The soil-moisture nudging technique suggests using model forecast errors in near-surface air temperature and relative humidity to re-initialize (update) soil moisture in atmospheric models. This study investigates the application of soil-moisture nudging using a single-column version of the European Centre for Medium-Range Weather Forecasts (ECMWF) model. The model was applied at 16 sites selected to sample a range of climates and land covers across the globe, with atmospheric forcing taken from the ECMWF operational analysis for 15 June 1994 and 15 December 1994. When observation errors are set to zero, the Optimal Interpolation technique for deriving estimates of nudging coefficients shows computational instability because strong correlation between near-surface temperature and near-surface relative humidity forecast errors makes the coefficient matrix of the linear equations ill-posed. Therefore, Principal Component Analysis (PCA) is used as a pre-processor to identify the independent and dependent components of temperature and relative humidity errors. With PCA, the soilmoisture nudging coefficients appropriate to the dominant principal component become stable and effective for correcting soil moisture within days. Such coefficients, when derived for the northern hemisphere summer over the 16 sites, are sufficiently consistent to propose using their all-site, daily-average values in a globally applicable soil-moisture analysis scheme. Tests of this method at the First ISLSCP (International Satellite Land Surface Climatology Program) Field Experiment (FIFE) site confirm that soil-moisture nudging can provide good estimates of near-surface weather variables and surface fluxes in a numerical weather prediction (NWP) model which gives poor simulation of precipitation (thereby poor soil moisture). However, the PCA method is unable to give an accurate determination of the location of the soil moisture within soil layers that are accessible to the atmosphere. Further, these tests show that, if the NWP model has good simulation of precipitation but poor simulation of surface radiation, soil-moisture nudging could wrongly vary the soil moisture so as to provide good estimates of the surface sensible-heat flux, but poor estimates of surface latent-heat flux. An approach to distinguish error sources from surface radiation and soil moisture is necessary for further improvement.

Abstract. We construct an infinite martingale sequence on the dual symbolic space from a uniformly quasisymmetric circle endomorphism preserving the Lebesgue measure. This infinite martingale sequence is uniformly bounded. Thus from the martingale convergence theorem, there is a limiting martingale which is the unique L 1 limit of this uniformly bounded infinite martingale sequence. Moreover, we prove that the classical Hilbert transform gives an almost complex structure on the space of all uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure. Furthermore, we discuss the complex manifold structure which is the integration of the almost complex structure. We further discuss the comparison between the global Kobayashi's metric and the global Teichmüller metric on the fiber of the forgetful map at the basepoint. We prove that these two metrics are not equivalent.

Let f f and g g be two circle endomorphisms of degree d ≥ 2 d\geq 2 such that each has bounded geometry, preserves the Lebesgue measure, and fixes 1 1 . Let h h fixing 1 1 be the topological conjugacy from f f to g g . That is, h ∘ f = g ∘ h h\circ f=g\circ h . We prove that h h is a symmetric circle homeomorphism if and only if h = I d h=Id .

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