Temporal Prediction of Landslide-Generated Waves Using a Theoretical–Statistical Combined Method

Round 1

Reviewer 1 Report

The manuscript "Temporal prediction of landslide-generated waves using a theoretical-statistical combined method" is well written by the authors and very useful data is presented. I appreciate the authors for the good contribution. Following are my suggestions/queries: 

 

1) Line 162 and Line 297, authors used Gaussian mixture model. Why only this model? Please give the justification and advantages as compared with other methods. 

2) Line 303, it is mentioned that "The am of T4 decrease sharply after 0.2 s". As per Figure 9(a) it is around 0.17 to 0.18 s. Please check. 

3) What is slide Froude number and its significance? 

4) Line 315, the 92 experiments are classified as 5 classes. What is the criteria for these groups? 

5) Figure 11 (a), does not show any trend. Is it possible to develop some trends to data in Figures 11(b) and 11(c)?

6) Line 334, Out of 92 experiments, 82 are used for training and 10 for validation. Why only this combination is selected? 82 for training will by overfitting. Please do justify. Also plot training and validation performances in the form of graphs. 

7) According Table 2, performance of experimental data are better than theoretical (R square being greater for experimental data). What is the reason? 

8) Carbopol is used as a slide material. Is there any other material that can be used? 

9) It is possible to add velocity distribution at different section along the centerline due to sliding of mass? It will enhance the scope of the paper. 

10) What is the rate of lift of the gate?

11) Figure 6 and 7 are the results of numerical solution. Please give the details of numerical analysis. 

 

Typographical mistakes at few places need to be corrected. 

Author Response

Reviewer 1:

The manuscript "Temporal prediction of landslide-generated waves using a theoretical-statistical combined method" is well written by the authors and very useful data is presented. I appreciate the authors for the good contribution.

Response: We would like to thank the reviewer for reviewing the manuscript. The inputs have been very helpful for improving the manuscript. We agree with the comments and we revised our manuscript accordingly. We provide a point-by-point response to the reviewer’s comments. All the changes are highlighted in the trackchanges file attached to this submission.

 

Following are my suggestions/queries: 

 

• Line 162 and Line 297, authors used Gaussian mixture model. Why only this model? Please give the justification and advantages as compared with other methods.

Response: For the panel data analysis, we first used the Gaussian mixture model to classify the experimental dataset, and then used the random coefficient model to quantify the temporal relation between the slide parameters and wave parameters. These two models selected in this study have been widely used in panel data analysis. As a first step to introducing panel data model into the analysis of landslide generated waves, our idea is to use models that have been commonly adopted.

• Line 303, it is mentioned that "The am of T4 decrease sharply after 0.2 s". As per Figure 9(a) it is around 0.17 to 0.18 s. Please check. 

 Response: Correction has been done in the revised manuscript.

 

• What is slide Froude number and its significance? 

Response: In the field of landslide generated waves, the slide Froude number was a dimensionless number that commonly used to scale the velocity of the sliding mass on impact. The parameter has been proposed and widely used since 1980s. In this manuscript, we introduced the parameter and its expression in Sec 2.2.

 

• Line 315, the 92 experiments are classified as 5 classes. What is the criteria for these groups? 

Response: The classification was conducted using the Gaussian mixture model, which was presented in Sec 3.1. The criteria of classification were introduced in Sec. 5.2. The text was as follows:

As shown in Sec. 5.1, the scaled maximum wave amplitude Am and scaled maximum 321 wave height Hm can be well estimated by the slide Froude number Fr, scaled slide thickness 322 S, and scaled effective slide mass M using empirical equations, with the coefficient of 323 determination R2 larger than 0.9. We thus selected these three dimensionless groups as the 324 criteria to evaluate the properties of experiments, and classified the experimental dataset 325 using the Gaussian mixture model based on these three indicators.

 

• Figure 11 (a), does not show any trend. Is it possible to develop some trends to data in Figures 11(b) and 11(c)?

Response: The general trends exhibited by Fig. 11 is that tests in each class generally gather together. It confirms that texts in each class have similar characteristics. However, the distribution of the dataset can not be quantified.

 

• Line 334, Out of 92 experiments, 82 are used for training and 10 for validation. Why only this combination is selected? 82 for training will by overfitting. Please do justify. Also plot training and validation performances in the form of graphs. 

Response: We followed the data allocation policy used in many previous studies, selected 90% of the dataset to train the model, and 10% of the dataset to verify the validity of the model. That is, 82 are used for training and 10 for validation.

 

• According Table 2, performance of experimental data are better than theoretical (R square being greater for experimental data). What is the reason? 

Response: For the prediction using experimental data, the temporal relation was developed between the experimental data of slide parameters on impact and the experimental data of wave parameters. For the prediction using theoretical data, we first estimated the slide parameters on impact using a theoretical model, and substituted the experimental data of slide parameters by the theoretical data, then predicted the experimental data of wave parameters from the theoretical slide parameters on impact. As shown in Fig. 8, the slide parameters estimated using the theoretical model has a slight deviation from the experimental data. This is why the accuracy of the temporal prediction using theoretical slide parameters is slightly lower than that using experimental slide parameters.

 

• Carbopol is used as a slide material. Is there any other material that can be used? 

Response: For the experimental investigations of landslide generated waves, previous studies commonly mimicked natural landslide with ideal materials to simplify its flow dynamics. Granular slide materials have been routinely used since 1990s. Some early

researches also used blocks to mimic landslides. We introduced Carbopol into the experimental studies of landslide-generated waves to mimic landslides whose rhological properties can be described by viscoplastic model. We also mentioned in Sec 4.1 that details of the material refer to our earlier publications (Meng, 2018; Meng and Ancey, 2019; Meng et al., 2020).

 

• It is possible to add velocity distribution at different section along the centerline due to sliding of mass? It will enhance the scope of the paper. 

Response: The time series data of the mean velocity used in this study was obtained from the velocity field measured using PIV technique. We agree with the reviewer that the velocity distribution is an interesting data. However, the velocity distribution of Carbopol moving along the slope as well as its theoretical analysis have been published in several previous article of our lab. This paper mainly focus on the temporal prediction of the landslide-waves using the theoretical-statistical combined model, thus we didn’t extend the details of the velocity distribution in this work.

 

• What is the rate of lift of the gate?

Response: The rate of lift is 2.5 m/s. The text is as follows:

Once the lock gate is released, the material accelerated energetically under gravity and reached velocities as high as 2.5 m/s.

 

11) Figure 6 and 7 are the results of numerical solution. Please give the details of numerical analysis. 

Response: Figure 6 and 7 were the numerical solutions of the theoretical model presented in Sec. 3.2. It was not a novel theoretical model. The model was initially proposed and detailedly presented in a previous work of our Lab (Ancey et al. 2012). Therefore, we mentioned in Sec. 3.2 that the details of the theoretical model as well as the numerical solutions refers to this previous work.

Reviewer 2 Report

Brief summary:

The manuscript continues the authors' previous works on landslide-generated waves. The authors have previously determined wave amplitudes and heights in experiments with viscoplastic material in laboratory conditions in [16-18], used a Gaussian mixture model for data classification in [23], used a random coefficient model for regression in [23], outlined theoretical solutions for the temporal slide thickness and depth-averaged velocity in [25], and used particle imaging velocimetry (PIV) techniques in [25]. In the present manuscript, the authors combine these techniques for determination of wave amplitudes and heights from slide thickness and depth-averaged velocity as a function of time with experiments in laboratory conditions. The results in Fig. 12 show that the temporal shape of the amplitude can be estimated approximately while the bumpy parts have been missed.

General comments:

The presentation of the model development in Section 3 must be improved. The idea is correct but the notational shortcomings must be corrected so that all the methods are presented in a unified style and each symbol has only a single definition throughout the manuscript. It looks like that the methods have been copied from the authors' previous papers without changing the original symbols. The manuscript is difficult to read if the same symbol has a different meaning in different subsections. Proper discussion of the results is missing.

Detailed comments:

Line 112: m without a subscript E has not been explained.

Line 120: It is confusing that m_E has two meanings. It is the immersed slide mass in Section 2.3 while it is the slide mass in Section 2.2.

Lines 133-136: Please mention also in the text that x_i are the slide parameters and y_j are the wave parameters. It is not clear why the slide parameters include the wave amplitude, which is clearly a wave parameter. What are the additional parameters referred to by "etc" in slide parameters and wave parameters? Only the listed ones are considered in the manuscript.

Lines 133-134: It is confusing that N has two meanings. It is the number of experiments in Section 3.1 while it is the number of the explanatory variables in Section 2.2.

Line 138: It is started with a 2-D case without the time variable. Does the whole Section 3.1 deal with the 2-D case so that the time variable is considered only in Section 3.2? Or is the time variable considered from line 146 onwards? 

Equation 4: The subscripts i and t are missing from u in the last equation. 

Line 141 and Eq. 4: Please clarify that the cross section index i refers to the wave parameters and the explanatory variables index k refers to the slide parameters. It is not clear why x_{kit} has also an index i.

Lines 142-143: If gamma is an operator, how does it operate on x_{kit} to produce beta_k, i.e., gamma(x_{kit})=..??..=beta_k, where b_k is the mean of b_{ki} over i?

Lines 143-144: It is confusing that the symbol u has two meanings. It is a random error in Section 3.1 while it is a velocity in Sections 2 and 3.2.

Line 145: The equation beta_i=beta+gamma_i does not make sense, because beta is a vector, beta_i a scalar, and gamma_i has not been defined.

Equation 5 and line 145: These are assumptions and not consequences. "obtain" is not a correct word.

Equation 6 and lines 146-148: The difference between N and T is not clear. N is the test number in Fig. 2b but here T is the test number. One would expect T to be the number of time steps. The matrix and vector dimensions do not match in Eq. 6 if NT is not N and if NK is not N. So, please clarify that X_1,...,X_N are T-vectors. It seems illogical that tilde X = u. Is N the number of a panel or the number of panels?

Line 152: "eliminate" is a misleading word here.

Equation 10: omega_1 and omega_0 have not been explained.

Equation 14 and line 179: Please clarify that an estimate of mu_k is obtained by finding the root of Eq. 12 using the Newton-Raphson method. The term added during iteration is given in Eq. 14. The notion of indices and primes is not clear. The iteration formula is missing. An initial mu_1 should not be denoted with the same symbol as the estimated mu_1.

Lines 185-187: sigma_i^(k) is not the variance but the standard deviation. x_i^(n) is not the variance but the data.

Equation 16 and lines 187-188: There is no iteration in Eq. 16. Or where is the iteration?

Lines 195-196: The dimensions do not match: if V_I is a volume (3-dimensional) and l_0 a length (1-dimensional), then V_I/l_0 is 2-dimensional while s_g is 1-dimensional. So, V_I is obviously a volume per unit width.

Equation 18: n has not been explained.

Line 202: It is confusing that K has two meanings. It is a parameter in Section 3.2 while it is the number of the explanatory variables in Section 3.1.

Line 202: mu has not been explained. It does not look like to be a mean value as in Section 3.1, but a consistency according to [25] and Section 4.1.

Line 203: tau_c has not been explained here but only later in Section 4.1.

Equation 19: A has not been explained.

Lines 209-213: The flume is clearly 3-dimensional and not 2-dimensional.

Line 225: It is confusing that gamma has two meanings. It is shear rate in Section 4.1 while it is an operator in Section 3.1.

Line 235: CW has not been explained. Is it continuous wave even if the laser is a pulsed one?

Line 284: There are two values for l_0, namely, l_0=0.4 m and l_0=0.3 m. Which one is the correct one?

Table 1: alpha has not been explained.

Lines 297-299 and Eqs. 4 and 6: It is not clear if the coefficients beta_{ki} and gamma_{ki} are the same for all the time steps or if they depend on time in the random coefficient model.

Lines 302-304 and Fig. 9: The claim that a_m of T4 has a turning point at 3.2 s does not hold according to Fig. 9. The turning point is later in time. The overall shape of a_m for T1 and T4 is quite similar, so it is too much to state that one decreases "sharply" and the other "flatly". Do you mean T3 instead of T4? The amplitude is inconsistently denoted by "a_m" in the text and by "a" in Fig. 9. Which one is correct? If the coefficients obtained from the random coefficient model are applied to the theoretical slide thickness and velocity, then the wave amplitude and height are continuous functions of time.

Lines 327-355: Results are shown only for the wave amplitude A(T) while the results for the wave height H(T) would be also interesting to see.

The English language is mainly fluent but there are a few misleading words mentioned in the detailed comments.

Author Response

Reviewer 2:

Brief summary:

The manuscript continues the authors' previous works on landslide-generated waves. The authors have previously determined wave amplitudes and heights in experiments with viscoplastic material in laboratory conditions in [16-18], used a Gaussian mixture model for data classification in [23], used a random coefficient model for regression in [23], outlined theoretical solutions for the temporal slide thickness and depth-averaged velocity in [25], and used particle imaging velocimetry (PIV) techniques in [25]. In the present manuscript, the authors combine these techniques for determination of wave amplitudes and heights from slide thickness and depth-averaged velocity as a function of time with experiments in laboratory conditions. The results in Fig. 12 show that the temporal shape of the amplitude can be estimated approximately while the bumpy parts have been missed.

 

General comments:

The presentation of the model development in Section 3 must be improved. The idea is correct but the notational shortcomings must be corrected so that all the methods are presented in a unified style and each symbol has only a single definition throughout the manuscript. It looks like that the methods have been copied from the authors' previous papers without changing the original symbols. The manuscript is difficult to read if the same symbol has a different meaning in different subsections. Proper discussion of the results is missing.

Response: We would like to thank the reviewer for the thoughtful review of the manuscript. The inputs have been very helpful for improving the manuscript. We agree with almost all comments and we revised our manuscript accordingly. We provide a point-by-point response to the reviewer’s comments. All the changes are highlighted in the trackchanges file attached to this submission.

 

Detailed comments:

Line 112: m without a subscript E has not been explained.

 Response: Explanations have been added to the manuscript:

$m$ is defined by the mass of the immersed slide material at the moment when the wave height reaches its maximum.

 

Line 120: It is confusing that m_E has two meanings. It is the immersed slide mass in Section 2.3 while it is the slide mass in Section 2.2.

 Response: Correction has been done. It should be $m$ instead of $m_E$ in section 2.2.

 

Lines 133-136: Please mention also in the text that x_i are the slide parameters and y_j are the wave parameters. It is not clear why the slide parameters include the wave amplitude, which is clearly a wave parameter. What are the additional parameters referred to by "etc" in slide parameters and wave parameters? Only the listed ones are considered in the manuscript.

 Response: The text has been revised. Now the sentence reads:

$\lbrace x_1, x_2, ..., x_i \rbrace$ denote the slide parameters (e.g., slide velocity, slide thickness,  slide mass, slide density, and etc), and  $\lbrace y_1, y_2, ..., y_j \rbrace$ denote the wave parameters (e.g., wave amplitude, wave height, wave period, wave length, and etc). The objective of this study was to quantify$\lbrace y_1, y_2, ..., y_j \rbrace$ from $\lbrace x_1, x_2, ..., x_i \rbrace$ .

 

Lines 133-134: It is confusing that N has two meanings. It is the number of experiments in Section 3.1 while it is the number of the explanatory variables in Section 2.2.

 Response: The symbol $N$ was defined repeatedly. We have corrected the text by defining the number of variables in section 3.1 as $I$.

 

Line 138: It is started with a 2-D case without the time variable. Does the whole Section 3.1 deal with the 2-D case so that the time variable is considered only in Section 3.2? Or is the time variable considered from line 146 onwards? 

 Response: Corrections have been done. The presentation was started from a variable coefficient statistical model, and then turns to the random coefficient model. The text now reads:

We first study the variable coefficient statistical model. Regression coefficients of panel data will not change with time but change with cross section. The model formula is as follows:

 

Equation 4: The subscripts i and t are missing from u in the last equation. 

 Response: Correction has been done.

 

Line 141 and Eq. 4: Please clarify that the cross section index i refers to the wave parameters and the explanatory variables index k refers to the slide parameters. It is not clear why x_{kit} has also an index i.

 Response: Corrections have been done.

 

Lines 142-143: If gamma is an operator, how does it operate on x_{kit} to produce beta_k, i.e., gamma(x_{kit})=..??..=beta_k, where b_k is the mean of b_{ki} over i?

 Response: Corrections have been done. gamma is the deviation degree that the individuals deviate from the common mean value

 

Lines 143-144: It is confusing that the symbol u has two meanings. It is a random error in Section 3.1 while it is a velocity in Sections 2 and 3.2.

 Response: Correction has been done. We redefined the random interference term here as ‘r’ instead of ‘u’ in the revised manuscript.

 

Line 145: The equation beta_i=beta+gamma_i does not make sense, because beta is a vector, beta_i a scalar, and gamma_i has not been defined.

Response: Correction has been done. The sentence now reads:

For the random coefficient model, β_i is set as a random variable. Swamy [44] put forward the following assumptions:

 

Lines 297-299 and Eqs. 4 and 6: It is not clear if the coefficients beta_{ki} and gamma_{ki} are the same for all the time steps or if they depend on time in the random coefficient model.

Response: This part has been re-organized. Eq. 4 refers to a variable coefficient statistical model, and Eq. 6 refers to the random coefficient model. They depend on time in the random coefficient model.

 

Equation 5 and line 145: These are assumptions and not consequences. "obtain" is not a correct word.

 Response: Correction has been done in the revised manuscript.

 

Equation 6 and lines 146-148: The difference between N and T is not clear. N is the test number in Fig. 2b but here T is the test number. One would expect T to be the number of time steps. The matrix and vector dimensions do not match in Eq. 6 if NT is not N and if NK is not N. So, please clarify that X_1,...,X_N are T-vectors. It seems illogical that tilde X = u. Is N the number of a panel or the number of panels?

Response: The symbols were mis-defined. Also, a part of the equation was lacking in the original manuscript. Corrections have been done in the revised manuscript.

 

Line 152: "eliminate" is a misleading word here.

 Response: Correction has been done.

 

Equation 10: omega_1 and omega_0 have not been explained.

Response: Corrections have been done.

 

Equation 14 and line 179: Please clarify that an estimate of mu_k is obtained by finding the root of Eq. 12 using the Newton-Raphson method. The term added during iteration is given in Eq. 14. The notion of indices and primes is not clear. The iteration formula is missing. An initial mu_1 should not be denoted with the same symbol as the estimated mu_1.

 Response: Correction has been done. The text now reads:

The initial $\mu_1$, $\mu_2$ are iterated to $\mu_1$, $\mu_2$ using the approximate Newton--Raphson steps.

 

Lines 185-187: sigma_i^(k) is not the variance but the standard deviation. x_i^(n) is not the variance but the data.

Response: Corrections have been done.

 

Equation 16 and lines 187-188: There is no iteration in Eq. 16. Or where is the iteration?

 Response: Correction has been done. The text now reads:

The formulas of the variance $\sigma_i^{(k)}$ and the weighting $\pi_k$ can be written as:

 

Lines 195-196: The dimensions do not match: if V_I is a volume (3-dimensional) and l_0 a length (1-dimensional), then V_I/l_0 is 2-dimensional while s_g is 1-dimensional. So, V_I is obviously a volume per unit width.

Response: The flume width is lacking here. Correction has been done in the manuscript.

 

Equation 18: n has not been explained.

Response: The text has been re-organized. We now introduce the H-B equation and the related parameters Section 3.2. See details in the manuscript.

 

Line 202: It is confusing that K has two meanings. It is a parameter in Section 3.2 while it is the number of the explanatory variables in Section 3.1.

Response: The text has been re-organized. We now introduce the H-B equation and the related parameters Section 3.2. See details in the manuscript.

 

Line 202: mu has not been explained. It does not look like to be a mean value as in Section 3.1, but a consistency according to [25] and Section 4.1.

Response: The text has been re-organized. We now introduce the H-B equation and the related parameters Section 3.2. See details in the manuscript.

 

Line 203: tau_c has not been explained here but only later in Section 4.1.

Response: The text has been re-organized. We now introduce the H-B equation and the related parameters Section 3.2. See details in the manuscript.

 

Equation 19: A has not been explained.

Response: Corrections have been done. A has been introduced in the revised manuscript.

 

Lines 209-213: The flume is clearly 3-dimensional and not 2-dimensional.

Response: The flume is a narrow flume with a width of 0.12 m, we only consider the propagation of waves in one direction. Thus, the flume was considered as a 2 dimensional flume. We have deleted the description of ‘2 dimensional’ in the revised manuscript to avoid ambiguity.

 

Line 225: It is confusing that gamma has two meanings. It is shear rate in Section 4.1 while it is an operator in Section 3.1.

Response: It was $\dot{\gamma}$ in the H-B equation in sec 4.1 (the equation has been moved to sec 3.2 in the revised manuscript). The symbol $\gamma$ was used in sec 3.1.

 

Line 235: CW has not been explained. Is it continuous wave even if the laser is a pulsed one?

Response: Correction has been done. It was a writing mistake here.

 

Line 284: There are two values for l_0, namely, l_0=0.4 m and l_0=0.3 m. Which one is the correct one?

Response: Correction has been done. Actually it was l_0=0.3 m.

 

Table 1: alpha has not been explained.

Response: Correction has been done. It should be $theta$ instead of $alpha$. $theta$ has been defined in the text.

 

Lines 302-304 and Fig. 9: The claim that a_m of T4 has a turning point at 3.2 s does not hold according to Fig. 9. The turning point is later in time. The overall shape of a_m for T1 and T4 is quite similar, so it is too much to state that one decreases "sharply" and the other "flatly". Do you mean T3 instead of T4? The amplitude is inconsistently denoted by "a_m" in the text and by "a" in Fig. 9. Which one is correct? If the coefficients obtained from the random coefficient model are applied to the theoretical slide thickness and velocity, then the wave amplitude and height are continuous functions of time.

Response: Corrections have been done. The text now reads:

The amplitude $a(t)$ of T1 decreases after t = 1.5 s without any rally. For the evolution curve of $a(t)$ of Test T2, T3, and T4, fluctuations can be observed during the decreasing period. For example, the $a(t)$ of T2 start to decrease at 0.15 s, it has a turning point approximately at 0.30 s.

 

Lines 327-355: Results are shown only for the wave amplitude A(T) while the results for the wave height H(T) would be also interesting to see.

Response: We agree with the reviewer’s comments. However, when a predictive model works well for predicting wave amplitude, it usually works well for predicting wave height. Thus, we didn’t extend the model to predict the wave height.

 

Reviewer 3 Report

The authors worked on a very important issue associated with natural disasters, i.e., landslides. Specifically, the authors study the landslide-generated waves problem by providing models that build the temporal relation between the wave characteristics and the slide features. Insights about impulsive waves generated by snow are gained with this investigation. The authors experimentally determined the slide’s temporal velocity field using the particle image velocimetry technique, and the temporal wave amplitude was estimated using panel data analysis. Finally, the authors validated the proposed theoretical-statistical combined predictive method with the support of experimental data. Congratulations on this work. It has all my support for publication. 

However, the reviewer recommends expanding the introductory section, citing some relevant studies on the topic, such as:

 

·         Yavari-Ramshe, S., Ataie-Ashtiani, B. Numerical modeling of subaerial and submarine landslide-generated tsunami waves—recent advances and future challenges. Landslides 13, 1325–1368 (2016). https://doi.org/10.1007/s10346-016-0734-2

·         Di Risio, M., & Sammarco, P. (2008). Analytical modeling of landslide-generated waves. Journal of waterway, port, coastal, and ocean engineering, 134(1), 53-60.

 

In the introductory section, the reviewer encourages the authors to detail the differences between the before-mentioned studies and the current study.

 

Comments for author File: Comments.pdf

The reviewer recommends minor editorial changes:

In lines 81-86: “Comparing with previous empirical equations which predicted the maximum wave characteristics from slide parameters on impact, the proposed statistical-theoretical combined model not only estimated the temporal parameters of the sliding mass on impact but also the predicted the temporal wave characteristics from these temporal slide parameters, which helps to understand how the slide features affects the wave generation.”  The reviewer considers the authors should delete the article “the” before “predicted” in line 84. Also, delete the “s” of the word “affects” in line 86.

 In line 297, it seems there is a typo: (see?)

 

Author Response

Reviewer 3:

The authors worked on a very important issue associated with natural disasters, i.e., landslides. Specifically, the authors study the landslide-generated waves problem by providing models that build the temporal relation between the wave characteristics and the slide features. Insights about impulsive waves generated by snow are gained with this investigation. The authors experimentally determined the slide’s temporal velocity field using the particle image velocimetry technique, and the temporal wave amplitude was estimated using panel data analysis. Finally, the authors validated the proposed theoretical-statistical combined predictive method with the support of experimental data. Congratulations on this work. It has all my support for publication. 

Response: We would like to thank the reviewer for reviewing the manuscript. The inputs have been very helpful for improving the manuscript. We agree with the comments and we revised our manuscript accordingly. We provide a point-by-point response to the reviewer’s comments. All the changes are highlighted in the trackchanges file attached to this submission.

 

However, the reviewer recommends expanding the introductory section, citing some relevant studies on the topic, such as:

 

• Yavari-Ramshe, S., Ataie-Ashtiani, B. Numerical modeling of subaerial and submarine landslide-generated tsunami waves—recent advances and future challenges. Landslides 13, 1325–1368 (2016). https://doi.org/10.1007/s10346-016-0734-2
• Di Risio, M., & Sammarco, P. (2008). Analytical modeling of landslide-generated waves. Journal of waterway, port, coastal, and ocean engineering, 134(1), 53-60.

Response: Corrections have been done. The suggested reference have been cited in the revised manuscript. We also expanded several other relevant studies. The following text is added in the introduction section:

The problem of impulse waves generated by subaerial landslides has attracted considerable attention in recent decades. Many of the physical insights into this phenomena have 30 come from laboratory scale-down experiments [4–10], and to a lesser extent from theoretical models [11,12], numerical simulations [13–17], and field data surveys [18–21].

 

In the introductory section, the reviewer encourages the authors to detail the differences between the before-mentioned studies and the current study.

Response: This study take the first step to the temporal prediction of landslide generated wave. The novelty and challenge of this study have been illustrated in the introduction section:

None has yet quantified the temporal relation 55 between the wave characteristics and slide features. The objective of this study was to determine the temporal relation between the wave characteristics and slide parameters.

Comparing with previous empirical equations which predicted the maximums of wave parameters, the challenge of this study was to model several variables at varying time points.

 

The reviewer recommends minor editorial changes:

In lines 81-86: “Comparing with previous empirical equations which predicted the maximum wave characteristics from slide parameters on impact, the proposed statistical-theoretical combined model not only estimated the temporal parameters of the sliding mass on impact but also the predicted the temporal wave characteristics from these temporal slide parameters, which helps to understand how the slide features affects the wave generation.”  The reviewer considers the authors should delete the article “the” before “predicted” in line 84. Also, delete the “s” of the word “affects” in line 86.

 Response: Corrections have been done.

 

 In line 297, it seems there is a typo: (see?)

 Response: Correction has been done. The mis-placed typo has been deleted.

Round 2

Reviewer 2 Report

General comments:

1) There are still notational issues in the description of the methods and also methodological problems. Splitting of coefficients beta_{ki} into beta_k and gamma_{ki} is questionable. The coefficients can be very different for different wave parameters, so that the mean value beta_k (average of beta_{ki} over wave parameters i (height, amplitude, wavelength, period, etc.)) can be far away from beta_{ki}. The GLS estimator in Eq. 7 estimates the mean values beta_k, although we would also need the deviations gamma_{ki} to be able to estimate the wave parameters. Moreover, the values of Delta and sigma_i in Eq. 7 are usually unknown and sample statistics should be used instead.

2) The methods are described for panel data including multiple wave parameters, which is also highlighted on lines 66-90 in the introduction. However, test results are presented only for one wave parameter (wave amplitude). The authors replied that "when a predictive model works well for predicting wave amplitude, it usually works well for predicting wave height", but we do not know it if the results are not shown. Moreover, what is the point of presenting and highlighting panel data if the experiments do not deal with panel data but time sequence data for prediction of a single dependent variable? The abstract on lines 7-9 and 11-14 and the conclusions on lines 411 and 414 are thus misleading, because true panel data were not used to quantify wave amplitude nor validate the theoretical-statistical method, but only panel data with one cross-sectional variable which are called time sequence data as explained in [30].

Detailed comments:

Equations 1 and 2 and lines 109-110: N was changed to I in Eq. 1, but it still appears on line 109. The subscript of Y is j in Eq. 1 but n in Eq. 2.

Lines 113-115 and 124-125: There is still confusion between m and m_E. It seems that the definition on lines 113-115 is not for m but for m_E.

Line 152: beta_k denotes both a vector and a component of the same vector. The vector should be denoted by beta in accordance with Eq. 6. Consider using boldface or an overline for vectors.

Line 156: It is not clear why the subscript of beta is now denoted by i while it was previously denoted by k. Is i=k or does i refer to the cross section index? Is beta_i a vector consisting of the coefficients averaged over i for all explanatory variables k or a vector consisting of the coefficients beta_{ki}, k=1,...,K, for cross section i? Or is beta_i a scalar random variable, which receives values from a set consisting of the coefficients beta_{ki}, k=1,...,K?

Lines 155-158: You replied that beta and gamma depend on time in the random coefficient model, so please mention it also in the manuscript. But are you sure they depend on time? Why would they depend on time?

Equation 5: There is still u in Eq. 5, although it was previously replaced by r. gamma_i with only one subscript, r_i with only one subscript, and T have not been explained. Does the single subscript refer to the test number? Is T=N? Is x_{it}=x_{kt}?

Line 164 and Eq. 6: The dimensions in Eq. 6 do not match if tilde X gamma + r is a diagonal matrix and the other terms are vectors of size NPx1. It seems that tilde X gamma + r is actually a vector and not a matrix.

Line 168: "removed" should be "obtained".

Equation 7: There is a typo in the placement of the exponent (-1) in the formula for W_i.

Lines 202-203 and Eq. 13: mu_i^k should be mu_i^(k), right? Eq. 13 does not include mu_i^(k) but mu_k', so it is not clear what the formula for mu_i^(k) is.

Equations 19 and 22: An expression for the depth-averaged velocity is given for a slightly nonuniform flow in Eq. 19 and for a steady uniform flow in Eq. 22. Which one is used in the experiments?

Equations 21 and 24: An expression for A is given twice.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

The authors have responded very well to the comments of the reviewer.

There are a few typos:

Line 110: beta_i should be beta_n.

Line 157: beta should be beta_i.

Equation 5: The term (beta_j - overline beta) should be transposed in the second assumption.

Line 169: The second element in y should be y_2'.

Equation 7: The upper left block of V should start with X_1.

An additional comment:

Lines 148-158: The reference to Swamy (1970) should be placed one paragraph earlier, because these lines are from Swamy (1970).

Author Response

There are a few typos:

Line 110: beta_i should be beta_n.

Response: Correction has been done.

 

Line 157: beta should be beta_i.

Response: Correction has been done.

 

Equation 5: The term (beta_j - overline beta) should be transposed in the second assumption.

Response: Correction has been done.

 

Line 169: The second element in y should be y_2'.

Response: Correction has been done.

 

Equation 7: The upper left block of V should start with X_1.

Response: Correction has been done.

 

Lines 148-158: The reference to Swamy (1970) should be placed one paragraph earlier, because these lines are from Swamy (1970).

Response: Correction has been done.