

Bananadoughnut
tomography – can it reveal plumes (better
than conventional ray theory)? 

2006 
Montelli
et al. (2006b) publish an Swave
finitefrequency global mantle tomography
model claiming to confirm the existence
of deep mantle plumes below a large number
of postulated hotspots. They propose deep
mantle plumes beneath Ascension, Azores,
Canary, Cape Verde, Cook Island, Crozet,
Easter, Kerguelen, Hawaii, Samoa and Tahiti.
They infer plumes beneath Afar, the Atlantic
Ridge, Bouvet (Shona), Cocos/Keeling, Louisville
and Reunion originating below the upper
mantle, and midmantle plumes beneath Bowie,
Hainan, Eastern Australia and Juan Fernandez.
They infer that plumes beneath Eifel and
the Seychelles are unambiguously confined
to the upper mantle. None have plume heads
which they interpret as suggesting a weak
temperaturedependent viscosity for lower
mantle minerals, and/or compositional variations.
Disagreements between the P and
Swave images cast doubt on their
preferred depth extent of the lowwavespeed
anomaly beneath Iceland. 
2006 
van
der Hilst & de Hoop (2006) reply
to Montelli
et al. (2006a) reiterating that the
higheramplitude anomalies obtained in the
finitefrequency inversion of Montelli
et al. (2006a) occurs mainly for small,
weak anomalies, whose resolution by long
period data has not been demonstrated. They
assert that the models with and without
banana doughnut kernels are statistically
similar. In the absence of a superior model,
comparisons of their and the Princeton models
do not provide insight into the efficacy
of the finitefrequency method in improving
the present data set, nor the accuracy of
the resulting models. The new Swave
model of Montelli
et al. (2006b) disagrees in detail with
the Pwave model of Montelli
et al. (2004a,b),
casting doubt on aspects of both (the P
and Swave models only support
plumes if one attributes the many differences
between them to poor resolution) and emphasising
that direct conversion of seismic wave speeds
to temperature is unsafe. 
2006 
Trampert
& Spetzler (2006) compare surface
wave tomography models obtained using finitefrequency
kernels and ray theory and show that models
from finitefrequency and raytheoretical
inversions are statistically similar. They
concluded that finitefrequency theory is
a better forward theory to represent the wavefield,
but the associated inverse problem is equally
ill posed as for ray theory, and that the
solution is dominated by the chosen regularization.
Resolution of the order of the Fresnel zone
or better is thus not achieved and the better
finitefrequency theory did not impart any
benefits. 
2006 
Boschi
et al. (2006) point out that:
 in
the inversions of Montelli
et al. (2004b), ray theory achieves
a better data fit than finitefrequency
theory for given model complexity, but
 comparisons
between ray and finitefrequency theory
solutions for the same inverse problem
might require a more sophisticated statistical
analysis. In particular, a question
that needs to be addressed is how to
regularize equivalently ray and finitefrequency
theoretical inversions. Application
of different theories results in different
distribution of energy in the tomographic
matrix to be inverted, and this requires
different regularization schemes and
sizes of the damping parameters.

2005 
Montelli
et al. (2005) comment on the research
note by van
der Hilst & de Hoop, 2005. They
state that the claim that finitefrequency
inversion does not result in measurable
improvements in tomographic images is
misguided, and whereas unmodelled finitefrequency
effects in crustal corrections may account
for slow anomalies of up to 0.3% beneath
very small island stations, those effects
are negligible for larger islands such
as Reunion and Kerguelen where plumes
remain the most probable explanation for
the observed low velocities.

2005 
In a reply, de
Hoop & van der Hilst (2005) point
out that Dahlen
& Nolet (2005) are mistaken in
thinking the primary concern of van
der Hilst & de Hoop (2005) is
the effect of uncertainty in the earthquake
source signature and origin time. They
reiterate that their concerns that, on
the basis of several arguments, the notion
of a "doughnut hole" in the
sensitivity field of a ray is irrelevant
and, given the need for arbitrary damping,
ray theory and finitefrequency theory
are likely to yield results that are practically
the same.
In order to illustrate their concerns
with the results of Montelli et al. (2004a,b)
in a way that is accessible to the general
reader, van der Hilst and de Hoop publish
a Research Note containing illustrative
material (van
der Hilst & de Hoop, 2005). They
claim that the effect of "bananadoughnut"
kernels on the pattern and amplitude of
seismic anomalies is smaller than that
of factors such as the level of damping.
They present illustrations that show that
subducting slabs are not resolved in the
upper mantle in the model of Montelli
et al. (2004a,b),
that lowwavespeed anomalies in the upper
mantle beneath the Indian ocean are essentially
the same in both inversions using ray
tracing and inversions using "bananadoughnut"
kernels, and that these anomalies correlate
onetoone with the locations of island
seismic stations. They argue that the
claim that bananadoughnut theory reveals
30 – 50% higher amplitudes than
ray theory for the plumelike lowvelocity
bodies, which feature prominently in the
reports by Montelli et al. (2004a,b),
is not justified. They conclude
that the use of "bananadoughnut"
kernels has not produced results that
are significantly better than the conventional
ray theory that has been used for many
years.

2005 
Julian
(2005) computes bananadoughnut kernals
for nearverticalincidence multiple ScS
phases at Hawaii and shows that the differential
times are too insensitive to detect a narrow
plume in the upper mantle. He also shows that
the locations of lowwavespeed bodies reported
by Montelli et al. (2004a,b),
and interpreted as plumes correlate with the
data distribution in the otherwise poorly
sampled oceans, and therefore are probably
artifacts. 
2005 
In
a comment, Dahlen
& Nolet (2005) reject the criticisms
by de
Hoop & van der Hilst (2005). They
counter, for example, that the existence
of caustics in a strongly heterogeneous
medium is irrelevant because the sensitivity
kernels developed by Dahlen
et al. (2000) fully account for
PP caustics. They reject the criticism
that their sensitivity kernels are inadequate
and do not properly account for uncertainties
in the unperturbed source pulse. They reiterate
their claim that finitefrequency ("bananadoughnut")
kernels are an appropriate tool for inverting
the dataset they use, which comprises the
traveltimes of nontriplicated P,
S, PP and SS
waves, measured by crosscorrrelation with
a synthetic pulse. 
2005 
de
Hoop & van der Hilst (2005)
study the results of Dahlen
et al. (2000) and claim that finitefrequency
seismic waves can have zero amplitude in
the case of an unperturbed ray in a simple
or quasihomogeneous medium (where there
are no caustics) and if the exact source
time function is known and used. However,
these conditions are not in general met
in the case of real data and heterogeneous
media. In real cases, it is thus unknown
where the oscillatory kernels have zero
values, which limits their use in tomography.
Their view is that the influence of the
bananadoughnut kernels has been overstated
and the differences between ray theory and
finite frequency theory overinterpreted.
The images suggested to represent
plumes thus have little to do with the use
of new theory. Some of the plumelike features
suffer from wellknown resolution problems
and some in the upper mantle may not be
as well imaged as Montelli et al. (2004 a, b)
claim. Much of the deeper signal may be
robust, however. 
2004 
Montelli
et al. (2004b) report their results
in more detail, presenting a comparison
of geometricray and finitefrequency traveltime
tomography for 86,405 longperiod P
and PP–P traveltimes measured
by crosscorrelation.They report that, depending
on depth and size, the amplitudes of the
velocity perturbations in the finitefrequency
tomographic images are 30 – 50% larger
than in the corresponding geometricray
images. Their results show plumelike anomalies
beneath Ascension, Azores, the Canary Islands,
Easter, Tahiti, Hawaii, Bouvet, Kerguelen,
Cape Verde, Tibesti, Kilimanjaro and Galapagos 
2004 
Montelli
et al. (2004a) (see also Supporting
Material) utilise the theory of Dahlen
et al. (2000) in global tomography.
They apply the theory to 88,739 longperiod
wave travel times measured using crosscorrelation
and invert them together with 1,496,025
shortperiod travel times. They report
that the amplitudes of deep, smallscale
velocity heterogeneities are underestimated
by 30 – 60% when using geometric
ray theory. They further report that the
results reveal the existence of six wellresolved
plumes extending into the lowermost mantle
beneath Ascension, Azores, the Canary
Islands, Easter, Samoa, and Tahiti, a
less wellresolved plume beneath Hawaii,
and additional plumes that are mostly
confined to the upper mantle.

2000 
Dahlen
et al. (2000) publish a paper showing
that for finitefrequency seismic traveltimes
measured by crosscorrelation of broadband
waveforms with spherical Earth synthetic
seismograms, the traveltime measurement
is only sensitive to the wave speed in a
hollow bananashaped region surrounding
the geometric ray. The sensitivity is zero
exactly along the ray itself. This suggests
that global tomography inversions that use
ray theory need to be recomputed to map
correctly the volumes sampled by the seismic
waves used. 
See
also Seismology:
The hunt for plumes. 

References

Boschi,
L., T.W. Becker, G. Soldati and A.M. Dziewonski,
On the relevance of Born theory in global
seismic tomography, Geophys. Res. Lett.,
33, , L06302, doi:10.1029/2005GL025063,
2006.





Julian,
B.R., 2005, What can seismology say about hot
spots?, in Foulger, G.R., J.H. Natland, D.C.
Presnall and D.L. Anderson, ed., Plates,
Plumes, and Paradigms, Special Paper 388,
Geological Society of America, p. 155170.

Montelli,
R., Nolet, G., Dahlen, F.A., Masters, G.,
Engdahl, E.R. & Hung, S.H., 2004a. Finite
frequency tomography reveals a variety of
plumes in the mantle, Science, 303,
338–343.

Montelli,
R., Nolet, G., Masters, G., Dahlen, F. &
Hung, S.H., 2004b. Global P and
PP traveltime tomography: rays versus
waves, Geophys. J. Int., 158,
637–654.


Montelli,
R., Nolet, G., Dahlen, F.& Masters, G.,
2006b, A catalogue of deep mantle plumes:
new results from finitefrequency tomography,
Geochem. Geophys. Geosys., in press.




last updated
12th December, 2008 


