the late 1990s, and the application proposed, while very elegant, is limited to a very narrow and little interesting case of steady groundwater flow in an aquifer with prescribed head at the boundary and with an underlying hydraulic conductivity field drawn from a multi Gaussian random function with an exponential covariance. For the paper to be worth to be considered for publication it should have addressed a tougher problem: transient, non-Gaussian field, larger variance, anisotropic, generic boundary conditions, addressing the issue of measurements taken at different supports. As is, the a paper is a very nice mathematical exercise with little practical interest.
As a minor point, there was an earlier benchmarking of inverse models not mentioned in the opening, which addressed a tougher problem than the one discussed here published in Water Resources Research by Zimmerman et al. In 1988.

Citation: https://doi.org/10.5194/egusphere-2024-337-RC2
- CC2:
  'Reply on RC2', James Hambleton, 17 Jun 2024
  
  Thanks heartily for the feedback. Can the Reviewer be more specific about what work in the early 1990's endeavoured (and perhaps succeeded) in addressing the aim of the paper, so succinctly identified in the Reviewer's appraisal?
  
  Citation: https://doi.org/10.5194/egusphere-2024-337-CC2
  - CC3: 'Reply on CC2', James Hambleton, 17 Jun 2024
    
    Apologies for the substantive error in my previous reply with respect to late 1990s versus early 1990s. Perhaps more importantly, apologies for the typographical error of writing "1990's" instead of "1990s," an error now immortalised on this platform.
    
    Citation: https://doi.org/10.5194/egusphere-2024-337-CC3
- AC1:
  'Authors' reply to RC2', Scott K. Hansen, 18 Jul 2024
  We appreciate the reviewer's remarks that the paper is well written and take comfort in the fact that they did not identify any methodological errors. That said, we take exception to the overall gist of their review and believe that there may have been confusion about the nature of our paper’s contribution.
  We view our work as basic science concerning the information content of point hydraulic measurements, but we think the reviewer may have understood it as a benchmarking study aimed at evaluating calibration method performance. This interpretation would explain some remarks that we otherwise find difficult to account for. The Zimmerman paper they mention (DOI: 10.1029/98WR00003) is a benchmarking study comparing the performance of geostatistical inversion schemes but does not otherwise address our key themes.
  Discussing two reviewer remarks in particular:
  A) The reviewer indicates that they struggle to see what is novel in our study. Here are two major original features:
  The principal novelty of our study, as we stress in our introductory material, is the investigation of the connection between spatial scale of features in the hydraulic conductivity field and our ability to characterize them from a given density of point measurements. As far as we know, this has never been systematically investigated and no one has previously quantified the relationship between amount learned and relative measurement density, as we have. Our key result---that feature identification becomes possible above 4x measurement scale, with quality increasing in power law fashion with increasing feature scale above this threshold---has no real precedents.
  
  The Karhunen-Loeve basis function framework we develop for assessing calibration accuracy without requiring the imposition of experimenter-specified a priori spatial zonation or subjective weights on the objective function is itself novel and can be applied (with suitable modifications) to analysis of a variety of different field recovery problems.
  
  B) The reviewer remarks that the flow regime we employ in our analysis---steady flow in a mildly heterogeneous multi-Gaussian conductivity field---is too straightforward for their liking, and they write that the manuscript would only be worthwhile to publish if it considered a "tougher problem" with, e.g., an esoteric non-Gaussian statistical correlation structure, or high variance, or transient flow. This seems strange and perhaps represents a misunderstanding of what we are studying.
  If our work were an algorithm benchmarking study where we were trying to select the most robust approach from amongst candidates, we could see the merit in choosing a "tough" benchmark. But our work is not a benchmarking study; rather, it concerns the information content of hydraulic measurements. This basic science question is best answered using a canonical flow model that exhibits the physics we wish to study, employing realistic parameters and standard assumptions but otherwise shedding needless complexity. Therefore, the more niche modeling scenarios suggested by the reviewer are counterproductive for our purposes.
  We stress: our study is not "about" any specific flow configuration; rather we are studying how reliably groundwater flow transports information about conductivity features to remote measurement locations. Given that the feature scale / identifiability relationship has never been quantified in any scenarios at all, we believe that choosing a common flow scenario as a basis for our analyses is a good choice.
  We were surprised to see the flow scenario chosen described as a “little interesting case”, given that the assumptions made regarding the flow model are all common ones in the literature used in, conservatively, hundreds of papers. Concerning two mentioned decisions:
  Multivariate Gaussian random fields are overwhelmingly used in hydrologic Monte Carlo studies, and the log-variance we employed is similarly well within the normal range of values considered. Such models are of course idealizations; the working hypothesis in any Monte Carlo analysis of this sort is that the precise details of the conductivity correlation structure are of at most secondary significance compared to the meta-parameters of interest (in our case correlation length). To the extent this is true, the specific choice is of no impact. To the extent structural details are impactful, why would selection of a less common structure be superior and render the work publishable? We cannot see the rationale behind this.
  
  Also consider the suggestion that we employ a transient model. In nature, at field scale, it is common to find flow that admits a quasi-steady model, and flow that requires a transient model. These lead to different calibration problems, and both are relevant; it is not clear why the reviewer considers the transient option superior. In any event it would not be feasible for us to study the latter systematically. Performing the high-quality systematic study we already present under steady conditions consumed 1000s of compute hours. Moving to a transient problem would require an explosion of computational resources, as we would now be inverting in the <conductivity> x <storativity> x <flow history> space, and with a much slower-to-run model. And we would still have to make arbitrary decisions constraining the flow history.
  
  Ultimately, we do not consider the negative comments well founded.
  
  Citation: https://doi.org/10.5194/egusphere-2024-337-AC1

Interactive discussion

Status: closed

CC1:
'Comment on egusphere-2024-337', Giacomo Medici, 13 Mar 2024

General comments
Good theoretical research with implication on groundwater flow modelling and the engineering of the reservoirs where the geological flow heterogeneities are of paramount importance. Please, follow my guidance to improve the manuscript.

Specific comments
Line 6. I suggest “This technique allows unbiased”. Add the word “technique”.
Lines 20-22. Mini-permeameter, slug, packer and pumping tests can be also used to identify flow heterogeneities and determine the hydraulic conductivity. Specify this point.
Line 22. “Point-to-point tracer tests” to detect flow heterogeneities. Please, add recent literature on the topic:
- Deleu, R., Frazao, S. S., Poulain, A., Rochez, G., & Hallet, V. (2021). Tracer Dispersion through Karst Conduit: Assessment of Small-Scale Heterogeneity by Multi-Point Tracer Test and CFD Modeling. Hydrology, ((4)
- Lorenzi, V., Banzato, F., Barberio, M. D., Goeppert, N., Goldscheider, N., Gori, F., Lacchini A., Manetta M., Medici G., Rusi S., Petitta, M. (2024). Tracking flowpaths in a complex karst system through tracer test and hydrogeochemical monitoring: Implications for groundwater protection (Gran Sasso, Italy). Heliyon, 10(2)
- Poulain, A., Rochez, G., Van Roy, J.P., Dewaide, L., Hallet, V. and De Sadelaer, G., 2017. A compact field fluorometer and its application to dye tracing in karst environments. Hydrogeology Journal, 25
Lines 48-88. The literature on the topic is much broader.
Line 84. Disclose the 3 to 4 specific objectives of your research by using numbers (e.g., i, ii and iii).
Line 90. “Feature scale”. This expression is difficult to understand. Do you mean “observation scale”?
Line 199. “theoretical observations”. Can you re-call the key equations instead?
Line 501. “Other geophysical fields”. (i) remind to the reader that the principal implications are in the calibration of groundwater flow models, (ii) which other implications/applications in geophysics? You can look at my general comments.
Line 514. Please, add relevant literature on the topic.

Figures and tables
Figure 1. Very nice figure, but it needs some changes. (i) Make the rectangles closer, (ii) make words and numbers larger, and (iii) thicker the black nodes.
Figures 2-4. Make words and figures larger on x and y axes.

Citation: https://doi.org/10.5194/egusphere-2024-337-CC1
- AC3: 'Reply on CC1', Scott K. Hansen, 18 Jul 2024
  
  We thank the commenter for his positive overall assessment.
  He makes many well-founded specific comments for improving the textual and graphic clarity, as well as the referencing. We will take them to heart.
  
  Citation: https://doi.org/10.5194/egusphere-2024-337-AC3
RC1:
'Comment on egusphere-2024-337', Anonymous Referee #1, 19 Apr 2024

The authors present an approach for using Karhunen-Loeve explanations to estimate spatial variability of properties and system states on observations of head, permeability, and velocity magnitude, both with and without regularization. They apply the adjoint approach for efficiently determining gradients of the objective function, allowing for more efficiency and, presumably, high-order approximations. The methods are explained thoroughly, including the assumptions and simplifications. I just have a few minor comments.
1. A substantial amount (about 20%) of the paper was devoted to developing the adjoint equation and the adjoint-based form of the derivatives of the objective function. Since that was just a tool to be used in the analysis, the detailed development seemed to detract from the main focus of the paper. Could Section 2 be moved to an appendix?
2. Figure 1 is very helpful as an example of the recovery of spatial distributions of head and permeability through the approach presented in the paper. All other figures show "error" between measures and fits, so Figure 1 is very useful as means of showing the reader the intermediate step. However, Figure 1 is out of place - it appears on p. 11, but it isn't mentioned in the text until page 20. Also more explanation can be provided to make the link between what appears in Figure 1 and how that is related to the data point plotted in the other figures. I wonder also why the two subplots of Figure 1 use different true ln K fields. If subplot a has regularization and subplot b does not, it would be more informative to see that results from the same true ln K field so that the reader can see the benefit of regularization. It would also be helpful to see the numerical value of "error" (the quantities that are plotted in the other figures) to get a sense for where these fits fall in those plots, compared to all other realizations that appear in the plots in Figures 2 and above.
3. I would like to see some explanation of the practicality of this method. What measurement density is needed? Is it different for measurements of different quantities?

Citation: https://doi.org/10.5194/egusphere-2024-337-RC1
- AC2: 'Authors' reply to RC1', Scott K. Hansen, 18 Jul 2024
  
  We thank the reviewer for their helpful comments.
  We agree with point 1: the paper would be well served by moving much of the adjoint-state derivation into an appendix.
  Regarding the comments on figures in point 2: we agree that it would be more useful for the regularized and unregularized examples to share the same ground truth; it is no problem to do this. We also agree about adding more linkage between these diagrams and the rest of the text: both by qualitative discussion and by reporting the various error metrics that correspond the fields shown, both before and after calibration. The figure should also be moved closer to its point of first reference.
  Regarding point 3: in our view one of the most interesting conclusions of our work is that measurements must be at least 4x as dense as the scale of contiguous features one wishes to delineate, with denser measurements providing increasingly better performance. And although some combinations of measurements performed somewhat better than others, this basic pattern was shared by all the different combinations of measurement types we considered, as shown in Fig. 4. Though this information was contained in the original draft, we think it could benefit from being communicated more clearly, as it is so important.
  
  Citation: https://doi.org/10.5194/egusphere-2024-337-AC2
RC2:
'Comment on egusphere-2024-337', Anonymous Referee #2, 17 Jun 2024

This is a well written paper (until the discussion section) on the study of the worth of different types of data for inverse modeling in hydrogeology. Its aim is to try to identify the necessary sampling density in order to identify the underlying hydraulic conductivity structure. While I enjoyed reading the paper and the detailed derivations of the different equations, I had a hard time trying to find the novelty of the work. This question has been studied as early as in the late 1990s, and the application proposed, while very elegant, is limited to a very narrow and little interesting case of steady groundwater flow in an aquifer with prescribed head at the boundary and with an underlying hydraulic conductivity field drawn from a multi Gaussian random function with an exponential covariance. For the paper to be worth to be considered for publication it should have addressed a tougher problem: transient, non-Gaussian field, larger variance, anisotropic, generic boundary conditions, addressing the issue of measurements taken at different supports. As is, the a paper is a very nice mathematical exercise with little practical interest.
As a minor point, there was an earlier benchmarking of inverse models not mentioned in the opening, which addressed a tougher problem than the one discussed here published in Water Resources Research by Zimmerman et al. In 1988.

Citation: https://doi.org/10.5194/egusphere-2024-337-RC2
- CC2:
  'Reply on RC2', James Hambleton, 17 Jun 2024
  
  Thanks heartily for the feedback. Can the Reviewer be more specific about what work in the early 1990's endeavoured (and perhaps succeeded) in addressing the aim of the paper, so succinctly identified in the Reviewer's appraisal?
  
  Citation: https://doi.org/10.5194/egusphere-2024-337-CC2
  - CC3: 'Reply on CC2', James Hambleton, 17 Jun 2024
    
    Apologies for the substantive error in my previous reply with respect to late 1990s versus early 1990s. Perhaps more importantly, apologies for the typographical error of writing "1990's" instead of "1990s," an error now immortalised on this platform.
    
    Citation: https://doi.org/10.5194/egusphere-2024-337-CC3
- AC1:
  'Authors' reply to RC2', Scott K. Hansen, 18 Jul 2024
  We appreciate the reviewer's remarks that the paper is well written and take comfort in the fact that they did not identify any methodological errors. That said, we take exception to the overall gist of their review and believe that there may have been confusion about the nature of our paper’s contribution.
  We view our work as basic science concerning the information content of point hydraulic measurements, but we think the reviewer may have understood it as a benchmarking study aimed at evaluating calibration method performance. This interpretation would explain some remarks that we otherwise find difficult to account for. The Zimmerman paper they mention (DOI: 10.1029/98WR00003) is a benchmarking study comparing the performance of geostatistical inversion schemes but does not otherwise address our key themes.
  Discussing two reviewer remarks in particular:
  A) The reviewer indicates that they struggle to see what is novel in our study. Here are two major original features:
  The principal novelty of our study, as we stress in our introductory material, is the investigation of the connection between spatial scale of features in the hydraulic conductivity field and our ability to characterize them from a given density of point measurements. As far as we know, this has never been systematically investigated and no one has previously quantified the relationship between amount learned and relative measurement density, as we have. Our key result---that feature identification becomes possible above 4x measurement scale, with quality increasing in power law fashion with increasing feature scale above this threshold---has no real precedents.
  
  The Karhunen-Loeve basis function framework we develop for assessing calibration accuracy without requiring the imposition of experimenter-specified a priori spatial zonation or subjective weights on the objective function is itself novel and can be applied (with suitable modifications) to analysis of a variety of different field recovery problems.
  
  B) The reviewer remarks that the flow regime we employ in our analysis---steady flow in a mildly heterogeneous multi-Gaussian conductivity field---is too straightforward for their liking, and they write that the manuscript would only be worthwhile to publish if it considered a "tougher problem" with, e.g., an esoteric non-Gaussian statistical correlation structure, or high variance, or transient flow. This seems strange and perhaps represents a misunderstanding of what we are studying.
  If our work were an algorithm benchmarking study where we were trying to select the most robust approach from amongst candidates, we could see the merit in choosing a "tough" benchmark. But our work is not a benchmarking study; rather, it concerns the information content of hydraulic measurements. This basic science question is best answered using a canonical flow model that exhibits the physics we wish to study, employing realistic parameters and standard assumptions but otherwise shedding needless complexity. Therefore, the more niche modeling scenarios suggested by the reviewer are counterproductive for our purposes.
  We stress: our study is not "about" any specific flow configuration; rather we are studying how reliably groundwater flow transports information about conductivity features to remote measurement locations. Given that the feature scale / identifiability relationship has never been quantified in any scenarios at all, we believe that choosing a common flow scenario as a basis for our analyses is a good choice.
  We were surprised to see the flow scenario chosen described as a “little interesting case”, given that the assumptions made regarding the flow model are all common ones in the literature used in, conservatively, hundreds of papers. Concerning two mentioned decisions:
  Multivariate Gaussian random fields are overwhelmingly used in hydrologic Monte Carlo studies, and the log-variance we employed is similarly well within the normal range of values considered. Such models are of course idealizations; the working hypothesis in any Monte Carlo analysis of this sort is that the precise details of the conductivity correlation structure are of at most secondary significance compared to the meta-parameters of interest (in our case correlation length). To the extent this is true, the specific choice is of no impact. To the extent structural details are impactful, why would selection of a less common structure be superior and render the work publishable? We cannot see the rationale behind this.
  
  Also consider the suggestion that we employ a transient model. In nature, at field scale, it is common to find flow that admits a quasi-steady model, and flow that requires a transient model. These lead to different calibration problems, and both are relevant; it is not clear why the reviewer considers the transient option superior. In any event it would not be feasible for us to study the latter systematically. Performing the high-quality systematic study we already present under steady conditions consumed 1000s of compute hours. Moving to a transient problem would require an explosion of computational resources, as we would now be inverting in the <conductivity> x <storativity> x <flow history> space, and with a much slower-to-run model. And we would still have to make arbitrary decisions constraining the flow history.
  
  Ultimately, we do not consider the negative comments well founded.
  
  Citation: https://doi.org/10.5194/egusphere-2024-337-AC1

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

ED: Reconsider after major revisions (further review by editor and referees) (25 Jul 2024) by Mauro Giudici

AR by Scott K. Hansen on behalf of the Authors (25 Sep 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (27 Sep 2024) by Mauro Giudici

RR by Roseanna Neupauer (12 Nov 2024)

RR by Philippe Ackerer (22 Nov 2024)

Suggestions for revision or reasons for rejection

The paper addresses an interesting issue, not new but not solved until now. It is a paper dedicated to a new methodology using a parameterization based on Karhunen-Loève expansions. The methodology is tested with a synthetic ‘model aquifer’ under simplified conditions. Using synthetic data set avoids wrong interpretation of the methodology performance due to uncertainty and measurement errors that are unavoidable when using real test cases. Of course, it questioned the feasibility of the method, but this is another issue.

The paper is well written, the methodology well described and the results convincingly discussed.

Some comments that can be discussed:
- Since the exact heads and hydraulic conductivities are known over the all domain, it may be interesting to analyze the interactions between head and hydraulic conductivities (using a cross variogram for example). This information could be included in the discussion about the spacing/density of the measurements.
- I was surprised by the use of measured velocities. They cannot be measured, to my knowledge. In wells, flow rates are measured and the width of the captured zone is required to estimate the velocity. In the field, at least 3 piezometers are necessary to estimate the head gradient, but hydraulic conductivity is needed to compute an average velocity.
- The head sampling is based on a regular grid. I do not see the interest of the regular grid for a heterogeneous aquifer.
- Only 50 random fields are used. This is very low compared to the variance of the LnK and no reliable statistics can be drawn with such a number. What is the interest of these 50? Why not testing random fields with other properties (smaller LnK variance, lower integral scales, …).
- Optimization is stopped after 500 steps. Why? The stopping criterion should be based on the value of the objective function and/or it gradient. The number of required steps may be different, depending on the measurement density for example.
- Typo Line 302, 411

Hide

ED: Publish subject to revisions (further review by editor and referees) (26 Nov 2024) by Mauro Giudici

AR by Scott K. Hansen on behalf of the Authors (03 Dec 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (06 Dec 2024) by Mauro Giudici

RR by Roseanna Neupauer (26 Dec 2024)

RR by Philippe Ackerer (16 Jan 2025)

ED: Publish subject to technical corrections (16 Jan 2025) by Mauro Giudici

AR by Scott K. Hansen on behalf of the Authors (20 Jan 2025) Author's response Manuscript

Journal article(s) based on this preprint

24 Mar 2025

Feature scale and identifiability: how much information do point hydraulic measurements provide about heterogeneous head and conductivity fields?

Scott K. Hansen, Daniel O'Malley, and James P. Hambleton

Hydrol. Earth Syst. Sci., 29, 1569–1585, https://doi.org/10.5194/hess-29-1569-2025,https://doi.org/10.5194/hess-29-1569-2025, 2025

Short summary

Scott K. Hansen, Daniel O'Malley, and James P. Hambleton

Viewed

Total article views: 610 (including HTML, PDF, and XML)

HTML	PDF	XML	Total	BibTeX	EndNote
436	135	39	610	32	30

HTML: 436
PDF: 135
XML: 39
Total: 610
BibTeX: 32
EndNote: 30

Views and downloads (calculated since 11 Mar 2024)

Month	HTML	PDF	XML	Total
Mar 2024	133	34	9	176
Apr 2024	61	8	6	75
May 2024	17	9	3	29
Jun 2024	70	18	9	97
Jul 2024	58	16	9	83
Aug 2024	23	3	1	27
Sep 2024	16	5	0	21
Oct 2024	5	4	0	9
Nov 2024	9	6	0	15
Dec 2024	4	8	0	12
Jan 2025	9	7	1	17
Feb 2025	21	8	1	30
Mar 2025	10	9	0	19

Cumulative views and downloads (calculated since 11 Mar 2024)

Month	HTML	PDF	XML	Total
Mar 2024	133	34	9	176
Apr 2024	61	8	6	75
May 2024	17	9	3	29
Jun 2024	70	18	9	97
Jul 2024	58	16	9	83
Aug 2024	23	3	1	27
Sep 2024	16	5	0	21
Oct 2024	5	4	0	9
Nov 2024	9	6	0	15
Dec 2024	4	8	0	12
Jan 2025	9	7	1	17
Feb 2025	21	8	1	30
Mar 2025	10	9	0	19

Viewed (geographical distribution)

Total article views: 622 (including HTML, PDF, and XML) Thereof 622 with geography defined and 0 with unknown origin.

Country	#	Views	%

Latest update: 24 Mar 2025

Download

The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.

Preprint (972 KB)
Metadata XML

Short summary

We consider how well one can identify hydraulic conductivity that varies from place to place by using only measurements obtained at a finite number of groundwater monitoring wells. In particular, we relate how accurately features (meaning connected high- or low-conductivity regions) are identified to their size and to well spacing, and examine which kinds of information are most valuable. When feature size exceeds four times the well spacing, better-than-random identification is possible.


Total:	0
HTML:	0
PDF:	0
XML:	0