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.

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.

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.

The GTL implemented in the USR was based on Ely et al. (2010) and uses the geology-based Vs30 maps of Wills and Clahan (2006) to specify velocity values at the Earths surface in the voxet. V P , and in turn density, are inferred from surface V S using the scaling laws of Brocher (2005). These values were parameterized to a depth of z T = 350 meters with the following formulations:where z ′ is depth, V ST and V P T are are S-and P-wave velocities extracted from the crustal velocity model at depth z T , P () is the Brocher (2005) P-wave velocity scaling law, andThe coefficient a controls the ratio of surface velocity to original 30 meter average, b controls overall curvature, and c controls near-surface curvature of the velocity profile. The coefficients a = 1/2, b = 2/3, and c = 3/2 were chosen to fit the generic rock profile of Boore and Joyner (1997) while also producing smooth and well-behaved profiles when combined with the underlying basin and crustal velocity models (Ely et al., 2010) ( Figure 7). S2 Model validation, comparison, and uncertaintyThe velocity model (CVM) component of the USR described here is assembled from several different data sets and models, and thus it is challenging to formally assess model resolution and uncertainties. One clear step for the sedimentary basins is to assess the variability in well data that is not represented in the final model. As we discussed, these data measure interval transit times over borehole distances of less than 1 m, whereas the velocity model uses smoothed (25 m sampled) versions of these data. To make this assessment, we compared observations directly with the velocity values represented at 108 well bore locations in the Los Angeles basin. Our analysis shows a standard deviation of 6.5% around a mean of 1.0 for the ratio between compressional wave slowness in logs and the model in a population of ca. 1.1 million samples. This corresponds to a standard deviation in V P of ±99 m/s at 2000 m/s.