The point-source method for 3D reconstructions for the Helmholtz and Maxwell equations

We use the point-source method (PSM) to reconstruct a scattered field from its associated far field pattern. The reconstruction scheme is described and numerical results are presented for three-dimensional acoustic and electromagnetic scattering problems. We give new proofs of the algorithms, based on the Green and Stratton-Chu formulae, which are more general than with the former use of the reciprocity relation. This allows us to handle the case of limited aperture data and arbitrary incident fields. Both for 3D acoustics and electromagnetics, numerical reconstructions of the field for different settings and with noisy data are shown. For shape reconstruction in acoustics, we develop an appropriate strategy to identify areas with good reconstruction quality and combine different such regions into one joint function. Then, we show how shapes of unknown sound-soft scatterers are found as level curves of the total reconstructed field.

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.

Estimating drizzle drop size and precipitation rate using two-colour lidar measurements

A method to estimate the size and liquid water content of drizzle drops using lidar measurements at two wavelengths is described. The method exploits the differential absorption of infrared light by liquid water at 905 nm and 1.5 μm, which leads to a different backscatter cross section for water drops larger than ≈50 μm. The ratio of backscatter measured from drizzle samples below cloud base at these two wavelengths (the colour ratio) provides a measure of the median volume drop diameter D0. This is a strong effect: for D0=200 μm, a colour ratio of ≈6 dB is predicted. Once D0 is known, the measured backscatter at 905 nm can be used to calculate the liquid water content (LWC) and other moments of the drizzle drop distribution. The method is applied to observations of drizzle falling from stratocumulus and stratus clouds. High resolution (32 s, 36 m) profiles of D0, LWC and precipitation rate R are derived. The main sources of error in the technique are the need to assume a value for the dispersion parameter μ in the drop size spectrum (leading to at most a 35% error in R) and the influence of aerosol returns on the retrieval (≈10% error in R for the cases considered here). Radar reflectivities are also computed from the lidar data, and compared to independent measurements from a colocated cloud radar, offering independent validation of the derived drop size distributions.

Estimating oceanic heat content change using isotherms

This paper presents a new analysis of ocean heat content changes over the last 50 yr using isotherms by calculating the mean temperature above the 148C isotherm and the depth of the 148C isotherm as separate variables. A new quantity called the ‘‘relative heat content’’ (‘‘RHC’’) is introduced, which represents the minimum local heat content change over time, relative to a fixed isotherm. It is shown how mean temperature and isotherm depth changes make separable and additive contributions to changes in RHC. Maps of RHC change between 1970 and 2000 show similar spatial patterns to a traditional fixed-depth ocean heat content change to 220 m. However, the separate contributions to RHC suggest a more spatially uniform contribution from warming above the isotherm, while isotherm depth changes show wind-driven signals, of which some are identifiable as being related to the North Atlantic Oscillation. The time series show that the warming contribution to RHC dominates the global trend, while the depth contribution only dominates on the basin scale in the North Atlantic. The RHC shows minima associated with the major volcanic eruptions (particularly in the Indian Ocean), and these are entirely contributed by mean temperature changes rather than isotherm depth changes. The depth change contributions to RHC are strongly affected by the recently reported XBT fall-rate bias, whereas the mean temperature contributions are not. Therefore, only the isotherm depth change contributions toRHCwill need to be reassessed as fall-rate-corrected data become available.