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.

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.