This paper studies a macroeconomic model in which financial experts borrow from less productive agents. We pursue four sets of results: (i) The economy is prone to instability and occasionally enters volatile episodes. As volatility spikes agents precautionary motive increases depressing prices even further. Log-linear approximations fail to capture these non-linear effects that can cause economies to be significantly depressed for long periods of time. (ii) Endogenous risk during volatile episodes increases asset price correlations. (iii) Financial experts impose a negative externality on each other and on the labor sector by not maintaining adequate capital cushion, and funding structure. (iv) While risk sharing within the financial sector (through securitization and derivatives contracts) reduces many inefficiencies, it can also amlify systemic risks.

/pdf/a-macroeconomic-model-with-a-financial-sector-1s2akgr5uv.pdf

A macroeconomic model with a financial sector

We have developed model S40RTS of shear-velocity variation in Earth's mantle using a new collection of Rayleigh wave phase velocity, teleseismic body-wave traveltime and normal-mode splitting function measurements. This data set is an order of magnitude larger than used for S20RTS and includes new data types. The data are related to shear-velocity perturbations from the (anisotropic) PREM model via kernel functions and ray paths that are computed using PREM. Contributions to phase delays and traveltimes from the heterogeneous crust are estimated using model CRUST2.0. We calculate crustal traveltimes from long-period synthetic waveforms rather than using ray theory. Shear-velocity perturbations are parametrized by spherical harmonics up to degree 40 and by 21 vertical spline functions for a total of 35 301 degrees of freedom. S40RTS is characterised by 8000 resolved unknowns. Since we compute the exact inverse, it is straightforward to determine models associated with fewer or more unknowns by adjusting the model damping. S40RTS shares many characteristics with S20RTS because it is based on the same data types and similar modelling procedures. However, S40RTS shows more clearly than S20RTS the abrupt change in the pattern of shear-velocity heterogeneity across the 660-km phase transition and it presents a more complex patern of shear-velocity heterogeneity in the lower mantle. Utilities to visualise S40RTS and software to analyse the resolution of S40RTS (or models for different damping parameters) are made available.

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S40RTS: A degree-40 shear-velocity model for the mantle from new Rayleigh wave dispersion, teleseismic traveltime and normal-mode splitting function measurements

SUMMARY

We 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 Frechet 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 Frechet derivatives of such objective functions may be written in the generic form , where δ ln m=δm/m denotes the relative model perturbation. The volumetric kernel Km 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 Km 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 Km 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. Frechet derivatives of the objective function with respect to topographic variations δh on internal discontinuities may be expressed in terms of 2-D kernels Kh and Kh in the form , where Σ denotes a solid-solid or fluid-solid boundary and ΣFS a fluid–solid boundary, and ∇Σ 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 Frechet derivatives of the objective function χ may in this case be written in the form , where e† denotes the adjoint strain tensor on the finite-fault plane Σ. In the case of a point source this result reduces further to the calculation of the time-dependent adjoint strain tensor e† 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.

Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels

[1] We document our tomographic method and present a new global model of three-dimensional (3-D) variations in mantle P wave velocity. The model is parameterized by means of rectangular cells in latitude, longitude, and radius, the size of which adapts to sampling density by short-period (1 Hz) data. The largest single data source is ISC/NEIC data reprocessed by Engdahl and coworkers, from which we use routinely picked, short-period P, Pg, Pn, pP, and pwP data (for earthquakes during the period 1964∼2007). To improve the resolution in the lowermost and uppermost mantle, we use differential times of core phases (PKPAB − PKPDF, PKPAB − PKPBC, Pdiff − PKPDF) and surface-reflected waves (PP-P). The low-frequency differential times (Pdiff, PP) are measured by waveform cross correlation. Approximate 3-D finite frequency kernels are used to integrate the long-period data (Pdiff, PP) and short-period (P, pP, PKP) data. This global data set is augmented with data from regional catalogs and temporary seismic arrays. A crust correction is implemented to mitigate crustal smearing into the upper mantle. We invert the data for 3-D variations in P wave speed and effects of hypocenter mislocation subject to norm and gradient regularization. Spatial resolution is ∼100 km in the best sampled upper mantle regions. Our model, which is available online and which will be updated periodically, reveals in unprecedented detail the rich variation in style of subduction of lithospheric slabs into the mantle. The images confirm the structural complexity of downwellings in the transition zone discussed in previous papers and show with more clarity the structure of slab fragments stagnant in the transition zone beneath east Asia. They also reveal low wave speed beneath major hot spots, such as Iceland, Afar, and Hawaii, but details of these structures are not well resolved by the data used.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2007GC001806

A new global model for P wave speed variations in Earth's mantle

S U M M A R Y The rapid expansion of broad-band seismic networks over the last decade has paved the way for a new generation of global tomographic models. Significantly improved resolution of global upper-mantle and crustal structure can now be achieved, provided that structural information is extracted effectively from both surface and body waves and that the effects of errors in the data are controlled and minimized. Here, we present a new global, vertically polarized shear speed model that yields considerable improvements in resolution, compared to previous ones, for a variety of features in the upper mantle and crust. The model, SL2013sv, is constrained by an unprecedentedly large set of waveform fits (∼3/4 of a million broad-band seismograms), computed in seismogram-dependent frequency bands, up to a maximum period range of 11– 450 s. Automated multimode inversion of surface and S-wave forms was used to extract a set of linear equations with uncorrelated uncertainties from each seismogram. The equations described perturbations in elastic structure within approximate sensitivity volumes between sources and receivers. Going beyond ray theory, we calculated the phase of every mode at every frequency and its derivative with respect to Sand P-velocity perturbations by integration over a sensitivity area in a 3-D reference model; the (normally small) perturbations of the 3-D model required to fit the waveforms were then linearized using these accurate derivatives. The equations yielded by the waveform inversion of all the seismograms were simultaneously inverted for a 3-D model of shear and compressional speeds and azimuthal anisotropy within the crust and upper mantle. Elaborate outlier analysis was used to control the propagation of errors in the data (source parameters, timing at the stations, etc.). The selection of only the most mutually consistent equations exploited the data redundancy provided by our data set and strongly reduced the effect of the errors, increasing the resolution of the imaging. Our new shear speed model is parametrized on a triangular grid with a ∼280 km spacing. In well-sampled continental domains, lateral resolution approaches or exceeds that of regionalscale studies. The close match of known surface expressions of deep structure with the distribution of anomalies in the model provides a useful benchmark. In oceanic regions, spreading ridges are very well resolved, with narrow anomalies in the shallow mantle closely confined near the ridge axis, and those deeper, down to 100–120 km, showing variability in their width and location with respect to the ridge. Major subduction zones worldwide are well captured, extending from shallow depths down to the transition zone. The large size of our waveform fit data set also provides a strong statistical foundation to re-examine the validity field of the JWKB approximation and surface wave ray theory. Our analysis shows that the approximations are likely to be valid within certain time–frequency portions of most seismograms with high signal-to-noise ratios, and these portions can be identified using a set of consistent criteria that we apply in the course of waveform fitting.

/pdf/global-shear-speed-structure-of-the-upper-mantle-and-20tz52hlnh.pdf

Global shear speed structure of the upper mantle and transition zone

Summary

Seismic traveltimes are the most widely exploited data in seismology. Their Frechet or sensitivity kernels are important tools in tomographic inversions based on the Born or single-scattering approximation. The current study is motivated by a paradox posed by two seemingly irreconcilable observations in the numerical calculations for the sensitivity kernels of the traveltime perturbations. Calculations of kernels for 2-D media by the normal-mode approach indicate that traveltimes are most sensitive to the structure on and around the geometrical ray paths corresponding to the seismic arrivals, whereas calculations for 3-D media by geometrical ray theory predict exactly zero sensitivities on the ray paths. In the current work, we employ these two completely different wave-propagation approaches, the more efficient geometrical ray theory and the more accurate normal-mode theory, to investigate the 3-D sensitivities of the delay times to shear-wave speed variations. Expressions for the delay-time Frechet kernels are presented for both methods, and extensive numerical experiments are conducted for various types of seismic phases as well as for different reference earth models. The results show that the contradictory observations are but two examples of a wide range of behaviours in the delay-time sensitivity. For most of the seismic phases in realistic reference models with multiple discontinuities, wave-speed gradients and low-velocity zones, the wavefields are highly complicated and ray theory, which describes the response by the contributions of a few geometrical rays between the source and receiver, produces qualitatively different delay-time kernels from those obtained by the normal-mode theory, which includes essentially all contributions.

Three‐dimensional Fréchet differential kernels for seismicdelay times

Seismic modeling, processing, and inversion often require the calculation of the seismic response resulting from a suite of closely related seismic models. Even though changes to the model may be restricted to a small subvolume, we need to perform simulations for the full model. We present a new finite‐difference method that circumvents the need to resimulate the complete model for local changes. By requiring only calculations in the subvolume and its neighborhood, our method makes possible significant reductions in computational cost and memory requirements. In general, each source/receiver location requires one full simulation on the complete model. Following these pre‐computations, recalculation of the altered wavefield can be limited to the region around the subvolume and its neighborhood. We apply our method to a 2-D time‐lapse seismic problem, thereby achieving a factor of 15 reduction in computational cost. Potential savings for 3-D are far greater.

An efficient method for calculating finite-difference seismograms after model alterations

Finite-difference (FD) techniques are widely used to model wave propagation through complex structures. Two main sources of error can be identified: (1) from numerical dispersion and numerical anisotropy and (2) by modeling the response of internal grid boundaries. Conventional discretization criteria to reduce the effects of numerical dispersion and numerical anisotropy have long been established (5-8 gridpoints per wavelength for a fourth-order accurate FD scheme). We analyze the second source of errors, comparing different staggered-grid FD solutions to the Cagniard-de Hoop solution in models with fluid-solid contacts. Our results confirm that it is sufficient to rely on conventional discretization criteria if the fluid-solid interface is aligned with the grid. If accurate modeling of the Scholte wave is required, then a new imaging method we propose should be used to allow for conventional sampling of the wavefield to minimize numerical dispersion. However, for an interface not aligned with the grid (irregular interfaces), a spatial sampling of at least 15 gridpoints per minimum wavelength is required to obtain acceptable results, particularly in seismic seabed applications where Scholte waves may need to be modeled more accurately.

Finite-difference modeling of wave propagation in a fluid-solid configuration

Described is a finite-difference methodology for efficiently computing the seismic response from a seismic model subject to several changes within sub-volumes. Initially, the response from an original model is calculated and the wave field is recorded at receivers and along closed surfaces around the sub-volumes. As changes occur, the recorded seismograms can be updated by simulating the response on small models encompassing the immediate neighborhood of the regions of change, the recording locations, and the parts of model that contribute to subsequent reflections and scattering recorded at the receivers. The recorded wave field can then be added to the reference wave field to obtain the seismic responses of alterations to the model without having to recalculate the full altered model.

Method of determining the response caused by model alterations in seismic simulations

Several staggered grid schemes have been suggested for performing finite-difference calculations for the elastic wave equations. In this paper, the dispersion relationships and related computational requirements for the Lebedev and rotated staggered grids for anisotropic, elastic, finite-difference calculations in smooth models are analyzed and compared. These grids are related to a popular staggered grid for the isotropic problem, the Virieux grid. The Lebedev grid decomposes into Virieux grids, two in two dimensions and four in three dimensions, which decouple in isotropic media. Therefore the Lebedev scheme will have twice or four times the computational requirements, memory, and CPU as the Virieux grid but can be used with general anisotropy. In two dimensions, the rotated staggered grid is exactly equivalent to the Lebedev grid, but in three dimensions it is fundamentally different. The numerical dispersion in finite-difference grids depends on the direction of propagation and the grid type and parameters. A joint numerical dispersion relation for the two grids types in the isotropic case is derived. In order to compare the computational requirements for the two grid types, the dispersion, averaged over propagation direction and medium velocity are calculated. Setting the parameters so the average dispersion is equal for the two grids, the computational requirements of the two grid types are compared. In three dimensions, the rotated staggered grid requires at least 20% more memory for the field data and at least twice as many number of floating point operations and memory accesses, so the Lebedev grid is more efficient and is to be preferred.

Chris Chapman

Papers

Three‐dimensional Fréchet differential kernels for seismicdelay times

An efficient method for calculating finite-difference seismograms after model alterations

Finite-difference modeling of wave propagation in a fluid-solid configuration

Method of determining the response caused by model alterations in seismic simulations

A comparison of the dispersion relations for anisotropic elastodynamic finite-difference grids