Several models of the P- and S-wave velocities have been published during the last four years. They have been based on various data sources, including the waveforms, differential and absolute travel times read from, mostly, digitally recorded seismograms and the arrival times reported in the Bulletins of the International Seismological Centre (ISC).

Morelli and Dziewonski [1991] give the details of their experiment in which they used the P (direct P-wave), PcP (P-wave reflected from the core-mantle boundary), PKP (P-wave with the turning point in the liquid core) and PKIKP (P-wave with the the turning point in the solid inner core) travel time data from the ISC Bulletins to simultaneously relocate the summary events, determine the laterally heterogeneous structure of the lower mantle, the topography of the core-mantle boundary and the heterogeneous structure of the inner core.

Fukao et al. [1992] used a finer grid size to map the velocity variations near subduction zones and a coarser grid for the global P-velocity variations. The regional aspects of their analysis agreed with some conclusions of van der Hilst et al. [1991] inversion for the structure near the subduction zones of the Western Pacific; there were also some discrepancies.

Pulliam et al. [1993] have investigated the stability of inversion of the ISC P-wave data base using different norms; the first is the Euclidean norm and leads to the least squares solution, the other involves minimization of the sum of the absolute values of the residuals). Also, they compared their results with those from the earlier studies and found a substantial similarity in the pattern of anomalies in the lowermost mantle but a significant differences in the amplitude.

Tanimoto [1990] used the mantle wave and long-period body wave waveforms to derive a whole mantle S-wave following the approach developed by Woodhouse and Dziewonski [1986, 1989]. He represented the lateral variations by spherical harmonics up to degree 6 and the radial variations by 11 shells with a the velocity perturbation constant as a function of depth within each shell.

Su and Dziewonski [1991] used the data set collected by Woodhouse and Dziewonski [1986], but relaxed the damping, which seemed to result in an attenuated level of heterogeneity in the lowermost mantle. Their model predicted the absolute SS travel time data and the differential travel time data (SS - S and ScS - S) of Woodward and Masters, [1991a, b] rather well.

Upper mantle tomographic models, including the effect of anisotropy, have been derved by Montagner and Tanimoto [1991, 1992] using the mantle surface waves. Zhang and Tanimoto [1992, 1993] developed a degree-36 model of the uppermost 500 km of the mantle using the surface wave data of Zhang [1991]. Geodynamic implications of this model are discussed by Anderson et al. [1992]. Su et al. [1992a] point out that the expression of mid-ocean ridges in model is inadequate; it predicts variation of the vertical S-wave anomaly as a function of the sea floor age that is 4 to 5 times smaller than observed. Ekström and Dziewonski [1994] developed a degree-20 S-velocity model with an enhanced resolution in the upper mantle, needed to satisfy the short-period surface wave data ( 35 s) together with the mantle wave and body wave waveforms and differential and absolute travel times. The conclusion is that the entire data set is compatible and that the best of the published global models do not require major revisions below the depth of 100 km or so.

Dziewonski and Woodward [1992] and Woodward et al. [1993] have combined the waveform inversion with the travel time inversion to improve the resolution and to remedy some of the problems with the path average approximation used in the waveform inversion [ Li and Tanimoto, 1992; Li and Romanowicz, 1995]. They constructed two models: one in which the mantle is split between the upper and lower mantle () and the other, in which the whole mantle is parameterized by a continuous expansion in Chebyshev polynomials (). The two models are compared in Dziewonski et al. [1993] and Woodward et al. [1994]. Except for some 100--150 km on either side of the 670 km discontinuity, the models show nearly perfect correlation. In the discontinuous model, there is a substantial change at 670 km in the character of the spectrum: from red---dominated by degree 2---above the discontinuity to pink (power still decreases with , but slowly), below it.

Su et al. [1992b, 1994] have expanded the data base by incorporating into the waveform collection seismograms for the years 1985--1990. This includes the data from the GEOSCOPE network, with many stations---particularly in the southern hemisphere---in previously unoccupied locations. In derivation of their model S12, they included the SS and S travel times measured by Su [1993]. The degree of expansion was increased to 12, which more than doubled the number of coefficients with respect to the degree-8 models.

In a parallel study, Masters et al. [1992] derived a degree-10 model using even larger set of travel times. For the control of the upper mantle structure they relied on the splitting coefficients, which have no resolution for the odd-degree spherical harmonics. The resolution of the model appears to be better than that of S12 [Su et al., 1994] in the lowermost mantle because of more numerous S-wave observations.

Substantial improvement in resolution in the lowermost mantle is being achieved by waveform measurements of S - SKS and SKKS - SKS travel times [ Liu and Dziewonski, 1994]. The most recent model that incorporates over 3,500 global measurements of these phases indicates negative velocity anomalies near core-mantle boundary as large as -4%. This is similar to the values reported by Wysession et al. [1994] near the CMB under the Pacific.

Studies involving the measurement and interpretation of the splitting of normal modes are important since they avoid approximations involved in great circle path representation used in the waveform and travel time methods. Li et al. [1991a] provided additional data and models. Also, Li et al. [1991b] have shown that the normal modes require the ratio of the relative shear velocity perturbations to the compressional velocity perturbations to be somewhat larger than 2.

An important study, even though regional in nature, has been published by Grand [1994], who is using S, SS, SSS and ScS data to obtain a 3-D structure under the Americas and the surrounding oceans. Comparison of a cross-section through North America, where the data coverage is the best, shows that this model and S12 have very similar large scale properties. However, the models differ more appreciably under the South America---a region with considerably fewer seismographic stations.

Vasco et al. [1994] have inverted the ISC Bulletin data for both P- and S-wave velocities, with the simultaneous relocation of the earthquakes and determination of station corrections. Their results show similarities with other P- or S-wave models.

A joint P- and S-wave inversion involving the waveform data, travel time data determined from the waveform analysis and extracted from the ISC Bulletins has been reported by Su and Dziewonski [1993]. They find that the P and S-wave solutions are highly correlated and that the ratio of the relative perturbation of the shear velocity to the compressional velocity is about 2, with some variation throughout the mantle.

In other related developments, there has been the first attempt to map the 670 km (or 660 km) discontinuity, using the differential travel times between the surface and underside reflections [ Shearer and Masters, 1992]. Also, in a very difficult undertaking, lateral variations in attenuation have been studied [ Romanowicz, 1990; Durek et al., 1993; Romanowicz, 1994]. The latter report specifies a 3-D model of the shear attenuation in the mantle using spherical harmonic expansion up to degree 6.

There is an ongoing debate about the reliability of the tomographic models. Pulliam and Stark [1994], for example, give a rather pessimistic assessment of the errors with which the structure can be determined. However, they base their conclusions on an approach which depends on the resolution of the structure within each---often small---volume element.

In view of the `red' nature of the spectrum of the heterogeneity, this may not be the most fruitful approach. For many applications, the spherical harmonic expansion coefficients are of particular interest. It might be possible to resolve the degree-2 component without having to resolve also the degree-15 coefficients, which is demanded by the volume element approach involving, for example, ... km discretization.

Stark and Hengartner [1993; see comment by Morelli and Dziewonski, 1995] calculate that the CMB topography could be determined with about 1 km accuracy if the topography is limited to degree 4, but that the errors would exceed 100 km if there was substantial power up to degree 20. Since the indication from the studies of the spectrum of heterogeneity is that the spectrum is red, we might be able to determine quite accurately the long wavelength field without being aliased by the presence of the relatively weak, high wavenumber component.

In other studies, damping is introduced to assure the stability of the solution. Therefore, a reliable assessment of the errors is not possible. The viability of the models must be established be comparing them with the patterns detected in the data and by investigating their implications for other geophysical observables and processes, such as the gravity field (geoid), plate motions, mantle flow and convection, mineral physics, geomagnetism and geochemistry. A number of such applications provide a growing body of evidence that the global mantle tomography is on the right track.