Full Waveform Inversion for Time-Distance Helioseismology

Inferring interior properties of the Sun from photospheric measurements of the seismic wavefield constitutes the helioseismic inverse problem. Deviations in seismic measurements (such as wave travel times) from their fiducial values estimated for a given model of the solar interior imply that the model is inaccurate. Contemporary inversions in local helioseismology assume that properties of the solar interior are linearly related to measured travel-time deviations. It is widely known, however, that this assumption is invalid for sunspots and active regions, and likely for supergranular flows as well. Here, we introduce nonlinear optimization, executed iteratively, as a means of inverting for the sub-surface structure of large-amplitude perturbations. Defining the penalty functional as the $L_2$ norm of wave travel-time deviations, we compute the the total misfit gradient of this functional with respect to the relevant model parameters %(only sound speed in this case) at each iteration around the corresponding model. The model is successively improved using either steepest descent, conjugate gradient, or quasi-Newton limited-memory BFGS. Performing nonlinear iterations requires privileging pixels (such as those in the near-field of the scatterer), a practice not compliant with the standard assumption of translational invariance. Measurements for these inversions, although similar in principle to those used in time-distance helioseismology, require some retooling. For the sake of simplicity in illustrating the method, we consider a 2-D inverse problem with only a sound-speed perturbation.Comment: 24 pages, 10 figures, to appear in Ap

Performance of Two 18-Story Steel Moment-Frame Buildings in Southern California During Two Large Simulated San Andreas Earthquakes

Using state-of-the-art computational tools in seismology and structural engineering, validated using data from the Mw=6.7 January 1994 Northridge earthquake, we determine the damage to two 18-story steel moment-frame buildings, one existing and one new, located in southern California due to ground motions from two hypothetical magnitude 7.9 earthquakes on the San Andreas Fault. The new building has the same configuration as the existing building but has been redesigned to current building code standards. Two cases are considered: rupture initiating at Parkfield and propagating from north to south, and rupture propagating from south to north and terminating at Parkfield. Severe damage occurs in these buildings at many locations in the region in the north-to-south rupture scenario. Peak velocities of 1 m.s−1 and 2 m.s−1 occur in the Los Angeles Basin and San Fernando Valley, respectively, while the corresponding peak displacements are about 1 m and 2 m, respectively. Peak interstory drifts in the two buildings exceed 0.10 and 0.06 in many areas of the San Fernando Valley and the Los Angeles Basin, respectively. The redesigned building performs significantly better than the existing building; however, its improved design based on the 1997 Uniform Building Code is still not adequate to prevent serious damage. The results from the south-to-north scenario are not as alarming, although damage is serious enough to cause significant business interruption and compromise life safety

Waveform modeling of the slab beneath Japan

The tomographic P wave model for the Japan subduction zone derived by Zhao et al. (1994) has two very striking features: a slab about 90 km thick with P wave velocities 3–6% higher than the surrounding mantle and a mantle wedge with −6% low-velocity anomalies. We study three-component seismograms from more than 600 Hi-net stations produced by two earthquakes which occurred in the downgoing Pacific Plate at depths greater than 400 km. We simulate body wave propagation in the three-dimensional (3-D) P wave model using 2-D finite difference (FDM) and 3-D spectral element (SEM) methods. As measured by cross correlation between synthetics and data, the P wave model typically explains about half of the traveltime anomaly and some of the waveform complexity but fails to predict the extended SH wave train. In this study we take advantage of the densely distributed Hi-net stations and use 2-D FDM modeling to simulate the P-SV and SH waveforms. Our 2-D model suggests that a thin, elongated low-velocity zone exists atop the slab, extending down to a depth of 300 km with an S wave velocity reduction of 14% if a thickness of 20 km is assumed. Further, 3-D SEM simulations confirm that this model explains a strong secondary arrival which cannot easily be imaged with standard tomographic techniques. The low-velocity layer could explain the relatively weak coupling associated with most subduction zones at shallow depths (<50 km), generally involving abundant volcanic activity and silent earthquakes, and it may also help to further our understanding of the water-related phase transition of ultramafic rocks, and the nature of seismicity at intermediate depths (~70–300 km)

Finite-frequency sensitivity of body waves to anisotropy based upon adjoint methods

We investigate the sensitivity of finite-frequency body-wave observables to mantle anisotropy based upon kernels calculated by combining adjoint methods and spectral-element modelling of seismic wave propagation. Anisotropy is described by 21 density-normalized elastic parameters naturally involved in asymptotic wave propagation in weakly anisotropic media. In a 1-D reference model, body-wave sensitivity to anisotropy is characterized by ‘banana–doughnut’ kernels which exhibit large, path-dependent variations and even sign changes. P-wave travel-times appear much more sensitive to certain azimuthally anisotropic parameters than to the usual isotropic parameters, suggesting that isotropic P-wave tomography could be significantly biased by coherent anisotropic structures, such as slabs. Because of shear-wave splitting, the common cross-correlation travel-time anomaly is not an appropriate observable for S waves propagating in anisotropic media. We propose two new observables for shear waves. The first observable is a generalized cross-correlation travel-time anomaly, and the second a generalized ‘splitting intensity’. Like P waves, S waves analysed based upon these observables are generally sensitive to a large number of the 21 anisotropic parameters and show significant path-dependent variations. The specific path-geometry of SKS waves results in favourable properties for imaging based upon the splitting intensity, because it is sensitive to a smaller number of anisotropic parameters, and the region which is sampled is mainly limited to the upper mantle beneath the receiver

Performance of 18-Story Steel Momentframe Buildings during a large San Andreas Earthquake - A Southern California-Wide End-to-End Simulation

The mitigation of seismic risk in urban areas in the United States and abroad is of major concern for all governments. Unfortunately no comprehensive studies have attempted to address this issue in a rigorous, quantitative manner. This study tackles this problem head-on for one typical class of tall buildings in southern California. The approach adopted here can be used as a template to study earthquake risk in other seismically sensitive regions of the world, such as Taiwan, Japan, Indonesia, China, South American countries (Chile, Bolivia, etc.), and the west coast of the United States (in particular, Seattle). In 1857 a large earthquake of magnitude 7.9 [1] occurred on the San Andreas fault with rupture initiating at Parkeld in Central California and propagating in a southeasterly direction over a distance of more than 360 km. Such a unilateral rupture produces signicant directivity toward the San Fernando and Los Angeles basins. Indeed, newspaper reports (Los Angeles Star [2, 3]) of sloshing observed in the Los Angeles river point to long-duration (1-2 min) and long-period (2-8 s) shaking, which could have a severe impact on present-day tall buildings, especially in the mid-height range. To assess the risk posing tall steel moment-frame buildings from an 1857-like earthquake on the San Andreas fault, a nite source model of the magnitude 7.9 November 3, 2002 Denali fault earthquake is mapped on to the San Andreas fault with rupture initiating at Parkeld in Central California and propagating a distance of about 290 km in a south-easterly direction. As the rupture proceeds down south from Parkeld and hits the big bend on the San Andreas fault, it sheds off a signicant amount of energy into the San Fernando valley, generating large amplitude ground motion there. A good portion of this energy spills over into the Los Angeles basin with many cities along the coast such as Santa Monica and Seal Beach and more inland areas going east from Seal beach towards Anaheim experiencing long-duration shaking. In addition, the tail-end of the rupture sheds energy from SH/Love waves into the Baldwin Park-La Puente region, which is bounded by a line of mountains that creates a mini-basin, further amplifying the ground motion. The peak velocity is of the order of 1 m.s in the Los Angeles basin, including downtown Los Angeles, and 2 m.s in the San Fernando valley. Signicant displacements occur in the basins but not in the mountains. The peak displacements are in the neighborhood of 1 m in the Los Angeles basin and 2 m in the San Fernando valley. The ground motion simulation is performed using the spectral element method based seismic wave propagation program, SPECFEM3D. To study the effects of the ground motion simulated at 636 sites (spread across southern California, spaced at about 3.5 km each way), computer models of an existing 18-story steel moment-frame building and a redesigned building with the same conguration (redesigned to current standards using the 1997 Uniform Building Code) are analyzed using the nonlinear structural analysis program, FRAME3D. For these analyses, the building Y direction is aligned with the geographical north direction. As expected, the existing building model fares much worse than the redesigned building model. Fracture occurs in at least 25% of the connections in this building when located in the San Fernando valley. About 10% of connections fracture in the building when located in downtown Los Angeles and the mid-Wilshire district (Beverly Hills), while the numbers are about 20% when it is located in Santa Monica, west Los Angeles, Inglewood , Alhambra, Baldwin Park, La Puente, Downey, Norwalk, Brea, Fullerton, Anaheim and Seal Beach. The peak interstory drifts in the middle-third and bottom-third of the existing building are far greater than the top-third pointing to damage being localized to the lower oors. The localization of damage in the lower oors rather than the upper oors could potentially be worse because of the risk of more oors pancaking on top of each other if a single story gives way. Consistent with the extent of fracture observed, the peak drifts in the existing building exceed 0.10 when located in the San Fernando valley, Baldwin Park and neighboring cities, Santa Monica, west Los Angeles and neighboring cities, Norwalk and neighboring cities, and Seal Beach and neighboring cities, which is well into the postulated collapse regime. When located in downtown Los Angeles and the mid-Wilshire district, the building would barely satisfy the collapse prevention criteria set by FEMA [4] with peak drifts of about 0.05. The performance of the newly designed 18-story steel building is signicantly better than the existing building for the entire region. However, the new building still has signicant drifts indicative of serious damage when located in the San Fernando valley or the Baldwin Park area. When located in coastal cities (such as Santa Monica, Seal Beach etc.), the Wilshire-corridor (west Los Angeles, Beverly Hills, etc.), the mid-city region (Downey, Norwalk, etc.) or the booming Orange County cities of Anaheim and Santa Ana, it has peak drifts of about 0.05, once again barely satisfying the FEMA collapse prevention criteria [5]. In downtown Los Angeles it does not undergo much damage in this scenario. Thus, even though this building has been designed according to the latest code, it suffers damage that would necessitate closure for some time following the earthquake in most areas, but this should be expected since this is a large earthquake and building codes are written to limit the loss of life and ensure "collapse prevention" for such large earthquakes, but not necessarily limit damage. Unfortunately, widespread closures such as this could cripple the regional economy in the event of such an earthquake. A second scenario considered in the study involves the same Denali earthquake source mapped to the San Andreas fault but with rupture initiating in the south and propagating to the north (with the largest amount of slip occurring to the north in Central California) instead of the other way around. The results of such a scenario indicate that ground shaking would be far less severe demonstrating the effects of directivity and slip distribution in dictating the level of ground shaking and the associated damage in buildings. The peak drifts in existing and redesigned building models are in the range of 0.02-0.04 indicating that there is no signicant danger of collapse. However, damage would still be signicant enough to warrant building closures and compromise life safety in some instances. The ground motion simulation and the structural damage modeling procedures are validated using data from the January 17, 1994, Northridge earthquake while the band-limited nature of the ground motion simulation (limited to a shortest period of 2 s by the current state of knowledge of the 3-D Earth structure) is shown to have no signicant effect on the response of the two tall buildings considered here with the use of observed records from the 1999 Chi Chi earthquake in Taiwan and the 2001 Tokachi-Oki earthquake in Japan

Finite-Frequency SKS Splitting: Measurement and Sensitivity Kernels

Splitting of SKS waves caused by anisotropy may be analyzed by measuring the splitting intensity, i.e., the amplitude of the transverse signal relative to the radial signal in the SKS time window. This quantity is simply related to structural parameters. Extending the widely used cross-correlation method for measuring travel-time anomalies to anisotropic problems, we propose to measure the SKS-splitting intensity by a robust cross-correlation method that can be automated to build large high-quality datasets. For weak anisotropy, the SKS-splitting intensity is retrieved by cross-correlating the radial signal with the sum of the radial and transverse signals. The cross-correlation method is validated based upon a set of Californian seismograms. We investigate the sensitivity of the SKS-splitting intensity to general anisotropy in the mantle based upon a numerical technique (the adjoint spectral-element method) considering the full physics of wave propagation. The computations reveal a sensitivity remarkably focused on a small number of elastic parameters and on a small region of the upper mantle. These fundamental properties and the practical advantages of the measurement make the cross-correlation SKS-splitting intensity particularly well adapted for finite-frequency imaging of upper-mantle anisotropy

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

Spectral-element simulations of wave propagation in porous media

We present a derivation of the equations describing wave propagation in porous media based upon an averaging technique which accommodates the transition from the microscopic to the macroscopic scale. We demonstrate that the governing macroscopic equations determined by Biot remain valid for media with gradients in porosity. In such media, the well-known expression for the change in porosity, or the change in the fluid content of the pores, acquires two extra terms involving the porosity gradient. One fundamental result of Biot's theory is the prediction of a second compressional wave, often referred to as 'type II' or 'Biot's slow compressional wave', in addition to the classical fast compressional and shear waves. We present a numerical implementation of the Biot equations for 2-D problems based upon the spectral-element method (SEM) that clearly illustrates the existence of these three types of waves as well as their interactions at discontinuities. As in the elastic and acoustic cases, poroelastic wave propagation based upon the SEM involves a diagonal mass matrix, which leads to explicit time integration schemes that are well suited to simulations on parallel computers. Effects associated with physical dispersion and attenuation and frequency-dependent viscous resistance are accommodated based upon a memory variable approach. We perform various benchmarks involving poroelastic wave propagation and acoustic–poroelastic and poroelastic–poroelastic discontinuities, and we discuss the boundary conditions used to deal with these discontinuities based upon domain decomposition. We show potential applications of the method related to wave propagation in compacted sediments, as one encounters in the petroleum industry, and to detect the seismic signature of buried landmines and unexploded ordnance

Even-degree lateral variations in the Earth's mantle constrained by free oscillations and the free-air gravity anomaly

The recent occurrence of several large earthquakes, in particular the 1994 June 9 Bolivia event, has motivated a re‐examination of the Earth's large‐scale heterogeneity from a normal‐mode or free‐oscillation perspective. Compared to earlier studies, the number of normal‐mode constraints on lateral variations in the mantle has increased five‐fold, and toroidal and cross‐coupled modes complement the traditional spheroidal mode data set. It is demonstrated that this large collection of mode data, combined with the free‐air gravity anomaly, can reliably constrain even‐degree lateral variations in wave velocities as well as density. We present the first whole‐mantle density model constrained by seismology. Our shear and compressional velocity models are consistent with existing models based upon traveltimes and waveforms, and are reasonably well correlated throughout the mantle. Shear and bulk sound velocity models exhibit a gradual decrease in correlation with depth, and are anti‐correlated near the core–mantle boundary. We find that lateral variations in density are poorly correlated with wave velocities, and are locally anti‐correlated with shear velocity in the lowermost mantle. The correlations between wave velocities and density suggest both a thermal and a compositional origin to lateral heterogeneity. In addition to traditional maps of lateral variations in wave velocity, we also present maps of lateral variations in shear and bulk moduli. The inversion puts weak constraints on even‐degree topographic variations on the core–mantle boundary, the 660 km discontinuity and dynamic free surface topography. Finally, we determine both radially and laterally varying scaling relationships, including Poisson's ratio