S U M M A R YWe draw connections between seismic tomography, adjoint methods popular in climate and ocean dynamics, time-reversal imaging and finite-frequency 'banana-doughnut' kernels. We demonstrate that Fréchet derivatives for tomographic and (finite) source inversions may be obtained based upon just two numerical simulations for each earthquake: one calculation for the current model and a second, 'adjoint', calculation that uses time-reversed signals at the receivers as simultaneous, fictitious sources. For a given model, m, we consider objective functions χ (m) that minimize differences between waveforms, traveltimes or amplitudes. For tomographic inversions we show that the Fréchet derivatives of such objective functions may be written in the generic form δχ = V K m (x)δ ln m(x) d 3 x, where δ ln m = δm/m denotes the relative model perturbation. The volumetric kernel K m is defined throughout the model volume V and is determined by time-integrated products between spatial and temporal derivatives of the regular displacement field s and the adjoint displacement field s † ; the latter is obtained by using time-reversed signals at the receivers as simultaneous sources. In waveform tomography the time-reversed signal consists of differences between the data and the synthetics, in traveltime tomography it is determined by synthetic velocities, and in amplitude tomography it is controlled by synthetic displacements. For each event, the construction of the kernel K m requires one forward calculation for the regular field s and one adjoint calculation involving the fields s and s † . In the case of traveltime tomography, the kernels K m are weighted combinations of banana-doughnut kernels. For multiple events the kernels are simply summed. The final summed kernel is controlled by the distribution of events and stations. Fréchet derivatives of the objective function with respect to topographic variations δh on internal discontinuities may be expressed in terms of 2-D kernels K h and K h in thewhere denotes a solid-solid or fluid-solid boundary and FS a fluid-solid boundary, and ∇ denotes the surface gradient. We illustrate how amplitude anomalies may be inverted for lateral variations in elastic and anelastic structure. In the context of a finite-source inversion, the model vector consists of the time-dependent elements of the moment-density tensor m(x, t). We demonstrate that the Fréchet derivatives of the objective function χ may in this case be written in the formwhere † denotes the adjoint strain tensor on the finite-fault plane . In the case of a point source this result reduces further to the calculation of the time-dependent adjoint strain tensor † at the location of the point source, an approach reminiscent of an acoustic time-reversal mirror. The theory is illustrated for both tomographic and source inversions using a 2-D spectral-element method.

Summary We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3‐D earth models. The method combines the flexibility of a finite element method with the accuracy of a spectral method. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wavefield on the elements is discretized using high‐degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss–Lobatto–Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simplifies the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/reflectivity method for layer‐cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.

T he Interior Exploration using Seismic Investigations, Geodesy and Heat Transport (InSight) mission landed on Mars on 26 November 2018 in Elysium Planitia 1,2 , 38 years after the end of Viking 2 lander operations. At the time, Viking's seismometer 3 did not succeed in making any convincing Marsquake detections, due to its on-deck installation and high wind sensitivity. InSight therefore provides the first direct geophysical in situ investigations of Mars's interior structure by seismology 1,4. The Seismic Experiment for Interior Structure (SEIS) 5 monitors the ground acceleration with six axes: three Very Broad Band (VBB) oblique axes, sensitive to frequencies from tidal up to 10 Hz, and one vertical and two horizontal Short Period (SP) axes, covering frequencies from ~0.1 Hz to 50 Hz. SEIS is complemented by the APSS experiment 6 (InSight Auxiliary Payload Sensor Suite), which includes pressure and TWINS (Temperature and Winds for InSight) sensors and a magnetometer. These sensors monitor the atmospheric sources of seismic noise and signals 7. After seven sols (Martian days) of SP on-deck operation, with seismic noise comparable to that of Viking 3 , InSight's robotic arm 8 placed SEIS on the ground 22 sols after landing, at a location selected through analysis of InSight's imaging data 9. After levelling and noise assessment, the Wind and Thermal Shield was deployed on sol 66 (2 February 2019). A few days later, all six axes started continuous seismic recording, at 20 samples per second (sps) for VBBs and 100 sps for SPs. After onboard decimation, continuous records at rates from 2 to 20 sps and event records 5 at 100 sps are transmitted. Several layers of thermal protection and very low self-noise enable the SEIS VBB sensors to record the daily variation of the

Summary We use a spectral‐element method to simulate seismic wave propagation throughout the entire globe. The method is based upon a weak formulation of the equations of motion and combines the flexibility of a finite‐element method with the accuracy of a global pseudospectral method. The finite‐element mesh honours all first‐ and second‐order discontinuities in the earth model. To maintain a relatively constant resolution throughout the model in terms of the number of grid points per wavelength, the size of the elements is increased with depth in a conforming fashion, thus retaining a diagonal mass matrix. In the Earth's mantle and inner core we solve the wave equation in terms of displacement, whereas in the liquid outer core we use a formulation based upon a scalar potential. The three domains are matched at the inner core and core–mantle boundaries, honouring the continuity of traction and the normal component of velocity. The effects of attenuation and anisotropy are fully incorporated. The method is implemented on a parallel computer using a message passing technique. We benchmark spectral‐element synthetic seismograms against normal‐mode synthetics for a spherically symmetric reference model. The two methods are in excellent agreement for all body‐ and surface‐wave arrivals with periods greater than about 20 s.

With the use of a large collection of free-oscillation data and additional constraints imposed by the free-air gravity anomaly, lateral variations in shear velocity, compressional velocity, and density within the mantle; dynamic topography on the free surface; and topography on the 660-km discontinuity and the core-mantle boundary were determined. The velocity models are consistent with existing models based on travel-time and waveform inversions. In the lowermost mantle, near the core-mantle boundary, denser than average material is found beneath regions of upwellings centered on the Pacific Ocean and Africa that are characterized by slow shear velocities. These anomalies suggest the existence of compositional heterogeneity near the core-mantle boundary.

S U M M A R YWe 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, m 16 , is described in terms of independent shear (V S ) and bulk-sound (V B ) 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.

Abstract. A new technique for making single-station phase velocity measurements is developed and applied to a large number of globally recorded Rayleigh and Love waves in the period range 35-150 s. The method is based on phase-matched filter theory and iteratively suppresses the effect of interfering overtones by minimizing residual dispersion. The model surface wave signal is described by its amplitude and apparent phase velocity, both of which are parameterized in terms of smooth B-spline functions of frequency. A misfit function is constructed which represents the difference between the model and observed waveforms, and the optimal spline coefficients are estimated in an iterative misfit minimization algorithm. In order to eliminate cycle skips in the measurements of phase at short periods, the waveforms are first matched at long periods, and the frequency range is gradually extended to include higher frequencies. The application of the algorithm to records from the Global Seismographic Network, using earthquakes in the Harvard centroid-moment tensor catalog, results in the determination of more than 50,000 high-quality dispersion curves. The observed variations in measured dispersion for pairwise similar paths are used to estimate realistic uncertainties in the data. Phase delays at discrete periods are inverted for global maps of variations in phase velocity expanded in spherical harmonics up to degree 40. A realistic resolution test indicates that structures are well recovered up to at least degree 20. The new phase velocity maps explain 70-96% of the observed variance in phase residuals, reflecting the high internal consistency of the dispersion measurements.

Using an inversion strategy based on adjoint methods, we developed a three-dimensional seismological model of the southern California crust. The resulting model involved 16 tomographic iterations, which required 6800 wavefield simulations and a total of 0.8 million central processing unit hours. The new crustal model reveals strong heterogeneity, including local changes of +/-30% with respect to the initial three-dimensional model provided by the Southern California Earthquake Center. The model illuminates shallow features such as sedimentary basins and compositional contrasts across faults. It also reveals crustal features at depth that aid in the tectonic reconstruction of southern California, such as subduction-captured oceanic crustal fragments. The new model enables more realistic and accurate assessments of seismic hazard.