I have written this somewhat simplified account
after many requests from colleagues and students to explain
what the series of articles, *Comments* and *Replies*
in Geophysical Journal is about. It is not intended
to replace that discussion, which should be the ultimate
source of information. We ourselves have
struggled to understand some of the issues raised by
our colleagues, and to clarify our own writings in a more
accessible language - we are
aware that the original literature is not easily accessible.

For more than a hundred years, seismologists have treated
seismic P and S waves as if they satisfy the same laws as
optical rays (*ray theory).* This approach has been extraordinarily
succesful: it enabled Beno Gutenberg to determine the radius of the
Earth's core in 1910, Inge Lehmann to discover the tiny inner core in
1936, and Harold Jeffreys and Keith Bullen to derive a
spherical Earth model that could accurately predict seismic travel
times and be used to locate earthquakes by reading the times of
P and S waves from seismograms located far away.

But even after those gigantic accomplishments, ray theory continued to help seismologists to make fundamental discoveries. In the 1970's, it had become evident that a spherical Earth model was not sufficient: some regions on Earth are slower or faster than others, basically because they are warmer or cooler. Researchers at MIT and Harvard, led by Keiti Aki and Adam Dziewonski, pioneered the technique of seismic tomography. This gave us the first tangible evidence that the deep seismicity near the ocean's trenches is actually caused by oceanic lithosphere sinking back in Earth's mantle. In 1990, an undergraduate at Utrecht, Suzan van der Lee - now at Northwestern University - came up with the first image of a slab (the Aegean) sinking in the lower mantle for her senior thesis project; because this image was counter the prevalent opinion that the less dense upper mantle did not mix with heavier lower mantle rock, it took a while to double-check it with her advisors and get into print [1].

At the same
time Rob van der Hilst - also at Utrecht, now at MIT - came up with
extensive evidence that the slabs around the Pacific were able to
sink into the lower mantle [2]. At the same
time, these findings seemed to make
minced meat of the so-called *two layered* convection model.
This model
states that Earth cools off with little or no
any exchange of mass between upper
and lower mantle. It was succesful in explaining
geochemical observations.
These required a hidden *reservoir* where
argon and helium could be held since Earth's formation
without escaping into
the atmosphere, and the lower mantle was the obvious candidate for
that as long as its rock stayed where it was: deep.

But after all these successes, limitations
of *ray theory* became
apparent. Wavelengths of seismic waves often measure in the
hundreds of kilometers, and this makes it difficult to see
small objects with seismic waves. This is very similar to the
limitations of the optical microscope which does not allow
us to see beyond a very small resolution, given by the Fresnel
zone. If we treat seismic waves like optical
rays, they must also have Fresnel zones. Because seismic
rays curve back to the surface of Earth, these zones
take on the shape of bananas, as is shown in the
following figure:

Figure 1. This banana-shaped curve shows the Fresnel zones for seismic waves: regions that influence a seismic P wave inside Earth. The outer curve is for a P wave with a dominant period of 4 sec, the innermost one for a period of 1 second. Objects smaller than the Fresnel zone cannot be seen using ray theory (Source: G. Nolet, Imaging the deep earth: technical possibilities and theoretical limitations, Proc. XXIIth Assembly ESC Barcelona 1990, ed. A. Roca, 107-115, 1991)

The simple idea of a Fresnel zone gave us an idea
of what we could resolve in the limit of *ray theory*, but it
gave us no way to improve on it. But independently
from the work on P and S waves, seismologists working with
very low-frequency waves (*surface waves* and *normal
modes*) had already figured out how complicated the actual
sensitivity is of seismic waves. Figure 2 shows
that the sensitivity for a normal mode is also spread out
over the surface of the Earth in a band-like structure (though
it looks more like a caterpillar than a banana).

Figure 2. At one (very low) frequency the sensitivity of a a normal mode to Earth's structure is band-like (Source: J. Woodhouse and T.P. Girnus, Geophys. J. Roy. astr. Soc., 68, 653-673, 1982).

In the mid 1980's, Roel Snieder at Utrecht (now at Colorado School
of Mines) was the first to derive
similiar sensitivities for surface waves
at much higher frequencies. He also
attempted to interpret perturbations in such seismic
waveforms tomographically,
using a first order perturbation theory.
But it was not until about ten years later that it became evident
that one could get a Fresnel zone-like *sensitivity kernel*
by summing over many of such surface wave modes, as shown in
Figure 3.

Figure 3. Summing normal modes yields a banana-like sensitivity for S waves (Source: X.-D. Li and T. Tanimoto, Geophys. J. Int., 112, 92-102, 1993)

These kernels still look very much like the Fresnel zone sketched
in Figure 1.
The *doughnut hole* made its appearance in the theory when
postdoc Henk Marquering at Princeton started to look at the
sensitivity of *arrival times* of seismic S waves, using
techniques very similar to those used earlier by Snieder
and by Li and Tanimoto.

Marquering's results were initially baffling and seemed to
contradict every seismology textbook published in the 20th century:
the location of the ray itself was shown to be
a region of *zero*
sensitivity for the travel time of the wave! In striking contrast,
*Ray theory* defines the ray as the *only*
region where the travel time is sensitive to Earth's structure,
and the two theories are in obvious conflict to each other.
But after grappling for a while with this paradox we understood
it (an explanation is provided in [18]), and extensive numerical
tests soon proved that it was, in fact, *ray theory* that
is deficient, not the new *finite frequency* theory.
Princeton's Tony Dahlen soon developed a very efficient, though
approximate, way to compute the sensitivity kernels for travel
time (named *banana-doughnut kernels* by Marquering because
the region of zero sentitivity creates a doughnut hole at the center
of the banana). Figure 4 shows two such a kernels,
computed with Dahlen's theory [3].

Figure 4. Sections from banana-doughnut kernels as seen from aside and across for a P wave at an epicentral distance of 60 degrees. For a dominant period of 20 sec, the doughnut hole is massive. For a shorter period (2 sec) the hole is much narrower and only clearly visible in the white line that denotes the amplitude of sensitivity across the midpoint of the kernel (Source: Dahlen et al., Geophys. J. Int., 141, 157-174, 2000).

The concept that travel times can not be influenced
by Earth structures located *on the ray itself*
is counterintuitive, and has met with resistance of
two kinds: many - even very esteemed - seismologists
initially expressed
disbelief, but that hesitancy
usually disappeared when we showed
how well the theory worked when tested on purely
synthetic seismograms computed with
the pseudospectral method (see the publications of
Shu-Huei Hung and Adam Baig in Geophys. J. Int. [4-8]). The
second resistance was of the *it doesn't make a
difference in practice* type.
This ignores that a short period delay, e.g. as reported by the ISC,
has a very narrow sensitivity kernel, whereas a long period delay
can have a sensitivity kernel with a width of 1000 km or more. If a delay
is visible in thre ISC data set but not in the long period data set,
this tells us something about the size of the heterogeneity. Montelli
et al. [24] used this succesfully to image lower mantle plumes.
Since then,
tomographic studies by several authors [21-23] have shown that one
can exploit the frequency-dependence of the sensitivity kernels
even further by using up to seven filter bands
and significantly increase resolution.
Indeed, the
kernels soon turned out to make a first decisive contribution
to the resolution of small features when Princeton
graduate student Raffaella Montelli started to test the
new theory on actual data and - to her own surprise -
discovered more than a dozen
plumes (hot uprisings under islands like Hawaii) in the
lower mantle of Earth [24]. By coincidence, this discovery
came at a time that the plume hypothesis itself was
under fire. I suspect that this has contributed to fuel
the debate about the banana-doughnut kernels (paradoxically,
plume opponents and I may soon find common ground since
analysis of the lower mantle plumes seems to raise
significant problems for the model of *whole mantle
convection* and bring us back to a very limited mass
exchange between upper- and lower mantle).

So the kernels do make a difference!
Not everyone agrees, however. Martijn de Hoop (now Purdue)
and Rob van der Hilst wrote a paper in Geophysical Journal [4]
in which they claim that though
"the sensitivity being identical zero on the unperturbed
source-receiver ray", that "the kernel itself does not have a
zero on the unperturbed ray". The discussion is very much about
mathematical niceties that an ordinary seismologist (such as me)
would not ever wish to worry about, but in the end this
is nonsense in our view, which we have detailed in
Dahlen, F.A. and G. Nolet, Geophys. J. Int. 163, 949-951, 2005 (click
here:)
.

One particular aspect of
the paper by de Hoop and van der Hilst may lead
readers astray and
for a proper understanding of finite frequency theory
we explore it in some detail.
They claim that one can *mollify* kernels to make
the zero sensitivity disappear. They state that this *mollifying*
happens implicitly when the matched filter that
is used in the measurement of the time delay is not exactly
equal to the observed P or S pulse.
Or when it is time-shifted, e.g. as
a consequence of an error in the earthquake's origin time.

First of all, this view reflects a basic misconception of
the role of
banana-doughnut kernels: they are *designed* to interpret the
difference between the matched and observed waveforms!
This has nothing to do with
the zero sensitivity on the ray. The argument
that origin time errors can annihilate the zero sensitivity
is correct, but * only if the tomographer allows origin time
bias to propagate into the model * - something I teach my
students to avoid at all cost.
The rather complicated mathematical arguments
can be paraphrased as follows:

Assume we have a tomographic system of equations with a model
**m**, data **d** and time errors **e** and
a matrix (with the banana-doughnut
kernels in some discretized version)
**A**:

** A m = d + e **

De Hoop and van der Hilst map the error **e** back
into the model by a transformation **e=Em**, and rephrase
the system as:

** (A-E) m = d **

to conclude that when one reads back the kernels in the new
*mollified*
matrix ** (A-E)**, these kernels have no zeroes on the ray.
Magic? Well,
this is correct in so far as one could
actually do the mapping **e=Em**. Which is a problem unless
the errors can be adequately modeled by **m** and most
importantly if
one actually knows the errors **e** - but why then not
subtract the error from the data? If the errors, on the other
hand, represent a systematic shift (e.g. because all origin times
are estimated too early) no one
would do tomography this way, because it maps the bias
explicitly into the model. In practice, tomographers center the
distribution of delay times around 0 just to avoid that the
model is affected by such bias.
This point waas not adequately addressed by de Hoop and van der Hilst
in their Reply [11].

The second attack by van der Hilst and de Hoop is in another
paper published in Geophysical Journal [12]. The argument is mainly
that the beneficial effects of banana-doughnut kernels are
limited to a small
magnification of amplitude anomalies that could easily have been
accomplished by reducing the *damping* (or *regularization*)
of the inversion that is at the basis of every tomographic
interpretation. We disagree with the authors that there is that
much freedom in choosing one's damping strategy because data errors
are known quite well and the misfit of the data
(or * chi-square* in the language of statistics) cannot be
too large. We reject the examples shown by the authors to
illustrate their point of view because they are selected for cases
where the Fresnel zone is narrow (near the surface) or where
the anomalies are much larger than the Fresnel zone (slabs). We
also reject their statistical analysis because it fails to
take into account that wavefront healing has both positive and
negative signs - resulting in a null result when averaging over
all effects, as the authors do. But these papers are much easier to
understand and we refer the interested reader to [12] and a
preprint of our response (click on Machiavelli):

As a footnote, there is also a paper that questions the
beneficial effects of finite frequency inversions for surface
wave phases [14]. Such kernels have been developed by Ying
Zhou (now at Virgina Tech) [15-17]. In contrast to Zhou,
Anne Sieminski does not invert for 3D structure of the Earth,
but attempts to retrieve a 2D map of *phase velocities*.
This approach has two limitations: first of all, inverting
data for one frequency only ignores the beneficial effects of
the different widths of sensitivity kernels at different
frequencies (which is correlated to depth of the sensitivity,
but that is no reason not to exploit it). But most importantly
there is again a conceptual misunderstanding: the concept
of *local phase velocity* implies the validity of
*ray theory* for surface waves! If wavefronts are deformed
by heterogeneities off the geometrical raypath, their local
speed is determined by their past history, and not unique.
Ying Zhou [15] has shown how large the approximations are
that one has to make to adopt a local velocity, and that
these approximations usually annihilate the beneficial effects
of finite frequency interpretations. If the path coverage is
dense, and the kernel width need not be exploited to get a good
resolution, even the local velocity approach - despite its
significant shortcomings - can yield a phenomenal increase
in resolution [19].

The original paper to study finite-frequency tomography for body
waves is
(Dahlen et al., GJI, 2000) [3]

However, many may find this a hard nut to crack. A simplified
derivation of the same results - together with a method to
compute kernels in local, 3D structures, can be found in [18]
(click here:)

[1] Spakman, W., van der Lee, S., & van der Hilst, R.,
Travel-time tomography of the European-Mediterranean mantle down to 1400 km,
Phys. Earth Plan. Int., 79, 3-74, 1993.

[2] van der Hilst, R., Engdahl E.R., Spakman W. & Nolet, G., Tomographic imaging of subducted lithosphere below northwest Pacific island arcs, Nature, 353, 37-43, 1991,

[3] Dahlen, F.A., Hung, S.-H. & Nolet, G., 2000. Fr'echet kernels for finite-frequency
travel times - I. Theory, Geophys. J. Int., 141, 157-174.

[4] Baig, A. & Dahlen, F., 2004. Statistics of traveltimes and amplitudes in random media, Geophys. J. Int., 158, 187-210.

[5] Baig, A., Dahlen, F., & Hung, S.-H., 2003. Traveltimes of waves in
three-dimensional random media, Geophys. J. Int., 153,
467-482.

[6] Hung, S.-H., Dahlen, F., & Nolet, G., 2000. Frechet kernels for
finite-frequency travel times - II. examples, Geophys. J.
Int., 141, 175-203.

[7] Hung, S.-H., Dahlen, F., & Nolet, G., 2001. Wavefront healing: a
banana-doughnut perspective, Geophys. J. Int., 146,
289-312.

[8] Baig, A. & Dahlen, F., 2004. Traveltime biases in random media and the s-wave discrepancy, Geophys. J. Int., 158, 922-938.

[9] de Hoop, M. & van der Hilst, R., 2005. On sensitivity kernels for wave equation transmission tomography, Geophys. J. Int., 160, 621-633.

[10] Dahlen, F. & Nolet, G., 2005. Comment on the paper on sensitivity kernels for wave equation transmission tomography by de Hoop and van der Hilst, Geophys. J. Int., 163, 949-951.

[11] de Hoop, M. & van der Hilst, R., 2005. Reply to a comment by F.A. Dahlen and G. Nolet on: On sensitivity kernels for wave equation tomography, Geophys. J. Int., 163, 952-955.

[12] van der Hilst, R. & de Hoop, M., 2005. Banana-doughnut kernels and mantle tomography, Geophys. J. Int., 163, 956-961.

[14] Sieminski, A., L've^eque, J.-J., & Debayle, E., 2004. Can finite-frequency effects be accounted for in ray theory surface wave tomography, Geophys. Res. Lett., 31, doi:10.029/2004GL021402.

[15] Zhou, Y., Dahlen, F., & Nolet, G., 2004. Three-dimensional sensitivity kernels for surface wave observables, Geophys. J. Int., 158, 142-168.

[16] Zhou, Y., Dahlen, F., Nolet, G., & Laske, G., 2005. Finite-frequency effects in global surface wave tomography, Geophys. J. Int., 163, 1087-1111.

[17] Zhou, Y., Nolet, G., Dahlen, F., & Laske, G., 2005. Global upper mantle structure from finite-frequency surface-wave tomography, J. Geophys. Res., in press.~

[18] Nolet, G., Dahlen, F., & R.Montelli, 2005. Traveltimes and amplitudes of seismic waves: a re-assessment, in: A. Levander and G. Nolet (eds.), Array analysis of broadband seismograms, AGU Monograph Ser., 157, 37-48.

[19] T. Yang and D. Forsyth,
Regional tomographic inversion of the amplitude and phase of Rayleigh waves with 2-D sensitivity kernels, Geophys. J. Int., in press, 2006.

[20] Nolet, G. and R. Montelli, Optimum parameterization of tomographic models, Geophys. J. Int., 161, 365-372, 2005.

[21] Yang, T., Y. Shen, S. van der Lee, S.C. Solomon and S.-H. Hung, Upper mantle beneath the Azores hotspot from finite-frequency seismic tomography, EPSL , 250, 11-26, 2006

[22] S.-H. Hung, Y. Shen and L.-Y. Chiao, Imaging seismic velocity structure beneath the Iceland hotspot: a finite frequency approach, JGR , 109, B08305, 2004

[23] K. Sigloch, N. McQuarrie and G. Nolet, Two-stage subduction history under North America inferred from multiple-frequency tomography, Nature Geosci., 1, 458-462, 2008

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