Finite-frequency Sensitivity Kernels Research Articles

TOMOGRAPHY OF PLASMA FLOWS IN THE UPPER SOLAR CONVECTION ZONE USING TIME-DISTANCE INVERSION COMBINING RIDGE AND PHASE-SPEED FILTERING

The consistency of time--distance inversions for horizontal components of the plasma flow on supergranular scales in the upper solar convection zone is checked by comparing the results derived using two k--\omega filtering procedures -- ridge filtering and phase-speed filtering -- commonly used in time--distance helioseismology. It is shown that both approaches result in similar flow estimates when finite-frequency sensitivity kernels are used. It is further demonstrated that the performance of the inversion improves (in terms of simultaneously better averaging kernel and lower noise level) when the two approaches are combined together in one inversion. Using the combined inversion I invert for horizontal flows in the upper 10 Mm of the solar convection zone. The flows connected with supergranulation seem to be coherent only in the upper ~5 Mm depth, deeper down there is a hint on change of convection scales towards structures larger than supergranules.

Finite-frequency sensitivity kernels for two-station surface wave measurements

SUMMARY We present a method for the computation offinite-frequency sensitivity kernels for two-station surface wave measurements. It is based on the combination of spectral-element modelling of seismic wave propagation, adjoint techniques and traveltime estimates from two-station crosscorrelations of seismograms. The analysis of sensitivity kernels for a 1-D earth model resulted in two major conclusions: (1) The finite-frequency sensitivity is zero along the interstation ray path for group velocity measurements obtained by cross-correlations. It follows that interstation group velocity measurements should not be made by cross-correlation if a raybasedinterpretationisused.(2)Althoughsensitivityalongtheinterstationraypathisdominant for phase velocity measurements, sensitivity far from the ray path can be large, depending on the details of the source–receiver geometry. The complexities of finite-frequency two-station sensitivity kernels should be taken into account to avoid misinterpretations and to improve the quality of tomographic inversions.

Optimized discrete wavelet transforms in the cubed sphere with the lifting scheme-implications for global finite-frequency tomography

SUMMARY Wavelets are extremely powerful to compress the information contained in finite-frequency sensitivity kernels and tomographic models. This interesting property opens the perspective of reducing the size of global tomographic inverse problems by one to two orders of magnitude. However, introducing wavelets into global tomographic problems raises the problem of computing fast wavelet transforms in spherical geometry. Using a Cartesian cubed sphere mapping, which grids the surface of the sphere with six blocks or ‘chunks’, we define a new algorithm to implement fast wavelet transforms with the lifting scheme. This algorithm is simple and flexible, and can handle any family of discrete orthogonal or bi-orthogonal wavelets. Since wavelet coefficients are local in space and scale, aliasing effects resulting from a parametrization with global functions such as spherical harmonics are avoided. The sparsity of tomographic models expanded in wavelet bases implies that it is possible to exploit the power of compressed sensing to retrieve Earth’s internal structures optimally. This approach involves minimizing a combination of a l2 norm for data residuals and a l1 norm for model wavelet coefficients, which can be achieved through relatively minor modifications of the algorithms that are currently used to solve the tomographic inverse problem.

Seismic waveform sensitivity to global boundary topography

SUMMARY We investigate the implications of lateral variations in the topography of global seismic discontinuities, in the framework of high-resolution forward modelling and seismic imaging. We run 3-D wave-propagation simulations accurate at periods of 10 s and longer, with Earth models including core–mantle boundary topography anomalies of ∼1000 km spatial wavelength and up to 10 km height. We obtain very different waveform signatures for PcP (reflected) and Pdiff (diffracted) phases, supporting the theoretical expectation that the latter are sensitive primarily to large-scale structure, whereas the former only to small scale, where large and small are relative to the frequency. PcP at 10 s seems to be well suited to map such a small-scale perturbation, whereas Pdiff at the same frequency carries faint signatures that do not allow any tomographic reconstruction. Only at higher frequency, the signature becomes stronger. We present a new algorithm to compute sensitivity kernels relating seismic traveltimes (measured by cross-correlation of observed and theoretical seismograms) to the topography of seismic discontinuities at any depth in the Earth using full 3-D wave propagation. Calculation of accurate finite-frequency sensitivity kernels is notoriously expensive, but we reduce computational costs drastically by limiting ourselves to spherically symmetric reference models, and exploiting the axial symmetry of the resulting propagating wavefield that collapses to a 2-D numerical domain. We compute and analyse a suite of kernels for upper and lower mantle discontinuities that can be used for finite-frequency waveform inversion. The PcP and Pdiff sensitivity footprints are in good agreement with the result obtained cross-correlating perturbed and unperturbed seismogram, validating our approach against full 3-D modelling to invert for such structures.

Imaging mantle plumes with instantaneous phase measurements of diffracted waves

In a synthetic tomographic experiment, we succeeded to recover an idealized narrow mantle plume reaching deep into the lower mantle by using a misfit based on the instantaneous phase difference. A misfit based on simple cross-correlation traveltime shifts leaves the lower mantle part of the plume largely unresolved, despite the use of finite-frequency sensitivity kernels. The time-continuous and amplitude-independent instantaneous phase misfit allows us to measure the interaction between direct and diffracted waves as a function of time, which is difficult to capture by simple cross-correlation traveltime measurements. The diffracted waves arriving later than the main phase are essential to improve the tomographic result. The measurement of diffracted waves yields the necessary information to recover the plume correctly even in the lower mantle. The instantaneous phase measurement is ideal to capture this interaction, but other time- or frequency-dependent measurements may give similar results. We also investigated the effect of wavefront healing on cross-correlation traveltime shifts for a range of differently sized idealized mantle plumes.We confirm that wavefront healing severely reduces traveltime shifts when the plume conduit is considerably thinner than the width of the first Fresnel zone. For plume conduits with a diameter on the order of 100 km, even traveltime shifts measured at periods as short as T = 1 s are affected.

Tomography of the 1995 Kobe earthquake area: comparison of finite-frequency and ray approaches

We determined a detailed 3-D crustal model in the 1995 Kobe earthquake (M 7.2) area in southwest Japan using both finite-frequency and ray tomography methods. Our finite-frequency tomography technique is based on the single-scattering theory. The finite-frequency sensitivity kernel derived in this study reflects correctly the sensitivity of the heterogeneity off the geometrical ray path and the existence of Fresnel volume, and the kernel depends on the dominant frequency of the observed wave. The dominant frequency is estimated directly from the earthquake magnitude based on a relation that is obtained by regressively analyzing the displacement spectra of 20 earthquakes in the study area. We used 141 118 P-wave and 133 648 S-wave high-quality arrival-time data from 2813 Kobe aftershocks and 3140 other local earthquakes during 1995–2010. The tomographic images obtained with the finite-frequency and ray tomography methods show a high level of similarity, which is verified quantitatively by adopting the structural similarity index. Our results show that the Kobe main shock hypocentre is located in a distinctive zone characterized by a high Poisson′s ratio and a low product VP×VS of P- and S-wave velocities, which is interpreted as a fluid-filled, fractured rock matrix that may have triggered the 1995 Kobe earthquake.

Shear wave velocity structure of the southern African upper mantle with implications for the uplift of southern Africa

SUMMARY Broad-band seismic data from the southern African seismic experiment and the AfricaArray network are used to investigate the seismic velocity structure of the upper mantle beneath southern Africa, and in particular beneath the Kaapvaal Craton. A two-plane approximation method that includes a finite frequency sensitivity kernel is employed to measure Rayleigh wave phase velocities, which are inverted to obtain a quasi-3-D shear wave velocity model of the upper mantle. We find phase velocities for the Kaapvaal Craton and surrounding mobile belts that are comparable to those reported by previous studies, and we find little evidence for variation from east to west across the Namaqua–Natal Belt, a region not well imaged in previous studies. A high-velocity upper-mantle lid is found beneath the Kaapvaal Craton and most of southern Africa. For the Kaapvaal Craton, the thickness of the lid (∼150–200 km) is consistent with the lid thicknesses reported in many previous studies. The cratonic lid is underlain by a ∼100-km thick low-velocity zone with a 3.9 per cent maximum velocity reduction. By comparing the velocity model to those published for other Archean cratons, we find few differences, and therefore conclude that there is little evidence in the shear wave velocity structure of the mantle to indicate that the southern African plateau is supported by an upper-mantle thermal anomaly.

Ambient noise tomography with a large seismic array

The emergence of large-scale arrays of seismometers across several continents presents the opportunity to image the Earth's structure at unprecedented resolution, but methods must be developed to exploit the capabilities of these deployments. The capabilities and limitations of a method called “eikonal tomography” applied to ambient noise data are discussed here. In this method, surface wave wavefronts are tracked across an array and the gradient of the travel time field produces estimates of phase slowness and propagation direction. Application data from more than 1000 stations from EarthScope USArray in the central and western US and new Rayleigh wave isotropic and anisotropic phase velocity maps are presented together with an isotropic and azimuthally anisotropic 3D Vs model of the crust and uppermost mantle. As a ray theoretic method, eikonal tomography models bent rays but not other wavefield complexities. We present evidence, based on the systematics of an observed 1 ψ component of anisotropy that we interpret as anisotropic bias caused by backscattering near an observing station, that finite frequency phenomena can be ignored in ambient noise tomography at periods shorter than ∼ 40 to 50 s. At longer periods a higher order term based on wavefront amplitudes or finite frequency sensitivity kernels must be introduced if the amplitude of isotropic anomalies and the amplitude and fast-axis direction of azimuthal anisotropy are to be determined accurately.

Acoustic, elastic and poroelastic simulations of CO2 sequestration crosswell monitoring based on spectral-element and adjoint methods

SUMMARY The key issues in CO2 sequestration involve accurate monitoring, from the injection stage to the prediction and verification of CO2 movement over time, for environmental considerations. ‘4-D seismics’ is a natural non-intrusive monitoring technique which involves 3-D time-lapse seismic surveys. Successful monitoring of CO2 movement requires a proper description of the physical properties of a porous reservoir. We investigate the importance of poroelasticity by contrasting poroelastic simulations with elastic and acoustic simulations. Discrepancies highlight a poroelastic signature that cannot be captured using an elastic or acoustic theory and that may play a role in accurately imaging and quantifying injected CO2. We focus on time-lapse crosswell imaging and model updating based on Frechet derivatives, or finite-frequency sensitivity kernels, which define the sensitivity of an observable to the model parameters. We compare results of time-lapse migration imaging using acoustic, elastic (with and without the use of Gassmann's formulae) and poroelastic models. Our approach highlights the influence of using different physical theories for interpreting seismic data, and, more importantly, for extracting the CO2 signature from seismic waveforms. We further investigate the differences between imaging with the direct compressional wave, as is commonly done, versus using both direct compressional (P) and shear (S) waves. We conclude that, unlike direct P-wave traveltimes, a combination of direct P- and S-wave traveltimes constrains most parameters. Adding P- and S-wave amplitude information does not drastically improve parameter sensitivity, but it does improve spatial resolution of the injected CO2 zone. The main advantage of using a poroelastic theory lies in direct sensitivity to fluid properties. Simulations are performed using a spectral-element method, and finite-frequency sensitivity kernels are calculated using an adjoint method.

Structure of North American mantle constrained by simultaneous inversion of multiple-frequencySH,SS, and Love waves

[1] We simultaneously invert for the velocity and attenuation structure of the North American mantle from a mixed data set: SH wave traveltime and amplitude anomalies, SS wave differential traveltime anomalies, and Love wave fundamental mode phase delays. All data are measured for multiple frequency bands, and finite frequency sensitivity kernels are used to explain the observations. In the resulting SH velocity model, a lower mantle plume is observed to originate at about 1500 km depth beneath the Yellowstone area, tilting about 40° from vertical. The plume rises up through a gap in the subducting Farallon slab. The SH velocity model confirms high-level segmentation of the Farallon slab, which was observed in the recent P velocity model. Attenuation structure is resolvable in the upper mantle and transition zone; in estimating it, we correct for focusing. High-correlation coefficients between δlnVS and δlnQS under the central and eastern United States suggest one main physical source, most likely temperature. The smaller correlation coefficients and larger slopes of the δlnQS − δlnVS relationship under the western United States suggest an influence of nonthermal factors such as the existence of water and partial melt. Finally, we analyze the influence of the different components of our data set. The addition of Love wave phase delays helps to improve the resolution of both velocity and attenuation, and the effect is noticeable even in the lower mantle.

Segmented African lithosphere beneath the Anatolian region inferred from teleseismic P-wave tomography

SUMMARY Lithospheric deformation throughout Anatolia, a part of the Alpine–Himalayan orogenic belt, is controlled mainly by collision-related tectonic escape of the Anatolian Plate and subduction roll-back along the Aegean Subduction Zone. We study the deeper lithosphere and mantle structure of Anatolia using teleseismic, finite-frequency, P-wave traveltime tomography. We use data from several temporary and permanent seismic networks deployed in the region. Approximately 34 000 P-wave relative traveltime residuals, measured in multiple frequency bands, are inverted using approximate finite-frequency sensitivity kernels. Our tomograms reveal segmented fast seismic anomalies beneath Anatolia that corresponds to the subducted portion of the African lithosphere along the Cyprean and the Aegean trenches. We identify these anomalies as the subducted Aegean and the Cyprus slabs that are separated from each other by a gap as wide as 300 km beneath Western Anatolia. This gap is occupied by slow velocity perturbations that we interpret as hot upwelling asthenosphere. The eastern termination of the subducting African lithosphere is located near the transition from central Anatolia to the Eastern Anatolian Plateau or Arabian–Eurasian collision front that is underlain by large volumes of hot, underplating asthenosphere marked by slow velocity perturbations. Our tomograms also show fast velocity perturbations at shallow depths beneath northwestern Anatolia that sharply terminates towards the south at the North Anatolian Fault Zone (NAFZ). The associated velocity contrast across the NAFZ persists down to a depth of 100–150 km. Hence, our study is the first to investigate and interpret the vertical extent of deformation along this nascent transform plate boundary. Overall, the resolved upper-mantle structure of Anatolia is directly related with the geology and tectonic features observed at the surface of the Anatolian Plate and suggest that the segmented nature of the subducted African lithosphere plays an important role in the evolution of Anatolia and distribution of its tectonic provinces.

On the connection between artifact filtering in reverse-time migration and adjoint tomography

Finite-frequency sensitivity kernels in seismic tomography define the volumes inside the earth that influence seismic waves as they traverse through it. It has recently been numerically observed that an image obtained using the impedance kernel is much less contaminated by low-frequency artifacts due to the presence of sharp wave-speed contrasts in the background model, than is an image obtained using reverse-time migration. In practical reverse-time migration, these artifacts are routinely heuristically dampened by Laplacian filtering of the image. Here we show analytically that, for an isotropic acoustic medium with constant density, away from sources and receivers and in a smooth background medium, Laplacian image filtering is identical to imaging with the impedance kernel. Therefore, when imaging is pushed toward using background models with sharp wave-speed contrasts, the impedance kernel image is less prone to develop low-frequency artifacts than is the reverse-time migration image, due to the implicit action of the Laplacian that amplifies the higher-frequency reflectors relative to the low-frequency artifacts. Thus, the heuristic Laplacian filtering commonly used in practical reverse-time migration is fundamentally rooted in adjoint tomography and, in particular, closely connected to the impedance kernel.

Noise cross-correlation sensitivity kernels

SUMMARY We determine finite-frequency sensitivity kernels for seismic interferometry based upon noise cross-correlation measurements. Under the assumptions that noise is spatially uncorrelated but non-uniform, we determine ensemble-averaged cross correlations between synthetic seismograms at two geographically distinct locations. By minimizing a measure of the difference between observed and simulated ensemble cross correlations—subject to the constraint that the simulated wavefield satisfies the seismic wave equation—we obtain ensemble sensitivity kernels. These ensemble kernels reflect the sensitivity of ensemble cross-correlation measurements to variations in model parameters, for example, mass density, shear and compressional wave speeds and the spatial distribution of noise. Ensemble kernels are calculated based upon the interaction between two wavefields: an ensemble forward wavefield and an ensemble adjoint wavefield. To obtain the ensemble forward wavefield, one first calculates a generating wavefield obtained by inserting a signal determined by the characteristics of the noise at the location of the first receiver, saving the results of this calculation at locations where noise is generated, that is, typically on (a portion of) the Earth's surface. Next, one uses this generating wavefield as the source of the ensemble forward wavefield associated with the first receiver. The ensemble adjoint wavefield is obtained by using measurements between simulated and observed ensemble cross correlations as a seismic source located at the second receiver. The interaction between ensemble forward and adjoint wavefields ‘paints’ ensemble sensitivity kernels. We illustrate the construction of ensemble kernels and their nature in two and three dimensions using a spectral-element method. In addition to a ‘banana-doughnut’ feature connecting the two receivers, as in traditional finite-frequency earthquake tomography, some noise cross-correlation sensitivity kernels exhibit hyperbolic ‘jets’ protruding from each receiver in a direction away from the other receiver. Ensemble sensitivity kernels for long-period (T > ∼ 50 s) non-uniform noise in global models exhibit sensitivity along the minor and major arcs. These kernels reflect the fact that measurements typically involve long time-series that include multi-orbit surface waves. Like free oscillations, such measurements are sensitive to structure along the great circle through the two receivers. From the perspective of noise cross-correlation tomography, we discuss strategies for inversions in terrestrial and helioseismology.

Empirically determined finite frequency sensitivity kernels for surface waves

SUMMARY We demonstrate a method for the empirical construction of 2-D surface wave phase traveltime finite frequency sensitivity kernels by using phase traveltime measurements obtained across a large dense seismic array. The method exploits the virtual source and reciprocity properties of the ambient noise cross-correlation method. The adjoint method is used to construct the sensitivity kernels, where phase traveltime measurements for an event (an earthquake or a virtual ambient noise source at one receiver) determine the forward wave propagation and a virtual ambient noise source at a second receiver gives the adjoint wave propagation. The interference of the forward and adjoint waves is then used to derive the empirical kernel. Examples of station‐station and earthquake‐station empirical finite frequency kernels within the western United States based on ambient noise and earthquake phase traveltime measurements across USArray stations are shown to illustrate the structural effects on the observed empirical sensitivity kernels. We show that a hybrid kernel constructed from the empirical kernel and the kernel for a reference model can be used to compute traveltimes accurate to second order in model perturbations for an earth-like model. A synthetic test demonstrates the application of such hybrid kernels to predict surface wave phase traveltimes.

Seismic tomography of the southern California crust based on spectral-element and adjoint methods

We iteratively improve a 3-D tomographic model of the southern California crust using numerical simulations of seismic wave propagation based on a spectral-element method (SEM) in combination with an adjoint method. The initial 3-D model is provided by the Southern California Earthquake Center. The data set comprises three-component seismic waveforms (i.e. both body and surface waves), filtered over the period range 2–30 s, from 143 local earthquakes recorded by a network of 203 stations. Time windows for measurements are automatically selected by the FLEXWIN algorithm. The misfit function in the tomographic inversion is based on frequency-dependent multitaper traveltime differences. The gradient of the misfit function and related finite-frequency sensitivity kernels for each earthquake are computed using an adjoint technique. The kernels are combined using a source subspace projection method to compute a model update at each iteration of a gradient-based minimization algorithm. The inversion involved 16 iterations, which required 6800 wavefield simulations. The new crustal model, m16, is described in terms of independent shear (VS) and bulk-sound (VB) wave speed variations. It exhibits strong heterogeneity, including local changes of ±30 per cent with respect to the initial 3-D model. The model reveals several features that relate to geological observations, such as sedimentary basins, exhumed batholiths, and contrasting lithologies across faults. The quality of the new model is validated by quantifying waveform misfits of full-length seismograms from 91 earthquakes that were not used in the tomographic inversion. The new model provides more accurate synthetic seismograms that will benefit seismic hazard assessment.

Principal component analysis of anisotropic finite-frequency sensitivity kernels

SUMMARY We use a principal component analysis to characterize the finite-frequency sensitivity of seismic observables to anisotropy. A general anisotropic medium may be described in terms of 21 independent elastic parameters, each of which has an associated ‘primary’ sensitivity kernel. Our principal component analysis ranks linear combinations of the primary kernels to ascertain the dominant anisotropic parameters associated with a particular seismic observable. The principal parameters are those to which a given data set is the most sensitive. We demonstrate the efficiency of the method for a single arrival associated with a particular source–receiver combination, and apply it to a small synthetic Love-wave data set with a simple source–receiver geometry. For direct body wave arrivals, such as P, S and SKS, and direct Love and Rayleigh surface waves, our principal component analysis finds the same small combinations of dominant anisotropic parameters previously identified based upon asymptotic methods. The analysis further confirms the importance of mode coupling in finite-frequency surface wave sensitivity kernels. Our approach can be directly incorporated into a tomographic inversion to automatically select the general anisotropic parameters which are best constraint, for example, without prescribing the model to be transversely isotropic with a particular symmetry axis. The computational overhead associated with the calculation of the 21 primary kernels and the subsequent principal component analysis is minimal relative to an isotropic calculation.

Multiple-frequencySH-wave tomography of the western US upper mantle

We estimate the SH-wave velocity and attenuation structures of the western US upper mantle using the dense network of the USArray and new techniques: we observe a multiple-frequency data set of both traveltime and amplitude anomalies, and interpret these with full 3-D finite-frequency sensitivity kernels. Amplitudes show stronger frequency dependence than traveltimes. We perform a joint inversion on the measured traveltime and amplitude anomalies, interpreting them in terms of velocity and attenuation heterogeneities. Aside from the expected clear division between the slow, tectonically active region in the west and the fast craton in the east, several interesting smaller velocity anomalies are observed. The subduction along the Cascades at 100–300 km depths shows lateral discontinuity, with a ‘slab hole’ (absence of fast anomalies) observed around 45°N. The delaminated Sierra Nevada Mountains root is observed to have sunk to 200 km depth. The Yellowstone plume seems to have an origin (weak slow velocity anomalies) near 1000 km depth, but the plume conduit seems to be interrupted by a fast anomaly, which is identified as a fragment of the Farallon slab. The S-velocity model shows a trench-perpendicular ‘slab gap’ (absence of fast anomalies) at almost the same location as in the P model recently published by Sigloch et al. (2008). The methodological improvements described in the first paragraph have several benefits. Amplitude data help to sharpen the edges of narrow velocity heterogeneities in the shallow upper mantle. The focusing effect from velocity heterogeneity dominates over that of attenuation and must be considered when interpreting amplitude anomalies. In general, velocity and attenuation heterogeneities correlate positively, suggesting that temperature plays a major role in forming the anomalies.

The finite-frequency sensitivity kernel for migration residual moveout and its applications in migration velocity analysis

We have derived a broadband sensitivity kernel that relates the residual moveout (RMO) in prestack depth migration (PSDM) to velocity perturbations in the migration-velocity model. We have compared the kernel with the RMO directly measured from the migration image. The consistency between the sensitivity kernel and the measured sensitivity map validates the theory and the numerical implementation. Based on this broadband sensitivity kernel, we propose a new tomography method for migration-velocity analysis and updating — specifically, for the shot-record PSDM and shot-index common-image gather. As a result, time-consuming angle-domain analysis is not required. We use a fast one-way propagator and multiple forward scattering and single backscattering approximations to calculate the sensitivity kernel. Using synthetic data sets, we can successfully invert velocity perturbations from the migration RMO. This wave-equation-based method naturally incorporates the wave phenomena and is best teamed with the wave-equation migration method for velocity analysis. In addition, the new method maintains the simplicity of the ray-based velocity analysis method, with the more accurate sensitivity kernels replacing the rays.

Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods

We determine adjoint equations and Fréchet kernels for global seismic wave propagation based upon a Lagrange multiplier method. We start from the equations of motion for a rotating, self-gravitating earth model initially in hydrostatic equilibrium, and derive the corresponding adjoint equations that involve motions on an earth model that rotates in the opposite direction. Variations in the misfit function χ then may be expressed as , where δ ln m = δm/m denotes relative model perturbations in the volume V, δ ln d denotes relative topographic variations on solid–solid or fluid–solid boundaries Σ, and ∇Σδ ln d denotes surface gradients in relative topographic variations on fluid–solid boundaries ΣFS. The 3-D Fréchet kernel Km determines the sensitivity to model perturbations δ ln m, and the 2-D kernels Kd and Kd determine the sensitivity to topographic variations δ ln d. We demonstrate also how anelasticity may be incorporated within the framework of adjoint methods. Finite-frequency sensitivity kernels are calculated by simultaneously computing the adjoint wavefield forward in time and reconstructing the regular wavefield backward in time. Both the forward and adjoint simulations are based upon a spectral-element method. We apply the adjoint technique to generate finite-frequency traveltime kernels for global seismic phases (P, Pdiff, PKP, S, SKS, depth phases, surface-reflected phases, surface waves, etc.) in both 1-D and 3-D earth models. For 1-D models these adjoint-generated kernels generally agree well with results obtained from ray-based methods. However, adjoint methods do not have the same theoretical limitations as ray-based methods, and can produce sensitivity kernels for any given phase in any 3-D earth model. The Fréchet kernels presented in this paper illustrate the sensitivity of seismic observations to structural parameters and topography on internal discontinuities. These kernels form the basis of future 3-D tomographic inversions.

Imaging mantle transition zone thickness withSdS-SSfinite-frequency sensitivity kernels

SUMMARY We invert differential SdS-SS traveltime residuals measured from stacked waveforms and finitefrequency sensitivity kernels for topography on the 410- and 660-km discontinuities. This approach yields higher resolution images of transition zone thickness than previous stacking methods, which simply average/smooth over topographic features. Apparent structure measured using simple stacking is highly dependent upon the bin size of each stack. By inverting for discontinuity topography with a variety of bin sizes, we can more accurately calculate the true structure. The inverted transition zone model is similar to simple stack models with an average thickness of 242 km, but the lateral variations in thickness are larger in amplitude and smaller in scale. Fast seismic velocities in 3-D mantle models such as SB4L18 correlate with areas of thicker transition zone. The elongated curvilinear regions of thickened transition zone that occur near subduction zones are narrow and high amplitude, which suggests relatively little lateral spreading and warming of subducted lithosphere within the transition zone. The anomalously thin transition zone regions are laterally narrow, and not broadly continuous. If these variations in transition zone thickness are interpreted as thermal in nature, then this model suggests significant temperature variations on small lateral scales.