NOAA Technical Memorandum OAR PMEL-116

Analytic Theory of Tsunami Wave Scattering

in the Open Ocean with Application

to the North Pacific

H.O. Mofjeld, V.V. Titov, F.I. Gonz ́alez, and J.C. Newman

Pacific Marine Environmental Laboratory

7600 Sand Point Way NE

Seattle, WA 98115

January 2000

Contribution 2168 from NOAA/Pacific Marine Environmental Laboratory
NOTICE

Mention of a commercial company or product does not constitute an endorsement by

NOAA/OAR. Use of information from this publication concerning proprietary products

or the tests of such products for publicity or advertising purposes is not authorized.

Contribution No. 2168 from NOAA/Pacific Marine Environmental Laboratory

For sale by the National Technical Information Service, 5285 Port Royal Road

Springfield, VA 22161

ii
Contents iii

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Basic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Lineal Step Topography: Normal Incidence . . . . . . . . . . 4

2.2 Continously Varying Topography . . . . . . . . . . . . . . . . 5

3. Lineal Escarpments . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Vertical Step Escarpment: Normal Incidence . . . . . . . . . 8

3.2 Vertical Escarpment: Non-Normal Incidence . . . . . . . . . . 11

3.3 Error Function Escarpment: Normal Incidence . . . . . . . . 14

4. Lineal Ridges and Trenches . . . . . . . . . . . . . . . . . . . 16

4.1 Rectangular Ridge: Normal Incidence . . . . . . . . . . . . . 17

4.2 Rectangular Ridge: Transmissivity . . . . . . . . . . . . . . . 18

4.3 Rectangular Ridge: Oblique Incidence . . . . . . . . . . . . . 22

4.4 Gaussian Ridge: Normal Incidence . . . . . . . . . . . . . . . 24

5. Circular Seamounts . . . . . . . . . . . . . . . . . . . . . . . . 27

6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.1 Ridges and S eamounts . . . . . . . . . . . . . . . . . . . . . . 34

6.2 Escarpments and Trenches . . . . . . . . . . . . . . . . . . . 34

6.3 Transient Tsunamis . . . . . . . . . . . . . . . . . . . . . . . 35

7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 36

9. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

List of Figures

1 Schematic diagram of a step escarpment . . . . . . . . . . . . . . . . 9

2 Amplitudes of reflected and transmitted tsunami waves resulting

from the interaction of a normally incident wave on a step escarpment, as a function of the depth parameter  . . . . . . . . . . . . . 10

3 Energy flux components for reflected and transmitted waves resulting

from the interaction of a normally incident wave on a step escarpment, as a function of  . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Angles of incident and refracted waves as a tsunami wave propagates

obliquely across a step escarpment . . . . . . . . . . . . . . . . . . . 12

5 Critical angle φcrit of incidence for a tsunami wave approaching a

step escarpment from the shallow side, as a function of  . . . . . . . 13

6 Amplitudes of reflected and transmitted tsunami waves resulting

from the interaction of an obliquely incident wave with a step escarpment, as a function of  . . . . . . . . . . . . . . . . . . . . . . . 14

7 Schematic diagram of an escarpment having an error-function shaped

transition in depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 Amplitudes of reflected tsunami waves resulting from the interaction

of a normally incident wave with a low error-function escarpment, as

a function of the width-scale k0σ . . . . . . . . . . . . . . . . . . . . 16

9 Schematic diagram of a rectangular ridge . . . . . . . . . . . . . . . 17

10 Transmissivity for tsunami waves that are normally incident on a

rectangular ridge, as a function of the cross-ridge phase shift θ . . . 19

11 Minimum transmissivity Tmin for tsunami waves that are normally

incident on a rectangular ridge, as a function of  . . . . . . . . . . . 20
iv Contents

12 Transmissivity for tsunami waves that are normally incident on a

rectangular ridge, as a function of  . . . . . . . . . . . . . . . . . . . 21

13 Schematic diagram showing the angles of incident, on-ridge, and

transmitted waves when an obliquely incident tsunami wave interacts with a rectangular ridge . . . . . . . . . . . . . . . . . . . . . . 23

14 Transmissivity factor λ, as a function of . . . . . . . . . . . . . . . . 24

15 Deviation of the minimum transmissivity from unity, as a function of  25

16 Schematic diagram of a ridge having a Gaussian cross-section . . . . 25

17 Amplitude of the reflected wave resulting from the interaction of

a normally incident tsunami wave with a low Gaussian ridge, as a

function of the ridge-width scale k0σ . . . . . . . . . . . . . . . . . . 27

18 Schematic diagram of a circular seamount having a summit of constant depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

19 Magnitudes of the off-seamount coefficients for scattered wave components, as a function of  . . . . . . . . . . . . . . . . . . . . . . . . 30

20 Same as Figure 19 but for the on-seamount coefficients . . . . . . . . 30

21 Depth of topography features as a function of the scattering index S 32

22 Smith-Sandwell topography for the North Pacific Ocean . . . . . . . 33

List of Tables

1 Feature depth H1, as a function of depth of the surrounding region

H0 and the scattering index S . . . . . . . . . . . . . . . . . . . . . . 31

2 Ridge width and seamount diameter corresponding to the lowest frequency resonances, as functions of tsunami wave period and depth

of the feature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Analytic Theory of Tsunami Wave Scattering in the Open Ocean

with Application to the North Pacific

H.O. Mofjeld, V.V. Titov, F.I. Gonz ́alez, and J.C. Newman

Abstract. A theory is developed to better understand the interaction of tsunami waves with

small-scale, submarine topography in the open North Pacific Ocean. The tsunamis are assumed to

be linear, long gravity waves at single frequencies. The topography is given simple forms from which

explicit formulas can be derived for transmitted, reflected, and scattered waves. This topography

includes linear features (escarpments and ridges) and circular seamounts. The theory shows that

the most important factor determining the intensity of scattering and reflection is the depth of a

feature compared with the depth of the surrounding region. A useful measure of this depth effect

is the scattering index S = 1 − Tmin, derived from the theory of lineal ridge scattering. Here

Tmin = 2/(1 + 2), where  ≡ H1/H0. For a regional depth of H0 = 5500 m, features with depth

shallower than H1 = 1500 m interact significantly with tsunamis and those shallower than 400 m

can have a major effect on these waves. The horizontal extent of a feature, compared with the

wavelength, and the angle of incidence also affect the amount of scattering and reflection. Based on

these criteria, the Emperor Seamount Chain, the Hawaiian Ridges, the Mid-Pacific Mountains, the

Aleutian/Komandorskiye and Kuril Island Arcs, and the Shatsky and Hess Rises scatter and reflect

transoceanic tsunamis. This interaction also increases the duration of these tsunamis. Simulating

these processes with numerical models requires sufficiently accurate topography and high enough

spatial resolution.

1. Introduction

The NOAA/PMEL Tsunami Research Program is carrying out a project to

improve the U.S. Tsunami Warning System and the capability of the Pacific

Disaster Center (Gonz ́alez et al., 1999). The project includes both modeling and observational components. The modeling component will lead to a

database of simulated Pacific-wide tsunamis. Using the database, simulated

tsunamis can be tuned quickly to earthquake information and to open-ocean

tsunami observations. The tuned model tsunamis will then provide forecasts of tsunami wave heights for coastal communities in Alaska, Hawaii,

California, Washington, Oregon, and the Pacific Insular States. Presently,

the source regions for the model tsunamis are limited to the Alaska/Aleutian

Subduction Zone (AASZ). However, plans are in place to expand the source

region to include the entire Pacific Rim. The open-ocean observations will

be reported in real time by an array of Deep-ocean Assessment and Reporting of Tsunamis (DART) buoys in the North Pacific Ocean and a station

near the Equator.

A goal of this work is to provide this guidance to the NOAA Tsunami

Warning Centers and the Pacific Disaster Center within an hour following

a tsunamigenic earthquake in the Pacific Region. Having a rapid procedure is required in order to give emergency managers as long a lead time

as possible before the tsunamis reach their regions of responsibility. It

is also intended that this guidance will lead to a reduction in the number of false tsunami warnings and unnecessary evacuations. Details of the

NOAA/PMEL Tsunami Program, the Method Of Splitting Tsunami (MOST)

numerical model database (Titov et al., 1999), and the DART buoy system
2 Mofjeld et al.

are given at the Tsunami Program’s Web site (URL: http://www.pmel.

noaa.gov/tsunami/).

While doing MOST model simulations of tsunamis in the North Pacific,

a number of issues arose concerning the effects of fine-scale topography on

Pacific-wide tsunamis. A study was then carried out to investigate these

issues. There were four main goals for the study: (a) understand how finescale topography affects tsunami wave propagation, (b) develop simple criteria for identifying topographic features that reflect and scatter significant

tsunami energy, (c) interpret the tsunami wave-height patterns seen in the

MOST model simulations, and (d) determine how smoothing the topography affects the model tsunamis. The preliminary results of the study were

presented at the IUGG 99 Tsunami Symposium by Mofjeld et al. (1999b).

In this technical memorandum, we present the analytic theory that was

developed to address the first two goals. It is an application of linear, longwave theory to the scattering of tsunami waves off idealized forms of topography. The analytic theory is meant to provide tools to use when interpreting

the results of the MOST numerical model. No claim is made that the bulk of

the theory is either original or comprehensive. Indeed, the theory is meant

to be as limited and simple as possible. Even so, it is still extensive enough

to unduly burden a publication in the peer-reviewed literature. This technical memorandum was therefore written to describe the theory in sufficient

detail to act as a reference for publications describing the effects of realistic

topography on Pacific-wide tsunamis. For a general overview and bibliography on tsunami prediction research, please see the recent review by Mofjeld

et al. (1999a). What is new in the theory is the concept of minimum transmissivity and a scattering index S. This index turns out to be very useful

in identifying major scattering features in the open ocean.

Although tsunami scattering is weak over low topographic features (Carrier, 1971), recent model studies show that a number of features in the North

Pacific do scatter substantial tsunami energy (Mofjeld et al., 1999b). Much

of the finer-scale topography in the North Pacific consists of lineal features

such as escarpments, ridges, and trenches. Many of these have cross-feature

scales of tens to hundreds of kilometers but extend in length over thousands

of kilometers. These are the cross-feature scales that are most relevant to

tsunami wave scattering in the open ocean. The large extent of these features

means that any tsunami propagating across the North Pacific will encounter

a number of these features. There are also extensive fields of seamounts in

the region.

The technical memorandum is organized as follows. After this introduction are four sections deriving the analytic theory: basic formulation, escarpments, ridges and trenches, and circular seamounts. These provide the

essential formulas and interpretative tools for the discussion section, which

identifies the major scattering features in the North Pacific. Concluding the

technical memorandum is a summary of results. Readers may find it efficient

to look over the next section on the basic formulation of the theory and then

go directly to the Discussion and Conclusions (Sections 6–7), referring back

to the detailed theoretical sections as needed.
Analytic Theory of Tsunami Wave Scattering 3

2. Basic Formulation

The analytic theory is based on long waves interacting with idealized topography in the open ocean. In the theory, we neglect the relatively small effects

of non-linear processes, dispersion and friction in the open ocean, as well as

the Coriolis effect. It turns out that the amount of scattering often depends

on a feature’s spatial extent relative to the tsunami wavelength or, more precisely, the phase shift across the feature. To include this dependence in the

theory, each tsunami is assumed to occur at a given frequency. Hence, the

theory may be thought of as a frequency analysis of tsunami wave scattering.

The focus of the theory is on scattering at locations that are well away from

the source and impact regions. The incident tsunami waves can then be

approximated by plane waves. While the long wave equations and formulas

for plane waves are readily available in many references and textbooks, it is

convenient to repeat them here in order to define the notation.

The long waves satisfy linearized equations of motion in the surface elevation η and the wave transport Q = H u, which is the product of the water

depth H and the horizontal water velocity u = (ux, uy)

∂ Q

∂ t = − g H  η (1)

∂ η

∂ t = − · Q (2)

where t is time and g is the acceleration of gravity.

For constant depth H, these equations have plane wave solutions of the

form



η

Q



= A

1

c kˆ



e i [k·x − ωt] (3)

where A is the complex amplitude, c = √gH is the long wave speed, kˆ is the

unit vector pointing in the same direction as the wavenumber k = (kx, ky),

x = (x, y) is the horizontal distance vector, and ω is the angular frequency.

Defining the incident angle φ relative to the x-axis, the unit direction

vector kˆ can be written

kˆ = (cos φ, sin φ) . (4)

In the following sections, analogous notation will be defined for reflected

waves scattered off lineal topography (rectangular coordinate system) and

for waves scattered off seamounts (cylindrical coordinate system).

A useful concept to apply to wave scattering problems is the conservation

of energy flux. This principle requires that without dissipation, a steady flux

into a region must be balanced by an equal flux out of the region. To apply

the principle to scattering and reflection of tsunami waves inside a region,

it is convenient to write the fluxes as integrals over the bounding surface

of the flux density. For a two-dimensional plane wave (3) the flux density,
4 Mofjeld et al.

integrated over depth, is defined as the energy propagating per unit time

and per unit width

F = ρgc kˆ |A|

2

2 . (5)

It is a vector in the direction of the wavenumber k and is proportional to the

water density ρ, the acceleration of gravity g, the long wave speed c, and the

square of the wave amplitude’s magnitude |A|. Since this form (5) has been

integrated over depth, each integral around the bounding surface becomes a

line integral of the normal component of the energy flux density.

For lineal topographic features (such as escarpments, ridges, and trenches),

rectangular regions provide simple forms for the flux components at the

bounding surfaces if two sides are oriented parallel to the features. When an

incident wave is planar, the transmitted and reflected waves are also planar.

Conservation of energy flux then reduces to a simple balance between flux

densities. In vector form,

FI + FR = FT (6)

where the subscripts refer to incident (I), reflected (R), and transmitted (T)

waves.

In the case of scattering off an isolated circular features, a circular region centered on the seamount provides simple formulas for the fluxes. For

instance, Lamb (1932) uses the flux out of such a region to estimate the

amplitudes of scattered waves resulting from the interaction of plane waves

with circular islands.

2.1 Lineal Step Topography: Normal Incidence

The theory uses stepwise topography to approximate the real topography.

These steps consist of constant-depth segments, separated by vertical transitions in depth. For lineal topography (escarpments, ridges and trenches),

we orient the coordinate system so that the topography varies only in the

x-direction. In this subsection, we concentrate mainly on normal incidence

(φ = 0◦). Then within the j-th segment (xj ,< x < xj + δxj ), the water

depth Hj is constant and the phase velocity cj and incident x-wavenumber

kj component (dropping the subscript x) are

cj = gHj and kj = ω

cj

(7)

For normal incidence the incident wave has the form



ηj

Qj



= Aj



1

cj



e i [θj (x)−ωt] (8)

where the cumulative phase θj (x) is given by

θj (x) = θ0 + kj (x − xj ) + 

j

j=1

kj−1(xj − xj−1) . (9)
Analytic Theory of Tsunami Wave Scattering 5

Here, the last term is the sum of the individual contributions to the phase

that are made as the wave propagates across each topographic step leading

to the j-th step.

For the reflected wave, we use ξj to denote the surface elevation, Rj

the water transport, and Bj the complex amplitude. Since this wave is

propagating in the opposite direction (in the negative x-direction) from the

incident wave, we reverse the signs of the transport (cj → −cj ) and the

phase (θj → −θj ) in (8)



ξj

Rj



= Bj

1

−cj



e i [−θj (x)−ωt] (10)

The location of the origin x = 0 and the phases of the complex amplitudes A(0) and B(0) at x = 0 are chosen to be the most convenient for the

particular topographic feature or the method of solution.

Linear wave theory requires that two quantities be continuous at an

abrupt change in water depth: the total surface elevation and the component

of the total water transport that is perpendicular to the transition in depth.

In the case of normal incidence on lineal topography varying only in the

x-direction, these conditions for the depth transition at x = xj are

ηj−1 + ξj−1 = ηj + ξj (11)

Qj−1 + Rj−1 = Qj + Rj (12)

Substituting for the normal transports in terms of the surface elevations and

using (7),

cj−1 [ηj−1 − ξj−1] = cj [ηj − ξj ] (13)

or

ηj−1 − ξj−1 = j [ηj − ξj ] (14)

where

j ≡

 Hj

Hj−1

 1

2

. (15)

When the matching conditions are applied to all the transitions in the

region, the resulting matching equations lead to the formulas describing the

wave scattering by the full topographic feature. When the subscript j is

dropped, the depth parameter  corresponds to the minimum depth of a

feature relative to the depth of the nearby region over which the incident

waves propagate to the feature. As we will see throughout the rest of this

memorandum,  plays a fundamental role in the scattering theory.

2.2 Continously Varying Topography

The theory for continuously varying topography is based on the matching

equations (11–14) in which the segment widths are allowed to go to zero
6 Mofjeld et al.

(δxj → 0). The derivation given below is for normal incidence. It is convenient to use the wave speeds cj in the analysis. Solving (11–14) for ηj and

ξj leads to symmetric equations for the two waves

ηj − ηj−1 = −δcj

2cj

[ηj−1 − ξj−1] (16)

ξj − ξj−1 = −δcj

2cj

[ξj−1 − ηj−1] (17)

where δcj ≡ cj − cj−1.

Writing the surface elevations ηj−1 and ξj−1, evaluated at the matching

location x = xj , as

ηj−1 = Aj−1 eiθj and ξj−1 = Bj−1 e−iθj (18)

and using the phase relation

θj = θj−1 + kj−1δxj (19)

where

kj−1 = ω

cj−1

and δxj = xj − xj−1 (20)

leads to difference equations for the wave amplitudes

Aj − Aj−1 = −δcj

2cj



Aj−1 − Bj−1 e−2iθj



(21)

Bj − Bj−1 = −δcj

2cj



Bj−1 − Aj−1 e2iθj



. (22)

Dividing (21–22) by δxj and taking the limit δxj → 0 leads in turn to

the differential equations in the amplitudes A(x) and B(x)

dA

dx = −βA + βB e−2iθ and dBdx = −βB + βA e2iθ (23)

where

β ≡

1

2c

dc

dx (24)

and the phase θ(x) is given by the integral of wavenumber over x

θ(x) = 	 x

0

k(x) dx . (25)

The boundary conditions on the amplitude A of the incident wave is that

it is equal to a known value A0 at a location x = 0 that is well before the

wave reaches the topography of interest

A = A0 at x = 0 . (26)
Analytic Theory of Tsunami Wave Scattering 7

The condition on the amplitude B is that the back-scattered wave has no

source on the far side of the topographic feature. Hence,

B → 0 as x → ∞ . (27)

Equations (23) can be simplified by using modified amplitudes A and

B that incorporate Green’s 1

4 th–law

A =

 H

H0

− 1

4

A and B =

 H

H0

− 1

4

B . (28)

The differential equations (23) then take the form

dA

dx = βB e−2iθ and dBdx = βA e2iθ (29)

which can be integrated to give

A = A

0 +

	 x

0

β(x) B e−2iθdx (30)

B = B

0 +

	 x

0

β(x) A e2iθdx (31)

where

θ =

	 x

0

k(x) dx .

The phase terms in the integrands of (30–31) contain factors of two in the

exponents because the total phase shift is the sum of two terms: the phase

shift θ due to forward propagation of the incident wave to a scattering site

and another shift of θ for the propagation of the scattered wave returning

from that site.

In this memorandum, the scattering by continuously varying topography

is limited to low relief. In this case, the topographic scattering is weak

enough that the incident amplitude A0 can be substituted into the equation

(31) for A

. For an isolated topographic feature that is limited in extent

(β = 0 for |x| > some x1),

B .= B

0 +

	 x

0

β(x) A

0 e2iθ

dx (32)

or

B .

= B

0 + A0

	 x

0

β(x) e2iθdx . (33)

Using the boundary condition (27) on B leads to a first approximation for

amplitude of the scattered wave

0 .= B

0 + A0

	 ∞

0

β(x) e2iθdx . (34)
8 Mofjeld et al.

Changing the origin (x = 0) so that it is centered on the topographic feature

and extending the lower limit (x → −∞) of the integer in (34), the amplitude

B0 of the scattered wave is given approximately by

B0 .

= −A0

	 ∞

−∞

β(x) e2iθdx . (35)

At the same order of approximation the incident wave is unaffected by

the small-slope topography (more precisely, topography where β/k0 	 1),

other than the Green’s law variations in amplitude and cumulative shifts in

phase away from k0x. This is the WKBJ approximation for the interaction

of a tsunami wave with topography

η(x) .= A0

 H

H0

− 1

4

eiθ , θ = θ0 +

	 x

x0

k(x) dx . (36)

A first correction to the amplitude of the transmitted wave, due to scattering,

can be made using the conservation of energy flux (5–6).

In the following sections, the theory for continuously varying topography

will be applied to two cases. The first case is an escarpment having an error

function profile in depth, and the second is a ridge having a Gaussian profile.

These cases provide a way of identifying those processes predicted with step

topography that are likely to occur when tsunami waves encounter more

realistic topography. They also identify three frequency regimes: where

the step-topography theory applies, a continuous theory is needed, and the

WKBJ approximation adequately describes the behavior of the waves.

3. Lineal Escarpments

This section on lineal (straight) escarpments focuses on scattering and reflection by single transitions in depth. The simplest case is a step transition

when the tsunami wave approaches at normal incidence (φ = 0◦). For this

case, the behavior of the wave is dependent only on the depth parameter 

and is independent of the wave frequency ω. At other incident angles φ, the

behavior is similar to the case of normal incidence except when the tsunami

waves approach the escarpment from the shallow side. Then, the waves can

be perfectly reflected for φ>φcrit, which depends on frequency ω. The last

part of the section deals with a continuous depth transition in the form of

an error function profile. The extension to non-normal incidence is straightforward. This section also provides essential background for the following

one on scattering by ridges and trenches.

3.1 Vertical Step Escarpment: Normal Incidence

Letting H0 be the depth on the side of a step escarpment (Figure 1) with

the incident wave and letting H1 be the depth on the other side, the matching equations (11 and 14) lead to the following relations between the wave
Analytic Theory of Tsunami Wave Scattering 9

Figure 1: Schematic diagram of a step escarpment having a vertical transition

between two constant-depth regions H0 and H1. View is edge-on.

amplitudes

AI + BR = AT (37)

AI − BR = AT (38)

where AI is the amplitude of the incident wave, BR is the amplitude of the

reflected wave, AT is the amplitude of the transmitted wave, and x = 0

at the escarpment. Here, the subscript has been dropped from the depth

parameter  ≡ 1.

Note that the matching conditions (37–38) do not change as the wave

frequency ω is changed. Hence, the results for the step escarpment are

independent of ω when the tsunami waves approach the escarpment with

normal incidence.

Solving (37–38) for the amplitude ratios RT ≡ AT /AI and RR ≡ BR/AI ,

RT = 2

1 + 

and RR = 1 − 

1 +  . (39)

Figure 2 shows how the amplitude ratios RT and RR vary with . When the

incident wave approaches from the deep side of the escarpment ( < 1), the

amplitude AT is larger than AI ; and the amplitude BR of the reflected wave

is positive. As the depth parameter  becomes smaller, the greater depth

contrast across the escarpment causes AT → 2 × AI and BR → AI .

Conversely, there is negative reflection (BR < 0) when the incident wave

approaches from the shallow side ( > 1 ). In this case, increasing depth contrast leads to smaller AT → 0 and BR → −1. Hence, large depth contrasts

(||  1) produce strong wave reflection (|RR| → 1) regardless of whether

the incident wave approaches the escarpment from the deep or the shallow

side.

The transition (Figure 2) between the two asymptotic regimes passes

through  = 1 where there is perfect transmission (RT = 1) and no reflection
10 Mofjeld et al.

Figure 2: Amplitudes of reflected and transmitted tsunami waves resulting from

the interaction of a normally incident wave (φ = 0◦) on a step escarpment, as a

function of the depth parameter  = H1/H0. The amplitudes are shown as ratios

relative to the amplitude A0 of the incident wave.

(RR = 0). The width of the transition can be measured by the end-values

of the range 1.9 ≥ T ≥ 0.1. Using this criterion, the transition scans 1/19 ≤

 ≤ 19 . The maximum rate of change of RT and RR occurs when  = 1

max 

dRT

d 

= max 

dRR

d 

= −2 . (40)

Using (5), the transmitted energy flux FT can then be written

FT = 4

(1 + )

2 FI and FR = −(1 − )

2

(1 + )

2 FI . (41)

As Figure 3 shows, the energy flux FT of the transmitted wave has a

maximum value of unity (perfect transmission) at  = 1 and decreases toward

zero energy transmission as the depth contrast increases. This includes the

case in which the transmitted wave AT → 2 × AI in which the escarpment

shoals to a very shallow depth on the side opposite the incident wave. The

energy flux FT of the transmitted wave is zero at  = 1 and increases with

the depth contrast in such a way that the total energy flux (6) is conserved.
Analytic Theory of Tsunami Wave Scattering 11

Figure 3: Energy flux components in the direction of wave propagation for the

reflected FR and transmitted FT tsunami waves resulting from the interaction of a

normally incident wave on a step escarpment, as a function of the depth parameter .

The fluxes are shown as ratios relative to the flux FI of the incident wave.

3.2 Vertical Escarpment: Non-Normal Incidence

When the incident wave (Figure 4) approaches a step escarpment from a

non-normal angle (φ0 = 0◦), the incident, reflected, and transmitted waves

all satisfy the dispersion relation

kx,j = kj cos φj and ky,j = kj sin φj where kj ≡

ω

cj

. (42)

For the step escarpment, Snell’s law of refraction (derived from the constancy of ky across the y-independent topography ky,1 = ky,0), determines

the direction φ1 of the transmitted wave in terms of the incident angle φ0

and the depth parameter 

φ1 = arcsin( sin φ0) . (43)

As shown in Figure 4, refraction causes the transmitted wave to turn more

normal (φ1 < φ0) when the incident wave approaches from the deep side of

the escarpment ( < 1).

Conversely, it is less normal (φ1 > φ0) when the incident wave approaches

from the shallow side ( > 1). Unlike normal incidence, it is possible for the

incident wave to be perfectly reflected by the escarpment when the incident

wave approaches from the shallow side of the escarpment. That is, there
12 Mofjeld et al.

Figure 4: Schematic diagram showing the angles of incident (φ0) and refracted (φ1)

waves as a tsunami wave propagates obliquely across a step escarpment. φ1 < φ0

indicates that the incident wave is approaching from the deep side of the escarpment.

is a critical incident angle [φ0]crit beyond which the transmitted wave is

evanscent (kx,1 imaginary). A formula for this angle can be derived from the

dispersion relation

(kx,j )

2 = ω2

c2

j

− (ky,j )

2 (44)

by noting that ky,1 = ky,0 and setting kx,1 = 0

0 = ω2

c2

1

− k2

y,0 . (45)

Using (42) leads to

[φ0]crit = arcsin 




−1 ,  ≥ 1 . (46)

Figure 5 shows that the critical angle [φ0]crit decreases rapidly as  increases past  = 1. Since tsunami waves are not likely to have exactly normal

incidence on an escarpment, the theory suggests that there is a strong tendency for tsunami waves to be reflected at a sharp transition from very

shallow water to the deep ocean. This tendency is much less pronounced

for oceanic escarpments since rarely is  > 1.1 for escarpments in the North

Pacific, except for the Mendocino Escarpment.

Formulas for the transmissivity and reflectivity can be derived for nonnormal incidence using modified versions of the matching conditions (37–

38). While the matching condition for surface elevation remains the same,
Analytic Theory of Tsunami Wave Scattering 13

Figure 5: Critical angle φcrit of incidence (above which waves are perfectly reflected) for a tsunami wave approaching a step escarpment from the shallow side,

as a function of the depth parameter .

the condition on water transport is on its x-component

AI + BR = AT (47)

cos φ0 [AI − BR] =  cos φ1 AT . (48)

We define a modified form of the depth parameter as



 = 

cosφ1

cosφ0

(49)

where φ1 is given be (43).

Dividing (48) by cos φ0 and using (49), the matching conditions (47–

48) then have the same form (39) as those for normal incidence, with 

substituted for . The amplitude ratios for non-normal incidence can then

be written down immediately

R

T = 2

1 +  and R

R = 1 − 

1 +  for <crit (50)

and, noting that perfect reflection occurs for larger values of ,

R

T = 2 and RR = 1 for  ≥ crit . (51)

Figure 6 shows that R

T and RR follow RT and RR closely when the

incident wave approaches the escarpment from the deep-water side ( ≤ 1).

As  increases, they continue to follow RT and RR until  is near its cutoff
14 Mofjeld et al.

Figure 6: Amplitudes of reflected B0 and transmitted A1 tsunami waves resulting

from the interaction of an obliquely incident wave with a step escarpment, as a

function of the depth parameter  = H1/H0. The amplitudes are shown as ratios

relative to the amplitude A0 of the incident wave, where the incident angle φ0 = 30◦.

 > 1 when the incident waves approach the escarpment from the shallow side.

crit. In that neighborhood, both R

T and RR increase rapidly toward the

values (51) expected for perfect reflection. They remain at these values for

>crit. As seen in Figure 5, the depth parameter  need not be much

greater than unity before this condition of perfect reflection is reached.

3.3 Error Function Escarpment: Normal Incidence

In this subsection, we consider scattering off a low escarpment having an

error function profile in the x-direction. The first order theory we will use

is valid for topography in which only small changes in depth occur relative

to the total depth. We assume normal incidence for the waves. When the

depth profile across an escarpment has this profile, we can write the depth

H(x) as

H(x) = H0



1 + 2 − 1

σ

√π

	 x

−∞

e

− x2

σ2 dx

. (52)

As shown in Figure 7, H0 is asymptotic depth (x → −∞) on the same side

of the escarpment as the incident wave. If H1 denotes the asymptotic depth

(x → ∞) on the other side of the escarpment, the depth parameter  we will
Analytic Theory of Tsunami Wave Scattering 15

Figure 7: Schematic diagram of an escarpment having an error-function shaped

transition in depth between two constant-depth regions H0 and H1.

use is the asymptotic value

 ≡



H1

H0

 1

2

. (53)

Since the fractional change in depth is small across the escarpment, it is

convenient to define δ as the deviation of  from unity

δ ≡  − 1 . (54)

Here, δ < 0 when the incident wave approaches the escarpment from the

deep-water side and δ > 0 when it approaches from the shallow side.

To discuss first-order scattering off this escarpment, it is convenient to

focus on amplitude B0 (35) of the back-scattered wave relative to the amplitude A0 of the incident wave. For normal incidence (φ = 0◦) and to first

order in δ, the scattering source term β (24) for an escarpment with an

error function profile is

β .= δ

2σ

√π

e

− x2

σ2 . (55)

Then the modified amplitude B

0 (35) for the back-scattered wave is

B

0

.

= −A

0

δ

2σ

√π

	 ∞

−∞

exp 2ik0 − x2

σ2



dx (56)

or

B0 .

= A0

(−δ)

2

e−k2

0σ2

. (57)

Consistent with the step topography, this expression shows that the amplitude of the back-scattered wave (Figure 8) is independent of frequency when
16 Mofjeld et al.

Figure 8: Amplitudes B0 of reflected tsunami waves resulting from the interaction

of a normally incident wave (φ = 0◦) with a low error-function escarpment, as a

function of the width-scale k0σ, for various values of the deviation δ of the depth

parameter from constant depth ( = 1). The amplitudes are given as ratios relative

to the amplitude A0 of the incident wave.

the transition in depth across the escarpment occurs over a short distance

compared with the tsunami wavelength

B0 .

= (−δ)

2

A0 for k0σ 	 1 . (58)

Conversely, the amplitude is negligible

B0 .

= 0 when k0σ > 2.5 . (59)

In between, there is a range (0.2 < k0σ < 2) in which the amplitude B0

of the back-scattered wave changes rapidly with changes in the wavenumber

k0, and therefore changes in the wave frequency ω. Hence, there is a range

of frequencies in which the scattering properties are dispersive (frequency

dependent). This is even though it is often said tsunami waves are nondispersive in the open ocean, in the shallow-water wave limit.

4. Lineal Ridges and Trenches

This section considers lineal (straight) ridges and trenches surrounded by an

abyssal plain of constant depth. Like the previous section on escarpments, it

contains, first, an analysis of scattering off step topography (normal and nonnormal incidence) and then an analysis for continuously varying topography
Analytic Theory of Tsunami Wave Scattering 17

Figure 9: Schematic diagram of a rectangular ridge having a summit of constant

depth H1 and surrounded by a region of constant depth H0. The sides of the ridge

are vertical.

(Gaussian profile). The tsunami waves approach these features from the

abyssal plain, rather than propagating as trapped waves along ridges. For

a discussion of the latter, please see the recent paper by Koshimura et al.

(1999).

The analysis for rectangular ridges leads the concept of minimum transmissivity, which leads in turn to a scattering index that can be used to assess

the potential for finer-scale topography to scatter tsunami waves in the open

ocean.

4.1 Rectangular Ridge: Normal Incidence

Figure 9 shows a tsunami wave (3) that is normally incident (φ = 0◦) on a

rectangular ridge of width L and depth of H1. The ridge is surrounded by

an abyssal plain with the same depth (H0 = H2) on either side of the ridge.

The matching conditions (11 and 14) at the two depth transitions (x1 =

0 and x2 = L) can be written as conditions on the amplitudes (Aj ,Bj ).

By assumption, B2 = 0 since this is the amplitude of a wave that would

be incident on the other side of the ridge. Four matching conditions then

determine the remaining four amplitudes in terms of the amplitude A0 of

the incident wave.

We take the segment width δx0 to be so small (δx0 → 0) that

η0 = A0 ξ0 = B0 at x = x1 . (60)

Letting the phase shift across the ridge θ and the depth parameter  ≡ 1 be

defined by

θ ≡ k1L and  ≡

H1

H0

, (61)
18 Mofjeld et al.

the four matching conditions are then

A0 + B0 = A1 + B1 (62)

A0 − B0 =  [A1 − B1] (63)

A1eiθ + B1e−iθ = A2 (64)





A1eiθ − B1e−iθ

= A2 . (65)

To solve (62–65) for the amplitudes in terms of the amplitude A0 of the

incident wave, it is convenient to first write B0, A1, and B1 in terms of the

amplitude A2 of the transmitted wave and then write A2 in terms of A0

B0 = −i


1 − 2

2



sin θ A2 (66)

A1 =


1 + 

2



e−iθ A2 (67)

B1 = −


1 − 

2



e iθ A2 (68)

where

A2 =



cos θ − i


1 + 2

2



sin θ

−1

A0 . (69)

The amplitude B0 (66) of the reflected wave is zero when either the ridge

has the same depth as the surrounding region ( = 1) or the width of the ridge

goes to zero (θ → 0◦). For a given value of , B0 is a maximum whenever

the phase shift θ across the ridge is a multiple of 90◦. Conversely, it is

zero whenever θ = 0◦, 180◦ ... . These two extrema correspond to minimum

transmission and perfect transmission of wave energy, respectively.

When  is close to unity, the amplitude A1 (67) is near A2 and B1 (68)

is small, so that the incident wave is relatively unchanged as it propagates

over the ridge. When the ridge is very shallow ( 	 1), there is an apparent

tendency for the on-ridge waves to have large amplitudes. However, we will

see that the rectangular ridge produces artificial results in this parameter

range, especially with regard to wave resonances on the ridge.

4.2 Rectangular Ridge: Transmissivity

Equation (69) leads to a formula for the magnitude of the ratio A2/A0 between the amplitudes of the transmitted and incident waves, which we will

call the transmissivity

T = Tmin 

T2

min cos2 θ + sin2 θ

− 1

2 (70)

where the minimum transmissivity Tmin is given by

Tmin = 2

1 + 2 . (71)
Analytic Theory of Tsunami Wave Scattering 19

Figure 10: Transmissivity (T = |A2/A0|) for tsunami waves that are normally

incident (φ = 0◦) on a rectangular ridge, as a function of the cross-ridge phase shift

θ = k1L for various values of the depth parameter  = H1/H0.

In the interpretation of the transmissivity given below, it is convenient

to discuss it in terms of its percentage relative to perfect transmission (no

topographic scattering by the ridge). Figure 10 shows that the transmissivity

T (70) is a periodic function of the phase shift θ across the ridge. The

maximum value (T = 100%) occurs at θ = 0◦, 180◦ and integer multiples

of the latter; while the minima (T = Tmin) occur at θ = 90◦ and its odd

multiples. When the ridge has low relief so that  is close to unity, T remains

close to 100%. This is regardless of the ridge width, as measured by the phase

shift θ, and hence the wave frequency. The minimum value of T (Figure 10)

decreases rapidly as  increases over the range  = 0.6 → 0.2, and the amount

of scattering therefore tends to be a sensitive function of both  and θ in this

parameter range. At smaller values of  (e.g., the  = 0.1 curve in Figure 10),

the transmissivity is low over a broad range of the phase shift θ. To rise from

low values to T = 100% at θ = 0◦, 180◦ and its multiples, the transmissivity

T increases rapidly in the vicinity of these θ values. This increase is most

pronounced when the depth parameter  is small, corresponding to high

ridges that reach close to the surface relative to the surrounding region.

The minimum transmissivity Tmin (71) is a very useful measure of the

potential that linear topographic features have to scatter tsunami waves in

the open ocean. One distinct advantage of using it is that it depends only on

the depth of the feature compared with the depth of the region immediately

surrounding the feature. As seen in Figure 11, Tmin forms a symmetric,

bell-shaped curve when plotted versus log(), with the peak of the curve at
20 Mofjeld et al.

Figure 11: Minimum transmissivity Tmin for tsunami waves that are normally

incident (φ = 0◦) on a rectangular ridge, as a function of the depth parameter .

 = 1. Besides the behavior of Tmin discussed in the previous paragraph, it

is worth noting that near  = 1 the minimum transmissivity is a parabolic

function of the deviation δ ≡  − 1

Tmin .

= 1 − 1

2 (δ)

2 . (72)

For rectangular trenches ( > 1), the minimum transmissivity Tmin (Figure 11) decreases to small values as  increases. However, the subsection

3.2 on escarpments shows that only a modest deviation from normal incidence is required to produce strong reflection, which corresponds to markedly

decreased transmission. Further, even with normal incidence on a trench,

increasing  tends to decrease the phase shift across a trench and therefore

diminishes the effect of the trench width on scattering. We must therefore regard scattering off trenches to be distinctly different in behavior from

scattering off ridges, when  deviates greatly from unity.

To understand how the transmissivity T varies with  for fixed ridge

width L and wave frequency ω, we note that the cross-ridge phase shift θ

for normal incidence is given by

θ = θ1



(73)

where θ1 is the cross-ridge phase shift when  = 1. Because of their inverse

relationship, θ increases as  decreases and vice versa.

For θ1 = 47◦, Figure 12 shows that as  decreases from unity, the transmissivity T is closely approximated by the minimum transmissivity Tmin for
Analytic Theory of Tsunami Wave Scattering 21

Figure 12: Transmissivity T for tsunami waves that are normally incident (φ = 0◦) on a rectangular ridge,

as a function of the depth parameter  for fixed frequency ω and ridge width L when the cross-ridge phase

shift θ
=1 = 47◦.

a considerable range of . As  decreases further, the phase shift θ (73) approaches 90◦; the transmissivity then rises above Tmin and reaches T = 100%

when the ridge width is a quarter wavelength (θ = 90◦). The reflectivity

(R ≡ |B0|/|A0|) is also plotted in Figure 12 to show the reciprocal relationship between T and R, in which T2 + R2 = 1.

As  decreases further, T oscillates rapidly between Tmin and 100% due

to on-ridge resonances. Subsection 4.4 below will show that the higher-order

resonances do not occur when the ridge has a Gaussian cross-section. Hence,

whether they exist or not depends on the details of the cross-ridge depth

profile; only the first case (θ = 90◦) is a robust feature of ridge scattering.

When  > 1, the behavior of T (Figure 12) is very different from the ridge

regime. The T curve deviates almost immediately from Tmin as  increases

from  = 1, decreasing smoothly toward a relatively high asymptotic value

T→∞ .

=



1 +

1

4 θ2

1

− 1

2

(74)

where θ1 < π

2 is in radians. The transmissivity T (Figure 12) decreases

monotonically toward T→∞ and there are therefore no passages of θ (73)

through 90◦ as  increases. For wide trenches, T may rise at least once to

perfect transmission T = 100% before decreasing to T→∞. However, T still
22 Mofjeld et al.

has the general properties of deviating quickly from unity and remains much

higher than Tmin as  increases further.

In conclusion, the analysis and examples given in this subsection show

that when tsunami waves are normally incident on rectangular ridges, the

transmissivity T (70) is closely approximated by Tmin (71) over substantial

ranges of the depth parameter  and the width of the ridges, as measured by

the cross-ridge phase shift θ. Hence, the minimum transmissivity Tmin is a

useful quantity for identifying those ridge-like features in the open ocean that

are likely to cause significant scattering of tsunami waves and, conversely,

the ones that allow the waves to pass without affecting the transmitted

amplitude.

In contrast, T deviates quickly from Tmin as  increases over  = 1 and

remains high, implying that rectangular trenches produce only weak scattering. This result is specific to normal incidence. The analysis of non-normal

incidence on step escarpment (Section 3.2), in which the wave approaches

obliquing from the shallow side (analogous to incidence on a wide trench),

shows that there is a marked tendency for strong reflection and hence T 	 1.

4.3 Rectangular Ridge: Oblique Incidence

The purpose of this subsection is to show that the transmissivities T (70)

and Tmin (71) are reasonable approximations for ridges over a substantial

range of incident angles, even though they were derived for normal incidence.

We assume that the lineal ridge is still aligned with the y-coordinate.

Then, letting φj be the angle in the jth segment that the wavenumber vector *

kj makes with the x-axis, the x- and y-components of the wavenumber

are

kx,j = kj cos φj and ky,j = kj sin φj , (75)

where

kj = ω

cj

.

For a rectangular ridge, refraction decreases the incident angle φ0 (Figure 13) to φ1. S ince H2 = H0, φ2 = φ0. The matching conditions then

take the same form as (62–65) except that the depth parameter  and the

cross-ridge phase shift θ1 are replaced by



 = 

cos φ1

cos φ0

(76)

and

θ

1 = θ1 sec φ1 . (77)

Since the forms of the matching conditions are the same, it follows that

the transmissivities for non-normal incidence have the same form as T (70)

and Tmin (71), when  and θ

1 are substituted for  and θ1

T = T

min 

T 2

min cos2 θ1 + sin2 θ1

− 1

2 (78)

T

min = 2

1 +  2 . (79)
Analytic Theory of Tsunami Wave Scattering 23

Figure 13: Schematic diagram showing the angles of incident φ0, on-ridge φ1, and

transmitted φ2 = φ0 waves when an obliquely incident tsunami wave interacts with

a rectangular ridge.

From Snell’s law of refraction (derived from the condition ky,0 = ky,1),

φ1 = arcsin ( sin φ0), (80)

or

φ1   φ0 (81)

for | | . 1 and | φ0| . 30◦ .

The decrease in the angle φ1 over the ridge, as compared with the incident

angle φ0, acts to decrease the effect of oblique incidence on the transmissivities. We can see this explicitly by studying an approximate formula for T

min

as a function of 1 and φ0. Noting that



  

cos ( φ0 )

cos ( φ0) (82)

or 

   + 




1 − 

2 2 φ2

0

2 , φ0 in radians, (83)

the minimum transmissivity T

min is approximately (within 10%)

T

min = Tmin + δTmin, δTmin  λ () φ20 (84)

where

λ () ≡ 




1 − 2 2

(1+ 2 )

2 . (85)
24 Mofjeld et al.

Figure 14: Transmissivity factor λ, as a function of the depth parameter .

As shown in Figure 14, the factor λ (85) is small over the full range of

 that is applicable to ridges (0 ≤  < 1). Its maximum value λmax = 0.214

occurs at  = 0.351.

The shapes of the deviations δT

min (Figure 15) follow that of λ (). However, they are scaled by an angular dependence φ2

0 that reduces the deviation

δT

min considerably (for φ0 ≤ 30◦). For instance, the maximum deviation

δT

min = 0.059 for φ0 = 30◦. This corresponds to <11% of typical values

for the normal-incidence Tmin. Hence, the formula (71), derived for normal incidence (φ0 = 0◦), is a reasonable approximation to the minimum

transmissivity over a substantial range of incidence angles φ0.

4.4 Gaussian Ridge: Normal Incidence

The first-order scattering theory for a low ridge having a Gaussian profile

(Figure 16) provides a useful contrast to scattering off the rectangular ridge.

The depth profile has the form

H(x) = H0



1+(

2 − 1) e

− x2

σ2



(86)

where we define the depth parameter  in terms of the minimum depth H1

over the crest of the ridge and the depth H0 of the surrounding region,

 ≡



H1

H0

 1

2

. (87)
Analytic Theory of Tsunami Wave Scattering 25

Figure 15: Deviation δT 

min of the minimum transmissivity from unity, as a function of the depth parameter  for various values of the incident angle φ0.

Figure 16: Schematic diagram of a ridge having a Gaussian cross-section with

minimum depth H1, surrounded by a region of constant depth H0.
26 Mofjeld et al.

Following the same procedure (Section 3.3) that was used to develop scattering formulas for the error function escarpment, the β–factor for normal

incidence (φ0 = 0) on a low Gaussian ridge is

β .= δ x

σ2 e

− x2

σ2 (88)

where δ ≡  − 1.

Substituting this form into equation (35) leads to the integral relation

for the amplitude B0 of the reflected wave

B0 .

= −A0

δ

σ2

	 ∞

−∞

x exp 2ik0 − x2

σ2



dx . (89)

Integration by parts then leads to the explicit formula

B0 .= iA0 (−δ)

√π k0σ e−k2

0σ2

. (90)

The factor of i = √−1 in (90) causes the reflected wave to lag the incident

wave in time by an additional 90◦. This phase shift is due to the offset in

space between the contributions that the up-slope and down-slope halves of

the Gaussian ridge (Figure 16) make to the scattering process.

To illustrate how the amplitude of the reflected wave depends on the

height of the Gaussian ridge and the wavelength, it is convenient to plot the

ratio B0 / iA0, as a function of k0σ for various values of δ. As Figure 17

shows, the amplitude ratio is small for both low and high values of k0σ,

with a single maximum (−δ)

π/2e that occurs between these regimes at

k0σ = 1/

√2. The domain in which the ratio is a significant fraction of the

maximum is roughly 0.05 < k0σ < 2. At low frequencies and therefore low

wavenumbers, the ratio is proportional to k0σ. At the high frequencies, the

cutoff is exponentially sharp.

The interpretation of the low-frequency regime is that the up-slope and

down-slope contributions to scattering nearly cancel. The net scattering

that does occur is due to the slight offset of these slope areas relative to

the wavelength. This spatial offset increases linearly with the product k0σ.

In the high-frequency regime (k0σ > 2), the bottom slopes are too gentle

to produce significant scattering. The wave behavior is then in the WKBJ

regime.

To understand why the optimal scattering occurs at k0σ = 1/

√2, we note

that the effective phase shift across the Gaussian ridge is about θ = 2k0σ.

For the rectangular ridge, the lowest frequency resonance in scattering occurs

when θ = 90◦, or θ = π / 2 radians. Then, k0σ = 0.785 which is within 12% of

1/

√2=0.707. Hence, the optimal scattering for the Gaussian ridge occurs

when the centers of the up-slope and down-slope regions are separated in

distance by a quarter of a wavelength, which is the same condition that

is needed for the lowest-frequency scattering resonance associated with the

rectangular ridge.

Since the lowest-frequency scattering resonance is very similar for the

Gaussian and rectangular ridges, this resonance appears to be a robust feature of the scattering process. Unlike the rectangular shape, however, the
Analytic Theory of Tsunami Wave Scattering 27

Figure 17: Amplitude B0 of the reflected wave resulting from the interaction of a

normally incident tsunami wave (φ0 = 0◦) with a low Gaussian ridge, as a function

of the ridge-width scale (k0σ) for various values of the deviation δ of the depth parameter  = H1/H0 from unity. The amplitudes are shown as the ratio B0/(iA0),

where A0 is the amplitude of the incident wave.

Gaussian shape does not lead to any higher-frequency resonances. Hence,

the existence of the higher resonances depends on the detailed shapes of the

topographic features.

The dependence (Figure 17) of the ratio B0 / iA0 on δ is linear for low

Gaussian ridges, in which the positive values of the ratio correspond to ridges

(δ < 0) and negative values correspond to Gaussian trenches (δ > 0). As

with low escarpments, an estimate of the amplitude of the transmitted wave

can be made by invoking conservation of energy flux (6).

5. Circular Seamounts

This section revisits a standard topic in wave scattering theory, the scattering of plane waves off circularly symmetric seamounts and islands (e.g.,

Lamb, 1932; Meyer, 1995). The derivation generally follows those for lineal

features, except that it is done in polar coordinates and involves sums of

Bessel function solutions to the equations of motion.

Of particular interest are the conditions under which radially symmetric resonant scattering is likely to occur. This interest is prompted by

model simulations showing concentric scattering features that occur when

tsunami waves pass by the southern end of the Emperor Seamount Chain

and the Mid-Pacific Mountains. The analysis is limited to seamounts hav-
28 Mofjeld et al.

Figure 18: Schematic diagram of a circular seamount of radius r0, having a summit

of constant depth H1 and surrounded by a region of constant depth H0. The sides

of the seamount are vertical.

ing constant-depth summits and vertical side-walls. For such seamounts,

Meyer (1995) provides a detailed analysis of multiple resonances that are

theoretically possible with this topography.

As shown in Figure 18, we use a polar coordinate system (r,φ) in which

the origin is at the center of the seamount and the φ = 0◦ axis is in the

direction that the incident wave is propagating. The seamount has a depth

of H1 and is surrounded by an abyssal plain having a constant depth H0.

The radius of the seamount is r0. In terms of the incident wave η, the

amplitudes of the scattered wave ξ and the wave ζ over the seamount are

determined by matching the surface elevation and the radial components of

water transport at the edge of the seamount r = r0

η + ξ = ζ (91)

Qr + Rr = Sr

or

c0



dη

dr + dξdr 

= c1

dζ

dr . (92)

Using the standard method of solution for 2-D scattering of plane waves

off cylindrical objects, each wave is expanded into sums of products in which

the angular dependences are cosine functions cos nφ. Letting n denote the
Analytic Theory of Tsunami Wave Scattering 29

index of the sums

η = AI

∞

n=0

anJn(k0r) cos nφ (93)

ξ = AI

∞

n=0

hnH(1)

n (k0r) cos nφ (94)

ζ = AI

∞

n=0

bnJn(k1r) cos nφ (95)

where Jn is the nth order Bessel function of the first kind and H(1)

n is the

nth order Hankel function of the first kind, corresponding to outward propagation away from the seamount.

The coefficients an for a plane wave are

an =



1 for n = 0

2 i n for n ≥ 1 . (96)

Since the angular functions cos nφ are linearly independent, the matching

conditions (91–92) reduce to equations in the coefficients

anJn(θ0) + hnHn(θ0) = bnJn(θ1) (97)

anJ



n(θ0) + hnHn(θ0) =  bnJ



n(θ1) (98)

where θ0 = k0r0 and θ1 = −1θ0. Here, the primes denote differentiation and

it is understood that Hn is a Hankel function of the first kind.

Solving (97–98) for bn and hn

bn = an



J



n(θ0)Hn(θ0) − Jn(θ0)Hn(θ0)



/ D (99)

hn = an



J



n(θ0)Jn(θ1) −  Jn(θ0)J



n(θ1)



/ D (100)

where

D ≡  J

n(θ1)Hn(θ0) − Jn(θ1)Hn(θ0) . (101)

The angular dependence and intensity of scattering that occurs as a

function of the depth parameter  can be seen by plotting the ratio |hn|/|an|.

Figure 19 shows that over a considerable range ( > 0.15), dipole scattering

(n = 1) is most important. The intensity of scattering in this regime is small

(|hn|/|an| < 0.08).

In contrast (Figure 19), the intensity of the monopole scattering (n=0)

increases rapidly as  decreases and dominates the scattering process around

the values of  and θ0 = k0r0 corresponding to the lowest-frequency resonance

over the seamount, where J0(θ1) .

= 0. This, and other resonances, can also

be seen in the plots (Figure 20) of on-seamount bn magnitudes. As with the

rectangular ridge, the presence (or absence) of higher resonances is likely to

depend very critically on the details of the topography. However, the lowest
30 Mofjeld et al.

Figure 19: Magnitudes of the off-seamount coefficients hn for scattered wave components, as a function of the depth parameter  = H1/H0 for the non-dimensional

radius θ0 = 0.2. The magnitudes are shown as ratios relative to corresponding coefficient an for the incident plane wave.

Figure 20: Same as Figure 19 but for the on-seamount coefficients bn.
Analytic Theory of Tsunami Wave Scattering 31

resonance is likely to be a robust feature that is insensitive to the details of

the topography.

From the analysis of scattering from circular seamounts with constantdepth summits and vertical side-walls, we see that the scattering tends to

be weak and dominated by dipole (n = 1) scattering. The exception is when

the wave frequency, seamount radius, and relative depth allow a monopole

(n = 0) resonance to occur over the seamount. In this case a strong, radially

symmetric pattern will be seen radiating from the seamount as a tsunami

wave passes.

6. Discussion

In this section, we use the analytic theory to address several important issues

concerning tsunami wave scattering in the North Pacific. The first issue is

how shallow must a topographic feature be to scatter significant tsunami

energy and, further, how shallow must it be before it will almost certainly

interact strongly with tsunami waves. Since the minimum transmissivity

Tmin (71) is large when the interaction of tsunami waves with topography is

small and small when the interaction is large, it is useful to define a scattering

index

S = 1 − Tmin . (102)

This index has the convenient property that it is small when the scattering

is small, and large when it is large. Since Tmin is bounded between zero

and unity, so is S. Based on the theory presented in this memorandum, this

index provides an effective tool for identifying those topographic features in

the ocean that are likely to scatter or reflect significant tsunami wave energy.

As shown in Figure 21 and Table 1, a feature needs to be shallower than

3000 m in order to produce even a small amount of scattering (S ≥ 0.01)

for the regional depths (H0 = 4000–6000 m) that occur in the open North

Pacific. It needs to be <1500 m to produce substantial scattering (S ≥ 0.2)

and will almost certainly have a major effect on tsunami waves (S ≥ 0.5) if

it is shallower than 400 m.

Hence, tsunami wave scattering between the Alaska-Aleutian Subduction

Zone and Hawaii (Figure 22) is small because of the large depths of topographic features in this region. Titov et al. (1999) verify this result by showTable 1: Feature depth H1 (m), as a function of depth H0 (m) of

the surrounding region and the scattering index S.

H0 S : 0.01 0.05 0.10 0.20 0.50

4000 3011 2096 1571 1000 287

4500 3387 2358 1768 1125 323

5000 3764 2620 1964 1250 359

5500 4140 2883 2161 1375 395

6000 4516 3145 2357 1500 431
32 Mofjeld et al.

Figure 21: Depth H1 of topography features as a function of the scattering index

S, for different values of the depth H0 of the surrounding region. Based on normally

incident tsunami waves interacting with rectangular ridges.

ing that tsunamis propagating southward from the AASZ are sensitive only

to the regional variations in topography. This is based on a one-dimensional

numerical model (single-frequency tsunamis propagating southward along

157◦W) and MOST model simulations for tsunamis generated in the AASZ.

There are, however, high features farther west and south that interact

strongly with tsunami waves. These are the Emperor Seamount Chain, the

Hess and Shatsky Rises, the Mid–Pacific Mountains, the Musician Seamounts,

and the Hawaiian Ridges themselves. In addition, the Aleutian–Komandorskiye and Kuril Island Arcs are major barriers to tsunamis. There are also

likely to be significant topographic interactions in the continental margin off

the U.S. West Coast, British Columbia, and Southeastern Alaska, due to the

complex topography associated with the Alaskan Seamounts, the Endeavour and Juan de Fuca Ridges, the Gorda Fracture Zone, and the Mendocino

Escarpment.

The geographic extent of a topographic feature is also important in terms

of how strongly it interacts with tsunami waves. A useful measure of this

extent is the width of a feature corresponding to its lowest frequency resonance. These resonances are the most likely to occur and the least sensitive

to the details of the topography. Also, many ridges and large seamounts

have flat tops due to erosion that occurred when these features extended

very close to the sea surface.
Analytic Theory of Tsunami Wave Scattering 33

Smith/Sandwell Topography 6.2

140oE 150oE 160oE 170oE 180o 170oW 160oW 150oW 140oW 130oW 120oW

140oE 150oE 160oE 170oE 180o 170oW 160oW 150oW 140oW 130oW 120oW

0o 0o

5oN 5oN

10oN 10oN

15oN 15oN

20oN 20oN

25oN 25oN

30oN 30oN

35oN 35oN

40oN 40oN

45oN 45oN

50oN 50oN

55oN 55oN

Emperor Seamount Chain

Hawaiian Ridges

Mid-Pacific Mountains

Aleutian/Komandorskye Islands

Kuril Islands

Shatsky Rise Hess Rise

Mendocino Escarpment

depth [meters]

-7000 -6500 -6000 -5500 -5000 -4500 -4000 -3500 -3000 -2500 -2000 -1500 -1000 -500 0

Figure 22: Smith-Sandwell topography for the North Pacific Ocean. Features in the open ocean with depths

<1500 m are likely to scatter a significant amount of tsunami wave energy, while features with depths <400 m

are likely to have a major impact on passing tsunami waves.
34 Mofjeld et al.

6.1 Ridges and Seamounts

For tsunami waves that have normal incidence φ = 0◦ on rectangular ridges

(Section 4.1), this resonance occurs when the cross-ridge width is a quarterwavelength (k1L1

4

= 90◦) as computed from the wave period τ and the ridge

depth H1

L1

4

= τ

4

gH1 (103)

For circular seamounts having flat tops, the corresponding resonance occurs

very near the first zero of the zero-order Bessel function [J0 (k1r0)=2.405].

The diameter of the seamount at this resonance is then

D0 = 2.405

π

τ

g H1 (104)

Over the range of tsunami wave periods τ = 5–50 min, there is a wide

range of the possible widths and diameters (Table 2) that allow the lowest frequency resonances to occur. However, the most tsunami energy will

be scattered by the shallower (H1 ≤ 400 m) seamounts. In this case, the

horizontal scales range from <10 km to 160 km. Here, the shorter scales

correspond to short-period tsunamis incident on shallow features, and the

longer scales to long-period tsunamis incident on deep features. The signatures of these resonances in observed tsunamis and numerical simulations

are strong reflections by ridges and strong circular patterns eminating from

seamounts.

Further application of the theory to ridges and seamounts in the North

Pacific is made difficult by the complex nature of these features. For instance, the Emperor Seamount Chain (Figure 22) has numerous short ridge

segments that are topped by seamounts and separated by deep passes. Such

irregular topography over a wide range of scales is also seen along the Aleutian/Komandorskiye and Kuril Island Arcs and the Hawaiian Ridges. The

other major features (Hess and Shatsky Rises, Mid-Pacific Mountains, Musician Seamounts and the eastern continental margin off the U.S. West Coast,

British Columbia, and Southeastern Alaska) has complicated topography as

well.

6.2 Escarpments and Trenches

Escarpments can cause strong reflections when the tsunami waves approach

these features obliquely from the shallow side. This is provided the depth

contrasts are large enough across the escarpments. Given the criteria developed in Section 3, the only escarpment in the North Pacific (Figure 22)

that has a large enough depth contrast to strongly reflect tsunami waves

is the eastern segment of the Mendocino Escarpment, near the U.S. West

Coast. The effect of the escarpment is enhanced by a ridge that runs westward along its crest from the coast to 129◦W. These tsunami waves need

to approach from the northern side at non-normal incidence. Hence, there

is a tendency for tsunamis generated in the AASZ and Kamchatka regions

to be partially reflected toward the north when they reach the easternmost
Analytic Theory of Tsunami Wave Scattering 35

Table 2: Ridge width (103) and seamount diameter (104) corresponding to the lowest frequency resonances, as functions of

tsunami wave period and depth of the feature. The diameters are

3.06 times the widths, for the same period and depth.

Ridge Seamount Ridge Seamount

Period Depth Width Diameter Period Depth Width Diameter

(min)(m) (km) (km) (min)(m) (km) (km)

5 100 2 7 30 100 14 43

500 5 16 500 32 96

1000 7 23 1000 45 136

1500 9 28 1500 55 167

10 100 5 14 40 100 19 58

500 11 32 500 42 129

1000 15 46 1000 59 182

1500 18 56 1500 73 223

20 100 9 29 50 100 23 72

500 21 64 500 53 161

1000 30 91 1000 74 227

1500 36 111 1500 91 278

segment of the Mendocino Escarpment. It will also produce some reflection

of tsunami waves generated in the Cascadia Subduction Zone, which lies

immediately north of the escarpment.

While the North Pacific (Figure 22) is bounded along its northern and

western margins by deep trenches, the theory presented in Section 4 indicates that these features produce very little scattering for a relatively wide

range of incident angles. However, they do contribute to wave trapping of

tsunamis on the adjacent continental shelves and tsunami propagation along

the shelves and continental slopes. They do this by decreasing the incident

angle necessary for trapping (evanescent off-shelf wave). The Emperor and

Chinook Trenches, located east of the Emperor Seamount Chain, do not

scatter significant tsunami wave energy because the surrounding region is so

deep.

6.3 Transient Tsunamis

Real tsunamis are transient events that can be thought of as the sum of

single-frequency components, where the dominant components of a real

tsunami tend to occur within a relatively narrow frequency band. Hence, the

general results of the single-band theory can be applied to these tsunamis.

Scattering and reflections extend the duration of real tsunamis in the

North Pacific. Since these processes depend on the wavelength of the tsunamis

relative to the horizontal scales of the topography, this topography also

causes dispersion in the tsunami waves. This is even when they satisfy

the long-wave equations, which are non-dispersive for constant depth. They

also contribute to the random appearance of tsunami time series observed

at tide gages as various decaying wave components interfere with each other

(Mofjeld et al., 1999a).
36 Mofjeld et al.

The theory has focused on tsunami wave scattering from isolated topographic features. However, the pattern from even a small number of scatterers located near each other is very difficult to interpret. Numerical models

(e.g., Mofjeld et al., 1999b; Titov et al., 1999) are then the appropriate tools

to study the combined effects of realistic topography. Numerical simulations

also go beyond a frequency analysis to an analysis of tsunami wave scattering when these waves have realistic tsunami patterns in time. The analytic

theory provides essential tools for interpreting the model simulations.

7. Conclusions

Based on the analytic theory presented in Sections 3–7, we are led to the

conclusions shown in Table 3.

The theory also leads to quantities, based on the water depth, that can

be used to assess whether topographic features are likely to cause significant

scattering and/or wave reflection. Of particular value is the scattering index

S. The application of this index to the North Pacific Ocean is given in the

Discussion Section.

Numerical tsunami models of the North Pacific that do not resolve these

features adequately, either by lacking spatial resolution or using smoothed

topography, will not properly simulate the interaction between these features

and tsunamis. The theory presented in this memorandum provides simple,

yet powerful, tools for interpreting these interactions.

8. Acknowledgments

This work was funded in part by NOAA, the U.S. National Tsunami Hazard Mitigation Program. Funding was also provided by Department of Defense through the

NASA Research Announcement NRA-98-OES-13.
Analytic Theory of Tsunami Wave Scattering 37

Table 3: Conclusions

Feature Type Depth Profile Incident Angle Conclusions

Basic Formulation

(lineal features)

Step Normal Scattering and reflection depend on the depth of a feature,

relative to the depth of the surrounding region, and on

wave frequency.

Oblique Wave direction becomes more normal when the feature is

shallower than its surroundings and more oblique when

propagating into deeper water (sometimes reflecting the

waves).

Continuous Normal Local scattering strength is proportional to the logarithmic gradient of the depth. Otherwise, wave amplitudes

are controlled by Green’s 1

4 th–law and phases by the spatial integral of the local phase gradient.

Escarpments Step Normal Reflection off an abrupt escarpment is independent of

wave frequency and is strongest at large depth contrasts.

Oblique Perfect reflection can occur as waves attempt to propagate

into deeper water (when incident angle is large enough).

Error

Function

Normal Narrow-width escarpments behave like the step escarpment; the behavior is a sensitive function of frequency

when the product of wavelength times width scale is order

unity; gently sloping features produce negligible scattering.

Ridges Rectangular Normal The scattering index, based on minimum transmissivity,

is a useful tool for identifying important scattering features. Strongest reflections (lowest transmission)occur

when the cross-ridge phase shift is an odd multiple of 90◦,

and weakest for a multiple of 180◦. Narrow ridges produce

only weak reflections, unless they are very high.

Oblique Refraction causes behavior to be much like the normalincidence case, over a substantial range of incident angles.

(Ridge-trapped waves are not considered in the theory.)

Gaussian Normal The minimum transmission matches closely the first minimum for the rectangular ridge, but the other (higher frequency)minima are missing; hence the first minimum is

a robust feature of ridge topography, but the others are

not. Smooth, broad ridges produce only weak reflections

(high transmissivity).

Trenches Rectangular Normal Trenches produce little reflection at normal incidence,

even when they are deep.

Oblique Trenches can enhance shelf/slope trapping of tsunami

waves near subduction zones.

Circular Flat-Top (Plane

waves)

The scattering by seamounts is weak and forms a dipole

pattern, except for the lowest-frequency monopole (radially symmetric)resonance. In this case, the amplitude on

the seamount is substantially larger than the amplitude

in the surrounding region.
38 Mofjeld et al.

9. References

Carrier, G.F. (1971): The dynamics of tsunamis. In Mathematical Problems in the

Geophysical Sciences, W.H. Reid, (ed.), American Mathematical Society, 157–

187.

Gonz ́alez, F.I., E.N. Bernard, H.B. Milburn, V.V. Titov, H.O. Mofjeld, M.C. Eble,

J.C. Newman, R.A. Kamphaus, and C.L. Hadden (1999): An integrated approach to improving tsunami warning and mitigation. Oceans ’99 MTS/IEEE

Abstracts.

Koshimura, S., F. Imamura, and N. Shuto (1999): Propagation of obliquely incident

tsunamis on a slope Part II: characteristics of on-ridge tsunamis. Coast. Eng.

J., 41(2), 165–182.

Lamb, H. (1932): Hydrodynamics. Dover, 738 pp.

Meyer, R.E. (1995): On the interaction of water waves with islands. In Long-Wave

Runup Models, H. Yeh, P. Liu and C. Synolakis (eds.), World Scientific, 3–24.

Mofjeld, H.O., F.I. Gonz ́alez, and J.C. Newman (1999a): Tsunami prediction in

U.S. coastal regions. Coastal Ocean Prediction, C.N.K. Mooers (ed.), American

Geophysical Union, 353–375.

Mofjeld, H.O., V.V. Titov, F.I. Gonz ́alez, and J.C. Newman (1999b): Tsunami

wave scattering in the North Pacific. IUGG 99 Abstracts, Week B, July 26–30,

1999, B.132.

Titov, V.V., H.O. Mofjeld, F.I. Gonz ́alez, and J.C. Newman (1999): Offshore forecasting of Alaska-Aleutian Subduction Zone tsunamis in Hawaii. NOAA Tech.

Memo. ERL PMEL-114, PMEL, Seattle, WA, 22 pp.