Geophysical Journal International

Transcription

1 A Bayesian Method to Quantify Azimuthal Anisotropy Model Uncertainties: Application to Global Azimuthal Anisotropy in the Upper Mantle and Transition Zone Journal: Manuscript ID: Draft Manuscript Type: Research Paper Date Submitted by the Author: n/a Complete List of Authors: Yuan, Kaiqing; University of California Berkeley, Earth and Planetary Science Beghein, Caroline; University of California Los Angeles, Earth, Planetary, and Space Sciences Keywords: Seismic anisotropy < SEISMOLOGY, Tomography < GEOPHYSICAL METHODS, Inverse theory < GEOPHYSICAL METHODS, Probability distributions < GEOPHYSICAL METHODS, Statistical seismology < SEISMOLOGY, Surface waves and free oscillations < SEISMOLOGY

2 Page of submitted to Geophys. J. Int. A Bayesian Method to Quantify Azimuthal Anisotropy Model Uncertainties: Application to Global Azimuthal Anisotropy in the Upper Mantle and Transition Zone K. Yuan, and C. Beghein Department of Earth, Planetary, and Space Sciences, University of California Los Angeles, Los Angeles, CA 00, USA now at Berkeley Seismological Laboratory, University of California Berkeley, Berkeley, CA 0, USA SUMMARY Seismic anisotropy is a powerful tool to constrain mantle deformation, but its existence in the deep upper mantle and topmost lower mantle is still uncertain. Recent results from higher mode Rayleigh waves have, however, revealed the presence of % azimuthal anisotropy between 00 km and 00 km depths, and changes in azimuthal anisotropy across the mantle transition zone boundaries. This has important consequences for our understanding of mantle convection patterns and deformation of deep mantle material. Here, we propose a Bayesian method to model depth variations in azimuthal anisotropy and to obtain quantitative uncertainties on the fast seismic direction and anisotropy amplitude from phase velocity dispersion maps. We then apply this new method to existing global fundamental and higher mode Rayleigh wave phase velocity maps to assess the likelihood of azimuthal anisotropy in the deep upper mantle and previously detected changes in anisotropy at the transition zone boundaries are robustly constrained by those data. We use a model space search approach, which yields more reliable uncertainty estimates than traditional inverse techniques, and discuss different ways to calculate the standard deviation on the fast axes direction and anisotropy amplitude. Our results confirm that deep

3 Page of Yuan, K. and Beghein, C. upper mantle azimuthal anisotropy is favored and well-constrained by the higher mode data employed. The fast seismic directions are in agreement with our previously published model and the data favor changes in azimuthal anisotropy at the top and bottom of the transition zone. Key words: Seismic anisotropy Tomography Inverse theory Probability distributions Statistical seismology Surface waves and free oscillations. INTRODUCTION The directional dependence of seismic wave velocity, or seismic anisotropy, is a powerful tool to investigate mantle deformation and geodynamics (Montagner ; Karato ; Becker et al. 00; Long 0). The lattice preferred orientation (LPO) of the crystallographic axes of elastically anisotropic material is generally assumed to be the cause of the seismic anisotropy detected in Earth s mantle, but it can also be caused by the shape preferred orientation (SPO) of isotropic structures with contrasting elastic properties such as layered structures, cracks, melt tubules, or lenses (Kendall & Silver ; Montagner ). In the mantle lithosphere, frozen-in seismic anisotropy is often attributed to olivine LPO related to past tectonic processes (Karato ; Nicolas & Christensen ; Silver ). In the asthenosphere, olivine LPO associated with present-day mantle deformation is often invoked to explain observations of seismic anisotropy because the fast seismic direction often aligns with the absolute plate motion (Nishimura & Forsyth ; Smith et al. 00; Debayle et al. 00; Marone & Romanowicz 00; Beghein et al. 0), and the preferred alignment of olivine can be used to determine the direction of mantle flow (Becker et al. 00). In the lowermost mantle, both SPO through horizontal layering or aligned inclusions (Kendall & Silver ) and LPO of the post-perovskite phase (Oganov 00) have been proposed to explain observations of anisotropy. Most tomographic models of seismic anisotropy are obtained by regularized inversion of seismic data such as surface waves, free oscillations, and/or long-period body waves, and they all provide ample evidence for the presence of seismic anisotropy in the uppermost 00 km of the mantle. Radial anisotropy, which represents differences in seismic wave velocity between the vertical and horizontal directions, is required in the uppermost mantle to simultaneously explain Love and Rayleigh wave dispersion data. It is included in the top 0 km of the one-dimensional (-D) Preliminary Reference Earth Model (PREM) of Dziewonski & Anderson ( ), and in several -D radially anisotropic global models of the uppermost mantle models (see Chang et al. ( 0) for a recent review). Azimuthal anisotropy, i.e. the dependence of seismic wave velocities with the azimuth of propagation, is

4 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy also present in the uppermost mantle at the global scale (Montagner & Tanimoto ; Trampert & Woodhouse 00; Debayle et al. 00; Ekström 0; Debayle & Ricard 0; Yuan & Beghein 0; Becker et al. 0; Moulik & Ekström 0; Yuan & Beghein 0). Most global models of azimuthal anisotropy display common features at these depths, such as the alignment of the fast axes with the plate motion direction at asthenospheric depths beneath ocean basins and with the paleospreading directions in the oceanic lithosphere. -D models of radial anisotropy generally agree with one another, but there are discrepancies in models of lateral variations in radial anisotropy (Chang et al. 0). The D layer is also known to be radially anisotropic at the regional scale from shear-wave splitting measurements (Kendall & Silver ), and regional azimuthal anisotropy has also been detected at these depths (Maupin et al. 00). A few global tomographic models suggest D radial anisotropy might be present at the global scale as well, though the effect of the crustal correction (Panning et al. 00), of prior scaling relationships between elastic parameters (Beghein & Trampert 00a; Beghein & Trampert 00b; Beghein et al. 00; Beghein 00), and trade-offs between isotropic and anisotropic structure (Kustowski et al. 00; Chang et al. 0) cast doubt on the global nature of radial anisotropy at these depths. To date, there is no global azimuthal anisotropy model of the lowermost mantle. The lack of evidence for seismic anisotropy in the rest of the mantle has long been interpreted as the result of deformation by diffusion creep (Karato et al. ). Evidence for radial anisotropy in the deep upper mantle and uppermost lower mantle has, however, been accumulating over the past decade. The first global model displaying anisotropy in the deep upper mantle and D was the -D model of Montagner and Kennett ( ), which was followed by multiple -D and -D global models (Beghein & Trampert 00b; Panning & Romanowicz 00; Beghein et al. 00; Panning & Romanowicz 00); Kustowski et al. 00; Visser et al. 00a; Panning et al. 00; Romanowicz & Lekić 0; Auer et al. 0; Chang et al. 0; French & Romanowicz 0). Shear-wave splitting studies have also suggested the presence of radial anisotropy near subduction zone in the mantle transition zone (TZ) and top of the lower mantle (Fouch & Fischer ; Wookey & Barruol 00; Chen & Brudzinski 00; Foley & Long 0). Azimuthal anisotropy may also be present at these depths. It has been shown to be compatible with higher mode Love waves (Trampert & van Heijst 00) and coupled free oscillation data (Beghein et al. 00; Hu et al. 0). A study combining surface wave and shear-wave splitting data beneath North America also suggested that azimuthal anisotropy is needed at greater depths than commonly assumed (Marone & Romanowicz 00), and similar conclusions were drawn by Kosarian et al. ( 0) for California. At the global scale, while early -D models did not show any significant azimuthal anisotropy below 0 km depth (Montagner & Tanimoto ; Debayle et al. 00), more recent studies present about % anisotropy in the deep upper mantle at least down to 00 km (Debayle & Ricard 0; Yuan & Beghein 0; Yuan & Beghein 0), and possibly even

5 Page of Yuan, K. and Beghein, C. deeper down to the bottom of the TZ and top of the lower mantle (Yuan & Beghein 0; Yuan & Beghein 0). There are still large discrepancies among models, but the increasing evidence for seismic anisotropy in the deep upper mantle challenges our understanding of mantle deformation (Figure ). Our previously published global azimuthal anisotropy model (Yuan & Beghein 0), hereafter referred to as YBSVani, not only displayed a non-negligible amount of azimuthal anisotropy ( to %) below 0 km depth, but it also revealed changes in the seismic fast direction at the TZ boundaries. The interpretation of these results is still non-unique due to the paucity of mineral physics data on TZ material anisotropy. They have, however, important consequences for our understanding of mantle convection and the anisotropy of deep upper mantle material as it could imply changes in mantle flow direction at the TZ, changes in volatile content, in slip system in TZ material, etc. It is thus essential to determine which model features are robust. In this paper, we present a Bayesian forward modeling method to quantify azimuthal anisotropy model parameter uncertainties and trade-offs. Like most tomographic models, YBSVani was obtained by regularized inversion of seismic data. In this particular case, the model was derived from fundamental and higher mode surface wave phase velocity maps. Estimating reliable model uncertainties from linear inversions is, however, not straightforward since most inversions yield a posterior model covariance smaller or equal to the prior covariance by construction (Tarantola ). If there is a large model null-space, the posterior covariance can be strongly underestimated (Trampert ), making both the interpretation and the uncertainty assessment of tomographic models difficult (Beghein & Trampert 00; Beghein 00). Model space search approaches are generally better suited to determine posterior model uncertainties as they can explore the entire model space, including the null-space, and map the range of models that can fit the data reasonably well. In some cases, this type of method can even find solutions to the problem that could not be found with traditional inverse methods (Beghein & Trampert 00). In this paper, we modeled azimuthal anisotropy in the upper mantle and topmost lower mantle and quantified associated parameter uncertainties and tradeoffs using the Neighbourhood Algorithm (Sambridge a; Sambridge b), hereafter referred to as the NA. Among many other applications, this direct search technique has previously been used successfully to model inner core anisotropy (Beghein & Trampert 00), regional and global mantle seismic velocities (Beghein et al. 00; Snoke & Sambridge 00), and radial anisotropy (Beghein & Trampert 00a; Beghein & Trampert 00b; Beghein et al. 00; Visser et al. 00a; Yao et al. 00; Beghein 00). Yao ( 0) recently proposed a two-step method using the NA to model azimuthal anisotropy from fundamental mode surface waves. However, as explained in section, the author introduced several approximations that may affect the outcome of the modeling and the model

6 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy uncertainty estimates, and did not display or discuss the posterior uncertainties on the fast seismic direction and the anisotropy amplitude. The method we present here solves the linear problem that relates azimuthal anisotropy elastic parameters to phase velocities using exact and laterally varying sensitivity kernels to account for variations in crustal structure, and quantifies model uncertainties for the azimuthal anisotropy amplitude and the fast axes directions. PHASE VELOCITY DATA In this study, we employed the fundamental and first six higher mode anisotropic Rayleigh wave phase velocity maps determined by Visser et al. ( 00b). There were fundamental modes between s and s, first overtones between s and s, second overtones between s and 0 s, third overtones between s and s, fourth overtones between s and s, fifth overtones between s and s, and sixth overtones between s and s. This type of seismic data is ideal to provide depth constraints on Earths internal structure due to their dispersive properties (Figure ). In addition, combining fundamental and higher mode surface wave data significantly increases the depth resolution of tomographic models. Contrary to fundamental mode surface waves, which can only resolve the top 00 km of the mantle, the set of higher modes employed here have sensitivity to azimuthal anisotropy well into the deep upper mantle and topmost lower mantle (Figure ). As explained by Visser et al. ( 00b), the lateral resolution of these phase velocity models generally decreases with increasing overtone number because the quality of the path azimuthal coverage (and thus the number of modes measured reliably) decreases for higher modes. Ray coverage was very good everywhere for the fundamental modes, and in most continental regions and the northwestern Pacific for the higher modes, but it was poorer for the third through sixth higher modes in the southeastern Pacific, southern Indian Ocean, and southern Atlantic. Another factor that affected the lateral resolution of the phase velocity maps was the choice of the damping made by the authors who opted for maintaining a constant relative model uncertainty for all modes. This too resulted in phase velocity maps of decreasing resolution with increasing overtone number. Visser et al. ( 00b) estimated that the fundamental mode Ψ terms are resolved up to spherical harmonic degree and degree for the higher modes, which corresponds to a resolving power of about 00 km near the surface, decreasing to 0 km near the TZ. Because the inferences made in this paper focus on largescale anisotropy, using data of varying resolution should not strongly affect our results. Trade-offs between the different terms of equation constitute another source of uncertainty when constructing anisotropic phase velocity maps from path-averaged measurements. One cannot completely separate the different terms because data coverage is imperfect owing to the uneven distribution of earthquakes and seismic stations over the globe. The resolution matrices calculated by Visser et al. ( 00b) showed

7 Page of Yuan, K. and Beghein, C. that these trade-offs were minimal and that there was therefore little mapping of lateral heterogeneities or topography at discontinuities into the anisotropic terms. Keeping these caveats in minds, the phase velocity maps employed here consist in a unique dataset of anisotropic higher mode Rayleigh waves with unprecedented vertical resolution that enabled us to image azimuthal anisotropy in the deep upper mantle. METHOD. Phase Velocity Anisotropy At any given point at Earth s surface, perturbations dc in surface wave phase velocity with respect to predictions from a reference Earth model can be expressed as a function of the azimuth of propagation Ψ as follows (Montagner & Nataf ): dc(t,ψ) = dc 0 (T) + dc (T)cos(Ψ)+dc (T)sin(Ψ) + dc (T)cos(Ψ)+dc (T)sin(Ψ) () where T is the period of the wave. dc 0 represents the phase velocity anomaly averaged over all azimuths and thedc i (i =,,) terms represent the azimuthal dependence of the phase velocity. TheΨ terms can help constrain depth variations in vertically polarized shear (SV) wave azimuthal anisotropy, and the Ψ terms can help determine horizontally polarized shear (SH) wave anisotropy. Equation is valid for fundamental and higher mode surface waves. Fundamental mode Rayleigh waves typically are not expected to have any Ψ dependence, as demonstrated by Montagner & Tanimoto ( ) for realistic petrological models. The same may not, however, be true for higher modes since they are sensitive to deeper structure (Figure ), and indeed Visser et al. ( 00b) determined that a Ψ dependence significantly improved the fit of their Rayleigh wave fundamental and higher mode phase velocity path-averaged measurements (Visser et al. 00b). Nevertheless, because the sensitivity of fundamental and higher mode Rayleigh waves to SH anisotropy is very small, here we only used the Ψ terms of equation to build a -D model of SV azimuthal anisotropy in the top 000 km of the mantle. The relation between Ψ phase velocity anisotropy and azimuthal anisotropy at depth is given by the following set of equations (Montagner & Nataf ): dc (T) = [G c (r)k G (T,r)+B c (r)k B (T,r)+H c (r)k H (T,r)]dr () dc (T) = [G s (r)k G (T,r)+B s (r)k B (T,r)+H s (r)k H (T,r)]dr () where elastic parameters G c (r) and G s (r) relate to V SV azimuthal anisotropy, and the other four parameters relate to P-wave azimuthal anisotropy. K G (r,t), K B (r,t), and K H (r,t) are the local

8 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy partial derivatives, or sensitivity kernels, for Rayleigh wave at period T and radius r, which can be calculated for a reference model using normal mode theory (Takeuchi ). Examples of kernels calculated using model PREM (Dziewonski & Anderson ) are shown in Figures and. The fast azimuth of propagation Θ and the anisotropy amplitude G of vertically polarized shearwaves are given by: Θ = arctan(g s/g c ) () and G = G s +G c () Similar relations exist forb c,s and H c,s.. Parameterization We divided Earth s surface into 0 0 cells, and the data were inverted by applying the NA to equations and at each grid cell. The reader should note, however, that 0 does not correspond to the lateral resolution of our models since it is directly controlled by the resolution of the phase velocity maps and is limited to larger wavelengths in the deep upper mantle than in the shallow mantle, as discussed in section. It is also important to note that the quantitative uncertainty analysis performed in this study doe snot account for uncertainties stemming from the non-uniform ray path coverage, but is solely focused on the model parameter resolution for a given set of dispersion curves and estimated data uncertainties. At every grid cell we parameterized G c (r) and G s (r) vertically using cubic spline functions S i (r) (i=,...,) of varying depth spacing (Fig. (A)): G c (r) = G s (r) = i= G i cs i (r) G i ss i (r) i= B c (r), B s (r), H c (r), and H s (r) were parameterized each with cubic spline functions (Fig. (B)), resulting in a total of unknowns fordc and fordc at each grid cell: B c (r) = BcS i i (r) () B s (r) = H c (r) = i= BsS i i (r) i= HcS i i (r) i= () () () (0)

9 Page of Yuan, K. and Beghein, C. H s (r) = HsS i i (r) () i= B c,s and H c,s are poorly resolved due to the similarity of their partial derivatives (Figure ), and are often either neglected in surface wave inversions (Marone & Romanowicz 00) or assumed to be proportional to G c,s (Yao 0). Here, we decided to solve equations and exactly for all three parameters for accuracy. Because they are poorly resolved independently of how fine a vertical parameterization we use, we chose to discretize them with less spline functions than for G c (r) and G s (r) to reduce the total number of unknowns and computational costs. We focus the discussion and interpretation onv SV anisotropy only (i.e.,g c (r) and G s (r)). Equations and therefore become: dc (T) = G i [ ci G,i (T)+ B i c I B,i (T)+HcI i H,i (T) ] () i= i= dc (T) = G i [ si G,i (T)+ B i s I B,i (T)+HsI i H,i (T) ] () i= i= where I G,i (T) = S i (r)k G (T,r)dr () I B,i (T) = S i (r)k B (T,r)dr () I H,i (T) = S i (r)k H (T,r)dr () In this work, we used a parameterization in terms of relative perturbationsdlng c,s = G c,s /L,dlnB c,s = B c,s /A, and dlnh c,s = H c,s /F, where L, A, and F are the so-called Love elastic parameter (Love ). L determines the wavespeed of vertically polarized shear-waves (V SV = L/ρ), A relates to the wavespeed of horizontally traveling P-waves, and F controls the velocity of waves propagating at intermediate angles. It should be noted that the spline functions used in this study differ slightly from the ones employed to obtain model YBSVani (Yuan & Beghein 0). We therefore cannot fairly compare YBSVani with our new model resulting from the NA. In this work, in addition to applying a model space search approach (see section.), we thus obtained an updated -D model using the new splines described above together with the same dataset and singular value decomposition (SVD) method as in Yuan & Beghein ( 0). This new model, hereafter referred to as YBSVaniSVD, is almost identical and display the same features as YBSVani.

10 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy. Effect of the Crust An important aspect of modeling lateral heterogeneities or anisotropy in the mantle relates to crustal structure. Many first generation -D velocity and anisotropy models were obtained using sensitivity kernels calculated based on the -D reference mantle model PREM (Dziewonski & Anderson ). However, crustal thickness, velocities, and density vary laterally and neglecting these variations can bias the model due to the mapping of crustal structure into the mantle (Boschi & Ekström 00; Marone & Romanowicz 00; Kustowski et al. 00; Bozdaǧ & Trampert 00). When inverting isotropic surface wave data for mantle velocities or radial anisotropy, it is essential to account for the effect of lateral crustal variations on the sensitivity kernels (Boschi & Ekström 00; Marone & Romanowicz 00) and to either correct the data with an a priori crustal model (Boschi & Ekström 00) or invert the data simultaneously for the Moho depth and mantle structure (Meier et al. 00; Visser et al. 00a; Chang et al. 0). It should, however, be noted that in this last case data uncertainties need to be small to resolve the Moho depth due to trade-offs with velocities (Lebedev et al. 0). In the case of azimuthal anisotropy,dc anddc do not depend on the Moho depth (equations and ), and there exists no global azimuthal anisotropy model of the crust that we can use to correct the data a priori. Moreover, because the data used here have little sensitivity to crustal depths, they are likely not sufficient to resolve azimuthal anisotropy in the crust (Figure ). We did, however, account for the effect of crustal structure and variations in Moho depth on the partial derivatives as in Yuan & Beghein ( 0). At each grid cell, we generated a local -D model composed of the PREM mantle to which we superimposed crustal model CRUST.0 (Bassin et al. 000), and calculated the corresponding partial derivatives (Takeuchi ). We refer to Yuan & Beghein ( 0) for examples of laterally varying sensitivity kernels. The approach taken here is slightly different from that of Yao ( 0) who used the isotropic part of the phase velocity maps together with the NA to generate a new local -D mantle model. However, we do not expect this to strongly influence our results since it was demonstrated that accounting for lateral variations in mantle structure to calculate the sensitivity kernels does not yield any significant difference in the -D azimuthal anisotropy model (Yuan & Beghein 0).. The Neighbourhood Algorithm The NA (Sambridge a; Sambridge b) is a direct search technique that identifies regions of relatively low and relatively high misfit, associated with high and low likelihoods, respectively. It allows us to map the entire model space (within selected boundaries), including the model null-space, and therefore obtain information on its approximate topology without having to introduce explicit a priori information or regularization on the model parameters. Model space search techniques are most often applied to non-linear problems, which can be highly non-unique and can have a non-gaussian

11 Page 0 of Yuan, K. and Beghein, C. cost function with multiple minima. In that case, the solution obtained by traditional inverse techniques, which generally assume Gaussian model parameter distributions (Tarantola ), is strongly dependent on prior assumptions and regularization. Forward modeling methods therefore offer a more robust way to solve non-linear problems. They are also useful to solve linear problems since these do not necessarily have Gaussian model parameter distributions (Beghein 00), and because they can sample the model null-space such techniques allow us to explore a larger part of the model space. The NA is composed of two stages. During a first stage the model space is sampled randomly, and a cost function is calculated to determined how well each model explains the data. At each iteration, the number of models generated increases in the vicinity of the best fitting regions of the model space. The first stage of the NA differs from many other Monte Carlo techniques in that its objective is not to locate a single optimal model, but to obtain an overview of the model space. It therefore keeps track of all the models generated instead of discarding the worse data fitting models at each iteration. In this work, we solved the problem fordc anddc separately at each grid cell, i.e. we ran the NA = times fordc and times fordc, searching the model space for parameters each time: spline parameters forg c (org s ), spline parameters forb c (orb s ) and spline parameters for H c (or H s ). Each model parameter (i.e., dlng i c,s, dlnbc,s, j and dlnhc,s j for i =,..., and j =,..., ) was allowed to vary uniformly between 0.0 and +0.0 around model YBSVaniSVD, which resulted from a regularized inversion as explained in section.. This range was selected to allow most parameters to change sign if required by the data. In a second stage, a Baysian appraisal of all the models is performed. Unlike other statistical techniques, such as importance sampling, that draw inferences on the models using only a subset of the ensemble of models generated, the NA makes use of all the models, good and bad, generated during the first stage. As pointed out by Sambridge ( b), in some cases one can learn from the models that fit the data poorly as much as from those that fit the data well. The entire family of models obtained in the first stage is thus converted into posterior probability density functions (PPDFs) by associating the relatively low and high misfit values to high and low likelihoods, respectively. These PPDFs can be used to assess the robustness of the model parameters. For a PPDF denoted by P(m) where m is a point in the model space, the posterior mean model for the ith parameter is given by the following integral performed over the model space (Sambridge b): < m i >= m i P(m)dm () The posterior variances of the model parameters can be obtained from the diagonals of the posterior model covariance matrix given by:

12 Page of C i,j = Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy m i m j P(m)d(m) < m i >< m j > () Because the entire model space was sampled, the model uncertainties inferred are more accurate than those resulting from regularized inversions which tend to underestimate posterior variance, especially in the presence of a large model the null-space (Trampert ; Beghein & Trampert 00; Beghein 00). The -D marginal distribution of a given model parameter m i can be obtained by integrating P(m) numerically over all other parameters (Sambridge b): d M(m i ) =... P(m) dm k () k=,k i wheredis the total number of model parameters. The shape and width of these -D marginals provide useful information on how well constrained a given parameter is and whether the model distribution is Gaussian, in which case the mean < m i > coincides with the peak of the distribution, i.e. the most likely value. Information about parameter trade-offs can be obtained from the off-diagonal terms of the posterior covariance matrix, and from the -D marginal distributions, which are calculated by integratingp(m) over all but two parameters. The -D marginal PPDF for theith andjth parameters is given by: M(m i,m j ) =... P(m) d k=,k i,k j dm k (0) Figures and display examples of -D and -D marginals at two different grid cells. Clear trade-offs are visible in Figure (c) between theg c spline parameters that peak at 00 km and 00 km depth, and in Figure (c) between the G c spline parameters that peak at 00 km and 0 km depth. While some spline parameters are relatively well constrained and nearly Gaussian (e.g. the -th dlng c spline in Figure (a)), others display double peaks (e.g. the -th dlng c and dlng s splines Figure (b) and (e), respectively), showing that not all model distributions are Gaussian. From the PPDFs of these G i c,s spline parameters, we can reconstruct probability distributions for G c and G s as a function of depth by: (i) Drawing 0,000 random values for each of the G i c,s (i =,...,) coefficients from their posterior -D marginal distributions; (ii) For each set of G i c,s values, calculate the G c,s (r) profile (equations for G c (r) and ), which results in0,000 G c,s (r) models This yields distributions ofdlng c (r) anddlng s (r) models drawn directly from theg i c andg i s PPDFs at each grid cell. An example is shown in Figure. We note that the mean models as identified by the NA do not necessarily coincide with the inversion results, and that the data generally favor larger amplitudes than obtained from a regularized inversion. This is, of course, to be expected since inverse

13 Page of Yuan, K. and Beghein, C. methods tend to damp the model toward the reference value especially where resolution is poor. It should also be pointed out that the mean model does not necessarily correspond to the best fitting model since not all PPDFs are Gaussian. This is why it is important to not discuss the mean model alone, but to account for its associated uncertainties.. Error Propagation To evaluate the robustness of the features observed in YBSVani (Yuan & Beghein 0), we ideally need to determine the mean amplitude dlng and fast axis direction Θ at each depth and grid cell and their associated uncertainties. However, the formulation of the linearized forward problem described in section. does not allow us to model dlng and Θ directly. Previous attempts at estimating model uncertainties on azimuthal anisotropy with the NA (Yao 0) have focused on the G c and G s uncertainties, and only discussed the azimuthal anisotropy (dlng and Θ) model resulting from the most likely or mean G c and G s only. Uncertainties on dlng and Θ can, however, be reconstructed from thedlng c anddlng s model distributions, as explained below. In this section we discuss and compare two possible methods to estimate uncertainties ondlng and Θ. A first approach consists in reconstructing dlng and Θ distributions by () drawing models from the PPDFs of the individual dlng c (r) and dlng s (r) profiles (Figure ), and () calculating dlng(r) and Θ(r) for each pair of dlng c (r) and dlng s (r) model drawn. This yields distributions of dlng(r) and Θ(r) models drawn directly from the dlng c and dlng s PPDFs. However, the synthetic tests we performed (Figure ) showed that the model corresponding to the mean of the reconstructed dlng or Θ distributions, hereafter referred to as dlng mean and Θ mean, can differ strongly from the model built with the mean of the individual dlng c and dlng s distributions (referred to as dlng c,mean and dlng s,mean ). In other words, dlng mean dlng s,mean +dlng c,mean and Θ mean arctan(g s,mean/g c,mean ). This is somewhat counter-intuitive since these synthetic tests assume Gaussian distributions. The model corresponding to the mean of theg i c andg i s PPDFs is thus the model that best explains the data, and one might therefore expect that same model to correspond to dlng mean and Θ mean. Our synthetic examples show that the two models differ when the dlng c and dlng s distributions are both centered on zero, i.e. the parameters are not resolved (Figure (f)), or when one the distributions is highly skewed (Figure (g) and (h)). We also see that the PPDFs for Θ are even more sensitive than dlng to the shape of the combined dlng s and dlng c PPDFs. The PPDFs forθcan be highly skewed (Figure (b)), or have multiple peaks (Figure (c)-(f)). Because of these potential issues, we opted for another approach instead, involving the propagation of the errors obtained from the individualdlng c (r) anddlng s (r) PPDFs. This approach assumes the PPDFs are Gaussian distributed, which is clearly an approximation for some parameters (Figures

14 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy to ), but it enables us to avoid possible artifacts such as those seen in the synthetic examples. Let us take a function f that depends on parameters x and y that are assumed to be Gaussian with standard deviationsσ x andσ y, respectively. If we further assume that thexandy variables have no covariance, the variance σ f of function f depends on the variances σ x and σ y of x and y as follows (Clifford ): σ f = ( f x ) σ x + ( ) f σy () y Therefore, forf = G = G s +G c, we can determine that : σ G = G sσ G s +G cσ G c G s +G c And for f = Θ = arctan(g s/g c ): σ Θ = G cσ G s +G sσ G c (G s +G c) Here, we used G c = G c,mean and G s = G s,mean as determined from equation and the variances σ G c and σ G s result from the off-diagonals of the posterior covariance matrix (equation ). The uncertainty maps displayed in section were determined from these error propagation calculations. We assumed no covariance between G s and G c, which is a reasonable approximation since Visser et al. ( 00b) showed there was little covariance between thedc anddc terms of equation. RESULTS AND DISCUSSION. Goodness of Fit Figure 0 compares some of the azimuthally anisotropic phase velocity maps measured by Visser et al. ( 00b) with predictions from the model resulting from a regularized inversion (YBSVaniSVD) and from the mean NA model, i.e. the model corresponding to the mean of the G c (r) and G s (r) distributions. We see that both models can generally reproduce the data well and that the discrepancies are mostly in the amplitudes and less in the fast axes directions. Following Yuan & Beghein ( 0), we calculated an average χ misfit by averaging the χ from theg c model (χ c) and from theg s model (χ s) over all grid cells: χ c,s = N ( ) di (Am) i () N i= σ i χ = N c ( χ N c,i +χ s,i) c i= where N is the total number of data, d i is the ith component of the data vector d, σ i is the ith component of the vector containing data errors, and(am) i is theith component of the data prediction () () ()

15 Page of Yuan, K. and Beghein, C. vector Am calculated using equations and. N c is the total number of grid cells. Table gives the average χ misfit for each model, and confirms that the two models can explain the data within uncertainties. Table compares the average variance reduction for the two models, using the following definition of the variance reduction: N i= VR = (d i (Am) i ) () N i= d i It shows that model YBSVaniSVD explains the data better than the mean NA model, which we attribute to the fact that the mean NA model does not correspond exactly to one of the best data-fitting models due to the non-gaussian nature of the posterior model -D distributions.. Global Averages Figure represents the root mean square (rms) of the relative anisotropy amplitude and of the vertical gradient of the fast axes (dθ/dr) calculated for the mean NA model and for YBSVaniSVD. As in Yuan & Beghein ( 0), we determined dθ/dr at each grid cell every 0 km depth with a 0 km window, after which we determined its rms as a function of depth. We see that even though the -D average of the mean NA model presents a few more oscillations below 00 km depth than the model obtained by regularized inversion, the two models display similar features. We observe. % to % anisotropy in the top 00 km and about % below, down to at least the bottom of the TZ. We also detect amplitude minima between 0 km and 00 km, around 0 km and 0 km, and near the boundaries of the TZ. These minima are associated with higher gradients in the fast axes direction, as observed in YBSVani, which could be interpreted as global changes in mantle flow direction. The interpretation of these results is, however, non-unique because too few mineral physics data are available on the anisotropy of mantle TZ material, and on the effect of pressure and/or composition on the relation between single crystal anisotropy and large-scale seismic anisotropy. Nevertheless, they do provide new constraints on deep upper mantle circulation.. -D Models Figures to are maps that compare model YBSVaniSVD and the mean NA model at different depths. In Figures and, both the mean NA model and the fast axes standard deviation are displayed. The fast axes standard deviation was estimated at each grid cell by error propagation (equation ). Figures and focus on the anisotropy amplitude and its standard deviation (equation ). The two models show very similar fast axes directions at most depths and comparable amplitudes in the upper 0 km of the mantle. They are also consistent with previous studies (Nishimura & Forsyth

16 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy ; Montagner & Tanimoto ; Debayle et al. 00) in the top 00 km. The model amplitudes are slightly stronger for the mean NA model at greater depths, as expected from Figures to, but the differences between the models are generally within the model uncertainties. The strongest and best resolved model amplitudes (of at least % to % anisotropy) in the top 0 km can be found in the youngest parts of the Pacific plate, at the Africa-Eurasia plate boundary, and around the South American subduction zone. Lower amplitudes are seen in the western Pacific at these depths. These low amplitudes were first detected by Nishimura & Forsyth (Nishimura & Forsyth ) who related them to changes in the horizontal direction of anisotropic fabric with depth rather than being due to a decrease of in situ anisotropy. In Yuan & Beghein ( 0), we showed, however, that the lower SV anisotropy amplitude in the western Pacific is close to the average amplitudes of other oceanic plates and is therefore not anomalously low with respect to other plates. We also note that while seismic anisotropy amplitudes are anomalously high in the shallow mantle in the middle of the Pacific plate, it is one of the places where the amplitude is the weakest at greater depths, sugesting a relatively shallow origin for this signal, such as asthenospheric mantle flow as previously suggested from radial anisotropy models (Ekström & Dziewonski ; Gaboret et al. 00). We also find a relatively strong signal of about % anisotropy at 00 km depth near the India-Eurasia convergence zone and in the Indonesian subduction region. Amplitude uncertainties are closer to the mean model amplitudes at greater depths, except in a few locations between 00 km and 0 km such as the Western part of the Pacific where subduction occurs, near the Arabian plate, and India. Below 0 km depth, a stronger, well-resolved signal appears in the northwestern part of the Pacific and Asia. Strong discrepancies exist between model YBSVani model (Yuan & Beghein 0) and the uppermost mantle model of Marone & Romanowicz ( 00) under North America. In Yuan & Beghein ( 0), we attributed this disagreement to differences in the horizontal resolution of the models. Here, we see that the uncertainties in the fast axes directions at 00 km are strong beneath this region, which would reconcile the differences between the models. Uncertainties on the fast axes are also a little stronger toward the western part of the Pacific. As we go deeper, more regions display larger standard deviations in the fast axes direction, but a few features appear well constrained. For instance, the fast seismic direction is close to the absolute plate motion (Gripp & Gordon 00) beneath the young and mid Pacific plate down to about 0 km depth (Figure ), though model YBSVaniSVD appears to reflect the plate motion slightly better than the mean NA model (see for instance in the younger parts of the Pacific plate). Some of the most interesting features detected in YBSVani (Yuan & Beghein 0) were the changes in the average azimuthal anisotropy fast directions associated with amplitude minima at about 0 km depth, and near the TZ boundaries. In Yuan & Beghein ( 0), we showed that the rms am-

17 Page of Yuan, K. and Beghein, C. plitude and the rms of dθ/dr were stable with respect to regularization, to the presence of seismic discontinuties in the reference model, and to the presence of lateral heterogeneities in mantle. However, the robustness of these features is better tested with quantitative model uncertainties, but these are difficult to determine and to display for rms(dlng) and rms(dθ/dr). Another question that can arise is whether these changes in azimuthal anisotropy are global or appear only at a few locations. If they occur globally, they might be interpreted purely by the effect of pressure on TZ material anisotropy. If they occur only in a few regions, compositional effects might come into play. To answer these questions, we calculated the difference Θ between the fast seismic direction at two selected depths (d and d ), which we divided by the uncertainty σ obtained from the variance of the fast axis directions (σd and σ d σ ) at each of these two depths:σ = d +σ d. The results are shown in Figure. We consider the change in azimuthal anisotropy to be robust if the difference between the fast axe directions is larger than the standard deviation. However, where the fast axes difference is lower than < 0, it may be wiser to consider the difference to not be significant even if the uncertainty is small since there could be other sources of errors in the model resulting from errors in the phase velocity maps that are not accounted for in the uncertainties estimated with the NA (see discussion in section ). From Figure, we see the fast axes direction does not appear to change significantly at all latitudes and longitudes around 0 km depth, and we find no clear correlation with plate tectonics. It is therefore not likely a global signal and may be reflect changes in composition and/or random errors or inconsistencies in the data. On the contrary, the fast axes direction changes significantly at most latitudes and longitudes near the top and the bottom of the TZ. Only a few locations (e.g., the Northwestern Pacific at the top of the TZ) are associated with small to no changes (< 0 to < 0 ) inθ, and there is no clear relation between places where Θ is small at the top and the bottom of the TZ. From this we could argue that the change in azimuthal anisotropy at the TZ boundaries is close to being a global signal and therefore relates to phase changes, though we cannot completely rule out laterally variable compositional effects as well. Figure shows the correlation coefficient between the two models as a function of depth. The correlation was calculated at various depths after expansion of each map in generalized spherical harmonics up to degree 0, following Yuan & Beghein ( 0). At all depths, the correlation is well above the % significance level as calculated by Becker et al. ( 00), confirming the general agreement between the models seen in the maps of Figures to. CONCLUSIONS The goal of this research was to present a new method to model and obtain quantitative uncertainties on -D azimuthal seismic anisotropy. It was then applied to global higher mode surface wave phase

18 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy velocity data to assess the likelihood of azimuthal anisotropy in the deep upper mantle. For this, we employed the Neighbourhood Algorithm developed by Sambridge ( a; b), a model space search approach that enables the user to search the whole model space, including the null space, without the need to introduce strong prior information, and to perform a Bayesian assessment of the models obtained. The resulting model parameter uncertainties are generally more robust than estimates from regularized inversions, especially in the presence of a large model null-space. The relative misfit highs and lows are converted into a posterior probability density function, which enables us to determine which parameters are well constrained by the data alone. Even though the linearized formulation of the problem relating phase velocities to azimuthal anisotropy at depth does not allow us to directly obtain uncertainties on the anisotropy amplitude and fast axes direction, we showed that that they can be determined a posteriori. We compared two approaches to calculate these uncertainties from the posterior model parameter distributions resulting from the NA. Of the two approaches, we concluded that a simple error propagation calculation yielded the most reliable estimates of the mean anisotropy amplitude and fast axes direction and of their respective variances. The model space search resulted in an distribution of models from which a mean model and standard deviation were obtained. The mean model was found to be overall consistent with a model obtained by regularized inversion with the same dataset and parameterization. The correlation between the two models was above the % significance level, and the fast axes directions were generally consistent between the two models. The mean NA model displayed slightly larger amplitudes than the model resulting from a regularized inverse method, which is to be expected since regularization tends to lower amplitudes. We confirmed that the change in average azimuthal anisotropy at the TZ boundaries characterizing our previously published model, YBSVani, is constrained by the data and did result from artefact or parameter trade-offs that can affect a model obtained by the linear inversion. We also showed that while the change in fast seismic direction at the TZ boundaries is approximately a global feature of the best fitting models, the change in anisotropy identified around 0 km depth is not significant at all latitudes and longitudes, but no clear correlation with surface plate tectonics was found. ACKNOWLEDGMENTS We wish to thank Karin Visser and Jeannot Trampert for making the phase velocity maps freely available online at and Malcom Sambridge for sharing his Neighbourhood Algorithm. Partial derivatives were calculated using program MINEOS (available on the CIG website athttp:// and fig-

19 Page of Yuan, K. and Beghein, C. ures were made using the Generic Mapping Tool, Gnuplot, and Xmgrace. This research was partially funded by NSF EAR grants 0 and 0. REFERENCES Auer, L., Boschi, L., Becker, T., Nissen-Meyer, T., & Giardini, D., 0. Savani: A variable resolution wholemantle model of anisotropic shear velocity variations based on multiple data sets, J. geophys. Res., (), 00-0, doi:0.00/0jb00 Bassin, C., Laske, G. & Masters, G., 000. The Current Limits of Resolution for Surface Wave Tomography in North America, EOS,, F Becker, T. W., Kellogg, J. B., Ekström, G., & O Connell, R. J., 00. Comparison of azimuthal seismic anisotropy from surface waves and finite strain from global mantle-circulation models, Geophys. J Int., (), -, doi:0.0/j.-x x Becker, T. W., Ekström, G., Boschi, L., & Woodhouse, J. D., 00. Length scales, patterns and origin of azimuthal seismic anisotropy in the upper mantle as mapped by Rayleigh waves, Geophys. J Int., (), -, doi:0./j.-x.00.0.x Becker, T. W., Conrad, C.P., Schaeffer, A.J., & Lebedev, S., 0. Origin of azimuthal seismic anisotropy in oceanic plates and mantle, Earth Planet. Sci. Lett., 0(), -0, doi:0.0/j.epsl Beghein, C., 00. Radial anisotropy and prior petrological constraints: A comparative study, J. geophys. Res.,, B00, doi:0.0/00jb00 Beghein, C. & Trampert, J., 00. Robust normal mode constraints on inner-core anisotropy from model space search, Science,, -, doi:0./science.0 Beghein, C. & Trampert, J., 00a. Probability density functions for radial anisotropy from fundamental mode surface wave data and the neighbourhood algorithm, Geophys. J Int.,, -, doi:0./j.- X.00.0.x Beghein, C. & Trampert, J., 00b. Probability density functions for radial anisotropy: Implications for the upper 00 km of the mantle, Earth Planet. Sci. Lett.,, -, doi:0.0/s00-x(0)00- Beghein, C., Resovsky, J., & Trampert, J., 00. P and S tomography using normal mode and surface wave data with a Neighbourhood Algorithm, Geophys. J Int., (), -, doi:0.0/j.- X.00.0.x Beghein, C., Trampert, J., & van Heijst, H. J., 00. Radial anisotropy in seismic reference models of the mantle, J. geophys. Res.,, B00, doi:0.0/00jb00 Beghein, C., Resovsky, J., & van der Hilst, R. D., 00. The signal of mantle anisotropy in the coupling of normal modes, Geophys. J Int., (), 0-, doi:0./j.-x x Beghein, C., Yuan, K., Schmerr, N., & Xing, Z., 0. Changes in seismic anisotropy shed light on the nature of the Gutenberg discontinuity, Science,, -0, doi:0./science. Boschi, L., & Ekström, G., 00. New images of the Earth s upper mantle from measurements of surface wave phase velocity anomalies, J. geophys. Res., 0, B, doi:0.0/000jb0000

20 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy Bozdaǧ, E. & Trampert, J., 00. Assessment of tomographic mantle models using spectral element seismograms, Geophys. J Int., 0(), -, doi:0./j.-x.00.0.x Chang, S., Ferreira, A., Ritsema, J., van Heijst, H., & Woodhouse, J. H., 0. Global radially anisotropic mantle structure from multiple datasets: A review, current challenges, and outlook, Tectonophysics,, -, doi:0.0/j.tecto Chen, W. P. & Brudzinski, M. R., 00. Seismic anisotropy in the mantle transition zone beneath Fiji-Tonga, Geophys. Res. Lett., 0(),, doi:0.0/00gl0 Clifford, A.A.,. Multivariate error analysis: a handbook of error propagation and calculation in manyparameter systems, Wiley, New York. Debayle, E. & Ricard, Y., 0. Seismic observations of large-scale deformation at the bottom of fast-moving plates, Earth Planet. Sci. Lett.,, -, doi:0.0/j.epsl Debayle, E., Kennett, B., & Priestley, K., 00. Global azimuthal seismic anisotropy and the unique platemotion deformation of Australia, Nature, (0), 0-, doi:0.0/nature0 Dziewonski, A. M. & Anderson, D. L.,. Preliminary reference Earth model, Phys. Earth Planet. Inter., (), -, doi:0.0/00-0()00- Ekström, G., 0. A global model of Love and Rayleigh surface wave dispersion and anisotropy, -0 s, Geophys. J Int., (), -, doi:0./j.-x.0.0.x Ekström, G., & Dziewonski, A. M.,. The unique anisotropy of the Pacific upper mantle, nature, (), -, doi:0.0/ Foley, B. J. & Long, M. D., 0. Upper and mid-mantle anisotropy beneath the Tonga slab, Geophys. Res. Lett., (), L00, doi:0.0/00gl0 Fouch, M. J. & Fischer, K. M.,. Mantle anisotropy beneath northwest Pacific subduction zones, J. geophys. Res., 0(B), -0, doi:0.0/jb00 French, S. W. & Romanowicz, B., 0. Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography, Geophys. J Int., (), 0-, doi:0.0/gji/ggu Gaboret, C., Forte, A.M., & Montagner, J.-P., 00. The unique dynamics of the Pacific Hemisphere mantle and its signature on seismic anisotropy, Earth Planet. Sci. Lett., 0(-), -, doi:0.0/s00- X(0)000- Gripp, A.E. & Gordon, R.G., 00. Young tracks of hotspots and current plate velocities, Geophys. J Int., 0(), -, doi:0.0/j.-x.00.0.x Hu, X.-G., Xue, X.-X., Liu, L.-T., & Sun, H.-P., 0. Normal mode coupling due to azimuthal anisotropy in the transition zone: an example from Taiwan Island, Geophys. J Int., 0(), -, doi:0./j.- X.0.0.x Karato, S.,. Seismic anisotropy in the deep mantle, boundary layers and the geometry of mantle convection, Pure Appl. Geophys.,(-), -, doi:0.00/s Karato, S., Zhang, S., & Wenk, H.-R.,. Superplasticity in Earth s lower mantle - evidence from seismic anisotropy and rock physics, Science, 0(), -, doi:0./science.0..

21 Page 0 of Yuan, K. and Beghein, C. Karato, S.-i.,. Seismic anisotropy: Mechanisms and tectonic implications, in Rheology of Solids and of the Earth, pp. -, eds Karato, S.-I. & Toriumi, M., Oxford University Press, New York. Kendall, J.-M. & Silver, P. G.,. Constraints from seismic anisotropy on the nature of the lowermost mantle, Nature, (), 0-, doi:0.0/0a0 Kosarian, M., and Davis, P. M., Tanimoto, T., & Clayton, R. W., 0. The relationship between upper mantle anisotropic structures beneath California, transpression, and absolute plate motions, J. geophys. Res., (B), B00, doi:0.0/00jb00 Kustowski, B., Dziewonski, A., & Ekström, G., 00. Nonlinear crustal corrections for normal-mode seismograms, Bull. seism. Soc. Am., (), -, doi:0./00000 Kustowski, B., Ekström, G., & Dziewonski, A., 00. Anisotropic shear-wave velocity structure of the Earth s mantle: A global model, J. geophys. Res., (B), B0, doi:0.0/00jb00 Lebedev, S., Adam, J.M.C., & Meier, T., 00. Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics, (C), -, doi:0.0/j.tecto Long, M. D., 0. Constraints on subduction geodynamics from seismic anisotropy, Rev. Geophys., (), -, doi:0.00/rog.000 Love, A. E. H.,. A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cambridge Marone, F. & Romanowicz, B., 00. The depth distribution of azimuthal anisotropy in the continental upper mantle, Nature, (), -0, doi:0.0/nature0 Matsu ura, M. and Hirata, N.,. Generalized least-squares solutions to quasi-linear inverse problems with a priori information, Journal of Physics of the Earth, 0(), -, doi:0./jpe.0. Maupin, V., Garnero, E. J., Lay, T., & Fouch, M. J., 00. Azimuthal anisotropy in the D layer beneath the Caribbean, J. geophys. Res., 0, B00, doi:0.0/00jb000 Meier, U., Curtis, A., & Trampert, J., 00. Global crustal thickness from neural network inversion of surface wave data, Geophys. J Int., (), B0-, doi:0./j.-x.00.0.x Montagner, J.-P.,. Can seismology tell us anything about convection in the mantle?, Rev. Geophys., (), -, doi:0.0/rg000 Montagner, J.-P., and Nataf, H.-C.,. A simple method for inverting the azimuthal anisotropy of surface waves, J. geophys. Res.,, -0, doi:0.0/jb0ib0p00 Montagner, J.-P. & Tanimoto, T.,. Global upper mantle tomography of seismic velocities and anisotropies, J. geophys. Res.,, B, 0-0, doi:0.0/jb00 Montagner, J.-P., and Kennett, B.L.N.,. How to reconcile body-wave and normal-mode reference Earth models, Geophys. J Int., (), -, doi:0./j.-x..tb0.x Moulik, P. & Ekström, G., 0. An anisotropic shear velocity model of the Earth s mantle using normal modes, body waves, surface waves and long-period waveforms, Geophys. J Int., (), -, doi:0.0/gji/ggu

22 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy Nicolas, A. & Christensen, N.,. Formation of anisotropy in upper mantle peridotites - a review, in Composition, Structure, and Dynamics of the Lithosphere-Asthenosphere System, eds Fuchs, A. & Froideveaux, C., American Geophysical Union, Washington, D. C.. doi: 0.0/GD0p0 Nishimura, C. E. & Forsyth, D. W.,. Rayleigh wave phase velocities in the Pacific with implications for azimuthal anisotropy and lateral heterogeneities, Geophys. J Int., (), -0, doi:0./j.- X..tb00.x Nishimura, C. E. & Forsyth, D. W.,. The anisotropic structure of the upper mantle in the Pacific, Geophysical Journal, (), 0-, doi:0./j.-x..tb0.x Oganov, A., 00. Anisotropy of Earth s D layer and stacking faults in the MgSiO post-perovskite phase, Nature,, -, doi:0.0/nature0 Panning, M. & Romanowicz, B., 00. Inferences on Flow at the Base of Earth s Mantle Based on Seismic Anisotropy, Science, 0(), -, doi:0./science.0 Panning, M. & Romanowicz, B., 00. A three-dimensional radially anisotropic model of shear velocity in the whole mantle, Geophys. J Int., (), -, doi:0./j.-x x Panning, M. P., Lekić, V., & Romanowicz, B., 00. Importance of crustal corrections in the development of a new global model of radial anisotropy, J. geophys. Res., (B), B, doi:0.0/00jb000 Romanowicz, B. & Lekić, V., 0. Inferring upper-mantle structure by full waveform tomography with the spectral element method, Geophys. J Int., (), -, doi:0./j.-x.0.0.x Sambridge, M., a. Geophysical inversion with a neighbourhood algorithm - I. Searching a parameter space, Geophys. J Int., (), -, doi:0.0/j.-x..00.x Sambridge, M., b. Geophysical inversion with a neighbourhood algorithm - II. Appraising the ensemble, Geophys. J Int., (), -, doi:0.0/j.-x x Silver, P. G.,. Seismic anisotropy beneath the continents: Probing the Depths of Geology, Ann. Rev. Earth Planet Sci., (), -, doi:0./annurev.earth... Smith, D. B., Ritzwoller, M. H., & Shapiro, N. M., 00. Stratification of anisotropy in the Pacific upper mantle, J. geophys. Res., 0(B), doi:0.0/00jb0000 Snoke, J. A. & Sambridge, M., 00. Constraints on the S wave velocity structure in a continental shield from surface wave data: Comparing linearized least squares inversion and the direct search Neighbourhood Algorithm, J. geophys. Res., 0(B), doi:0.0/00jb000 Takeuchi, H. & Saito, M.. Seismic surface waves, in Methods in Computational Physics, pp. -, ed. Bolt B., Academic Press, New York. Tarantola, A.,. Inverse Problem Theory and Methods for Model Parameter Estimation, Elsevier, New York. Trampert, J.,. Global seismic tomography: the inverse problem and beyond, Inverse Problems,, -, doi:0.0/0-///00 Trampert, J. & van Heijst, H. J., 00. Global Azimuthal Anisotropy in the Transition Zone, Science, (), -, doi:0./science.00 Trampert, J. & Woodhouse, J. H., 00. Global anisotropic phase velocity maps for fundamental mode surface

23 Page of Yuan, K. and Beghein, C. waves between 0 and 0 s, Geophys. J Int., (), -, doi:0.0/j.-x.00.0.x Visser, K., Trampert, J., and Kennett, B. L. N., 00b. Global anisotropic phase velocity maps for higher mode Love and Rayleigh waves, Geophys. J Int., (), 0-0, doi:0./j.-x.00.0.x Visser, K., Trampert, J., Lebedev, S., and Kennett, B. L. N., 00a. Probability of radial anisotropy in the deep mantle, Earth Planet. Sci. Lett., 0, -0, doi:0.0/j.epsl Wookey, J. & Barruol, G., 00. Mid-mantle deformation inferred from seismic anisotropy, Nature,, - 0, doi:0.0/a Yao, H., 0. A method for inversion of layered shear wavespeed azimuthal anisotropy from Rayleigh wave dispersion using the Neighborhood Algorithm, Earthq. Sci. (), -, doi:0.00/s Yao, H., Beghein, C., & van der Hilst, R.D., 00. Surface-wave array tomography in SE Tibet from ambient seismic noise and two-station analysis: II Crust and upper mantle structure, Geophys. J Int., (), 0-, doi:0./j.-x.00.0.x Yuan, K., & Beghein, C., 0. Seismic anisotropy changes across upper mantle phase transitions, Earth Planet. Sci. Lett.,, -, doi:0.0/j.epsl Yuan, K., & Beghein, C., 0. Three-dimensional variations in Love and Rayleigh wave azimuthal anisotropy for the upper 00 km of the mantle, J. geophys. Res., (), -, doi:0.00/0jb00

24 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy Table. Averageχ misfit for the model obtained by regularized inversion and for the mean NA model. Model n = 0 n = 0 n = n = n = n = n = n = YBSVaniSVD Mean NA model Table. Average variance reduction for the model obtained by regularized inversion and for the mean NA model. Model n = 0 n = 0 n = n = n = n = n = n = YBSVaniSVD Mean NA model

25 Yuan, K. and Beghein, C. DPK00 0 Page of 0 DR0 YBSVani SL0SVA % 00km % 00km % 00km % 00km % 0km % 0km % 0km % 0km % 00km % 00km % 00km % 00km % 00km % 00km % 00km % 00km dlng (%) 0 dlng (%) 0 dlng (%) 0 dlng (%) Figure. Azimuthal anisotropy models in the top 00 km of the mantle from previous studies: DPK00 (Debayle et al. 00), DR0 (Debayle & Ricard 0), SL0SVA (Becker et al. 0), and YBSVani (Yuan & Beghein 0).

26 Page of (a) (c) Ψ - T = s % S dc/c (%) Ψ - T = s % S dc/c (%) Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy Depth [km] Depth [km] G B H Kernels [/km] G Kernels [/km] B H (b) (d) Ψ - T = s % S dc/c (%) Ψ - T = s % S dc/c (%) Depth [km] Depth [km] G Kernels [/km] B H G B H ] Kernels [/km] Figure. Ψ azimuthally anisotropic Rayleigh wave phase velocity maps (Visser et al. 00b) at s period for the fundamental mode (a), the first (b), second (c), and fifth (d) overtone, and associated partial derivatives. The sensitivity kernels were calculated using reference model PREM (Dziewonski & Anderson ) for relative perturbations in parameters G, B, and H.

27 Page of Yuan, K. and Beghein, C n=0 0 n= Depth [km] n= n= n= n= n= G/L Partial derivative [/km] Figure. Phase velocity partial derivatives for relative perturbations in vertically polarized shear-wave azimuthal anisotropy. These sensitivity kernels were calculated using model PREM for the fundamental modes and first six higher modes employed in this study.

28 Page of Depth [km] Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy (a) (b) Spine amplitude Spine amplitude Figure. Splines functions employed (a) to parameterize G c (r) and G s (r) and (b) to parameterize H c (r), H s (r),b c (r), and B s (r).

29 Page of Yuan, K. and Beghein, C. D E F G c QGVSOLQHSDUDPHWHUNP G H I G s QGVSOLQHSDUDPHWHUNP Figure. Examples of -D and -D marginal distributions at the grid cell located at longitude and 0 latitude. The -D marginals are represented by the thick grey curves. The 0 % and % confidence contours are shown by the solid white lines in the -D marginal plots. A 0 % contour means that the area inside the contour equals to 0 % of the total area. Both the -D and the -D marginals were normalized to. The black cross indicates the location of the regularized inversion result around which the model space seacrh was performed.

30 Page of D E F G c Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy 0. G H I G s Figure. Examples of -D and -D marginal distributions at the grid cell located at longitude and 0 latitude. The -D marginals are represented by the thick grey curves. The 0 % and % confidence contours are shown by the solid white lines in the -D marginal plots. A 0 % contour means that the area inside the contour equals to 0 % of the total area. Both the -D and the -D marginals were normalized to. The black cross indicates the location of the regularized inversion result around which the model space seacrh was performed.

31 Page 0 of Yuan, K. and Beghein, C. Depth [km] (a) (b) G c /L (%) G s /L (%) Normalized PPDF Figure. Example of normalized PPDFs for (a) G c /L, (b) G s /L, (c) relative anisotropy amplitude G/L, and (d) fast direction Θ at longitude and 0 latitude. L is the Love ( ) elastic coefficient that controls the speed of vertically polarized shear waves. The G/L and Θ distributions were obtained by drawing random models within the PPDFs determined by the NA for the G c and G s spline coefficients. The solid black lines represent the mean values of the distributions, the dashed black lines represent one standard deviation, and the solid white line is from model YBSVaniSVD at that same location. The dashed white line in panels (c) and (d) represents the relative anisotropy amplitude and fast direction, respectively, obtained from the mean G c /L and the meang s /L models.

32 Page of (a) (c) (e) (g) mean = 0. σ = 0.0 Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy mean = -0. σ = dlng s dlng c dlng O mean = 0 σ = dlng s dlng c dlng mean = 0. σ = 0.00 mean = 0. σ = dlng mean = -0. σ = O dlng s dlng c dlng dlng s dlng c 0 mean = 0 σ = O O mean = -0. σ = 0.0 (b) (d) (f) (h) mean = 0. σ = 0.0 mean = 0. σ = dlng s dlng c dlng O mean = 0 σ = dlng s dlng c dlng mean = 0 σ = 0.0 dlng mean = 0. σ = O dlng s dlng c mean = -0. σ = 0.0 mean = 0 σ = O mean = 0. σ = dlng s dlng c dlng Figure. Synthetic examples of resampled dlng c and dlng s PPDFs to calculate dlng and Θ distributions. Each of the six panels corresponds to different dlng c and dlng s distributions, with different means and variances as indicated in the legends. In each panel, the black vertical line in the dlng c and dlng s indicates the mean of the dlng c and dlng s PPDFs. The black vertical line in the reconstructed dlng and Θ distributions indicates the value ofdlng andθcalculated from the mean of thedlng c and dlng s distributions. O

33 Page of Yuan, K. and Beghein, C Depth [km] Correlation coefficient Figure. Correlation between YBSVaniSVD and the mean NA model obtained from the mean G s (r) and mean G c (r) distributions.

34 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy Data YBSVaniSVD Prediction Mean NA model Prediction (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (k) (l) S 0 T=s S T=s S T=s S 0 T=s dc/c (%) dc/c (%) dc/c (%) Figure 0. Phase velocity maps measured by Visser et al. ( 00b) (left) compared to predictions from the mean NA model (middle) and from the model obtained by singular value decomposition (right) the s Rayleigh wave fundamental mode ((a)-(c)), first overtone ((d)-(f)), second overtone ((g)-(i)), and fifth ((j)-(l)) overtone..

35 Page of Yuan, K. and Beghein, C Depth [km] YBSVaniSVD NA dlng [%] Fast Axis Vertical Gradient [$/km] Figure. -D average model amplitude (left) and gradient of the fast axis direction (right) calculated from the mean G s and meang c distributions. Model YBSVaniSVD is shown by the black curves for comparison.

36 Page of

37 Page of

38 Page of

39 Page of

40 Page of Bayesian Uncertainties in Upper Mantle Azimuthal Anisotropy % peak to peak anisotropy mm/yr (a) (b) (c) 00 km 0 km 00 km (d) (e) (f) 00 km 0 km 00 km Figure. Comparison between the APM directions (yellow arrows) and the fast seismic directions (red bars) for the mean NA model ((a)-(c)) and YBSVaniSVD ((d)-(f)). The APM was calculated using the no-net rotation reference model NNR-NUVEL A (Gripp & Gordon 00). The mean NA model fast direction is plotted with its standard deviation (blue). Plate boundaries are shown by black lines, and continents are delimited by grey lines.

41 Page 0 of Yuan, K. and Beghein, C. Between 00 km and 0 km (a) (d) Between 00 km and 0 km (b) (e) Between km and 0 km (c) (f) Θ (degrees) Θ/σ Figure. (a)-(c): Difference in fast axes direction Θ between 00 km and 0 km, between 00 km and 0 km, and between km and 0 km depth; (d)-(f): Ratio of the difference in fast axes direction to standard deviation σ at these same depths depth. These depth windows were chosen to encompass the depths at which high gradients in fast seismic direction were visible in both the rms of the NA mean model and YBSVaniSVD.