JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. B4, PAGES 6321-6333,APRIL 10, 1991
Dispersion
of P Wavesin Subducted
Lithosphere'
Evidence for an Eclogite Layer
DAVID
GUBBINS
Department
ofEarthSciences,
University
ofLeeds,
Leeds,
United
Kingdom
i•OEL
SNIEDER,
Department
of Theoretical
Geophysics,
University
of Utrecht,
Utrecht,
Netherlands
Cold,subducted
lithosphere
hasrelatively
fastseismic
velocity
which
leadstoearlyarrivals
for
some
event-station
paths.Theeffectisverylargeforevents
in theTonga-Kermadec
deepseismic
zonerecordedat certainNew Zealandstations.Theseparticulararrivalsare very high-frequency
(3 Hzorgreater)
andsometimes
resemble
twodistinct
phases,
thelaterarrival
appearing
at about
thetimepredicted
by Jeffreys-Bullen
tables.DatafromthedigitalstationSNZOin Wellington
confirmthe traveltime resultsof the analogstationsand furthermoreshowfrequencies
above5
Hz, muchhigherthancanbeseenonanalog
records,
andupto 4%dispersion
in therange1-8
Hz. Energy
in thesecond
phase
(which
isoftenabsent
at SNZO)ismainly1-2 Hz. Thedigital
datasupport
theidea,proposed
earlier,thattheeffectis caused
bypropagation
through
a thin
slabwhichpasses
onlyshort-wavelength
waves.
Theessential
features
ofthewavepropagation
are
modeled
byacoustic
waves
in a one-dimensional
high-velocity
slab;thewaveforms
produced
by
the modelare discussed
in termsof the leakymodesof the systemand calculatedby a reflectivity
method.A verythin (< 15 km) uniformslabprovides
the required
dispersion,
but the waves
areheavily
attenuated
andwouldneverbeobserved
at teleseismic
distances;
a thicker
slaballows
theenergy
through
butdoesnotgiveenough
dispersion.
Altering
thevariation
ofvelocity
across
theslabprovides
therequired
dispersion
if a thickhigh-velocity
layer,withwavespeed
increasing
gradually
withheight,
isoverlain
bya thinlidofevenhigher
velocity.
Forthemodels
considered
the lid thickness
mustlie in the range6-15 km andbe continuous
froma depthof about50 km
to the bottomof the earthquake
zone.The thicklayercouldarisefromthe thermalanomalyin
thesubducted
lithosphere;
thethinlid maybethegabbroic
partof thesubducted
crustthat has
transformed to eclogite.
seismograms
forwaves
that havepassed
throughtypicalslab
structures,
and Cormlet[1989]hascomputed
diffraction
ef-
INTRODUCTION
Deepseismic
zonesoccurwhena subducting
plate de- fectsfrom slab structuresusinga Gaussianbeammethod,
scendsinto the mantle. The plate graduallywarms and
but otherwise rather little has been done with full wave the-
undergoes
minerslogical
phasechanges
as it descends;
re- ory.
viewsofsimplethermalmodels
aregivenbySleep[1973],and
Ansell and Gubbins[1986], hereinafterreferredto as
modelsof the mineralogy
aregivenby Ringwood
[1982]and AG, studiedsomeremarkablearrivalsat stationsin the
Helffrichet al. [1989].Seismological
studiesof subducted New Zealand network. Waves from events in the Tongaslabshaveinvolvedsourcemechanisms
(for example,seeGi- Kermadecdeepseismiczoneexhibitresiduals
in excessof
ardiniand Woodhouse
[1984]for the Tonga-Kermadec
zone 10 s relativeto Jeffreys-Bullen
(JB) tablesand the appeardiscussed
in thispaper);phaseconversions
at the interface anceof two distinctP phases:a•uearly,high-frequency,
arbetweenthe slaband surrounding
mantle[Snokeet al., 1977; rival and a later phase,with lowerfrequency,arriving at
Fukaoet al., 1978;Nakanishi,1980;Nakanishiet al., 1981; aboutthe JB time. FigureI givesa map of the regionand
Matsuzawaet al., 1986];propagation
effectssuchas travel showsthe geometry.AG's resultscan be summarized
as
timeanomalies
[DaviesandMcKenzie,
1969];andfocusing follows:
1. Stations on the east of the North Island of New Zealand
effectsof the slab[Daviesand Julian,1972]. Recentwork
on tomographic
inversion
for the P wavespeedin the sub- and the northernend of SouthIsland (their zone3) exhibit
duction zone has included refinement of the source locations
the fast double arrivals. The correspondingray paths lie
[e.g.,Spencer
andGubbins,
1980;Creager
andJordan,1984; closeto the subducted slab for most of their length due to
Engdahland Gubbins,
1987].In manyof thesestudiesthe a fortuitous bend in the zone.
fast zonebecomesvery thin and in mostcasesthe width is
limitedby the coarse
parameterization
ofthe velocitymodel.
The fast zonemay be so thin that ray theory breaksdown
andfrequency-dependent
wavepropagation
mayoccur. Vidale[1987]hasuseda finitedifference
methodto compute
2. Stations on the west side of North Island (their zones
1 and2) are eitherslowrelativeto JB (showing
that early
arrivals on the east side are not simply causedby bias in
the traveltime curve)or are heavilyattenuatedby passage
through the Taupo volcaniczone.
3. Arrivalsthroughout
therestofSouthIsland(theirzone
4) give variable results.
Copyright
1991bytheAmerican
Geophysical
Union
4. Countingpeakssuggests
the earlyphasehasa dominant frequencyof 3 Hz, but the analogrecordingsystem
Paper number 90JB02741.
may mask higher frequencies.
0148-0227/91/90J$-02741$05.00
6321
6322
GUBBINSAND SNIEDER: DISPERSIONOF SUBDUCTEDLITHOSPHERE
to some extent by the quiet borehole installation and large
dynamic range of the digital recording system.
The Wellington site is relatively noisy compared with
someother North Island stations suchas Mangahao (MNG),
and it is not as well placed relative to the slab to observe
these early arrivals. JB residuals at the Willmore seismometer from
suitable
sources show a bimodal
distribution
with
one peak around the JB time and one about 4 s early, suggestingthe early phaseis often missedaltogether: the distribution at the quieter site of MNG has a single peak several
secondsearly (AG). JB residualsfor WEL from the International SeismologicalCentre (ISC) Bulletin are shownin
Table 1, which can be comparedwith the similar table for
MNG given by AG. The largest residuals are from events
in the latitude range 25-35øS abovedepth 300 km or thereabouts, dependingon latitude. The pattern is similar for
many New Zealand sites. Ray tracing confirms that the
correspondingray paths lie 10-50 km below the surfacedefined by the deep earthquakes,supportingthe view that the
anomaly is caused by fast subducted lithosphere. Station
NZO
SNZO is only a few kilometersfrom Wellington(WEL) and
should exhibit a similar pattern.
We approachthe digital data with four specificquestions
in mind:
1. How high is the frequency of the early arrival?
2. Does the high-frequencycontinueto arrive in late parts
of the wave train
50S
161ZlE
171Zl
181Zl
1 71Z1•4
Fig. 1. Events used for this study. SNZO is a standard $RO site
near Wellington, New Zealand.
5. At Wellington, where two phasesare seenfrequently,
two seismometershave operated side by side. The Willmorehasgoodhigh-frequencyresponseand recordsthe early
phasewell, but the other seismometer,a Benioffwith poorer
high-frequencyresponse,doesnot record it. Comparisonof
the known frequency responsesof the two instruments suggeststhe early phasehas dominant frequencywell above 3
or does it have the form
of an isolated
pulse?
3. Are there two distinct phasesor a dispersedwave train?
4. What kind of structures are responsiblefor the observed precursor?
FAST ARRIVALS AT THE WELLINGTON
SRO
STATION
SNZO
Data were selectedusing the National Earthquake Information Center (NEIC) CD-roms distributed by the Orfeus
data center. The archive includesall events above magnitude 5.5 from
1980 to 1984.
Suitable
events were selected
Hz.
from the latitude-longitude box 20ø-40øS, 170øE-170øW.
These resultswere interpreted as slab effects: the thin slab The searchyielded twenty-eightevents,three of which were
allows short-wavelengthwavesto travel at the high-velocity unsuitablebecausethe SNZO instrumenttriggeredlate (this
but the longer wavelengthsdo not "see"the slab and travel only becameapparentafter scrutinyof the record). Twentyat normal mantle speed. Slow arrivals in zone I have passed five eventsremained;their hypocentersare listed in Table 2.
through the slow wedge above the slab; variable arrivals in They cover the full range of events discussedby AG; three
the south have left the slab and been subjected to a variety
of attenuation and scattering effects so that sometimesthe
two-phase nature of the arrival is preserved and sometimes
the early phase is lost.
AG's study waslimited by the data quality, which wasre-
are deep events, someat the extreme north of the zone where
the early phaseis observed,and severalvery closeevents.
The short-period record was plotted for each event, the
time of the first arrival picked,and the residualrelative to JB
travel time tablescalculated.The resultwasfoundto agree,
strictedby analogrecording.Estimatinghigh frequencyby to a few tenths of a second, with the residual published
countingpeaksis very unsatisfactory,and result (5) above, by the ISC for WEL or SNZO (in this time window, ISC
from Wellington, suggestsvery high frequenciesare present appears to report WEL in preferenceto SNZO unlessthe
that cannot be seenon the paper record. Digital recordingis formerstationfailsto record). The results,shownin Table 2,
required for a more quantitative study. The instruments of confirmthe findingsof AG for other stationsin the region.
the New Zealandnetwork are now being replacedand digital The residuals are somewhat smaller than for MNG but still
recordswill eventuallybecomeavailable. In this preliminary exceed 10 s in two cases. The long-period trace showed
study we use the only existing digital station (SNZO), the no evidence of a precursor, and often the arrival was late
SeismicResearchObservatory (SRO) site at Wellington. It
has only one short-period, vertical component, instrument,
the filter is not good for our purpose, and there are problems with the trigger so the first few secondsof the arrival
are often lost, but these inadequaciesare compensatedfor
relative to Jeffreys-Bullen times. These records were not
used.
Each trace was filtered to extract specific frequency
bands. Resultsfor event 13 are shownin Figure 2. A highpass causal filter with corner frequency at 2 Hz gives the
GUBBINSAND $NIEDER: DISPERSIONOF SUBDUCTEDLITHOSPHERE
TABLE
1. ISC
Residuals
for Station
WEL
from
6323
1964 to 1974
Depth
Latitude
0 km
100 km
200 km
15øS
16øS
17øS
18øS
19øS
20øS
21øS
22øS
23øS
24øS
25øS
26øS
27øS
28øS
29øS
30øS
31øS
32øs
-1.1(32)
-1.9(21)
-2.1(3)
1.8(3)
0.1(6)
-0.9(11)
1.3(19)
-1.9(11)
-2.9(9)
-1.5(5)
-6.8(2)
-1.5(3)
-1.9(8)
-1.1(4)
-3.6(15)
-1.6(20)
-0.8(10)
-3.5(10)
33øS
34øS
-6.6(9)
-3.2(2)
......
......
35øS
36øS
37øS
38øS
39øS
-4.8(1)
-2.0(4)
-3.4(3)
-1.8(2)
2.5(1)
-4.1(1)
......
......
-1.3(1)
0.1(1)
4008
-0.3(4)
-3.3(1)
-1.8(3)
-0.5(4)
-1.7(4)
-4.4(6)
-2.1(5)
-1.5(3)
-1.6(6)
0.7(5)
-3.7(2)
-3.6(3)
-3.3(3)
-9.6(2)
-5.8(1)
-4.1(1)
300 km
-2.4(4)
-3.0(1)
-1.8(2)
-0.8(2)
-1.2(8)
-0.4(7)
-1.0(5)
-2.8(1)
-1.6(1)
8.3(1)
-6.1(2)
-7.1(2)
-6.8(2)
-5.6(1)
400 km
-3.0(1)
-2.4(2)
-0.8(2)
-2.1(1)
0.6(3)
-1.3(1)
-1.8(1)
-1.8(1)
-4.5(1)
-5.1(1)
....
-4.8(3)
-4.4(4)
....
-2.6(3)
-2.3(2)
-2.6(1)
-2.4(2)
-2.1(1)
-0.2(3)
-1.1(3)
-1.8(1)
-0.7(2)
-1.4(5)
-0.8(2)
-3.8(1)
-
500 km
600 km
-2.0(24)
-1.7(5)
-1.6(5)
-2.0(10)
-1.8(14)
-1.9(9)
-0.2(10)
-
-2.2(6)
-1.0(4)
-2.2(3)
-2.1(10)
-2.3(14)
-1.3(1)
-
-
-
-
.....
.....
.....
.......
41øS
2.1(4)
......
Time anomalies are relative to Jeffreys-Bullen travel time tables. The numbers in parentheses denote the number of observations in each latitude-depth cell. Very fast arrivals occur for
depths < 300 km and the latitude range 25ø-35øS.
clearest signal and makes the pick easier. A lowpasscausal
filter with a corner frequency at 1 Hz was used to search
for "normal" arrivals: one appears on the second trace in
Figure 3 about 2 s before the JB time. At these close distancesthe seismogramcan be quite complexevenfor a sire-
pie Earth model. The theoretical arrivals were calculated
for model PREM and compared with the lowpass-filtered
traces to search for conventional second arrivals. In many
casesthe first significant energy was a depth phase pP or
sP, and one is marked in Figure 2 (trace f < 1 Hz).
TABLE 2. Events Used for the Study of Digital Data
Aø
JB,s
r0, km
Latitude
Longitude
Depth, km
Mb
Event
Year
Month
Day
Hour
Minutes
21.47
19.02
18.56
18.08
17.15
15.90
-4.0*
-5.2*
-4.1'
-11.1'
-7.0*
-8.5*
596
513
592
322
287
260
-21.466
-23.466
-23.627
-25.096
-25.795
-27.134
-175.451
-177.297
-178.368
-175.569
-176.014
-176.474
68.0
79.0
336.0
37.0
26.8
27.0
6.3
6.7
5.9
5.9
6.4
5.5
16
3
20
13
26
6
1983
1980
1984
1982
1984
1980
3
4
I
3
9
12
21
13
19
7
28
2
7
18
16
15
0
13
44
4
15
41
3
17
Seconds
17.79
31.90
16.74
57.11
35.46
3.30
15.04
13.47
-8.7
-9.7
218
191
-28.159
-29.516
-176.320
-177.228
30.7
33.0
5.6
5.6
19
10
1983
1981
7
11
28
18
1
17
40
37
33.32
48.71
12.99
12.90
-8.2*
-7.6*
394
191
-29.335
-29.934
-179.039
-177.741
323.0
33.0
6.0
6.1
8
11
1981
1981
9
12
28
26
17
17
56
5
18.09
32.52
12.15
-9.7
170
-30.573
-178.208
33.0
5.7
7
1981
3
7
23
30
8.40
12.12
11.92
11.16
-2.6
-8.5*
-11.2
172
288
159
-30.536
-30.383
-31.485
-178.375
-179.339
-178.660
33.0
238.0
79.0
5.5
6.0
6.0
27
15
14
1984
1983
1982
10
1
3
19
26
28
19
16
3
59
2
52
56.24
21.35
34.59
10.77
-6.6*
132
-32.068
-178.370
33.0
5.7
23
1984
9
17
9
8
52.73
10.74
10.65
9.93
9.76
9.22
8.40
7.28
5.72
4.48
3.72
-9.0
-5.5
-8.2
-5.4
-3.8
-7.0
-6.0
-4.4
-2.0
-0.3
131
128
116
110
90
81
79
51
48
68
-32.091
-32.204
-32.835
-32.788
-33.380
-33.866
-35.346
-37.112
-37.307
-38.220
-178.235
-178.376
-178.791
-179.306
-179.348
179.649
179.960
179.712
177.301
177.399
33.0
33.0
45.8
33.0
33.0
25.2
33.0
34.6
33.0
68.2
5.5
5.5
5.6
5.6
6.0
5.8
5.7
5.7
5.7
5.9
24
25
18
21
22
17
I
9
2
28
1984
1984
1983
1984
1984
1983
1980
1981
1981
1984
9
9
6
8
8
5
11
12
3
3
22
22
25
4
30
5
29
1
5
8
10
11
10
14
16
4
6
17
12
0
15
43
3
59
6
43
48
46
53
40
16.06
16.51
17.42
34.36
13.92
50.46
46.90
43.71
29.50
47.89
Events are arranged in order of distancefrom the station SNZO. Numbering order is arbitrary. A ø is the distance, JB is the
Jeffreys-Bullenresidual, LF is the time of a secondlow-frequencyphase relative to JB (when one was observed), and, and r0 is
the maximum depth of the ray path. Latitude and longitude are measuredin degreesand the minus signsindicate south and west
respectively.
6324
GUBBINS AND SNIEDER: DISPERSION OF SUBDUCTED LITHOSPHERE
Even!•20• :13],recorded
in S/VZO
Event 7 is shownin Figure 3. Here the lowpassarrival
is very difficult to detect, although energy clearly begins
to arrive about 2 s after the JB time and builds to significantly abovethe noiseby 185 s. Note that the arriving wave
recorded at SNZO doesnot have an impulsive character but
rather builds up slowly to an extended wave train. This
is not the case at other stations
for the same event.
The
bottom trace in Figure 3 showsthe recordingof the same
event at a secondstation, CTAO, in Australia. The waveform is very clean and impulsive, showingthe complexityof
pP•
the arrival at SNZO is a path and not a sourceeffect.
All events were studied and arrivals picked, where possible, on both highpassand lowpasstraces. The picks were
based on amplitude rather than frequency changes. The
high-frequencyresults reproducedthe picks of the original
records; a low-frequency pick was often impossible. The
results confirm the general observationthat a 1 Hz signal
f> 5 tIz .•
arrives near the JB time at SNZO.
335 240 245 JBt
255
frequencyof i Hz, one highpassfiltered with a corner frequency
of 2 Hz, and one highpassedwith a corner frequency of 5 Hz.
Picks are shown for the first arrival, JB travel time, low-frequency
"second arrival", and depth phase.
The fourth tracein Figure 2 showsthe resultof a highpass
filter with cornerfrequencyat 5 Hz. The clear arrival shows
that the precursorcontainsvery high-frequencyindeed. In
this study the upper limit of the observedfrequencyis often set by the instrument responseand the sampling rate
(20 times per second). This suggestsan instrument more
sensitiveto high-frequencywould yield better information.
From this study we conclude:
1. The initial phase includes frequencies above 5 Hz.
It is remarkable that such high frequenciesshould survive
long propagationpaths in the mantle (up to 20ø). Der et
al. [1982]find that Q increaseswith increasingfrequency
and dependson tectonic setting. Furthermore, Bache et al.
[1985]find that Q increases
with frequencyand is independent of mantle path above 3 Hz. "Normal" mantle may well
have very low attenuation for high-frequencybody waves.
2. The initial high-frequencyis not in the form of a distinct pulse but continueswell into the coda, indicating reverberationswithin structure along the path.
3. Later, low-frequency,arrivals do occur but are often
submergedin the noise and may be confusedwith depth
phasesor other conventionalsecondarrivalsdue to a layered
crustal
structure.
4. The JB travel time tables give a generally accurate
prediction of the low-frequencyarrival. The deep event 8
(Table 2) is an exception:this may reveal an inaccuracyin
the tables for deep events, or mislocation.
5. The waveformsare a path rather than a sourceeffect
becauseother stations, suchas CTAO, give clear impulsive
arrivals.
These conclusionsreinforce those made by AG and show
that the frequency
of the firstarrivalis evenhigherthan
could be measuredon the analog recordings.
EVIDENCE
FOR DISPERSION
Are there two distinct phaseswith different frequencies
or a single dispersedwave train? Two phasescan coexist
within the framework of ray theory only if one ray propagatesthrough the slab and the other propagatesoutsidethe
slab independently. If the structure has significantvariationson the length scaleof the first Fresnelzone,the concept
CTAO,unfH[ered
of ray theory breaks down and frequency-dependenteffects,
such as dispersion,become operative. For example, depth
phaseson these seismogramshave relatively low-frequency
(i.e., normal 1 Hz energy), they take off with a very different
angleof incidencefrom the direct P wave, do not travel near
Fig. 3. Short-period seismogramsfor event 7 in Table 2; as in the slab, and can thereforebe regardedas rays independent
Figure 2 except for the last which showsthe short-period vertical
of direct P. They do not exhibit dispersion. AG suggest
componentseismogramat the SRO CTAO in Charters Towers,
Australia; the simple waveform showsthe complexity at SNZO is that the P wave splits and part travels through the slab at
a path and not a source effect. The P arrival of the CTAO record high speed,retaining high frequencies,while the other part
is aligned with the JB time of the SNZO record.
travels in "normal" mantle beneath the slab, losinghigh fre-
i
-
ldO ld5 I70jBt 175
GUBBINSAND SNIEDER: DISPERSIONOF SUBDUCTEDLITHOSPHERE
6325
quenciesby attenuation. This view of two distinct geometrical ray paths will be valid only if the width of the Fresnel
zone of the rays is much smaller than the length scale of
variation in the structure, otherwisethe assumptionsof geometrical optics breaks down. Any observeddispersionwill
lack of energy in the bottom left-hand corner of every plot.
Figures 4a and 4b give events 13 and 7, for comparisonwith
seismogramsin Figures 2 and 3 respectively. Each show
indicate
The last pair, 3 and 15, are both deep events, with only
that
this is indeed
the case.
Dispersion will manifest itself in the seismogramsby a
gradual changein frequencyin the precursor,with lower-
frequencyenergy arriving after the first onset of highfrequencyenergy. It was impossibleto discern any such
dispersionin the analog records. We now examine the digital seismogramsfrom SNZO for evidenceof dispersionin
the very early part of the waveform. Each waveform was
filtered to pass the restricted frequencybands 0.5-1.5, 1.52.5, 2.5-3.5, 4.5-5.5, and 5.5-6.5 Hz, using a one-passBessel
filter which is causal and therefore preservesthe onset time
of the signal. An exampleof the resultsis shownin Figure 4
for event 15, a deep event. Dispersion is clearly visible in
the filtered
traces.
Most
of it in this case occurs between
1
3-4% dispersion.Events 11 and 26 in 5c and 5d give two
more examplesof shalloweventswith about 2% dispersion.
1% dispersion.Ray tracing for the latter two eventsshows
that the waves pass below the deepest earthquakes in the
zone and, presumably, leave the slab relatively quickly. The
dispersionis correspondinglysmall.
This range of earthquakesshowsthat dispersionis always
present, although the delay is not uniform with frequency.
Most of the time-frequencyplots suggestthat the dispersion
is prominent between 1 and 3 Hz; there are also "holes" in
the arriving energy (at 3.5 Hz in event 15 and at 4 Hz,
168 s, in event 7 for example). This fine structure can arise
from detailed layering in the subductedslab as explained in
section 5, but no attempt has been made to model it. The
energy arriving before the precursorat both low and highfrequency is noise. The dispersion is rather lessthan the JB
and 3 Hz.
An alternative way to display the dispersionis similar to
the multiple-filter techniqueof Dziewonskiet al. [1969],in
which time windowsare passedthrough a suite of zero-phase
Gaussianfilters with a range of frequenciesand the resulting
time-frequency window is contoured. The filters are acausal
but the effect on the onset time of the signal is negligible.
The plots in Figure 5 have been scaledto equalizethe energy
in eachfrequencyband. Figure 5fcorrespondsto event 15 in
Figure 4. Frequencyis plotted alongthe horizontal axis, and
time is plotted up the vertical axis for an interval slightly
exceedingthe length of the precursor. A vertical swath of
Figure 5f centered on, for example, 3 Hz, gives the power
in the band-passedtrace centeredon 3 Hz in Figure 4. The
onset on each trace in Figure 4 can be seen to correspond
to the start of significantenergy in Figure 5f. The Gaussian
filtering techniqueaveragesin time as well as frequency,so
the time-frequency plots give only a smoothed estimate of
the seismicenergy. Dispersionappearsin Figure 5f through
A?ent
#3026'/15f,
recorded
•h 5W•0
residual or the burst of 1 Hz energy that is often seenon
SNZO or the analog records.
MODELING
OF THE
DISPERSION
These wavestravel along the strike of the slab for a considerable distance. The variation in seismicvelocity along
the slab is assumed to be smooth, like that in normal mantle, and any possible sudden changesalong the ray path,
for example due to a break in the slab, are ignored. Velocity variations acrossthe slab are, however,rapid becauseof
the compositional anomalies and temperature perturbation
associatedwith the subducted lithosphere. Under such conditions we can hope to separate the deflection of the propagation path causedby slowvariations along the slab from
dispersion caused by rapid variations acrossthe slab and
treat the wave field as a set of (leaky) modes propagating
within the slab. The WKBJ approximationfor guidedwaves
[Bretherton,1968]justifiesthis approach.We are interested
only in the dispersioninducedby rapid variationsof properties acrossthe slab, and not in the path of propagationalong
the slab. We therefore model the slab with a plane-layered
medium. The data will undoubtedly contain complicatedef-
fects arisingfrom any three-dimensionalstructure through
which it passes,but generalpropertiesof the dispersionwill
depend predominantly on the structure acrossthe path and
the distance traveled through that structure.
We also choose to restrict the initial modeling to the
acoustic case; no S waves are included. In this way we
shall isolate dispersioneffectson the P wave and eliminate
complicationsin the synthetic seismogramsassociatedwith
conversionto $. We do not expect conversionto be high
or important becausethese wavestravel at close-to-grazing
incidence with the slab surfaces and because we are only
interested in the first part of the waveform: any conversion
to $ wave energy will travel more slowly and will not arrive
•Hz
in the first few seconds. The acoustic calculation
152
Fig. 4. Band-passfiltered tracesfor event 15. •ach trace has a
bandwidthof 1 Hz, centeredon (kom top down) 1, 2, 3, 4, 5, and
count properly for amplitudes. Energy will be interchanged
at interfacesby conversionto and from $ and this will not be
includedin our synthetics. We therefore draw no conclusions
from amplitudesor relative amplitudesin the synthetics.A
more completestudy of the elastic caseis reservedfor later
6 Hz.
work.
15t
15•
15•
I•0
I•2
I•4
cannot ac-
6326
GUBBINSAND SNIEDER: DISPERSIONOF SUBDUCTEDLITHOSPHERE
Event (82066) (13)
i
,
i
,
i
,
•
Event (81066) (7)
,
i
,
i
' ,•.:i•
• ;-."
' :•.-.-.:;-;
' ' ......
ß'•
.'!,•':;•!•i,•:::s::
..s:.-:•::..:.,..•..+...
.... .........
.....
...........
........
't i -:':-.-•i• .........•i: .-:?' ..:•,':•
....
....
2.
3.
4.
5.
:•
•.
3.
4.
Frequency
(Hz)
Frequency
(Hz)
Event (81360) (11)
,
5.
(b)
Event (84272) (26)
-.-:.:.:-:.:.:-:.
i
2.
3.
4.
5.
,
2.
Frequency
(Hz)
(C)
Event (80104) (3)
Frequency
(Hz)
(d)
Event (83026) (15)
ii•.:.ii•i•!i!i,'....:'
:::'..h.-:':::..'
'..,:,',•,•,:•
' ......
.......
. .•f
......
i
1.
i
2.
i
3.
I
4.
i
J
5.
Frequency
(Hz)
1.
('e)
2.
3.
4.
5.
Frequency
(Hz)
(]9
Fig. 5. Time-frequency
plotsfor six differenteventsshowing
dispersion.Comparethe plot for event15 with the
band-passedtraces in Figure 4.
GUBBINSAND SNIEDER;DISPERSIONOF SUBDUCTEDLITHOSPHERE
Our simplestmodel is that of a high-velocityhorizontal
slab, with thicknessH and wave speedc•, embeddedin an
infinite medium with wave speed co. Both materials have
density p. The acousticcaseis equivalentto a fluid layer
6327
The vertical wave number in the slab, r/x, is given asymptotically by
.
J,
7,")t
2
•7F
d = wH
*/• -
embedded in an infinite fluid medium. The source is placed
(7)
at variousheightswithin the slab at a•= 0, where (m,z) are
whichfollows
from(4) and(23). Thewavelength
forther,th
Cartesian coordinates with a• horizontal and z downward,
overtoneis therefore2H/n: the modesare forcedto fit into
with the top of the layer at z - 0. Positioningof the source
hasa largeeffecton the amplitudeof the response,but we do
the slab.
not wish to discussthis in the context of an acoustic model,
which will not providerealistic amplitudes.The responseis
calculated
at a horizontal
distance L from the source and
Note that frequency appears everywheremultiplied by
H, the slab thickness, so that reducing the slab thicknessis equivalent to consideringa lower frequency. Taking
co- 8 km s-•, c• - 8.4 km s-• and a 5% velocitycontrast
just above the slab.
First considerthe modes of the system. Their frequencies
are derived from the condition that energy radiates away
between the slab and the surroundingmedium, (5) gives
a 1% differencebetweeninfinite frequency•,(= w/2•r) and
•,H • 30 for r, = 1, the fundamental, correspondingto 1 Hz
from both upper and lower interfaceof the slab. The period
equation, derived in the appendix, is
waves and a slab thicknessof 30 kin. The small parameter,
ß
tan•-•7 2= 1- 27
(1)
where a is definedby (20) and 7 by (21), which may be
rewritten
as
•
•
(• -•)
•, is in this case0.14. Equation (6) giveskill - 0.125, showing that the waves decay exponentially along the slab by a
factor of e approximately every eight slab widths, or by a
factor of 64 in a typical slab length of 1000 km (corresponding to angular distance10ø, typical of the eventsin Table 2)
for 1 Hz waves and H - 30 km. The damping reflects the
decay of 1 Hz energy relative to high-frequencyenergy and
is severe.
where c = w/k is the phasespeedof the mode.
Equation (1) determines7 and hencethe phasevelocity
of the mode in terms of wH. We are interested in highkequency modes that propagate like body waves close to
the phasespeedof the fast slab. These modeshave
We require more than 1% dispersionto explain the observations. Equation (5) gives 3% dispersionat 2 Hz, as
requiredby the data, for an 8 km thick slab. For a 5% velocity contrast we have e - 0.32. However, the attenuation
length is now much shorter, closeto 2H, or 16 kin. These
frequenciesare therefore completelyeliminated by passage
alongthe slab;the attenuationfactorat 2 Hz is 7 10-2s in
1000 kin! The asymptotic approximation is not very accurate at this value of e, but a numerical solutionfor the roots
and 7 << 1. Hence solutionsof (1) e•st with, to leading of (1) reinforcesthe conclusionof very heavy attenuation.
order in the small par•eter e,
We concludethat although the uniform slab can produce
the required dispersionif it is thin enough, the attenuation
;
i•=d
=
(3) associatedwith leakageof energy out of the slab is so great
•H
e
=
d
'>>1
that the waves would never be observed.
This result is con-
where• is an integer, the overtonenumber. The phasespeed firmed by the synthetic seismogramsdescribedin the next
section.
is, kom (2),
Our model is a very simple one, and it might be argued
that the slab will retain more energy in the real situation.
c-c• 1+2• alia
(4) However,there is everythingto indicate the contrary: twists
Note that the phasespeedis higher than in either of the two in the slab to a lessideal geometrywould losemore energy;
P-to-S conversions,not accounted for in the model, would
media and is in this limit independentof co.
The group velocity is obtained kom (4) by. setting also drain energy from the P wave.
The only aspect of the model we can adjust in order to
k = •/c, where k is the wave number, •d differentiating
satisfy
the twin constraints of high dispersionand low atwith respectto •. After somemanipulation this gives,again
tenuation is the profile of the slab. We are thereforeforced
to first order in e,
to considera more complicated structure in order to retain
the energy inside the slab.
c•=c• 1 2•aH•
(5)
MODELING
The group velocity is, as expected, slower than the wave
speedin the fastermedium,and it decreases
with decreasing
kequency: the high frequenciestravel the fastest.
The wave number
no attenuation.
is real to this order in e and there is
Second-order terms must be considered in
determiningthe imaginarypart of the wavenumber. They
give
2na=ac• d
THE WAVEFORMS
BY A SIMPLE
REFLECTIVITY
METHOD
Consider a more complex slab structure. The same approach applies in the determination of leaky mode wave
numbers, but the period equation must be solved numerically: we losethe simpleanalytical approachof the previous
section. Furthermore, the frequencyno longer scalessimply
with the slab thickness,although scaleinvariancewith the
total
thickness
of the slab remains.
The
derivation
of the
period equationbecomesrather intricate and specializedto
6328
GUBBINS AND SNIEDER: DISPERSIONOF SUBDUCTED LITHOSPHERE
each slab profile; it is more efficient to use a numerical reflectivity approach.
Consider a stack of N homogeneousplane layers with velocity c,, density p, and thicknessHn. The stack is surrounded on both sidesby a homogeneousspacewith velocity
c0 and density p0. An explosivesourceis presentsomewhere
in the slab, and a radiation condition is imposedoutside the
modelsuntil the syntheticsexhibited the main featuresseen
in the data.
A secondhigh-velocity layer was added to give a staircasestructure as shownin Figure 6. The wavespropagating
in the surroundingmedium arrive at 125 s. Those in the
high-velocity lid and those in the bottom of the slab are vis-
ible as two distinctprecursors(119 s and 122 s). The total
slab. The generatedwavefield is computedusingthe reflec- thicknesswas kept at 60 km and the width of the upper lid,
tivity method [Fuchsand Muller, 1971]. The wavefield is H, wasvaried.The lid wasgivenvelocity8.4 km s-x, the
expressedas a doubleintegral over slowness
p and frequency intermediatelayer8.2 km s-x, and the surrounding
medium
8.0 km s-x. The resultingsyntheticseismograms
areshown
u(r,z,t) -- •
d•
fo
dpAf(a•,p,z)expia•(pr-t)
(8)
in Figure 6. For H < 6 km there is no precursorarriving with the velocity of the lid: the energy is not retained,
as with the case of a uniform
where Af has the form
Af(•o,
p,z)- G
ox
,...,•oHNo,¾,
px
,...,p•,P,z) (9)
A(•oH•
(•Hx•x,...,•H•,p•,...,p•,p)
•d A = 0 is the period equation. In this expression,r is
the dist•ce along the slab (the length of the ray path), z is
the dist•ce from the receiver from the •op of the slab, and
• is given by
I
•
""-(c•-P:)
slab.
For H > 20 km there
is no dispersion:energyof all frequenciesarrives at the fast
speedcorresponding
to the lid. At intermediatewidthsthere
is dispersion,as the time-frequencyplots in Figure 7 show.
The seismograms
havethe appearanceof two distinctpulses
rather than a dispersedwave train. The best compromise
between dispersionand a substantial precursor is provided
by the model with H-10 kin.
A more complicated set of models had 10 layers to approximate a smooth velocity variation acrossthe slab. A
smooth structure is expected from the temperature variation in the subductedplate. The overall width of the fast
(10)zoneis againkeptat 60 km, andthe high-velocity
lid is
In the numerical examplespresentedin •his paper, the slownessintegral in (8) is performedby a s•raightforwardnumerical quadrature and the frequency integral is realized by a
varied in thickness.The lid has velocity8.4 km s-x and
the ramp variesfrom 8.0 to 8.33 km s-x. The resultsare
f•t
is no dispersion,
andmostof the energyarrivesat 8 km s-x ,
Fourier
transform.
The integrand can be derived using propagator matrices
shownin Figure 8. When H = 0 (i.e., there is no lid) there
showingthat a fast lid is neededfor dispersionin combina-
[Gilbert and Backus,1966]. The readeris referredto Aki tion with the thicker, slower,zone to retain the energy. The
and Richar• [1980]for details concerningthe reflectivity precursor,travelingat the lid velocity,is virtually absentup
method and the use of propagators. The excitation of the to and includingH = 6 kin. There is almost no dispersion
•ave field influencesonly the numeratorG in (9), while the for H >_15 km (the energypropagatesat the fast lid speed)
period function • in (9) dependsonly on the material prop- and very little for H > 10 kin. The syntheticswith the most
erties of the medium. Assumingthe frequency• to be real, realistic dispersionare obtained with lid thicknessesof 8-10
the zeroes of • on the real p a•s correspondto trapped
modes, while the zeroes on the unphysical Riemann sheet
ia the complexp plane correspondto leakymodes[Watson,
7•o-layer modelsfor differen•U]ic•ness
H
1972]. Note that the frequencyand the layer •hicknessen8 km/s
Hdkm
ter the integrand in the reflectivity integral (8) only in the
dimensionlesscombinationwH• V•. For the simplestcaseof
a single layer with the same density p • the surrounding
60 km
material •d a sourceat the top of the layer, the wave field
at a distance -z kom the top of the slab is given by
T
u(r,z,t) -- -•e i'/'
Z(p,w)&dp
(11)
8 knVs
where
--
\7rT/
)V'(a],p)pei'"(r"-"ø•-')
(12)
and
r•0sin •or• H + irh cos•or•xH
•(•'P)= •-,, •o•,S- i(,• +,) •i.•,/
(1•)
The period equation(1) followsfrom settingthe denominator in (13) to zero.
Numerical resultsfor a uniform high-velocityslab were as Fig. 6. Synthetic seismogramsfor two-layer modelsof the fast
predicted by the modal analysis in the previoussection: no slab. Total thicknessis 60 km; thicknessof the high-velocitylid
fast arrival could be seenfor a thin slab, and no dispersion H varies. Note the appearanceof three phasescorrespondingto
for a thick slab. We studiedprogressively
more complicated
the three velocities
in the model.
GUBBINSAND SNIEDER:DISPERSION
OF SUBDUCTED
LITHOSPHERE
6329
Two-layer model for H=O km
I
•
I
•
I
•
I
•
I
•
I
ß"'":J••.:.....;
.........
•........
2.
3.
4.
5.
Frequency (Hz)
Two-layer model for H=8 km
Two-layer
model
for
H=6
kmI
I
•
I
•
I
•
I
•
I
•
I • I , I • I , I , Il
I
....
i
i
'
2.
i
'
3.
i
'
4.
i
'
'
1
2.
5.
Two-layer modelfor H= 10 km
•
•
I
•
I
,
I
4.
1
5.
Frequency (Hz)
Frequency (Hz)
I
3.
•
.•:•..•.:..•
I
Two-layer model for H=15 km
I
•
I
•
I
•
I
,
I
•
I
ß
..• •.•
i
i
œ.
$.
•.
2.
3.
4.
5.
Frequency (Hz)
Fig. 7. Time-frequency
plots corresponding
to the synthetics
in Figure6.
t
6330
GUBBINSAND SNIEDER:DISPERSIONOF SUBDUCTEDLITHOSPHERE
.
H
•o
8 4 k•s
•.•s
//-
•
8•s
II-•km
•
tively, a high-velocity lid above a smoothly varying, slowervelocity layer (again faster than the surroundingmantle)
produces the observed dispersionwithout the appearance
of two distinct phases. The smooth model is preferred becauseit matchesthe seismograms
best and is physicallymost
plausible. With better data it might be possibleto relate
persistent "holes"in the time-frequencyplots to layeringin
the subductedslab, but existing data doesnot justify any
further interpretation. A firm result of the theory is the existenceof a very thin high-velocitylayer. The width is constrained by the model to lie between 6 and 15 kin, probably
between 8 and 10 km. The one-dimensionalapproximation
may make this an overestimate, but the acoustic approxi-
H - 10 •m
mation is not believedto affect it sincethe wavelengthsof
acousticand elastic wavesare the samefor a givenvelocity.
The acoustic approximation is unlikely to predict amplitudes accurately and we have avoided drawing conclusions
based on amplitude. However, we believe sourceposition
relative to the slab to be critical in determiningamplitudes
I14
II•
II•
I•0
I•
1•4
I•
I•
in the real Earth, as it is in the acoustic calculations.
Fig. 8. Syntheticseismograms
for rampmodels.The smoothly The same conclusionswould apply to models with the
varyingbaseis represented
by 10 layers.Notethe appearance
of high-velocity "lid" on the bottom rather than the top, but
a uniformlydispersed
wavetrain and the absence
of earlyhigh- it is unlikely that such a high-velocity region should lie at
frequencyenergy for fir = 0 and fir = 6 kin.
the base of the subducted lithosphere; it is more plausible
to have a thin layer on top. The lid has similar thickness
to oceanic crust, much thinner than either the lithosphere
kin. The time-frequencyplots in Figure 9 showdispersion or the width generally assumedfor the fast seismiczone.
rather than separate arrivals.
Subductedgabbroiccrust will have low seismicvelocity;it
The two-layermodelsdo not fit the data well because must have transformed to eclogite in order to explain the
of the appearance
of two distinctphases.However,three- observedhigh-velocity, and we think this is the most likely
dimensionalstructurealongthe slab may smearout these explanation of the high-velocity lid.
H-15
•m ••
Helffrich et al. [1989]have performedtheoreticalcalcuarrivalsandproducea verisimilitude
of dispersion
in thereal
data,sothat it is hardto discriminate
betweenthe two-layer lations, usingthe Birch-Murnaghanlaw and experimental
and ramp models.
CONCLUSIONS
This study of digital data has given simple answersto
the first two questionsgiven in the introduction:the early
signal is very high frequency,up to 8 Hz on this instrument, and it continues to arrive well into the wave train.
The value of 8 Hz is based on an instrument
with
20 times
per secondsampling, with a Nuyquist frequencyof 10 Hz,
and it is quite possiblethat even higher frequencyenergyis
present. The third question, whether there is dispersionor
two distinct phases,cannot be answeredwith the same clarity. The band-pass-filteredseismograms
show1-3% dispersion,which is rather lessthan both the JB residual(5-7%)
data from the laboratory, for the seismicvelocitiesof materials at elevatedpressureand temperature. They conclude
that eclogitecannotproducethe requiredvelocityanomaly:
they attribute only 0.5% to variation in compositionand
1.75% to temperatureeffectsin the slab. The thin top layer
of the slab is expectedto warm up relativelyquicklyand
the temperatureanomalywill be very small at depth. If
thesetheoreticalcalculationsare right, we must seekan alternative explanation for the high-velocitylid. Stressassociatedwith subductionmay causemineral orientationof
olivine in the mantle above the slab, producing anisotropy
ond pulse of low-frequencyenergy (usually at about the JB
time). This secondpulse is not alwayspresent. We conclude the propagation path produces some dispersion,but
that other effectsmay also be present to causemultipathing.
with the fastest direction aligned with the shear and the
slowestdirectionnormalto the slab[McKenzie,1979].This
anisotropywill be difficultto distinguishfrom heterogeneity:
seismicwaveswill be travelinghorizontallynear the deepest pointson the ray and thereforemore slowlyfrom both
the anisotropiceffectand the decayin temperatureanomaly
with depth. The two effectsarisefrom differentregions,one
in the top of the slab and one abovethe slab. Somemore sophisticatedobservation,like shearwavesplitting, is needed
The dispersionis considerablymore severethan that studied
to discriminate between the two. We prefer the eclogite in-
and the time
between
first onset and the arrival
of a sec-
by Vidale [1987]and Vidale and Garcia-Gonzales
[1988]in
terpretationbecauseof the similaritybetweenthe thickness
broad slablike structures and demands a sharper variation
in seismic velocity than they used.
The theory showsthat a simple uniform fast slab cannot
explain the observations because the energy leaks away. A
more complicated structure is needed to retain the energy.
A high-velocity thin lid above a thicker layer of slowervelocity (but still faster than the surroundingmantle) produces
essentially two arrivals, the first being confined to high-
of the lid and that
frequency,depending on the thicknessof the lid. Alterna-
of oceanic
crust.
Dispersion requiring about 200 km of propagation
throughthe fast layer is observedfrom two eventsbelow
300 km (events8 and 20). Their ray pathslie closeto the
slab until the level of the deepestevents,when, presumably,
they pass out the bottom. The lid must thereforepenetrate to at least 500 kin. The dispersioncould be caused
by propagationbeneaththe receiver,but this wouldrequire
energyto reenterthe high-velocitylid after passagethrough
GUBBINSAND SNIEDER:DISPERSION
OF SUBDUCTED
LITHOSPHERE
Ramp model for H-O km
I
•
I
•
I
•
I
•
I
•
6331
I
•.•i!•i•.•..;i!:;•:.j:.......!•:•;::•.•.::•:•::•i.•i.:•;/•.:•...•;•:.•:.:.;.`•:•.:•.
.......................
.....
:::qiii•i:?:?
.....................................
i
i
i
i
i
2.
3.
4.
5.
Frequency (Hz)
Ramp model for H=6 km
I
•
I
•
I
•
I
•
I
,
Ramp model for H=8 km
I
I
•
I
•
I
•
I
•
I
•
I
....................................
............
i
i
i
i
i
2.
3.
4.
5.
I
i
Frequency (Hz)
I
•
I
'
i
'
3.
i
4.
'
i
5.
Frequency (Hz)
Ramp modelfor H=10 km
•
i
2.
•
I
•
I
Ramp
model
for• H=15
km,
I
,
I
•
I
I
•
I
::::•
.:?'-•
•..
??..............
..... . .:
'
2.
3.
4.
'
5.
Frequency (Hz)
i
I
2.
3.
4.
5.
Frequency (Hz)
Fig. 9. Time-frequency
plotsfor the ramp models.Note the evendispersionfor the ramp models.
6332
GUBBINS AND SNIEDER: DISPERSION OF SUBDUCTED LITHOSPHERE
the mantle, which seems unlikely. This also means the lid
must be continuous, without any breaks or faults, for the
dispersionto be observedfor so many events.
Our model is rather different from that of tIuppert and
sent a 10 km fast layer nor, in some cases,a 60 km thick
lithosphere.It is generallyhoped that a coarseparameterisation, combinedwith a restriction to 1 Hz frequency,eliminates the effects of small-scale structure, but there is no
Frohlich[1981],who studiedarrivalsfrom stationsAFI and guarantee of this, particularly when ISC data is used withRAO, which lie above the Tonga-Kermadecslab, and pre- out examining the original seismogramsfor their frequency
ferred a fast lower regionand thick slowerlid, as expected content. The presenceof thin slabswill alsogive the appearfrom the temperature anomaly. It is possibletheir data ance of anisotropy,with rays traveling in the slab yielding
could be explained by a model similar to ours, with the fast faster times than those traveling acrossthe slab. Coarse
lid on top, but it seemsiraplausible to have the very thin lid samplingof sucha structurecan easily(and erroneously)be
interpreted as evidencefor anisotropy.
required by our data on the underside.
Converted phases require a sharp interface on the slab
surface[Fukao et al., 1978; Matsusawaet al., 1986; Nakan- APPENDIX: DERIVATION OF THE PERIOD EQUATION FOR A
HIGH-VELOCITY
SLAB
ishi et al., 1981], and it is difficult to explain sucha sharp
contrast by temperature alone, becausesharp temperature
Consider the elastic fluid medium described in section 3.
changeswould diffuseaway relatively quickly [Sleep,1973;
The verticaldisplacementobeysthe waveequation[Kennett,
Helffrich et al., 1989]. A basalticlayer wouldform a waveg1983]. Considera planewavewith frequencyw and horizonuide and propagate high-frequencyenergy at low seismic
tal wave number k. The vertical slownessin the slab, r/•, is
velocity. Such arrivals would be late and therefore difficult
to detect in the seismogramsstudied in this paper, although
the absenceof a precursorfor the closesteventsin Table 2
suggeststhe fast path is absentat shallowdepthscloseto the
receiver. Comparing residualsand deepestpoints on the ray
paths in Table 2 showsthat clear precursorsaxe produced
by structure at and 80 kin. If the fast zone is eclogite the
transformationfrom gabbro thereforeoccur above 80 kin.
This compareswell with the resultsof Hori et al. [1985],
who find evidencefor a low-velocity channel, interpreted as
subducted basaltic crust, to a depth of at least 50-60 kin,
basedon P-S convertedphases,but is considerablyshallower
defined by
V•= c• w2
(14)
with a similar equation for the slownessoutside the
slab, r/0. In two dimensions
(x,z) the verticalmotion
wexp[-i (wt - kx)] satisfies
the equation
0 2w
Oz 2
+ V2a•aw
=0
(15)
with solutionsproportionalto exp [+ir/wz].
Let the slab occupythe regionbetweenz = 0 and z = H.
We expectleaky modesin the high-velocityslaband representthem with real frequency,allowingthe verticalslowness
to be complex. The matching conditionsat the interfaces
z = 0 and z = H axe continuityof vertical displacement,
than depthsfoundby Nakanishiet al. [1981]and Matsuzawa
et al. [1986]for a low-velocitychannelin the Japanesearc.
Either the convertedphasessamplea differentdiscontinuity
or the gabbro-eclogite
transitionoccursat a differentdepth
beneathJapan. To investigatethe Tonga-Kermadec
struc- w, and pressure.The latter conditionimpliescontinuity
ture furtherwerequireinstruments
situatedabovethe slab of (p/r/a) OwlOz. The periodequationfor leakymodesin
in order to detect a waveguideeffect;suchan experiment
in New Zealandmight well be effectivein determiningthe
depth to the transition.
Our resultshave consequences
for tomographicstudies
the slabis obtainedby imposingradiationconditionson the
upper and lowersurfaces:solutionsin the infinite medium
must take the form of decayingwavestravelingaway from
the slab. Applyingcontinuityof w and pressureand elimi-
and waveform studies based on ray theory of Gaussian hating w0 givesthe conditions
beams. Strong variations of the structure acrossthe slab
cause
the
breakdown
ofray
theory
and
related
asymptotic
methods.Conclusions
drawnabout the structureof the slab
Ow•
=+iw
r/---•2w•
Oz
(16)
from
ray-geometric
methods,
including
some
based
onGauswhere
theplus
sign
ischosen
atz - H andtheminus
sign
sianbeams,
maytherefore
bewrong.Theresults
ofHa atz = 0,sothewaves
decay
away
fromtheslab.
[1978],
using
Gaussian
beam
theory,
maypredict
theam- Thegeneral
solution
intheslab
may
bewritten
as
plitude of the arrival accurately but do not predict the fre-
quencydependence.However, Cormier's [1989]method is
wl = A coswr/lz+ B sinwr/lz
(17)
capable of predicting dispersion.
wherethe constants
A and B may be complex.DifferenThere are also implications for tomographic inversions.
The residuals
for these events at New Zealand
stations
are
very large and will be omitted from any tomographicstudy,
most of which remove "outliers"
of more than a few seconds.
tiating and substituting
into the two matchingconditions
expressed
in (16) givesthe periodequationfor the slowness
i (r/•!r/o) sinwr/•H+ i (r/,!r/o) coswr/xH
Many tomographic studies also use station correctionsand
X i(r/•/r/o)sin•or/•H-cos•or/•H
- 0 (iS)
New Zealand sites have large correctionsfor azimuths along
which reduces to
the Tonga-Kermadec trench. There is therefore a danger
that slab effects will be mapped elsewherein the mantle:
2ir/0r/x
(19)
the information on deep structure comesmainly from deep
events, all of which reside in subduction zones, all of which
We define the reducedwave speedd by
have anomalous structure. Tomographic studies, such as
tanwr/•
H - - r/0•
+ r/•.
those by Creagerand Jordan [1984]usinga coarseparameterization,or even thoseof $pakman[1988],cannotrepre-
1
1
=
1
d
(20)
GUBBINSAND SNIEDER:DISPERSIONOF SUBDUCTEDLITHOSPHERE
and the dimensionless
parameter7 to replacethe (complex)
phase speed as
w2
so the vertical
c•
wave numbers
r/0-
c•
become
1d•'
6333
Giardini, D., and 3. H. Woodhouse, Deep seismicity and modes
of deformation in Tonga subduction zone, Nature, 307, 505509, 1984.
Gilbert, F., and G. E. Backus, Propagator matrices in elastic wave
and vibration problems, Geophysics,31,326-332, 1966.
Ha, J., A parabolic propagation model for the propagation of
precursory signals through the subducted lithosphere, MSc
thesis, Victoria Univ., Wellington, New Zealand, 1978.
Helffrich, G. R.., S. Stein, and B. J. Wood, Subduction zone thermal structure and mineralogy and their relationship to seis-
mic wave reflectionsand conversionsat the slab/mantle in-
and
When expressedin terms of 7 (19) givesthe form of the
period equation(1) usedin section4.
terface. J. Geophys. Res., 96, 753-763, 1989.
Hori, S., H. Inoue, Y. Fukao, and M. Ukawa, Seismicdetection of
the untransformed "basaltic" oceanic crust subducting into
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Acknowledgments.D.G. thanksthe Universityof Utrecht for
342 pp., Cambridge University Press, New York, 1983.
support when this work was started. The work was partially
supportedby NEP•C grant GP•3/7488. We are grateful to Nico Matsuzawa, T., N. Umino, A. Hasegawa,and A. Takagi, UpVlaar for numerous discussionsand for making us aware of the
possiblerole of the eclogitelayer.
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