the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 Apr 2024
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 seismically-induced tsunami for a rapid, assumed instantaneous, vertical seafloor displacement is given by the Kajiura low-pass 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 outer-rise – 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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Supplement
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Status: open (until 14 May 2024)
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RC1: 'Comment on nhess-2024-41', Anonymous Referee #1, 03 Apr 2024
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See the enclosed PDF file.
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AC1: 'Reply on RC1', Alice Abbate, 09 Apr 2024
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The comment was uploaded in the form of a supplement: https://nhess.copernicus.org/preprints/nhess-2024-41/nhess-2024-41-AC1-supplement.pdf
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AC1: 'Reply on RC1', Alice Abbate, 09 Apr 2024
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RC2: 'Comment on nhess-2024-41', Anonymous Referee #2, 24 Apr 2024
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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 pseudo-code 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 in-line 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 box-car 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 unit-slip 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 sea-surface 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 sea-floor 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 big-O notation, but perhaps not little-o. Helpful to indicate in the Supplement its meaning and how it is derived. Curious that the little-o 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.
- L122-123: 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.
- L199-203: 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.
- L242-243: It would be helpful indicate the pertinent Laplace equation near the beginning of Section 2.
Citation: https://doi.org/10.5194/nhess-2024-41-RC2 -
RC3: 'Comment on nhess-2024-41', Anonymous Referee #3, 27 Apr 2024
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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/nhess-2024-41-RC3
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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