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For the time stationary global geomagnetic field, a new modelling concept is presented. A Bayesian non-parametric approach provides realistic location dependent uncertainty estimates. Modelling related variabilities are dealt with systematically by making little subjective apriori assumptions. Rather than parametrizing the model by Gauss coefficients, a functional analytic approach is applied. The geomagnetic potential is assumed a Gaussian process to describe a distribution over functions. Apriori correlations are given by an explicit kernel function with non-informative dipole contribution. A refined modelling strategy is proposed that accommodates non-linearities of archeomagnetic observables: First, a rough field estimate is obtained considering only sites that provide full field vector records. Subsequently, this estimate supports the linearization that incorporates the remaining incomplete records. The comparison of results for the archeomagnetic field over the past 1000 yr is in general agreement with previous models while improved model uncertainty estimates are provided.
In a previous study, a new snapshot modeling concept for the archeomagnetic field was introduced (Mauerberger et al., 2020, ). By assuming a Gaussian process for the geomagnetic potential, a correlation-based algorithm was presented, which incorporates a closed-form spatial correlation function. This work extends the suggested modeling strategy to the temporal domain. A space-time correlation kernel is constructed from the tensor product of the closed-form spatial correlation kernel with a squared exponential kernel in time. Dating uncertainties are incorporated into the modeling concept using a noisy input Gaussian process. All but one modeling hyperparameters are marginalized, to reduce their influence on the outcome and to translate their variability to the posterior variance. The resulting distribution incorporates uncertainties related to dating, measurement and modeling process. Results from application to archeomagnetic data show less variation in the dipole than comparable models, but are in general agreement with previous findings.
We introduce a technique for the modeling and separation of geomagnetic field components that is based on an analysis of their correlation structures alone. The inversion is based on a Bayesian formulation, which allows the computation of uncertainties. The technique allows the incorporation of complex measurement geometries like observatory data in a simple way. We show how our technique is linked to other well-known inversion techniques. A case study based on observational data is given.
We construct a family of admissible analysis reconstruction pairs of wavelet families on the sphere. The construction is an extension of the isotropic Poisson wavelets. Similar to those, the directional wavelets allow a finite expansion in terms of off-center multipoles. Unlike the isotropic case, the directional wavelets are not a tight frame. However, at small scales, they almost behave like a tight frame. We give an explicit formula for the pseudodifferential operator given by the combination analysis-synthesis with respect to these wavelets. The Euclidean limit is shown to exist and an explicit formula is given. This allows us to quantify the asymptotic angular resolution of the wavelets.
Context. The theoretically studied impact of rapid rotation on stellar evolution needs to be compared with these results of high-resolution spectroscopy-velocimetry observations. Early-type stars present a perfect laboratory for these studies. The prototype A0 star Vega has been extensively monitored in recent years in spectropolarimetry. A weak surface magnetic field was detected, implying that there might be a (still undetected) structured surface. First indications of the presence of small amplitude stellar radial velocity variations have been reported recently, but the confirmation and in-depth study with the highly stabilized spectrograph SOPHIE/OHP was required.
Aims. The goal of this article is to present a thorough analysis of the line profile variations and associated estimators in the early-type standard star Vega (A0) in order to reveal potential activity tracers, exoplanet companions, and stellar oscillations.
Methods. Vega was monitored in quasi-continuous high-resolution echelle spectroscopy with the highly stabilized velocimeter SOPHIE/OHP. A total of 2588 high signal-to-noise spectra was obtained during 34.7 h on five nights (2 to 6 of August 2012) in high-resolution mode at R = 75 000 and covering the visible domain from 3895 6270 angstrom. For each reduced spectrum, least square deconvolved equivalent photospheric profiles were calculated with a T-eff = 9500 and log g = 4.0 spectral line mask. Several methods were applied to study the dynamic behaviour of the profile variations (evolution of radial velocity, bisectors, vspan, 2D profiles, amongst others).
Results. We present the discovery of a spotted stellar surface on an A-type standard star (Vega) with very faint spot amplitudes Delta F/Fc similar to 5 x 10(-4). A rotational modulation of spectral lines with a period of rotation P = 0.68 d has clearly been exhibited, unambiguously confirming the results of previous spectropolarimetric studies. Most of these brightness inhomogeneities seem to be located in lower equatorial latitudes. Either a very thin convective layer can be responsible for magnetic field generation at small amplitudes, or a new mechanism has to be invoked to explain the existence of activity tracing starspots. At this stage it is difficult to disentangle a rotational from a stellar pulsational origin for the existing higher frequency periodic variations.
Conclusions. This first strong evidence that standard A-type stars can show surface structures opens a new field of research and ask about a potential link with the recently discovered weak magnetic field discoveries in this category of stars.
We use a dynamic scanning electron microscope (DySEM) to map the spatial distribution of the vibration of a cantilever beam. The DySEM measurements are based on variations of the local secondary electron signal within the imaging electron beam diameter during an oscillation period of the cantilever. For this reason, the surface of a cantilever without topography or material variation does not allow any conclusions about the spatial distribution of vibration due to a lack of dynamic contrast. In order to overcome this limitation, artificial structures were added at defined positions on the cantilever surface using focused ion beam lithography patterning. The DySEM signal of such high-contrast structures is strongly improved, hence information about the surface vibration becomes accessible. Simulations of images of the vibrating cantilever have also been performed. The results of the simulation are in good agreement with the experimental images.
In this study we analyse the error distribution in regional models of the geomagnetic field. Our main focus is to investigate the distribution of errors when combining two regional patches to obtain a global field from regional ones. To simulate errors in overlapping patches we choose two different data region shapes that resemble that scenario. First, we investigate the errors in elliptical regions and secondly we choose a region obtained from two overlapping circular spherical caps. We conduct a Monte-Carlo simulation using synthetic data to obtain the expected mean errors. For the elliptical regions the results are similar to the ones obtained for circular spherical caps: the maximum error at the boundary decreases towards the centre of the region. A new result emerges as errors at the boundary vary with azimuth, being largest in the major axis direction and minimal in the minor axis direction. Inside the region there is an error decay towards a minimum at the centre at a rate similar to the one in circular regions. In the case of two combined circular regions there is also an error decay from the boundary towards the centre. The minimum error occurs at the centre of the combined regions. The maximum error at the boundary occurs on the line containing the two cap centres, the minimum in the perpendicular direction where the two circular cap boundaries meet. The large errors at the boundary are eliminated by combining regional patches. We propose an algorithm for finding the boundary region that is applicable to irregularly shaped model regions.
We discuss to what extent a given earthquake catalog and the assumption of a doubly truncated Gutenberg-Richter distribution for the earthquake magnitudes allow for the calculation of confidence intervals for the maximum possible magnitude M. We show that, without further assumptions such as the existence of an upper bound of M, only very limited information may be obtained. In a frequentist formulation, for each confidence level alpha the confidence interval diverges with finite probability. In a Bayesian formulation, the posterior distribution of the upper magnitude is not normalizable. We conclude that the common approach to derive confidence intervals from the variance of a point estimator fails. Technically, this problem can be overcome by introducing an upper bound (M) over tilde for the maximum magnitude. Then the Bayesian posterior distribution can be normalized, and its variance decreases with the number of observed events. However, because the posterior depends significantly on the choice of the unknown value of (M) over tilde, the resulting confidence intervals are essentially meaningless. The use of an informative prior distribution accounting for pre-knowledge of M is also of little use, because the prior is only modified in the case of the occurrence of an extreme event. Our results suggest that the maximum possible magnitude M should be better replaced by M(T), the maximum expected magnitude in a given time interval T, for which the calculation of exact confidence intervals becomes straightforward. From a physical point of view, numerical models of the earthquake process adjusted to specific fault regions may be a powerful alternative to overcome the shortcomings of purely statistical inference.
This paper is concerned with localization properties of coherent states. Instead of classical uncertainty relations we consider "generalized" localization quantities. This is done by introducing measures on the reproducing kernel. In this context we may prove the existence of optimally localized states. Moreover, we provide a numerical scheme for deriving them.
We explore fluctuations of the horizontal component of the Earth's magnetic field to identify scaling behaviour of the temporal variability in geomagnetic data recorded by the Intermagnet observatories during the solar cycle 23 (years 1996 to 2005). In this work, we use the remarkable ability of scaling wavelet exponents to highlight the singularities associated with discontinuities present in the magnetograms obtained at two magnetic observatories for six intense magnetic storms, including the sudden storm commencements of 14 July 2000, 29-31 October and 20-21 November 2003. In the active intervals that occurred during geomagnetic storms, we observe a rapid and unidirectional change in the spectral scaling exponent at the time of storm onset. The corresponding fractal features suggest that the dynamics of the whole time series is similar to that of a fractional Brownian motion. Our findings point to an evident relatively sudden change related to the emergence of persistency of the fractal power exponent fluctuations precedes an intense magnetic storm. These first results could be useful in the framework of extreme events prediction studies.