Minimum relative entropy and probabilistic inversion in groundwater hydrology

The similarity between maximum entropy (MaxEnt) and minimum relative entropy (MRE) allows recent advances in probabilistic inversion to obviate some of the shortcomings in the former method. The purpose of this paper is to review and extend the theory and practice of minimum relative entropy. In this regard, we illustrate important philosophies on inversion and the similarly and differences between maximum entropy, minimum relative entropy, classical smallest model (SVD) and Bayesian solutions for inverse problems. MaxEnt is applicable when we are determining a function that can be regarded as a probability distribution. The approach can be extended to the case of the general linear problem and is interpreted as the model which fits all the constraints and is the one model which has the greatest multiplicity or “spreadout” that can be realized in the greatest number of ways. The MRE solution to the inverse problem differs from the maximum entropy viewpoint as noted above. The relative entropy formulation provides the advantage of allowing for non-positive models, a prior bias in the estimated pdf and `hard' bounds if desired. We outline how MRE can be used as a measure of resolution in linear inversion and show that MRE provides us with a method to explore the limits of model space. The Bayesian methodology readily lends itself to the problem of updating prior probabilities based on uncertain field measurements, and whose truth follows from the theorems of total and compound probabilities. In the Bayesian approach information is complete and Bayes' theorem gives a unique posterior pdf. In comparing the results of the classical, MaxEnt, MRE and Bayesian approaches we notice that the approaches produce different results. In␣comparing MaxEnt with MRE for Jayne's die problem we see excellent comparisons between the results. We compare MaxEnt, smallest model and MRE approaches for the density distribution of an equivalent spherically-symmetric earth and for the contaminant plume-source problem. Theoretical comparisons between MRE and Bayesian solutions for the case of the linear model and Gaussian priors may show different results. The Bayesian expected-value solution approaches that of MRE and that of the smallest model as the prior distribution becomes uniform, but the Bayesian maximum aposteriori (MAP) solution may not exist for an underdetermined case with a uniform prior.     

Hot accretion discs with thermal Comptonization and advection in luminous black hole sources

We solve for the structure of a hot accretion disc with unsaturated thermal Comptonization of soft photons and with advection, generalizing the classical model of Shapiro et al. The upper limit on the accretion rate due to advection constrains the luminosity to ≲ 0.15 y^3/5 α^7/5 of the Eddington limit, where y and α are the Compton and viscosity parameters, respectively. The characteristic electron temperature and Thomson optical depth of the inner flow at accretion rates within an order of magnitude of that upper limit are ∼ 10⁹ K and ∼ 1, respectively. The resulting spectra are then in close agreement with the X-ray and soft γ-ray spectra from black hole binaries in the hard state and Seyferts. At low accretion rates, bremsstrahlung becomes the dominant radiative process.     

Periodic rotations of a rigid body located at the libration point

Periodic rotations of a rigid body close to the flat motions were found. Their orbital stability was investigated. Analysis was done up to second order of the small parameter. It was proved that solutions found are orbitally stable except of the third order resonance case. This resonance do not appear if terms up to the first order of small parameter are considered only.     

A model of the gravitational field of Amalthea

Utilizing the topographic model of Jovian moon Amalthea (Stooke, 1994) and supposing that its mass density is constant we derived its basic geometrical and dynamical characteristics. For calculations the harmonic model of topography of the degree and order 18 was selected. The model appears to fit the entire surface to a mean accuracy of a few hundred meters, except in the regions localized around longitudes 0° and 180°. On the basis of the harmonic expansion of the topography we estimated the volume (V = 2.43 ± 0.02 km³) and the mean radius of topographyr ₀ = (79.7 ± 0.2) km. Generalized moments of inertia up to the order 2, principal moments of inertia and orientation of the principal axes with respect to the original reference frame were also calculated. The results show that although Amalthea has extremely irregular shape it may be treated dynamically as an almost symmetric body (B C). Finally, the set of the Stokes coefficients up to the degree and order 9 was evaluated. The results are verified by direct numerical integration.     

Measuring visual pollution by outdoor advertisements in an urban street using intervisibilty analysis and public surveys

Debates on the encroaching commercialization of public space by outdoor advertising highlight its possible negative impact on local quality of life and enjoyment of public spaces. These overstimulating outdoor advertisements are often considered a source of visual pollution, but cities have no standard way of measuring where it exists and its local impact, and thus cannot regulate it effectively. This study illustrates that visual pollution can be measured in a useful way by relating public opinion to the number of visible advertisements (intervisibility analysis). Using a 2.5D outdoor advertisement (OA) dataset (location and height) of a busy urban street in Lublin, Poland, this preliminary experiment translates visibility into visual pollution. It was found that streetscape views with more than seven visible OAs created visual pollution in this case study. The GIS-based methodology proposed could provide Lublin officials with a basic tool to assess and manage visual pollution, by informing permitting decisions on OAs.     

Strontium isotope composition of Middle Miocene primary gypsum (Badenian of the Polish Carpathian Foredeep Basin): evidence for continual non‐marine inflow of radiogenic strontium into evaporite basin

Strontium isotope compositions of ancient sulphate deposits not only provide chemostratigraphic information but also offer insight into the system in which the evaporites precipitated. Primary gypsum from two Middle Miocene (Badenian) sections in southern Poland shows steadily higher ⁸⁷Sr/⁸⁶Sr ratios than those expected from a marine‐derived formation. The ratios are interpreted as the result of increasing inflow into the basin at the time of gypsum precipitation. Palaeogeographic reconstructions suggest that riverine runoff sources were situated in the West and East European platforms (to the north and east, respectively) and the Carpathians (to the south), which are mostly composed of Mesozoic sedimentary rocks; their dissolution cannot be responsible for the higher ⁸⁷Sr/⁸⁶Sr ratios recorded. We conclude that Archaean and Palaeoproterozoic igneous and supracrustal rocks of the Ukrainian Shield were the source of the higher ⁸⁷Sr/⁸⁶Sr ratios recorded in the Badenian primary gypsum. A distinctive decreasing trend of ⁸⁷Sr/⁸⁶Sr ratios from western Ukraine to southern Poland is explained by a consistent direction of brine inflow during gypsum crystallization (typical cyclonic circulation controlled by the Coriolis effect).     

Differential calibration of seismometers for measurement of rotation and strain

A method of correcting the seismometers’ responses discrepancy for differential measurements of strains and rotations in the seismic far field is proposed. The method concerns differential calibration of the whole seismometers’ array by electric current. A model of corrective filtering of the obtained differential signals in the Z domain is given. Two methods of the filter parameters’ estimation are introduced; they are based on the seismometers response to the calibration. A practical test of the methods — an analysis of a recorded seismic event — is added. Significant reduction of differences between simultaneous seismic recordings was obtained, which is interpreted as cleaning of differential signal from spurious elements.     

Environmental responses to Lateglacial climatic fluctuations recorded in the sediments of pre‐Alpine Lake Mondsee (northeastern Alps)

Investigation of the sedimentary record of pre‐Alpine Lake Mondsee (Upper Austria) focused on the environmental reaction to rapid Lateglacial climatic changes. Results of this study reveal complex proxy responses that are variable in time and influenced by the long‐term evolution of the lake and its catchment. A new field sampling approach facilitated continuous and precisely controlled parallel sampling at decadal to sub‐annual resolution for µ‐XRF element scanning, carbon geochemistry, stable isotope measurements on ostracods, pollen analyses and large‐scale thin sections for microfacies analysis. The Holocene chronology is established through microscopic varve counting and supported by accelerator mass spectrometry ¹⁴C dating of terrestrial plant macrofossils, whereas the Lateglacial age model is based on δ¹⁸O wiggle matching with the Greenland NGRIP record, using the GICC05 chronology. Microfacies analysis enables the detection of subtle sedimentological changes, proving that depositional processes even in rather large lake systems are highly sensitive to climate forcing. Comparing periods of major warming at the onset of the Lateglacial and Holocene and of major cooling at the onset of the Younger Dryas reveals differences in proxy responses, reflecting threshold effects and ecosystem inertia. Temperature increase, vegetation recovery, decrease of detrital flux and intensification of biochemical calcite precipitation at the onset of the Holocene took place with only decadal leads and lags over a ca. 100 a period, whereas the spread of woodlands and the reduction of detrital flux lagged the warming at the onset of the Lateglacial Interstadial by ca. 500–750 a. Cooling at the onset of the Younger Dryas is reflected by the simultaneous reaction of δ¹⁸O and vegetation, but sedimentological changes (reduction of endogenic calcite content, increase in detrital flux) were delayed by about 150–300 a. Three short‐term Lateglacial cold intervals, corresponding to Greenland isotope substages GI‐1d, GI‐1c2 and GI‐1b, also show complex proxy responses that vary in time. Copyright © 2011 John Wiley & Sons, Ltd.