Green’s Function for Two-Dimensional Waves in a Radially Inhomogeneous Elastic Solid

A first-order differential equation of Green's function, at the origin G(0), for the one-dimensional lattice is derived by simple recurrence relation. Green's function at site (m) is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's function at the site (m, n) and the first derivative of the lattice Green's function at the site (m ± 1, n) is presented for the two-dimensional lattice, a differential equation of second order in G(0, 0) is obtained. By making use of the latter recurrence relation, lattice Green's function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross-section are evaluated analytically and numerically for one- and two-impurities.

Download Full-text

Green's Function for Two-Dimensional Magnetohydrodynamic Waves. II

The Physics of Fluids ◽

10.1063/1.1706203 ◽

1961 ◽

Vol 4 (10) ◽

pp. 1246

Author(s):

Harold Weitzner

Keyword(s):

Green’S Function ◽

Green's Function ◽

Two Dimensional ◽

Magnetohydrodynamic Waves

Download Full-text

Chemical Source Inversion Using Assimilated Constituent Observations in an Idealized Two-Dimensional System

Monthly Weather Review ◽

10.1175/2009mwr2775.1 ◽

2009 ◽

Vol 137 (9) ◽

pp. 3013-3025

Author(s):

Andrew Tangborn ◽

Robert Cooper ◽

Steven Pawson ◽

Zhibin Sun

Keyword(s):

Kalman Filter ◽

Green’S Function ◽

Green's Function ◽

Transport Model ◽

Chemical Constituents ◽

Gaussian Model ◽

Satellite Observation ◽

Dimensional System ◽

Two Dimensional ◽

Source Inversion

Abstract A source inversion technique for chemical constituents is presented that uses assimilated constituent observations rather than directly using the observations. The method is tested with a simple model problem, which is a two-dimensional Fourier–Galerkin transport model combined with a Kalman filter for data assimilation. Inversion is carried out using a Green’s function method and observations are simulated from a true state with added Gaussian noise. The forecast state uses the same spectral model but differs by an unbiased Gaussian model error and emissions models with constant errors. The numerical experiments employ both simulated in situ and satellite observation networks. Source inversion was carried out either by directly using synthetically generated observations with added noise or by first assimilating the observations and using the analyses to extract observations. Twenty identical twin experiments were conducted for each set of source and observation configurations, and it was found that in the limiting cases of a very few localized observations or an extremely large observation network there is little advantage to carrying out assimilation first. For intermediate observation densities, the source inversion error standard deviation is decreased by 50% to 90% when the observations are assimilated with the Kalman filter before carrying out the Green’s function inversion.

Download Full-text