ADVANCES IN GEOPHYSICS docx

JOHNSON An alternative to using seismic data for studying the source of an earthquake is to use tsunami waveforms.. In a similar manner, when an earthquake generates a tsunami, the waves

ADVANCES IN

G E O P H Y S I C S

VOLUME 39

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BARRY SALTZMAN

Depaflment of Geology and Geophysics

Yale University New Haven, Connecticut

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CONTENTS

CONTRIBUTORS ix

Heterogeneous Coupling along Alaska-Aleutians as Inferred from Tsunami Seismic and Geodetic Inversions JEAN M JOHNSON 1 Introduction

2 Generation Computation and Inversion of Tsunami Waveforms

2.1 Generation Propagation and Observation of Tsunamis

2.3 Inversion of Tsunami Waveforms

and Tsunami Wave Inversions

3.1 Introduction

3.2 The 1965 Rat Islands Earthquake

3.3 Tsunami Study

3.4 Comparison of Seismic and Tsunami Results

3.5 Conclusions

4 The 1957 Great Aleutian Earthquake

4.1 Introduction

4.2 Previous Seismic Studies

4.3 Tsunami Source Area

4.4 Tsunami Waveform Inversion

4.5 Comparison of Seismic and Tsunami Results

4.6 The 1986 Andreanof Islands Earthquake

5 Rupture Extent of the 1938 Alaskan Earthquake as Inferred from Tsunami Waveforms

5.1 Introduction

5.2 Previous Studies of the 1938 Earthquake

5.3 Tsunami Waveform Inversion

5.4 Conclusions

1 April 1946 Aleutian Tsunami Earthquake

6.1 Introduction

6.2 Previous Seismic Analysis

6.3 Tsunami Analysis

6.4 Discussion

2.2 Forward Computation of Tsunamis

3 The 1965 Rat Islands Earthquake: A Critical Comparison of Seismic

6 Estimation of Seismic Moment and Slip Distribution of the

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77

vi CONTENTS

6.5 Seismic and Tsunami Hazards

6.6 Conclusions

and Geodetic Data

7.1 Introduction

7.2 Previous Seismic Studies

7.3 Previous Geodetic Studies

7.4 Previous Tsunami Studies

7.5 Joint Inversion

7.7 Discussion

8 Conclusions

References

7 The 1964 Prince William Sound Earthquake: Joint Inversion of Tsunami 7.6 Comparison with Previous Studies

Appendix: Notes on the Tsunami Waveform Inversion Method

79 81 82 82 84 84 86 87 98 99 101 105 110 Local Tsunamis and Earthquake Source Parameters ERIC L GENT 1 Introduction 117

2.1 General Approaches 121

2.2 Coseismic Surface Deformation 123

2.3 Tsunami Propagation 126

2.4 Tsunami Run-up 130

3 Local versus Far-Field Tsunamis 133

3.1 Source Parameters Affecting Far-Field Tsunamis 133

3.2 Coseismic Displacement near a Coastline 134

3.3 Wave Evolution over the Source Area 135

4 Tectonic Setting of Tsunamigenic Earthquakes 138

4.1 Types of Subduction Zone Faulting 138

4.2 Nature of Rupture along the Interplate Thrust 139

141 2 Tsunami Theory 120

5 Effect of Static Source Parameters on Tsunamis

5.1 Fault Geometry 145

5.2 Fault Slip 153

5.3 Slip Direction 155

5.5 Summary o f Static Source Parameter Effects 164

164 6.1 Slip Variations 165

6.2 Triggered and Compound Earthquakes 171

5.4 Physical Properties 160

6 Effect of Spatial Variations in Earthquake Source Parameters

CONTENTS vii

7 Effect of Temporal Variations in Earthquake Source Parameters 175

7.1 RiseTime 175

7.2 Rupture Velocity 178

7.3 Dynamic Overshoot of Vertical Displacements 181

8 Local Effects of Tsunami Earthquakes 182

8.1 Characteristics of Tsunami Earthquakes 184

8.2 Results from Broadband Analysis of Recent Tsunami Earthquakes 187

8.3 Mechanics of Shallow Thrust Faults Related to Local Tsunamis 189

8.4 Outstanding Problems 191

192 9.1 Geometric and Physical Parameters 192

9.2 Temporal Progression of Rupture 194

9.3 Magnitude and Distribution of Slip 194

10 Conclusions 195

Appendix 197

References 198

9 Case History: 1992 Nicaragua Earthquake and Tsunami

INDEX 211

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CONTRIBUTORS

Nuni hers in purentheses indicate the puges on which !he authors’ contributions begin

ERIC L GEIST (1171, U S Geological Survey, Menlo Park, California JEAN M JOHNSON (l), Division of Natural Sciences, Shorter College,

94025

Rome, Georgia 30165-4298

ix

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HETEROGENEOUS COUPLING ALONG

FROM TSUNAMI, SEISMIC, AND

GEODETIC INVERSIONS ALASKA-ALEUTIANS AS INFERRED

JEAN M JOHNSON

Division of Natural Sciences Shorter College Rome, Georgia

1 INTRODUCTION

The Alaskan-Aleutian arc has a history of rupturing in large and great earthquakes The most recent sequence began in 1938 and has ruptured almost the entire arc from southern Alaska to the western Aleutians (Figure 1) This sequence includes five great earthquakes: the 1938 Alaskan,

1946 Aleutian, 1957 (Central) Aleutian, 1964 Prince William Sound (or Alaskan), and 1965 Rat Islands earthquakes Three of these five-the

1957, 1964, and 1965 earthquakes-are among the 10 largest earthquakes

of the 20th century

These earthquakes are clearly important to those who assess seismic hazards These five earthquakes caused hundreds of deaths and millions of dollars of damage, both from the earthquakes themselves and from the tsunamis they generated In most instances, the tsunamis caused more deaths than the earthquakes, not only near the earthquake source, but far across the ocean on distant shores to which the tsunamis propagated For this reason, it is extremely important to understand these earthquakes in order to save lives and property in future earthquakes

As great subduction zone earthquakes, these five events are also of interest to seismologists who wish to understand the mechanics of earth- quake rupture and earthquake recurrence Detailed knowledge of these earthquakes is important to understanding the physics of how these events occurred, the subduction process in the Alaskan-Aleutian subduction zone, and how future earthquakes will occur In order to address these larger issues, the most fundamental parameters of the earthquakes must first be ascertained

For seismologists, one of the most important source parameters of an earthquake is the seismic moment, which is a measure of the earthquake size Seismic moment is related to how much movement, or slip, occurs on the fault during the rupture process By modern seismological methods,

1

Copyright 0 1999 by Academic Press All rights of reproduction in any form reserved

the moment of an earthquake can be well determined from the seismic waves recorded on seismometers Recent studies (Ruff and Kanamori, 1983; Kikuchi and Fukao, 19871, however, have shown that the slip is not uniform on a fault, but has variations across the rupture surface In other words, some patches of the fault have high slip and others have low slip The areas of high slip are interpreted according to the asperity model (Kanamori, 1978) An asperity on a fault is where the two sides are held together by an area of higher strength than the areas surrounding it When the stress on the fault exceeds the strength of the asperity, the asperity fails as an earthquake High slip occurs at the asperity, and lower slip occurs in the surrounding areas This leads to variations of moment release along the fault and is expressed as complexity in the seismic waves that are generated Asperities can fail individually, or they can fail with other asperities in complex, multiple rupture events The same asperity can rerupture over many earthquake cycles

Lay and Kanamori (1981) proposed an asperity model for the world's subduction zones, including the Alaskan-Aleutian zone They suggested that for the eastern end near southern Alaska, the asperity distribution is uniform over the entire fault contact zone, and rupture always occurs in great events, with rupture zones of hundreds of kilometers For the central and western parts of the subduction zone in the Aleutians, they suggested that the asperities are smaller, and rupture over several cycles can be variable Sometimes an asperity may fail individually, with a rupture length

of approximately 100 km; at other times, several asperities may fail in one event, with a rupture length of hundreds of kilometers

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 3

Where earthquakes have occurred is sometimes not as important as where they have not occurred Several sections of the Alaskan-Aleutian arc have not ruptured in the great earthquakes of this century These segments are called seismic gaps (Sykes, 1971) The seismic gap theory

(McCann et al., 1979) suggests that the seismic gaps have a higher potential to rupture in earthquakes than do segments that have recently experienced large earthquakes If a seismic gap of a few hundred kilome- ters were to fail in one earthquake, it could cause extensive damage and generate a destructive trans-Pacific tsunami Figure 1 shows that the gaps are delineated by the ends of the adjacent earthquake aftershock zones If the aftershock zone is longer than the areas of high slip, the seismic gaps may be longer than presently believed Therefore, it is important to determine the rupture length of the large earthquakes correctly

The asperity model of Lay and Kanamori for the Alaskan-Aleutian arc must be tested and the seismic gaps must be identified Do asperities exist? Are the slip distributions of these earthquakes highly variable? Do they conform to the asperity model? Can the results of seismic studies for moment release distributions (where they exist) be correlated to the slip distributions? Are the seismic gaps larger than suggested by the aftershock zones that bound them? These questions can be answered by determining the slip distributions of the great 20th century earthquakes This is important both for scientific understanding of these past earthquakes and for making predictions concerning future events If asperities persist through many earthquake cycles, as suggested by the asperity model, it should be possible to predict the locations of future great earthquakes, or

at least to predict where slip will be highest If the seismic gap hypothesis

is correct, the present seismic gaps of the Alaskan-Aleutian subduction zone may be the sites of large earthquakes in the near future This information is extremely important for seismic and tsunami hazard plan- ning, such as developing building codes in Alaska and managing land in coastal areas where earthquakes and tsunamis are likely to strike

Modern seismological methods can determine where on a fault the moment release is highest, but these methods cannot determine if these areas are also the areas of highest slip Also, these methods require the use of high-quality seismic data For the Alaskan-Aleutian earthquakes, such data do not always exist The global network of high-quality instru-

ments, the World Wide Standard Seismograph Network (WWSSN), started

in 1964 This means that for several of these earthquakes, the seismologi- cal methods cannot be used to determine the source parameters of interest The slip distributions, rupture lengths, and seismic moments are unknown or poorly estimated This means that the asperity model cannot

be tested for these earthquakes, nor the seismic gaps identified

4 JEAN M JOHNSON

An alternative to using seismic data for studying the source of an earthquake is to use tsunami waveforms All the Alaskan-Aleutian earth- quakes generated tsunamis that were observed in many locations around the Pacific Ocean Figure 2 compares the use of seismic and tsunami data When an earthquake occurs, seismic waves radiate through the solid body

of the earth and are recorded on seismometers as waveforms The wave- forms contain information about the earthquake source, but are also a function of the structure of the earth through which they pass and the instrument on which they are recorded In a similar manner, when an earthquake generates a tsunami, the waves propagate across the ocean and are recorded as waveforms on tide gauges in bay and harbors Just like the seismic waveforms, the tsunami waveforms carry information about the earthquake source, the effects of propagation over the ocean, and the instrument on which they are recorded For seismic waves, the most important effect on propagation is the velocity structure of the earth; for tsunami waveforms, the most important effect on propagation is the depth

of the water Of these two, the depth of the oceans is better known than the velocity structure of the earth; therefore, the propagation effects can

be simulated more precisely by computational methods for tsunamis than for seismic waves Once the effects of propagation and the instrument have been accounted for, the tsunami waveforms can be used to study the source parameters of the earthquake

We here discuss the uses of tsunami waveforms to determine the source parameters of the five great Alaskan-Aleutian earthquakes Section 2.2 reviews the generation, propagation, and observation of tsunamis It also explains the method of tsunami waveform inversion used in this study In Section 2.3 we determine the slip distribution and seismic moment of the

FIG 2 Comparison of seismic and tsunami wave propagation and recording

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 5

1965 Rat Islands earthquake and compare results of seismic and tsunami wave inversions Sections 2.4 and 2.5 then detail the application of this method to the 1957 Aleutian and 1938 Alaskan earthquakes to determine their slip distribution, rupture area, and seismic moment Section 2.6

concerns the 1946 Aleutian earthquake, an extremely unusual seismic event that generated one of the largest tsunamis of the century Tsunami waveform inversion can be used for earthquakes that occur under the ocean, but naturally they cannot be used for earthquakes that occur on land Section 2.7 explains an expansion of the tsunami waveform inversion method to include geodetic data for the study of the 1964 Prince William Sound earthquake, the second largest earthquake of the 20th century, which occurred on the continental margin Section 2.8 states the conclu- sions derived from these various individual studies

2 GENERATION, COMPUTATION, AND INVERSION OF

Using tsunami waveforms to estimate source parameters of a tsunami- genic earthquake involves both a forward and an inverse problem The forward problem consists of the generation, propagation, and recording of the tsunami waveforms The inverse problem consists of using a Green’s function technique to invert the waveforms to determine some number of source parameters The forward problem is discussed first

2.1 Generation, Propagation, and Observation of Tsunamis

2.1.1 Generation of Tsunamis

Crustal deformation of the earth due to internal faulting is generally modeled using the elastic theory of dislocation The earth is treated as a homogeneous, isotropic, and elastic material that obeys the laws of classi- cal linear elastic theory Steketee (1958) first applied dislocation theory from crystal physics to fault models Steketee showed that internal strains are caused by dislocation across an internal displacement surface The strain field within the body and on the surface of the body depends on the size, shape, and orientation of that displacement surface and the distribu- tion of offset on it Steketee’s solution for the displacement field at any point within the strained body is

h JEAN M JOHNSON

where u k is the displacement at some point in the body, u is the slip on the displacement surface, A and p are elastic moduli, v is the direction cosine normal to the fault, and the integration is carried out over the displacement surface Z

Equation (1) must be evaluated on the surface of a body like the earth because this is where we can observe the displacement or deformation due

to the internal dislocation or faulting

The movement on an internal or buried fault produces characteristic patterns of deformation-uplift, subsidence, and offset-of the earth’s surface (Kasahara, 1981) These patterns are a function of the fault parameters, shown in Figure 3 The amount of deformation is a linear function of the amount of slip; i.e., twice the slip on the fault creates twice

the deformation of the surface Figure 4 shows the typical uplift and

subsidence pattern due to a shallow-dipping thrust fault Numerous studies (Chinnery, 1961; Ben-Menahem and Gillon, 1970; Mansinha and Smylie,

1971) have developed analytical formulas to determine the surface defor- mation given the necessary fault parameters In this analysis of the Alaskan-Aleutian earthquakes, the deformation of the earth’s surface is computed from the equations of Okada (1985) The fault parameters necessary to determine the deformation are fault area (length and width), location (latitude, longitude, and depth), strike, dip, rake, and amount of fault motion

When the deformation due to an earthquake occurs under water, in a subduction zone for example, the uplift and subsidence of the ocean floor causes displacement of the ocean surface away from its equilibrium

length L

FIG 3 Definition of fault parameters L is length of fault, W is width Strike is measured

in degrees clockwise from North, dip is measured in degrees downward from the horizontal plane, rake is measured counterclockwise in degrees from the horizontal Slip u has a strike-slip us and dip-slip ud component The position of the reference point at the top edge

of the fault is given in latitude, longitude and depth

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 7

position, thus generating a tsunami The problem of determining the

actual uplift of the ocean surface from a pattern of ocean bottom deforma-

tion is not trivial (Kajiura, 1963), but Abe (1973) showed that the general

pattern and magnitude of uplift and subsidence due to faulting are

reflected in the wave shapes and amplitudes of the tsunami that is

generated Abe also showed that the general fault parameters could be

estimated from the tsunami waves

Kajiura (1970) discussed the energy transfer between the uplifted solid

earth and the ocean water He showed that for rapid deformation occur-

ring in less than a few minutes, the uplift could be considered to occur

instantaneously with respect to tsunamis Great earthquakes of the

Alaskan-Aleutian subduction zone typically have rupture durations of

several minutes (a maximum of 4 minutes); therefore, the deformation is

here treated as instantaneous The displacement of the ocean surface from

FIG 4 Surface deformation pattern due to buried thrust fault The fault parameters are

listed The contour interval is in centimeters Each line represents 7 cm The greatest uplift is

92 cm; the greatest subsidence is 24 cm Solid lines represent uplift, dashed lines represent

subsidence X indicates the reference point of the fault

8 JEAN M JOHNSON

its equilibrium position is assumed to match exactly the vertical compo- nent of the ocean floor deformation due to faulting This uplift of the ocean surface is the initial condition of the tsunami for computational purposes

been generated by an earthquake, it propagates across the ocean as a wave The restoring force is gravity Thus, a tsunami is a gravity wave just

as the ocean tides are; however, a tsunami has nothing to d o with the tides This discussion treats the water body as a uniform, inviscid, incompress- ible liquid that has a free surface and upon which the only body force acting is gravity We consider propagation of waves with wavelength A in one dimension, as shown in Figure 5 The z axis is vertical upwards and the wave travels in the positive x direction Euler’s equation of motion is

Du du

Dt at + ( u * V)U

_ - - -

FIG 5 Geometry of a one-dimensional tsunami propagation problem The water depth is

d , the water height is h , and the wavelength is A

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 9

The total derivative term on the left-hand side of (2) represents the local acceleration and the nonlinear advection term Resolving (2) into its components gives

We now assume that the second order nonlinear advection terms of

D u / D t are small and can be ignored This gives the equation of motion

If we consider the conservation of mass across a small region with length

dr, the volume change per unit time must be equal to the flow rate of water out of the region; thus

Eliminating u from (7) and (10) gives the one-dimensional wave equa-

If the water depth d is constant or varying slowly, (11) becomes

d t 2 dX

where c = @ In this case, the velocity of the wave is determined solely

by the water depth

We now justify the neglect of the vertical acceleration The time T taken for a wave of wavelength A to pass a specified point is A/c Hence, the

O ( h c 2 / A d ) The ratio of the vertical to the horizontal components o f

acceleration is of O ( d / A ) If d << A, the vertical acceleration is negligible

For tsunamis, the water depth is an average of about 5 km and the source area is tens or hundreds of square kilometers; therefore, neglecting the vertical acceleration is appropriate

In summary, the assumptions made to reach Eqs (7) and (10) are: ( 1 )

the fluid is uniform, inviscid, and incompressible, (2) the horizontal scale of motion is much larger than the water depth, (3) the nonlinear advection

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 11

term is small and can be ignored, and (4) the amplitude of the waves is small compared to the water depth

The preceding equations utilize the velocity field u , but they can also be written in terms of the flux rate vector Q, defined by

Q = lh udz = /_odudz = ud

These are alternate expressions for linear long waves

2.1.2.2 The Boussinesq equation The linear long wave equation derived

in the preceding section is generally appropriate for tsunami propagation; however, several possibly important effects have been neglected, including the nonlinear advection terms, bottom friction, the Coriolis force, and

dispersion A more general equation for tsunami propagation is the

two-dimensional Boussinesq equation (Peregrine, 1967) including bottom friction and the Coriolis force Thus,

These equations can be rewritten in terms of the flow rate vector Q

12 JEAN M JOHNSON

The terms in (18) are (a) local acceleration, (b) advection, (c) the

Coriolis force, (d) linear pressure gradient, (e) nonlinear pressure gradient, (f) experimentally determined bottom friction, and (8) dispersion

We analyze each of the terms in (18) to determine its order of magni-

tude for a typical transoceanic tsunami Assuming a water height of

h = h,sin(kr - w t ) , where k is the wavenumber and w is the angular frequency, the flux becomes Q = c o h , where c , = w / k Term (a) in (18) becomes wc,,h = gdkh Each term in (18) is normalized by this term The

results are listed in Table 1 The orders of magnitude are for a tsunami with wavelength of 200 km and wave height 1 m traveling over an ocean of depth 4500 m

From Table 1, we see that all the terms except the local acceleration, linear pressure gradient, and the Coriolis force are of order or smaller; therefore they can be ignored for a transoceanic tsunami The

TABLE 1 MAGNITUDE OF TERMS I N LINEAR BOUSSINESQ EQUATION FOR TSUNAMI WITH WAVELENGTH 200 KM AND WAVE HEIGHT 1 M TRAVEIJNG OVER AN O C E A N WITH DEPTH

4500 M

Normalized

10-4

(f) Bottom friction"

(g) Dispersion

"The coriolis force is given by F,c'"' = -2Clrf cos 8, where R is the angular frequency

of the earth's rotation and 8 is the colatitude (90" - latitude) The coordinate system is

x = East, y = South, and z = vertical up The equation for the Coriolis force reduces to

+ J , where f is 1.45 X s - ' The cosine of the colatitude for a tsunami traveling over the northern Pacific ranges from nearly 1 to nearly 0 Also, cu = @ = O(10') and

k = O(10-5)

'Cf is an experimentally determined coefficient with value W 4 for a tsunami in the deep ocean

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 13

equation of motion can be reduced to

14 JEAN M JOHNSON

where cp is longitude, 0 is colatitude, R is the radius of the earth, and Here we use Equations (23) and (24) to simulate the propagation of tsunamis The equations are solved by a finite difference scheme discussed

less than 1 m (Lander and Lockridge, 1989) However, several recordings

of tsunamis in the deep ocean have been made by ocean bottom pressure gauges in both the Gulf of Alaska (Gonzalez et uf., 1991) and off the coast

of Japan (Okada, 1995)

Historically, observations of tsunamis have been confined to coastal

areas, typically bays and harbors (Indeed, the word tsunami comes from the Japanese and means “harbor wave.”) Records of the run-up, or the

height of tsunami inundation on land, can be found in historical docu- ments going back hundreds, even thousands, of years in countries such as

China, Japan, and Greece (Soloviev and Go, 1974; Iida et al., 1967) These observations of run-up, though highly interesting and useful for general estimation of earthquake magnitude, cannot be used at present to deter- mine the source parameters of the generating earthquake Run-up is a highly nonlinear process and is extremely sensitive not only to the incom- ing wave but also to the local topography where it occurs The run-up problem is mainly the realm of coastal engineers, and much research effort

is directed to understanding and modeling the phenomenon (Shuto, 1991;

Titov and Synolakis, 1993; Abe, 1995; Liu et a f , 1995; Briggs el al., 1995) Tsunami observations most useful for estimating the source parameters

of an earthquake are the recordings of the tsunami on tide gauges Tide gauges are situated in harbors and bays and, as their name suggests, are meant to record the tides These instruments, however, also record the waveforms of tsunamis as continuous time series

There are numerous types of tide gauges The most common is the stilling-well gauge, shown in Figure 6 Though this type is no longer used

in the United States, all the tsunamis used in this study were recorded on

this type of instrument These tide gauges are generally those that are a

part of the U.S Coast and Geodetic Survey system meant to monitor the

tides and changes in sea level

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 15

recorded trace

water intake

L

FIG 6 Sketch of a stilling-well tide gauge

The stilling-well gauge consists of a well with a water intake and a float that is attached to a recording system As the water level in a bay or harbor increases or decreases, the float in the stilling well rises or falls This motion is recorded on a paper record as a continuous trace Each recording system has an amplification factor determined by the normal high and low water levels of the tides in each particular harbor or bay The typical amplification factor is 1:12; i.e., 1 inch of vertical displacement on the record equals 12 inches of vertical water movement The paper record

is set to turn at a speed of about 1 inch per hour Local-time hour marks are recorded at the beginning of each hour At tide gauge stations, a tide observer has the duty of periodically (usually daily) checking the recording system and making corrections to the timing system

There are several complications to using the tide gauge records in determining the source parameters of earthquakes The first of these is that the precise instrument response of the tide gauges is unknown Stilling-well gauges are meant to monitor ocean tides, which have a period

of about 12 hours, and are designed to reduce the noise of wind waves, which have much shorter periods Tsunamis along the coast have a wide range of periods, typically 10-30 minutes (Salsman, 19591, but can have

periods as short as 5 minutes or as long as 60 minutes The typical period

of a tsunami is much less than that of the ocean tides, and the tsunami recording may be affected by the instrument response Satake et al (1988)

made a survey of the tide gauges in Japan and found a wide variety of responses to water level changes that depended on the actual site and

16 JEAN M JOHNSON

design of each instrument The gauges did not affect actual amplitudes and periods greatly, except for gauges specifically designed to reduce wind wave noise severely in regions that are seasonally very stormy No similar study has been performed in situ on gauges in the United States, but Noye (1976) showed that for the typical U.S stilling-well gauge, the amplitude of

a recorded tsunami is reduced and the period lengthened This problem becomes severe for very-short-period waves but is minor for longer peri- ods For example, a 25-cm wave of period 15 minutes has amplitude reduction of approximately 20% and period increased about 10% On the other hand, a 50-cm wave of period 5 minutes has amplitude reduced 50% and period increased 40% As a result, tsunami waveforms used here are filtered to reduce waves with periods shorter than 10 minutes This reduces

complications due to instrument response, but no other corrections for

response are made The tides are also filtered from the data, in various ways explained in the following sections

Another complication of the tide gauge records is the response of the harbor or bay in which the tide gauge is situated When some frequency in the spectrum of an incoming tsunami matches one of the normal modes of the bay or harbor, resonance amplification may occur (Henry and Murty, 1995) In addition, the multiple reflections of a wave in an enclosed space such as a harbor usually form standing waves that can persist for days These phenomena complicate the recording of the tsunami, especially in its later phases These phenomena can be treated by normal mode theory (Satake and Shimizaki, 19881, but this is not useful for studying the source parameters of the earthquake Here, therefore, only the beginning of the tide gauge records are used The first few pulses of wave motion are generally due almost exclusively to the incoming tsunami and are uncon- taminated by the resonance effects of harbors

The last complication in using tide gauge records is in the assumption that the recorded tsunami can be treated as a linear long wave In the open oceans, the tsunami behaves as a simple shallow-water wave, as discussed in the last section When it reaches land, however, the nonlinear effects dominate and the linear long-wave equation is no longer appropri- ate Tide gauges are situated in bays and harbors In these environments, the applicability of the linear long-wave equation begins to break down; however, if the incoming wave is still small compared to the water depth, the long-wave equation is still generally applicable The water depth in harbors is usually o n the order of 10 m For a wave height of 1 m or less, the linear long wave is still appropriate In most cases of transoceanic tsunamis, the recorded wave height is about a meter or less However, even when the wave height of a recorded tsunami used here is higher, the wave is still assumed to be a linear long wave

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 17 2.2 Forward Computation of Tsunamis

2.2.1 Finite Difference Computation

Solutions to the linear long-wave Equations (23) and (24) can be

obtained by a finite difference numerical method, which is based on the Taylor series expansion The continuous domain equations are discretized, then the derivatives are approximated by differences resulting in algebraic equations Many finite difference schemes are summarized in Weiyan

(1992) The numerical method employed here is the staggered leapfrog

scheme (Press et af., 19861, an explicit second-order scheme in which only

one unknown appears in the difference equation and is evaluated in a grid system in terms of known or previously calculated quantities

Figure 7 shows the grid system In the leapfrog scheme, the equation of motion and the equation of continuity are solved on grid systems shifted one-half grid space with respect to each other In other words, the water

computed at the edges

in Cartesian coordinates can be written in terms of a leapfrog difference

I Flow Rate (a,)

FIG 7 Sketch of a grid system for computation of the two-dimensional propagation of tsunamis in the Cartesian coordinate system The equation of motion is solved at the edges of the grids; the equation of continuity is solved at the center of the grids AS is the grid spacing

18 JEAN M JOHNSON

equation where time is written as t = 1At and space as x = iAx, At is the

time step, Ax is the spatial grid space, and i and 1 are integers Thus

Equation (25) is staggered in both time and space

The leapfrog scheme is a second-order scheme, implying that the trunca- tion errors are on the order of (AxI2 For the leapfrog scheme, however, the grid space is 1/2(Ax); therefore, the truncation errors are 1 / 4 ( A ~ ) ~

There is no overlap between the two grids; therefore, the two spatial

grid points i and i + 1/2 can be treated by the same argument (21, and the

time points 1 - 1/2 and 1 can be treated by the same variable ( i t ) For the full two-dimensional equations, including the Coriolis force (Eqs 23 and

24), we can write the components of Q and h as three-dimensional arrays

with two spatial dimensions i and j and one temporal dimension if Q

becomes Q&, j , i t ) and Q&, j , i t ) for the flux in the cp and 8 directions

on a spherical earth The water height h becomes h(i, j, it) The equations

of motions and continuity become

(27)

Equations (26) and (27) are the equations in the finite difference code used here In the computation, the initial water height in all grid spaces is known In the tsunami source region, the initial condition is given by the

A t [

-

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 19

ocean floor uplift, as discussed in Section 2.1.1 During the first time step, the flux is computed and then the wave heights recomputed This contin- ues for each time step as the tsunami propagates out over the finite difference grid system Eventually, the tsunami will propagate to the observation points The water heights at each observation point are com- puted at each time step, stored, and, at the end of the computation, given

as a synthetic waveform

2.2.2 Stability

In using any numerical method, it is important to analyze the stability to determine if the numerical solution is an accurate approximation of the differential equations Stability of a finite difference scheme has variable meanings (Weiyan, 19921, but it is often based on the definition of Courant

the domain of the difference solution be wholly inside the domain of the differential solution In practical terms, this requires that the errors of the difference solution remain bounded This can be achieved by choosing appropriate grid sizes and time steps according to the CFL number

c A t

A x

where c is the velocity of the wave For a one-dimensional linear long

Ensuring that the CFL condition is met may produce amplitude stability, but it cannot guarantee it (Weiyan, 1992)

For the equations in two dimensions, this becomes

A t

A x

In a finite difference scheme for which the spatial grid size is fixed, the

depth the tsunami crosses In the case of the Alaskan-Aleutian tsunamis in this study, the smallest grid size is less than 1 km and the deepest water depth d within this grid size is approximately 4000 m; therefore, the optimum time step to ensure stability is 3 s

20 JEAN M JOHNSON

Another stability condition that must be met is phase stability Satake (1987) performed numerical experiments to determine the optimum grid size to ensure numerical stability of the phase of a numerical waveform Satake found that at least eight spatial grid points per wavelength are required to ensure phase stability The wavelength of tsunamis generated

by a great earthquake is tens to hundreds of kilometers The largest grid size used in the computation is approximately 10 km Tsunamis of wave- length 100 km or larger in this grid area will fulfill the phase stability condition

There are two boundary conditions imposed in the finite difference computation: the land-ocean boundary and the open ocean boundary at the edge of the computation area

The land-ocean boundary condition is total reflection at the coast At a land-ocean boundary, the flux Q is zero In reality, this condition does not hold entirely Wave energy is absorbed at the coast in the run-up process

of waves on shores This affects the waveforms generated at the observa- tion points, because these are generally close to land; therefore, the tsunami computation is more accurate at the beginning and degenerates as time passes and reflected energy begins to dominate the waveform The other boundary is at the edge of the computational area Here, a radiation condition (Hwang and Divoky, 1975) is employed, as shown in Figure 8 As the wave passes out of the computation area, it retains the same slope Thus

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 21

or, in the computational notation,

This equation is used to calculate the wave height at the boundary

2.2.4 Bathymetry Data

The equation of motion for a linear long wave indicates that the tsunami velocity depends only on the water depth; therefore, the synthetic wave- form is very sensitive to the bathymetry The more accurate the bathymetry, the more accurate the computation This suggests that a fine grid size be adopted in which to calculate the tsunami propagation; however, very fine grid spacing of the entire northern Pacific Ocean would be impractical due

to the enormous computational effort For the majority of the deep Pacific Ocean, the bathymetry changes slowly For this area, the grid spacing need not be any finer than 5' (approximately 10 km) The 5' bathymetry data used here are the ETOPO5 data provided by the National Geophysical Data Center

In coastal areas, the bathymetry changes much more rapidly than in the deep ocean Also, islands and harbors where tide gauges are located cannot be adequately represented by a 5' grid Figure 9 shows this for the Aleutian island of Unalaska In the 5' grid, the island is severely distorted and Dutch Harbor, where the tide gauge is located, disappears entirely Therefore, in coastal areas, 1' (less than 2-km) grid size is used These data

22 JEAN M JOHNSON

are provided by the U.S National Ocean Service (NOS) at 15'' intervals

We averaged this data to 1' and manually corrected it using NOS bathy- metric maps These 1' bathymetry data are used along the west coast of the United States, around the Aleutian and Hawaiian islands, and around Japan

The addition of areas of smaller grid size nested within the larger computational area with 5' grid size requires some modification to the finite difference scheme discussed in Section 2.2.1 Figure 10 shows an example of such a finer grid system adjacent to a coarser grid system In this example the ratio of the large to the small grid size is 1:2 The sum of the flux q, into the smaller grids must equal the flux Q, out of the larger grids To compute the flux into one smaller grid, the water height at a fictitious grid point h , is computed by linear interpolation between the water heights in the larger grids

The largest 5' computational area over which we simulate tsunami propagation is from 130"E to ll0"W and from 10"N to 65"N, covering the

majority of the North Pacific Ocean The computer memory required for this simulation is approximately 10 times the number of grid points, or, in bytes, 40 times the number of grid points For the grid system just given, the number of grid points is nearly 1 million For this system, the computer memory must be 40 Mbytes For the 5' and 1' nested grid system, the

computation of a synthetic tsunami traveling for 15 hours in this grid space, the computation time required on a Hewlett-Packard 735 is approxi- mately 12 hours

FIG 10 Sketch o f the nested grid system involved in finite difference computation using

bdthymetry data sets

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 23

2.3.1 Inversion Method

The computation of tsunami waveforms using the finite difference scheme described in the previous section is the forward problem It requires specifying the initial condition of the tsunami, including the amount of slip on the fault, to generate synthetic waveforms In reality, the initial condition of the tsunami is unknown; only the tsunami waveforms are available We invert the tsunami waveforms to determine the initial condition of the tsunami, thereby determining some source parameters of the generating earthquake

There are many source parameters of an earthquake (focal mechanism, fault area, depth, and slip) for which we could invert the tsunami wave-

forms (see Fig 3) However, if we attempt to invert for all of them, the model space would be very complex The problem would be underdeter- mined and the model parameters could not be determined uniquely (Menke, 1989) Here, we choose to invert for one parameter only-the amount of slip on the fault We assume all other parameters are known This assumption is a reasonable one Even though there is no well- determined focal mechanism for several of the Alaska-Aleutian earth- quakes in this study, all these earthquakes are subduction zone events and occurred on the plate interface, where many similar, though smaller, events occur often These events are recent and are therefore recorded on high-quality modern instruments The focal mechanisms of these smaller events can be assumed to be similar to the mechanisms of the great earthquakes The direction of underthrusting can also be estimated by the

global plate motion models (DeMets et al., 1990) In addition, the smaller

underthrusting events highlight seismically the plate interface The great earthquakes must lie along the same boundary, giving the fault dip and depth of the earthquakes The fault area of the great earthquakes can be estimated from the size of the aftershock zone (Sykes, 1971; Kanamori, 1977) This leaves only the amount of slip as the unknown parameter This

is an extremely important parameter because it is directly related to the seismic moment of the earthquake Seismic moment is one of the most fundamental parameters of an earthquake It is the magnitude of a shear double-couple source or the moment of one component couple (Kasahara, 1981) The seismic moment is thus a measure of the size of an earthquake The slip on the fault is related to the seismic moment by

M" = p s D , ( 3 2 )

where Mu is the seismic moment, p is the rigidity, S is the fault area, and

D is the average slip on the fault Therefore, the slip gives a direct

24 JEAN M JOHNSON

estimate of the seismic moment of the earthquake, if the slip area is

known

As mentioned in Section 1, modern seismic wave analysis has shown that

the slip on a fault is not uniform but that there are some patches where

slip is large (e.g., Ruff and Kanamori, 1983; Kikuchi and Fukao, 1987) The

method used here for tsunami waveform inversion can locate these areas

of high slip

This technique was originally applied to Japanese earthquakes by Satake

(1987, 1989) using near-field and regional waveforms recorded on tide

gauges in Japan The method in this study uses regional and far-field

tsunami waveforms The method is as follows

The fault area of the earthquake is divided up into several smaller

subfaults (Fig 11) The fault parameters, such as depth and focal mecha-

Heterogeneous slip distribution on the fault plane

uplift of ocean floor and ocean surface

n

A

/ actual bathyrnetry Green’s functions

obsetvation at station I

_I

Ajj(t) xj = bj(t)

FIG 11 Green’s function technique for tsunami inversion The upper figure shows slip

heterogeneity on the fault plane The lower figure shows the method used to estimate the slip

distribution Unit slip on suhfault j produces uplift of the ocean floor and ocean surface The

propagation is numcrically computed on the actual hathymethry of the ocean Each subfault

produces ii Green’s function The linear superposition of the Green’s functions produces the

ohservation at station I The slip distrihution is obtained hy solving t h e linear equation for x j ,

the slip on suhFault j

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 25

nism for each subfault, correspond to the fault parameters of the earth- quake The ocean bottom deformation due to each subfault is computed

for unit slip (1 m) as described in Section 2.1 This displacement is used as

the initial condition for the synthetic tsunami waveform, or Green’s function, from each subfault The tsunami propagation to each tide gauge

is computed for each subfault by the finite difference scheme discussed in Section 2.2 The result is one Green’s function for each subfault for each observation point-the tide gauge location For example, for six subfaults, there would be six Green’s functions for each tide gauge

The observed tsunami waveforms from the tide gauge record are as- sumed to be a linear superposition of the Green’s functions; thus the unknown slip, or weighting value for each Green’s function, can be determined by solving the linear equation

where A , , ( t ) is the matrix of Green’s functions for tide gauge i due to slip

on subfault j , b , ( t ) is the observed tsunami waveform at tide gauge i , and

x, is the unknown weighting value for subfault j This equation is shown in matrix representation in Figure 12 This is an overdetermined problem (Menke, 1989)’ because there are more data than unknowns The equation

is solved by the least squares method, which minimizes the misfit between the observed and synthetic waveforms The weighting values give the slip

on each subfault, thereby giving the slip or asperity distribution of the earthquake

Although the least squares problem is well defined and a solution is guaranteed, this solution may not represent something that is physical For example, all the earthquakes in this study are considered to be under- thrusting earthquakes; in other words, the slip is unidirectional The unconstrained least squares solution may contain both positive and nega- tive weighting values for x, We consider this to be unrealistic To avoid this physical absurdity, we use a nonnegative least squares method (Law- son and Hanson, 1974) This method requires that all values of the solution be positive or zero Whenever we use this method, the least squares solution is also presented so the effects of imposing the nonnega- tive constraint may be assessed

26 JEAN M JOHNSON

slip on

Fit, 12 Matrix representation of Equation (2.33)

geophysical problems, the variance of the data cannot be estimated, with the result that the formal statistical errors are not a good measure of the true errors Also, for the nonnegative least squares method, there is no analytical expression for the errors In this situation, we must use some other technique to estimate the errors

The method we have adopted is similar to the jackknifing technique described by Tichelaar and Ruff (1989) Jackknifing is a resampling tech- nique designed to avoid the problems inherent in determining the errors from the least squares method In jackknifing, a fixed number j of random data points is dropped from the dataset of n members (see Fig 13) This produces a resample that is then inverted for the model parameters This procedure is repeated many times, theoretically until all possible combina- tions of [ n - j ] resamples have been inverted, though, for large n, this becomes computationally very intensive The errors are then simply the standard deviation of the resample inversion results, times a scale factor

FIG 13 Schematic representation of the jackknife technique The original data vector has

five components, waveforms 1-5 The data are resampled by deleting from the original data a fixed number of waveforms (one in this example) to form multiple jackknife resamples (in this

case k ) Each resample defines a model estimate The multiple model estimates are then

combined to give a best model and its standard error Modified from Tichelaar and Ruff,

1989

The scale factor is d ( k - p + l ) / ( n - k ) , where k = n - j and p is the number of model parameters For a “delete-half’ jackknife, in which half the data are deleted to make each resample, the scale factor is 1

The method we use to estimate the errors of the tsunami waveforms inversions is similar; however, the resamples are made up by dropping an entire waveform from the dataset, rather than a number of random points

In this way, all the waveforms are dropped successively from the dataset, and the errors are estimated from a number of jackknife inversions equal

to the number of waveforms used in the inversion In some cases where there are few waveforms available, this means the errors are determined from a statistically small sample Also, the jackknife inversion can be strongly influenced by the presence or absence of a particular waveform, especially when it is important to maintaining a good station distribution around the source area These two factors can cause the errors to be very large, but they are probably the maximum errors for the inversions The errors are always listed with the inversion results in the following parts

28 JEAN M JOHNSON

COMPARISON OF SEISMIC AND TSUNAMI WAVE INVERSIONS*

3.1 Introduction

locate areas of high moment release on an earthquake fault zone Numer-

Kikuchi and Fukao, 1985; Beck and Christensen, 1991; Christensen and

high moment release during earthquake rupture These areas of high moment release are interpreted as asperities: areas on the fault that are locked prior to the earthquake and have high slip during rupture (Lay and

Kanamori, 1981) It has been rarely possible, however, to verify these

moment release distributions by independent means In a few cases, geodetic data are available and can be inverted directly for the slip

distribution (Miyashita and Matsu’ura, 1978, for the 1964 Prince William Sound event; Barrientos and Ward, 1990, for the 1960 Chile event; Holdahl and Sauber, 1994, for the 1964 Prince William Sound event)

More often, though, the geodetic data are extremely limited and the entire slip distribution cannot be reconstructed

Tsunami waveform inversion is another independent means to deter- mine the slip distribution of an earthquake, but there are only a few tsunami studies of earthquakes for which high-quality seismic data also

exist Satake (1989) studied both the 1968 Tokachi-Oki and 1983 Japan Sea

earthquakes using tsunami data and compared the results to seismic wave studies Satake showed that the tsunami and seismic data produce compat- ible, but not identical, results Tanioka et al (1995) studied the 1993

Hokkaido Nansei-oki earthquake using tsunami waveforms recorded in Japan and Korea The slip distribution obtained from the tsunami inver- sion agrees in general with the results from inversion of strong ground

motion and teleseismic P-waves (Mendoza and Fukuyama, 1996), but the

specifics of the two results are different From these few studies using both high-quality seismic and tsunami data, it can be concluded that the results

of inversions using either dataset will give similar, but not identical, results

local and regional tsunami data There is no study comparing inversion results from far-field tsunami waveforms to seismic data We wish to test

*This section adapted from Johnson and Satake (1996) The Rat Islands earthquake: A

critical comparison of seismic and tsunami wave inversions Bull Seismol SOC Am 8 6 6 )

Reproduced with permission

HETEROGENEOUS COUPLING ALONG ALASKA-ALEUTIANS 29

using high-quality data An excellent candidate for this comparison is the

1965 Rat Islands earthquake It occurred during the WWSSN era, so it was recorded on high-quality seismic instruments The tsunami it generated was also recorded on tide gauges around the Pacific We invert tsunami data from the 1965 Rat Islands earthquake and compare those results to results from seismic wave analyses

3.2 The 1965 Rat Islands Earthquake

The Rat Islands earthquake occurred on 4 February 1965 at 5:01:28 G.M.T Its epicenter is 51.3"N, 178.6"E (Stauder, 1968), and its aftershock area extends from 171" to 179.5"E, a distance of approximately 650 km Despite its large magnitude, the earthquake occurred in such a remote and sparsely inhabited area that there were no fatalities, either from t h e earthquake or from the tsunami it generated, and only minor damage and flooding occurred in the Aleutians The tsunami was recorded around the Pacific Ocean, but it caused no damage in any far-field area

3.2.1 Focal Mechanism

The focal mechanism of the earthquake inferred from P-wave first motions and S-wave polarizations is of a shallowly dipping thrust fault (Stauder, 1968) Wu and Kanamori (1973) determined the faulting parame- ters as strike = 290" and dip = 18" The slip angle measured clockwise from the strike direction is 41.4"

The tectonic setting of the 1965 earthquake is shown in Figure 14 The North American plate overrides the Pacific plate, and subduction becomes increasingly oblique from east to west along the subduction zone It eventually becomes entirely right-lateral strike-slip at the western end of the Aleutians, near Kamchatka From 180" to 170"E, convergence is partitioned approximately 40% strike-slip in the east to nearly 80% strike- slip in the west

From 170" to 180"E, the overriding North American plate consists of large tectonic blocks (the Rat, Buldir, and Near blocks, from east to west) separated by deep submarine canyons (Murray and Heck canyons) These structures are well defined in the western Aleutians, but have no corollary

in the eastern Aleutians Geist et al (1988) have suggested that the blocks are formed by clockwise rotation due to the obliquity of subduction in the western Aleutians

Mogi (1969) showed that the aftershocks of the Rat Islands earthquake were strongly correlated to the tectonic blocks Beck and Christensen (1991) have suggested that the asperity distribution of the 1965 rupture