The accuracy in an X-ray pulsar-based navigation system depends mainly on the accuracy of the pulse phase estimation. In this paper, a novel method is proposed which combines an epoch folding process and a cross-correlation method with the idea of “averaging multiple measurements”. In this method, pulse phase is estimated multiple times on the sampled subsets of arriving photons' time tags, and a final estimation is obtained as the weighted average of these estimations. Two explanations as to how the proposed method can improve accuracy are provided: a Signal to Noise Ratio (SNR)-based explanation and an “error-difference trade-off” explanation. Numerical simulations show that the accuracy in pulse phase estimation can be improved with the proposed algorithm.

We propose some new weighted averaging methods for gradient recovery, and present analytical and numerical investigation on the performance of these weighted averaging methods. It is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in two-dimension. Numerical results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.

The development of weighted averaging models for diatoms and lake-water temperature is reviewed in relation to other factors controlling diatom abundance in lakes. It is argued that diatom–temperature models are weaker than those developed for salinity, pH and phosphorus. Moreover, even where a statistically significant model is developed for water temperature, its application over long time periods requires caution. For example, over long time periods (such as the Holocene), soil and vegetation will have changed considerably, altering lake-catchment hydrochemistry and hence the lake biota. Applications of diatom-temperature models in these instances may result in inferred temperatures that are merely artefacts of changing lake-water chemistry. However, palaeolimnological studies using variance partitioning analysis of diatom profiles in sediment cores and independent climate data show that climate signals are recorded in lake sediments. It is suggested, therefore, that the link between changing climate and diatoms be addressed via an alternative method than that of temperature-inference models. The situation demands a more complete understanding of the interactions between lake hydrodynamics and nutrient inputs from catchments over a variety of timescales.