There are mistakes and typographical errors in Remark on Algorithm 723: Fresnel Integrals , which appeared in ACM Transactions on Mathematical Software 22, 4 (December 1996). This remark corrects those errors. The software provided to Collected Algorithms of the ACM was correct.

A method to compute explicit solutions of homogeneous triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients is described. It is suitable for the limited case of well separated eigenvalues, or for multiple zero eigenvalues provided the entire column corresponding to a zero eigenvalue is zero. The solution for the case of constant inhomogeneity is described. The method requires only the computation of a constant matrix using a simple recurrence. Computing the solutions of the system from that matrix, for values of the independent variable, requires one to exponentiate only the diagonal of a matrix. It is not necessary to compute the exponential of a general triangular matrix. Although this work was motivated by a study of nuclear decay without fission or neutron absorption, which is used throughout as an example, it has wider applicability.

Abstract Complicated and comprehensive software that is meant to execute in a non‐interactive or semi‐interactive mode needs to be configured to carry out the desired tasks, needs to carry out those tasks efficiently, needs to be extensible to take on additional ambitions, and needs to be maintainable. To reduce costs, it is helpful if experts in the discipline to which the program applies can configure and operate the program without needing to become expert software engineers and without needing to become familiar with the internal details of the program, and if software engineers who develop and maintain the program need not become experts in its target discipline. Progress toward these goals can be advanced by posing the software as a language interpreter. We describe the application of this principle to ground‐based data analysis software for the Microwave Limb Sounder instrument on the NASA Earth Observing System Aura satellite, but we believe the principle has substantially broader applicability. Copyright © 2007 John Wiley & Sons, Ltd.

The retrieval algorithms for the Earth Observing System Microwave Limb Sounder (MLS) on the Aura spacecraft, launched on July 15, 2004, are described. These algorithms are used to produce estimates of geophysical parameters such as vertical profiles of atmospheric temperature and composition ("Level 2" data) from the calibrated MLS observations of microwave limb radiance ("Level 1" data). The MLS algorithms are based on the standard optimal estimation approach, a weighted nonlinear least squares optimization with a priori constraints. New aspects include adaptation to a two-dimensional system, and an approach to the issues of retrieval "phasing" and error propagation that differs from that taken for previous similar instruments. Important new aspects of the software that implements these algorithms are also described, along with the algorithm configuration for the "version 1.5" dataset. Some examples are shown from MLS in-orbit observations.

This paper describes the Earth Observing System (EOS) Aura Microwave Limb Sounder (MLS) forward model for a clear-sky atmosphere emitting unpolarized radiation. This model is used for the majority of the EOS MLS radiance calculations. The orbital and viewing geometry of MLS on Aura is such that a two-dimensional (vertical and line-of-sight, horizontal) forward model provides the most accurate basis for the inversion performed in retrievals. The model consists of atmospheric radiative transfer convolved with the instrumental field of view and the spectral response. An additional model relating the pointing heights supplied by the Aura satellite operations and MLS scan pointing encoders to the limb tangent pressure and atmospheric temperature is described. The analytic computation of Jacobians needed for retrievals is given.

This work supplements the Earth Observing System (EOS) Microwave Limb Sounder (MLS) clear-sky unpolarized forward model with algorithms for modeling polarized emission from the Zeeman-split 118.75-GHz O/sub 2/ spectral line. The model accounts for polarization-dependent emission and for correlation between polarizations with complex, 2/spl times/2 intensity and absorption matrices. The oxygen line is split into three Zeeman components by the interaction of oxygen's electronic spin with an external magnetic field, and the splitting is of order /spl plusmn/0.5 MHz in a typical geomagnetic field. Zeeman splitting is only significant at pressures low enough that collisional broadening (/spl sim/1.6 MHz/hPa) is not very large by comparison. The polarized forward model becomes significant for MLS temperature retrievals at pressure below 1.0 hPa and is crucial at pressures below /spl sim/0.03 hPa. Interaction of the O/sub 2/ molecule with the radiation field depends upon the relative orientation of the radiation polarization mode and the geomagnetic field direction. The model provides both limb radiances and the derivatives of these radiances with respect to atmospheric temperature and composition, as required by MLS temperature retrievals. EOS MLS views the atmospheric limb at 118.75 GHz with a pair of linear-cross-polarized, 100-kHz-resolution, 10-MHz-wide spectrometers. The antennas of the associated receivers are scanned to view rays with tangent heights from the Earth's surface to 0.001 hPa. Comparisons of the modeled MLS radiances with measurements show generally good agreement in line positions and strengths, however residuals in the line centers at the highest tangent heights are larger than desired and still under investigation.

The Earth Observing System Microwave Limb Sounder measures several atmospheric chemical species (OH, HO/sub 2/, H/sub 2/O, O/sub 3/, HCl, ClO, HOCl, BrO, HNO/sub 3/, N/sub 2/O, CO, HCN, CH/sub 3/CN, volcanic SO/sub 2/), cloud ice, temperature, and geopotential height to improve our understanding of stratospheric ozone chemistry, the interaction of composition and climate, and pollution in the upper troposphere. All measurements are made simultaneously and continuously, during both day and night. The instrument uses heterodyne radiometers that observe thermal emission from the atmospheric limb in broad spectral regions centered near 118, 190, 240, and 640 GHz, and 2.5 THz. It was launched July 15, 2004 on the National Aeronautics and Space Administration's Aura satellite and started full-up science operations on August 13, 2004. An atmospheric limb scan and radiometric calibration for all bands are performed routinely every 25 s. Vertical profiles are retrieved every 165 km along the suborbital track, covering 82/spl deg/S to 82/spl deg/N latitudes on each orbit. Instrument performance to date has been excellent; data have been made publicly available; and initial science results have been obtained.

Algorithm 723: Fresnel Integrals" has been improved to provide more precise results for x Ͼ Ͼ 0.

An implementation of approximations for Fresnel integrals and associated functions is described. The approximations were originally developed by W. J. Cody, but a Fortran implementation using them has not previously been published.

A method is presented to reduce the number of coefficients necessary to represent Patterson's quadrature formulae, to reduce the amount of storage necessary for storing function values, and to produce slightly smaller error in evaluating the formulae.