Roadmap | The review process | Home
   Banana doughnuts
Banana-doughnut tomography – can it reveal plumes (better than conventional ray theory)?
Montelli et al. (2006b) publish an S-wave finite-frequency 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 mid-mantle 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 temperature-dependent viscosity for lower mantle minerals, and/or compositional variations. Disagreements between the P- and S-wave images cast doubt on their preferred depth extent of the low-wave-speed anomaly beneath Iceland.

van der Hilst & de Hoop (2006) reply to Montelli et al. (2006a) reiterating that the higher-amplitude anomalies obtained in the finite-frequency 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 finite-frequency method in improving the present data set, nor the accuracy of the resulting models. The new S-wave model of Montelli et al. (2006b) disagrees in detail with the P-wave model of Montelli et al. (2004a,b), casting doubt on aspects of both (the P- and S-wave 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 finite-frequency kernels and ray theory and show that models from finite-frequency and ray-theoretical inversions are statistically similar. They concluded that finite-frequency 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 finite-frequency theory did not impart any benefits.

Boschi et al. (2006) point out that:

  1. in the inversions of Montelli et al. (2004b), ray theory achieves a better data fit than finite-frequency theory for given model complexity, but
  2. comparisons between ray- and finite-frequency 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 finite-frequency 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.

Montelli et al. (2005) comment on the research note by van der Hilst & de Hoop, 2005. They state that the claim that finite-frequency inversion does not result in measurable improvements in tomographic images is misguided, and whereas unmodelled finite-frequency 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.


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 finite-frequency 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 "banana-doughnut" 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 low-wave-speed anomalies in the upper mantle beneath the Indian ocean are essentially the same in both inversions using ray tracing and inversions using "banana-doughnut" kernels, and that these anomalies correlate one-to-one with the locations of island seismic stations. They argue that the claim that banana-doughnut theory reveals 30 – 50% higher amplitudes than ray theory for the plume-like low-velocity bodies, which feature prominently in the reports by Montelli et al. (2004a,b),  is not justified. They conclude that the use of "banana-doughnut" 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 banana-doughnut kernals for near-vertical-incidence 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 low-wave-speed 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.
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 finite-frequency ("banana-doughnut") kernels are an appropriate tool for inverting the dataset they use, which comprises the traveltimes of non-triplicated P, S, PP and SS waves, measured by cross-corrrelation with a synthetic pulse.
de Hoop & van der Hilst (2005) study the results of Dahlen et al. (2000) and claim that finite-frequency seismic waves can have zero amplitude in the case of an unperturbed ray in a simple or quasi-homogeneous 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 banana-doughnut 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 plume-like features suffer from well-known resolution problems and some in the upper mantle may not be as well imaged as Montelli et al. (2004a,b) claim. Much of the deeper signal may be robust, however.

Montelli et al. (2004b) report their results in more detail, presenting a comparison of geometric-ray and finite-frequency traveltime tomography for 86,405 long-period P and PP–P traveltimes measured by cross-correlation.They report that, depending on depth and size, the amplitudes of the velocity perturbations in the finite-frequency tomographic images are 30 – 50% larger than in the corresponding geometric-ray images. Their results show plume-like anomalies beneath Ascension, Azores, the Canary Islands, Easter, Tahiti, Hawaii, Bouvet, Kerguelen, Cape Verde, Tibesti, Kilimanjaro and Galapagos


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 long-period wave travel times measured using cross-correlation and invert them together with 1,496,025 short-period travel times. They report that the amplitudes of deep, small-scale velocity heterogeneities are underestimated by 30 – 60% when using geometric ray theory. They further report that the results reveal the existence of six well-resolved plumes extending into the lowermost mantle beneath Ascension, Azores, the Canary Islands, Easter, Samoa, and Tahiti, a less well-resolved plume beneath Hawaii, and additional plumes that are mostly confined to the upper mantle.

Dahlen et al. (2000) publish a paper showing that for finite-frequency seismic traveltimes measured by cross-correlation of broadband waveforms with spherical Earth synthetic seismograms, the travel-time measurement is only sensitive to the wave speed in a hollow banana-shaped 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.


last updated 12th December, 2008