The Last Interglacial (LIG) stage (ca. 130-115 ka), with polar temperatures likely 3-5 • C warmer than today, serves as a partial analogue for low-end future warming scenarios. Multiple indicators suggest that LIG global sea level (GSL) was higher than at present; based upon a small set of local sea level indicators, the Intergovernmental Panel on Climate Change (IPCC)'s Fourth Assessment Report inferred an elevation of approximately 4-6 m. While this estimate may be correct, it is based upon overly simplistic assumptions about the relationship between local sea level and global sea level. Sea level is often viewed as a simple function of changing global ice volume. This perspective neglects local variability, which arises from several factors, including the distortion of the geoid and the elastic and isostatic deformation of the solid Earth by shifting ice masses. Accurate reconstruction of past global and local sea levels, as well as ice sheet volumes, therefore requires integrating globally distributed data sets of local sea level indicators. To assess the robustness of the IPCC's global estimate and search for patterns in local sea level that are diagnostic of meltwater sources, we have compiled a comprehensive database that includes a variety of local sea level indicators from 47 localities, as well as a global sea level record derived from oxygen isotopes. We generate a global synthesis from these data using a novel statistical approach that couples Gaussian process regression to Markov Chain Monte Carlo simulation of geochronological errors. Our analysis strongly supports the hypothesis that global sea level during the Last Interglacial was higher than today, probably peaking between 6-9 m above the present level. This level is close to that expected from the complete melting of the Greenland Ice Sheet, or from major melting of both the Greenland and West Antarctic Ice Sheets. In the period when sea level was within 10 m of the modern value, the fastest rate of sea level rise sustained for a 1 ky period was likely about 80-110 cm per century. Combined with the evidence for mildly higher temperatures during the LIG, our results highlight the vulnerability of ice sheets to even relatively low levels of sustained global warming.

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

S U M M A R YIt is often advantageous to investigate the relationship between two geophysical data sets in the spectral domain by calculating admittance and coherence functions. While there exist powerful Cartesian windowing techniques to estimate spatially localized (cross-)spectral properties, the inherent sphericity of planetary bodies sometimes necessitates an approach based in spherical coordinates. Direct localized spectral estimates on the sphere can be obtained by tapering, or multiplying the data by a suitable windowing function, and expanding the resultant field in spherical harmonics. The localization of a window in space and its spectral bandlimitation jointly determine the quality of the spatiospectral estimation. Two kinds of axisymmetric windows are here constructed that are ideally suited to this purpose: bandlimited functions that maximize their spatial energy within a cap of angular radius θ 0 , and spacelimited functions that maximize their spectral power within a spherical harmonic bandwidth L. Both concentration criteria yield an eigenvalue problem that is solved by an orthogonal family of data tapers, and the properties of these windows depend almost entirely upon the space-bandwidth product N 0 = (L + 1) θ 0 /π . The first N 0 − 1 windows are near perfectly concentrated, and the bestconcentrated window approaches a lower bound imposed by a spherical uncertainty principle. In order to make robust localized estimates of the admittance and coherence spectra between two fields on the sphere, we propose a method analogous to Cartesian multitaper spectral analysis that uses our optimally concentrated data tapers. We show that the expectation of localized (cross-)power spectra calculated using our data tapers is nearly unbiased for stochastic processes when the input spectrum is white and when averages are made over all possible realizations of the random variables. In physical situations, only one realization of such a process will be available, but in this case, a weighted average of the spectra obtained using multiple data tapers well approximates the expected spectrum. While developed primarily to solve problems in planetary science, our method has applications in all areas of science that investigate spatiospectral relationships between data fields defined on a sphere.

Abstract. Gravity and topography provide important insights regarding the degree and mechanisms of isostatic compensation. The azimuthally isotropic coherence function between the Bouguer gravity anomaly and topography evolves from high to low for increasing wavenumber, a diagnostic that can be predicted for a variety of lithospheric loading models and used in inversions for flexural rigidity thereof. In this study we investigate the isostatic response of continental Australia. We consider the effects of directionally anisotropic plate strength on the coherel•ce. The anisotropic coherence function is calculated for regions of Australia that have distinctive geological and geophysical properties. The coherence estimation is performed by the Thomson multiple-Slepian-taper spectral analysis method extended to two-dimensional fields. Our analysis reveals the existence of flexural anisotropy in central Australia, indicative of a weaker N-S direction of lower Te. This observation is consistent with the suggestion that the parallel faults in that area act to make the lithosphere weaker in the direction perpendicular to them. It can. also be related to the N-S direction of maximum stress and possibly the presence of E-W running zones weakened due to differential sediment burial rates. We also demonstrate that the multitaper method has distinct advantages for computing the isotropic coherence function. The ability to make many independent estimates of the isostatic response that are minimally affected by spectral leakage results in a coherence that is more robust than with modified periodogram methods, particularly at low wavenumbers. Our analysis elucidates the reasons for discrepancies in previous estimates of effective elastic thickness Te of the Australian lithosphere. In isotropic inversions for Te, we obtain values that are as much as a factor of 2 less than those obtained in standard inversions of the periodogram coherence using Bouguer gravity and topography but greater than those obtained by inversions that utilize free-air rather than Bouguer gravity and ignore the presence of subsurface loads. However, owing to the low spectral power of the Australian topography, the uncertainty on any estimate of Te is substantial.

Summary We present an azimuthally anisotropic 3‐D shear‐wave speed model of the Australian upper mantle obtained from the dispersion of fundamental and higher modes of Rayleigh waves. We compare two tomographic techniques to map path‐average earth models into a 3‐D model for heterogeneity and azimuthal anisotropy. Method I uses a rectangular surface cell parametrization and depth basis functions that represent independently constrained estimates of radial earth structure. It performs an iterative inversion with norm damping and gradient regularization. Method II uses a direct inversion of individual depth layers constrained by Bayesian assumptions about the model covariance. We recall that Bayesian inversions and discrete regularization approaches are theoretically equivalent, and with a synthetic example we show that they can give similar results. The model we present here uses the discrete regularized inversion of independent path constraints of Method I, on an equal‐area grid. With the exception of westernmost Australia, we can retrieve structure on length scales of about 250 km laterally and 50 km in the radial direction, to within 0.8 per cent for the velocity, 20 per cent for the anisotropic magnitude and 20° for its direction. On length scales of 1000 km and longer, down to about 200 km, there is a good correlation between velocity heterogeneity and geologic age. At shorter length scales and at depths below 200 km, however, this relationship breaks down. The observed magnitude and direction of maximum anisotropy do not, in general, appear to be correlated to surface geology. The pattern of anisotropy appears to be rather complex in the upper 150 km, whereas a smoother pattern of fast axes is obtained at larger depth. If some of the deeper directions of anisotropy are aligned with the approximately N–S direction of absolute plate motion, this correspondence is not everywhere obvious, despite the fast (7 cm yr−1) northward motion of the Australian plate. More research is needed to interpret our observations in terms of continental deformation. Predictions of SKS splitting times and directions, an integrated measure of anisotropy, are poorly matched by observations of shear‐wave birefringence.