the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Modeling tsunami initial conditions due to rapid coseismic seafloor displacement: efficient numerical integration and a tool to build unit source databases
Abstract. The initial condition for the simulation of a seismicallyinduced tsunami for a rapid, assumed instantaneous, vertical seafloor displacement is given by the Kajiura lowpass filter integral. This work proposes a new efficient and accurate approach for its numerical evaluation, valid when the sea floor displacement is discretized as a set of rectangular contributions. We compare several truncated quadrature formulae, selecting the optimal one. We verify that we can satisfactorily approximate the initial sea level perturbation as a linear combination of those induced by the elementary sea floor displacements. The methodology is tested on the tsunamigenic Kuril earthquake doublet – a megathrust and an outerrise – occurred in the Central Kuril Islands in late 2006 and early 2007. We also confirm the importance of the horizontal contribution to the tsunami generation and we consider a simple model of the inelastic deformation of the wedge, on a realistic bathymetry. The proposed approach results accurate and fast enough to be considered relevant for practical applications, and a tool is provided to create tsunami unit source databases for a given region of interest.
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RC1: 'Comment on nhess202441', Anonymous Referee #1, 03 Apr 2024

AC1: 'Reply on RC1', Alice Abbate, 09 Apr 2024
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202441/nhess202441AC1supplement.pdf

AC1: 'Reply on RC1', Alice Abbate, 09 Apr 2024

RC2: 'Comment on nhess202441', Anonymous Referee #2, 24 Apr 2024
The NHESS manuscript “Modeling tsunami initial conditions due to rapid coseismic seafloor displacement: efficient numerical integration and a tool to build unit source databases” by Abbate et al. develops and describes a computationally efficient procedure to calculate the attenuation of vertical displacement in the water column during tsunami generation. This is an important study that provides an accurate and efficient method to determine this phenomenon: too often the water column Green’s function (“Kajiura filter”) is ignored, leading to an overestimation of onshore wave heights, runup, and inundation. When implemented in the past, it has often been calculated assuming a constant water depth in the source region. Overall, the study is well conceived and the manuscript is well organized and written. The detailed description of the algorithm and pseudocode in the supplement is appreciated for future applications. General suggestions are provided below to revise the text for the NHESS readership as well as specific inline comments and corrections. These should all be easily addressed by the authors.
General comments:
 I very much appreciate the mathematical rigor of the analysis, so often lacking in many geophysical papers (my own included). The Abstract reads well, but some of the introductory text could be made more engaging to a natural hazards and geophysical audience. For the Introduction, it would be good to describe the objective of the study closer to the top of the section, particularly in terms of implications for tsunami hazard assessment. For Section 2, I would very much encourage describing the geophysical problem and associated approximations first, before jumping straight into the mathematics.
 The rationale for using boxcar source is unclear to me. Is it because of specific analytic/spectral properties? Alternatively, it would be more harmonious with existing tsunami modeling practice to use vertical seafloor displacements from unitslip dislocations (i.e., unit fault sources), although granted, this would have to be regional/subduction zone specific.
 It would be particularly informative to determine the effect on seasurface elevation profiles of earthquake ruptures that reach to the sea floor and form a scarp. The scarp displacement is obviously attenuated through the water column, but it has been unclear in previous studies what the resulting sea elevation profile is and the effect on the maximum amplitude. Related to this, I’m assuming the 2006 earthquake was not a seafloor rupturing event but the 2007 earthquake was? It would be helpful to indicate this in the manuscript explicitly.
Specific comments:
 L64: Which authors are referred to?
 L97: It would be helpful to describe the “Laplacian problem”/equation for the readers here. Referred to later in the manuscript as well.
 4: I suspect most readers are familiar with bigO notation, but perhaps not littleo. Helpful to indicate in the Supplement its meaning and how it is derived. Curious that the littleo term is not included in the 2D equation (9) (or supplement eqn. 27).
 L120: Because it is used as a reference solution, it would be helpful to know more about the GAQ method, either in the main text or supplement. What properties does it have that makes it more accurate? It would also be helpful to have more description of the Filon quadrature method.
 L122123: Please indicate specifically how small “u” is related to big “U”.
 L148: Please specify how “numerical integration” is performed. (using each quadrature method?)
 3: It’s a little confusing to have the bars in the chart ordered differently than the table directly beneath the chart.
 L199203: Again, it’s confusing why an equivalent Heaviside function is used for sea floor displacement rather than directly using the elastic dislocation equations (Okada) with the source parameters as described.
 L242243: It would be helpful indicate the pertinent Laplace equation near the beginning of Section 2.
Citation: https://doi.org/10.5194/nhess202441RC2 
AC2: 'Reply on RC2', Alice Abbate, 30 Apr 2024
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202441/nhess202441AC2supplement.pdf

RC3: 'Comment on nhess202441', Anonymous Referee #3, 27 Apr 2024
Since the other reports are already available, I am only providing additional comments.
Mathematical questions:
1. In equation (4), why is it \epsilon^3?
2. In equation (5), the last parenthesis should be after dm.
3. I don't understand equation (6).
4. I don't understand equation (7).
5. Figure 3: it is misleading because in the text the authors mention two quadrature formulas and they mention three in the figure. Why is GAQ the groundtruth?
6. In equation (9), why is it \epsilon^4?
There are several awkward sentences. Examples are:
1. The last sentence of the abstract
2. The sentence on lines 58/59
The last author is missing in the reference Kervella and Dutykh (2007). In the main text, it should read Kervella et al. (2007). Please replace >> and << by their LaTeX notation: \gg and \ll. I would replace the first sentence of Section 2 by: Let R denote the set of real numbers. We consider a domain D \in R. Trigonometric functions inside equations should be written \cosh, \cos, \sin, \max, etc.
Citation: https://doi.org/10.5194/nhess202441RC3 
AC3: 'Reply on RC3', Alice Abbate, 13 May 2024
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202441/nhess202441AC3supplement.pdf

AC3: 'Reply on RC3', Alice Abbate, 13 May 2024

RC4: 'Comment on nhess202441', Kirill Sementsov, 01 May 2024
[general comments]
The authors of Abbate et al. developed a longtime desired tool to calculate the initial perturbation of the water surface in the tsunami source (Laplacian Smoothing Tool). The LST considers the smoothing effect of the water layer, and therefore significantly improves the accuracy of the input data for numerical tsunami modelling. The linear recombination of the unit sources for the Central Kuril Islands has been solved by in just 9 min, which allows us to hope that the developed tool will be in demand not only in retrospective tsunami studies, but also in realtime tsunami forecast. The paper and its supplementary materials describe the approach underlying LST and the details of the implementation of this approach. The material is well organized, written in clear language, and deserves publication after minor revisions (see comments below).
[specific comments]
The only weakness of the paper is the absence of a detailed comparison of LST with the more accurate methods of initial perturbation calculation. The amplitudes of the Kuril tsunamis calculated using LST are compared with similar amplitudes obtained by Rabinovich et al. 2008 and Nosov & Kolesov 2011. But it is difficult to get any insights from such a comparison of amplitudes alone, especially since the bathymetric and bottom deformation data were different in all these works.... However, considering that the main purpose of the paper was to describe and demonstrate LST, comparison of LST with methods of other authors can be postponed for further research.
The theoretical background of LST is based on Abrahams et al 2023, Davies & Griffin 2018 and Nosov & Kolesov 2011. It is not clear from Section 1 whether any of these papers compared the Kajiuratype filter (with the average ocean depth) and the solution of the full 3D Laplace problem (in the ocean with variable depth). I recommend the authors to emphasize the presence/absence of such a comparison, and to mention the paper by Sementsov & Nosov 2023 (https://doi.org/10.20948/mm20230206), in which the comparison of the Kajiura filter and the full Laplace problem solution was carried out for a 2D case (0XZ).
In Figure 2, tolerance is shown in colour (without units) and is also plotted on the vertical axis (in %). In the text of the article, the formula for MAE is given first, and the subsequent analysis is carried out in terms of tolerance. I recommend the authors to check the figure once again and briefly describe the connection between MAE and tolerance.
Section 3. The integration limit U and the optimal quadrature method (GLQ) for the 2D case were chosen based on the tests for 1D. A comment may need to be added that the 1D results can indeed be extended to 2D.
145146: ‘It should be recalled that the approximation is valid when both the bathymetry and coseismic displacement vary slowly within such a radius (4H0)’. Are there any quantitative limitations for this ‘slowly’? If these limitations are violated, can the result be improved by reducing the cell size? (these questions can be discussed in the Discussion section up to the authors decision).
301302: ‘The LST appears thus to smooth about threetimes more the uplifted sea surface than the subsided one for this event’. Why? Probably, because the uplift peak is located in shallow water while the subsidence peak is located in deep water?
305: It is also interesting to note that the filtered and unfiltered peaks are slightly shifted horizontally one relative to the other. Up to the authors decision this fact could be mentioned in the text.
338, 400: “nine models”. Why nine, but not seven?
Kuril 2006: Vertical, A, B;
Kuril 2007, northwest dipping: Vertical, A;
Kuril 2007, southeast dipping: Vertical, A.
Seven in total!
[technical corrections]
Figure 5.
In the Figure: panel e is labeled f by mistake.
In the caption (3rd line): replace (b) with (d).
I guess, this Figure can be improved if the authors sign each panel 1D or 2D, respectively, indicate the depth H in panels (a)(c), and indicate the value of a in panels (d)(f). The reader can find all this information in the text, but it would be easier to perceive the figure if this information was shown on it directly.
253: typo, remove the ‘a’.
298: Replace Fig.7 with Fig.6.
321: The sentence ‘Findings...Fig.12’ should be moved to the next paragraph.
Revised by Kirill Sementsov

AC4: 'Reply on RC4', Alice Abbate, 13 May 2024
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202441/nhess202441AC4supplement.pdf

AC4: 'Reply on RC4', Alice Abbate, 13 May 2024
Status: closed

RC1: 'Comment on nhess202441', Anonymous Referee #1, 03 Apr 2024

AC1: 'Reply on RC1', Alice Abbate, 09 Apr 2024
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202441/nhess202441AC1supplement.pdf

AC1: 'Reply on RC1', Alice Abbate, 09 Apr 2024

RC2: 'Comment on nhess202441', Anonymous Referee #2, 24 Apr 2024
The NHESS manuscript “Modeling tsunami initial conditions due to rapid coseismic seafloor displacement: efficient numerical integration and a tool to build unit source databases” by Abbate et al. develops and describes a computationally efficient procedure to calculate the attenuation of vertical displacement in the water column during tsunami generation. This is an important study that provides an accurate and efficient method to determine this phenomenon: too often the water column Green’s function (“Kajiura filter”) is ignored, leading to an overestimation of onshore wave heights, runup, and inundation. When implemented in the past, it has often been calculated assuming a constant water depth in the source region. Overall, the study is well conceived and the manuscript is well organized and written. The detailed description of the algorithm and pseudocode in the supplement is appreciated for future applications. General suggestions are provided below to revise the text for the NHESS readership as well as specific inline comments and corrections. These should all be easily addressed by the authors.
General comments:
 I very much appreciate the mathematical rigor of the analysis, so often lacking in many geophysical papers (my own included). The Abstract reads well, but some of the introductory text could be made more engaging to a natural hazards and geophysical audience. For the Introduction, it would be good to describe the objective of the study closer to the top of the section, particularly in terms of implications for tsunami hazard assessment. For Section 2, I would very much encourage describing the geophysical problem and associated approximations first, before jumping straight into the mathematics.
 The rationale for using boxcar source is unclear to me. Is it because of specific analytic/spectral properties? Alternatively, it would be more harmonious with existing tsunami modeling practice to use vertical seafloor displacements from unitslip dislocations (i.e., unit fault sources), although granted, this would have to be regional/subduction zone specific.
 It would be particularly informative to determine the effect on seasurface elevation profiles of earthquake ruptures that reach to the sea floor and form a scarp. The scarp displacement is obviously attenuated through the water column, but it has been unclear in previous studies what the resulting sea elevation profile is and the effect on the maximum amplitude. Related to this, I’m assuming the 2006 earthquake was not a seafloor rupturing event but the 2007 earthquake was? It would be helpful to indicate this in the manuscript explicitly.
Specific comments:
 L64: Which authors are referred to?
 L97: It would be helpful to describe the “Laplacian problem”/equation for the readers here. Referred to later in the manuscript as well.
 4: I suspect most readers are familiar with bigO notation, but perhaps not littleo. Helpful to indicate in the Supplement its meaning and how it is derived. Curious that the littleo term is not included in the 2D equation (9) (or supplement eqn. 27).
 L120: Because it is used as a reference solution, it would be helpful to know more about the GAQ method, either in the main text or supplement. What properties does it have that makes it more accurate? It would also be helpful to have more description of the Filon quadrature method.
 L122123: Please indicate specifically how small “u” is related to big “U”.
 L148: Please specify how “numerical integration” is performed. (using each quadrature method?)
 3: It’s a little confusing to have the bars in the chart ordered differently than the table directly beneath the chart.
 L199203: Again, it’s confusing why an equivalent Heaviside function is used for sea floor displacement rather than directly using the elastic dislocation equations (Okada) with the source parameters as described.
 L242243: It would be helpful indicate the pertinent Laplace equation near the beginning of Section 2.
Citation: https://doi.org/10.5194/nhess202441RC2 
AC2: 'Reply on RC2', Alice Abbate, 30 Apr 2024
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202441/nhess202441AC2supplement.pdf

RC3: 'Comment on nhess202441', Anonymous Referee #3, 27 Apr 2024
Since the other reports are already available, I am only providing additional comments.
Mathematical questions:
1. In equation (4), why is it \epsilon^3?
2. In equation (5), the last parenthesis should be after dm.
3. I don't understand equation (6).
4. I don't understand equation (7).
5. Figure 3: it is misleading because in the text the authors mention two quadrature formulas and they mention three in the figure. Why is GAQ the groundtruth?
6. In equation (9), why is it \epsilon^4?
There are several awkward sentences. Examples are:
1. The last sentence of the abstract
2. The sentence on lines 58/59
The last author is missing in the reference Kervella and Dutykh (2007). In the main text, it should read Kervella et al. (2007). Please replace >> and << by their LaTeX notation: \gg and \ll. I would replace the first sentence of Section 2 by: Let R denote the set of real numbers. We consider a domain D \in R. Trigonometric functions inside equations should be written \cosh, \cos, \sin, \max, etc.
Citation: https://doi.org/10.5194/nhess202441RC3 
AC3: 'Reply on RC3', Alice Abbate, 13 May 2024
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202441/nhess202441AC3supplement.pdf

AC3: 'Reply on RC3', Alice Abbate, 13 May 2024

RC4: 'Comment on nhess202441', Kirill Sementsov, 01 May 2024
[general comments]
The authors of Abbate et al. developed a longtime desired tool to calculate the initial perturbation of the water surface in the tsunami source (Laplacian Smoothing Tool). The LST considers the smoothing effect of the water layer, and therefore significantly improves the accuracy of the input data for numerical tsunami modelling. The linear recombination of the unit sources for the Central Kuril Islands has been solved by in just 9 min, which allows us to hope that the developed tool will be in demand not only in retrospective tsunami studies, but also in realtime tsunami forecast. The paper and its supplementary materials describe the approach underlying LST and the details of the implementation of this approach. The material is well organized, written in clear language, and deserves publication after minor revisions (see comments below).
[specific comments]
The only weakness of the paper is the absence of a detailed comparison of LST with the more accurate methods of initial perturbation calculation. The amplitudes of the Kuril tsunamis calculated using LST are compared with similar amplitudes obtained by Rabinovich et al. 2008 and Nosov & Kolesov 2011. But it is difficult to get any insights from such a comparison of amplitudes alone, especially since the bathymetric and bottom deformation data were different in all these works.... However, considering that the main purpose of the paper was to describe and demonstrate LST, comparison of LST with methods of other authors can be postponed for further research.
The theoretical background of LST is based on Abrahams et al 2023, Davies & Griffin 2018 and Nosov & Kolesov 2011. It is not clear from Section 1 whether any of these papers compared the Kajiuratype filter (with the average ocean depth) and the solution of the full 3D Laplace problem (in the ocean with variable depth). I recommend the authors to emphasize the presence/absence of such a comparison, and to mention the paper by Sementsov & Nosov 2023 (https://doi.org/10.20948/mm20230206), in which the comparison of the Kajiura filter and the full Laplace problem solution was carried out for a 2D case (0XZ).
In Figure 2, tolerance is shown in colour (without units) and is also plotted on the vertical axis (in %). In the text of the article, the formula for MAE is given first, and the subsequent analysis is carried out in terms of tolerance. I recommend the authors to check the figure once again and briefly describe the connection between MAE and tolerance.
Section 3. The integration limit U and the optimal quadrature method (GLQ) for the 2D case were chosen based on the tests for 1D. A comment may need to be added that the 1D results can indeed be extended to 2D.
145146: ‘It should be recalled that the approximation is valid when both the bathymetry and coseismic displacement vary slowly within such a radius (4H0)’. Are there any quantitative limitations for this ‘slowly’? If these limitations are violated, can the result be improved by reducing the cell size? (these questions can be discussed in the Discussion section up to the authors decision).
301302: ‘The LST appears thus to smooth about threetimes more the uplifted sea surface than the subsided one for this event’. Why? Probably, because the uplift peak is located in shallow water while the subsidence peak is located in deep water?
305: It is also interesting to note that the filtered and unfiltered peaks are slightly shifted horizontally one relative to the other. Up to the authors decision this fact could be mentioned in the text.
338, 400: “nine models”. Why nine, but not seven?
Kuril 2006: Vertical, A, B;
Kuril 2007, northwest dipping: Vertical, A;
Kuril 2007, southeast dipping: Vertical, A.
Seven in total!
[technical corrections]
Figure 5.
In the Figure: panel e is labeled f by mistake.
In the caption (3rd line): replace (b) with (d).
I guess, this Figure can be improved if the authors sign each panel 1D or 2D, respectively, indicate the depth H in panels (a)(c), and indicate the value of a in panels (d)(f). The reader can find all this information in the text, but it would be easier to perceive the figure if this information was shown on it directly.
253: typo, remove the ‘a’.
298: Replace Fig.7 with Fig.6.
321: The sentence ‘Findings...Fig.12’ should be moved to the next paragraph.
Revised by Kirill Sementsov

AC4: 'Reply on RC4', Alice Abbate, 13 May 2024
The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess202441/nhess202441AC4supplement.pdf

AC4: 'Reply on RC4', Alice Abbate, 13 May 2024
Data sets
Laplacian Smoothing Tool (LST) and related data Alice Abbate https://doi.org/10.5281/zenodo.10786626
Model code and software
Laplacian Smoothing Tool (LST) and related data Alice Abbate https://doi.org/10.5281/zenodo.10786626
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