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Consider a multidimensional diffusion process X = {X (t) : t ∈ [0, 1]}. Let ε > 0 be a deterministic, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of X, we construct a probability space, supporting both X and an explicit, piecewise constant, fully simulatable process Xε such that sup 0≤t≤1with probability one. Moreover, the user can adaptively choose ε ∈ (0, ε) so that X ε (also piecewise constant and fully simulatable) can be constructed conditional on Xε to ensure an error smaller than ε with probability one. Our construction requires a detailed study of continuity estimates of the Itô map using Lyons' theory of rough paths. We approximate the underlying Brownian motion, jointly with the Lévy areas with a deterministic ε error in the underlying rough path metric.MSC 2010 subject classifications: Primary 34K50, 65C05, 82B80; secondary 97K60

A new method of gas fuelling has been introduced in the HL-1M tokamak. The method consists of a pulsed high speed molecular beam formed by a Laval type nozzle. The velocity of the well collimated hydrogen beam is about 500 m/s. About 6 × 10 19 molecules pass through the nozzle and into the vacuum chamber in each pulse. A series of helium pulses was injected into the HL-1M low density (ne = 4 × 10 18 m −3 ) hydrogen plasma. With penetration depth up to 12 cm, the ramp-up rate of the electron density, dne/dt, was as high as 3.1 × 10 20 m −3 • s −1 at steady state, and the resulting plasma density reached ne = 5.6 × 10 19 m −3 . The profile peaking factor of the electron density, Qn = ne(0)/ ne of about 100 ms after helium molecular beam injection (MBI) reached a maximum value of more than 1.51. The energy confinement time τE measured by diamagnetism is 26 ms, which is over 30% longer than that of the gas puffing (GP) results under the same operational conditions. The improvement of τE and increase of Qn for MBI were comparable to those of small pellet injection (PI) in HL-1M, as well as those of slow PI in ASDEX (Kaufmann, M., et al., Nucl. Fusion 28 (1988) 827). It is argued that the peaked density profile induced by the deepened particle injection is a factor essential for the confinement improvement apart from the isotope effect of helium particles, because the density peaking factor Qn is normally less than 1.4 for GP plasma in HL-1M. The particle confinement time with MBI increased sixfold in comparison with that before injection.

Abstract.A novel approach for the joint retrieval of aerosol optical depth (AOD) and aerosol type, using Meteosat Second Generation -Spinning Enhanced Visible and Infrared Imagers (MSG/SEVIRI) observations in two solar channels, is presented. The retrieval is based on a Time Series (TS) technique, which makes use of the two visible bands at 0.6 µm and 0.8 µm in three orderly scan times (15 min interval between two scans) to retrieve the AOD over land. Using the radiative transfer equation for plane-parallel atmosphere, two coupled differential equations for the upward and downward fluxes are derived. The boundary conditions for the upward and downward fluxes at the top and at the bottom of the atmosphere are used in these equations to provide an analytic solution for the AOD. To derive these fluxes, the aerosol single scattering albedo (SSA) and asymmetry factor are required to provide a solution. These are provided from a set of six pre-defined aerosol types with the SSA and asymmetry factor. We assume one aerosol type for a grid of 1 • ×1 • and the surface reflectance changes little between two subsequent observations. A k-ratio approach is used in the inversion to find the best solution of atmospheric properties and surface reflectance. The k-ratio approach assumes that the surface reflectance is little influenced by aerosol scattering at 1.6 µm and therefore the ratio of surface reflectances in the solar band for two subsequent observations can be wellapproximated by the ratio of the reflectances at 1.6 µm. A further assumption is that the surface reflectance varies only slightly over a period of 30 min. The algorithm makes use of numerical minimisation routines to obtain the optimal solution of atmospheric properties and surface reflectance by selection of the most suitable aerosol type from pre-defined sets.A detailed analysis of the retrieval results shows that it is suitable for AOD retrieval over land from SEVIRI data. Six AErosol RObotic NETwork (AERONET) sites with different surface types are used for detailed analysis and 42 other AERONET sites are used for validation. From 445 collocations representing stable and homogeneous aerosol type, we find that >75 % of the MSG-retrieved AOD at 0.6 and 0.8 µm values compare favourably with AERONET observed AOD values, within an error envelope of ± 0.05 ± 0.15τ and a high correlation coefficient (R>0.86). The AOD datasets derived Published by Copernicus Publications on behalf of the European Geosciences Union. L. Mei et al.: Retrieval of aerosol optical depth over landusing the TS method with SEVIRI data is also compared with collocated AOD products derived from NASA TERRA and AQUA MODIS (The Moderate-resolution Imaging Spectroradiometer) data using the Dark Dense Vegetation (DDV) method and the Deep Blue algorithms. Using the TS method, the AOD could be retrieved for more pixels than with the NASA Deep Blue algorithm. This method is potentially also useful for surface reflectance retrieval using SEVIRI observations. The current paper focuses on AOD retrieval ...