the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
AdaHRBF v1.0: GradientAdaptive HermiteBirkhoff Radial Basis Function Interpolants for Threedimensional Stratigraphic Implicit Modeling
Baoyi Zhang
Linze Du
Umair Khan
Yongqiang Tong
Lifang Wang
Hao Deng
Abstract. Threedimensional (3D) stratigraphic modelling is capable of modeling the shape, topology, and other properties of strata in a digitalized manner. The implicit modeling approach is becoming the mainstream approach for 3D stratigraphic modelling, which incorporates both the offcontact attitudes and the oncontact occurrence information of stratigraphic interface to estimate the stratigraphic potential field (SPF) to represent the 3D architectures of strata. However, the magnitudes of SPF gradient controlling variation trend of SPF values cannot be directly derived from the known stratigraphic attribute or attitude data. In this paper, we propose an HermiteBirkhoff radial basis function (HRBF) formulation, AdaHRBF, with an adaptive gradient magnitude for continuous 3D SPF modeling of multiple stratigraphic interfaces. In the linear system of HRBF interpolant constrained by the scattered oncontact attribute points and offcontact attitude points of a set of strata in 3D space, we add a novel optimizing term to iteratively obtain the true gradient magnitude. The case study shows that the HRBF interpolants can consistently establish accurate multiple stratigraphic interfaces and fully express the internal stratigraphic attribute and attitude. To ensure harmony of the variation of stratigraphic thickness, we adopt the relative burial depth of stratigraphic interface to the Quaternary as the SPF attribute value and propose a new stratigraphical thickness index (STI) to represent the variation trend of stratigraphic thickness in SPF. In addition, the proposed stratigraphic potential field modeling by HRBF interpolants can provide a suitable basic model for subsequent geosciences numerical simulation.

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Interactive discussion
Status: closed

RC1: 'Comment on egusphere20221304', Italo Goncalves, 17 Feb 2023
In this work, the authors introduce an iterative procedure to deal with the uncertainty in the gradient magnitudes of attitude data in implicit geological modeling, which represent local inverse thickness. The results are very solid, and I believe the manuscript is suitable for publication after the corrections pointed below.
Line 175: if the cubic function is the only one used, I see no reason the present other types in a table. You can point to a reference that lists them and save space. Also, in 3 dimensions the cubic function is the one that minimizes the curvature (eq. 1), the others do not necessarily do so. See Chapter 6 in Rasmussen and Williams (2006) and Wendland (2005).
Line 241: isn’t there a risk of obtaining a negative λ_{t+1} with eq. 8? Have you tried using something like λ_{t+1} = (l_{t}  l_{t1})^{2}? Also, I suppose the same update is applied to the 3 directions, but it is important to emphasize this.
If contact data is somehow unavailable or unreliable in part of the space, would the model be able to benefit from offcontact point data (such as indicators for stratum A, stratum B, etc.)? They could be useful to improve the classification accuracy close to the DTM where this information is available. See Hillier et al. (2014).
This might already be published elsewhere, but I could not see the difference between HRBF and basic RBF. The equations presented seem the same as the basic RBF equations seen in the books. Please clarify the difference and point to the reference that introduced HRBF.
A few points to improve the discussion. Please elaborate on how the present work compares to these (which have already been cited):
 This approach is very similar to von Harten et al. (2021), with the difference that here the diagonal matrix is applied to the gradients instead of the contacts.
 Gonçalves et al. (2017) use the strike and dip vectors as zerogradient directions in order to avoid assigning an arbitrary magnitude to the normals. Have you tried this approach?
 By extending the inequality constraints by Hillier et al. (2014) to the gradients, it is possible to obtain the same results presented here, in principle.
I did not find the STI to be very informative of the stratigraphic characteristics of the strata. It seems to be a little erratic and can vary from the minimum to maximum value within the same stratum, which seems to defeat its very purpose. Perhaps trying to assign a geological meaning to the gradients is not a good idea, as they can be very dependent of the specific data points that were used and are a result of the minimumcurvature characteristic of RBF. I think the manuscript would not suffer with the removal of this section.
Minor points:
The manuscript seems to suffer from a compilation error. See pages 6 and 7. All the R symbols are displaced.
Lines 59, 348: “true” gradient magnitudes seem a rather strong term. I would call it an optimized gradient.
Line 184: a line break after the semicolon would improve readability.
Line 200: it might be worth mentioning that θ_{2} = θ_{1} + 90º.
Figure 2: if the vectors g and n have the same direction but not necessarily the same magnitude, I think the figure could be improved by adding a vector n with a different length.
Line 212: please rephrase to avoid starting a new section with “however”.
Line 218: suppress “the”.
Figure 3: was it handdrawn or computed? A computed example might make the point clearer.
Line 231: it is worth emphasizing that n is a unit vector.
Figure 7: I think this example is unnecessary given the previous two, but I leave it at the authors’ discretion.
Figure 12: is this field value the burial depth that was mentioned before? How was it measured? Is it constant within a given contact?
Line 349: “The changes of gradient magnitude are shown…”
Line 475: “attitude”
References:
Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian processes for machine learning. MIT Press. https://doi.org/10.1142/S0129065704001899
Wendland, H. (2005). Scattered Data Approximation. Cambridge University Press.
Citation: https://doi.org/10.5194/egusphere20221304RC1 
AC1: 'Reply on RC1', Baoyi Zhang, 08 Mar 2023
Dear Dr. Italo Goncalves:
Thanks for your effort to review our manuscript titled " AdaHRBF v1.0: GradientAdaptive HermiteBirkhoff Radial Basis Function Interpolants for Threedimensional Stratigraphic Implicit Modeling ", and now we have just revised this manuscript according to your good suggestions. The details are in the attached document.
Best regards,
Baoyi Zhang and coauthors.

RC2: 'Comment on egusphere20221304', Lachlan Grose, 20 Feb 2023
The paper presents AdaHRBF (adaptive hermitebirkhoff radial basis function), a new interpolation method for building implicit geological models. The main contribution of this the iterative process for adapting the gradient of the implicit function to prevent artefacts due to inconsistent gradient magnitude norms.
The paper is generally well written with a logical structure and I believe that it is a good contribution towards the field. In general, the authors have referenced some of the appropriate literature however I believe a deeper analysis of the different implicit modelling techniques would improve the paper – for instance when reviewing discrete smooth interpolation that papers by Mallet 1980/1992 were presented as implicit methods where these papers actually discuss the method applied to 2D surfaces.
The authors should thoroughly review the paper “ThreeDimensional Modelling of Geological Surfaces Using Generalized Interpolation with Radial BasisFunctions” as there are a lot of parallels with the presented works that seem to be missed.
If the method is presented as an approach to tackle the issues with fold geometries it would be worthwhile reviewing the relevant literature around fold modelling: Laurent et al., 2016, Grose et al., 2017,2019, 2020, Hillier et al., 2014.
Don’t change between strike and dip data and attitude data, keep it consistent. Preferably something meaningful for geologists.
What is wrong with the formatting of the pages with equations? All of the equations need to be carefully checked to ensure that they are readable and there are no extra symbols!
The figure captions are brief and difficult to follow. They should be stand alone and provide a description and brief interpretation of the contents.
I am not sure what the stratigraphic index adds, it is hard to interpret. For example figure 23 section 14 near D1y there is an odd geometry. What causes this? It is orthogonal to the expected orientation of stratigraphy?
Major comments:
From reading the manuscript the main contribution is the ability of the interpolator to adapt to variations in the magnitude of the gradient norm. This is a problem that was discussed by Gautier Laurent in doi: 10.1007/s110040169637y but his paper does not appear to be referenced. In this paper an iterative approach for updating the gradient norm was presented in a discrete modelling approach. This paper should be discussed and compared with the work presented here, I would strongly encourage the authors to provide a comparison between the two methods.
I am not convinced that adding the constant to the diagonal component of the second derivative matrix actually changes the magnitude of the gradient norm. I believe that it will just allow for a larger misfit between the orientation observations and the implicit function – which will have the same result but means that any information in the gradient direction will not be incorporated. If the method is actually just removing outlier data then should the message of the paper be changed to this rather than for adapting the magnitude of the gradient norm?
I would be interested in seeing a comparison between this method and a discrete approach where the regularisation contribution can locally change, see LoopStructural paper in GMD for a comparison between discrete interpolation and RBF. I would also be interested in seeing the model without any orientation data and just interpolating from the contact locations and also when constraining the direction of the gradient using tangent constraints.
Details comments:
Line 26: replace significance with importance
Line 29: delete “and has garnered extensive attention from geologists”
Line 30: Implicit/explicit definition should refer to the approaches as ways of representing surfaces not as methods for building models
Line 3946: A lot of the mentioned studies are not reliant on implicit modelling, you could do the same thing with explicit models. E.g. implicit function is not combined directly with geophysics the model is discretized first which means it could be replaced with a model defined by explicit surfaces. Same point for uncertainty analysis. I would replace this section with relevant references to implicit modelling not just a list of all studies that use implicit modelling
Line 48: delete “in HRBF method”, you can do the same in discrete as well
Line 60: What was adaHRBF method compared with? It is presented as better than an alternative
Line 63: “Distribution of attribute” what do you refer to here it is unclear
Line 70 73: Mallet reference is for DSI applied to nodes of triangular surface, do you mean to reference mallet 2004 or frank 2007/ caumon 2013?
Line 7680: This paragraph started as discrete modelling and then jumps to rbf methods, perhaps keeping them separate will make it easier for the reader
Line 9294: Renaudeau and Irakarma are discrete or somewhat discrete methods
Line 118121: “Moreover, RBF/HRBFbased methods construct implicit field functions separately for each geological interface and extract the zero value equipotential surfaces to locate the geological interface. Therefore, it is difficult to maintain topological consistency between geological bodies, let alone to represent their internal attributes and structural attitudes.”
This is not true, the surfe library by Hillier et al 2014 can do all of these points…
Line 126: Reference first sentence of paragraph
Line 126: What do you mean by geological maps? Do you mean the outcrop pattern of contacts?
Line 130: Change annotation of f1/f2 to something that can’t be confused with faults
Line 153: optional? Should be optimal?
Line 162: Can you not change the order of the polynomial trend? Hillier et al can?
Line 165: define the meaning of f*
Line 174178: delete table and reference to other basis functions. If you only include r3 then why introduce the others. You could refer the reader to Hillier et al
Line 179180: The explanation of the construction of the matrices is not clear, it is not obvious what each component represents. Either leave this information for supplementary material if its not necessary for understanding or add more explanation about the different terms.
Line 196: “added into modelling process” add references to all of the work that already does this e.g. hillier et al 2014, caumon 2013 etc
Line 212: “it is difficult to obtain the gradient magnitude through any geological observation.” Delete
Line 222: “the same gradient magnitudes” what do the red circles indicate
Line 224: “to the Eq. 4” change to “to Eq. 4”
Line 224: “used” replace with “and used”
Line 227: There are similarities to this diagonal block to the smoothing parameter in Surfe – this was used in Grose et al., 2020 to show a comparison between smoothing regularisation in rbf to the regularisation used by discrete interpolation
Line 238239: My understanding of adding a constraint to the diagonal of the matrix is it allows for the interpolant to have a larger misfit to the constraint. So this means that by iteratively adjusting the diagonal for specific gradient constraint you are actually changing how well those constraints are honoured by the interpolant which includes not just the gradient magnitude but also the orientation constraint.
Line 247: replace convergency with “ when convergence is reached”
Line 268: “distribution of attribute and attitude points;”
Are the attribute points constraining the value of the implicit field or are they "interface" points as per calcagno where they set the implicit field to be constant along all points related to a single contact? I don't understand how if the points aren't constraining the value of the implicit field this results in a scalar field with the same range as the original dataset when the gradient norms are unit vectors.
Line 330: explain the cross section more
Line 336: “attitude points”
Personally I don't think attitude points speaks to me as a geologist, could you refer to orientation observations or structural observations. At least make it consistent with the geological map, you have angle of strike and dip vector as the legend
Line 358: “he” replace with “the”
Line 470: “However, existing RBF and HRBF interpolants implicitly reconstruct a single geological interface and extract it as the zerovalue equipotential surface.”
Not true, read and reference Hillier et al., 2014
Line 472: “Moreover, existing RBF and HRBF interpolants need several independent scalar fields to simulate geological interfaces”
This is also not true, Hillier et al use a single scalar field. You also use several scalar field because you represent each fault block independently. If you see the fault modelling method in Grose et al., 2021, this can be used with Surfe and would allow for a single scalar field for stratigraphy.
Line 484: “they are incapable of interpolating or extrapolating a fold series within a continuous structural style” this point by Jessell 2014 was addressed by a few publications Laurent et al., 2016, and Grose et al., 2017,2018,2019,2020 as well as Hillier et al., 2014
Line 486489: How have you addressed point 1)? You don't extrapolate a fold series in this manuscript, you interpolate a fold shape from gradient constraints but that is not the same. I would remove this section as it is not consistent with the literature.
Line 490496: This section makes no sense, needs revisiting.
I don't see using the burial depth as being a new contribution from this paper, it is the same method used by various authors Caumon 2013, Hillier 2014, Grose et al., 2020
Line 505: “Because 3D stratigraphic potential fields can be coupled with various geoscience numerical simulation methods, they have a broad prospect for application in related fields such as metallogenic prediction.” Delete or move to discussion, don’t introduce a new idea in the conclusion
Line 510: “A goal for future work is to introduce a drift function in the model to accommodate discontinuity of fault planes. In addition, the uncertainty of the model should be considered in the modeling process, and additional geophysical exploration data and geological interpretation should
be incorporated into the modeling constraints.”
Move to discussion, but also please ensure that you reference the limitations of a drift function.
Citation: https://doi.org/10.5194/egusphere20221304RC2 
AC3: 'Reply on RC2', Baoyi Zhang, 27 Mar 2023
Dear Dr. Lachlan Grose:
Thanks for your effort to review our manuscript titled " AdaHRBF v1.0: GradientAdaptive HermiteBirkhoff Radial Basis Function Interpolants for Threedimensional Stratigraphic Implicit Modeling ", and now we have just revised this manuscript according to your good suggestions. The details are in the attached document.
Best regards,
Baoyi Zhang and coauthors.

AC3: 'Reply on RC2', Baoyi Zhang, 27 Mar 2023

RC3: 'Comment on egusphere20221304', Michal Michalak, 03 Mar 2023
I decided to review this paper from a more optimization perspective since authors use "optimization" terms very often throughout the manuscript. And my field is rather data science in geology where optimization problems are common. But I'm glad to see proper implicit interpolation guys among reviewers who can better evaluate the contribution to their field.
My general opinion is that in the present form it is difficult to evaluate the contribution because it seems that the key methods are not referenced properly. For example, a textbook about Functional Analysis (pure mathematics) is referenced to support applications of interpolation concepts in geology  an unlikely source of information, where in fact there already exist very specific papers about HermiteBirkhoff interpolation. Moreover, the authors use standard terms (optimization) in a nonstandard context which makes the paper difficult to read: I am looking for an optimization criterion but I can't find it.
A positive note: the paper has a logical structure and a neglible overlap with previous work of the author.
Line 20: What do you mean by optimization? Optimization is usually considered either as minimizing something bad (e.g. misfit function) or maximizing something good (e.g. profit). The Wikipedia definition says: "mathematical optimization is the selection of a best element, with regard to some criterion, from some set of available alternatives". Despite many occurrences of "optimization" throughout the manuscript, I cannot find a criterion that is optimized. Instead, I can hypothesize that by "optimizing" the authors mean learning the true value of something. So I would argue that this research is not about optimization. If the authors do not agree, I would like to see an explanation of:
1) why should we consider the results obtained by authors as optimal, i.e. why any other candidate solution is worse than the results proposed by authors in relation to some criterion
2) a thorough description of methods assumed to give optimal resultsAs a side note, I was once requested by a reviewer regarding why calculating eigenvectors from an orientation matrix should give optimal results. You can see how it was done in the 4.4.2 section of the below paper (the content rather irrelevant for your paper):
Michalak, M. P., Kuzak, R., Gładki, P., Kulawik, A., & Ge, Y. (2021). Constraining uncertainty of fault orientation using a combinatorial algorithm. Computers and Geosciences, 154, 104777.
(https://doi.org/10.1016/j.cageo.2021.104777) , or here (free access): https://github.com/michalmichalak997/3GeoCombine/blob/master/Michalak_2021_combinatorial_accepted.pdf
Line 36: "Speed"  when I build a triangulated surface using 800 points, it doesn't take more than two seconds. But when I try to do a similar thing using interpolation methods in GemPy, it takes really long  so I would argue that speed may not be the best marketing candidate for implicit interpolation methods. Moreover, in cokriging methods it is not enough to add surface points  you need to add 3D orientations. But if it is a subsurface terrain, then how do you get an independent orientation measurement? To sum up, I would like to see a discussion about limitations of implicit methods.
Line 48: This is the first occurrence of HRBF in the manuscript so it should be preceded with full name. However, here you point to some weaknesses of HRBF and in line 57 you propose HRBF as your main contribution. I'm confused with this presentation.
Line 57: what is actually HermiteBirkhoff interpolation? The concept should be explained. In the paper, I can see only one reference (except rather inadequate one about functional analysis) about using Hermite interpolation theory in geology (Wang et al. 2018). I would say that the referenced paper better presents the foundational aspect of the method.
Lines 155157: I can see three components of the energy function E (two sums and one integral).
1) What do these components represent and how they can be interpreted?
2) I can see that a textbook about functional analysis is referenced to support the equation (Eq.1). Where exactly in this book did you find information about "minimizing smoothness and unevenness of the energy function"? It seems that it is a general mathematical textbook so I would be surprised to see there notions such as "energy function of stratigraphic potential field" or "degree of unevenness". In fact, I have checked the 1972 edition of the Bachman&Narici book and I could not find such concepts. If you found them in 2000 edition, please provide a scan.
3) can you reference other works where Eq. 1 is used?Line 511: does your work address the problem of subjectivity in implicit methods mentioned by Grose et al. 2021 (text below)? Please discuss.
"The fundamental reasoning behind our approach is that the subjective constraints that are required to capture the geological features with standard implicit algorithms will be one of the greatest sources of uncertainty in the model." (https://doi.org/10.5194/gmd1439152021)
Title: I would suggest to change the title so that it presents the main value of the research. As of now, the first part of the title contains some technical terms but in my opinion it should point to the added value for threedimensional stratigraphic implicit modelling. So if it is optimization, then I would like to see the reflection of optimization in the title.
Citation: https://doi.org/10.5194/egusphere20221304RC3 
AC4: 'Reply on RC3', Baoyi Zhang, 28 Mar 2023
Dear Dr. Michal Michalak:
Thanks for your effort to review our manuscript titled " AdaHRBF v1.0: GradientAdaptive HermiteBirkhoff Radial Basis Function Interpolants for Threedimensional Stratigraphic Implicit Modeling ", and now we have just revised this manuscript according to your good suggestions. The details are in the attached document.
Best regards,
Baoyi Zhang and coauthors.

AC4: 'Reply on RC3', Baoyi Zhang, 28 Mar 2023

CC1: 'Comment on egusphere20221304', Michael Hillier, 17 Mar 2023
I won't make any detailed comments as the other reviewers did a great job. However, I would like to post a few comments quickly.
While our earlier GRBF paper (Hillier et al. 2014) did not model multiple conformal stratigraphic horizons simultaneously, it can be easily adapted to do so as we showed in Hillier et al. 2016. All that is needed is the increments idea of Lajaunie et al. 1997. All these functionalities and more are developed in the open source library (surfe)  https://github.com/MichaelHillier/surfe. Therefore, statements like "...existing
RBF and HRBF interpolants need several independent scalar fields to simulate geological interfaces..." are wrong.Below is the reference.
Hillier, M., de Kemp, E. A., and Schetselaar, E. M.: Implicitly modelled stratigraphic surfaces using generalized
interpolation, in: AIP conference proceedings, 1738, 050004, International Conference of Numerical Analysis and
Applied Mathematics, 22–28 September 2015, Rhodes, Greece, https://doi.org/10.1063/1.4951819, 2016.
Also, I would consider this work more as generalized interpolation than HermiteBirkhoff interpolation (see Wendland 2005, Macêdo 2011).Citation: https://doi.org/10.5194/egusphere20221304CC1 
EC1: 'Reply on CC1', Thomas Poulet, 20 Mar 2023
Thank you Michael for the information.
Best regards,
Thomas Poulet.
Citation: https://doi.org/10.5194/egusphere20221304EC1 
AC2: 'Reply on CC1', Baoyi Zhang, 20 Mar 2023
Dear Michael Hillier:
Thanks for your effort to review our manuscript, and we will revise our manuscript according to your suggestions.
CC1: 'Comment on egusphere20221304', Michael Hillier, 17 Mar 2023
I won't make any detailed comments as the other reviewers did a great job. However, I would like to post a few comments quickly.
While our earlier GRBF paper (Hillier et al. 2014) did not model multiple conformal stratigraphic horizons simultaneously, it can be easily adapted to do so as we showed in Hillier et al. 2016. All that is needed is the increments idea of Lajaunie et al. 1997. All these functionalities and more are developed in the open source library (surfe)  https://github.com/MichaelHillier/surfe. Therefore, statements like "...existing RBF and HRBF interpolants need several independent scalar fields to simulate geological interfaces..." are wrong.
Below is the reference.
Hillier, M., de Kemp, E. A., and Schetselaar, E. M.: Implicitly modelled stratigraphic surfaces using generalized interpolation, in: AIP conference proceedings, 1738, 050004, International Conference of Numerical Analysis and Applied Mathematics, 22–28 September 2015, Rhodes, Greece, https://doi.org/10.1063/1.4951819, 2016.
Response: Thank you for providing your excellent reference (Hillier et al. 2016) and open source library (surfe), and we will revise our statement comparing to your works.
Also, I would consider this work more as generalized interpolation than HermiteBirkhoff interpolation (see Wendland 2005, Macêdo 2011).Response: Besides constraints of scattered multivariate HermiteBirkhoff data, the Generalized RBF, proposed by you in (Hiller et al. 2014), reconstructs an implicit function with more constraints of lithologic markers (inequality) and lineations (tangent). A goal for future work is to integrate these constraints in our solution to utilize more kinds of modeling data.
Citation: https://doi.org/10.5194/egusphere20221304AC2

EC1: 'Reply on CC1', Thomas Poulet, 20 Mar 2023
Interactive discussion
Status: closed

RC1: 'Comment on egusphere20221304', Italo Goncalves, 17 Feb 2023
In this work, the authors introduce an iterative procedure to deal with the uncertainty in the gradient magnitudes of attitude data in implicit geological modeling, which represent local inverse thickness. The results are very solid, and I believe the manuscript is suitable for publication after the corrections pointed below.
Line 175: if the cubic function is the only one used, I see no reason the present other types in a table. You can point to a reference that lists them and save space. Also, in 3 dimensions the cubic function is the one that minimizes the curvature (eq. 1), the others do not necessarily do so. See Chapter 6 in Rasmussen and Williams (2006) and Wendland (2005).
Line 241: isn’t there a risk of obtaining a negative λ_{t+1} with eq. 8? Have you tried using something like λ_{t+1} = (l_{t}  l_{t1})^{2}? Also, I suppose the same update is applied to the 3 directions, but it is important to emphasize this.
If contact data is somehow unavailable or unreliable in part of the space, would the model be able to benefit from offcontact point data (such as indicators for stratum A, stratum B, etc.)? They could be useful to improve the classification accuracy close to the DTM where this information is available. See Hillier et al. (2014).
This might already be published elsewhere, but I could not see the difference between HRBF and basic RBF. The equations presented seem the same as the basic RBF equations seen in the books. Please clarify the difference and point to the reference that introduced HRBF.
A few points to improve the discussion. Please elaborate on how the present work compares to these (which have already been cited):
 This approach is very similar to von Harten et al. (2021), with the difference that here the diagonal matrix is applied to the gradients instead of the contacts.
 Gonçalves et al. (2017) use the strike and dip vectors as zerogradient directions in order to avoid assigning an arbitrary magnitude to the normals. Have you tried this approach?
 By extending the inequality constraints by Hillier et al. (2014) to the gradients, it is possible to obtain the same results presented here, in principle.
I did not find the STI to be very informative of the stratigraphic characteristics of the strata. It seems to be a little erratic and can vary from the minimum to maximum value within the same stratum, which seems to defeat its very purpose. Perhaps trying to assign a geological meaning to the gradients is not a good idea, as they can be very dependent of the specific data points that were used and are a result of the minimumcurvature characteristic of RBF. I think the manuscript would not suffer with the removal of this section.
Minor points:
The manuscript seems to suffer from a compilation error. See pages 6 and 7. All the R symbols are displaced.
Lines 59, 348: “true” gradient magnitudes seem a rather strong term. I would call it an optimized gradient.
Line 184: a line break after the semicolon would improve readability.
Line 200: it might be worth mentioning that θ_{2} = θ_{1} + 90º.
Figure 2: if the vectors g and n have the same direction but not necessarily the same magnitude, I think the figure could be improved by adding a vector n with a different length.
Line 212: please rephrase to avoid starting a new section with “however”.
Line 218: suppress “the”.
Figure 3: was it handdrawn or computed? A computed example might make the point clearer.
Line 231: it is worth emphasizing that n is a unit vector.
Figure 7: I think this example is unnecessary given the previous two, but I leave it at the authors’ discretion.
Figure 12: is this field value the burial depth that was mentioned before? How was it measured? Is it constant within a given contact?
Line 349: “The changes of gradient magnitude are shown…”
Line 475: “attitude”
References:
Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian processes for machine learning. MIT Press. https://doi.org/10.1142/S0129065704001899
Wendland, H. (2005). Scattered Data Approximation. Cambridge University Press.
Citation: https://doi.org/10.5194/egusphere20221304RC1 
AC1: 'Reply on RC1', Baoyi Zhang, 08 Mar 2023
Dear Dr. Italo Goncalves:
Thanks for your effort to review our manuscript titled " AdaHRBF v1.0: GradientAdaptive HermiteBirkhoff Radial Basis Function Interpolants for Threedimensional Stratigraphic Implicit Modeling ", and now we have just revised this manuscript according to your good suggestions. The details are in the attached document.
Best regards,
Baoyi Zhang and coauthors.

RC2: 'Comment on egusphere20221304', Lachlan Grose, 20 Feb 2023
The paper presents AdaHRBF (adaptive hermitebirkhoff radial basis function), a new interpolation method for building implicit geological models. The main contribution of this the iterative process for adapting the gradient of the implicit function to prevent artefacts due to inconsistent gradient magnitude norms.
The paper is generally well written with a logical structure and I believe that it is a good contribution towards the field. In general, the authors have referenced some of the appropriate literature however I believe a deeper analysis of the different implicit modelling techniques would improve the paper – for instance when reviewing discrete smooth interpolation that papers by Mallet 1980/1992 were presented as implicit methods where these papers actually discuss the method applied to 2D surfaces.
The authors should thoroughly review the paper “ThreeDimensional Modelling of Geological Surfaces Using Generalized Interpolation with Radial BasisFunctions” as there are a lot of parallels with the presented works that seem to be missed.
If the method is presented as an approach to tackle the issues with fold geometries it would be worthwhile reviewing the relevant literature around fold modelling: Laurent et al., 2016, Grose et al., 2017,2019, 2020, Hillier et al., 2014.
Don’t change between strike and dip data and attitude data, keep it consistent. Preferably something meaningful for geologists.
What is wrong with the formatting of the pages with equations? All of the equations need to be carefully checked to ensure that they are readable and there are no extra symbols!
The figure captions are brief and difficult to follow. They should be stand alone and provide a description and brief interpretation of the contents.
I am not sure what the stratigraphic index adds, it is hard to interpret. For example figure 23 section 14 near D1y there is an odd geometry. What causes this? It is orthogonal to the expected orientation of stratigraphy?
Major comments:
From reading the manuscript the main contribution is the ability of the interpolator to adapt to variations in the magnitude of the gradient norm. This is a problem that was discussed by Gautier Laurent in doi: 10.1007/s110040169637y but his paper does not appear to be referenced. In this paper an iterative approach for updating the gradient norm was presented in a discrete modelling approach. This paper should be discussed and compared with the work presented here, I would strongly encourage the authors to provide a comparison between the two methods.
I am not convinced that adding the constant to the diagonal component of the second derivative matrix actually changes the magnitude of the gradient norm. I believe that it will just allow for a larger misfit between the orientation observations and the implicit function – which will have the same result but means that any information in the gradient direction will not be incorporated. If the method is actually just removing outlier data then should the message of the paper be changed to this rather than for adapting the magnitude of the gradient norm?
I would be interested in seeing a comparison between this method and a discrete approach where the regularisation contribution can locally change, see LoopStructural paper in GMD for a comparison between discrete interpolation and RBF. I would also be interested in seeing the model without any orientation data and just interpolating from the contact locations and also when constraining the direction of the gradient using tangent constraints.
Details comments:
Line 26: replace significance with importance
Line 29: delete “and has garnered extensive attention from geologists”
Line 30: Implicit/explicit definition should refer to the approaches as ways of representing surfaces not as methods for building models
Line 3946: A lot of the mentioned studies are not reliant on implicit modelling, you could do the same thing with explicit models. E.g. implicit function is not combined directly with geophysics the model is discretized first which means it could be replaced with a model defined by explicit surfaces. Same point for uncertainty analysis. I would replace this section with relevant references to implicit modelling not just a list of all studies that use implicit modelling
Line 48: delete “in HRBF method”, you can do the same in discrete as well
Line 60: What was adaHRBF method compared with? It is presented as better than an alternative
Line 63: “Distribution of attribute” what do you refer to here it is unclear
Line 70 73: Mallet reference is for DSI applied to nodes of triangular surface, do you mean to reference mallet 2004 or frank 2007/ caumon 2013?
Line 7680: This paragraph started as discrete modelling and then jumps to rbf methods, perhaps keeping them separate will make it easier for the reader
Line 9294: Renaudeau and Irakarma are discrete or somewhat discrete methods
Line 118121: “Moreover, RBF/HRBFbased methods construct implicit field functions separately for each geological interface and extract the zero value equipotential surfaces to locate the geological interface. Therefore, it is difficult to maintain topological consistency between geological bodies, let alone to represent their internal attributes and structural attitudes.”
This is not true, the surfe library by Hillier et al 2014 can do all of these points…
Line 126: Reference first sentence of paragraph
Line 126: What do you mean by geological maps? Do you mean the outcrop pattern of contacts?
Line 130: Change annotation of f1/f2 to something that can’t be confused with faults
Line 153: optional? Should be optimal?
Line 162: Can you not change the order of the polynomial trend? Hillier et al can?
Line 165: define the meaning of f*
Line 174178: delete table and reference to other basis functions. If you only include r3 then why introduce the others. You could refer the reader to Hillier et al
Line 179180: The explanation of the construction of the matrices is not clear, it is not obvious what each component represents. Either leave this information for supplementary material if its not necessary for understanding or add more explanation about the different terms.
Line 196: “added into modelling process” add references to all of the work that already does this e.g. hillier et al 2014, caumon 2013 etc
Line 212: “it is difficult to obtain the gradient magnitude through any geological observation.” Delete
Line 222: “the same gradient magnitudes” what do the red circles indicate
Line 224: “to the Eq. 4” change to “to Eq. 4”
Line 224: “used” replace with “and used”
Line 227: There are similarities to this diagonal block to the smoothing parameter in Surfe – this was used in Grose et al., 2020 to show a comparison between smoothing regularisation in rbf to the regularisation used by discrete interpolation
Line 238239: My understanding of adding a constraint to the diagonal of the matrix is it allows for the interpolant to have a larger misfit to the constraint. So this means that by iteratively adjusting the diagonal for specific gradient constraint you are actually changing how well those constraints are honoured by the interpolant which includes not just the gradient magnitude but also the orientation constraint.
Line 247: replace convergency with “ when convergence is reached”
Line 268: “distribution of attribute and attitude points;”
Are the attribute points constraining the value of the implicit field or are they "interface" points as per calcagno where they set the implicit field to be constant along all points related to a single contact? I don't understand how if the points aren't constraining the value of the implicit field this results in a scalar field with the same range as the original dataset when the gradient norms are unit vectors.
Line 330: explain the cross section more
Line 336: “attitude points”
Personally I don't think attitude points speaks to me as a geologist, could you refer to orientation observations or structural observations. At least make it consistent with the geological map, you have angle of strike and dip vector as the legend
Line 358: “he” replace with “the”
Line 470: “However, existing RBF and HRBF interpolants implicitly reconstruct a single geological interface and extract it as the zerovalue equipotential surface.”
Not true, read and reference Hillier et al., 2014
Line 472: “Moreover, existing RBF and HRBF interpolants need several independent scalar fields to simulate geological interfaces”
This is also not true, Hillier et al use a single scalar field. You also use several scalar field because you represent each fault block independently. If you see the fault modelling method in Grose et al., 2021, this can be used with Surfe and would allow for a single scalar field for stratigraphy.
Line 484: “they are incapable of interpolating or extrapolating a fold series within a continuous structural style” this point by Jessell 2014 was addressed by a few publications Laurent et al., 2016, and Grose et al., 2017,2018,2019,2020 as well as Hillier et al., 2014
Line 486489: How have you addressed point 1)? You don't extrapolate a fold series in this manuscript, you interpolate a fold shape from gradient constraints but that is not the same. I would remove this section as it is not consistent with the literature.
Line 490496: This section makes no sense, needs revisiting.
I don't see using the burial depth as being a new contribution from this paper, it is the same method used by various authors Caumon 2013, Hillier 2014, Grose et al., 2020
Line 505: “Because 3D stratigraphic potential fields can be coupled with various geoscience numerical simulation methods, they have a broad prospect for application in related fields such as metallogenic prediction.” Delete or move to discussion, don’t introduce a new idea in the conclusion
Line 510: “A goal for future work is to introduce a drift function in the model to accommodate discontinuity of fault planes. In addition, the uncertainty of the model should be considered in the modeling process, and additional geophysical exploration data and geological interpretation should
be incorporated into the modeling constraints.”
Move to discussion, but also please ensure that you reference the limitations of a drift function.
Citation: https://doi.org/10.5194/egusphere20221304RC2 
AC3: 'Reply on RC2', Baoyi Zhang, 27 Mar 2023
Dear Dr. Lachlan Grose:
Thanks for your effort to review our manuscript titled " AdaHRBF v1.0: GradientAdaptive HermiteBirkhoff Radial Basis Function Interpolants for Threedimensional Stratigraphic Implicit Modeling ", and now we have just revised this manuscript according to your good suggestions. The details are in the attached document.
Best regards,
Baoyi Zhang and coauthors.

AC3: 'Reply on RC2', Baoyi Zhang, 27 Mar 2023

RC3: 'Comment on egusphere20221304', Michal Michalak, 03 Mar 2023
I decided to review this paper from a more optimization perspective since authors use "optimization" terms very often throughout the manuscript. And my field is rather data science in geology where optimization problems are common. But I'm glad to see proper implicit interpolation guys among reviewers who can better evaluate the contribution to their field.
My general opinion is that in the present form it is difficult to evaluate the contribution because it seems that the key methods are not referenced properly. For example, a textbook about Functional Analysis (pure mathematics) is referenced to support applications of interpolation concepts in geology  an unlikely source of information, where in fact there already exist very specific papers about HermiteBirkhoff interpolation. Moreover, the authors use standard terms (optimization) in a nonstandard context which makes the paper difficult to read: I am looking for an optimization criterion but I can't find it.
A positive note: the paper has a logical structure and a neglible overlap with previous work of the author.
Line 20: What do you mean by optimization? Optimization is usually considered either as minimizing something bad (e.g. misfit function) or maximizing something good (e.g. profit). The Wikipedia definition says: "mathematical optimization is the selection of a best element, with regard to some criterion, from some set of available alternatives". Despite many occurrences of "optimization" throughout the manuscript, I cannot find a criterion that is optimized. Instead, I can hypothesize that by "optimizing" the authors mean learning the true value of something. So I would argue that this research is not about optimization. If the authors do not agree, I would like to see an explanation of:
1) why should we consider the results obtained by authors as optimal, i.e. why any other candidate solution is worse than the results proposed by authors in relation to some criterion
2) a thorough description of methods assumed to give optimal resultsAs a side note, I was once requested by a reviewer regarding why calculating eigenvectors from an orientation matrix should give optimal results. You can see how it was done in the 4.4.2 section of the below paper (the content rather irrelevant for your paper):
Michalak, M. P., Kuzak, R., Gładki, P., Kulawik, A., & Ge, Y. (2021). Constraining uncertainty of fault orientation using a combinatorial algorithm. Computers and Geosciences, 154, 104777.
(https://doi.org/10.1016/j.cageo.2021.104777) , or here (free access): https://github.com/michalmichalak997/3GeoCombine/blob/master/Michalak_2021_combinatorial_accepted.pdf
Line 36: "Speed"  when I build a triangulated surface using 800 points, it doesn't take more than two seconds. But when I try to do a similar thing using interpolation methods in GemPy, it takes really long  so I would argue that speed may not be the best marketing candidate for implicit interpolation methods. Moreover, in cokriging methods it is not enough to add surface points  you need to add 3D orientations. But if it is a subsurface terrain, then how do you get an independent orientation measurement? To sum up, I would like to see a discussion about limitations of implicit methods.
Line 48: This is the first occurrence of HRBF in the manuscript so it should be preceded with full name. However, here you point to some weaknesses of HRBF and in line 57 you propose HRBF as your main contribution. I'm confused with this presentation.
Line 57: what is actually HermiteBirkhoff interpolation? The concept should be explained. In the paper, I can see only one reference (except rather inadequate one about functional analysis) about using Hermite interpolation theory in geology (Wang et al. 2018). I would say that the referenced paper better presents the foundational aspect of the method.
Lines 155157: I can see three components of the energy function E (two sums and one integral).
1) What do these components represent and how they can be interpreted?
2) I can see that a textbook about functional analysis is referenced to support the equation (Eq.1). Where exactly in this book did you find information about "minimizing smoothness and unevenness of the energy function"? It seems that it is a general mathematical textbook so I would be surprised to see there notions such as "energy function of stratigraphic potential field" or "degree of unevenness". In fact, I have checked the 1972 edition of the Bachman&Narici book and I could not find such concepts. If you found them in 2000 edition, please provide a scan.
3) can you reference other works where Eq. 1 is used?Line 511: does your work address the problem of subjectivity in implicit methods mentioned by Grose et al. 2021 (text below)? Please discuss.
"The fundamental reasoning behind our approach is that the subjective constraints that are required to capture the geological features with standard implicit algorithms will be one of the greatest sources of uncertainty in the model." (https://doi.org/10.5194/gmd1439152021)
Title: I would suggest to change the title so that it presents the main value of the research. As of now, the first part of the title contains some technical terms but in my opinion it should point to the added value for threedimensional stratigraphic implicit modelling. So if it is optimization, then I would like to see the reflection of optimization in the title.
Citation: https://doi.org/10.5194/egusphere20221304RC3 
AC4: 'Reply on RC3', Baoyi Zhang, 28 Mar 2023
Dear Dr. Michal Michalak:
Thanks for your effort to review our manuscript titled " AdaHRBF v1.0: GradientAdaptive HermiteBirkhoff Radial Basis Function Interpolants for Threedimensional Stratigraphic Implicit Modeling ", and now we have just revised this manuscript according to your good suggestions. The details are in the attached document.
Best regards,
Baoyi Zhang and coauthors.

AC4: 'Reply on RC3', Baoyi Zhang, 28 Mar 2023

CC1: 'Comment on egusphere20221304', Michael Hillier, 17 Mar 2023
I won't make any detailed comments as the other reviewers did a great job. However, I would like to post a few comments quickly.
While our earlier GRBF paper (Hillier et al. 2014) did not model multiple conformal stratigraphic horizons simultaneously, it can be easily adapted to do so as we showed in Hillier et al. 2016. All that is needed is the increments idea of Lajaunie et al. 1997. All these functionalities and more are developed in the open source library (surfe)  https://github.com/MichaelHillier/surfe. Therefore, statements like "...existing
RBF and HRBF interpolants need several independent scalar fields to simulate geological interfaces..." are wrong.Below is the reference.
Hillier, M., de Kemp, E. A., and Schetselaar, E. M.: Implicitly modelled stratigraphic surfaces using generalized
interpolation, in: AIP conference proceedings, 1738, 050004, International Conference of Numerical Analysis and
Applied Mathematics, 22–28 September 2015, Rhodes, Greece, https://doi.org/10.1063/1.4951819, 2016.
Also, I would consider this work more as generalized interpolation than HermiteBirkhoff interpolation (see Wendland 2005, Macêdo 2011).Citation: https://doi.org/10.5194/egusphere20221304CC1 
EC1: 'Reply on CC1', Thomas Poulet, 20 Mar 2023
Thank you Michael for the information.
Best regards,
Thomas Poulet.
Citation: https://doi.org/10.5194/egusphere20221304EC1 
AC2: 'Reply on CC1', Baoyi Zhang, 20 Mar 2023
Dear Michael Hillier:
Thanks for your effort to review our manuscript, and we will revise our manuscript according to your suggestions.
CC1: 'Comment on egusphere20221304', Michael Hillier, 17 Mar 2023
I won't make any detailed comments as the other reviewers did a great job. However, I would like to post a few comments quickly.
While our earlier GRBF paper (Hillier et al. 2014) did not model multiple conformal stratigraphic horizons simultaneously, it can be easily adapted to do so as we showed in Hillier et al. 2016. All that is needed is the increments idea of Lajaunie et al. 1997. All these functionalities and more are developed in the open source library (surfe)  https://github.com/MichaelHillier/surfe. Therefore, statements like "...existing RBF and HRBF interpolants need several independent scalar fields to simulate geological interfaces..." are wrong.
Below is the reference.
Hillier, M., de Kemp, E. A., and Schetselaar, E. M.: Implicitly modelled stratigraphic surfaces using generalized interpolation, in: AIP conference proceedings, 1738, 050004, International Conference of Numerical Analysis and Applied Mathematics, 22–28 September 2015, Rhodes, Greece, https://doi.org/10.1063/1.4951819, 2016.
Response: Thank you for providing your excellent reference (Hillier et al. 2016) and open source library (surfe), and we will revise our statement comparing to your works.
Also, I would consider this work more as generalized interpolation than HermiteBirkhoff interpolation (see Wendland 2005, Macêdo 2011).Response: Besides constraints of scattered multivariate HermiteBirkhoff data, the Generalized RBF, proposed by you in (Hiller et al. 2014), reconstructs an implicit function with more constraints of lithologic markers (inequality) and lineations (tangent). A goal for future work is to integrate these constraints in our solution to utilize more kinds of modeling data.
Citation: https://doi.org/10.5194/egusphere20221304AC2

EC1: 'Reply on CC1', Thomas Poulet, 20 Mar 2023
Peer review completion
Journal article(s) based on this preprint
Baoyi Zhang et al.
Data sets
AdaHRBF Zhang et al. https://github.com/csugeo3d/AdaHRBF
Model code and software
AdaHRBF Zhang et al. https://doi.org/10.5281/zenodo.7340093
Baoyi Zhang et al.
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