Abstract. This paper examines how the resolution of small-scale geological density models is improved through the fusion of information provided by gravity measurements and density muon radiographies. Muon radiography aims at determining the density of geological bodies by measuring their screening effect on the natural flux of cosmic muons. Muon radiography essentially works like medical X-ray scan and integrates density information along elongated narrow conical volumes. Gravity measurements are linked to density by a 3-D integration encompassing the whole studied domain. We establish the mathematical expressions of these integration formulas – called acquisition kernels – and derive the resolving kernels that are spatial filters relating the true unknown density structure to the density distribution actually recovered from the available data. The resolving kernels approach allows to quantitatively describe the improvement of the resolution of the density models achieved by merging gravity data and muon radiographies. The method developed in this paper may be used to optimally design the geometry of the field measurements to perform in order to obtain a given spatial resolution pattern of the density model to construct. The resolving kernels derived in the joined muon/gravimetry case indicate that gravity data are almost useless to constrain the density structure in regions sampled by more than two muon tomography acquisitions. Interestingly the resolution in deeper regions not sampled by muon tomography is significantly improved by joining the two techniques. The method is illustrated with examples for La Soufrière of Guadeloupe volcano.
AbstractIn this paper, energy slope averaging in the one-dimensional steady gradually varied flow model is considered. For this purpose, different methods of averaging the energy slope between cross-sections are used. The most popular are arithmetic, geometric, harmonic and hydraulic means. However, from the formal viewpoint, the application of different averaging formulas results in different numerical integration formulas. This study examines the basic properties of numerical methods resulting from different types of averaging.
This article analyses and compares two approaches related to automating modeling of valve electric circuits with piecewise linear approximation of valve characteristics. The first approach is based on operator equivalent circuits’ analytical formulas and on analytical expressions programming which describe equivalent circuits on each conduction interval of valve elements. The second approach provides implementation of a system for modeling valve converters based on implicit numerical integration formulas.
Some approximate integration formulas of statistical interest