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.

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A macroeconomic model with a financial sector

We have developed and implemented a robust and practical scheme for anisotropic 3D acoustic full-waveform inversion (FWI). We demonstrate this scheme on a field data set, applying it to a 4C ocean-bottom survey over the Tommeliten Alpha field in the North Sea. This shallow-water data set provides good azimuthal coverage to offsets of 7 km, with reduced coverage to a maximum offset of about 11 km. The reservoir lies at the crest of a high-velocity antiformal chalk section, overlain by about 3000 m of clastics within which a low-velocity gas cloud produces a seismic obscured area. We inverted only the hydrophone data, and we retained free-surface multiples and ghosts within the field data. We invert in six narrow frequency bands, in the range 3 to 6.5 Hz. At each iteration, we selected only a subset of sources, using a different subset at each iteration; this strategy is more efficient than inverting all the data every iteration. Our starting velocity model was obtained using standard PSDM model bui...

https://library.seg.org/doi/pdf/10.1190/geo2012-0338.1

Anisotropic 3D full-waveform inversion

Computational Science and Engineering (CSE) is the multi-disciplinary field of computer-based modelling and simulation for studying scientific phenomena and engineering designs. Modelling and simulation helps to validate theory, and makes it possible to analyse scenarios that would otherwise be too time-consuming, expensive, or dangerous to study by experiment. Data exploration helps to turn numbers into insight which is especially challenging in times of Big Data.

Computational Science and Engineering

We present wave equations for modeling and reverse-time migration in acoustic VTI media, derived directly from Hooke’s law and the equations of motion. The resulting set of five first-order differential equations in the three particle velocity components and two independent stress components involves no approximations apart from the acoustic VTI approximation itself and allows handling variable density in a natural way.

Acoustic VTI Wave Equations And Their Application For Anisotropic Reverse-time Migration

Full waveform inversion (FWI) delivers high-resolution quantitative images and is a promising technique to obtain macroscale physical property model of the subsurface. In most geophysical applications, prior information, such as that collected in wells, is available and should be used to increase the image reliability. For this, we propose to introduce three terms in the definition of the FWI misfit function: the data misfit itself, the first-order Tikhonov regularization term acting as a smoothing operator, and a prior model norm term. This last term is the way to smoothly introduce prior information into the FWI workflow. On a selected target of the Marmousi synthetic example, significant improvement was obtained when using the prior model term for noise-free and noisy synthetic data. The prior model term may significantly reduce the inversion sensi tivity to incorrect initial conditions. The limited range of spatial wavenumber sampling by the acquisition may be compensated with the prior model information, for multiple-free and multiplecontaminated data. Prior and initial models play different roles in the inversion scheme. The starting model is used for wave propagation and therefore drives the data-misfit gradient, whereas the prior model is never explicitly used for solving the wave equation and only drives the optimization step as an additional constraint to minimize the total objective function. Thus, the prior model is not required to follow kinematic properties as precisely as the initial model, except in zones of poor illumination. In addition, we investigate the influence of a simple dynamic decreasing weighting of the prior model term. Once the cycle-skipping problem has been solved, the impact of the prior model term is gradually reduced within the misfit function to be driven by seismic-data only.

https://users.isterre.fr/brossier/Asnaashari_2013_GEO.pdf

Regularized seismic full waveform inversion with prior model information

A new time-space domain high-order finite-difference method for the acoustic wave equation

Spatial finite-difference (FD) coefficients are usually determined by the Taylor-series expansion (TE) or optimization methods. The former can provide high accuracy on a smaller wavenumber or frequency zone, and the latter can give moderate accuracy on a larger zone. Present optimization methods applied to calculate FD coefficients are generally gradient-like or global optimization-like algorithms, and thus iterations are involved. They are more computationally expensive, and sometimes the global solution may not be found. I examined second-order spatial derivatives and computed the optimized spatial FD coefficients over the given wavenumber range using the least-squares (LS) method. The results indicated that the FD accuracy increased with increasing operator length and decreasing wavenumber range. Therefore, for the given error and operator length, globally optimal spatial FD coefficients can be easily obtained. Some optimal FD coefficients were given. I developed schemes to obtain optimized LS-...

Globally optimal finite-difference schemes based on least squares

SUMMARY We derive explicit and new implicit staggered-grid finite-difference (FD) formulas for derivatives of first order with any order of accuracy by a plane wave theory and Taylor’s series expansion. Furthermore, we arrive at a practical algorithm such that the tridiagonal matrix equations are formed by the implicit FD formulas derived from the fractional expansion of derivatives. Our results demonstrate that the accuracy of a (2N + 2)th-order implicit formula is nearly equivalent to or greater than that of a (4N)th-order explicit formula. The new implicit method only involves solving tridiagonal matrix equations. We also demonstrate that a( 2N + 2)th-order implicit formulation requires nearly the same amount of memory and computation as those of a (2N + 4)th-order explicit formulation but attains the accuracy achieved by a (4N)th-order explicit formulation when additional cost of visiting arrays is not considered. Our analysis of efficiency and numerical modelling results for elastic wave propagation demonstrates that a high-order explicit staggered-grid method can be replaced by an implicit staggered-grid method of some order, which will increase the accuracy but not the computational cost.

An implicit staggered‐grid finite‐difference method for seismic modelling

/pdf/optimal-staggered-grid-finite-difference-schemes-based-on-2gpuwhhu0g.pdf

Optimal staggered-grid finite-difference schemes based on least-squares for wave equation modelling

We propose an efficient scheme to absorb reflections from the model boundaries in numerical solutions of wave equations. This scheme divides the computational domain into boundary, transition, and inner areas. The wavefields within the inner and boundary areas are computed by the wave equation and the one-way wave equation, respectively. The wavefields within the transition area are determined by a weighted combination of the wavefields computed by the wave equation and the one-way wave equation to obtain a smooth variation from the inner area to the boundary via the transition zone. The results from our finite-difference numerical modeling tests of the 2D acoustic wave equation show that the absorption enforced by this scheme gradually increases with increasing width of the transition area. We obtain equally good performance using pseudospectral and finite-element modeling with the same scheme. Our numerical experiments demonstrate that use of 10 grid points for absorbing edge reflections attains nearly perfect absorption.

Yang Liu

Papers

A new time-space domain high-order finite-difference method for the acoustic wave equation

Globally optimal finite-difference schemes based on least squares

An implicit staggered‐grid finite‐difference method for seismic modelling

Optimal staggered-grid finite-difference schemes based on least-squares for wave equation modelling

A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation