The relative density-to-shear velocity scaling (ζ) provides a diagnostic for the presence of compositional variations in the mantle. We invert shear-wave velocity from a recent 3-D model and gravity anomalies for radial profiles of ζ of the uppermost mantle beneath Australia. We performed calculations for the three major tectonic provinces that constitute the continent, and found significant differences between them. The ζ profile for the Phanerozoic region can be explained by thermal variations alone. In contrast, negative values of ζ suggest that variations in composition are important between ∼75 and ∼150 km depth in the Proterozoic continental lithosphere (central Australia). It is likely that chemical variations are also required to explain the inferences for the Archean craton (west Australia), but poor tomographic resolution precludes a definitive conclusion. The scaling factors found are consistent with chemical depletion of deep Precambrian lithosphere, which supports a tectosphere model for the Australian continental roots.
 To determine if a tectosphere is present beneath Australia we investigate the ratio of relative changes in density and shear speed (ζ), a diagnostic for variations in composition:
Predominance of thermal effects produces positive values of ζ. The precise effect of changes in bulk chemistry on ζ is not well-defined, but depletion in dense elements (e.g., iron or basaltic components) would decrease the amplitude of density anomalies. Small and negative values of ζ therefore indicate that effects of composition are significant. Global scale investigations have determined depth profiles of ζ for continental and oceanic regions [e.g., Forte et al., 1994; Deschamps et al., 2001], and presented evidence for chemical depletion in the tectosphere [Forte and Perry, 2000; Deschamps et al., 2002]. More recently, Perry et al.  proposed a detailed regional study for North America.
 The Australian continent can be divided in three main tectonic provinces (Figure 1a) with a westward age progression from Phanerozoic (<500 Ma), to Proterozoic (500–2500 Ma), and Archean (>2500 Ma). If parts of the Australian continent are underlain by a tectosphere one would expect lateral variations in the depth profiles ζ(r). Indeed, research presented here suggests that depth profiles for these domains differ from one another and that they can be used to constrain the depth range of chemical depletion.
 We invert gravity (δg) and shear-wave velocity anomalies (δVS) for radial changes in the scaling factor ζ(r,θ, ϕ). To model gravity anomalies we integrate density anomalies δρ vertically and laterally following
where R and are the Earth's radius and mean density, respectively, rCMB the radius of the core-mantle boundary, g0 the surface acceleration of gravity, Kg(Δ, r) the local gravity kernels, and Δ the angular distance between the locations (θ, ϕ) and (θ′, ϕ′). This equation is similar to that used by Kogan and McNutt .
 We perform the analysis in the part of the spectral domain that is sensitive to the uppermost mantle. Spherical harmonic degrees lower than ℓ = 10 sample the whole mantle and are removed to avoid smearing effects. To focus the study on the depth interval 50–300 km and reduce the influence of the crust we also removed degrees higher than ℓ = 60. For the band-pass filtering, ℓ1 ≤ ℓ ≤ ℓ2, the local gravity kernels are given by
where Pℓ0(cos Δ) are Legendre polynomials and Gℓ(r) the radial geoid kernels. The latter depend strongly on the radial viscosity profile and boundary topography and are computed using the method proposed by Forte and Peltier  and viscosity profile MF2 due to Mitrovica and Forte . Interestingly, geoid kernels for degrees higher than ℓ = 10 depend only slightly on the viscosity profile. An important consequence is that within error bars, the scaling factor for the uppermost mantle (z < 400 km) is not sensitive to viscosity [Deschamps et al., 2001].
 Radial geoid kernels (and therefore local gravity kernels) for degrees ℓ = 10 to 60 are sensitive to the upper mantle but have negligible values throughout the lower mantle. The depth integration in equation (2) can then be reduced to the layer 35 ≤ z ≤ 670 km. For different depths, Figure 2 shows the local gravity kernels for ℓ = 10 − 60. The amplitude of local gravity kernels decreases rapidly with increasing angular distance. This justifies the use of truncated local kernels, which are equal to the local kernels for angular distances smaller than a cut-off angular distance (Δc) and zero elsewhere. Here we present results for Δc = 12°.
 Substitution of equation (1) into equation (2) gives a linear relation between δg, δVS, and ζ. We use a generalized inversion to estimate ζ from δg and δVS. We performed simultaneous inversions for each age-region represented in Figure 1a. The regularization is controlled by a damping factor ɛ. Increasing ɛ enhances the smoothness of ζ(r) but leads to higher misfit of the reconstructed data with respect to the observed data. We inferred the optimal value of ɛ from the trade-off curves shown in Figure 3 and investigated the sensitivity of the solution to the choice of ɛ.
 The (isotropic) shear-wave velocity model AUS04-Vs is defined on a 1° × 0.5° lateral grid and for 10 depth layers [see, e.g., Zielhuis and Van der Hilst, 1996]. The lateral resolution is ∼300 km in eastern and central Australia but degrades toward the Archean west owing to reduced wave path coverage. Figure 1b shows AUS04-Vs at 140 km depth. Figure 1c depicts the tide-free, non-hydrostatic free-air gravity anomalies from EGM96, filtered for the spectral window ℓ = 10 − 60. We assumed that the crust is in hydrostatic equilibrium and we used CRUST2.0 [Bassin et al., 2000] to remove its contribution to gravity anomalies.
 The main sources of uncertainty are the errors in the tomographic and crustal models. In absence of formal estimates of error in wavespeed, we studied the effect of random noise generated from a Gaussian probability distribution with zero mean and a standard deviation (σerr) equal to 50% of the RMS of the AUS04-Vs values at the depth considered. For each layer of CRUST2.0, the σerr of the error distribution used is 1% of the layer density, but near layer boundaries we use 5% to mitigate effects of incorrect interface depths. We then produced 1000 perturbed VS and crust models, and we inverted each of them for ζ as a function of depth. At each depth, the mean and standard deviation of the 1000 perturbed profiles thus produced are used as estimates of the scaling factor ζ(r) and its uncertainty, σζ.
 The solid curves in Figures 4a–4c represent the average scaling factor ζ(r) for each region, and the error bars represent one standard deviation uncertainty, σζ. Fortunately, even the generous a priori errors described above do not lead to very large uncertainty in ζ. For reference, we depict with grey shading the thickness of the Phanerozoic and Proterozoic CL as estimated from the high wavespeed lid in AUS04-Vs; the Archean CL is poorly resolved by the seismic data used [Simons et al., 2002]. The average scaling factor for Phanerozoic is positive throughout the CL (Figure 4a). It is perhaps slightly negative between 130 and 190 km depth, coincident with the seismic low velocity zone [Zielhuis and van der Hilst, 1996], but the large error bars may render this insignificant. The scaling factor profile for the Proterozoic is more complicated. Within error, ζ is positive to a depth of 75 km, negative between 75 and 150 km depth, and statistically zero below. Finally, the average scaling factor for Archean is smaller in amplitude than both others and is negative to a depth of ∼200 km (Figure 4c). The negative values appear statistically significant, but we recall that resolution of wavespeed anomalies is poor in this part of the model [e.g., Simons et al., 2002].
 It is important to assess the robustness of our results. First, the resolution matrices (Figures 4d–4f) demonstrate that the scaling factors for the Phanerozoic and Proterozoic are well resolved down to ∼300 km. Below that depth (not shown here) substantial smearing occurs due to decreasing values of the gravity and surface wave sensitivity kernels. The scaling factor for Archean is poorly resolved even for depths less than 300 km. Second, we computed profiles of scaling factors for several values of the damping factor ɛ. For low values of ɛ, profiles are less smooth and uncertainties are larger. However, the shapes of the profiles and, in particular, the negative values between 75 and 150 km depth for the Proterozoic, are fairly robust. Finally, to investigate the influence of the spectral window, we performed calculations for ℓ = 10 − 40, ℓ = 20 − 60, and ℓ = 10 − 80, but we did not find significant changes. We thus conclude that the ζ(r) inferred for the Phanerozoic and Proterozoic are robust down to 300 km depth.
4. Discussion and Concluding Remarks
 The Phanerozoic CL extends down to at most 120 km depth [e.g., Zielhuis and van der Hilst, 1996], and the positive scaling factor across this top layer can be explained with thermal effects alone. Small values of ζ might signal compositional changes at larger depth, perhaps associated with the ocean-continent difference. Indeed, AUS04-Vs suggests very slow shear wave velocities between 100 and 200 km depth beneath the easternmost part of Australia, which appear continuous to the slow velocities observed beneath the oceanic domain at the east of Australia (Figure 1b). In contrast, variations in composition within the CL are required to explain the inferred scaling factors beneath the Proterozoic shield of central Australia. Purely thermal anomalies can explain ζ(r) for z ≤ 50 km, but the sharp decrease of ζ below this depth suggests that chemical depletion is present between 90 and 140 km depth. Similarly, chemical depletion seems to have occurred beneath the Archean province. We cannot propose a robust conclusion, however, because the tomographic model is poorly resolved and shows significant smearing in this region. An outstanding question is the nature and strength of chemical depletion. The scaling factor alone cannot resolve the trade-off between thermal and compositional effects. The amplitude of thermal and compositional variations can be inferred from appropriate statistical analysis of VS and density distributions [Deschamps and Trampert, 2003], but independent information would be needed to constrain density better (e.g., the surface topography, provided it is correctly converted into dynamic topography) and the temperature (e.g., compressional-wave velocity model and heat flow data).
 The radial scaling factors that we infer are generally consistent with the tectosphere model due to Jordan . Beneath Precambrian terrains (here, restricted to the Proterozoic) the continental roots appear chemically distinct, whereas data for the Phanerozoic regions can be explained without significant compositional variations in the lithosphere. It is likely that the chemical depletion of the Proterozoic continental lithosphere contributes to the long term stability of the Australian continental roots, but from the work described in this paper it is yet not clear if chemical depletion by itself would be sufficient to prevent gravitational instability due to cooling. Rheology probably plays an important role [Shapiro et al., 1999a; Lenardic and Moresi, 1999], but more systematic studies are necessary to understand the stability of a thermal boundary layer in a chemically depleted fluid.
 We are grateful to Eric Debayle and Alessandro Forte for constructive reviews. This research was funded by ISES (grant 627) (FD) and the Nederlands Organization for Scientific Research (NWO, grant VICI 86503001) (RvdH).