Tsunami generation and propagation caused by stochastic seismic fault driven by two Gaussian white noises in the - and -directions are investigated. This model is used to study the tsunami amplitude amplification under the effect of the noise intensities, spreading uplift length and rise times of the three-dimensional stochastic fault source model. Tsunami waveforms within the frame of the linearized shallow-water theory for constant water depth are analyzed analytically by transform methods (Laplace in time and Fourier in space). The amplification of tsunami amplitudes builds up progressively as time increases during the generation process due to wave focusing while the maximum wave amplitude decreases with time during the propagation process due to the geometric spreading and also due to dispersion. The maximum amplitude amplification is proportional to the propagation length of the stochastic source model and inversely proportional to the water depth. The increase of the normalized noise intensities on the bottom topography leads to an increase in oscillations and amplitude in the free surface elevation. We derived and analyzed the mean and variance of the random tsunami waves as a function of the propagated uplift length, noise intensities, and the average depth of the ocean along the generation and propagation path.

1. Introduction

A tsunami is most often triggered by undersea earthquakes that cause massive changes to the ocean floor. Tsunami can also be caused by other undersea events such as volcanoes or landslides. When an undersea earthquake or another major disturbance causes a section of the ocean floor to suddenly rise or sink, the mass of water above the affected area also rises or sinks. This unexpected movement of the water creates a series of powerful waves. Undersea earthquakes that cause massive changes to the ocean floor and the displacement of a large volume of water are the most common cause of a tsunami. In recent years, there have been a number of subduction zone earthquakes that have generated unexpectedly large local tsunamis, for example, Papua New Guinea tsunami, 17 July, 1998, Sumatra earthquake and tsunami, 26 December, 2004, Solomon Islands tsunami, 2 April, 2007, Samoa tsunami, 29 September, 2009, the Chile tsunami, 27 February, 2010, and the Fukushima Prefecture, Japan, 11 March, 2011. The investigation of the tsunami waves has become of great practical interest that attracts nowadays a lot of attention due in part to the intensive human activity in coastal areas. The evaluation of a local tsunami threat is useful to get a more effective measure for tsunami warning systems and for protection works. The massive destruction and loss of life associated with the recent tsunamis has underscored the need to develop and implement tsunami hazard mitigation measures. In recent years, significant advances have been made in developing mathematical models to describe the entire process of tsunami event generated by seismic seafloor deformation caused by an underwater earthquake; see [13]. Numerical models based on the nondispersive shallow-water equations are often used to simulate tsunami propagation and run-up (e.g., [4, 5]). The propagation of these waves is strongly influenced by the shape of the bottom. The problem of tsunami source reconstruction is a key concern of tsunami research. Certainly, the best way to solve this problem is to measure tsunami waves in the open ocean near to the source area and far away from the distorting influence of the near-shore topography. The sea bottom deformation following an underwater earthquake is a complex phenomenon. This is why, for theoretical or experimental studies, researchers have often used uniform bottom motions such as the vertical motion of a box. Most investigations of tsunami generation and propagation used integral solution (in space and time) for an arbitrary bed displacement based on a linearized description of wave motion in either a two- or three-dimensional fluid domain of uniform depth. The complexity of the integral solutions developed from the linear theory even for the simplest model of bed deformation prevented many authors from determining detailed wave behavior, especially near the source region. In reality the sea bottom deformations following an underwater earthquake are characterized by some rugosity. In practice, the available data about the geometries of the source model are always subject to some uncertainties. The missing information can be modeled by the inclusion of random effects. These circumstances have led several authors to consider water wave propagation in random media; see [68]. In this paper, a stochastic model is proposed to describe the bottom irregularity and its effect on the generation and propagation of the tsunami waves. Moreover, the tsunami wave resulting from the random bottom source model is compared with the tsunami wave due to simplified uniform bottom topography in the form of sliding Heaviside step function. Using a stochastic source model, it is demonstrated that local tsunami amplitudes vary by as much as a factor of two or more, depending on the local bathymetry. If another earthquake source parameter such as focal depth is varied in addition to the slip distribution patterns, even greater uncertainty in local tsunami amplitude is expected.

Most previous research has modeled tsunami waves under the effect of deterministic source model. Hammack [9] studied experimentally the generated waves by raising or lowering a box at one end of a channel. He considered two types of time histories: an exponential and a half-sine bed movement. Todorovska and Trifunac [10] investigated the generation and propagation of waves by a slowly spreading uplift of the bottom in linearized shallow-water wave theory. They showed that the effects of the spreading of the ocean floor deformation (faulting, submarine slides, or slumps) on the amplitudes and periods of the generated tsunamis are the largest when the spreading velocity of uplift and the tsunami velocity are comparable. Todorovska et al. [11] investigated tsunami generation by a slowly spreading uplift of the sea floor in the near field considering the effects of the source finiteness and directivity. They described mathematically various two-dimensional kinematic models of submarine slumps and slides as combinations of spreading constant or slopping uplift functions. Their results show that, for given constant water depth, the peak amplitude depends on the ratio of the spreading velocity of the sea floor to the long wavelength tsunami velocity; see [12]. Hayir [13] investigated the motion of a submarine block slide with variable velocities and its effects on the near-field tsunami amplitudes. He found that the amplitudes generated by the slide are almost the same as those created by its average velocity. Both Trifunac et al. [12] and Hayir [13] used very simple kinematic source models represented by Heaviside step functions for representing the generation of tsunami. Dutykh et al. [14] studied the generation of long wave through the ocean by a moving bottom. They demonstrated the differences between the classical approach (passive generation) and the active generation under the effect of the bottom motion. Abou-Dina and Hassan [1] constructed a numerical model of tsunami generation and propagation depending on a nonlinear theory under the effect of a variable bed displacement with constant water depth. They considered nonlinearities and omitted the linear effects of frequency dispersion. Dutykh and Dias [15] studied theoretically the generated waves by multiplying the static deformations caused by slip along a fault by various time laws: instantaneous, exponential, trigonometric, and linear. Kervella et al. [16] performed a comparison between three-dimensional linear and nonlinear tsunami generation models. They observed very good agreement from the superposition of the wave profiles computed with the linear and fully nonlinear models. In addition, they found that the nonlinear shallow-water model was not sufficient to model some of the waves generated by a moving bottom because of the presence of frequency dispersion. Moreover, they suggested that for most events the linear theory is sufficient. Hassan [2] discussed the solution of the nonlinear problem of propagation of waves generated in a homogeneous fluid, occupying an infinite channel, by the bounded motion of the bottom. He demonstrated that the predictions of the linear theory are in good agreement with those of the nonlinear theory for sufficiently small amplitude of the bottom’s motion. Hassan et al. [17] investigated the tsunami evolution during its generation under the effect of the variable velocities of realistic submarine landslides based on a two-dimensional curvilinear slide model. They described the tsunami generation from submarine gravity mass flows in three stages: the first stage represented by a rapid curvilinear down and uplift faulting with rise time, the second stage represented by a unilaterally propagation in the positive direction to a significant length to produce curvilinear two-dimensional models represented by a depression slump and a displaced accumulation slide model, and the last stage represented by the time variation in the velocity of the accumulation slide (block slide) by using transforms method. Ramadan et al. [18] studied the nature of the tsunami build-up and propagation during and after realistic curvilinear source models represented by a slow uplift faulting and a spreading slip-fault model. They studied the tsunami amplitude amplification as a function of the spreading velocity and rise time. They also analyzed the normalized peak amplitude as a function of the propagated uplift length, width, and the average depth of the ocean. Ramadan et al. [19] investigated the tsunami generation and propagation resulting from a spreading of a seismic faulting in two orthogonal directions in linearized shallow-water wave theory. They found that the normalized peak amplitudes were smaller when the slip-fault spreads in two orthogonal directions than the numerical values for one-dimensional solution because of the interaction of the velocities.

It is becoming widely recognized that bottom topography is irregular and difficult to predict and its aspects are best revealed through random source models which are more realistic, because sources of tsunamis are generally uncorrelated. Despite this, a few of the analytical and numerical studies considered stochastic source models for the investigation of the generation and propagation of tsunami waves. This is due to the complexity in the mathematical modeling and analysis of the stochastic case compared to the deterministic case. The main complexity in the stochastic case arises from the difficulty in the derivation of the integral solution of the random profile. Probability of tsunami occurrence frequency on the coast of California was evaluated first by Wiegel [20]. Rascón and Villarreal [21] carried out a statistical study to estimate a stochastic possibility of tsunami hit on the Pacific coast of Mexico. Successively, similar studies have been undertaken to know the local property of the tsunami occurrence frequency. Nakamura [22] considered an extended Poisson process in order to get a better fit to the exceedance probability of local tsunami. Geist [23] determined the effect that ruptures complexity, by way of different slip distribution patterns on the local tsunami wave field under the effect of stochastic source model. Geist [24] developed rapid tsunami models to forecast far-field tsunami amplitudes from initial earthquake information using stochastic source model. de Bouard et al. [7] studied the motion of the free surface of a body of fluid over a variable bottom, in a long wave asymptotic regime. They focused on the two-dimensional case, assuming that the bottom of the fluid region can be described by a stationary random process whose variations take place on short length scales and which are decorrelated on the length scale of the long waves. Craig et al. [25] gave a new derivation and an analysis of long wave model equations for the dynamics of the free surface of a body of water over random variable bottom and illustrated several numerical simulations of nonlinear waves. They considered the case where the bottom is random with uniform statistical properties. They described the asymptotic regime of small amplitude long waves, for which the correlation length of the bottom is short compared to the wavelength. Manouzi and Seaïd [26] developed an efficient numerical method for solving stochastic water waves of Wick type driven by white noise using combination of the Wiener-Itô chaos expansion with a Galerkin finite element method. Their method transforms the stochastic shallow-water equations into a system of deterministic shallow-water equations that are solved for each chaos coefficient in the solution expansion. Dutykh et al. [27] estimated the maximum wave run-up height on a random slope plain shore by solving the classical nonlinear shallow-water equations using finite volume schemes. They compared the results of the random bottom model with the more conventional approaches. Omar et al. [28] studied the nature of the tsunami generation and propagation resulting from unilateral spreading in the -direction of a stochastic source model driven by a Gaussian white noise. Ramadan et al. [29] investigated the tsunami generation and propagation caused by stochastic submarine landslides and slumps driven by a Gaussian white noise in the - and -direction. They found that the increasing in the normalized noise intensities increases the oscillations of the free surface elevation and an increase in the difference in the peak amplitudes of the waveforms in the generation process, while in the propagation process, shows more dispersion and oscillation in the propagated free surface elevation.

In this paper, we investigate the tsunami wave in the near- and far-field using the transform methods (Laplace in time and Fourier in space). A linearized solution for constant water depth is derived for a random sea floor uplift represented by a sliding Heaviside step function under the influence of two independent Gaussian white noise processes in the - and -directions. The appearance of the Gaussian white noise (an irregular stochastic process) in our bottom topography leads to complications in the integrals resulting from the transform methods. These integrals were found to be of a stochastic type, which are not solvable as a Riemann or Lebesgue integral. Stochastic integrals are mainly classified as Itô or Stratonovich integrals; see [3034]. In our work, we consider the resultant integrals in Itô sense, which can be solved by many numerical methods; see [31, 33, 34]. The objective of this study is to determine the effect of the stochastic bottom topography on the generation and propagation of the tsunami wave form and discuss aspects of tsunami generation that should be considered in developing this model as well as the propagation wave after the formation of the source model has been completed. Due to the stochastic nature of the bottom topography, the free surface elevation is considered to be a stochastic process; thus, for each realization (sample path) of the bottom topography, there exists a corresponding realization for the free surface elevation. Studying just one realization of the bottom topography has no significance and cannot be used to analyze the behavior of all possible solutions. Therefore, we derive the mean and variance of the free surface elevation. Moreover, we study the fluid wave motion above the finite source, with different noise intensities of the stochastic source model and with variable propagation of the ocean floor uplift, for constant spreading velocity. The effects of dispersion due to the random bottom topography on the tsunami waveforms are treated analytically by means of measuring the variance and illustrating it graphically. In addition, we investigate the mean and the variance of the tsunami waveforms in the near-field and far-field as function of the spreading length, noise intensities, and the water depth.

2. Mathematical Formulation of the Problem

Consider a three-dimensional fluid domain as shown in Figure 1. It is supposed to represent the ocean above the fault area. It is bounded above by the free surface of the ocean and below by the rigid ocean floor , where is the free surface elevation, is the water depth, and is the sea floor displacement function. The domain is unbounded in the horizontal directions and and can be written as . For simplicity, is assumed to be a constant. Before the earthquake, the fluid is assumed to be at rest; thus the free surface and the solid boundary are defined by and , respectively. Mathematically, these conditions can be written in the form of initial conditions: . At time , the bottom boundary moves in a prescribed manner which is given by . The deformation of the sea bottom is assumed to have all the necessary properties needed to compute its Fourier transform in and and its Laplace transform in . The resulting deformation of the free surface is to be found as part of the solution. It is assumed that the fluid is incompressible and the flow is irrotational. The former implies the existence of a velocity potential which fully describes the flow and the physical process. By definition of , the fluid velocity vector can be expressed as . Thus, the potential flow must satisfy Laplace’s equation The potential must satisfy the following kinematic and dynamic boundary conditions on the free surface and the solid boundary, respectively: where is the acceleration due to gravity. As described above, the initial conditions are given by

2.1. Linear Shallow-Water Theory

Various approximations can be considered for the full water wave equations. One is the system of nonlinear shallow-water equations that retains nonlinearity but no dispersion. Solving this problem is a difficult task due to the nonlinearities and the a priori unknown free surface. The concept of shallow water is based on the smallness of the ratio between water depth and wave length. In the case of tsunamis propagating on the surface of deep oceans, one can consider that shallow-water theory is appropriate because the water depth (typically several kilometers) is much smaller than the wave length (typically several hundred kilometers), which is reasonable and usually true for most tsunamis triggered by submarine earthquakes, slumps, and slides; see [9, 15]. Hence, the problem can be linearized by neglecting the nonlinear terms in the boundary condition (2) and applying the boundary conditions on the nondeformed instead of the deformed boundary surfaces (i.e., and on instead of and ). The linearized problem in dimensional variables can be written as subjected to the following boundary conditions: The linearized shallow-water solution can be obtained by the Fourier-Laplace transform.

2.2. Solution of the Problem

Our interest is the resulting uplift of the free surface elevation . An analytical analysis is carried out to illustrate the generation of a tsunami for a given bed profile . Mathematical modeling is carried out to obtain waves generated by vertical and lateral displacements of underwater earthquake using the combined Fourier-Laplace transform of the Laplace equation analytically. All our studies took into account constant depths for which the Laplace and fast Fourier transform (FFT) methods could be applied. Equations (4), (5a), (5b), and (5c) can be solved by using the method of integral transforms. We apply the Fourier transform in as follows: with its inverse transform and the Laplace transform in time Combining (5a) and (5c) yields the single free surface condition After applying the transforms and using the property , the initial conditions (3), (4), (5b), and (9) become The transformed free surface elevation can be obtained from (5c) as A general solution of (10) will be given by where . The functions and can be easily found from the boundary conditions (11) and (12): Substituting (15a) and (15b) in the general solution (14) yields where is the circular frequency of the wave motion. The free surface elevation can be obtained from (13) as The circular frequency describes the dispersion relation of tsunamis and implies phase velocity and group velocity . Hence, , and . Since , hence as , both and , which implies that the tsunami velocity for wavelengths long compared to the water depth . The free surface elevation can be evaluated for a specified by obtaining its transform , then substituting it into (17), and inverting analytically to obtain which is further converted to by using double inverse Fourier transform. The above linearized solution is known as the shallow-water solution. We considered the stochastic source model for the sea floor displacement as a variable slip-fault, propagating unilaterally in the positive -direction for time with finite velocity . In the -direction, the model randomly propagates instantaneously.

The free surface elevation can be evaluated by using inverse Laplace transform of from the Convolution theorem as follows: The sea floor displacement is taken as where where represents the Heaviside function, denotes the initial uplift of the bottom topography, and denote two independent Gaussian white noise processes with two real valued parameters that control the strength of the induced noises in the - and -directions, respectively, and is the rapture velocity of the stochastic bottom in the -direction.

Laplace and Fourier transforms can now be applied to the bottom topography (19) to obtain where where and are the propagated length and width of the stochastic bottom topography in the - and -direction, respectively.

The transformed free surface elevation, , can be obtained by substituting (22) into (17) as where Substituting (24) into (26) yields The free surface elevation can be evaluated by using inverse Laplace transform of as follows: where . The integrals in (28) are stochastic integrals that can be considered as Itô integrals. Such integrals can be solved by various methods; see [31, 34]. The solution for can be obtained from (23) and (26) as which can be transformed to the time domain as

This completes the solution of the problem, , in the double Fourier transform domain.

Finally, is evaluated using the inverse fast Fourier transform (IFFT). The IFFT is a fast algorithm for efficient implementation of the inverse discrete Fourier transform (IDFT). In this paper, this inversion is done using the Matlab IFFT algorithm.

In order to implement the algorithm efficiently, singularities should be removed by finite limits as follows.(a)As implies , , and , then and have the following limits: (b)As implies and , then and have the following limits: (c)As implies and , then and have the following limits:

2.3. Mathematical Derivation of the Mean and Variance of the Stochastic Tsunami Waveforms

The presence of Gaussian white noise in the model of the bottom topography leads to randomness in the generated waveforms. These waveforms are considered to be a stochastic process, which has an infinite number of trajectories. All these trajectories are considered as solutions for our stochastic model, but some trajectories may be more probable than others. Thus, instead of dealing with only one of these trajectories, we derive mathematically the mean and variance of the stochastic tsunami waveforms to get a better insight of the overall behavior of the stochastic tsunami waveforms; see [35].

First, we derive the mean of the tsunami waveforms by expressing the transformed free surface elevation, , using (27) and (29) as where Taking the expectation of (34) and using the property that the expectation of an Itô integral equals zero, see [32], yield As the free surface elevation is a continuous function in , thus Fubini’s theorem, see page in Kloeden and Platen [31], can be applied to obtain the mean of the tsunami waveforms as follows: Second, to derive the variance of the tsunami waveforms we need to introduce the following operators.(1)For a function , the double Laplace transform is defined by (2)Let represent the Fourier transform of a function . By , we mean .(3)For a function , we define by In addition to the previous operators, the following two lemmas are needed for the derivation of the variance.

Lemma 1. Consider where denotes the conjugate of the function .

Proof. Consider

Lemma 2. Consider

Proof. Consider Using Lemma 1, we obtain

Let denote the autocorrelation function of the process . It can be shown that Assuming that is the difference between the transformed free surface elevation (34) and its mean (36), then Recalling the fact, see [30, Theorem 4.5, page 99], that if and are regular adapted processes, such that then we have and hence (47) becomes Applying Lemma 2 to (50), we obtainFrom (45) and (51), we can deduce thatFinally, the variance can be obtained by taking , , and in the autocorrelation function (52) as follows:

3. Results and Discussion

In this section, we present numerical results to illustrate the nature of the tsunami build-up and propagation during and after the uplift process of a stochastic bottom topography. This model was used to evaluate tsunami waveforms up to distances of several source dimensions, for various combinations of the model parameters (depth to the sea bed, , fault length and width, and , and time since rupture initiation, ). The numerical results demonstrate the waveform amplification resulting from source spreading in the -direction and wave focusing in the near-field and the tsunami spreading in the far-field under the effect of different factors. When the source process is completed and for rapid lateral spreading, the displacement of the free surface above the source resembles the displacement of the ocean floor; see [10, 18]. For velocities of spreading smaller than , the tsunami amplitudes in the direction of the source propagation become small with high frequencies. As the velocity of the spreading approaches , the tsunami waveform has progressively larger amplitude, with high frequency content, in the direction of the slip spreading; see [10, 18]. These large amplitudes are caused by wave focusing (i.e., during slow earthquakes). Examples of such slow earthquakes are the June , , Chile earthquakes which ruptured as a series of earthquakes for about an hour, see [36], and the February , , Banda Sea earthquake; see [37].

Tsunami generally occurs due to vertical movement of the seafloor that vertically displaces the water column. Large vertical displacement of the sea bottom ground causes a corresponding large motion at the sea surface. The generation of tsunami by vertical displacements of the ocean floor depends on the characteristic size (length and width ) of the displaced area and on the time it takes to spread the motion over the entire source region. Therefore, researchers presented kinematic source models in the form of Heaviside functions to describe the generation of tsunamis; see [912, 38, 39].

It is difficult to estimate, at present, how often the amplification may occur during actual faulting, sliding, or slumping, because of the lack of detailed knowledge about the ground deformations in the source area of past tsunamis. Therefore, we constructed mathematically a reasonable random tsunami source model represented by a sliding Heaviside step function under the influence of two independent Gaussian white noise processes in the - and -directions. Due to the stochastic nature of the tsunami waveforms, we mainly examine the mean and variance of the tsunami generation and propagation under the effect of different factors.

3.1. Time Evolution during Tsunami Generation

The effects of variations of the uplift of the considered stochastic source model in the - and -directions are studied as a function of time on the generation of tsunamis. These effects are studied through the investigation of the generation of tsunamis by unilateral displacements of the ocean floor and under the effect of the characteristic size length of the displaced area, water depth , and normalized noise intensities and , where and . The velocity of the sea floor spreads, , is chosen to be equal to the long wave tsunami velocity , (i.e., maximum amplification). The tsunami waveforms generated for the stochastic source model with normalized noise intensities (deterministic case), , and are illustrated in Figures 2, 3, and 4, respectively. These waveforms are generated at constant water depth , propagated length , and source width at rise time , and , where and . It can be seen from these figures that the amplitude of the wave builds up progressively as increases and the focusing and the amplification of tsunami amplitudes occur above the spreading edge of the source model as seen in Figure 5. This amplification occurs above the source progressively, as the source evolves, by adding uplifted fluid to the fluid displaced previously by uplifts of preceding source segments. As the spreading length in the fault increases, the amplitude of the tsunami wave increases. Moreover, it can be observed from Figure 5 that, at , the deterministic waveform is in complete agreement with the aspect of the tsunami generated by a slowly spreading uplift of the ocean bottom presented by Todorovska and Trifunac [10] who considered a very simple kinematic source model on the form of Heaviside step function.

3.1.1. Effect of Normalized Noise Intensities and on Tsunami Generation Waveform

We studied the effect of the normalized noise intensities on the tsunami generation in Figures 2, 3, and 4. It can be observed from these figures that the increase in the normalized noise intensity leads to an increase in the deformation in the bottom topography, while the wave behavior is mainly the same. Figure 6 presents a collection of the waveforms from Figure 5 at to get a better insight of the effect of the normalized noise intensity on the tsunami generation, where it can be observed that the increase of the normalized noise intensities on the bottom topography leads to slight difference in the peak amplitude of the waveforms in addition to an increase in oscillations in the free surface elevation. These differences can be further investigated by discussing the variance of the generated tsunami waveforms. We investigated the normalized variance () of the tsunami generation in Figures 7 and 8 at , , and at time , and . It can be seen in Figure 8 that the variance increases with the increase in time and the maximum amplitude of the variance occurs above the spreading edge of the bottom topography. The maximum variance occurs at (maximum amplification of the wave). Thus we can conclude that the variance and the tsunami waveforms behave similarly during the generation process under the effect of the normalized noise intensities which is also clear in Figure 7.