Tsunami Excitation by Submarine Slides in Shallow-water Approximation

-- Landslide-induced tsunamis are receiving increased attention since there is evidence that recent large devastating events have been caused by underwater mass failures. Normally, numerical models are used to simulate tsunami excitation, most of which are based on shallow water, known also as long wave, approximation to the full equations of hydrodynamics. Analytical studies may handle only simplified problems, but help understand the basic features of physical processes. This paper is an analytical investigation of long-water waves excited by rigid bodies sliding on the sea bottom, based on the shallow-water approximation, which is here derived by properly scaling Euler equations for an inviscid, incompressible and irrotational ocean. In one-dimensional (1-D) cases (where motion depends only on one horizontal coordinate), under the further assumptions of small-height slide, which permits the recourse to linear theory, and of flat ocean floor, a solution for arbitrary body shape and velocity is deduced by applying the Duhamel theorem. It is also shown that this theorem can be advantageously used to obtain a general solution in case of a non-flat ocean floor, when the sea bottom follows a special power law, that can be adapted to study reasonable bottom profiles. The characteristics of the excited tsunamis are then evaluated by computing solutions in numerous examples, with special focus on wave pattern and wave evolution. The energy of the wave system is shown to depend on time: it grows expectedly in the initial phase of tsunami generation, when the moving body transfers energy to the water, but it may also diminish later, implying that a certain amount of energy may pass back from water waves to the slide.     

Effect of basal guided waves on landslides

A landslide model riding on basal guided waves is investigated to explain lower net frictions at high slide velocities from the wave-theoretical point of view. It is shown that there is a wave propagated along the basal layer at the phase velocity equal to the slide velocity, as well as a guided wave with considerably higher phase velocities propagated likewise along the basal layer as a leaking mode at low slide velocities. With increasing slide velocity the phase velocity of the guided wave decreases until it is equal to that of the slide mass. Over this threshold slide velocity, a sonic boom is generated around the basal layer, and the shock contributes to a loosening of the slide mass into a fluidized state. Landslides on long slide-ways are more liable to exceed this threshold velocity since their slide velocities tend to be higher than those on short slide-ways of a similar shape. Hence, the reduction of net friction of landslides can possibly be better correlated with the lengths of slide-ways than with the volumes of landslides as is widely maintained.     

Analysis of effects of active sources on observed phase velocity based on the thin layer method

In the wave field induced by active sources, the observed phase velocity of surface waves is influenced by both mode incompatibility (i.e. non-planar spread of surface waves is idealized as plane waves) and body waves. Effects of sources are usually investigated based on numerical simulations and physical models. Several methods have been proposed to mitigate the effects. In application, however, these methods may also have difficulties since the energy of the body waves depends on soil stratification and parameters. There are multiple modes of surface waves in layered media, among which the higher modes dominate the wave field for soils with the irregular shear velocity profiles. Considering the mode incompatibility and the higher modes, we derive analytical expressions for the effective phase velocity of the surface waves based on the thin layer stiffness method, and investigate the effects of the body waves on the observed phase velocity through the phase analysis of the vibrations of both the surface waves and the body waves. The results indicate that the effective phase velocity of the surface waves in layered media varies with the frequency and the spread distance, and is underestimated compared to that of the plane surface waves in the spread range less than about one wavelength. The oscillations that appeared in the observed phase velocity are due to the involvement of the body waves. The mode incompatibility can be ignored in the range beyond one wavelength, while the influence range of the body waves is far beyond one wavelength. The body waves have a significant influence on the observed phase velocity of the surface waves in soils with a soft layer trapped between the first and the second layers because of strong reflections.     

Scattered Surface Wave Energy in the Seismic Coda

One of the many important contributions that Aki has made to seismology pertains to the origin of coda waves (Aki, 1969; Aki and Chouet, 1975). In this paper, I revisit Aki's original idea of the role of scattered surface waves in the seismic coda. Based on the radiative transfer theory, I developed a new set of scattered wave energy equations by including scattered surface waves and body wave to surface wave scattering conversions. The work is an extended study of Zeng et al. (1991), Zeng (1993) and Sato (1994a) on multiple isotropic-scattering, and may shed new insight into the seismic coda wave interpretation. The scattering equations are solved numerically by first discretizing the model at regular grids and then solving the linear integral equations iteratively. The results show that scattered wave energy can be well approximated by body-wave to body wave scattering at earlier arrival times and short distances. At long distances from the source, scattered surface waves dominate scattered body waves at surface stations. Since surface waves are 2-D propagating waves, their scattered energies should in theory follow a common decay curve. The observed common decay trends on seismic coda of local earthquake recordings particular at long lapse times suggest that perhaps later seismic codas are dominated by scattered surface waves. When efficient body wave to surface wave conversion mechanisms are present in the shallow crustal layers, such as soft sediment layers, the scattered surface waves dominate the seismic coda at even early arrival times for shallow sources and at later arrival times for deeper events.     

Application of Analytical Modeling to the Far-field Investigation of Tsunami and Oceanic Rayleigh Waves from the 1998 Papua New Guinea Earthquake

We analyze far-field Rayleigh and tsunami waves generated by the 1998 Papua New Guinea (PNG) earthquake. Using the normal mode theory and Thomson-Haskell matrix formalism we calculate synthetic mareograms of oceanic surface waves excited by finite-dimensional line source and propagated in a flat, multilayered oceanic structure. Assuming that the source of destructive sea waves was the main shock of the PNG event and based on the expression for seismic wave displacement in the far-field zone, we compute the energy of the seismic and tsunami waves and the E_z /E_ts ratio. The results of our modeling are generally consistent with those obtained empirically for events with magnitude 7. Also, treating the results of a submarine slide as a single solitary wave and using the theoretical arguments of Striem and Miloh (1976) we estimate the energy of the tsunami induced by a landslide. The difference between the energy of the seismic tsunami and of the aseismic one is about one order of magnitude.The results of our theoretical modeling show that surface sea waves in the far-field zone account well for seismic origin, although additional tsunami energy from a landslide source could be required to explain the local massive tsunami in the Sissano Lagoon.     

Some Aspects of Energy Balance and Tsunami Generation by Earthquakes and Landslides

— Tsunamis are generated by displacement or motion of large volumes of water. While there are several documented cases of tsunami generation by volcanic eruptions and landslides, most observed tsunamis are attributed to earthquakes. Kinematic models of tsunami generation by earthquakes — where specified fault size and slip determine seafloor and sea-surface vertical motion — quantitatively explain far-field tsunami wave records. On the other hand, submarine landslides in subduction zones and other tectonic settings can generate large tsunamis that are hazardous along near-source coasts. Furthermore, the ongoing exploration of the oceans has found evidence for large paleo-landslides in many places, not just subduction zones. Thus, we want to know the relative contribution of faulting and landslides to tsunami generation. For earthquakes, only a small fraction of the minimum earthquake energy (less than 1% for typical parameter choices for shallow underthrusting earthquakes) can be converted into tsunami wave energy; yet, this is enough energy to generate terrible tsunamis. For submarine landslides, tsunami wave generation and landslide motion interact in a dynamic coupling. The dynamic problem of a 2-D translational slider block on a constant-angle slope can be solved using a Green's function approach for the wave transients. The key result is that the largest waves are generated when the ratio of initial water depth above the block to downslope vertical drop of the block H ₀ /W sin δ is less than 1. The conversion factor of gravitational energy into tsunami wave energy varies from 0% for a slow-velocity slide in deep water, to about 50% for a fast-velocity slide in shallow water and a motion abruptly truncated. To compare maximum tsunami wave amplitudes in the source region, great earthquakes produce amplitudes of a few meters at a wavelength fixed by the fault width of 100 km or so. For submarine landslides, tsunami wave heights — as measured by b, block height — are small for most of the parameter regime. However, for low initial dynamic friction and values of H ₀ /W sin δ less than 1, tsunami wave heights in the downslope and upslope directions reach b and b/4, respectively.Wavelengths of these large waves scale with block width. For significant submarine slides, the value of b can range from meters up to the kilometer scale. Thus, the extreme case of efficient tsunami generation by landslides produces dramatic hazards scenarios.     

无限饱水孔隙弹性空间中由运动源所产生的位移场

基于Ｂｉｏｔ的饱和孔隙弹性介质的运动方程和利用复变函数方法，本文研究了无限饱水孔隙弹性空间中由常速运动源所产生的位移场。考虑了两种类型源：ａ．沿无限空间水平轴运动的斜向集中力源；ｂ．运动双力偶源。关于力源的运动速度，考虑了４种情形：ａ．力源的运动速度Ｕ小于饱水孔隙弹性介质的三种体波速度一亚音速情形；ｂ．速度Ｕ小于介质的第一纵波过度和横波速度，但大于第二纵波速度─—弱跨音速情形；ｃ．速度Ｕ小于第一纵波速度，但大于横波速度和第二级波速度─—强跨音速情形；ｄ．速度Ｕ大于介质的所有三种体波速度─—超音速情形。结果表明，在跨音达和超音速情形里，解呈现出与力源相联系的平面冲击波特征，位移出现了相应的跳跃。     

Analytical Solutions for Tsunami Waves Generated by Submarine Landslides in Narrow Bays and Channels

Analytical theory of tsunami wave generation by submarine landslides is extended to the case of narrow bays and channels of different geometry, in the shallow-water theory framework. New analytical solutions are obtained. For a number of bottom configurations, the wave field can be found explicitly in the form of the Duhamel integral. It is described by three waves: one forced wave propagating together with the landslide and two free waves propagating in opposite directions. The cases for bays with triangular (V-shaped bay), parabolic (U-shaped bay), and rectangular cross-sections are discussed in detail. The dynamics of the offshore-propagating wave in linearly inclined bays of different cross-section are also studied asymptotically for the resonant moving landslide. Different cases of landslides of increasing and decreasing volume are considered. It is shown that even if the landslide is moving under fully resonant conditions, the amplitude of the propagating tsunami wave may still be bounded, depending on the type of the landslide.     

On the interaction of rayleigh surface waves with structures

A two-dimensional soil-structure interaction analysis is carried out for transient Rayleigh surface waves that are incident on a structure. The structure is modelled by a three-degree of freedom rigid basemat to which is attached a flexible superstructure, modelled by a single mass-spring system. The structural responses to a given Rayleigh wave train are compared with those that would have been obtained if the free-field acceleration-time history had been applied as a normally incident body wave. The results clearly exhibit the [frequency filtering] effects of the rigid basemat on the incident Rayleigh waves. It is shown that, if seismic excitation of a structure is, in fact, due to Rayleigh surface waves, then an analysis assuming normally incident body waves can considerably over-estimate structural response, both at basemat level for horizontal and vertical motions and for vertical oscillations of the superstructure. However, in the examples considered here, relatively large rocking effects were induced by the Rayleigh waves, thus giving maximum horizontal accelerations in the superstructure that were of comparable magnitude for Rayleigh and normally incident body waves.     

A note on tsunami amplitudes above submarine slides and slumps

Tsunami generated by submarine slumps and slides are investigated in the near-field, using simple source models, which consider the effects of source finiteness and directivity. Five simple two-dimensional kinematic models of submarine slumps and slides are described mathematically as combinations of spreading constant or slopping uplift functions. Tsunami waveforms for these models are computed using linearized shallow water theory for constant water depth and transform method of solution (Laplace in time and Fourier in space). Results for tsunami waveforms and tsunami peak amplitudes are presented for selected model parameters, for a time window of the order of the source duration.The results show that, at the time when the source process is completed, for slides that spread rapidly (c_R/c_T≥20, where c_R is the velocity of predominant spreading), the displacement of the free water surface above the source resembles the displacement of the ocean floor. As the velocity of spreading approaches the long wavelength tsunami velocity the tsunami waveform has progressively larger amplitude, and higher frequency content, in the direction of slide spreading. These large amplitudes are caused by wave focusing. For velocities of spreading smaller than the tsunami long wavelength velocity, the tsunami amplitudes in the direction of source propagation become small, but the high frequency (short) waves continue to be present. The large amplification for c_R/c_T1 is a near-field phenomenon, and at distances greater than several times the source dimension, the large amplitude and short wavelength pulse becomes dispersed.A comparison of peak tsunami amplitudes for five models plotted versus L/h (where L is characteristic length of the slide and h is the water depth) shows that for similar slide dimensions the peak tsunami amplitude is essentially model independent.     

Lituya Bay Landslide Impact Generated Mega-Tsunami 50th Anniversary

On July 10, 1958, an earthquake M_w 8.3 along the Fairweather fault triggered a major subaerial landslide into Gilbert Inlet at the head of Lituya Bay on the southern coast of Alaska. The landslide impacted the water at high speed generating a giant tsunami and the highest wave runup in recorded history. The mega-tsunami runup to an elevation of 524 m caused total forest destruction and erosion down to bedrock on a spur ridge in direct prolongation of the slide axis. A cross section of Gilbert Inlet was rebuilt at 1:675 scale in a two-dimensional physical laboratory model based on the generalized Froude similarity. A pneumatic landslide tsunami generator was used to generate a high-speed granular slide with controlled impact characteristics. State-of-the-art laser measurement techniques such as particle image velocimetry (PIV) and laser distance sensors (LDS) were applied to the decisive initial phase with landslide impact and wave generation as well as the runup on the headland. PIV provided instantaneous velocity vector fields in a large area of interest and gave insight into kinematics of wave generation and runup. The entire process of a high-speed granular landslide impact may be subdivided into two main stages: (a) Landslide impact and penetration with flow separation, cavity formation and wave generation, and (b) air cavity collapse with landslide run-out and debris detrainment causing massive phase mixing. Formation of a large air cavity — similar to an asteroid impact — in the back of the landslide is highlighted. A three-dimenional pneumatic landslide tsunami generator was designed, constructed and successfully deployed in the tsunami wave basin at OSU. The Lituya Bay landslide was reproduced in a three-dimensional physical model at 1:400 scale. The landslide surface velocities distribution was measured with PIV. The measured tsunami amplitude and runup heights serve as benchmark for analytical and numerical models.     

Modelling the hysteresis in the velocity pattern of slow‐moving earth flows: the role of excess pore pressure

This paper describes the velocity pattern of a slow‐moving earth flow containing a viscous shear band and a more or less rigid landslide body on top. In the case of small groundwater fluctuations, Bingham's law may describe the velocity of these slow‐moving landslides, with velocity as a linear function of excess shear stress. Many authors have stated that in most cases a non‐linear version of Bingham's law best describes the moving pattern of these earth flows. However, such an exponential relationship fails to describe the hysteresis loop of the velocity, which was found by some authors. These authors showed that the velocity of the investigated earth flows proved to be higher during the rising limb of the groundwater than during the falling limb. To explain the hysteris loop in the velocity pattern, this paper considers the role of excess pore pressure in the rheological behaviour of earth flows by means of a mechanistic model. It describes changes in lateral internal stresses due to a change in the velocity of the earth flow, which generates excess pore pressure followed by pore pressure dissipation. Model results are compared with a hysteresis in the velocity pattern, which was measured on the Valette landslide complex (French Alps). Copyright © 2005 John Wiley & Sons, Ltd.     

基于伴随方法的线性台阵背景噪声面波和远震体波联合成像研究

伴随层析成像（Adjoint Tomography）通过求解全波方程来准确模拟地震波在复杂介质中的传播,并利用波形信息来反演地下结构,是新一代的高分辨率成像方法.其中3-D伴随层析成像需要庞大的计算资源,而2-D反演相对更具计算效率.面波和远震体波是研究地壳上地幔速度结构的重要方法,它们对S波速度及Moho面的敏感度不同,通过联合反演,可以得到更为准确的S波速度结构及Moho面.通过两种数据的高度互补性,本文提出基于伴随方法的线性台阵背景噪声面波和远震体波联合成像方法,同时约束台阵下方S波速度结构及Moho面形态.我们将该方法应用到符合华北克拉通岩石圈典型结构特征的理论模型上,测试结果表明联合反演方法优势明显,相比于面波伴随层析成像,能获得更高分辨率的S波速度结构,同时能精准约束Moho面形态.相比于体波伴随层析成像,联合反演能有效压制高频假象,降低波形反演过程中的非线性化程度.本研究有望提供一种更为高效精准的线性台阵成像方法,搭建联合伴随层析成像理论框架,提升岩石圈成像分辨率,并为后续其他类型波形数据的引入提供思路和方法.     

A note on specific variability of long surface gravity waves and drag coefficient in coastal upwelling zone

In this paper we solve analytically wave kinematic equations and the wave energy transport equation, for basic long surface gravity wave in the coastal upwelling zone. UsingGent andTaylor's (1978) parameterization of drag coefficient (which includes interaction between long surface waves and the air flow) we find variability of this coefficient due to wave amplification and refraction caused by specific surface water current in the region. The drag coefficient grows towards the shore. The growth is faster for stronger current. When the angle between waves and the current is less than 90° the growth is mainly connected with the waves steepness, but when the angle is larger, it is caused by relative growth of the wave phase velocity.     

Numerical simulation of the landslide-induced tsunami of 1988 on Vulcano Island, Italy

 On 20 April 1988 a landslide of approximately 200,000 m³ occurred on the northeastern flank of the volcano La Fossa on the island of Vulcano. The landslide fell into the sea, producing a small tsunami in the bay between Punte Nere and Punta Luccia that was observed locally in the neighbouring harbour called Porto Levante. The slide occurred during a period of unrest at the volcano that was monitored very accurately. The study of this event is composed of two parts, the simulation of the landslide and the simulation of the ensuing tsunami; the former is studied by means of a Lagrangian-type numerical model in which the landslide is seen as a multibody system, an ensemble of material-deforming blocks interacting together during their motion; the latter is simulated according to the Eulerian view by solving the shallow-water approximation to Navier-Stokes equations of fluid dynamics, with the incorporation of a forcing term depending on the slide motion. Technically, the slide evolution is computed first, and this result is then used to evaluate the excitation term of the hydraulic equations and to calculate the tsunami propagation. Computed wave fronts radiate both toward the open sea, with rapid amplitude decay, and along the shore, in the form of edge waves that lose energy slowly. Comparison between model outputs and observations can be carried out only in a qualitative way owing to the absence of tide-gauge records, and results are satisfactory. Received: 14 September 1998 / Accepted: 18 December 1998     

Wave-induced steady streaming,mass transport and net sediment transport in rough turbulent ocean bottom boundary layers

The interaction between two important mechanisms which causes streaming has been investigated by numerical simulations of the seabed boundary layer beneath both sinusoidal waves and Stokes second order waves, as well as horizontally uniform bottom boundary layers with asymmetric forcing. These two mechanisms are streaming caused by turbulence asymmetry in successive wave half-cycles (beneath asymmetric forcing), and streaming caused by the presence of a vertical wave velocity within the seabed boundary layer as earlier explained by Longuet-Higgins. The effect of wave asymmetry, wave length to water depth ratio, and bottom roughness have been investigated for realistic physical situations. The streaming induced sediment dynamics near the ocean bottom has been investigated; both the resulting suspended load and bedload are presented. Finally, the mass transport (wave-averaged Lagrangian velocity) has been studied for a range of wave conditions. The streaming velocities beneath sinusoidal waves (Longuet-Higgins streaming) is always in the direction of wave propagation, while the streaming velocities in horizontally uniform boundary layers with asymmetric forcing are always negative. Thus the effect of asymmetry in second order Stokes waves is either to reduce the streaming velocity in the direction of wave propagation, or, for long waves relative to the water depth, to induce a streaming velocity against the direction of wave propagation. It appears that the Longuet-Higgins streaming decreases as the wave length increases for a given water depth, and the effect of wave asymmetry can dominate, leading to a steady streaming against the wave propagation. Furthermore, the asymmetry of second order Stokes waves reduces the mass transport (wave-averaged Lagrangian velocity) as compared with sinusoidal waves. The boundary layer streaming leads to a wave-averaged transport of suspended sediments and bedload in the direction of wave propagation.     

Tunneling of seismic body waves through thin high-velocity layers in complex structures

Summary The hybrid ray-reflectivity method is applied to the problem of the transmission of the reflected wave field through a thin high-velocity layer (or through a thin stack of high velocity layers), situated in the overburden of the reflector. In the hybrid ray-reflectivity method, the standard ray method is applied in the smooth parts of the model, and the reflectivity method is used locally at the thin high-velocity layer. With the exception of small epicentral distances, the standard ray method itself fails in such computations. The reason is that a considerable part of the energy for overcritical angles of incidence may be tunneled through the thin high-velocity layer along complex ray-paths, corresponding to inhomogeneous waves. The reflectivity method, applied locally at the thin high-velocity layer, automatically includes all inhomogeneous wave contributions. Thus, the hybrid ray-reflectivity method removes fully the limitations of the standard ray method, but still retains its main advantages, such as its applicability to 2-D and 3-D complex layered structures, flexibility, and low-cost computations. In the numerical examples, the hybrid ray-reflectivity synthetic seismograms are compared with standard ray synthetic seismograms and with full reflectivity computations. The numerical examples show that the hybrid ray-reflectivity method describes the tunneling of seismic energy through a thin high-velocity layer with sufficient accuracy.     

多孔介质中水位变化对波形、波速和波谱影响的实验研究

为了研究地下水和地震波之间的关系，设计了一套水位实验装置，用注水方法研究了多孔介质中水位变化对P波和S波的波速、波形和波谱的影响。实验结果表明：波垂直水面传播时，速度与时间平均方程的结果基本一致；波平行水面传播时，虽有叠加效应但并非简单的叠加。从干燥到注满水，P波速度明显增大，波幅和波谱能量显下降，主频明显降低；在接近注满水时，出现特殊的高频初至小波，而S波速度略有减小 ，波幅和波谱能量则一般是先降后升，主频基本没有变化。比较了注子和饱水的不同，并引入三相气体包裹体模型进行了解释。     

区域性的地震面波层析方法

本文提出了一个利用平面上长方形区域內的面波频散资料,求区域的三维速度结构的方法。将地震波慢度表示成二重Fourier级数,反演其系数为深度的函数,最后合成速度值,并以Love面波为例作了数值计算试验,绘制出了深度为150km的速度平面分布图。      

Composition of high frequency ambient noise from cross-correlation:A case study using a small aperture array

Long-time cross correlation of ambient noise has been proved as a powerful tool to extract Green's function between two receivers.The study of composition of ambient noise is important for a better understanding of this method.Previous studies confirm that ambient noise in the long period (3 s and longer) mostly consists of surface wave,and 0.25-2.5 s noise consists more of body waves.In this paper,we perform cross correlation processing at much higher frequency (30-70 Hz) using ambient noise recorded by a small aperture array.No surface waves emerge from noise correlation function (NCF),but weak P waves emerge.The absence of surface wave in NCF is not due to high attenuation since surface waves are strong from active source,therefore probably the high ambient noise mostly consists of body wave and lacks surface wave.Origin of such high frequency body waves in ambient noise remains to be studied.