Computing Fresnel integrals via modified trapezium rules

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement.

Simplified estimation of ray-path mirroring height for HF radiowaves reflected from the ionospheric F-region

From Milsom's equations, which describe the geometry of ray-path hops reflected from the ionospheric F-layer, algorithms for the simplified estimation of mirror-reflection height are developed. These allow for hop length and the effects of variations in underlying ionisation (via the ratio of the F2- and E-layer critical frequencies) and F2-layer peak height (via the M(3000)F2-factor). Separate algorithms are presented which are applicable to a range of signal frequencies about the FOT and to propagation at the MUF. The accuracies and complexities of the algorithms are compared with those inherent in the use of a procedure based on an equation developed by Shimazaki.

SPARE-ICE: synergistic Ice Water Path from passive operational sensors

This article presents SPARE-ICE, the Synergistic Passive Atmospheric Retrieval Experiment-ICE. SPARE-ICE is the first Ice Water Path (IWP) product combining infrared and microwave radiances. By using only passive operational sensors, the SPARE-ICE retrieval can be used to process data from at least the NOAA 15 to 19 and MetOp satellites, obtaining time series from 1998 onward. The retrieval is developed using collocations between passive operational sensors (solar, terrestrial infrared, microwave), the CloudSat radar, and the CALIPSO lidar. The collocations form a retrieval database matching measurements from passive sensors against the existing active combined radar-lidar product 2C-ICE. With this retrieval database, we train a pair of artificial neural networks to detect clouds and retrieve IWP. When considering solar, terrestrial infrared, and microwave-based measurements, we show that any combination of two techniques performs better than either single-technique retrieval. We choose not to include solar reflectances in SPARE-ICE, because the improvement is small, and so that SPARE-ICE can be retrieved both daytime and nighttime. The median fractional error between SPARE-ICE and 2C-ICE is around a factor 2, a figure similar to the random error between 2C-ICE ice water content (IWC) and in situ measurements. A comparison of SPARE-ICE with Moderate Resolution Imaging Spectroradiometer (MODIS), Pathfinder Atmospheric Extended (PATMOS-X), and Microwave Surface and Precipitation Products System (MSPPS) indicates that SPARE-ICE appears to perform well even in difficult conditions. SPARE-ICE is available for public use.

Systematic and random errors between collocated satellite ice water path observations

There remains large disagreement between ice-water path (IWP) in observational data sets, largely because the sensors observe different parts of the ice particle size distribution. A detailed comparison of retrieved IWP from satellite observations in the Tropics (!30 " latitude) in 2007 was made using collocated measurements. The radio detection and ranging(radar)/light detection and ranging (lidar) (DARDAR) IWP data set, based on combined radar/lidar measurements, is used as a reference because it provides arguably the best estimate of the total column IWP. For each data set, usable IWP dynamic ranges are inferred from this comparison. IWP retrievals based on solar reflectance measurements, in the moderate resolution imaging spectroradiometer (MODIS), advanced very high resolution radiometer–based Climate Monitoring Satellite Applications Facility (CMSAF), and Pathfinder Atmospheres-Extended (PATMOS-x) datasets, were found to be correlated with DARDAR over a large IWP range (~20–7000 g m -2 ). The random errors of the collocated data sets have a close to lognormal distribution, and the combined random error of MODIS and DARDAR is less than a factor of 2, which also sets the upper limit for MODIS alone. In the same way, the upper limit for the random error of all considered data sets is determined. Data sets based on passive microwave measurements, microwave surface and precipitation products system (MSPPS), microwave integrated retrieval system (MiRS), and collocated microwave only (CMO), are largely correlated with DARDAR for IWP values larger than approximately 700 g m -2 . The combined uncertainty between these data sets and DARDAR in this range is slightly less MODIS-DARDAR, but the systematic bias is nearly an order of magnitude.

Optimized hash for network path encoding with minimized false positives

The Bloom filter is a space efficient randomized data structure for representing a set and supporting membership queries. Bloom filters intrinsically allow false positives. However, the space savings they offer outweigh the disadvantage if the false positive rates are kept sufficiently low. Inspired by the recent application of the Bloom filter in a novel multicast forwarding fabric, this paper proposes a variant of the Bloom filter, the optihash. The optihash introduces an optimization for the false positive rate at the stage of Bloom filter formation using the same amount of space at the cost of slightly more processing than the classic Bloom filter. Often Bloom filters are used in situations where a fixed amount of space is a primary constraint. We present the optihash as a good alternative to Bloom filters since the amount of space is the same and the improvements in false positives can justify the additional processing. Specifically, we show via simulations and numerical analysis that using the optihash the false positives occurrences can be reduced and controlled at a cost of small additional processing. The simulations are carried out for in-packet forwarding. In this framework, the Bloom filter is used as a compact link/route identifier and it is placed in the packet header to encode the route. At each node, the Bloom filter is queried for membership in order to make forwarding decisions. A false positive in the forwarding decision is translated into packets forwarded along an unintended outgoing link. By using the optihash, false positives can be reduced. The optimization processing is carried out in an entity termed the Topology Manger which is part of the control plane of the multicast forwarding fabric. This processing is only carried out on a per-session basis, not for every packet. The aim of this paper is to present the optihash and evaluate its false positive performances via simulations in order to measure the influence of different parameters on the false positive rate. The false positive rate for the optihash is then compared with the false positive probability of the classic Bloom filter.

The signature of a rough path: uniqueness

In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly–Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.

Economic Transformation in Ghana: Where Will the Path Lead?

In the context of the Ghanaian government’s objective of structural transformation with an emphasis on manufacturing, this paper provides a case study of economic transformation in Ghana, exploring patterns of growth, sector transformation, and agglomeration. We document and examine why, despite impressive growth and poverty reduction figures, Ghana’s economy has exhibited less transformation than might be expected for a country that has recently achieved middle-income status. Ghana’s reduced share of agriculture in the economy, unlike many successfully transformed countries in Asia and Latin America, has been filled by services, while manufacturing has stagnated and even declined. Likely causes include weak transformation of the agricultural sector and therefore little development of agro-processing, the emergence of “consumption cities” and consumption-driven growth, upward pressure on the exchange rate, weak production linkages, and a poor environment for private-sector-led manufacturing.