This proof-of-concept study presents a parameter-free, linear Backus–Gilbert inversion scheme, tractable for seismic tomography problems. It leads to efficient computations of unbiased tomographic ...images, accompanied by meaningful resolution and uncertainty infor- mations. Moreover, as there is no need to parametrize the model space in this parameter-free approach, it enables numerically accurate data sensitivity kernels to be effectively exploited in tomographic inversions. This is a major benefit over discrete tomographic methods, for which data sensitivity kernels are often inaccurate, as they are projected on a given model parametrization prior to be exploited in the inversion, and these parametrizations are usually coarse to limit the number of parameters and keep tractable the problems of model estima- tion and/or appraisal. Therefore, this new tomographic scheme fuels great hopes on better constraining multiscale seismic heterogeneities in the Earth’s interior by exploiting accurate data sensitivity kernels, that is, taking full advantage of known wave-propagation physics, and enabling quantitative appraisals of tomographic features. As a remark, since its computational cost grows as a function of the total number of data squared, it may be better suited to han- dle moderate-size data sets, typically encountered in regional-scale tomography. Theoretical developments are illustrated within a finite-frequency physical framework. A set of 27 070 teleseismic S -wave time residuals is inverted, with focus on imaging and appraising shear- wave velocity anomalies lying in the mantle below Southeast Asia, in the 350–1410 km depth range.
The appraisal of tomographic models, of fundamental importance towards better understanding the Earth's interior, consists in analysing their resolution and covariance. The discrete theory of ...Backus–Gilbert, solving all at once the linear problems of model estimation and appraisal, aims at evaluating weighted averages of the true model parameters. Contrary to damped least-squares techniques, one key advantage of Backus–Gilbert inversion is that no subjective regularization is needed to remove the non-uniqueness of the model solution. Indeed, it is often possible to identify unique linear combinations of the parameters even when the parameters themselves are not uniquely defined. In other words, the non-uniqueness can be broken by averaging rather than regularizing. Over the past few decades, many authors have considered that, in addition to a high computational cost, it could be a clumsy affair in the presence of data errors to practically implement the Backus–Gilbert approach to large-scale tomographic applications. In this study, we introduce and adapt to seismic tomography the Subtractive Optimally Localized Averages (SOLA) method, an alternative Backus–Gilbert formulation which retains all its advantages, but is more computationally efficient and versatile in the explicit construction of averaging kernels. As a leitmotiv, we focus on global-scale S-wave tomography and show that the SOLA method can successfully be applied to large-scale, linear and discrete tomographic problems.
In this study, we focus on Northwest Iran and exploit a dataset of Rayleigh-wave group-velocity measurements obtained from ambient noise cross-correlations and earthquakes.We build group-velocity ...maps using the recently developed SOLA Backus-Gilbert linear tomographic scheme as well as the more traditional Fast-marching Surface-wave Tomography method.The SOLA approach produces robust, unbiased local averages of group velocities with detailed information on their local resolution and uncertainty; however, it does not as yet allow ray-path updates in the inversion process. The Fast-marching method, on the other hand, does allow ray-path updates, although it does not provide information on the resolution and uncertainties of the resulting models (at least not without great computational cost) and may suffer from bias due to model regularisation.The core of this work consists in comparing these two tomographic methods, in particular how they perform in the case of strong vs. weak seismic-velocity contrasts and good vs. poor data coverage. We demonstrate that the only case in which the Fast-marching inversion outperforms the SOLA inversion is for strong anomaly contrasts in regions with good path coverage; in all other configurations, the SOLA inversion produces more coherent anomalies with fewer artefacts.
Seismic tomography allows us to image the interior of the Earth. In general, to determine the nature of seismic anomalies, constraints on more than one seismic parameter are required. For example, ...the ratio R between perturbations in vs and vp (dlnvs and dlnvp, respectively) is studied extensively in the lowermost mantle and interpreted in terms of thermal and/or chemical anomalies. However, to jointly interpret tomographic models of variations in vs and vp or their ratio R, it is essential for them to share the same local resolution. Most existing models do not provide resolution information, and thus cannot guarantee to honour this condition. In addition, uncertainties are typically not provided, making it difficult to robustly interpret the ratio R=dlnvs/dlnvp. To overcome these issues, we utilise the recently developed SOLA tomographic method, a variant of the linear Backus–Gilbert inversion scheme. SOLA retrieves local-average model estimates, together with information on their uncertainties, whilst it also provides direct control on model resolution through target kernels. In this contribution, we apply SOLA to normal-mode data with sensitivity to both vs and vp, as well as density throughout the mantle. Specifically, we aim to develop models of both vs and vp with the same local resolution. We test our methodology and approach using synthetic tests for various noise cases (random noise, data noise or also additional 3D Earth noise due to variations in other physical parameters than the one of interest). We find that the addition of the 3D noise increases the uncertainties in our model estimates significantly, only allowing us to find model estimates in six or four layers for vs and vp, respectively. While the synthetic tests indicate that no satisfactory density models can be obtained, we easily manage to construct models of dlnvs and dlnvp with almost identical resolution, from which the ratio R can be robustly inferred. The obtained values of R in our synthetic experiments significantly depend on the noise case considered and the method used to calculate it, with the addition of 3D noise always leading to an overestimate of R. When applying our approach to real data, we obtain values of R in the range of 2.5–4.0 in the lowest 600 km of the mantle, which are consistent with previous studies. Our model estimates with related resolving kernels and uncertainties can be used to test geodynamic model predictions to provide further insights into the temperature and composition of the mantle.
•SOLA inversions enable us to directly control the resolution of seismic tomography.•To robustly interpret R, vs and vp models require the same local resolution.•Normal-mode inversions using SOLA constrain R robustly throughout the mantle.•Knowledge of resolution and uncertainty is crucial for quantitative interpretations.
Geophysical tomographic studies traditionally exploit linear, damped least squares inversion methods. We demonstrate that the resulting models can be locally biased toward lower or higher amplitudes ...in regions of poor data illumination, potentially causing physical misinterpretations. For example, we show that global model S40RTS is locally biased toward higher amplitudes below isolated receivers where raypaths are quasi‐vertical, such as on Hawaii. This leads to questions on the apparent low‐velocity structure interpreted as the Hawaii hot spot. We prove that a linear Backus‐Gilbert inversion scheme can bring the Earth's interior into focus through unbiased tomographic lenses, as its model estimates are constrained to be averages over the true model. It also efficiently computes the full generalized inverse required to infer both model resolution and its covariance, enabling quantitative interpretations of tomographic models.
Key Points
Damped least squares tomographic models can be locally biased in poorly sampled regions
Slow velocity anomalies below Hawaii are biased toward higher amplitudes in model S40RTS
We show how to efficiently compute unbiased models including their full resolution and covariance
SUMMARY
Since most tomographic problems deal with imperfect data coverage and noisy data, an estimate of the seismic velocity in the Earth can only be a local average of the ‘true’ velocity with some ...attached uncertainty. We use the SOLA (subtractive optimally localized averages) method, a Backus–Gilbert-type method based on the resolution–uncertainty trade-off, to build a range of models of Rayleigh-wave velocities in the Pacific upper mantle. We choose one solution and show how to analyse the model using its resolution and uncertainties. We exploit the model statistics to evaluate the significance of deviations from a theoretical prediction: a half-space cooling model of the Pacific lithosphere. We investigate a slow-velocity anomaly located northeast of Hawaii, at about 200 km depth, and a pattern of alternatively slow- and fast-velocity bands, aligned approximately northwest to southeast, between 200 and 300 km depth. According to our resolution and uncertainty analyses, both features seem to be resolved.
SUMMARY A major challenge in seismic tomography consists in quantifying and representing model resolution and uncertainty, particularly at global scales. This information is crucial for ...interpretations of tomographic images and their technical application in geodynamics. However, due to large computational costs, there have been only few attempts so far to coherently analyse the spatially varying resolving power for a complete set of model parameters. Here, we present a concept for an effective evaluation and global representation of the 3-D resolution information contained in a full set of averaging kernels. In our case, these kernels are constructed using the ‘Subtractive Optimally Localized Averages’ (SOLA) method, a variant of classic Backus-Gilbert inversion suitable for global tomography. Our assessment strategy incorporates the following steps: (1) a 3-D Gaussian function is fitted to each averaging kernel to measure resolution lengths in different directions and (2) we define a classification scheme for the quality of the averaging kernels based on their focus with respect to the estimated 3-D Gaussian, allowing us to reliably identify whether the inferred resolution lengths are robust. This strategy is not restricted to SOLA inversions, but can, for example, be applied in all cases where point-spread functions are computed in other tomographic frameworks. Together with model uncertainty estimates that are derived from error propagation in the SOLA method, our concept reveals at which locations, resolution lengths and interpretations of model values are actually meaningful. We finally illustrate how the complete information from our analysis can be used to calibrate the SOLA inversion parameters—locally tunable target resolution kernels and trade-off parameters—without the need for visual inspection of the individual resulting averaging kernels. Instead, our global representations provide a tool for designing tomographic models with specific resolution-uncertainty properties that are useful in geodynamic applications, especially for linking seismic inversions to models of mantle flow.
We present a graphical user interface to facilitate the processing of teleseismic shear-wave splitting observations. In contrast to a fully automated technique, we present a manual, per-event ...approach that maintains user control during the sequence of processing. The SplitLab environment is intended to undertake the repetitive processing steps while enabling the user to focus on quality control and eventually the interpretation of the results. Pre-processing modules of SplitLab create a database of events and link the corresponding seismogram files. The seismogram viewer tool uses this database to perform the measurement interactively. Post-processing of the combined results of such a project includes a viewer and export option. Our emphasis lies in the application to teleseismic shear-wave splitting analysis, but our code can be extended easily for other purposes. SplitLab can be downloaded at http://www.gm.univ-montp2.fr/splitting/.
SUMMARY
We investigate the influence of crust on time residual measurements made by cross-correlation in the 10–51 s filtering period range on a global scale, considering two crustal models: CRUST2.0 ...and CRUST1.0. This study highlights, in a quantitative way, crust-related time corrections. One part of this correction is directly linked to the body wave traveltime through the crust as predicted by the ray theory, whereas a second part is related to interferences with multiple crustal reflections. This second component, called finite-frequency (FF) crustal correction, is frequency-dependent unlike the ray-theory based correction. We show that if this frequency-dependent crust-related correction is not taken into account in cross-correlation measurements, it may lead to a dispersive effect in S-wave delay-times that could ultimately bias tomographic models. On average, this FF correction increases with the filtering period. Comparisons between the two crustal models highlight the significant dispersive effect of the crust, which has complex patterns depending on geological contexts, with an important role of the sediment thickness. Although ray crustal corrections remain important, FF crustal effects may lead to a bias in measurements if not properly taken into account; on average they may reach 0.9–1.6 s for CRUST2.0 and 0.5–1.6 s for CRUST1.0, for period ranging from 10 to 51 s, respectively.
SUMMARY
Tomographic-geodynamic model comparisons are a key component in studies of the present-day state and evolution of Earth’s mantle. To account for the limited seismic resolution, ‘tomographic ...filtering’ of the geodynamically predicted mantle structures is a standard processing step in this context. The filtered model provides valuable information on how heterogeneities are smeared and modified in amplitude given the available seismic data and underlying inversion strategy. An important aspect that has so far not been taken into account are the effects of data uncertainties. We present a new method for ‘tomographic filtering’ in which it is possible to include the effects of random and systematic errors in the seismic measurements and to analyse the associated uncertainties in the tomographic model space. The ‘imaged’ model is constructed by computing the generalized-inverse projection (GIP) of synthetic data calculated in an earth model of choice. An advantage of this approach is that a reparametrization onto the tomographic grid can be avoided, depending on how the synthetic data are calculated. To demonstrate the viability of the method, we compute traveltimes in an existing mantle circulation model (MCM), add specific realizations of random seismic ‘noise’ to the synthetic data and apply the generalized inverse operator of a recent Backus–Gilbert-type global S-wave tomography. GIP models based on different noise realizations show a significant variability of the shape and amplitude of seismic anomalies. This highlights the importance of interpreting tomographic images in a prudent and cautious manner. Systematic errors, such as event mislocation or imperfect crustal corrections, can be investigated by introducing an additional term to the noise component so that the resulting noise distributions are biased. In contrast to Gaussian zero-mean noise, this leads to a bias in model space; that is, the mean of all GIP realizations also is non-zero. Knowledge of the statistical properties of model uncertainties together with tomographic resolution is crucial for obtaining meaningful estimates of Earth’s present-day thermodynamic state. A practicable treatment of error propagation and uncertainty quantification will therefore be increasingly important, especially in view of geodynamic inversions that aim at ‘retrodicting’ past mantle evolution based on tomographic images.
PDF
