Magnitude-based scaling of tsunami propagation

Magnitude‐based scaling of tsunami propagation

M. Arthur Simanjuntak1 and Diana J. M. Greenslade1

Received 16 August 2010; revised 12 April 2011; accepted 25 April 2011; published 28 July 2011.

[1] Most current operational tsunami prediction systems are based upon databases ofprecomputed tsunami scenarios, where some form of linear scaling is applied to theprecomputed model runs in order to represent specific earthquake magnitudes. This canintroduce errors due to assumptions made about the rupture width and possible effects ondispersion. In this paper, we perform a series of numerical experiments on uniformdepth domains, using the Method of Splitting Tsunamis (MOST) model, and developestimates of the maximum error that an assumed discrepancy in the width of a rupture willproduce in the resulting field of maximum tsunami amplitude. This estimate was producedfrom fitting the decay of maximum amplitude with normalized distance for variousresolutions of the source widths to the grid size, resulting in a simple power law whosecoefficients effectively vary with wavelength resolution. This provides a quantificationof the effect that linear scaling of precomputed scenarios will have on forecasts of tsunamiamplitude. This estimate of scaling bias is investigated in relation to the scenario databasethat is currently in use within the Joint Australian Tsunami Warning Centre.

Citation: Simanjuntak, M. A., and D. J. M. Greenslade (2011), Magnitude‐based scaling of tsunami propagation, J. Geophys.Res., 116, C07027, doi:10.1029/2010JC006599.

1. Introduction

[2] Forecasting the impact of tsunamis generated by earth-quakes remains a challenge to evaluate in real time, despitecurrent computational capability. Notwithstanding the com-putational demand of high‐resolution coastal inundationmodels, there are still challenges associated with predictingthe initial surface elevation from a limited set of earthquakeparameters, and the computational demands of simulating thewave propagation on a global scale. It is no surprise that all ofthe current operational tsunami prediction systems involvedatabases of precomputed scenarios [e.g., Gica et al., 2008;Greenslade et al., 2010]. Within this strategy, there are anumber of approaches that may be employed to handle theuncertainty in specifying the initial surface elevation. The“unit source” approach consists of wave solutions fromsmall ruptures of identical dimensions (unit sources) locatedalong the length and width of subduction zones; in a tsunamievent, these solutions are combined and linearly scaled inorder to represent a specific magnitude earthquake, or tomatch the observed sea level variability from tsunameters.The “scaling” approach, on the other hand, consists of solu-tions from earthquakes with a range of predefined magni-tudes and rupture dimensions; in an event, if the observedearthquake magnitude does not match any of the magnitudesof the precomputed scenarios then the solution from thenearest scenario (in space and in magnitude) is scaled to

match the observed earthquake magnitude and/or sea levelobservations.[3] The Joint Australian Tsunami Warning Centre

(JATWC), operated jointly by the Bureau of Meteorologyand Geoscience Australia, bases its current prediction strategyon the latter approach using the T2 scenario database[Greenslade et al., 2010]. The T2 database currently consistsof 521 potential earthquake source locations and four differentmagnitudes (Mw= 7.5, 8.0, 8.5 and 9.0) at each source location,resulting in 1866 individual scenarios. The magnitude (Mw)is related to seismic moment (Mo) as

Mw ¼ 2

3log10Mo � 9:1ð Þ; ð1Þ

and seismic moment is related to the rupture characteristicsof the earthquake as

Mo ¼ �LWuo; ð2Þ

where m is the shear modulus [Nm−2], L is the length of therupture [m], W is the width [m] and uo is the slip [m]. Herem is taken to be 4.5 × 1010 Nm−2. It can be seen that themagnitude of the earthquake is therefore directly related tothe volume of the initial displacement.[4] A convenient way to obtain the solution for an

earthquake with a magnitude intermediate to the scenarios isto linearly scale the wave solution from the nearest scenarioaccording to the rupture geometry. This can be achieved byassuming a fixed correlation between the earthquake momentmagnitude and the total volume of the seafloor displacement.If this strategy is adopted, scaling of the solution with any (orall) of the width, slip, or length of the rupture is possible.Linear scaling with respect to slip is the most easily exploited

1Centre for Australian Weather and Climate Research, AustralianBureau of Meteorology, Melbourne, Victoria, Australia.

Published in 2011 by the American Geophysical Union.

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due to the linearity of the wave solution in deep water.However, slip is not as strongly correlated with momentmagnitude as are width and length [Wells and Coppersmith,1994].[5] If it is assumed that the change in the earthquake

moment magnitude is correlated with the change in the slipand the width only, then the error in linear scaling due to theeffect of the width needs to be taken into account. We canexpect the wave signals from two initial conditions of thesame volume but different shape to vary from each other inthe vicinity of the source, and become more similar withdistance from the source [Kajiura, 1963;Hammack and Segur,1978], at least in the uniform depth, two‐dimensional case,where the effect of rupture length can be ignored.[6] The decay of the wave due to dispersion inversely

scales with the width of the rupture, because wavelength ingeneral is proportional to the rupture width; it is this dif-ference in decay rate that tends to eventually iron out thesignal difference due to initial conditions [Hammack andSegur, 1978]. There is also the influence of nonlinearity,which is not addressed in this work. Therefore, although theeffect of width discrepancy can be safely ignored after asufficient distance from the source, we need to address theerror that it introduces closer to the source, where the wavesfrom the ‘scaled’ solution tend to have different initialshapes and decay rates. The effects of dispersion withintsunami models have previously been noted, for example,by Grilli et al. [2007].[7] The questions that are addressed in this study are:

what is the influence of an assumed change in rupture width,and at what distance from the source does this cease to havean impact on the solution. In this paper, we perform a seriesof numerical experiments in order to develop estimates ofthe maximum error that an assumed discrepancy in thewidth of a rupture will produce in the resulting field ofmaximum amplitude. This provides a quantification of theeffect that linear scaling of precomputed scenarios will haveon forecasts of tsunami amplitude. These results can beincorporated into numerical forecasts as a guide to the levelof uncertainty.[8] The outline of the paper is as follows. In section 2 the

numerical model MOST is described, focusing on the relevantparameters and their typical scales in the T2 scenario data-base. These scales form the basis of the numerical experi-ments, which are described in section 3. The results of theexperiments are presented in section 4, summarized in termsof a power law decay of the maximum amplitude. Thissection also presents an expression for scaling bias derivedfrom the results. In section 5, we investigate this scalingbias for typical cases from the scenario database and lastlyour conclusions are presented and discussed in section 6.

2. The MOST Model

[9] The model used for the numerical experiments is theMOST model [Titov and Synolakis, 1998], predominantlybecause this is the model used for the T2 scenario databasethat is implemented within the JATWC. MOST solves thedepth‐integrated shallow water equations in 2D using themethod of fractional steps, in which the 2D problem isreduced to two 1D problems, that are then solved serially.This is achieved by solving the 1D continuity and momentum

equations along characteristic lines, by decomposing theproblem into its Riemann invariants. These are then solvedusing the method of undetermined coefficients [Titov andSynolakis, 1998].[10] The model is efficient because the dispersion term is

not modeled explicitly, but is replaced by numerical disper-sion introduced by the finite difference method [Naganoet al., 1991]. The numerical dispersion term is as follows:

Snum ¼ �C2oDx2

121� K2� � @4h

@x4; ð3Þ

where h is water depth, Co = (gh)0.5,Dx is the horizontal gridsize, and K = DtCo/Dx is the Courant number. This can becompared with the physical dispersion term in the linearBoussinesq equation

Sphys ¼ �C2oh

2

3

@4h

@x4ð4Þ

to yield the Imamura number

Im ¼ SnumSphys

� �0:5

¼ Dx

2h1� K2� �0:5

; ð5Þ

and numerical dispersion exactly matches the physical dis-persion when Im = 1. That is, for small Courant number,K, the horizontal grid spacing Dx must be twice the waterdepth h. This property has been exploited in the past toreplace physical dispersion [e.g., Nagano et al., 1991], andto correct numerical dispersion [e.g., Yoon, 2002]. Burwellet al. [2007] derive the numerical dispersion term in MOSTfrom their filter function, and show that this yields the sameresult as expanding the linear long wave equation as derivedby Imamura and Goto [1988] and Nagano et al. [1991]. Thismeans that within the MOST model, propagation (at least in1D) can be approximated by a kind of linear Boussinesqequation that replaces physical dispersion with numericaldispersion.[11] Burwell et al. [2007] show that in the linear, uniform

depth and uniform grid size case, the evolution of the Riemanninvariants are second‐order accurate in time and space, andcan be envisaged as repeated application of space filtertransfer functions to initial conditions. The amplitude andphase characteristics of the filter function, in turn, correspondto the effect of numerical diffusion and dispersion, respec-tively. Burwell et al. [2007] show that these two mechanismscompete with each other in MOST. They demonstrate thatfor K < 1/

ffiffiffi2

pdispersive effects are dominant over diffusion,

which implies that decay in the leading wave group ismainly due to shorter, slower waves being left behind, butthese poorly resolved waves are not damped and thereforepersist artificially. For the opposite (high diffusion, lowdispersion) case, shorter waves are damped quickly. Withinthe T2 scenario database, K is low enough, at least in thelow latitudes, to qualify most scenarios as being in the high‐dispersive, low‐diffusive regime. The numerical experimentsin this paper therefore focus on low Courant number, whichis very dominantly dispersive.[12] For the typical simulation setup in T2, it is of interest

to gauge the effect of the discrepancy between grid size and

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water depth on the propagation of tsunamis in the openocean. Kajiura [1963] suggested that if

pa ¼ Rh2=613

W> 0:25; ð6Þ

where R = (gh)0.5t is the distance over which the tsunami haspropagated, then the dispersion effect is not negligible. Itcan be shown that this is consistent with the lower bound inwhich the asymptote of linear dispersive theory may bevalid [Hammack and Segur, 1978]. The existing rupturewidths in the T2 scenario database range from 50 km (forMw = 7.5 ruptures) to 100 km (for Mw = 9.0 ruptures) Thismeans that for a typical depth h = 3000 m, the distanceat which dispersion becomes important ranges from R =1000 km for W = 50 km, to R = 10,000 km for W = 100 km.These comprise the bounds of typical travel distances forthe T2 scenario database, suggesting that the dispersioneffect cannot be ruled out.

3. Numerical Experiments

[13] The numerical experiments were designed to testthe dispersive behavior of tsunamis in an idealized setting.According to the Buckingham Pi Theorem [Kundu andCohen, 2004], the three physical dimensions and the 9dimensional variables plus the acceleration of gravity makeup 7 dimensionless variables, shown in Table 1. It is worthnoting that the Imamura number arises naturally from this

dimensional analysis. A consequence of this is that thephysical length scale H has been replaced by its numericalcounterpart Dx(1 − K2)0.5. Hence the dimensional variablesthat are conventionally normalized using H are instead nor-malized by Dx(1 − K2)0.5 to reflect the fact that dispersion iscontrolled by the horizontal grid resolution.[14] Notable from Table 1 is the absence of wavelength

as a parameter. For example, for small Courant number K,the dimensionless parameter relevant for wave dispersionis (Dx/W)2 (Table 1). We have used the rupture width W toscale the wavelength, which can be interpreted as: all otherfactors being equal, the wavelength increases as the rupturewidth increases, and vice versa. Information on the rupturewidth is typically more readily available than informationon the wavelength, therefore the scaling variables werechosen with this practical point in mind.[15] The tsunami warning scheme used by the JATWC is

based on values of the maximum amplitude of sea levelelevation [Allen and Greenslade, 2010]. We therefore inves-tigate the dispersive behavior of tsunamis in terms of thedecay of the maximum amplitude with linear distance Ralong the line of maximum energy. A depiction of thedomain and setup of the experiments is shown in Figure 1.These consist of eastward propagating tsunamis on anequatorial beta plane. The sources and mareogram locationsare oriented so that they capture the maximum variations inwave energy decay as W/Dx is varied. Following Hammackand Segur [1978], we normalize the maximum amplitude interms of the total volume of the initial condition, which inturn can be conveniently parameterized by WLuo rather thandirectly measured. The dip angle of the rupture is kept fixedat 30°, which means the scaling of our results is specific tothis angle. However, this can be linearly adjusted within thetypical range of dip variation. A total of 43 simulations wereperformed. The specific values of all relevant parameters,including the dimensionless parameters are shown inTable 2. When applicable, the variables are first converted toCartesian length [m] before nondimensionalization.

4. Results

4.1. Evolution and Scaling of Maximum Amplitude

[16] If it is assumed that the evolution of tsunami wavesis strictly linearly dispersive, then the maximum amplitude

Table 1. Dimensionless Variables

Variablea

Courant number (K) (gH)12Dt/Dx

Imamura number (Im) (Dx/2H)(1 − K2)0.5

Numerical dispersion factor (Dx/W)2(1 − K2)Normalized volume Vo/(WLuo)Normalized maximum amplitude hmaxDx2/V0

Normalized distance (gH)12t/Dx

Nonlinearity factor uo/H

aHere g is the acceleration of gravity (m/s2), H is the mean waterdepth (m), Dt is the time step (s), Dx is the horizontal grid spacing (m),W is the rupture width (m), L is the rupture length (m), uo is the ruptureslip (m), Vo is the volume of the initial disturbance (m3) and hmax (m) isthe maximum surface elevation.

Figure 1. Schematic of the domain of the numerical experiments. The dimension of the surface initialcondition (in gray) is not shown to scale with the domain. The circles show the locations of the mareo-grams. The solid circles, denoted a, b, c, and d, are at a distance of 500, 2000, 5000, and 10,000 km fromthe source, respectively. The time series taken at these four points are shown in Figure 2.

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should be proportional to the volume of the initial deforma-tion, and it should decay with distance or time at a constantrate [Hammack and Segur, 1978]. The scaling strategydescribed in section 1 implies that the maximum amplitudeof waves resulting from the deformation of a specificvolume can be obtained by linearly scaling the wave solutionfrom deformation of another volume. Figure 2 demonstratesthe validity this assumption with respect to variation inrupture width W. Time series of the water elevation at about500 km, 2000 km, 5000 km, and 10,000 km from the source,are presented in Figure 2 in descending order (circles a‐d). Ineach row, the red time series corresponds to the unscaledsolution from a wave generated with a deformation thathas a rupture width of W = 100 km, while other colors cor-respond to solutions from deformations with different rupturewidths and fixed uo, which have been scaled according to thedeformation volume. Figure 2 shows that the assumptionthat maximum amplitude is directly proportional to the vol-

ume of the initial deformation only applies after a sufficientlylong distance from the source, when the wave signals havesignificantly weakened. This example suggests that scalinga solution from a lower magnitude to a higher magnituderesults in an overestimation of the first or maximum ampli-tude of the wave at intermediate distance, because it effec-tively replaces the ‘true’ solution with one whose source hasa narrower width and higher slip.[17] As mentioned before, we shall not attempt an

exhaustive analysis of the wave characteristics from thenumerical experiments. Instead we attempt to summarize theresults in terms of the decay of maximum wave amplitude,which has direct relevance to tsunami warnings and strongprecedence from past theoretical analyses. As a starting pointwe will adhere to linear dispersive theory [e.g., Kajiura,1963; Hammack and Segur, 1978], in which the asymp-totic solution for the decay of nondimensional wave

Table 2. Values of the Parameters Used in the Numerical Experiments

Experiment

Setup VariablesDimensionless

Variables

DispersionFactor

NonlinearityFactor

H(m)

W(km)

L(km)

uo(m)

Dx(arcmin)

Dt(s) K Im

1 1000 100 400 1 4 8 0.107 3.69 5.44E‐03 1.00E‐062 1000 100 400 2.5 4 8 0.107 3.69 5.44E‐03 2.50E‐063 1000 100 400 5 4 8 0.107 3.69 5.44E‐03 5.00E‐064 1000 100 400 7.5 4 8 0.107 3.69 5.44E‐03 7.50E‐065 1000 100 400 10 4 8 0.107 3.69 5.44E‐03 1.00E‐056 1000 80 400 1 4 8 0.107 3.69 8.51E‐03 1.00E‐067 1000 60 400 1 4 8 0.107 3.69 1.51E‐02 1.00E‐068 1000 40 400 1 4 8 0.107 3.69 3.40E‐02 1.00E‐069 1000 20 400 1 4 8 0.107 3.69 1.36E‐01 1.00E‐0610 3000 100 400 1 4 8 0.185 1.22 5.32E‐03 3.33E‐0711 3000 100 400 2.5 4 8 0.185 1.22 5.32E‐03 8.33E‐0712 3000 100 400 5 4 8 0.185 1.22 5.32E‐03 1.67E‐0613 3000 100 400 7.5 4 8 0.185 1.22 5.32E‐03 2.50E‐0614 3000 100 400 10 4 8 0.185 1.22 5.32E‐03 3.33E‐0615 3000 80 400 1 4 8 0.185 1.22 8.31E‐03 3.33E‐0716 3000 60 400 1 4 8 0.185 1.22 1.48E‐02 3.33E‐0717 3000 40 400 1 4 8 0.185 1.22 3.32E‐02 3.33E‐0718 3000 20 400 1 4 8 0.185 1.22 1.33E‐01 3.33E‐0719 6000 100 400 1 4 8 0.261 0.597 5.13E‐03 1.67E‐0720 6000 100 400 2.5 4 8 0.261 0.597 5.13E‐03 4.17E‐0721 6000 100 400 5 4 8 0.261 0.597 5.13E‐03 8.33E‐0722 6000 100 400 7.5 4 8 0.261 0.597 5.13E‐03 1.25E‐0623 6000 100 400 10 4 8 0.261 0.597 5.13E‐03 1.67E‐0624 6000 80 400 1 4 8 0.261 0.597 8.02E‐03 1.67E‐0725 6000 60 400 1 4 8 0.261 0.597 1.43E‐02 1.67E‐0726 6000 40 400 1 4 8 0.261 0.597 3.21E‐02 1.67E‐0727 6000 20 400 1 4 8 0.261 0.597 1.28E‐01 1.67E‐0728 6000 100 400 1 2 8 0.523 0.264 1.00E‐03 1.67E‐0729 6000 80 400 1 2 8 0.523 0.264 1.56E‐03 1.67E‐0730 6000 60 400 1 2 8 0.523 0.264 2.78E‐03 1.67E‐0731 6000 40 400 1 2 8 0.523 0.264 6.25E‐03 1.67E‐0732 6000 20 400 1 2 8 0.523 0.264 2.50E‐02 1.67E‐0733 6000 100 400 1 1 2 0.261 0.149 3.21E‐04 1.67E‐0734 6000 80 400 1 1 2 0.261 0.149 5.01E‐04 1.67E‐0735 6000 60 400 1 1 2 0.261 0.149 8.91E‐04 1.67E‐0736 6000 40 400 1 1 2 0.261 0.149 2.00E‐03 1.67E‐0737 6000 20 400 1 1 2 0.261 0.149 8.02E‐03 1.67E‐0738 6000 80 400 1 5 8 0.209 0.756 1.29E‐02 1.67E‐0739 6000 40 400 1 5 8 0.209 0.756 5.14E‐02 1.67E‐0740 6000 20 400 1 5 8 0.209 0.756 2.06E‐01 1.67E‐0741 6000 100 400 1 4 8 0.261 0.597 5.13E‐03 1.67E‐0742 6000 100 200 1 4 8 0.261 0.597 5.13E‐03 1.67E‐0743 6000 100 800 1 4 8 0.261 0.597 5.13E‐03 1.67E‐07

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amplitude, to the first order, is commonly presented, in ageneral sense, as

�n ¼ C1" Rnð Þ��*T �ð Þ; ð7Þ

where hn is the nondimensional amplitude, C1 is a constant," is a small quantity which defines the order of the per-turbation, Rn is a nondimensional linear travel distance ortravel time of the leading wave, a* > 0 is a decay constant,and T is usually an Airy‐like or Bessel‐like integralinvolving z, which is the position of the leading wave rel-ative to the linear travel distance. This relationship appliesfor Rn → ∞ and z → 0. In Kajiura’s theory, the constant a*varies with the source width and travel distance. Noting thatthe maximum amplitude hmax can be found by T(zmax),which in turn is assumed to be of O(1), and define b ≡ C1",then by using Table 1 we can attempt to describe the decayof the maximum amplitude using

�maxDx2

V¼ �

R

Dx

� ��

; ð8Þ

for which we seek to find, experimentally, the functionsb = f (W/Dx) and a = g(W/Dx). Here, (W/Dx) is related to thenumerical dispersion factor in Table 1, but for simplicity –and because K ≪ 1, we omit the factor (1 − K2). In other

words, we seek to find an expression for the decay of themaximum amplitude of a 2D tsunami wave as mainly afunction of the horizontal resolution of the dominant wave-lengths which in turn can be characterized by W/Dx. Thefunctions f and g are found by fitting the results from eachexperiment in the form of

log10�maxDx2

V

� �¼ log10 � þ � log10

R

Dx

� �: ð9Þ

[18] Here, the fitting applies for R > L, where L is thelength of the rupture. The fitted values of a and log10b areplotted against W/Dx in Figure 3. The symbols do not dif-ferentiate between experiments of the same parameter setbut different slip, hence due to the normalization, the effectof slip between experiments of otherwise identical param-eter set cannot be distinguished (see Table 2). We can seethat the behaviors of both parameters can be classified intotwo distinct regimes. In the first of these, a and log10b arenot constant but vary with W/Dx, and their behaviors can bedescribed approximately by

� ¼ �1:30þ 0:042W

Dx

log10 � ¼ �0:17W

Dx

9>>=>>;;W

Dx< 15: ð10aÞ

Figure 2. Time series of surface elevation at increasing distances from the source. Colors other than redcorrespond to sources withW < 100 km, from which the surface elevation has been scaled by (100 km/W).In descending order, the time series were taken from 500, 2000, 5000, and 10,000 km from the source.These correspond to points a, b, c, and d, respectively, in Figure 1.

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Figure 3. The parameters of equation (8) as a function of W/Dx.

Figure 4. Locations of the hypothetical South Java and Puysegur events.

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In the second regime, they are constant:

� ¼ �0:67

log10 � ¼ �2:5

9=;;

W

Dx� 15: ð10bÞ

[19] To summarize, a indicates the rate of decay of maxi-mum amplitude with respect to normalized distance. Theinterpretation of the result is that the decay rate of maximumamplitude due to dispersion decreases as the wavelengthbecomes better resolved, consistent with theoretical models[e.g., Kajiura, 1963], and eventually the decay goes with(R/Dx)−2/3 when W/Dx ≥ 15. This horizon point beyondwhich decay becomes asymptotic is in agreement with thenumerical experiments of Shuto [1986], in which it wasshown that the numerical decay in wave height is minimizedwhen the wavelength is covered by more than 20 grid points.

We have clarified here that the decay rate is invariantwith respect to wavelength resolution for W/Dx ≥ 15, butbecomes dependent on the resolution as the wavelength getsshorter. The interpretation of b for W/Dx < 15 is that fortwo initial conditions with the same volume but differentdimensions, the one with the larger slip would generatewaves with higher initial maximum amplitude. For W/Dx ≥15, only the volume matters. In other words, for poorer‐resolved waves, the maximum amplitude should be scaledby the slip; for well‐resolved waves, by the volume.

4.2. Scaling Bias

[20] In the previous section it was shown that f and g arenot constants but functions of W/Dx, for W/Dx < 15.Therefore scaling of maximum amplitudes based only onslip values will introduce a bias:

Bias x; yð Þ ¼ 1� �2

�scaled¼ 1� C x; yð Þ; ð11Þ

where hscaled is obtained by scaling the solution from anexisting event, i.e., a database scenario with rupture widthand slip: (W, uo) and h2 is from an alternate event (W2, uo2).C(x, y) denotes the ratio of the two amplitudes. Unfortu-nately, there is no convenient expression for C(x, y) for allx and y. However, using equations (9) and (10a), we canobtain an approximate expression for C along the line ofmaximum energy:

C xm; ymð Þ � C Rð Þ � 10aDWDx

R

Dx

� �bDWDx

; ð12Þ

where a = −0.17 and b = 0.042, DW = W − W2, and R ={(xm − x0)

2 + (ym − y0)2}0.5, with (x0, y0) being the coordinates

of the rupture’s centroid, and (xm, ym) the coordinates ofpoints along the line of maximum energy.[21] This suggests that the bias near the source increases

with the discrepancy in the width, but this bias decreaseswith distance due to the dispersive effect. Moreover, it canbe seen in Figure 3 that the bias has an extremely weakdependence on the mean water depth due to the nature of thenumerical dispersion.

5. Application to T2 Scenarios

[22] As discussed previously, scaling the hmax from aparticular scenario introduces a bias for which, given thecomplexity of the bathymetry and the complex geometry ofthe initial conditions, a universal expression is not available.However, it is possible to use equation (12) to assess thebias along the line of maximum energy. In this section, weassess the validity of equation (12) within the practicalenvironment of the T2 scenario database. It should be notedthat the validation was not conducted by comparing T2scenarios with a specific simulation among the numericalexperiments; in most cases there is no matching simulationthat bears a resemblance to any T2 scenario. Rather, itshould be emphasized that the results from the numericalexperiments are generalized in equation 12, which in turn isapplied to the selected T2 scenarios.[23] A relevant feature of the T2 scenario database is the

way in which the ruptures for each scenario have been

Figure 5. Two fields of maximum amplitude for a Mw =8.47 event south of Java: (top) hscaled has been obtainedby scaling the Mw = 8.5 T2 scenario, and (bottom) ha hasbeen obtained from rerunning the model with a rupturewidth of 70 km.

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constructed. Ruptures for large earthquakes are representedas the combination of a number of smaller 100 km longrupture elements, each of which has their strike aligned withthe local subduction zone. For example, for a Mw = 8.5scenario, four adjacent rupture elements are combined tocreate one rupture with length approximately 400 km. Notethat this length will not be exact due to the curvature of thesubduction zone. Further details on the rupture constructionare given by Greenslade et al. [2010].[24] We now consider two hypothetical scenarios: a South

Java event and a Puysegur event. The locations of theirruptures are shown in Figure 4. It should be noted that theselocations were not chosen to represent any past tsunami-genic earthquakes that have occurred in the vicinities, butsimply as locations that are relatively near to the Australiancoastline.

5.1. South Java

[25] Our first hypothetical example is based on a Mw =8.5 scenario along the Sunda Arc in the Indian Ocean,

located just to the south of Java (see Figure 4). For thiscase, rupture dimensions for the Mw = 8.5 scenarios in theT2 scenario database are L ≈ 400 km, W = 80 km and uo =5 m. We create an “alternate” scenario by rerunning theMOST model with a new rupture width in order to achievea slightly different magnitude to the original scenarios. Forthe example presented here, we use the width of thealternate scenario as 70 km (hence DW = −10 km), whichproduces a Mw = 8.47 event. All other rupture and modeldetails are kept fixed. The appropriate scaling [Greensladeet al., 2010] was then applied to the original T2 scenario(in this case, a factor of 0.89) to match this magnitude. Thescaled maximum amplitudes (hscaled) are then subtractedfrom the alternate maximum amplitudes (ha) at each gridpoint in the domain to obtain the bias:

�bias ¼ �a � �scaled : ð13Þ

[26] The two fields of maximum amplitude are shown inFigure 5 and the bias in Figure 6. As can be seen from

Figure 6. Difference (bias) between the two fields of maximum amplitude shown in Figure 4.

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Figure 5, the maximum wave amplitudes arising from theJava earthquake mostly occur perpendicular to the strike(i.e., perpendicular to the length of the rupture), a featuresometimes described as “beaming”. The exception to thisis the area to the south of 17°S, where the influence of theExmouth Plateau is apparent. In the lower panel of Figure 5,where the rupture has a width of 70 km (10 km less than thatof the upper panel), greater transmission of the wave energyonto and beyond the Exmouth Plateau is apparent. Figure 6shows that, apart from the high values along the earthquakerupture, the maximum bias occurs near the center of the“beaming” region and in general decreases with distance fromthe source. This general trend is defied by the presence ofcomplex topography such as the Exmouth Plateau.[27] In order to compare this bias with the theoretically

expected value in equation (12), we first multiply (hscaled) by

the theoretical bias, with respect to the closest distance of apoint to any of the rupture elements:

�C x; yð Þ ¼ �scaled 1� C R*ð Þð Þ

R* ¼ min Rnð Þ

Rn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix� xnð Þ2þ y� ynð Þ2

q;

ð14Þ

where (xn, yn) is the location of the nth rupture element. Wegroup the grid points into bins perpendicular to the strike,with length L and width 20 grid points. This defines the lineof maximum energy, i.e., (xm, ym) in equation (12). Fromeach bin, we calculate the mean and standard deviationsfrom the computed bias (equation (13)) and the theoretical

Figure 7. Comparison between the theoretical bias and the computed bias as a function of normalizeddistance from the source for the South Java Mw = 8.47 case: (top) raw bias values and (bottom) the biasvalues normalized by hscaled.

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bias (equation (12)), in both dimensional and percentageform (as a percentage of hscaled). These are plotted againstR* in Figure 7. The general trend of decreasing bias withdistance from the source can be seen more quantitatively inFigure 7. Although the dimensional value of the bias variessignificantly, the percentage value follows a more slowlyvarying trend. Comparing the computed biases with thetheoretical values (red line), this trend can be explained bythe dispersive decay that is described by equation (12).

5.2. Puysegur

[28] A second hypothetical example we consider here isbased on a Mw = 8.5 scenario on the Puysegur subductionzone, located to the southwest of New Zealand (see Figure 4).In this case, we increase the width of the rupture for thealternate model run by DW = 20 km which provides a Mw =8.57 event and compare this to the relevant Mw = 8.5 T2

scenario scaled by a factor of 1.25. The two fields of maxi-mum amplitude are shown in Figure 8. The wave energy inthis case receives little topographical influence until it arrivesat the continental shelf of southeast Australia. The bias(Figure 9) shows a pattern consistent with that of the Javacase, in that the bias decreases away from the source andoccurs mostly within the “beaming” region. The secondarybias “lobe” located just to the north of this region suggests theinfluence of the source geometry and bathymetry.[29] Unlike the South Java case, the initial general direction

of the wave energy here is significantly more oblique to thegrid arrangement. In order to normalize the distance R inequation (12) and to evaluate the bias as a function of distancefrom the source and we use a proxy for effective grid size:

Dx* ¼ Dx sin� þDy cos�; ð15Þ

where � is the strike angle. The bin‐averaged values inFigure 10 show substantially larger standard deviations ofthe bias compared to those in Figure 7, consistent with theeffect of the larger width discrepancy. Comparison with thetheoretical values shows that the general trend can still beexplained by equation (12).

5.3. Summary

[30] Although only two cases have been presented indetail here, a total of 8 alternate scenarios were generatedand compared to their equivalent scaled T2 scenarios. Foreach of the two locations (South Java and Puysegur), thesealternate cases were (1) a Mw = 8.42 with rupture width of60 km i.e., DW = −20 km, (2) a Mw = 8.47 with rupturewidth of 70 km, i.e., DW = −10 km, (3) a Mw = 8.54 withrupture width of 90 km, i.e., DW = 10 km, and (4) a Mw =8.57 with rupture width of 100 km, i.e., DW = 20 km.[31] To summarize the results, the mean and standard

deviation of the percentage bias from the bin located 240grid points (about 15°) away from the source are plottedagainst the normalized width discrepancy in Figure 11. Thisdistance was chosen since it represents a typical propagationdistance from these sources to the Australian coastline.Figure 11 suggests that for this typical propagation distancethe bias lies within 10%. It can be seen that there is aninvariance in the percentage bias, although it should beborne in mind that only two source regions have beenexamined here. Also consistent with the theory is that apositive width discrepancy is almost a mirror image of thediscrepancy of the opposite sign.

6. Conclusions

[32] In this work we have constructed a simple decay lawof maximum amplitude with respect to the source widthresolution in the T2 scenario database. For typical resolu-tions, this law applies for ‘intermediate distances’, that is,ranging from a few wavelengths from the source up toapproximately 5000 km, which is the maximum practicaldistance for transoceanic tsunami propagation. It is shownthat within these parameters, wave signals from sources ofthe same volume but different width and slip will exhibitdifferent shapes and amplitudes initially but will converge along distance from the source due to the differential decay

Figure 8. Two fields of maximum amplitude for a Mw =8.57 event on the Puysegur subduction zone: (top) hscaledhas been obtained by scaling the Mw = 8.5 T2 scenario,and (bottom) ha has been obtained from rerunning the modelwith a rupture width of 100 km.

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rate influenced by the grid‐dependent dispersive effect. Thisimplies that a scaled solution (based on slip) from a sourcewith a narrower width would initially have an overestimatedmaximum amplitude, and would subsequently decay faster.[33] Kajiura [1963] suggested several regimes for maxi-

mum amplitude decay, delineated by the parameter pa (seeequation (6). The typical configuration and travel distances ofthe T2 scenarios, and those in the numerical experiments ofsection 3 correspond to pa of O(1). For sufficiently widesources (about 15 grid spaces), the decay rate converges to the−2/3 decay law, which corresponds to the two‐dimensionaltheory of Kajiura [1963] for pa < 1. Note, however, thatwe have not attempted to theoretically reproduce the decaylaw for W/Dx < 15, so there is still a question on whetherequation (10) corresponds to the physical world. Hammackand Segur [1978] showed that the initial maximum amplitudeis inversely proportional to the Ursell number, in an expo-nential sense. This is qualitatively similar to the results here.Finally, it should be stressed that it is the initial maximumamplitude that contributes most to the scaling bias, whereas

the dispersive effect attenuates this bias with distance by aslowly varying function.[34] In the application to the T2 scenarios, it is shown that

although the mean of the bias with distance can be charac-terized by equation (12), the distribution of this bias aroundthe mean differs considerably between scenarios, and incases where complex topography is involved, between dis-crepancy in width. In all cases, a discrepancy in width of thesame magnitude but opposite sign produced bias distribu-tions which can be characterized as mirror images of eachother. The slowly varying nature of the bias with distancemeans that in most transoceanic propagation cases, the dis-tribution of the bias due to scaling can be characterized bythe distance‐independent mean and standard deviation.Ideally, this should be examined on a case‐by‐case basis.It should be borne in mind that this scaling law is not validclose to the source (within a few wavelengths), but shouldbe applicable further from the source up to the scales onthe order of oceanic basin widths. Closer to the coast,nonlinearity becomes important as the depth becomes

Figure 9. Difference (bias) between the two fields of maximum amplitude shown in Figure 7.

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Figure 10. Comparison between the theoretical bias and the computed bias as a function of normalizeddistance from the source for the Puysegur Mw = 8.57 case: (top) the raw bias values and (bottom) the biasvalues normalized by hscaled.

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shallower. It should be emphasized that this is not coveredin this work, nor is the T2 resolution fine enough toaccurately resolve the nonlinear effect toward the coast.

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Yoon, S. B. (2002), Propagation of distant tsunamis over slowly varyingtopography, J. Geophys. Res. , 107(C10), 3140, doi:10.1029/2001JC000791.

D. J. M. Greenslade and M. A. Simanjuntak, Centre for AustralianWeather and Climate Research, Australian Bureau of Meteorology, GPOBOX 1289, Melbourne, VIC 3001, Australia. ([email protected])

Figure 11. Percentage bias from eight alternate scenarios as a function of normalized width discrepancy.

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