Let us consider one-dimensional propagation of a tsunami from the deep ocean to coastal water, which is initially at the stationary state, as illustrated in Fig. . We define the x* axis as the positive seaward horizontal coordinate with an origin at the still-water shoreline. There is a wave observation point at x*=l, where water surface displacements driven by the incoming tsunami are continuously measured. The seabed slope landward of the observation point is assumed to be uniform, while the seabed on its sea side may have an arbitrary form as long as it smoothly connects to the landward uniform slope. The measured wave profile contains both landward-propagating waves from the source and seaward-propagating waves reflected from the shore. Our problem is to determine a coastal wave profile under the observed-wave boundary condition. This is different from the previous formalism in which coastal wave evolution is predicted under the incident-wave boundary condition by assuming a horizontal bed seaward of the boundary .

Here, we use linear shallow-water equations to formulate the problem. Wave non-linearity becomes significant in shallow water, and the non-linear effect distorts the time and spatial axes. However, it is well known that the wave amplitude is not significantly affected by non-linearity unless the non-linear wave distortion leads to wave breaking . The non-linear shoreline motion can be readily derived from the linear solution via the hodograph transform, and the run-up height is unchanged from the linear case if the boundary value assignment is linearised e.g.. Furthermore, the occurrence of wave breaking, which limits the applicable range of the present approach, can be predicted as a breakdown point of the hodograph transform under the same condition. While the linearised boundary value assignment potentially affects the run-up height and the wave breaking condition, non-linear modifications are minor as long as the ratio of wave amplitude to water depth is small at the boundary . Therefore, the main process of practical interest can be described by linear equations when we place the boundary in deep water. Another important fact, which has often been neglected in analytical models, is that a tsunami decays on its path to the shore by different factors. Therefore, we incorporate a damping term into the momentum equation to account for the tsunami decay. Accordingly, the damped propagation of tsunamis over a uniform slope can be described by the following equations: 1∂η*∂t*+βu*+βx*∂u*∂x*=0,2∂u*∂t*+g∂η*∂x*=d*. Here, the asterisk superscript represents a dimensional variable. Therefore, t* represents time, η*(x*,t*) is the water surface elevation above the still-water level, u*(x*,t*) is the positive seaward horizontal velocity, β is the seabed slope, d* is the damping term and g is the gravitational acceleration.

The quadratic law is often used to model tsunami damping (d*∝u*u*), but this introduces non-linearity into the problem, which makes the analytical work infeasible. Instead, we employ the linear damping term as follows: 3d*=-α*u*, where α* (≥0) is the damping parameter, which has a dimension of the reciprocal of time. The linear damping term introduces an exponential wave decay with the characteristic time corresponding to α*-1 into the non-dissipative solution . In virtue of its simple formulation, the linear model has been used to represent damped propagation of tsunamis in deep water. Many researchers have suggested the damping parameter of O(10-5) s-1 for deep-water tsunami propagation through comparisons of the model and observations e.g.. However, the damping parameter could be much higher for nearshore tsunami dissipation. analytically derived a dissipative run-up formula of monochromatic long waves over a uniform slope using the linear damping term. They suggested the α* value of O(10-2) s-1 in a typical situation of calculating tsunami run-up.

Tsunamis decay in shallow water by different factors, such as skin and form drag over the seabed as well as local topography and coastal structures, the scales of which are much smaller than the tsunami wavelength. Here, we attempt to represent the tsunami decay due to different factors collectively by the linear damping term; thus, the parameter α* works as an empirical parameter rather than as a physical parameter. This treatment makes the resulting kernel representation a semi-empirical one. We will discuss the variability in this parameter through real-world tsunami applications in a later section.

The governing equations are non-dimensionalised by the following dimensionless variables: x*=xl,t*=tl/gβ,η*=ηβl,u*=ugβl,4α*=αgβ/l. Here, the spatial scales and timescales are chosen to be the horizontal slope length l and half of the wave travel time over the entire slope l/gβ, respectively. Namely, the horizontal dimensionless coordinate takes the value of x=0 at the shoreline and x=1 at the offshore boundary. The dimensionless time of t=2 equals the one-way travelling time from/to the boundary to/from the shore. Of note is that the shallow-water approximation requires l≪L0/β, where L0 is the representative wavelength.

Substituting Eq. () into Eqs. () and () yields the dimensionless forms of the governing equations: 5∂η∂t+u+x∂u∂x=0,6∂u∂t+∂η∂x+αu=0. Eliminating η from Eqs. () and () yields a single-variable wave equation of u as 7∂2u∂t2+α∂u∂t-2∂u∂x-x∂2u∂x2=0. We solve this equation for wave evolution over initially stationary water on a uniform slope given an offshore wave profile of arbitrary form at x=1. For this purpose, we apply Laplace transform to Eq. (). Then, we have 8(s2+αs)u^-2∂u^∂x-x∂2u^∂x2=0, where u^ is the Laplace transform of u, which is given by 9u^(x,s)=∫0∞u(x,t)e-stdt with a complex number frequency parameter s. Equation () is the modified Bessel's equation and has a bounded solution at the shoreline (x=0) as 10u^(x,s)=G(s)x-12I12ss+αx12, where In is the modified Bessel function of the first kind and the nth order and G(s) is an arbitrary function that will be determined by the offshore boundary condition.

Accordingly, the water surface elevation in the Laplace domain, η^(x,s), can be obtained from Eqs. () and () such that η^(x,s)=-1su^+x∂u^∂x11=-ss+αsG(s)I02ss+αx12.

To determine G(s), we use the boundary condition, η^(1,s)=η^0(s), where η^0(s) is the Laplace transform of the observed time series of water surface elevation at x=1. Then, we can express the solutions for water surface elevation and horizontal velocity as 12η^(x,s)=sΨ^0(x,s)η^0(s),13u^(x,s)=-sΨ^1(x,s)η^0(s), where Ψ^0(x,s) and Ψ^1(x,s) are the transfer functions for η^ and u^ in the Laplace domain, respectively, which are given as Ψ^0(x,s)=I02ss+αx12sI02ss+α,14Ψ^1(x,s)=x-12ss+αI12ss+αx12I02ss+α. Here, η^ and u^ are expressed as the product of each transfer function and sη^0 in the Laplace domain. The factor of s was isolated in Eqs. () and () such that the transfer functions are invertible to the time domain. Consequently, η and u in the time domain are commonly expressed by the convolution of the rate of water surface displacement at the offshore boundary with the inverse Laplace transforms of the transfer functions as follows: 15η(x,t)=Ψ0(x,t)*dη0dt,16u(x,t)=-Ψ1(x,t)*dη0dt, where the asterisk denotes the convolution operation. Ψn(x,t) results from the inverse Laplace transform of Ψ^n(x,s), which is expressed using the Bromwich integral such that 17Ψn(x,t)=12πi∫γ-i∞γ+i∞Ψ^n(s)estds. Here, γ is a positive real number greater than every singularity of Ψ^n(s). The common form of solutions in Eqs. () and () suggests that the convolution kernels accommodate the general processes of wave transformation under the observed-wave boundary condition. An exception arises at the shoreline (x=0), where Ψ^1 is not well defined. The two transfer functions in Eq. () can be related at the shoreline as Ψ^1(0,s)=sΨ^0(0,s), which turns into their relation in the time domain as Ψ1(0,t)=∂Ψ0(0,t)/∂t. With this relation, the kinematic condition at the shoreline results from Eqs. () and (): 18us=-∂ξ∂t, where us(t)≡u(0,t) represents the shoreline velocity and ξ(t)≡η(0,t) expresses the vertical shoreline displacement. This suggests that the choice of the bounded solution at the shoreline in Eq. () is equivalent to imposing the linearised kinematic condition. It should be emphasised that the present solutions are based on the moving boundary condition at the shoreline. The shoreline velocity can be obtained from Eqs. () and ().

The next step is to derive the kernel functions by evaluating the Bromwich integrals in Eq. (). The residue theorem can be used to determine the inverse Laplace transform of the transfer function. Both Ψ^0(x,s) and Ψ^1(x,s) have an infinite number of poles at s=sk such that I0(2sk(sk+α))=0. Additionally, Ψ^0(x,s) has a pole at s=0, while the singularity of Ψ^1(x,s) at s=0 is removable. The complex zeros of the modified Bessel function are distributed on the imaginary axis; thus, they can be associated with positive real zeros of the Bessel function of the first kind and zeroth order, ck, that satisfies J0(ck)=0. All poles of the two functions can be summarised as follows: s0=0,sk=-α+ick2-α22and19s‾k=-α-ick2-α22, where k is a natural number (k=1,2,3…) and ck is the positive zeros of J0 in ascending order of magnitude, as graphed in Fig. . Equation () suggests that the damping factor has an upper limit such that α<c1≈2.405. When k is large, the positive zeros of the Bessel function can be approximated as 20ck≈k-14π. This approximation suggests that ck linearly increases with k, as shown in Fig. .

Because both functions are holomorphic elsewhere, the Bromwich integrals can be evaluated using the theory of residue such that Ψ0(x,t)=Res[Ψ^0(s0)es0t]+∑k=1∞Res[Ψ^0(sk)eskt]+Res[Ψ^0(s¯k)es¯kt],21Ψ1(x,t)=∑k=1∞Res[Ψ^1(sk)eskt]+Res[Ψ^1(s‾k)es‾kt], where Res[Ψ^(s)est] is the residue of Ψ^(s)est at the point s. Substituting Eqs. () and () into Eq. () yields 22Ψ0(x,t)=1+ψ0(x,t),Ψ1(x,t)=x-12ψ1(x,t) with ψn(x,t)=-2e-αt2∑k=1∞Jnckx12λkJ1cksin⁡λk2t+nπ2+θk,23λk=ck2-α2,θk=atan2λk,α. Both kernels consist of infinite series of the combination of the Bessel function of x12 and the sinusoidal function of t. In addition, the kernels contain the negative exponential function of time and decay with time at a higher rate for larger α. For α=0 without the damping effect, the kernels exhibit oscillatory behaviour in the infinite time length; thus, the kernel convolution involves a full wave history from the initial time at the offshore boundary. Introduction of the damping effect confines the causal relation to the near past. Kernel convolution deals with the separation of incident and reflected waves because it is formulated with the observed-wave profile containing two-way components. Separation requires a longer observed-wave history for a smaller damping factor.

Figures  and show graphs of Ψ0(x,t) and Ψ1(x,t), respectively, on the time axis for different values of the damping factor; namely, α=0, α=0.2 and α=1.0. Ψ0(x,t) is shown at x=0, x=1/8 and x=1/2, while Ψ1(x,t) is graphed at x=1/8, x=1/2 and x=1. Of note is that Ψ0(1,t)=1 and Ψ1(0,t) are not well defined. The initial forms of the two kernels agree with those derived under the incident-wave boundary condition because the initial part of observed-wave data represents a shoreward-propagating wave. They deviate from the incident-wave kernels in the large time domain exhibiting cyclic singularities. Cyclic singularities occur at the following timing: 24tm±=21±(-1)mx12+4(m-1), where m is the natural number. Both Ψ0 and Ψ1 diverge to positive or negative infinity at tm- and form slope discontinuities at tm+. Of note is that 2(1-x12) corresponds to the propagation time from the offshore boundary to the reference point, whereas 2(1+x12) represents the propagation time between the two locations via reflection on the shore. Hence, the two kernels have cyclic singularities representing arrivals of past seaward- and shoreward-propagating waves with a period of t=4, which equals the wave round-trip time over the entire slope. Equation () can be alternatively described as a series of characteristic curves in the (x,t) plane, x=t-2-4(m-1)2/4, which represent forth and back propagation of a wave signal on the slope. Kernel convolution generates a coastal waveform by processing the two-way propagating waves at the offshore boundary. At the shoreline (x=0) where tm+=tm-, the divergence and slope discontinuity merge with each other and the bimodal kernel structure disappears from Ψ0. A similar merging of singularities also happens to Ψ1 at the offshore boundary (x=1). The infinite series in Eq. () is convergent except at tm-, where it turns into a harmonic series for large k; the convergence properties of the infinite series are more rigorously discussed in Appendix A. Therefore, the infinite series can be numerically calculated until the point of convergence except at the singular points.

Substituting Eq. () into Eqs. () and () yields 25η(x,t)=η0(t-t0)+∫t0tψ0(x,τ)dη0dtt-τdτ,26u(x,t)=-x-12∫t0tψ1(x,τ)dη0dtt-τdτ, where t>t0≡2(1-x12) and η=u=0 otherwise. These representations suggest that η(x,t)≈η0(t-t0) and u(x,t)≈0 hold when the incoming wave is much longer than the slope length. This can be viewed as the seabed slope working like a vertical wall against a relatively long wave. The convolution describes wave deformation over the slope under the observed-wave boundary condition. Despite cyclic singularities, the kernel convolution is well defined, and the convolution can be numerically performed, which will be presented in the next section.

The kernels have cyclic singularities at tm+ and tm-. For the convenience of presenting the convolution method, the singular time points are redefined as follows: 27tjm=21-(-1)jx12+4(m-1), where j is 0 or 1 (tm-=t0m and tm+=t1m). To avoid midpoint singularities, the integration interval is divided into multiple time segments: [t01,t11],[t11,t02],[t02,t12],…,[t0m,t1m],[t1m,t0m+1],… Then, based on the kernel form, different variable changes are applied in [t0m,t1m] and [t1m,t0m+1] based on the double exponential integration formula by via τ0m(p0)=22m-1+x12tanh⁡π2sinh⁡p028for[t0m,t1m],τ1m(p1)=22m+(1-x12)tanh⁡π2sinh⁡p129for[t1m,t0m+1]. These changes of variables will concentrate integration points of time near the singular points using the double exponential function. Consequently, each segment is mapped to an infinite interval of pj with vanishing endpoint singularities. For shoreline motions (x=0), the segments of j=0 disappear because t0m=t1m. Because the integrand decays with a double exponential rate, actual numerical integration is performed within a finite interval of pj. Therefore, the convolutions in Eqs. () and () are rewritten as 30η(x,t)=η0(t-t0)+∑m=1M∑j=01∫-∞∞ψ0(x,τjm(pj))dη0dtt-τjm(pj)dτjmdpjdpj,31u(x,t)=-x-12∑m=1M∑j=01∫-∞∞ψ1(x,τjm(pj))dη0dtt-τjm(pj)dτjmdpjdpj, where M is an integer ceiling of (t-t0)/4 when α=0 and can be a smaller number when α>0 because the kernels decay on the time axis. The convolution is numerically performed by applying the segment-by-segment integration following the trapezoidal rule. The convolution is efficiently implemented by the advance tabulation of ψndτjm/dpj. We read the kernel table and then perform computation of the infinite integrals in several segments to predict a coastal waveform.

The convolution method is demonstrated and validated by comparing numerical results with an analytical solution for a simple problem. Here, we consider a simple problem in which a monochromatic wave is observed away from the shore that contains both incident and reflected waves. The observed wave is now given as 32η0(t)=sin⁡(λt), where λ is the dimensionless angular frequency. It is assumed that the coastal wave is in an equilibrium state, and a partial standing wave is formed owing to wave decay over the uniform slope.

To derive the equilibrium solution, we first take the Laplace transform of Eq. () and substitute it into Eq. (). Then, we have 33η^(x,s)=λI0(2s(s+α)x12)I0(2s(s+α))(s+iλ)(s-iλ). This can be inversely transformed into the time domain by the residue theorem. However, the residues associated with the zeros of the modified Bessel function, sk and s‾k in Eq. (), yield a transient solution because they accommodate the decaying exponential function of time with the damping factor α. Therefore, we need to consider only the residues of s=±iλ for the equilibrium solution in the case of α>0.

Consequently, the damped equilibrium solution for water surface elevation, η̃(x,t), is given by 34η̃(x,t)=a(x)sin⁡λt+ϕ(x), where the wave amplitude a(x) and the phase ϕ(x) are given by a(x)=AC(x)+BD(x)2+AD(x)-BC(x)2A2+B2,35ϕ(x)=atan2AD(x)-BC(x),AC(x)+BD(x) with A=ReI0(2iλ(iλ+α),B=ImI0(2iλ(iλ+α),C(x)=ReI0(2iλ(iλ+α)x,D(x)=ImI0(2iλ(iλ+α)x. The wave amplitude on the slope cannot be represented by simple mathematical functions and exhibits drastic changes with varying λ because the observation point moves between the node and anti-node of the partial standing wave.

To validate the numerical convolution method, the wave amplitude is computed for non-zero α values using Eq. (). Because the damping effect confines the causal relation in a finite time, we set the total time length of the convolution such that the kernel value sufficiently decays (t=12/α). The numerical convolution was performed over a sufficiently large interval (-3<pj<3) for both j=0 and 1 with a common subinterval Δpj. To investigate the sensitivity of the result to Δpj, the numerical solutions were obtained with three different subintervals, i.e. Δpj=0.05, 0.1, and 0.25.

Figure  shows the comparison of wave amplitude from numerical convolution and the exact solution at three different locations (x=0, 1/2 and 1/8). The two cases with different values of the damping parameter, α=0.1 and α=1.0, are shown in the left and right columns, respectively. In each figure, exact and numerical solutions of wave amplitude are compared on the λ axis. The dimensionless angular frequency can be expressed as λ=2πl/L0 with the representative wavelength L0; thus, λ=2π represents the situation in which the wavelength is equal to the slope length. In the case of α=0.1, wave amplitude shows large fluctuations on the λ axis owing to considerable reflection. The coastal wave amplitude becomes very large when a partial node is formed at x=1 where wave amplitude is fixed to be unity. The fluctuations are smeared out in the case of α=1.0 owing to the stronger damping effect. Numerical results agree well with the exact values in this range of λ when Δpj<0.1. Numerical error appears from the high-frequency side when Δpj=0.25. These results confirm that numerical convolution produces an exact solution with a sufficiently small subinterval relative to the tsunami wave period.

In the case of α=0, the monochromatic wave creates a complete node at x=1 when λ=ck/2. Therefore, we have no wave signal at x=1 in the equilibrium state, and the wave amplitude given by Eq. () goes to infinity because the boundary value given by Eq. () is not appropriate. The kernel method works even in such a case as long as the full standing wave is formed from an initially stationary state. To demonstrate this, we consider a transient incident wave given by 36ηi(t)=tanh⁡λt10sin⁡(λt), where the tanh function was introduced for a smooth transition from the stationary water surface to a monochromatic wave. As illustrated in Fig. a, the incident wave grows to a monochromatic wave over several wave periods. When we set λ=c1/2≈1.2024, a node will be formed at x=1 under the incident wave. Using the incident-wave kernel previously proposed by the author , we can obtain the water surface elevation at x=1, η0, as shown in Fig. b. The null elevation datum occurs after the initial transient phase due to the formation of a full standing wave. Figure c compares shoreline elevation profiles, ξ, computed from ηi using the incident-wave kernel and that computed from η0 using the present kernel. The two results show a perfect agreement, and the wave amplitude converges to the analytical solution of the monochromatic-wave run-up height by : 2/J0(c1)2+J1(c1)2≈3.85. Even if the null datum occurs at x=1 as a result of the superposition of incident and reflected waves, the kernel convolution can predict a waveform at any location on the slope from the initial transient part of the wave history at x=1. It is worth emphasising that the present kernel works for such a problem because it is constructed for the initial-boundary value problem. Without the initial condition, we could not derive an incident-wave signal from offshore wave data when a full node is formed at the boundary.

Case 1 is located at the coast of Mutsu-Ogawara, where relatively small waves of similar amplitude repeatedly attack the coast in a period of approximately 30 min. The offshore wave data were recorded by an ultrasonic wave gauge installed on the seabed at a depth of 43.8 m, approximately 3.4 km off the coast (P1 in Fig. b). In addition, an onshore wave record is available from a tide station inside Mutsu-Ogawara Port (P2 in Fig. b). Because the offshore wave data contained high-frequency fluctuations that may not be real waves propagating shoreward, the following analyses were performed with the low-pass-filtered time series (<0.003 Hz) as shown in Fig. a. The numerical convolution was performed with the filtered wave data with three different values of the damping parameter, α=0.1, α=0.5 and α=1.0, on the assumption of a uniform slope (β=43.8/3400≈0.013). The numerical integration in each segment was implemented for -3<pj<3 with Δpj=0.05. The numerical error was confirmed to be negligibly small.

Figure b and c compare the computed waveforms at the shoreline, ξ*, and the observed-wave data at P2 with the different values of damping parameters, respectively. The first term in Eq. (), η0*(t-t0), is also plotted to highlight the slope effect described by kernel convolution. The comparison of ξ* and η0* shows that relatively short wave components are significantly amplified over the slope. The computed results with α=0.1 overestimate the short waves on the shore. The observed-wave data at P2 are in good agreement with the computed result of α=0.5, but the initial waveforms are reproduced slightly better with α=1.0. This result suggests that the damping factor varies with wave amplitude, and the non-linear formulation of the damping term may be required to achieve higher accuracy. Nonetheless, despite the crude assumptions, the kernel method works well to predict the coastal waveform from the observed data a few kilometres off the coast when the α value is appropriately chosen.

Case 2 is located on the coast ranging from Kamaishi to Ryoishi, areas which were highly devastated during the 2011 tsunami event. Two bottom pressure sensors (TM1 and TM2) captured the initial waveform of the deep-water tsunami propagating towards the coast . As shown in Fig. , TM1 was located 76 km off the coast at a depth of approximately 1600 m, while TM2 was 47 km off the coast at a depth of approximately 1000 m. In addition, a GPS buoy (GPS802) deployed 15 km off the coast recorded a full profile of the tsunami propagating at a depth of approximately 200 m . Because three observation points are aligned on a cross-shore line, the dataset can be used to validate the kernel method for the long-distance propagation of the real-world tsunami with a large amplitude. Coastal wave gauges in shallow water were destroyed by the tsunami; thus, there are no data for onshore waveforms. Nevertheless, intensive post-event surveys provide the range of coastal run-up heights in this area . The ria coast exhibits an intricate coastline, but the dimension of coastline variations is smaller than the tsunami wavelength. Therefore, tsunami propagation is expected to be described with the one-dimensional kernel approach to a large extent.

Figure a shows the observed waveform at TM1. The leading part of the tsunami in this region was characterised by a short, impulsive wave of large amplitude riding on a relatively long wave. We predict the waveforms at the locations of TM2 and GPS803 via the kernel convolution of the wave data at TM1, assuming the water depth to be linearly decreasing from TM1 to the shore (β=0.02). Figure b and c compare the computed waveform for α=0.1, α=0.5 and α=1.0 with the observed data at TM2 and GPS802, respectively. The computed profile agrees with the observed one at both TM1 and GPS802 when α=0.5. However, the water surface elevation is predicted to be higher after the peak at both locations, and this discrepancy probably occurs because the observed wave at TM1 is located far from the coast and contains wave components that do not propagate shoreward. The resulting run-up height on the shore is sensitive to the choice of the damping parameter, as shown in Fig. d. The measured run-up heights after the event significantly varied owing to the intricate coastline; however, the maximum run-up height was up to 20 m in the coastal areas. Therefore, the α value of 0.5–1.0 should be employed for the purpose of predicting the run-up of the tsunami in this area.

Next, we predict the waveform at the location of GPS803 using the wave data at TM2 graphed in Fig. e. The computed results at GPS802 show better agreement with the observation than those predicted from TM1, as shown in Fig. f. The results in Fig. c and f confirm that the observed wave at TM1 did not fully propagate shoreward and suggest that TM1 is too far from the shore for the kernel application. The lower value of α realises better agreement at GPS802, but the maximum run-up height is overestimated with the same value (Fig. c). This poses the limitation of the linear formulation of the damping effect that cannot account for relatively high tsunami attenuation in shallow water owing to the quadratic dependence on flow velocity. In particular, the local topographic elements along the intricate coastline caused additional damping effects in the coastal area. Therefore, the empirical damping parameter should be optimised for the target location of the prediction. Figure  shows another prediction of the wave profile at the shoreline from the wave data at GPS802 that were available for a much longer period than at TM2. The shoreline displacement shows a similar form to the previous predictions, and the low damping parameter of α=0.1 produces successive high waves that are not observed in this area. The onshore waveform of the leading wave in the case of α=0.5–1.0 agrees well with the previous model result based on the two-dimensional non-linear shallow-water equations .

The kernel representation of coastal tsunami evolution provides an efficient method for predicting onshore tsunami waveforms based on a single wave observation away from the shore. Coastal waveforms are almost simultaneously obtained with the observed profile because the computation time for the numerical integration is negligibly small compared to the tsunami timescale. In this case, the lead times for the real-time prediction of shoreline motions are 16 and 8 min when predicted from TM2 and GPS802, respectively. The prediction accuracy is limited by the crude assumptions underlying kernel representation that makes the fast prediction possible: one-dimensional propagation over a uniform slope with the linear damping effect. Nonetheless, the damping parameter can be optimised for the prediction of tsunamis at a specific coastal location from an offshore observation point; this can be done with the help of a numerical model. The two cases suggest that coastal run-up is well predicted when the damping parameter is set in the range of 0.5–1.0.