Extended Multidimensional Integration Formulas on Polytope Meshes

2019 ◽  
Vol 41 (5) ◽  
pp. A3152-A3181
Author(s):  
Allal Guessab ◽  
Boris Semisalov
Keyword(s):  
Integration Formulas ◽  
Multidimensional Integration

scholarly journals Improvement of density models of geological structures by fusion of gravity data and cosmic muon radiographies

2015 ◽  
Vol 5 (1) ◽  
pp. 83-116 ◽  
Author(s):  
K. Jourde ◽  
D. Gibert ◽  
J. Marteau
Keyword(s):  
Gravity Data ◽  
Field Measurements ◽  
Small Scale ◽  
Density Structure ◽  
Spatial Filters ◽  
Muon Tomography ◽  
Mathematical Expressions ◽  
Integration Formulas ◽  
Muon Radiography ◽  
Gravity Measurements

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.

scholarly journals Impact of Energy Slope Averaging Methods on Numerical Solution of 1D Steady Gradually Varied Flow

2015 ◽  
Vol 62 (3-4) ◽  
pp. 101-119 ◽  
Author(s):  
Wojciech Artichowicz ◽  
Dzmitry Prybytak
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Numerical Integration ◽  
Cross Sections ◽  
Flow Model ◽  
One Dimensional ◽  
Averaging Methods ◽  
Integration Formulas ◽  
Different Types ◽  
The One

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.

A quasi‐Monte Carlo approach to 3-D migration: Theory

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 918-928 ◽  
Author(s):  
Yonghe Sun ◽  
Gerard T. Schuster ◽  
K. Sikorski
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Dynamic Range ◽  
Seismic Energy ◽  
Regular Array ◽  
Quasi Monte Carlo ◽  
Kirchhoff Migration ◽  
Grating Lobe ◽  
Random Array ◽  
Multidimensional Integration

A mathematical breakthrough was recently achieved in understanding the tractability of multidimensional integration using nearly optimal quasi‐Monte Carlo methods. Inspired by the new mathematical insights, we have studied the feasibility of applying quasi‐Monte Carlo methods to seismic imaging by 3-D prestack Kirchhoff migration. This earth imaging technique involves computing a large [Formula: see text] number of 3-D or 4-D integrals. Our numerical studies show that nearly optimal quasi‐Monte Carlo migration can produce the same or better quality earth images using only a small fraction (one fourth or less) of the data required by a conventional Kirchhoff migration. The explanation is that an image migrated from a coarse quasi‐random array of seismic data is less likely, on average, to be aliased than an image migrated from a regular array of data. In migrating these data, the geophones act as an incoherent arrangement of loud‐speakers that broadcast the reflected wavefield back into the earth; the broadcast will produce reinforcement or cancellation of seismic energy at the diffractor or grating lobe locations, respectively. Thus quasi‐Monte Carlo migration contains an inherent anti‐aliasing feature that tends to suppress migration artifacts without losing bandwidth. The penalty, however, is a decrease in the dynamic range of the migrated image compared to an image from a regular array of geophones. Our limited numerical results suggest that this loss in dynamic range is acceptable, and so justifies the anti‐aliasing benefits of migrating a random array of data.

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