Spectral-Element and Adjoint Methods in Seismology

We provide an introduction to the use of the spectral-element method (SEM) in seismology. Following a brief review of the basic equations that govern seismic wave propagation, we discuss in some detail how these equations may be solved numerically based upon the SEM to address the forward problem in seismology. Examples of synthetic seismograms calculated based upon the SEM are compared to data recorded by the Global Seismographic Network. Finally, we discuss the challenge of using the remaining differences between the data and the synthetic seismograms to constrain better Earth models and source descriptions. This leads naturally to adjoint methods, which provide a practical approach to this formidable computational challenge and enables seismologists to tackle the inverse problem

Double-difference adjoint seismic tomography

We introduce a `double-difference' method for the inversion for seismic wavespeed structure based on adjoint tomography. Differences between seismic observations and model predictions at individual stations may arise from factors other than structural heterogeneity, such as errors in the assumed source-time function, inaccurate timings, and systematic uncertainties. To alleviate the corresponding nonuniqueness in the inverse problem, we construct differential measurements between stations, thereby reducing the influence of the source signature and systematic errors. We minimize the discrepancy between observations and simulations in terms of the differential measurements made on station pairs. We show how to implement the double-difference concept in adjoint tomography, both theoretically and in practice. We compare the sensitivities of absolute and differential measurements. The former provide absolute information on structure along the ray paths between stations and sources, whereas the latter explain relative (and thus higher-resolution) structural variations in areas close to the stations. Whereas in conventional tomography a measurement made on a single earthquake-station pair provides very limited structural information, in double-difference tomography one earthquake can actually resolve significant details of the structure. The double-difference methodology can be incorporated into the usual adjoint tomography workflow by simply pairing up all conventional measurements; the computational cost of the necessary adjoint simulations is largely unaffected. Rather than adding to the computational burden, the inversion of double-difference measurements merely modifies the construction of the adjoint sources for data assimilation.Comment: 21 pages, 17 figures, accepted for publication by the Geophysical Journal Internationa

Segmentation of Fault Networks Determined from Spatial Clustering of Earthquakes

We present a new method of data clustering applied to earthquake catalogs, with the goal of reconstructing the seismically active part of fault networks. We first use an original method to separate clustered events from uncorrelated seismicity using the distribution of volumes of tetrahedra defined by closest neighbor events in the original and randomized seismic catalogs. The spatial disorder of the complex geometry of fault networks is then taken into account by defining faults as probabilistic anisotropic kernels, whose structures are motivated by properties of discontinuous tectonic deformation and previous empirical observations of the geometry of faults and of earthquake clusters at many spatial and temporal scales. Combining this a priori knowledge with information theoretical arguments, we propose the Gaussian mixture approach implemented in an Expectation-Maximization (EM) procedure. A cross-validation scheme is then used and allows the determination of the number of kernels that should be used to provide an optimal data clustering of the catalog. This three-steps approach is applied to a high quality relocated catalog of the seismicity following the 1986 Mount Lewis ($M_l=5.7$) event in California and reveals that events cluster along planar patches of about 2 km$^2$, i.e. comparable to the size of the main event. The finite thickness of those clusters (about 290 m) suggests that events do not occur on well-defined euclidean fault core surfaces, but rather that the damage zone surrounding faults may be seismically active at depth. Finally, we propose a connection between our methodology and multi-scale spatial analysis, based on the derivation of spatial fractal dimension of about 1.8 for the set of hypocenters in the Mnt Lewis area, consistent with recent observations on relocated catalogs

Imagerie de milieux salifères aux échelles crustales et expérimentales par méthodes de migration sismique et méthode de l'adjoint : applications marines

La découverte de structures géologiques salines est une raison économique importante pour l'exploration dans le monde car elles constituent un piège naturel pour diverses ressources. Cependant, l'imagerie de ces structures est un grand défi. En raison des propriétés du sel, dont les vitesses de propagation sont beaucoup plus élevées que celles des strates adjacentes, les ondes sismiques sont piégées dans ces structures, produisant un grand nombre d'artefacts numériques parasites, tels que des multiples. Cela interfère avec le signal sismique primaire, ce qui empêche de voir clairement ce qui se trouve sous les structures salifères. Parmi toutes les méthodes d'exploration géophysique, la méthode de migration par renversement temporel (RTM), qui fait partie des méthodes qui utilisent la résolution de la forme d'onde sismique complète, est un outil d'imagerie très puissant, même dans les régions à géologie complexe. Dans ce travail, nous utilisons la méthode RTM basée sur l'adjoint, qui consiste essentiellement en trois étapes : la solution de l'équation des ondes, la solution de l'équation des ondes adjointe et la condition d'imagerie, qui consiste en la corrélation des champs d'ondes directs et adjoints. Ce travail peut être divisé en deux cas d'étude : le premier cas consiste en un modèle synthétique bidimensionnel d'un dôme de sel, issu de la migration finale d'une étude réelle dans le Golfe du Mexique. Le second cas consiste en un modèle tridimensionnel expérimental (WAVES), élaboré par le laboratoire LMA de Marseille, qui simule une structure saline, structures sédimentaires environnantes, et un socle. Le modèle a été immergé dans l'eau pour recréer un sondage marin réaliste. Deux types de données différents ont été obtenus dans cette expérience : des données à décalage nul et des données à décalage multiple. Pour résoudre l'équation des ondes impliquée dans la méthode RTM basée sur l'adjoint, nous utilisons des différences finies d'ordre 4 dans les deux cas. De plus, dans le second cas, nous avons utilisé le code UniSolver, qui résout la méthode RTM basée sur l'adjoint en utilisant des différences finies d'ordre 4 et un parallélisme basé sur MPI. Nous avons mis en œuvre les équations viscoélastiques pour simuler l'effet de l'atténuation. Pour cette raison, le schéma "Checkpointing" est introduit pour calculer la condition d'imagerie et assurer la stabilité physique et numérique. Dans le premier cas d'étude, nous analysons la reconstruction de l'image du dôme de sel que produisent différents noyaux de sensibilité. Nous calculons ces noyaux en utilisant différentes paramétrisations (densité - vitesse P), ou (densité - constantes de Lamé) pour une rhéologie acoustique. Nous étudions également comment l'utilisation de différents modèles a priori affecte l'image finale en fonction du type de noyau calculé. En utilisant les résultats obtenus en 2D, nous calculons des noyaux synthétiques tridimensionnels en utilisant une rhéologie élastique. Dans le second cas, nous effectuons d'abord une calibration des propriétés du modèle pour des données à décalage nul, et une fois que les données synthétiques et réelles s'ajustent bien, nous calculons les noyaux tridimensionnels. Nous résolvons le problème direct pour le cas à décalage multiple avec et sans effets d'atténuation. Nous comparons les données synthétiques calculées avec des rhéologies élastiques ou viscoélastiques avec les données réelles. Cela permet ainsi de voir l'impact de l'atténuation dans les signaux. Cela ouvrira la voie à de la RTM et des simulations de la forme d'onde complète viscoélastique dans des contextes tectoniques salifères. Enfin, nous avons implémenté l'algorithme LSRTM pour les données acoustiques synthétiques bidimensionnelles et pour les données viscoélastiques réelles tridimensionnelles, qui est un processus d'inversion itératif, nous avons suivi l'approche du gradient conjugué, en vérifiant que les conditions de Wolfe sont satisfaites.Finding salt geological structures is an important economic reason for exploration in the world because they constitute a natural trap for various resources such as oil, natural gas, water, and also the salt itself can be exploitable. However, the imaging of these structures is a great challenge. Due to the properties of salt, with propagation velocities much higher than the adjacent strata, seismic waves are trapped within these structures, producing a large number of spurious numerical artifacts, such as multiples. This interferes with the primary seismic signal, making it impossible to see clearly what is underneath the salt structures (salt domes for instance). Among all the geophysical exploration methods, the Reverse Time Migration method (RTM), which is part of the methods that solve the complete seismic waveform, is a very powerful imaging tool, even in regions of complex geology. In this work we use the adjoint-based RTM method, which basically consists of three stages: the solution of the wave equation (forward problem), the solution of the adjoint wave equation (adjoint problem), and the imaging condition, which consists in the correlation of the forward and adjoint wavefields. This work can be divided in two cases of study: the first case consists in a two-dimensional synthetic model of a salt dome, taken from the final migration of a real survey in the Gulf of Mexico. The second case consists in an experimental three-dimensional model (WAVES), elaborated by the LMA laboratory in Marseille (France), which simulates a salt structure (with surrounding sedimentary structures), and a basement. The model was immersed in water to recreate a reallistic marine survey. Two different data types were obtained in this experiment: zero-offset and multi-offset data. To compute the adjoint-based RTM method we use fourth-order finite differences in both cases. Furthermore, in the second case we used the UniSolver code, which solves the adjoint-based RTM method using fourth-order finite differences and MPI-based parallelism. It was also necessary to implement the viscoelastic equations to simulate the effect of attenuation. Because of this, the Checkpointing scheme is introduced to calculate the imaging condition and ensures physical and numerical stability in the migration procedure. In the first case study we analyze the recovery of the salt dome image that different sensitivity kernels produce. We calculate these kernels using different parametrizations (density - P velocity), (density - Lamé constants), or (density - P impedance) for an acoustic rheology. We also study how the use of different a priori models affects the final image depending on the kind of kernel computed. Using the results obtained previously in 2D, we calculate synthetic three-dimensional kernels using an elastic rheology. In the second case (the realistic/experimental case), we perform a calibration of the model properties for zero-offset data, and once the synthetic and real data fit well, we calculate the three-dimensional kernels. We compute the forward problem of the multi-offset data with and without attenuation effects. We compare the synthetic data computed with elastic or viscoelastic rheologies with the real data. This will allow to see the impact of attenuation in the signals. This will pave the way to viscoelastic full waveform in salt tectonic context. Finally, we implemented the Least Squares Reverse Time Migration (LSRTM) algorithm for the two-dimensional acoustic synthetic data and for the 3-dimensional viscoelastic real data, which is an iterative inversion process, we have followed the Conjugated Gradient (CG) approach, checking that Wolfe conditions (convergence and curvature of the misfit functions are satisfied

Seismic anisotropy of Precambrian lithosphere : Insights from Rayleigh wave tomography of the eastern Superior Craton

The seismic data used in this study are freely available from the CNDC (Canadian National Data Centre for Earthquake Seismology and Nuclear Explosion Monitoring) and IRIS DMC (Data Management Center) via their data request tools. The Leverhulme Trust (grant RPG-2013-332) and National Science Foundation are acknowledged for financial support. L.P. is supported by Janet Watson Imperial College Department Scholarship and the Romanian Government Research Grant NUCLEU. F.D. is supported by NSERC through the Discovery Grants and Canada Research Chairs program. We also thank two anonymous reviewers and the Associate Editor for insightful comments that helped improve the manuscript.Peer reviewedPublisher PD

Application of Support Vector Machine Modeling for the Rapid Seismic Hazard Safety Evaluation of Existing Buildings

The economic losses from earthquakes tend to hit the national economy considerably; therefore, models that are capable of estimating the vulnerability and losses of future earthquakes are highly consequential for emergency planners with the purpose of risk mitigation. This demands a mass prioritization filtering of structures to identify vulnerable buildings for retrofitting purposes. The application of advanced structural analysis on each building to study the earthquake response is impractical due to complex calculations, long computational time, and exorbitant cost. This exhibits the need for a fast, reliable, and rapid method, commonly known as Rapid Visual Screening (RVS). The method serves as a preliminary screening platform, using an optimum number of seismic parameters of the structure and predefined output damage states. In this study, the efficacy of the Machine Learning (ML) application in damage prediction through a Support Vector Machine (SVM) model as the damage classification technique has been investigated. The developed model was trained and examined based on damage data from the 1999 Düzce Earthquake in Turkey, where the building’s data consists of 22 performance modifiers that have been implemented with supervised machine learning