• Journal of the Korean Mathematical Society
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• v.38 no.4
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• pp.711-722
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• 2001

We survey recent results on the inverse kinematic problem arising in geophysics. The question is whether one can determine the sound speed (index of refraction) of a medium by measuring the travel times of the corresponding ray paths. We emphasize the anisotrpic case.

• Proceedings of the KSEEG Conference
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• 2000.04b
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• pp.308-310
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• 2000

Sampling rates become inconsistent when spatial data in spherical coordinate are re-sampled with respect latitudinal or longitudinal degree for mathematical processes such as Fourier Transfrom, and this results in the distrtions of the processed data in the wavenmber domain. This distortions are more evident in the polar regions. An example is presented to show such distortions during the recovery process of free-air gravity anomalies from ERSI radar atimeter data in Russian Arctic Barents Sea, and a method is present to minimize the distortion using Lambect Conformal Conic map projection.

• Korean Journal of Remote Sensing
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• v.18 no.3
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• pp.171-179
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• 2002

Sampling rates become inconsistent when spatial data in the spherical coordinate are resampled with respect to latitudinal or longitudinal degree for mathematical processes such as Fourier Transform, and this results in distortions of the processed data in the wavenumber domain. These distortions are more evident in the polar regions. An example is presented to show such distortions during the recovery process of free-air gravity anomalies from ERS-1 satellite radar altimeter data from the Barents Sea in the Russian Arctic, and a method is presented to minimize the distortion using the Lambert Conformal Conic map projection. This approach was found to enhance the free-air gravity anomalies in both data and wavenumber domains.

• Geomechanics and Engineering
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• v.23 no.4
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• pp.327-338
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• 2020

The role of precise prediction of subsurface fluids and discrimination among them cannot be ignored in reservoir characterization and petroleum prospecting. A suitable rock physics model should be build for the extraction of valuable information form seismic data. The main intent of current work is to present a rock physics model to analyze the characteristics of seismic wave propagating through a cracked porous rock saturated by a three phase fluid. Furthermore, the influence on wave characteristics due to variation in saturation of water, oil and gas were also analyzed for oil and water as wet cases. With this approach the objective to explore wave attenuation and dispersion due to wave induce fluid flow (WIFF) at seismic and sub-seismic frequencies can be precisely achieved. We accomplished our proposed approach by using BISQ equations and by applying appropriate boundary conditions to incorporate heterogeneity due to saturation of three immiscible fluids forming a layered system. To authenticate the proposed methodology, we compared our results with White's mesoscopic theory and with the results obtained by using Biot's poroelastic relations. The outcomes reveals that, at low frequencies seismic wave characteristics are in good agreement with White's mesoscopic theory, however a slight increase in attenuation at seismic frequencies is because of the squirt flow. Moreover, our work crop up as a practical tool for the development of rock physical theories with the intention to identify and estimate properties of different fluids from seismic data.

• Geophysics and Geophysical Exploration
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• v.23 no.2
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• pp.89-96
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• 2020

This tutorial summarizes the coordinate transforms for formulating geophysical problems. To ensure mathematical consistency, this discussion begins with the right-hand rule. Further, the concepts of active and passive transforms are introduced. By extending these concepts, the coordinate transform and its inverse between two coordinates are related to the matrix transpose. The yaw-pitch-roll rotation and the azimuth-deviation-tool face rotation transforms are described as the most frequently used schemes, and the relation between the Rodrigues' rotation formula and these two transforms are mathematically explained. The "Gimbal Lock" problem inherent in yaw-pitch-roll rotation is schematically presented and mathematically derived. As a useful tool overcome this problem, the principle and usage of the quaternion is also described.

• Korean Journal of Remote Sensing
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• v.10 no.2
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• pp.37-48
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• 1994

In spatial information processing, particularly in non-renewable resource exploration, the spatial data sets, including remote sensing, geophysical and geochemical data, have to be geocoded onto a reference map and integrated for the final analysis and interpretation. Application of a computer based GIS(Geographical Information System of Geological Information System) at some point of the spatial data integration/fusion processing is now a logical and essential step. It should, however, be pointed out that the basic concepts of the GIS based spatial data fusion were developed with insufficient mathematical understanding of spatial characteristics or quantitative modeling framwork of the data. Furthermore many remote sensing and geological data sets, available for many exploration projects, are spatially incomplete in coverage and interduce spatially uneven information distribution. In addition, spectral information of many spatial data sets is often imprecise due to digital rescaling. Direct applications of GIS systems to spatial data fusion can therefore result in seriously erroneous final results. To resolve this problem, some of the important mathematical information representation techniques are briefly reviewed and discussed in this paper with condideration of spatial and spectral characteristics of the common remote sensing and exploration data. They include the basic probabilistic approach, the evidential belief function approach (Dempster-Shafer method) and the fuzzy logic approach. Even though the basic concepts of these three approaches are different, proper application of the techniques and careful interpretation of the final results are expected to yield acceptable conclusions in cach case. Actual tests with real data (Moon, 1990a; An etal., 1991, 1992, 1993) have shown that implementation and application of the methods discussed in this paper consistently provide more accurate final results than most direct applications of GIS techniques.

• Structural Engineering and Mechanics
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• v.61 no.5
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• pp.563-576
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• 2017

This paper is a continuation of the investigations started in the paper by Akbarov, S.D., Guliyev, H.H and Yahnioglu, N. (2016) "Natural vibration of the three-layered solid sphere with middle layer made of FGM: three-dimensional approach", Structural Engineering and Mechanics, 57(2), 239-263, to the case where the three-layered sphere is a hollow one. Three-dimensional exact field equations of elastodynamics are employed for investigation and the discrete-analytical method is employed for solution of the corresponding eigenvalue problem. The FGM is modelled as inhomogeneous for which the modulus of elasticity, Poison's ratio and density vary continuously through the inward radial direction according to power law distribution. Numerical results on the natural frequencies are presented and discussed. These results are also compared with the corresponding ones obtained in the previous paper by the authors. In particular, it is established that for certain harmonics and for roots of certain order, the values of the natural frequency obtained for the hollow sphere can be greater (or less) than those obtained for the solid sphere.

• Structural Engineering and Mechanics
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• v.57 no.2
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• pp.239-263
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• 2016

The paper studies the natural oscillation of the three-layered solid sphere with a middle layer made of Functionally Graded Material (FGM). It is assumed that the materials of the core and outer layer of the sphere are homogeneous and isotropic elastic. The three-dimensional exact equations and relations of linear elastodynamics are employed for the investigations. The discrete-analytical method proposed by the first author in his earlier works is applied for solution of the corresponding eigenvalue problem. It is assumed that the modulus of elasticity, Poisson's ratio and density of the middle-layer material vary continuously through the inward radial direction according to power law distribution. Numerical results on the natural frequencies related to the torsional and spheroidal oscillation modes are presented and discussed. In particular, it is established that the increase of the modulus of elasticity (mass density) in the inward radial direction causes an increase (a decrease) in the values of the natural frequencies.

• Geophysics and Geophysical Exploration
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• v.23 no.4
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• pp.243-252
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• 2020

To share an understanding of trajectory measurement in surveys using borehole, this tutorial summarizes the relevant mathematical principles of the borehole deviation survey based on coordinate transform. For uncased or open holes, calculations of the azimuth-deviation-tool face rotation using three-component accelerometer and magnetometer measurements are summarized. For the steel-cased holes, calculations are based on the time-derivative formula of the coordinate transform matrix; yaw-pitch-roll angles through time are mathematically determined by integrating the threecomponent angular velocity measurements from the gyroscope while also removing the Earth's rotation effect. Sensor and data fusion to increase the accuracy of borehole deviation survey is explained with an example of the method. These principles of borehole deviation surveys can be adapted for attitude estimation in air-borne surveys or for positioning in tunnels where global positioning system (GPS) signals cannot be accessed. Information on the optimization filter that must be incorporated in sensor fusion is introduced to help future research.

• Geophysics and Geophysical Exploration
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• v.1 no.1
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• pp.8-18
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• 1998

In exploration seismology, the Kirchhoff hyperbola has been successfully used to migrate reflection seismo-grams. The mathematical basis of Kirchhoff hyperbola has not been clearly defined and understood for the application of prestack or poststack migration. The travel time from the scatterer in the subsurface to the receivers (exploding reflector model) on the surface can be a kinematic approximation of Green's function when the source is excited at position of the scatterer. If we add the travel time from the source to the scatterer in the subsurface to the travel time of exploding reflector model, we can view this travel time as a kinematic approximation of the partial derivative wavefield with respect to the velocity or the density in the subsurface. The summation of reflection seismogram along the Kirchhoff hyperbola can be evaluated as an inner product between the partial derivative wavefield and the field reflection seismogram. In addition to this kinematic interpretation of Kirchhoff hyperbola, when we extend this concept to shallow refraction seismic data, the stacking of refraction data along the straight line can be interpreted as a measurement of an inner product between the first arrival waveform of the partial derivative wavefield and the field refraction data. We evaluated the Kirchhoff hyperbola and the straight line for stacking the refraction data in terms of the first arrival waveform of the partial derivative wavefield with respect to the velocity or the density in the subsurface. This evaluation provides a firm and solid basis for the conventional Kirchhoff migration and the straight line stacking of the refraction data.