Ultrasound wave propagation through rough interfaces: Iterative methods

Two iterative methods for the calculation of acoustic transmission through a rough interface\ud between two media are compared. The methods employ a continuous version of the conjugate\ud gradient technique. One method is based on plane-wave expansions and the other on boundary\ud integral equations and Green’s functions. A preconditioner is presented which improves the\ud convergence for spectra that include evanescent modes. The methods are compared with regard to\ud computational efficiency, rate of convergence, and residual error. The sound field differences are\ud determined for a focused ultrasound beam distorted by surfaces having a Gaussian roughness\ud spectrum. The differences are evaluated from the root-mean-square differences on the rough surface\ud and in the focal plane

Simulation of wave propagation through aberrating layers of biological media

Two iterative methods for the calculation of acoustic reflection and transmission at a rough interface between two media are compared. The methods are based on a continuous version of the conjugate gradient technique. One method is based on plane-wave expansions while the other method is based on boundary integral equations and Green's functions. The methods are compared with regard to computational efficiency, rate of convergence, and residual erro

Iterative calculation of reflected and transmitted acoustic waves at a rough interface

A rigorous iterative technique is described for calculating the acoustic wave reflection and transmission at an irregular interface between two different media. The method is based upon a plane-wave expansion technique in which the acoustic field equations and the radiation condition are satisfied analytically, while the boundary conditions at the interface are satisfied numerically. The latter is accomplished by an iterative minimization of the integrated squared error in the boundary conditions by a conjugate gradient technique, leading to a converging and relatively simple scheme. The plane interface result can be used as starting value. Although in principle the method is rigorous, numerical examples show that in practice there is a lower bound on the error in the boundary conditions which can be achieve

Radial profiles of temperature and viscosity in the Earth's mantle inferred from the geoid and lateral seismic structure

In the framework of dynamical modelling of the geoid, we have estimated basic features of the radial profile of temperature in the mantle. The applied parameterization of the geotherm directly characterizes thermal boundary layers and values of the thermal gradient in the upper and lower mantle. In the inverse modelling scheme these parameters are related to the observables (geoid and seismic structure of the mantle) through the viscosity profile which is parameterized as an exponential function of pressure and temperature. We have tested 104 model geotherms. For each of them we have found proper rheological parameters by fitting the geoid with the aid of a genetic algorithm. The geotherms which best fit the geoid show a significant increase of temperature (600-800ºC) close to the 660-km discontinuity. The value of the thermal gradient in the mid-mantle is found to be sub-adiabatic. Both a narrow thermal core-mantle boundary layer and a broad region with a superadiabatic regime can produce a satisfactory fit of the geoid. The corresponding viscosity profiles show similarities to previously presented models, in particular in the viscosity maximum occurring in the deep lower mantle. The best-fitting model predicts the values of activation volume V and energy E which are in a good agreement with the data from mineral physics, except for V in the lower mantle which is found somewhat lower than the estimate based on melting temperature analysis. An interesting feature of the viscosity profiles is a local decrease of viscosity somewhere between 500 and 1000 km depth which results from the steep increase of temperature in the vicinity of the 660-km discontinuity

Modelling planetary dynamics by using the temperature at the core-mantle boundary as a control variable: effects of rheological layering on mantle heat transport

In planetary convection, there has been a great emphasis laid on the usage of the Rayleigh number as a control parameter for describing the vigor of convection. However, realistic mantle rheology not only depends on temperature, pressure, strain-rate and composition, but also on the nature of the dominant creep mechanism, which varies with pressure and also with temperature. It is difficult to study the effects of varying influences from the convective strength without also changing the mantle flow law in the process. We have adopted the approach of using as the sole control parameter, the temperature at the core mantle boundary, T , in modelling planetary dynamics with a composite non-Newtonian and Newtonian CMB rheology, which is temperature-dependent in the upper mantle and both temperature- and pressure-dependent in the lower mantle. Increasing the T strengthens convective vigor and leads to a non-linear increase of averaged temperature, CMB heat-flow and root-mean-squared velocity. The interior viscosity decreases strongly with T and internal heating due to CMB radioactivity. A viscosity maximum is found in the horizontally averaged viscosity profile at a depth around 2000 km. This viscosity hill moves downward with diminishing amplitude in the face of increasing dissipation number and internal heating. The bottom third of the lower mantle appears to be superadiabatic as a consequence of the stiff lower-mantle rheology. The . scaling relationship between the Nusselt Nu number and T shows a relatively insensitive increase of Nu with T .In CMB CMB . terms of an effective Rayleigh number of the whole system, Ra , the power-law exponent of the Nu Ra relationship is E E very low, around 0.12. Strong pressure-dependence of lower-mantle rheology and its large volume relative to the entire mantle would induce a much lower cooling rate of the planet than previous models based on parameterized convection with a temperature-dependent viscosity. q1998 Elsevier Science B.V. All rights reserved

Using firm data to assess the performance of equilibrium search models of the labor market

Equilibrium search models are useful tools for the evaluation of labor market policies. Recently developed equilibrium search models of the labor market are able to fit the wage distribution perfectly with longitudinal labor supply data, by estimating an appropriate distribution of labor productivity across firms. This paper formally compares such structural estimates to their directly observed counterparts in firm data. More generally, we investigate the extent to which these models are able to explain the observed distributions of wages, productivities and firm sizes across firms, as well as the extent to which they are able to explain the observed relationships between these variables across firms. The parameters that capture search frictions are estimated with worker data that are matched to the data

Early formation and long-term stability of continents resulting from decompression melting in a convecting mantle

The origin of stable old continental cratonic roots is still debated. We present numerical modelling results which show rapid initial formation during the Archaean of continental roots of ca. 200 km thick. These results have been obtained from an upper mantle thermal convection model including differentiation by pressure release partial melting of mantle peridotite. The upper mantle model includes time-dependent radiogenic heat production and thermal coupling with a heat reservoir representing the Earths lower mantle and core. This allows for model experiments including secular cooling on a time-scale comparable to the age of the Earth. The model results show an initial phase of rapid continental root growth of ca. 0.1 billion year, followed by a more gradual increase of continental volume by addition of depleted material produced through hot diapiric, convective upwellings which penetrate the continental root from below. Within ca. 0.6 Ga after the start of the experiment, secular cooling of the mantle brings the average geotherm below the peridotite solidus thereby switching off further continental growth. At this time the thickness of the continental root has grown to ca. 200 km. After 1 Ga of secular cooling small scale thermal instabilities develop at the bottom of the continental root causing continental delamination without breaking up the large scale layering. This delaminated material remixes with the deeper layers. Two more periods, each with a duration of ca. 0.5 Ga and separated by quiescent periods were observed when melting and continental growth was reactivated. Melting ends at 3 Ga. Thereafter secular cooling proceeds and the compositionally buoyant continental root is stabilized further through the increase in mechanical strength induced by the increase of the temperature dependent mantle viscosity. Fluctuating convective velocity amplitudes decrease to below 10 mma Õ 1 and the volume average temperature of the sub-continental convecting mantle has decreased ca. 340 K after 4 Ga. Surface heatflow values decrease from 120 to 40 mW m Õ 2 during the 4 Ga model evolution. The surface heatflow contribution from an almost constant secular cooling rate was estimated to be 6 mW m Õ 2, in line with recent observational evidence. The modelling results show that the combined effects of compositional buoyancy and strong temperature dependent rheology result in continents which overall remain stable for a duration longer than the age of the Earth. Tracer particles have been used for studying the patterns of mantle differentiation in greater detail. The observed ( p, T, F, t)-paths are consistent with proposed stratification and thermo-mechanical history of the depleted continental root, which have been inferred from mantle xenoliths and other upper mantle samples. In addition, the particle tracers have been used to derive the thermal age of the modelled continental root, defined by a hypothetical closing temperature. © 2000 Elsevier Science B.V. All rights reserved