On mathematical and physical principles of transformations of the coherent radar backscatter matrix

The congruential rule advanced by Graves for polarization basis transformation of the radar backscatter matrix is now often misinterpreted as an example of consimilarity transformation. However, consimilarity transformations imply a physically unrealistic antilinear time-reversal operation. This is just one of the approaches found in literature to the description of transformations where the role of conjugation has been misunderstood. In this paper, the different approaches are examined in particular in respect to the role of conjugation. In order to justify and correctly derive the congruential rule for polarization basis transformation and properly place the role of conjugation, the origin of the problem is traced back to the derivation of the antenna height from the transmitted field. In fact, careful consideration of the role played by the Green’s dyadic operator relating the antenna height to the transmitted field shows that, under general unitary basis transformation, it is not justified to assume a scalar relationship between them. Invariance of the voltage equation shows that antenna states and wave states must in fact lie in dual spaces, a distinction not captured in conventional Jones vector formalism. Introducing spinor formalism, and with the use of an alternate spin frame for the transmitted field a mathematically consistent implementation of the directional wave formalism is obtained. Examples are given comparing the wider generality of the congruential rule in both active and passive transformations with the consimilarity rule.

SHIMMER (1.0): a novel mathematical model for microbial and biogeochemical dynamics in glacier forefield ecosystems

SHIMMER (Soil biogeocHemIcal Model for Microbial Ecosystem Response) is a new numerical modelling framework designed to simulate microbial dynamics and biogeochemical cycling during initial ecosystem development in glacier forefield soils. However, it is also transferable to other extreme ecosystem types (such as desert soils or the surface of glaciers). The rationale for model development arises from decades of empirical observations in glacier forefields, and enables a quantitative and process focussed approach. Here, we provide a detailed description of SHIMMER, test its performance in two case study forefields: the Damma Glacier (Switzerland) and the Athabasca Glacier (Canada) and analyse sensitivity to identify the most sensitive and unconstrained model parameters. Results show that the accumulation of microbial biomass is highly dependent on variation in microbial growth and death rate constants, Q10 values, the active fraction of microbial biomass and the reactivity of organic matter. The model correctly predicts the rapid accumulation of microbial biomass observed during the initial stages of succession in the forefields of both the case study systems. Primary production is responsible for the initial build-up of labile substrate that subsequently supports heterotrophic growth. However, allochthonous contributions of organic matter, and nitrogen fixation, are important in sustaining this productivity. The development and application of SHIMMER also highlights aspects of these systems that require further empirical research: quantifying nutrient budgets and biogeochemical rates, exploring seasonality and microbial growth and cell death. This will lead to increased understanding of how glacier forefields contribute to global biogeochemical cycling and climate under future ice retreat.

A mathematical model for the dynamics of Banana Xanthomonas Wilt with vertical transmission and inflorescence infection

A mathematical model for Banana Xanthomonas Wilt (BXW) spread by insect is presented. The model incorporates inflorescence infection and vertical transmission from the mother corm to attached suckers, but not tool-based transmission by humans. Expressions for the basic reproduction number R0 are obtained and it is verified that disease persists, at a unique endemic level, when R0 > 1. From sensitivity analysis, inflorescence infection rate and roguing rate were the parameters with most influence on disease persistence and equilibrium level. Vertical transmission parameters had less effect on persistence threshold values. Parameters were approximately estimated from field data. The model indicates that single stem removal is a feasible approach to eradication if spread is mainly via inflorescence infection. This requires continuous surveillance and debudding such that a 50% reduction in inflorescence infection and 2–3 weeks interval of surveillance would eventually lead to full recovery of banana plantations and hence improved production.