1. Introduction
Tsunamis are ocean surface waves which originate after sudden displacement of the sea bottom. A tsunami propagates long distances across oceans in the form of a series of long gravity waves [1]. Upon arriving at a coast, it pushes huge amount of water onto the shore, referred to as tsunami run-up. Earthquakes are the most significant source of tsunamis [2]. Thus and rightly so, tsunami research has concentrated mostly on the hazard posed by seismic sources. Nevertheless, there occurred hazardous tsunamis triggered by near-shore submarine landslides along subduction zones as well [3]. There are also several well-documented cases with complex generation mechanisms, involving landslides as a secondary hidden mechanism alongside an earthquake, and resulted in a totally unexpected tsunami or an extraordinary event for which abnormally high run-up values were recorded at some sites. Arguably the most typical and well-known example for such an event is the 1998 Papua New Guinea tsunami [4].
In terms of tsunami generation potential, it is known for earthquake-generated (i.e. tectonic) tsunamis that approximately only 1% of the energy of a typical shallow earthquake is converted into tsunami wave energy, although even such a small fraction is enough to generate strong tsunamis [3]. In the case of landslide-tsunamis, on the other hand, there is a dynamic interaction between landslide motion and tsunami wave generation, and the amount of energy release is usually small compared to earthquakes; however, landslides have shown their potential as a ubiquitous and costly natural hazard. The 1958 Lituya Bay, Alaska (USA) landslide is one of the most spectacular examples of a subaerial landslide that caused large localized waves. It was set off from the great strike-slip earthquake in south eastern Alaska on 10 July 1958 [5], [6], [7], [8], [9]. The landslide, which fell into a narrow fjord, caused a slosh wave which reached about 520 m height, then continued out the fjord where it was at about 10 m' height when it first met the ocean and decreased quickly as it spread out in the open ocean [3]. Even though the total displaced water volume was too small to cause a destructive trans-oceanic tsunami, the Lituya Bay event indicated that landslides can generate localized tsunamis that can be extremely hazardous.
Shortly afterwards, another notable event occurred in Europe. On 9 October 1963 a block of nearly 270 million cubic meters was detached from the reservoir of the Vajont Dam in Northern Italy, generated an impulse wave which overtopped the dam and caused an unconfirmed number of more than 2000 casualties, making it the most deadly landslide in Europe [10]. With its 270 m height, the Vajont event is also the second highest tsunami run-up in recorded history that was generated by a landslide [11], [12].
Catastrophic events also occurred recently. In 2007, a rock fell into Lake Lucerne, Switzerland, created an impulse wave that caused slight damage on the opposite shore of the lake when it flowed up into the village of Weggis [13]. In 2010, a tsunami that ran up nearly 40 m on the opposite shore and destroyed trees, roads and campsite facilities at Chehalis Lake, Canada [14], [15]. Many other examples of landslide-tsunamis and impulse waves are summarized in the literature. While [16] addressed events occurred in the Americas, Huber [17] focused on events that occurred in Switzerland in the last 600 years, and [18] examined events occurred in Italy. Review of these studies exposed that landslide tsunamis are more frequent than thought.
The subject attracted researchers especially after the devastating landslide-induced tsunami of 1998 Papua New Guinea and therefore interest and, in direct proportion to this, number of research focusing on analytical [19], [20], [21], [22], [23], [24], [25], [26], [27], numerical [28], [29], [30], [31] and physical [29], [30], [32], [33], [34], [35], [36], [37], [38], [39] aspects of landslide generated tsunamis has increased in the literature in the last two decades. Propagation of landslide generated impulse waves in reservoirs is also studied [40], [41].
It is obviously vital to predict period, run-up height and inundation distance of tsunamis in order to facilitate timely evacuation of coastal populations and minimize economical damage. Physical model studies play an important role in revealing dependence of maximum tsunami run-up on certain landslide parameters. However, they are extensive studies which require infrastructure, time and money. In the present work we suggest artificial neural networks (ANNs) framework to successfully predict the maximum tsunami run-up and to investigate its relationship with the geometrical parameters of the landslide. For this purpose, first, we generated a dataset by calculating the maximum run-up for pre-determined ranges of landslide parameters through an analytical solution. We then proposed ANN-based prediction models equipped with a feature selection algorithm to predict the maximum run-up of landslide-generated tsunami waves and ultimately used these models to determine the parameters that are dominant on the maximum run-up. We note that such an analysis involving many parameters is extremely challenging using an analytical or even a numerical approach, and, to the best of authors' knowledge, this study is among the first to apply machine learning methods to the landslide tsunami problem.
2. Methodology of ANN-based prediction models
2.1. Dataset generation
The role of landslide parameters on the maximum run-up will be investigated through ANN-based prediction models, which apparently require high-quality datasets. However, especially for landslide tsunamis, finding field or even laboratory data is a hard task. Available field data is mostly limited to landslide scarps or deposits, or far field recordings from tide gauges. The datasets needed for prediction of maximum run-up are therefore generated by utilizing an analytical model.
The analytical model utilized for dataset generation is a generalization proposed by Aydın [42] for the analytical solution of Liu et al. [20] (see the Appendix section for the details). For the bottom disturbance we considered a Gaussian profile defined as
which is a translation by an amount of the profile proposed by Liu et al. [20], and a soliton-type profile
recently proposed by Aydın [42]. It is noted that both and are expressed in terms of the distorted spatial variable introduced by Liu et al. [20]. Here is the aspect ratio of the landslide and is the beach slope angle, see the definition sketch provided in the Appendix (Fig. 10). The hyperbolic bottom profile in Eq. (2) is the well-known solitary wave and its shape is very similar to that of the Gaussian profile (1); both profiles assume the same maximum amplitude, but the hyperbolic profile is broader, occupying a larger area by approximately 10% (Fig. 1).
which is a translation by an amount of the profile proposed by Liu et al. [20], and a soliton-type profile
recently proposed by Aydın [42]. It is noted that both and are expressed in terms of the distorted spatial variable introduced by Liu et al. [20]. Here is the aspect ratio of the landslide and is the beach slope angle, see the definition sketch provided in the Appendix (Fig. 10). The hyperbolic bottom profile in Eq. (2) is the well-known solitary wave and its shape is very similar to that of the Gaussian profile (1); both profiles assume the same maximum amplitude, but the hyperbolic profile is broader, occupying a larger area by approximately 10% (Fig. 1).
Fig.1 Comparison of the bottom profiles given in Eqs. (1) and (2), respectively, at initial time , (a) on the slope as functions of , and, (b) on a horizontal plane as functions of . The initial slide submergence is chosen as . |
In Fig. 2 we plotted the tsunami run-up height, which is defined as the time series of free surface elevation at the initial shoreline ( ), i.e. calculated from the analytical solution for a landslide triggered by (1) for three different values of the initial slide submergence. The horizontal slide length ( ) is varied in the left panels and the vertical slide thickness ( ) is varied in the right panels. The maximum tsunami run-up ( ), which is arguably the most important quantity of interest in coastal engineering practice, along with the inundation distance, and is defined as the maximum value of , i.e.
is also indicated in each panel by colored circle. An immediate conclusion that can be drawn from Fig. 2 is that, the maximum run-up variation with is much more significant than with . However, assessing the role of other parameters and determining at least the most and the least dominant parameters on maximum run-up becomes a challenging duty if the analytical solution is desired to be used.
Fig.2 Dependence of run-up on (left) landslide horizontal length for a fixed slide thickness of m, and, (right) slide thickness for a fixed slide length of m. Results for three different values of the initial slide submergence ( , 3, 6) are presented and the maximum run-ups are also indicated by colored circles. The slope angle is chosen as . |
We therefore proceed to utilize the analytical model summarized in the Appendix in order to generate a dataset and to construct ANN-based prediction models so that we can investigate the role of landslide parameters on the maximum run-up. For the purpose of clarification, we remark that we do not intend here to either discuss the qualitative behavior of the analytical model or simulate the analytical model results using ANNs. We rather use the analytical solution in order to provide the required input for the ANN-based prediction models that are suggested to be used when analytical or numerical models are not available or not feasible.
The input data is generated for the two bottom profiles introduced in Eqs. (1) and (2), imposed over the sea bottom as sliding mass. Accordingly, in the dataset the code indicates the cross-section of sliding mass: designates the exponential profile defined in (1), while designates the hyperbolic profile defined in (2).
The other features considered in the prediction model are beach slope angle , initial slide submergence , landslide vertical thickness , landslide horizontal length , and the instant when the maximum run-up is observed. The beach slope angle is varied from 2 to 20 to scatter mild to moderate slopes. The choice of initial slide submergence determines whether the landslide is subaerial or submarine. Since we focus on submarine landslides here, we varied between 1.5 and 6 (dimensionless). Although the spatial variable is later replaced by for the sake of a convenient analytical solution and is accordingly used in the definition of the bottom profiles (1) and (2), in our dataset we preserved the dimensional form of the slide submergence parameter in the -coordinate, defined by
We also remark that in our methodology the landslide thickness is calculated by multiplying the slide length , which is randomly selected between 10 m and 500 m, with the slide aspect ratio , the range of which is determined in accordance with the thin-slide assumption .
The output or response variable is the maximum tsunami run-up, . Fig. 2 shows dependence of on the predictors defined above. Although we observed greater values within the respective ranges of the predictors, the maximum run-up is narrowed down to vary between 0.5 and 25 m, so that our dataset reflects a realistic range observed in the measurements from historical tsunamis [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58]. This limitation eventually reduced the number of rows in our dataset from 57,926 to 54,952.
2.2. Fundamentals of ANNs
2.2.1. General structure
Today's technology leads us to find and use better computation and modeling tools. The term soft computing is a recently coined term describing the symbiotic use of many emerging computing disciplines [59]. described it to be tolerant of imprecision, and uncertainty, in contrast to the traditional hard computing. Soft computing represents the combination of relatively new methods such as fuzzy logic, probabilistic reasoning, genetic algorithms, and ANNs in an incredibly wide variety of application areas, by providing us complementary methods to solve complex real-world problems with cost and time economy. A review of application of soft computing techniques for coastal studies can be found in Dwarakish and Nithyapriya [60].
ANNs are computational structures that can be trained to learn patterns from examples. They were firstly explored by Rosenblatt [61] and [62] at the end of 1950s. Artificial neurons, which are designed to imitate biological neurons, are the core components of ANNs, so that the learning process of human brain can be simulated artificially using mathematical methods. With complex architecture capable of finding correlations between observed data without any assumption on the physics of the considered model or phenomenon [63], they recently started replacing classical mathematical models in an attempt to simulate complex multi-parameter engineering problems. Furthermore, with the massive and ever increasing amount of data becoming available, ANNs are gradually becoming more important and widely implemented in every discipline. In proportion to this, it is possible to find applications of ANN-based models within different aspects of the field of coastal and ocean engineering in many recent studies, e.g. prediction of wind and wave climate [64], [65], minimizing the loss of submarine transmission lines [66], energy efficiency in sea freight transportation [67], just to name a few.
There are several types of ANNs introduced in the literature. Among these, the multi-layer perceptron (MLP) is a well-known and probably the most used network in which every node in a layer connects to each node in the following layer and exits on the output nodes. An MLP consists of at least three layers: an input layer accepting the inputs (also called features), one or more hidden layers processing the inputs, and an output layer producing the result (Fig. 3). MLP employs activation functions, also known as transfer functions, in order to find the mathematical relationship between the input and the output patterns. Nonlinear activation functions such as logistic (sigmoid) and hyperbolic tangent (tanh) are usually utilized for datasets that cannot be separated linearly. Usually, networks with one hidden layer consisting of sufficient number of neurons can adequately fit most finite input-output mapping problems [68]. We therefore constructed an MLP-based prediction model with one hidden layer, employing the linear and the logistic activation functions in the hidden and the output layers, respectively. The Levenberg-Marquardt back-propagation algorithm is used for the network training (learning) algorithm and the maximum number of passes of the training dataset through the network, also called epochs, is limited to 100 in the model. Number of neurons in the hidden layer is decided by a trial-error approach, as detailed in Section 2.3.
Fig.3 Architecture of an MLP network model. |
2.2.2. Feature selection
Features of the data that are used to train the models have great influence on the model's performance. Irrelevant or less relevant features might negatively impact the model [69]. While feature selection is accepted as an essential algorithm for reducing number of features in a dataset in order to increase the speed of the learning process, it also helps understand and manage the dataset more easily. Reduction of computational cost and over-fitting possibility and improvement of accuracy are among the other advantages of feature selection [70]. Many systematic methods can be found to handle the selection process. In this study we use the ReliefF algorithm for regression in order to rank the predictors according to their importance, which is determined based on their scores [71].
2.2.3. K-fold cross-validation
Cross-validation is used to assure the generalization ability of the prediction model. In k-fold cross-validation the data is first segmented into equally-sized partitions randomly, out of which folds are used for learning and the remaining fold is reserved for testing. Subsequently, iterations of training and testing are performed and the accuracy of the prediction model is assumed to be the average accuracy of these iterations [72]. Fig. 4 illustrates the algorithm for .
Fig.4 Implementation of the 10-fold cross-validation algorithm. For each of the 10 test sets, 10-fold cross-validation is employed on the training set. The accuracy of each prediction is the average of the folds. |
2.2.4. Performance metrics
While constructing and testing an appropriate MLP-based prediction models are real issues, evaluating the outputs is another matter. The model performance can not be measured precisely unless the results are evaluated correctly. Performance metrics are the tools that evaluate the statistical performance of forecasting ability of the prediction model. [73] suggested the use of various metrics such as Mean Error, Mean Absolute Error (MAE), Mean Percentage Error, Mean Absolute Percentage Error (MAPE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Standard Deviation of Errors, and Multiple Correlation Coefficient (R). Among these we will consider MSE, RMSE, MAE, MAPE, and R as performance measures in our study. While MSE will be considered in the optimization phase, the others will be used to test the performance of the models.
MSE is the average of squares of the differences between predicted and actual values. Therefore, for a test data of rows, if is the actual value and is the predicted value, then MSE is calculated as
RMSE is the square root of MSE: MSE/RMSE measures the average magnitude of the error; hence, as expected, smaller MSE/RMSE is preferred.
MAE represents the average absolute difference between the actual and predicted values, i.e.
while MAPE represents the percentage of average absolute relative error between the actual and predicted values, i.e.
As in the case for MSE/RMSE, smaller MAE/MAPE is, more successful the prediction is.
The multiple correlation coefficient R indicates the strength of the relationship between actual and predicted values. It is given by
where is the average of the actual values. It is desired that R .
2.3. Implementation of prediction models
As stated in Section 2.2.1 above, MLP is preferred in the present study due to its ability to adequately fit finite input-output mapping problems. The flowchart of MLP-based prediction models combined with feature selection is presented in Fig. 5. The number of inputs is six ( , , , , , and ), and the maximum run-up ( ) is the only output of the dataset (Section 2.1).
Fig.5 (a) Flowchart showing implementation steps of the MLP-based prediction models. (b) Flowchart showing implementation steps of the feature selection algorithm. |
Implementation steps of the MLP-based prediction model are illustrated in Fig. 5(a). The cross-validation algorithm with is first used to randomly split the dataset into ten subsets, nine of which are used as training data and the remaining tenth is used as test data as in Fig. 4. Then, for each fold, the training data is used to determine the optimum hidden layer size, i.e. the optimum number of neurons used in the hidden layer, which is a crucial step in the MLP. Afterwards, the prediction model constructed with the optimized number of neurons in the hidden layer is tested against the test data and RMSE, MAE, MAPE and R are calculated for the fold. Finally, the performance of the prediction model is determined by calculating each metric's average over the folds.
In a similar fashion, while optimizing the hidden layer size, each training dataset is further split into four subsets (i.e. 4-fold cross-validation), three of which are used for training and the remaining quarter is separated for test. The number of neurons is then changed between 2 and 20, the average MSE is recorded in each case and the neuron number with minimum average MSE is decided to be the optimum number of neurons in the hidden layer. It is important here to note that the fold's test data is not used in this optimization phase.
In Fig. 5(b) we illustrate implementation of the backward feature elimination algorithm. Six models are constructed for the six predictors, disregarding the feature with the worst score in each following model. The details are explained in Section 3.
All computations are performed in the MATLAB™ environment (Release 2019a, Update 4) on a 64-bit Windows 10 workstation.
3. Results and discussion
In order to reduce the computational cost, subsets of various size are sampled and a preliminary dataset selection study is performed. For this purpose, eight datasets having 3000 to 24,000 randomly chosen rows with equal increment are prepared and a representative dataset is selected based on the MSE value, since MSE is a good indicator of the model performance in the optimization phase (see Section 2.3 for details). The average MSE values from five runs are plotted against the size of the dataset in Fig. 6. The results show that MSE is not improving for datasets larger than 9000 rows (hereinafter the dataset including 9000 rows will be referred to as 9K).
Fig.6 Variation of average MSE with the dataset size. |
We compare in Fig. 7 the maximum run-up distribution, i.e. scatter plot, for the whole 54,952-row dataset and the 9K dataset with respect to each feature, excluding those with discrete values, namely the bottom profile and the beach angle. We observe that the structure of the whole dataset is preserved in the 9K subset for all features. The descriptive statistics of the variables in the 9K dataset is tabulated in Table 1 and a box-whiskers plot is also provided in Fig. 8. The mean and the standard deviation of the 9K dataset are in great agreement with those of the whole dataset, as shown in the relevant columns of Table 1.
Fig.7 Scatter plots of the output variable ( ) for (a) the whole dataset of 54,952 rows, and (b) the 9K dataset. The predictors from top to bottom are the initial slide submergence, the maximum slide thickness, the maximum slide length, and the time of the maximum run-up. |
Table 1 Descriptive statistics of the variables in the 9K dataset except for the bottom profile code. The means and the standard deviations for the whole dataset are also provided. |
| Variable | Minimum | Maximum | Mean | Standard deviation | ||
|---|---|---|---|---|---|---|
| Empty Cell | 9K | 9K | 9K | Whole data | 9K | Whole data |
| (m) | 15.29 | 2999.84 | 836.48 | 841.70 | 680.30 | 684.05 |
| (degrees) | 2 | 20 | 9.86 | 9.84 | 5.87 | 5.88 |
| (m) | 0.76 | 65.57 | 18.33 | 18.42 | 12.58 | 12.60 |
| (m) | 10.14 | 499.99 | 218.94 | 220.17 | 131.23 | 131.29 |
| (sec) | 7.42 | 292.51 | 72.63 | 73.03 | 39.25 | 39.43 |
| (m) | 0.50 | 24.99 | 9.64 | 9.69 | 6.84 | 6.85 |
Table 2 Predictors of the 9K dataset are ranked according to their ReliefF scores ( left), based on which six models (M6-M1) are constructed using backward feature elimination ( right). |
| Predictor | ReliefF score | Model | Features |
|---|---|---|---|
| -0.0042 | M6 | , , , , , | |
| -0.0016 | M5 | , , , , | |
| -0.0010 | M4 | , , , | |
| 0 | M3 | , , | |
| 0.0011 | M2 | , | |
| 0.0268 | M1 |
Fig.8 Box-whiskers plot for the descriptive statistics of the 9K dataset. |
Based on these observations, we consequently decided to perform calculations on the 9K dataset with the aim of proposing prediction models with low computational cost. This approach brought great time economy since we ran our models multiple times to assure stable results.
In an effort to see the influence of each feature on the maximum run-up, the features are ranked by using the ReliefF algorithm (Table 2, left-hand side). This algorithm ranks the predictors according to their influence on the output by assigning a score to each predictor; predictor with a smaller score indicates that it has less effect on the output compared to another predictor with higher score. Six models, labeled as M6 through M1, are then constructed by means of backward feature elimination. While M6 employs all six features, we disregard the feature with the worst score one by one in the subsequent models (Table 2, right-hand side).
Results of the prediction models are summarized in Table 3 and the performance metrics of each model are plotted in Fig. 9. In an attempt to minimize any possible abnormal increase or decrease in the performance metrics due to randomness of data and to present stable results, we reported the average results of 50 runs for each prediction model.
Table 3 The performance metrics (Mean Standard deviation of the 10-folds, 4-subfolds and 50 runs) of the models M6-M1 with predictors given in Table 2. |
| Model | RMSE (m) | MAE (m) | MAPE (%) | R |
|---|---|---|---|---|
| M6 | 0.051 0.005 | 0.037 0.004 | 1.109 0.095 | 0.999 |
| M5 | 0.055 0.005 | 0.039 0.004 | 1.194 0.115 | 0.999 |
| M4 | 1.425 0.009 | 0.986 0.009 | 14.082 0.179 | 0.978 |
| M3 | 1.581 0.002 | 1.123 0.002 | 14.300 0.091 | 0.973 |
| M2 | 1.774 0.002 | 1.249 0.002 | 15.380 0.093 | 0.966 |
| M1 | 1.851 0.002 | 1.292 0.003 | 15.817 0.145 | 0.963 |
Fig.9 Performance metrics of the models as tabulated in Tables 3 and 4. |
Performance metrics in model M6, which includes all features, and model M5, which disregards the beach slope angle, are at the same order, confirming that the beach slope is the least effective parameter on the maximum run-up of landslide-generated tsunamis. When the landslide horizontal length is further excluded (M4), however, there is a significant increase in the error metrics and decrease in R. Performances of the models get worse when we further ignore the initial slide submergence in M3, the bottom profile in M2, and the time when the maximum run-up is recorded in M1. These parameters can therefore be said to moderately affect . Based on the scores presented in Table 2 and the metric results tabulated in Table 3, the landslide vertical thickness is apparently the predictor that has greatest impact on the maximum run-up.
In an attempt to validate and generalize the methodology developed here, we tested the models created above with an independent dataset. For this purpose, we first generated a new dataset with 1000 rows, randomly selected from the whole dataset, separate from the 9K dataset used above. This new dataset, named as 1K for convenience, is then given as test data to the optimized models and the resulting performance metrics are reported in Table 4 and plotted in Fig. 9. As seen, the models produce similar results for both datasets, indicating the accuracy and precision of the proposed models.
Table 4 The performance metrics (Mean Standard deviation of the 50 runs) of the models M6-M1 for the 1K independent dataset. |
| Model | RMSE (m) | MAE (m) | MAPE (%) | R |
|---|---|---|---|---|
| M6 | 0.052 0.004 | 0.037 0.003 | 1.155 0.087 | 1.000 |
| M5 | 0.059 0.004 | 0.040 0.003 | 1.360 0.094 | 0.999 |
| M4 | 1.380 0.007 | 0.952 0.007 | 14.016 0.171 | 0.980 |
| M3 | 1.520 0.002 | 1.083 0.001 | 14.341 0.067 | 0.976 |
| M2 | 1.743 0.002 | 1.218 0.001 | 15.570 0.086 | 0.968 |
| M1 | 1.837 0.003 | 1.262 0.001 | 16.096 0.098 | 0.964 |
Consequently, based on our methodology, the slide thickness ( ) and the beach slope angle ( ) have been respectively the most and the least effective parameters on the maximum run-up of landslide tsunamis, while the time of maximum run-up ( ), the cross-section of the slide profile ( ), the initial slide submergence ( ), and the slide length ( ) have had moderate effects (Tables 2 and 3).
In spite of the simple character of the analytical model employed to generate the data source, the results presented above have shown good agreement with the literature. Comprehensive experimental studies for solid-block [29] and deformable granular landslides [35] sliding over an inclined plane revealed the significance of slide thickness as a primary factor governing the maximum run-up, while it is influenced secondarily by slide volume, implying the role of slide length (and width in the case of three dimensional analysis). A similar conclusion was also drawn for circular geometry (i.e. conical island) by Romano et al. [38], who found out that slide thickness significantly affects the wave amplitudes in the near field.
We would like to make a final remark about the beach slope, which is referred to as the least effective parameter on the maximum run-up. Analytical [20], [42] and experimental studies [29] have shown that the maximum run-up has no dependency on the beach slope in the case of subaerial slides, unlike the case of submarine slides. This does not contradict with the results presented above; our findings summarized in Tables 2 and 3 can rather be interpreted as an indication of a limited impact of beach slope on the maximum run-up, compared to the other parameters considered in the analysis. However, it should be noted that the conclusions above remain valid under the plane beach assumption. Also, irregularities in the ocean bathymetry should for sure be considered for a more realistic modeling approach (see the discussion in Bandyopadhyay et al. [74]).
4. Conclusions
The aim of this paper was to propose MLP-based prediction models for the maximum run-up of landslide-generated tsunamis and to determine the hierarchy of the parameters affecting the maximum run-up through these models. For this purpose, the data generated from a linear analytical solution was analyzed by means of MLP utilized with a feature selection algorithm. A preliminary analysis on randomly selected multiple datasets opted us into a 9,000-row set, which is smaller than one sixth of the original dataset and hence is economical in terms of computational load. The feature selection algorithm ReliefF is then used to score the features, guiding us to construct six models by eliminating the features one by one according to their scores. The model performances are compared by calculating RMSE, MAE, MAPE, and R. The results indicated that the model with all features can predict the maximum run-up with a MAPE value of 1.109%, while the model excluding the beach slope is still very accurate, with a MAPE of 1.194%. The thickness of the slide ( ) is found to be the most effective parameter on the maximum tsunami run-up, while the angle of the beach slope ( ) is found as the least effective. The results obtained here are also shown to be in accordance with the results reported in the literature.
In conclusion, the methodology developed here can serve for a fast, flexible and reliable alternative when there is lack of a qualitative tool as such as an analytical model, or when there are too many parameters so that ranking them based on their effectiveness on the maximum run-up through an analytical model becomes a difficult task.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Appendix A. The analytical model and its limitations
In this section we provide a short summary of the analytical model used to generate the data required for ANN-based prediction models.
Although nonlinearity and frequency dispersion play important roles during propagation of landslide-generated tsunamis, the resulting governing equations become challenging in terms of an analytical solution. Also given that our focus is on the tsunami run-up, rather than the propagation characteristics of the subsequent waves, and that, once constructed, the methodology proposed here can be tested on different datasets obtained from nonlinear and/or dispersive models, we used the readily available linear nondispersive analytical solution of Liu et al. [20] adopted to the case of submarine landslides [42].
Assuming small amplitude waves (i.e. , ) propagating over a plane beach with slope , [20] formulated the forced linear shallow-water wave equation in nondimensional form for thin slides (i.e. ) as
where stands for the slide aspect ratio (Fig. 10). Liu et al. [20] introduced the change of variables and rewrote the above equation as
Fig.10 Definition sketch for the landslide problem (not to scale). and respectively indicate the maximum vertical thickness and the maximum horizontal length of the sliding mass at its initial location ( ). The undisturbed water depth is , where is the beach angle with the horizontal. SWL stands for the still water level. (Modified from [42]). |
Analytical solution of this equation can uniquely be determined given two initial conditions defining an initially quiet sea: zero initial wave height, , and zero initial wave velocity, , or . As expected, the total solution of this nonhomogeneous initial value problem is the sum of homogeneous and particular solutions, i.e. . Utilizing the Hankel integral transform of order zero, [20] obtained the homogeneous and particular solutions as in is the Bessel function of order zero. The versatility of the above solution is that the particular solution is derived in terms of the bottom perturbation . The coefficients and are found from the initial conditions as
Liu et al. [20] then analyzed evolution and run-up of a Gaussian-type subaerial seafloor deformation, . Very recently, Aydın [42] generalized [20]'s approach to submarine landslides by translating the bottom disturbance by an amount , i.e. , which apparently does not spoil the particular solution . Aydın [42] further demonstrated that with profiles of the form remains as particular solution of (3) and proposed a second bottom profile through .
[20] further employed a nonlinear numerical model to compare with and discuss the limitations of the proposed analytical model. They concluded that the analytical solution provides an accurate representation of the physics of the problem for large (i.e. ), and that even for smaller the difference between the linear analytical and nonlinear numerical models is less than 10%. Hence, the analytical solution used here can be considered as a simple yet proper model.
Supplementary material
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.joes.2022.05.00710.1016/j.joes.2022.05.007.
